IEEE MTT-V066-I04 (2018-04)

APRIL 2018

VOLUME 66

NUMBER 4

IETMAB

(ISSN 0018-9480)

Editorial ............................................................................................ L. Perregrini and J. C. Pedro

1681

PAPERS

EM Theory and Analysis Techniques Modal Analysis of Corrugated Plasmonic Rods for the Study of Field Localization, Conductor Attenuation, and Dielectric Losses .............................................................................. M. Ng Mou Kehn and J. Y. Li A General Scheme for the Discontinuous Galerkin Time-Domain Modeling and S-Parameter Extraction of Inhomogeneous Waveports ................................. G. Chen, L. Zhao, W. Yu, S. Yan, K. Zhang, and J.-M. Jin A Straightforward Updating Criterion for 2-D/3-D Hybrid Discontinuous Galerkin Time-Domain Method Controlling Comparative Error .................................... W. Mai, P. Li, C. G. Li, M. Jiang, W. Hao, L. Jiang, and J. Hu VoxHenry: FFT-Accelerated Inductance Extraction for Voxelized Geometries ............................................... ............................................... A. C. Yucel, I. P. Georgakis, A. G. Polimeridis, H. Ba˘gcı, and J. K. White Devices and Modeling Characterization of PMMA/BaTiO3 Composite Layers Through Printed Capacitor Structures for Microwave Frequency Applications ........................................................................................ O. Gbotemi, S. Myllymäki, J. Kallioinen, J. Juuti, M. Teirikangas, H. Jantunen, M. M. Kržmanc, D. Suvorov, M. Sloma, and M. Jakubowska Design and Large-Signal Characterization of High-Power Varactor-Based Impedance Tuners ............................ ...................... C. Sánchez-Pérez, C. M. Andersson, K. Buisman, D. Kuylenstierna, N. Rorsman, and C. Fager Bandwidth Optimization Method for Reflective-Type Phase Shifters ......................................................... .................................... D. Müller, A. Haag, A. Bhutani, A. Tessmann, A. Leuther, T. Zwick, and I. Kallfass Radiative Quality Factor in Thin Resonant Metamaterial Absorbers .......................................................... ............................. N. Fernez, L. Burgnies, J. Hao, C. Mismer, G. Ducournau, D. Lippens, and É. Lheurette Nonlinear Effects of SiO2 Layers in Bulk Acoustic Wave Resonators ........................................................ .................... C. Collado, J. Mateu, D. Garcia-Pastor, R. Perea-Robles, A. Hueltes, S. Kreuzer, and R. Aigner

1684 1701 1713 1723

1736 1744 1754 1764 1773

(Contents Continued on Back Cover)

(Contents Continued from Front Cover) A Direct Matrix Synthesis for In-Line Filters With Transmission Zeros Generated by Frequency-Variant Couplings .................................. Y. He, G. Macchiarella, G. Wang, W. Wu, L. Sun, L. Wang, and R. Zhang Design and Optimization of Tunable Silicon-Integrated Evanescent-Mode Bandpass Filters .............................. ....................................................................................... Z. Yang, D. Psychogiou, and D. Peroulis Synthesis-Applied Highly Selective Tunable Dual-Mode BPF With Element-Variable Coupling Matrix ................ .............................................................................. D. Lu, X. Tang, N. S. Barker, M. Li, and T. Yan Stability and Bifurcation Analysis of Multi-Element Non-Foster Networks .................. A. Suárez and F. Ramírez Effects of Noisy and Modulated Interferers on the Free-Running Oscillator Spectrum .................................... ........................................................................................... S. Sancho, M. Ponton, and A. Suarez Strongly Enhanced Sensitivity in Planar Microwave Sensors Based on Metamaterial Coupling .......................... .......................................................................... M. Abdolrazzaghi, M. Daneshmand, and A. K. Iyer Passive Circuits Asymmetric Compact OMT for X-Band SAR Applications ............ M. A. Abdelaal, S. I. Shams, and A. A. Kishk Ortho-Mode Sub-THz Interconnect Channel for Planar Chip-to-Chip Communications ................................... ................................................................... B. Yu, Y. Ye, X. Ding, Y. Liu, Z. Xu, X. Liu, and Q. J. Gu Novel Planar and Waveguide Implementations of Impedance Matching Networks Based on Tapered Lines Using Generalized Superellipses ...................................... S. Cogollos, J. Vague, V. E. Boria, and J. D. Martínez Arbitrary Prescribed Wideband Flat Group Delay Circuits Using Coupled Lines ........ G. Chaudhary and Y. Jeong Broadband High-Efficiency Input Coupler With Mode Selectivity for a W-Band Confocal Gyro-TWA ................. .......................................................................... J. Wang, Y. Yao, Q. Tian, H. Li, G. Liu, and Y. Luo A Broadband Wilkinson Power Divider Based on the Segmented Structure .......................................... T. Yu Coupling Coefficient Reconfigurable Wideband Branch-Line Coupler Topology With Harmonic Suppression ......... ............................................................................... Z. Qamar, S. Y. Zheng, W. S. Chan, and D. Ho A Coupled-Line Coupling Structure for the Design of Quasi-Elliptic Bandpass Filters ...................... C.-J. Chen Compact Multiband Bandpass Filter Using Low-Pass Filter Combined With Open Stub-Loaded Shorted Stub ........ .............................................................................................. Q. Yang, Y.-C. Jiao, and Z. Zhang Hybrid and Monolithic RF Integrated Circuits Class-X—Harmonically Tuned Power Amplifiers With Maximally Flat Waveforms Suitable for Over One-Octave Bandwidth Designs ............................................................................ X. Li, M. Helaoui, and X. Du Postmatching Doherty Power Amplifier With Extended Back-Off Range Based on Self-Generated Harmonic Injection ....................................................... X. Y. Zhou, S. Y. Zheng, W. S. Chan, X. Fang, and D. Ho Broadband Nonreciprocal Phase Shifter Design Technique ..................................................................... .................................................. M. Palomba, D. Palombini, S. Colangeli, W. Ciccognani, and E. Limiti A W-Band LNA/Phase Shifter With 5-dB NF and 24-mW Power Consumption in 32-nm CMOS SOI ................. ................................................................................................... M. Sayginer and G. M. Rebeiz A Low Phase Noise 210-GHz Triple-Push Ring Oscillator in 90-nm CMOS ......... C.-L. S. Hsieh and J. Y.-C. Liu Highly Isolating and Broadband Single-Pole Double-Throw Switches for Millimeter-Wave Applications Up to 330 GHz .................................................................................. F. Thome and O. Ambacher A G-Band Monolithically Integrated Quasi-Optical Zero-Bias Detector Based on Heterostructure Backward Diodes Using Submicrometer Airbridges ..................... S. M. Rahman, Z. Jiang, Md. I. B. Shams, P. Fay, and L. Liu Zero-Bias 50-dB Dynamic Range Linear-in-dB V-Band Power Detector Based on CVD Graphene Diode on Glass ..................................... M. Saeed, A. Hamed, Z. Wang, M. Shaygan, D. Neumaier, and R. Negra Instrumentation and Measurement Techniques De-Embedding Method for Strongly Coupled Cavities ....................................... R. Peter and G. Fischerauer Measurement Technique to Emulate Signal Coupling Between Power Amplifiers .... D. Nopchinda and K. Buisman Reflection of Coherent Millimeter-Wave Wavelets on Dispersive Materials: A Study on Porcine Skin .................. ............................................................................... S. Heunisch, L. Ohlsson, and L.-E. Wernersson

1780 1790 1804 1817 1831 1843 1856 1864 1874 1885 1895 1902 1912 1921 1926

1939 1951 1964 1973 1983 1998 2010 2018 2025 2034 2047

IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY The Microwave Theory and Techniques Society is an organization, within the framework of the IEEE, of members with principal professional interests in the field of microwave theory and techniques. All members of the IEEE are eligible for membership in the Society upon payment of the annual Society membership fee of $24.00 and obtain electronic access, plus $50.00 per year for print media. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only. ADMINISTRATIVE COMMITTEE T. B RAZIL, President A. A BUNJAILEH G. LYONS

D. S CHREURS, President Elect

A. Z HU, Secretary

IEEE Transactions on Microwave Theory and Techniques J. C. P EDRO L. P ERREGRINI

IEEE Microwave Magazine R. C AVERLEY

Editors-in-Chief of the publications of the IEEE Microwave Theory and Techniques Society IEEE Microwave and Wireless IEEE Transactions on Terahertz IEEE Journal of Electromagnetics, RF Components Letters Science and Technology and Microwaves in Medicine and Biology N. S COTT BARKER J. S TAKE J.-C. C HIAO

Honorary Life Members J. BARR T. I TOH

A. A BUNJAILEH , Treasurer

Elected members M. B OZZI T. B RAZIL N. C ARVALHO G. C HATTOPADHYAY W. C HE K. G HORBANI R. G UPTA R. H ENDERSON P. K HANNA M. M ADIHIAN J. NAVARRO D. PASQUET G. P ONCHAK S. R AMAN J. E. R AYAS -S ANCHEZ S. R EISING A. S ANADA D. S CHREURS

D. K ISSINGER S. KOUL M. S TEER

IEEE Journal on Multiscale and Multiphysics Computational Techniques Q. H. L IU

Distinguished Lecturers

R. S PARKS P. S TAECKER

2016–2018 C. C AMPBELL P. ROBLIN T. NAGATSUMA N. S HINOHARA

2017–2019 W.Y. A LI -A HMAD N. B. C ARVALHO

Past Presidents 2018–2020 S. BASTIOLI M. G ARDILL J. E VERARD

D. W ILLIAMS (2017) K. W U (2016) T. L EE (2015)

MTT-S Chapter Chairs Albuquerque: E. FARR Argentina: F. J. D I V RUNO Atlanta: W. W ILLIAMS Austria: W. B OESCH Bahia: M. TAVARES D E M ELO Baltimore: R. C. PARYANI Bangalore/India: V. V. S RINIVASAN Beijing: W. C HEN Belarus: S. M ALYSHEV Benelux: G. VANDENBOSCH Boston: A. Z AI Bombay/India: A. D. JAGATIA/ Q. H. BAKIR Brasilia/Amazon: M. V. A. N UNES Buenaventura: C. S EABURY Buffalo: M. R. G ILLETTE Bulgaria: M. H RISTOV Canada, Atlantic: C. J. W HITT Cedar Rapids/Central Iowa: M. K. ROY Central & South Italy: L. TARRICONE Central North Carolina: F. S UCO Central Texas: A. A LU Centro-Norte Brasil: A. P. L ANARI B O Chengdu: B.-Z. WANG Chicago: K. J. K ACZMARSKI Cleveland: M. S CARDELLETTI Columbus: C. C AGLAYAN Connecticut: C. B LAIR Croatia: S. H RABAR Czech/Slovakia: V. Z AVODNY Dallas: R. PANDEY Dayton: A. J. T ERZUOLI

Delhi/India: S. K. KOUL Denver: T. S AMSON/ W. N. K EFAUVER Eastern North Carolina: T. N ICHOLS Egypt: E. A. E L H. H ASHEESH Finland: K. H ANEDA /V. V IIKARI Florida West Coast: J. WANG Foothills: M. C HERUBIN France: A. G HIOTTO /P. D ESCAMPS Germany: P. K NOTT Greece: S. KOULOURIDIS Guadalajara: Z. B RITO Gujarat/India: M. B. M AHAJAN Guangzhou: Q.-X. C HU Harbin: Q. W U Hawaii: A. S INGH Hiroshima: K. O KUBO Hong Kong: H. W. L AI /K. M. S HUM Houston: S. A. L ONG Hungary: L. NAGY Huntsville: B. J. W OLFSON Hyderabad/India: Y. K. V ERMA Indonesia: M. A LAYDRUS Islamabad: H. C HEEMA Israel: S. AUSTER Kansai: T. K ASHIWA Kingston: C. E. S AAVEDRA/ Y. M. A NTAR Kitchener-Waterloo: R. R. M ANSOUR Kolkata/India: D. G UHA Lebanon: E. H UIJER Lithuania: I. NAIDIONOVA

Long Island/New York: S. PADMANABHAN Los Angeles, Coastal: H.-Y. PAN Los Angeles, Metro/San Fernando: J. C. W EILER Macau: W.-W. C HOI Madras/India: V. A BHAIKUMAR Malaysia: F. C. S EMAN Malaysia, Penang: P. W. W ONG Mexican Council: R. L M IRANDA Milwaukee: S. S. H OLLAND Montreal: K. W U Morocco: M. E SSAAIDI Nagoya: T. S EKINE Nanjing: W. H ONG New Hampshire: D. S HERWOOD New Jersey Coast: A. AGARWAL New South Wales: R. M. H ASHMI New Zealand: A. W ILLIAMSON North Italy: G. O LIVERI North Jersey: A. P ODDAR Northern Australia: J. M AZIERSKA Northern Canada: A. K. I YER/ M. DANESHMAN Northern Nevada: B. S. R AWAT Norway: Y. T HODESEN Orange County: H. J. D E L OS S ANTOS Oregon: K. M AYS Orlando: M. S HIRAZI Ottawa: Q. Z ENG Philadelphia: A. S. DARYOUSH Peru: G. R AFAEL -VALDIVIA

Phoenix: C. S COTT /S. ROCKWELL Pikes Peak: K. H U Poland: W. J. K RZYSZTOFIK Portugal: R. F. S. C ALDEIRINHA Princeton/Central Jersey: A. K ATZ Queensland: M. S HAHPARI Rio de Janeiro: J. R. B ERGMANN Rochester: J. D. M AJKOWSKI Romania: T. P ETRESCU Russia, Moscow: V. A. K ALOSHIN Russia, Novosibirsk: A. B. M ARKHASIN Russia, Saratov/Penza: M. D. P ROKHOROV Russia, Tomsk: D. Z YKOV San Diego: T. E. BABAIAN Santa Clara Valley/San Francisco: O. E. L ANEY Seattle: D. H EO /M. P. A NANTRAM Seoul: J.-S. R IEH Serbia and Montenegro: Z. M ARINKOVIC Shanghai: J. F. M AO Singapore: X. C HEN South Africa: D. D E V ILLIERS South Australia: C. O. F UMEAUX South Brazil: C. K RETLY Southeastern Michigan: A. G RBIC Spain: M. F ERNANDEZ BARCIELA Sri Lanka: A. W. G UNAWARDENA St. Louis: D. BARBOUR Sweden: M. G USTAFSSON

Switzerland: N. PARRA M ORA Syracuse: M. C. TAYLOR Taegu: Y.-H. J EONG Tainan: C.-L. YANG Taipei: Y.-J. E. C HEN Thailand: T. A NGKAEW Tiblisi, Rep. of Georgia: K. TAVZARASHVILI Tokyo: M. NAKATSUGAWA Toronto: G. V. E LEFTHERIADES Tucson: H. X IN /M. L I Tunisia: N. B OULEJFEN ¨ E RG UL ¨ Turkey: O. Twin Cities: C. F ULLER UK/RI: A. R EZAZADEH Ukraine, East: K. V. I LYENKO Ukraine, Kiev: Y. P ROKOPENKO Ukraine, Vinnytsya: O. O. KOVALYUK Ukraine, West: M. I. A NDRIYCHUK United Arab Emirates: N. K. M ALLAT Uttar Pradesh/India: A. R. H ARISH Vancouver: D. G. M ICHELSON Venezuela: J. B. P ENA Victoria: E. V INNAL Virginia Mountain: G. W ILLIAMS Washington DC/Northern Virginia: R. R. B ENOIT Western Saudi Arabia: A. S HAMIM Winnipeg: P. M OJABI Xian: X. S HI

Editorial Board of IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Editors-In-Chief Editorial Assistants J OSE C ARLOS P EDRO Universidade de Aveiro Aveiro, Portugal

L UCA P ERREGRINI Univ. of Pavia Pavia, Italy

L AURA G RIMOLDI Italy

A NA R IBEIRO Portugal

Associate Editors S T E´ PHANE B ILA XLIM Limoges, France X UDONG C HEN Nat. Univ. of Singapore Singapore TA -S HUN C HU National Tsing Hua University Hsinchu, Taiwan

A LESSANDRA C OSTANZO Univ. of Bologna Bologna, Italy C HISTIAN DAMM Univ. Ulm Ulm, Germany

A NDREA F ERRERO Keysight Technol. Santa Rosa, CA, USA J OSE A NGEL G ARCIA Universidad de Cantabria Santander, Spain

K AMRAN G HORBANI RMIT Univ. Melbourne, Vic., Australia J USEOP L EE Korea Univ. Seoul, South Korea

A RUN NATARAJAN Oregon State Univ. Corvallis, OR, USA H ENDRIK ROGIER Univ. of Ghent Ghent, Belgium

C HRISTOPHER S ILVA The Aerospace Corporation El Segundo, CA, USA M ARTIN VOSSIEK Univ. of Erlangen-N¨urnberg Erlangen, Germany

PATRICK FAY Univ. of Notre Dame Notre Dame, IN, USA

XUN GONG University of Central Florida Orlando, FL, USA

T ZYH -G HUANG M A NTUST Taipei, Taiwan

M IGUEL A NGEL S ANCHEZ S ORIANO University of Alicante Alicante, Spain

J OHN W OOD Obsidian Microwave, LLC Raleigh-Durham, NC, USA

IEEE Officers JAMES A. J EFFERIES, President J OS E´ M. F. M OURA, President-Elect W ILLIAM P. WALSH, Secretary J OSEPH V. L ILLIE, Treasurer K AREN BARTLESON, Past President

W ITOLD M. K INSNER, Vice President, Educational Activities S AMIR M. E L -G HAZALY, Vice President, Publication Services and Products M ARTIN J. BASTIAANS, Vice President, Member and Geographic Activities F ORREST D. “D ON ” W RIGHT, President, Standards Association S USAN “K ATHY ” L AND, Vice President, Technical Activities S ANDRA “C ANDY ” ROBINSON, President, IEEE-USA J ENNIFER T. B ERNHARD, Director, Division IV—Electromagnetics and Radiation

IEEE Executive Staff S TEPHEN P. W ELBY, T HOMAS S IEGERT, Business Administration J ULIE E VE C OZIN, Corporate Governance D ONNA H OURICAN, Corporate Strategy JAMIE M OESCH, Educational Activities E ILEEN M. L ACH, General Counsel & Chief Compliance Officer VACANT, Human Resources C HRIS B RANTLEY, IEEE-USA

Executive Director & Chief Operating Officer C HERIF A MIRAT, Information Technology K AREN H AWKINS, Marketing C ECELIA JANKOWSKI, Member and Geographic Activities M ICHAEL F ORSTER, Publications KONSTANTINOS K ARACHALIOS, Standards Association M ARY WARD -C ALLAN, Technical Activities

IEEE Periodicals Transactions/Journals Department

Senior Director, Publishing Operations: DAWN M ELLEY Director, Editorial Services: K EVIN L ISANKIE Director, Production Services: P ETER M. T UOHY Associate Director, Editorial Services: J EFFREY E. C ICHOCKI Associate Director, Information Conversion and Editorial Support: N EELAM K HINVASARA Managing Editor: C HRISTOPHER P ERRY Journals Coordinator: C HRISTINA M. R EZES IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES (ISSN 0018-9480) is published monthly by the Institute of Electrical and Electronics Engineers, Inc. Responsibility for the contents rests upon the authors and not upon the IEEE, the Society/Council, or its members. IEEE Corporate Office: 3 Park Avenue, 17th Floor, New York, NY 10016-5997. IEEE Operations Center: 445 Hoes Lane, Piscataway, NJ 08854-4141. NJ Telephone: +1 732 981 0060. Price/Publication Information: Individual copies: IEEE Members $20.00 (first copy only), nonmember $217.00 per copy. (Note: Postage and handling charge not included.) Member and nonmember subscription prices available upon request. Copyright and Reprint Permissions: Abstracting is permitted with credit to the source. Libraries are permitted to photocopy for private use of patrons, provided the per-copy fee of $31.00 is paid through the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923. For all other copying, reprint, or republication permission, c 2018 by The Institute of Electrical and Electronics Engineers, Inc. write to Copyrights and Permissions Department, IEEE Publications Administration, 445 Hoes Lane, Piscataway, NJ 08854-4141. Copyright  All rights reserved. Periodicals Postage Paid at New York, NY and at additional mailing offices. Postmaster: Send address changes to IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES, IEEE, 445 Hoes Lane, Piscataway, NJ 08854-4141. GST Registration No. 125634188. CPC Sales Agreement #40013087. Return undeliverable Canada addresses to: Pitney Bowes IMEX, P.O. Box 4332, Stanton Rd., Toronto, ON M5W 3J4, Canada. IEEE prohibits discrimination, harassment and bullying. For more information visit http://www.ieee.org/nondiscrimination. Printed in U.S.A.

Digital Object Identifier 10.1109/TMTT.2018.2811848

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

1681

Editorial

M

ORE than a year has passed since we began our duties as Editors-in-Chief of this T RANSACTIONS, the flagship publication of the IEEE Microwave Theory and Techniques Society (MTT-S). Since then, we have worked hard to provide a better service to the microwave community, and it is time to make a first assessment. Next month, we will be celebrating the first anniversary of our J OURNAL - WITHIN - A -J OURNAL (J W J) ON M ICROWAVE S YSTEMS AND A PPLICATIONS, the JwJ on MSA. This new publication is the most visible result of the “Systems Initiative,” undertaken by the IEEE MTT-S Administrative Committee, and whose primary objective is to restate our long tradition and interests on any systems that involve some form of microwave technology. The JwJ on MSA had its inaugural issue in May 2017 and has since been issued on a bi-monthly basis. Having as Editor-in-Chief, José C. Pedro, it has been printed as a standalone part, but bound together with the other MTT T RANSACTIONS regular papers. In a year, the JwJ on MSA published 151 original manuscripts, which correspond to nearly 27% of the total 567 papers published in this T RANSACTIONS. These systems’ papers are organized in the following five different topics: 1) wireless communication systems; 2) wireless power transfer and RFID systems; 3) microwave imaging and radar applications; 4) microwave sensors and biomedical applications; 5) microwave photonics. These are assigned to dedicated Associate Editors and expert reviewers on each of the fields. Besides this special initiative, we continued the tradition of publishing Special Issues on emerging and/or popular topics. In 2017, we published a “Special Issue on RF Frontends for Mobile Radio Terminal Applications,” and in June 2018 we will have a “Special Issue on Substrate Integrated Waveguide Circuits and Systems for MHz-Through-THz Wireless and Photonics Applications.” As usual, this T RANSACTIONS publishes in December a Special Issue dedicated to extended papers of the International Microwave Symposium of the current year. Moreover, we routinely publish Mini-Special Issues dedicated to conferences sponsored by the MTT-S. In 2017, we published six of them dedicated to RFIC 2016, WPTC 2016, NEMO 2016, LAMC 2016, and RWW 2017, and in 2018, we have planned another seven mini-special issues dedicated to RFIC 2017, NEMO 2017, IMWS-AMP 2017, IMaRC 2017, APMC 2017, LAMC 2017, and RWW 2018.

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2018.2815878

Another significant achievement is the reduction of the review time. As shown in the above plot, the mean time from submission to first decision in 2017 was reduced down to 56 days, and the mean time to final decision dropped to 70 days. This great improvement would not have been possible without the dedication and professionalism of the Associate Editors, who worked tirelessly to guarantee the high technical level of this transaction, to ensure enough reviewers for each paper, and who issued their paper recommendations in a timely manner. Therefore, we sincerely thank our Associate Editors, Stéphane Bila, Xudong Chen, Alessandra Costanzo, Christian Damm, Patrick Fay, Andrea Ferrero, Kamran Ghorbani, Juseop Lee, Tzyh-Ghuang Ma, Arun Natarajan, Hendrik Rogier, Christopher Silva, Martin Vossiek, and John Wood. To continue this positive trend, we decided to enlarge the team, by adding four new Associate Editors, namely, Ta-Shun Chu, José Angel García, Xun Gong, and Miguel Ángel Sánchez Soriano, who started handling papers in 2018. They were selected because of their expertise and thus to help deal with the large number of papers submitted on the most popular topics (e.g., passive components and filters, device modeling, power amplifiers, mixed-signal circuits, receivers, and transmitters). The biographies of the new Associate Editors are reported in this editorial. Coming to the future plans, we intend to further develop the system initiative, and to open the journal to cutting-edge technologies. To this aim, we already planned a special issue to appear in 2019 on “5G Hardware and System Technologies,” and another special issue for 2020 on “Broadband MillimeterWave Power Amplifiers.” Stay tuned and submit your stateof-the-art research to these special issues! We would like to thank all the authors for submitting their research findings to this T RANSACTIONS, and the reviewers for their commitment and constructive comments. Finally, our gratitude goes to the President of the IEEE MTT-S, Prof. Tom Brazil, the Past Presidents Dr. Dylan William and Prof. Ke Wu, the Publication Committee Chair, Prof. Michael Steer, for the continuous support of the IEEE MTT-S to the success of this T RANSACTIONS.

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

1682

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

L UCA P ERREGRINI, Editor-in-Chief Department of Electrical, Computer, and Biomedical Engineering University of Pavia 27100 Pavia, Italy

J OSÉ C ARLOS P EDRO, Editor-in-Chief Department of Electronics, Telecommunications, and Informatics Engineering University of Aveiro 3810-193 Aveiro, Portugal

Ta-Shun Chu (S’06–M’10) received the B.S. degree in civil engineering and M.S. degree in applied mechanics from National Taiwan University, Taipei, Taiwan, in 2000 and 2002, respectively, and the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, CA, USA, in 2010. In 2010, he joined the Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan, where he is currently an Associate Professor and the Director of the NTHU EECS 820 Laboratory. His current research interests include millimeter wave, RF, analog, and mixed-mode integrated circuit development for radar systems. Dr. Chu was a recipient of the Alfred E. Mann Innovation in Engineering Doctoral Fellowship from 2008 to 2009.

José Angel García (S’98–A’00–M’02) received the B.Sc. degree in telecommunications engineering from the Instituto Superior Politécnico “José A. Echeverría” (currently the Technological University of Havana, ISPJAE), Havana, Cuba, in 1988, and the Ph.D. degree from the University of Cantabria, Santander, Spain, in 2000. From 1988 to 1991, he was a Radio System Engineer with the High-Frequency (HF) Communication Center, where he designed antennas and HF circuits. From 1991 to 1995, he was an Instructor Professor with the Telecommunication Engineering Department, ISPJAE, Havana. From 1999 to 2000, he was a Radio Design Engineer with the Thaumat Global Technology Systems, focused on base-station arrays. From 2000 to 2001, he was a Microwave Design Engineer/Project Manager with TTI Norte, Santander, Spain, during which he was in charge of the research line on software defined radio while involved with active amplifying antennas. In 2002, he joined the University of Cantabria, as a Senior Research Scientist “Ramón y Cajal,” where he is currently an Associate Professor. From 2003 to 2004, he was a Post-Doctoral Researcher with the Telecommunications Institute, University of Aveiro, Aveiro, Portugal. In 2011, he was a Visiting Researcher with the Microwave and RF Research Group, University of Colorado, Boulder, CO, USA. He has been leading coordinated projects on linearization and high efficiency transmitters, funded by the Spanish Research, Development, and Innovation Program, since 2003. His current research interests include the nonlinear characterization and modeling of GaN HEMTs, as well as the design of efficient RF/microwave power amplifiers, rectifiers, dc/dc power converters, and wireless transmission architectures. Dr. García was a recipient of the MUTIS Grant for doctoral studies from the Spanish Ministry of Foreign Affairs. He has been a Reviewer of several IEEE journals and a TPRC member of the IEEE MTT-S International Microwave Symposium and European Microwave Week.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

1683

Xun Gong (S’02–M’05–SM’11) received the B.S. and M.S. degrees in electrical engineering from FuDan University, Shanghai, China, in 1997 and 2000, respectively, and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 2005. He was with the Air Force Research Laboratory, Hanscom, MA, USA, in 2009, under the support of the Air Force Office of Scientific Research Summer Faculty Fellowship Program. In 2005, he was with the Birck Nanotechnology Center, Purdue University, West Lafayette, IN, USA, as a Post-Doctoral Research Associate. In 2005, he joined the University of Central Florida (UCF), Orlando, FL, USA, as an Assistant Professor, where he is currently a Professor of electrical and computer engineering and the Director of the Antenna, RF, and Microwave Integrated Systems Laboratory. His current research interests include microwave passive components and filters, sensors, antennas and arrays, flexible electronics, and packaging. Dr. Gong has served on the Editorial Boards of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES, the IEEE T RANSACTIONS ON A NTENNAS AND P ROPAGATION, IEEE M ICROWAVE AND W IRELESS C OMPONENT L ETTERS, and IEEE A NTENNAS AND W IRELESS P ROPAGATION L ETTERS. He is an Associate Editor of the T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES. He has been the Associate Editor of IEEE M ICROWAVE A ND W IRELESS C OMPONENT L ETTERS since 2013. He was the Technical Program Committee Chair of the 2013 IEEE AP-S/URSI International Symposium on Antennas and Propagation and the Operations Chair of the 2014 IEEE MTT-S International Microwave Symposium. He was the General Chair of the 2016 IEEE International Workshop on Antenna Technology and the 2012 IEEE Wireless and Microwave Technology Conference. He was the IEEE AP/MTT Orlando Chapter Chair from 2007 to 2010. He was a recipient of the NSF Faculty Early CAREER Award in 2009, the UCF Reach for the Stars Award in 2016, the Research Incentive Award at UCF in 2011 and 2016, the Teaching Incentive Program Award at UCF in 2010 and 2015, and the Outstanding Engineer Award from the IEEE Florida Council and the Orlando Section, respectively, in 2009.

Miguel Ángel Sánchez-Soriano (S’09–M’13) was born in Murcia, Spain, in 1984. He received the degree in telecommunications engineering (with a Special Award) and Ph.D. degree in electrical engineering from Miguel Hernandez University (UMH), Elche, Spain, in 2007 and 2012, respectively. In 2007, he joined the Radio Frequency Systems Group, UMH, as a Research Assistant. He was a Visiting Researcher with the Microwaves Group, Heriot-Watt University, Edinburgh, U.K., in 2010. From 2013 to 2015, he was with the LabSTICC Group, Université de Bretagne Occidentale, Brest, France, as a Post-Doctoral Researcher. In 2015, he was a “Juan de la Cierva” Research Fellow with the Grupo de Aplicaciones de Microondas, Technical University of Valencia, Valencia, Spain. Since 2015, he has been an Assistant Professor with the University of Alicante, Alicante, Spain. His current research interests include the analysis and design of microwave planar devices, especially filters and their reconfigurability, and the multiphysics study of high-frequency devices. Dr. Sánchez-Soriano was a recipient of the Runner-Up HISPASAT Award for the Best Spanish Doctoral Thesis in New Applications for Satellite Communications from the Spanish Telecommunication Engineers Association (COIT/AEIT) and the Extraordinary Ph.D. Award from Miguel Hernández University. He is an Associate Editor of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES and the IET M ICROWAVES , A NTENNAS , AND P ROPAGATION , and a regular Reviewer for more than ten journals and several IEEE international conferences, including IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS, IET Electronics Letters, and IEEE ACCESS.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Modal Analysis of Corrugated Plasmonic Rods for the Study of Field Localization, Conductor Attenuation, and Dielectric Losses Malcolm Ng Mou Kehn , Senior Member, IEEE, and Jen Yung Li Abstract— By virtue of the strong confinement of surface-wave fields that it provides, the transverse corrugated conducting rod is an important structure in the fields of plasmonics and metasurfaces, finding applications in focusing, sensing, imaging, spectroscopy, and subwavelength optics, among others. This paper presents an analytical modal method for treating such grated rods with generally dielectric-filled grooves, one which offers rapid yet accurate surface-wave modal solutions. Based on the asymptotic corrugation boundary conditions, the formulation is simple and elegant, providing not only the dispersion relationship between the frequency and wavenumber but also the explicit functional forms of the fields. Dispersion and modal field results obtained by the proposed method are validated with an independent fullwave solver. Because of its candidacy for microwave applications at high powers and high frequencies, as well as transmissions over long distances, studies of dielectric and conductor losses are also carried out, for both the grooved rod and its likewise corrugated circular waveguide counterpart. Parametric studies are conducted on three aspects, namely, the degree of field localization on the surface of the rod, as well as attenuation due to dielectric and metal losses. Measurements of dispersion and field decay properties conducted on a manufactured rod that concur with theory are also reported. Index Terms— Asymptotic boundary conditions, corrugated rod, modal analysis, plasmonics, surface waves, vector potential.

I. I NTRODUCTION

T

HE interaction of light with a metal surface is dominated by the free electrons that behave like a plasma, whereby coherent fluctuations of free electrons are bound at the interface separating a metal from a dielectric (typically air) medium at optical frequencies [1] known as surface plasmon polaritons (SPPs). These charge oscillations represent surface modes with field strengths being the strongest at the interface and decaying exponentially into both media [2], [3]. This property means that SPPs have the innate ability to spatially confine fields within distances vertically from the surface which are significantly smaller than the wavelength in the visible regime, thereby achieving excellent transport and subwavelength confinement of electromagnetic energy in the direction perpendicular to the surface. In 1950, Goubau [4]

Manuscript received August 22, 2017; revised October 31, 2017 and December 17, 2017; accepted December 27, 2017. This work was supported by the Ministry of Science and Technology, Taiwan. (Corresponding author: Malcolm Ng Mou Kehn.) The authors are with the Department of Electrical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2018.2791935

investigated the applicability of nonradiating surface waves for transmission lines. In particular, that seminal paper showed that cylindrical surface waves can be generated with a high efficiency and used for power transmissions. The spoof SPP concept was first realized experimentally on planar geometries [5], [6], but later also extended to wire geometries [7], [8] where high transverse confinement of terahertz and microwave fields has been demonstrated. The simplest example of a textured metallic surface is the so-called domino plasmon [9]–[11], the ability of which in guiding waves that graze across the ridges of the metal-grounded conducting bricks but yet tightly confining fields close to the surface has been successfully demonstrated. Inspired by [4] and [7], the structure with a modulated surface that supports spoof SPPs which is investigated in this paper is the transversely corrugated cylindrical conducting rod with generally dielectric-filled grooves. It has been found that such corrugated cylindrical rods are good surface-wave guiding structures for long-distance microwave transmission [12] and find applications also as endfire antennas [13]. Also addressing major challenges in photonics, optoelectronics [14], as well as material and biological sciences [15], this ability of energy concentration into subwavelength domains offers promising applications in numerous sectors of industries such as medical, security, pharmaceutics [16], astronomy, earth sciences, and even fabrication technologies. Providing resolutions up to the nanometer or subwavelength scales, these applications include sensing [17]–[22], imaging [23]–[25], spectroscopy [14], [23], [26], near-field optics, microscopy [27], diagnostics, probing, spectroscopic fingerprinting for identification (typically of chemical composition), detection (even of single molecule), and amongst many others. Typical techniques for treating such a periodic structure like the corrugated rod including the method of moments involving Floquet theorem as well as brute-force mesh-upand-solve numerical approaches such as the finite-difference time-domain technique and the finite-element method, all of which being full-wave rigorous solutions. These approaches, despite being accurate, are computationally intensive in terms of both processor speed and computer memory. Moreover, they are complicated and thus difficult for any general out-of-field researcher without the relevant background to formulate and construct the program code for his own investigations. The elegant modal method presented in this paper, on the other hand, is simple to formulate and code up, but yet highly accurate and computationally efficient. It can thus be readily

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

picked up by most investigators. Based on the classical vector potential analysis [28] entailing standard boundary conditions (continuity of tangential fields across interfaces), the key aspect that distinguishes this approach lies with the employment of an additional set of vital boundary conditions known as the asymptotic corrugations boundary conditions (ACBCs), defined originally in [29] and extended thereafter in [11]. The ACBCs have also been applied in [30] for the calculation of scattering from corrugated cylindrical rods. This paper shall take this structure further into the context of plasmonics, metasurfaces, and surface-waveguides. Because of its ability in concentrating fields on the surface, the physical extent of which may be made arbitrarily small by reducing the diameter, the corrugated rod is a suitable candidate for high-powered, high frequency (HPHF) applications such as electron cyclotron resonance heating systems for achieving nuclear fusion [31], plasma heating and diagnostics [32], particle accelerators or colliders in high-energy physics [33], [34], as well as cardiac catheter ablation for the medical treatment of arrhythmias by ohmic heating of heart tissues [35], [36]. However, these also raise the need to investigate the pitfalls associated with such HPHF scenarios that often also involve microwave transmissions over long distances. Namely, the power losses arising from nonperfect metals of finite conductivities and lossy dielectric fillings of grooves are the two issues of concern. All these shall be investigated here. The closest traditional relative, or rather, contender to this corrugated rod would be the likewise transversely corrugated circular waveguide, which unlike its open-natured rod counterpart, is a closed structure instead. Comparisons between these two versions in terms of their propagation and attenuation shall also be carried out. Besides the involvement of a different coordinate system and dissimilar kinds of mathematical functions in the modal analysis, Ng Mou Kehn [11] studied neither conductor nor dielectric losses, unlike this paper which looks into all these aspects and has compared them with those of a contemporary alternative. Moreover, no experimental work was reported in [11], whereas results of measurements on a fabricated prototype of the corrugated rod shall be presented herein. A brief outline of this paper is as follows. Sections II–VIII shall be dedicated to the presentation of the formulation for treating both the transversely corrugated open rod as well as its likewise concentrically grated closed waveguide counterpart. Validation with an independent full-wave commercial software then ensues in Section IX. Parametric studies on the dispersion properties are conducted in Section X. The studies of dielectric attenuation and conductor losses are carried out, respectively, in Sections XI and XII. Outcomes of measurements carried out on a manufactured corrugated rod that corroborate the theory are reported in Section XIV. These are then followed by a conclusion that discusses and summarizes the main aspects of this paper. II. V ECTOR P OTENTIAL A NALYSIS Before commencing, the parameters that describe the geometries of the two types of waveguiding structures,

Fig. 1. Geometry of the transverse corrugated (a) conducting cylindrical rod and circular waveguide. (b) Perspective. (c) Lateral view. Both have grooves generally filled by dielectric with parameters (εgrv , μgrv ). For both, inner radius is a, outer radius b (corrugation depth = b − a), period dz , and groove width g( b) = L m κρ  ρ TE TM  prop

prop

× Cφ  cos(mφ) + Dφ  sin(mφ) ×e

TE TM prop − jβz z

TE TM

(2)

in all of which  = { FA }, and whereby the upper and lower items within the two-element column vectors within curly braces correspond to each other throughout all expressions. The modal integer m denotes the number of full-cycle variations along the circumferential φ coordinate whereas the likewise integer q indicates how many half-cycle field variations are there along the axial z-direction but just within the groove. The other symbols in (2) are defined as follows: K α rod exterior: ρ>b

prop = ext core , L = J , κ = β ; for waveguide core: 0 − kprop (4) κρ  = ± βz < k prop TE TM

with kprop = ω(μprop εprop)1/2 . In this way, the attenuation constant αρext along the radial ρ direction in the exterior region of the rod, being common to both modal groups, yields a real and positive value, thus being the characteristic of slow surface waves that propagate without attenuation along the axial z-direction with real phase constant βzext but decay in amplitude with increasing radial distance from the surface of the cylindrical rod structure.

3

C. Standard Boundary Conditions: Vanishing Tangential E-Fields Over Conducting Surfaces The next step is the enforcement of standard boundary conditions, which for the present situation, only involves the requirement that the tangential electric fields are to vanish over all conducting surfaces within the groove, stated as follows:   grv  grv  grv E φT ζ ρ = ab = 0; E z T ζ ρ = ab = 0; E φT ζ (z = −g/2) = 0 grv

grv

(6) whereby the upper and lower items in the triangular brackets pertain, respectively, to the open rod and closed waveguide versions, a rule that shall henceforth hold throughout this paper. These relations are all to be satisfied separately and individually for both ζ ≡ E & M, i.e., both modal groups. Upon defining the following:  Bm (βρ), w = 0, {w} Bm (βρ) = B = J or Y or K (7) Bm (βρ), w = 1, in which the prime attached to the Bessel function bears the usual meaning of the first derivative with respect to the argument, then applying the above boundary conditions separately for the TEz and TMz modal groups, the z-components of the vector potentials for the groove region become grv

z

⎡

  grv ⎢ grv = CTE  ⎢ ⎣Jm βρ  ρ − TE TM

TM



The various field components of both modal groups are then written for the two regions of each waveguiding type, as follows: reg

1 ∂z reg ρTE  = {∓} εreg  TM μreg ρ ∂φ   reg reg z  = 00 ρ  = TE TM

TE TM

reg

1 ∂z reg φ  = {±} εreg  TE ∂ρ μreg TM

reg

∂ 2 z ; TE j ωμreg εreg ρ ∂φ∂z TM

2  ∂ 1 reg reg 2 z   = + kreg z TE j ωμreg εreg ∂z 2 TM reg

φ  =

1

reg

∂ 2 z ; j ωμreg εreg ∂ρ∂z 1

(5)

where  = { EH },  = { EH },  = { FA }, and the subscript “reg” may represent either “grv,” “ext,” or “core” (there is thus a set of these above relations for each of these three scripts denoting the region).

⎤  ⎥ grv ⎥ a  Ym βρTE  ρ ⎦

  {1 } grv Jm0 βρ  ab TE TM

{10 }

Ym

 grv βρTE 

b

TM

grv

TM



grv

× Cφ  cos(mφ) + Dφ  sin(mφ) ×



TE TM

sin cos

 qπ z g

TE

+

qπ 2

 TM (8)

with grv grv CTE  = CρTE  TM

B. Field Components

grv

E ρT ζ (z = −g/2) = 0; E φT ζ (z = g/2) = 0; E ρT ζ (z = g/2) = 0

TM



grv  Dgrv z TE . C zTM

These updated potentials then lead to correspondingly refreshed field components within the groove according to (5) for reg ≡ grv. III. A SYMPTOTIC C ORRUGATION B OUNDARY C ONDITIONS With the platform established by the vector potential analysis of the foregoing section, the ACBC, as stated below, are ready to be employed   grv  grv  E φTE ρ = ba + E φTM ρ = ba = 0  (9)  prop  prop  E φTE ρ = ba + E φTM ρ = ba = 0   grv  grv  E zTE ρ = ba + E zTM ρ = ba   prop  prop  = E zTE ρ = ba + E zTM ρ = ba   grv  grv  HφTE ρ = ba + HφTM ρ = ba 

  prop prop = HφTE ρ = ba + HφTM ρ = ba .

(10) (11) (12)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Assuming q = 0 in the grooves for both TEz and TMz modes, constituting an intrinsic attribute of the ACBC, it is readily found that this condition results in a null TEz modal set in the groove region. The prevailing TMz modal fields are then given by grv  grv  grv (13) E ρTM q=0 ≡ E φTM q=0 ≡ HzTM ≡ 0 ⎧ grv  ⎫ ⎡ $% $% ⎤  0  grv a  0 ⎪ ⎨ E zTM q=0 ⎪ ⎬ 0  0    β J ρ grv  m grv grv TM HρTM q=0 =⎣ Jm 1 βρTM ρ −  grv ba  Ym 1 βρTM ρ ⎦ ⎪ ⎪ Ym βρTM b ⎩ H grv  ⎭ φTM q=0 ⎫ ⎧ − jω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m ⎪ ⎬ ⎨ grv μgrv ρ × CTM × grv ⎪ ⎪ ⎪ βρTM ⎪ ⎪ ⎪ ⎪ ⎪ − ⎭ ⎩ μgrv ⎧ grv ⎫ ⎡⎧ grv ⎫ ⎤ ⎪ ⎪ ⎨ CφTM ⎪ ⎨ DφTM ⎪ ⎬ ⎬ grv ⎢ grv ⎥ × ⎣ DφTM cos(mφ)+ −CφTM sin(mφ)⎦ ⎪ ⎪ ⎪ ⎪ ⎩ C grv ⎭ ⎩ D grv ⎭ φTM φTM (14) whereby this time, every curly brace contains three elements, each of which again corresponds to one another throughout the expression. It is reminded that the integers in the superscripted curly braces of the Bessel functions denote the order of the derivatives. It is then easily seen that (9) is satisfied, on one hand, by the absence of TEz modal fields within the grooves as already grv asserted, and on the other hand, via the null E φTM component from (13). As for (10), using the potentials of (2) in (5) to obtain the E φ component of the exterior region for both modal types, this condition may be written as   prop  prop prop prop κρTE L m κρTE ba CφTE cos(mφ) + DφTE sin(mφ) prop  b  κz m L m κρprop TM a b ωμprop   prop a prop × CφTM sin(mφ) − DφTM cos(mφ) = 0.

+

(15)

Likewise using (14) and applying (2) to (5) to acquire the E z components in the grooves as well as the exterior (for the rod) and core (for the waveguide) regions (entailing just the TM mode though since E z = 0 for the TE mode in all regions), the condition of (11) is stated by ⎧ ⎫   grv  a ⎪ ⎪ 

 

 J β ⎨ ⎬ m ρ grv grv grv & − j ωC b − b  TM b  Y β J β TM

i groove index ↓

m

ρTM a

grv

m

ρTM a

Ym βρTM b ⎪  ⎩  grv univ grv × CφTM cos(mφ) + DφTM sin(mφ) e− j kz idz

∼ = ∓

dz →0

a

⎪ ⎭

 prop 2  prop  κρTM L m κρTM ba j ωμpropεprop

×

' () *

prop CφTM cos(mφ)

+

prop  prop DφTM sin(mφ) e− j βz z

Based on the same arguments (for the same reasons), the final condition of (12) is readily converted to the following:   grv a  grv 

 

 β J ρ m grv βρTM grv grv TM b  grv  Ym βρTM ba Jm βρTM ba − −CTM μgrv Ym βρTM ab   grv grv × CφTM cos(mφ) + DφTM sin(mφ) prop

=

βz

m

 b  L m κρprop TE a b

ωμpropεprop a  prop  prop × CφTE sin(mφ) − DφTE cos(mφ) prop

−

 κρTM   prop b  prop prop L m κρTM a CφTM cos(mφ) + DφTM sin(mφ) . μprop (18)

At this juncture, the pertinent ACBC equations are (15), (17), and (18).

kzuniv



i dz − g/2 < z < i dz + g/2, vanishing elsewhere along the axial coordinate. This summation thus represents a stepwise staircase approximation that discretely models a continuous variation with z, one which matches with the exponential term on the right side of the equation. Notice also the introduction of a so-called universal axial phase constant k zuniv to the groove fields (on the left-hand side of the equation) that shall be common throughout the structure, particularly, for both the groove as well as the exterior and core regions in order prop for phase continuity (explaining the brace that equates βz univ to k z on the right-hand side), and also to be shared by both modal groups. The equality between the left- and righthand sides of this (16) approaches exactness as the period dz tends to zero. This relation thus expresses the asymptotic homogenization of the groove region as the unit cell becomes infinitesimally small. Focusing within the i th groove only, i.e., idz − g/2 < z < i dz + g/2, the summation over the groove index i in (16) is removed and the exponential term on the right-hand side univ is replaced by e− j kz z |idz −g/2 1), the impedance −1 matrix, Z P , (Z P = Y P ) tabulating the self- and mutual resistances and inductances of all ports is obtained by L number of simulations. In each simulation, the voltage source is connected to one port while the nodes in the remaining ports are connected to the ground. Then, the currents on all ports are computed and one column of the admittance matrix Y P for the structure is obtained by multiplying the column vector of the solution [I; ] with the row vector VT pertinent to the considered port at each simulation. In a succinct way, this operation can be expressed as ⎤ ⎡ .. ⎡ ⎤ −1 . T ⎥ ⎢ Z −A R ⎢ Vl ⎥ (11) Y P = ⎣ · · · VlT · · · ⎦ ⎦ ⎣ AR 0 .. . where l = 1, . . . , L, Y P is complex symmetric since A R and V are real and Z is complex symmetric. For the structures with single conductor, defining a port and connecting at least one node in the port to the ground eliminates the static null space of dimension one, which gives

YUCEL et al.: VoxHENRY: FFT-ACCELERATED INDUCTANCE EXTRACTION FOR VOXELIZED GEOMETRIES

rise to slow convergence of the iterative solution of the linear system of equations in (9). For the structures comprising of several separate conductors, the potential of one node on each conductor has to be set to zero to eliminate the nullspaces associated with each conductor. III. N UMERICAL S OLUTION A. Sparse Preconditioner To ensure the rapid convergence of iterative solution of (9), FastHenry utilizes a mesh-based approach in conjunction with a local-inversion preconditioner [9] along with other positive definite extensions. That said, implementation of mesh-based approach for the VoxHenry appears to be nontrivial due to the existence of the piecewise-linear basis functions in the basis function set. Therefore, we here propose a sparse preconditioner to accelerate the iterative convergence of VoxHenry, which reads   T −1 −1 Y −A R P=O = (12) AR 0 where Y is a diagonal matrix with entries corresponding to magnitudes of diagonal entries of Z. During the iterative solution of (9), the sparse preconditioner P is applied to both sides of (9) as       T −1   T −1 T   V Y −A R Y −A R Z −A R I = . 0  AR 0 AR 0 AR 0 (13) To perform the multiplications with sparse preconditioner, the standard application of Schur complement [16] to the solution of linear systems of equations can be leveraged. To do that, let the column vector [a; b] represents the result of MVM at the right-hand side of (9) at each iteration. The multiplication of the column vector with the sparse preconditioner results in a column vector [c; d], i.e.,     T −1   a c Y −A R . (14) = b d AR 0 The resulting column vector [c; d] is obtained in two steps: in the first step, d is obtained as −1

d=S

(b − A R Y

−1

a)

(15)

where S is the Schur complement of the block Y of the −1 T matrix O, i.e., S = A R Y A R . In the second step, c is obtained using d as c=Y

−1

T

(a + A R d).

(16)

In (15) and (16), computing the inverse of Y is trivial as Y is a diagonal matrix. In addition, inversion of Schur complement S requires minimal computational resources when performed with LDLT decomposition in Suitesparse package [17] since S is a sparse, positive definite, and symmetric real matrix. The effectiveness of the proposed sparse preconditioner is demonstrated in the numerical example in Section IV-A4, in which the number of iterations is a few at low frequencies while that is around twenty at the maximum frequency.

1727

B. FFT Acceleration As discussed in Section II-B, the computational cost of β,α MVMs involving the blocks in Z, Z , can be reduced 2 from O(K ) to O(K t log K t ) by using FFTs. To do that, the block Toeplitz tensors G β,α , β, α = {x, y, z, 2D, 3D}, β,α corresponding to the blocks of MoM matrix in (10), Z , β, α ∈ {x, y, z, 2D, 3D} are obtained. Next, each block Toeplitz tensor G β,α with dimensions K x × K y × K z is embedded in a block circulant tensor C β,α with dimensions 2K x × 2K y × 2K z ; the procedure to obtain each block Toeplitz and circulant tensor is expounded in Appendix B. Once the β,α are FFTs of all circulant tensors C β,α corresponding to Z β,α computed and stored in C˜ , those are used to compute MVM β,α β,α involving Z , which is CCβ = Z Iα , as    β,α α C˜ (17) ∗ I˜ . C C β = cβ I β + IFFT α

Here ∗ and IFFT stand for tensor–tensor multiplication and inverse FFT operator, I α is the tensor with dimensions 2K x × 2K y × 2K z that contains the unknown current coefficients α I α }. corresponding the basis function f α (r), and I˜ = FFT{I β c is a constant which equals to 1/(σ x) for β = {x, y, z}, 1/(σ 6x) for β = 2D, and 1/(σ 2x) for β = 3D. Before tensor–tensor multiplication, I α should be properly filled via Iα and zeros. After the multiplication, the result CCβ should be selected from the entries of C C β . It is worth noting that β,α the memory required to store all C˜ scales as O(K t ). On the other hand, if the conventional MoM scheme was β,α would scale as O(K 2 ), used, the memory for storing all Z which is prohibitive for large-scale problems. To validate the computational complexity and memory scaling of the FFT accelerated simulator, the computational time required for the MVMs involving Z and the memory requirement for storing all circulant tensors in each numerical example of Section IV are shown in Fig. 5(a) and (b). Needless to say, wall time for the MVMs for all numerical examples scales with O(K t log K t ) β,α [Fig. 5(a)], while the memory required to store all C˜ for all numerical examples scales with O(K t ) [Fig. 5(b)]. IV. N UMERICAL R ESULTS This section presents several numerical results that demonstrate the accuracy, efficiency, and applicability of the VoxHenry simulator. In all examples below, the conductivity of the conductors is set to 5.8 × 107 S/m and the structures are excited by voltage sources. When applicable, the results obtained by VoxHenry are compared with those obtained by FastHenry or analytical expressions. The discrepancy between results are quantified through L 2 norm error, which is defined as   N f N f ˜ f i )|2 ˜ fi )|2 (18) err = |F( f i ) − F( | F( i=1

i=1

where F denotes the quantity of interest (e.g., inductance and resistance) computed by VoxHenry while F˜ represents the quantity of interest computed either by FastHenry or analytical expression, unless stated otherwise. All simulations are performed on an Intel Xeon CPU E5-2680 v4 processor.

1728

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

Fig. 5. (a) Computational time for MVM involving Z and (b) memory requirement for storing all circulant tensors in numerical examples in Section IV.

A. Validation Examples 1) Straight Conductor With Square Cross Section: VoxHenry and FastHenry are used to obtain frequency-dependent resistance and inductance of a straight conductor with dimensions 10 × 10 × 30 μm (width × height × length). The structure is connected to a port from both ends and excited at frequencies ranging from 1 Hz to 10 GHz (four equally spaced frequency points at each decade). The structure is discretized by voxels of size 0.25 μm, which gives rise to N = 960 000 and M = 587 200. In Fig. 6(a) and (b), the frequency-dependent resistance and inductance obtained by VoxHenry are compared with those obtained by FastHenry. Perfect agreement between results is observed; the L 2 norm error between the results is computed as 0.010 and 0.0013 for the resistance and inductance, respectively. Furthermore, the current distribution on the structure at 2.5 GHz is shown in Fig. 6(c) and the skin effect is clearly observed. It should be noted that the VoxHenry computes the frequency-dependent inductance and resistance at 41 frequency points in 20.5 min, which shows the efficiency of VoxHenry for a structure discretized with 192 000 voxels. 2) Wire: Next, the frequency-dependent resistance and inductance of a wire with length 50 μm and radius 5 μm are computed using VoxHenry and FastHenry. The structure is connected to a port from both ends and excited at frequencies ranging from 1 Hz to 10 GHz (four equally spaced frequency points at each decade). The structure is discretized by voxels of size 0.2 μm (N = 2 470 000 and M = 1 508 976), 0.25 μm (N = 1 264 000 and M = 775 664), 0.5 μm (N = 158 000 and M = 99 116), and 1.0 μm (N = 20 000 and M = 13 080). The frequencydependent resistance and inductance of the wire discretized with voxels of size 0.25 μm are computed by the VoxHenry, FastHenry, and analytical expressions [18] [Fig. 7(a) and (b)]. The L 2 norm error (with respect to the analytical solution) in resistance and inductance computed by the VoxHenry is 0.063 and 0.0062, and in those computed by the FastHenry is 0.062 and 0.006, respectively. It should be noted here that

Fig. 6. Frequency-dependent (a) resistance and (b) inductance of a straight conductor with square cross section computed via VoxHenry and FastHenry. (c) Current distribution on the structure at 2.5 GHz.

for both simulators, L 2 norm error, which is computed with respect to analytical solution, does not decrease when the voxel size is reduced to 0.2 μm. This can be explained by the fact that the accuracy of the analytical expressions is limited due to the approximations used during their derivation. To this end, resistance and inductance values obtained by the VoxHenry and FastHenry for the wire discretized with voxels of size 0.2 μm are used as the reference solutions [ F˜ in (18)] to compute L 2 norm errors. For the wire discretized with voxels of size {1.0, 0.5, 0.25} μm, L 2 norm errors in resistance and inductance are found to be {3.02, 0.13, 0.11} × 10−2 and {4.18, 0.36, 0.04} × 10−3 for the VoxHenry and {3.18, 1.33, 0.11} × 10−2 and {4.74, 2.64, 0.06} × 10−3 for the FastHenry. The accuracy of the solutions obtained by both of the simulators exhibit a similar convergence rate. Simulation times required by the VoxHenry and FastHenry to compute the resistance and inductance at 41 frequency points are {0.20, 2.26, 33.12, 112.21} and {0.18, 3.43, 64.63, 202.45} min, respectively. Furthermore, the memory requirements of the VoxHenry and FastHenry are {0.03, 0.45, 11.12, 30.65} and {0.06, 0.56, 5.45, 11.97} GB, respectively. It should be noted here that the two simulators have implementation differences. While the VoxHenry executes a multithreaded MATLAB code, the FastHenry is built on a serial C code. It should also be noted that the FastHenry is not applicable to all voxelized structures (such as the one in Section IV-A6)

YUCEL et al.: VoxHENRY: FFT-ACCELERATED INDUCTANCE EXTRACTION FOR VOXELIZED GEOMETRIES

1729

Fig. 7. Frequency-dependent (a) resistance and (b) inductance of a wire computed via VoxHenry, FastHenry, and analytical formula. (c) Current distribution on the wire at 2.5 GHz.

since it cannot assign nodes on all faces of the voxels and model the rotating currents on voxels due to corner issue explained in Section II-B. Finally, Fig. 7(c) shows the current distribution computed by the VoxHenry inside the wire discretized with voxels of size 0.25 μm. Fig. 7(c) clearly shows the skin depth phenomenon. 3) Parallel Straight Conductors: Two parallel conductors with dimensions 10 × 5 × 30 μm (width × height × length) positioned 10 μm apart from each other (edge-to-edge) are considered. The conductors lie parallel to the xy plane and are connected to the ports from their ends. The structure is excited at frequencies ranging from 1 Hz to 10 GHz (four equally spaced frequency points at each decade) and discretized by voxels of size 0.25 μm (N = 1 600 000 and M = 985 600). In Fig. 8(a)–(d), the frequency-dependent self-resistance, mutual resistance [19], self-inductance, and mutual inductance of a conductor obtained by VoxHenry and FastHenry are compared. Apparently, an excellent match between the results is obtained; L 2 norm error between the results is computed as 0.01, 0.008, 0.001, and 0.0009 for the self-resistance, mutual resistance, self-inductance, and mutual inductance, respectively. In Fig. 8(e), the current distribution on the structure at 2.5 GHz is plotted when it is excited from one port; the skin and proximity effects [20] on the conductors are clearly observed. Note that the VoxHenry completes the simulation at 41 frequency points in 45.75 min.

Fig. 8. In a two parallel conductor structure, the frequency-dependent (a) self-resistance, (b) mutual resistance, (c) self-inductance, and (d) mutual inductance of a conductor computed via VoxHenry and FastHenry. (Note that the frequencies below 10 kHz are not considered while plotting mutual resistance as the mutual resistance at these frequencies is less than the machine precision.) (e) Current distribution on both conductors at 2.5 GHz.

4) Square Coil: A square coil with length 100 μm, which is formed by conductors with square cross section 5 × 5 μm, is considered. The square coil lies on the xz plane and its lower arm is formed by two conductors with length 49 μm. The structure is connected to a port defined in the spacing between the conductors in the lower arm. The structure is excited at frequencies ranging from 1 Hz to 10 GHz (four equally spaced frequency points at each decade) and discretized by voxels of size 0.25 μm (N = 3 024 000 and M = 1 875 280).

1730

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

Fig. 9. Frequency-dependent (a) resistance and (b) inductance of a square coil computed via VoxHenry and FastHenry. (c) Current distribution on the square coil at 2.5 GHz. (d) Frequency versus number of iterations required to reach to relative residual error 10−8 during iterative solution of linear system of equations (with N = 3 024 000 and M = 1 875 280).

In Fig. 9(a) and (b), the frequency-dependent resistance and inductance of the square coil obtained by VoxHenry and FastHenry are compared. Clearly, a very good match between the results is observed; L 2 norm error between the results is computed as 0.019 and 0.0038 for the resistance and inductance, respectively. In Fig. 9(c), the current distribution on the structure at 2.5 GHz clearly shows the skin and proximity effects on the conductors. VoxHenry obtains the resistance and inductance at 41 frequency points in 47 min. For each frequency point, the number of iterations required to reach to relative residual error of 10−8 during the iterative solution of (9) is given in Fig. 9(d). Apparently, a couple of iterations are needed at lower frequencies while 22 iterations are need to obtain accurate solution at the highest frequency. The fast convergence of the iterative solution shows the effectiveness of the sparse preconditioner.

Fig. 10. Frequency-dependent (a) self-resistance, (b) self-inductance, and (c) mutual inductance of a square coil computed via VoxHenry and FastHenry. (d) Current distribution on both square coils at 3 GHz.

5) Two Parallel Square Coils: Two square coils with the specifications given in the previous numerical example are positioned parallel to each other and the spacing between them (center-to-center) is varied from 6 μm to 600 μm. Both coils are connected to ports defined in the spacing between the conductors in their lower arms and excited at 3 GHz. The structure is discretized by voxels of size 0.5 μm (N = 756 000 and M = 484 040). The frequency-dependent self-resistance, self-inductance, and mutual inductance of one square coil obtained by VoxHenry and FastHenry are shown in Fig. 10(a)–(c). An excellent match between results is observed; L 2 norm error between the results is calculated as 0.0029, 0.0084, and 0.0058 for the self-resistance, self-inductance, and mutual inductance, respectively. Furthermore, the current distribution on the structure is plotted when the distance between coils is 20 μm [Fig. 10(d)]. For this configuration, VoxHenry obtains the current distribution, resistance, and inductances in 4.85 min.

YUCEL et al.: VoxHENRY: FFT-ACCELERATED INDUCTANCE EXTRACTION FOR VOXELIZED GEOMETRIES

1731

Fig. 11. (a) Frequency-dependent self-inductance of a circular coil computed via VoxHenry with different voxel sizes and analytical formula. (b) Current distribution on circular coil at 3 GHz.

6) Circular Coil: In the final validation example, a circular coil with loop radius 150 μm and tube radius 5 μm is considered. The coil is connected to a port defined in a spacing formed between x = 155.0 μm and x = 155.0 + a μm in the lower part of the coil, where a is the voxel size, and excited at frequencies ranging from 1 Hz to 1 THz (four equally spaced frequency points at each decade). The structure is discretized by the voxels of size 0.5 μm (N = 2 969 300 and M = 1 867 528), 1 μm (N = 373 400 and M = 245 488), and 2 μm (N = 47 695 and M = 33 988). The frequencydependent inductance of circular coil obtained by VoxHenry with different voxel sizes and analytical formula [18] are compared in Fig. 11(a). Perfect agreement between the VoxHenry and analytical results is observed for the voxel size of 0.5 μm. L 2 norm error between the analytical result and VoxHenry result for the voxel size of 0.5 μm, 1 μm, and 2 μm is obtained as 8.09 × 10−4 , 1.9 × 10−3 , and 5.4 × 10−3 , respectively. Furthermore, the current distribution on the coil obtained by VoxHenry for voxel size of 0.5 μm is shown in Fig. 11(b). It is worth mentioning that VoxHenry computes the inductance on 49 frequency points in 144, 13.5, and 2.7 min and requires 25.5, 2.64, and 0.3-GB memory for the voxel size of 0.5 μm, 1 μm, and 2 μm, respectively. B. Large-Scale Examples 1) Square Coil Array: A 3-by-3 array of square coils with center-to-center spacing 30 μm is considered [Fig. 12(a)]. Each square coil with length 20 μm is formed by conductors with square cross section 4 × 4 μm and lies on the xy plane; its lower arm parallel to x-axis is formed by two conductors with length 9 μm and a port is defined in the spacing between these conductors for each coil. The coils are excited (one at a time) at frequencies ranging from 1 Hz to 10 GHz (four equally spaced frequency points at each decade). Two different scenarios are considered. In the first scenario, the coils are discretized by voxels of size 0.25 μm (N = 2 856 960 and M = 1 787 904). In the second scenario, the coils are situated over a square ground plane with dimensions 100×100×2 μm (width × length × height) and whole structure is discretized by

Fig. 12. (a) Geometry of square coil array. The frequency-dependent (b) self-resistance and (c) self-inductance of the fifth coil when the ground plane does not exist and exists. (d) Frequency-dependent mutual inductances of fifth coil with the second, fourth, and sixth coils when the ground plane does not exist and exists. The current distribution on the structure excited from the fifth port at 2.5 GHz when the ground plane (e) does not exist and (f) exists.

voxels of size 0.25 μm (N = 9 256 960 and M = 5 794 304); the spacing between coils and ground plane is 2 μm. The frequency-dependent self-resistance and self-inductance of the fifth coil [i.e., the coil in the middle of array

1732

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

TABLE I D ETAILED B REAKDOWN OF THE M EMORY U SAGE AND WALL T IME FOR THE S QUARE C OIL A RRAY E XAMPLE . U NITS FOR M EMORY AND WALL T IME ARE GB AND s.

in Fig. 12(a)] obtained in both scenarios are shown in Fig. 12(b) and (c). Furthermore, the mutual inductances between fifth coil and second, fourth, and sixth coils computed in both scenarios are shown in Fig. 12(d). In Fig. 12(b) and (d), the effect of ground plane on the quantities of interest is apparent. For the analysis at 2.5 GHz, the current distributions on the structure obtained in both scenarios are shown [Fig. 12(e) and (f)]. In the same analysis, the detailed breakdown of the memory usage and wall time is provided in Table I. Apparently, wall time for MVM involving sparse preconditioner P is nearly the same as that for MVM involving MoM matrix Z in both scenarios. 2) RF Coil Array: A 2-by-2 array of RF coils with centerto-center spacing 50 μm is analyzed by VoxHenry [Fig. 13(a)]. Each square RF coil with maximum length 40 μm is formed by conductors with square cross section 3 × 3 μm; its arm parallel to x-axis and below z = 0 is formed by two conductors with length 7 μm and a port is defined in the spacing between these conductors for each coil. The coils are excited (one at a time) at frequencies ranging from 1 Hz to 10 GHz (four equally spaced frequency points at each decade). Again, two different scenarios are considered. In the first scenario, the RF coil array is discretized by voxels of size 0.25 μm (N = 4 089 600 and M = 2 590 656). In the second scenario, a square ground plane with dimensions 110 × 110 × 2 μm (width × length × height) is placed 2 μm below the bottom of coils and whole structure is discretized by voxels of size 0.25 μm (N = 11 833 600 and M = 7 437 696). The frequency-dependent self-resistance and inductance of the first RF coil [i.e., the coil on the left bottom of the array in Fig. 13(a)] as well as the mutual inductances between the first RF coil and the remaining RF coils in the array obtained in both scenarios are shown in Fig. 13(b)–(d). The effect of ground plane on the quantities of interest is clearly

Fig. 13. (a) Geometry of RF coil array. The frequency-dependent (b) self-resistance and (c) self-inductance of the first RF coil. (d) Frequencydependent mutual inductances of first RF coil with remaining coils when the ground plane does not exist and exists. The current distribution on the structure excited from the first port at 2.5 GHz when the ground plane (e) does not exist and (f) exists.

seen in the figures. Furthermore, the current distributions on the structure obtained in both scenarios are plotted for the analysis at 2.5 GHz [Fig. 13(e) and (f)]. For the same analysis, the detailed breakdown of the memory usage and wall time is given in Table II. In the first scenario, wall time for MVM involving P is the half of that for MVM involving Z. In the second scenario, it is comparable with that for MVM involving Z. V. C ONCLUSION VoxHenry, an FFT-accelerated inductance extractor, was presented for computing frequency-dependent inductances and

YUCEL et al.: VoxHENRY: FFT-ACCELERATED INDUCTANCE EXTRACTION FOR VOXELIZED GEOMETRIES

TABLE II D ETAILED B REAKDOWN OF THE M EMORY U SAGE AND WALL T IME FOR THE RF C OIL A RRAY E XAMPLE . U NITS FOR M EMORY AND WALL T IME ARE GB AND s.

1733

To obtain the linear system of equations in (9), first the current density J(r) in (1) is expanded in terms of basis functions fkα (r), then the resulting equation is tested with β fl (r), where α, β ∈ {x, y, z, 2D, 3D}. The resulting entries β,α of the blocks in Z and Z , are   δlk x,x −w Zlk = G(r, r )d V  d V (20) σ x Vl Vk 2D,2D

Zlk

=

δlk σ (6x)   w − (x −x ) (x  −x k )G(r, r )d V  d V l (x)2 Vl Vk       + (y − yl ) (y − yk )G(r, r )d V d V Vk

Vl

(21) 3D,3D

Zlk

δlk = σ (2x) w − (x)2



 (x −xl ) Vl

Vl

A PPENDIX A T E NTRIES OF Z, A , AND I y

Assume that the basis functions fkx (r), fk (r), fkz (r), fk2D (r), are numbered between 1 and K , K +1 and 2K , 2K +1 and 3K , 3K + 1 and 4K , and 4K + 1 and 5K , respectively. Then, the entries of unknown current coefficient vector IIk , k = 1, . . . , 5K , (N = 5K ) are ⎧ x ⎪ ⎪ Ik , 1 ≤ k ≤ K ⎪ ⎪ ⎪ ⎪ Iky , K + 1 ≤ k ≤ 2K ⎨ Ik = Ikz , 2K + 1 ≤ k ≤ 3K (19) ⎪ ⎪ 2D ⎪ Ik , 3K + 1 ≤ k ≤ 4K ⎪ ⎪ ⎪ ⎩ I 3D , 4K + 1 ≤ k ≤ 5K . k fk3D (r)

 +4

x,2D

y,2D

Zlk

z,3D

Zlk

2D,3D

Zlk

Vk

     (z −z l ) (z −z k )G(r, r )d V d V

Vl

Zlk

(x  −x k )G(r, r )d V  d V

 (y − yl ) (y  − yk )G(r, r )d V  d V



+ resistances of the structures discretized by voxels. VoxHenry solves the electric volume integral and current conservation equations after discretizing the currents on structures by a carefully selected basis function set, which allows accurately modeling the currents on arbitrarily shaped voxelized structures. During the iterative solution of such equations, VoxHenry leverages FFTs to accelerate MVMs and a sparse preconditioner to ensure the rapid convergence of iterative solution. Numerical results showed the applicability, accuracy, and efficiency of the proposed VoxHenry inductance extractor. In specific, VoxHenry computed the frequencydependent inductance of a circular coil with three digit accuracy, showing its accuracy for arbitrarily shaped voxelized structures. Furthermore, VoxHenry completed the simulation of an RF coil array with ground plane for one frequency point in 62 min, which involve the solution of a linear system of equations with 19 271 296 unknowns; this demonstrates the efficiency of the proposed VoxHenry-inductance extractor.

Vk

Vk

(22)

 

w (x  − x k )G(r, r )d V  d V (23) x Vl Vk   w = (y  − yk )G(r, r )d V  d V (24) x Vl Vk   2w = (z  − z k )G(r, r )d V  d V (25) x Vl Vk   w = (x − xl ) (x  − x k )G(r, r )d V  d V (x)2 Vl Vk    =−

−

(y − yl ) Vl

(y  − yk )G(r, r )d V  d V

Vk

(26) where w = j ωμ/(x)4 , δlk = 1 for l = k, otherwise δlk = 0. We should note here that some blocks can be obtained from 2D,x x,2D T 2D,y = (Z ) , Z = the transpose of other blocks: Z y,2D T 3D,z z,3D T (Z ) , and Z = (Z ) . Furthermore, many blocks x,x y,y z,z x,2D x,3D y,2D are the same: Z = Z = Z , Z =Z ,Z = y,3D 2D,x 3D,x 2D,y 3D,y 3D,2D , Z = Z , Z = −Z , and Z = −Z 2D,3D Z . The integrals in (20)–(26) are evaluated using the methods described in [21]–[23]. T β The entries of A are obtained by testing ∇(r) with fl (r), β ∈ {x, y, z, 2D, 3D}. Consider the volume integral resulting from this testing operation  β   fl (r) 1 β , ∇(r) = f (r) · ∇(r)d V . (27) x 2 x 2 Vl l

1734

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

TABLE III S IGNS OF B LOCK T OEPLITZ T ENSOR FOR C ONSTRUCTING E ACH B LOCK IN E ACH C IRCULANT T ENSOR

Toeplitz blocks used to construct the blocks of circulant tensor are assigned as in Table III. Note that this table corresponds to of [3, Table V] and the blocks in circulant tensor are labeled by L, M, N, LM, LN, MN, and LMN, as in [3]. R EFERENCES

Applying the divergence theorem and invoking the divergenceβ free property of fl (r) yields  β  6  fl (r) 1  β , ∇(r) = (r)fl (r) · nˆ l,i d S (28) x 2 x 2 Sl,i i=1

where Sl,i is the i th surface of Vl with outward pointing unit normal nˆ l,i and houses mth node, m ∈ {1, . . . , M}. Evaluation of each surface integral (with nonzero integrand) yields the value of (r) on mth node and a constant. While the former T is the mth entry of , the latter is an entry of A as ⎧ ⎪ ±1, β = x and nˆ l,i = ±ˆx ⎪ ⎪ ⎪ ⎪ ⎪ β = y and nˆ l,i = ±ˆy ⎨±1, T Alm = ±1, β = z and nˆ l,i = ±ˆz ⎪ ⎪ ⎪ {0.5, −0.5}, β = 2D and nˆ l,i = {±ˆx, ±ˆy} ⎪ ⎪ ⎪ ⎩{0.5, 0.5, −1}, β = 3D and nˆ = {±ˆx, ±ˆy, ±ˆz}. l,i (29) A PPENDIX B B LOCK T OEPLITZ AND C IRCULANT T ENSORS To obtain a block Toeplitz tensor G β,α corresponding a β,α in (10), the voxels (empty and nonempty ones) block Z are numbered with indices k = (k x , k y , k z ), k x = 1, . . . , K x , k y = 1, . . . , K y , and k z = 1, . . . , K z . Then, the basis function in the first voxel with the indices (1, 1, 1) is assigned as the basis function. Finally, the entries of the Toeplitz tensor β,α G kx ,k y ,kz is obtained by sweeping over all testing functions

on all voxels with the indices (k x , k y , k z ), k x = 1, . . . , K x , k y = 1, . . . , K y , and k z = 1, . . . , K z , and evaluating the corresponding integral in (20)–(26). [Note that δlk = 0 in (20)–(22) for (k x , k y , k z ) = (1, 1, 1).] For example, x,x the entries of G x,x corresponding to the block Z are obtained by evaluating (20) after setting δlk = 0. A block circulant tensor C β,α corresponding to a block in β,α Z in (10) is obtained by properly embedding the related block Toeplitz tensor G β,α in C β,α . To this end, the procedure explained in Appendix B of [3] is followed. The signs of the

[1] K. Gala, V. Zolotov, R. Panda, B. Young, J. Wang, and D. Blaauw, “On-chip inductance modeling and analysis,” in Proc. 37th Annu. Design Autom. Conf., Los Angeles, CA, USA, 2000, pp. 63–68. [2] M. Fujishima and S. Amakawa, “Recent progress and prospects of terahertz CMOS,” IEICE Electron. Exp., vol. 12, no. 13, pp. 1–7, 2015. [3] A. G. Polimeridis, J. F. Villena, L. Daniel, and J. K. White, “Stable FFT-JVIE solvers for fast analysis of highly inhomogeneous dielectric objects,” J. Comput. Phys., vol. 269, pp. 280–296, Jul. 2014. [4] M. F. Catedra, E. Gago, and L. Nuno, “A numerical scheme to obtain the RCS of three-dimensional bodies of resonant size using the conjugate gradient method and the fast Fourier transform,” IEEE Trans. Antennas Propag., vol. 37, no. 5, pp. 528–537, May 1989. [5] H. Gan and W. C. Chew, “A discrete BCG-FFT algorithm for solving 3D inhomogeneous scatterer problems,” J. Electromagn. Waves Appl., vol. 9, pp. 1339–1357, Apr. 1995. [6] J. R. Phillips and J. K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. Comput.Aided Design Integr. Circuits Syst., vol. 16, no. 10, pp. 1059–1072, Oct. 1997. [7] E. Bleszynski, M. Bleszynski, and T. Jaroszewic, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, no. 5, pp. 1225–1251, Sep./Oct. 1996. [8] R. L. Wagner, J. Song, and W. C. Chew, “Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces,” IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 235–245, Feb. 1997. [9] M. Kamon, M. J. Tsuk, and J. K. White, “FASTHENRY: A multipoleaccelerated 3-D inductance extraction program,” IEEE Trans. Microw. Theory Techn., vol. 42, no. 9, pp. 1750–1758, Sep. 1994. [10] A. C. Yucel, I. P. Georgakis, A. G. Polimeridis, H. Bagci, and J. K. White, “VoxHenry: FFT-accelerated inductance extraction for voxelized geometries,” in Proc. 38th Prog. Electromagn. Res. Symp., Saint Petersburg, Russia, May 2017, p. 91. [11] A. E. Ruehli, “Survey of computer-aided electrical analysis of integrated circuit interconnections,” IBM J. Res. Develop., vol. 23, no. 6, pp. 626–639, 1979. [12] A. C. Cangellaris, J. Prince, and L. P. Vakanas, “Frequency-dependent inductance and resistance calculation for three-dimensional structures in high-speed interconnect systems,” IEEE Trans. Compon., Hybrids, Manuf. Technol., vol. 13, no. 1, pp. 154–159, Mar. 1990. [13] C.-T. Tsai, H. Massoudi, C. H. Durney, and M. F. Iskander, “A procedure for calculating fields inside arbitrarily shaped, inhomogeneous dielectric bodies using linear basis functions with the moment method,” IEEE Trans. Microw. Theory Techn., vol. MTT-34, no. 11, pp. 1131–1139, Nov. 1986. [14] W. T. Weeks, L. L.-H. Wu, M. F. McAllister, and A. Singh, “Resistive and inductive skin effect in rectangular conductors,” IBM J. Res. Develop., vol. 23, no. 6, pp. 652–660, 1979. [15] B. Krauter and L. T. Pileggi, “Generating sparse partial inductance matrices with guaranteed stability,” in Proc. IEEE/ACM Int. Conf. CAD, San Jose, CA, USA, Nov. 1995, pp. 45–52. [16] R. A. Horn and F. Zhang, “Basic properties of the schur complement,” in The Schur Complement and Its Applications. Boston, MA, USA: Springer, 2005, pp. 17–46. [17] T. A. Davis, Direct Methods for Sparse Linear Systems. Philadelphia, PA, USA: SIAM, 2006. [18] Radio Instruments and Measurements—Circular C74, Washington, DC, USA, U.S. Government Printing Office, 1924. [19] S. C. Thierauf, High-Speed Circuit Board Signal Integrity. Norwood, MA, USA: Artech House, 2004. [20] L. Daniel, A. Sangiovanni-Vincentelli, and J. White, “Proximity templates for modeling of skin and proximity effects on packages and high frequency interconnect,” in Proc. IEEE/ACM Int. Conf. CAD, San Jose, CA, USA, Nov. 2002, pp. 326–333. [21] A. G. Polimeridis, F. Vipiana, J. R. Mosig, and D. R. Wilton, “DIRECTFN: Fully numerical algorithms for high precision computation of singular integrals in Galerkin SIE methods,” IEEE Trans. Antennas Propag., vol. 61, no. 6, pp. 3112–3122, Jun. 2013.

YUCEL et al.: VoxHENRY: FFT-ACCELERATED INDUCTANCE EXTRACTION FOR VOXELIZED GEOMETRIES

[22] A. G. Polimeridis, J. F. Villena, L. Daniel, and J. K. White, “Robust J-EFVIE solvers based on purely surface integrals,” in Proc. Int. Conf. Electromagn. Adv. Appl. (ICEAA), Turin, Italy, 2013, pp. 379–381. [23] I. P. Georgakis and A. G. Polimeridis, “Reduction of volume-volume integrals arising in Galerkin JM-VIE formulations to surface-surface integrals,” in Proc. Eur. Conf. Antennas Propag. (EuCAP), Paris, France, 2017, pp. 324–326.

Abdulkadir C. Yucel received the B.S. degree (summa cum laude) in electronics engineering from the Gebze Institute of Technology, Kocaeli, Turkey, in 2005, and the M.S. and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 2008 and 2013, respectively. From 2005 to 2006, he was a Research and Teaching Assistant at the Electronics Engineering Department, Gebze Institute of Technology. From 2006 to 2013, he was a Graduate Student Research Assistant with the Radiation Laboratory, University of Michigan, where he was a Research Fellow from 2013 to 2015. Since 2016, he has been a PostDoctoral Researcher with the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA, and affiliated with the Computational Electromagnetics Group, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia. His current research interests include computational electromagnetics with emphasis on uncertainty quantification for electromagnetic analysis on complex platforms, electromagnetic compatibility and interference analysis, nature-based design of electromagnetic devices, integral equation-based frequency, and time-domain solvers and their accelerators. Dr. Yucel was a recipient of the Fulbright Fellowship in 2006, the Electrical Engineering and Computer Science Departmental Fellowship of the University of Michigan in 2007, and the Student Paper Competition Honorable Mention Award of the IEEE International Symposium on Antennas and Propagation Symposium in 2009. He has been serving as an Associate Editor for the International Journal of Numerical Modelling: Electronic Networks, Devices, and Fields and as a Reviewer for various technical journals.

Ioannis P. Georgakis received the Diploma degree in electrical engineering and computer science from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2014. He is currently pursuing the Ph.D. degree at the Center for Computational Data-Intensive Science and Engineering, Skolkovo Institute of Science and Technology, Moscow, Russia. His current research interests include computational electromagnetics, with an emphasis in volume integral equation methods for magnetic resonance imaging modeling. Mr. Georgakis was a recipient of three scholarships of academic excellence by the State Scholarship Foundation during his undergraduate studies (2010–2012).

Athanasios G. Polimeridis (SM’16) was born in Thessaloniki, Greece, in 1980. He received the Diploma and Ph.D. degrees in electrical engineering and computer science from the Aristotle University of Thessaloniki, Thessaloniki, in 2003 and 2008, respectively. From 2008 to 2012, he was a Post-Doctoral Research Associate with the Laboratory of Electromagnetics and Acoustics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. From 2012 to 2015, he was a PostDoctoral Research Associate with the Massachusetts Institute of Technology, Cambridge, MA, USA, where he was a member of the Computational Prototyping Group, Research Laboratory of Electronics. He is currently an Assistant Professor with the Skolkovo Institute of Science and Technology, Moscow, Russia. His current research interests include computational methods for problems in physics and engineering (classical electromagnetics, quantum and thermal electromagnetic interactions, and magnetic resonance imaging) with an emphasis on the development and implementation of integral-equationbased algorithms. Dr. Polimeridis was a recipient of the Swiss National Science Foundation Fellowship for Advanced Researchers in 2012.

1735

Hakan Ba˘gcı (S’98–M’07–SM’14) received the B.S. degree in electrical and electronics engineering from Bilkent University, Ankara, Turkey, in 2001, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Illinois at Urbana–Champaign (UIUC), Urbana, IL, USA, in 2003 and 2007, respectively. From 1999 to 2001, he was an Undergraduate Researcher with the Computational Electromagnetics Group, Bilkent University. From 2001 to 2006, he was a Research Assistant with the Center for Computational Electromagnetics and Electromagnetics Laboratory, UIUC. From 2007 to 2009, he was a Research Fellow with the Radiation Laboratory, University of Michigan, Ann Arbor, MI, USA. In 2009, he joined the Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia, as an Assistant Professor of electrical engineering. He has authored or co-authored nine finalist papers of the Student Paper Competition of the 2005, 2008, 2010, and 2014 IEEE Antennas and Propagation Society International Symposiums and 2013 and 2014 Applied Computational Electromagnetics Society Conferences. His current research interests include computational electromagnetics with emphasis on timedomain integral equations and their fast marching-on-in-time-based solutions, well-conditioned integral-equation formulations, and the development of fast hybrid methods for analyzing statistical electromagnetic compatibility/ electromagnetic interference phenomena on complex and fully loaded platforms. Dr. Ba˘gcı was a recipient of the 2008 International Union of Radio Scientists Young Scientist Award and the 2004–2005 Interdisciplinary Graduate Fellowship of the Computational Science and Engineering Department, UIUC. His paper “Fast and Rigorous Analysis of EMC/EMI Phenomena on Electrically Large and Complex Structures Loaded with Coaxial Cables” was one of the three finalists (with honorable mention) of the 2008 Richard B. Schulz Best Transactions Paper Award given by the IEEE Electromagnetic Compatibility Society.

Jacob K. White (F’08) received the B.S. degree in electrical engineering and computer science (EECS) from the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA, in 1980, and the S.M. and Ph.D. degrees in EECS from the University of California at Berkeley, Berkeley, CA, USA, in 1983 and 1985, respectively. He was with the IBM T. J. Watson Research Center. He joined EECS, MIT, as an Analog Devices Career Development Assistant Professor in 1987. He is currently the C. H. Green Professor with the EECS Department, MIT, where he is involved in simulation and optimization techniques for problems in medical technology, nanophotonics, and electrical circuits and interconnect; and experimenting with blended computationand maker-centric strategies for teaching control, machine learning, and electromagnetics. Dr. White was a co-recipient of the 2013 A. R. Newton Technical Impact Award in EDA with Keith Nabors for their fast capacitance extraction program FastCap. He was a Presidential Young Investigator in 1988, an Associate Editor of the IEEE T RANSACTIONS ON C OMPUTER -A IDED D ESIGN OF I NTEGRATED C IRCUITS AND S YSTEMS from 1992 to 1996, a member of the Spectre/SpectreRF development team from 1989 to 1999, a Chair of the International Conference on Computer-Aided Design in 1999, served as an Associate Director of the Research Laboratory of Electronics, MIT, from 2001 to 2006, an Academic Research Fellow of Ansoft/Ansys from 2010 to 2016, and served as the MIT EECS Co-Education Officer from 2011 to 2014.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Characterization of PMMA/BaTiO3 Composite Layers Through Printed Capacitor Structures for Microwave Frequency Applications Omodara Gbotemi , Sami Myllymäki, Jani Kallioinen, Jari Juuti, Merja Teirikangas, Heli Jantunen, Marjeta Maˇcek Kržmanc, Danilo Suvorov, Marcin Sloma, and Malgorzata Jakubowska

Abstract— This paper presents the extraction of microwave properties of low-temperature cured inorganic composite materials based on barium titanate (BaTiO3 ). These composite materials exhibit attractive features such that when the volume fraction of the filler contents varied, its electrical properties of high permittivity and moderately low loss tangent can be manipulated to suit different areas of applications. For the extraction of the permittivity and the loss tangent, three different ink particles were developed and printed on the top of interdigitalshaped microwave capacitor. The properties of the inks were extracted from measured results through computer simulations. The obtained results were verified with several types of interdigital capacitor structures of different fingers and linewidths. The effect of the thickness of the ink layer materials on the top of the capacitor structures was likewise investigated. The results show relative permittivity (ε r ) values of 30, 25, and 27 for composite layers printed using inks with Pr. A shape at 67.4 wt% (percentage by weight), Pr. B shape at 66.3 wt%, and Pr. C shape at 67.1 wt% of BaTiO3 , respectively, at 2 GHz. Corresponding loss tangents (tan δ) were 0.065, 0.040, and 0.025. The dielectric properties of the composite materials are influenced by the thickness variation of the ink layers on the capacitor structures. This novel capacitor composite materials would be a promising candidate for printed application in mobile telecommunication operations, especially in the frequency range of 0.5–3 GHz. Index Terms— Barium titanate (BaTiO3 ), microwave, printed interdigital capacitor, RF passives, screen printing.

Manuscript received May 9, 2017; revised October 23, 2017; accepted November 12, 2017. This work was supported in part by Tekes, Finland, in part by Sachtleben Pigments Oy, Finland, in part by Pulse Finland Oy, Finland, in part by NOF Corporation Japan, and in part by the Academy of Finland under Grant 267573 and Grant 273663. (Corresponding author: Omodara Gbotemi.) O. Gbotemi, S. Myllymäki, J. Juuti, M. Teirikangas, and H. Jantunen are with the Microelectronics Research Unit, University of Oulu, Oulu, Finland (e-mail: [email protected]; [email protected]; jajuu@ ee.oulu.fi; [email protected]; [email protected]). J. Kallioinen is with Tioxide Europe Ltd. (e-mail: [email protected]). M. M. Kržmanc and D. Suvorov are with the Jožef Stefan Institute, Ljubljana, Slovenia (e-mail: [email protected]; [email protected]). M. Sloma and M. Jakubowska are with the Department of Silicon Technonoly, Institute of Electronic Materials Technology, 01-919 Warsaw, Poland (e-mail: [email protected]; marcin.sloma@ mchtr.pw.edu.pl). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2781694

I. I NTRODUCTION

T

HE use of printing as a technique for the realization of large-area and low-cost electronic systems has recently gained a remarkable level of interest in the microelectronics industry. Printed electronics offers several benefits over traditional electronics such as transparency, flexibility, and less costly fabrication, and it is growing more rapidly. Printing is expected to further support the realization of electronics on flexible and considerably cheap substrates such as metal foils and plastic [1]. Essentially, the increased need for energyefficient products with tremendous performance and costeffective manufacturing is stimulating the advancement of overall printed electronics market [2]. The technology is deployed and under investigations in several applications such as sensors [3]–[5], radio frequency circuits [6]–[9], solar cells [10], antennas [11], [12], and other electronic components like displays [13]. To realize these systems, it is necessary to develop printable materials with low loss and sufficient permittivity values, especially for high-frequency applications [14]. Functional materials with suitable low loss and high relative permittivity do already exist, but they are high-temperature sintered [15], [16]. The high-temperature sintering or sputtering methods limit their practical usage on the flexible organic substrate. Thus, low curing temperature materials with considerable electrical properties are needed. Lately, several studies have been carried out in the field of dielectric properties of polymer–ferroelectric ceramic composites [17]–[23]. Some of the possible commercially available of polymer–ceramic composites are polystyrene, polyethlene, peek, PTFE as matrix materials with fillers such as alumina, glasses, titania, and silica [15]. Wang et al. [24] prepared polyimide/barium titanate (BaTiO3 ) composite with different BaTiO3 contents from 0 to 90 wt% (0–67.5 vol.%), which yielded relative permittivity ranging from 3.53 to 46.50 and loss tangent from 0.005 to 0.015 at the frequency of 10 kHz. Balaraman et al. [25] utilized hydrothermal process ( Vfwd . The return losses and transducer gain for a matched source impedance are obtained as S12 S21  L in = S11 + (1) 1 − S22  L |S21 |2 (1 − | L |2 ) (2) GT = |1 − S22  L |2 where Si j are the tuner scattering parameters and  L is the load impedance reflection coefficient. In this paper, we extend the coverage definition to account for the above-mentioned high-power considerations. To distinguish it from other works, the coverage-defined items 1–4 will be called large-signal Smith chart coverage, C L S . On the other hand, the term smallsignal Smith chart coverage C S S is used for the conventional coverage, which only considers items 1 and 2. B. Optimization Process Given an impedance tuner topology, the voltages and currents at the circuit nodes are easy to obtain using a basic circuit theory. However, when tuning components are added to the circuit, and the performance needs to be evaluated under different load conditions, analytic approaches become very complex. For this reason, we instead propose an optimizationbased process to obtain the circuit design parameters for a given impedance tuner topology. The goal function to be optimized is the large-signal Smith chart coverage defined previously. Considering a generic impedance tuner topology, the optimization process is schematically described in the flowchart

Fig. 1. Flowchart for the large-signal optimization process which is used to design high-power impedance tuners.

of Fig. 1: a set of possible solutions is generated by the optimization algorithm in Step 0. In Step 1, a load impedance is selected from a large set of impedances uniformly distributed on the Smith chart. In Step 2, the voltages and currents in the circuit are solved for all the possible combinations of bias voltages. Transducer gain, return losses, and voltages across the varactors are calculated in Step 3 for every varactor bias condition. If the coverage requirements defined in Section II-A are met, then this impedance belongs to the coverage region (see Step 4). When all load impedances are evaluated, the Smith chart coverage is calculated (Step 5). This is the goal function to be maximized by the optimization algorithm. The optimization will stop when either the desired Smith coverage or the maximum number of iterations is reached. Otherwise, a new set of circuit parameters are chosen in Step 6 and the process starts over. During all the process, a 50- source impedance is considered. The optimization speed is significantly improved by using a simplified, linear simulation method. For the varactors, it is assumed that the RF voltage swing does not affect the capacitance value. Hence, the varactor is modeled as a capacitance whose value only depends on the bias voltage. This assumption drastically simplifies the simulation process, but its accuracy needs to be verified against nonlinear harmonic balance (HB) simulations. It will be shown later that the difference between the linear approach and HB is not significant, which confirms the validity of the proposed linear simulation methodology. The linear simulations allow us to efficiently predict varactor voltage swing and gain as needed for an approximate evaluation of the large-signal coverage conditions. Note that in the flowchart of Fig. 1, the large-signal Smith chart coverage is used as a goal function. This situation is

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PÉREZ et al.: DESIGN AND LARGE-SIGNAL CHARACTERIZATION OF HIGH-POWER VARACTOR-BASED IMPEDANCE TUNERS

3

referred to as large-signal optimization. If the RF voltage across the varactors is not considered, i.e., the small-signal Smith chart coverage is used as a goal function, the process is called small-signal optimization. III. H IGH -P OWER I MPEDANCE T UNER C IRCUIT D EMONSTRATOR The design methodology presented in Section II is here applied to the design of a varactor-based impedance tuner at 2.45 GHz, targeting 25-W RF power handling. A description of the selected topology and varactor devices is first presented. The effect and necessity of considering the highpower conditions are illustrated with several examples. Finally, the optimization process and the details of the demonstrator implementations are given.

Fig. 2. Schematic of the chosen impedance tuner topology. An antiseries configuration with a different number of parallel stacking is used.

A. SiC Varactors The impedance tuner design use Chalmers 14-finger SiC varactors [21] as tuning elements. They feature a maximum capacitance of 14 pF, a tuning range ≈6 and a quality factor ≈40 at 1 GHz and zero bias. The tuning is not as abrupt as in typical silicon varactors, which, together with the high breakdown voltage (Vbd = 100 V), makes them very suitable for high-power applications. Their excellent performance has already been demonstrated in class-E pulsewidth modulation [32] and class-J dynamic load modulation (DLM) [22], [24] PAs. An antiseries configuration is used to further improve power handling [33]. More details of the nonlinear model, including the parasitics, can be found in [34]. B. Topology Selection To keep a low design complexity, a topology with two control voltages is chosen. After some preliminary simulations and given the design frequency, a transmission-line-based double-stub topology is selected, where the stubs are loaded with the SiC varactors. The double-stub topology has been successfully used before in varactor [35] and MEMS switchbased [36] impedance tuners, although not yet at high-power conditions. However, its potential for high-power conditions was recently demonstrated in a dual-band DLM PA [23]. Having the varactors in shunted configuration to ground also simplifies the biasing of the varactors. A schematic of the double-stub impedance tuner topology is presented in Fig. 2. Five transmission lines with characteristic impedances and electrical lengths Z i , θi (i = 1 . . . 5) compose the circuit. Still with two control voltages, a different number of varactors can be stacked both in series and in parallel to improve both linearity and the power handling capabilities [37]. In this paper, an antiseries configuration [33] is used, and different parallel-stacking options will be analyzed. A large inductor (L gnd ) connected to ground is used as the ground reference of the upper varactors. C. High-Power Considerations It is well known that the effective tuning range of a varactor is reduced as the RF power increases [21]. Moreover,

Fig. 3. Varactor safe operation area for two different load impedance conditions at the 25-W available input power. The black region comprises the bias combinations (V1 and V2 ) where the voltage across the varactor reaches breakdown or forward conduction. (a) Z L = 50 . (b) Z L = 120 + 24 j .

in impedance tuner applications, the RF voltage levels can increase dramatically depending on the load impedance. This means that the effective tuning range is also highly dependent on the load impedance conditions. An example based on the simulations is presented to illustrate how the large-signal conditions limit the tuning range. The double-stub topology in Fig. 2 with 2 antiseries ×2 parallel varactors and 25-W available input power is considered. The circuit is optimized for large-signal conditions using the procedure described in Section II-B. For any arbitrary load impedance, only a limited range of the two varactor bias voltages can be used without exceeding the breakdown voltage (100 V) or forward conduction limits. Fig. 3 shows the usable bias ranges for the two varactors in two load impedance situations. For a 50- load impedance, most of the bias voltage combinations are valid, excluding those closest to the lower and upper limits for both varactors [see Fig. 3(a)]. However, for a different load impedance (here 120 + 24 j ), Fig. 3(b) shows that the effective usable bias ranges are reduced significantly. To further illustrate the importance of the large-signal optimization process, another example is presented. Three different optimized impedance tuners with the same topology, a number of varactors (two parallel and two antiseries) and carrier frequency are considered. The first tuner one is optimized for small-signal conditions (as defined in Section II-A), i.e., the voltage swings across the varactors is considered negligible. The other two circuits are optimized

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

Fig. 4. Large-signal Smith chart coverage versus available input power for three different optimized impedance tuner circuits.

for 25- and 50-W available input power, respectively, using the procedure described in Section II-B. Then, the largesignal Smith chart coverage for different power conditions is reevaluated for the three circuits. The results are shown in Fig. 4. For the small-signal optimized circuit, the maximum Smith chart coverage is obtained at very low input power conditions. However, the effect of increasing the power leads to a steep decrease in the large-signal Smith chart coverage. At 25 W, the coverage is barely 35%, and at 50 W, it is further reduced down to 10%. On the other hand, for the other two circuits that are optimized under large-signal restrictions, significantly improved power handling is observed. For example, the coverage for the 50-W optimized circuit at the small-signal conditions is only 50%. However, increasing the power up to 50 W just decreases the coverage down to 40%. A similar trend is observed for the circuit optimized for 25 W. These results clearly illustrate the importance of large-signal considerations in the design of high-power impedance tuners. D. Demonstrator Circuit Optimization and Implementation A 25-W impedance tuner design at 2.45 GHz is implemented to validate the proposed tuner design concept. Different varactor configurations are initially considered. A 2 × 2 configuration (two antiseries and two parallel varactors) per stub offers a good tradeoff in terms of performance and complexity/cost. A 4 × 2 configuration (four antiseries and two parallel varactors) increases the large-signal Smith chart coverage by around 7 percentage units, but the complexity associated with the assembly process makes the 2×2 alternative more suitable. The large-signal optimization process described in Section II-B is applied at a single frequency, 2.45-GHz and 25-W available power. The limits for the return losses and transducer gain are set considering typical values in other publications [38], [39], i.e., in,max = −10 dB and G T ,min = −2 dB, respectively. Breakdown voltage Vbd = 100 V and forward conduction voltage Vfwd = 0 V are values determined by the SiC varactor technology.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

The 511 load impedance states, evenly distributed across the Smith chart and bias voltages from 0 to 70 V with a 1-V step for both varactors, are then evaluated. Therefore, in total, more than 2.5 million settings are evaluated. Performing HB simulations under these circumstances is relatively time-consuming but manageable. However, since an optimization process of the circuit parameters, e.g., transmission-line lengths and widths, is also performed, the total simulation process slows down and becomes unfeasible in terms of computing time if HB simulations are used. With the linear simulation method described in Section II-B, the simulations and optimization time is dramatically reduced. Different optimization algorithms can be used to maximize the large-signal coverage. In this paper, a genetic algorithm is chosen, as such algorithms have been widely studied and successfully applied in many fields in engineering, e.g., in electromagnetics [40] and also in similar impedance tuner circuits [39]. A flowchart showing the optimization process is presented in Fig. 1. The parameters to be optimized are the electrical lengths θi , i = 1, . . . , 5 and characteristic impedances Z i , i = 1, . . . , 5 of the transmission lines as in the selected tuner topology (see Fig. 2). The optimization returns a large-signal Smith chart coverage of 60% at 2.45 GHz. The T-junctions and tapers were not included in the linear simulations, since the linewidths can take quite large and small values compared with the substrate thickness, which makes the accuracy of the models not so good. The circuit obtained through this optimization process needs to be cross-verified with nonlinear simulations, where all the T-junctions and tapers will be included. HB simulations are performed using Keysight Advanced Design System (ADS), including the nonlinear models of the varactors, the bonding wires, and the complete bias networks. The large-signal Smith chart coverage obtained with the HB simulations is 57%. The reduction in coverage is not significant and confirms that the linear approach that was used in the optimization process is valid. Note that if the whole optimization process would be carried out using the HB simulator, the required time would be several orders of magnitude longer. This would make the entire process unfeasible and thereby validates the proposed design methodology. The final layout is completed by adding T−junctions and tapers to avoid large discontinuities in the transmission lines. The prototype is built using a Rogers 4350B substrate (r = 3.66, tan δ = 0.0037, and thickness = 0.508 mm) mounted on a brass fixture. To improve the thermal stability, the varactors are soldered to the fixture and connected to the circuit using gold bond wires. The varactor bias network comprises a 40-nH Coilcraft inductor 0402HP, both for the feeding and the dc grounding. A picture of the fabricated prototype, which measures 39 mm × 44 mm, can be seen in Fig. 5. IV. E XPERIMENTAL R ESULTS This section presents the experimental results for the fabricated impedance tuner. Small-signal, continuous-wave (CW) large-signal load–pull and two-tone large-signal load–pull measurements are performed.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PÉREZ et al.: DESIGN AND LARGE-SIGNAL CHARACTERIZATION OF HIGH-POWER VARACTOR-BASED IMPEDANCE TUNERS

5

Fig. 5. Photograph of the fabricated SiC impedance tuner prototype (39 mm× 44 mm). The prototype features 2 × 2 varactors per stub. A zoomed-in view of the varactors is also presented.

Fig. 8. Load impedances that belong to the small-signal Smith chart coverage at 2.2 GHz.

Fig. 6. Measured and simulated tuner S-parameters at two different bias conditions with Z L = Z S = 50 . (a) V1 = 1 V and V2 = 1 V. (b) V1 = 70 V and V2 = 70 V.

Fig. 9.

Fig. 7. Measured and simulated small-signal Smith chart coverage versus frequency.

A. Small-Signal Measurements Small-signal measurements are performed using a vector network analyzer. Scattering parameters are measured for all combinations of varactor bias voltages, and swept between 0 and 72 V. A comparison between ADS small-signal simulations and measured S11 and S21 at two different bias points is presented in Fig. 6. Except for a minor frequency shift, good agreement is, in general, observed across the entire frequency range from 1.5 to 3.5 GHz. Using the measured S-parameters and assuming linearity, the transducer gain and return losses can be obtained for different load impedance conditions, using the expressions in Section II-A. The small-signal Smith chart coverage C S S is then used, since the power level is considered low enough not to cause either breakdown or forward conduction. In Fig. 7, the simulated and measurement-based small-signal coverage

LSNA-based setup used for CW measurements.

is presented in the 1–4-GHz frequency range. Although the measurement-based coverage is slightly lower than predicted by simulations, the overall agreement is good across the entire frequency range. The design was centered at 2.45 GHz, but the highest coverage value is obtained at 2.2 GHz. Henceforth, the results will, therefore, be related to 2.2 GHz. We believe that the frequency discrepancy can be due to the modeling accuracy of the simulator, since the circuit includes some large discontinuities, i.e., connection of transmission lines (and the corresponding T-junctions) with rather high and low impedances. Finally, it should be noted that even if the circuit was optimized for a single frequency, a coverage higher than 50% is maintained across a 500-MHz bandwidth. The impedances that comprise the small-signal Smith chart coverage region at 2.2 GHz are presented in Fig. 8. The coverage region is centered in the Smith chart, as expected, with a maximum tuning range of | L | ≈ 0.7. As shown in Fig. 8, the simulations and measurements agree well. B. Large-Signal CW Measurements Large-signal CW measurements are performed under different load conditions at 2.2 GHz, using a large-signal network analyzer (LSNA; Maury/NMDG MT4463) and a passive load– pull tuner (Maury MT986B02). The test setup is shown in Fig. 9. A PA is used to provide up to 25-W available power at the impedance tuner [device under test (DUT)] input reference plane. Reflectometers at the DUT input and output ports couple the incident and reflected signals to the LSNA sampling module. A passive load–pull tuner is used to present

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

different load impedances at the DUT output reference plane. The load–pull tuner is located after the reflectometer and its losses, therefore, limit the maximum reflection coefficient that can be presented to the DUT. Still max ≈ 0.8 was obtained in the measurements. A total of 311 different load impedances, uniformly distributed across the Smith chart, are used for the load–pull measurements. It should be noted that only the fundamental load impedance termination can be controlled in this setup. The LSNA is calibrated at the DUT reference planes. The parameters of interest, i.e., return losses (RL), power delivered to the load (PL ), and transducer gain (G T ), are, therefore, easily obtained from the calibrated voltage waves: a1 , a2 , b1 , and b2 . The available input power (Pav ) is calculated as 1 |a1 |2 (3) 2Z 0 where Z 0 is the reference impedance, here 50 . The power delivered to the load is 1 PL = (|b2 |2 − |a2 |2 ). (4) 2Z o The transducer gain can be then obtained as the ratio of PL and Pav , i.e., PL GT = . (5) Pav The reflection coefficient at the input is calculated as Pav =

b1 in = a1

(6)

and the return loss will be RL = 20 log10 (|in |). The load impedance presented to the DUT can also be obtained from the load reflection coefficient a2 (7) L = . b2 A comprehensive measurement procedure is established to cope with the high-power conditions. A flowchart describing the procedure is shown in Fig. 10. In Step 1, a load impedance Z L is fixed at low-power conditions with the passive load– pull tuner. The varactor bias voltages are set in Step 2, using the small-signal measurement data presented in Section IV-A. Note that it could happen that more than one varactor bias voltage combination leads to the same return losses. Thus, the varactor bias voltages are chosen so that maximum transducer gain is obtained. In Steps 3 and 4, a fine sweep is performed around the bias points, and the varactor voltages are retuned if a better bias point is found. In Step 5, the input power is increased slowly. Initially, the power step is 1 dB, but as power increases, the step is reduced down to 0.25 dB. The voltage waves are measured in Step 6. The varactor current is checked to prevent forward conduction and breakdown. The voltage is increased or decreased, respectively, until there is negligible varactor current. The power is increased until an input power of 25 W is reached. Then, the next load impedance is set with the load–pull tuner, and the procedure is repeated again. The whole measurement process is finished when all the load impedances are measured. To avoid reaching the breakdown voltage for the varactors, we rely on the simulation

Fig. 10. Flowchart for the large-signal load–pull measurement process, here illustrated for a single load–pull impedance.

Fig. 11. Maximum transducer gain (G T ) contours in dB at 2.2 GHz at 25-W available input power for different load impedance conditions. (a) HB simulation. (b) LSNA measurements.

data where all the varactor voltage swings are calculated and stored for the different load conditions and input powers. In a real scenario of a high-power application, it will be important to monitor and track the impedance changes so that minimum loss and safe operating conditions are maintained. G T and RL measured with the LSNA under different load conditions are presented in Figs. 11 and 12. Measurements are compared with Keysight ADS HB simulations under the same fundamental load impedance and power conditions. Transducer gain contours of Fig. 11 show a gain better than −2 dB for almost all the loads measured. Good agreement is observed between measurements and simulations, although the losses are slightly higher in the measurements. Good agreement is also observed in terms of return loss.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PÉREZ et al.: DESIGN AND LARGE-SIGNAL CHARACTERIZATION OF HIGH-POWER VARACTOR-BASED IMPEDANCE TUNERS

7

Fig. 12. Return losses RL for the maximum G T configuration at 2.2 GHz at 25-W available input power for different load impedance conditions. (a) HB simulation. (b) LSNA measurements.

Fig. 14. Transducer gain versus available input power measured with the LSNA at two different load impedances.

Fig. 15. Phase change measured with the LSNA at two different load impedances.

Fig. 13. (a) Simulated optimum varactor bias voltages V1 for maximum transducer gain at 25 W and 2.2 GHz. (b) Simulated optimum varactor bias voltages V2 for maximum transducer gain at 25 W and 2.2 GHz. (c) Measured optimum varactor bias voltages V1 for maximum transducer gain at 25 W and 2.2 GHz. (d) Measured optimum varactor bias voltages V2 for maximum transducer gain at 25 W and 2.2 GHz.

Instead of showing the large-signal coverage in a similar way as in Fig 8, the representation of G T and RL is preferred, since some impedances that would belong to the coverage region were not able to be measured. It should also be noted that at the edge of the Smith chart, the power dissipation in the varactors can be in the range of 8–9 W, considering the values of G T and return losses. Thus, a good heat transfer is necessary to avoid thermal stability problems. In the impedance tuner, this is achieved by mounting the varactors directly on the brass fixture. The two varactor bias voltages are optimized for maximum G T at each tested load impedance. The optimized varactor bias settings are shown in Fig 13 both for measurements and simulations. It is also interesting to see how the gain is affected by increased input power. A power sweep from 32 to 44 dBm

(1.5–25 W) for two different load impedance terminations is shown in Fig. 14. For the 50- load impedance case, an almost negligible gain compression of around 0.05 dB is observed. For a load termination of 9.4 + 5.6 j , the gain compression is more noticeable. Around 0.2 dB of gain variation is observed between the 32- and 44-dBm input power. Although this variation is not very significant, it should be noted that, according to our definition of the coverage, impedances with transducer gain values below −2 dB do not belong to the coverage region. Hence, this specific impedance would belong to the coverage region under low-power conditions (below 10 W approximately), but not when the power is 25 W. This highlights, again, the difference and the importance of accounting for the power conditions when defining the Smith chart coverage. Another important metric to observe is the phase distortion. This can be especially important if the impedance tuner is used together with a PA in a linear transmitter. Fig. 15 shows the AM/PM curve for the same two load impedances that were shown before. At 50- load impedance, no phase distortion is observed as the input power is increased. However, for

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 16. LSNA setup for two-tone measurements. f 0 = 2.2 GHz and  f = 1 MHz.

Fig. 18. Delivered output power at the fundamental (2200 MHz) and thirdorder intermodulation versus available input power, measured with the LSNA at three different load impedances.

Fig. 17. IMD3 levels 2 f 2 − f 1 (dBc) (a) measured with the LSNA under different load conditions when the SiC tuner is driven with a two-tone at 20-W total input power and 1-MHz tone separation and (b) simulated in the same conditions.

a load termination of 9.4 + 5.6 j , a phase change of around 12◦ is observed. The average phase change among the 313 impedances is around 4◦ . C. Two-Tone Measurements Two-tone measurements are performed to evaluate the tuner linearity under high-power conditions. The test setup is shown in Fig. 16. Since the PA used as a driver is operated close to compression, the outputs from two different generators and PA drivers are combined. This enables us to generate a twotone signal with negligible intermodulation distortion at the DUT input reference plane. Two tones at f 1 = 2200 MHz and f 2 = 2201 MHz are fed into the PAs. At peak power, each tone is around 10 W so at the DUT input, and so the total average input power at the impedance tuner calibrated reference plane is 20 W. All intermodulation as well as harmonic components can be measured both at the input and the output using the LSNA. The same measurement procedure as explained in Section IV-B is then used. The power at the fundamental frequencies f1 and f 2 as well as the third-order components 2 f 1 − f2 and 2 f 2 − f 1 are measured with the LSNA for all the 313 different load impedances. The third-order intermodulation distortion (IMD3) is calculated as the dB difference between the fundamental and third-order intermodulation components. Similar to the transducer gain and return loss, the levels of IMD3 are presented for different load impedances in Fig. 17(a) at 20-W input power. Two-tone HB simulations are also presented in Fig 17(b). It should be noted that, both in LSNA measurements and HB simulations, the varactor bias voltages are optimized to get maximum G T at each load impedance. Moreover, the second-harmonic terminations measured with the LSNA have been included in the HB simulations to get a

fair comparison. An overall good agreement is observed between measurement and simulations. The optimum load impedance, from an IMD3 point of view, is very well predicted by the simulations although the absolute distortion levels are lower in the LSNA measurements, particularly in the region with the lowest distortion. This difference is mainly due to dynamic range limitations in the LSNA measurements. The transducer gain is also measured for the two-tone case, showing almost the same values as for single-tone measurements (see Fig. 11). Finally, a power sweep showing the fundamental and the third-order components for three different load impedances is shown in Fig. 18. Around 20 dB of IMD3 difference is observed among the three different load impedance terminations. The third-order intercept point (IIP3) ranges from 51 to 61 dBm between these three impedances. These results confirm the excellent linearity of the impedance tuner, even under high-power conditions. V. C ONCLUSION A detailed design methodology and comprehensive largesignal load–pull characterization of a varactor-based highpower impedance tuner has been presented. The importance of considering the high-power conditions in the design of impedance tuners and the negative consequences of neglecting these conditions has been highlighted. Following an optimization process that relies on simple linear simulations, yet considering the high-power conditions, an impedance tuner demonstrator capable of handling up to 25 W for a wide range of load impedances has been built. A comprehensive large-signal load–pull characterization has been carried out showing an overall very good performance and agreement with the simulations. Losses below −2 dB were measured at 25-W input power at load reflection coefficients  L < 0.8. Measured return losses are below −10 dB for most of the loads. The gain compression observed is almost negligible (around 0.1 dB on average), whereas phase distortion is more significant but still below 5◦ in most cases. Two-tone measurements revealed IMD levels from 30 up to 70 dBc,

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PÉREZ et al.: DESIGN AND LARGE-SIGNAL CHARACTERIZATION OF HIGH-POWER VARACTOR-BASED IMPEDANCE TUNERS

and input IIP3 in excess of 50 dBm for a wide range of load impedances. In summary, these favorable properties make the proposed impedance tuner suitable for use in a wide range of applications. Additionally, improving the quality factor of the varactors is an important goal as future work to reduce losses and hence improve the efficiency which is of extreme importance in high-power applications. The results presented clearly demonstrate the importance and potential of considering high-power operating conditions in the design and characterization of impedance tuner circuits. Moreover, the presented design procedure can be extended to other topologies, other tuning, or switching devices and to broadband or multiband designs. R EFERENCES [1] Y. Yoon, H. Kim, H. Kim, K.-S. Lee, C.-H. Lee, and J. Kenney, “A 2.4-GHz CMOS power amplifier with an integrated antenna impedance mismatch correction system,” IEEE J. Solid-State Circuits, vol. 49, no. 3, pp. 608–621, Mar. 2014. [2] A. van Bezooijen et al., “RF-MEMS based adaptive antenna matching module,” in Proc. IEEE Radio Freq. Integr. Circuits (RFIC) Symp., Jun. 2007, pp. 573–576. [3] Z. Zhou and K. L. Melde, “Frequency agility of broadband antennas integrated with a reconfigurable RF impedance tuner,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 56–59, 2007. [4] H. Song, B. Bakkaloglu, and J. T. Aberle, “A CMOS adaptive antennaimpedance-tuning IC operating in the 850 MHz-to-2 GHz band,” IEEE Int. Solid-State Circuits Conf. (ISSCC) Dig. Tech. Papers, Feb. 2009, pp. 384–385. [5] H. M. Nemati, C. Fager, U. Gustavsson, R. Jos, and H. Zirath, “Design of varactor-based tunable matching networks for dynamic load modulation of high power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 5, pp. 1110–1118, May 2009. [6] M. de Jongh, A. van Bezooijen, K. Boyle, and T. Bakker, “Mobile phone performance improvements using an adaptively controlled antenna tuner,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2011, pp. 1–4. [7] C. Sanchez-Perez, J. de Mingo, P. Garcia-Ducar, and P. L. Carro, “Performance improvement of mobile DVB-H terminals using a reconfigurable impedance tuning network,” IEEE Trans. Consum. Electron., vol. 55, no. 4, pp. 1875–1882, Nov. 2009. [8] N. J. Smith, C.-C. Chen, and J. L. Volakis, “Adaptive tuning topologies to overcome losses in matching circuits for small antennas,” in Proc. 7th Eur. Conf. Antennas Propag. (EuCAP), 2013, pp. 3017–3018. [9] A. V. Bezooijen et al., “A GSM/EDGE/WCDMA adaptive series-LC matching network using RF-MEMS switches,” IEEE J. Solid-State Circuits, vol. 43, no. 10, pp. 2259–2268, Oct. 2008. [10] A. S. Morris, Q. Gu, M. Ozkar, and S. P. Natarajan, “High performance tuners for handsets,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2011, pp. 1–4. [11] J. De Luis, A. Morris, Q. Gu, and F. De Flaviis, “Tunable antenna systems for wireless transceivers,” in Proc. IEEE Int. Symp. Antennas Propag. (APSURSI), Jul. 2011, pp. 730–733. [12] B.-K. Kim, T. Lee, D. Im, D.-K. Im, B. Kim, and K. Lee, “Design methodology of tunable impedance matching circuit with SOI CMOS tunable capacitor array for RF FEM,” in Proc. Asia–Pacific Microw. Conf. (APMC), Nov. 2013, pp. 7–9. [13] R. Whatley, T. Ranta, and D. Kelly, “RF front-end tunability for LTE handset applications,” in Proc. IEEE Compound Semiconductor Integr. Circuit Symp. (CSICS), Oct. 2010, pp. 1–4. [14] R. B. Whatley, Z. Zhou, and K. L. Melde, “Reconfigurable RF impedance tuner for match control in broadband wireless devices,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 470–478, Feb. 2006. [15] C. Sanchez-Perez, J. de Mingo, P. Garcia-Ducar, P. L. Carro, and A. Valdovinos, “Design of a varactor-based matching network using antenna input impedance variation knowledge,” in Proc. IEEE Veh. Technol. Conf. (VTC Fall), 2011, pp. 1–5. [16] J. D. Mingo, A. Valdovinos, A. Crespo, D. Navarro, and P. Garcia-Ducar, “An RF electronically controlled impedance tuning network design and its application to an antenna input impedance automatic matching system,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 2, pp. 489–497, Feb. 2004.

9

[17] F. Ali, E. Lourandakis, R. Gloeckler, K. Abt, G. Fischer, and R. Weigel, “Tunable multiband power amplifier using thin-film BST varactors for 4G handheld applications,” in Proc. IEEE Radio Wireless Symp. (RWS), Jan. 2010, pp. 236–239. [18] A. Wiens et al., “Tunable in-package impedance matching for high power transistors based on printed ceramics,” in Proc. Eur. Microw. Conf. (EuMC), Sep. 2015, pp. 1236–1239. [19] C. Patel and G. Rebeiz, “A high-reliability high-linearity high-power RF MEMS metal-contact switch for DC–40-GHz applications,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 10, pp. 3096–3112, Oct. 2012. [20] R. Whatley, T. Ranta, and D. Kelly, “CMOS based tunable matching networks for cellular handset applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2011, pp. 1–4. [21] C. M. Andersson et al., “A SiC varactor with large effective tuning range for microwave power applications,” IEEE Electron Device Lett., vol. 32, no. 6, pp. 788–790, Jun. 2011. [22] C. Andersson et al., “Theory and design of Class-J power amplifiers with dynamic load modulation,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 12, pp. 3778–3786, Dec. 2012. [23] C. Sanchez-Perez, M. Ozen, C. M. Andersson, D. Kuylenstierna, N. Rorsman, and C. Fager, “Optimized design of a dual-band power amplifier with sic varactor-based dynamic load modulation,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 8, pp. 2579–2588, Aug. 2015. [24] W. Hallberg, D. Gustafsson, M. Ozen, C. M. Andersson, D. Kuylenstierna, and C. Fager, “A class-J power amplifier with varactor-based dynamic load modulation across a large bandwidth,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2015, pp. 1–4. [25] S. Preis, A. Wiens, H. Maune, W. Heinrich, R. Jakoby, and O. Bengtsson, “Reconfigurable package integrated 20 W RF power GaN HEMT with discrete thick-film MIM BST varactors,” Electron. Lett., vol. 52, no. 4, pp. 296–298, Apr. 2016. [26] D. Kienemund et al., “A fully-printed MIM varactor for high power application,” in Proc. 46th Eur. Microw. Conf. (EuMC), Oct. 2016, pp. 623–626. [27] K. Entesari and G. M. Rebeiz, “RF MEMS, BST, and GaAs varactor system-level response in complex modulation systems,” Int. J. RF Microw. Comput.-Aided Eng., vol. 18, no. 1, pp. 86–98, 2008. [28] J. Ferretti, S. Preis, W. Heinrich, and O. Bengtsson, “VSWR protection of power amplifiers using BST components,” in Proc. German Microw. Conf. (GeMiC), Mar. 2016, pp. 445–448. [29] W. N. Allen and D. Peroulis, “Three-bit and six-bit tunable matching networks with tapered lines,” in Proc. IEEE Topical Meeting Silicon Monolithic Integr. Circuits RF Syst. (SiRF), Jan. 2009, pp. 1–4. [30] S. Fouladi, F. Domingue, N. Zahirovic, and R. R. Mansour, “Distributed MEMS tunable impedance-matching network based on suspended slowwave structure fabricated in a standard CMOS technology,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 4, pp. 1056–1064, Apr. 2010. [31] C. Sanchez-Perez, J. de Mingo, P. Garcia-Ducar, P. L. Carro, and A. Valdovinos, “Figures of merit and performance measurements for RF and microwave tunable matching networks,” in Proc. Eur. Microw. Integr. Circuits Conf. (EuMIC), 2011, pp. 402–405. [32] M. Ozen, R. Jos, C. M. Andersson, M. Acar, and C. Fager, “Highefficiency RF pulsewidth modulation of class-E power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 11, pp. 2931–2942, Nov. 2011. [33] K. Buisman et al., “Distortion-free varactor diode topologies for RF adaptivity,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 157–160. [34] M. Özen et al., “High efficiency RF pulse width modulation with tunable load network class-E PA,” in Proc. IEEE 12th Annu. Wireless Microw. Technol. Conf. (WAMICON), Apr. 2011, pp. 1–6. [35] R. Quaglia, C. M. Andersson, C. Fager, and M. Pirola, “A double stub impedance tuner with SiC diode varactors,” in Proc. (APMC) Asia– Pacific Microw., 2011, pp. 267–270. [36] J. Papapolymerou, K. L. Lange, C. L. Goldsmith, A. Malczewski, and J. Kleber, “Reconfigurable double-stub tuners using MEMS switches for intelligent RF front-ends,” IEEE Trans. Microw. Theory Techn., vol. 51, no. 1, pp. 271–278, Jan. 2003. [37] K. Buisman et al., “Varactor topologies for RF adaptivity with improved power handling and linearity,” in IEEE MTT-S Int. Microw. Symp. Dig., 2007, pp. 319–322. [38] C. Hoarau, N. Corrao, J. D. Arnould, P. Ferrari, and P. Xavier, “Complete design and measurement methodology for a tunable RF impedancematching network,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 11, pp. 2620–2627, Nov. 2008.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

[39] C. Sanchez-Perez, J. de Mingo, P. L. Carro, and P. Garcia-Ducar, “Design and applications of a 300–800 MHz tunable matching network,” IEEE J. Emerg. Sel. Topics Circuits Syst., vol. 3, no. 4, pp. 531–540, Dec. 2013. [40] J. M. Johnson and V. Rahmat-Samii, “Genetic algorithms in engineering electromagnetics,” IEEE Antennas Propag. Mag., vol. 39, no. 4, pp. 7–21, Aug. 1997.

César Sánchez-Pérez received the M.Sc. and Ph.D. degrees in electrical engineering and microwave electronics from the University of Zaragoza, Zaragoza, Spain, in 2004 and 2012, respectively. From 2012 to 2014, he held a post-doctoral position with the Microwave Electronics Laboratory, Chalmers University of Technology, Gothenburg, Sweden. Since 2015, he has been with Qamcom Research and Technology AB, Gothenburg. His current research interests include wireless communications systems, with an emphasis on tunable matching networks and high-efficiency transmitters.

Christer M. Andersson received the M.Sc. degree in engineering nanoscience from Lund University, Lund, Sweden, in 2009, and the Ph.D. degree in microwave electronics from the Chalmers University of Technology, Gothenburg, Sweden, in 2013. From 2013 to 2015, he was a member of the Information Technology Research and Development Center, Amplifier Group, Mitsubishi Electric Corporation, Kamakura, Japan. Since 2015, he has been with Qamcom Research and Technology AB, Gothenburg. His current research interests include wide bandgap devices, compact modeling, and the design of high-efficiency power amplifiers.

Koen Buisman (S’05–M’09) received the M.Sc. and Ph.D. degrees in microelectronics from the Delft University of Technology, Delft, The Netherlands, in 2004 and 2011, respectively. From 2004 to 2014, he was with the Delft Institute of Microsystems and Nanoelectronics, Delft University of Technology. In 2014, he joined the Chalmers University of Technology, Gothenburg, Sweden, where he is currently an Assistant Professor with the Microwave Electronics Laboratory, Department of Microtechnology and Nanoscience. He has authored or co-authored over 45 refereed journals and conference papers. He holds two patents. His current research interests include varactors for RF adaptivity, nonlinear device characterization, and technology optimization for wireless systems.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Dan Kuylenstierna (S’04–M’07) was born in Gothenburg, Sweden, in 1976. He received the M.Sc. degree in engineering physics and Ph.D. degree in microtechnology and nanoscience from the Chalmers University of Technology, Gothenburg, in 2001 and 2007, respectively. He is currently an Associate Professor with the Microwave Electronics Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology. His current research interests include the monolithic microwave integrated circuit design, reconfigurable circuits, frequency generation, and phase-noise metrology. Dr. Kuylenstierna was a recipient of the IEEE Microwave Theory and Techniques Society Graduate Fellowship Award in 2005.

Niklas Rorsman (M’10) received the M.Sc. degree in engineering physics and Ph.D. degree in electrical engineering from the Chalmers University of Technology, Gothenburg, Sweden, in 1988 and 1995, respectively. In 1998, he joined the Chalmers University of Technology, where he is currently leading the microwave wide bandgap technology activities and investigating the application of graphene in microwave electronics.

Christian Fager (S’98–M’03–SM’15) received the M.Sc. and Ph.D. degrees in electrical engineering and microwave electronics from the Chalmers University of Technology, Gothenburg, Sweden, in 1998 and 2003, respectively. He is currently a Professor with the Microwave Electronics Laboratory, Chalmers University of Technology. He has authored or co-authored over 100 papers in international journals and conferences. His current research interests include the design and modeling of linear and energy-efficient transmitters for future wireless systems. Dr. Fager was a recipient of the Best Student Paper Award of the IEEE MTT-S International Microwave Symposium in 2002. He is currently an Associate Editor for the IEEE Microwave Magazine.

1754

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

Bandwidth Optimization Method for Reflective-Type Phase Shifters Daniel Müller , Alexander Haag, Akanksha Bhutani, Axel Tessmann, Arnulf Leuther, Thomas Zwick, Senior Member, IEEE, and Ingmar Kallfass

Abstract— This paper discusses a novel optimization technique for broadband reflective-type phase shifters (RTPS). The performance characteristics of an RTPS, including its achievable maximum phase shift and the corresponding average loss, are calculated on the basis of simulated or measured varactor data. In addition, the root-mean-square (rms) errors are evaluated to determine the broadband performance of the device. To extract the characteristics, a reciprocal, lossless matching network with arbitrary output impedance is connected in series to the varactor. The results are plotted on Smith charts, which yield an intuitive representation of the matching network area to achieve the design goals. The technique is verified by realizing a WR-3 band (220–325 GHz) RTPS microwave monolithic integrated circuit realized in a 50-nm metamorphic high electron mobility transistor InGaAs technology. The measured phase shift is 118° at the center frequency of 240 GHz with a bandwidth of 62 GHz. The rms phase and amplitude errors are lower than 5.6° and 0.67 dB, respectively. Index Terms— Beamsteering, InGaAs, metamorphic high electron mobility transistor (mHEMT), monolithic microwave integrated circuit (MMIC), phased array, reflective-type phase shifter (RTPS), WR-3 band (220–325 GHz).

I. I NTRODUCTION

M

OBILE traffic is increasing drastically and is predicted to exceed 49 exabytes per month by 2021. This corresponds to more than an eightfold increase compared with the mobile traffic in 2016, i.e., 7.2 exabytes per month [1]. In order to address the demand of high data rates, communication systems are being extended into the millimeterwave (mmW) frequency range (30 to 300 GHz), where high absolute bandwidths are available. In this frequency range, the small wavelengths lead to small circuit sizes and the feasibility of antenna array integration, which enables electronically steerable antenna arrays with high gain [2]. Due to the generally high atmospheric loss in this frequency range, Manuscript received September 19, 2017; revised November 13, 2017; accepted November 17, 2017. Date of publication December 15, 2017; date of current version April 3, 2018. This work was supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) in the framework of the MilliPhase Project under Grant KA3062/10-1. (Corresponding author: Daniel Müller.) D. Müller, A. Haag, A. Bhutani, and T. Zwick are with the Institute of Radio Frequency Engineering and Electronics, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany (e-mail: [email protected]). A. Tessmann and A. Leuther are with the Fraunhofer Institute for Applied Solid State Physics, D-79108 Freiburg, Germany. I. Kallfass is with the Institute of Robust Power Semiconductor Systems, University of Stuttgart, D-70569 Stuttgart, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2779156

these systems usually operate in the so-called atmospheric windows, which are located at the local minima of the atmospheric attenuation. One of these windows is located in the frequency range of 200–300 GHz [3] and has attracted several systems in recent years, showing data rates up to 100 Gbit/s [4]–[6]. Yet all of these systems are limited to fixed point-to-point links and are incapable of changing the direction of radiation electronically, i.e., they do not provide any kind of electronic beamsteering. To enhance such systems with beamsteering capabilities, integrated phase shifters are key elements. Phase shifters are used to introduce a variable phase shift between two adjacent antennas of an antenna array, which results in a change of the main lobe direction. Besides the capability of introducing a certain phase shift at a given center frequency, phase shifters should also provide low insertion loss and a certain bandwidth. Well-established figure of merits for comparing the bandwidth of different phase shifters are root-mean-square (rms) phase and amplitude errors. Practical realizations of phase shifters comprise many different concepts, such as the vector modulator [7]–[9], and distributed [10], [11], switched [12], [13], or reflective-type phase shifter (RTPS) [14]–[23]. While switched-type phase shifters are restricted to discrete phase steps, the other concepts are able to vary the phase shift virtually stepless, only dependent on the accuracy of the control voltages. Therefore, to achieve sufficiently small phase steps, switched-type phase shifters need multiple stages, which results in a large chip size. Likewise, the vector modulator and the distributed phase shifter usually need a large chip area too. In contract, the RTPS when designed in a proper manner can be realized with small dimensions. In terms of controlling effort, concepts, such as vector modulators or multiple stage discrete phase shifters, depend on many bias voltages, while an RTPS needs only one control voltage. This simplifies the controlling scheme drastically, especially when it comes to multichannel systems. Therefore, the RTPS by virtue of its small size, simple design, and simple controlling scheme is very well suited for future phased array systems. The main drawback of the RTPS is the limited bandwidth in terms of rms amplitude and phase error, which depend on the available reflective loads and parasitic effects. To overcome this limitation, this paper presents a novel design method, which gives an immediate insight into the maximum achievable performance depending on the reflective loads that are available in a particular technology. Furthermore, the design goals of the matching networks for achieving specific characteristics are directly given in Smith charts.

0018-9480 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

MÜLLER et al.: BANDWIDTH OPTIMIZATION METHOD FOR RTPSs

1755

Fig. 1. Simplified schematic of an RTPS showing the principle of operation.

Fig. 2. Schematic showing the principle of the varactor matching and the reference planes used in the calculations.

II. R EFLECTIVE -T YPE P HASE S HIFTER P RINCIPLE

coefficient  M . The equivalent Z 0 signal source is a representation of the couplers output port, which usually has an impedance close to the system impedance. Since the matching network is done using reactive elements and transmission lines, it is assumed to be reciprocal and lossless. Even though matching networks at mmW frequencies have significant loss, the presented technique is still applicable, as will be verified by measurements in Section VI-C. The scattering matrix of such a matching network has to satisfy the following condition [26]:

The basic RTPS, shown in Fig. 1, consists of a 3-dB 90° hybrid coupler and two identical, adjustable reflective loads Z L . Between the hybrid coupler and the reflective load, a matching network M is used for bias supply and for extending the phase control range of the RTPS. The input signal passes the coupler and the matching network M and is reflected at the load Z L , which introduces a certain phase shift. After passing the matching network and the coupler again, the reflected signal will add constructively at the output while canceling out at the input port. Usually, the reflective load is realized using a varactor diode, which has a voltagedependent capacitance Cv , steerable between Cmin and Cmax . The capacitance ratio Cmax/Cmin of the varactor limits the phase control range of the overall phase shifter [24]. Especially in the high mmW frequency range where only poor varactor diodes with capacitance ratios of approximately 2 are available, the relative phase shift has to be extended by a suitable matching network M. One way to extent the phase control range was presented in [24] and leads to realizing the matching network simply by a series inductance L. However, this technique is based on simple equivalent circuits and is therefore limited to frequencies where parasitic effects are negligible. Another technique, considering the parasitic effects, was presented in [25] and is based on the design of generic lossless matching networks to achieve the desired phase control range. While the referenced methods may lead to the desired phase control range, they do not consider or provide any information about the resulting bandwidth of the phase shifter or more specifically the reflective load. Hence, these methods might be sufficient for narrowband systems but broadband systems require a more sophisticated design method. III. M ETHOD FOR BANDWIDTH O PTIMIZATION To achieve broadband designs, the method presented in this paper extends the approach presented in [25] and translates it into Smith charts domain, leading to intuitive design goals. Since most practical RTPS realizations use varactors as the reflective load, in the following context, the terms reflective load and varactor are used interchangeably. The presented technique does not depend on simplified equivalent circuits but allows to directly use measured or simulated data sets of varactor diodes. To investigate the achievable performance, the matching network M located between the source and the varactor is analyzed [25] (see Fig. 2). The method utilizes the phase- and amplitude behavior dependence of the input reflection coefficient C to the output reflection

ST S∗ = U

(1) T

∗

where U is the unitary matrix, and S and S are the transpose and complex conjugate matrix of S, respectively. Solving this equation for a two-port network results in |S11 |2 + |S21 |2 = 1 2

2

|S12 | + |S22 | = 1 ∗ ∗ S12 + S21 S22 = 0. S11

(2) (3) (4)

While (2) and (3) are the necessary conditions of a lossless two port, (4) can be used to relate the phase and amplitude of the different elements. Evaluating (2)–(4) results in the following conditions: |S11 | = |S22 | |S12 | = |S21 |  |S21 | = 1 − |S22 |2 −ψ11 + ψ12 = −ψ21 + ψ22 + π

(5) (6) (7) (8)

where ψkj represents the phase of the matrix element Skj . Using these conditions, the scattering matrix of a two port embedded in a Z 0 system, which fulfills (5)–(8), can be written as [27]    |S22 |e j ψ11 1 − |S22 |2 e j ψ21  SM = . (9) 1 − |S22 |2 e j ψ21 |S22 |e j ψ22 Since the focus is on the reflection coefficient as seen by the varactor toward the source, the independent variable is the reflection coefficient  M , which is generally calculated by [26]  M = S22 +

S12 S21 s . 1 − S11 s

(10)

In a Z 0 system, s = (Z s − Z 0 )/(Z s + Z 0 ) vanishes, which simplifies (10) to  M = S22 . Together with (8) and due to the reciprocity ψ21 = ψ12 , (9) can be rewritten as    |M |e j (2ψ21−ψ M −π) 1 − | M |2 e j ψ21 SM = (11) 1 − | M |2 e j ψ21 | M |e j ψ M

1756

Fig. 3.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

Resistive and capacitive components of the varactor at 240 GHz.

where | M | and ψ M represent the magnitude and argument of the reflection coefficient  M , respectively. Therefore, we are now able to obtain the input reflection coefficient C , which is given by C = | M |e j (2ψ21−ψ M −π) +

(1 − | M |2 )e j 2ψ21  L . 1 − | M |e j ψ M  L

Fig. 4. 50- Smith chart showing an overlay of the resulting mean amplitude of C at the corresponding reflection coefficient  M .

(12)

This can be rewritten using the relation | M |2 =  M  ∗M to    L −  ∗M j 2ψ21 . (13) C = e 1 − L M In an RTPS, only the relative phase difference is important, and therefore, the absolute phase of C is of no interest. To simplify (12), the phase ψ21 = 0 is arbitrarily chosen, which without limitation of generality results in  L −  ∗M . (14) C = 1 − L M Equation (14) is the basis for subsequent investigations on how to achieve certain design goals for the RTPS by choice of a proper matching network M with a specific output reflection coefficient  M . IV. MMIC VARACTOR D IODE E VALUATION In order to get a detailed view on the behavior of the varactor in combination with the matching network, a fine grid of reflection coefficients ranging from | M | = 0 . . . 1 with angles ψ M between 0° . . . 360° was created. These parameters were chosen to cover the full smith chart and therefore represent the achievable performance of the varactor for all passive reflection coefficients  M . Since the resulting data are dependent not only on the reflection coefficient  M but, due to the capacitance behavior of the varactor, also on frequency, it is not possible to show a complete result set in a 2-D graph. Therefore, the following plots refer to the intended center frequency of the RTPS at 240 GHz. The resulting characteristics of C in dependence of the reflection coefficient  M is plotted to the corresponding position of  M . In the following plots, an integrated varactor diode, based on a single finger transistor, which is described in more detail in Section VI-A, is evaluated. A capacitance ratio of 2.2 is achieved by exploiting the voltage-dependent behavior of the gate-to-source capacitance. The extracted resistive and capacitive components versus the varactor voltage, which can be adjusted in the range from −1 to 0.5 V without damaging the device, are shown in Fig. 3 at the intended center frequency of 240 GHz.

Fig. 5. 50- Smith chart showing an overlay of the resulting relative phase shift of C at the corresponding reflection coefficient  M .

Evaluating the magnitude of the varactor in combination with the matching network using (14) results in Fig. 4, which shows the mean magnitude of C , calculated over the control voltage sweep of the varactor. The magnitude decreases toward reflection coefficients  M , which are close to the complex conjugate of the varactor reflection coefficient  L and therefore have small resistive and large positive reactive components. Since the matching network is assumed to be lossless, the decrease in magnitude is solely dependent on the resistive components of the varactor. The minimum magnitude results for a conjugate match of the varactor, e.g.,  M =  ∗L , which leads to maximum power absorption inside the resistive elements of the varactor. To calculate the relative phase, the varactor voltage was varied between −1 and 0.5 V and referenced to the highest voltage. The resulting relative phase difference is shown in Fig. 5, showing that approximately the same reflection coefficients  M , which result in a low magnitude, offer a high phase control range. In addition, the maximum relative phase shift shown in Fig. 5 predicts that a relative phase shift of more than 150° cannot be achieved without severe loss. These results indicate that in the design process, a tradeoff between relative phase shift and loss has to be found. When it comes to broadband designs, the frequencydependent behavior of the varactor has great influence on the overall bandwidth performance. To clarify this behavior, Fig. 6 shows the relative phase at the C reference plane of the matching network for two different matching network output

MÜLLER et al.: BANDWIDTH OPTIMIZATION METHOD FOR RTPSs

1757

Fig. 6. Comparison of the relative phase shift for two matching network output reflection coefficients  M = 0.43 71° (solid lines) and  M = 0.61 97° (dashed lines) at different varactor control voltages.

Fig. 8. 50- Smith chart showing a contour overlay of the available bandwidth for different maximum rms phase errors. (a) 22.5°. (b) 11.2°. (c) 5.6°.

Fig. 7.

RMS phase errors for three different reflection coefficients  M .

reflection coefficients  M = 0.43 71° and  M = 0.61 97°, both selected from the 100° contour line in Fig. 5. As can be seen in Fig. 6, both configurations yield to 100° of relative phase shift at 240 GHz while, above and below 240 GHz, the resulting phase differs strongly. The matching network output reflection coefficient  M = 0.43 71° yields to a relatively flat response over the band, whereas the relative phase for  M = 0.61 97° increases strongly from 70° to 160°. These results clearly show that a broadband RTPS design using only Figs. 4 and 5 is not feasible. To extend the presented method to enable broadband designs, for each reflection coefficient  M , the rms phase and amplitude errors are calculated over the full frequency range of the varactor data set. Fig. 7 shows the rms phase error versus frequency for different reflection coefficients  M . To calculate the rms errors for each matching network, the varactor control voltage is swept from −1 to 0.5 V with a fine step size, and ten curves with equidistant relative phase shift are extracted. This is necessary, since the capacitance–voltage behavior of the varactor is nonlinear, and therefore, using fixed voltage steps would result in nonequidistant phase measurements, which would distort the rms phase and amplitude error calculation. The resulting bandwidths are extracted by setting an upper boundary, which is exemplary set to 11.25°, and finding the symmetric bandwidth. The symmetric bandwidth leads to more meaningful results for a given center frequency, e.g., for  M = 0.43 71° results in a symmetric bandwidth of 180 GHz instead of more than 240 GHz for a conventional bandwidth. The extracted bandwidths for a center frequency of 240 GHz are shown in Figs. 8 and 9 for the rms phase

Fig. 9. 50- Smith chart showing a contour overlay of the available bandwidth for different maximum rms amplitude errors. (a) 3 dB. (b) 1.5 dB. (c) 0.75 dB.

and amplitude error, respectively. The lower half of the Smith chart, where only limited phase shifts are possible, results in very high bandwidths and is therefore not shown in Figs. 8 and 9. Different limits of the rms phase and amplitude error are shown in individual plots to show the bandwidth dependence. The results indicate that for relatively relaxed rms errors such as 22.5° and 3 dB, very high bandwidths can be achieved and focus can be set on the relative phase control range or the mean amplitude. Increasing the requirements on rms errors, the area to achieve high bandwidths drastically narrows, and a considerate design is necessary to trade off rms errors, relative phase shift, and mean amplitude. While interpreting these results, it has to be kept in mind that, for example, a low rms phase error, without consideration of the actual relative phase shifting range, has only limited use.

1758

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

V. I NFLUENCE OF THE C OUPLER C HARACTERISTICS TO THE RTPS The coupler, used for splitting and combining the incoming signal, has large influence on the overall performance of the RTPS. The following section investigates the overall impact of the coupler loss as well as the amplitude and phase imbalance of the through and coupled port. In general, the S-parameter matrix of a perfectly matched and a perfectly isolated coupler is given by [28] ⎡ ⎤ 0 α jβ 0 ⎢α 0 0 jβ ⎥ ⎥ Scpl,ideal = ⎢ (15) ⎣ jβ 0 0 α⎦ 0 jβ α 0

Fig. 10. Evaluation of the impact of coupler amplitude imbalance to the coefficient B for different values of coupler loss L.

where α and β are the coefficients for the through and coupled path. The absolute loss of the coupler is represented by the condition |α|2 + |β|2 = L with L ∈ {0, 1}. A valid solution to this condition is √  (16) α = L 1 − c2 √ − j φ (17) β = Lce where c is defined as

c=

1 pr + 1

(18)

with pr being the power ratio and φ is the phase imbalance between the direct and coupled port. For pr = 0 and φ = 0, this results in the scattering matrix of an ideal 90° hybrid coupler. If the coupler is used in a reflective-type architecture, the combined behavior can be calculated by setting up the vector aRTPS for the incoming waves according to Fig. 1 as ⎡ ⎤ ⎡ ⎤ a1 1 ⎢a2 ⎥ ⎢c b2 ⎥ ⎥ ⎢ ⎥ aRTPS = ⎢ (19) ⎣a3 ⎦ = ⎣c b3 ⎦ a4 0 where a1 is the incoming wave with a normalized amplitude and a2 and a3 are the reflected waves c b2 and c b3 of the identically terminated ports 2 and 3, respectively. Port 4 is assumed to be perfectly matched, which yields to no reflected wave and therefore a4 = 0. Solving bRTPS = Scpl,ideal · aRTPS

(20)

for b1 and b4 results in the S-parameter matrix of an RTPS   2   j 2αβ α − β2 A B = c · SRTPS = c · . (21) B A j 2αβ α2 − β 2 Based on this result, it is possible to investigate the influence of the couplers amplitude and phase imbalance on the RTPS. Due to the reciprocal behavior of the RTPS, only two parameters have to be evaluated S11 = S22 = C (α 2 − β 2 )   

(22)

S12 = S21 = C j 2αβ   

(23)

=A

=B

Fig. 11. Evaluation of the coefficient A, which is the minimum RTPS return loss versus phase and magnitude imbalance of the coupler.

where the coefficients A and B can be interpreted as distortion factors for the reflection coefficient C . Inserting (16) and (17) into A and B, we end up with pr − e− j 2φ pr + 1 √ pr − j φ B = j 2L · e . pr + 1 A= L·

(24) (25)

The absolute loss L of the coupler decreases both coefficients A and B, resulting in an increase of return loss as well as insertion loss. For equal power division pr = 1 and zero phase imbalance φ = 0, the magnitude of the distortion factor |A| vanishes while |B| equals the loss L. Therefore, the RTPS is perfectly matched, and the insertion loss depends only on the coupler loss L and the loss of the combination of matching network and varactor C . Furthermore, the magnitude of the additional insertion loss |B| is only dependent on the coupler’s amplitude imbalance and not on the phase imbalance. The couplers impact to the insertion loss of the RTPS performance is shown in Fig. 10 and shows the achievable performance for a lossless varactor combined with a lossless matching network, i.e., |C | = 1. The coefficient B is plotted versus the magnitude imbalance pr of the coupler for several values of coupler loss L. It can be observed that even a rather high magnitude imbalance of 4 dB increases the overall RTPS insertion loss less than 1 dB. Considering the matching of the RTPS, both, the amplitude and phase imbalance of the coupler, show an effect on the RTPS. This behavior is shown in Fig. 11, where the resulting magnitude is an upper boundary of the input and output matching of the RTPS, which represents a lossless

MÜLLER et al.: BANDWIDTH OPTIMIZATION METHOD FOR RTPSs

1759

Fig. 13. Additional insertion loss of the RTPS depending on the type of the 90° coupler.

Fig. 12.

Simplified schematic of the broadband RTPS.

coupler (L = 1) and lossless matching network and reflective load (i.e., |C | = 1). As an example, if a return loss of 10 dB is required, for a lossless coupler, matching network and reflective load, the magnitude and phase imbalance have to be below 2.8 dB and 17°, respectively. VI. B ROADBAND WR-3 RTPS R EALIZATION A. Technology To verify the design concept, a WR-3 frequency range (220–325 GHz) RTPS phase shifter was manufactured in a 50-nm InGaAs mHEMT technology provided by the Fraunhofer Institute for Applied Solid State Physics (IAF). The technology features high-speed transistors with an unity current gain frequency f t and a maximum oscillation frequency f max of 380 and 600 GHz, respectively. To achieve these high frequencies, a high indium content of 80% is used in the In0.80 Ga0.20 As/In0.53 Ga0.47 As composite channel. Since the process does not offer a special doping profile for varactor diodes, a conventional one-finger mHEMT transistor is used with shortened drain and source connection. Therefore, the voltage-dependent gate–source capacitance acts as a varactor and offers a capacitance control ratio Cmax/Cmin of roughly 2.2. The passive circuitry makes use of two available metal layers with a sheet resistance of 100 and 8 m/ for the first and galvanic metal, respectively. Metal–insulator–metal capacitors are realized by embedding the passivation silicon nitride (SiN) layer in between the metal layers. Additional features of the process are airbridges, for suppressing unwanted modes on the coplanar waveguide transmission lines, as well as a backside process featuring backside metallization and thru-substrate vias. Further information about the utilized process can be found in [29]. B. Reflective-Type Phase Shifter Design While Fig. 1 showed only the basic schematic of an RTPS, the actual design is more complex, as shown in Fig. 12. The matching network is realized by a T-network consisting of transmission lines with an RF-short using a parallel capacitance CRF-short . The varactor diode bias is applied over Rbias and CRF-block . To prevent dc voltage at the input and output of the RTPS, series blocking capacitors Cdc-block were inserted between the coupler and the matching network.

Fig. 14. 50- Smith chart showing the relative phase shift with an overlay of the filtered area for different rms errors, which fulfill an rms error bandwidth > 40 GHz, a relative phase shift > 90°, and a mean amplitude of at least −3 dB.

1) 3-dB Coupler Design: Before designing the coupler, it is necessary to take a look at important key characteristics, such as phase- and amplitude imbalance. Several realizations of 90° couplers, such as branch line or Lange couplers, are available and proven at mmW frequencies [26]. To verify the impact of the different couplers, two layouts were designed and verified by electromagnetic simulations. The results of these two couplers were evaluated using (24) and (25), and the impact of the coupler to the overall return loss and insertion loss of the RTPS was calculated. While the predicted return loss is better than 10 dB for both couplers, the additional insertion loss differs significantly. The results of the coefficient B are shown in Fig. 13. It is obvious that the Lange coupler offers lower loss as well as less amplitude variation versus frequency and was therefore selected for the integration. 2) Varactor Matching Network Design: The plots showing the mean amplitude of C in Fig. 4 and the relative phase shift of C in Fig. 5 built the foundation of the matching network design for the varactor. In addition, Figs. 8 and 9 showing the achievable bandwidth with a certain maximum rms phase and amplitude error make it obvious that for achieving low rms phase and amplitude error with low loss and a high relative phase shift, a certain tradeoff in terms of the matching network output reflection coefficient  M has to be found. To visualize this, in Fig. 14, the relative phase shift of C is overlayed with contour lines, fulfilling the condition of a mean amplitude higher than −3 dB, a relative phase shift higher than 90°, and a minimum bandwidth of 40 GHz with rms errors lower than the indicated values. The different rms phase error limits were chosen to be consistent with discrete phase shifters where a maximum rms phase error of half a least significant bit (smallest phase step) is commonly used as limit [30], [31]. With this knowledge, it is now possible to design the matching

1760

Fig. 15.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

Chip photograph of the RTPS. The chip size is 0.5 × 0.5 mm2 .

Fig. 16. Measured (solid lines) and simulated (dashed lines) gain of the RTPS for different bias voltages Vvctrl .

network in a way that the rms error requirements are fulfilled and a maximum phase control range is achieved. The matching network itself consists of transmission lines and capacitors, as shown in Fig. 12. Simulated results of the reflection coefficients  M for port 2 (M,P2 ) and port 3 (M,P3 ) are shown as overlay in Fig. 14, indicating rms errors of less than 0.75 dB and 5.6° with a bandwidth higher than 40 GHz. The difference between both reflection coefficients is due to small variations in the output reflection coefficient  S of the embedded Lange coupler. C. Measurement All measurements were performed with an on-wafer setup using WR-3 Picoprobes manufactured by GGB, WR-3 extension modules by OML, Inc., and a Keysight PNA-X vector network analyzer. The setup was calibrated in the extended WR-3 frequency range from 200 to 330 GHz using an impedance standard substrate and the thru-reflect-line calibration method. Unless otherwise noted, all measurements were performed at a room temperature of 25 °C. A microscopic photograph of the RTPS microwave integrated circuit (MMIC) is shown in Fig. 15, and the chip size is 0.5 × 0.5 mm2 , including the RF- and dc probing pads. Without probing pads, the phase shifter core area is only 0.3 × 0.2 mm2 . The control voltage of the phase shifter was swept between −1.0 and +0.5 V. To evaluate the data, the bias voltages were chosen to result in ten curves with equidistant relative phase at 240 GHz. Measurements of the gain and relative phase behavior of the RTPS are shown in Figs. 16 and 17, respectively. As can be seen, the amplitude shows only little frequency dependence, and the mean amplitude has less than ±0.7-dB variation over the full frequency range. At the

Fig. 17. Measured (solid lines) and simulated (dashed lines) relative phase of the RTPS for different bias voltages Vctrl . In addition, the analytically predicted relative phase of C for both matching network output coefficients is shown.

Fig. 18. Measured (solid lines) and simulated (dashed lines) relative phase and gain tuning characteristic of the RTPS at 240 GHz for different ambient temperatures.

center frequency, the amplitude variation is below ±0.6 dB. Compared with the simulation, the mean loss of RTPS shows an increase of 1.5 dB to 7 dB. Part of this deviation is due to loss of the probe-to-pad interconnect, which was not accounted for in the simulation. The measured relative phase is in close agreement to the simulation, as shown in Fig. 17. The relative phase control range results between 135° at 200 GHz and 100° at 280 GHz with an almost linear decrease in-between. In addition, Fig. 17 shows the maximum phase control range of C predicted by Fig. 5 for both matching network output reflection coefficients   M,P2 and   M,P3 , which are slightly different due to asymmetries of the Lange coupler. The lossless prediction is in close agreement to the measured and simulated results, which verifies that the presented approach is applicable even at mmW frequencies with nonnegligible losses. The amplitude and phase tuning characteristics of the phase shifter at the center frequency for different ambient temperatures are shown in Fig. 18. While the relative phase shift has only minor dependence on the ambient temperature, the loss increases by 1.4 dB at 125 °C. The simulation models were extracted at 25 °C, and therefore, no simulation data at higher temperatures are available. Measurement and simulation of the group delay are shown in Fig. 19 for different control voltages. Due to inevitable noise in mmW measurements, the group delay was smoothed using a moving average filter with a span size of 10 GHz. Measured and simulated results of the input and output return loss are shown in Fig. 20. The RTPS shows an input return loss of better than 11 dB and an output return

MÜLLER et al.: BANDWIDTH OPTIMIZATION METHOD FOR RTPSs

1761

TABLE I C OMPARISON TO P UBLISHED RTPS MMIC P HASE S HIFTERS

Fig. 19. Measured (solid lines) and simulated (dashed lines) group delay of the RTPS.

Fig. 20. Measured (solid lines) and simulated (dashed lines) input and output return loss of the reflective-phase shifter for control voltages Vctrl from −1 to 0.5 V.

loss better than 12 dB over the full WR-3 frequency range. The measured behavior is in close consistency to the simulation. Finally, the simulated and measured rms errors of the RTPS, which were optimized for a center frequency of 240 GHz, are shown in Fig. 21. The rms phase error is below 5.6° in the frequency range from 214 to 276 GHz. In this frequency range, the rms amplitude error stays below 0.67 dB. These results fulfill the design goals as well as the analytically predicted behavior. Again, both measurements are in very good agreement to the simulated results. Due to limitations on available power sources in the WR-3 frequency range, the linearity of the RTPS could only be evaluated by simulations. The simulated mean amplitude and relative phase shift at the center frequency for input powers ranging from −20 to 9 dBm and control voltages from −1 to 0.5 V are shown in Fig. 22. RMS gain and phase

Fig. 21. Measured (solid lines) and simulated (dashed lines) rms amplitude and phase error of the RTPS. In addition, the design goal limits of Section VI-B2 are overlayed.

Fig. 22. Simulated mean amplitude and relative phase shift at the center frequency versus input power.

Fig. 23. Simulated rms amplitude (solid lines) and phase (dashed lines) error of the RTPS for different input power levels.

errors for different input power levels are shown in Fig. 23. The upper input power is limited by the convergence of the transistor models. Up to 5 dBm, only minor degradations in

1762

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

performance are visible. At higher input powers, the relative phase shift decreases down to 91°, whereas the mean loss increases by approximately 1 to −6.5 dB. While the rms gain error is also increasing toward higher input powers, the rms phase error decreases, which is a result from the lower phase shifting range. VII. S TATE - OF - THE -A RT C OMPARISON A comparison to other published MMIC RTPS is done in Table I. Toward higher frequencies, the available capacitance ratio of the varactor diodes decreases, which makes it challenging to realize a high phase shift over a large bandwidth. However, at frequencies up to 32 GHz, higher capacitance ratios are available, and there are several examples with relative bandwidth above 40% [14], [15], [22]. Furthermore, these broadband realizations are restricted to phase shifts below 123°. Different RTPSs presenting phase shifts above 165° exist [16], [17], [21]–[23], but these are limited to relative bandwidths below 14.3%. The presented phase shifter represents a tradeoff between phase shift and bandwidth and achieves excellent results with a relative bandwidth of 25.3% and a phase shift of 118°, despite the limited capacitance ratio of the varactor diode. Compared with [22] and [23], which operate at comparable center frequencies, the presented RTPS achieves the highest relative bandwidth with comparable loss and lower rms errors. VIII. C ONCLUSION In this paper, a novel concept for designing a broadband RTPS has been presented. The performance of an RTPS in the high mmW frequency range is predominantly dependent on the varactor matching network design. Investigations on the impact of the matching network toward the relative phase shift, the insertion loss, as well as broadband characteristics, such as the achievable bandwidth depending on different limits for the rms phase and amplitude errors, are carried out. In addition, to the prediction of the maximum achievable performance, the corresponding design goals for the matching network are directly given as a target area for the output matching coefficient of the corresponding varactor matching network. This enables the direct evaluation of the maximum performance of an RTPS, depending on the measured or the simulated data of the available varactors, independent of the technology. The measured results fulfill the design goals in terms of bandwidth and rms errors and successfully verify the presented optimization method. To overcome the limited phase shift, it is either possible to cascade the presented phase shifter, at the cost of insertion loss and bandwidth, or to realize a hybrid integration with a discrete 2-bit phase shifter. ACKNOWLEDGMENT The authors would like to thank their colleagues with the IAF Technology Department for their excellent contributions during epitaxial growth and wafer processing. R EFERENCES [1] CISCO VNI Mobile Forecast (2015–2020). Accessed: Feb. 2017. [Online]. Available: http://www.cisco.com/c/en/us/solutions/collateral/ service-provider/visual-networking-index-vni/mobile-white-paper-c11520862.html

[2] S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter-wave cellular wireless networks: Potentials and challenges,” Proc. IEEE, vol. 102, no. 3, pp. 366–385, Mar. 2014. [3] R. J. Emery and A. M. Zavody, “Atmospheric propagation in the frequency range 100-1000 GHz,” Radio Electron. Eng., vol. 49, nos. 7–8, pp. 370–380, Jul./Aug. 1979. [4] S. Koenig et al., “Wireless sub-THz communication system with high data rate,” Nature Photon., vol. 7, no. 12, pp. 977–981, Dec. 2013. [5] N. Sarmah, P. R. Vazquez, J. Grzyb, W. Foerster, B. Heinemann, and U. R. Pfeiffer, “A wideband fully integrated SiGe chipset for high data rate communication at 240 GHz,” in Proc. 11th Eur. Microw. Integr. Circuits Conf. (EuMIC), 2016, pp. 181–184. [6] H.-J. Song et al., “Demonstration of 20-Gbps wireless data transmission at 300 GHz for KIOSK instant data downloading applications with InP MMICs,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2016, pp. 1–4. [7] D. S. McPherson, H.-C. Seo, Y.-L. Jing, and S. Lucyszyn, “110 GHz vector modulator for adaptive software-controlled transmitters,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 1, pp. 16–18, Jan. 2001. [8] D. Müller, A. Tessmann, A. Leuther, T. Zwick, and I. Kallfass, “A h-band vector modulator mmic for phase-shifting applications,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2015, pp. 1–4. [9] C. Quan, S. Heo, M. Urteaga, and M. Kim, “A 275 GHz active vectorsum phase shifter,” IEEE Microw. Wireless Compon. Lett., vol. 25, no. 2, pp. 127–129, Feb. 2015. [10] M. Parlak and J. F. Buckwalter, “A low-power, W -band phase shifter in a 0.12 μm sige biCMOS process,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 11, pp. 631–633, Nov. 2010. [11] Y. Du, W. Su, X. Li, Y. Huang, and J. Bao, “A novel MEMS distributed phase shifter for D-band application,” in Proc. IEEE Int. Conf. Microw. Millimeter Wave Technol. (ICMMT), vol. 1. Jun. 2016, pp. 542–544. [12] H.-S. Lee and B.-W. Min, “W-band CMOS 4-bit phase shifter for high power and phase compression points,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 62, no. 1, pp. 1–5, Jan. 2015. [13] D. Müller et al., “D-band digital phase shifters for phased-array applications,” in Proc. German Microw. Conf., 2015, pp. 205–208. [14] S. Lucyszyn and I. D. Robertson, “Two-octave bandwidth monolithic analog phase shifter,” IEEE Microw. Guided Wave Lett., vol. 2, no. 8, pp. 343–345, Aug. 1992. [15] S. Lucyszyn and I. D. Robertson, “Analog reflection topology building blocks for adaptive microwave signal processing applications,” IEEE Trans. Microw. Theory Techn., vol. 43, no. 3, pp. 601–611, Mar. 1995. [16] J.-C. Wu, C.-C. Chang, S.-F. Chang, and T.-Y. Chin, “A 24-GHz full-360° CMOS reflection-type phase shifter MMIC with low lossvariation,” in Proc. IEEE Radio Freq. Integr. Circuits Symp., Apr. 2008, pp. 365–368. [17] R. Garg and A. S. Natarajan, “A 28-GHz low-power phased-array receiver front-end with 360° RTPS phase shift range,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 11, pp. 4703–4714, Nov. 2017. [18] S. Lee, J.-H. Park, H.-T. Kim, J.-M. Kim, Y.-K. Kim, and Y. Kwon, “A 15-to-45 GHz low-loss analog reflection-type MEMS phase shifter,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 3. Jun. 2003, pp. 1493–1496. [19] F. Meng, K. Ma, K. S. Yeo, S. Xu, C. C. Boon, and W. M. Lim, “Miniaturized 3-bit phase shifter for 60 GHz phased-array in 65 nm CMOS technology,” IEEE Microw. Wireless Compon. Lett., vol. 24, no. 1, pp. 50–52, Jan. 2014. [20] B. Biglarbegian, M. R. Nezhad-Ahmadi, M. Fakharzadeh, and S. Safavi-Naeini, “Millimeter-wave reflective-type phase shifter in CMOS technology,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 9, pp. 560–562, Sep. 2009. [21] T. Liu, K. Han, and J. Zhang, “Design and realization of E-band 4-bit phase shifter MMIC,” in Proc. IEEE 16th Int. Conf. Commun. Technol. (ICCT), Oct. 2015, pp. 661–663. [22] A. Natarajan, A. Valdes-Garcia, B. Sadhu, S. K. Reynolds, and B. D. Parker, “W -band dual-polarization phased-array transceiver frontend in SiGe BiCMOS,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 6, pp. 1989–2002, Jun. 2015. [23] D. Müller, S. Reiss, H. Massler, A. Tessmann, A. Leuther, T. Zwick, and I. Kallfass, “A h-band reflective-type phase shifter MMIC for ISMband applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2014, pp. 1–4. [24] F. Ellinger, R. Vogt, and W. Bachtold, “Compact reflective-type phaseshifter MMIC for C-band using a lumped-element coupler,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 5, pp. 913–917, May 2001.

MÜLLER et al.: BANDWIDTH OPTIMIZATION METHOD FOR RTPSs

[25] P. Alcón, N. Esparza, L. F. Herrán, and F. L. Heras, “On the design of generic matching networks in reflective-type phase shifters for antennas,” in Proc. 9th Eur. Conf. Antennas Propag. (EuCAP), 2015, pp. 1–5. [26] D. M. Pozar, Microwave Engineering, 3rd ed. Hoboken, NJ, USA: Wiley, 2005. [27] D. M. Kerns and R. W. Betty, Basic Theory of Waveguide Junctions and Introductory Microwave Network Analysis, 1st ed. Oxford, U.K.: Pergamon, 1967. [28] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (Electromagnetic Wave Theory). New York, NY, USA: IEEE Press, 1992. [29] A. Leuther et al., “50 nm MHEMT technology for G- and H-band MMICs,” in Proc. IEEE 19th Int. Conf. Indium Phosphide Rel. Mater. (IPRM), May 2007, pp. 24–27. [30] D. W. Kang, J. G. Kim, B. W. Min, and G. M. Rebeiz, “Single and four-element K a-band transmit/receive phased-array silicon RFICs with 5-bit amplitude and phase control,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 12, pp. 3534–3543, Dec. 2009. [31] B. Cetindogan, E. Ozeren, B. Ustundag, M. Kaynak, and Y. Gurbuz, “A 6 bit vector-sum phase shifter with a decoder based control circuit for X-band phased-arrays,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 1, pp. 64–66, Jan. 2016. Daniel Müller received the Dipl.-Ing. (M.S.E.E) degree in electrical engineering from the Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany, in 2012, where he is currently pursuing the Dr.-Ing. (Ph.D.) degree at the Institute of Radio Frequency Engineering and Electronics (IHE). In 2012, he joined IHE as a Research Associate and a Teaching Assistant. His current research interests include the design of monolithic integrated circuits for radar imaging and broadband communication systems in the millimeter-wave frequency range. Alexander Haag was born in Germersheim, Germany, in 1994. He received the bachelor’s degree in electrical engineering from the Karlsruhe Institute of Technology, Karlsruhe, Germany, in 2016, where he is currently pursuing the master’s degree in electrical engineering at the Institute of Radio Frequency Engineering and Electronics (IHE). His bachelor’s thesis on the topic of millimeter-wave phase shifters for high-speed communication systems was carried out at IHE.

Akanksha Bhutani received the B.Tech. degree in electronics and communication engineering from the Maulana Azad National Institute of Technology, Bhopal, India, in 2007, and the M.Sc. degree in information and communication engineering from the Karlsruhe Institute of Technology, Karlsruhe, Germany, in 2012, where she is currently pursuing the Ph.D. degree at the Institute of Radio Frequency Engineering and Electronics with a focus on antenna-in-package solutions for millimeter-wave systems. Axel Tessmann received the Dipl.-Ing. and Ph.D. degrees in electrical engineering from the Karlsruhe Institute of Technology, Karlsruhe, Germany, in 1997 and 2006, respectively. In 1997, he joined the High Frequency Devices and Circuits Department, Fraunhofer Institute for Applied Solid State Physics (IAF), Freiburg, Germany, where he is currently involved in the development of monolithically integrated circuits and subsystems for high data-rate wireless communication and high-resolution imaging systems. He is currently the Group Manager of the Millimeter-Wave Packaging and Subsystem Group, Fraunhofer IAF. His current research interests include the design and packaging of millimeter-wave and submillimeter-wave ICs using high electron mobility transistors on GaAs, GaN, and InP, and circuit simulation and linear and nonlinear device modeling.

1763

Arnulf Leuther received the Dipl.Phys. and Ph.D. degrees in physics from RWTH Aachen University, Aachen, Germany. He has been with the Fraunhofer Institute for Applied Solid State Physics, Freiburg, Germany, since 1996. His current research interests include the development of high electron mobility transistor technologies for sensor and communication systems up to 600 GHz.

Thomas Zwick (S’95–M’00–SM’06) received the Dipl.-Ing. (M.S.E.E.) and Dr.-Ing. (Ph.D.E.E.) degrees from the Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany, in 1994 and 1999, respectively. From 1994 to 2001, he was a Research Assistant with the Institut fr Hchstfrequenztechnik und Elektronik (IHE), KIT. In 2001, he joined the IBM T. J. Watson Research Center, Yorktown Heights, NY, USA, as a Research Staff Member. From 2004 to 2007, he was with Siemens AG, Lindau, Germany. During this period, he managed the RF Development Team for automotive radars. In 2007, he became a Full Professor with KIT, where he is currently the Director of IHE. He has authored or co-authored over 300 technical papers and holds 20 patents. His current research interests include wave propagation, stochastic channel modeling, channel measurement techniques, material measurements, microwave techniques, millimeter-wave antenna design, wireless communication, and radar system design. Dr. Zwick served on the Technical Program Committees (TPCs) of several scientific conferences. His research team received over ten Best Paper Awards of international conferences. He was the General Chair of the International Workshop on Antenna Technology in Karlsruhe in 2013 and the IEEE MTT-S International Conference on Microwaves for Intelligent Mobility in Heidelberg, Germany, in 2015. He was also the TPC Chair of the European Microwave Conference 2013. From 2008 to 2015, he was the President of the Institute for Microwaves and Antennas. He was a Distinguished IEEE Microwave Lecturer from 2013 to 2015 with his lecture on QFN-based packaging concepts for millimeter-wave transceivers.

Ingmar Kallfass received the Dipl.-Ing. degree in electrical engineering from the University of Stuttgart, Stuttgart, Germany, in 2000, and the Dr.-Ing. degree from the University of Ulm, Ulm, Germany, in 2005. In 2001, he was a Visiting Researcher with the National University of Ireland, Ireland, Dublin. In 2002, he joined the Department of Electron Devices and Circuits, University of Ulm, as a Teaching and Research Assistant. In 2005, he joined the Fraunhofer Institute for Applied Solid-State Physics (IAF), Freiburg, Germany, where he was involved in nonlinear millimeter-wave integrated circuit design. From 2009 to 2012, he was a Professor with the Karlsruhe Institute of Technology, Karlsruhe, Germany, where he was involved in the field of high-speed integrated circuits in a shared professorship with the Fraunhofer IAF in the frame of the German Excellence Initiative. Since 2013, he is the Director of the Institute of Robust Power Semiconductor Systems, University of Stuttgart, as part of the Robert Bosch Center for Power Electronics, where he has been involved in compound semiconductor-based circuits and systems for microwave and power electronics.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Radiative Quality Factor in Thin Resonant Metamaterial Absorbers Nicolas Fernez, Ludovic Burgnies , Jianping Hao, Colin Mismer, Guillaume Ducournau, Didier Lippens, and Éric Lheurette Abstract— Perfect electromagnetic absorption in an array of thin resonators is analyzed by means of the quality factor involving separately the contribution of losses and a coupling with free space. An equivalent electrical circuit based on an open-resonator model is introduced for an absorber made of dielectric cubes arranged in a square lattice deposited onto a metallic ground plane. From full-wave simulations, two regimes depending on the lattice period are pointed out. A quadratic dependence of the radiative quality factor is shown for largest periods, whereas the radiative Q-factor is governed by a coupling between resonators at small periods. It results that an optimal period for obtaining a unitary absorption can be deduced from a single simulation and that a maximal absorption bandwidth emerges from a tradeoff between these two regimes. Finally, the radiative Q-factor concept is applied to analyze an absorber made of patch resonators with the goal to broaden the absorption bandwidth. Particularly, the performances of a multisized patch absorber are experimentally evaluated in the W-band (75–110 GHz) with a good agreement when compared to simulations. Such an analysis of Q-factor appears as a powerful tool for designing singlesized and multisized resonator absorbers targeting a specific absorption spectrum. Index Terms— Dielectric resonator, electromagnetic absorber, metamaterials, perfect metamaterial absorbers.

I. I NTRODUCTION

P

REVIOUSLY devoted to military applications as stealth [1], electromagnetic absorbers have been extended to more general civil and scientific applications in electromagnetic shielding [2], in reduction of radar cross section [3], in people protection to ambient electromagnetic radiations [4] or to improve performances of photodetectors and microbolometers [5]. Depending on the targeted operating frequency, various technologies can be found in the literature. At the lowest frequency bands, electromagnetic absorption can be achieved by means of a Dallenbach screen [6] made of a single layer of metal-backed ferrite materials, with a

Manuscript received July 13, 2017; revised September 29, 2017 and December 5, 2017; accepted December 12, 2017. This work was supported in part by DGA, French Ministry of Defense, in part by the Agence Nationale de la Recherche, the French Ministry of Research, through the ASTRID Project 3DRAM, and in part by PIC-3D METASPACE. (Corresponding author: Éric Lheurette.) N. Fernez, J. Hao, C. Mismer, G. Ducournau, D. Lippens, and É. Lheurette are with the Univ. Lille, CNRS, Centrale Lille, ISEN, Univ. Valenciennes, UMR 8520—IEMN, F-59000 Lille, France (e-mail: eric.lheurette@ iemn.univ-lille1.fr). L. Burgnies is with the University Lille, CNRS, Centrale Lille, ISEN, University Valenciennes, UMR 8520—IEMN, F-59000 Lille, France, and also with the Université du Littoral Côte d’Opale, rue Ferdinand Buisson, 62228 Calais, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2784808

heavy weight drawback. Charged polymer foam can be used to lighten structures, but they have to be thicker in counterpart. An alternative solution is to consider metamaterials to reduce thickness and weight of absorbers. Moreover, such a metamaterial technology is suitable at higher frequency when ferrite can suffer from material relaxation. The perfect metamaterial absorber concept has been introduced by Landy et al. [7] as a stack of two metallic resonators separately excited by electric and magnetic fields. By manipulating resonances of the equivalent material parameters ε and μ independently, a matching condition ε = μ was achieved. Such a condition is equivalent to impedance matching between metamaterials and the wave impedance in air Z 0 = 377 , and it causes a perfect absorption ( A = 1). Due to their resonant nature, metamaterial absorbers are narrowband and solutions to extend the absorption bandwidth are required. On the other hand, larger bandwidths can be observed for absorbers made of lossy frequency-selective surface (FSS) [8]. The FSS located on the top of a back-grounded dielectric substrate can produce a high-impedance surface (HIS) which can be further impedance matched to Z 0 by connecting lumped resistors between FSS patterns, by using resistive material to define the FSS, or by inserting a lossy dielectric substrate. The operating principle of such absorbers can be modeled by an equivalent RLC parallel resonant circuit, where the capacitance corresponds to the FSS operating under its own intrinsic series resonance, the inductance to the impedance of the ground plane through the thickness of the dielectric substrate, and the resistance to losses which has to be matched to Z 0 for a perfect absorption [9]. However, this simple circuit model is appropriate for substrate such as d/D > 0.3, where d and D are the substrate thickness and the FSS period, respectively [9]. For thinner substrate, proximity of the ground plane with the FSS has to be taken into account by correcting the value of the capacitance and the inductance. If the capacitance for thinner substrate can be evaluated [9], values for the inductance and the resistance due to dielectric loss are missing, even if some trends can be favorably predicted by the model. Actually for thinner substrate, the structure does not work anymore as an HIS absorber but as an array of resonators. Resonator-based absorbers are a third type of absorbers found in the literature, especially at higher frequency from THz [10] up to optics [11]. Depending on the targeted frequency, absorbers can be produced by dielectric resonators or they can be formed by metallic patterns deposited onto a dielectric substrate whose operating mechanism is similar to the one of a patch antenna array [12].

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

For the former, absorption can result from a Mie resonance with an electromagnetic field concentration inside dielectric resonators, whereas for the latter energy is concentrated beneath metallic patterns at the resonant frequency. Then energy is both dissipated by dielectric loss and coupled to free space, and a perfect absorption can be achieved as a result of a tradeoff between these two contributions. The thickness of absorbers can be commonly reduced by using high permittivity materials. A Ferroelectric material such as BaSrTiO3 (BST) with a dielectric constant of a few hundreds and a relatively low dielectric loss (tan(δ) ∼ 0.02) is a good candidate with an expected thickness reduction of around ten for dielectric resonators operating at microwaves [13]. But the dielectric relaxation frequency measured at around 800 GHz for a BST thin film could permit to consider such a ferroelectric material for millimeter wavelength band absorbers as well [14]. However, in THz and optics other materials such as Silicon or TiO2 are currently considered [10], [11], [15], [16]. For the latter, metallic patterns deposited onto a conventional back-grounded substrate are considered [12], [17]–[19]. This technology opens additional degrees of freedom. Multiband and wideband absorbers can be produced by combining patches with different sizes and topologies [20], or by introducing disordered structures [21], [22], while the angular robustness of the absorption under oblique incidence can be enhanced by increasing the complexity of the metallic patterns in multilayer arrays [19]. Despite the fact that the resonant frequency of resonators can be estimated by a metallic cavity approximation for dielectric resonators [23] or from a microstrip antenna formula for patch resonators [12], [24], time-consuming parametric numerical computations are required to design resonator-based absorbers with the objective of a perfect absorption ( A = 1) [13]. Moreover, the absorption bandwidth is not easy to predict and conditions for achieving the largest bandwidth for a specific arrangement of resonators are missing. Here, we propose to analyze resonator-based absorbers in terms of quality factors with three objectives: 1) designing perfect absorbers by estimating an optimal lattice period of a resonator array; 2) evaluating the largest bandwidth which can be achieved for such absorbers; and 3) considering a quality factor analysis and an equivalent electrical circuit in a multisized patch arrangement in order to broaden the absorption bandwidth. This paper is organized as follows. In Section II, the equivalent electrical circuit of an array of resonators is introduced, as well as definitions of involved quality factors. Section III is dedicated to a dielectric resonator absorber with an analysis of the electrical circuit and of the radiative Q-factor, whereas Section IV is devoted to the theory and modeling of a patch absorber. Finally, experiments are presented for a multisized patch absorber and compared to equivalent circuit modeling results and to finite-element numerical simulations. II. E QUIVALENT E LECTRICAL C IRCUIT The investigated structure was made of an array of dielectric or patch resonators deposited onto a metallic ground plane

Fig. 1. (a) Illustration of an array of cubic dielectric resonators. (b) Equivalent electrical circuit of an array of resonators. (c) Elementary cell considered in simulations.

in a square lattice arrangement of period p. For introducing an equivalent electrical circuit, we first considered an array of cubic dielectric resonators of dimension a with a high dielectric constant εr ∼ 130 as illustrated in Fig. 1(a). Such a high permittivity value can be achieved by a ferroelectric material as BST, and a perfect absorber operating in X-band with a low thickness around λ/18 has already been demonstrated experimentally [13]. The reflection coefficient S11 ( f ) of the absorber was calculated with the software HFSS by Ansys. An elementary cell including one dielectric resonator deposited onto a square metallic plane was considered with master–slave boundary conditions, and a normal incident plane wave produced through a Floquet port as illustrated in Fig. 1(c). Then, the absorption and the equivalent impedance of the absorber were calculated as follows: A( f ) = 1 − |S11 ( f )|2 1 + S11 ( f ) Z eq ( f ) = Z 0 · 1 − S11 ( f )

(1) (2)

with Z 0 = 377 , the wave impedance in air. Note that Z eq is the impedance calculated at the input plane of the absorber, at a distance a from the ground plane. By way of illustration, real and imaginary parts of the equivalent impedance Z eq calculated for an array of dielectric resonators are plotted in Fig. 2. They correspond to the absorber shown in Fig. 1(a) and studied in Section III for p = 12 mm and tan(δ) = 0.02, but they can be considered as a general frequency dependence of impedance. At low frequency Z eq is inductive with an imaginary part linearly increasing, and the resonance feature occurring around 10 GHz can be attributed to a Mie-type magnetic resonance by analyzing E- and H-field maps [13]. The frequency behavior of the impedance plotted in Fig. 2 can be modeled by the equivalent electrical circuit shown in Fig. 1(b). The RLC parallel circuit describes the resonators while L gnd is an inductance produced by the bare part of the ground plane surrounding resonators through a distance a equivalent to the thickness of dielectric resonators.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. FERNEZ et al.: RADIATIVE QUALITY FACTOR IN THIN RESONANT METAMATERIAL ABSORBERS

Fig. 2. Real and imaginary parts of the equivalent impedance Z eq of an array of dielectric resonators and of the impedance Z res in the inset.

3

Fig. 3. Absorption of an array of cubic dielectric resonators (a = 1.8 mm and tan(δ) = 0.01) with the lattice period p = 1.8 to 20 mm as a parameter. The inset is a zoomed-in view of the absorption around the resonance for p = 2.5 to 20 mm.

Then, equivalent impedance Z eq was decomposed as Z eq = Z res + i L gnd ω and each element was determined by Im(Z eq ) f R R = Re(Z eq ) f R L gnd = 2π f R 1 1 R( f+ − f − ) 1 L= C = 2π R( f + − f − ) 2π f R2

(3)

(4)

where f R , f − , and f + correspond to the resonant frequency of Z res , and frequencies with maximum and minimum imaginary part of Z res , respectively. Note that the real parts of Z res and Z eq are equal and f R in (4) is determined first when Re(Z eq ) is maximum. Previously to the determination of f − and f + , the inductance L gnd and the impedance Z res were calculated from (4) and (3), respectively. Frequencies f+ and f− are explicitly specified in the inset of Fig. 2, which represents the impedance Z res around the resonance. As the absorber can be modeled by an electrical circuit, three quality factors can be defined as   C C Q rad = Z 0 (5) Qd = R L L  −1  −1 Q tot = Q d + Q −1 . (6) rad The quality factor Q d is related to the Q-factor of Z res and it reflects resonators losses, whereas Q rad is the radiative Qfactor which describes the coupling of resonators with free space. Q tot is the resultant Q-factor of the open resonator model including both these contributions. From a circuit description viewpoint, Q d , Q rad , and Q tot are equivalent to unloaded Q 0 , external Q e , and loaded Q L quality factors, respectively, defined in [23] with the wave impedance in air Z 0 acting as a resistive load. Finally, the maximum of absorption can be calculated by [25]  −1 2 Q − Q −1 rad Amax = 1 − |S11 |2f R = 1 −  d (7) −1 2 Q −1 + Q rad d at the resonant frequency fR =

2π

1 √

LC

.

(8)

It is worth mentioning that a perfect absorption ( Amax = 1) is achieved for a condition Q d = Q rad which is equivalent to an impedance matching condition Z eq = Z 0 . We propose to analyze the behavior of resonator-based absorbers by means of the radiative Q-factor. For this purpose, the reflection coefficient S11 ( f ) was first determined by simulation. Then, the absorption A( f ), the equivalent impedance Z eq , each element of the equivalent electrical circuit and Q d were calculated by (1), (2), (4), and (5), respectively. From A( f ), the total Q-factor was determined by Q tot =

fR f

(9)

where  f is the full-width at half-maximum around the resonance of A( f ). Finally, the radiative Q-factor was calculated by inverting (6) and by considering previously determined Q tot and Q d values. Note that the radiative Q-factor can be calculated by (5) as well, and a comparison between Q rad values calculated by (5) and (6) is done in Section III. III. A RRAY OF C UBIC D IELECTRIC R ESONATORS A. Periodicity and Loss Effects Both the lattice period p and the dielectric losses, represented by the loss tangent tan(δ), are the key factors in the design process, the effects of which are studied in this section. Here, we consider an array of cubic dielectric resonators [see Fig. 1(a)] with a high permittivity εr = 130 (1 − i tan(δ)). Size of the cubes was fixed to a = 1.8 mm. The absorption A( f ) was calculated with HFSS by varying the lattice period p. Each element of the equivalent circuit and the radiative Q-factor were calculated following the procedure described in Section II. Fig. 3 shows the absorption simulated around the resonance for tan(δ) = 0.01 and with the lattice period as a parameter. When p increases from 1.8 up to 10 mm, absorption broadens and the absorption peak decreases until p equals 1.85 mm, whereas above 1.9 mm the absorption band becomes narrower with an increase of the peak value. For p = 10 mm absorption

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 4. Lattice period dependence of the inductance L gnd calculated by (4) in circle marks and by (10) in solid line for dielectric loss tan(δ) = 0.005.

is maximum with A ∼ 1 and decreases for higher periods. It can be predicted that the total quality factor of the absorber decreases from p = 1.8 to around 1.9 mm, and increases above. Note that a period p = a = 1.8 mm corresponds to a Dallenbach absorber made of a back-grounded lossy dielectric slab. This case was simulated to complete analysis at lowest periods. The inductance L gnd calculated with (4) is plotted with circle marks in Fig. 4 for tan(δ) = 0.005. Fluctuation of L gnd appearing when p < 4 mm stems from a frequency sensitivity of the imaginary part of Z eq at the resonance, especially for low dielectric loss yielding a very sharp variation. This poorly defined L gnd value introduces a weak asymmetry in the frequency dependence of the imaginary part of Z res with Im[Z res ( f − )] = −Im[Z res ( f + )]. However, as the RLC elements rely on frequencies f R , f + , and f − , and not on the value of Im[Z res], they are not very sensitive to this weak asymmetry. Moreover, L gnd does not suffer from this deviation for higher dielectric loss with tan(δ) ≥ 0.02 when the resonance is broader, and a smooth period dependence of L gnd is observed. By introducing the impedance Z gnd due to the bare ground plane through a distance a corresponding to the thickness of resonators, L gnd can be approximated by L gnd = with

Im(Z gnd) f R 2π f R

 2

a = Z 0 tanh(γ a) · 1 − χ p

(10)



Z gnd

(11)

and where γ = α + iβ is the propagation constant in the surrounding medium of resonators (air in the current absorber, γ = i ω/c). In (11), χ is a fitting parameter introduced to ensure that the value of L gnd computed from (10) when p = a is equal to that from the following equation corresponding to a Dallenbach absorber L gnd =

π2

Z0 √ . f R εR

(12)

Fig. 5. Lattice period dependence of the inductance, capacitance, and resistance of the equivalent circuit of resonators calculated with the dielectric loss tan(δ) = 0.001, 0.005, and 0.02 as a parameter.

Then equality is achieved for χ = 0.6. On the other hand, for large periods ( p  a), Z gnd tends to the impedance of a bare ground plane without resonator with L gnd = 2.38 nH, in good accordance with the limit value of L gnd in Fig. 4. Finally, as depicted by solid line in Fig. 4 the value of L gnd calculated by (10) perfectly fits the inductance calculated by (4), even for p < 4 mm. Note that (10) is invariant with dielectric loss, and it can be chosen for the determination of L gnd whatever the tan(δ) value. RLC elements of resonators are plotted in Fig. 5 as a function of p with the dielectric loss tan(δ) = 0.001, 0.005, and 0.02 as a parameter. For a period p > 4 mm, a quadratic dependence of each element can be pointed out with a capacitance increasing in p 2 , while inductance and resistance decrease in 1/ p2 . We can note that when C increases the impedance (iCω)−1 decreases similar to all the impedances of the RLC parallel circuit and to Z res . The decrease of Z res can be explained by the energy rate of the incident wave interacting with resonators. As the dimension of resonators a stays constant, by increasing the period p the part of the incident energy interacting with resonators follows a (a/ p)2 law producing the decrease of Z res and the quadratic dependences observed in Fig. 5. Moreover, the values of L and C are invariant with dielectric loss which is solely reflected in the values of R. Opposite quadratic variations of L and C are consistent with the observation of a resonant frequency f R quasi-constant for p > 4 mm, as shown in Fig. 3. By combining these variations with the quadratic dependence of R, Q d is constant and it depends on the dielectric loss. As losses in the absorber are produced by the dielectric material, Q d can be defined as follows: 1 . (13) Qd = tan(δ) Finally, the radiative quality factor Q rad calculated by means of (6) or (5) is plotted in Fig. 6. As expected, it appears that Q rad is invariant with the dielectric losses and the value of Q rad resulting from (5) is slightly smaller than the value calculated by (6). Even if the difference is weak, it seems better

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. FERNEZ et al.: RADIATIVE QUALITY FACTOR IN THIN RESONANT METAMATERIAL ABSORBERS

5

coefficient q, and the resonant frequency of the symmetric mode of coupled resonators can be estimated by [25]   2 −2 (14) f R ( p) = f R (∞) · 1 − q( p) − Q 0 ( p)

Fig. 6. Lattice period dependence of the radiative quality factor calculated by (5) for tan(δ) = 0.02, and calculated by (6) for the dielectric loss tan(δ) = 0.001, 0.005, and 0.02 as a parameter.

to use (6) than (5) for calculating Q rad because RLC elements in (5) are post processed data, whereas Q tot in (6) is directly deduced from the absorption bandwidth  f . Thereafter, Q rad is calculated by (6) but periodicity effects can be analyzed by keeping (5) in mind. Resulting from the L and C variations plotted in Fig. 5, a quadratic dependence of Q rad is observed for p > 4 mm. It can be seen that a perfect absorption condition Q rad = Q d is achievable by changing solely the array period if the dielectric loss tan(δ) is previously known. When Q rad < Q d the equivalent impedance of the absorber at the resonance is too high to be impedance matched to Z 0 , and when Q rad > Q d the equivalent impedance is too low. For p < 2 mm, we can see that variations of RLC elements and of Q rad differ from quadratic dependences (see Fig. 6). For low periods, coupling between resonators strongly increases and it modifies the total quality factor Q tot . As the coupling factor between resonators is not introduced in the model, the radiative Q-factor is affected by a strong coupling at low periods as discussed in Section III-B. B. Discussion Absorption phenomenon results from a magnetic Mie resonance with a magnetic field concentrated in dielectric resonators which act as magnetic dipoles. By reducing the period, the resonators become closer and the coupling between equivalent dipoles increases. Depending on dipoles arrangement, transverse and longitudinal couplings with parallel and series dipoles can be defined, respectively. It was shown that a longitudinal coupling predominates in an array of BST cubes producing a lowering of the resonant frequency [13]. This frequency shift can be attributed to the symmetric mode of the longitudinal coupling whereas for the asymmetric mode an increasing resonant frequency is expected [26]. Actually, as all the resonators are excited by the magnetic field of the incident wave, all the magnetic dipoles are oriented in the same direction and only the symmetric mode of the longitudinal coupling can be observed. Moreover, the coupling level between resonators can be characterized by a coupling

where Q 0 and f R (∞) are, respectively, the Q-factor and the resonant frequency of an uncoupled resonator. The derivation of (14) shows that a more important increase for q( p) than for Q −1 0 ( p) is required to involve a decrease of the resonant frequency observed in Fig. 3. In such a case, an array of coupled resonators operates with the condition q > Q −1 0 and the total Q-factor of coupled resonators is determined by the coupling q and no more by dielectric loss [25]. As Q rad is calculated by (6) from the total Q-factor without considering the resonators intercoupling, increase in Q tot due to this coupling is reported in Q rad as shown in Fig. 6. The tradeoff between total Q-factor determined by q for low periods and by Q rad for large periods limits the minimal value of Q tot for an absorber made of resonators. This minimal value can predict the highest achievable absorption bandwidth  f for a given structure. Moreover by choosing a specific material such as tan(δ) = (Q rad )−1 min , an optimal absorber combining perfect absorption ( A = 1) and the largest bandwidth  f can be designed. Finally, from the quadratic dependence of Q rad with p observed in Fig. 6, together with its invariance with dielectric loss, an optimal period p A=1 leading to a perfect absorption can be deduced from a single simulation by 1 (15) p A=1 = psimu tan(δ) A=1 · Q rad ( psimu) where Q rad ( psimu) is determined by a simulation of the absorber with a period psimu > pthresh (where pthresh is 4 mm in this study) whatever the dielectric loss value. tan(δ) A=1 in (15) is the dielectric loss of resonators constituting the targeted absorber. From a practical point of view, a period psimu = 10a and a value of the dielectric loss tan(δ) = 0.02 could be considered for simulation. As an illustration, starting with Q rad = 311 determined from a single simulation performed with psimu = 18 mm and tan(δ) = 0.02, an optimal period with perfect absorption p A=1 = 10.2 mm can be estimated by (15) for a targeted dielectric loss tan(δ) = 0.01 (Q d = 100). This optimal period is consistent with the maximum of absorption observed in Fig. 3 for p = 10 mm. Note that previous results have been performed for dielectric resonators with specific dimension and dielectric constant. It could be interesting to normalize results with respect to the free space wavelength or the cube dimension a. However, it is worth mentioning that the results strongly depend on the real part of the dielectric constant which influences not only the resonant frequency but also the electromagnetic fringing field around resonators. Especially, it is expected that the coupling between resonators is affected by the fringing field and the values of (Q rad )min and pthresh should change as a consequence, whereas a quadratic dependence should be preserved for p > pthresh. Therefore, to generalize results to another cube dimension or dielectric constant two simulations should be performed at large periods (i.e., p > 2a) in order

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

to check that the optimal period p A=1 estimated by (15) is in the quadratic dependence of Q rad . IV. A RRAY OF PATCH R ESONATORS A resonator-based absorber can be achieved by an array of metallic patterns deposited onto a metallic-backed dielectric slab. As for a patch antenna, the resonant frequency strongly depends on the geometric size of the metallic pattern, and absorber dimensions can be theoretically adjusted for any targeted frequency. By combining patches with different sizes, multiband and wideband absorbers can be achieved [20], [27]. Here, we first analyze an array of single-sized patch by means of the Q-factor approach before to model a multisized patch absorber by impedance combination. A. Periodicity and Patch Size Effects As for dielectric resonators absorber the lattice period p, losses and the resonator size are the key factors in the design process. In this section, we will focus on periodicity and patch size effects. The reflection coefficient of the absorber was simulated by considering the structure illustrated in the inset of Fig. 7(b) with periodic master–slave conditions and a Floquet port as illustrated in Fig. 1(c). A square patch array is located on top of a 0.1-mm-thick polyethylene terephthalate (PET) back-grounded substrate. The dielectric constant of the substrate was εr = 2.9 (1 − i 0.025) and four sizes of the square patch were considered for analysis: w = 0.85, 0.89, 0.945, and 1 mm. After simulations the radiative Q-factor was extracted following the methodology developed in previous sections with some additional cautions. The first one concerns the material surrounding resonators which is responsible for complex impedance Z gnd . Moreover for a patch absorber, electromagnetic fields are concentrated beneath the patch at the resonant frequency, and the relevant transversal dimension of resonators is w. To take these specific parameters into account, the effect of the ground plane not restricted to L gnd has to be described by the complex impedance   2

w Z gnd = Z 0 tanh(γ h) · 1 − χ (16) p with

√ ω εr . (17) c In (16) and (17), h and εr are the thickness and complex dielectric constant of the substrate, respectively. A second caution is related to the behavior of a patch absorber for small periods. Actually when the period p decreases down to the patch size w, the absorber behaves as a plain metallic plane covering the dielectric slab which corresponds to a plain reflector. Then the impedance Z gnd has to be zero for p = w and χ = 1 is consecutively required in (16). Equivalent RLC elements of a patch absorber are plotted in Fig. 7(a) as a function of the lattice period with the patch size w as a parameter. It can be seen that R and C values are quasi-invariant with w, while an increasing inductance L yields the expected decrease of the resonant frequency when γ =i

Fig. 7. Lattice period dependence of (a) RLC elements and of (b) radiative Q-factor for a patch array of single size w = 0.85, 0.89, 0.945, and 1 mm.

w increases. At small periods, the capacitance C increases due to the coupling between patches. For such small periods, the resonator intercoupling is more important for the widest patch, as it can be encountered in case of an FSS. For the radiative Q-factor plotted in Fig. 7(b), coupling between patch resonators is observed for small periods followed by a quasi-quadratic increase. This variation is similar to the one predicted for full dielectric structures (see Fig. 6). A limit value of p between coupling and uncoupling regimes relies on the patch size with an uncoupling regime occurring at a larger period for the widest patch. Finally, we can see that a perfect absorption A = 1 achieved when Q rad = Q d = 1/tan(δ) = 40 is a function of the patch size. A perfect absorption is obtained when p A=1 = 2.47 mm for w = 0.85 mm and when p A=1 = 2.72 mm for w = 1 mm. It should be mentioned that these period values are a little smaller than p = 2.8 mm considered to design the multisized patch absorber. This point will be commented in Section IV-B. Real and imaginary parts of the simulated impedance for a patch array of period p = 2.8 mm with a single patch size w = 0.85, 0.89, 0.945, and 1 mm as a parameter are plotted in Fig. 8(a) and (b), respectively. As expected, the resonant frequency decreases when the patch is enlarged, and resonances at f R = 80.7, 84.7, 89.2, and 92.6 GHz are close to each other. It can be seen that the impedance is

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. FERNEZ et al.: RADIATIVE QUALITY FACTOR IN THIN RESONANT METAMATERIAL ABSORBERS

Fig. 9.

7

Free-space experimental setup in millimeter band.

single-sized patch absorber is required in order to achieve an impedance matching between Z 4 p and Z 0 . Here by combining multisized patches with a period p = 2.8 mm, an impedance balance occurs with the real part of Z 4 p becoming higher than Z 0 for the first two resonances, and staying lower for the last ones. Furthermore, we can see in Fig. 8(b) that the imaginary part of Z 4 p is inductive for the two first resonances, whereas it becomes capacitive for the last ones. V. E XPERIMENTS

Fig. 8. (a) Real part and (b) imaginary part of the impedance Z eq (i) for a patch array of single size w(i) = 0.85, 0.89, 0.945, and 1 mm, and of the equivalent impedance Z 4 p calculated by (18).

relatively well matched to Z 0 for w = 1 mm while it slightly mismatched for smaller patch size in good accordance with the condition Q rad > Q d observed in Fig. 7(b) at p = 2.8 mm. B. Model of a Multisized Patch Absorber By combining impedances calculated for arrays of single patch size, the equivalent impedance of a multisized patch absorber can be calculated by Z4p =

4

(Z res (i ) + Z gnd(i ))

(18)

i=1

with

   Z 0 tanh(γ h) w(i ) 2 Z gnd(i ) = · 1− √ 4 εr p

(19)

where Z res (i ) is the equivalent impedance of patch resonators of size w(i ) and γ is calculated by (17). The impedance Z 4 p is plotted in Fig. 8. While each impedance Z eq (i ) was lower than 377 , it appears that the close vicinity of resonance frequencies increases the real part of Z 4 p at each resonance. Therefore to design the multisized patch absorber, a slightly larger period than the highest p A=1 value determined for a

Four square patches were considered with different sizes w = 0.85, 0.89, 0.945, and 1 mm. They were arranged in a supercell as illustrated in the insets of Figs. 9 and 10. Resonators were patterned under the shape of square metallic patches deposited onto a 0.1-mm-thick (∼λ0 /34) flexible PET substrate by means of an inkjet printing technology (Ceraprinter X-series of Ceradrop with a printing head SE128 AA of Fujifilm Dimatrix). The back side of the dielectric substrate was previously metallized with an aluminum layer by the supplier (Goodfellow Corporation). A silver nanoparticle ink EMD5714 (Suntronic Solsys) with 40 wt% was used for printing with a pulse voltage of 58 V and a jetting frequency of 1500 Hz. Subsequently a sintering of the sample during 40 min at 150 °C was performed to obtain a good conductivity of printed patterns. A photograph of a 30 × 30 mm2 absorber with ten supercells in transverse directions is placed in the experimental setup shown in Fig. 9, with a zoomed-in view region containing one supercell in the inset. Even though flexibility of the device is illustrated in the upper left inset of Fig. 9, the device was put onto a 2-in semiconductor holder for measurements and absorption under curvature is over the scope of the present paper. The free space experimental setup is shown in Fig. 9. It consisted of a Rohde & Schwarz ZVA 24 vector network analyzer (VNA) with transmit & receive external frequency extenders in the frequency band (75–110 GHz) connected to a circular horn antenna to achieve free space S-parameter measurements. Then the radiated beam at horn output was collimated by a dielectric lens in order to insure a plane wave illumination of the device.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

Fig. 10. Comparison of measured absorptions (M_x, M_y), simulated absorptions (S_x, S_y), and absorption calculated from Z 4 p . Measured and simulated absorptions are plotted for an electric field oriented in the x-(M_x, S_x) and y-(M_y, S_y) directions shown in the inset.

Prior to measurements VNA and external frequency extenders were calibrated in a waveguide configuration by using a standard through–reflect–match method, and the reflection coefficient of a perfectly reflective mirror in place of the device was measured as a reference for a de-embedding process. Experimental absorptions achieved by a differential measurement between the reflection coefficient of the device and the mirror are plotted in Fig. 10 and compared to simulated absorptions for two orthogonal E-field polarizations. These simulations (S_x, S_y) were performed by using the supercell illustrated in the inset of Fig. 10 with periodic master–slave boundary conditions and a period p = 2.8 mm. A weak sensitivity of absorption to polarization is observed in simulation with a lower absorption around 0.6 at 83 GHz for a polarization oriented in the x-direction. Sensitivity to polarization stems from the asymmetry of the structure due to the patch arrangement and to a coupling between patches. As for BST cubes, patch resonators act as magnetic dipoles with a predominance of the longitudinal coupling between patches. When polarization is oriented in the x-direction (S_x), the largest patches (1 and 0.945 mm) are longitudinally coupled with each other and they resonate almost at the same frequency. Then, resonances interfere and produce a low absorption around 0.6 observed at 83 GHz in Fig. 10. Similarly, the two other longitudinally coupled patches (0.89 and 0.85 mm) interfere and produce a low absorption around 0.7 at 91.5 GHz. On the other hand, when polarization is oriented in the y-direction, longitudinally coupled patches are (1 and 0.89 mm) and (0.945 and 0.85 mm). However, for each couple of patches, resonance frequencies are more separated and when one resonator is bright the second one is dark. Therefore, absorption can be viewed as a superposition of resonances for y-polarization. Polarization sensitivity is also observed in the experimental responses but with flatter shapes in the absorption frequency bands and with larger bandwidths. Broadening of experimental bandwidths originates from higher losses of the fabricated resonators. Let us recall that patches were patterned by inkjet

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

printing of a silver nanoparticles solution. Such a metallic ink has a lower conductivity [28] than bulk silver considered in simulation. Consecutively, large resonances can be observed in the experimental absorption bands resulting in larger absorption bandwidths and in flatter shapes. Moreover simulations were performed with nominal patch sizes and not with measured sizes of the fabricated patches which were a little different. It results a shift of the measured absorption bands compared to the simulated ones. However, it can be seen that wideband absorption is experimentally achieved with a bandwidth of 20% around 85 GHz in good accordance with simulated results. Finally, the absorption calculated by inverting (2) and by considering the equivalent impedance Z 4 p is plotted in Fig. 10. It can be shown that Z 4 p correctly predicts the absorption simulated by considering a supercell of four patches, especially for the polarization oriented in the y-direction while a weak mismatch occurs for the second polarization. As previously described, absorption can be viewed as a superposition of resonances for y-polarization, which is well depicted by (18). Note that this trend is specific to the structure considered here. More generally, as Z 4 p does not take into account the patch arrangement in the supercell, it is not suitable for analyzing the sensitivity of a multisized patch absorber to polarization. However, the radiative Q-factor concept can help designers by giving a method for combining multisized resonators to produce a broadband absorber. VI. C ONCLUSION Absorption in resonator-based structures has been analyzed by means of a radiative quality factor approach. A simple equivalent electrical circuit has been introduced. It consisted of a RLC parallel circuit for resonators in series with an inductive impedance taking into account the material surrounding resonators. Quadratic dependences of the equivalent RLC circuit as well as of the radiative Q-factor have been observed in an array of dielectric resonators when the lattice period is greater than twice the cube dimension with dielectric loss invariance. Such dependences can be turned to advantage for limiting parametric studies aimed at defining an optimal period for perfect absorption. For small periods, the radiative Q-factor revealed a coupling between resonators responsible for the decrease of the resonant frequency as well as for the limitation of the absorption bandwidth. From this assessment, a perfect absorption with the largest achievable bandwidth can be designed by separately adjusting the lattice period for the absorption level and the dielectric loss for the bandwidth. The Q-factor approach has been investigated for a patch absorber as well, with a final goal to broaden the absorption bandwidth by combining multisized patches. The analysis of equivalent impedances revealed that the combination of patches designed to resonate at close frequencies contributes to an improvement of the impedance matching condition at resonances and consecutively produces a better absorption. Finally, the equivalent impedance Z 4 p of a multisized patch absorber was calculated by introducing the contribution of each patch by equivalent impedances in series. The equivalent

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. FERNEZ et al.: RADIATIVE QUALITY FACTOR IN THIN RESONANT METAMATERIAL ABSORBERS

impedance Z 4 p led to a correct prediction of the experimental absorption. Complementary to the full-wave numerical method, this Q-factor analysis appears as a powerful tool in order to predict the absorbing capabilities of many structures. In particular, it should be helpful for the design of large area supercells, including disordered arrangement of resonators and aperiodic patterns based on fractal curves. ACKNOWLEDGMENT Author N. Fernez would like to thank the Direction Générale de l’Armement and Lille University for their support for Doctoral Fellowship. Author J. Hao would like to thank the China Scholarship Council for her Ph.D. Fellowship. R EFERENCES [1] W. F. Bahret, “The beginnings of stealth technology,” IEEE Trans. Aerosp. Electron. Syst., vol. 29, no. 4, pp. 1377–1385, Oct. 1993. [2] T. Khalid, L. Albasha, N. Qaddoumi, and S. Yehia, “Feasibility study of using electrically conductive concrete for electromagnetic shielding applications as a substitute for carbon-laced polyurethane absorbers in anechoic chambers,” IEEE Trans. Antennas Propag., vol. 65, no. 5, pp. 2428–2435, May 2017. [3] H. B. Baskey, E. Johari, and M. J. Akhtar, “Metamaterial structure integrated with a dielectric absorber for wideband reduction of antennas radar cross section,” IEEE Trans. Electromagn. Compat., vol. 59, no. 4, pp. 1060–1069, Aug. 2017. [4] J. Tak and J. Choi, “A wearable metamaterial microwave absorber,” IEEE Antennas Wirel. Propag. Lett., vol. 16, pp. 784–787, 2017. [5] S. Venkatachalam, G. Ducournau, J.-F. Lampin, and D. Hourlier, “Netshaped pyramidal carbon-based ceramic materials designed for terahertz absorbers,” Mater. Des., vol. 120, pp. 1–9, Apr. 2017. [6] A. Fernandez and A. Valenzuela, “General solution for singlelayer electromagnetic-wave absorber,” Electron. Lett., vol. 21, no. 1, pp. 20–21, Jan. 1985. [7] N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett., vol. 100, no. 20, p. 207402, 2008. [8] B. A. Munk, Frequency Selective Surfaces: Theory and Design. Hoboken, NJ, USA: Wiley, 2005. [9] F. Costa, S. Genovesi, A. Monorchio, and G. Manara, “A circuit-based model for the interpretation of perfect metamaterial absorbers,” IEEE Trans. Antennas Propag., vol. 61, no. 3, pp. 1201–1209, Mar. 2013. [10] X. Liu, K. Fan, I. V. Shadrivov, and W. J. Padilla, “Experimental realization of a terahertz all-dielectric metasurface absorber,” Opt. Express, vol. 25, no. 1, pp. 191–201, Jan. 2017. [11] C. Zou et al., “Nanoscale TiO2 dielectric resonator absorbers,” Opt. Lett., vol. 41, no. 15, pp. 3391–3394, Aug. 2016. [12] H. Yoshiizumi, R. Suga, O. Hashimoto, and K. Araki, “A design of circular patch array absorber based on patch antenna theory,” in Proc. Eur. Microwave Conf. (EuMC), 2015, pp. 1100–1103. [13] J. Hao, V. Sadaune, L. Burgnies, and D. Lippens, “Ferroelectrics based absorbing layers,” J. Appl. Phys., vol. 116, no. 4, p. 043520, Jul. 2014. [14] G. Houzet et al., “Ionic polarization occurrence in BaSrTiO3 thin film by THz-time domain spectroscopy,” Ferroelectrics, vol. 430, no. 1, pp. 36–41, Jan. 2012. [15] S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nature Nanotechnol., vol. 11, no. 1, pp. 23–36, Jan. 2016. [16] P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nature Commun., vol. 3, p. 692, Feb. 2012. [17] Y. Ishii, Y. Takida, Y. Kanamori, H. Minamide, and K. Hane, “Fabrication of metamaterial absorbers in THz region and evaluation of the absorption characteristics,” Electron. Commun. Jpn., vol. 100, no. 4, pp. 15–24, Apr. 2017. [18] P. Bouchon, C. Koechlin, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Wideband omnidirectional infrared absorber with a patchwork of plasmonic nanoantennas,” Opt. Lett., vol. 37, no. 6, pp. 1038–1040, Mar. 2012. [19] M. Yoo, H. K. Kim, and S. Lim, “Angular- and polarization-insensitive metamaterial absorber using subwavelength unit cell in multilayer technology,” IEEE Antennas Wireless Propag. Lett., vol. 15, pp. 414–417, 2016.

9

[20] B. Mulla and C. Sabah, “Multi-band metamaterial absorber topology for infrared frequency regime,” Phys. E, Low-Dimensional Syst. Nanostruct., vol. 86, pp. 44–51, Feb. 2017. [21] J. Hao, É. Lheurette, L. Burgnies, É. Okada, and D. Lippens, “Bandwidth enhancement in disordered metamaterial absorbers,” Appl. Phys. Lett., vol. 105, no. 8, p. 081102, Aug. 2014. [22] A. Bhati, A. Shukla, A. Jain, K. R. Hiremath, and V. Dixit, “Numerical study of randomly distributed wire based metamaterial absorber,” in Proc. Int. Conf. Microw., Opt. Commun. Eng. (ICMOCE), 2015, pp. 92–95. [23] D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ, USA: Wiley, 2012. [24] C. A. Balanis, Antenna Theory: Analysis and Design, 3th ed. Hoboken, NJ, USA: Wiley, 2005. [25] K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, S. Savel’ev, and F. Nori, “Colloquium?: Unusual resonators: Plasmonics, metamaterials, and random media,” Rev. Mod. Phys., vol. 80, no. 4, pp. 1201–1213, Oct. 2008. [26] N. Liu and H. Giessen, “Coupling effects in optical metamaterials,” Angew. Chem. Int. Ed., vol. 49, no. 51, pp. 9838–9852, Dec. 2010. [27] J. Hao, “Broad band electromagnetic perfect metamaterial absorbers,” Ph.D. dissertation, IEMN, Univ. Lille, Villeneuve d’Ascq, France, 2016. [28] A. Chahadih, P. Y. Cresson, C. Mismer, and T. Lasri, “V-band vialess GCPW-to-microstrip transition designed on PET flexible substrate using inkjet printing technology,” IEEE Microw. Wireless Compon. Lett., vol. 25, no. 7, pp. 436–438, Jul. 2015.

Nicolas Fernez, photograph and biography not available at the time of publication.

Ludovic Burgnies, photograph and biography not available at the time of publication.

Jianping Hao, photograph and biography not available at the time of publication.

Colin Mismer, photograph and biography not available at the time of publication.

Guillaume Ducournau, photograph and biography not available at the time of publication.

Didier Lippens, photograph and biography not available at the time of publication.

Éric Lheurette, photograph and biography not available at the time of publication.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Nonlinear Effects of SiO2 Layers in Bulk Acoustic Wave Resonators Carlos Collado , Senior Member, IEEE, Jordi Mateu , Senior Member, IEEE, David Garcia-Pastor, Student Member, IEEE, Rafael Perea-Robles, Alberto Hueltes, Susanne Kreuzer, and Robert Aigner Abstract— This paper presents the development of a comprehensive distributed circuit model to account for the existing nonlinear effects in bulk acoustic wave (BAW) resonators. The comprehensiveness of the model and its distributed implementation allow for the inclusion of the nonlinear effects occurring in any layer of the BAW configuration, not only the piezoelectric layer. The model has been applied to evaluate the nonlinear contribution of the piezoelectric layer and silicon dioxide (SiO2 ) layer in the Bragg reflector. The nonlinear manifestations are a function of the frequency of the driving fundamental tones. Accurate measurements of state-of-the-art resonators validate the model proposed and confirm the contribution of the SiO2 layer in the overall nonlinear performance. Index Terms— Bulk acoustic wave (BAW), electroacoustic, nonlinearities, second harmonic (H2), SiO2 , third harmonic (H3), third-order intermodulation (IMD3) product.

I. I NTRODUCTION

B

ULK acoustic wave (BAW) technology is becoming the main solution for the complex RF filtering of current and future mobile devices [1]. The increasing amount of services, frequency band, and worldwide interoperability requirements for the handsets force the industry to place more and more high-performance filters in a reduced space. Besides small footprint, power handling, high selectivity, and low insertion losses, nonlinearities in BAW resonators have become a hot issue in the last years as the regulations and standards demand higher linearity. Electroacoustic dynamics of BAW resonators are intrinsically nonlinear. Nonlinear relations can be stablished between field magnitudes, and it is not trivial to unambiguously identify what are the most important contributions to observable effects [2]. A circuital model is critical to understand the origin of nonlinearities. The model must be able to relate nonlinear properties of material that are independent of geometry

Manuscript received June 13, 2017; revised August 23, 2017 and October 6, 2017; accepted November 29, 2017. This work was supported by the Catalan Government under Grant 2014 SGR 1103. (Corresponding author: Carlos Collado.) C. Collado, D. Garcia-Pastor, R. Perea-Robles, and A. Hueltes are with the Signal Theory and Communications Department, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain (e-mail: [email protected]). J. Mateu is with the Signal Theory and Communications Department, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain, and also with the Centre Tecnologic de Telecomunicacions de Catalunya, 08060 Barcelona, Spain. S. Kreuzer and R. Aigner are with Orvo, Inc., Apopka, FL 32703 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2783377

with experimental observations. After the first nonlinear distributed equivalent circuit of acoustic devices published in 1993 [3], extensive work has been done in the last 10 years. Collado et al. [4] published a nonlinear distributed model for BAW resonators, which was based in the KLM circuit. This model was a phenomenological approach to describe the second harmonic (H2) generation and was surpassed by the nonlineal distributed Mason model [5], [6], which was based in the constitutive nonlinear equations. This later model was completed in [7] adding the thermal domain to the constitutive equations. Feld et al. [2], Collado et al. [4], Shim and Feld [5], [6], and Rocas et al. [7] outlined the importance of measuring harmonics and intermodulation generation sweeping the frequency of the fundamental/s signal/s. The frequency pattern of a given observable must be consistent with the distributed model and it constitutes a fingerprint of the origin of the nonlinearity. In addition, different manifestations of nonlinearity, through different experiments, must be also consistent and explained with a unique model. References [4] and [7] using solidly mounted resonators (SMRs) and [2], [5], and [6] using thin-film bulk acoustic wave resonators (FBARs) concluded that H2 and third-order intermodulation (IMD3) was generated in the piezoelectric layer. Accordingly, all models developed were focused on the nonlinearities in the piezolayer and none of them included the effects other layers could have in the nonlinearities. Specifically, the silicon dioxide (SiO2 ) layers usually present in the Bragg Reflector of SMRs can play a significant role depending of the stack configuration. This paper experimentally validates the latter statement by performing three different experiments: 1) generation of second-order harmonics H2 and thirdorder harmonics H3; 2) IMD3 products; 3) detuning with a dc voltage bias applied to the electrodes. II. C ONSTITUTIVE E QUATIONS A. Piezoelectric Layers The constitutive nonlinear electroacoustic equations for the piezoelectric layer are extensively explained in [5]–[7]. Those equations relate the field magnitude stress T , strain S, electric field E, and electric displacement D to each other using the constants c E , e, and ε S (stiffness, and piezoelectric and dielectric constants, respectively). The equations according to

0018-9480 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

the nomenclature described in [6] are as follows: T = c E S − eE + T D = eS + ε S E + D

(1)

where the nonlinear terms T and D are S2 S3 E2 + c3E − ϕ3 + ϕ5 S E 2 6 2 2 3 2 E E S D = ε2S + ε3S − ϕ5 + ϕ3 S E 2 6 2 T = c2E

(2)

which depend on several second-order (c2E , ϕ3 , ϕ5 , ε2S ) and third-order (c3E , ε3S ) derivative constants. Using (1) and (2), we can define the nonlinear constants as   c2E c3E 2 ϕ5 E E cNL = c · 1 + E S + E S + E E 2c 6c 2c  ϕ3  ϕ5 S+ E eNL = e · 1 − 2e  2e  ε2S ε3S 2 ϕ3 S S εNL = ε · 1 + S E + S E + S S (3) 2ε 6ε 2ε in order to help a better understanding of the role of each parameter. The model will be finally implemented using the strain S and the electrical displacement D as independent variables, and adding the nonlinear voltage sources Tc and Vc to a conventional distributed Mason model [5], [6]. As done in [2] and [7], the piezoelectric layer is discretized into many unit cells of small thickness z, where the strength of the electric field is constant e T = c D S − S D + TC ε D − eS E = − VC (4) εS where e TC = T + S D ε D (5) VC = S z ε and the stiffened elasticity is defined as cD = cE +

e2 . εS

(6)

Fig. 1.

Nonlinear unit cell of the piezoelectric layer [2], [7].

III. N ONLINEAR M ODEL A. Piezoelectric The circuit model of the piezoelectric layer is the same than the one used in [2] and [5]–[7]. Equations (4) and (5) are implemented using the distributed Mason model, in which the equivalences force = voltage and particle velocity = current allow for an equivalent circuit implementation. This model discretizes the piezoelectric layer into many unit cells (slabs of thickness z), which account for electroacoustic interactions (4) and the acoustic wave propagation in the thickness direction. Each cell obeys the telegrapher equations for an acoustic transmission line with distributed parameters [8] z Cz = Ac D L z = z ρ A z G z = (9) A ηD in which A, c D , ρ, η D are the resonators’ area, stiffened elasticity, mass density, and viscosity, respectively. Fig. 1 shows the nonlinear unit cell [2], [7] of the circuit model. The number of unit cells that must be cascaded depends on the highest frequency of interest. In our case, we will discretize the piezoelectric in 60 unit cells.

B. Nonpiezoelectric Layers

B. Nonpiezoelectric

For the nonpiezoelectric layers, the nonlinear relation between stress and strain, truncated up to a third order, can be written as

Two different approaches have been used to model a nonpiezoelectric layer depending on the potential nonlineal behavior of the material. 1) Linear Layers: Acoustic wave propagation can be modeled with a conventional T-network circuit [Fig. 2(a)] for an acoustic transmission line with propagation constant γ and characteristic impedance Z 0 ω γ =α+ j Z0 = ρ · A · v A (10) vA v A being the phase velocity and α the attenuation constant [8] √ 1 2 ρc α = ηω . (11) 2 c2

T = cNP S + TC 1 1 TC = c2,NP S 2 + c3,NP S 3 (7) 2 6 in which the subscript NP indicates the material and c2,NP , c3,NP are the nonlinear derivatives of the elastic constant cNP,NL   c2,NP c3,NP 2 cNP,NL = cNP · 1 + S+ S . (8) 2cNP 6cNP

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. COLLADO et al.: NONLINEAR EFFECTS OF SIO2 LAYERS IN BAW RESONATORS

3

Fig. 2. Circuit models of the nonpiezoelectric layers. T-network equivalent circuit of an (a) acoustic transmission line and (b) nonlinear unit cell of a discretized transmission line. Fig. 3.

Stack configuration of the measured SMR and BAW resonator.

2) Discretized Model for Nonlinear Layer: For the nonlinear case, the field magnitudes must be calculated at each point along the thickness of the layer. A discretized model [8] is again used [Fig. 2(b)] with distributed inductance, capacitance, and conductance according to (9) and nonlinear voltage sources Tc according to (7) IV. D EVICES AND L INEAR M EASUREMENTS Distributed nonlinear models can reproduce the standingwave pattern of the fields along the thickness direction of the stack at a given frequency. The local nonlinear contribution to a given measurable magnitude, such as the H2 power at the electrodes of the resonator can be simulated. The frequency pattern of this measurable quantity is like a fingerprint of the origin of the nonlinearities since each nonlinear constant will contribute in a different way as a function of frequency. This is because some of the constants, those appearing in T and D in (2), determine the sources Tc through (5), and those appearing in D affect to Vc and Tc . Furthermore, as those constant are the results of derivatives with regard different independent magnitudes, S or E, the effect on the frequency pattern will be also different according to the different frequency pattern of S and E. The main advantage of using distributed models is that it provides valuable information [2], [4]–[7] to characterize the materials and determine their nonlinear constants. In addition, a given set of nonlinear constant must be consistent with all the experimental results that those constants can determine. In this paper, we discuss the experiments we have performed to unambiguously identify and quantify nonlinear constants in BAW resonators. A. Description of the Devices The measured device is an Aluminum Nitride (AlN)-based SMR formed by more than ten layers. The thickness l of the AlN is 0.9 μm and the series resonance frequency is around 2.3 GHz. The layers of the electrodes and Brag reflector are shown in Fig. 3. The stack used is similar to [9]. B. Linear Fitting A very good match between the linear measurements and the model is critical for an accurate model of the nonlinearities. Not only the main resonance must be accurately modeled, but also the out-of-band resonances appearing at higher frequency as we will discuss in the following sections.

Fig. 4. Simulated (blue dashed line) and measured (red dotted line) narrowband (top) and broadband (bottom) input impedance of a BAW resonator.

Fig. 4 shows the measured input impedance and the simulated response showing very good agreement. V. N ONLINEAR M EASUREMENTS A main concern in BAW resonators is about the secondorder nonlinearities, which are usually characterized by H2 generation [2], [4]–[7], [10]. A. Narrowband Second-Harmonic Measurements Fig. 5 shows the measured H2 with a DUT input power of 21 dBm. As it was expected, there is a maximum at a frequency just above the series resonance of 2.32 GHz. At this frequency, the stress is maximum inside the piezoelectric layer [4] and there is consensus [2], [5]–[7] on the origin of the nonlinearity: the parameter ϕ5 , (δ1 in [5] and [6]) which sets the strain-dependent piezoelectric constant, or the electric field-dependent elasticity due to equivalent secondorder interactions between physical domains. In fact, despite ϕ5 having an effect on Vc and Tc according to (2)–(5), the main contributor to the H2 level at this frequency is the nonlinear source Vc. Fig. 5 shows the H2 simulated with ϕ5,AlN = −17 · eAlN. This value is exactly the same we obtained in [7].

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

Fig. 5. Measured (black dashed line) and simulated H2 for the following E cases: only ϕ5,AlN (green asterisks), ϕ5,AlN and c2,AlN (red circles), and ϕ5,AlN and c2,SiO (blue squares).

The smooth ripple appearing is due to the measurement system and its measured effect is included in the simulation [11]. The simulation matches the measured response related to the main resonance (peak at 2.34 GHz), but fails to model the additional peak appearing at 2.245 GHz. To understand where this peak is coming from, we have to take a look at the out-ofband resonances present in the broadband input impedance of the resonator. As it can be seen in Fig. 4, a small resonance shows at twice that frequency 4.49 GHz. Apparently there is an acoustic mode at 4.49 GHz, which is able to interact with H2 of the driving tone at 2.245 GHz. As the mode couples to the electrical domain, the H2 measurement shows a high peak in Fig. 5. As the high frequency out-of-band resonances are strongly dependent on the stack configuration, the linear response must be accurately modeled to reproduce this effect in simulations, as introduced in Section IV. At a first glance, the term c2E of the piezoelectric constant can produce this high peak, as shown in Fig. 5. The pair of values E E = −32.2 · cAlN and ϕ5,AlN = −23.8 · eAlN reproduces c2,AlN the measured frequency pattern. Nevertheless, the nonlinear term of the elastic constant c2E is not a uniquely able to produce this effect. A nonlinear elastic constant for the SiO2 with c2,SiO = −6.4 · cSiO can produce the same effect, as shown in Fig. 5. In the latter case, the  nonlinearities of the AlN are set to ϕ5,AlN = −18.7 · eAlN and E c2,AlN = 0. In this case, we have checked through simulations that the SiO2 layer placed just underneath the bottom electrode is the predominant SiO2 layer causing this H2 peak. This makes sense since the standing-wave pattern of the strain in SiO2 is dominated by the first layer at that frequency. It is clear that additional measurements must be done to disE ) cern between those two working hypotheses: (ϕ5,AlN, c2,AlN  and (ϕ5,AlN, c2,SiO ). B. Broadband Second-Harmonic Measurements Broadband H2 measurements can help since the standingwave pattern at higher frequencies changes considerably within the stack. Fig. 6 shows the measured H2 from 2.1 to 3.2 GHz. As it can be seen, high H2 peaks appear

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 6. Measured (black solid line) and simulated broadband H2 for the E following cases: ϕ5,AlN and c2,AlN (red dashed line) and ϕ5,AlN and c2,SiO (dotted blue line).

Fig. 7. Measured (solid black line) and simulated broadband H2 for the cases ϕ5,AlN and c2,SiO with ε2S = 0 (blue dotted line). E at 2.8 and 3 GHz. Note that the AlN nonlinear term c2,AlN cannot reproduce them, yet the SiO2 term c2,SiO creates excellent agreement with measurements, as can be seen in Fig. 6. It is worth mentioning that in both scenarios we have introduced a nonlinear term for the AlN dielectric constant. E must be taken in consideration The term ε2S = 20·ε S eAlN/cAlN to reproduce the out-of-resonance plateau of the H2, as shown in Fig. 7.

C. Narrowband IMD3 and H3 We have also performed IMD3 and H3 measurements. Two tones were applied with a tone spacing of  f = f 2 − f 1 = 1 MHz and DUT input power of 30 dBm. Fig. 8 shows the measurements and simulations for the hypotheE ). This hypothesis overestimates the IMD3 sis (ϕ5,AlN, c2,AlN and H3 at resonance. Fig. 9 shows the measurements and  simulations for the second hypothesis (ϕ5,AlN , c2,SiO ). This matches the measurements better. Note that no additional third-order terms have been included. Therefore, third-order intermodulation products and harmonics are coming from second-order nonlinear terms by remix of the second-order  , c2,SiO ) with the fundamental signals. terms (ϕ5,AlN Sticking to the confirmed second hypothesis, we investigate which term is the main contributor to third-order

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. COLLADO et al.: NONLINEAR EFFECTS OF SIO2 LAYERS IN BAW RESONATORS

Fig. 8. Measured (dotted lines) and simulated (solid lines) IMD3 (2 f 1 − f 2 in blue circles and 2 f 2 − f 1 red squares) and H3 (3 f 1 in black diamonds) E . for the case ϕ5,AlN and c2,AlN

Fig. 9. Measured (dotted lines) and simulated (solid lines) IMD3 (2 f 1 − f 2 in blue circles and 2 f 2 − f 1 red squares) and H3 (3 f 1 in black diamonds) for the case ϕ5,AlN and c2,SiO .

 nonlinearities: ϕ5,AlN or c2,SiO. Fig. 10 shows simulations with c2,SiO = 0, and Fig. 11 shows simulations with ϕ5,AlN = 0.  As it can be seen, ϕ5,AlN slightly underestimates the IMD3 and fails to simulate the H3 below resonance. On the contrary c2,SiO overestimates the IMD3 and predict the H3 very well, which is completely dominated by this term. In our previous work on SMR and BAW resonator [6], we stated that the IMD3 (with tone spacing big enough to avoid thermal effects) was due to c3E . H3 was not measured because the measurement bandwidth was limited to 50 MHz around resonance. Because the stack was different, we did not discover the contribution of the SiO2 layers on the nonlinearities and we wrongly postulated that the IMD3 was E = directly generated by a third-order nonlinear term c3,AlN E . Here, we have performed simulations with this −111 · cAlN value and setting c2,SiO = 0 and the IMD3 peak at resonance is E , as shown exactly the same than the one resulting from c2,SiO in Fig. 12. However, IMD3 and H3 out of resonance, between 2.23 and 2.3 GHz, are not consistent, as shown in Fig. 12. E is not the main It now seems clear than third-order term c3,AlN contribution to the measured IMD3 for this stack.

5

Fig. 10. Measured (dotted lines) and simulated (solid lines) IMD3 (2 f 1 − f 2 in blue circles and 2 f 2 − f 1 red squares) and H3 (3 f 1 in black diamonds) for the case only ϕ5,AlN .

Fig. 11. Measured (dotted lines) and simulated (solid lines) IMD3 (2 f 1 − f 2 in blue circles and 2 f 2 − f 1 red squares) and H3 (3 f 1 in black diamonds) for the case only c2,SiO .

Fig. 12. Measured (dotted lines) and simulated (solid lines) IMD3 (2 f 1 − f 2 in blue circles and 2 f 2 − f 1 red squares) and H3 (3 f 1 in black diamonds) E for the direct generation due to only c3,AlN .

D. DC Detuning The third experiment is an S-parameters measurement while a dc bias voltage is applied at the electrodes. This must provide consistent results for the hypothesis to be confirmed. This experiment is of remarkable interest since the SiO2 layers are

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

Fig. 13. Input impedance with +25-V dc voltage: measured (red solid line), HB simulation (black dotted line), and closed-form expression (blue dashed line).

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

−25 to 25 V that was measured, and simulations using (12) for E ) overestimates both hypotheses. The hypothesis (ϕ5,AlN, c2,AlN the frequency shift in comparison with measurements. On  E , c2,SiO ) is the other side, when the hypothesis of (ϕ5,AlN assumed, simulations and measurements perfectly overlap. This is the third and simplest independent experiment that  E , c2,SiO ). Using the provided confirms the hypothesis (ϕ5,AlN  formulation, it is a fast test to set the magnitude of ϕ5,AlN and does require neither complex measurements nor nonlinear simulations. However, this experiment by itself does not give any information about the nonlinearities coming from the SiO2 layers, which play a significant role in the generation of harmonics and intermodulation products as shown earlier. VI. N ONLINEARITIES IN OTHER L AYERS

Fig. 14. Frequency shift of series (red squares) and shunt (blue circles) resonances in parts per million of measured series resonance (black plus sign) and E shunt resonance (black times sign), and simulations under (ϕ5,AlN , c2,AlN )  E hypothesis (dashed lines) and (ϕ5,AlN , c2,SiO ) hypothesis (solid lines).

not subject to the static electric field between electrodes, nor will static strain exist in the SiO2 layers. Although, the nonlinear model we are using is able to simulate the effects of a dc bias applied voltage, we have also derived new closed-form expressions, whose derivation is detailed in the Appendix. As a consequence, linear simulations may account for an applied dc voltage just by replacing the E , e S constants cAlN AlN , and ε by the terms  E e c2,AlN E,DC E · 1 − E D VDC cAlN = cAlN cAlN cAlNl  E 2 1 c3,AlNe ϕ5 2 +  V − E VDC 2 c E c D l· 2 DC lcAlN AlN AlN   ϕ5 DC eAlN = eAlN · 1 + E VDC cAlNl   ε2S S,DC S (12) = ε · 1 − S VDC ε ε ·l l being the piezoelectric thickness. In order to validate the closed-form equations, Fig. 13 shows the measured and simulated input impedance for an applied voltage of +25 V. HB simulations using the nonlinear model and linear simulations using the constants described in (12) overlap perfectly. Fig. 14 shows the frequency shifts of the series and shunt resonances in parts per million for a voltage ranging from

Other materials, such as the aluminum (Al) or the tungsten (W), could also contribute on the nonlinearities. In fact, both, Al and W, can reproduce the narrowband H2 behavior of Fig. 5 E E or c2,W terms are chosen. However, those if appropriate c2,Al hypotheses fail to model the broadband H2 response and they overestimate by far the IMD3. It is also worth to mention that other third-order nonlinear terms might also contribute to the direct IMD3 generation. This is for instance the case of the third-order terms of elastic constant of the layers in the Bragg reflector, such as SiO2 E E , respectively), which could be or W layers (c3,SiO and c3,W considered to improve the match between simulations and measurements. Those terms, with the proper sign, might cancel out the IMD3 rising from remixing effects at frequencies around 2.32 GHz, and shift the simulated maximum IMD3 toward higher frequencies, giving a more accurate fitting to the measured IMD3. However, further experiments and research need to be done to properly discern and quantify the contribution of these layers SiO2 and/or W. VII. C ONCLUSION Unexpected H2 peaks may appear at some frequencies in BAW resonator and FBAR. When the device is driven at a frequency close to the resonance frequency, such that an outof-band resonance exists at twice that frequency, the resonating conditions enhance the H2 and the output H2 power may become significant. For the particular stack studied in this paper, we have unambiguously demonstrated that the silicon dioxide layer below the bottom electrode is responsible for this effect by three different experiments: broadband H2, IMD3 and H3, and dc detuning. Once the origin of this anomalous effect is well established, it may be corrected by modifying the design of the Bragg reflector and electrodes slightly to avoid out-of-band resonances close to twice the main resonance. For the resonator we have measured, the IMD3 and H3 were also dominated from the second-order nonlinear silicon dioxide through remixing effects. The harmonic and intermodulation experiments were consistent with linear measurements of the input impedance of the resonator under a dc bias voltage. We also provide new closedform expressions which agree perfectly with the nonlinear simulations.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. COLLADO et al.: NONLINEAR EFFECTS OF SIO2 LAYERS IN BAW RESONATORS

7

A PPENDIX

R EFERENCES

Using (1) and (2), and considering negligible the term according to the simulations of Section V, we can write the nonlinear relation between stress, strain, and electric field

[1] A. Tag and C. Ruppel, “RF acoustic for mobile communication: Challenges and modern solutions,” IEEE Microw. Mag., vol. 16, no. 7, pp. 22–24, Jul. 2015. [2] D. A. Feld, D. S. Shim, S. Fouladi, and F. Bayatpur, “Advances in nonlinear measurement & modeling of bulk acoustic wave resonators,” in Proc. IEEE Int. Ultrason. Symp., Chicago, IL, USA, Sep. 2014, pp. 264–272. [3] Y. Cho and J. Wakita, “Nonlinear equivalent circuits of acoustic devices,” in Proc. IEEE Ultrason. Symp., vol. 2. Baltimore, MD, USA, Oct. 1993, pp. 867–872. [4] C. Collado, E. Rocas, J. Mateu, A. Padilla, and J. M. O’Callaghan, “Nonlinear distributed model for bulk acoustic wave resonators,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 12, pp. 3019–3029, Dec. 2009. [5] D. S. Shim and D. A. Feld, “A general nonlinear mason model of arbitrary nonlinearities in a piezoelectric film,” in Proc. IEEE Int. Ultrason. Symp., San Diego, CA, USA, Oct. 2010, pp. 295–300. [6] D. S. Shim and D. A. Feld, “A general nonlinear mason model and its application to piezoelectric resonators,” Int. J. RF Microw. Comput.Aided Eng., vol. 21, no. 5, pp. 486–495, Sep. 2011. [7] E. Rocas, C. Collado, J. Mateu, N. D. Orloff, J. C. Booth, and R. Aigner, “Electro-thermo-mechanical model for bulk acoustic wave resonators,” IEEE Trans. Ultrason., Ferroelect., Freq. Control, vol. 60, no. 11, pp. 2389–2403, Nov. 2013. [8] B. A. Auld, Acoustic Fields and Waves in Solids, vol. 1. Malabar, FL, USA: Krieger, 1990. [9] A. Tajic et al., “No-drift BAW-SMR: Over-moded reflector for temperature compensation,” in Proc. IEEE Int. Ultrason. Symp., Tours, France, Sep. 2016, pp. 1–4. [10] E. Rocas, C. Collado, J. C. Booth, E. Iborra, and R. Aigner, “Unified model for bulk acoustic wave resonators’ nonlinear effects,” in Proc. IEEE Int. Ultrason. Symp., Rome, Italy, 2009, pp. 880–884. [11] C. Collado et al., “First-order elastic nonlinearities of bulk acoustic wave resonators,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 5, pp. 1206–1213, May 2011.

1 1 T = c E S − eE + c2E S 2 + c3E S 3 + ϕ5 S E. 2 6

(13)

In order to characterize the dc feed contribution, the electric field and strain can be modeled as the sum of the contributions of the static magnitudes E DC and SDC produced by the dc voltage and the electric field and strain produced by the fundamental signal with peak amplitudes E 0 and S0 S = SDC + S0 cos(ωt) E = E DC + E 0 cos(ωt).

(14)

By use of (13), and selecting the terms that affect the strain and electric field at the fundamental frequency, we can write   1 1 2 + c3E S02 + ϕ5 E DC T = c E + c2E SDC + c3E SDC 2 8 × S0 cos(ωt) − (e − ϕ5 SDC )E 0 cos(ωt). (15) The saturation term (1/8)c3E S02 that does not depend on the dc bias voltage can be neglected for low-power levels, and (15) can be written as T = c E,DC S0 cos(ωt) − eDC E 0 cos(ωt)

(16)

where the stiffness and piezoelectric constants modified by the dc voltage are, respectively   c2E 1 c3E 2 ϕ5 E,DC E c = c · 1 + E SDC + S + E DC c 2 c E DC c E   ϕ5 (17) SDC . eDC = e · 1 − e The strain produced by the dc voltage is   e e −VDC SDC = E E DC E c c l

(18)

l being the thickness of the piezoelectric layer. Therefore, we can define and effective elastic c E,DC and piezoelectric eDC constants under dc bias voltage as   c2E e 1 c3E e2 ϕ5 E,DC E 2 VDC = c · 1+ E 2 VDC + V + c (c ) l 2 (c E )3l 2 DC c E l   ϕ5 (19) eDC = e · 1 − E VDC . c l

Carlos Collado (A’02–M’03–SM’10), photograph and biography not available at the time of publication.

Jordi Mateu (A’02–M’04–SM’10), photograph and biography not available at the time of publication.

David Garcia-Pastor (S’17), photograph and biography not available at the time of publication.

Rafael Perea-Robles, photograph and biography not available at the time of publication.

Alberto Hueltes, photograph and biography not available at the time of publication.

The contribution of the nonlinear term ε2S can be taking into account straightforward just replacing   ε2S S,DC S (20) ε = ε · 1 − S VDC . ε ·l

Susanne Kreuzer, photograph and biography not available at the time of publication.

The modified terms of (19) and (20) can be used instead of c E , e, and ε S in linear simulations.

Robert Aigner, photograph and biography not available at the time of publication.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

A Direct Matrix Synthesis for In-Line Filters With Transmission Zeros Generated by Frequency-Variant Couplings Yuxing He, Giuseppe Macchiarella, Fellow, IEEE, Gang Wang , Member, IEEE, Wentao Wu, Liguo Sun , Lu Wang , and Rong Zhang, Student Member, IEEE Abstract— A direct matrix approach is presented for the first time to synthesize high selectivity in-line topology filters where multiple transmission zeros are generated and independently controlled by a set of frequency-variant couplings. As the resultant network only involves resonators cascaded one by one without any auxiliary elements (such as cross-coupled or extractedpole structures), this paper provides the best synthesis solution in configuration simplicity for narrowband filters. Considering both the couplings and capacitances of a traditional low-pass prototype, a generalized transformation on the admittance matrix is introduced as the basis of the synthesis, which allows more than one cross-coupling to be annihilated in a single step, while generating a frequency-variant coupling simultaneously. It is then shown that the in-line topology as well as some other unique topologies can be determined by applying a specific sequence of the transformations. For the validation, a group of examples with synthesis as well as experimental results are demonstrated. Index Terms— Coupling matrix, filter synthesis, frequencyvariant coupling, microwave filters, transmission zero (TZ).

I. I NTRODUCTION

I

MPLEMENTING finite transmission zeros (TZs) are crucial in the modern microwave filter design for the distinct improvement of out-of-band frequency selectivity. With the adaptability to realizing all kinds of (i.e., real, imaginary, and complex) TZs and the diversity of topology constructions, the use of cross-coupled structures is considered as one of the most prevalent ways for the TZ generation [1]–[6]. However, this mechanism considerably raises the complexity of the design and tuning process due to the multiple coupling pathways (i.e., a number of cross couplings other than the mainline Manuscript received June 30, 2017; revised October 17, 2017; accepted December 17, 2017. This work was supported by the National Natural Science Foundation of China under Grant 61671421. (Corresponding author: Liguo Sun.) Y. He was with the Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei 230027, China. He is now with the Institute of Advanced Sciences, Yokohama National University, Yokohama 240-8501, Japan (e-mail: [email protected]). G. Macchiarella is with the Dipartimento di Elettronica Informazione e Bioingegneria, Politecnico di Milano, 20133 Milan, Italy (e-mail: [email protected]). G. Wang, W. Wu, L. Sun, L. Wang, and R. Zhang are with the Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei 230027, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2018.2791940

coupling route). An alternative manner for TZ realization is proposed by utilizing the extracted-pole sections [7]–[11]. As extra nonresonating nodes are adopted, this topology still brings in more circuit elements and a larger size during the fabrication. To make better advantage for the configuration simplicity, a new concept of in-line topology filters has been recommended in recent years [12]–[19]. In [12] and [13], an in-line configuration is produced by carefully implementing the bypass couplings between nonadjacent resonators (via the change of orientation for selected resonators). As the nature of these works are still based on cross-coupled topologies, the design complexity remains unchanged and the TZs still cannot be controlled independently. A more attractive solution is then proposed in [14]–[19]. By replacing the ideal constant inverters with frequency-variant couplings (supposed linear and with strong frequency dependence), it is observed that TZs can be generated and independently controlled without any cross-coupled or extracted-pole structures. In result, the derived network only involves resonators serially cascaded one after another, providing the simplest configuration for high selectivity filters. However, it is noticed that most of the related works are based on optimizations [14]–[17]. In [18], secondand third-order in-line filters are theoretically discussed by separately analyzing the capacitive and inductive couplings of each mixed-coupled structure. Nevertheless, this mechanism requires a number of manipulations and a general approach for in-line filters with higher orders and more frequencyvariant couplings is still unsolved. Recently, a favorable direct synthesis method is reported in [19], but is available for a very particular in-line condition where only one TZ can be realized. Lacking a general direct synthesis approach, whether a selected in-line network can realize required frequency response is still not predictable. In this paper, a general direct synthesis approach is presented for synthesizing in-line filters with multiple TZs (as shown in Fig. 1) for the first time [20]. Concerning the capacitance matrix and coupling matrix together, a new transformation for the admittance matrix is proposed, which presents new possibilities that can transform a triplet section into a frequency-variant coupling and annihilate more than one cross coupling in a single step. With a specific procedure of transformations, an in-line topology containing a set of frequency-variant couplings is ultimately decided

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

admittance matrix is introduced as [Yn+1 ] = [Tn+1 ]T [Yn ][Tn+1 ] = [Tn+1 ]T ([Mn ] + [Cn ] − j [G])[Tn+1 ] = [Tn+1 ]T [Mn ][Tn+1 ] + [Tn+1 ]T [Cn ][Tn+1 ] − j [G] (3) = [Mn+1 ] + [Cn+1 ] − j [G] Fig. 1. Ideal prototype for a class of in-line topology filters with multiple frequency-variant couplings, which can generate TZs at f z1 , f z2 , . . . , f z,NZ , independently.

from the traditional transversal array network with constant coupling elements. Combining the proposed technique with cross-coupled and extracted-pole structures, this paper also proposes other unique topologies that expand the flexibility of direct matrix synthesis for narrowband high selectivity filters. In the following, basic theory on the admittance matrix transformation process is detailed in Section II. The design procedure for determining the in-line and other unique topologies is presented in Section III. Thereafter, a series of illustrative examples, together with the synthesis results, is demonstrated in Section IV. In Section V, a fourth-order coaxial resonator in-line filter, containing two frequency-variant couplings (generating TZs at 1.64 and 1.97 GHz separately), is designed, fabricated, and tested for the experimental validation. Finally, Section VI provides the conclusion. II. S YNTHESIS T HEORY A. Admittance Matrix Transformation According to the well-established theory in [6], the proposed approach starts with the N +2 admittance matrix [Y ] (suppose the row/column numbers as S, 1, 2, . . . , N, L in the following) for an Nth order low-pass cross-coupled prototype [Y ] = [M] + [C] − j [G]

(1)

where [M] and [C] are the coupling and capacitance matrices, respectively, while specify matrix [G] structures the terminal conductances (assumed as G SS = G LL = 1) at source and load. Note that [C] is diagonal with the entries CSS = CLL = 0 and C11 = C22 = · · · = C N,N = 1, implying that all the coupling elements are constant here. Therefore, S-parameters for the prototype can be obtained by the following [6]:

[R]

⎡

1 ⎢... ⎢ ⎢ 0 ⎢ =⎢ ⎢... ⎢ 0 ⎢ ⎣... 0

... 0 ... ... . . . cos θr ... 0 . . . sin θr ... ... ... 0

... 0 ... ... . . . − sin θr ... 0 . . . cos θr ... ... ... 0

⎤ ... 0 ... ...⎥ ⎥ ... 0 ⎥ ⎥ ... ...⎥ . ⎥ ... 0 ⎥ ⎥ ... ...⎦ . . . 1 N+2,N+2 (4)

Stipulating the rotation pivot as [i, j ] (i , j = 1, 2, . . . , N) and the rotation angle as θr , [R] is observed to be unity except for the entries [R]ii = [R] j j = cos θr , [R]i j = − sin θr , and [R] j i = sin θr . In this case, the coupling values are changed in concordance with the well-known rotation process given in [6], but the rotation should now affect the capacitance matrix as well; i.e., [Cn+1 ] = [R]T [Cn ][R]

(5)

where the relevant capacitance values in [Cn+1 ] become (k = S, 1, 2, . . . , N, L; k = i , j )  = cos θr Cik − sin θr C j k Cik

C j k = sin θr Cik + cos θr C j k  Cki = cos θr Cki − sin θr Ckj

 Ckj = sin θr Cki + cos θr Ckj

Cii = cos2 θr Cii + sin2 θr C j j − 2 sin θr cos θr Ci j

C j j = sin2 θr Cii + cos2 θr C j j + 2 sin θr cos θr Ci j Ci j = Ci j (cos2 θr − sin 2 θr ) + sin θr cos θr (Cii − C j j ). (6)

S21 () = −2 j [Y ]−1 L1

S11 () = −1 − 2 j [Y ]−1 11

S22 () = −1 − 2 j [Y ]−1 L ,L

where n = 0, 1, 2, . . . , represents the corresponding parameters of the nth transform operation. The proposed transformation [T ] is then discussed in two conditions. 1) Similarity transformation where [T ] refers to the rotation matrix [R]

(2)

−1 −1 where [Y ]−1 L1 , [Y ]11 , and [Y ] L ,L refer to the relevant entries of the inverse matrix [Y ]−1 . To obtain a practical filter configuration, it is known that transformations on coupling matrix are always required, such as the manipulations discussed in [2] and [4]. While in [19], the transformations on capacitance matrix are also taken into account. Concluding both situations, in this paper, a generalized transformation (represented by matrix [T ]) on the

Moreover, it should be stated that [Yn+1 ] shares the same eigenvalues with [Yn ]. 2) Rescaling transformation where [T ] is represented by the rescaling matrix [U ] ⎤ ⎡ 1 0 0 ... ... 0 √ ⎢ 0 α1 0 ... ... 0 ⎥ ⎥ ⎢ √ ⎢ 0 α2 0 ... 0 ⎥ 0 ⎥ [U ] = ⎢ ⎢... ... 0 ... ... ...⎥ ⎥ ⎢ √ ⎣ 0 αN 0 ⎦ ... ... ... 0 0 0 ... 0 1 N+2,N+2 (7)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. HE et al.: DIRECT MATRIX SYNTHESIS FOR IN-LINE FILTERS WITH TZs

3

while α1 , α2 , . . . , and α N are the rescaling factors for 1st, 2nd, . . . , and Nth resonator, separately. Since [U ] is defined as a diagonal matrix, this transformation derives [Mn+1 ] = [U ]T [Mn ][U ] = [U ][Mn ][U ] [Cn+1 ] = [U ]T [Cn ][U ] = [U ][Cn ][U ].

(8)

Accordingly, the corresponding entries of [Mn+1 ] and [Cn+1 ] are modified by (i , j = 1, 2, . . . , N) √ M j j = α j M j j Mij = αi α j Mi j √ √ M S j = α j M S j M L j = α j M L j √ C j j = α j C j j Ci j = αi α j Ci j C S S

=

C L L

= C S S = C L L = 0.

(9)

Note that the topology configuration is not affected by the rescaling transformation. It should be emphasized that after this transformation, the eigenvalues of admittance matrix [Yn+1 ] will be changed as well. Nevertheless, it can still be obtained that the entries in the inverse matrices [Yn ]−1 and [Yn+1 ]−1 satisfy −1 [Yn+1 ]−1 11 = [Yn ]11

−1 [Yn+1 ]−1 L1 = [Yn ] L1

−1 [Yn+1 ]−1 L L = [Yn ] L L

(10)

revealing that S-parameters for the low-pass prototype remain the same. Since the capacitances C11 , C22 , . . . , CNN are always supposed to be unity in traditional synthesis, it is easy to obtain from (6) that matrix [Cn ] (n = 0, 1, 2, . . .) keep unchanged during the similarity rotations. Consequently, only operations on [M] are considered during the synthesis. However, this special condition is no more satisfied as long as rescaling transformation is utilized; and thus providing new possibilities for the topology determination. B. Fundamental Transformation Process Based on the proposed admittance matrix transformation, fundamental process to obtain an in-line topology can be introduced. The process first considers a basic structure depicted in Fig. 2(a), which contains a triplet (resonators 1–3, realizing a TZ at f z ) as well as another direct-coupled resonator 4. Based on the process given in [6], a specific similarity transformation is implemented to annihilate the coupling M23 by selecting the rotation pivot [2], [3] and rotation angle θa = 0.5 tan−1 (2M23 /(M33 − M22 )), which results in a lattice network exhibited in Fig. 2(b). Note that the corresponding coupling elements are modified [represented as Mij , i , j = 1–4 in Fig. 2(b)], while the capacitances keep the same. In order to annihilate the cross couplings between resonators 2 and 4 as well as resonators 1 and 3 of the lattice network simultaneously, the synthesis continues by successively applying a rescaling transformation on resonator 2 (with rescaling factor α2 ) and another similarity transformation with pivot [2], [3] (assuming the rotation angle as θb ). Note that these operations are in accordance with the ones recommended

Fig. 2. Proposed fundamental transformation process. (a) Basic triplet section with constant couplings. (b) Equivalent lattice network. (c) Equivalent lattice topology after rescaling on resonator 2 (with rescaling factor α2 ). (d) Eventual in-line topology containing frequency-variant coupling.

in [19]. Particularly, considering (3)–(9) and the derivations in [19] together, α2 and θb can be represented as  M M13 34 (11)  M M12 24  √  M α2 M24 = tan−1 . (12) θb = tan−1 − √ 13   α2 M12 M34

α2 = −

The relevant coupling and capacitance values after the manipulations are illustrated in Fig. 2(c) and (d). In consequence, it is noticed that an in-line topology is determined. Note that the similarity transformation should be implemented on the capacitance matrix as well, which obtains [in terms of (5) and (6)]     C22 = cos2 θb C22 + sin2 θb C33 − 2 sin θb cos θb C23 2 2 = cos θb α2 + sin θb (13a)     C33 = sin2 θb C22 + cos2 θb C33 + 2 sin θb cos θb C23 = sin2 θb α2 + cos2 θb (13b)

while creating a frequency slope

     + C23 C23 = sin θb cos θb C22 − C33 (cos2 θa − sin2 θa ) (14) = sin θb cos θb (α2 − 1).  + As a result, a frequency-variant coupling, namely, C23  that  represents the low pass frequency, while C23 is the frequency slope), is produced between resonators 2 and 3. Also, it should be stated that the desired TZ f z is thus generated and completely determined by the frequency-variant coupling. Comparing the resulting in-line topology with the triplet structure in Fig. 2(a), it is apparent that the number of coupling elements for realizing a TZ is reduced; this shows exactly the benefits of in-line topologies in configuration simplicity.  (note M23

III. D ESIGN P ROCEDURE A. Direct Synthesis Procedure for an Nth Order In-Line Prototype Starting from an Nth order transversal array prototype with constant couplings, as shown in Fig. 3(a), the initial admittance

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 4. Proposed unique topology that contains in-line frequency-variant couplings, extracted-pole sections, and cross-coupled structures as an entity.

This operation thus produces new admittance matrix [Y2 ] by the following: [M2 ] = [U ]T [M1 ][U ] = [U ][M1 ][U ] [C2 ] = [U ]T [C1 ][U ] = [U ][C1 ][U ] [Y2 ] = [U ]T [Y1 ][U ] = [U ][Y1 ][U ] = [M2 ] + [C2 ] − j [G]. Fig. 3. Synthesis procedure for an N th order in-line filter. (a) N th order transversal array network. (b) Network with multiple triplet sections. (c) Network with a lattice structure. (d) Rescaling of relevant resonator and coupling (marked by red dotted circles) in the first lattice structure. (e) Network with one in-line frequency-variant coupling. (f) Realization of the second lattice structure. (g) Network with two in-line frequencyvariant couplings. (h) Final in-line topology with multiple-frequency-variant couplings.

matrix [Y0 ] is described by [Y0 ] = [M0 ] + [C0 ] − j [G]

(15)

where [M0 ] refers to the initial coupling matrix and [C0 ] is the initial diagonal capacitance matrix. By applying the proposed fundamental process in Section II, the procedure to synthesize an Nth order in-line prototype is provided as follows, where both the network transformations and the equivalent mathematical operations are detailed. Step 1: Utilizing similarity transformations, i.e., [T1 ] = [R1 ], to obtain a basic lattice structure, which acquires networks exhibited in Fig. 3(b) and (c) successively. Note that the multiple-triplet sections in Fig. 3(b) (in dotted circles) are connected directly or by a series of direct-coupled resonators. It can be observed that this step modifies only the relevant coupling elements, without changing the capacitance matrix. The resulting coupling matrix [M1 ], capacitance matrix [C1 ], and admittance matrix [Y1 ] are therefore given by [M1 ] = [R1 ] [M0 ][R1 ] T

[C1 ] = [C0 ] [Y1 ] = [R1 ][Y0 ][R1 ]T = [M1 ] + [C0 ] − j [G]. (16) Step 2: Applying rescaling transformation, i.e., [T2 ] = [U ], to redefine the capacitance element in the first basic triplet section. Taking into account the rescaling factor of resonator i as αi , as Fig. 3(d) shows, the entry Uii in [U ] can be evaluated from (7) and (11) by   Mi−1,i+1 Mi+1,i+2 √ Uii = αi = − (17)   Mi−1,i Mi,i+2 where all the coupling elements at the right side belong to [M1 ].

(18) In addition, note that the topology of the network is not influenced in this step. Step 3: Implementing a further similarity transformation on the rescaled network, which ultimately realizes an in-line frequency-variant coupling as Fig. 3(e) illustrates. Representing the involved transformation by a rotation matrix [R2 ], this step can be described mathematically by [M3 ] = [R2 ]T [M2 ][R2 ] [C3 ] = [R2 ]T [C2 ][R2 ] [Y3 ] = [R2 ]T [Y2 ][R2 ] = [M3 ] + [C3 ] − j [G]

(19)

where the resultant capacitance matrix [C3 ] possesses nonzero entries out of the main diagonal, representing the frequency slope of the produced frequency-variant coupling. Step 4: Afterward, the same matrix manipulations in steps 1–3 are adopted for the other triplet sections, where a final in-line topology with multiple-frequency-variant couplings can be acquired, as depicted from Fig. 3(f)–(h). Note that a frequency-variant coupling is transformed from a triplet section in the presented approach, the number of TZs (denoted as NZ here) thus depends on the triplets that can be realized during the transformations. For an Nth order in-line filter, NZ is investigated from Fig. 3 that satisfies N (20) 2 which indicates that the maximum number of TZs for an Nth order in-line filter would be N/2 in this paper. NZ ≤

B. Direct Synthesis Procedure for Unique Topology Being an absolute direct matrix technique, the proposed synthesis can also be combined with [21]–[23], resulting in a new kind of unique topology as Fig. 4 illustrates, which contain cross-coupled structures, extracted-pole sections, and in-line frequency-variant couplings as an entity. The design procedure for synthesizing an Nth order unique topology filter with NZ TZs is briefly provided as follows. Step 1: Applying the technique in [23] to generate a couple of extracted-pole structures at the source and load,

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. HE et al.: DIRECT MATRIX SYNTHESIS FOR IN-LINE FILTERS WITH TZs

5

Fig. 6.

Fig. 5. Routing schematic and relevant coupling values for the fifth-order prototype. (a) Network with triplet structures. (b) Network with lattice structures. (c) Network with lattice structures after rescaling for resonators 1 and 4 (marked by red dotted circles). (d) Final in-line network.

separately. Assuming P1 extracted-pole sections appear at the two terminations, it is noticed that P1 TZs are created accordingly. Step 2: Using the approach detailed in this paper to realize several in-line frequency-variant couplings. As a result, another P2 TZs are generated once a number of P2 frequencyvariant couplings are determined. Step 3: Implementing further similarity transformations to realize more practical cross-coupled configurations; these configurations can generate the rest P3 (=NZ–P1 –P2 ) TZs. Since TZs can be realized by different types of structures in the unique topology, it is certain that the proposed approach expands the flexibility of direct matrix synthesis for narrowband high selectivity filters, and may be conveniently adopted in the future microwave/RF systems. IV. I LLUSTRATIVE E XAMPLES Based on the above discussions, two synthesis examples are detailed to verify the effectiveness of the presented approach. A. Fifth-Order In-Line Prototype With Two Frequency-Variant Couplings The first example demonstrated here is a fifth-order 22-dB return loss low-pass prototype that contains two frequency-variant couplings, which can generate finite TZs at f z1 = −1.5 j and f z2 = 2.1 j , respectively. The transversal array network for the prototype is first determined in terms of [6]. With a group of similarity transformations, networks with triplet and lattice structures are obtained successively as Fig. 5(a) and (b) illustrate, where the two TZs can be individually realized. Note that all the coupling elements are constant in these networks. The transformation continues by rescaling resonators 1 and 4 based on (17). Capacitances of the two resonators thus turn into 5.4913

Synthesis S-parameters of the fifth-order in-line prototype.

and 2.7123, respectively. In addition, relevant coupling values are changed as well as marked in red and italic in Fig. 5(c). Finally, another two similarity rotations are utilized according to (19). By selecting rotation pivot [1], [2] (with rotation angle 32.6458°) and pivot [4], [5] (with rotation angle 51.9149°), an in-line topology is eventually decided in Fig. 5(d), together with the ultimate coupling values. The resulting capacitances for the resonators are determined as C11 = 4.1843, C22 = 2.3070, C33 = 1, C44 = 1.6515, and C55 = 2.0608. It should be stated that two frequencyvariant couplings appear between resonators 1 and 2 (i.e., M12 = 2.0400  + 3.0601) as well as resonators 4 and 5 (i.e., M45 = 0.8313  − 1.7458) here. The corresponding S-parameters are shown in Fig. 6, performing the desired response, which well validates the presented approach. Moreover, note that M12 | f z1 = 2.0400 f z1 + 3.0601 = 0 M45 | f z2 = 0.8313 f z2 − 1.7458 = 0

(21)

from which it can be observed that f z1 and f z2 are generated and controlled by the two frequency-variant couplings independently. B. Eighth-Order Prototype With Unique Topology An eighth-order 22-dB return loss prototype in a unique topology that contains three extracted-pole sections (generating TZs at 3 j , 2 j , and −2.6 j , respectively), an in-line frequency-variant coupling (generating a TZ at 1.7 j ), as well as a triplet (generating a TZ at −1.8 j ) is synthesized to show the design flexibility of the presented approach. The extracted-pole sections are first realized by means of the principle in [23]. Admittance matrix transformations detailed in this paper are then adopted, which can realize an inline frequency-variant coupling between resonators 3 and 4. Thereafter, further similarity transformations are utilized to obtain a triplet for the rest cross-coupled subnetwork. The final routing schematic for the eighth-order prototype, together with the coupling values, is given in Fig. 7. Note that the capacitances of two particular resonators (marked as resonators 3 and 4 in Fig. 7) are C33 = 1.6635 and C44 = 1.5106, separately; while the other resonator capacitances are unity. In addition, Fig. 8 shows the corresponding synthesized S-parameters for the eighth-order example.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 7. Routing schematic and coupling values for the eighth-order example in a unique topology.

Fig. 8. Synthesis S-parameters of the eighth-order prototype in a unique topology.

V. E XPERIMENTAL VALIDATION For the experimental validation, a fourth-order in-line bandpass filter is designed and manufactured by using coaxial resonator structures. The filter is expected to operate at f 0 = 1.80 GHz with bandwidth BW = 80 MHz and 22-dB return loss. Moreover, two TZs are required at f z1 = 1.64 GHz and f z2 = 1.97 GHz, which are separately generated by two frequency-variant couplings.

Fig. 9. Corresponding low-pass prototypes for the fourth-order in-line filter. (a) Network with triplet structures. (b) Network after generating the first in-line frequency-variant coupling. (c) In-line network with different capacitances. (d) Final in-line network with equal and unity capacitances.

A. Synthesis for the Corresponding Low-Pass Prototype After the frequency is normalized to the low pass domain, the desired TZ positions turned out to be −4.2 j and 4 j , respectively. Therefore, the corresponding low-pass prototype with two triplets is first determined and is shown in Fig. 9(a). It is noticed that the TZs are realized by the first and the second triplets individually. According to our proposed approach, Fig. 9(b) and (c) is thus obtained successively. Note that the capacitances of the resonators in Fig. 9(c) are derived as C11 = 1.3329, C22 = 1.2721, C33 = 0.7827, and C44 = 0.8247. To facilitate the physical implementation, another rescaling transformation is adopted here by using rescaling matrix ⎤ ⎡ 1 0 0 0 0 √0 ⎢ 0 1/ C11 0 0 0⎥ ⎥ ⎢ √0 ⎥ ⎢0 0 1/ C 0 0 0 22 ⎥ ⎢ √ [U ] = ⎢ ⎥ C 0 0 0 0 0 1/ 33 ⎥ ⎢ √ ⎣0 0 0 0 1/ C44 0 ⎦ 0 0 0 0 0 1 (22) which can normalize the capacitances to unity again  = C  = C  = C  = 1 after the rescaling trans(i.e., C11 22 33 44 formation). The relevant coupling values are thus modified

Fig. 10.

Synthesis S-parameters for the fourth-order low-pass prototype.

Fig. 11. Resonating frequencies as well as circuit couplings for the fourthorder in-line filter.

accordingly, as expressed in Fig. 9(d). The response of the final low-pass prototype is illustrated in Fig. 10, which performs the desired functions. B. Physical Implementation Based on the derived prototype, resonating frequencies and circuit couplings for the filter are thus determined by applying frequency and impedance renormalizations in [24], which results in a network as Fig. 11 gives. It should be stated

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. HE et al.: DIRECT MATRIX SYNTHESIS FOR IN-LINE FILTERS WITH TZs

7

that the two frequency-variant couplings in Fig. 9(d) thus become coefficients with values (represented by k12 and k34 , respectively)  BW f f0 − 0.9707 − k12 = −0.2311 f0 f f0  f f0 BW k34 = −0.2429 − (23) + 0.9716 f0 f f0 which are correlated with the frequency f . Despite, it is difficult to look for a physical structure that is able to fit k12 or k34 within the whole frequency range, we can approximately solve this problem by considering only a small range around the center frequency f 0 . From a practical point of view, a frequency-variant coupling should thus possesses two important characteristics. First, it exhibits particular value at the center frequency f 0 . For instance, we observe that k12 | f0

BW = −0.9707 = −0.0431 f0

k34 | f0 = 0.9716

BW = 0.0432. f0

Fig. 12. (a) Schematic (left) and EM model in HFSS (right, with shell box removed) for a capacitive-inductive mixed coupled structure. (b) Ideal mixed coupling.

(24)

Second, its value turns into 0 (vanish) at a particular frequency. As an example here, k12 or k34 is correlated with f z1 or f z2 by  f z1 f0 BW k12 | f z1 = −0.2311 − =0 − 0.9707 f0 f z1 f0  BW f z2 f0 k34 | f z2 = −0.2429 + 0.9716 − = 0 (25) f0 f z2 f0 which yields ⎛ fz1

f z2

BW = ⎝−M12 + f0





BW M12 f0

2

⎞  2C12 2 ⎠ + 4C12 f0

= 1.64 GHz ⎛ ⎞   2 BW BW 2C34 2 ⎠ = ⎝−M34 + + 4C34 M34 f0 f0 f0 = 1.97 GHz.

(26)

It is then noticed that these two demands can be practically achieved by using an appropriate mixed-coupled structure in Fig. 12(a), as [18], [19], and [25] demonstrate. Based on [18] and [26], a mixed coupling is thus generated as Fig. 12(b) shows, whose value at f 0 (referred as K in the following) can be altered via different physical dimensions, such as diameter dc of the opposite metal disks [which produce the negative (capacitive) part (i.e., capacitance Cm ) inside the mixed coupling], and height h l of the horizontal metal rod [which realizes the positive (inductive) part (i.e., inductance L m ) of the mixed coupling]. Moreover, since capacitance Cm and inductance L m construct a resonator inherently, the mixed coupling will vanish at its inherent resonant frequency f z (which can also be altered with the capacitance and inductance values).

Fig. 13. Simulated results versus different h l and dc of a mixed coupled structure. (a) Coupling coefficient at f 0 (denoted as K ). (b) TZ f z .

In consequence, a specific frequency-variant coupling can thus be approximately realized by properly selecting the combination of diameter dc and height h l in a mixed-coupled structure. To explain the design more precisely, the derivation of K and the resonant frequency f z versus different values of dc and h l (under assumption of a symmetric model) are simulated in terms of [18] and illustrated in Fig. 13. It is apparent from Fig. 13(a) that the mixed coupling can be dominated as positive (inductive) or negative (capacitive) at f0 by a similar mixed-coupled structure, with different physical dimensions. Stipulated as a symmetric model here, the result

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 15. Original simulated S-parameters of the fourth-order filter (before adjustment), with dimensions dc1 = 5.8 mm, h l1 = 7.7 mm, dc2 = 5.8 mm, and h l2 = 10.2 mm.

Fig. 14. (a) Top view schematic for the fourth-order in-line filter, with dimensions d1 = 12 mm, g = 2 mm, w1 = 20 mm, w2 = 16 mm, and w3 = 20 mm. Note that the dotted boxes (marked as F-r C) refer as frequency-variant couplings between resonators 1 and 2, and resonators 3 and 4, separately. (b) Ideal EM module (in HFSS) with indicated resonator number.

may not be exactly accurate and some further adjustments may still be required. However, it is believed that Fig. 13 reveals the relationship between desired couplings and physical structures. On the basis of these results, we choose dc1 = 5.8 mm and h l1 = 7.7 mm for generating the frequency-variant coupling k12 , and dc2 = 5.8 mm and h l2 = 10.2 mm for generating the frequency-variant coupling k34 in the fourth-order filter, respectively. Moreover, the other constant couplings between source and resonators 1–4 and load in Fig. 11 are decided by using traditional extraction mechanisms in the literature. With above discussions, top view schematic for the whole coaxial cavity filter is reported in Fig. 14(a), where all the corresponding dimensions are listed along. It can be observed that the four coaxial resonators are composed of metal cylinders (with diameter d1 = 12 mm and 21 mm in height) and individual square boxes (30 mm × 30 mm × 25 mm), which are direct-coupled one after another via inductive windows (with thickness of 2 mm). Note that the proposed frequency-variant couplings (marked by dotted box F-r C) are implemented between resonators 1 and 2, as well as resonators 3 and 4, separately. Besides, the constant input coupling (between source and resonator 1) and output coupling (between resonator 4 and load) are defined by using tapped-in lines with the diameter of 1.2 mm and respective heights of 9.4 and 7.1 mm. The electromagnetic (EM) module (in HFSS) for the fourth-order example is then given in Fig. 14(b). The simulated S-parameters without adjustments are presented in Fig. 15, showing acceptable responses. Note that both f z1 and f z2 are above the required positions (i.e., 1.64 and

Fig. 16. Photograph of the fabricated fourth-order in-line coaxial resonator filter (with top lid).

Fig. 17. EM simulation and measured results for the fourth-order in-line coaxial resonator filter.

1.97 GHz), it infers that dc1 and dc2 should be increased or h l1 and h l2 should be reduced during the simulation, according to the concepts presented in Fig. 13. With minimal adjustments, the ultimate dimensions are selected as dc1 = dc2 = 6 mm, h l1 = 7.6 mm, and h l2 = 9.8 mm for this fourth-order filter. Accordingly, the fourth-order in-line coaxial resonator filter is fabricated as Fig. 16 shows, with dimensions 67 mm × 67 mm × 29 mm. The final EM simulation and the measured S-parameters are reported in Fig. 17. Note that the locations of two TZs f z1 and f z2 , the center frequency f 0 , as well as the bandwidth in the simulation results are perfectly matched to the requirements (i.e., f z1 = 1.64 GHz, f z2 = 1.97 GHz, f 0 = 1.80 GHz, and BW = 80 MHz) to satisfy our application in a special wireless system. Moreover, the simulated S11 is less than −15 dB within the entire operating band, which is considered acceptable. By some further tuning of the fabricated filter (via the top screws between the resonators

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. HE et al.: DIRECT MATRIX SYNTHESIS FOR IN-LINE FILTERS WITH TZs

in Fig. 16), S11 is improved to less than −22 dB in the measured results, along with a very slight change on the operating bandwidth and TZs. (The measured operating band now becomes 1.755–1.847 GHz, i.e., BW = 92 MHz, while f z1 turns out to be 1.645 GHz, and f z2 turns out to be 1.976 GHz.) VI. C ONCLUSION This paper presents a direct matrix approach for synthesizing in-line topology filters that contain a group of frequencyvariant couplings for the first time. It is noticed that multiple TZs can thus be generated and independently controlled by the frequency-variant couplings without any auxiliary crosscoupled or extracted-pole structures. Involving only directcoupled resonators in the resultant network, the proposed approach provides the simplest synthesis solution for high selectivity filters. Combining the proposed approach with techniques developed in previous works, new topologies with in-line frequency-variant couplings, extracted-pole sections, and cross-coupled structures can be derived, expanding the design flexibility for narrowband high performance filters. R EFERENCES [1] A. Atia, A. Williams, and R. Newcomb, “Narrow-band multiplecoupled cavity synthesis,” IEEE Trans. Circuits Syst., vol. CS-21, no. 5, pp. 649–655, Sep. 1974. [2] R. J. Cameron, “General coupling matrix synthesis methods for Chebyshev filtering functions,” IEEE Trans. Microw. Theory Techn., vol. 47, no. 4, pp. 433–442, Apr. 1999. [3] R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microw. Theory Techn., vol. 51, no. 1, pp. 1–10, Jan. 2003. [4] S. Tamiazzo and G. Macchiarella, “An analytical technique for the synthesis of cascaded N-tuplets cross-coupled resonators microwave filters using matrix rotations,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 5, pp. 1693–1698, May 2005. [5] P. Kozakowski, A. Lamecki, P. Sypek, and M. Mrozowski, “Eigenvalue approach to synthesis of prototype filters with source/load coupling,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 2, pp. 98–100, Feb. 2005. [6] R. J. Cameron, C. Kudsia, and R. Mansour, “Coupling matrix synthesis of filter networks,” in Microwave Filters for Communication Systems, 1st ed. Hoboken, NJ, USA: Wiley, 2007. [7] S. Amari and U. Rosenberg, “Synthesis and design of novel in-line filters with one or two real transmission zeros,” IEEE Trans. Microw. Theory Techn., vol. 52, no. 5, pp. 1464–1478, May 2004. [8] S. Amari and G. Macchiarella, “Synthesis of inline filters with arbitrarily placed attenuation poles by using nonresonating nodes,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 10, pp. 3075–3081, Oct. 2005. [9] G. Macchiarella and S. Tamiazzo, “Synthesis of microwave duplexers using fully canonical microstrip filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 721–724. [10] O. Glubokov and D. Budimir, “Extraction of generalized coupling coefficients for inline extracted pole filters with nonresonating nodes,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 3023–3029, Dec. 2011. [11] P. Zhao and K.-L. Wu, “A direct synthesis approach of bandpass filters with extracted-poles,” in Proc. Asia–Pacific Microw. Conf., Seoul, South Korea, Nov. 2013, pp. 25–27. [12] Y. Wang and M. Yu, “True inline cross-coupled coaxial cavity filters,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 12, pp. 2958–2965, Dec. 2009. [13] S. Bastioli and R. V. Snyder, “Inline pseudoelliptic TM01δ -mode dielectric resonator filters using multiple evanescent modes to selectively bypass orthogonal resonators,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 12, pp. 3988–4001, Dec. 2012. [14] S. Amari, M. Bekheit, and F. Seyfert, “Notes on bandpass filters whose inter-resonator coupling coefficients are linear functions of frequency,” in IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA, USA, Jun. 2008, pp. 1207–1210.

9

[15] S. Amari and J. Bornemann, “Using frequency-dependent coupling to generate finite attenuation poles in direct-coupled resonator bandpass filters,” IEEE Microw. Guided Wave Lett., vol. 9, no. 10, pp. 404–406, Oct. 1999. [16] M. Politi and A. Fossati, “Direct coupled waveguide filters with generalized Chebyshev response by resonating coupling structures,” in Proc. Eur. Microw. Conf. (EuMC), Sep. 2010, pp. 966–969. [17] U. Rosenberg, S. Amari, and F. Seyfert, “Pseudo-elliptic direct-coupled resonator filters based on transmission-zero-generating irises,” in Proc. Eur. Microw. Conf. (EuMC), Sep. 2010, pp. 962–965. [18] H. Wang and Q.-X. Chu, “An inline coaxial quasi-elliptic filter with controllable mixed electric and magnetic coupling,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 3, pp. 667–673, Mar. 2009. [19] S. Tamiazzo and G. Macchiarella, “Synthesis of cross-coupled filters with frequency-dependent couplings,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 3, pp. 775–782, Mar. 2017. [20] Y. He, G. Wang, L. Sun, L. Wang, R. Zhang, and G. Rushingabigwi, “Direct matrix synthesis for in-line filters with transmission zeros generated by frequency-variant couplings,” in IEEE MTT-S Int. Microw. Symp. Dig., Hawaii, HI, USA, Jun. 2017, pp. 356–359. [21] S. Tamiazzo and G. Macchiarella, “Synthesis of cross-coupled prototype filters including resonant and non-resonant nodes,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 10, pp. 3408–3415, Oct. 2015. [22] G. Macchiarella, M. Oldoni, and S. Tamiazzo, “Narrowband microwave filters with mixed topology,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 12, pp. 3980–3987, Dec. 2012. [23] Y. He, G. Wang, X. Song, and L. Sun, “A coupling matrix and admittance function synthesis for mixed topology filters,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 12, pp. 4444–4454, Dec. 2016. [24] G. Macchiarella, “Generalized coupling coefficient for filters with nonresonant nodes,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 12, pp. 773–775, Dec. 2008. [25] S. Bastioli, R. Snyder, and P. Jojic, “High power in-line pseudoelliptic evanescent mode filter using series lumped capacitors,” in Proc. 41st Eur. Microw. Conf. (EuMC), Oct. 2011, pp. 87–90. [26] Q.-X. Chu and H. Wang, “A compact open-loop filter with mixed electric and magnetic coupling,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 2, pp. 431–439, Feb. 2008.

Yuxing He received the B.S. and Ph.D. degrees in electrical engineering from the University of Science and Technology of China, Hefei, China, in 2008 and 2017, respectively. In 2017, he joined the Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milan, Milan, Italy, as a Visiting Researcher. He is currently working as a Postdoctoral Researcher at the Institute of Advanced Sciences, Yokohama National University, Yokohama, Japan. His current research interests include microwave and RF passive circuit design, advanced synthesis techniques for high performance filters and multiplexers, and RF system integration. Dr. He serves on the Review Board of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES and IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS .

Giuseppe Macchiarella (F’15) is currently a Professor of microwave engineering with the Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milan, Italy. He is currently a Scientific Coordinator of PoliEri, a Research Laboratory on monolithic microwave integrated circuit, which was jointly supported by the Politecnico di Milan and Ericsson Company, Milan. He is responsible for several contracts and collaborations with companies operating in the microwave industry. His current research interests include microwave engineering: microwave acoustics (SAW devices), radio wave propagation, and numerical methods for electromagnetic, power amplifiers, linearization techniques, and passive devices, and the development of new techniques for the synthesis of microwave filters and multiplexers. He has authored or coauthored over 150 papers on journals and conferences proceedings. Mr. Macchiarella is the Chair of the IEEE MTT-8 Technical Committee (filters and passive components).

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Gang Wang (M’98) received the B.S. degree from the University of Science and Technology of China, Hefei, China, in 1988, and the M.S. and Ph.D. degrees from Xidian University, Xi’an, China, in electrical engineering, in 1991 and 1996, respectively. From 1996 to 1998, he was a Post-Doctoral Research Fellow with Xi’an Jiaotong University, Xi’an, supported by the Chinese Government, where he was an Associate Professor from 1998 to 2000. In 2001, he joined the ITM Department, Mid-Sweden University, as a Visiting Researcher. From 2002 to 2003, he was a Post-Doctoral Research Associate with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FI, USA. From 2003 to 2010, he was the Chair Professor with Jiangsu University, Zhenjiang, China. He is currently a Full Professor with the University of Science and Technology of China. His current research interests ultra-wideband electromagnetics, near-field target detection and imaging, RFID/sensor design, and microwave circuit and antenna design. Dr. Wang is a Senior Member of the Chinese Institute of Electronics.

Lu Wang received the B.S. degree from the University of Science and Technology of China, Hefei, China, in 2013. He is currently pursuing the Ph.D. degree in electrical engineering at the University of Science and Technology of China, Hefei. His current research interests include isolation structures design for multi-in multi-out antennas, fragment-type microwave circuits, and antenna design.

Wentao Wu received the B.S. degree from the Hefei University of Technology, Hefei, China, in 2000, and the M.S. degree from the University of Science and Technology of China, Hefei, in 2005. He is currently an Engineer with the University of Science and Technology of China. His current research interests include electronic device design and related software development.

Liguo Sun received the B.S. and Ph.D. degrees in electrical engineering from the University of Science and Technology of China, Hefei, China, in 1982 and 1991, respectively, and the M.S. degree in electrical engineering from the China Research Institute of Radio Wave Propagation, Beijing, China, in 1985. From 1985 to 1988, he was an Engineer with the China Research Institute of Radio Wave Propagation, Xinxiang, China. From 1991 to 1995, he was an Associate Research Fellow with the Institute of Remote Sensing Application, Chinese Academy of Sciences, Beijing. From 1995 to 1996, he was with the Massachusetts Institute of Technology, Cambridge, MA, USA, and Northeast University, Boston, MA, USA, as a Senior Visiting Scholar. From 1996 to 2000, he was a Senior Engineer and a Manager with DSC Ltd., Tyco International Ltd. From 2000 to 2009, he was a Senior Staff Engineer with Sychip Inc. and Murata Manufacturing Company, Ltd. Since 2009, he has been a Full Professor with the University of Science and Technology of China. His current research interests include RF circuits and devices, RF system integration, and signal integrities.

Rong Zhang (S’15) received the B.S. degree from Anhui University, Hefei, China, in 2013. She is currently pursuing the Ph.D. degree in electronical engineering at the University of Science and Technology of China, Hefei. Her current research interests include compact ultra-wideband antenna and passive circuit designs, microwave near-field imaging systems, and radar signal processing.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Design and Optimization of Tunable Silicon-Integrated Evanescent-Mode Bandpass Filters ZhengAn Yang , Member, IEEE, Dimitra Psychogiou , Member, IEEE, and Dimitrios Peroulis, Fellow, IEEE

Abstract— This paper reports on the design and optimization of MEMS-tunable evanescent-mode cavity-based bandpass filters with continuously variable center frequency within an octave tuning range. The devised filters are manufactured using silicon-micromachining techniques that enable their actualization for frequencies located in the millimeter-wave (30–100 GHz) regime. An RF design methodology that takes into consideration all microfabrication-induced constrains—e.g., nonvertical wall profiles and finite MEMS deflection—enables high unloaded factor ( Q u ) and also minimizes bandwidth (BW) variation within the octave tuning range is reported. Furthermore, a new passively compensating package-integrated input/output feeding structure that enables optimal impedance matching over the entire tuning range is also presented. In order to evaluate the devised RF design methodology, a filter prototype was manufactured and measured at Ka-band. It exhibits a measured frequency tuning between 20 and 40 GHz (2:1 tuning range), relative BW between 1.9 and 4.7%, insertion loss between 3.1 and 1.1 dB, and input reflection below 15 dB. This paper also explores important tradeoffs between mechanical stability and insertion loss by comparing creep-resistant to pure-Au tuning diagrams. Index Terms— Bandpass filter (BPF), evanescent-mode filter, high quality factor, MEMS-filter, millimeter-wave filter, reconfigurable filter, tunable filter.

I. I NTRODUCTION

H

IGH quality factor (Q u ) widely tunable bandpass filters (BPFs) are highly desirable in the preselect frontend stage of microwave and millimeter-wave systems to enable additional functionality, reduce receiver complexity, and allow mass deployment in frequency-congested scenarios [1]–[3]. The considered RF filter architectures include discrete lumpedelement (LE)-based resonators [4], [5], planar transmission line (TL)-based geometries [6]–[8], and 3-D waveguide cavities [9]. Whereas LE-based filters can be monolithically integrated to the RF receiver chip, their low quality factors (Q u less than 30) render them unsuitable for high-frequency applications particularly above 10 GHz. On the other

Manuscript received June 21, 2017; revised October 26, 2017; accepted December 17, 2017. (Corresponding author: ZhengAn Yang.) Z. Yang and D. Peroulis are with the Birck Nanotechnology Center, School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]; [email protected]). D. Psychogiou is with the Department of Electrical, Computer, and Energy Engineering, University of Colorado at Boulder, Boulder, CO 80309 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2018.2799575

hand, planar TL-based filters exhibit moderate Q u (less than 100–200) and can be readily co-integrated with surfacemount tuning elements. They are suitable for low-to-mid frequency applications but still result in unacceptable high losses in high frequencies. Nonplanar (3-D) filters outperform the aforementioned technologies with Q u in the order of 1000–10 000 depending on the employed technology. Nevertheless, they come at the expense of higher volume, weight, and cost. Tunability is another important factor that needs to be considered. This is typically achieved by co-integrating reactive elements such as MEMS devices [6], semiconductor varactors [7], [10], and ferroelectric capacitors [11] within the filter volume. Although semiconductor-based varactors feature compact size and high tuning range, they suffer from low Q u for frequencies higher than 10 GHz. On the other hand, ferromagnetic varactors exhibit a good compromise among tuning range, speed, and power consumption. However, the requirement for specific substrate materials limits their integration potential. Although they are still at a research stage, MEMS-based tuners have demonstrated excellent RF loss, high tunability, and are scalable to over 100 GHz [12]–[21]. Several recent research efforts have successfully demonstrated co-integration of MEMS tuners and 3-D waveguides for frequency-agile low-loss transfer functions. However, the actualization of a widely tunable response remains a great challenge. Pelliccia et al. [22] and [23] report on a tunable waveguide BPF whose center frequency is reconfigured by integrating ohmic cantilever RF-MEMS switches in the waveguide sidewalls. Whereas high Q u (larger than 750) is obtained over the entire tuning range, its center frequency is tuned by only 10%. In yet another approach, dielectric-loaded waveguide BPFs are tuned by means of MEMS actuators. They exhibit Q u larger than 500 and center frequency tuning of less than 3% [18], [24]. Widely tunable waveguide-based BPFs with more than an octave tuning range, wide spurious-free response, and Q u larger than 350 have been reported in [12] and [25]–[33] and summarized in [34] and [35]. They are based on capacitively loaded evanescent-mode (EVA) cavity resonators whose resonant frequency is tuned by MEMS actuators placed above a critical capacitive gap found between the apex of the resonators loading post and the cavity’s upper wall. In [36] and [37], piezoelectric disks are used instead of MEMS

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

TABLE I G ENERAL D ESIGN C ONSTRAINTS OF MEMS T UNABLE EVA F ILTER

actuators. These type of tuners provide large deflections with moderate-to-high actuation voltage (0 to ±220 V). However, they suffer from low tuning speed (ms) and hysteresis. EVA resonators with faster response (10 s of μs), lower actuation voltage (10 s of V), and higher immunity to vibrations have been reported in [25]–[27]. Instead of diagrams, they employ MEMS beams above the resonators post. However, their tuning range is severely limited by the small MEMS deflection. Silicon-on-insulator-based MEMS diaphragms were reported in [28]–[30] as an effective solution to mitigate the hysteresis response. Such technology enables the realization of tunable EVA-based filters at frequencies as high as 10 s of GHz. However, they need excessively high dc biasing up to 500 V. Yang and Peroulis [12], Zeng et al. [31], and Yang and Peroulis [32] reported on micromachined Au-based microcorrugated diaphragms (MCDs) with stateof-the-art mechanical and electrical performance in terms of reduced dc biasing (less than 140 V) and enhanced stability by introducing nonuniform microcorrugations and creep-resistive Au-V material. Furthermore, bidirectional tuners are introduced in [33] to enable filter corrective tuning without compromising its RF performance. In this paper, an optimal RF design flow for all-silicon MEMS-tunable EVA BPFs is extensively discussed. It consider all technology constrains and allows the actualization of a high Q u transfer function with minimum bandwidth (BW) variation in an octave tuning range. Moreover, a new fully passive input/output feeding mechanism that facilitates impedance matching over the entire tuning range is presented. The suggested RF devised methodology is validated through the electromagnetic (EM) design and experimental testing of a two-resonator BPF. RF measurements demonstrate a center frequency tuning between 20 and 40 GHz (2:1 tuning range), relative BW between 1.9% and 4.7%, and input reflection less than 15 dB over the entire tuning range. Furthermore, the incorporation of the Au-V tuner results in a significantly improved frequency stability with a 7× slower drift rate (phase II creep) compared to MEMS tunable EVA filters based on pure Au MCD tuners. However, this comes with additional insertion loss of 0.15–0.9 dB loss due to the increased resistivity of Au-V. The content of this paper is organized as follows. Section II introduces the devised MEMS-enabled all-silicon tunable filter concept. In Section III, the RF design details of the allsilicon EVA resonator are expounded. Sections IV and V present the design of the interresonator and external couplings, respectively. A specific RF design example of a two-resonator Ka-band BPF is discussed in Section VI. Its RF performance is experimentally validated through the manufacturing and

Fig. 1. Conceptual drawing of the all-silicon evanescent-mode cavity filter with tunable center frequency.

testing of a tunable filter prototype (Section VII). Finally, Section VIII compares the RF performance of the devised allsilicon technology with state-of-the-art results. II. A LL -S ILICON T UNABLE F ILTER C ONCEPT A. General Geometrical Configuration The geometrical details of the tunable all-silicon filter concept are depicted in Fig. 1 for an example case of a tworesonator transfer function. The top silicon substrate is sputter coated with a 1 μm Au-V film. It consists of two flexible Au-V MCDs that act as flexible cavity upper walls and control the resonant frequency of each resonator. A pair of electrodes—Au-coated silicon-etched posts—is placed above the tuning diaphragms through appropriately etched openings on the back side of the diaphragms. A thin layer of SiO2 is deposited on the biasing electrodes in order to electrically isolate them from the silicon substrate. This allows us to independently control each MCD. As an additional design detail to be noted, the height of the bias electrode determines the initial electrostatic bias gap—approximately 48 μm in this implementation—which accordingly defines the maximum applied actuation voltage for a given MCD deflection. As will be further discussed in Section VII, the filter assembly is enclosed within a Au-coated metallic package in order to facilitate its RF excitation through coaxial-type connectors. They are located at the bottom side of the package with their center pins inserted into the filter cavities through the etched via holes and couple to the cavity field by means of magnetic coupling. It should be noted that the feeding pins always maintain dc contact with the cavity upper ceiling for optimal mechanical stability, lower fabrication uncertainty, and stronger magnetic field coupling. B. Practical Realization Aspects Table I presents the most critical practical realization aspects of the MEMS tunable all-silicon technology and the chosen

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. YANG et al.: DESIGN AND OPTIMIZATION OF TUNABLE SILICON-INTEGRATED EVANESCENT-MODE BPFs

geometrical details for the presented development. They are summarized as follows: 1) maximum MEMS deflection; 2) minimum capacitive gap; 3) slope of the cavity walls; 4) minimum radius of the cavity post. The selected microfabrication and assembly processes dictate most of these. Specific constraints are described as follows. The loading capacitance of the EVA resonator dictates the filter’s tuning range. This can be increased by achieving a larger MEMS displacement g and by starting from a smaller initial RF gap g L . We note, however, that larger g requires higher dc bias and induces more tensile stress that makes the MEMS diaphragm susceptive to creep. This is particularly critical when viscoelastic metals such as Au is employed [38]. Such issues can be mitigated by introducing corrugations on the MEMS diaphragm that alleviate the stress and creep as discussed in [12]. As such the maximum acceptable dc bias voltage and diaphragm creep rate set the limit for the maximum diaphragm deflection. This is quantified around 15 μm for the MCD used in this paper. The minimum initial gap g L is mostly limited by manufacturing issues and mostly substrate roughness and postrelease diaphragm bending. For silicon substrates roughness is negligible compared to the postrelease diaphragm bending. As a result, the fabrication process needs to be well tuned to minimize the diaphragm’s initial bending. In this paper, we achieve this by performing a prerelease MEMS-to-cavity bonding that prevents diaphragm buckling during the bonding process. The alternative process of postrelease bonding induces severe thermal stresses that typically lead to unacceptably high postrelease diaphragm deflection. Using this process, we can minimize g L to about 1.2 μm with an uncertainty of less than 0.2 μm as further explained in Section VII. Employing silicon as a structural material for this filter technology also brings some practical realization considerations. In order to realize high-Q u air-filled cavities, their bulk micromachined sidewalls need to exhibit low surface roughness as well as to be coated with at least a 1-μm thick metallization layers (i.e., thicker than the skin depth: 0.43 μm at 30 GHz). Smooth cavity walls are achieved with anisotropic wet-etching based on a TMAH + triton X-100 solution. This process though results in undercuts that in turn limit the minimum achievable center post top radius a0 as discussed in [39]. After performing a set of in-house microfabrication experiments, the minimum post top surface radius was found to be around 55 μm without post shape deterioration. However, such an etching recipe creates 47° sidewalls that need to be considered during the RF design of these filters. This is discussed in detail in Section VII. III. R ESONATOR A NALYSIS AND D ESIGN A. Analytical Resonator Model Fig. 2(a) shows a conceptual drawing of the EVA resonator in all-silicon technology. As mentioned before, wet-cavity etching results in inclined cavity walls as opposed to the vertical ones of a conventional coaxial resonator. The EM field

3

Fig. 2. Conceptual drawing of the EVA cavity resonator in all-silicon technology. (a) Cross-sectional view of the resonator. (b) Equivalent circuit model.

distribution within the cavity resembles the TEM mode of a pure coaxial line. As such, the all-silicon resonator architecture can be modeled as a capacitively loaded tapered coaxial line as shown in the circuit schematic in Fig. 2(b). Its characteristic impedance Z 0 at position x can be calculated as    x b0 − tan(θ) 1 μo Z 0 (x) = (1) ln x 2π εo a0 + tan(θ) in which a0 and b0 are the inner and outer conductor radii at x = 0 [Fig. 2(a)]. Such a linear variation of the inner/outer conductor radius results in a nonlinear impedance change. To better model its response, we follow the design approach depicted in Fig. 3(a). Specifically, the tapered line segment is discretized into N electrically short nontapered coaxial lines whose inner/outer conductor ratio equals the mean ratio of the corresponding tapered line section. For each section, we compute its equivalent ABC D matrix leading to a straight-forward calculation of the overall ABC D matrix. The overall ABC D matrix represents an equivalent π-type network [40] with a series inductance L coax and two parallel capacitances C1 , C2 to ground [see Fig. 3(a)]. To demonstrate this model, we apply the aforementioned analysis to an EVA tunable resonator with a0 = 0.11 mm, b0 = 2.4 mm, h = 1.5 mm, and θ = 60°. Fig. 3(b) shows the calculated L coax , C1 , C2 , as a function of the number of TL segments N. They are plotted normalized to the N = 1 value. As can be seen, all equivalent circuit parameters converge for N ≥ 4. The values of the modeled circuit parameters are also compared to HFSS simulations. While L coax agrees well to the HFSS results, both capacitances are underestimated. This is because the proposed model ignores the vertical electric field caused by the inclined side walls. Nevertheless, this is not a serious limitation, because 1) C2 is shorted and 2) C1 is in parallel with and much smaller than Cpost . As shown below, even a large error in C1 hardly affects the computed resonant frequency since it is much smaller than Cpost . Besides the TEM mode, part of the electric field energy is stored in an electric field distribution similar to the TM010 mode. This part of electric energy is modeled as Cplate as shown in Fig. 4(a). As expected, this mode is significant only when Cpost is small, i.e., only when the cavity is not

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 4. (a) Equivalent circuit of an all-silicon EVA resonator. (b) Resonance frequency versus capacitive gap.

Fig. 3. Tapered coaxial line. (a) 1-D discretization approach and its resulting equivalent π -type circuit model. (b) Equivalent circuit parameters for different number (N ) of TL segments.

heavily loaded. This is clearly shown in Fig. 4(b). To quantify the significance of Cplate , the resonance frequency ω0 is calculated assuming Cplate is negligible so as ω0 ≈ 1/(L coax (Cpost + C1 ))1/2 and shown in Fig. 4(b). Comparing these to HFSS results show that we may get a 10% error if Cplate is ignored. While it is not easy to analytically model Cplate , HFSS simulations can easily estimate it if higher than 10% modeling/design accuracy is desired. To summarize, the parallel combination of Cpost , C1 , and Cplate forms the total capacitance of the resonator Cresonator = Cpost + C1 + Cplate .

(2)

For micrometer-scale capacitive gaps (1 to 20 μm in this case), Cpost in parallel with Cplate tend to dominant as they are typically 5× larger than C1 . As such, a 20% error in C1 will not generate significant error on the center frequency estimation [see Fig. 4(b)]. In other words, we observe no significant difference in the calculated ω0 by increasing the modeling section number N from 1 to 4. For simplicity, therefore, we can model the tapered line by using just a single section (N = 1 in Fig. 4). Based on the above analysis,

L coax and C1 can be approximated as    μ0 1 b L coax ≈ ln h 8 6π × 10 ε0 a 2π  C1 ≈  h. 6 × 108 με00 ln ab

(3) (4)

B. Tuning Range Analysis According to the schematic shown in Fig. 4(a) and assuming Cpost is significantly larger than C1 , the frequency tuning ratio can be calculated as  1  C1 + Cplate − 2 ω0H gL = + (5) ω0L g L + gmax CpostL in which ω0H and ω0L are the highest and lowest frequencies of the tuning range. The capacitance CpostL represents the initial post capacitance when the MEMS tuner is at its unbiased position. This can be approximated as CpostL ≈

ε0 πa02 . gL

(6)

Note that the first term in (5) is controlled by the MEMS tuner deflection, while the second term is mostly related to the cavity dimensions. A larger tuning ratio can be achieved for a smaller initial gap g L . Consequently, we fix g L to the smallest initial RF gap g Lmin that can be manufactured or tolerated from a power handling point of view [29]. With a fixed g L , CpostL is only related to a0 . The second term in (5) also shows that larger CpostL (i.e., a0 ) leads to higher tuning range. We should

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. YANG et al.: DESIGN AND OPTIMIZATION OF TUNABLE SILICON-INTEGRATED EVANESCENT-MODE BPFs

5

Fig. 5(b) summarizes the Cresonator and Cplate values at the ω0H tuned state as extracted by the values shown in Fig. 5(a). The analytically calculated C1 , L coax and FEMsimulated Cpost values have been used here. As expected, there exists an optimal b value that maximizes the tuning range. C. Quality Factor Analysis and Optimization The analytical equation for Q u of an EVA cavity resonator has been derived in [41] as Q u = (ω0 μ0 /Rs )(ln((b/a))h/ ((1/a) + (1/b))h + 2 ln((b/a))) where Rs is the sheet resistance of the cavity sidewalls. In [41], the optimum relationship for maximum Q u is obtained when the ratio between the inner and outer radii equals 0.28. However, this is only valid for static resonator designs. A new design approach is needed for a tunable design particularly when many practical design and fabrication considerations have been to be taken into account. Four independent design variables (g L , a, b, h) along with the required tuning range and fabrication limitations dominate Q u . There is no easily derived analytical equation that directly leads to optimized cavity dimensions. Numerically exhaustive searching is also impractical. Nevertheless, we obtain useful insights by reorganizing the aforementioned Q u expression as Qu = Fig. 5. (a) Simulation results of unloaded quality factor and tuning range for a given RF gap variation of 19 μm, center post radius of 90 μm, and initial gap of 1 μm. (b) Extracted capacitances versus different cavity designs with various b values.

also keep in mind that for each selected CpostL (or a0 ), there exists an optimal set of b and h that minimize the numerator—C1 + Cplate —of the second term in (5) leading to the highest possible tuning ratio. We can obtain further insight on these aspects by inserting (3) and (6) into ω0L = 1/(L coax CpostL )1/2 . This yield h=

g L (6 × 108 ) 1 b . 2 a 2 √μ  ω0L 0 0 ln a 0

(7)

Equation (7) defines the b-h relationship for given a0 , g L , and ω0L . Also, this equation for h can be used in (4) to calculate C1 . This allows us to make the following observations. First, a low C1 requires a large b; h will be calculated in (7). On the other hand, a large b value increases Cplate — please see design example in Fig. 5(b). As a result, there is an optimal set of b and h to minimize the sum of C1 + Cplate . This optimization is better performed with the aid of an FEM simulator. Let us demonstrate this optimization process for a resonator with vertical sidewalls shown in Fig. 5(a). In this example, g L = 1 μm, gmax = 19 μm, and ω0L = 20 GHz. If the post radius is chosen as a0 = 90 μm, (7) will return various possible cavity designs with different aspect ratios. As shown in Fig. 5(a), the maximum tuning ratio is obtained for h = 0.6 mm and b = 0.83 mm. In general, high-aspect ratio cavities should be avoided when a large tuning range is required.

ω0 μ0 L coax   . Rs (2 × 10−7 ) a1 + b1 h + 2 ln ab

(8)

It is interesting to note that (∂(Q u )/∂(L coax )) is significantly higher than any of the other partial derivatives ((∂(Q u )/∂(a)), (d∂(Q u )/∂(b)), and (∂(Q u )/∂(h))). Thus, Q u is dominated by L coax . Consequently, as shown in (5) and (8), although a large L coax leads to high Q u , it also leads to a small CpostL and reduced tuning ratio. It is clear then that Q u and tuning ratio is an important tradeoff pair. The right strategy is to find a design that meets the tuning range requirement with the smallest possible CpostL and thus the highest possible Q u . The iterative optimization procedure is summarized as follows. 1) Decide on the minimum gap g L = g Lmin based on fabrication limitations and the needed power handling/ linearity requirements [29]. 2) Start by setting a0 to its smallest fabrication-achievable value a0min . This will result to the minimum CpostL for the given gap g L = g Lmin and therefore the highest Q u . 3) Find the largest possible tuning range by optimizing the cavity aspect ratio with the procedures detailed in Section III-B. 4) If the achieved tuning range is smaller than the target, increase a0 until the obtained tuning ratio reaches the target. If the achieved tuning range is higher than the target, increase g L (so to increase Q u as well as power handling) and repeat steps 2 and 3. A design example of a 20–40 GHz resonator is illustrated in Fig. 6. The design constraints for this paper are listed in Table I. By following the above design procedures, we see that for designs with a0 less than 100 μm, there is no practical solution that fulfills the tuning requirements. The red solid

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

Fig. 6. Quality factor optimization of a 20–40 GHz resonator that fulfills all tuning requirements under design constraints.

line in Fig. 6 is the best possible design in the sense that all remaining solutions will have larger than required tuning range at a cost of lower Q u . The design implemented in this paper is shown with the green dashed line in Fig. 6 (a0 = 110 μm, b0 = 2.4 mm, h = 0.4 mm). Such design was necessary due to the following: 1) the bottom area requirement for the coaxial probe feed; 2) limited choices of available wafer thicknesses; 3) in-house fabrication capabilities. IV. I NTERRESONATOR C OUPLING D ESIGN The general interresonator coupling structure of an all-silicon EVA tunable filter is shown in Fig. 7. Note that adjacent EVA resonators are synchronously or asynchronously tuned and coupled through a coupling iris whose cross section is a trapezoid due to anisotropic silicon etching. This is represented in Fig. 7(a) with a solid line. As reported in [42], such waveguide can be approximated as an equivalent rectangular waveguide with width W shown in Fig. 7(a) (dashed line). The depth of the iris—defined by silicon wet-etching—is the same as the resonator depth h. The coupling strength is determined by the resonator separation D and the iris width W . While for a single-frequency design, a range of D and W values provide the required coupling strength, the situation is more complex for a tunable filter. In the latter case, it is important to select the optimal D and W values that minimize BW variation over the tuning range while simultaneously providing the needed coupling strength. To this end, it is preferable to have an analytical model that guides the interresonator coupling structure design process. Significant research results [43]–[45] have been reported in the literature regarding the design and modeling of evanescent mode waveguide BPFs. The interresonator coupling section is defined as the area between the two cavity posts. Capacitive loading on the cavity post drives the coupling section under evanescent mode operation. In [43], an evanescentmode coupling section without step discontinuity (W = 2b) is modeled as a TL assuming only the fundamental TE10 mode

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 7. (a) Equivalent rectangular waveguide for a tapered waveguide, (b) Top view of interresonator coupling structure, (c) Side view of interresonator coupling section.

within the coupling section. It is also capacitively loaded as shown in Fig. 8(c). The evanescent-mode characteristic impedance j X 0 of the TL and its propagation constant γ0 can be calculated as 465h j X 0 = j  (9) λ 2 W 2W − 1

  λ 2 2π γ0 = − 1. (10) λ 2W In this paper, the diameter of the loading post is small compared to the outer conductor. As such, the inductance associated with the coaxial structure is not negligible. The resonator inductance—derived in Section III as L coax —is included in parallel to Cpost . In addition, the model is further generalized for a coupling structure with step discontinuities as shown in 8(b). The shunt susceptance generated by the step discontinuity assuming the TE10 mode has been derived in [46]. Such susceptance behaves as a lumped inductor L step whose value is negative within the frequency range under its cutoff frequency. The analytical expression of L step based on the geometry shown in Fig. 7 can be written as L step = −

h (4b)(1 − ρ/2) 465 W  2π(3 × 108 ) ρ 2 (1 + ρ) ln ρ2

(11)

where ρ = 1 − W/2b is the geometrical ratio of the step discontinuity. The evanescent-mode TL in Fig. 8(c) can be transformed into an equivalent π-network and rearranged as the circuit shown in the blue box in Fig. 9 with the following parameters: X 0 sinh γ0 D (12) Lm = ω and X 0 tanh γ0 D Le = . (13) ω

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. YANG et al.: DESIGN AND OPTIMIZATION OF TUNABLE SILICON-INTEGRATED EVANESCENT-MODE BPFs

7

Fig. 8. (a) Top view of interresonator coupling structure. (b) Equivalent interresonator coupling structure. (c) Equivalent circuit of the coupled resonators.

Fig. 10. (a) Model validation of coupling structure (a) without iris and (b) with iris.

for L e , L m as 

K (ω) = sinh(γ0 D)

Fig. 9.

Further derived equivalent circuit of the coupled resonators.

Thus, the circuit shown in Fig. 9 is in the form of a coupled resonator and the coupling coefficient can be calculated as K (ω) = (L e  L coax  L step )/L m by linear circuit analysis. However, both L m and L e are nonlinear in the sense that both inductance values are frequency dependent. Such disagreement is fixed by multiplying a correction factor  which is derived in [43, Appendix]. In this paper, the model is expanded with two extra elements, L coax and L step . Consequently, the original formula of  is rederived into 2  (14)  = L e L coax L step L L coax L step 1 + L coax L step + 1 2 1 − eL coax L step 1−

λc λ

in which λc = 2W is the cut-off wavelength. Equation (14) is reduced to the original  = 2(1 + (1/1 − (λc /λ)2 ))−1 by setting L coax = L step = ∞. The coupling coefficient K (ω) can be derived by using the aforementioned expressions

 X0 ω(L coax L step )

+1

.

(15)

The validity and significance of the proposed circuit model is demonstrated by comparing it to HFSS simulations as shown in Fig. 10. To extract K from HFSS simulations, we follow the process outlined in [47] for magnetically coupled synchronously tuned resonators and write K = ( f e2 − f m2 )/( f e2 + f m2 ). The frequencies f e and f m represent the even- and odd-mode resonant frequencies that correspond to the lowest two resonant frequencies obtained by HFSS eigenmode simulations. In Fig. 10(a), we validate L coax with a coupled EVA resonator pair with no step discontinuity in the coupling section. The predicted K from the proposed model is in close agreement with the simulation results. In Fig. 10(b), we introduce a step discontinuity. The proposed model that includes both L step and L coax agrees well with the simulation results. The aforementioned analysis and equivalent circuit assume that only the evanescent TE10 mode exists within the interresonator coupling region. Such approximation is valid only when the iris step discontinuity is small (W/2b close to 1) or far from its adjacent resonator posts (D/2b larger than 1). Otherwise, the above analysis underestimates the coupling strength due to excluding higher order modes generated by the step discontinuity. Full-wave simulation is necessary for accurate results when those higher order modes are significant.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

Fig. 11.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Coupling structure design with frequency dispersion optimization.

One way to make the model capture such effect is to replace the physical iris width W with an effective iris width We given by We /W = m(D/2b)−n (1 − W/2b)2 + 1. The coefficients m = n = 1.5 are extracted from numerical FEM experiments. As expected for large D/2b or when W/2b ≈ 1, We is equal to W . As mentioned before, a widely tunable filter imposes additional requirements on its coupling coefficient in order to minimize frequency dispersion. This can be quantified by calculating the derivative of (15). By keeping only the dominant terms, this derivative can be approximated as  0  D − ∂γ ∂ω ∂K  .  (16) X0 ∂ω sinh(γ0 D) ω(L coax + 1 L step ) Comparing to (15), (16) can be further simplified as

∂γ0 ∂K

K. ≈ D

∂ω ∂ω

Fig. 12. (a) Basic probe-coupling scheme. (b) Equivalent circuit model. (c) Simulated reflection coefficient shown on the Smith chart for calculating L phase . (d) Simulated external quality factor as a function of the probe position.

(17)

Typically, (17) is minimized at the center of the band ω0M where the optimal coupling is designed to be K |ω0M = FBWm i j where FBW is the desired fractional BW and m i j is the synthesized normalized coupling coefficient between the i th and j th resonators. Thus, at the center of the band, we have

∂γ0 ∂K

(FBWm i j ).

= D (18) ∂ω |ω ∂ω 0M

Based on the expression of γ0 and (15), choosing a small W reduces (∂γ0 /∂ω) while it also results in a small D to maintain the same center-frequency condition K |ω0M = F BW m i j . Consequently, limiting frequency dispersion requires us to reduce W to the geometric limit 2b ≥ W ≥ 2(b 2 − D 2 /4)1/2 . This coupling design procedure is illustrated on a two-pole tunable filter example with FBW = 0.025 and m 12 = 0.707. In Fig. 11, we have graphed the normalized coupling coefficient for different design choices. At the center of the tuning band (30 GHz), we have maintained K |ω0M = 0.017 for all designs. As expected, the worst results occur for the maximum possible width W = 2b. Optimal results are obtained for the smallest possible width W = 2(b 2 − (D 2 /4))1/2 . We choose the coupling design with the lowest frequency dispersion achieved by D = 4.3 mm and W = 1.9 mm to proceed.

V. E XTERNAL C OUPLING D ESIGN The external coupling mechanism scheme is shown in Fig. 12(a). We employ a magnetic coupling mechanism since nearly all electric field is concentrated on top of the post. Nevertheless, the presented design technique is not limited to the selected geometry. The external coupling coefficient is mainly controlled by the distance between the center pin of the coaxial probe and the center post (Dprobe ). This coupling structure can be represented by the equivalent circuit model shown in Fig. 12(b). In particular, the coupling to the resonator is modeled as an ideal transformer whose turn ratio n is a function of frequency and Dprobe . A TL with characteristic impedance Z 0 and length L phase is included to represent the phase shift that is generated by the coaxial probe discontinuity—calculated at the defined reference plane. Due to the nature of this geometry, there exists no closedform equation to accurately determine Q e and L phase . Hence numerical simulations need to be performed in order to extract the aforementioned parameters. As an example, we design the external coupling for the previously selected resonator dimensions a0 = 0.11 mm, b0 = 2.4 mm, and h = 0.4 mm. Both Q e and L phase are extracted by HFSS simulations. Specifically, L phase is

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. YANG et al.: DESIGN AND OPTIMIZATION OF TUNABLE SILICON-INTEGRATED EVANESCENT-MODE BPFs

Fig. 13.

9

External matching network structure and its circuit model.

determined from the phase offset of the simulated reflection coefficient shown in the Smith chart of Fig. 12(c). The value of L phase (0.85 mm in this case) does not depend on Dprobe which confirms the validity of the suggested equivalent circuit model. With the simulated reflection coefficient S11 , Q e can be extracted as Q e = ω0 (τ S11(ω0 )/4) in which τ S11(ω0 ) is the reflection group delay at the tuned resonance frequency ω0 . The extracted Q e for various Dprobe is shown in Fig. 12(d). It can be seen that Q e varies monotonically with Dprobe and frequency. There is a unique Dprobe value that yields the desired Q e at a single tuned frequency (often taken as the center of the band ω0M ). Similarly to the interresonator coupling design, the frequency dependence of the external coupling is important for widely tunable filters since it determines the available impedance matching. While active compensation is possible, it comes at the cost of additional loss and nonlinearity. In this paper, we opt for passive compensation as shown in Fig. 13. This technique includes two TL sections with lengths L 1 and L 2 and impedances Z 0 and Z t cascaded in series. This mechanism is explained as follows. The original unmodified external quality factor without the two-line matching network in Fig. 12(b) can be calculated as Q e0 = (n 2 ωCresonator )/Re(Yin0 ) where Yin0 —typically (1/50 )—is the input admittance. The effect of the two-line matching network can be understood by comparing the real parts of the input admittances Re(Yin0 ) Q eNEW (19) = Q e0 Re(YinNEW ) where Q eNEW is the tailored external quality factor achieved with the addition of the two-line matching network. The ratio of Yin0 over YinNEW can be understood as a correction factor calculated as Z 0 + j Z j tan(β(L 1 + L phase )) 1 (20) = Zj YinNEW Z j + j Z 0 tan(β(L 1 + L phase )) where Z j = Z 0 (Z t + j Z 0 tan(β L 2 )/(Z 0 + j Z t tan(β L 2 )) is the input impedance observed at the connection point between the lines and β is the propagation constant.

Fig. 14. (a) Example of external quality factor tailored by matching network. (b) Example of external coupling tailored by matching network for filter maintains FBW and absolute BW.

The main idea then is to design the two-line network so the external quality factor achieves its optimal value. This value is determined by perfect impedance matching over the entire tuning range. This requires an external quality factor equal to 1 mi j (21) Q eIdeal (ω) = K i j (ω) m 2si in which, K i j is predetermined by the interresonator coupling design and m s1 , m i j represent the needed denormalized external and interresonator coupling coefficients, respectively. To realize an external quality factor as close as Q eIdeal , we can follow a two-step design process. First, vary Dprobe until we achieve the best implementable probe position that makes Q e0 as close as possible to Q eIdeal . Second, we choose the two-line section parameters to further tune the coupling coefficient until the obtained Q eNEW becomes nearly equal to Q eIdeal . To illustrate this process with an example, we design the external coupling of a two-pole filter with m s1 = 0.78, m 21 = 0.707 as shown in Fig. 14(a). Here, we have selected K 21 (ω) to have the least frequency dispersion shown in Fig. 11 with the red curve. The calculated Q eIdeal and the best possible Q e0 are shown in Fig. 14(a). The designed Q eNEW approximates Q eIdeal quite well with Z t = 78 , L 1 = 0.3 mm, and L 2 = 0.6 mm. Furthermore, in Fig. 14(b), we demonstrate that the presented passive compensation scheme can successfully modify Q e0 to realize tunable filters with constant FBW or constant

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

Fig. 15.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Design and optimization flow of the EVA all silicon filter.

BW over an octave tuning range. This is achieved by appropriately selecting the dimensions of the two cascaded coaxial lines. VI. F ILTER D ESIGN Fig. 15 graphically summarizes the overall filter design process discussed in detail in the previous sections. By following these design steps, we obtain the main filter design parameters, i.e., Q u , K, Q e over the entire tuning range. Consequently, a frequency dependent N+2 coupling matrix m(ω) can be constructed and the scattering filter parameters can be calculated using the method in [47]. Fig. 16 graphically represents these for the designed filter along with the ones obtained by HFSS simulations. The calculated response is in very good agreement with the HFSS response further validating the presented approach. VII. E XPERIMENTAL VALIDATION A two-pole all-silicon filter prototype was fabricated at the Birck Nanotechnology Center at Purdue University, West Lafayette, IN, USA, to experimentally validate the presented theory. The fabrication process described in [48] was employed. Filter prototypes with both pure Au and Au-V diaphragms were built. Fig. 17(b)–(d) includes photographs of the fabricated MEMS diaphragm dies, filter cavity, and dc electrodes. In addition to the individual fabrication steps, the final filter assembly is also conducted in the cleanroom. Prior to this assembly, an RCA clean is performed to remove particle-caused contamination. The critical assembly steps can be summarized as follows. First, we perform the die-to-die bonding of the filter cavity with the top ceiling wafer with special attention paid to the MEMS diaphgram-cavity post alignment. Second, the MEMS diaphragm is dry-etch released (Xenon Difluoride). The postrelease RF gap can be determined by a laser conformal microscope through the back side dc biasing via hole. It is possible to finely adjust the initial RF gap by adding extra etching cycles to slightly change the postrelease bending of the diaphragm. These techniques allow us to achieve the desired gap within about 0.2 μm. Third, we place the bonded MEMS die to the metal fixture shown

Fig. 16. Simulated (HFSS) and calculated (coupling matrix) filter responses.

in Fig. 17(e). This metal fixture was manufactured by CNC machining. Two miniaturized coaxial ports were inserted and soldered within the fixture body. The insertion depth of the coaxial port is well controlled by the fixture recess depth. Slight adjustments to center the coaxial pins were conducted under a microscope. The fourth and last steps are to insert and bond the dc actuation electrodes over the MEMS diagrams. Fig. 17 depicts the employed characterization setup. The S-parameters were measured using an Agilent E836lA vector network analyzer through a pair of coaxial connectors that are placed underneath the Au-metalized fixture. Each resonator is individually controlled by a separate dc bias voltage applied through a dc probe connected to a Keithley 2400 SourceMeter. Fig. 18 shows the measured filter performance for several bias states. The center frequency of the filter can be tuned between 20 and 40 GHz for an applied dc bias voltage between 0 and 180 V. This corresponds to an effective diaphragm deflection of 17 μm. This was determined by fitting the measured RF response to the EM model. The measured insertion loss is less than 3 dB and the return loss is less than 15 dB. The FBW was measured between 1.9% and 4.7% which is a bit larger than the calculated range (1.7%–3.9%). This is attributed to the undercut of the wet etching that enlarged the width of the coupling iris. Considering the measured insertion loss and BW, we can estimate Q u throughout the tuning range between 264 and 540 for the Au-V case and 420 and 700 for the Au case. Fig. 19 summarizes these results. The Au-V

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. YANG et al.: DESIGN AND OPTIMIZATION OF TUNABLE SILICON-INTEGRATED EVANESCENT-MODE BPFs

11

Fig. 19. (a) Measured insertion loss with connector loss de-embedded. (b) Extracted unloaded quality factor.

Fig. 17. (a) RF characterization setup of the two-resonator all-silicon BPF. (b) Front view of the manufactured die of the diaphragm. (c) Front view of the manufactured die of the cavity. (d) Side view of the manufactured die of the dc biasing electrode. (e) Cross-sectional view of the packaged device.

coefficient over time. To simplify the process, only the resonance of one of the resonators is recorded. The resonance of the other resonator is placed at a distance larger than 10 GHz from the resonator under test. The characterization procedure can be summarized as follows. The frequency of the resonator is initially recorded at a zero-biasing level (relaxed membrane) which corresponds to a frequency of 20 GHz. Subsequently, a dc bias voltage is applied to drive the resonance to the designated frequency (24 GHz for this measurement). The obtained response for the two diaphragm cases (Au and Au-V) is summarized in Fig. 20. The recorded drift is due to phase II creep also known as the second stage creep. More details are available in [54]. In this stage, the creep rate is approximately constant. The Au-V diaphragm exhibits a 7× slower phase II creep rate than the pure Au case that corresponds to a frequency drift rate around 2.7 MHz/h. Summarizing the above, it is apparent that there is a tradeoff between mechanical stability and RF performance when pure Au and Au-V composites are employed. A closed feedback control loop can fully eliminate the drift rate [33], [55]. VIII. C OMPARISON W ITH S TATE - OF - THE -A RT R ESULTS

Fig. 18. Measured filter S-parameters and tuning performance over its 20–40 GHz tuning range.

diaphragm results in higher insertion loss and lower quality factor due to its higher resistivity as also discussed in [53]. Frequency stability measurements were performed on the same characterization setup by recording the reflection

The MEMS enabled all-silicon EVA tunable filter design of this paper builds upon the all-silicon concept that was first presented in [56] and was further advanced into a tunable filter in [12]. However, due to the proposed filter design methodology, this paper exhibits 22% wider tuning range, 1.9× lesser BW variation, and 2× wider impedance matching as well as acceptable matching over the entire tuning range. Furthermore, the frequency stability of this paper is significantly improved by employing creep-resistive Au-V diaphragm that exhibits

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 12

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

TABLE II P ERFORMANCE C OMPARISON OF K-BAND MEMS T UNABLE F ILTERS

filter architecture in this paper. Last, the filter of this paper demonstrates the highest FOM among all other filter technologies. Note that the FoM of the tunable filter is defined as in [57] and is summarized as fH − fL FoM = √ BW H BW L IL H IL L

(23)

where f H and f L are the highest and lowest achieved filter center frequencies, BW H , BW L , IL H , and IL L are BWs and insertion losses at the highest and lowest tuned frequencies. Fig. 20. Measured frequency drift after phase II creep is initiated. The measured center frequency is 24 GHz.

7× slower frequency drift rate than pure Au diaphragms [12]. A more complete comparison with alternative 3-D tunable BPFs is summarized in Table II. As can be seen, the filter presented in this paper exhibits the highest tunability (100%) which is significantly higher than filters based on conventional cavities (see [22], [49]–[51]) whose tunability is less than 5%. Such tunability also outperforms other EVA tunable filers, such as for example, the ones in [25] and [51]. This is because the integrated MEMS tuner is capable of providing capacitance variation of over 10–1. Moreover, the presented filter offers the most compact cavity volume. This conclusion is based on calculating the wavelength-normalized resonator volume at the lowest achievable frequency Resonator Volume Volume R = (22) (λ L )3 where λ L is the wavelength at the lower edge of the band (20 GHz in our case). We can also observe that the filter in this paper outperforms all filters listed in the table especially for the nonEVA cavity implementations [22], [24], [49]–[51]. In addition, Arnold et al. [49] demonstrated by manual—potentially motorized—tuning. In addition, the tuning of [50] and [51] involves air-pressure actuation and step-motor system, respectively. These actuation solutions typically occupy larger volumes making the overall filter size much larger than the

IX. C ONCLUSION This paper reports on a detailed design technique for achieving optimal MEMS-tunable evanescent-mode cavitybased BPFs with continuously variable center frequency over an octave tuning range. The devised filters are manufactured using silicon-micromachining techniques that enable their actualization for frequencies located in the millimeterwave (30–100 GHz) regime. For the first time, we present an RF design methodology that takes into consideration all microfabrication-induced constrains—e.g., nonvertical wall profiles and finite MEMS deflection—and allows for high unloaded quality factor (Q u ) and minimum BW variation to be achieved over the desired tuning range. A new fully passive input/output compensating structure integrated within the filter’s package controls the filter’s external coupling and further enhances the filter’s performance. A two pole 20–40 GHz tunable EVA filter is designed, optimized, and experimentally demonstrated to validate the presented concepts. The fabricated filter incorporates electrostatically tuned MCD with enhanced tuning stability by co-sputtered Au-V material. Measurements demonstrated a tuning range between 20 and 40 GHz, relative BW of 1.9%–4.7%, and impedance matching over the entire tuning range which is 22%, 1.9×, and 2× better than previously reported state-of-the-art MEMS tunable filters of this type [12]. Furthermore, the Au-V tuner results in significantly improved frequency stability with a 7× slower drift rate (phase II creep) compared to MEMS tunable filters based on Au MCD tuners. However, this comes at a cost of an additional 0.15–0.9 dB loss (30%–35% penalty of Q u ) due to the higher resistivity of Au-V.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. YANG et al.: DESIGN AND OPTIMIZATION OF TUNABLE SILICON-INTEGRATED EVANESCENT-MODE BPFs

R EFERENCES [1] N. Bhushan et al., “Network densification: The dominant theme for wireless evolution into 5G,” IEEE Commun. Mag., vol. 52, no. 2, pp. 82–89, Feb. 2014. [2] P. Bahramzy et al., “A tunable RF front-end with narrowband antennas for mobile devices,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 10, pp. 3300–3310, Oct. 2015. [3] R. R. Mansour, “High-Q tunable filters for wireless base station applications,” in Proc. IEEE Int. Conf. Microw, Commun. Antennas Electron. Syst., Tel Aviv, Isarel, Nov. 2015, pp. 1–5. [4] T.-C. Lee, J. Lee, and D. Peroulis, “Dynamic bandpass filter shape and interference cancellation control utilizing bandpass–bandstop filter cascade,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 8, pp. 2526–2539, Aug. 2015. [5] Y. Shim, Z. Wu, and M. Rais-Zadeh, “A high-performance continuously tunable MEMS bandpass filter at 1 GHz,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 8, pp. 2439–2447, Aug. 2012. [6] C.-C. Cheng and G. M. Rebeiz, “High-Q 4–6-GHz suspended stripline RF MEMS tunable filter with bandwidth control,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 10, pp. 2469–2476, Oct. 2011. [7] W. Tang and J.-S. Hong, “Varactor-tuned dual-mode bandpass filters,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 8, pp. 2213–2219, Aug. 2010. [8] J. Ni and J.-S. Hong, “Compact varactor-tuned microstrip high-pass filter with a quasi-elliptic function response,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 11, pp. 3853–3859, Nov. 2013. [9] R. Stefanini, M. Chatras, A. Pothier, C. Guines, and P. Blondy, “HighQ 3D tunable RF MEMS filter with a constant fractional bandwidth,” in Proc. 8th Eur. Microw. Integr. Circuits Conf., Nuremberg, Germany, Oct. 2013, pp. 1–4. [10] J.-R. Mao, W.-W. Choi, K.-W. Tam, W. Q. Che, and Q. Xue, “Tunable bandpass filter design based on external quality factor tuning and multiple mode resonators for wideband applications,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 7, pp. 2574–2584, Jul. 2013. [11] Y. Zheng, M. Sazegar, H. Maune, X. Zhou, J. R. Binder, and R. Jakoby, “Compact substrate integrated waveguide tunable filter based on ferroelectric ceramics,” IEEE Microw. Wireless Compon. Lett., vol. 21, no. 9, pp. 477–479, Sep. 2011. [12] Z. Yang and D. Peroulis, “A 23–35 GHz MEMS tunable all-silicon cavity filter with stability characterization up to 140 million cycles,” in IEEE MTT-S Int. Microw. Symp. Dig., Tampa, FL, USA, Jun. 2014, pp. 1–3. [13] M. D. Hickle, M. D. Sinanis, and D. Peroulis, “Tunable high-isolation W-band bandstop filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, USA, Jun. 2015, pp. 1–4. [14] A. J. Alazemi and G. M. Rebeiz, “A low-loss 1.4–2.1 GHz compact tunable three-pole filter with improved stopband rejection using RF-MEMS capacitors,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, USA, May 2016, pp. 1–4. [15] M. A. El-Tanani and G. M. Rebeiz, “High-performance 1.5–2.5-GHz RF-MEMS tunable filters for wireless applications,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 6, pp. 1629–1637, Jun. 2010. [16] K.-Y. Lee and G. M. Rebeiz, “A miniature 8–16 GHz packaged tunable frequency and bandwidth RF MEMS filter,” in Proc. IEEE Int. Symp. Radio-Freq. Integr. Technol. (RFIT), Jan. 2009, pp. 249–252. [17] S. S. Attar, S. Setoodeh, P. D. Laforge, M. Bakri-Kassem, and R. R. Mansour, “Low temperature superconducting tunable bandstop resonator and filter using superconducting RF MEMS varactors,” IEEE Trans. Appl. Supercond., vol. 24, no. 4, Aug. 2014, Art. no. 1500709. [18] F. Huang, S. Fouladi, and R. R. Mansour, “High-Q tunable dielectric resonator filters using MEMS technology,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 3401–3409, Dec. 2011. [19] S. Fouladi, F. Huang, W. D. Yan, and R. R. Mansour, “High-Q narrowband tunable combline bandpass filters using MEMS capacitor banks and piezomotors,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 393–402, Dec. 2012. [20] D. Peroulis, E. Naglich, M. Sinani, and M. Hickle, “Tuned to resonance: Transfer-function-adaptive filters in evanescent-mode cavity-resonator technology,” IEEE Microw. Mag., vol. 15, no. 5, pp. 55–69, Jul. 2014. [21] P. Blondy and D. Peroulis, “Handling RF power: The latest advances in RF-MEMS tunable filters,” IEEE Microw. Mag., vol. 14, no. 1, pp. 24–38, Jan. 2013. [22] L. Pelliccia, F. Cacciamani, P. Farinelli, P. Ligander, and O. Persson, “High-Q MEMS-tunable waveguide filters in K-band,” in Proc. 42nd Eur. Microw. Conf., Amsterdam, The Netherlands, Nov. 2012, pp. 273–276.

13

[23] L. Pelliccia, P. Farinelli, V. Nocella, F. Cacciamani, F. Gentili, and R. Sorrentino, “Discrete-tunable high-Q E-plane filters,” in Proc. 43rd Eur. Microw. Conf., Nuremberg, Germany, Oct. 2013, pp. 1215–1218. [24] W. D. Yan and R. R. Mansour, “Tunable dielectric resonator bandpass filter with embedded MEMS tuning elements,” IEEE Trans. Microw. Theory Techn., vol. 55, no. 1, pp. 154–159, Jan. 2007. [25] D. Psychogiou, D. Peroulis, Y. Li, and C. Hafner, “V-band bandpass filter with continuously variable centre frequency,” IET Microw., Antennas Propag., vol. 7, no. 8, pp. 701–707, Jun. 2013. [26] J. Small, M. S. Arif, A. Fruehling, and D. Peroulis, “A tunable miniaturized RF MEMS resonator with simultaneous high Q (500–735) and fast response speed (< 10–60 μs),” J. Microelectromech. Syst., vol. 22, no. 2, pp. 395–405, Apr. 2013. [27] R. Stefanini, J. D. Martinez, M. Chatras, A. Pothier, V. E. Boria, and P. Blondy, “Ku band high-Q tunable surface-mounted cavity resonator using RF MEMS varactors,” IEEE Microw. Wireless Compon. Lett., vol. 21, no. 5, pp. 237–239, May 2011. [28] X. Liu, L. P. B. Katehi, W. J. Chappell, and D. Peroulis, “High-Q tunable microwave cavity resonators and filters using SOI-based RF MEMS tuners,” J. Microelectromech. Syst., vol. 19, no. 4, pp. 774–784, Aug. 2010. [29] X. Liu, L. Katehi, W. J. Chappell, and D. Peroulis, “Power handling of electrostatic MEMS evanescent-mode (EVA) tunable bandpass filters,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 2, pp. 270–283, Feb. 2012. [30] M. S. Arif and D. Peroulis, “All-silicon technology for high-Q evanescent mode cavity tunable resonators and filters,” J. Microelectromech. Syst., vol. 23, no. 3, pp. 727–739, Jun. 2014. [31] J. Zeng, Z. Yang, and D. Peroulis, “Thermally stable nonuniform microcorrugated capacitive MEMS tuner,” J. Microelectromech. Syst., vol. 24, no. 3, pp. 522–524, Jun. 2015. [32] Z. Yang and D. Peroulis, “A 20–40 GHz tunable MEMS bandpass filter with enhanced stability by gold-vanadium micro-corrugated diaphragms,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, USA, Jun. 2016, pp. 1–3. [33] Z. Yang and D. Peroulis, “A 19–40 GHz Bi-directional MEMS tunable all silicon evanescent-mode cavity filter,” in IEEE MTT-S Int. Microw. Symp. Dig., Honolulu, HI, USA, Jun. 2017, pp. 1–4. [34] D. Psychogiou and D. Peroulis, “Advances in high-Q tunable filter technologies,” Int. J. Adv. Eng. Sci. Appl. Math., vol. 7, no. 4, pp. 170–176, Dec. 2015. [35] D. Peroulis and D. Psychogiou, “MEMS-tunable silicon-integrated cavity filters,” in Proc. IEEE 16th Top. Meeting Silicon Monolithic Integr. Circuits RF Syst., Austin, TX, USA, Jan. 2016, pp. 1–4. [36] H. Joshi, H. H. Sigmarsson, D. Peroulis, and W. J. Chappell, “Highly loaded evanescent cavities for widely tunable high-Q filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Honolulu, HI, USA, Jun. 2007, pp. 1–4. [37] E. J. Naglich, J. Lee, H. H. Sigmarsson, D. Petroulis, and W. J. Chappell, “Intersecting parallel-plate waveguide loaded cavities for dual-mode and dual-band filters,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 5, pp. 1829–1838, May 2013. [38] M. McLean, W. L. Brown, and R. P. Vinci, “Temperature-dependent viscoelasticity in thin Au films and consequences for MEMS devices,” J. Microelectromech. Syst., vol. 19, no. 6, pp. 1299–1308, Dec. 2010. [39] P. Pal et al., “Surfactant adsorption on single-crystal silicon surfaces in TMAH solution: Orientation-dependent adsorption detected by in situ infrared spectroscopy,” J. Microelectromech. Syst., vol. 18, no. 6, pp. 1345–1356, Dec. 2009. [40] D. M. Pozar, Microwave Engineering. New York, NY, USA: Wiley, 2005. [41] H. H. Sigmarsson, “Widely tunable, high-Q, evanescent-mode cavity filters: Fabrication, control, and reconfigurability,” Ph.D. dissertation, School Elect. Comput. Eng., Purdue Univ., West Lafayette, IN, USA, 2010. [42] E. M. Popescu and S. Song, “Trapezoidal waveguides: First-order propagation equivalence with rectangular waveguides,” J. Phys. A, Math. Theor., vol. 40, no. 48, pp. 14555–14574, Nov. 2007. [43] G. F. Craven and C. K. Mok, “The design of evanescent mode waveguide bandpass filters for a prescribed insertion loss characteristic,” IEEE Trans. Microw. Theory Techn., vol. MTT-19, no. 3, pp. 295–308, Mar. 1971. [44] C. K. Mok, “Design of evanescent-mode waveguide diplexers,” IEEE Trans. Microw. Theory Techn., vol. MTT-21, no. 1, pp. 43–48, Jan. 1973. [45] J.-C. Nanan, J.-W. Tao, H. Baudrand, B. Theron, and S. Vigneron, “A two-step synthesis of broadband ridged waveguide bandpass filters with improved performances,” IEEE Trans. Microw. Theory Techn., vol. 39, no. 12, pp. 2192–2197, Dec. 1991.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 14

[46] N. Marcuvitz, Waveguide Handbook. Stevenage, U.K.: IET, 1951. [47] J.-S. G. Hong, Microstrip Filters for RF/Microwave Applications. New York, NY, USA: Wiley, 2011. [48] J. Li, Z. Yang, D. Psychogiou, M. D. Sinanis, and D. Peroulis, “Creep-resistant nanocrystalline gold-vanadium alloyed microcorrugated diaphragms (MCDS),” in Proc. Transducers, Anchorage, AK, USA, Jun. 2015, pp. 1–4. [49] C. Arnold, J. Parlebas, and T. Zwick, “Reconfigurable waveguide filter with variable bandwidth and center frequency,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 8, pp. 1663–1670, Aug. 2014. [50] F. Sammoura and L. Lin, “Micromachined W-band polymeric tunable iris filter,” Microsyst. Technol., vol. 17, no. 3, pp. 411–416, Mar. 2011. [51] B. Yassini, M. Yu, and B. Keats, “A K a-band fully tunable cavity filter,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 12, pp. 4002–4012, Dec. 2012. [52] S. J. Park, I. Reines, C. Patel, and G. M. Rebeiz, “High-Q RF-MEMS 4–6 GHz tunable evanescent-mode cavity filter,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 2, pp. 381–389, Feb. 2010. [53] J. Li, Z. Yang, M. D. Hickle, D. Psychogiou, and D. Peroulis, “Electrical properties of creep-resistant nanocrystalline gold-vanadium thin films at millimeter-wave frequencies,” in Proc. IEEE 16th Top. Meeting Silicon Monolithic Integr. Circuits RF Syst., Austin, TX, USA, Jan. 2016, pp. 1–3. [54] J. Li and A. Dasgupta, “Failure-mechanism models for creep and creep rupture,” IEEE Trans. Rel., vol. 42, no. 3, pp. 339–353, Sep. 1993. [55] M. A. Kharter and D. Peroulis, “Real-time feedback control system for tuning evanescent-mode cavity filters,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 9, pp. 2804–2813, Sep. 2016. [56] M. S. Arif and D. Peroulis, “A 6 to 24 GHz continuously tunable, microfabricated, high-Q cavity resonator with electrostatic MEMS actuation,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2012, pp. 1–3. [57] V. Pleskachev and I. Vendik, “Figure of merit of tunable ferroelectric planar filters,” in Proc. 33rd Eur. Microw. Conf. (EuMC), vol. 1. Munich, Germany, Oct. 2003, pp. 191–194.

ZhengAn Yang (S’10–M’18) received the B.S. degree in electrical engineering from Purdue University, West Lafayette, IN, USA, in 2009, where he is currently pursuing the Ph.D. degree in electrical engineering. His current research interests include RF and microwave device and system design. MEMS design, microfabrication, and reliability enhancement technologies. Mr. Yang is a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He was a recipient of the IEEE MTT-S IMS 2016, Second Place Award of the Student Design Competition on Switchable RF-MEMS Filter, and the IEEE MTT-S IMS 2014 Third Place Award of the Student Paper Competition.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Dimitra Psychogiou (S’10–M’14) received the Dipl.-Eng. degree in electrical and computer engineering from the University of Patras, Patras, Greece, in 2008, and the Ph.D. degree in electrical engineering from the Swiss Federal Institute of Technology (ETH) Zürich, Switzerland, in 2013. From 2013 to 2016, she was with Purdue University, West Lafayette, IN, USA, as a Research Scientist. She is currently an Assistant Professor of electrical, computer and energy engineering with the University of Colorado at Boulder, Boulder, CO, USA. Her current research interests include RF design and characterization of reconfigurable microwave and millimeter-wave passive components, RF-MEMS, acoustic wave resonator-based filters, tunable filter synthesis, and frequency-agile antennas. Dr. Psychogiou is currently an Associate Editor of IET Microwaves, Antennas, and Propagation.

Dimitrios Peroulis (S’99–M’04–SM’15–F’17) received the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 2003. He has been with Purdue University, West Lafayette, IN, USA, since 2003, where he is currently a Professor of electrical and computer engineering, the Deputy Director with the Birck Nanotechnology Center, the Graduate Admissions Director with the School of Electrical and Computer Engineering, and a Faculty Scholar. He has co-authored over 300 journal and conference papers. His current research interests include reconfigurable electronics, RF MEMS, and wireless sensors. He has been a key contributor on developing high-quality reconfigurable filters and filter synthesis techniques based on tunable miniaturized high-Q resonators. He is also leading unique research efforts in high-power multifunctional RF electronics. Dr. Peroulis was a recipient of the National Science Foundation CAREER Award in 2008, the Outstanding Young Engineer Award of the IEEE Microwave Theory and Techniques Society, in 2014, and the Outstanding Paper Award from the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society (Ferroelectrics Section), in 2012. He was also the recipient of ten teaching awards including the 2010 HKN C. Holmes MacDonald Outstanding Teaching Award and the 2010 Charles B. Murphy award, which is Purdue University’s highest undergraduate teaching honor. His students have been the recipients of numerous Student Paper Awards and other student research-based scholarships.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Synthesis-Applied Highly Selective Tunable Dual-Mode BPF With Element-Variable Coupling Matrix Di Lu , Student Member, IEEE, Xiaohong Tang, Member, IEEE, N. Scott Barker, Senior Member, IEEE, Mei Li, Member, IEEE, and Tengfei Yan Abstract— This paper proposes a unified synthesis methodology and the related architecture for fully passband-tunable dual-mode filters with wide tuning range. The methodology and architecture relate the well-developed frequency-fixed synthesis technologies to tunable filters, and can be applied to any tunable filter design and analysis by an element-variable coupling matrix. This general methodology accounts for bandwidth (BW), center frequency (CF), and return loss (RL) (passband ripple or stopband rejection) tuning, and helps to implement the passbandtunable filter. For validation, a novel tunable dual-mode bandpass filter (BPF) based on the architecture is presented. The proposed filter has the ability to arbitrarily construct the twopole passband within the frequency tuning range as predicted. In addition, three transmission zeros (TZs) are generated by the proposed structure, which results in a highly selective passband and a reconfigurable stopband. A tunable filter with a tuning range from 0.8 to 1.2 GHz is designed and fabricated. The measured filter presents an elliptic response during the frequency tuning. The 50% CF tuning range (0.8–1.42 GHz) with a constant absolute BW1 dB and a BW1 dB tuning range from 50 to 500 MHz at the fixed CF are measured. Good agreement among matrix calculation, circuit model simulation and measurement demonstrates the validity of the proposed method and structure. In addition, different states of the stopband with the same passband are obtained to further broaden its application. Index Terms— Bandwidth (BW) tuning, dual-mode resonator, fully tunable bandpass filter (BPF), fully tunable filter synthesis, stopband reconfiguration, three transmission zeros (TZs), TZ tuning.

I. I NTRODUCTION

E

MERGING microwave wireless systems require multiband and multifunctional receiver subsystems to meet

Manuscript received March 20, 2017; revised July 29, 2017 and October 14, 2017; accepted November 29, 2017. This work was supported by the National Natural Science Foundation of China under Grant 61573180. (Corresponding author: Di Lu.) D. Lu is with the EHF Key Laboratory of Science, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China, and also with the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904 USA (e-mail: [email protected]; [email protected]). X. Tang and T. Yan are with the EHF Key Laboratory of Science, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China. N. S. Barker is with the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904 USA. M. Li is with the College of Communication Engineering, Chongqing University, Chongqing 400044, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2783376

the demands of modern telecommunications for various modern communications. To handle such different operational states, a number of filters and other microwave circuits are utilized to form the switched filter banks, resulting in the separated bands and bandwidth (BW) operations with good isolation. Unfortunately, this solution tends to be bulky and lossy. A good way to cope with this problem is to replace the filter banks with low loss and miniaturized electronically tunable components. Hence, a number of tunable filters have been reported using different electronic tuning devices, such as p-i-n diodes [1], [2], varactor diodes [3], [4], yttrium–iron–garnet [5], and RF microelectromechanical systems [6], [7]. Among them, high-selectivity planar filters with tuning elements, especially the tunable filter with the center frequency (CF), BW, and transmission zero (TZ) tuning, plays an irreplaceable role for its easy integration into wireless circuits and good adaptation to various microwave systems. Although many high-selectivity planar tunable filters based on λ/2 or λ/4 resonators have been reported, only a small number of them simultaneously achieve CF, BW, and RL (passband ripple level or stopband rejection) control [8]–[13]. However, their BW tuning range is narrow, and CF, BW, and RL can not be arbitrarily controlled. In contrast with conventional counterparts (based on λ/2 or λ/4 resonators), dual-mode tunable resonators are attractive because each of the dual-mode resonators can be employed as a doubly tuned resonant circuit, and thereby, the number of resonators required for a given degree of filter is reduced by half, thus resulting in a compact filter configuration. In addition, the dual-mode resonator is easier to be electrically tuned since there is no coupling between the two modes. Dual-mode electrically tunable filters were first reported in [14], further implemented with constant BW in [15], and designed via synthesis in [16]. In addition, Tsai et al. [17], [18] presented a CF-and BW-controllable BPF using a loop-shaped dual-mode resonator, and Zhu and Abbosh [19] reported an inner-coupled dual-mode resonator to achieve fully tunable filter. On the other hand, benefiting from the unique property that two modes can be individually controlled without coupling, two coupled dual-mode resonators were employed to realize a dual-band tunable filter [20]. Aside from the dual-mode resonator-based tunable filter, the tri-mode resonator was adopted to design high-selectivity tunable filter, studied in [21], which can provide high-order filtering with better stopband rejection, wider BW3 dB tuning range and higher selectivity. The reported filter

0018-9480 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

exhibited the CF tuning range from 600 to 1450 MHz and BW3 dB tuning range from 120to 950 MHz with two TZs, but the larger circuit size, higher loss and more complex design and control process of the higher order filter limited its applications. On the other hand, with the rapid development of tunable filter techniques, the general approach for tunable filter design by using the coupling matrix or J-/K-inverter is demanded. Many direct and advanced synthesis techniques based on frequency-dependent or frequency-independent coupling matrix or J-/K-inverters have been proposed and deeply studied [22]. However, they have only been applied to the frequency-fixed filter and are difficult to be used for the tunable designs. The biggest challenge is that all elements of the synthesized coupling matrix or J-/K-inverters vary simultaneously with BW tuning and CF tuning, which leads the to each tuning response corresponding to different groups of coupling matrix or J-/K-inverters. Thus, the frequency mapping for the tunable filter is impossible, and the tunable elements cannot be represented by the coupling matrix or J-/K-inverter directly. In addition, independently controlling the individual element in the coupling matrix by a physical structure is still difficult. Moreover, using frequency-fixed coupling matrix to design the tunable filter is inaccurate because the coupling coefficients between two resonant modes are frequency dependent in practice which can be omitted for the frequency-fixed filter design [23], [24] and will become an big error for wide tuning range tunable filter design. To meet such demands, at first, a synthesis methodology with fully element-variable coupling matrix is proposed in this paper to synthesize the passband of the fully tunable dual-mode filter. Benefiting from such a general methodology, the coupling matrix and frequency-fixed filter synthesis can be applied to the tunable filter design, and the CF and BW tuning range can be synthesized by the variable elements of the coupling matrix. The frequency mapping is achievable by specifying the CF of the tuning frequency range, and wide frequency tuning range synthesis is available and more accurate since the coupling coefficients between two resonant modes are zeros. To confirm the validity, second, a new fully tunable dual-mode filter is also presented. The tuning elements of the filter are directly related to the variable elements in the matrix. Thus, a direct analysis of the effect of the element tuning is performed, and the comparison between the theoretical response and model simulation is conducted. Then, the synthesis methodologies for frequency tuning and BW tuning are verified to be practical and reliable by EM simulation. Finally, for further demonstration, the resultant filter is implemented and measured. The measured results show that: 1) The tunable filter possesses the CF tuning range (from 0.8 to 1.42 GHz, i.e., over 50%) and BW1 dB tuning range from 5% to 50% with three TZs, which cover the prescribed CF and BW tuning range. 2) Both stopband BW and rejection level reconfiguration are achieved as predicted. 3) Two self-adaptive TZs allow high-selectivity response with the variation of CF, BW, or different

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 1. Architecture of the proposed high-selectivity tunable dual-mode BPF.

stopband states, as calculated by the proposed structure analysis. 4) All-OFF state is available as predicted by the matrix and circuit model. The good agreement between coupling matrix, transmission line model, and experimental results, as a proof-of-concept, validates the proposed synthesis methodology. The performance comparison with other tunable filters also confirms the superiority of the proposed tunable filtering structure. It is pointed out that, in this paper, the proposed coupling matrix synthesize the tunable passband response. However, the TZs introduced by the proposed structures are not predicted by the synthesis methodology because of the unpredictability and uncontrollability of the source–load coupling and nonresonating mode in the proposed design. II. M ETHODOLOGY A. Coupling Matrix Analysis Fig. 1 shows the proposed topology architecture which consists of a tunable resonator (in dashed box) with two tuning elements and two external quality factor (Q e ) tuning mechanisms. In the resonator, even mode fe , and odd mode f o are independently tunable. The Q e tuning structure is used to control RL (or the passband ripple level or stopband rejection level), thus achieving the source or load impedance match. Suggested in [16], the frequency-fixed response of the dualmode filter architecture can be represented by a general coupling symmetric matrix (1), where Mee and Moo are expressed as Mee/oo = ( fd / f e/o − f e/o / f d )/FBWd , and f d and FBWd are denoted by the frequency mapping elements. f d is the center frequeny, RLd is the return loss (RL), and FBWd is the fractional BW. They are prescribed by the designer as the filter design goals to extract the frequency-fixed filter prototype (e.g., coupling matrix). For simplicity, the symmetrical structure is usually utilized, which leads to the equality of source/load-even/odd mode couplings, MSe/So = MeL/oL . The source/load–even/odd mode couplings can be expressed as (MSe/So )2 = 1/(FBWd × Q exe/exo). Capacitive MSL is placed between input and output ports to introduce TZs [16], which is not included in Fig. 1 because it is typically not tunable ⎡ ⎤ 0 MSe MSo MSL ⎢ MSe Mee 0 MeL ⎥ ⎥. (1) M = ⎢ ⎣ MSo 0 Moo MoL ⎦ MSL MeL MoL 0 It is verified in [16] that when variable Mee/oo and MSe/So (MeL/oL ) are under control within a specified tuning range, all filter passbands can be synthesized by an elementvariable coupling matrix, and the bandpass response can be

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LU et al.: SYNTHESIS-APPLIED HIGHLY SELECTIVE TUNABLE DUAL-MODE BPF

arbitrarily formed in the corresponding tuning range. However, the self-coupling coefficients Mee/oo are dependent on both the f d and FBWd , which leads to the difficulty of the Mee/oo control and physical realization. For instance, different groups of f d and FBWd may result in the same Mee/oo , but most of them can not be used to form the tunable passband. This obstacle also limits the frequency tuning range to a narrow band. In order to better represent the proposed architecture for a wide tuning range, the denormalized coupling matrix is considered for passband synthesis, which is given in the following equations:   m ee m eo x 0 = (2) [m  ] = m oe m oo 0 y (Q exe + Q exo ) Q eSe = Q eLe = Q eSo = −Q eLo = Q e = 2 (3) (4) x = fd / fe − fe / fd (5) y = fd / fo − fo / fd f exe/exo f exe/exo 1 ≈ = . (6) Q exe/exo =  f ±90 FBWexe/exo BWexe/exo As can be seen, all tuning elements of the proposed architecture are directly represented by variable elements of the coupling matrix (2)–(6), and Q e is directly related to the architecture. Also, x, y, and Q e are FBWd -independent. For the frequency-fixed filter, the prescribed CF ( f d ), FBWd , and RLd are used to extract the frequency-fixed coupling matrix, but for tunable filter, f d , RLd , and the tuning r ange based on the proposed architecture can be realized according to the denormalized coupling matrix as follows [22]–[24]: ⎧ 1/Q eSo 1/Q eSe ⎪ ⎪ ⎨ y11 (s) = s + j x − s + j y d √ √d (7) 1/Q eSe · Q eLe 1/Q eSo · Q eLo ⎪ ⎪ ⎩ y21 (s) = − sd + j x sd + j y where sd is low-pass prototype angular frequency (in s domain) and Y0 is the terminal admittance (normalized to 1 for the following calculations). The formula derivation is outlined in the Appendix. It is found from (2)–(7) that any passband of the dualmode filter can be uniquely modeled by x, y, and Q e , when the frequency mapping element fd is given. This is important for CF- and BW-tunable filter design and synthesis, since the value of these parameters do not vary with the prescribed FBWd of the frequency-fixed filter prototype. When the frequency mapping element f d is determined, the one-toone correspondence, between the tunable frequency response and the modeled variable parameters, is predefined, so that these variable parameters can be physically achieved and easily tuned. The only mapping element f d is also defined as the center point of the frequency tuning range, and can be arbitrarily set by the designer. The x, y, and Q e , thereby, can be used to synthesize and control the frequency response independently and directly related to the physical structures. B. Matrix Response Analysis To analyze the tuning behavior, an element-fixed coupling matrix with f d = 1 GHz is modeled by the parameters x, y,

3

Fig. 2.

Theoretical curves for the element-fixed coupling matrix.

Fig. 3.

Theoretical curves for varying x.

Fig. 4.

Theoretical curves for varying y.

Fig. 5.

Theoretical curves for varying Q e .

and Q e at first, and its frequency response is given in Fig. 2. By separately tuning the model parameters (x, y, and Q e ), the even mode, odd mode, and RL of the filter can be adjusted independently, as provided in Figs. 3–5. Figs. 3 and 4 show that when the x and y are adjusted from 0.6 to −0.6, the evenmode and odd-mode frequencies move from 0.75 to 1.25 GHz. Fig. 5 illustrates the calculated frequency response when Q e is tuned from 10 to 55. The impedance matching behavior can be observed, in which the RL varies independently. Thus, when the tuning ranges of x, y, and Q e are specified, the CF, BW, and RL can be arbitrarily configured within a certain tuning

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

Fig. 6. Theoretical curves for tuning the CF with 130-MHz BW3 dB and 20-dB RL.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 9. Resonant structure used in the proposed high-selectivity tunable BPF I. (a) Transmission line circuit. (b) Odd mode. (c) Even mode.

Fig. 7. Theoretical curves for tuning the BW with 1-GHz CF and 20-dB RL.

Fig. 10. (a) Calculated odd-mode resonant frequency versus Co with different Z 1 = 1/Y1 , when θ1 = 25° is used. (b) Calculated even-mode resonant frequency versus Ce with different Z 1 = 1/Y1 , when θ1 = 25°θ2 = 55°, Z 2 = 100 , and Ce2 = 0 pF are used.

Fig. 8. Theoretical curves for the stopband reconfiguration (RL or ripple tuning) with 1-GHz CF and 130-MHz BW3 dB .

range, which results in a CF-, BW-, and RL-tunable (fully tunable) filter. To demonstrate the advantages of the tunable filter design, CF tuning and BW tuning behaviors are performed by purposely changing the variable elements in the matrix, represented by x, y, and Q e . As depicted in Figs. 6 and 7, the CF tuning behavior with a constant 130-MHz BW3 dB and BW-tuning behavior with a fixed 1-GHz CF are performed, respectively. The CF is tuned from 0.8 to 1.2 GHz, when x ranges from 0.058 to −0.028, y ranges from 0.034 to −0.044, and Q e ranges from 18.765 to 27.777. The BW3 dB is tuned from 130 to 300 MHz, when x ranges from 0.095 to 0.2, y ranges from −0.095 to 0.24, and Q e ranges from 10.2 to 27.777. Moreover, the stopband (RL or ripple level) reconfiguration with a fixed CF and BW3 dB can be achieved by tuning Q e . As shown in Fig. 8, the variation of Q e results in the passband ripple level or RL change. The higher the Q e , the sharper rolloff has the passband. Note that, when it is operating for a specified polynomial prototype, there are different conclusions can be drawn. For instance, when it is operating for a Chebyshev polynomial, lowering Q e , the RL tends to be stronger and stopband rejection decreases. In the meanwhile, when it is operating for a Butterworth polynomial, as the Q e decreases, both the RL and stopband rejection of the filter will deteriorate. In addition, all-OFF state (S21 = 0 or S11 = 1) can be obtained by solving (2)–(7), when x is equal to y(x = y) with any Q e , which also provides the theoretical foundation for the switch-OFF state of this type of dual-mode filters.

III. T RANSMISSION L INE M ODEL A. Resonator Analysis To achieve the proposed architecture with the desired element-variable coupling matrix, a new varactor loaded resonator is presented. The transmission line model of the proposed resonator is illustrated in Fig. 9(a), which can be simply treated as a four-variable-capacitor-loaded modified E-shaped resonator. To characterize the resonator, even-odd mode analysis is carried out, as shown in Fig. 9(b) and (c). The odd mode equivalent circuit is a conventional varactor loaded λ/4 resonator. According to the resonance condition Im(Yino) = 0, the resonant frequency ( f o ) can be calculated as Yino = j 2π f o Co − j Y1 cot θ1,d .

(8)

f d is the reference frequency for the transmission line model and is also denoted by a frequency for frequency mapping as well as the center point in the frequency tuning range ( f d = 1 GHz for this design and utilized in the following sections). The even-mode circuit (Z 2 = 1/Y2 ) is shown in Fig. 9(c). Under the similar resonance condition, Im(Yino) = 0, the even mode can be calculated from ⎧ Yine1 = j Y1 tan θ1,d ⎪ ⎪ ⎨ j 2π f Ce + j Y2 tan θ2,d Yine2 = Y2 (9) ⎪ Y2 − 2π f Ce tan θ2,d ⎪ ⎩ Yine = Yine1 + Yine2 + j 2π f Ce2 . The observation implies the odd resonant mode is controlled by Co , and meanwhile, the even resonant mode is controlled by Ce and Ce2 simultaneously. Figs. 10 and 11 show the resonant frequencies f e and f o as a function of different characteristic impedances Z 1 and Z 2 . As shown, as the characteristic

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LU et al.: SYNTHESIS-APPLIED HIGHLY SELECTIVE TUNABLE DUAL-MODE BPF

5

Fig. 13. Calculated odd-mode resonant frequency and TZs versus Co by (6) and (7), when θ1 = 25°, θ2 = 55°, Z 1 = 50 , Z 2 = 100 , Ce = 5 pF, Ce2 = 0.25 pF, and Cm = 2.4 pF are used. Fig. 11. Calculated even-mode resonant frequency versus Ce with different Z 2 = 1/Y2 when θ1 = 25°, θ2 = 55°, Z 1 = 50  and Ce2 = 0 pF are used.

Fig. 14. Calculated odd-mode resonant frequency and TZs versus Co by (6) and (7), when θ1 = 25°, θ2 = 55°, Z 1 = 50 , Z 2 = 100 , Co = 3.4 pF, Ce2 = 2.5 pF and Cm = 1 pF are used.

Fig. 12. Transmission line model of the proposed BPF with taping capacitive transformer. (a) Transmission line circuit. (b) Odd mode. (c) Even mode. (d) Typical coupling topology with its corresponding responses when it is considered as a fixed filter. (NRN stands for nonresonating node.)

impedance Z 1 decreases, the odd-mode frequency increases and the even-mode frequency decreases, while the shape of their frequency-capacitance curve is kept almost unchanged. Compared to the impedance Z 1 , Z 2 has a predominant effect on the even-mode frequency. When Z 2 is increased, the evenmode tuning range moves toward the high-frequency area. As a conclusion, the designer can specify the frequency location of the even and odd mode tuning ranges by controlling the associated characteristic impedances without affecting their frequency tuning range. B. Filter Analysis The Q e tuning structures are realized by the matching varactor-based taping transformer, and applied to the proposed resonator with the parameters θ1 = 25°, Z 1 = 50 , θ2 = 45°, and Z 2 = 50 , so that the predefined tuning frequency from 0.8 to 1.2 GHz can be covered. The transmission line model of the resultant filter is given in Fig. 12(a), where two capacitors Cm with the loading position θt = 2° constitute the Q e tuning structures. The loading position is considered as the total parasitic effect of the loaded varactor with the loading width (small taps) and the very short transmission line between

varactors Cdm and Cdm (will be shown in Fig. 17). The associated odd- and even-mode circuits are obtained by applying the even-odd mode analysis, as shown in Fig. 12(b) and (c). Fig. 12(d) presents the extracted coupling matrix of the frequency-fixed filter according to the proposed transmission line model. It is seen two transmission poles form the passband and three TZs effectively enhance the selectivity and stopband performance. Since the source–load coupling and NRN, generated three TZs, are not controllable with the CF, BW or RL tuning, they are not taken into consideration by the proposed tunable filter synthesis. To characterize the overall transmission line circuit [Fig. 12(a)], the total odd-mode and even-mode input admittances, Ytino , and Ytine are derived as ⎧ 1 ⎪ ⎪ Ytino1 = − j Y1 ⎪ ⎪ tan(θ ⎪ 1,d − θt,d ) ⎪ ⎨ j 2π f Co + j Y1 tan θt,d Ytino2 = Y1 (10) ⎪ Y1 − 2π f Co tan θt,d ⎪ ⎪ ⎪ j 2π f Cm × (Ytino1 + Ytino2) ⎪ ⎪ ⎩Ytino = j 2π f Cm + (Ytino1 + Ytino2) ⎧ j 2π f Ce + j Y2 tan θ2,d ⎪ ⎪ Ytine1 = Y2 ⎪ ⎪ ⎪ Y2 − 2π fe Ce tan θ2,d ⎪ ⎪ ⎪ ⎪ Ytine2 = Ytine1 + j 2π f Ce2 ⎪ ⎪ ⎨ Ytine2 + j Y1 tan(θ1,d − θt,d ) Ytine3 = Y1 (11) ⎪ Y1 + j Ytine2 tan(θ1,d − θt,d ) ⎪ ⎪ ⎪ ⎪ Ytine4 = j Y1 tan θt,d ⎪ ⎪ ⎪ ⎪ (Ytine3 + Ytine4 ) + j 2π f Cm ⎪ ⎪ . ⎩Ytine = (Ytine3 + Ytine4 ) × j 2π f Cm Under the resonance conditions, Ytino = ∞, Ytine = ∞, and Ytino = Ytine odd-mode resonant frequency f o , and the even-mode resonant frequency f e and TZ f z are rigorously calculated by solving (10) and (11). The calculated fo , f e , and f z responses are depicted in Figs. 13–15 by tuning Co , Ce , and Ce2 . As can be shown in Fig. 13, with the increase

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 15. Calculated odd-mode resonant frequency and TZs versus Ce2 by (6) and (7), when θ1 = 25°, θ2 = 55°, Z 1 = 50 , Z 2 = 100 , Co = 8.2 pF, Ce = 5 pF and Cm = 2.4 pF are used.

of Co , fo moves from 1.413 to 0.76 GHz, while TZ2 and TZ3 emerge and then split afterward. The self-adaptive TZ2 shifts with fo within the prescribed tuning range (0.8–1.2 GHz). Meanwhile, f e and TZ1 are kept unchanged. On the other hand, as Ce increases, f e decreases from 1.324 to 0.736 GHz and the self-adaptive TZ1 moves with f e simultaneously, as shown in Fig. 14, while f o is kept unchanged. Fig. 15 illustrates that TZ3 decreases as Ce2 decreases while f o , fe , and the other TZs are almost fixed, which demonstrates that TZ3 is nearly independently controlled by Ce2 as well. Note that, the small impact on f e by tuning TZ3 is positive. In order to maintain a satisfactory selectivity level, reducing TZ3 at low CF or increasing TZ3 at high CF is inevitable, which leads to f e decreases at low frequency and increases at high frequency. Since the tuning frequency range of f o is wider than f e , this impact dynamically extends the CF tuning range. Moreover, specifying TZ3 at desired frequency also help to yield a good rejection level at upper stopband. When Co , Ce , Ce2 , and Cm are designated by the designer, the resultant frequency response can be directly calculated by ⎧ Y0 (Yino − Yine ) ⎪ ⎪ ⎨ S21 = (Y + Y )(Y + Y ) 0 ine 0 ino (12) 2−Y Y ⎪ Y ine ino ⎪ 0 ⎩ S11 = . (Y0 + Yine )(Y0 + Yino ) C. External Q (Qe ) For this type of tunable filter, the tapping transformer based on the matching varactor, Cm , is employed. By adding the tuning capacitor Cm between the resonator and I/O port, as shown in Fig. 12(a), the coupling strength can be controlled resulting in a tunable Q e . The observation of (10) and (11) implies the odd- and even-mode input impedances vary with Ce and Co , but are mainly determined by Cm . This is because the value of the odd- and even-mode input impedances remains much larger than the other term, i.e., j 2π f Cm . With different Ce and Co , the designer can, herein, obtain different curves of the odd-mode external quality factor (Q exe /Q exo ) by changing Cm . For verification, Q exo and Q exe are extracted using the formula (13), as shown in Fig. 16. It is observed that the Q exo and Q exe are controlled by Cm , and their tuning ranges are fairly broad ∂[ϕs11( f e/o )] f e/o Q exe/exo = − × . (13) 2 ∂f Q exe /Q exo also evaluates coupling energy to even-/odd-mode resonators. For instance, when Q exe /Q exo increases within an

Fig. 16. Extracted Q exo and Q exe with different frequencies (Z 1 = 50 , θ1 = 25°, θ2 = 55°, and Z 2 = 100 ).

Fig. 17.

Basic configuration of the proposed filter.

acceptable range, the ripple level will be changed, the rejection level of the filter will increase, and the selectivity of the passband will be improved. This phenomenon is also predicted by the matrix response in Fig. 8, where Q e = (Q exe + Q exo )/2. IV. R EALIZATION A NALYSIS AND E LEMENT D ETERMINATION A. Realization Analysis In order to implement the circuit using the same type varactor, SMV1234 (Ct = 1.2–10 pF, Rs = 0.80 , L s = 0.7 nH), the physical structure is designed, as shown in Fig. 17. Z 1 = 50 , Z 2 = 100 , θ1 = 25°, and θ2 = 55° (at f d = 1 GHz) are chosen as the design parameters so that the odd-mode tuning can cover the even-mode tuning range to achieve the widest CF and BW tuning range reconfiguration with the 1-GHz center tuning frequency ( f d = 1 GHz). The calculated even-mode and odd-mode tuning ranges, as shown in Figs. 13 and 14, are 0.736–1.324 GHz (Ce = 5–0.5 pF) and 0.76–1.413 GHz (Co = 10–1 pF). In the coupling matrix (1) with −15 dB RL, 10% FBWd , and 1-GHz CF, the associated m ee (x) and m oo (y) tuning ranges are 0.622−0.57 and 0.555−0.705, respectively. The individual external Q exe and Q exo tuning ranges are extracted in Fig. 16. They are 4.62–24 and 1.7–79.8, respectively. The total external Q e tuning range can be calculated as Q e = (Q exe + Q exo )/2 [24], which is 3.16–51.7. The corresponding coefficients are m Se/eL = m So/oL = 0.44–0.126 ( f d = 1 GHz, FBWd = 10%, RL = −15 dB). Thus, the proposed tunable

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LU et al.: SYNTHESIS-APPLIED HIGHLY SELECTIVE TUNABLE DUAL-MODE BPF

7

TABLE I D IMESIONS FOR FABRICATED F ILTER , R EFER TO F IG . 17

Fig. 18. CF tuning response comparison between transmission line model and coupling matrix model.

Fig. 19. BW tuning response comparison between transmission line model and coupling matrix model.

Fig. 20. Stopband (RL or ripple)-tuning response comparison between transmission line model and coupling matrix model.

filter is represented by the coupling matrix (17). (Re /Ro indicates the even-/odd-mode resonator) [m  ] = Re Ro

Re x 0

Ro 0 , y

f d = 1 GHz −0.57 < x < 0.622 −0.705 < y < 0.555 3.16 < Q e < 51.7.

(14)

As discussed in Section II, any two-pole passband within the two mode tuning range can be configured by manipulating the elements in the matrix, represented by x, y, and Q e . For illustration, the response comparison between coupling matrix and transmission model for CF tuning, BW tuning, and RL-(stopband or ripple) reconfiguration are presented in Figs. 18–20. The minimum BW1 dB is 20 MHz at 750 MHz and the maxmum BW1 dB is 690 MHz at 1.075 GHz. The center freqeuncy tuning range is from 780 to 1370 MHz

(at 100-MHz BW1 dB ) and from 880 to 1270 MHz (at 300-MHz BW1 dB ) It is evident that a good matching between the synthesized matrix and transmission line model in the tunable passbands is achieved by tuning the associated capacitors, which implies that the tunable passband of the proposed filter is synthesized by the element-variable coupling matrix. For instance, for the CF increasing with a constant absolute BW (ABW), x and y are decreased proportionally by reducing Ce and Co , while Q e is decreased by increasing Cm . Fig. 18 presents the 0.8–1.2 GHz CF tuning behavior with 130-MHz ABW3 dB . Similarly, for the BW3 dB increasing, x and y are tuned in the inverse direction by increasing Ce and decreasing Co , while Q e is decreased by increasing Cm . Frequency responses of the 130–300 MHz BW3 dB tuning with 1-GHz CF is given in Fig. 19. RL (stopband rejection and ripple level in passband) is mainly controlled by Q e (Cm ). For RL tuning, the little CF shift due to the Cm change is compensated by fine adjusting of Ce and Co . The typical RL tuning response is shown in Fig. 20, where the rejection level (roll-off) and RL reconfiguration behavior is evident. The discrepancy between the matrix and transmission line model on the skirts of the passband in Figs. 18–20 is caused by the TZs which are generated by the proposed resonator only. Thus, the tunable passband of the filter is synthesized by the element-variable coupling matrix. The overall filter is designed using Ansoft-HFSS and Agilent-ADS. The final filter dimensions after optimization between HFSS and ADS are presented in Table I. The circuit substrate is RT/Duroid 5880 substrate with εr = 2.2 and h = 0.508 mm. All the variable capacitors are the models of the silicon hyperabrupt junction varactor diodes SMV1234. The lumped capacitors are 0402 ceramic capacitors, Cm1 = 10 pF, and the resistors are 0402 chip resistors, R = 10 k. The tuning elements Ce , Ce2 , and Co (SMV1234) in Fig. 12, are realized using the back-to-back diode configuration due to the ease of biasing at the common node. Two lumped capacitors are used to block dc, and meanwhile, connect in series with the varactors to form Cm . As shown in Fig. 17, Cdm , Cdo , Cde , and Cde2 denote the equivalent diode capacitors in the practical filter. To validate the tuning behavior in correspondence with the theory analysis, the full wave simulation analysis is carried out. As depicted in Fig. 21, the even mode, odd mode, and TZ3 can be separately tuned by changing the associated capacitors, and the tuning behaviors are the same as the matrix and transmission model analysis. The even mode and f z1 , as well as the odd mode and f z2 , are tuned synchronously resulting in a highly selective passband during the frequency tuning process. Also depicted in Fig. 21, as the capacitor Ce2 increases, the TZ3 ( f z3 ) shifts toward the low-frequency area,

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 23. Fig. 21. EM simulated results for only varying Ce , Co , and Ce2 using weak coupling method.

Fig. 22. EM simulated filter response for only changing Cm to reconfigure rejection level.

thus leading to the upper stopband narrower and its rejection stronger. As shown in Fig. 22, tuning Cm within a certain range reconfigures the stopband rejection in both upper and lower stopband. It indicates the stopband rejection can be increased/decreased by decreasing/increasing the passband ripple or RL when the CF and BW3 dB are fixed. Thus, the Q e (stopband rejection RL or passband ripple) is independently controlled by Cm . This result is also in agreement with the theoretical analysis in Section III. B. Determination of Tuning Elements Based on the analysis discussed above, tuning method of the proposed filter can be summarized as follows. 1) For the given m ee and m oo , tune Ce and Co to achieve the desired x and y, respectively. Meanwhile, the nonindependent source–load coupling m S L is determined by Co as well. 2) Determine Q exe and Q exo by tuning Cm to implement the designed Q e . Thus, the matrix-based passband is realized.

Filter tuning procedure of the proposed filter.

3) Fine-tune Cm to yield the desired lower stopband rejection (passband ripple level or RL). 4) Tune Ce2 to determine the desired upper stopband rejection and upper stopband BW. 5) Finally, fine tune f e and f o , to compensate the BW and CF shift caused by stopband reconfiguration. Fig. 23 presents the corresponding flowchart for determination of the tuning elements based on the general coupling matrix and frequency mapping element, fd . Note that, f e need to be set lower than f o (x > y) to maintain the elliptic response. When f e is higher than f o , all TZs will disappear which has been discussed in [16] and [17] and is not repeated in this paper. V. M EASUREMENT The fabricated filter is shown in Fig. 24(a). By applying different voltages to all varactors, passband’s CF, RL, and BW along with stopband can be controlled as predicted by the element-variable coupling matrix and transmission model. The measured S-parameters in comparison with the EM simulation results and coupling matrix responses are shown in Fig. 24(a) and (b), when only the CF (from 0.8 to 1.2 GHz) or BW (from 130 to 400 MHz) is tuned. The measurements are in good in-band agreement with the matrix and EM model. The measured out-band responses also match well with EM simulation. In order to demonstrate its flexibility, several typical operation modes are measured and discussed. A. Center Frequency Tuning With Constant 1-dB Absolute BW (ABW) First, Ce2 needs to be kept at the minimum value to get a wide stopband. Then, change Co and Ce proportionally, based on x and y in the coupling matrix, to obtain a constant BW. Finally, tune Cm to match the I/O impedance according to calculated Q ex , and tune Ce2 to approach the desired stopband.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LU et al.: SYNTHESIS-APPLIED HIGHLY SELECTIVE TUNABLE DUAL-MODE BPF

9

Fig. 25. Measured filter S-parameters for filter with (a) 90-MHz ABW1 dB with 25-dB stopband rejection and (b) 90-MHz ABW1 dB and 20-dB stopband rejection.

Fig. 24. f d = 1-GHz configuration of the proposed filter. (a) Coupling matrix, EM simulated, and measured results with x = 0.51, y = 0.27, Q e = 20.4, Cdm = 3 pF, Cdo = 5.6 pF, Cde = 9.1 pF, Cde2 = 5.6 pF (low frequency) and x = −0.29, y = −0.44, Q e = 24.4, Cdm = 1.35 pF, Cdo = 2.55 pF, Cde = 2.95 pF, Cde2 = 1.6 pF (high frequency). (b) x = 0.095, y = −0.095, Q e = 23.67, Cdm = 1.9 pF, Cdo = 4.54 pF, Cde = 5.94 pF, Cde2 = 4.85 pF (narrow BW) and x = 0.367, y = −0.28, Q e = 8, Cdm = 3.3 pF, Cdo = 3.39 pF, Cde = 8.58 pF, Cde2 = 2.25 pF (wide BW).

As a demonstration, the measured S-parameter curves are given and depicted in Fig. 25(a). As shown, the CF of the measured filter varies from 0.8 to 1.21 GHz with a constant 90-MHz ABW1 dB (BW3 dB ≈ 130 MHz), and 25-dB stopband S21 is kept over the stopband. The insertion loss (IL) is less than 3 dB, while three TZs results a highly selective passband over the tuning range. The measurement experimentally verifies the analysis discussed before. Additionally, to fully exploit the used varactors, the tuning range of the filter with a 20-dB stopband rejection is measured and shown in Fig. 25(b). It is observed, when all varactors SMV1234 are fully tuned, that the CF varies from 0.8 to 1.42 GHz with 90-MHz ABW1 dB , and a highly selective passband and good stopband rejection (S21 over 20 dB) are kept over the entire tuning range.

Fig. 26.

Measured BW tuning from 50 to 500 MHz of the proposed filter.

B. 1-dB BW Tuning With Fixed Center Frequency The proposed filter can also operate in BW tuning mode, in which only the BW is tuned while the CF is kept unchanged. Fig. 26 shows the measured results for the filter in BW tuning mode. When CF is fixed at 1 GHz, the BW1 dB can be tuned from 50 to 500 MHz and the IL changes from 2.8 to 0.8 dB. Consistent with the prediction in Section IV, the BW1 dB can change from 50 to 300 MHz while the high selectivity is kept due to the three TZs. The maximum BW of the filter can

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

Fig. 27. Measured stopband reconfiguration of the filter. (a) Different upper stopbands. (b) Different both stopband rejections.

achieve is 500 MHz, when the tuning elements are fully tuned, which is plotted in a magenta dashed line in Fig. 26. C. Filter Stopband Reconfiguration at Fixed Center Frequency Aside from the synthesized tunable response extracted from the coupling matrix, TZ tuning can be implemented by controlling the on-load voltages of the filter as discussed in Section IV. While keeping Co , Ce , and Cm unchanged, the TZ3 in the upper stopband can be separately tuned by tuning Ce2 . The measured results are depicted in Fig. 27(a). It can be seen that by reducing Ce2 , the BW of the upper stopband is widened from 130 MHz (30 dB) to 800 MHz (10 dB), but the rejection level (S21 ) is decreased from better than 30 dB to about 10 dB, when CF and BW1 dB of the passband are set to 0.86 GHz and 140 MHz, respectively. It is noted that the stopband tuning behavior affects the CF and BW a little (about 10 MHz). Nevertheless, these influences can be compensated by slightly changing Co , Ce , and Cm , which is also in accordance with the transmission line model analysis in Section IV. On the other hand, through tuning passband ripple level (or RL) and TZ3, controlled by Cm , and Ce2 , both BW and rejection level of the stopband can be adjusted. As can be seen from the measured results of the whole stopband reconfiguration [Fig. 27(b)], the S21 of the lower and upper stopband

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 28.

Measured all- OFF state of the filter.

Fig. 29.

Measured IIP3 of the filter.

is adjusted from 35 to 15 dB, when the CF and BW1 dB are fixed at 1 GHz and 100 MHz, respectively. Meanwhile, the symmetrical quasi-elliptic response, high selectivity and three finite TZs can be observed clearly. The ILs are 2.7 dB at 35-dB rejection level (Chebyshev in-band response), 2 dB at 25-dB rejection level (Butterworth in-band response), and 2.5 dB at 20-dB rejection level (Butterworth in-band response). The variation tendency of the IL, RL, and stopband change are in reasonable agreement with the prediction of the coupling matrix and transmission line model. D. Filter With All-Off State Operation Apart from the BPF application, the presented filter can allow the all-OFF state operation, which is also predicted by the coupling matrix, discussed in Section II. By appropriately setting Co and Ce , matrix elements x = y is fulfilled, and the pole locations f e ≈ f o is achieved, thus leading to an all-OFF state. By choosing the appropriate Q e and tunable TZ location, a good rejection level and BW of the stopband are obtained. The measured results are shown in Fig. 28. The S21 is maintained less than −20 dB. There is a peak of S21 (around 1.3 GHz), which is mainly because two resonant modes are not rejected by TZs completely in order to yield the better S21 rejection over the tuning range. E. Linearity and Power Handling Fig. 29 presents the measured input third-order intercept point (IIP3) with different CF and BW1 dB , when the two-tone signal, spaced by 2 MHz, is used for the IIP3 test. As seen,

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LU et al.: SYNTHESIS-APPLIED HIGHLY SELECTIVE TUNABLE DUAL-MODE BPF

11

TABLE II P ERFORMANCE C OMPARISONS OF M EASURED R ESULTS W ITH O THER T UNABLE F ILTERS

the minimum IIP3 of the filter is 17 dBm. The IIP3 increases as BW1 dB and CF increases, and is mainly dominated by BW1 dB . As suggested in [25], the 1-dB compression point at does not truly represent the large-signal performance of the filter response. However, the 8-dBm input power level is also used for all the filter performance measurement without a considerable distortion. Thereby, the input power handing level of the proposed filter is at least 8 dBm. F. Filter Performance Comparison Table II lists the performance comparisons with other tunable filters. It is found that the proposed work realizes the stopband rejection level/BW control and three TZs, while keeping a wide CF and BW tuning range as well as good IL performance. It has to be admitted that the tradeoff between the strong rejection and wide BW of the upper stopband is needed to be considered for the proposed filter design, due to the wide frequency tuning range and TZ control. However, this shortcoming can be solved by simply introducing the defected ground structure to extend the stopband significantly without a considerable size and IL increase, such as [26] and [27]. VI. C ONCLUSION This paper has demonstrated a general synthesis methodology and the related architecture for the wide tuning range fully tunable dual-mode filter design and analysis. The fully tunable dual-mode filter is modeled and can be synthesized by an element-variable coupling matrix. By adjusting the matrix elements, the CF, BW, and RL are independently tuned resulting in a fully tunable passband. A new microstrip tunable filter has been proposed to validate the synthesis methodology. The tuning elements in the proposed filter are directly related to the variable elements in the coupling matrix, and the tuning behaviors of the matrix and filter have been analyzed. Response comparison between the matrix and the filter has verified the feasibility. In addition,

the inherent advantage of the proposed filter such as high selectivity and tunable TZ in the filter response has been exhibited and investigated. Finally, a fully tunable dual-mode filter example has been fabricated and measured, and a wide tuning range (800–1420 MHz) has been obtained which has covered the prescribed frequency tuning range. The CF tuning range from 800 to 1420 MHz with a constant BW (90-MHz BW1 dB or around 130-MHz BW3 dB ) and BW tuning range from 50 to 500 MHz with 1-GHz CF have been given to demonstrate the validity of the proposed synthesis methodology. The all-OFF state and reconfigurable stopband, in a good agreement with coupling matrix and simulation analysis, have been presented, and further exhibited the broad development perspective of the proposed methodology and filter. A PPENDIX We derive the synthesis formula for the multimode tunable filter herein, from the conventional coupling matrix model. The formula derivation is given as follows. For the lossless (N + 2) × (N + 2) and multimode resonator-based (or transversal filter structure based) symmetrical coupling matrix model (without mutual coupling) given in [22] and [28] with the same termination admittance Y0 , we can derive the Y-parameter as ⎡ N ⎤ N   y11k y12k ⎥ ⎢ ⎢ k=1 ⎥ k=1 ⎢ ⎥ [ys ] = ⎢ N ⎥ N  ⎣ ⎦ y21k y22k ⎡

k=1 N 

k=1 2 M Sk

⎢ ⎢ k=1 s + j Mkk =⎢ ⎢ N M Sk M Lk ⎣ k=1

s + j Mkk

⎤ N  M Sk M Lk ⎥ s + j Mkk ⎥ k=1 ⎥ ⎥ N 2  M Lk ⎦ s + j Mkk k=1

(A1)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 12

where

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

⎧ (M Sk/Lk )2 = 1/(FBWd · Q eSk/Lk ) ⎪ ⎪ ⎨ Mkk =( f d / f kk −f kk / f d )/FBWd ⎪ f fd ⎪ ⎩s = j − /FBWd . fd f

(A2)

s is normalized low-pass prototype angular frequency (in s domain). By manipulating (A1) and (A2), we can obtain ⎡ N ⎤ N   m 2Sk m Sk m Lk ⎢ ⎥ s + j m kk ⎥ ⎢ k=1 sd + j m kk k=1 ⎢ ⎥ (A3) [ys ] = ⎢ N ⎥ N  m 2Lk ⎣  m Sk m Lk ⎦ sd + j m kk sd + j m kk k=1

where

k=1

⎧ (m Sk/Lk )2 = 1/(Q eSk/eLk ) ⎪ ⎪ ⎨ ( f d / f kk −  f kk / f d ) m kk =  ⎪ f fd ⎪ ⎩sd = j . − fd f

(A4)

sd is actual low-pass prototype angular frequency. m Sk/Lk denotes the denormalized coupling coefficients between kth resonating node and the source or load. R EFERENCES [1] J.-X. Xu and X. Y. Zhang, “Single- and dual-band LTCC filtering switch with high isolation based on coupling control,” IEEE Trans. Ind. Electron., vol. 64, no. 4, pp. 3137–3146, Apr. 2017. [2] S.-C. Weng, K.-W. Hsu, and W.-H. Tu, “Compact and switchable dualband bandpass filter with high selectivity and wide stopband,” Electron. Lett., vol. 49, no. 20, pp. 1275–1277, Sep. 2013. [3] J.-X. Chen, Y. Ma, J. Cai, L.-H. Zhou, Z.-H. Bao, and W. Che, “Novel frequency-agile bandpass filter with wide tuning range and spurious suppression,” IEEE Trans. Ind. Electron., vol. 62, no. 10, pp. 6428–6435, Oct. 2015. [4] J. Xu and Y. Zhu, “Tunable bandpass filter using a switched tunable diplexer technique,” IEEE Trans. Ind. Electron., vol. 64, no. 4, pp. 3118–3126, Apr. 2017. [5] R. L. Fjerstad, “Some design considerations and realizations of iriscoupled YIG-tuned filters in the 12–40 GHz region,” IEEE Trans. Microw. Theory Techn., vol. MTT-18, no. 4, pp. 205–212, Apr. 1970. [6] M. M. Shalaby, M. A. Abdelmoneum, and K. Saitou, “Design of spring coupling for high-Q high-frequency MEMS filters for wireless applications,” IEEE Trans. Ind. Electron., vol. 56, no. 4, pp. 1022–1030, Apr. 2009. [7] K. Ma, R. M. Jayasuriya, and D. R. L. C. Siong, “Fully integrated highisolation low-loss digitally controlled MEMS filters,” IEEE Trans. Ind. Electron., vol. 58, no. 7, pp. 2690–2696, Jul. 2011. [8] J. Long, C. Li, W. Cui, J. Huangfu, and L. Ran, “A tunable microstrip bandpass filter with two independently adjustable transmission zeros,” IEEE Microw. Wireless Compon. Lett., vol. 21, no. 2, pp. 74–76, Feb. 2011. [9] Y.-C. Chiou and G. M. Rebeiz, “A tunable three-pole 1.5–2.2-GHz bandpass filter with bandwidth and transmission zero control,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 11, pp. 2872–2878, Nov. 2011. [10] D. Psychogiou, R. Gómez-García, and D. Peroulis, “Fully-reconfigurable bandpass/bandstop filters and their coupling-matrix representation,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 1, pp. 22–24, Jan. 2016. [11] T. Yang and G. M. Rebeiz, “Tunable 1.25–2.1-GHz 4-pole bandpass filter with intrinsic transmission zero tuning,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 5, pp. 1569–1578, May 2015. [12] P.-L. Chi, T. Yang, and T.-Y. Tsai, “A fully tunable two-pole bandpass filter,” IEEE Microw. Wireless Compon. Lett., vol. 25, no. 5, pp. 292–294, May 2015. [13] A. Anand and X. Liu, “Reconfigurable planar capacitive coupling in substrate-integrated coaxial-cavity filters,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 8, pp. 2548–2560, Aug. 2016.

[14] W. Tang, J.-S. Hong, and Y.-H. Chun, “Compact tunable microstrip bandpass filters with asymmetrical frequency response,” in Proc. 38th Eur. Microw. Conf. (EuMC), Oct. 2008, pp. 599–602. [15] W. Tang and J.-S. Hong, “Varactor-tuned dual-mode bandpass filters,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 8, pp. 2213–2219, Aug. 2010. [16] A. L. C. Serrano, F. S. Correra, T.-P. Vuong, and P. Ferrari, “Synthesis methodology applied to a tunable patch filter with independent frequency and bandwidth control,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 3, pp. 484–493, Mar. 2012. [17] H.-J. Tsai, N.-W. Chen, and S.-K. Jeng, “Center frequency and bandwidth controllable microstrip bandpass filter design using loop-shaped dual-mode resonator,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 10, pp. 3590–3600, Oct. 2013. [18] H.-J. Tsai, B.-C. Huang, N.-W. Chen, and S.-K. Jeng, “A reconfigurable bandpass filter based on a varactor-perturbed, T-shaped dualmode resonator,” IEEE Microw. Wireless Compon. Lett., vol. 24, no. 5, pp. 297–299, May 2014. [19] H. Zhu and A. Abbosh, “Compact tunable bandpass filter with wide tuning range using ring resonator and short-ended coupled lines,” Electron. Lett., vol. 51, no. 7, pp. 568–570, Apr. 2015. [20] X. Huang, L. Zhu, Q. Feng, Q. Xiang, and D. Jia, “Tunable bandpass filter with independently controllable dual passbands,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 9, pp. 3200–3208, Sep. 2013. [21] J.-R. Mao, W.-W. Choi, K.-W. Tam, W. Q. Che, and Q. Xue, “Tunable bandpass filter design based on external quality factor tuning and multiple mode resonators for wideband applications,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 7, pp. 2574–2584, Jul. 2013. [22] R. J. Cameron, R. Mansour, and C. M. Kudsia, Microwave Filters for Communication Systems: Fundamentals, Design and Applications. Hoboken, NJ, USA: Wiley, 2007. [23] A. E. Atia and A. E. Williams, “New types of waveguide bandpass filters for satellite transponders,” Comsat Tech. Rev., vol. 1, no. 1, pp. 1–117, 1971. [24] J.-S. Hong, Microstrip Filters for RF/Microwave Applications. New York, NY, USA: Wiley, 2001. [25] M. A. El-Tanani and G. M. Rebeiz, “A two-pole two-zero tunable filter with improved linearity,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 4, pp. 830–839, Apr. 2009. [26] G. Chaudhary, Y. Jeong, and J. Lim, “Harmonic suppressed dual-band bandpass filters with tunable passbands,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 7, pp. 2115–2123, Jul. 2012. [27] G. Chaudhary, Y. Jeong, and J. Lim, “Dual-band bandpass filter with independently tunable center frequencies and bandwidths,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 107–116, Jan. 2013. [28] R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microw. Theory Techn., vol. 51, no. 1, pp. 1–10, Jan. 2003.

Di Lu (S’14) was born in Kunming, China, in 1987. He received the M.S. degree from the Electronic Engineering School, Chengdu University of Information and Technology, Chengdu, China, in 2013. He is currently pursuing the Ph.D. degree at the University of Electronic Science and Technology of China, Chengdu. Since 2015, he has been a Visiting Student with the University of Virginia. His current research interests include design microwave filters, tunable filters, frequency multipliers, mixers, millimeter-wave circuits, and RF microelectromechanical systems. Xiaohong Tang (M’08) received the B.S. and Ph.D. degrees in electromagnetism and microwave technology from the University of Electronic Science and Technology of China (UESTC), Chengdu, China. He is currently a Professor with UESTC. He has authored over 100 journal and conference papers. His current research interests include microwave and millimeter-wave communication and computational electromagnetics. Dr. Tang was a recipient of several national and provincial awards.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LU et al.: SYNTHESIS-APPLIED HIGHLY SELECTIVE TUNABLE DUAL-MODE BPF

N. Scott Barker (S’94–M’99–SM’13) received the B.S.E.E. degree from the University of Virginia, Charlottesville, VA, USA, in 1994, and the M.S.E.E. and Ph.D. degrees in electrical engineering from the University of Michigan at Ann Arbor, Ann Arbor, MI, USA, in 1996 and 1999, respectively. From 1999 to 2000, he was a Staff Scientist with the Naval Research Laboratory, Washington, DC, USA. In 2001, he joined the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, where he is currently a Professor. He recently co-started the company Dominion Micro Probes Inc., Charlottesville, to develop the terahertz frequency wafer probe technology co-invented by his group at the University of Virginia. He has authored or co-authored over 60 publications. His current research interests include applying microelectromechanical systems and micromachining techniques to the development of millimeter-wave and terahertz circuits and components. Prof. Barker was a recipient of the Charles L. Brown Department of Electrical and Computer Engineering New Faculty Teaching Award 2006, the Faculty Innovation Award 2004, the 2003 National Science Foundation CAREER Award, the 2000 IEEE Microwave Prize, and First and Second Place of the Student Paper Competition of the IEEE MTT-S IMS. He has been served on the MTT-21 Technical Committee on RF-MEMS since 2000 and was the Committee Chair from 2008 to 2011. He has also served for many years on the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium Technical Program Review Committee. In 2011, he served on the Steering Committee, IEEE MTT-S IMS, Baltimore, MD, USA. He was the Technical Program Committee Vice Chair of the 2014 IEEE MTT-S IMS, Tampa, FL, USA. He was an Associate Editor of IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS 2008–2010. He is currently an Associate Editor of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES .

13

Mei Li (M’16) received the Ph.D. degree in radio physics from the University of Electronic Science and Technology of China, Chengdu, China in 2016. From 2014 to 2016, she was a Visiting Graduate with the Applied Electromagnetics Research Group, University of California at San Diego, La Jolla, CA, USA. She is currently with Chongqing University, Chongqing, China. Her current research interests include metasurfaces, antennas, arrays, and tunable devices.

Tengfei Yan is currently pursuing the Ph.D. degree with the Institute of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, China. He is currently a Researcher with the Shenzhen Key Laboratory of Millimeter-Wave and Wideband Wireless Communications, City University of Hong Kong Shenzhen Research Institute, Shenzhen, China.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

1817

Stability and Bifurcation Analysis of Multi-Element Non-Foster Networks Almudena Suárez , Fellow, IEEE, and Franco Ramírez , Senior Member, IEEE Abstract— A stability and bifurcation analysis of multi-element non-Foster networks is presented, illustrated through its application to non-Foster transmission lines. These are obtained by periodically loading a passive transmission line with negative capacitors, implemented with negative-impedance converters (NICs). The methodology takes advantage of the possibility to perform a stability analysis per subintervals of the perturbation frequency. This will allow an independent analytical study of the low-frequency instability, from which simple mathematical criteria will be derived to prevent bias-network instabilities at the design stage. Then, a general numerical method, based on a combination of the Nyquist criterion with a pole-zero identification of the individual NIC, will be presented, which will enable the detection of both low- and high-frequency instabilities. A bifurcation analysis of the multi-element non-Foster structure will also be carried out, deriving the bifurcation condition from a matrix-form formulation of the multi-element structure. The judicious choice of the observation ports will enable a direct calculation of all the coexisting bifurcation loci, with no need for continuation procedures. These bifurcation loci will provide useful insight into the global-stability properties of the whole NIC-loaded structure. Index Terms— Bifurcation, non-Foster network, non-Foster transmission line, stability.

I. I NTRODUCTION

N

ON-FOSTER circuits [1]–[6] are based on the use of negative capacitors or inductors, implemented through transistor-based negative-impedance converters (NICs) [1]–[3], [7], [8]. The non-Foster circuits have been applied for broadband impedance matching of electrically small antennas [1], [2], [5], [9], [10] and other system components. They also enable the realization of fast-wave transmission lines [6], [11]–[13], which allow the control of the phase delay of guided-wave devices, applied on squint-free beamforming of antenna arrays [13]–[16]. Other interesting metamaterial realizations can be found in [17]–[20]. The fast-wave propagation is achieved by periodically loading [6], [11]–[14] the transmission line with negative capacitors, implemented with

Manuscript received July 5, 2017; revised October 27, 2017; accepted December 25, 2017. Date of publication February 1, 2018; date of current version April 3, 2018. This work was supported in part by the Spanish Ministry of Economy and Competitiveness and in part by the European Regional Development Fund (ERDF/FEDER) under Project TEC2014-60283-C3-1-R and Project TEC2017-88242-C3-1-R. This paper is an expanded version from the 2017 IEEE MTT-S International Microwave Symposium Conference, Honolulu, HI, USA, June 4–9, 2017. (Corresponding author: Franco Ramírez.) The authors are with the Departamento de Ingeniería de Comunicaciones, Universidad de Cantabria, 39005 Santander, Spain (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2018.2793893

transistor-based configurations. However, a problem inherent to circuits based on non-Foster elements is the likelihood of unstable behavior, since any NIC is a potentially unstable twoport network [1]–[6]. Using the NIC h-parameter matrix [3], it is shown that any practical two-port NIC is always shortcircuit stable (SCS) and open-circuit unstable at one port (i.e., when terminating that port in a short-circuit and an open circuit, respectively), and open-circuit stable and short-circuit unstable at the other port. In fact, the great potentiality of non-Foster circuits may be severely limited by their unstable behavior. Several previous works have addressed the stability analysis of non-Foster circuits [3]–[5], [17], [21]–[24]. Some of them rely on simplified equivalent networks of the loaded NIC [2], represented as a negative capacitor (or inductor), which is unrealistic. When considering the transistor-based NIC, most of the stability analyses presented are based on time-domain integration, applying a Gaussian pulse [5], or on the Nyquist criterion [4], [22]. The time-domain analysis may be unreliable, since transients can be very long in NIC circuits, as demonstrated in [25]. On the other hand, when using the Nyquist criterion, care must be taken in the selection of the complex function to which the Nyquist criterion is applied. This function should not contain any poles of the right-hand side (RHS) of the complex plane [26]–[29]. This is because the number Ncir of clockwise encirclements of the Nyquist plot around the origin provides the difference between the number N Z of RHS function zeroes (agreeing with the number of RHS system poles) and the number N P of RHS function poles, i.e., Ncir = N Z − N P [26]. Thus, unless N P = 0, the result will be inconclusive. The condition N P = 0 is ensured through the use of the normalized determinant function (NDF) [28], [29]. The NDF is calculated at the intrinsic device terminals. In this way, all the possible feedback effects, including those resulting from parasitic elements inside the device models [30], are taken into account. If the analysis is not performed at the intrinsic terminals, the determinant may exhibit RHS poles, resulting from the interaction of the passive and purely active elements. In the case of black-box models, circuit-simulator models may be used. However, this may be problematic if the original black box includes multiple reactive effects, since the analysis must be performed in a large frequency interval, only limited by the device filtering effects. In this paper, we propose the use of pole-zero identification [31]–[33] to ensure the stable behavior of the active components or blocks. This method relies on the fact that all the transfer functions [31]–[33] defined in the same linear system share the same denominator. It is based on the fitting of a transfer function with a quotient of polynomials, which

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

1818

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

provides the zeroes and poles with good accuracy. Thus, it is not necessary to calculate the transfer functions at the intrinsic device terminals, so the method is fully compatible with the use of black-box models for the active devices. Nevertheless, exact pole-zero cancellations may occur in a particular transfer function, so two or more functions must be generally tested. Pole-zero identification [31]–[33] has recently been applied to non-Foster circuits [22], presenting pole-zero maps for particular circuit-component values, without a detailed investigation of the causes for the instability phenomena. The recent work [25] presents an in-depth stability analysis of a typical non-Foster matching network, applicable for antenna matching. Using analytical derivations as well as polezero identification, two different instability mechanisms are detected. The first mechanism is associated with the presence of real poles of low magnitude, caused by feedback effects inherent to the NIC transistor configuration. The second is associated with poles of higher magnitude, detectable in a higher frequency interval. This instability was found to be due to the dominance of the negated impedance over the circuit passive-loading effects, in a manner similar to that in [21] and [34]. However, the analytical formulations and numerical methodologies presented in [25] were restricted to a nonFoster circuit based on a single NIC. The aim of this paper is to extend the investigation to networks containing multiple NICs, as in the case of fast-wave transmission lines [6], [11]–[15], which are periodically loaded with NIC elements. Several procedures will be presented with two different objectives: getting insight into the instability mechanisms and providing useful methodologies to predict and suppress instability at the design stage. The procedures take advantage of the possibility to perform the stability analysis per subintervals of the perturbation frequency, as is usually done in the numerical pole-zero identification [31]–[33]. This will allow distinguishing the mechanism for low-frequency instabilities from other instabilities detectable at higher frequencies. In fact, the transistor cross-coupled configuration is prone to give rise to unstable real poles [25], which can be predicted with an analytical formulation. The formulation will enable a convenient derivation of stability criteria, in terms of the transistor transconductances, biasing elements, and the number of cells, of easy evaluation at the design stage. The bifurcation phenomenon [35]–[41] that gives rise to the circuit instability will also be determined, deriving the bifurcation condition that defines the low-frequency stability boundary. Regarding the high-frequency instabilities, the insightful work [42] is based on the derivation of analytical expressions to describe the frequency response of the non-Foster transmission line. However, ideal negative elements are considered instead of transistor-based NICs. Here, the stability analysis will be carried out in a numerical manner, using complete and realistic models of all the elements in the NIC-loaded line. This is preferred due to the complexity of the transistor-device models and the impact of parasitics. The method will be based on a combination of pole-zero identification [31]–[33] and the Nyquist criterion [26]. The bifurcation phenomenon [35]–[37] that gives rise to the higher frequency instabilities will also be identified, and

the bifurcation condition will be derived from a matrixform formulation of the multi-element structure. The judicious choice of the observation ports will enable a direct calculation of all the coexisting bifurcation loci, with no need for continuation procedures. Note that the goal of this paper is not the investigation on the fast-wave transmission line, which has been done in [6] and [11]–[14], but on their stability and bifurcation behavior, as well as the derivation of accurate and easy-to-use stability-analysis methodologies. This paper is organized as follows. Section II summarizes the methodology to obtain a non-Foster transmission line, based on the topology proposed in [11] and [12]. Section III presents the analytical formulation to predict instabilities associated with the bias network, as well as the design criteria. Section IV presents the general numerical methodology for stability analysis. Section V describes a practical technique to obtain the instability boundary. II. N ON -F OSTER T RANSMISSION L INE Following the methodology in [6] and [11]–[14], the nonFoster transmission line is implemented through the periodic loading of this line with negative capacitors. Neglecting resistive contributions, the phase constant of the original passive transmission line is given as  ω√ β p = ω l x cx = εeff (1) C where l x and cx are the inductance and capacitance per unit length, respectively, c is the speed of light, and εe f f is the effective dielectric constant. The phase constant can be decreased by reducing the effective line capacitance through the periodical connection of negative capacitors. Assuming the parallel connection of one capacitor per length x, the phase constant becomes [15]    Cneg (2) βn F (ω) = ω l x cx − x where Cneg has been defined as a positive quantity. Note that the radicand is frequency dependent, since the negative capacitance can only be achieved in a certain frequency band. In order to avoid backward propagation, the total capacitance must be positive [7], [8], so the following condition must be fulfilled: Cneg >0 (3) cx − x In practical realization, the non-Foster transmission line will contain a limited number N of x cells, corresponding to a total length L. To achieve fast-wave propagation in a certain bandwidth, the phase delay through the line of length L must be smaller than that resulting from light passing through the vacuum [6], [11]–[14]. The negative capacitance is obtained by using NICs, terminated with a positive capacitor. The NIC chosen here is based on bipolar transistors, with the topology proposed in [11] and [12]. It uses two cross-coupled transistors, with a capacitor Cneg , connected between their emitter terminals [Fig. 1(a)]. According to [11] and [12], this topology, which

SUÁREZ AND RAMÍREZ: STABILITY AND BIFURCATION ANALYSIS OF MULTI-ELEMENT NON-FOSTER NETWORKS

Fig. 1. Schematic of the fast-wave propagation transmission line, adapted from [7] and [8]. (a) NIC circuit, loaded with Cneg and R. The resistance R may be used only in an auxiliary manner, for analysis purposes. The connection nodes to the transmission line correspond to the collector of the first transistor TC1 and ground. (b) Transmission line periodically loaded with the NICs. (c) and (d) Top and bottom views of the fabricated prototype [FR-4 (εr = 4.4, h = 1.6 mm) substrate, using AVAGO AT41511 transistors].

also includes base, collector, and emitter resistors, reduces the impact of discrepancies between the transistor elements. The NIC is connected to the transmission line through the collector terminal TC1 . As previously reported [1], when the NIC input port is defined at the collector terminals, it is SCS, or stable when terminating the port in an impedance smaller than the

1819

input one. As argued in [11] and [12], this should be the case when introducing the NIC terminated in the capacitor Cneg into the transmission line, due to the condition (3). However, these stability considerations, in terms of open and short-circuit terminations, derive from a number of simplifying assumptions, with the two-port network described with a constant h-parameter matrix [3]. Therefore, a rigorous stability analysis at circuit level is essential. Since, as stated, the aim of this paper is not the fast-wave transmission line design but the investigation of its stability properties, the test-bench is very similar to the one presented in [11] and [12]. The transmission line is implemented on the substrate FR-4 (εr = 4.4, h = 1.6 mm). In a similar way as [11] and [12], N = 3 transmission line sections are considered [Fig. 1(b)], though extensions to an arbitrary number N will also be assumed at some instances. As in [11] and [12], the length and width of the distributed transmission line sections are 33.3 and 10 mm, respectively. The negative capacitors −Cneg are implemented through the NIC shown in Fig. 1(a). Biasing inductors have been avoided since, as verified with the formulation presented in Section III, they make the circuit stable behavior even more critical, in agreement with previous reports [2], [9], [10]. Accurate models of all the active elements, provided by Modelithics, will be used in all the simulations, though in this section simplified analytical models will also be considered for an insightful investigation of the low-frequency instability. The biasing circuitry consists of the dc blocking capacitor C2 , in series with the collector terminal TC1 , and Cdc , in the cross-coupling branches, and the resistors R Bi , RCi, and R Ei , where i = 1, 2. Note that different values of these resistors are used, in principle, for the two transistors, since the NIC (with its input port defined between the collector terminal TC1 and the ground) is not symmetric. An additional resistor R, connected in series with the capacitor Cneg , is also considered, as in some previous works [3]. The resistor R may not be necessary, but will facilitate the calculation of the instability boundaries in Section IV, even if the actual design has R = 0 . After a preliminary design of the NIC circuit, the whole non-Foster transmission line has been optimized in order to achieve a superluminal phase delay in the frequency band comprised between 1 and 200 MHz. This has been done by considering M evenly spaced frequency points, f m = f o + m f , where m = 0 to M − 1. The objective is to get a nondispersive phase delay, fulfilling φ = −τ ω, where τ is the time delay, which must satisfy τ < L / c, under the constraint of a total positive capacitance [as in (3)]. The optimization is carried out through M single-point scatteringparameter simulations of the same 3-cell transmission line (with common element values), at the frequencies fm , where m = 0 to M − 1. The M goals are φ = −τ 2π f m . The value of the delay τ is progressively reduced, in a sequence of M-goal optimizations. The optimized elements are the capacitor Cneg , the resistor R, and the biasing resistors. The results for the values VCC = 9 V, RC1 = 300 , RC2 = 47 , R E1 = 100 , R E2 = 257 , Cneg = 10 pF, and R = 0  are shown in Fig. 2(a).

1820

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

Fig. 2. Optimization of the fast-wave transmission line. (a) Variation of the phase delay versus the input frequency obtained through multiple single point optimizations. (b) Stability analysis through pole-zero identification. The real part of the dominant poles has been represented versus VBB . There are three pairs of critical complex-conjugate poles and two critical real poles. (c) Expanded view of the pole locus in the low-frequency range. (d) Measured mixer-like spectrum, with two fundamental frequencies.

The stability analysis of the resulting circuit has been carried out through pole-zero identification [31]–[33]. As stated in the introduction, this method is based on the fact that all the transfer functions that can be defined in a linear system share the same denominator, which formally agrees with the

system characteristic determinant. For the stability analysis, the circuit is terminated with 50  impedances, although any other arbitrary termination may equally be considered. In most cases, the transfer function used is of impedance type, though an admittance type transfer function [33] can also be considered and will be used later in this paper. In the impedance-type case, a small-signal current generator at the frequency f is introduced at a sensitive circuit node, such as those corresponding to the active devices. The transfer function Z ( f ) is defined as the ratio between the node voltage and the current introduced. The poles, which are common to all the possible transfer functions, are obtained by applying pole-zero identification to Z ( f ). The procedure in the case of an admittance-type transfer function [31]–[33] would be analogous. Pole-zero identification has been applied under variations of the bias voltage VBB . In Fig. 2(b), the variation of the imaginary and real part of the poles has been represented versus VBB . There are three pairs of critical complex-conjugate poles and two critical real poles, which cross the imaginary axis at different VBB values. In fact, the detection of the unstable real poles is demanding and requires an identification in the extremely low frequency interval 100 Hz ≤ f ≤ 100 kHz. Fig. 2(c) presents an expanded view of the pole locus in the low-frequency range, showing how the two real poles merge into a pair of complex-conjugate poles just before crossing the imaginary axis. Then, they split again into two real poles, one of which approaches that exist but never crosses to the left-hand side (LHS). The presence of both real and high-frequency poles indicates a complex unstable behavior, which will be difficult to predict and control through a standard application of the available methodologies. The design was measured at VBB = 0.85 V and found to be unstable [Fig. 2(d)], in agreement with the predictions of Fig. 2(b). The measured spectrum is shown in Fig. 2(c). It is a mixer-like spectrum, with two fundamental frequencies. The incommensurate spectral lines with the highest power are at 364 and 406 MHz. For VBB = 0.85 V, the poles at about 320 MHz in the simulation of Fig. 2(b) have not crossed the imaginary axis yet, since the Hopf bifurcation occurs at VBB = 0.87 V. However, small discrepancies in the component models would justify the observation of the corresponding oscillation at the lower bias voltage VBB = 0.85 V. As stated, the second most significant incommensurate frequency in the measured spectrum is slightly above 400 MHz. This should correspond to the one above 500 MHz in the pole analysis. Although the two frequencies do not agree, inspection of Fig. 2(b) shows the high sensitivity of the frequency of this second pair of critical poles to VBB . In addition, one should take into account that, in general, poles only predict the oscillation frequency at the initial start-up transient, when the oscillator still behaves in a linear manner. As the oscillation amplitude continues to grow, there can be a significant variation of the oscillation frequency, due to nonlinear effects. This will be more remarkable under a low quality factor, as is the case of this circuit, mostly composed of capacitive and resistive elements. The frequency corresponding to the third pair of critical poles does not appear in the spectrum as an independent fundamental.

SUÁREZ AND RAMÍREZ: STABILITY AND BIFURCATION ANALYSIS OF MULTI-ELEMENT NON-FOSTER NETWORKS

1821

in Z o = 50 . However, the analysis can be easily extended to arbitrary termination impedances. Provided that there are no cancellations of RHS zeroes and RHS poles in the function Yin (s) (which should be checked in a separate manner), the stability properties of the whole configuration in Fig. 3(a) will be determined by the roots of the following characteristic polynomial: 2Gden(s) + Nnum(s) = 0

Fig. 3. Equivalent circuit used for the stability analysis in the low-frequency range. (a) Transmission line. (b) Circuit used to calculate Yin , as the ratio between Iin and Vin .

This is because the quasi-periodic solution measured has stabilized through inverse bifurcations [35]–[41], at which the additional critical poles have crossed to the LHS of the complex plane. As stated, the low-amplitude real poles and the complexconjugate poles are obtained through identification in a lowfrequency band and a higher frequency band, respectively. This property will enable the distinction of the mechanism giving rise to the low-amplitude real poles from other instabilities, detectable at higher frequencies, including those associated with the impedance negation. In the following section, a detailed analytical study of the low-frequency instability is presented. III. L OW-F REQUENCY I NSTABILITY Initially, the instability due to real poles on the RHS will be studied. This should be detectable by considering a small perturbation frequency, so most of the transistor parasitic elements can be neglected, as well as the pair RCneg , behaving basically as an open circuit, which is due to the high value of the impedance associated with Cneg in the low-frequency range. On the other hand, the transmission line sections can be considered as short circuits. Under these simplifications, in the case of an N-cell transmission line, the low-frequency equivalent circuit approximately consists of the parallel connection of N one-port blocks with the input admittance Yin and two termination impedances, as shown in Fig. 3(a). The results will be validated through comparison with complete circuit-level simulations using the models from Modelithics and with measurements. For the stability analysis of the circuit in Fig. 3(a), a small perturbation of complex frequency s is considered. Applying Kirchoff’s laws to the equivalent circuit in Fig. 3(a), one obtains the following characteristic equation: 2 num(s) 2 =0 (4) + N Yin (s) = +N Zo Zo den(s) where num(s) and den(s) are the numerator and denominator of Yin (s). In (4), the line has been assumed to be terminated

(5)

where G = 1/Z o . All the roots of (5), agreeing with the system poles, should have a negative real part. Equation (5) is of general application, regardless of the particular bias network, defining the form of Yin (s). In fact, the bias network considered in Fig. 1 is just a particular case, considered for illustration. The NIC input admittance at low frequencies Yin (s) will be calculated using simplified low-frequency models of the transistor devices, in which all the capacitive and inductive elements can be neglected. Resistive elements might be needed in some cases, but here a very good prediction of the stability properties has been achieved without these elements, so they will also be neglected for a better insight. Thus, the transistors are modeled with ideal linear transconductance functions gmi Vi , where i = 1, 2. For each bias point (of each transistor device), the gm in the simplified model of the bipolar transistor is estimated as gm = Ic /25.9 mS [43]. This provides the simplified equivalent circuit of Fig. 3(b). Note that this is just an example of equivalent low-frequency circuit, since the method can be extended to other bias-network topologies. To calculate the low-frequency Yin (s), the equivalent circuit in Fig. 3(b) is excited with the AC voltage source Vin . Applying Kirchoff’s laws, one obtains the following linear system:   1 1 q I2 + Vin = Iin − p+ C s C2 s  dc  1 I1 + a 1 + b1 I2 = Iin Cdc s   1 b 2 I1 + I2 = 0 (6) a2 + Cdc s where Iin is the current through the voltage source and the following parameters have been defined: p = R E2 + 1/gm2 q = (R E2 + 1/gm2 )G B2 a1 = (R E2 + 1/gm2 )(G B2 + G C1 ) a2 = (R E1 + 1/gm1 )(G B1 + G C2 ) b1 = (R E2 + 1/gm2 )G B2 G C1 b2 = (R E1 + 1/gm1 )G B1 G C2

(7)

−1 Note that the conductances G B = R −1 B , G C = RC , and −1 G E = R E have been defined. The input admittance Yin is obtained as the ratio Yin = Iin /Vin . It is easily seen that the resulting expression has no inherent cancellations of zeroes and poles. Replacing the expression for Yin (s) into (5), one obtains the following third-order characteristic polynomial:

α3 S 3 + α2 S 2 + α1 S + α0 = 0

(8)

1822

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

the coefficients of the characteristic polynomial. They can be easily checked after each variation of the biasing network elements and transistor-bias point. An analytical formulation can also be derived for the case of bias networks based on inductive elements. This will give rise to a higher system order, as a result of the higher number of reactive elements. For simplicity, the circuit considered in the derivation has no resistors, and DC feeds are used for a direct biasing of the transistor terminals. The resistor R E2 is eliminated, grounding the corresponding terminal, and the resistors R E1 , R B1 , R B2 , RC1, and RC2 are replaced with DCfeeds. In the most usual case of equal values for the DCblock capacitors and DC-feed inductors, given by L and C, the coefficients of the characteristic polynomial are as follows: Fig. 4. Evolution of the three roots of (8) versus the resistor RC1 . The results are compared with those obtained with pole-zero identification applied to two different circuits, simulated in microwave commercial software. One is the same equivalent circuit in Fig. 3 and the other is the realistic nonFoster transmission line, with full accurate models of all the passive and active elements, provided by Modelithics.

α6 = (−G Lgm1 gm2 + 12Cgm1 + 2C G)L 3 C 2 α5 = (7G Lgm1 + 12C)C 2 L 2 α4 = (12gm1 + 7G)C 2 L 2 α3 = (5G Lgm1 + 12C)C L α2 = (3gm1 + 5G)C L α1 = G Lgm1 + 3C

where all the coefficients are constant quantities, given as 2 α3 = C2 Cdc (Ga2 p + Na1 a2 − N)   C2 G(a2 q + b2 p) + NC2 (a1 b2 + a2 b1 ) α2 = Cdc + Cdc G(a1 a2 − 1) α1 = C2 Gb2 q + NC2 b1 b2 + Cdc G(a1 b2 + a2 b1 )

α0 = Gb1 b2

α7 = (2G − 3gm2 )gm1 C 3 L 4

α0 = G

(9)

In view of the positive sign of α0 , a necessary condition for stability is α1 , α2 , α3 > 0. From the inspection of (9), the coefficients α1 , α2 can be either positive or negative. A critical quantity is a1 a2 −1, which may become negative for high transistor gain and different combinations of (low) emitter resistors and (high) collector resistors. However, even if the condition on the equal sign of the polynomial coefficients is fulfilled, the system may be unstable since the Routh–Hurtwiz criterion [26] involves the evaluation of other parameters, depending on the polynomial coefficients. Fig. 4 presents the evolution of the three roots of (8) versus the resistor RC1 . The results are compared with those obtained with pole-zero identification applied to two different circuits, simulated in microwave commercial software. One is the same equivalent circuit in Fig. 3 and the other is the realistic nonFoster transmission line, with full accurate models of all the active and passive elements, provided by Modelithics. As stated, the two have been simulated with microwave commercial software, applying pole-zero identification [31]–[33] to the impedance function Z ( f ) extracted from this software. As shown in Fig. 4, the analytical results overlapped with the simulations of the simplified model and exhibit an excellent agreement with those corresponding to the accurately modeled circuit. In the latter case, to detect the negative real pole with very small magnitude, it has been necessary to consider a small frequency interval, going from zero to a few hundred Hertz. The advantage of the analytical formulation is the possibility of a direct evaluation of the Routh–Hurtwiz stability conditions at the design stage, since these conditions depend only on

(10)

According to the Routh–Hurwitz criterion, if 3gm2 > 2G(α7 < 0), a sufficient condition for circuit instability is fulfilled. This is the case in the design considered here, having gm2 = 32 mS and G = 20 mS. Our results agree with the previous observations [2], [9], [10], stating that biasing networks based on inductors are more likely to give rise to instability. As gathered from Fig. 4, the circuit becomes unstable from a certain value of RC1 , in consistency with the reduction of the damping effect of this resistor, in parallel at the input port. The qualitative change of the stability is due to the crossing of a pair of complex-conjugate poles through the imaginary axis, which corresponds to a Hopf bifurcation [35]–[41]. Despite this, one should note that for most of the unstable intervals, there are two real poles on the RHS, which should lead to a relaxation oscillation, in agreement with the results of the previous work [25]. In addition to the critical pair of complex conjugate poles, both the analytical formulation and polezero identification predict a negative real pole of very small magnitude. This will give rise to long transients, so the timedomain simulation of this kind of circuit should be extremely inefficient and/or inconclusive. As gathered from our simplified system (8), instability cannot be due to the direct crossing through zero of a real pole. Mathematically, this would imply the fulfillment of (8) for s = 0 (real pole crossing the imaginary axis), leading to the impossible condition α0 = 0 [see definitions in (9)]. The bifurcation can only be due to a pair of complex-conjugate poles crossing the imaginary axis. This is also seen in Fig. 2(c), even though the analysis in this figure corresponds to the full circuit, loaded with R and Cneg . As shown in Fig. 4, after crossing the imaginary axis, the pair of complex-conjugate poles splits into two real poles,

SUÁREZ AND RAMÍREZ: STABILITY AND BIFURCATION ANALYSIS OF MULTI-ELEMENT NON-FOSTER NETWORKS

1823

Fig. 6. Hopf-bifurcation locus in terms of the two resistors RC1 , RC2 , delimiting the region giving rise to instabilities detectable in the lower frequency interval. Measurements have been superimposed with good agreement.

Fig. 5. Measured spectra for different RC values. (a) RC1 = 100 . Stable. (b) RC1 = 150 . Stable. (c) RC1 = 180 . Unstable. (d) RC1 = 220 . Unstable.

which is indicative of a relaxation oscillation. These oscillations are associated with the periodical charging and discharging of circuit capacitors. They can be reasonably expected in a circuit based on resistive and capacitive elements. For validation, the circuit has been experimentally characterized in the absence of the loading elements Cneg and R. Fig. 5 shows the measured spectra for different RC values. The experimental results exhibit a very good agreement with the analytical and simulation predictions of Fig. 4. The dense spectral content observed in Fig. 5(c) and (d) is consistent with this kind of oscillation. There is an additional independent fundamental, which should have arisen from a secondary Hopf bifurcation of the oscillation predicted by the analysis in Fig. 4. The previous identification of the bifurcation phenomenon that gives rise to the low-frequency instability will allow a direct calculation of the stability boundary. Obtaining these boundaries is useful since they provide global information on the circuit stability properties and their evolution versus relevant circuit parameters. The qualitative stability change is due to a Hopf bifurcation [35]–[41], so the stability boundary in terms of any two convenient parameters can be calculated by replacing s with j ω in (8). Splitting the resulting complex equation into real and imaginary parts, one obtains the following system of two real equations: −α2 ω2 + α0 = 0 (a) −α3 ω2 + α1 = 0 (b)

(11)

ω2

Solving [11(b)] for and replacing in [11(a)], the Hopfbifurcation condition is derived α0 α3 − α2 α1 = 0

(12)

Useful parameters to analyze the Hopf-bifurcation condition are RC1 , RC2 , gm1 , and gm2 , since in the bipolar model the transconductance is approximate independent of the collector resistors. As an example, Fig. 6 presents the Hopf bifurcation locus in terms of RC1 and RC2 . As gathered from the figure, the circuit is stable for sufficiently small value of any of these

two resistors. The Hopf locus is in total agreement with the predictions of the stability analysis in Fig. 4, which was carried out for RC2 = 75 . Measurements for different pairs of resistors have been carried out. Points with stable behavior are indicated with asterisks. Points with unstable behavior are indicated with squares. There is a very good agreement with the analytical predictions by (12). IV. H IGH -F REQUENCY I NSTABILITY The high-frequency stability analysis will only be performed considering the realistic models of all the active and passive elements in the non-Foster transmission line. This is due to the relevant impact of parasitic elements at high frequency. The stability analysis will be numerical and based on a combination of pole-zero identification and the Nyquist criterion. This should be able to predict both the low-frequency instabilities considered in Section III and the high-frequency instabilities, associated with the impedance negation and other effects. Two different components will be distinguished in the full circuit configuration: the passive transmission line, bounded by Z o = 50 (or any other impedance values), and the N NICs that periodically load this transmission line. Note that there can be changes affecting the passive line only, e.g., a variation in the number N of sections or an alteration of the termination impedances due to mismatch. For instance, instability arising with increasing N is reported in [42] and [44]. However, since the circuit operates in a linear regime, the input admittance Yin (s), exhibited by the NIC will remain the same, under this line modification. To derive the characteristic system resulting from a small perturbation at the frequency s, the schematic in Fig. 7 will be considered. The terminated passive line is described with an N × N admittance matrix [Y L (s)], defined at the N ports where the N NICs are connected to this line. On the other hand, the N NICs are represented with an N × N diagonal matrix, with equal diagonal components given by the input admittance Yin (s), which is indicated as diag[Yin (s)]. Then, the perturbed circuit must fulfill the following characteristic system: {[Y L (s)] + diag[Yin (s)]}V = 0

(13)

1824

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

Fig. 7. Setup for the numerical stability analysis of the non-Foster transmission line. (a) Linear matrix definition. (b) Numerical calculation of the input admittance of the individual NIC: Yin ( jω). Pole-zero identification will be applied to this admittance-type transfer function.

Fig. 8. Verification of the NIC stability prior to the stability analysis through the Nyquist plot. (a) Pole-zero locus obtained through the identification of Yin (s) for Cneg = 10 pF. (b) Evolution of the real part of the dominant poles versus Cneg , indicating stable behavior for all the Cneg values.

where V is the vector composed by the voltage increments at the NIC connection nodes, resulting from the small perturbation. Provided that there are no exact cancellations of RHS poles and zeroes in the function Yin (s) (which will be checked in a separate manner), the poles of the NIC-loaded transmission line will be given by the zeroes or roots of the following characteristic equation:

not exhibit any RHS poles, or, equivalently, as derived in the next paragraph, the NIC must be stable under an SCS. The NIC stability analysis under a short-circuit termination is carried out through pole-zero identification. To identify the function Yin (s), a series-voltage source is introduced [Fig. 7(b)] between the two nodes used for the NIC connection to the transmission line. Then, the ratio between the current circulating through this source and the voltage delivered provides a transfer function of admittance type. The absence of RHS poles in Yin (s) would indicate stability under a shortcircuit termination at the input port (SCS). As discussed, care must also be taken to check that there is no cancellation of RHS poles with RHS zeroes in the Yin (s) function, which is done by performing this identification at different locations of the NIC, under a short-circuit termination at the node where it is connected to the transmission line. For RC1 = 150 , one obtains a stable NIC behavior under a short-circuit termination. The pole-zero locus obtained through the identification of Yin (s) is shown in Fig. 8(a). All the poles are on the LHS of the complex plane. To check for any possible RHS pole-zero cancellations, the identification has been applied to several transfer functions, defined at different NIC nodes. No RHS poles have been obtained in any case. The procedure has also been applied under variations of the capacitance Cneg [Fig. 8(b)]. The poles remain on the LHS in all cases, so the Nyquist criterion should be applicable for any Cneg . To obtain the Nyquist plot, the complex frequency s should be replaced by j ω in (14), which provides the function

det(s) = det{[Y L (s)] + diag[Yin (s)]} = 0

(14)

As an example, in the case of the line with N = 3 NIC elements, the above formulation particularizes as ⎧⎡ ⎤ ⎫ Y11 (s) Y12 (s) 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ Y12 (s) Y22 (s) Y12 (s) ⎦ + ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ (s) Y (s) 0 Y 12 11 ⎤ ⎡ det[Y (s)] = det =0 0 0 Yin (s) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 ⎪ 0 ⎦ ⎪ Yin (s) ⎪ ⎪ ⎪ ⎩ ⎭ 0 0 Yin (s) (15) As stated, due to the possible cancellation of RHS zeroes and poles in Yin (s), one must ensure that Yin (s) does not exhibit any RHS poles, as these would give rise to instabilities internal to the NIC that would not be detectable with (14). In practice, the number of unstable circuit poles can be directly calculated in the microwave commercial software through an application of the Nyquist criterion [4], [22] to the determinant in (14). For this criterion to be usable, the function det(s) in (14) should not contain any RHS poles, as discussed in the introduction, Ncir = N Z − N P . Because the matrix [Y L (s)] representing the terminated transmission line is passive, N P = 0 may only come from possible RHS poles in the function Yin (s). Thus, Yin (s) should

det( j ω) = det{[Y L ( j ω)] + diag[Yin ( j ω)]}

(16)

Then, a sweep must be carried out in ω from zero to infinite, though in practice, it is sufficient to sweep ω from zero to a

SUÁREZ AND RAMÍREZ: STABILITY AND BIFURCATION ANALYSIS OF MULTI-ELEMENT NON-FOSTER NETWORKS

1825

Fig. 10. Validation of the results of the Nyquist plots in Fig. 9 with pole-zero identification. The real part of the dominant pair of complex-conjugate poles has been represented versus Cneg . This pair of complex-conjugate poles cross to the RHS at Cneg = 15.5 pF, in full agreement with the prediction of the Nyquist plot. Fig. 9. Nyquist analysis in the case of a non-Foster transmission line with N = 3 elements and RC1 = 150 . (a) Global view when using the function (16). (b) Expanded view when using the function (16). Unstable behavior is obtained for Cneg ≥ 15.5 pF. (c) Nyquist plots obtained when using the feedback formulation in (17) for the same set of Cneg values. (d) Expanded view.

maximum value, depending on the device gain and low-pass characteristic. Because the complex function considered [given by in (13)] contains no RHS poles, the number of unstable poles of the non-Foster transmission line will agree with the number of clockwise encirclements of the Nyquist plot about the origin of the complex plane: Ncir = N Z . The described procedure has been initially applied in the case of RC1 = 150  and N = 3. In Fig. 9, the Nyquist plot has been traced for different values of Cneg , comprised between 5 and 16 pF. One can notice a high phase sensitivity about the point (0, 0), which requires several zooms to evaluate the number of encirclements. As gathered from the expanded views of Fig. 9(b), the Nyquist plot does not encircle the origin for Cneg values below approximately 15.5 pF. However, it does encircle the origin for Cneg > 15.5 pF. This can be associated with the dominance of the negative capacitance in the nonFoster transmission line from certain Cneg . The validity of the predictions obtained with the Nyquist plot has been checked with pole-zero identification applied to the entire non-Foster transmission line. In Fig. 10, the real part of the dominant pair of complex-conjugate poles has been represented versus Cneg . This pair of complex-conjugate poles crosses to the RHS at Cneg = 15.5 pF, in full agreement with the prediction of the Nyquist plot. The Nyquist criterion could not be used to detect the lowfrequency instability, analyzed in Section III. This is because the standalone NIC is unstable under short-circuit terminations for some values of its resistive and capacitive components. Using the analysis in Section III, these components are selected in order to prevent the low-frequency instability. Then, as verified in Fig. 8, the standalone NIC is stable under short-circuit terminations in the whole frequency range, so the Nyquist criterion can be applied to analyze the stability of the whole non-Foster transmission line.

The difficulties associated with the significant variations in the phase sensitivity of the Nyquist plot with respect to ω can be avoided using a formulation of feedback type, based on the multiplication of the characteristic system (13) by the inverse of the passive matrix [Y L (s)]−1 (provided that the matrix is nonsingular). In the general case of N × N matrices, the resulting Nyquist determinant, of feedback type [32], [33], is as follows: det f b ( j ω) = det{[U ] + [Y L ( j ω)]−1diag[Yin ( j ω)]} = det{[U ] + [Z L ( j ω)]diag[Yin ( j ω)]} (17) where [U ] is the N × N diagonal matrix and [Z L ( j ω)] is the passive-impedance matrix, directly calculated with the commercial software. The feedback formulation involves a normalization effect, as the one demonstrated in [28] and [29]. The Nyquist plots obtained with the feedback formulation (17) for the same Cneg values are shown in Fig. 9(c) and (d). The same stability predictions are obtained, though the phase sensitivity is more even versus the perturbation frequency. Note that the Nyquist analysis and pole-zero identification are used here in a complementary manner. Pole-zero identification is applied only to the individual NIC to ensure the absence of RHS poles of Yin (s). This allows application of the Nyquist criterion to the whole non-Foster transmission line, which can be done in the microwave commercial software through a simple computation of the matrices and determinant in (17), without any postprocessing of the data. This enables a direct checking of the stability properties at the design stage. One advantage of the Nyquist criterion with respect to the polezero identification is the fact that the whole stability analysis is carried out through a single frequency sweep. It does not require splitting the perturbation frequency range (from dc to a very high value) into subintervals. We believe that this cooperative use of pole-zero identification and the Nyquist criterion can be of practical interest in other circuits with complex topologies. As another example, the Nyquist criterion has been applied to a non-Foster transmission line containing N = 10 cells, using the feedback formulation (17). Under variations of Cneg , instability arises from much smaller value than in the case of

1826

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

Fig. 13. Calculation of the impedance/admittance matrices with ports defined between the nodes where the elements R and Cneg should be connected to the whole configuration. Fig. 11. Application of the Nyquist criterion to a non-Foster transmission line containing N = 10 cells, with RC1 = 150 , using the feedback formulation (17). (a) Cneg = 1 pF, global view of Nyquist plot, predicting the stable beavior. (b) Cneg = 1 pF, expanded view. (c) Cneg = 3 pF, global view of Nyquist plot, predicting the unstable behavior. (d) Cneg = 3 pF, expanded view.

Fig. 12. Validation of the Nyquist plots through pole-zero identification applied to the entire non-Foster transmission line. The real part of the dominant poles has been represented versus Cneg . The circuit becomes unstable at about Cneg = 2.5 pF. Two pairs of complex-conjugate poles cross to the RHS one after the other.

only N = 3 cells. This is shown in Fig. 11, which compares the results obtained for Cneg = 1 pF and Cneg = 3 pF. For Cneg = 1 pF, the Nyquist plot does not encircle the origin, so the non-Foster transmission line is stable. For Cneg = 3 pF, the Nyquist plot encircles the origin twice, so the non-Foster transmission line should contain two pairs of unstable of poles on the RHS. This has been validated through application of pole-zero identification to the whole transmission line. The results are shown in Fig. 12, where the real part of the dominant poles has been represented versus Cneg . In agreement with the predictions of the Nyquist plot, the circuit is stable at Cneg = 1 pF. In the neighborhood of Cneg = 2.5 pF, two pairs of complex-conjugate poles cross to the RHS, one after the other. For Cneg = 3 pF, the circuit is unstable with two pairs of complex-conjugate poles on the RHS, in agreement with the predictions of the Nyquist plot in Fig. 11. V. H OPF -B IFURCATION L OCUS As has been shown in the various stability analyses versus relevant circuit parameters, the qualitative change of stability

is due to the crossing of a pair of complex-conjugate poles through the imaginary axis, which corresponds to a Hopf bifurcation [37]–[40], [45]. This is, in fact, the most usual instability mechanism in a circuit operating in small signal, since a real pole crossing through zero would imply either a turning point or a branching point in the DC solution curve [35]–[37], which are relatively rare. On the other hand, the observation of several consecutive crossing of complex-conjugate poles through the imaginary axis is indicative of the possible existence of several Hopf locus sections or even disconnected loci. Here, a methodology for the direct tracing of Hopf bifurcation locus, with no need for continuation methods, is proposed. This is based on a matrix formulation, with a judicious choice of the reference ports, as many as the number N of NICs. In fact, the N ports will be defined between nodes loaded with equivalent passive branches, containing elements that do not affect the transistor biasing, since those elements will be considered as unknowns of the bifurcation equation. The non-Foster transmission line will be opened at ports defined between the nodes where the capacitance Cneg , in series with the resistor R, is connected to the structure (Fig. 13). As will be shown, this facilitates the calculation of the instability boundaries, even in the case in which the resistor R is absent, which would simply correspond to the particular situation R = 0 . The N × N impedance matrix defined as indicated in Fig. 13 will be denoted as [Z cir (η, s)]. Note that this matrix contains both passive and active elements. The stability boundary defined by the Hopf-bifurcation locus is calculated in terms Cneg and R, plus an additional parameter η. This parameter will correspond to any quantity with a relevant impact on the stability properties, such as a bias voltage. For the calculation of the Hopf locus, the frequency s in the characteristic determinant should be replaced with j ω. In terms of [Z cir (η, j ω)], the Hopf bifurcation condition is given as det( j ω, R, Cneg , η)   = det [Z cir (η, j ω)] + diag R +

1 Cneg j ω

 = 0 (18)

SUÁREZ AND RAMÍREZ: STABILITY AND BIFURCATION ANALYSIS OF MULTI-ELEMENT NON-FOSTER NETWORKS

1827

where the second impedance matrix is diagonal, with equal diagonal elements, given by the impedance of the series connection of R and Cneg . For illustration, the formulation will be particularized to the case N = 3, which provides det( j ω, R, Cneg , η) ⎧ ⎡ Z 11 (η, j ω) ⎪ ⎪ ⎪ ⎣ Z 12 (η, j ω) ⎪ ⎪ ⎪ ⎪ ⎪ Z 13 (η, j ω) ⎪ ⎪ ⎡ ⎪ ⎪ 1 ⎨ ⎢ R+ Cneg j ω = det ⎢ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪+⎢ 0 ⎪ ⎢ ⎪ ⎪ ⎢ ⎪ ⎪ ⎣ ⎪ ⎪ ⎩ 0 =0

⎤ Z 12 (η, j ω) Z 13 (η, j ω) Z 22 (η, j ω) Z 12 (η, j ω) ⎦ Z 12 (η, j ω) Z 11 (η, j ω) 0 R+

0 1

0

Cneg j ω 0

R+

1 Cneg j ω

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎤⎪ ⎪ ⎪ ⎬ ⎥ ⎥⎪ ⎥⎪ ⎪ ⎥⎪ ⎪ ⎥⎪ ⎪ ⎥⎪ ⎪ ⎦⎪ ⎪ ⎪ ⎭ (19)

For each value of the parameter η, the complex equation (18) provides a system of two real equations in the three unknowns ω, R, and Cneg , which will give rise to a curve or locus in the plane defined by R and Cneg . In practice, the locus is calculated in a simple manner by sweeping ω from zero to a maximum value, and solving the two real equations obtained for each ω in terms of R and Cneg . Note that splitting (18) into real and imaginary parts provides two coupled polynomial equations in R and Cneg . The system is solved by exporting the matrix [Z cir (η, j ω)] from the commercial microwave software and using a program such as MATLAB to compute the roots in R and Cneg for each ω value. Under a sufficiently fine sweep in η, it is also possible to obtain the Hopf-bifurcation locus (or loci) in the plane η and Cneg for a constant R value (R = 0, for instance). The analysis described has been applied to the non-Foster transmission line in Fig. 1, containing N = 3 NICs. The extra parameter η is the base-bias voltage VBB . Solving (19), one obtains the ensemble of Hopf bifurcation loci in Fig. 14(a), in the plane defined by R and Cneg . As can be seen, there are two distinct Hopf loci for each VBB , which is consistent with the observation of two pairs of complex-conjugate poles in Figs. 2(b) and 11. The loci in the solid line correspond to VBB = 0.8 V. The loci in the dashed line correspond to VBB = 0.85 V, and the loci in the dotted line correspond to VBB = 0.95 V. The unstable regions are shadowed, using a different color for each VBB . Note that obtaining the Hopf bifurcation loci through continuation plus parameter switching [38] would have been cumbersome, due to the presence of turning points and the mentioned existence of two disconnected loci. In fact, when using continuation to trace the loci, and due to its local quality, one of the two disconnected loci could have been missed. Fig. 14(b) shows in more detail the two Hopf bifurcation loci obtained for VBB = 0.8 V, which has been thoroughly validated through pole-zero identification. At each R value, the circuit is stable for Cneg values below the Hopf loci, which corresponds to a smaller impact of the negated impedance on the non-Foster transmission line. Note that this negated impedance is introduced in parallel at the NIC locations, so a smaller effect is obtained for smaller Cneg and larger R,

Fig. 14. Ensemble of Hopf-bifurcation loci in the plane defined by R and Cneg . (a) Results obtained by solving (19), with the additional parameter η = VBB . The loci in the solid line correspond to VBB = 0.8 V. The loci in the dashed line correspond to VBB = 0.85 V and the loci in the dotted line correspond to VBB = 0.95 V. The unstable regions are shadowed, using a different color for each VBB . (b) Hopf bifurcation loci obtained for VBB = 0.8 V. The points A, B, and H indicate the Hopf bifurcation points validated through pole-zero identification in Fig. 15. Measurements for different pairs (Cneg , R) are superimposed. Capacitors used belong to the ATC Ceramics 600S series (Design Kits 26 T and 27 T).

proving impedances with larger magnitude, connected in parallel. The predictions of the two disconnected loci in Fig. 14(b) have been validated through a detailed pole-zero identification of the whole structure. The points A, B, to H in Fig. 14(b) indicate the Hopf bifurcation points validated through polezero identification in Fig. 15, where they are denoted in the same manner. In a first attempt, the bifurcation corresponding to the upper locus was not detected when performing the identification at the base terminal of the left NIC transistor in Fig. 7(b). It was necessary to change the observation node to TC1 [Fig. 7(b)], the one corresponding to the connection between the NIC and the transmission line. Fig. 15 shows the variation of the real part of the poles versus Cneg for different values of R. The four vertical paths followed in the identification of Fig. 15 are indicated in Fig. 14(b). As can be seen, there is an excellent agreement between the predictions of the loci in Fig. 14(b) and the results of the pole-zero identification in Fig. 14. In consistency with Fig. 12, for the lower R values, there are two distinct pairs of complex-conjugate poles crossing to the RHS.

1828

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

Fig. 15. Variation of the real part of the dominant poles represented versus Cneg for different values of R. The four vertical paths followed in the identification are indicated in Fig. 14(b). (a) R = 5 . (b) R = 10 . (c) R = 20 . (d) R = 40 .

Fig. 16. (a) Experimental validation of the Hopf loci shown in Fig. 14(a). (b) Spectra for R = 39 , Cneg = 10 pF, VBB = 0.8 V (stable), and VBB = 0.82 V (unstable), respectively. (c) and (d) Spectra for R = 22 , Cneg = 5.6 pF, VBB = 0.8 V (stable), and VBB = 0.85 V (unstable), respectively.

The stability boundary in Fig. 14 has been validated experimentally. Fig. 14(b) shows the results corresponding to VBB = 0.8 V. Squares indicate unstable points, whereas crosses correspond to stable points. Note that standard discrete capacitor (ATC Ceramics 600 series) and resistor values were used for these measurements. Taking into account the effect of VBB , additional validations have been performed. A pair of values of R and Cneg in the stable zone is selected, and the bias voltage is increased until the circuit becomes unstable, in agreement with Fig. 14(a). At the stability change, the pair of values should belong to the Hopf locus for the particular VBB . The procedure has been repeated for different pairs of R and Cneg . Fig. 16 shows the results of this procedure for

two pairs of element values (R = 39 , Cneg = 10 pF) and (R = 22 , Cneg = 5.6 pF). As seen in Fig. 14(a), at VBB = 0.8 V, the circuit should be stable for the two pairs of values. The corresponding spectra are shown in Fig. 16(a) and (c). In each case, when increasing VBB from 0.8 V, the loci will cross the pair of values Cneg and R giving rise to the onset of an oscillation at VBB = 0.82 V [Fig. 16(b)] and VBB = 0.85 V [Fig. 16(d)], respectively. Note that the Hopf loci in Fig. 14(a) enable a straightforward determination of the stability properties and stability margin for R = 0 , which is just a particular point of the horizontal axis. The Hopf loci for R = 0  in the plane defined by VBB and Cneg are calculated as follows. The bias voltage VBB is swept (the step used is 50 mV), solving the complex equation (19) at each sweep step, in terms of R and Cneg . This gives the ensemble of solution points, in terms of VBB , Cneg , and R, that fulfill Re(det) = 0 and Im(det) = 0. Interpolation of these data for R = 0 provides two different contours Re[det(VBB , Cneg )/R = 0 ] = 0

(20a)

Im[det(VBB , Cneg )/R = 0 ] = 0

(20b)

Then, the Hopf locus or loci (since there can be more than one) are given by the intersection points of the two contours. Following the procedure described, it has been possible to directly obtain the three Hopf loci in Fig. 17(a), which have been validated with pole-zero identification and experimental measurements (superimposed). Fig. 17(b)–(d) presents the results of pole-zero identification for three different values of the bias voltage VBB (0.76, 0.8, and 0.82 V). In each case, there is full agreement between the predictions by the loci in Fig. 17(a) and the crossings of the various pairs of complexconjugate poles through the imaginary axis. Regarding the measurement results superimposed in Fig. 17(a), there is also a very good agreement. Only the first Hopf bifurcation, i.e., the one that occurs from a stable DC solution can be measured, since the other ones occurring from an unstable DC solution are unphysical. We would like to emphasize that with the new method, the Hopf loci are calculated in an efficient manner, which demands neither optimization nor continuation techniques. It requires just a frequency sweep in commercial microwave software, together with the resolution of the two real equations obtained by splitting (18) into real and imaginary parts in a mathematical program. The loci will enable an insightful understanding of the evolution of the circuit stability properties. As a final comment, one can gather some information on the tolerance effects from inspection of the bifurcation loci in Fig. 6, corresponding to the low-frequency instability, and in Fig. 14, corresponding to the high-frequency instability (although this assumes equal values of R and Cneg ). To ensure robustness against tolerances, one should choose an operation point far from the Hopf bifurcation loci that constitute the stability boundaries. Additional loci can be traced in terms of other parameters with a likely impact on the stability properties. A more detailed way to predict the impact of

SUÁREZ AND RAMÍREZ: STABILITY AND BIFURCATION ANALYSIS OF MULTI-ELEMENT NON-FOSTER NETWORKS

1829

locus, derived from the analytical expressions. Instabilities detectable at higher frequencies are predicted with a numerical method that combines the pole-zero identification of the individual NIC element with the Nyquist criterion applied to a feedback formulation of the non-Foster transmission line. The stability boundaries are obtained through the root calculation of a determinant function, obtained from an N-port impedance matrix, easily extracted from commercial microwave software. This enables a direct computation of all the coexisting bifurcation loci, with no need for continuation techniques. All the results of the new methodologies have been rigorously validated with pole-zero identification and with measurements. R EFERENCES

Fig. 17. Hopf loci of the non-Foster transmission line for R = 0 . (a) Loci in the plane defined by the bias voltage VBB and Cneg . Measurement points at the instability boundaries are superimposed. Validation through pole-zero identification for (b) VBB = 0.76 V, (c) VBB = 0.8 V, and (d) V B B = 0.82 V.

tolerances is to perform pole-zero identifications combined with a Monte Carlo analysis, as presented in [46]. VI. C ONCLUSION A series of methodologies for the in-depth stability analysis and stabilization of non-Foster circuits containing multiple NIC elements have been presented. An analytical formulation has been derived to predict instabilities detectable in the lower frequency range, which can be avoided through a direct evaluation of the Routh–Hurwitz criterion at the design stage. The instability boundaries in terms of two significant parameters are calculated through a computation of the Hopf

[1] S. E. Sussman-Fort and R. M. Rudish, “Non-foster impedance matching of electrically-small antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 8, pp. 2230–2241, Aug. 2009. [2] C. R. White, J. S. Colburn, and R. G. Nagele, “A non-foster VHF monopole antenna,” IEEE Antennas Wireless Propag. Lett., vol. 11, pp. 584–587, 2012. [3] C. K. Kuo, “Realization of negative-immittance converters and negative resistances with controlled sources,” Ph.D. dissertation, School Elect. Comput. Eng., Georgia Inst. Technol., Atlanta, GA, USA, 1967. [4] H. Mirzaei and G. V. Eleftheriades, “A resonant printed monopole antenna with an embedded non-foster matching network,” IEEE Trans. Antennas Propag., vol. 61, no. 11, pp. 5363–5371, Nov. 2013. [5] A. M. Elfrgani and R. G. Rojas, “Successful realization of NonFoster circuits for wide-band antenna applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, USA, May 2015, pp. 1–4. [6] H. Mirzaei and G. V. Eleftheriades, “Unilateral non-Foster elements using loss-compensated negative-group-delay networks for guided-wave applications,” IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, USA, Jun. 2013, pp. 1–4. [7] J. G. Linvill, “Transistor negative impedance converters,” Proc. IRE, vol. 41, pp. 725–729, Jun. 1953. [8] J. G. Linvill and R. L. Wallace, Jr., “Negative impedance converters employing transistors,” U.S. Patent 2 726 370, Dec. 6, 1955. [9] R. Nealy, “Investigation of a negative impedance converter for wide band antenna matching,” Dept. Elect. Comput. Eng., Virginia Tech, Blacksburg, VA, USA, Tech. Rep. 28, Sep. 2012. [10] K. S. Song, “Non-foster impedance matching and loading networks for electrically small antennas,” Ph.D. dissertation, Dept. Elect. Comput. Eng., The Ohio State Univ., Columbus, OH, USA, 2011. [11] J. Long, M. M. Jacob, and D. F. Sievenpiper, “Broadband fast-wave propagation in a non-foster circuit loaded waveguide,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 4, pp. 789–798, Apr. 2014. [12] J. Long and D. Sievenpiper, “Stable multiple non-foster circuits loaded waveguide for broadband non-dispersive fast-wave propagation,” Electron. Lett., vol. 50, no. 23, pp. 1708–1710, Nov. 2014. [13] H. Mirzaei and G. V. Eleftheriades, “Arbitrary-angle squint-free beamforing in series-fed antenna arrays using non-Foster elements synthesized by negative-group-delay networks,” IEEE Trans. Antennas Propag., vol. 63, no. 5, pp. 1997–2010, May 2015. [14] H. Mirzaei and G. V. Eleftheriades, “Squint-free beamforming in seriesfed antenna arrays using synthesized non-foster elements,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. (APSURSI), Orlando, FL, USA, Jul. 2013, pp. 2209–2210. [15] H. Mirzaei and G. V. Eleftheriades, “An active artificial transmission line of squint-free series-fed antenna array applications,” in Proc. 41st Eur. Microw. Conf., Manchester, U.K., 2011, pp. 503–506. [16] M.-C. Tang, N. Zhu, and R. W. Ziolkowski, “Augmenting a modified egyptian axe dipole antenna with non-foster elements to enlarge its directivity bandwidth,” IEEE Antennas Wireless Propag. Lett., vol. 12, pp. 421–424, 2013. [17] E. Ugarte-Muñoz, S. Hrabar, D. Segovia-Vargas, and A. Kiricenko, “Stability of non-Foster reactive elements for use in active metamaterials and antennas,” IEEE Trans. Antennas Propag., vol. 60, no. 7, pp. 349–490, Jul. 2012. [18] S. Hrabar, I. Krois, A. Kiricenko, and I. Bonic, “Homogenization of active transmission-line-based ENZ metamaterials,” in Proc. APSURSI, Jul. 2010, pp. 1–4, doi: 10.1109/APS.2010.5562256.

1830

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 66, NO. 4, APRIL 2018

[19] M. Barbuto, A. Monti, F. Bilotti, and A. Toscano, “Design of a nonFoster actively loaded SRR and application in metamaterial-inspired components,” IEEE Trans. Antennas Propag., vol. 61, pp. 1219–1227, 2013. [20] Z. Ning and R. Ziolkowski, “Active metamaterial-inspired broadbandwidth, efficient, electrically small antennas,” IEEE Antennas Wireless Propag. Lett., vol. 10, pp. 1582–1585, 2011. [21] S. D. Stearns, “Circuit stability theory for non-Foster circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, USA, Jun. 2013, pp. 1–3. [22] A. M. Elfrgani and R. G. Rojas, “Stabilizing non-Foster circuits for electrically small antennas,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. (APSURSI), Memphis, TN, USA, Jul. 2014, pp. 464–465. [23] M. Jacob and D. Sievenpiper, “Gain and noise analysis of non-Foster matched antennas,” IEEE Trans. Antennas Propag., vol. 64, no. 12, pp. 4993–5004, Dec. 2016. [24] J. Brownlie, “On the stability properties of a negative impedance converter,” IEEE Trans. Circuit Theory, vol. CT-13, no. 1, pp. 98–99, Mar. 1966. [25] A. Suárez and F. Ramírez, “Circuit-level stability and bifurcation analysis of non-Foster circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., Honolulu, HI, USA, Jun. 2017, pp. 340–343. [26] K. Ogata, Modern Control Engineering. Upper Saddle River, NJ, USA: Prentice-Hall, 2001. [27] V. Rizzoli and A. Lipparini, “General stability analysis of periodic steady-state regimes in nonlinear microwave circuits,” IEEE Trans. Microw. Theory Techn., vol. MTT-33, no. 1, pp. 30–37, Jan. 1985. [28] A. Platzker and W. Struble, “Rigorous determination of the stability of linear n-node circuits from network determinants and the appropriate role of the stability factor K of their reduced two-ports,” in Proc. 3rd Int. Workshop Integr. Nonlinear Microw. Millimeterwave Circuits, Oct. 1994, pp. 93–107. [29] W. Struble and A. Platzker, “A rigorous yet simple method for determining stability of linear N-port networks [and MMIC application],” in Proc. 15th Gallium Arsenide Integr. Circuit Symp., 1993, pp. 251–254. [30] R. W. Jackson, “Rollett proviso in the stability of linear microwave circuits-a tutorial,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 3, pp. 993–1000, Mar. 2006. [31] J. Jugo, J. Portilla, A. Anakabe, A. Suarez, and J. M. Collantes, “Closedloop stability analysis of microwave amplifiers,” Electron. Lett., vol. 37, no. 4, pp. 226–228, Feb. 2001. [32] A. Anakabe et al., “Analysis and elimination of parametric oscillations in monolithic power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, USA, Jun. 2002, pp. 2181–2184. [33] N. Ayllon, J. M. Collantes, A. Anakabe, I. Lizarraga, G. Soubercaze-Pun, and S. Forestier, “Systematic approach to the stabilization of multitransistor circuits,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 8, pp. 2073–2082, Aug. 2011. [34] C. R. White, J. W. May, and J. S. Colburn, “A variable negativeinductance integrated circuit at UHF frequencies,” IEEE Microw. Wireless Compon. Lett., vol. 22, no. 1, pp. 35–37, Jan. 2012. [35] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York, NY, USA: Springer-Verlag, 1990. [36] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. New York, NY, USA: Springer-Verlag, 1990. [37] J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos, 2nd ed. Hoboken, NJ, USA: Wiley, 2002. [38] A. Suárez, Analysis and Design of Autonomous Microwave Circuits. Hoboken, NJ, USA: Wiley, Jan. 2009. [39] R. Quere, E. Ngoya, M. Camiade, A. Suarez, M. Hessane, and J. Obregon, “Large signal design of broadband monolithic microwave frequency dividers and phase-locked oscillators,” IEEE Trans. Microw. Theory Techn., vol. 41, no. 11, pp. 1928–1938, Nov. 1993. [40] A. Suárez and R. Quéré, Stability Analysis of Nonlinear Microwave Circuits. Boston, MA, USA: Artech House, 2003.

[41] V. Rizzoli and A. Neri, “State of the art and present trends in nonlinear microwave CAD techniques,” IEEE Trans. Microw. Theory Techn., vol. MTT-36, no. 2, pp. 343–356, Feb. 1988. [42] S. Rengarajan and C. R. White, “Stability analysis of superluminal waveguides periodically loaded with non-Foster circuits,” IEEE Antennas Wireless Propag. Lett., vol. 12, pp. 1303–1306, 2013. [43] C. C. Hu, Modern Semiconductor Devices for Integrated Circuits. Upper Saddle River, NJ, USA: Prentice-Hall, 2010. [44] J. Long, “Non-Foster circuit loaded periodic structures for broadband fast and slow wave propagation,” Ph.D. dissertation, School Elect. Comput. Eng., Univ. California at San Diego, La Jolla, CA, USA, 2015. [45] A. Suarez, S. Jeon, and D. Rutledge, “Stability analysis and stabilization of power amplifiers,” IEEE Microw. Mag., vol. 7, no. 10, pp. 51–65, Oct. 2006. [46] J. M. Collantes, N. Otegi, A. Anakabe, N. Ayllón, A. Mallet, and G. Soubercaze-Pun, “Monte-Carlo stability analysis of microwave amplifiers,” in Proc. WAMICON, Clearwater, FL, USA, 2011, pp. 1–6.

Almudena Suárez (M’96–SM’01–F’12) was born in Santander, Spain. She received the degree in electronic physics and Ph.D. degree from the University of Cantabria, Santander, in 1987 and 1992, respectively, and the Ph.D. degree in electronics from the University of Limoges, Limoges, France, in 1993. She is currently a Full Professor with the Communications Engineering Department, University of Cantabria. She has authored or co-authored Stability Analysis of Nonlinear Microwave Circuits (Artech House, 2003) and Analysis and Design of Autonomous Microwave Circuits (IEEE–Wiley, 2009). Prof. Suárez is a member of the Technical Committees of the IEEE Microwave Theory and Techniques Society International Microwave Symposium and European Microwave Conference. She was an IEEE Distinguished Microwave Lecturer from 2006 to 2008. She is a member of the Board of Directors of the European Microwave Association. She was the Coordinator of the Communications and Electronic Technology Area for the Spanish National Evaluation and Foresight Agency from 2009 to 2013. In 2014 and 2015, she was the Co-Chair of the IEEE Topical Conference on RF Power Amplifiers. She is the Editor-in-Chief of the International Journal of Microwave and Wireless Technologies from Cambridge University Press journals and an Associate Editor of IEEE Microwave Magazine.

Franco Ramírez (S’03–A’05–M’05–SM’16) was born in Potosí, Bolivia. He received the degree in electronic systems engineering from the Military School of Engineering, La Paz, Bolivia, in 2000, and the Ph.D. degree in communications engineering from the University of Cantabria, Santander, Spain, in 2005. From 1999 to 2000, he was with Ericsson de Bolivia Telecomunicaciones, La Paz, where he was involved in several projects related to GSM and time-division multiple access technologies. From 2005 to 2009, he was a Research Fellow of the “Ramón y Cajal” Programme, funded by the Spanish Ministry of Science and Innovation at the Communications Engineering Department, University of Cantabria, where he is currently an Associate Professor. His current research interests include phase noise, stability, and the development of nonlinear techniques for the analysis and design of autonomous microwave circuits.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Effects of Noisy and Modulated Interferers on the Free-Running Oscillator Spectrum Sergio Sancho , Member, IEEE, Mabel Ponton , Member, IEEE, and Almudena Suarez , Fellow, IEEE

Abstract— A new methodology for the prediction of oscillator phase dynamics under the effect of an interference signal is presented. It is based on a semianalytical formulation in the presence of a noisy or modulated interferer, using a realistic oscillator model extracted from harmonic-balance simulations. The theoretical analysis of the phase process enables the derivation of key mathematical properties, used for an efficient calculation of the interfered-oscillator spectrum. The resulting quasi-periodic spectrum is predicted, as well as the impact of the interferer phase noise and modulation over each spectral component, in particular over the one at the fundamental frequency. It is demonstrated that under some conditions, the phase noise at this component is pulled to that of the interference signal. Resonance effects at multiples of the beat frequency are also predicted. In addition, the effects of interferer phase and amplitude modulation on the oscillator phase dynamics have been studied and compared. For that analysis, efficient simulation techniques have been developed. The analyses have been validated with experimental measurements in an FET-based oscillator at 2.5 GHz, obtaining excellent agreement. Index Terms— Amplitude modulation, frequency-domain analysis, injection pulling, interferer, microwave oscillator, phase modulation, phase noise.

I. I NTRODUCTION

S

EVERAL are the possible effects of an interferer on the local oscillator of a communication system. In general, the oscillation frequency will be pulled toward that of the interferer, so a quasi-periodic spectrum is obtained, with the oscillation frequency affected by the interferer [1]–[3]. Synchronization to the interferer may also occur if the interferer frequency is close enough to the oscillation frequency. Even when the oscillator is inside a phase-locked loop (PLLs), the interferer may pull the voltage-controlled oscillator free-running frequency, shifting the whole hold-in range [4]. Besides these frequency-pulling effects, the oscilla-

Manuscript received July 1, 2017; revised October 25, 2017; accepted November 26, 2017. This work was supported in part by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under research projects TEC2014-60283-C3-1-R and TEC2017-88242-C3-1-R, in part by the Juan de la Cierva Research Program under Grant IJCI-2014-19141, and in part by the Parliament of Cantabria through the project Cantabria Explora under Grant 12.JP02.64069. This paper is an expanded version from the 2017 IEEE MTT-S International Microwave Symposium Conference, Honolulu, HI, USA, June 4–9, 2017. (Corresponding author: Sergio Sancho.) The authors are with the Departamento de Ingeniería de Comunicaciones, Universidad de Cantabria, 39005 Santander, Spain (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2786700

tor phase dynamics can be altered by the interferer noise [5] or modulation. In this paper, the formulation presented in [5] for the study of the interferer influence on the oscillator phase noise has been extended to include the effect of the interferer phase and amplitude modulations on the oscillator phase dynamics. We have limited the investigation to low amplitude interferers, since the strong pulling effects due to a high-power interferer would disrupt the communication. The study of the impact of the interferer on the phase dynamics of the free-running oscillator is the first stage of our investigation. Even if the frequency-pulling effect can be prevented by an afterward phase locking or injection locking of this oscillator, the oscillator phase dynamics can be modified by the interferer. This is because beyond a specific frequency offset from the carrier, the phase-noise spectrum under locked conditions is approximately similar to that of the free-running oscillator [6, chapter 2], [7, chapter 7]. As presented in this paper, the analysis may be of practical interest in some systems where the local oscillator operates in free-running conditions, like homodyne FMCW or Doppler radars [8], based on a self-injectionlocked oscillator. These systems actually behave in freerunning conditions due to the absence of independent sources. Their performance may be preserved when the local oscillator frequency is weakly pulled by an interferer, since the operation principle is based on the frequency difference between two signals (local and returned), generated by the same oscillator. In contrast, the perturbations on the free-running oscillator phase in the form of phase noise or undesired modulation can significantly degrade the circuit behavior [9]. This paper focuses on the case in which a low-power interferer gives rise to small frequency pulling but degrades the phase-noise spectrum. Predicting the oscillator phase degradation in the presence of an interferer will help correct the prototypes at the design stage, so as to make them more robust against the interferer action. To have an impact on the oscillator spectrum, the interferer frequency must be close to that of the interfered oscillator, as otherwise the two signals will have independent phase variations. The analysis is involved since in the general case of an oscillator that is not locked to the interferer, the solution will be quasi-periodic, with two fundamental frequencies: the oscillation frequency, affected by the interferer, and the interferer frequency [1], [10]. In [7], a rigorous general formulation for the noise analysis of nonautonomous circuits with multiple inputs is provided. The formulation is derived in terms of the circuit state variables

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

and stochastic analysis techniques are applied to obtain the power spectral density (PSD) of the circuit output variables. This is different from the present work, where emphasis is placed on the autonomous behavior of the interfered oscillator. However, an efficient multirate method is used in [7] to solve separately the variable components corresponding to different time scales. Such a separation will also be necessary in the interferer problem tackled in this paper, which considers a circuit with an autonomous frequency component due to the oscillation and a nonautonomous component due to the interferer. From a harmonic-balance (HB) simulation viewpoint [11], the only way to address the oscillator phase-noise analysis in the presence of an additional noisy fundamental is through a two-stage procedure. The first stage should be a twotone simulation, with one autonomous fundamental, able to account for the small pulling effects. The second stage should be a phase-noise analysis of the oscillator carrier. However, under a small difference between the interferer and oscillation frequencies, the HB Jacobian matrix is usually ill-conditioned, leading to convergence problems and an inaccurate prediction of the phase-noise spectrum. On the other hand, in the case of a modulated interferer, in ordinary envelope-transient simulations of autonomous circuits, there is usually a shift between the actual oscillation frequency f osc and the fundamental frequency of the Fourier series, fixed by the user [12]. This frequency shift introduces a phase ramp that contaminates the spurious modulation effects. To obtain the oscillation signal, the envelope time step must be sufficiently small to account for this frequency shift. In the presence of a narrowband modulation signal, the relatively small step will give rise to a high computational cost. Here, in order to circumvent these problems, a semianalytical method is proposed. This method allows the derivation of a stochastic differential equation for the autonomous phase, which acts as the single state variable of the system. A realistic model of the interfered oscillator is used, based on derivatives obtained through finite differences in HB, calculated about the free-running point [13]–[15]. The analysis relies on the determination of the phase perturbation of the interfered oscillator in the presence of the undesired signal, as well as its own noise sources. Limit forms of the stochastic analysis in the frequency domain are applied to approach the interferer influence on the far carrier phase-noise spectrum. In addition, a multirate method similar to the one in [7] is used to solve numerically the transference of the interferer phase modulation to the oscillation autonomous component. In Section II, the equation governing the oscillation phase in the presence of a noisy or modulated interferer will be derived. In Section III, a formulation based on this equation will be developed to analyze the effect of the interferer phase noise on the oscillation phase-noise characteristic. Finally, in Section IV, the effects of phase and amplitude modulated interferers on the oscillation phase will be studied and compared. The mechanisms producing spurious modulation of the oscillator phase will be explained in detail. All the analytical and simulation results will be compared with measurements in an FET-based oscillator at 2.5 GHz.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 1. Schematic of the interfered-oscillator operating at f osc = 2.5 GHz based on the FET NE3210S01. The interference is modeled with a current source and introduced to the oscillator through a circulator.

II. G ENERAL F ORMULATION FOR THE A NALYSIS OF THE I NTERFERER I NFLUENCE A. Derivation of the Nonlinear Model The analysis will be illustrated through its application to the FET-based oscillator at f osc = 2.5 GHz in Fig. 1. In the proposed experiment, an interference signal with carrier frequency f in enters the output port through a circulator. The interferer is modeled by its Norton equivalent with a current source i g (t). Interference signal entering through the output port is a realistic situation in front-end systems. However, the formulation can be equally applied for other locations of the interference equivalent source. It will be derived in terms of the voltage signal v(t) at the transistor drain terminal, although, in general, the analysis can be performed for any other node of observation. Note that the analysis can be performed at a node different from the one where the interferer current source is connected, since its effect is modeled by means of transfer functions [13]–[15]. In this circuit, the white noise sources are the thermal sources associated with resistive elements and the channel noise generated by the dc transconductance, whereas the colored noise source is the flicker source associated with the FET device. Using the technique in [16], the effect of all the white and colored [17], [18] noise sources existing in the circuit has been modeled with an equivalent current generator i n (t) connected in parallel at the transistor drain. The PSD of this current source is extracted from HB commercial software simulations of the free-running oscillator. The technique does not require the explicit knowledge of all the noise sources present in the oscillator. It is based on the fitting of a single noise source i n (t), located at the node where the interferer signal enters the circuit. In a circuit-level phase-noise analysis of the noninterfered oscillator, this source is fitted so as to provide the same phase-noise spectrum as the complete set of oscillator noise sources in the resistive elements and active devices. The interferer and noise equivalent sources is expressed as i g (t) = 2Re{U (t)e j ωin t } U (t) = Ig (t)e j [ψ

(t )+θg ]

i n (t) = 2Re{In (t)e j ωin t }

(1)

with ωin = 2π fin and the amplitude and phase components Ig (t) ∈ R and ψ (t) ∈ R accounting in general for modulation or noise processes. The noise current source i n (t) has

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. SANCHO et al.: EFFECTS OF NOISY AND MODULATED INTERFERERS ON FREE-RUNNING OSCILLATOR SPECTRUM

been modeled by means of a single time-varying harmonic component In (t). The introduction of the source i g (t) perturbs each qth harmonic component of the signal v(t) v(t) =

Q 

v q (t)

q=−Q

v q (t) = X q (t)e j qωin t X q (t) = [Vq + Vq (t)]e j ϑq (t )

(2)

where each v q (t) is a narrowband process centered at the frequency component q f in . While the interferer produces small amplitude perturbation components Vq (t), the phase components ϑq (t) are unbounded in the nonsynchronized state [1]. Now, following the semianalytical formulation (SAF) technique proposed in [13]–[15], the Kirchhoff current law is applied to the observation node focusing on the equation corresponding to the first-harmonic component. Then, the implicit function theorem is applied to obtain an envelope-domain model of the oscillator dynamics in the form of a single outertier equation Y [V1 + V1 (t), j (ωosc + ω) + s, U r (t), U i (t)] In (t)e− j φ (t ) = V1 + V1 (t)

(3)

where Y is the first-harmonic total admittance at the observation node, and ω = 2π f = ωosc − ωin , with ωosc = 2π f osc being the free-running frequency. The super indexes r andi mean real and imaginary parts of the corresponding signals. The phase φ (t) ≡ ϑ1 (t) represents the first-harmonic phase component at the interferer frequency fin . In this paper, an interference phenomenon perturbing the steady-state oscillation up to the first order will be considered. This case is quite realistic since the interferer usually enters the oscillator core after crossing filtering paths or electromagnetic shields, which makes its influence on (3) small. Then, (3) can be approximated by a first-order Taylor series about the freerunning state. Following a similar procedure as in [15], this approximation yields the following equation for the phase shift: φ˙ = ω + K s (t)sin[φ − ψ (t)] + K c (t)cos[φ − ψ (t)] + ε (t)

(4)

ε (t) = H r Inr (t) + H i Ini (t) with Bs (t) · YV Yω ×Y V Bc (t) × YV K c (t) = Yω ×Y V 1 YVi − j YVr H = V1 Yω ×Y V K s (t) =

(5)

where the simplifying relations a × b = a r br + a i bi and ab = a r bi − a i br have been introduced. The coefficients K c (t), K s (t), H r , and H i are provided in terms of the derivatives of the admittance function (3) evaluated at the

3

free-running solution ∂Y (V1, ωo , 0, 0) ∂V ∂Y (V1, ωo , 0, 0) Yω = ∂ω Ig (t) YU r Bs (t) = j V1 Ig (t) Bc (t) = − Y i V1 U ∂Y (V1, ωo , 0, 0) YU r,i = . (6) ∂U r,i According to (4), a constant shift of the interferer phase ψ (t) will produce a shift of the same amount in the phase φ(t), which will not affect the spectrum of v 1 (t). Then, without loss of generality, in the following θg = 0 will be assumed to simplify the formulation. Note that, as a difference from [15], the coefficients K c (t), K s (t) are time dependent to model the effect of the interferer amplitude modulation Ig (t). The admittance derivatives YV , Yω , YU r,i in (6) can be calculated through finite differences in commercial HB following the technique described in [15]. Equation (4) provided by the SAF technique can be used to calculate the oscillator phase dynamics in the presence of a noisy and modulated interferer. YV =

B. Validity of the Proposed Model The first-order model is composed of the derivatives of the oscillator total-admittance function, calculated with respect to the node amplitude, frequency, and interference signal, evaluated at the (noninterfered) free-running point. All the rest of circuit variables and harmonic frequencies are taken into account when applying finite differences to the auxiliary generator [13]–[15] used for the practical calculation of these derivatives. The modeling procedure is totally general and independent of the particular oscillator circuit and its operation frequency. However, there is a limitation on the interferer power, which should be sufficiently small for the first-order Taylor-series expansion to be applicable. In fact, the analysis presented is most relevant for a low interferer power. For too high power, frequency-pulling effects would be so strong that the oscillator would significantly impair the communication system. To evaluate the validity of the proposed approach, the oscillator is assumed to be injection locked by the interference of a particular amplitude Ig , even though this regime is not the object of this investigation. The synchronization bandwidth is calculated both through the semianalytical method, based on the first-order Taylor series approximation, and using a circuit-level HB simulation, with the aid of one auxiliary generator [13]–[15]. The two analyses are computationally undemanding. If the synchronization bands agree with the two different methods, the first-order approximation must be a valid one, so it will be applicable in unlocked conditions for the interferer amplitude Ig . The accuracy evaluation under injection-locked conditions takes advantage of the oscillator operation in a periodic regime, much simpler to analyze than the unlocked quasi-periodic one that is the object of this investigation.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

interferer, (4) gets simplified and the noise current source i n (t) produces the same phase-noise spectrum as the HB commercial software applied to the free-running oscillator. In order to calculate the phase noise of the interfered oscillator, in the first place, the steady-state solution in the absence of interferer modulation and noise sources must be analyzed. A. Phase Dynamics in the Absence of Interferer Modulation and Noise Sources In this case, (4) represents an autonomous system that can be written as φ˙ = ω + K s sinφ + K c cosφ = g(φ) Fig. 2. Synchronization range (SR). Comparison between the SAF model (7) and simulation in commercial HB software.

Let us consider the case of a noise and modulation-free interferer of amplitude Ig and phase ψ. As stated in [1], the oscillation can synchronize to the interferer provided the frequency difference | f | = | f osc − f in | is below a bifurcation limit, with f osc being the oscillation frequency in the absence of the interferer. In the synchronized state, the frequency ˙ difference φ(t) between the oscillation and the interferer vanishes. In that case, model (4) predicts the set of interferer frequency values providing synchronized solutions 1 (K s sinα + K c cosα), α ∈ [0, 2π] (7) 2π with α = φ − ψ. This prediction can be validated by comparison with the synchronization range (SR) using a circuit-level HB simulation with the aid of an auxiliary generator [19], [20]. The two analyses are computationally undemanding. If the synchronization bands agree with the two different methods, the first-order approximation must be a valid one, so it will be applicable in unlocked conditions for the interferer amplitude Ig . As an example, this comparison is carried out in Fig. 2 for two values of the interferer amplitude. In the first case, for Ig = 0.3 mA, the SR predicted with (7) agrees with the one obtained in commercial HB software. This indicates that model (4) is valid for this interferer amplitude. In the second case, for Ig = 1.4 mA, the disagreement between the results of (7) and the commercial HB software indicates that the interferer amplitude is too high for model (4) to be accurate. In this paper, the nonsynchronized case, i.e., the case of an interferer with frequency f in out from the SR, will be analyzed. Note that once model (4) is validated for a given value of the interferer amplitude, the interferer influence will decrease and eventually become negligible as the frequency difference | f | = | f osc − f in | grows. The narrowband noisy or modulated interferer considered in the subsequent sections will introduce small amplitude or frequency perturbations, with negligible influence on the model validity. f in = fosc +

III. P HASE -N OISE A NALYSIS The phase noise is given by the perturbation component of the phase variable φ (t) due to the effect of the noise sources present in i n (t) and i g (t). In the absence of the

(8)

with K c andK s being constant coefficients. The phase shift becomes a constant value φ = φs , fulfilling g(φ s ) = 0, only when the free-running frequency synchronizes to the interferer. Then, in the more general unsynchronized conditions g(φ) = 0, ∀φ must be fulfilled, implying that φ (t) grows or decreases monotonically. This fact, together with the system autonomy, produces an accumulation effect of the numerical error associated with the time-integration resolution of (8). As it is explained in Section III-B, this error may produce qualitative changes on the predicted spectrum about the free-running frequency in the presence of the interferer. To overcome this problem, the following procedure is followed. Let us assume that g(φ) > 0, ∀φ. The case g(φ) < 0, ∀φ is symmetric. The time required by the phase variable to pass through an arbitrary interval [φa , φb ] is given by  φa dφ T (φa , φb ) = φa g(φ) = u 0 (φb − φa ) + q(φb ) − q(φa ) 1 = u 0 + u(φ) u(φ) = ˜ g(φ)  2π 1 u0 = u(φ)dφ 2π 0 q(φ) ≡

u(φ)dφ ˜

q(φ) = q(φ ± 2π).

(9)

Note that the φ-periodic function u(φ) has been separated into its dc component u 0 and the term u(φ) ˜ fulfilling  2π u(φ)dφ ˜ = 0. (10) 0

From the definition of g(φ), the phase shift fulfills Tb = T (φ, φ + 2π) = u 0 2π + q(φ + 2π) − q(φ) = u 0 2π, ∀φ.

(11)

˙ Property (11) implies that φ(t) is periodic since ˙ + Tb ) = g[φ (t + Tb )] φ(t = g[φ (t) + 2π] = g[φ (t)] ˙ = φ(t).

(12)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. SANCHO et al.: EFFECTS OF NOISY AND MODULATED INTERFERERS ON FREE-RUNNING OSCILLATOR SPECTRUM

5

Considering (11) and (12), the phase variable can be expressed as φ (t) = ωb t +

N 

Pn e j nωb t , ωb = 2π f b = 2π/Tb . (13)

n=−N

The components { f b , P−N , . . . , PN } can be calculated in the frequency domain by introducing expression (13) in (8) and solving the resulting HB system ¯ ωb ), n = −N, . . . , N ωb + j nωb Pn = G n ( P,

(14)

where P¯ is the vector containing the harmonic components Pn and each G n is the nth harmonic component of the Tb -periodic signal g[φ (t)]. System (14) contains 2N + 1 equations and ¯ It can be balanced by making use 2N + 2 unknowns { f b , P}. of the autonomy of (8). Due to this autonomy, solution (13) fulfills (8) for any arbitrary constant time shift τ of the form N 

φ (t + τ ) = ωb (t + τ ) +

Pn e j nωb (t +τ )

n=−N N 

= ωb t +  Pn (τ ) =

Pn (τ )e j nωb t

n=−N

e j nωb τ ,

Pn n = 0 P0 + ωb τ, n = 0.

(15)

Then, we can select τ to fix P1i (τ ) = 0, where the superindex i means imaginary part. This additional equation is combined with system (14) to balance the number of equations and unknowns, yielding an algebraic system solvable by a number of optimization techniques, like Newton–Raphson. ¯ are calculated, expression (13) is Once the unknowns { f b , P} used to obtain the process v 1 (t) in (2), providing the spectrum nearby the interference frequency f in . The envelope of this process is X 1 (t), which can be expanded in Fourier series as X 1 (t) ≈ V1 e j φ = V1 e

(t )

j ωb t



exp

N 

 j Pn e

j nωb t

n=−N

=

M 

X 1k e j kωb t

(16)

k=−M

where expression (13) has been applied. Then, the spectrum of v 1 (t) has the form v 1 (t) = X 1 (t)e j ωin t M  = X 1k e j ωk t k=−M

ωk = 2π fk = 2π ( fin + k f b ).

(17)

Equation (17) implies that v 1 (t) is a multitone signal containing the components X 1k at the intermodulation frequencies f k = f in + k f b . For this reason, fb will be called the beat frequency. The oscillator free-running frequency is pulled from  ≡ f1 = f in + f b . This theoretical result has f osc to f osc been verified for the case of an interferer with amplitude

Fig. 3. Spectrum of the first-harmonic component X 1 (t) of the drain voltage v(t) in the presence of an interferer with Ig = 0.2 mA and  f = −10 MHz.

Ig = 0.2 mA and frequency f in = fosc + 10 MHz. In the ¯ have been calculated by first place, the unknowns { f b , P} solving system (14) with N = 5 harmonics, together with the additional equation P1i = 0. Then, the phase shift φ (t) has been properly sampled and introduced in the function X 1 (t) ≈ V1 e j φ (t ) to obtain M = 10 coefficients X 1k of the Fourier series (16) at the frequencies k f b . The spectrum in Fig. 3 contains the frequency components of the interfered-oscillator spectrum at f k = fin +k f b , with f in and f b being the interferer and beat frequencies, respectively. In the SAF, the value of the beat frequency f b is calculated through numerical resolution of system (14). For comparison, the spectrum of the time-varying first-harmonic component X 1 (t) has been obtained in HB commercial software using the envelope transient technique [21]–[23]. The beat frequency fb calculated through both techniques may differ slightly due to fact that the envelope-transient method takes into account the interferer effect at all the harmonic terms q f in with −Q ≤ q ≤ Q, and the SAF only takes into account the interferer effect at the fundamental frequency f in . This discrepancy becomes more evident for the high-order components f k = f in + k f b , as k gets increased. In Fig. 3, the harmonic components X 10 and X 11 corresponding, respectively, to the interferer and the free-running component are indicated. Observe that the freerunning frequency has been pulled from its unperturbed value  ≡ f 1 = f in + f b , with f osc = f in − 10 MHz to fosc f b ≈ −8 MHz. Note that the frequency-pulling phenomenon can be observed even in the presence of a small amplitude interferer, provided that the frequency difference | f in − fosc | is small enough, as demonstrated in [1]. The frequency domain technique (13) and (14) represents a significant advance from [14], where (8) was solved through a time integration technique. Apart from avoiding the already mentioned accumulation of numerical error, the new method is less computationally costly due to the small number of ¯ to be calculated. unknowns { f b , P} Note that in the case of a PLL or an injection-locked oscillator the situation would be different. In those cases, the admittance function Y in (3) has a dependence on the amplitude and phase components of the reference or external generator signal [16], [24]. This dependence removes the

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

system autonomy of the interfered system quasi-periodic solution. Therefore, the interfered system does not remain invariant under phase shifts, providing different phase-noise results than the autonomous case, which will be the one analyzed here. B. Phase Perturbation Due to the Noise Sources Let φ (t) = φ0 (t) be the phase shift in absence of noise sources. The interferer phase noise will be modeled by the input source time-varying phase ψ (t), considered as a small amplitude stochastic process, whereas its amplitude Ig remains constant. The noisy components introduced by the local sources and the interferer perturb the steady state φ0 (t) in the form φ (t) = φ0 (t) + φ (t). Provided φ (t) is small, (4) can be linearized about φ0 (t) obtaining ˙ φ(t) = b0 (t)[φ (t) − ψ (t)] + ε (t) dg[φ0 (t)] b0 (t) ≡ dφ = K s sinφ0 (t) + K c cosφ0 (t).

(18)

Equation (18) shows that the phase perturbation is governed by a linear time-variant stochastic differential equation determined by the Tb -periodic function b0 (t) and the noise sources ε (t) and ψ (t). Here, this equation will be solved by means of a time-frequency formulation, expressing the variables in a Fourier series with fundamental frequency f b , in terms of time-varying harmonic components. Following this procedure, the term of local noise sources ε (t) is expressed as [25]: ε (t) = εw (t) + εc (t) εw (t1 )εw (t2 )∗  = δ (t1 − t2 ) N  E n (t)e j nωb t εw (t) = E n (t) = E k ( f )El∗ ( f )

n=−N  B/2 −B/2

n (t) =

−B/2

n ( f )e j 2π f t d f

N 

Bn−l l (t) − Bn ψ (t)

l=−N

+ E n (t) + δn0 εc (t), n = −N, . . . N (21) with each Bn being the harmonic component of b0 (t) at the frequency n f b . These harmonics can be numerically obtained ¯ The noisy process through (18) from the components { f b , P}. v 1 (t) is obtained by introducing expression (20) in expansion (16)  N  M   j nωb t X 1 (t) = exp n (t)e X 1k e j kωb t

n =0

(19)

n (t)e j nωb t

n=−N  B/2

˙ n (t) = j nωb n (t) + 

E n ( f )e j 2π f t d f

l δ = 2N k

N 

differential equations is obtained:

⎫ k=−M ⎧n=−N M ⎬  ⎨ j nωb t = exp n (t)e X 1k e j [kωb t +0 (t )] ⎭ ⎩

where δkl is the Kronecker delta, εw (t) and εc (t) represent, respectively, the contributions of the local white and colored noise sources, and E n (t), ψ (t) and εc (t) are uncorrelated baseband stochastic processes with bandwidth B. The structure of the PSD function |ε( f )|2  depends on the type of colored sources present in the circuit. Due to the structure of the noise sources, the phase perturbation is a cyclo-stationary process that can be expressed as φ (t) =

Fig. 4. Phase noise in the presence of an interferer with Pin = −29 dBm  is pulled and f in = f osc + 3 MHz. The phase-noise characteristic about f osc toward the interferer phase-noise characteristic and noise resonances appear at k f b frequency offsets.

(20)

where each time-varying harmonic n (t) is a baseband process with bandwidth B. Introducing expression (20) in (18) and equating the terms that correspond to the same harmonic order, the following linear time-invariant system of stochastic

≈ X 1k (t)

=

M 

k=−M

X 1k (t)e j kωb t

k=−M j 0 (t ) X 1k e

(22)

where high-order terms in the perturbation harmonics n (t) have been neglected. Equation (22) shows that the harmonic components X 1k (t) become time varying due to the effect of the noise sources. In particular, the phase noise about  ≡ f 1 is given by the PSD S1 ( f ) of the normalf osc ized process X 11 (t)/|X 11 | = e j 0(t ) [7], [18], [25]–[27]. As stated in [18], [26], and [28], far enough from the carrier, the PSD S1 ( f ) can be approached by the second-order moment |0 ( f )|2 . This moment is obtained by translating system (21) to the frequency domain. Assuming small interferer amplitude, harmonic components Bn with |n| > 1 are neglected, obtaining the system j 2π (− f b + f )−1 ( f ) = B−1 0 ( f )− B−1ψ ( f )+ E −1( f ) j 2π f 0 ( f ) = B−1 1 ( f )+ B0 0 ( f )+ B1−1 ( f ) + εc (t) + E 0 ( f ) − B0 ψ ( f )

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. SANCHO et al.: EFFECTS OF NOISY AND MODULATED INTERFERERS ON FREE-RUNNING OSCILLATOR SPECTRUM

j 2π ( f b + f )1 ( f ) = B1 0 ( f ) − B1 ψ ( f ) + E 1 ( f ). (23) The second equation of system (23) shows that the direct conversion between the interferer phase noise ψ (t) and 0 (t) is due to the harmonic component B0 . In unsynchronized conditions, this component can be calculated as  Tb 1 B0 = b0 (t)dt Tb 0  Tb 1 dg[φ0(t)] = dt Tb 0 dφ  g(Tb ) 1 dg = Tb g(0) g 1 g(Tb ) = log Tb g(0) 1 φ˙ 0 (Tb ) = log Tb φ˙ 0 (0) =0 (24) where g = φ˙0 has been applied. Consequently, there is no direct conversion from the interferer phase noise ψ (t) to 0 (t). This is a key result that is in contrast with the synchronized case, where φ0 becomes constant and b0 (t) in (18) agrees with its dc value B0 . The application of the property B0 = 0 to system (23) allows, after some algebra, the extraction of the following intuitive equation that models the phase noise at the pulled free-running frequency in the presence of the interferer:

2 |ε( f ) |2  α 2 2 |ψ | + |0 ( f )|  ≈ ( f ) 2 4π 2 f 2 α + 4π 2 ( f 2 − fb ) (25) where α = 2|B1 |2 acts as a sensitivity coefficient that determines the influence of the interferer phase noise, and f is the frequency offset from the carrier at f 1 = f osc and the second term is the oscillator’s own phase noise. Equation (25) is valid for large enough values of the frequency offset f . In order to obtain (25), small interferer amplitude has been assumed. In other cases, system (21) should be translated to the frequency domain considering a higher number N of harmonic coefficients Bn . In the absence of an interferer α becomes zero and (25) agrees with the free-running oscillator phase noise. The introduction of the interferer provides α > 0, increasing the level of the phase-noise characteristic, since it adds a term proportional to the interferer phase noise. The interferer influence is most noticeable when its phase-noise characteristic is higher than that of the free-running oscillator. In that case, (25) predicts that the phase-noise characteristic about  is pulled toward the interferer phase-noise curve. This f osc behavior is verified in Fig. 4, where the measured phase-noise spectrum and the one predicted by (25) (SAF) are compared. In the measurement, the direct spectrum technique has been applied, using the phase-noise measurement personality in an Agilent PSA Spectrum Analyzer E4446A (option 226).The resolution bandwidth and video bandwidth of the measured spectra is RBW = 10 kHz and VBW = 10 kHz, respectively.

7

TABLE I P HASE N OISE AT 1 MH Z V ERSUS I NTERFERER P OWER

For the analysis, an interferer with Pin = −29 dBm and f in = f osc + 3 MHz has been introduced. Due to the pulling  = f − f , effect, the oscillation frequency is shifted to f osc in b with f b ≈ 2 MHz. The interferer is not represented. Instead, its measured phase noise from zero offset frequency is traced for comparison. As predicted by (25), the oscillator phase noise is pulled to that of the interferer. Equation (25) also demonstrates a resonance effect, with maximum phase noise at an offset frequency about f b . Note that the singularity predicted by (25) at f b is fully consistent with the presence of a steady-state  + f . In the measurement, several spectral line at f2 = f osc b resonances at frequency offsets k f b for k > 1 are observed, corresponding to the intermodulation components X 1k of the interfered spectrum predicted in Section III-A. In order to obtain an intuitive result, in the derivation of (25) only the terms Bk up to k = 1 have been considered. Therefore, only the first resonance at f b can be predicted. The rest of resonances could be obtained by considering the corresponding coefficients Bk for k > 1 when translating system (21) to the frequency domain. Note that the key property (24) has enabled the prediction of a phase-noise spectrum that is qualitatively different to that of the synchronized case, where the phase-noise characteristic agrees with that of the interferer up to a frequency offset from the carrier [5]. The result B0 ≈ 0 is numerically obtained when using φ0 (t) calculated from the frequency domain technique (13) and (14), while this result is often unobserved when solving (8) through time integration, due to the cumulative numerical error. Table I shows that as expected from the theoretical analysis, the interfered oscillator measured phase noise is pulled toward that of the interferer as Pin increases. The measurements have been compared with the results of (25). IV. A NALYSIS OF THE E FFECT OF THE I NTERFERER M ODULATION A. Phase Modulation In this section, the influence of a phase-modulated (PM) interferer on the free-running oscillation is analyzed. The modulation signal is modeled by the input source time-varying phase ψ (t), considered as a stochastic process, whereas the amplitude component Ig is fixed to a constant value. As a difference from the previous phase-noise analysis, in this case the magnitude of ψ (t) is not always small, and therefore linearization (18) is not applicable. In the presence of the interferer phase modulation, (4) becomes φ˙ = ω + K s sin[φ − ψ (t)] + K c cos[φ − ψ (t)] ≡ g[φ − ψ (t)].

(26)

In order to study the modulation effect, the components corresponding to the noise sources have been removed from (26).

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

In the following technique, these components can be included to analyze the noise-modulation-combined effect, in a straightforward way. In the case that the modulation signal ψ (t) contains frequency components of the order of fb , (26) can be directly solved through time integration, as it has been done in [14] using a less elaborate SAF. Nevertheless, that method may become computationally demanding when the modulation signal ψ (t) is a narrowband process with bandwidth B f b . In such cases, the use of time-frequency methods is advisable [21]–[23]. Here, the solution φ (t) to (26) has been expressed considering that the process ψ (t) modulates ¯ in (13). These components are the components { f b , P} narrowband processes whose time variation is slow when compared with Tb = 1/ f b φ (t) = ωb (t)t +

N 

Pn (t)e j nωb (t )t

(27)

n=−N

with ωb (t) = 2π f b (t). Introducing expression (27) in (16), the spectrum in the presence of a PM interferer is obtained X 1 (t) ≈ V1 e j φ

(t )

=

M 

X 1k (t)e j kωb0 t

(28)

k=−M

with ωb0 being the beat frequency in the absence of modulation (ψ = 0). The harmonic components X 1k (t) become time varying due to the effect of the modulation signal. In particular, as stated in the previous section, the PM  ≡ f is given spectrum about the oscillation frequency f osc 1 by the term X 11 (t). To obtain this term, in the first place, a quasi-static analysis of the effect of the narrowband PM interferer on the phase shift φ (t) will be carried out. Using this analysis, the expression of X 11 (t) in terms of the time¯ varying components { f b (t), P(t)} will be obtained. Finally, a technique to simulate these components will be derived. 1) Quasi-Static Approach: To illustrate the effect of the PM interferer on the oscillation component X 11 (t), the case of a step phase modulation signal has been considered. In this case, the process ψ (t) is given by the deterministic signal  0, t < ts ψ (t) = (29) ψ1 , t ≥ ts . To clarify the effect of the step modulation, the evolution ˙ is shown in of the system trajectory in the phase space (φ, φ) Fig. 5. For t < ts , the trajectory lies on the curve (φ, g(φ)), and the time domain expression for the phase variable is given by (13)  Pn e j nωb0 t , t < ts . (30) φ (t) ≡ φ0 (t) = ωb0 t + P0 +

Fig. 5. Quasi-static approach. Effect of the step interferer on the evolution of the system trajectory in the phase space.

where τ is a constant time shift due to the system autonomy and with Pn (τ ) defined in (15). When compared to expression (30), (31) shows that the effect of the step PM is to shift the phase of the components Pn for n = 0 and to make the amplitude of the dc component jump from P0 to P0 (τ ) + ψ1 . The order of magnitude of this jump can be bounded by applying the continuity property at t = ts φ0 (ts ) = φ1 (ts ) → |P0 | = |P0 (τ ) + ψ1 − P0 |       j nωb ts j nωb τ   = Pn e (1 − e ) n=0   |Pn |. ≤2

Equation (32) shows that the phase jump |P0 | due to an arbitrary modulation step is limited by the amplitude components |Pn |. The amplitude of the components Pn for n = 0 is directly proportional to the interferer power. Therefore, in the case of a low-power interferer, |P0 | will remain small for any arbitrary value ψ1 of the step PM. In the following, these results will be applied to obtain an expression of the modulated harmonic component X 11 (t) at the oscillation frequency. 2) Effect of the PM Interferer at the Oscillation Frequency: The multitone spectrum in the vicinity of the oscillation  is obtained as in (16) from the time-varying frequency f osc harmonic component X 1 (t) which, in the presence of a PM interferer, becomes   N  j ωb (t )t j nωb (t )t X 1 (t) ≈ V1 e exp j Pn (t)e

n =0

n=−N

At t = ts , the trajectory jumps to the curve (φ, g(φ − ψ1 )). The time domain expression for the phase variable beyond this point is φ˙ = g(φ − ψ1 ) → φ (t) ≡ φ1 (t) = φ0 (t + τ ) + ψ1 = ωb0 t + P0 (τ ) + ψ1 +

 n =0

(32)

n =0

= V1 e j [ωb (t )t +P0(t )] exp

⎧ ⎨ ⎩

n =0

= V1 e j [ωb (t )t +P0(t )] + O(α)  α= j Pn (t)e j nωb (t )t

j Pn (t)e j nωb (t )t

⎫ ⎬ ⎭

(33)

n =0

Pn (τ )e

j nωb0 t

(31)

where expression (27) for the phase shift φ (t) has been applied. In (33), the signal X 1 (t) is generated by the addition

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. SANCHO et al.: EFFECTS OF NOISY AND MODULATED INTERFERERS ON FREE-RUNNING OSCILLATOR SPECTRUM

9

of two terms. Comparing this result with (28) it is seen that the first term V1 e j [ωb (t )t +P0(t )] is contributes entirely to X 11 (t). The second term O(α) produces PM harmonic components at f k , for k = −M, . . . , M. Derivation (29)–(31) showed that the amplitudes of the components Pn for n = 0 remain small for a narrowband low-power PM interferer. Then, the contribution of O(α) to the term X 11 (t) can be neglected, obtaining X 11 (t) ≈ V1 e j ϕ

(t )

, ϕ (t) = [ωb (t) − ωb0 ]t + P0 (t). (34)

Approximation (34) shows that the PM interferer modulates the phase ϕ(t) of the harmonic component X 11 at the oscillation  . Note that in the previous quasi-state analysis, frequency f osc the effect of a step phase modulation of arbitrary magnitude ψ1 on the oscillation phase is ϕ = P0 (τ ) + ψ1 − P0 = P0

(35)

where |P0 | is limited by the small amplitude components |Pn |, as described in (32). Considering this case as a quasistatic approximation of a narrowband PM interferer, result (35) suggests that such modulation will produce a small perturbation on the phase ϕ (t). In order to verify this effect, in the following, a technique to simulate the time-varying phase ϕ (t) in the presence of the PM interferer will be derived. 3) Time-Frequency Simulation Technique: In the first place, the function φ (t) in the presence of a narrowband PM interferer is expressed in terms of a bi-variate function ˆ t) φ (t) = φ(t, ˆ 1 , t2 ) = ωb (t1 )t2 + φ(t

N 

Pn (t1 )e j nωb (t1 )t2

(36)

with ωb (t1 ) = 2π f b (t1 ) = 2π/Tb (t1 ) and the time variables t1 and t2 representing the slow and fast scales, respectively. Applying the same procedure to the nonlinear function g[φ − ψ (t)] in (26), we obtain g[φ − ψ (t)] = g(t) = g(t, ˆ t) ˆ 1 , t2 ) − ψ (t1 )] g(t ˆ 1 , t2 ) = ω + K s sin[φ(t ˆ 1 , t2 ) − ψ (t1 )]. + K c cos[φ(t

(37)

In [7] and [22], the method to obtain the equation governˆ 1 , t2 ) associated with a generic ing the bi-variate process φ(t ordinary differential equation (ODE) is derived. When applying this derivation to (26), the following partial differential equation is obtained:   ˆ 1 , t2 )  ˆ 1 , t2 )  ∂ φ(t ∂ φ(t = g(t ˆ 1 , t2 )|t1 =t2 . (38) t1 =t2 +   ∂t1 ∂t2  t1 =t2

In order to get the equation associated with the slow time scale t1 , function g(t ˆ 1 , t2 ) is expressed as N 

¯ containing 2N + 2 with n = −N, . . . , N. The set { f b , P} components is the set of state variables of system (40). In accordance with analysis (15), the additional equation P1i = 0 is included to balance the number of equations and variables. System (40) can be solved through backward-Euler time integration. Using this technique, the ODEs in (40) are discretized at each time value t1 , providing an algebraic system of the form Hn [ f b (t1 ), Pn (t1 )] = G n (t1 ), n = −N, . . . , N

n=−N

g(t ˆ 1 , t2 ) =

Fig. 6. Comparison between the simulated and measured spectra of the sinusoidal PM interferer and the harmonic component X 11 (t). An interferer with Pin = −25 dBm,  f = −3.6 MHz has been introduced. The modulation frequency and index are f m = 100 kHz and ψ = 0.3 rad, respectively.

G n (t1 )e j nωb (t1 )t2 .

(39)

n=−N

Now, introducing expressions (36), (37), and (39) in (38) and equating the terms that correspond to the same harmonic order the following system of first-order ODEs is obtained:   [ωb (t1 ) + ω˙ b (t1 )t1 ] j n Pn (t1 ) + δn0 + P˙n (t1 ) = G n (t1 ) (40)

(41)

¯ 1 ) must be solved for each t1 . where the unknowns f b (t 1 ), P(t The components G n (t1 ) are calculated in three steps. ˆ 1 , t2 ) is constructed from 1) The function φ(t ¯ 1 )} as indicated in (36). The time variable { f b (t 1 ), P(t t2 takes 2N + 1 equally spaced samples in the interval [0, Tb (t1 )]. ˆ 1 , t2 ) as 2) The function g(t ˆ 1 , t2 ) is constructed from φ(t in (37). 3) The harmonic components G n (t1 ) are calculated as N {G n (t1 )}n=−N = Ft2 {g(t ˆ 1 , t2 )}

(42)

with the operator Ft2 representing the fast Fourier transform in the t2 variable. Once system (40) is solved, the set of ¯ is introduced in (35) to obtain state variables { f b (t), P(t)} the slow-varying phase ϕ (t) providing the phase modulation  . This technique has been at the oscillation frequency f osc applied to simulate the effect of a sinusoidal PM interferer with with Pin = −25 dBm,  f = −3.6 MHz. The modulation frequency and index are f m = 100 kHz and ψm = 0.3 rad, respectively. In Fig. 6, the simulated and measured spectra of the PM interferer and the harmonic component X 11 (t) are compared. The resolution bandwidth and video bandwidth of the measured spectra is RBW = 10 kHz and VBW = 10 kHz, respectively. As predicted by the quasi-static analysis, the sideband  due to the PM are highly attenuated components about f osc with respect of those of the interferer. If these sidebands are too small, their observation in the measured spectrum can be tricky due to the noisy spectrum.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

In the analysis of Fig. 6, the simulation technique predicts a 12-dB attenuation of the PM sidebands, whereas in the measurement 17-dB attenuation is observed. The discrepancy in the PM case is attributed to a combination of several facts. The phase perturbation due to the PM interferer is simulated by integrating the envelope transient (40) in the slow time scale t1 . The oscillation is not phase locked to the interferer. This means that system (26) is autonomous, and therefore, for each time value, it remains invariant under a perturba˙ As a tion along a particular direction of the space (φ, φ). consequence, the possible components of the numerical noise along this direction are accumulated, generating discrepancies. As observed in Fig. 8, these discrepancies are much smaller in the amplitude modulation (AM) case, where no numerical integration is required.

Fig. 7. Variation of the beat frequency f b (Ig ) for an interferer at f in = f osc + 3.6 MHz.

B. Amplitude Modulation The case of an AM interferer will be analyzed by expressing the equivalent current source i g (t) as in (1), with the amplitude Ig (t) being a narrowband process modeling the AM. Without loss of generality, in the absence of PM the interferer phase ψ (t) can be set to zero. As a result, (4) becomes φ˙ = ω + K s (t) sin φ + K c (t) cos φ.

(43)

The phase shift φ (t) can be expressed using structure (13) ¯ being modulated by the interferer with the components { f b , P} amplitude Ig (t) N 

φ (t) = ωb (t)t +

Pn (t)e j nωb (t )t

n=−N

ωb (t) = 2π f b [Ig (t)] Pn (t) = Pn [Ig (t)].

(44)

Assuming that the interferer amplitude Ig (t) remains small, approximation (33) and (34) is applicable to obtain the term X 11 (t) providing the component at the oscillation frequency. Note that, as a difference from the case of the PM interferer, in this case the magnitude of the resulting phase modulation ϕ (t) at the oscillation frequency is not limited by condition (32). The time evolution of the phase shift φ(t) can be simulated for a realization of Ig (t) using the technique described in Section IV-A3 and then calculate the term X 11 (t) from (34) to obtain the spectrum X 11 ( f ) about the oscillation  . frequency f osc Deeper insight on the effect of the AM interferer can be gained by taking into account that, in the case of a narrowband AM interferer, the phase modulation ϕ (t) in (34) produces a spurious frequency modulation (FM) of the oscillation component that can be approximated by ϕ(t) ˙ ≈ ωb [Ig (t)] − ωb0 →  t ϕ(t) ≈ ωb [Ig (s)]ds − ωb0 t

(45)

0

with ωb0 being the beat frequency value for a given arbitrary value Ig0 of the interferer amplitude. In (45), the contri˙ have been bution of the terms ω˙ b (t) and P˙0 (t) to ϕ(t) neglected. This is because, due to the system autonomy, the

Fig. 8. Comparison between the simulated and measured spectra of the sinusoidal AM interferer and the harmonic component X 11 (t). An interferer with Pin = −25 dBm,  f = −3.6 MHz has been introduced. The modulation frequency and index are f m = 100 kHz and Ig /Ig0 = 1%, respectively.

beat frequency f b [Ig (t)] may vary freely with the interferer amplitude, whereas ωb (t) and P0 (t) are slow-varying signals whose time derivatives are comparatively small. The function f b (Ig ) can be easily calculated with the frequency domain technique described in Section III. This function is represented in Fig. 7 for the case of an interferer with  f = −3.6 MHz. For the analysis, system (14) has been solved with N = 5 harmonics. In order to compare the effect of AM and PM interferers, the function fb (Ig ) of Fig. 7 has been used to analyze the FM produced at the oscillation frequency by a sinusoidal AM interferer of the form Ig (t) = Ig0 + Ig sin 2π f m t

(46)

where Ig0 produces an interferer power Pin = −25 dBm, f m = 100 kHz, and Ig has been set to produce the same modula as in the previous PM example tion power spectrum about f osc of Fig. 6. Introducing the signal (46) into expression (45), the sidebands of the FM component X 11 (t) ≈ V1 e j ϕ (t ) at f m have been obtained, with the result of Fig. 8. The resolution bandwidth and video bandwidth of the measured spectra is RBW = 10 kHz and VBW = 10 kHz, respectively. The mean amplitude value Ig0 is marked in Fig. 7. Comparing

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. SANCHO et al.: EFFECTS OF NOISY AND MODULATED INTERFERERS ON FREE-RUNNING OSCILLATOR SPECTRUM

11

R EFERENCES

Fig. 9. Phase-noise spectrum in the presence of a sinusoidal PM interferer with Pin = −25 dBm,  f = −3.6 MHz. The modulation frequency and index are f m = 1 MHz and ψ = 1 rad, respectively.

Figs. 6 and 7, it is observed that the relative sidebands of the AM interferer are much lower than in the PM case and produce a similar spurious modulation at the oscillation component. This indicates that, in this case, the oscillation phase is more sensitive to the AM interferer than to the PM one due to the free variation of the beat frequency with the interferer amplitude. Note that the discrepancies between simulation and measurements appearing in Fig. 6 are much smaller in the AM modulation case, where no numerical integration is required. Finally, in Fig. 9, the measured phase-noise spectrum in the presence of a sinusoidal PM interferer with Pin = −25 dBm,  f = −3.6 MHz is introduced. The modulation frequency and index are f m = 1 MHz and ψ = 1 rad, respectively. As predicted by the theoretical analyses of Sections III and IV, the components of the PM interferer perturb the phasenoise spectrum in the vicinity of the frequency offsets f m  − f |. For comparison, the phase-noise simulation and | f osc in using (25) has been superimposed. If the PM modulation index is low enough, superposition principle applies and the PSD of the phase perturbation resulting from simulation (41) can be added to the phase-noise PSD, considering that both processes are uncorrelated. The spurious PM components are better observed in the spectrum analyzer than in the phasenoise characteristic. For this reason, we consider more accurate to compare the simulated PM effect with the signal power spectrum, as in Fig. 6. V. C ONCLUSION The influence of the interferer phase noise and modulation on the free-running oscillator spectrum has been analyzed. An SAF has been derived for the oscillation phase noise. This leads to a simple and insightful equation describing the pulling effect of the interfered-oscillator phase noise toward that of the interference signal. The cases of PM and AM interferers have been studied, providing efficient simulation techniques to predict their effects on the oscillation spectrum. The mechanisms producing spurious modulation of the oscillator phase have been explained in detail. All the predictions have been verified by measurements in an FET-based oscillator at 2.5 GHz.

[1] B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuits, vol. 39, no. 9, pp. 1415–1424, Sep. 2004. [2] S.-C. Yen and T.-H. Chu, “An Nth-harmonic oscillator using an N-push coupled oscillator array with voltage-clamping circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 2169–2172. [3] Z. Li and K. Wu, “On the leakage of FMCW radar front-end receiver,” in Proc. Global Symp. Millim. Waves, 2008, pp. 127–130. [4] S. Sancho, M. Ponton, A. Suarez, and F. Ramirez, “Analysis of injection pulling in phase-locked loops with a new modeling technique,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 3, pp. 1200–1214, Mar. 2013. [5] S. Sancho, M. Ponton, and A. Suarez, “Nonlinear technique for the analysis of the free-running oscillator phase noise in presence of an interference signal,” in IEEE MTT-S Int. Microw. Symp. Dig., Honolulu, HI, USA, Jun. 2017, pp. 79–82. [6] U. L. Rohde, Microwave and Wireless Synthesizers: Theory and Design. Hoboken, NJ, USA: Wiley, 1997. [7] A. Mehrotra and A. Sangiovanni-Vincentelli, Noise Analysis of Radio Frequency Circuits. Norwell, MA, USA: Kluwer, 2004. [8] F.-K. Wang, T.-S. Horng, K.-C. Peng, J.-K. Jau, J.-Y. Li, and C.-C. Chen, “Single-antenna Doppler radars using self and mutual injection locking for vital sign detection with random body movement cancellation,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 3577–3587, Dec. 2011. [9] K. Siddiq, R. J. Watson, S. R. Pennock, P. Avery, R. Poulton, and B. Dakin-Norris, “Phase noise analysis in FMCW radar systems,” in Proc. Eur. Radar Conf. (EuRAD), 2015, pp. 501–504. [10] T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems. Berlin, Germany: Springer-Verlag, 1989. [11] V. Rizzoli, F. Mastri, and D. Masotti, “General noise analysis of nonlinear microwave circuits by the piecewise harmonic-balance technique,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 5, pp. 807–819, May 1994. [12] A. Suárez and S. Sancho, “Application of the envelope-transient method to the analysis and design of autonomous circuits,” Int. J. RF Microw. Comput.-Aided Eng., vol. 15, no. 6, pp. 523–535, 2005. [13] A. Suarez, Analysis and Design of Autonomous Microwave Circuits. Hoboken, NJ, USA: Wiley, 2009. [14] J. Dominguez, A. Suarez, and S. Sancho, “Semi-analytical formulation for the analysis and reduction of injection-pulling in frontend oscillators,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 1589–1592. [15] S. Sancho, F. Ramírez, and A. Suárez, “Stochastic analysis of cycle slips in injection-locked oscillators and analog frequency dividers,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 12, pp. 3318–3332, Dec. 2014. [16] F. Ramirez, M. Ponton, S. Sancho, and A. Suarez, “Phase-noise analysis of injection-locked oscillators and analog frequency dividers,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 2, pp. 393–407, Feb. 2008. [17] F. X. Kaertner, “Analysis of white and f −α noise in oscillators,” Int. J. Circuit Theory Appl., vol. 18, no. 5, pp. 485–519, 1990. [18] A. Demir, “Phase noise and timing jitter in oscillators with colored-noise sources,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 12, pp. 1782–1791, Dec. 2002. [19] R. Quere, E. Ngoya, M. Camiade, A. Suarez, M. Hessane, and J. Obregon, “Large signal design of broadband monolithic microwave frequency dividers and phase-locked oscillators,” IEEE Trans. Microw. Theory Techn., vol. 41, no. 11, pp. 1928–1938, Nov. 1993. [20] A. Suárez and R. Quéré, Stability Analysis of Nonlinear Microwave Circuits. Boston, Germany: Artech-House, 2003. [21] E. Ngoya and R. Larcheveque, “Envelop transient analysis: A new method for the transient and steady state analysis of microwave communication circuits and systems,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1996, pp. 1365–1368. [22] H. G. Brachtendorf, G. Welsch, R. Laur, and A. Bunse-Gerstner, “Numerical steady state analysis of electronic circuits driven by multi-tone signals,” Electr. Eng., vol. 79, no. 2, pp. 103–112, 1996. [23] J. C. Pedro and N. B. Carvalho, “Simulation of RF circuits driven by modulated signals without bandwidth constraints,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, pp. 2173–2176. [24] M. Ponton, E. Fernandez, A. Suarez, and F. Ramirez, “Optimized design of pulsed waveform oscillators and frequency dividers,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 3428–3440, Dec. 2011.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 12

[25] S. Sancho, A. Suarez, J. Dominguez, and F. Ramirez, “Analysis of nearcarrier phase-noise spectrum in free-running oscillators in the presence of white and colored noise sources,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 3, pp. 587–601, Mar. 2010. [26] J. A. Mullen and D. Middleton, “Limiting forms of FM noise spectra,” Proc. Inst. Radio Eng., vol. 45, pp. 874–877, Jan. 1957. [27] S. Sancho, A. Suarez, and F. Ramirez, “General phase-noise analysis from the variance of the phase deviation,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 472–481, Jan. 2013. [28] A. Suarez, S. Sancho, S. V. Hoeye, and J. Portilla, “Analytical comparison between time- and frequency-domain techniques for phasenoise analysis,” IEEE Trans. Microw. Theory Techn., vol. 50, no. 10, pp. 2353–2361, Oct. 2002.

Sergio Sancho (A’04–M’04) received the bachelor’s degree in physics from Basque Country University, Leioa, Spain, in 1997, and the Ph.D. degree in electronic engineering from the Communications Engineering Department, University of Cantabria, Santander, Spain, in 2002. He is currently an Associate Professor with the Communications Engineering Department, University of Cantabria. His current research interests include the nonlinear analysis of microwave autonomous circuits and frequency synthesizers, including stochastic and phase-noise analysis.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Mabel Ponton (S’08–M’11) was born in Santander, Spain. She received the bachelor’s degree in telecommunication engineering, master’s degree in information technologies and wireless communications systems, and Ph.D. degree from the University of Cantabria, Santander, in 2004, 2008, and 2010, respectively. In 2006, she joined the Communications Engineering Department, University of Cantabria. From 2011 to 2013, she was with the Group of Electronic Design and Applications, Georgia Institute of Technology, Atlanta, GA, USA, as a Post-Doctoral Research Fellow. Her current research interests include the nonlinear analysis and simulation of radiofrequency and microwave circuits, with an emphasis on phase-noise, stability, and bifurcation analysis of complex oscillator topologies.

Almudena Suarez (M’96–SM’01–F’12) was born in Santander, Spain. She received the bachelor’s degree in electronic physics and Ph.D. degree from the University of Cantabria, Santander, Spain, in 1987 and 1992, respectively, and the Ph.D. degree in electronics from the University of Limoges, Limoges, France, in 1993. She is currently a Full Professor with the Communications Engineering Department, University of Cantabria. She co-authored Stability Analysis of Nonlinear Microwave Circuits (Artech House, 2003) and Analysis and Design of Autonomous Microwave Circuits (IEEE–Wiley, 2009). Prof. Suarez is a member of the Technical Committees of the IEEE MTT-S International Microwave Symposium and the European Microwave Conference. She was an IEEE Distinguished Microwave Lecturer from 2006 to 2008, with her lecture “Stability Analysis and Stabilization of Power Amplifiers.” She is the Editor-In-Chief of the International Journal of Microwave and Wireless Technologies from Cambridge journals. She is an Associate Editor of IEEE Microwave Magazine. She is a member of the Board of Directors of the European Microwave Association. She was the Coordinator of the Communications and Electronic Technology Area for the Spanish National Evaluation and Foresight Agency (ANEP) from 2009 to 2013.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Strongly Enhanced Sensitivity in Planar Microwave Sensors Based on Metamaterial Coupling Mohammad Abdolrazzaghi , Student Member, IEEE, Mojgan Daneshmand, Senior Member, IEEE, and Ashwin K. Iyer, Senior Member, IEEE

Abstract— Limited sensitivity and sensing range are arguably the greatest challenges in microwave sensor design. Recent attempts to improve these properties have relied on metamaterial (MTM)-inspired open-loop resonators coupled to transmission lines (TLs). Although the strongly resonant properties of the resonator sensitively reflect small changes in the environment through a shift in its resonance frequency, the resulting sensitivities remain ultimately limited by the level of coupling between the resonator and the TL. This paper introduces a novel solution to this problem that employs negative-refractive-index TL MTMs to substantially improve this coupling so as to fully exploit its resonant properties. A MTM-infused planar microwave sensor is designed for operation at 2.5 GHz, and is shown to exhibit a significant improvement in sensitivity and linearity. A rigorous signal-flow analysis of the sensor is proposed and shown to provide a fully analytical description of all salient features of both the conventional and MTM-infused sensors. Full-wave simulations confirm the analytical predictions, and all data demonstrate excellent agreement with measurements of a fabricated prototype. The proposed device is shown to be especially useful in the characterization of commonly available high-permittivity liquids as well as in sensitively distinguishing concentrations of ethanol/methanol in water. Index Terms— Artificial materials, couplers, metamaterials (MTMs), MTM transmission lines (TLs), permittivity measurement, resonators, sensors, signal-flow analysis.

I. I NTRODUCTION

M

ICROWAVE-BASED sensors have demonstrated enticing functionalities in many applications such as chemical, agricultural, medical, oil, and microfluidic systems [1]–[6]. They are attractive for their low cost, CMOS compatibility, easy fabrication, and design flexibility compared to other types of optical, thermal,ab, cavitybased, and MEMS-based sensors. The real-time response of microwave sensors is a primary reason for their superiority over expensive and labor-intensive chemical procedures for in situ applications [7], [8]. Manuscript received July 4, 2017; revised October 15, 2017 and December 12, 2017; accepted December 17, 2017. This work was supported in part by Alberta Innovates and Technology Futures, in part by the Natural Sciences and Engineering Research Council of Canada, in part by Canadian Microsystems Corporation, and in part by Canada Research Chair. (Corresponding author: Mohammad Abdolrazzaghi.) The authors are with the Donadeo Innovation Centre for Engineering, Electrical and Computer Engineering Department, University of Alberta, Edmonton, AB T6G 1H9, Canada (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2018.2791942

In dielectric-constant measurements of pure/composite aqueous solutions, resonant methods [9]–[11] are preferred to broadband methods [12]–[14] because of their high accuracy. Several microwave resonator-based sensors, whose operation is principally based on dielectric-constant variation in the material under test (MUT), have been reported. For example, a probe tip loaded with a dielectric resonator has been used to detect concentrations of sodium chloride [15] and glucose [16] in deionized water. A microwave biosensor based on a cavity resonator has been developed for measuring pig-blood D-glucose [17]. In addition, relatively high concentrations (1 μL/mL) of biomolecules (e.g., streptavidin) were measured using planar biosensors [18], which also prompted DNA detection using the same process [19]. A relatively recent development in planar microwave-sensor design is the use of resonant structures known as open-loop resonators (OLRs) [20] or their modified configurations as split-ring resonators (SRRs) [21]–[26]. These elements, which were initially deployed as constituent elements of electromagnetic metamaterials (MTMs), are compact, exhibit high quality factors, and offer noncontact, robust sensing suited to harsh environments and applications involving small analyte volumes. These attributes have made them increasingly popular as transducers in various sensing applications [27]–[31]. As an added advantage, manipulation of the resonators’ spatial configuration due to their flexible planar feature offers more convenience in practical setups compared with bulky and rigid waveguides/cavities [32], [33]. Many approaches, including that employed in this paper, focus on the planar implementation of these resonators, in which they are typically edge coupled to a microstrip (MS) transmission line (TL), as shown in [34]. In this arrangement, the strongly resonant properties of the resonator sensitively reflect small changes in the environment through a shift in its resonance frequency. However, the resulting sensitivities and sensing ranges are ultimately limited by the degree of coupling between the arms of the resonator and the adjacent TL, which affords only as much coupling as a typical MS coupled-line coupler, which in turn, can be simply modeled by a coupling capacitor in most planar resonator circuit models [27], [35]. This equivalent coupling capacitance Cc is well known to have a significant effect on the resonance frequency of the coupled system [36], [37]. In fact, Cc is also directly impacted by the MUT’s permittivity; however, this effect is not prominent due to the typically low coupling of conventional coupled-line couplers. The optimum conventional coupling length of λ/4 is too long to work very

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

well for λ/2 resonators due to size and layout restrictions, and shorter lengths offer typically weaker coupling. Ideally, we require short couplers providing a high degree coupling. Although the above is not achievable using conventional coupled-line couplers, high coupling is entirely feasible with MTM-based couplers. Recently, negative-refractive-index TL (NRI-TL or MTM-TL) [38], [39], in which a host TL is periodically loaded with lumped capacitors and inductors to achieve a backward-wave response, have been used to realize novel couplers that exhibit nearly unity coupling with steeper phase-lead over extremely short lengths by exploiting the continuous contradirectional leakage of power between an MTM and MS [39], [40]. Inspired by the MTM coupledline coupler, we introduce a novel planar microwave-sensor implementation in which the input MS is replaced by an MTM and edge-coupled to the long arm of the resonator [41]–[43]. Therefore, the first major contribution of this paper is to show that the resulting strong, contradirectional leakage of power dramatically enhances Cc , which in turn, imbues the planar sensor with enhanced sensitivity and linearity. The design of such sensors and their responses often begins with full-wave simulations. This paper employs the finiteelement-method simulator Ansys HFSS. However, as all constituent components of the proposed resonator are based on TL elements, the second major contribution of this paper is to propose a novel analytical technique employing microwave network analysis and signal-flow analysis, which is inspired by approaches previously employed to describe optical ring resonators [24], [26], [27]. This approach is inherently suited to the microwave regime and is particularly elegant and powerful in predicting the response of the proposed MTM-infused sensors, where the constituent MTM couplers and lumped elements can be easily represented through their scattering or transfer matrices. Using the developed signal-flow analysis, we are able to predict large enhancements in sensitivity as well as dramatically improved dynamic range for large permittivity variations. As the third major contribution of this paper, the signal-flow analysis is closely validated by both full-wave simulations and measurements. Furthermore, we show that the proposed MTM-based sensor is superior to its conventional MS counterpart in discriminating lossy MUTs—a feature that results from its improved dynamic range. Section II establishes the theory of operation and design process for both the conventional and MTM-infused sensors, which are validated using a signal-flow analysis and corroborated using full-wave simulations. Section III presents experimental results in two sections; the first presents the proposed sensor’s calibration fitting function for liquid-sensing applications while the second demonstrates the sensor’s efficacy and sensitivity in detecting concentrations of methanol/ethanol in deionized water. Section IV provides the conclusion. II. T HEORY AND D ESIGN In this section, we establish analytically the operation of the conventional sensor architecture by viewing it as a microwave-coupler problem as described above. Thereafter, we explain how substituting conventional microwave couplers

Fig. 1. Schematic of conventional planar resonator sensor depicted as a cascade of couplers and an OLR separated by a gap.

with MTM-based couplers, which offer much higher coupling over shorter lengths, enables dramatic enhancements in sensitivity (S =  f /εr , where  f is the shift in resonance frequency), and dynamic range (maximum detectable range of permittivity variation). For the purposes of analysis, we interpret the sensor shown in Fig. 1 to consist of a combination of three components: a resonator, a gap, and two planar MS couplers. The core resonator is an MS half-wavelength resonator, which is bent into a rectangular shape to reduce its size. The small gap separating two open ends of the resonator establishes a strong capacitance at resonance. The coupling between the input– output TLs and the resonator is provided by the coupledline couplers, which include the arms of the resonator. The resonance frequency of the sensor depends on the effective permittivity of the surrounding medium. Therefore, knowing a priori the nominal (i.e., vacuum) frequency response of the sensor, it becomes possible to determine the permittivity of the MUT. This determination may be made accurately provided that the quality factor of the resonator is sufficiently high. Commercially available full-wave electromagnetic simulation software used in the analysis of these types of structures can require extremely large meshes, particularly for finely featured structures such as those involving MTMs, resulting in very long simulation times. To address this issue, we propose an analytical approach based on a signal-flow analysis, which is fast, simple, and accurate in yielding the sensor’s frequency response. A. Signal-Flow Analysis A half-wavelength resonator consisting of TLs with Z 0 = 50  is established on Rogers 5880 substrate (εr = 2.2, tan(δ) = 0.0009, h = 0.8 mm) and the dimensions are given in the caption of Fig. 2. The effective permittivity of the MS substrate is computed with accuracy better than 0.2% [44] and includes dispersive characteristics of the TL allowing the possibility of higher order harmonic generation [45]–[47]. In addition, the contributions of various sources of loss are included [48]–[50]. Fig. 2(a) presents the proposed sensor as a resonator that is edge coupled to a split TL, a design that enables the possibility of a peak in the transmission spectrum at resonance. Each of the open ends of the split TL presents a unity voltage reflection coefficient. The power is inserted from the left and is coupled to the resonator over a coupling region, and then circulates inside the resonator and gap, before the power is coupled again to the output TL and collected at the output.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ABDOLRAZZAGH et al.: STRONGLY ENHANCED SENSITIVITY IN PLANAR MICROWAVE SENSORS

Fig. 4.

Fig. 2. (a) Dimensions of conventional sensor as: X length = 20 mm, Ylength = 7 mm, Tgap = 1 mm, w = 2.4 mm, s = 0.4 mm, l B = 4 mm, l A = 10 mm. (b) Representation as a cascade of two- and four-port networks.

3

Comparison between the proposed signal-flow analysis and HFSS.

portion is considered as an isolated TL of length l R with propagation constant β and S-parameters as follows:

 0 T e jβl R SOLR = (4) T e jβl R 0 where T = e−(αc +αd )l R accounts for attenuation due to conductor (αc ) and dielectric losses (αd ). The gap in the resonator is separately considered as a series capacitance Cgap with S-parameters as follows: ⎡ ⎤ Z 2Z 0

SGap Fig. 3.



Coupler port numbering convention.

The equivalent network for signal-flow analysis is shown in Fig. 2(b). This approach requires modeling each constituent component in the network using, for example, an S-parameter representation. With port references as indicated in Fig. 3, the S-matrix of a conventional (forward) coupled-line coupler may be written as follows [49]: ⎡ ⎤ 0 S12 S13 0 ⎢ S21 0 0 S24 ⎥ ⎥ (1) SM S−M S = ⎢ ⎣ S31 0 0 S34 ⎦ 0 S42 S43 0 where S21 = S12 = S43 = S34 = − j e S13 = S31 = S24 = S42 = e

− j (βe +βo )l 2



(βe − βo )l cos 2  (βe − βo )l sin 2

− j (βe +βo )l 2

⎢ 0+Z = ⎣ 2Z2Z

(2) (3)

where βe and βo are even and odd mode propagation constants and l is the length of the coupled lines. Matching for all ports (S11 = S22 = S33 = S44 = 0) as well as perfect isolation (S14 = S41 = S23 = S32 = 0) are assumed in this analysis; these may be practically achieved through proper selection of the MS dimensions, which control the even- and odd-mode impedances [35]. Representation of the input and output coupling through a four-port coupler description has already accounted for a portion of the path through the resonator. The remaining

0

2Z 0 + Z

2Z 0 + Z ⎥ ⎦ Z 2Z 0 + Z

(5)

where Z = 1 j ωCGap (ω) and Z 0 = 50 . The value of Cgap can be derived using various analytical techniques as well as curve fitting of empirical data (see [51]) within the ranges (e.g., of MS width and substrate height) for which the equations are valid. In this paper, Cgap is extracted from simulation using a commercial method-of-moments software to be approximately 12.6 fF. For the conventional MS sensor, a comparison between the proposed analytical method and full-wave simulation using HFSS shows a very good consistency (see Fig. 4), revealing both resonance peaks and antiresonances, over a wide range of frequencies from 1 to 5 GHz. Minor discrepancies at higher frequencies may be attributed to several simplifying assumptions, such as that of TEM mode propagation and the absence of higher order TL modes. Also, not considered for simplicity are parasitics introduced through MS bends, although equivalent lumped networks for these types of discontinuities may easily be included. B. Sensitivity Analysis This section investigates the effect on the sensor’s transmission spectrum of an MUT introduced into the resonator gap. Whereas it is clear that the electrical characteristics of the MUT affect the gap capacitance Cgap , there is also an effect on the coupling levels of the input and output coupledline couplers, which cannot be very easily incorporated into the coupler S-parameters. We therefore employ a well-known and effective representation of these couplers as equivalent capacitances Cc [27], [35], [51], [52] series connected to

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 5. (a) Coupling capacitors Cc enabling two-port resonator model. (b) Effect of Cc on resonance frequency. (c) Effect of MUT on resonance frequency with respect to the ratio Cc /Cgap .

the resonator, which is now represented as a TL segment in parallel with the gap capacitance Cgap [Fig. 5(a)]. The determination of Cc (and its equivalent two-port transmission matrix TCc ) from the S-parameters of the corresponding fourport coupler is detailed in the Appendix. The effect of an MUT placed at the resonator gap may now be effectively modeled as additional capacitances proportional to its permittivity, causing an increase in both Cgap and Cc . This simple interpretation enables us to use a transmission (ABCD or T )-matrix formalism to determine the overall transfer function, as follows:

 A B Ttotal = TCc × TOLR+gap × TCc = (6) C D where TCc =

⎡ ⎣1 0

⎤

1 cos(βl R ) ⎦ TOLR = j ωCc j Y0 sin(βl R ) 1

j Z 0 sin(βl R ) cos(βl R )



(7) and (8), shown at the bottom of this page. The corresponding transmission parameter S21 is therefore given by the following expression:



2

. S21 = 20 log

2C  (9)

B C

j ωCc + Z 0 + Z ( j ωC )2 + C  Z 0 0

c

The resonance frequency ( f res ) is defined as the local maximum of the transmission profile, which can be analyt ically derived by evaluating ∂ S21 ∂ f = 0. As shown in the denominator of (9), Cc (now including the effect of the MUT) ⎡ TOLR+gap

−1

Fig. 6. Sensitivity analysis. (a) Signal-flow analysis approach for a representative variation in Cgap and Cc (Cgap from 12.5 to 362.5 fF in 70-fF increments, with Cc = 0.25 Cgap ). (b) Permittivity variation in MUT simulated in HFSS.

has a significant role in determining f res . The variation of f res with Cc [shown for a representative case in Fig. 5(b)] is more pronounced for large capacitive coupling, which suggests a distinct benefit to improving this coupling mechanism. The shifts in resonance frequency for MUT permittivity values covering the range εr = 1 : 100 are presented in Fig. 5(c), where the effect of the coupled-line coupler is incorporated in the capacitance ratio Cc /Cgap in the range of 0.1–0.4. The simulation for different coupling-capacitance ratios shows not only that larger Cc yields larger variation in f res for a given variation in MUT permittivity, but also that the sensitivity and dynamic range is improved considerably for larger Cc . Having established analytically that the degree of (capacitive) coupling of the coupler determines the resonance downshift, we now study the practical example described in the caption of Fig. 2. Combining the sensor components using the proposed signal-flow analysis and sweeping over the additional capacitances introduced by the MUT (Cgap and Cc starting from 300 fF in 70-fF increments) results in the resonance downshifts shown in Fig. 6(a). Both the resonance frequencies and their quality factors are reduced, indicating the coupling between TLs and resonator, which causes more power to be trapped in the resonator than released to appear at the output [48].

⎢ = ⎣ (2 j ωC + 1 + cos(βl ))(1 − cos(βl )) gap R R j Z 0 sin(βl R )

j ωCgap +

⎤ 1

j Z 0 sin(βl R ) ⎥ = 1 ⎦ C 1

B 1

 (8)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ABDOLRAZZAGH et al.: STRONGLY ENHANCED SENSITIVITY IN PLANAR MICROWAVE SENSORS

5

curve (e.g., in the region from approximately 1 to 3.5 GHz), in contrast to the forward-wave (conventional) dispersion of the MS. Contradirectional coupling occurs where the two TLs are phase matched, i.e., where their isolated dispersion curves intersect. The resulting coupled dispersion curves (computed analytically using a transmission-matrix formalism for an infinite periodic structure and also using HFSS) exhibit a complex-mode region, describing the leakage of power from the forward MS mode to the backward MTM mode, which is manifested as a bandgap. As a result, the coupled port of the MTM/MS coupler is adjacent to the input port. Wideband impedance matching of the MTM is possible using a design condition (also known as a “balanced” condition) in which the Bloch impedance of the periodic structure is matched to that of the underlying TL, which additionally ensures that the stopband between the backward-wave and forward-wave passbands is closed [39]. In order to find the values for the loading elements L and C, which impact the frequency range and dispersion of the complex-mode region, we should be mindful that the frequency range of interest be centered about the intersection of the MS and MTM dispersion curves; i.e., that Fig. 7. (a) Equivalent circuit of MTM coupler. (b) Isolated- and coupledmode dispersion diagrams.

Since full-wave simulation affords the option of modeling MUTs of practical dimensions and permittivities, the sensor is now constructed in HFSS using an analyte with size l A ×l B × h A (l A = 10 mm, l B = 4 mm, h A = 2 mm) placed on the resonator’s gap, as shown in Fig. 2(a). This location offers the highest sensitivity given its very high electric field strength. A sweep over a selection of complex MUT permittivities (εr = 1 (Bare), 10, 20, 30, tan(δ) = 0.01) is conducted in HFSS [Fig. 6(b)], where the assumed loss tangent is typical of realistic MUTs (higher losses are examined later in this paper). As a result of increased capacitive loading due to the MUTs, the resonant spectrum undergoes a downshift, which matches the trend predicted by the signal-flow analysis. This result shows that signal-flow analysis captures the general behavior of the sensor with less complexity than time-consuming numerical methods. C. Introduction and Design of the MTM/MS Coupler The above analytical results validated by full-wave simulation affirm the value of seeking methods of improving the coupling represented by Cc , and therefore by the coupled-line couplers. Here, we may invoke the MTM/MS coupler [40], which is known precisely for its nearly 0-dB contradirectional coupling over electrically very short lengths. The MTM coupler geometry is constructed by placing an MTM adjacent to an MS, as described in [40], for which the unit cell is depicted schematically in Fig. 7(a). The MTM portion consists of an MS periodically loaded using series capacitors C and shunt inductors L. The isolated dispersion characteristics of representative MTM and MS designs are indicated in Fig. 7(b). The backward-wave behavior of the MTM is evident from the negative slope of its dispersion

βdMS = βdMTM

(10)

at f res . As a starting point, we select fres = 2.5 GHz. The loading elements L and C may be found with respect to the band-gap closure condition for a given value of d (length of unit cell) using design guidelines outlined in [40]. Loading elements satisfying the above-noted f res and of values that may be easily obtained using off the shelf, surfacemount components are determined to be C = 1.1 pF and L = 2.73 nH. The S-parameters of the coupler in the complex-mode band with propagation constant γ = α + jβ can be expressed as follows [40]: ⎡ ⎤ S13 0 0 S12 ⎢ S21 0 0 S24 ⎥ ⎥ (11) SMTM/MS = ⎢ ⎣ S31 0 0 S34 ⎦ 0 S42 S43 0 e+ jβ Nd (12) cosh(α Nd) + j sinh(α Nd) cot(ϕ) 1 = S24 = S42 = cos(ϕ)− j sin(ϕ) coth(α Nd) (13) e− jβ Nd (14) = cosh(α Nd) + j sinh(α Nd) cot(ϕ)

S43 = S34 = S31 = S13

S21 = S12

where the MTM is assumed to have N unit cells, each of length d, and φ is the transverse voltage phase difference between the two TLs (i.e., V0 on the MS and V0 e j φ on the MTM). Matched conditions at all ports (S11 = S22 = S33 = S44 = 0) and perfect isolation (S14 = S41 = S23 = S32 = 0) are assumed, and may be straightforwardly designed as outlined in [40]. The coupling performance of the MTM/MS is compared with that of the conventional MS/MS over total length of 40 mm. Fig. 8 illustrates a considerably higher (backward)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 8. S-parameters of MTM/MS and MS/MS couplers (simulated in HFSS).

Fig. 10. Simulation of variation in transmission spectrum due to increase in (a) Cgap and Cc (Cgap from 12.5 to 362.5 fF in 70-fF increments, with Cc = 0.40 Cgap ) with signal-flow analysis and (b) permittivity of MUT in HFSS.

Fig. 9. (a) Proposed MTM-based sensor’s dimensions as: X length = 20 mm, Ylength = 7 mm, Us = 1 mm, w = 2.4 mm, s = 0.4 mm, Tgap = 1 mm, l A = 10 mm, Ulength = 6.3 mm, l B = 4 mm, l1 = 1 mm, l2 = 2 mm, l3 = 3 mm, l4 = 1.4 mm, h A = 2 mm. (b) Loading-element models including loss.

coupling of −5.6 dB at 2.5 GHz for the MTM/MS coupler (configured akin to that shown in Fig. 3), while the MS/MS coupler only provides −24.5 dB (of forward coupling) at the same frequency. Having established the superiority of the MTM/MS coupler in providing a high coupling level, this coupler may now be integrated into the sensor architecture. The embedded MTM/MS coupler in the coupling stage of the sensor [Fig. 2(a)] is designed with the dimensions given in the caption of Fig. 9 for an impedance level of Z 0 = 50 , such that the resonance frequency would be slightly higher than the intersection point in Fig. 7(b), in order to allow downshifted resonances to reside entirely inside the complex-mode region. The loading capacitors C are connected in series with MS TLs, and inductors L are supported by narrow strips terminating in vias [Fig. 9 (a)] to establish the shunt connection to ground. Sources of loss in the loading elements are considered

Fig. 11. (a) Fabricated MTM-based sensor. (b) Conventional sensor. (c) MUT-sensing setup.

according to the equivalent series resistance of C and quality factor of L from datasheets, as shown in Fig. 9(b). It has already been confirmed that the operating frequency is well below the natural self-resonance frequencies of the elements selected for fabrication. It has also been observed (although not shown here) that, whereas the conventional MS/MS coupler’s coupling level varies with the MUT properties, the MTM/MS coupler retains a largely invariant coupling level, allowing it to operate predictably in a wider range of environmental conditions and applications. The signal-flow analysis proposed earlier to describe the conventional sensor may be used with equal confidence for the MTM-/MS-based sensor, given that the latter’s scattering matrix (and therefore, its equivalent transmission matrix)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ABDOLRAZZAGH et al.: STRONGLY ENHANCED SENSITIVITY IN PLANAR MICROWAVE SENSORS

7

Fig. 13. Dielectric properties of materials using Cole–Cole equation and measurement using dielectric probe kit.

Fig. 12. Fitting surface for (a) Permittivity and (b) Loss tangent extraction from shifts in resonance frequency and amplitude of resonance.

is known. Permittivity variations are once again incorporated through modifying Cgap . Fig. 10(a) shows the effect of incrementing Cgap on the transmission response of the sensor, where downshifts in f res are relatively large compared with their corresponding range for the conventional MS/MS sensor, shown in Fig. 5(a). This result confirms the higher sensitivity of the MTM-based sensor as a result of higher coupling afforded by the MTM/MS configuration. The sensitivity performance of the sensor is examined by applying an MUT (analyte), as shown in Fig. 9(a). An analyte with volume l A × l B × h A (l A = 10, l B = 4, h A = 2 [mm]) is placed on the resonator’s gap. A sweep over the MUT’s dielectric and loss properties (εr = 1, 10, 20, 30, tan(δ) = 0.01) is performed in HFSS [Fig. 10(b)]. The resulting dramatic resonance shifts for the MTM-based sensor once again confirm the impact of enhanced coupling, which demonstrates nearly doubled sensitivity with respect to the conventional MS/MS, in qualitative agreement with the results shown for the signalflow analysis in Fig. 10(a). In Section II-B, it was established that the four-port conventional coupled-line coupler could be reduced to an equivalent coupling capacitance Cc , and it was further shown that higher

values of Cc implied greater sensitivities. Given the strongly enhanced sensitivity observed for the MTM-based sensor, it is worthwhile to examine whether it can be associated with a commensurately high coupling capacitance. This analysis is also conducted in the Appendix. A simplified expression for coupling is achieved by approximating the MTM coupler’s properties using a two-port ABCD-matrix approach. The ABCD analysis of the whole MTM-based sensor is then shown to agree very well with full-wave HFSS results, establishing the suitability of the MTM-based sensor for applications requiring high sensitivity. In Section III, the proposed MTM/MS sensor and the conventional MS/MS sensor are fabricated and their performance parameters are compared in a number of scenarios. D. Performance Comparison So as to place the present work in context, the performance of the proposed MTM sensor is compared with the prevailing planar MS sensors in the literature. Table I presents the performance characteristics of several conventional microwave sensors with various configurations, resonance frequencies, and sample volumes. The measured results in the literature are provided for differing ranges of permittivity; hence, for the sake of fair comparison, the reported mean sensitivity values are computed with respect to the incremental frequency shift ( f εr2 − f εr1 ) for an incremental permittivity shift (εr2 − εr1 ). This metric corrects for the resonator size by normalizing to the nominal (bare) resonance frequency f 0 , but due to the variety of configurations, assumes optimization for sample volume. From this comparison, it is clear that the proposed MTM

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

TABLE I C OMPARISON OF C ONVENTIONAL AND P ROPOSED MTM S ENSORS

sensor exhibits superior mean sensitivity when exposed to known high-permittivity materials even when compared with the typically highly sensitive submersible sensors (#2 and #3).

TABLE II C OEFFICIENTS OF F ITTING F UNCTIONS FOR εr AND tan(δ)

III. E XPERIMENTAL R ESULTS Both the MTM/MS and MS/MS sensors are fabricated on Rogers 5880 substrate with dimensions as listed in the captions of Figs. 2 and 9, and in the former case, the capacitors are soldered in series and inductors are shorted using vias [Fig. 11(a) and (b)]. Here, we examine the functionality of the sensors in two scenarios. First, the sensor is calibrated for monitoring liquids flowing inside a plastic tube placed along the coupling region [Fig. 11(c)]. In Section III-B, the sensor is employed to detect extremely small concentrations of methanol/ethanol in deionized water in the same setup.

is suggested as follows: εr =

Pi j ( f res )i (S21 ) j

(15)

Q i j ( f res )i (S21 ) j

(16)

i=0 j =0

A. Measurement of Dielectric Properties In this section, simulation of the sensor using the proposed experimental configuration (material injection through a narrow tube), while varying the complex permittivity (εr = εr − j εr , tan(δ) = εr /εr ) of the MUT, is conducted in HFSS. Generally, variations in the real part of the permittivity εr contribute to tuning of f res , whereas the imaginary part εr primarily affects the magnitude of the transmission (S21 ). However, in practice both the real and the imaginary parts of εr contribute to f res and S21 at f res . The PTFE tube has inner and outer diameters of 1/32 × 1/16 , respectively, and is placed along the resonator’s gap [Fig. 11(c)]. MUTs with εr ranging from 1 to 100 and tan(δ) from 0 to 1 are passed through the tube, resulting in the variations in  f res and S21 that are depicted in Fig. 12(a) and (b). It is worth examining that the error introduced by considering only one of the two parameters ( f res or S21 ) in inferring εr and tan(δ). For example, it is determined that inferring εr using only  f res (i.e., neglecting S21 ) can lead to an error of up to 15%, mostly noticeable for lower εr . Conversely, if only S21 is used (i.e., neglecting  f res ), the error in the loss tangent can be as high as 95%, mainly for MUTs with higher tan(δ). In order to have a robust calibration that can cope with a full range of materials, a mixed relation between these parameters

M  N 

tan δ =

M  N  i=0 j =0

where Pi j and Q i j are coefficients. A simplified version of this general type of polynomial fitting has been used in [23] with M = N = 1. However, the need for high accuracy over a wide span of permittivities and loss tangents in the present application demands a higher order. We choose M = N = 3, yielding the parameters listed in Table II. These result in a regression coefficient R 2 = 0.99 (a measure of the success of the curve-fitting result) and enable an accurate discrimination of small variations in MUT properties. To validate the MTM-based sensor’s fitting function, we now compare the results of Fig. 12(a) and (b) with the known dielectric properties of some commonly available materials [IPA (C3 H8 O), ethanol (C2 H6 O), methanol (CH3 OH), water (H2 O), and toluene (C7 H8 )]. These are obtained using the Cole–Cole equations [59], which we further corroborate with measurements obtained using an open-ended dielectric probe (Fig. 13), demonstrate the accuracy of the proposed sensor, and suggest that this approach will prove useful for the inference of the dielectric properties of a wide range of materials when employing the MTM-based sensor as well (Table III).

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ABDOLRAZZAGH et al.: STRONGLY ENHANCED SENSITIVITY IN PLANAR MICROWAVE SENSORS

9

TABLE III A NALYTICAL AND M EASURED D IELECTRIC P ROPERTIES OF K NOWN M ATERIALS AND C OMPARING W ITH THE P ROPOSED F ITTING F UNCTION

Fig. 15. Comparison of frequency shifts  f and the bulk sensitivity S for both sensors (full tube versus empty tube) with change in dielectric constant of MUT (tan (δ) = 0).

Fig. 14. (a) Experimental setup for concentration measurement. (b) Timebased stability of the sensor. (c) Frequency shifts for various concentrations of ethanol/methanol in water.

B. Measurement of Ethanol/Methanol Concentrations Fig. 14(a) presents another setup wherein the sensitivity and dynamic range of the sensor is demonstrated using minute

samples of ethanol/methanol in deionized water. It is also instructive to examine the resolution of the sensor in dealing with incremental permittivity variations. Therefore, the bare sensor was studied over a period of 4 h, and the ambient humidity was purged out of the sensing box. Each data point of the scatter plot in Fig. 14(b) is an average of three vector network analyzer measurements recorded with LabView in 40-s increments. These data demonstrate a long-term measurement stability to within ±100 kHz, which establishes a highly accurate measurement resolution. Next, the experiment over mixing ethanol/methanol with DI water is administered according to the following process: two pumps are prepared to inject water and ethanol simultaneously into PTFE tubes that are joined together with a Y junction. Flow rates are designed to achieve the desired concentrations within the range of 50–700 ppm, while the mixture is passing over the resonator gap [see Fig. 14(a)] and the waste goes to a paper mug bin. Dry air flow is used continuously in order to reduce the destructive effect of humidity in the experiment, such that a relative humidity of 0% is maintained throughout the experiment. The ethanol concentration in the mixture is manually increased by 50 ppm at intervals of approximately 5–8 min, causing an upward shift in the resonance frequency; this may be expected from the fact that the increase in the volume fraction of ethanol (which has lower dielectric constant), and hence, the effective permittivity of the mixture is reduced, accordingly increasing f res . Once again, the transmission profiles are recorded through LabView using a vector network analyzer. The resonance frequencies are extracted with methanol as the solvent over both sensors and the results are shown in Fig. 14(c). The differences between the data corresponding to the MTM-based sensor and those of its conventional counterpart are indeed dramatic, demonstrating considerable improvements in dynamic range and sensitivity. Moreover, it is evident that the MTM-based sensor can detect concentrations as low as 50 ppm, far exceeding the conventional sensor’s detection limit. The simulated resonance frequency shifts, particularly for bulk materials with larger permittivity (only the tan(δ) = 0 case is shown for

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

Fig. 16.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Comparison of transmission magnitudes and phases between HFSS simulations and signal-flow analysis with measurements of both sensors.

Fig. 17. Capacitance approximation from analytical examination of NRI-TL/MS coupler.

simplicity) filling the tube, are presented in Fig. 15. The extracted sensitivities S (%) in this case are with respect to the bare sensor, i.e., change in permittivity εr with respect to a vacuum, and are defined as follows: f − f0 S(%) = × 100 (17) f0 (εr − 1) which also reveals that the proposed MTM-based sensor is especially suited to the sensitive discrimination of higher permittivity materials, consistent with the trends observed in Fig. 14(c). The transmission magnitude and phase response of the sensors obtained from HFSS coincide with the analytical response predicted by the signal-flow analysis and are shown to be in good agreement with measurement results, as shown in Fig. 16. This consistency shows the capability of the

Fig. 18. (a) Sensor model employing four-port couplers. (b) Simplified model employing equivalent two-port representation of couplers. (c) Comparison of transmission response predicted by simplified model and HFSS simulations.

proposed signal-flow-analysis-based theory in predicting the full frequency response of the sensor quite accurately as a function of the dielectric properties of applied MUTs. Nevertheless, it is worth noting potential sources of discrepancy, such as the presence/propagation of quasi-TEM or higher order modes in the MS structure, as well as radiation

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ABDOLRAZZAGH et al.: STRONGLY ENHANCED SENSITIVITY IN PLANAR MICROWAVE SENSORS

losses, ohmic losses and inherent tolerances in the loading components. IV. C ONCLUSION In this paper, we proposed a novel planar microwave sensor based on MTMs coupled to resonators. The MTM is dispersion engineered to enable a wide frequency range supporting transmission resonances of the MUT-loaded resonator. A signalflow analysis is developed to predict the transmission response of the MTM-based sensor by cascading the S-parameters of each of the sensor’s components. The proposed sensor is shown to exhibit very high sensitivity and dynamic range versus the conventional MS sensor with respect to introduction of external MUTs exhibiting a wide range of complex permittivities. The same analysis is done in HFSS and is confirmed by the measurement results of transmission profile (magnitude and phase) for both sensors. The sensors are fabricated and tested with various commonly available and well-characterized chemicals as a means for calibration, and thereafter, fitting curves are extracted to be used for permittivity characterization for unknown MUTs. Concentration measurements of ethanol/methanol in water validated the superior performance of the proposed sensor in sensitively differentiating concentrations, particular for high-permittivity materials, in a water host medium. The proposed methodology for fabricating planar sensors can be applied universally to many other configurations and be used in applications requiring high sensitivity, such as blood glucose monitoring or biomolecule detection. A PPENDIX In order to analyze the frequency dependence of the coupling levels of MS/MS and MTM/MS couplers when incorporated into the sensor, a two-port π-model of the coupling behavior (the inset of Fig. 17) is studied. The coupling element of –Y21 is found to be purely capacitive (Fig. 17). It is evident that the larger capacitance at lower frequencies demonstrated by the MTM/MS coupler is responsible for the downshift of resonant frequencies for MUTs with higher permittivity values, in contrast with the lower capacitance values for the MS/MS coupler, which also do not tend to vary much over frequency. This is an explicit advantage of MTM-based couplers over conventional ones due to their higher backward coupling levels. In pursuit of a closed-form transmission response of the MTM-based sensor, an ABCD-matrix description of the circuit in Fig. 6(a) is employed. In this approach, the input coupler, resonator, and the output coupler are considered to be in series, instead of the more physically accurate interconnections employed in the signal-flow analysis. This approximation is validated by the fact that the high directivity of the MTM coupler yields a negligible leakage of power to P4 in Fig. 18(a). Cascading the components in this manner prompts us to convert the four-port coupler into a two-port [Fig. 18(b)], after which the whole sensor’s ABCD matrix may be easily calculated. Accordingly, P4 (for high directivity of MTM coupler) and P2 (split gap in TL) are terminated with open circuits, such that the ABCD matrix of the MTM-based

11

coupler may be reduced from T4_port following equation: ⎡ a11 a12 a13 ⎢ a21 a22 a23 T4−port = ⎢ ⎣ a31 a32 a33 a41 a42 a43 ⎡ a − a a31 33 23 a21 a ⎢ ⎢ a13 − a11 23 ⎢ aa21 T2−Port = ⎢ ⎢ a43 − a13 a31 11 ⎣ a13 − a11 aa23 21

(17) into T2_port in the ⎤ a14 a24 ⎥ ⎥ a34 ⎦ a44

(18) a

a33 −a13 a31

⎤

11

a21 ⎥ ⎥ a11 ⎥. (19) ⎥ a a43 −a41 a13 ⎥ 11 ⎦ a21 a23 −a13 a11 Thus, the total ABCD matrix is evaluated using the following relation:

 a b Ttotal = T2−Port × TOLR+gap × T2−Port = (20) c d a23 −a13

where the coupled-line coupler is modeled as T2−port . This matrix encapsulates the coupling mechanism in the form of a two-port element, which is well known to closely mimic a (generally frequency dependent) series capacitance Cc (ω), as shown in Fig. 17. Furthermore, T2−port may easily be converted into a scattering matrix if necessary [48]. The transmission of the complete resonator described by Ttotal is easily defined as follows:



2

(21) S21 = 20 log

. b

a + + cZ 0 + d Z0

The resonance frequency occurs where S21 achieves a maximum; in other words, where the denominator of S21 achieved a minimum. The resulting transmission profile demonstrates generally excellent agreement with HFSS simulations [shown for a representative case in Fig. 18(c)]. ACKNOWLEDGMENT The authors would like to thank the Rogers Corporation, Coilcraft, and American Technical Ceramics for free substrate, inductor, and capacitor samples, respectively. The authors would also like to thank Dr. J. Pollock for his constructive guidance in HFSS simulations. R EFERENCES [1] M. S. Boybay and O. M. Ramahi, “Material characterization using complementary split-ring resonators,” IEEE Trans. Instrum. Meas., vol. 61, no. 11, pp. 3039–3046, Nov. 2012. [2] A. Ebrahimi, W. Withayachumnankul, S. Al-Sarawi, and D. Abbott, “High-sensitivity metamaterial-inspired sensor for microfluidic dielectric characterization,” IEEE Sensors J., vol. 14, no. 5, pp. 1345–1351, May 2014. [3] O. L. Bo and E. Nyfors, “Application of microwave spectroscopy for the detection of water fraction and water salinity in water/oil/gas pipe flow,” J. Non-Cryst. Solids, vol. 305, nos. 1–3, pp. 345–353, Jul. 2002. [4] K. K. Joshi and R. D. Pollard, “Sensitivity analysis and experimental investigation of microstrip resonator technique for the in-process moisture/permittivity measurement of petrochemicals and emulsions of crude oil and water,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, pp. 1634–1637. [5] K. Saeed, A. C. Guyette, I. C. Hunter, and R. D. Pollard, “Microstrip resonator technique for measuring dielectric permittivity of liquid solvents and for solution sensing,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2007, pp. 1185–1188.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 12

[6] K. H. Theisen and T. Diringer, “Microwave concentration measurement for process control in the sugar industry,” in Proc. SIT, vol. 60. 2000, pp. 79–92. [7] S. Andreescu and J.-L. Marty, “Twenty years research in cholinesterase biosensors: From basic research to practical applications,” Biomol. Eng., vol. 23, no. 1, pp. 1–15, Mar. 2006. [8] P. Serodio and J. M. F. Nogueira, “Considerations on ultra-trace analysis of phthalates in drinking water,” Water Res., vol. 40, no. 13, pp. 2572–2582, Jul. 2006. [9] R. Inoue, Y. Odate, E. Tanabe, H. Kitano, and A. Maeda, “Data analysis of the extraction of dielectric properties from insulating substrates utilizing the evanescent perturbation method,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 2, pp. 522–532, Feb. 2006. [10] L. F. Chen, C. K. Ong, C. P. Neo, V. V. Varadan, and V. K. Varadan, Microwave Electronics: Measurement and Materials Characterization. Hoboken, NJ, USA: Wiley, 2004. [11] Y.-H. Kim, K. Jang, Y. J. Yoon, and Y.-J. Kim, “A novel relative humidity sensor based on microwave resonators and a customized polymeric film,” Sensors Actuators B, Chem., vol. 117, no. 2, pp. 315–322, Oct. 2006. [12] M. D. Migliore, “Partial self-calibration method for permittivity measurement using truncated coaxial cable,” Electron. Lett., vol. 36, no. 15, p. 1275, Jul. 2000. [13] D. Misra, M. Chabbra, B. R. Epstein, M. Microtznik, and K. R. Foster, “Noninvasive electrical characterization of materials at microwave frequencies using an open-ended coaxial line: Test of an improved calibration technique,” IEEE Trans. Microw. Theory Techn., vol. 38, no. 1, pp. 8–14, Jan. 1990. [14] M. Hofmann, G. Fischer, R. Weigel, and D. Kissinger, “Microwavebased noninvasive concentration measurements for biomedical applications,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 5, pp. 2195–2204, May 2013. [15] A. Babajanyan, J. Kim, S. Kim, K. Lee, and B. Friedman, “Sodium chloride sensing by using a near-field microwave microprobe,” Appl. Phys. Lett., vol. 89, no. 18, p. 183504, Oct. 2006. [16] A. Bababjanyan, H. Melikyan, S. Kim, J. Kim, K. Lee, and B. Friedman, “Real-time noninvasive measurement of glucose concentration using a microwave biosensor,” J. Sensors, vol. 2010, Dec. 2010, Art. no. 452163. [17] S. Kim et al., “Noninvasive in vitro measurement of pig-blood D-glucose by using a microwave cavity sensor,” Diabetes Res. Clin. Pract., vol. 96, no. 3, pp. 379–384, Jun. 2012. [18] H.-J. Lee and J.-G. Yook, “Biosensing using split-ring resonators at microwave regime,” Appl. Phys. Lett., vol. 92, no. 25, p. 254103, Jun. 2008. [19] H.-J. Lee, H.-S. Lee, K.-H. Yoo, and J.-G. Yook, “DNA sensing using split-ring resonator alone at microwave regime,” J. Appl. Phys., vol. 108, no. 1, p. 14908, Jul. 2010. [20] C. Bernou, D. Rebière, and J. Pistré, “Theory, design and experimental results of a chemical sensor at ultrahigh frequencies,” in Proc. Sensors Microsyst., 2000, pp. 145–149. [21] C.-L. Yang, C.-S. Lee, K.-W. Chen, and K.-Z. Chen, “Noncontact measurement of complex permittivity and thickness by using planar resonators,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 1, pp. 247–257, Jan. 2016. [22] C.-S. Lee and C.-L. Yang, “Single-compound complementary split-ring resonator for simultaneously measuring the permittivity and thickness of dual-layer dielectric materials,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 6, pp. 2010–2023, Jun. 2015. [23] T. Chretiennot, D. Dubuc, and K. Grenier, “A microwave and microfluidic planar resonator for efficient and accurate complex permittivity characterization of aqueous solutions,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 2, pp. 972–978, Feb. 2013. [24] J. S. Bobowski, “Using split-ring resonators to measure the electromagnetic properties of materials: An experiment for senior physics undergraduates,” Amer. J. Phys., vol. 81, no. 12, pp. 899–906, Dec. 2013. [25] C.-S. Lee and C.-L. Yang, “Thickness and permittivity measurement in multi-layered dielectric structures using complementary splitring resonators,” IEEE Sens. J., vol. 14, no. 3, pp. 695–700, Mar. 2014. [26] L. Su, J. Mata-Contreras, P. Velez, and F. Martin, “Splitter/combiner microstrip sections loaded with pairs of complementary split ring resonators (CSRRs): Modeling and optimization for differential sensing applications,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 12, pp. 4362–4370, Dec. 2016.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

[27] M. H. Zarifi, M. Fayaz, J. Goldthorp, M. Abdolrazzaghi, Z. Hashisho, and M. Daneshmand, “Microbead-assisted high resolution microwave planar ring resonator for organic-vapor sensing,” Appl. Phys. Lett., vol. 106, no. 6, p. 62903, Feb. 2015. [28] M. Abdolrazzaghi and M. Daneshmand, “Compelling impact of intermodulation products of regenerative active resonators on sensitivity,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2017, pp. 1018–1021. [29] M. Abdolrazzaghi and M. Daneshmand, “Dual active resonator for dispersion coefficient measurement of asphaltene nano-particles,” IEEE Sensors J., vol. 17, no. 22, pp. 7248–7256, Nov. 2017. [30] M. H. Zarifi et al., “A microwave ring resonator sensor for early detection of breaches in pipeline coatings,” IEEE Trans. Ind. Electron., vol. 65, no. 2, pp. 1626–1635, Feb. 2018. [31] M. Abdolrazzaghi, M. H. Zarifi, W. Pedrycz, and M. Daneshmand, “Robust ultra-high resolution microwave planar sensor using fuzzy neural network approach,” IEEE Sensors J., vol. 17, no. 2, pp. 323–332, Jan. 2017. [32] H. Choi et al., “Design and in vitro interference test of microwave noninvasive blood glucose monitoring sensor,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 10, pp. 3016–3025, Oct. 2015. [33] A. A. Abduljabar, D. J. Rowe, A. Porch, and D. A. Barrow, “Novel microwave microfluidic sensor using a microstrip split-ring resonator,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 3, pp. 679–688, Mar. 2014. [34] M. Abidi, A. Elhawil, J. Stiens, R. Vounchx, J. B. Tahar, and F. Choubani, “Sensing liquid properties using split-ring resonator in Mm-wave band,” in Proc. 36th Annu. Conf. IEEE Ind. Electron. Soc. (IECON), Nov. 2010, pp. 1298–1301. [35] I. Bahl and P. B. R. Mongia, “RF and microwave coupled-line circuits,” Microw. J., vol. 44, no. 5, p. 390, 2001. [36] P. Vagner and M. Kasal, “A novel bandpass filter using a combination of open-loop defected ground structure and half-wavelength microstrip resonators,” Radioengineering, vol. 19, no. 3, pp. 392–396, 2010. [37] C.-Y. Hsiao and Y.-C. Chiang, “A miniaturized open-loop resonator filter constructed with floating plate overlays,” Prog. Electromagn. Res. C, vol. 14, pp. 131–145, 2010, doi: 10.2528/PIERC10051405. [38] G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications. Hoboken, NJ, USA: Wiley, 2005. [39] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Techn., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [40] R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow,” Electron. Lett., vol. 40, no. 5, pp. 315–317, Mar. 2004. [41] C. Chaichuay, P. P. Yupapin, and P. Saeung, “The serially coupled multiple ring resonator filters and Vernier effect,” Opt. Appl., vol. 39, no. 1, pp. 175–194, 2009. [42] D. Mahmudin, T. T. Estu, P. Daud, N. Armi, Y. N. Wijayanto, and G. Wiranto, “Environmental liquid waste sensors using polymer multicoupled ring resonators,” in Proc. Int. Conf. Smart Sens. Appl. (ICSSA), 2015, pp. 88–91. [43] J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Exp., vol. 12, no. 1, pp. 90–103, 2004. [44] E. Hammerstad and O. Jensen, “Accurate models for microstrip computer-aided design,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 80. May 1980, pp. 407–409. [45] M. Kobayashi, “A dispersion formula satisfying recent requirements in microstrip CAD,” IEEE Trans. Microw. Theory Techn., vol. 36, no. 8, pp. 1246–1250, Aug. 1988. [46] M. Kirschning and R. H. Jansen, “Accurate wide-range design equations for the frequency-dependent characteristics of parallel coupled microstrip lines (corrections),” IEEE Trans. Microw. Theory Techn., vol. MTT-33, no. 3, p. 288, Mar. 1985. [47] M. Kirschning and R. H. Jansen, “Accurate wide-range design equations for the frequency-dependent characteristic of parallel coupled microstrip lines,” IEEE Trans. Microw. Theory Techn., vol. MTT-32, no. 1, pp. 83–90, Jan. 1984. [48] D. M. Pozar, Microwave Engineering. Hoboken, NJ, USA: Wiley, 2005. [49] R. Mongia, I. J. Bahl, and P. Bhartia, RF and Microwave Coupled-Line Circuits. Norwood, MA, USA: Artech House, 1999. [50] K. Chang and L.-H. Hsieh, Microwave Ring Circuits and Related Structures. 2004.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ABDOLRAZZAGH et al.: STRONGLY ENHANCED SENSITIVITY IN PLANAR MICROWAVE SENSORS

[51] R. Garg, I. Bahl, and M. Bozzi, Microstrip Lines and Slotlines, 3rd ed. Norwell, MA, USA: Artech House, 2013. [52] J.-S. Hong and M. J. Lancaster, “Couplings of microstrip square openloop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Techn., vol. 44, no. 11, pp. 2099–2109, Nov. 1996. [53] G. Galindo-Romera, F. Javier Herraiz-Martínez, M. Gil, J. J. Martínez-Martínez, and D. Segovia-Vargas, “Submersible printed split-ring resonator-based sensor for thin-film detection and permittivity characterization,” IEEE Sensors J., vol. 16, no. 10, pp. 3587–3596, May 2016. [54] C. Liu and Y. Pu, “A microstrip resonator with slotted ground plane for complex permittivity measurements of liquids,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 4, pp. 257–259, Apr. 2008. [55] N. Wiwatcharagoses, K. Y. Park, J. A. Hejase, L. Williamson, and P. Chahal, “Microwave artificially structured periodic media microfluidic sensor,” in Proc. IEEE 61st Electron. Compon. Technol. Conf. (ECTC), Jun. 2011, pp. 1889–1893. [56] K. Saeed, A. C. Guyette, I. C. Hunter, and R. D. Pollard, “Microstrip resonator technique for measuring dielectric permittivity of liquid solvents and for solution sensing,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2007, pp. 1185–1188. [57] V. Sekar, W. J. Torke, S. Palermo, and K. Entesari, “A self-sustained microwave system for dielectric-constant measurement of lossy organic liquids,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 5, pp. 1444–1455, May 2012. [58] F. Buckley and A. A. Maryott, “Tables of dielectric dispersion data for pure liquids and dilute solutions,” U.S. Dept. Comerce, Nat. Bureau Standards, Washington, DC, USA, Tech. Rep., 1958.

Mohammad Abdolrazzaghi (S’14) received the B.Sc. degree from the Iran University of Science and Technology, Tehran, Iran, in 2009, and the M.Sc. degree from the Electrical and Computer Engineering Department, University of Alberta, Edmonton, AB, Canada, in 2017. He is currently a Research Assistant with the University of Alberta. His current research interest include active and passive sensors, microwave oscillators, superregenerative amplifiers, microwave mixers, microwave resonators, wideband antennas, metamaterials, biosensors, bioelectromagnetis, wave propagation in inhomogeneous medium, and machine learning algorithms (FNN, ANN, and SVM). Mr. Abdolrazzaghi was awarded the Alberta Innovates Technology Futures Scholarship from the University of Alberta in 2015, nominated for the Best Student Paper in the IMS 2016 conference and the IEEE Sensor 2016. He was a recipient of the First Prize in the 2015 CMC Microsystems’s National Research Council Industrial Collaboration Award and the Graduate Student Teaching Award from the University of Alberta in 2017. He has also served as an Invited Reviewer at the IEEE Sensors Conference.

13

Mojgan Daneshmand (SM’14) received the B.Sc. degree in electrical engineering from the Iran University of Science and Technology, Tehran, Iran, in 1999, the M.Sc. degree in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 2001, and the Ph.D. degree in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 2006. She is currently an Associate Professor with the University of Alberta and the Canada Research Chair Tier II in Radio Frequency (RF) Microsystems for Communication and Sensing. She is involved in applying RF and nanotechnology to wireless and satellite communication, energy, and biomedical applications. She has authored or co-authored more than 100 papers, including many in prestigious journals such as IEEE T RANSACTIONS and J OURNALS , N ANOSCALE, The Journal of Physical Chemistry, and Applied Physics. She holds several patents. Her publications have been cited more than 600 times. Her current research interests include high-resolution noncontact microwave sensing, waveguide switches, and switch matrices for satellite communication. Dr. Daneshmand was a recipient of the 2016 IEEE AP-S Lot Shafai Distinguished Mid-Career Award. She was awarded the Natural Sciences and Engineering Research Council of Canada and also the Canadian Space Agency (CSA) Postdoctoral Fellowships. Her group has received a range of awards, including the 2014 IEEE International Microwave Symposium (IMS) Graduate Student Design Competition Award and the 2015 CMC Microsystems’s National Research Council Industrial Collaboration Award. She has also received several paper awards including the IEEE IMS and APS Conference Student Paper Awards. She is currently an Associate Editor of the IEEE T RANSACTIONS ON A NTENNAS AND P ROPAGATION and the IEEE C ANADIAN J OURNAL OF E LECTRICAL AND C OMPUTER E NGINEER ING . She is currently the Co-Chair of the award winning IEEE Joint AP-S/MTT-S Northern Canada Chapter. She contributes broadly to international communities through roles on the IEEE IMS Technical Program Review Committee, IEEE AP-S Conference Organization Committee, and ANTEM Steering Committee. Ashwin K. Iyer (S’01–M’09–SM’14) received the B.A.Sc. (Hons.), M.A.Sc., and Ph.D. degrees in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 2001, 2003, and 2009, respectively, where he was involved in the discovery and development of the negative-refractiveindex transmission-line approach to metamaterial design and the realization of metamaterial lenses for free-space microwave subdiffraction imaging. He is currently an Associate Professor with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, where he leads a team of graduate students investigating novel RF/microwave circuits and techniques, fundamental electromagnetic theory, antennas, and engineered metamaterials, with an emphasis on their applications to microwave and optical devices, defense technologies, and biomedicine. He has co-authored a number of highly cited papers and four book chapters on the subject of metamaterials. Dr. Iyer is a registered member of the Association of Professional Engineers and Geoscientists of Alberta. He was a recipient of several awards, including the 2008 R. W. P. King Award and the 2015 Donald G. Dudley Jr. Undergraduate Teaching Award, presented by the IEEE AP-S, and the 2014 University of Alberta Provost’s Award for Early Achievement of Excellence in Undergraduate Teaching. His students are also the recipients of several major national and international awards for their research. He serves as the Co-Chair of the IEEE Northern Canada Section’s joint chapter of the AP-S and MTT-S societies. Since 2012, he has been serving as an Associate Editor of the IEEE T RANSACTIONS ON A NTENNAS AND P ROPAGATION.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Asymmetric Compact OMT for X-Band SAR Applications Mohamed A. Abdelaal , Student Member, IEEE, Shoukry I. Shams , Member, IEEE, and Ahmed A. Kishk, Fellow, IEEE

Abstract— A compact asymmetric orthomode transducer (OMT) with high isolation between the vertical and horizontal Ports is developed for the X-band synthetic aperture radar application. The basic idea of the design is to deploy the combined E- and H-plane bends within the common arm. Moreover, an offset between each polarization axis is introduced to enhance the isolation and decrease the size to be around one-third of most of the existing asymmetric OMTs. The OMT achieves better than 22.5-dB matching level and 65-dB isolation level between the two modes. Good agreement is obtained between measurements and full-wave simulations. Index Terms— Asymmetric orthomode transducer (OMT), compact OMT, synthetic aperture radar (SAR), X-band OMT, X-band SAR applications.

I. I NTRODUCTION

S

YNTHETIC aperture radar (SAR) systems are broadening in several applications. The basic concept of SAR was first proposed in 1952 by Carl Wiley that was patented in 1965 [1]. Russia, Europe, and Canada had also launched their own satellite carrying SAR system in 1987, 1991, and 1995, respectively [2]. SAR systems are widely used in remote sensing and other geological applications, as they offer good resolution over radar by utilizing movement of the antenna with respect to the object. These systems are expected to be integrated with other communication systems. Communication system requires a wide bandwidth to maximize the number of channels [3]. In addition, to duplicate the number of channels within the same spectrum, dual polarization antennas are used. An orthomode transducer (OMT) must be deployed to use the same antenna for both polarizations [4]. OMT is a three-port passive device that consists of a common port and two single-mode Ports [4]. The common port is the one that carries the two degenerate orthogonal modes while each of the other two Ports is coupled to one of the degenerate modes in the common port. OMTs have been proposed in 1956 by Tompkins [5] for the first time. Depending on the required power capability and other needs for the desired application, the OMT Ports can be implemented with any technology such as rectangular waveguides [6],

Manuscript received September 27, 2017; accepted December 17, 2017. (Corresponding author: Mohamed A. Abdelaal.) The authors are with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC H3G 2W1, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2018.2791960

microstrip lines [7], and coaxial lines [8]. OMTs reduce the size and weight of the antenna feeding systems, which is very critical in several applications [4], [9], [10]. OMTs have many classifications, which can be divided into three classes depending on the structure type of symmetry [11]. The first class is the twofold symmetry OMTs: they have two pairs of Ports each is coupled to one polarization, which makes the structure larger due to the need of power dividers. The symmetric OMTs have wideband operations since the higher order modes are suppressed [12]–[16]. The second class is the onefold symmetry OMTs [6], [17]–[19]. They utilize differential feeding for one polarization that requires a power divider. Onefold symmetry OMTs are smaller in size than the twofold. However, their size still is relatively large. Such structure was reversed in some designs [20]–[23]. Finally, the third class is the asymmetric OMTs [6], [24]–[28], which does not exploit any symmetry. Therefore, they are the most compact class among OMTs. However, they have limited matching and isolation levels. Nevertheless, there are many examples of published asymmetry OMTs for higher frequency bands that achieve proper matching and isolation levels. They utilized transformers or transitions from rectangular to the square waveguide to achieve the best matching. These OMTs have one polarization waveguide that is in line with the common port [29], [30]. As we are targeting to reduce the overall weight of the SAR system, a compact OMT is designed. Therefore, the launching cost will be reduced for the satellite applications, which exceeds U.S. $18 000 per kg. This paper presents a compact OMT that falls in the asymmetry class. The proposed design utilizes the combined E- and H-plane bends within the common arm. Moreover, an offset between the polarization Ports has been introduced to enhance the isolation. Compared to most of the published asymmetric OMTs, the proposed OMT size is around one-third. Nevertheless, the main contributions in this paper can be summarized as follows. The OMT design methodology of the combined E- and H-plane bends is introduced in Section III. The isolation is enhanced and the size is reduced by introducing an offset between the center axis of each polarization in Section III. In Section IV, we evaluate the mechanical and electrical characteristics of the present OMT with respect to other published OMTs. We validate the proposed methodology experimentally through two setups, the back-to-back and the matched load, in Section V.

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 2. Bends sectional view. (a) E-plane. (b) H-plane. (c) E- and H-planes.

Fig. 3.

Fig. 1. Traditional OMT structure. (a) Side view of V-Pol. inline. (b) Side view of H-Pol. inline. (c) Plane view of Noninline.

II. P HYSICAL A LIGNMENT OF OMT P ORTS The physical alignment of the OMT Ports may have different forms, as shown in Fig. 1, depending on the required mechanical consideration of the system where this OMT is going to be utilized. Fig. 1(a) and (b) shows the most commonly used configurations of OMTs where one of the polarization Ports is in line with the common port and the other is orthogonally transverse to the other two ports. Physically, the port that is in line with the common port can hold either the horizontal or vertical polarization. However, usually the vertical polarization is preferred to this port [6], [21]. These structures are easier to be matched due to the physical alignment of the ports. In some systems, the vertical and horizontal polarization Ports need to be placed in the opposite directions, as shown in Fig. 1(c). In this case, there are two solutions. The first one is using the traditional configurations shown in Fig. 1(a) and (b) and adding the E- or H-plane bend, which results in a relatively larger structure. Meanwhile, the second solution is to deploy both of the E- and H-plane bends in the common arm in a combined form, which provides a compact structure. III. C OMBINED OMT D ESIGN M ETHODOLOGY The problem discussed in Section II has two solutions. The first solution is to add the E- and/or H-plane bends, such bends are shown in Fig. 2(a) and (b), respectively. They can be added to the traditional OMT structure, which makes the design size larger. The second solution is to combine

Block diagram of the proposed OMT methodology.

both the E- and H-bends in the OMT core itself, as shown in Fig. 2(c). The second solution makes the structure more compact, but complicates achieving the required matching and isolation levels. On the other hand, such structure provides more freedom in aligning each polarization port independently. It should be stated that the proposed physical alignment of the OMT Ports is the most difficult and challenging task. The design methodology of the combined OMT is summarized in the block diagram shown in Fig. 3. A. OMT Description The proposed OMT is composed of three ports, common port, horizontal port, and vertical port. The common port has a square cross section of side length 0.8 . Each of the other Ports is a WR-112 standard rectangular waveguide of dimensions 1.112 × 0.497 (28.50 mm × 12.62 mm). The basic idea is using both of the E- and H-plane bends in the structure’s core as explained in the previous section. Moreover, multiple-section impedance transformers are added to each port in order to improve the matching level. The 3-D sectional view and the planar horizontal sectional view of the designed OMT are shown in Fig. 4. This structure achieves better than 25-dB matching level for both modes and isolation level better than 60 dB, as shown in Fig. 4(c). The proposed design is investigated and simulated using CST simulator; the simulation results are shown in Fig. 4(c). B. Offset OMT Design The proposed structure has an excellent isolation. However, this isolation can be improved by introducing a small offset between the center axes of each polarization. Introducing this

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ABDELAAL et al.: ASYMMETRIC COMPACT OMT FOR X-BAND SAR APPLICATIONS

Fig. 4. Proposed OMT without offset. (a) 3-D horizontal sectional view. (b) Plane horizontal sectional view. (c) Simulation results.

little offset in the proposed design improves the isolation level by more than 5 dB with a slight reduction of the OMT size. This offset is placed intentionally between the centers of Ports 2 and 3 to reduce the coupling level between them. This offset avoids having the maximum electric field of each waveguide from being on the same axis. If the maxima are on the same axes, strong undesired modes due to the discontinuities in the transformers are excited, which eventually increase the coupling that reduces the isolation. In addition, the common port is shifted more toward the direction of Port 3, which is introduced intentionally to satisfy the geometry requirements. Other specs requirements may lead to different configurations. The selected offset value is 0.2λ0 at the center frequency of the targeted band. The modified design after introducing the offset is shown in Fig. 5. The simulation results of both matching and transmission are shown in Fig. 5(c). The comparison between the isolation levels with and without

3

Fig. 5. Proposed OMT with offset. (a) 3-D horizontal sectional view. (b) Plane horizontal sectional view. (c) Simulation results.

the offset is shown in Fig. 6. The final dimensions of the impedance transformers after introducing the offset are shown in Table I. IV. E VALUATION OF P ROPOSED OMT A comparison between the proposed OMT and other published OMTs in terms of different specifications is presented. The published OMTs are classified into three classes depending on the structure symmetry, the first one is the twofold symmetry, the second is the onefold symmetry, and finally the asymmetric OMTs. We start with the twofold symmetry OMTs [14], [31], [32]. This design has many junctions including Y-junctions, E- or H-plane bends, and turnstile junction. These structures have improved the matching level compared to other classes. They have a comparable isolation with the second class at the expense of size and complexity.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

TABLE II C OMPARING THE P ROPOSED OMT W ITH D IFFERENT C LASSES

Fig. 6. Isolation level simulation results comparison between OMT with and without offset. TABLE I P ROPOSED OMT D IMENSIONS

TABLE III C OMPARING THE P ROPOSED OMT W ITH A SYMMETRIC OMTs

The second class of OMTs is the onefold symmetry, many examples of such class have been published [12], [33], [34]. This class has almost half the complexity and the size of the first class. The main advantage of this structure is the freedom of adding shorting pins and matching sections that were a problem before for such class of OMTs. This design provides higher isolation on the expense of the size and complex structure that uses many junctions. Meanwhile, the proposed design has only one junction. The third class is the asymmetric OMT, which is featured with its compact size and simple structure among other OMT classes [6], [21], [24]–[26]. This class has a fair comparison with the proposed OMT as they both belong to the same class of asymmetry. The structure is simple, but still needs some tuning stubs. This class achieves a comparable matching level with the present OMT. A comparison between the three classes is summarized in Table II. The asymmetric OMTs either have a T-junction in their core [6], [21], [24], [25] or a Y-junction [26], [35]. These

OMTs deploy either circular or square waveguide common port. The presented alignment offset makes it different from the traditional asymmetric OMT. In addition, all Ports share the same center plane cut and both the polarization Ports are orthogonal with the common port. Although none of the polarization Ports are in line with the common port that usually used to achieve better matching, the presented OMT overcomes the matching and isolation problems without compromising the compact size. The proposed design is compact compared to most of the published OMTs. The proposed design has many advantages over the other published OMTs. These advantages are at the expense of a slight sacrifice of matching bandwidth. Nerveless, the achieved matching level and bandwidth already satisfies the SAR application requirements. The comparison between the asymmetric OMTs and the present OMT is shown in Table III. V. VALIDATION AND E XPERIMENTAL R ESULTS The proposed OMT structure with an offset is validated experimentally by three different setups. The back-to-back setup by connecting two OMTs to each other without any

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ABDELAAL et al.: ASYMMETRIC COMPACT OMT FOR X-BAND SAR APPLICATIONS

5

Fig. 8. Proposed OMT with offset in back-to-back setup simulation and measured results of (a) matching and transmission absolute coefficients and (b) isolation absolute coefficients. Fig. 7. Fabricated OMT. (a) 3-D view. (b) Sectional view. (c) Measurement setup of the back-to-back setup.

rotation, then same setup with 90° rotation angle between the two OMTs, and finally measuring only one OMT that is connected to a nonstandard matched load through the common port. Sections V-A–V-C explain each setup and present the measurements compared to the simulation results. The proposed OMT is fabricated using a CNC machine that has a tolerance value of ±0.001 . The fabricated OMT and the backto-back measurement setup are shown in Fig. 7. A. Back-to-Back This setup is simple and reliable since the measuring Ports are four ports. The Ports numbering in this setup are: Ports 2 and 4 are for the horizontal polarization while Ports 3 and 5 are for the vertical polarization. Moreover, the common ports (Ports 1 and 6) are not existing since they are connected to each other. This setup is simulated and measured. Note that the transmission and isolation between

the Ports of the opposite sides exchange will be seen from the results later. The advantage of this method is providing information regarding the transmission coefficients of each polarization and the cross-polarization level that might be due to the OMT itself. It should also be stated that because the square cross-sectional waveguide is not a standard one and it is difficult to measure the transmission from Ports 2 and 3 to Port 1 without having a transition from square to rectangular waveguide to be connected to the common port. It can be seen that this method of using two similar OMTs back-to-back is providing the needed transitions from the common port to the two polarizations. The measurements are compared to the simulation results and are shown in Fig. 8. There is a good agreement between the measurements and the simulations. B. Back-to-Back With 90° This setup is also simple and reliable as the previous one, the only difference is that one of the two OMTs is rotated by 90°. In this setup the Ports numbering are changed to be: Ports 2 and 5 are for the horizontal polarization while

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 9. Proposed OMT with offset in back-to-back setup with 90° rotation angle simulation and measured results of (a) matching and transmission absolute coefficients and (b) isolation absolute coefficients.

Ports 3 and 4 are for the vertical polarization. The advantage of this setup over the previous one is to indicate if unwanted modes are propagating. Such modes can exist in the previous setup and being canceled by the length of the common port. Moreover, this setup also shows the tolerance of the fabricated components. Good agreement between simulation and measured results is shown in Fig. 9. C. Nonstandard Matched Load This measurement setup is based on connecting the common port (Port 1) of the proposed OMT to a matched load that has low reflections. If only standard components are available, the conventional method is to use a matched load that is mounted inside one of the standard rectangular waveguides. The matched load should be connected to the common port so that the two polarization signals are absorbed. Since the common port is a square waveguide and the load is a rectangular waveguide, a transition must be used. In this situation, the matched load and the transition will have two setups. The first one for the vertical polarization and the second for the horizontal polarization. A more convenient way is to design a

Fig. 10. Designed matched load. (a) Sectional view of the fabricated prototype. (b) Simulation results. (c) Connected to the OMT measured results (m) compared to simulation results (s).

matched load that has a square cross section with the same side length as the common port. This matched load is not a standard one, but it is preferred because no transitions are needed. However, no information regarding the actual transmission coefficients from Ports 2 and 3 to Port 1 is possible since it is terminated with the matched load. Since Port 1 is terminated by a custom-made matched load, the S-parameters measured for both polarizations are the S-parameters of the single OMT. The matched load is designed deploying four double tapered pyramids of Resin material (RS4200CHP) that is ceramic based with a dielectric constant of

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ABDELAAL et al.: ASYMMETRIC COMPACT OMT FOR X-BAND SAR APPLICATIONS

12.6 and tan δ = 0.76. The prototype of the fabricated matched load is shown in Fig. 10(a), and its simulation results are shown in Fig. 10(b). Comparison between the measured and simulation results based on this setup is shown in Fig. 10(c). The slight deviation between the simulation and measured results may occur due to the possible difference between the ideal simulated geometry and the fabricated geometry, which affect the actual machining level of the load since it consists of four pyramids that are supposed to be identical, but difficult to achieve. In addition, the two fabricated OMTs may not be identical. Not to mention that each OMT prototype consists of two halves in which each of them is machined separately, and then connected to each other. All of these are factors that add up causing such deviations. VI. C ONCLUSION A novel compact asymmetric OMT has been developed in this paper. The proposed structure is an asymmetric configuration that does not suffer from matching problems as the most asymmetric OMTs. The isolation between polarizations has been enhanced by introducing a slight deviation between the central axes of waveguides of each polarization. Moreover, the proposed structure size is compact comparing it with the most compact published ones. This structure was simulated, fabricated, and measured. The measurements have a very good agreement with the simulation results. The measurement setup used to validate the design in the back-to-back setup with rotation angles of 0◦ and 90°. Furthermore, the proposed design achieved matching and isolation levels better than 22.5 and 60 dB, respectively, over the frequency band from 8.5 to 9.6 GHz. R EFERENCES [1] C. A. Wiley, “Pulsed Doppler radar methods and apparatus,” U.S. Patent 3,196,436, Jul. 20, 1965. [Online]. Available: https://www. google.com/patents/US3196436 [2] A. Moreira, P. Prats-Iraola, M. Younis, G. Krieger, I. Hajnsek, and K. P. Papathanassiou, “A tutorial on synthetic aperture radar,” IEEE Geosci. Remote Sens. Mag., vol. 1, no. 1, pp. 6–43, Mar. 2013. [3] S. H. Van Wambeck and A. H. Ross, “Performance of diversity receiving systems,” Proc. IRE, vol. 39, no. 3, pp. 256–264, Mar. 1951. [4] H. Schlegel and W. D. Fowler, “The ortho-mode transducer offers a key to polarization diversity in EW systems,” Microw. Syst. News, vol. 9, pp. 65–70, Sep. 1984. [5] R. D. Tompkins, “A broad-band dual-mode circular waveguide transducer,” IRE Trans. Microw. Theory Techn., vol. 4, no. 3, pp. 181–183, Jul. 1956. [6] J. M. Rebollar, J. Esteban, and J. De Frutos, “A dual frequency OMT in the Ku band for TT&C applications,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., vol. 4. Jun. 1998, pp. 2258–2261. [7] R. W. Jackson, “A planar orthomode transducer,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 12, pp. 483–485, Dec. 2001. [8] S. J. Skinner and G. L. James, “Wide-band orthomode transducers,” IEEE Trans. Microw. Theory Techn., vol. 39, no. 2, pp. 294–300, Feb. 1991. [9] J. Lahtinen, J. Pihlflyckt, I. Mononen, S. J. Tauriainen, M. Kemppinen, and M. T. Hallikainen, “Fully polarimetric microwave radiometer for remote sensing,” IEEE Trans. Geosci. Remote Sens., vol. 41, no. 8, pp. 1869–1878, Aug. 2003. [10] G. L. James, “Wideband feed systems for radio telescopes,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 3. Jun. 1992, pp. 1361–1363. [11] A. M. Bøifot, E. Lier, and T. Schaug-Pettersen, “Simple and broadband orthomode transducer (antenna feed),” Proc. Inst. Elect. Eng.—Microw., Antennas Propag., vol. 137, no. 6, pt. H, pp. 396–400, Dec. 1990.

7

[12] A. Navarrini and R. Nesti, “Symmetric reverse-coupling waveguide orthomode transducer for the 3-mm band,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 1, pp. 80–88, Jan. 2009. [13] A. Tribak, J. L. Cano, A. Mediavilla, and M. Boussouis, “Octave bandwidth compact turnstile-based orthomode transducer,” IEEE Microw. Wireless Compon. Lett., vol. 20, no. 10, pp. 539–541, Oct. 2010. [14] A. Navarrini and R. L. Plambeck, “A turnstile junction waveguide orthomode transducer,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 1, pp. 272–277, Jan. 2006. [15] S.-G. Park, H. Lee, and Y.-H. Kim, “A turnstile junction waveguide orthomode transducer for the simultaneous dual polarization radar,” in Proc. Asia–Pacific Microw. Conf. (APMC), Dec. 2009, pp. 135–138. [16] U. Rosenberg, A. Bradt, M. Perelshtein, and P. Bourbonnais, “Broadband ortho-mode transducer for high performance modular feed systems,” in Proc. 40th Eur. Microw. Conf. (EuMC), Sep. 2010, pp. 807–810. [17] J. Uher and J. Bornemann, Waveguide Components for Antenna Feed Systems: Theory and CAD. Norwood, MA, USA: Artech House, 1993. [18] M. Ludovico, B. Piovano, G. Bertin, G. Zarba, L. Accatino, and M. Mongiardo, “CAD and optimization of compact ortho-mode transducers,” IEEE Trans. Microw. Theory Techn., vol. 47, no. 12, pp. 2479–2486, Dec. 1999. [19] P. Sarasa, A. Baussois, and P. Regnier, “A compact single-horn C/X dual band and circular polarized Tx & Rx antenna system,” in Proc. IEEE Antennas Propag. Soc. Symp., vol. 3. Jun. 2004, pp. 3039–3042. [20] J. A. Ruiz-Cruz, J. R. Montejo-Garai, and J. M. Rebollar, “Optimal configurations for integrated antenna feeders with linear dual-polarisation and multiple frequency bands,” IET Microw., Antennas Propag., vol. 5, no. 8, pp. 1016–1022, 2011. [21] A. Dunning, S. Srikanth, and A. R. Kerr, “A simple orthomode transducer for centimeter to submillimeter wavelengths,” in Proc. Int. Symp. Space Terahertz Technol., 2009, pp. 1–4. [22] S. Asayama and M. Kamikura, “Development of double-ridged wavegide orthomode transducer for the 2 MM band,” J. Infr., Millim., Terahertz Waves, vol. 30, no. 6, pp. 573–579, 2009. [23] N. Reyes, P. Zorzi, J. Pizarro, R. Finger, F. P. Mena, and L. Bronfman, “A dual ridge broadband orthomode transducer for the 7-mm band,” J. Infr., Millim., Terahertz Waves, vol. 33, no. 12, pp. 1203–1210, 2012. [24] N. Yoneda, M. Miyazaki, M. Tanaka, and H. Nakaguro, “Design of compact-size high isolation branching OMT by the mode-matching technique,” in Proc. 26th Eur. Microw. Conf., Sep. 1996, pp. 848–851. [25] U. Rosenberg and R. Beyer, “Compact T-junction orthomode transducer facilitates easy integration and low cost production,” in Proc. 41st Eur. Microw. Conf. (EuMC), Oct. 2011, pp. 663–666. [26] P. N. Choubey and W. Hong, “Novel wideband orthomode transducer for 70–95 GHz,” in Proc. IEEE Int. Wireless Symp. (IWS), Mar./Apr. 2015, pp. 1–4. [27] C. A. Leal-Sevillano, Y. Tian, M. J. Lancaster, J. A. Ruiz-Cruz, J. R. Montejo-Garai, and J. M. Rebollar, “A micromachined dual-band orthomode transducer,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 1, pp. 55–63, Jan. 2014. [28] A. Dunning, “Double ridged orthogonal mode transducer for the 16–26 GHz microwave band,” in Proc. Workshop Appl. Radio Sci., 2002, pp. 1–3. [29] T. J. Reck and G. Chattopadhyay, “A 600 GHz asymmetrical orthogonal mode transducer,” IEEE Microw. Compon. Lett., vol. 23, no. 11, pp. 569–571, Nov. 2013. [30] C. A. Leal-Sevillano et al., “Compact duplexing for a 680-GHz radar using a waveguide orthomode transducer,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 11, pp. 2833–2842, Nov. 2014. [31] D. Dousset, S. Claude, and K. Wu, “A compact high-performance orthomode transducer for the atacama large millimeter array (ALMA) band 1 (31–45 GHz),” IEEE Access, vol. 1, pp. 480–487, 2013. [32] G. Engargiola and A. Navarrini, “K-band orthomode transducer with waveguide Ports and balanced coaxial probes,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 5, pp. 1792–1801, May 2005. [33] G. Narayanan and N. Erickson, “Full-waveguide band orthomode transducer for the 3 mm and 1 mm bands,” in Proc. 14th Int. Symp. Space Terahertz Technol., 2003, pp. 1–5. [34] G. Narayanan and N. R. Erickson, “A novel full waveguide band orthomode transducer,” in Proc. 13th Int. Space Terahertz Symp., 2002, pp. 1–10. [35] Y. Li, “A simple waveguide OMT designed for millimeter wave application,” in Proc. IEEE Int. Conf. Microw. Millim. Wave Technol. (ICMMT), Jun. 2016, pp. 576–578.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

Mohamed A. Abdelaal (S’12) received the B.Sc. degree (Hons.) from the Faculty of Engineering, Department of Electronics and Communications Technology, Modern Academy, Cairo, Egypt, in 2008, and the M.Sc. degree from the Faculty of Engineering, Department of Electronics and Communications, Arab Academy for Science, Technology & Maritime Transport, Cairo, in 2013. He is currently pursuing the Ph.D. degree at the Electrical and Computer Engineering Department, Concordia University, Montreal, QC, Canada. From 2008 to 2014, he was a Teaching and Research Assistant with the Faculty of Engineering, Department of Electronics and Communications Technology, Modern Academy. From 2009 to 2013, he was a Teaching and Research Assistant with the Faculty of Engineering, Department of Electronics and Communications, Arab Academy for Science, Technology & Maritime Transport. Since 2014, he has been a Research Assistant with Concordia University. His current research interests include the orthomode transducer design and analysis, microwave reciprocal/nonreciprocal device design and analysis, ferrite materials-based microwave device design and analysis, dielectric resonators antenna design, millimeter wave antennas and devices, antenna design, and ridge gap waveguide technology. Mr. Abdelaal was ranked the first over 600 students in the bachelor’s degree achieving a grade of 95.92%. He was the recipient of the Concordia University International Tuition Fee Remission Award, the Graduate Student Support Program Award in 2014 for his excellence, and the Canadian National Committee-International Union of Radio Science Student Travel Award in 2015. He served as a Reviewer for The Institute of Engineering and Technology Journal.

Shoukry I. Shams (S’04–M’08) received the B.Sc. (Hons.) and M.Sc. degrees in electronics and communications engineering from Cairo University, Giza, Egypt, in 2004 and 2009, respectively, and the Ph.D. degree in electrical and computer engineering from Concordia University, Montreal, QC, Canada, in 2016. From 2005 to 2006, he served as a Teaching and Research Assistant with the Department of Electronics and Communications Engineering, Cairo University. From 2006 to 2012, he was a Teaching and Research Assistant with the IET Department German University, Cairo, Egypt. From 2012 to 2016, he was a Teaching and Research Assistant with Concordia University. His current research interests include the microwave reciprocal/nonreciprocal design and analysis, high-power microwave subsystems, antenna designs, and material measurements. Dr. Shams was the GUC-IEEE Student Branch Chair from 2010 to 2012. He was the recipient of the Faculty Certificate of Honor in 1999 and the Distinction with Honor in 2004 from Cairo University. He was the recipient of the Concordia University Recruitment Award in 2012 and the Concordia University Accelerator Award in 2016.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Ahmed A. Kishk (S’84–M’86–SM’90–F’98) received the B.Sc. degree in electronic and communication engineering from Cairo University, Giza, Egypt, in 1977, the B.Sc. degree in applied mathematics from Ain-Shams University, Cairo, Egypt, in 1980, and the M.Eng. and Ph.D. degrees from the University of Manitoba, Winnipeg, MB, Canada, in 1983 and 1986, respectively. From 1977 to 1981, he was a Research Assistant and an Instructor with the Faculty of Engineering, Cairo University. From 1981 to 1985, he was a Research Assistant with the Department of Electrical Engineering, University of Manitoba, where he was a Research Associate Fellow from 1985 to 1986. In 1986, he joined the Department of Electrical Engineering, University of Mississippi, Oxford, MS, USA, as an Assistant Professor. He was on sabbatical leave at the Chalmers University of Technology, Gothenburg, Sweden, from 1994 to 1995 and 2009 to 2010. He was a Professor with the University of Mississippi, from 1995 to 2011, where he was the Director of the Center for Applied Electromagnetic System Research from 2010 to 2011. He has been a Professor with Concordia University, Montreal, QC, Canada, since 2011. He has authored over 310-refereed journal articles and 450 conference papers. He has co-authored four books and several book chapters. He is currently the Tier 1 Canada Research Chair in advanced antenna systems with Concordia University. He is an editor of three books. He offered several short courses in international conferences. His current research interests include millimeterwave antennas for 5G applications, analog beamforming network, dielectric resonator antennas, microstrip antennas, small antennas, microwave sensors, RFID antennas for readers and tags, multifunction antennas, microwave circuits, EBG, artificial magnetic conductors, soft and hard surfaces, phased array antennas, computer-aided design for antennas, reflect/transmit arrays, wearable antennas, and feeds for Parabolic reflectors. Dr. Kishk is a Fellow of the Electromagnetic Academy and the Applied Computational Electromagnetics Society (ACES). He is a member of the Antennas and Propagation Society (AP-S), the Microwave Theory and Techniques Society, the Sigma Xi Society, the U.S. National Committee of International Union of Radio Science Commission B, the Phi Kappa Phi Society, the Electromagnetic Compatibility Society, and ACES. He and his students were the recipients of several awards. He was the recipient of the 1995 and 2006 Outstanding Paper Awards for papers published in the ACES Journal. He was the recipient of the 1997 Outstanding Engineering Educator Award from the Memphis Section of the IEEE, the Outstanding Engineering Faculty Member of the Year in 1998 and 2009, the Faculty Research Award for outstanding performance in research in 2001 and 2005, the Award of Distinguished Technical Communication for the entry of IEEE Antennas and Propagation Magazine in 2001, and the Valued Contribution Award for outstanding invited presentation, EM Modeling of Surfaces with STOP or GO Characteristics–Artificial Magnetic Conductors and Soft and Hard Surfaces from ACES. He was the recipient of the Microwave Theory and Techniques Society, Microwave Prize in 2004 and the 2013 Chen-To Tai Distinguished Educator Award of the IEEE AP-S. He was an Associate Editor of the Antennas and Propagation Magazine from 1990 to 1993. He was a Distinguished Lecturer of the AP-S from 2013 to 2015. He was an Editor of the Antennas and Propagation Magazine from 1993 to 2014. He was a Co-Editor of the Special Issue on “Advances in the Application of the Method of Moments to Electromagnetic Scattering Problems,” in the ACES Journal. He was also an Editor of the ACES Journal during 1997. He was an Editor-in-Chief of the ACES Journal from 1998 to 2001. He was the Chair of the Physics and Engineering Division, Mississippi Academy of Science, from 2001 to 2002. He was a Guest Editor of the Special Issue on “Artificial Magnetic Conductors, Soft/Hard Surfaces, and Other Complex Surfaces,” in the IEEE T RANSACTIONS ON A NTENNAS AND P ROPAGATION in 2005. He was a Technical Program Committee member for several international conferences. He was a member of the AP AdCom from 2013 to 2015. He was the 2017 AP-S President.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Ortho-Mode Sub-THz Interconnect Channel for Planar Chip-to-Chip Communications Bo Yu, Student Member, IEEE, Yu Ye, Member, IEEE, Xuan Ding, Yuhao Liu , Zhiwei Xu, Senior Member, IEEE, Xiaoguang Liu, Member, IEEE, and Qun Jane Gu , Senior Member, IEEE Abstract— This paper presents for the first time the design, fabrication, and demonstration of a dielectric waveguide (DWG)based ortho-mode sub-THz interconnect channel for planar chip-to-chip communications. By combining the proposed new transition of microstrip line with DWG orthogonally, the orthomode transition is constructed to form an ortho-mode channel. The measured minimum insertion losses for the E y11 mode and the E x11 mode are 6.6 dB with 20.3-GHz 3-dB bandwidth and 6.5 dB with 55.2-GHz 3-dB bandwidth, respectively. The simulation and measurement results agree well with each other. Index Terms— Channel, chip-to-chip, communication, dielectric waveguide (DWG), interconnect, micromachined, microstrip line (MSL), ortho-mode transition (OMT), space division multiplexing (SDM), sub-THz, THz.

I. I NTRODUCTION

F

OR chip-to-chip interconnect, there are two most important factors, energy efficiency, defined as the dc power divided by the data rate, and bandwidth density, defined as the data rate divided by the channel area. On the one hand, the power consumption and cooling cost keep increasing with higher data rate. To mitigate this issue, high-energy-efficiency interconnects are required. On the other hand, the CPU pin number increases slowly compared with the CPU bus bandwidth per wire. To satisfy the requirement of the total bus bandwidth, high-bandwidth-density interconnects are highly demanded. One key factor determining energy efficiency is the channel loss. The transmission line has relatively large loss at high frequencies [1], [2]. The metallic waveguide (MWG) also has higher loss than the dielectric waveguide (DWG) at high frequencies [3]–[8], which can be as high as 0.15 dB/mm for MWGs at 600 GHz [4]. Meanwhile, the attenuation of silicon (Si) is reported as 0.017–0.034 dB/mm at

Manuscript received July 1, 2017; revised September 15, 2017, October 27, 2017, and November 9, 2017; accepted November 19, 2017. This work was supported by the National Science Foundation. (Corresponding author: Qun Jane Gu.) B. Yu, Y. Ye, X. Ding, X. Liu, and Q. J. Gu are with the Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA 95616 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; jgu@ ucdavis.edu). Z. Xu is with Zhejiang University, Hangzhou, Zhejiang 310058, China (e-mail: [email protected]). Y. Liu is with Skyworks Solutions, Irvine, CA 92617 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2779496

Fig. 1.

Proposed DWG-based ortho-mode interconnect architecture.

300–1000 GHz [6], [8]. Therefore, Si DWG is a good candidate for the high-energy-efficiency sub-THz/THz interconnect. High permittivity of Si supports small channel dimensions, which thus results in high bandwidth density. Besides, the space division multiplexing (SDM) technique can further boost the bandwidth density by sharing multiple logical channels through the same physical DWG link. Thus, the total bandwidth density with N logical channels is given by ρBW =

BW × N ACHNL

(1)

where the BW is the channel bandwidth for each mode, N is the number of the propagated modes, and ACHNL is the effective channel area. The SDM can also combine with the frequency division multiplexing, the time division multiplexing, and/or the code division multiplexing to further boost the data rate and bandwidth density as illustrated in Fig. 1. There are two major research areas, optical interconnect and electrical interconnect, to address the interconnect issue. Optical interconnects [9]–[12] have the advantages of low loss and wide bandwidth, whereas the integration of high-efficiency light sources with current CMOS processes is still very challenging. Electrical interconnects [3], [13]–[26] have the merits of compatibility and scalability with silicon processes while with the drawback of high loss at high frequencies. Therefore, both schemes have their own limitations to completely address the interconnect issue. The Si DWG-based sub-THz interconnect, using the spectrum sandwiched between optical and microwave frequencies, had been proposed to solve the interconnect issue by leveraging the advantages of both optical and electrical interconnect approaches: low-loss quasi-optical channels as well as advanced high-speed semiconductor devices [27]–[30]. With this sub-THz channel, the high-energy-efficiency and highbandwidth-density single-mode sub-THz interconnect had been demonstrated in [31].

0018-9480 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

Fig. 2.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Four-port OMT model.

To further boost the bandwidth density, the ortho-mode channel has been studied in this paper. The ortho-mode transition or transducer (OMT), known as a polarization duplexer, is a device combining or separating orthogonally polarized signals. The ideal OMT can be modeled as a four-port device as shown in Fig. 2. The 4 × 4 scattering matrix S is given by ⎤ ⎡ 0 0 1 0 ⎢0 0 0 1⎥ ⎥ (2) S=⎢ ⎣ 1 0 0 0 ⎦. 0 1 0 0 For an ortho-mode channel, the majority research focuses on the MWG-based OMT channel, which requires expensive machining and are not compatible with planar structures [32]–[34]. In this paper, we have demonstrated a Si DWG-based OMT channel with a small channel area and compatible with planar micromachine processes. The OMT channel consists of a square waveguide and two OMTs. Compared with our previous work [35], this paper presents a discussion about a thorough analysis of a single-mode and OMTs design, including the difference between single-ended and differential transition feeding and the planar integration considerations, including a bending structure and a backside trench. Besides, the loss distribution for the complete structure is analyzed. With the optimization of two important design parameters, the channel performances are also significantly boosted compared with [35]. First, the length of the microstrip line (MSL) is reduced. Second, the metal thickness is increased from 100 to 300 nm. Therefore, the channel performance is significantly improved from 8–9 dB to 6.6 dB. This paper is organized as follows. Section II reviews and presents the design method of the OMT channel and planar integration considerations. Section III discusses the fabrication, measurement, and nonideality of the OMT subTHz interconnect channel, which is followed by the conclusion in Section IV. II. OMT C HANNEL D ESIGN Before discussing the transition design, a DWG design is reviewed. The design of DWG includes several aspects, mode, loss, bandwidth, isolation, and so on. For a rectangular DWG, the fundamental mode is E y11 or E x11 . The loss of a DWG is determined by several factors, including the dielectric loss of the material, the geometry of the waveguide, and possibilities of mode conversion. Material loss is one of the most critical loss source. Reference [27] has

Fig. 3. Differential MSL-to-DWG single-mode transition. (a) Perspective view. (b) Side view.

investigated that high-resistivity silicon is a good candidate for the DWG. Besides, the bending and discontinuity structures could cause the reflection loss and radiation loss. In addition, the additional loss could be from the mode conversion, which is indicated by the effective index of the first several modes [27]. The bandwidth of a DWG is primarily determined by the dispersion characteristics of the chosen mode of the propagating wave and the orthogonality and/or isolation from other modes. For a 500 μm × 500 μm Si DWG with the fundamental mode, the simulated cutoff frequency, which is 3 dB lower than the minimum insertion loss, is 94.7 GHz. Channel isolation is another key factor for high-bandwidthdensity and high-energy-efficiency communication systems. Higher channel isolation leads to smaller channel space for higher bandwidth density and higher energy efficiency due to smaller coupling noise. A. Transition Design The conventional MSL-to-MWG transition can confine EM waves inside thanks to the metal wall, but it is hard to be integrated in planar processes. To solve this issue, the MSL-toDWG transition could be used. Since the DWG does not have the metal wall, the reference ground is located in infinite for the single-ended MSL-to-DWG transition, which causes large loss. Therefore, a differential probes-based MSL-to-DWG transition is proposed to form a differential mode intrinsically to match E y11 or E x11 mode in the DWG. Fig. 3(a) and (b) shows the configuration of the proposed differential MSLto-DWG single-mode transition. It consists of a DWG, two MSLs, a pair of microstrip probes for the transition, and a backshort on the bottom of the DWG. Since the center operation frequency is set at 175 GHz, the length of each side of the DWG is about 500 μm [27]. With a substrate of 20-μm bisbenzocyclobutene (BCB) (εr = 2.65), the line width of the 50- MSL is 56 μm. With the center opening window between MSLs, the EM waves propagate in both directions, up and down, along the DWG. The backshort is placed λ/4 away from the feeding position to provide out-of-phase signal cancellation and only allow the EM waves propagate along the updirection of the DWG. The differential microstrip probes are used to form the differential mode. Fig. 4(a) and (b) shows the vectors of the E-field distributions of the transition for the E y11 and E x11 modes,

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. YU et al.: ORTHO-MODE SUB-THz INTERCONNECT CHANNEL FOR PLANAR CHIP-TO-CHIP COMMUNICATIONS

3

Fig. 4. Simulated vector of the E-field distribution of the transition. (a) E y11 . (b) E x11 . Cross-sectional view of the E-field distribution. (c) MSL. (d) Transition interface. (e) DWG.

Fig. 5.

Schematic of the input impedance Z in . (a) Side view. (b) Top view.

respectively, which is based on full-wave simulation in ANSYS high-frequency structure simulator. The EM waves are gradually transitioned from the quasi-TEM in the MSL as shown in Fig. 4(c), to the hybrid mode in the transition as shown in Fig. 4(d) and then to the E y11 or E x11 mode in the DWG as shown in Fig. 4(e). The essential mechanism of the transition design is the impedance matching. The input impedance of the transition Z in , shown in Fig. 5, is determined by the probe length. Fig. 6(a) shows the input impedance versus the probe length and Fig. 6(b) shows the input impedance versus the frequency. To match with 50 , the probe length is 180–200 μm at the frequencies of around 175 GHz. Simulated S11 versus the backshort-to-probe distance at 175 GHz is shown in Fig. 6(c). At the design frequency, λ/4 in silicon is approximated as 140 μm. With the differential MSL-to-DWG single-mode transition, an OMT is proposed in this paper as shown in Fig. 7. It consists of two single-mode transitions, which are placed orthogonally. The magnitude of the E-field distributions for the in-phase mode and quadrature-phase mode combinations is shown in Fig. 8(a) and (b), respectively. A back-to-back structure is shown in Fig. 9(a). The two pairs of the differential ports P1+ and P1− and P3+ and P3−

Fig. 6. (a) Simulated input impedance versus the probe length when frequency = 175 GHz. (b) Input impedance versus the frequency when the probe length = 200 μm. (c) Simulated S11 versus the backshort-to-probe distance with the frequency = 175 GHz and probe length = 200 μm.

Fig. 7.

Structure of the proposed OMT. (a) Perspective view. (b) Top view.

are for E y11 mode and that of ports P2+ and P2− and P4+ and P4− are for E x11 mode. As shown in Fig. 9(b), the minimum insertion loss is about 2.6 dB with 47.7-GHz 3-dB bandwidth referred to the minimum insertion loss for both modes. S11 is better than −10 dB at the range of 140–200 GHz. The S21 is better than −30 dB in the range of 145–187 GHz, and S41 is better than −30 dB in the range of 140–200 GHz.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 8. Simulated magnitude of the E-field distribution with two combination modes. (a) In phase. (b) Quadrature phase.

Fig. 11. (a) Perspective view of the planar structure of the OMT channel. (b) Side view of the channel. (c) Simulated transmission loss versus the bending radius r.

Fig. 9.

(a) Back-to-back OMT channel. (b) Simulated S-parameters.

Fig. 10. (a) Schematic of the rectangular planar rat-race balun. (b) Simulated phase difference between S21 and S31 . (c) Simulated magnitudes of S21 and S31 .

B. Planar Integration To convert the differential signals to single-ended signal, the rectangular planar rat-race balun [36], [37] is utilized as shown in Fig. 10(a). With the characteristic impedance of 70 , the line width of the balun is 34 μm. The ratio of the two individual branch lengths is 1:3 to form a 180° phase difference at the center frequency. The lengths of two branches are 295 and 975 μm through the optimization.

The simulated phase difference is less than 7° at the range of 140–190 GHz as shown in Fig. 10(b). The simulated minimum insertion loss for each branch is 3.5 dB as shown in Fig. 10(c). The simulated minimum insertion loss of backto-back baluns is about 1.0 dB. To implement the planar chip-to-chip structure, a bended DWG is the most intuitive method as shown in Fig. 11(a) and (b). The DWG bend is a bridge between the straight DWG and planar OMT. The bending radius r is the inner radius of the bend. The cost of a DWG bend is the introduced bending loss for both modes [27]. The bending loss includes two parts, radiation loss and mode conversion loss [27]. For small r compared with the cross-sectional area, both radiation loss and mode conversion loss are severe. As shown in Fig. 11(c), the larger the r value, the smaller the bending loss is. The insertion loss for each 500 μm × 500 μm bend is about 0.3 dB. Also, the bending losses for two modes are different due to the bending direction. To overcome this issue, a large r value is preferred. On the other hand, to enhance the reliability of the structure and reduce the profile of the structure, a small r value is preferred. By trading off these considerations, r is set as 400 μm. The side view of the transition is shown in Fig. 12(a), which consists of a DWG bend, a λ/4 DWG, and a signal feeding structure. However, the size of the feeding structure is much larger than the λ/4 DWG so that the transition has reliability issue. To solve the mechanical reliability issue, the substrate of the λ/4 DWG with a whole piece instead of a single stub is used as shown in Fig. 12(b). However, a whole piece of substrate will introduce the EM wave leakage issue due to the enlarged size of the λ/4 DWG as shown in Fig. 13(a).

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. YU et al.: ORTHO-MODE SUB-THz INTERCONNECT CHANNEL FOR PLANAR CHIP-TO-CHIP COMMUNICATIONS

5

Fig. 12. Side view of the transition with (a) single stub as a substrate, (b) whole piece as the substrate, or (c) whole piece with the backside trench.

Fig. 13. Magnitude of the E-field distributions. (a) Without backside trench. (b) With backside trench.

Fig. 15. (a) Bottom view of the OMT. (b) Simulated transmission loss versus the trench depth when the trench width = 400 μm. (c) Simulated transmission loss versus the trench width when the trench depth = 100 μm.

Fig. 14. S-parameter comparisons between without and with backside trench. (a) Transmission. (b) Reflection.

The backside trench is designed to overcome the leakage issue as shown in Fig. 12(c). With the trench, the leakage is significantly reduced as shown in Fig. 13(b). Both performance of the transmission and reflection are improved as shown in Fig. 14. In terms of the electrical performance, a large trench is required; in terms of the mechanical reliability, a small trench size is preferred. The leakage loss will be minimized if the depth and the width of the trench are larger than 60 and 400 μm, respectively, as shown in Fig. 15. The completed OMT channel for planar chip-to-chip communications is illustrated in Fig. 16. It consists of a straight

DWG, two DWG bends, two transitions, MSLs, four baluns, four GSG-to-MSL transitions, two backshorts, and two backside trenches. To simplify the demonstration, the joined substrate is used instead of two separated substrates. Besides, the overpass trace and vias are utilized to allow a crossover line. To maintain the symmetry of the differential inputs, a dummy overpass trace and two vias are employed in the path without crossing over as shown in Fig. 16(b). The width and length of the overpass trace are 28 and 200 μm, respectively. The size of the vias is 40 μm × 40 μm with a 10-μm depth. To assemble the DWG onto the OMT, the BCB bonding technique is used. The transmission loss versus the BCB thickness is plotted in Fig. 17. In this design, the bonding thickness is , ρ > 1000  · cm, εr = 11.9, and tan δ = 0.001) is adhered on a silicon handle wafer by cool grease. After that, the HR silicon wafer is etched through by deep reactive ion etching (DRIE) process to generate the DWG with bends. Finally, the DWG is picked up and two bending ends are faced down. The OMT fabrication process is summarized in Fig. 20(b). The 150-μm HR silicon is used as handling wafer and the BCB 4026-57 (εr = 2.65 and tan δ = 0.015 [38]) is used for a thin-film substrate. The coupling structure includes three metal layers and two dielectric layers. Metal 1 (50-nm/300-nm Ti/Au) is deposited on the top of the Si wafer. After that, Metal2 is sandwiched between two 10-μm BCB followed by the electrical-plating for Metal 3 (2-μm Au). After Metal 3 metallization, the wafer is flipped for the backshorts.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. YU et al.: ORTHO-MODE SUB-THz INTERCONNECT CHANNEL FOR PLANAR CHIP-TO-CHIP COMMUNICATIONS

7

Fig. 21. Photographs of the top view and bottom view of the OMT channel. Fig. 23. Measured eye diagram for mode of S42 at PRBS 231 − 1 and BER < 1 × 10−12 .

The measured bit-error rates (BERs) versus data rate for the mode of S42 are shown in Fig. 23, together with the eye diagram in the inset. The 231 − 1 PRBS pattern is generated from Anritsu MP2011B. The data rate is up to 10 Gb/s. Due to the limited frequency response, the eye diagram cannot be measured for the mode of S31 . To evaluate the performance of sub-THz interconnect channel, we define a figure of merit (FoM) as FoM = Fig. 22. Comparisons of the simulated and measured S-parameters for the OMT channel.

Finally, the trenches around the backshort are etched by the DRIE process. With the prepared DWG and OMT, a pick-and-place tool (Finetech Fineplacer PICO A4) is used to bond the DWG to the OMT by an extra BCB layer. The device photographs are shown in Fig. 21. The total OMT channel size is 16 mm × 5 mm with a 10-mm straight DWG. Using the center opening window between MSLs in the OMT, the alignment accuracy is achieved less than ±5 μm. B. Measurement The measurement setup consists of an Agilent network analyzer (PNA-X N5247A), a pair of Virginia Diodes frequency converter modules (VDI WR5.1-VNAX), WR-5 (140–220 GHz) S-bend waveguides, and a pair of WR-5 probes. The short, open, load, thru calibration method is employed to set the reference plane at the probe tip. Limited by our test equipment, only two ports are connected each time, whereas the other two ports are floated thanks to the good isolation between the orthogonal paths. Fig. 22 shows the comparisons of simulated and measured S-parameters for the OMT channel. The measured minimum insertion losses for the mode E y11 and the mode E x11 are 6.6 dB with 20.3-GHz 3-dB bandwidth from 151.1 to 171.4 GHz and 6.5 dB with 55.2-GHz 3-dB bandwidth from 150.8 to 206 GHz, respectively. S21 and S41 are better than −20 dB in the range of 140–210 GHz.

BW/ACHNL ρBW = Loss| Unit Length Loss/lCHNL

(3)

where BW is the channel bandwidth, ρBW is the channel bandwidth density, ACHNL is the effective channel area, Loss is the channel loss, and lCHNL is the channel length. Higher operating frequency leads to the increasing of FoM as discussed in [27]. Table I summarizes the proposed channel performance and makes comparisons with the state-of-the-arts. Baseband-based interconnect channel [21] has very high bandwidth density due to the small effective channel area, but it is not scalable to higher frequencies. This paper has the bandwidth density of 33.3 GHz/mm2 and the FoM of 832 GHz/mm/dB for the mode E y11 , and the bandwidth density of 90.5 GHz/mm2 and the FoM of 2262 GHz/mm/dB for the modes E x11 . For the whole channel, the bandwidth density is 123.8 GHz/mm2 and the FoM is 3094 GHz/mm/dB with the total bandwidth of 75.5 GHz, since these two modes are independent. Compared with the other interconnect channels, this paper has higher bandwidth density and achieves the best FoM thanks to the low-loss wide-bandwidth channel and the small effective channel area. Reference [3] has a good FoM, but the channel loss is relatively high and the integration for planar technologies is relatively complicated. To analyze the discrepancy between simulation and measurement, the surface roughness, assembly accuracy, and nonideal shape for the DWG have been investigated. Since the signal waves are mainly confined inside the DWG, the surface roughness and assembly accuracy have negligible effects on the performance. The most possible factor is the nonideal fabrication of the DWG with bends as shown in Fig. 24. The shape of the cross section of

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

TABLE I C OMPARISON A MONG D IFFERENT H IGH -S PEED I NTERCONNECT C HANNELS

Fig. 24. Photograph of the DWG. (a) Bottom view. (b) View of the left transition interface. (c) Front view. (d) Back view.

Fig. 26. Simulated group delays for the channel with a square or trapezoid DWG. (a) P1 → P3. (b) P2 → P4. Insets: ortho-mode excitations for both square and trapezoid shapes.

Fig. 25. Comparisons of the simulated and measured S-parameters for E y11 mode of the OMT channel with the updated shape.

the DWG becomes polygon instead of square. The front width of the DWG is about 470 μm. The back width is 390–430 μm. To simplify the modeling, a trapezoid shape is used instead of the polygon shape. With the updated shape, the simulated S31 matches better with the measured result as shown in Fig. 25. The difference between simulation and measurement in Fig. 25 could be caused by the inaccurate modeling. The polygon shape for the DWG could be caused by the overetching. To guarantee that all exposed areas are etched through, the etching time is set as about 120% compared with the normal etching time. Since the device wafer is loosely

bonded on the handle wafer by cool grease with several points, somewhere under the device wafer is not closely attached with the handle wafer. After the expected etching areas are etched through, the etching gas could be stored in the bonding air gap. This could be solved by characterizing the etching time and/or using SOI substrates. In addition, to quantify the dispersion, the simulated group delays for both paths are shown in Fig. 26 with the average group delay of about 140 ps. The nonideal fabrication effect causes large group delay variations of the mode of S31 . IV. C ONCLUSION This paper, for the first time, presents the design, analysis, and demonstration of a DWG-based ortho-mode sub-THz

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. YU et al.: ORTHO-MODE SUB-THz INTERCONNECT CHANNEL FOR PLANAR CHIP-TO-CHIP COMMUNICATIONS

interconnect channel for planar chip-to-chip communications. The detailed analyses of the single-mode transition design and the OMT design, the planar integration methods, including the dimension selection of the bending structure and backside trench, are conducted. This approach opens a new direction for high-energy-efficiency high-bandwidth-density chipto-chip communications by providing multiple logical links through the same physical channel. The bandwidth density could be further boosted. Moreover, this technique can be readily scaled up to THz frequencies by scaling down the channel dimension. This results in a better energy efficiency and bandwidth density at higher frequencies. R EFERENCES [1] S. Amakawa et al., “Design of well-behaved low-loss millimetre-wave CMOS transmission lines,” in Proc. IEEE 18th Workshop Signal Power Integr. (SPI), May 2014, pp. 1–4. [2] K. H. K. Yau, I. Sarkas, A. Tomkins, P. Chevalier, and S. P. Voinigescu, “On-wafer S-parameter de-embedding of silicon active and passive devices up to 170 GHz,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2010, pp. 600–603. [3] G. Gentile et al., “Silicon-filled rectangular waveguides and frequency scanning antennas for mm-wave integrated systems,” IEEE Trans. Antennas Propag., vol. 61, no. 12, pp. 5893–5901, Dec. 2013. [4] K. M. K. H. Leong et al., “WR1.5 silicon micromachined waveguide components and active circuit integration methodology,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 4, pp. 998–1005, Apr. 2012. [5] M. N. Afsar, H. Chi, and X. Li, “Millimeter wave complex refractive index, complex dielectric permittivity and loss tangent of high purity and compensated silicon,” in Proc. Conf. Precis. Electromagn. Meas., Jun. 1990, pp. 238–239. [6] D. Grischkowsky, S. Keiding, M. Van Exter, and C. Fattinger, “Farinfrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 7, no. 10, pp. 2006–2015, 1990. [7] S. L. Smith and V. Dyadyuk, “Measurement of the dielectric properties of Rogers R/flex 3850 liquid crystalline polymer substrate in V and W band,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., vol. 4B. Jul. 2005, pp. 435–438. [8] J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz timedomain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 21, no. 7, pp. 1379–1386, 2004. [9] N. Holonyak and M. Feng, “The transistor laser,” IEEE Spectr., vol. 43, no. 2, pp. 50–55, Feb. 2006. [10] M. A. Green, J. Zhao, A. Wang, P. J. Reece, and M. Gal, “Efficient silicon light-emitting diodes,” Nature, vol. 412, no. 6849, pp. 805–808, 2001. [11] F. E. Doany et al., “Terabit/sec VCSEL-based 48-channel optical module based on holey CMOS transceiver IC,” J. Lightw. Technol., vol. 31, no. 4, pp. 672–680, Feb. 15, 2013. [12] S. Mishra, N. K. Chaudhary, and K. Singh, “Overview of optical interconnect technology,” Int. J. Sci. Eng. Res., vol. 3, no. 4, pp. 390–396, 2012. [13] J.-D. Park, S. Kang, S. V. Thyagarajan, E. Alon, and A. M. Niknejad, “A 260 GHz fully integrated CMOS transceiver for wireless chip-tochip communication,” in IEEE VLSI Circuits Symp. Dig., Jun. 2012, pp. 48–49. [14] F. Zhu et al., “A low-power low-cost 45-GHz OOK transceiver system in 90-nm CMOS for multi-Gb/s transmission,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 9, pp. 2105–2117, Sep. 2014. [15] C. W. Byeon, C. H. Yoon, and C. S. Park, “A 67-mW 10.7-Gb/s 60-GHz OOK CMOS transceiver for short-range wireless communications,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 9, pp. 3391–3401, Sep. 2013. [16] W.-H. Chen et al., “A 6-Gb/s wireless inter-chip data link using 43-GHz transceivers and bond-wire antennas,” IEEE J. Solid-State Circuits, vol. 44, no. 10, pp. 2711–2721, Oct. 2009. [17] H. Wu et al., “A 60 GHz on-chip RF-interconnect with λ/4 coupler for 5 Gbps bi-directional communication and multi-drop arbitration,” in IEEE Custom Integr. Circuits Conf. Dig., Sep. 2012, pp. 1–4.

9

[18] S. Moghadami, F. Hajilou, P. Agrawal, and S. Ardalan, “A 210 GHz fully-integrated OOK transceiver for short-range wireless chip-to-chip communication in 40 nm CMOS technology,” IEEE Trans. THz Sci. Technol., vol. 5, no. 5, pp. 737–741, Sep. 2015. [19] J. W. Holloway, L. Boglione, T. M. Hancock, and R. Han, “A fully integrated broadband sub-mmWave chip-to-chip interconnect,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 7, pp. 2373–2386, Jul. 2017. [20] J. Han, Y. Lu, N. Sutardja, K. Jung, and E. Alon, “Design techniques for a 60 Gb/s 173 mW wireline receiver frontend in 65 nm CMOS technology,” IEEE J. Solid-State Circuits, vol. 51, no. 4, pp. 871–880, Apr. 2016. [21] B. Dehlaghi and A. C. Carusone, “A 0.3 pJ/bit 20 Gb/s/wire parallel interface for die-to-die communication,” IEEE J. Solid-State Circuits, vol. 51, no. 11, pp. 2690–2701, Nov. 2016. [22] P. A. Francese et al., “A 30 Gb/s 0.8 pJ/b 14 nm FinFET receiver datapath,” in IEEE Int. Solid-State Circuits Conf. (ISSCC) Dig. Tech. Papers, Jan./Feb. 2016, pp. 408–409. [23] T. Shibasaki et al., “A 56 Gb/s NRZ-electrical 247 mW/lane serial-link transceiver in 28 nm CMOS,” in IEEE Int. Solid-State Circuits Conf. (ISSCC) Dig. Tech. Papers, Jan./Feb. 2016, pp. 64–65. [24] M. Fujishima, M. Motoyoshi, K. Katayama, K. Takano, N. Ono, and R. Fujimoto, “98 mW 10 Gbps wireless transceiver chipset with D-band CMOS circuits,” IEEE J. Solid-State Circuits, vol. 48, no. 10, pp. 2273–2284, Oct. 2013. [25] N. Weissman, S. Jameson, and E. Socher, “A packaged 106–110 GHz bi-directional 10 Gbps 0.11 pJ/bit/cm CMOS transceiver,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2015, pp. 1–4. [26] J. Li, Z. Xu, W. Hong, and Q. J. Gu, “A Cartesian error feedback architecture,” IEEE Trans. Circuits Syst. I, Reg. Papers, to be published. [27] B. Yu, Y. Liu, Y. Ye, J. Ren, X. Liu, and Q. Gu, “High-efficiency micromachined sub-THz channels for low-cost interconnect for planar integrated circuits,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 1, pp. 96–105, Jan. 2016. [28] B. Yu, Y. Liu, Y. Ye, X. Liu, and Q. J. Gu, “Low-loss and broadband G-band dielectric interconnect for chip-to-chip communication,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 7, pp. 478–480, Jul. 2016. [29] B. Yu, Y. Ye, X. L. Liu, and Q. J. Gu, “Microstrip line based sub-THz interconnect for high energy-efficiency chip-to-chip communications,” in Proc. IEEE Int. Symp. Radio-Freq. Integr. Technol. (RFIT), Aug. 2016, pp. 1–3. [30] B. Yu, Y. Ye, X. L. Liu, and Q. J. Gu, “Sub-THz interconnect channel for planar chip-to-chip communication,” in Proc. IEEE Int. Symp. Electromagn. Compat. (EMC), Jul. 2016, pp. 901–905. [31] Y. Ye, B. Yu, X. Ding, X. Liu, and Q. J. Gu, “High energy-efficiency high bandwidth-density sub-THz interconnect for the ‘last-centimeter’ chip-to-chip communications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2017, pp. 805–808. [32] R. D. Tompkins, “A broad-band dual-mode circular waveguide transducer,” IRE Trans. Microw. Theory Techn., vol. 4, no. 3, pp. 181–183, Jul. 1956. [33] A. M. Boifot, E. Lier, and T. Schaug-Pettersen, “Simple and broadband orthomode transducer (antenna feed),” Proc. Inst. Elect. Eng.—Microw. Antennas Propag., vol. 137, no. 6, pp. 396–400, Dec. 1990. [34] M. Ludovico, B. Piovano, G. Bertin, G. Zarba, L. Accatino, and M. Mongiardo, “CAD and optimization of compact ortho-mode transducers,” IEEE Trans. Microw. Theory Techn., vol. 47, no. 12, pp. 2479–2486, Dec. 1999. [35] B. Yu, Y. Ye, X. Ding, Y. Liu, X. Liu, and Q. J. Gu, “Dielectric waveguide based multi-mode sub-THz interconnect channel for high data-rate high bandwidth-density planar chip-to-chip communications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2017, pp. 1750–1752. [36] K. S. Ang, Y. C. Leong, and C. H. Lee, “Analysis and design of miniaturized lumped-distributed impedance-transforming baluns,” IEEE Trans. Microw. Theory Techn., vol. 51, no. 3, pp. 1009–1017, Mar. 2003. [37] Y. Ye, L.-Y. Li, J.-Z. Gu, and X.-W. Sun, “A bandwidth improved broadband compact lumped-element balun with tail inductor,” IEEE Microw. Wireless Compon. Lett., vol. 23, no. 8, pp. 415–417, Aug. 2013. [38] H.-M. Heiliger, M. Nagel, H. G. Roskos, H. Kurz, F. Schnieder, and W. Heinrich, “Thin-film microstrip lines for MM and sub-MM/wave on-chip interconnects,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 2. Jun. 1997, pp. 421–424.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Bo Yu (S’14) received the B.S. degree in electrical engineering from Sichuan University, Sichuan, China, in 2007, the M.S. degree in electrical engineering from Peking University, Beijing, China, in 2010, and the M.S. degree in electrical engineering from International Technological University, San Jose, CA, USA, in 2012. He is currently pursuing the Ph.D. degree in electrical engineering at the University of California at Davis, Davis, CA, USA. His current research interests include RF and microwave integrated circuit design and sub-THz/THz transceiver and interconnects.

Yu Ye (S’12–M’14) received the B.S. degree in physics from Nanjing University, Nanjing, China, in 2009, and the Ph.D. degree in electrical engineering from the Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai, China, in 2014. In 2014, he joined the University of California at Davis, Davis, CA, USA, where he is currently a Post-Doctoral Researcher involved in silicon-based millimeter/terahertz integrated circuit design. His current research interests include RF integrated circuit design and system architectures for wired and wireless communications.

Xuan Ding received the B.S. and M.S. degrees in electrical engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 2010 and 2013, respectively. He is currently pursuing the Ph.D. degree in electrical engineering at the University of California at Davis (UC Davis), Davis, CA, USA. He is currently with the High Speed Integrated Circuits and Systems Lab, UC Davis. His current research interests include RF, microwave, and THz integrated circuits and systems.

Yuhao Liu received the B.Eng. degree in electrical engineering from McMaster University, Hamilton, ON, Canada, in 2011, and the Ph.D. degree in electrical engineering from the University of California at Davis, Davis, CA, USA, in 2016. He is currently a Senior Electrical Engineer with Skyworks Solutions, Irvine, CA, USA. His current research interests include radio frequency microelectromechanical devices, RF acoustic resonators and filters, tunable filters, and THz technologies.

Zhiwei Xu (S’97–M’03–SM’10) received the B.S. and M.S. degrees from Fudan University, Shanghai, China, and the Ph.D. degree from the University of California at Los Angeles, Los Angeles, CA, USA, all in electrical engineering. He has held various industrial positions with G-Plus Inc., SST communications, Conexant Systems, NXP Semiconductors, and HRL Laboratories, where he was involved in development for wireless LAN and SoC solution for proprietary wireless multimedia systems, CMOS cellular transceivers, multimedia over cable systems, and TV tuners, various aspects of millimeter- and submillimeter-wave integrated circuits and systems, software-defined radios, high-speed ADCs, and ultralow power analog VLSI. He is currently with Zhejiang University, Hangzhou, China, as a Professor, where he is involved in integrated circuits and systems for Internet-of-Things and communication applications.

Xiaoguang Liu (S’07–M’10) received the B.S. degree from Zhejiang University, Hangzhou, China, in 2004, and the Ph.D. degree from Purdue University, West Lafayette, IN, USA, in 2010. He is currently an Associate Professor with the Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA, USA. His current research interests include radio frequency microelectromechanical devices and other reconfigurable high-frequency components, high-frequency integrated circuits, and biomedical and industrial applications of high-frequency communication and sensing systems.

Qun Jane Gu (M’07–SM’15) received the Ph.D. degree from the University of California at Los Angeles, Los Angeles, CA, USA, in 2007. From 2010 to 2012, she was an Assistant Professor with the University of Florida, Gainesville, FL, USA. Since 2012, she has been with the University of California at Davis, Davis, CA, USA. Her current research interests include high-efficiency, low-power interconnect, millimeter-wave, and submillimeterwave integrated circuits, system-on-a-chip design techniques, and integrated terahertz circuits and systems for communication, radar, and imaging. Dr. Gu was a recipient of the NSF CAREER Award, the 2015 COE Outstanding Junior Faculty Award, and the 2017 Qualcomm Faculty Award. She is a co-author of several Best Paper Awards, including the Best Conference Paper Award of the 2014 IEEE Wireless and Microwave Technology Conference, the Best Student Paper Award of the 2015 IEEE APMC, the Best Student Paper Award of the 2016 IEEE MTT-S IMS (Second Place), the Best Student Paper Award of the 2016 IEEE RFIT, and the Best Student Paper of the 2017 IEEE MTT-S IMS (Third Place).

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Novel Planar and Waveguide Implementations of Impedance Matching Networks Based on Tapered Lines Using Generalized Superellipses Santiago Cogollos , Member, IEEE, Joaquin Vague, Vicente E. Boria, Senior Member, IEEE, and Jorge D. Martínez, Member, IEEE

Abstract— For the practical implementation of RF and microwave impedance matching networks, a widely employed solution—alternative to the use of classical impedance transformers—is based on tapered lines. This paper shows a simple method to design smooth tapers that take into account the dispersion of the line and the required design bandwidth simultaneously. A planar taper has been designed in microstrip technology with the same length of classical ones but improving their performances. A waveguide prototype has also been designed with similar performance to a commercial one but with one third of its length. Both tapered structures have been obtained through the optimization of very few parameters using the same design strategy. As a result, the reflection coefficient of the tapers can be optimally adapted to a given specific mask using the prescribed value of physical length. Experimental results for both tapers are included for the validation of the proposed topologies and the related design method. Index Terms— Impedance matching, microwave circuits, nonuniform transmission lines, planar circuits, waveguides.

I. I NTRODUCTION MPEDANCE matching is an old problem in RF and microwave technology, arising when two sections of different transmission lines have to be connected properly (i.e., minimizing potential return losses due to the mismatch of associated characteristic impedances). The first approach and the simplest one is the single- or multisection transformer consisting of quarter-wavelength sections of transmission lines [1]. Here, the goal is usually an equirriple reflection coefficient over a certain bandwidth, which is directly related to the considered number of sections. However, soon appeared some reasons to avoid this solution. As a first reason, the higher the bandwidth is, the higher the number of required matching sections is; therefore, the total length is the number of sections multiplied by λg /4 (λg being the wavelength on each transmission-line section). These methods do not allow

I

Manuscript received June 29, 2017; revised October 24, 2017; accepted December 17, 2017. This work was supported by MINECO (Spanish Government) through R&D under Project TEC2016-75934-C4-1-R. (Corresponding author: Santiago Cogollos.) S. Cogollos, J. Vague, and V. E. Boria are with the Department of Communications, Universitat Politècnica de València, 46022 Valencia, Spain (e-mail: [email protected]; [email protected]; [email protected]). J. D. Martínez is with the Department of Electronic Engineering, Universitat Politècnica de València, 46022 Valencia, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2018.2791952

fixing a certain length as a design parameter. Another reason to avoid the multisection transformer is the inconvenience of the impedance steps. Adding edges, corners, or abrupt changes inside the structure, avoiding a smooth profile of the involved transmission line or waveguide, is a challenge in the manufacturing process at high frequencies. In addition, the impedance steps introduce reactances which may be compensated, but complicating the design. Moreover, these sharp corners produce the excitation of higher order modes hard to predict at the design stage. In the quest for the optimal solution, tapered lines were designed using simple curve approximations only for planar designs. Linear, triangular, and exponential tapers provided a simple profile for a given length [2], [3]. The clear advantage of a typical taper is the matching, not in a given bandwidth but from a certain minimum frequency fmin . Finally, the Klopfenstein (Dolph–Chebyshev) taper appeared giving the optimal solution (in terms of equirriple response) for planar realizations [4]. The same solution, but using a different approach was obtained in the same year [5], pointing to a slight flaw in Klopfenstein’s solution that was definitely corrected in [6]. However, this optimal solution shows certain limitations and drawbacks. The first drawback is the lack of smoothness because there are two steps (at the input and at the output) inherent to this design [see Fig. 1 (top)]. When this ideal design is built, the steps cannot be easily manufactured as shown in Fig. 1 (bottom). Steps are not desired if excitation of spurious modes must be avoided. The second issue is the final design formula in terms of impedances. This is very convenient if the relationship impedance–frequency is constant. However, one of the main assumptions in [5] is that the characteristic impedance is fixed with frequency, and the impedance formula is given for f min . These facts can be tolerable for microstrip designs [7], but unacceptable for waveguide structures where characteristic impedance depends on the height–width ratio b/a (see [5], [8]). In [9], the degradation at high frequencies was handled, focusing on the fact that, in practice, an infinite bandwidth is never needed. Therefore, from certain frequency upward, the degradation is unimportant. Using this line of thought, a shorter taper than the optimum one can be designed for the same performance within a required bandwidth. In the line of avoiding inconveniences due to the steps of Klopfenstein’s taper near-optimum designs were developed.

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

Fig. 1. Top: Klopfenstein (Dolph–Chebyshev) taper of length L used to match lines with different widths w1 and w2 . The steps at the input and output, inherent to this taper, are clearly shown. Bottom: detail of a prototype of a Klopfenstein taper where the input step is shown.

A smooth solution was given in [10], whose only drawback was related to the loss of the equirriple behavior. Another remarkable work, based on a solution in series form similar to [5] but with a controllable equirriple response, was given in [11]. Its final aim was to place the reflection zeros in desired locations to control the ripple according to the designers’ criteria. However, theoretical designs such as [4], [5], [10], and [11] were developed with some assumptions: TEM mode propagation and characteristic impedance constant with frequency. Empirically, it is well known that this fact is not seen in practice. Here, the dispersion starts to play an important role in several aspects. 1) Spatial Dispersion: Permittivity  or conductivity σ has dependence on the wavenumber [12] (in guided signals β takes the role of the wavenumber k). In general, any guided signal varies with the term e j (ωt −βz). If  = (β) and σ = σ (β) this phenomenon can be an issue. 2) Frequency Dispersion: Phase velocity has dependence on the signal frequency. It can be either monomode or multimode. Controlling the value of the permittivity of the material along the length of the taper, a matching is also possible even with classical techniques [5]. It is remarkable that new advances controlling the dispersion (mainly in planar technologies) based on metamaterials are still in the early stages. These structures are based on cells with quasi-periodic

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

arrangements, where losses may be still high enough, but promising results were recently shown in [13] and [14]. In general, the main objective of this paper is to consider in a simple manner any type of dispersion and practical issues that can be included in an electromagnetic (EM) simulator. For instance, it is widely known that the characteristic impedance of a microstrip line is frequency dependent, and shows a positive increase with an increase in frequency [15]. Inclusion of dispersion starts to be important in microwave designs [16], with the main goal of approximating the real behavior and EM simulations as much as possible. Further advances have been carried out in this field. In the synthesis framework, Park [17] included loss and dispersion, but the response is computed solving the approximate Ricatti’s equation instead of using EM solvers. In addition, specific tapers have been analyzed by solving numerically Ricatti’s equation including realistic modeling (e.g., fin-line tapers [18]). Very recently, an efficient method has been developed to analyze tapers using circuit models at the expense of complicating the related theory [19]. Another strong motivation of this paper was the potential implementation of taper solutions in [4], [5], [10], and [11] with waveguide technology, which is not easy since the characteristic impedance in such technology involves one additional degree of freedom (when compared with planar technologies). The width a and the height b of the waveguide simultaneously affect the waveguide characteristic impedance [20], and although the previous methods are extensible in terms of impedances, there is no clue about the best relationship between a and b to achieve both a smooth profile and the best performance in terms of frequency at the same time. Therefore, the aim of this paper is to provide an alternative smooth taper profile (in both planar and waveguide technologies) accounting for the dispersion, the bandwidth, and a flexible reflection requirement (not necessarily equirriple) using a simple optimization algorithm. The algorithm provides the parameters (usually two for planar and four for waveguide designs) of a generalized superelliptic curve that will be the profile of the proposed novel taper. This new design method can be applied not only to planar topologies, as it was successfully shown in [21], but also to waveguide technology where modal dispersion is an issue. In fact, it is for waveguide technology where this method provides a higher performance, using the same design procedure with only subtle changes in its implementation that are fully described in this paper. Traditionally, waveguide multisection transformers (see Fig. 2) are designed using several methods. Among them, it is worth mentioning the most popular ones: i.e., classical methods [22], optimization for multiband responses [23], iterative methods [24] for short-step transformers, and transformed variable methods using Richards’ variable [25], [26]. Of course, all of them have to deal with the problems of the involved impedance steps using full-wave EM simulations. All these responses can be easily translated to tapered waveguides using the technique shown in this paper, with the only change of using a different specification mask. The advantage is that a tapered line can achieve a similar performance to a

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. COGOLLOS et al.: NOVEL PLANAR AND WAVEGUIDE IMPLEMENTATIONS OF IMPEDANCE MATCHING NETWORKS

Fig. 2. 3-D view schematic of a classical four-section transformer in waveguide technology.

3

Fig. 4. Generalized superellipse with semiaxes a = 3 and b = 1.5 and exponents m = 2.5 and n = 0.5. The shadowed quadrant will be used for the taper profile.

Fig. 5. Taper profile formed mirroring and offsetting the generalized superellipse quadrant of Fig. 4. Fig. 3. of r.

Superellipses with semiaxes a = 4 and b = 2 and different values

multisection transformer with a substantially shorter device. An additional advantage of the design method proposed in this paper is that there is no need of performing any EM optimization, since there is always a circuital model accounting for the modal dispersion and the change of the waveguide shape. Our approach tries to find the optimum performance using simpler methodology than the previous works. Moreover, the shortest possible length can be easily found according to specifications. Finally, easier fabrication is achieved because the proposed profile is a simple curve. II. TAPER P ROFILE U SING S UPERELLIPSES The idea of using superellipses is not new in engineering since it was used in architecture, typography, and furniture design [27] among others. A superellipse is also known as a Lamé curve after Gabriel Lamé (well known for his general theory of curvilinear coordinates, and his notation and study of classes of ellipse-like curves). The superellipse is a general curve (see Fig. 3) with only one parameter r ranging from r → 0 (curve approaches the cartesian axes), and r → ∞ (a rectangle). The general equation of a superellipse is  x r  y r + =1 (1) a b

where a and b are the semiaxes of the superellipse, and r is the parameter controlling the roundness of the ellipse. r = 1 provides the rhombus (linear curve) and r = 2 is a true elliptic profile. Fig. 3 shows the shape of the superellipses for a = 2b and several values of r . However, the curvature controlled with the only parameter r does not give enough freedom to generate a profile with competitive behavior. Further generalizations are used in several fields such as modeling in botany or metamaterials (see [28]–[30]). The equation we are going to use for tapers is a quadrant of a generalized superellipse, whose expression is  x m  y n + =1 (2) a b where m, n ∈ R+ are constants controlling the curvature of each dimension. The advantage of using the quadrant of a superellipse is twofold: matching the curve at both ends is automatic, and second the curve does not change the sign of its tangent (true tapers are monotonic). A typical example of generalized superelliptic profile is shown in Fig. 4, where m = 2.5 and n = 0.5. The shadowed quadrant will be the profile of the taper. By mirroring and offsetting this quadrant, we obtain the required taper, as shown in Fig. 5. Extreme values for m (i.e., m = 0 or m → ∞) lead to a sharp transition in the input side of the taper. The same occurs with n and the output of the taper. In addition, values of m greater than one produce an outward-curved (convex) junction with

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 7. Reflection coefficient computed varying the number of sections chosen to divide the geometrical shape of the taper. These curves correspond to the superelliptic example considered in Section VI-A. Fig. 6. Multisection transformer and its equivalent transmission-line circuit representation.

the y-axis. Values of m lesser than one produce an inwardcurved (concave) junction with the y-axis. The same applies to n and the x-axis junction. Therefore, the generalized superellipse can deal with impedance matching at least as well as a linear taper (generalized superellipse with m = n = 1). Curves like generalized superellipses with real-valued exponents provide a better fitting feature than polynomials. This is because polynomials have integer order by definition. In addition, using polynomial approximations, it is quite difficult to avoid derivative changes that would produce a wiggly tapered line. The typical approach of dividing the curve in splines will increase the unknowns of the problem as the number of the splines increases. Therefore, the use of just two parameters for the whole profile provides a simple solution with excellent performance. III. D ESIGN S TRATEGY FOR P LANAR TAPERS Let us assume that the taper profile is given. The physical geometry of a taper is continuous, like the one shown in Fig. 5, but it can be studied as a discrete multisection transformer where the number of sections N tends to infinity (see Fig. 6). This structure can be simulated with available EM tools. However, the solver has to be harnessed to an optimizer to obtain the optimal parameters of the superelliptic profile. Many simulations need to be performed during the optimization and a fast equivalent circuit simulator is required. Therefore, a segmentation strategy is chosen, being quite accurate and fast enough for our purpose. Let L i = L/N be the length of each section, where L is the total length of the taper and N is the number of sections. For the sake of simplicity, the sections will have the same length L 1 = L 2 = · · · L N . The number of sections N has to be high enough to neglect the steps between sections, namely, each i th section has width wi , and N is chosen so that |wi − wi+1 | <  ∀i = 1, . . . , N − 1

(3)

where  is the maximum allowed step that can be neglected in the analysis (e.g.,  < λg /100 being λg is the guided wavelength at the minimum frequency of interest). This can be done because the effect of a change in width is very small

for planar structures. The physical model can be analyzed using the electrical model shown in Fig. 6 (bottom). This model is a cascade of N sections of transmission lines, where each section has a characteristic impedance Z i given by the relationship between the width and the impedance for the considered substrate [7], [31]. Here, the dispersion can be taken into account since εeff (and therefore Z i ) is not constant with frequency in applications with wide bandwidth. The procedure to obtain the reflection coefficient starts with the computation of the input impedance of the first line Z in1 as Z in1 = Z 1

Z L + j Z 1 tan φ1 Z 1 + j Z L tan φ1

(4)

and the result is stored for each frequency point. Needless to say that the electrical length φ1 = β1 L 1 is computed assuming dispersion, that is included in the computation of β1 . Now, Z in1 becomes the load impedance when Z in2 is computed and in general, at the step i th the input impedance of section i is computed as Z ini = Z i

Z ini−1 + j Z i tan φi Z i + j Z ini−1 tan φi

(5)

and the process is repeated up to section N where Z inN is obtained. Finally, the reflection coefficient can be computed as Z inN − Z 0 . (6) Z inN + Z 0 A typical set of curves obtained to check the convergence of the proposed analysis method against the number of sections is shown in Fig. 7. in =

IV. O PTIMIZATION A LGORITHM Once the number of steps N has been properly chosen, the width for each section can be obtained using a superelliptic profile with parameters m and n using a simple function. The input parameters of the function are (N, m, n, w N , w1 ). The length of the taper is L = a (a being the semimajor axis of the generalized superellipse) and b = (w N − w1 )/2 (b being the semiminor axis), as can be deduced from Figs. 5 and 6. In order to avoid any step at the beginning or at the end of the taper, we assume that w N = w0 , w0 being the width of the line with Z 0 we are trying to match to the line with Z L . For the

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. COGOLLOS et al.: NOVEL PLANAR AND WAVEGUIDE IMPLEMENTATIONS OF IMPEDANCE MATCHING NETWORKS

same reason, the width of the first section w1 = w L , where w L is the width of the line with characteristic impedance Z L . Therefore, the taper starts with w N and smoothly changes its width to finally reach w1 . If we assume w N > w1 as it is drawn in Fig. 6, the function giving the width of i th section is   m 1/n i wi = w1 + (w N − w1 ) 1 − (7) N where we have assumed all sections having the same length for the sake of simplicity. Once the curve is chosen, two things are required: first, an EM simulator capable of dealing with such profile with a fair accuracy and second, an optimization algorithm providing a good convergence. The former is the main problem. However, for microstrip structures, a transmission-line approach can be used with intensive segmentation, and neglecting the small steps in the structure as explained in Section III. If the number of segments is high, the simulated structure behaves as a smooth and continuous device. Therefore, the designer has to choose the maximum impedance step that can be neglected, and enforce a partition with such small steps. After a convergence study, the number of segments needed for accurate simulations can be safely fixed. A typical convergence study is shown in Fig. 7, where a number of steps higher than 100 (i.e., N ≥ 100) do not produce a high impact on the response for this particular case. The optimization part of the problem has a wide range of possibilities. Since the number of involved optimization parameters is small, the authors chose a nonlinear least-squares method Nfreq

min E22 = min m,n

m,n



ei2

(8)

i=1

where E is the error function with Nfreq components ei (the number of points in the frequency sweep), and each component is defined as S11 ( f i ) − M( fi ), if S11 ( f i ) − M( fi ) > 0 (9) ei = 0, otherwise where M( fi ) and S11 ( f i ) are the values of the mask and of the reflection coefficient (in logarithmic units) at the i th frequency point, respectively. The reflection coefficient S11 changes not only with frequency but also with the shape of the taper that ultimately depends on m and n. The algorithm used to solve the minimization problem is quasi-Newton, and the Jacobian matrix can be obtained both numerically or analytically (if a segmentation of the problem with uniform transmission lines is used). The auxiliary matrix needed as the Jacobian in this algorithm is the Jacobian of the reflection coefficient   ∂ S11 ∂ S11 . (10) J= ∂m ∂n The computation of J for each frequency point speeds up the optimization algorithm if it is computed efficiently. Obviously it will depend on the approach used for the analysis of the structure, but in any case it can be used as a good

5

Fig. 8. Dependence among different variables used to apply the multivariate chain’s rule to obtain the Jacobian matrix.

estimation if computed using approximate formulas. Applying wise approximations in the EM simulation, the computation of S11 and its Jacobian can be performed at very high speed as follows. Let in = S11 be the reflection coefficient of the taper. If the goal is to obtain an approximation of the Jacobian with respect to m and n, we first notice that there is a dependence of the width with m and n, namely, w(m, n) given by (7) for each section. The next step is to obtain the electrical length φ, the characteristic impedance Z 0 , and the load impedance Z L from w using the classical formulas applied to each uniform i th section [7], [31]; and recalling that the load impedance of the i th section is the input impedance of the (i − 1)th section as explained before. In our case, we used formulas accounting for finite thickness of the microstrip conductor and dispersion effects, with the aim of obtaining the highest accuracy. It is clear that from the electrical length and the characteristic impedance of the i th section φi , Z i , and the input impedance of the (i − 1)th section Z ini−1 , the input impedance i th section Z ini can be obtained using (5) and ini using (6). Therefore, the whole chain is established and its diagram is shown in Fig. 8. Applying the chain’s rule and dropping the subindex i to avoid cumbersome notation, we have   ∂ in ∂ in J= ∂m ∂n ⎛ ∂φ ⎞ ∂w     ⎟ ∂w ∂w ∂ Z in ∂ Z in ∂ Z in ⎜ ∂ in ∂ Z0 ⎟ ⎜ · = · ⎝ ∂w ⎠ · (11) ∂ Z in ∂φ ∂ Z 0 ∂ Z L ∂m ∂n ∂ ZL ∂w

where each term can be computed analytically for each section and for each frequency point. Now, we are ready to write the direction v of the maximum variation of the error function [32] to seek the extremum using (8) and (9)   Nfreq 2 ∂E2  ∂E 2 2 2 =2 Ji ei . (12) v = ∇E2 = ∂m ∂n i=1

Namely, the direction of maximum variation is proportional to Ji (Jacobian at the i th frequency point) given by (11) as expected. It is important to note that all functions involved (e.g., Z in and φ) have to be computed for each section and for each frequency point. Therefore, if the functions are stored, the memory needed for each function is Nfreq × N complex numbers in general.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

V. S TRATEGY FOR WAVEGUIDE TAPERS Usually, an adaptation between two standard rectangular waveguides is needed for certain applications. Here, we cannot apply the typical planar approaches, since the waveguide geometry depends on the two dimensions (width and height) of the waveguide. There are several typical definitions of the characteristic impedance (power–voltage, power–current, and voltage–current), all of them depending on a and b and the modal impedance [20]. However, using the optimization technique for planar tapers to both dimensions separately, waveguide tapers can be designed without further difficulty choosing any of the aforementioned waveguide impedance definitions. The optimization problem is formally the same as the one described for planar tapers, with the only difference that there are two variables (m, n) for each rectangular waveguide dimension (four in total). This sole difference does not mean a great problem, since a four-variable optimization problem can be handled comfortably using the same optimization technique used for planar tapers. In order to avoid complex analytical expressions for the Jacobian matrix, in this case the computation has been carried out numerically. The main problem arises in the analysis part of the proposed waveguide taper. Here, we cannot neglect the steps when the taper is divided in a multisection structure, each one having a different rectangular cross section. The solution is to take into account such transitions involving a change in terms of height and width. Applying the superposition principle, we are going to assume that a waveguide step in height and width can be decomposed into two simpler steps (one in width and one in height). Since the partition provides small steps, we can use the asymptotic expressions given in [8]. Let be a step from a waveguide with dimensions a (width) and b (height), guided wavelength λg , and characteristic impedance Z 0 to a waveguide with parameters a , b , λ g , and Z 0 . The change in width leads to a change in the impedance following the expression:   λ g a Z 0 γ2 1+γ + ∀γ  1 (13) ≈ Z0 λg a 2 where a . (14) a The step itself can be modeled as a parallel reactance Z step = j X whose asymptotic expression for γ  1 is  2  2 λg γ (1 + γ ) ln γ 27 Q + Q Z0 1− (15) ≈ X 2a 8 1 + 8 ln γ2 1 − γ2 γ =1−

where Q =1−



 1−

2a 3λ

2



Q =1−

 1−

2a 3λ

2 (16)

and λ is the free-space wavelength. The step in height changes the characteristic admittance as Y0 b =1−δ (17) = Y0 b

Fig. 9. Equivalent transmission-line circuit representation of a multisection transformer.

Fig. 10. Linear waveguide taper WR-90 to WR-62. Blue dashed line: EM simulation using CST. Red dotted line: EM simulation using HFSS. Green solid line: simulation using marcuvitz formulas.

and the step itself can be modeled tance Ystep = j B, whose asymptotic (small step) is    B 2b δ 2 2 ln 2δ +1+ ≈ Y0 λg 2 1−δ

as a parallel suscepexpression for δ  1 17 16



b λg

2  .

(18)

The formulas are really accurate provided the step is small [8], and the model is extremely fast because the computation effort of the previous formulas for each step is negligible. The model is shown in Fig. 9, where the only difference with the planar model are the shunt reactances (modeling the corresponding waveguide steps or transitions). In order to check the simulator built with Marcuvitz’s formulas against commercial simulators, a linear taper between WR-90 and WR-62 standard waveguides has been used as a benchmark. The taper length chosen for this example is 150 mm, which is the typical length of a commercial taper. Fig. 10 shows the simulation of the linear taper using HFSS and CST compared with the results given by Marcuvitz’s formulas. The results agree very well taking into account that Marcuvitz’s formulas provide the results in less than 1 s (with a standard dual-core PC at 2.2 GHz with 4 GB of RAM), even though a partition with 400 sections has been used. It should be noted that the way the commercial EM simulators discretize the volume leads to different solutions, depending on whether the discretization follows the true shape or not. Therefore, instead using mesh sizes, a convergence criteria based on the solution stability is normally used. VI. D ESIGN E XAMPLES A. Planar Taper In order to check the performance of the novel impedance matching structure, a taper based on a generalized superellipse

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. COGOLLOS et al.: NOVEL PLANAR AND WAVEGUIDE IMPLEMENTATIONS OF IMPEDANCE MATCHING NETWORKS

7

Fig. 11. Reflection response of the linear taper compared with the superelliptical solution. The dotted line shows the analytic equirriple solution (Klopfenstein) and the dashed-dotted line shows the simulation of the nearoptimum solution given in [10].

Fig. 12. Comparison between the linear profile and the optimized solution. Dotted line: analytic solution showing the impedance steps. Dashed-dotted line: near-optimum profile. Only one side of the taper is plotted. The other side is symmetric with respect to the horizontal axis.

is designed to provide a return loss level RL > 25 dB in the range from 1 to 3 GHz between two impedances of Z 0 = 50 and Z L = 100 . For this example a RO4003 substrate is used. The minimum matching provides the minimum value for the length L of the taper [4]

the improvement is obtained in the whole frequency range. The limit for the performance is the equirriple solution given by Klopfenstein’s taper. However, (23) is analytical and does not consider more realistic issues. The superelliptic solution has been obtained after optimization using a real profile segmented in 2000 sections. In each section, the effective dielectric constant and the phase constant have been computed using empirical formulas [7], [31], assuming their true dependence with the frequency variation to better approximate the final response. The profile of Klopfenstein’s taper can be computed using the well-known formulas [3] ⎧ ln Z 0 , z ≤ 0 ⎪ ⎪ ⎪ ⎪ ⎨   2 ln Z (z) = 1 ln(Z 0 Z L )+ A 0 φ 2z − 1, A , 0 < z < L ⎪ ⎪ 2 cosh A L ⎪ ⎪ ⎩ lnZ L , z ≥ L

m = 10−R L/20 Z L − Z0 1 ZL ≈ ln 0 = Z L + Z0 2 Z0 A = cosh−1 ( 0 / m ) Ac0 L = √ 2π f min εreff

(19) (20) (21) (22)

where m is the maximum reflection coefficient allowed in the equirriple design, c0 is the speed of light in the vacuum, and f min is the minimum frequency at which the specification m is met. Equations (19)–(22) provide a length of L = 72.6 mm. However, it is expected that the Klopfenstein taper will not provide |S11 | < −25 dB in the EM simulation, because of all the assumptions in the related design method. The first one is the nondispersive media assumed in the calculations, the second one is the approximations of the equation (Ricatti’s equation) of the reflection coefficient (see [2], [4]), and the third one is that the solution is given in terms of an impedance profile (24) which is synthesized at a given frequency (not broadband). The design method proposed in this paper requires a mask (we used a constant mask to obtain a solution close to Klopfenstein’s), and an initial guess of the variables to be optimized (we used m = n = 1 giving the linear profile as the starting point). Optimizing the superelliptical profile, the method will provide (in less than 1 s) the optimal parameters: m = 0.94084 and n = 0.76022. Fig. 11 shows the difference in performance of the linear, the superelliptical, and the near-optimum tapers with the ideal Klopfenstein’s response given by the analytic formula  cos (β L)2 − A2 . (23) (β L) = 0 cosh A As expected, the superelliptical profile provides an improvement over the linear taper, which is the starting point of the optimization we have performed. In this example,

where



x

φ(x, A) = −φ(−x, A) = 0

 I1 (A 1 − y 2 ) d y ∀|x| < 1  A 1 − y2 (24)

where I1 (x) is the modified Bessel function. φ(x, A) can be computed very rapidly, since there is an efficient algorithm detailed in [33]. The profile of the linear taper (initial point of the optimization) is compared versus the superelliptic taper in Fig. 12. In Fig. 12, Klopfenstein’s solution shows the steps inherent to the equirriple solution. As a curiosity, Hecken’s solution appears in the same plot showing strong similarity with the superelliptic profile in the second half of the taper. In the first half of the taper, the superellipse rapidly approaches the Klopfenstein profile. It is worth stressing that the optimization algorithm seeks m and n values in each iteration; then, the profile is simulated and the error E referred to the mask is computed. In the following iteration, the new m and n values will be sought in order to decrease the error function. Therefore, a fast simulator is strongly recommended. The segmentation technique already explained before (see Section III) gives a good approximation, and a transmission-line simulator has been implemented for this purpose.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 14. Measured reflection response of the three tapers built in a back-toback configuration.

Fig. 13. Manufactured prototypes (circuit board at the top) and the TRL calibration kit (circuits at the bottom). The circuit at the top in the board is the superelliptic taper (near the letter S), the circuit at the middle is the linear taper, and the circuit at the bottom is the Klopfenstein taper (near the letter K).

The code for the analysis of the taper has been written in the MATLAB environment. For the optimization process, the nonlinear least-squares algorithm has been used. Upper and lower bounds for the values of m, n are set to practical values (lower bound 0.1 and upper bound 10 will suffice). For the initial values in the optimization, as explained, m = n = 1 (linear taper) is used as the starting point. Each taper (linear, superelliptic, and Klopfenstein) simulated in Fig. 11 has been manufactured in a cascaded back-to-back configuration (see Fig. 13, where the employed TRL calibration kit is also shown). Using this setup, the measurement can be carried out with a 50- calibrated vector network analyzer (VNA). This method allows to fairly compare the performances of the three tapers using the same calibration kit. Ideally, the performance of a back-to-back configuration provides a 3-dB higher reflection level. The measured reflection of manufactured prototypes is shown in Fig. 14. The results show how the superelliptic taper is clearly superior with respect to Klopfenstein’s at almost all frequencies. The linear taper is also worse below 2.1 GHz. It is important to note that the impedance steps in the Klopfenstein taper has a negative impact in its performance (at some frequencies, even the linear taper is better). B. Waveguide Taper The proposed solution has been validated with two manufactured prototypes in waveguide technology. Both are WR-90 to WR-62 waveguide tapers with length L = 50 mm. One is manufactured as a single piece using the wire electrical discharge machining (EDM) technique. The other one has been

Fig. 15. Manufactured tapers in two pieces using milling (left) and in one single piece using wire EDM (right). Both are 50-mm-long tapers matching WR-90 to WR-62 standard waveguides.

manufactured in two equal pieces using milling (see Fig. 15). The final goal of the design is to find the best possible solution with S11 < −40 dB in the band 11–12.5 GHz, which is common to both standard waveguides. With these goals, the solution will improve the response of the commercial 150-mm taper considered in Fig. 10. The optimization procedure gives as the optimal point the following values: m 1 = 0.6587 n 1 = 0.5259 m 2 = 0.7132 n 2 = 0.6845 where m 1 and n 1 are the parameters of the superellipse matching the waveguide widths, and m 2 , n 2 are the parameters related with the height. The response of the taper using the three simulators (two EM full-wave commercial tools and one circuit simulator using Marcuvitz’s formulas) is shown in Fig. 16. All simulations reached S11 < −40 dB. However, the observed differences rely on the 3-D meshing of the commercial simulators that do not exactly follow the profile due to the employed discretization methods, and to the fact that the circuit simulator uses the approximations outlined in Section V. Nevertheless, it is important to stress that the observed difference is small in linear units, since the low level of reflection is quite extreme. Once the two prototypes were built (as shown in Fig. 15), they have been measured. However, the measurement technique is not unique. If the taper is connected to the VNA with the WR-90 port, the load has to be connected to the WR-62 side of the taper. Conversely, if the VNA is

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. COGOLLOS et al.: NOVEL PLANAR AND WAVEGUIDE IMPLEMENTATIONS OF IMPEDANCE MATCHING NETWORKS

Fig. 16. Three simulations of the optimal waveguide taper with L = 5 mm. CST and HFSS give the same results with 3-D meshing. The circuit simulator using Marcuvitz’s formulas used 400 cascaded lines with the corresponding steps.

9

Fig. 18. Measurements of the reflection coefficient of the waveguide tapers from the WR-62 port. A WR-90 matching load has been used.

Fig. 19. Back-to-back configuration joining the tapers from the WR-62 ports. Commercial tapers with length L = 150 mm are also measured for comparison purposes. Fig. 17. Measurements of the reflection coefficient of the waveguide tapers from the WR-90 port. A WR-62 matching load has been used. EM and circuit simulations are also shown.

connected to the WR-62 port, the load has to be connected to the WR-90 side of the taper. The loads are different and the response differs depending on the used loads. The results in Fig. 17 show a good performance of both tapers, whose reflection has been measured from the WR-90 port using a WR-62 matching load at the other port of the tapers. It also shows how the predictions of the simulators are quite different. The circuit simulator predicts the reflection zero, whereas the EM simulation approaches the average level. The difference between the measured tapers and the simulations are due to two main sources: the frequency response of the matching load connected to the taper (it has a response near the level of our goal for the reflection coefficient) and the mechanization imperfections. In order to see the out-of-band performance, measurements of the tapers from the WR-62 port having loaded the WR-90 ports are shown in Fig. 18. The calibration kit used is different as well as the frequency range. Therefore, the differences with respect to Fig. 17 are easily justifiable. The reflection level is clearly maintained. There are alternative measurement techniques for tapers, as it has been shown in the microstrip example. If two equal tapers are available, they can be connected in a back-toback configuration and we can remove the load from the measurements. However, there are two possible configurations: joining the tapers from the WR-90 ports or from the

Fig. 20. Back-to-back configuration joining the tapers from the WR-90 ports. Commercial tapers with length L = 150 mm are also measured for comparison purposes.

WR-62 ones. The frequency range will be different and the results may also vary. We have two tapers with the only difference of the fabrication technique. Moreover, two equal commercial tapers with the same ports as our prototypes, but with L = 150 mm each one, are also available. For these measurements in back-to-back configuration we used a two-port calibration and measurement setup, thus removing the matching load and its influence from the measurements themselves. Figs. 19 and 20 show a comparison of experimental results for the back-to-back configurations with the commercial tapers, which are 3 times as long as the designed tapers (see both sizes in Fig. 21). The reflection level is similar despite the difference in length.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 21. Commercial taper (top) compared with one of the manufactured prototypes (bottom).

VII. C ONCLUSION A novel method to design tapered lines based on superellipses has been proposed. In microstrip technology, the superelliptic taper has a superior performance in most of the designed bandwidth, once compared with the equirriple optimal solution and the traditionally used linear taper. The enhanced performance and the related short design time justify the method proposed in this paper. The measured responses show the importance of smooth profiles with optimized behavior in tapered lines. For waveguide tapers, two prototypes based on the generalized superellipse geometry have been also manufactured. Their performance is similar when compared with commercial tapers 3 times longer. R EFERENCES [1] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures. Norwood, NJ, USA: Artech House, 1980. [2] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. Norwood, NJ, USA: Wiley, 2000. [3] D. M. Pozar, Microwave Engineering, 3rd ed. Norwood, NJ, USA: Wiley, 2005. [4] R. W. Klopfenstein, “A transmission line taper of improved design,” Proc. IRE, vol. 44, no. 1, pp. 31–35, Jan. 1956. [5] R. E. Collin, “The optimum tapered transmission line matching section,” Proc. IRE, vol. 44, no. 4, pp. 539–548, Apr. 1956. [6] D. Kajfez and J. O. Prewitt, “Correction to ‘a transmission line taper of improved design’ (letters),” IEEE Trans. Microw. Theory Techn., vol. MTT-21, no. 5, p. 364, May 1973. [7] K. C. Gupta, R. Garg, I. J. Bahl, and P. Bhartia, Microstrip Lines and Slotlines, 2nd ed. Norwood, MA, USA: Artech House, 1996. [8] N. Marcuvitz, Waveguide Handbook. Stevenage, U.K.: Peregrinus, 1986. [9] L. Solymar, “A note on the optimum design of non-uniform transmission lines,” Proc. IEE C, Monogr., vol. 107, no. 11, pp. 100–104, Mar. 1960. [10] R. P. Hecken, “A near-optimum matching section without discontinuities,” IEEE Trans. Microw. Theory Techn., vol. 20, no. 11, pp. 734–739, Nov. 1972. [11] J. P. Mahon and R. S. Elliott, “Tapered transmission lines with a controlled ripple response,” IEEE Trans. Microw. Theory Techn., vol. 38, no. 10, pp. 1415–1420, Oct. 1990. [12] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Course of Theoretical Physics), vol. 8, 2nd ed. New York, NY, USA: Elsevier, 1984. [13] R. Marqués, F. Martín, and M. Sorolla, Metamaterials With Negative Parameters (Microwave and Optical Engineering). New York, NY, USA: Wiley, 2008. [14] F. Martín, Artificial Transmission Lines for RF and Microwave Applications (Microwave and Optical Engineering). New York, NY, USA: Wiley, 2015.

[15] E. Hammerstad and O. Jensen, “Accurate models for microstrip computer-aided design,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1980, pp. 407–409. [16] M. Kobayashi, “A dispersion formula satisfying recent requirements in microstrip CAD,” IEEE Trans. Microw. Theory Techn., vol. MTT-36, no. 8, pp. 1246–1250, Aug. 1988. [17] E. J. Park, “An efficient synthesis technique of tapered transmission line with loss and dispersion,” IEEE Trans. Microw. Theory Techn., vol. 44, no. 3, pp. 462–465, Mar. 1996. [18] P. Pramanick and P. Bhartia, “Analysis and synthesis of tapered finlines,” in IEEE MTT-S Int. Microw. Symp. Dig., May/Jun. 1984, pp. 336–338. [19] A. Mantzke, S. Südekum, and M. Leone, “Broadband equivalent-circuit model for non-uniform transmission lines,” in Proc. IEEE Int. Symp. Electromagn. Compat. (EMC), Aug. 2015, pp. 600–605. [20] J. Helszajn, Ridge Waveguides and Passive Microwave Components (IEE Electromagnetic Waves Series), vol. 49. Stevenage, U.K.: IET, 2000. [21] S. Cogollos, J. Vague, V. E. Boria, and J. D. Martinez, “New design method of impedance matching networks based on tapered lines using generalized superellipses,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2017, pp. 1–4. [22] S. B. Cohn, “Optimum design of stepped transmission-line transformers,” IRE Trans. Microw. Theory Techn., vol. 3, no. 3, pp. 16–20, Apr. 1955. [23] U. Rosenberg, J. Bornemann, and S. Amari, “Design of dual-band waveguide transformers,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1215–1218. [24] R. J. Cameron, C. M. Kudsia, and R. R. Mansour, Microwave Filters for Communication Systems: Fundamentals, Design and Applications. Norwood, NJ, USA: Wiley, 2007. [25] H. J. Riblet, “General synthesis of quarter-wave impedance transformers,” IRE Trans. Microw. Theory Techn., vol. 5, no. 1, pp. 36–43, Jan. 1957. [26] S. Cogollos, V. E. Boria, and J. D. Martinez, “Generalized short step transformers for multi-band impedance matching,” in Proc. 42nd Eur. Microw. Conf., Oct. 2012, pp. 380–383. [27] M. Gardner, Mathematical Carnival. New York, NY, USA: Vintage Press, 1977. [28] J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Amer. J. Botany, vol. 90, no. 3, pp. 333–338, 2003. [29] J. Gielis, B. Beirinckx, and E. Bastiaens, “Superquadrics with rational and irrational symmetry,” in Proc. 8th ACM Symp. Solid Modeling Appl. (SM), New York, NY, USA, 2003, pp. 262–265. [Online]. Available: http://doi.acm.org/10.1145/781606.781647 [30] K. Yao, C. Li, and F. Li, “Design of electromagnetic cloaks for generalized superellipse using coordinate transformations,” in Proc. Int. Workshop Metamaterials, Nov. 2008, pp. 122–125. [31] B. C. Wadell, Transmission Line Design Handbook. Norwood, MA, USA: Artech House, 1991. [32] J. A. Snyman, Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms (Applied Optimization), vol. 97. New York, NY, USA: Springer, 2005. [33] M. A. Grossberg, “Extremely rapid computation of the Klopfenstein impedance taper,” Proc. IEEE, vol. 56, no. 9, pp. 1629–1630, Sep. 1968. Santiago Cogollos (M’07) was born in Valencia, Spain, in 1972. He received the degree in telecommunication engineering and Ph.D. degree from the Universitat Politècnica de València, Valencia, in 1996 and 2002, respectively. In 2000, he joined the Communications Department, Universitat Politècnica de València, where he was an Assistant Lecturer from 2000 to 2001, a Lecturer from 2001 to 2002, and became an Associate Professor in 2002. He has collaborated with the European Space Research and Technology Centre, European Space Agency, Noordwijk, The Netherlands, where he focused on the development of modal analysis tools for payload systems in satellites. In 2005, he held a post-doctoral research position at the University of Waterloo, Waterloo, ON, Canada, where he was involved in the new synthesis techniques in filter design. His current research interests include applied electromagnetics, mathematical methods for electromagnetic theory, analytical and numerical methods for the analysis of microwave structures, and design of waveguide components for space applications.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. COGOLLOS et al.: NOVEL PLANAR AND WAVEGUIDE IMPLEMENTATIONS OF IMPEDANCE MATCHING NETWORKS

Joaquin Vague was born in Valencia, Spain, in 1979. He received the electronic engineering degree from the Universitat Politècnica de València (UPV), Valencia, in 2003. He is currently pursuing the Ph.D. degree in electronic engineering. He is currently a Technical Researcher with the Department of Communications, UPV, where he is in charge of several laboratories. His current research interests include the design of microwave passive devices, in particular filters and nonreciprocal devices, as well as the improvement in manufacturing processes of RF and microwave components.

Vicente E. Boria (S’91–A’99–SM’02) was born in Valencia, Spain, in 1970. He received the Ingeniero de Telecomunicación degree (with first-class Hons.) and the Doctor Ingeniero de Telecomunicación degree from the Universidad Politécnica de Valencia, Valencia, in 1993 and 1997, respectively. In 1993, he joined the Departamento de Comunicaciones, Universidad Politécnica de Valencia, where he has been a Full Professor since 2003. In 1995 and 1996, he was holding a Spanish trainee position with the European Space Research and Technology Centre, European Space Agency, Noordwijk, The Netherlands, where he was involved in the EM analysis and design of passive waveguide devices. He has authored or co-authored ten chapters in technical textbooks, 160 papers in refereed international technical journals, and over 200 papers in international conference proceedings. His current research interests include the analysis and automated design of passive components, left-handed and periodic structures, as well as on the simulation and measurement of high-power effects in passive waveguide systems. Dr. Boria has been a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and the IEEE Antennas and Propagation Society since 1992. He is a member of the Editorial Boards of the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES , IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS , the IET Microwaves, Antennas and Propagation, and the IET Electronics Letters and Radio Science. He is currently an Associate Editor of IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS and IET Electronics Letters. He is also a member of the Technical Committees of the IEEE-MTT International Microwave Symposium and the European Microwave Conference.

11

Jorge D. Martínez (M’09) was born in Murcia, Spain, in 1979. He received the degree in telecommunication engineering and Ph.D. degree from the Polytechnic University of Valencia (UPV), Valencia, Spain, in 2002 and 2008, respectively. He joined the Department of Electronic Engineering, UPV, as a Research Fellow in 2002, where he became an Assistant Professor in 2009. During 2007, he was a Research Visitor at XLIM; CNRS, Paris, France; and the University of Limoges, Limoges, France, where he focused on the design and fabrication of RF MEMS components under the advice of Prof. Pierre Blondy. He has been an Associate Professor with the School of Telecommunication Engineering, UPV, since 2012, where he is currently a Researcher of the I3M R&D Institute and actively collaborates with the Microwave Applications Group. At I3M premises, he is the Technical Responsible of the Laboratory for High Frequency Circuits Fabrication, UPV, where he focused on lowtemperature co-fired ceramics and other related multilayered technologies. His current research interests include emerging technologies for reconfigurable microwave components with emphasis on tunable filters and RF MEMS, and the design and fabrication of advanced microwave filters in planar and substrate integrated waveguide technologies, as well as the application of multilayer fabrication technologies to RF/microwave and millimeter-wave applications.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Arbitrary Prescribed Wideband Flat Group Delay Circuits Using Coupled Lines Girdhari Chaudhary , Member, IEEE, and Yongchae Jeong, Senior Member, IEEE

Abstract— This paper presents the analytical design of transmission-type arbitrary prescribed wideband flat group delay (GD) circuits (type-I and type-II) using λ/4 coupled lines. The GD circuit type-I consists of two sections of coupled lines, whereas the GD circuit type-II consists of λ/4 transmission lines (TLs) at input and output, in addition to coupled lines. The additional λ/4 TLs at input–output ports in GD circuit type-II provide more freedom to obtain larger GD, as compared to type-I, without fabrication difficulties. The analytical analysis shows that the wideband flat GD response can be obtained by selecting the appropriate even- and odd-mode impedances of coupled lines and the characteristic impedance of TLs. To obtain the arbitrary prescribed wideband flat GD response, the closed-form analytical design equations are provided. For experimental validation of the proposed structures, prototypes of GD circuits (type-I and type-II) are designed and fabricated at the center frequency of 2 GHz. The measurement results agree well with the simulation and theoretical predicted results. Index Terms— Analog radio-signal processing, arbitrary wideband flat group delay (GD) response, coupled lines, signal cancellation, transmission-type circuit.

I. I NTRODUCTION

M

ICROWAVE circuits/filters that provide the desired group delay (GD) response with respect to frequency have various applications in communication systems including real-time analog radio-signal processing, RF self-interference cancellation in-band full-duplex radio, and signal cancellation in feed-forward amplifier [1]–[5]. The GD can be investigated by examining the frequency-dependent phase variation of transmitting scattering parameter, which can be mathematically defined as dϕ . (1) dω The flat GD in filters can be achieved by two approaches. The first one is to use an external all-pass GD circuits cascaded with a filter [6]–[21], which can increase circuit size and τg = −

Manuscript received June 27, 2017; revised August 30, 2017, October 25, 2017, and November 13, 2017; accepted November 19, 2017. This work was supported by the Korean Research Fellowship Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Science and ICT under Grant 2016H1D3A1938065 and in part by the Basic Science Research Program through the NRF of Korea funded by Ministry of Education, Science and Technology under Grant 2016R1D1A1B03931400. (Corresponding author: Yongchae Jeong.) The authors are with the Division of Electronics Engineering, IT Convergence Research Center, Chonbuk National University, Jeonju 54896, South Korea (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2779515

insertion loss. All-pass GD circuits can be categorized into reflection and transmission types. Reflection-type all-pass GD circuits can be realized by terminating coupled and through ports of a 3-dB hybrid coupler with one-port GD circuit, and the GD response of resulting two-port network is added up GD responses of the hybrid and one-port terminating circuit [6], [7]. A coupled-resonator network approach was utilized to design the GD equalizers using optimization procedures in [8] and [9]. A narrowband one-port circuits with an arbitrary prescribed GD response was synthesized in [10]. Later, this technique was improved and applied to the design of reflective GD circuits based on shunt stubs and stepped-impedance lines [11], alternating K /J inverters and λ/4 transmission line (TL) resonators [12], [13]. However, these techniques require complex iterative procedures for transformation of the prescribed GD problem from band-pass domain to the low-pass domain using a one-port ladder network, and again, the transformation of the synthesized low-pass network back to the band-pass domain, for implementation in a specific technology. Transmission-type all-pass GD circuits are realized using multisection coupler-based superconductive delay [14], cascading of all-pass networks [15], [16], and noncommensurate coupled lines by multiconductor TL technique [17]. However, this structure requires genetic optimization technique to implement final structure. Similarly, wideband all-pass GD circuits were synthesized by using uniform and nonuniform commensurate C- and D-sections [18], [19], multilayer broadside coupled lines GD circuit for analog signal processing [20]. However, these works require an iterative design procedure to map arbitrary prescribed GD problem from band-pass domain to low-pass domain and again back to bandpass domain for obtaining the optimum circuit parameters. Similarly, the transmission-type GD lines had been realized by employing a transversal filter structure, were realized with the reverse of a distributed amplifier [21], and required complex iterative procedures to obtain a large number of tap coefficients for specified GD response. The second approach is to design self-equalized or linear-phase filters which can be achieved by imposing linear-phase requirement in addition to the amplitude requirement [22]–[27]. However, these techniques require crosscoupling among nonadjacent resonators to produce more than one signal path or controlling sign of cross-coupling to place the transmission zeros on the right half plane. In addition, it is also difficult to specify and design arbitrarily prescribed flat GD filters directly in band-pass domain using these techniques.

0018-9480 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

Fig. 1. type-I.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

(a) Coupled line and (b) proposed structure of wideband GD circuit

In this paper, arbitrarily prescribed wideband flat GD circuits are presented based on coupled lines. This paper shows an alternative way to realize flat GD filters which are very simple and do not require any cross-coupling among nonadjacent resonators or controlling sign of cross-coupling or any kind of transformation to obtain the required circuit parameters. The closed-form analytical design equations are provided to obtain the required circuit parameters for the arbitrarily prescribed wideband flat GD response without any optimization procedure.

Fig. 2. Equivalent circuits of the proposed GD circuit type-I under (a) evenand (b) odd-mode excitations.

Similarly, Fig. 2(b) shows the equivalent odd-mode excitation circuit. The odd-mode input impedance of the subcircuit shown in Fig. 2(b) can be determined as (7) by using (4) and (5), with z L = 0

 zc πf πf Type-I 2 z ino = − j  − 2Ceff csc cot . (7) 2 f0 f0 2 1 − Ceff The S-parameters of the wideband GD circuit type-I shown in Fig. 1(b) can be found as (8) using (6) and (7) S11_Type-I = S22_Type-I

II. M ATHEMATICAL A NALYSIS Fig. 1(a) shows the structure of λ/4 coupled lines, where the coupled and through ports are open circuited [28]. The evenand odd-mode impedances of coupled line are normalized with port impedance Z 0 and denoted by z 0e and z 0o , respectively. √ Assuming z c = z 0e z 0o , normalized z 0e and z 0o are expressed as follows [28], [29]:   1 + Ceff 1 − Ceff z 0o = z c (2) z 0e = z c 1 − Ceff 1 + Ceff z 0e − z 0o (3) Ceff = z 0e + z 0o where Ceff is coupling coefficient of the coupled line. The z-parameters of the coupled lines shown in Fig. 1(a) can be expressed in terms of z c and Ceff as ⎡ ⎤ zc πf z c Ceff πf −j cot −j csc ⎢ 2 f0 2 f0 ⎥ 2 2 ⎢ ⎥ 1 − Ceff 1 − Ceff ⎢ ⎥ [z] = ⎢ z c Ceff πf zc π f ⎥ (4) ⎣− j  ⎦ csc −j cot 2 f0 2 f0 2 2 1 − Ceff 1 − Ceff where f and f 0 are the operating and the design center frequency, respectively. Based on z-parameters, the input impedance of coupled lines with termination load z L can be calculated as z 12 z 21 . (5) z in = z 11 − z 22 + z L A. Wideband Flat Group Delay Circuit Type-I Fig. 1(b) shows the proposed structure of the wideband GD circuit type-I. Since the structure is symmetrical, the evenand odd-mode analyses can be applied to find S-parameters. From the equivalent circuit of the even-mode excitation shown in Fig. 2(a), the even-mode input impedance can be expressed as (6) using (4) and (5), with z L = ∞ zc πf Type-I z ine = − j  cot . (6) 2 f0 1 − C2 eff

=

Type-I Type-I z ino − 1



Type-I Type-I z ine + 1 z ino + 1

z ine

x3 (1 − j x 1) (1 − j x 2) = S12_Type-I

= S21_Type-I

Type-I

= = where x1 = 

zc 2 1 − Ceff

zc

x2 =  x3 =

2z c 2 1 − Ceff

Type-I

−z

ino

Type-I Type-I z ine + 1 z ino + 1 z ine

j 2x 4 (1 − j x 1) (1 − j x 2)

(9)

 πf πf 2 cot sin − 2Ceff 2 f0 f0 cot

(10)

πf

2 f0 2 1 − Ceff

z c2 πf cot2 2 2 f0 1 − Ceff

x4 = 

(8)

(11)

sin

πf πf 2 − Ceff cot f0 2 f0

 πf πf 2 sin − Ceff cot . 2 f0 f0

 − sin

πf f0 (12) (13)

Similarly, the GD of the circuit type-I can be calculated as   2

z 1 − Ceff  c 1 d S21_Type-I 1 x5 = τType-I = − + 2π df 4 f0 x6 x7 (14) where πf πf πf 2 sin2 − 4Ceff cos (15) 2 f0 f0 f0

2

πf πf πf 2 2 sin2 + z c2 cot sin − 2Ceff x 6 = 1 − Ceff f0 2 f0 f0 (16)

x 5 = csc2

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. CHAUDHARY AND JEONG: ARBITRARY PRESCRIBED WIDEBAND FLAT GD CIRCUITS USING COUPLED LINES

3

TABLE I C ALCULATED C IRCUIT PARAMETERS OF C IRCUIT T YPE -I W ITH P OSITIVE AND N EGATIVE S IGNS IN (19)

Fig. 3.

Calculated results of the wideband flat GD circuit type-I.

πf πf 2 1 − Ceff + z c2 cos2 . (17) 2 f0 2 f0 As seen from (14)–(17), the GD depends on z c and Ceff of the coupled lines. Further, the GD at f0 can be simplified as x 7 = sin2

Type-I

τ f = f0 =

2 + z 2C 2 1 − Ceff  c eff . 2 2 4 f 0 z c Ceff 1 − Ceff

(18)

An arbitrarily specified flat GD response can be obtained by selecting appropriate z c and Ceff . For this purpose, we need to solve the value of z c using (18). Therefore, the solution of z c in terms of specified GD and Ceff can be found as     2  Type-I 2 2 2 f τ Type-I ± 2 f 0 τ f = f0 − 1 Ceff . z c = 1 − Ceff 0 f = f0 (19) As noted from (19), z c has two unique solutions, and the values are real and positive if  Type-I  Ceff (dB) = α − 20 log 2 f 0 τ f = f0 (20) where α is a positive factor which provides a degree of freedom to control Ceff and GD ripples. In this paper, in-band GD ripple (τripple) is defined as τripple =

τmax − τ f = f0 × 100%. τ f = f0

(21)

Here, τmax is the maximum GD at passband-edge frequency. The required value of Ceff can be obtained using (20) for the specified GD with acceptable τripple. Based on the above analytical analysis, Fig. 3 shows the calculated magnitudes/GD responses for circuit type-I. Table I gives the calculated circuit parameters. As shown in Fig. 3, z c with a positive sign in (19) provides the wideband return loss characteristics with three poles; however, the GD at band edge has a high peak. Similarly, it is observed that GD is flat over the wideband when negative sign assigned in (19) for the solution of z c . However, return loss is slightly narrow due to only one pole. Similarly, Fig. 4 shows the calculated magnitude/GD responses with a variation of α. In the case of positive sign in (19), τripple at in-band and band-edge frequencies are increased as α increased; however, impedance matching decreases. On the contrary, when α increased, τripple is decreased and GD approaches toward flat in the case of the negative sign in (19).

Fig. 4. Calculated GD/magnitude response of the proposed circuit type-I with different α and according to (a) positive sign (case 1) and (b) negative sign (case 2) in (19) for the solution of z c . Here, f 1 and f 2 are lower and upper cutoff frequencies when passband-edge GD approximately equals to GD at f 0 .

Meanwhile, the return loss bandwidth decreases, therefore, a tradeoff occurs between wideband flat GD and return loss. Also, it is concluded from this analysis that for wideband flat GD response and minimum τripple, the negative sign in (19) is preferable. Fig. 4(b) also shows the comparison results of the proposed circuit with conventional Butterworth and Chebyshev filters for the same GD at f 0 and return loss bandwidth specification. As shown in Fig. 4(b), the proposed circuit can provide the flat GD response as compared to conventional filters. To investigate the effect of selecting α, Fig. 5(a) shows the calculated τripple from (14) and (21) using MATLAB when

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 6. Calculated GDs fractional bandwidth (Type-I ) of type-I circuit for different α. Color bar represents different α values.

Fig. 5. (a) Calculated in-band GD ripples from (14) and (21) using MATLAB for different α. (b) Circuit parameters of wideband flat GD of circuit type-I with a negative sign in (19) and α = 0.8.

negative sign is selected in (19). As observed from Fig. 5(a), the τripple is decreased as α increased. In general, higher α is preferable for minimum τripple. Fig. 5(b) shows design graphs to calculated circuits parameters for specified flat GD. As seen from this graph, the z c increases and Ceff decreases as the GD increases. Therefore, the circuit type-I could have difficulty in practical realization in single-layer PCB technology for higher GD. This problem can be solved by circuit type-II which will be discussed in later section in detail. Based on the above analysis, the negative sign in (19) is preferable for minimum τripple. Once z c and Ceff are determined for the specified GD at f 0 and minimum τripple, we should find the cutoff frequencies where the GD is approximately equal to GD at f 0 , as shown in Fig. 4(b). Using (14), the GD fractional bandwidth of type-I circuit (Type-I) can be calculated as Type-I =

f2 − f1 = f (z c , Ceff ) f0

Fig. 7. Design flowchart to calculate circuit parameters of wideband GD circuit type-I.

(22)

where f 1 and f 2 are lower and upper cutoff frequencies in between which the GD is equal to desired GD at f 0 , as shown in Fig. 4. Since it is complicate to find f 1 and f 2 using (14), analytically, we used numerical method in MATLAB by sweeping frequency to find Type-I for mathematical simplicity. Fig. 6 shows the calculated Type-I for different α. As GD is increased, Type-I decreased. In addition, Type-I is slightly higher when α is small; however, the small α increases τripple. The design steps to calculate circuit parameters of circuit type-I with specified GD can be described with the flowchart shown in Fig. 7.

Fig. 8.

Proposed structure of the wideband GD circuit-II.

B. Wideband Flat Group Delay Circuit Type-II To overcome the limitation of wideband GD circuit type-I, λ/4 TLs with a characteristic impedance of z 1 are added at the input and output ports of GD circuit type-I, and Fig. 8 shows the proposed structure. Due to λ/4 TLs, the proposed circuit type-II provides the extra degree of freedom, so that realizable circuit parameters can be obtained for high GD. Fig. 9 shows the equivalent circuits of the proposed circuit type-II under even- and odd-mode excitations. Therefore, the even- and oddmode impedances of circuit type-II can be obtained from Fig. 9

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. CHAUDHARY AND JEONG: ARBITRARY PRESCRIBED WIDEBAND FLAT GD CIRCUITS USING COUPLED LINES

5

where πf π csc2 2 f0 2 f0

 z1 π πf  2 πf x9 = −z c cot  csc f0 z 1 − C 2 + z 2 f0 2 f0 1 c eff

  πf π  2 +z x 10 = − z 1 1 − Ceff c sin f0 f0 ⎧  2 ⎪ 2 ⎪ π z 2 1 − Ceff π z c z 1 Ceff ⎪ πf πf ⎨ 1 cos − csc2  f0 f0 f0 2 f0 x 11 = ⎪ πf πf π zc z1 2π z c z 1 ⎪ 2 2 ⎪ ⎩+ cot + cos f0 2 f0 f0 2 f0 

x8 = −

Fig. 9. Equivalent circuits of the proposed GD circuit type-II under (a) evenand (b) odd-mode excitations.

and are expressed as (23) and (24), shown at the bottom of this page. The S-parameter of circuit type-II can be found as (25) and (26) by using (23) and (24)

= S21_Type-II

Type-II Type-II z ino −1  Type-II   Type-II + 1 z ino +1 z ine x 9 x 11 − x 8 x 10

z ine

(25)

(x 8 − j x 9)(x 10 + j x 11) = S12_Type-II Type-II

Type-II

− z ino  Type-II  Type-II z ine + 1 z ino +1 j (x 9 x 10 + x 8 x 11 ) = (x 8 − j x 9)(x 10 + j x 11) z ine

= 

x9 = x 10 x 11

πf 2 f0



(27) z1

z c cot2



πf 2 − z 1 1 − Ceff 2 f0

2 +z z 1 1 − Ceff c    2 + z cos2 π f − 2z C 2 = 2 z 1 1 − Ceff c c eff 2 f0  2 sin π f − 2z z cos2 π f cot π f = z 12 1 − Ceff c 1 f0 2 f0 2 f0 π f 2 + 2z c z 1 Ceff cot . 2 f0

 (28) (29)

Type-II

z ino

j z1

As seen from (31)–(36), the circuit type-II has more degree of freedom in circuit parameters (such as z 1 ) than that of type-I; therefore, the limitation in the practical implementation of type-I for higher GD can be overcome with the proposed structure of type-II. The additional TLs mean that z 0e and z 0o can be reduced to a lower value for higher GD. The required value of z c with specified flat GD can be found by solving (36) in terms of Ceff and GD at f 0 as      z1 2 b ± b 2 − 4z 2 C 2 1 − Ceff (37) zc = 1 eff 2 where Type-II

(30)

1 d  S12_Type-II 1 d  S21_Type-II =− =− 2π df 2π df       x 10 x 11 − x 10 x 11 1 x8 x9 − x8 x9 = + (31) 2 + x2 2π x 82 + x 92 x 10 11

Type-II z ine

Furthermore, the GD at f0 can be simplified in the following equation:  ⎛ ⎞ 2 +z z 2 2 z c2 Ceff 1 c 1 − Ceff Ceff  ⎜ ⎟ ⎜ +z 3 1 − C 2 C 2 z + z 4 1 − C 2  ⎟ ⎜ c 1 ⎜ 1 eff eff 1 eff ⎟ Type-II ⎟.  τ f = f0 = ⎜ ⎟ 4 f0 ⎜ 2 1 − C2 C2 z ⎟ z 1 eff eff c ⎝ ⎠

b = 4 f 0 τ f = f0 z 1 − z 12 − 1.

Finally, the GD of circuit type-II can be expressed as (31) by using (26)–(30) τType-II

(34) ⎫ ⎪ ⎪ ⎪ ⎬ . ⎪ ⎪ ⎪ ⎭

(36) (26)

where x 8 = cot

(33)

(35)

S11_Type-II = S22_Type-II =

(32)

From (37), the value of z c is real and positive only if 

2z 1 (39) Ceff (dB) = β + 20 log b where β is a positive factor that controls the τripple. Like the circuit type-I, the β must be positive value which will provide a degree of freedom to control τripple. The required value of Ceff can be calculated using (39) when a designer specifies GD at f 0 and z 1 . In addition, z c has two unique solutions depending on positive and negative sign in (37).

  πf πf 2 − z c cot z 1 1 − Ceff tan 2 f0 2 f0 + zc

 2 z 1 1 − Ceff  2 sin π f − 2z z cos2 π f cot π f + 2z z C 2 cot z 12 1 − Ceff c 1 c 1 eff f0 2 f0 2 f0 = j    2 + 2z cos2 π f − 2z C 2 2z 1 1 − Ceff c c eff 2 f0 =

(38)

(23) πf 2 f0

(24)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

Fig. 10.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Synthesized magnitude/GD response of circuit type-II.

Fig. 10 shows the calculated magnitude/GD responses of circuit type-II. As like the circuit type-I, the solution of z c with a positive sign in (37) provides wide return loss characteristics with three poles; however, peaks occur in GD characteristics at band-edge frequencies. These peaks in GD can be minimized by obtaining a solution of z c with a negative sign in (37), as shown in Fig. 10. However, only one pole occurs at the center frequency when the GD is flat. In general, the negative sign in (37) is preferable for wideband flat GD with minimum ripple. Fig. 11(a) shows the synthesized magnitude/GD response of type-II with different Ceff and positive sign in (37) for the solution of z c . As shown in Fig. 11(a), ripples in GD at in-band and band-edge frequencies increase as β increases. However, the return loss bandwidth increases with the decrease of maximum achievable return loss magnitude. Similarly, Fig. 11(b) shows the calculated magnitude/GD responses of type-II with different β and negative sign in (37) for the solution of z c . The responses of the proposed circuit type-II are also compared with conventional Butterworth and Chebyshev filter response for the same GD at f0 and return loss bandwidth specification. The value of τripple is decreased and approached toward flat GD characteristics as β increases in the proposed circuit. However, the return loss bandwidth slightly decreases. Therefore, a tradeoff occurs between wideband flat GD and return loss bandwidth. To investigate the effect of selecting β, Fig. 12 shows numerically calculated τripple of circuit type-II with different z 1 and β using (21) and (31) in MATLAB. As shown in Fig. 12, τripple can be minimized by increasing the value of β. In general, higher β is preferable for minimum τripple. Fig. 13 shows the design graph to calculate the circuit parameters of circuit type-II with specified GD. From this graph, the value of z c increases and Ceff decreases as GD increases. It is also clear from this graph that practical realizable circuit parameters of coupled lines can be obtained by appropriately selecting z 1 . Specifically, the low z 1 is preferable for high GD and practical realizable z c and Ceff . Like the type-I circuit, once the circuit parameters such as z 1 , z c , and Ceff are determined based on specifications (GD, f0 , and τripple), we sweep frequency in (31) under Type-II condition τType-II = τ f = f0 , as shown in Fig. 11, for finding

Fig. 11. Calculated results of wideband flat GD circuit type-II with different β and according to (a) positive sign (case 1) and (b) negative sign (case 2) in (37) for the solution of z c . Here, f 1 and f 2 are lower and upper cutoff frequencies when passband-edge GD approximately equals to GD at f 0 .

Fig. 12. Calculated in-band group ripples of type-II circuit from (21) and (31) using MATLAB for different z 1 and β with a negative sign in (37). Color bar represents different β values.

lower and upper cutoff frequencies. Finally, the GD fractional bandwidth (Type-II ) can be found as Type-II =

f2 − f1 = f (z 1 , z c , Ceff ) f0

(40)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. CHAUDHARY AND JEONG: ARBITRARY PRESCRIBED WIDEBAND FLAT GD CIRCUITS USING COUPLED LINES

7

Fig. 13. Calculated z c and Ceff for flat wideband GD characteristics of circuit type-II with β = 0.9 and negative sign in (37).

Fig. 15. Design flowchart to calculate circuit parameters of wideband GD circuit type-II.

Fig. 14. Calculated GD fractional bandwidth (Type-II ) of type-II circuit for different β and z 1 . Color bar represents different β values.

Fig. 16. (a) Layout of the wideband flat GD circuit-I with physical dimensions. (b) Photograph of the fabricated circuit. Physical dimensions: L 11 = 28.6, L 12 = 3.9, L 12 = 2.5, W11 = 0.43, and g11 = 0.48 (unit: mm).

where f 1 and f 2 are lower and upper cutoff frequencies in between which the GD is within the desired GD at f 0 . Fig. 14 shows the calculated Type-II for different values of z 1 and β. From these graphs,  is decreased as GD increases. And  is slightly higher with small β. As like the type-I, Fig. 15 summarizes the design steps to calculate circuit parameters of circuit type-II with specified GD and the minimum τripple.

the calculated circuit parameters for given specification are determined as α = 0.85, z c = 2.2239 , Ceff = −11.1912 dB, z 0e = 2.9514 , and z 0o = 1.6757 . Using these circuit parameters, the calculated Type-I of the designed circuit is given as 19.1%. The renormalized circuit parameters of coupled lines with respect to 50- port impedances are Z 0e = 147.5699  and Z 0o = 83.7849 . Fig. 16(a) shows the layout and physical dimensions while Fig. 16(b) shows a photograph of fabricated circuit. In this layout, small length L 12 is added for connecting port. For minimizing the effect of L 12 , the same length is also added in another side of coupled lines [30]. Fig. 17 shows the simulated and measured GDs and S-parameter magnitudes. The measurement results agree well with simulation and theoretical predicated results. From the measurement, the GD and magnitude of transmission coeffiType-I cient at f 0 = 2 GHz are determined as τ f = f0 = 0.998 ns and |S21 | = −0.42 dB, respectively. In addition, the GD is flat in frequency extending from 1.78 to 2.15 GHz, with flat GD fractional bandwidth of 18.5%. Similarly, the input and output return losses at f 0 are measured as |S11 | = −26.87 dB and |S22 | = −32.32 dB, respectively.

III. I MPLEMENTATION AND R ESULTS For experimental demonstration, prototypes of GD circuit type-I and type-II were designed and fabricated on RT/Duroid 5880 substrate with a dielectric constant (εr ) of 2.20 and a thickness (h) of 0.787 mm. In this paper, we designed prototype circuits for flat (minimum ripples) GD. The simulation was performed using ANSYS HFSS 15. A. Results of Wideband Flat Group Delay Circuit Type-I For experimental validation, we set the design goal of circuit Type-I type-I as τ f = f0 = 1 ns at f 0 = 2 GHz with acceptable τripple of less than 1.3%. Using design flowchart shown in Fig. 7,

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

Fig. 17.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Simulation and measured results of GD circuit type-I.

Fig. 18. (a) Layout of the wideband flat GD circuit type-II with physical dimensions. (b) Photograph of the fabricated circuit.

B. Results of Wideband Flat Group Delay Circuit Type-II For experimental validation of circuit type-II, the prototype circuit was designed and fabricated at f 0 = 2 GHz with Type-II τ f = f0 = 2.5 ns and τripple ≤ 2%. Assuming z 1 = 0.72 , the calculated circuit parameters for the given specification are obtained as β = 1.1, z c = 2.4240 , and Ceff = −17.9321 dB using the proposed design method described previously. The calculated Type-II of the design circuit is given as 10.1% using above circuit parameters (z 1 , z c , and Ceff ). The renormalized circuit parameters of designed prototype are Z 1 = 36 , Z 0e = 137.6923 , and Z 0o = 106.6857 . Fig. 18 shows the layout and photograph of the fabricated circuit type-II. The physical dimensions of fabricated circuit are given as L 21 = 24.90, L 22 = 28.2, W21 = 0.38, W22 = 3.80, and g21 = 1.17 (unit: mm). Fig. 19(a) shows the measured and simulated S-parameters and GDs. These results show that the measurement results are in good agreement with the simulation and theoretical predicated values. From experiment, the GD is 2.498 ns at f 0 and flat GD extends from 1.90 to 2.10 GHz. Therefore, the flat GD fractional bandwidth of the fabricated circuit type-II is 10%, which is slightly narrower than that of circuit type-I because of higher GD than circuit type-I. Similarly, the measured S-parameters at f 0 = 2 GHz are determined as |S21 | = −0.89 dB, |S11 | = −28.18 dB, and |S22 | = −26.28 dB. To demonstrate the prototype for higher GD, an extra circuit was designed for GD of 4 ns at f 0 = 2 GHz with τripple ≤ 8%. Assuming z 1 = 0.4 , the calculated parameters are given as β = 0.9, z 0e = 1.4215 , and z 0o = 1.2201 . Once circuit parameters are obtained,

Fig. 19. Simulation and measured results of type-II circuit with GD of (a) 2.5 and (b) 4 ns. TABLE II C ALCULATED C IRCUIT PARAMETERS OF C IRCUIT T YPE -I W ITH P OSITIVE AND N EGATIVE S IGNS IN (37)

Type-II of circuit is estimated as 7.3% using (31) and (40). Fig. 19(b) shows the measured and simulation results of prototype. The measured GD is 3.96 ns which extends from 1.93 to 2.074 GHz with  = 7.20%. Similarly, the measured |S21 |, |S11 |, and |S22 | at f0 are determined as −1.10, −27.2, and −24.94 dB, respectively. Table III shows performance comparison of the proposed work with state of the arts. As seen from Table III, the proposed work provides wide flat GD. In addition, the proposed

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. CHAUDHARY AND JEONG: ARBITRARY PRESCRIBED WIDEBAND FLAT GD CIRCUITS USING COUPLED LINES

TABLE III P ERFORMANCE C OMPARISON

structure is very simple and does not require cross-coupling or controlling sign of cross-coupling. IV. C ONCLUSION In this paper, we present transmission-type wideband flat GD circuits using coupled lines. The analytical design equations are obtained using symmetrical even- and odd-mode analyses. The circuit parameters for arbitrarily specified flat GD are obtained analytically and do not require any optimization procedures. In addition, the proposed circuits are very simple to design and provide a wideband flat GD without any cross-coupling or controlling sign of cross-coupling. For experimental validation of the proposed structures, the prototypes of wideband GD circuits are fabricated at the center frequency of 2 GHz. The experimental results indicate that the proposed structures provide a wideband flat GD response and are applicable to various RF/microwaves circuits such as wideband analog RF self-interference cancellation, RF amplifier linearization, and real-time analog radio-signal processing. R EFERENCES [1] C. Caloz, S. Gupta, Q. Zhang, and B. Nikfal, “Analog signal processing: A possible alternative or complement to dominantly digital radio schemes,” IEEE Microw. Mag., vol. 14, no. 6, pp. 87–103, Sep./Oct. 2013. [2] S. Gupta, S. Abielmona, and C. Caloz, “Microwave analog real-time spectrum analyzer (RTSA) based on the spectral–spatial decomposition property of leaky-wave structures,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 12, pp. 2989–2999, Dec. 2009. [3] S. Abielmona, S. Gupta, and C. Caloz, “Compressive receiver using a CRLH-based dispersive delay line for analog signal processing,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 11, pp. 2617–2626, Nov. 2009. [4] K. E. Kolodziej, B. T. Perry, and J. G. McMichael, “Multitap RF canceller for in-band full-duplex wireless communications,” IEEE Trans. Wireless Commun., vol. 15, no. 6, pp. 4321–4334, Jun. 2016. [5] Y.-C. Jeong, D. Ahn, C.-D. Kim, and I.-S. Chang, “A feed-forward amplifier using an equal group-delay signal cancellation technique,” Microw. J., vol. 50, no. 4, pp. 126–134, Apr. 2007. [6] S. Lucyszyn and I. D. Robertson, “Analog reflection topology building blocks for adaptive microwave signal processing applications,” IEEE Trans. Microw. Theory Techn., vol. 43, no. 3, pp. 601–611, Mar. 1995.

9

[7] G. Chaudhary, H. Choi, Y. Joeng, J. Lim, and C. D. Kim, “ Design of group delay time controller based on a reflective parallel resonator,” ETRI J., vol. 34, no. 2, pp. 210–215, Apr. 2012. [8] M. H. Chen, “The design of a multiple cavity equalizer,” IEEE Trans. Microw. Theory Techn., vol. MTT-30, no. 9, pp. 1380–1383, Sep. 1982. [9] H.-T. Hsu, H.-W. Yao, K. A. Zaki, and A. E. Atia, “Synthesis of coupledresonators group-delay equalizers,” IEEE Trans. Microw. Theory Techn., vol. 50, no. 8, pp. 1960–1968, Aug. 2002. [10] Q. Zhang, S. Gupta, and C. Caloz, “Synthesis of narrowband reflectiontype phasers with arbitrary prescribed group delay,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 8, pp. 2394–2402, Aug. 2012. [11] T. Guo, Q. Zhang, Y. Chen, R. Wang, and C. Caloz, “Shunt-stub and stepped-impedance broadband reflective phasers,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 10, pp. 807–809, Oct. 2016. [12] W. Liao, Q. Zhang, Y. Chen, S. Wong, and C. Caloz, “Compact reflection-type phaser using quarter-wavelength transmission line resonators,” IEEE Microw. Wireless Compon. Lett., vol. 25, no. 6, pp. 391–393, Jun. 2015. [13] T. Guo, Q. Zhang, Y. Chen, R. Wang, and C. Caloz, “Single-step tunable group delay phaser for spectrum sniffing,” IEEE Microw. Wireless Compon. Lett., vol. 25, no. 12, pp. 808–810, Dec. 2015. [14] R. Withers, A. Anderson, P. Wright, and S. Reible, “Superconductive tapped delay lines for microwave analog signal processing,” IEEE Trans. Magn., vol. MAG-19, no. 3, pp. 480–484, May 1983. [15] W. J. D. Steenaart, “The synthesis of coupled transmission line all-pass networks in cascades of 1 to n,” IEEE Trans. Microw. Theory Techn., vol. MTT-11, no. 1, pp. 23–29, Jan. 1963. [16] K. Keerthan and K. J. Vinoy, “Design of cascaded all pass network with monotonous group delay response for broadband radio frequency applications,” IET Microw., Antenna Propag., vol. 10, no. 7, pp. 808–815, May 2016. [17] S. Gupta, A. Parsa, E. Perret, R. V. Snyder, R. J. Wenzel, and C. Caloz, “Group-delay engineered noncommensurate transmission line all-pass network for analog signal processing,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 9, pp. 2392–2407, Sep. 2010. [18] Q. Zhang, S. Gupta, and C. Caloz, “Synthesis of broadband dispersive delay structures formed by commensurate C- and D-sections,” Int. J. RF Microw. Comput.-Added Eng., vol. 24, no. 3, pp. 322–331, May 2014. [19] S. Taravati, S. Gupta, Q. Zhang, and C. Caloz, “Enhanced bandwidth and diversity in real-time analog signal processing (R-ASP) using nonuniform C-section phasers,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 9, pp. 663–665, Sep. 2016. [20] Y. Horii, S. Gupta, B. Nikfal, and C. Caloz, “Multilayer broadsidecoupled dispersive delay structures for analog signal processing,” IEEE Microw. Wireless Compon. Lett., vol. 22, no. 1, pp. 1–3, Jan. 2012. [21] B. Xiang, X. Wang, and A. B. Apsel, “A reconfigurable integrated dispersive delay line (RI-DDL) in 0.13-μm CMOS process,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 7, pp. 2610–2619, Jul. 2013. [22] J. D. Rhodes, “The design and synthesis of a class of microwave bandpass linear phase filters,” IEEE Trans. Microw. Theory Techn., vol. MTT-17, no. 4, pp. 189–204, Apr. 1969. [23] H. C. H. Cheung, F. Huang, M. J. Lancaster, R. G. Humphreys, and N. G. Chew, “Improvements in superconducting linear phase microwave delay line bandpass filters,” IEEE Trans. Appl. Supercond., vol. 5, no. 2, pp. 2675–2677, Jun. 1995. [24] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York, NY, USA: Wiley, 2001, pp. 350–360. [25] L. Szydlowski, N. Leszczynska, and M. Mrozowski, “A linear phase filter in quadruplet topology with frequency-dependent couplings,” IEEE Microw. Wireless Compon. Lett., vol. 24, no. 1, pp. 32–34, Jan. 2014. [26] X. Chen, W. Hong, T. Cui, J. Chen, and K. Wu, “Substrate integrated waveguide (SIW) linear phase filter,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 787–789, Nov. 2005. [27] G. Pfitzenmaier, “Synthesis and realization of narrow-band canonical microwave bandpass filters exhibiting linear phase and transmission zeros,” IEEE Trans. Microw. Theory Techn., vol. MTT-30, no. 9, pp. 1300–1311, Sep. 1982. [28] H.-R. Ahn, Asymmetric Passive Components in Microwave Integrated Circuits. New York, NY, USA: Wiley, 2006. [29] G. Chaudhary and Y. Jeong, “Transmission-type negative group delay networks using coupled line doublet structure,” IET Microw., Antennas Propag., vol. 9, no. 8, pp. 748–754, Jun. 2015. [30] H.-R. Ahn and M. M. Tentzeris, “Novel generic asymmetric and symmetric equivalent circuits of 90° coupled transmission-line sections applicable to marchand baluns,” IEEE Trans. Microw. Theory Techn., vol. 65, no. 3, pp. 746–759, Mar. 2017.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

Girdhari Chaudhary (S’10–M’13) received the B.E. degree in electronics and communication engineering from Nepal Engineering College, Kathmandu, Nepal, in 2004, the M.Tech. degree from MNIT, Jaipur, India, in 2007, and the Ph.D. degree in electronics engineering from Chonbuk National University, Jeonju, South Korea, in 2013. He is currently an Assistant Research Professor with the Division of Electronics Engineering, Chonbuk National University. His current research interests include multiband tunable passive circuits, in-band full-duplex systems and high-efficiency power amplifiers, and negative group delay circuits and its applications. Dr. Chaudhary was a recipient of the BK21 PLUS Research Excellence Award 2015 from the Ministry of Education, South Korea. He worked as a Principal Investigator of the independent Project through the Basic Science Research Program of the National Research Foundation (NRF), funded by the Ministry of Education Korea. He received the Korean Research Fellowship through the NRF of Korea, funded by the Ministry of Science and ICT. He has served as a Reviewer for the IEEE T RANSACTION ON M ICROWAVE T HEORY AND T ECHNIQUES , IEEE M ICROWAVE AND W IRELESS C OMPONENT L ETTERS , the IEEE T RANSACTION ON C IRCUIT AND S YSTEMS —I: R EGULAR PAPERS , and the IEEE T RANSACTION ON I NDUSTRIAL E LECTRONICS .

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Yongchae Jeong (M’99–SM’10) received the B.S.E.E., M.S.E.E., and Ph.D. degrees in electronics engineering from Sogang University, Seoul, South Korea, in 1989, 1991, and 1996, respectively. From 1991 to 1998, he was a Senior Engineer with Samsung Electronics. In 1998, he joined the Division of Electronics Engineering, Chonbuk National University, Jeonju, South Korea, where he is currently a Professor, director of IT Convergence Research Center and the HOPE-IT Human Resource Development Center of BK21 PLUS. From 2006 to 2007, he was a Visiting Professor with the Georgia Institute of Technology, Atlanta, GA, USA. He is currently teaching and conducting research in the area of microwave passive and active circuits, mobile and satellite base-station RF system, design of periodic defected transmission line, negative group delay circuits and its applications, in-band full-duplex radio, and RFIC design. He has authored or co-authored over 210 papers in international journals and conference proceedings. Prof. Jeong is a member of the Korea Institute of Electromagnetic Engineering and Science.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Broadband High-Efficiency Input Coupler With Mode Selectivity for a W-Band Confocal Gyro-TWA Jianxun Wang, Yelei Yao , Qizhi Tian, Hao Li, Guo Liu , and Yong Luo

Abstract— A 0.1-THz input coupler for HE04 mode confocal gyrotron traveling-wave amplifiers is presented. A high-order HE04 mode in confocal waveguide is efficiently excited from a TE10 mode. The coupler consists of a Y-type power-divider and a mode-converting section. The wave is coupled with the confocal waveguide via two nonstandard rectangular apertures to maximize the coupling efficiency. The mode-converting section is enclosed with mode-selective grooves to suppress the unwanted modes and thus achieve a high efficiency over a broad frequency band. The prototype is fabricated and measured. Measurements are conducted utilizing two identical back-to-back connected couplers, and the measured performances show excellent agreement with the numerical one. Experimental results show that the coupler achieves a transmission of better than −3 dB in the frequency band from 92 to 110 GHz and a maximum transmission of −1 dB. The coupler exhibits high conversion efficiency, broad bandwidth, and flat transmission throughout the 1-dB bandwidth, which can also be used as a mode converter in cold test experiments of the interaction circuits. Index Terms— Broadband, confocal gyrotron traveling-wave amplifier (gyro-TWA), high efficiency, input coupler, mode selectivity.

I. I NTRODUCTION

G

YROTRON traveling wave amplifiers (gyro-TWAs) from millimeter-wave to terahertz frequency band have an urgent demand for both military and civilian applications due to its merits such as high power, high gain, and broad bandwidth [1], [2]. Confocal gyro-TWA is a novel structure first proposed and experimentally demonstrated at MIT [3]. It can operate at a high-order mode (HOM) stably while avoiding the severe mode competition problem owing to its inherent merits of diffractive loss [4], [5]. This is a solution for conventional cylindrical gyro-TWA with loss-loaded circuits [6]–[10], which is difficult to operate at high frequency with high average power due to the severe limitation on thermal capability caused by ohmic loss and the arising of Manuscript received April 9, 2017; revised August 10, 2017 and October 13, 2017; accepted November 29, 2017. This work was supported by the National Natural Science Foundation of China under Grant 61671129. (Corresponding author: Jianxun Wang.) The authors are with the School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China (e-mail: [email protected]; [email protected]; 201521040623@ std.uestc.edu.cn; [email protected]; [email protected]; yong_luo_77@ sina.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2785235

beam interception. Moreover, it can ease the design of output system since the output of confocal waveguide is a quasiGaussian beam. Thus, it is possible for gyro-TWA to operate at W-band and beyond with high average power while avoiding the concomitant problem of mode competition by applying the novel mode-selective circuit. The input coupler is an essentially crucial component of a gyro-TWA [11]–[13], which should be able to efficiently convert a fundamental mode into the operating mode of the interaction circuits. Generally, there are two solutions for the input couplers of the confocal gyro-TWA, i.e., the insertedwaveguide configurations [4], [14]–[19] and the quasi-optical waveguide configurations [17]. The inserted waveguide configuration applies a fundamental circular/rectangular waveguide that is inserted through one of the mirrors (single-fed) or both two mirrors (dual-fed) of the confocal waveguide to inject the microwave to the interaction circuit. Note that the bandwidth of the inserted-waveguide configurations is not enough to sustain the excellent bandwidth property of the whole amplifier system. MIT innovatively proposed a three-mirror quasi-optical input coupler to convert the HE11 mode to the HE06 confocal waveguide mode. Due to the relative insensitivity of Gaussian beam to frequency, the calculated bandwidth of this coupler could easily exceed 5 GHz. The quasi-optical input coupler achieves a wide bandwidth as well as a low insert loss compared to the inserted waveguide configurations, whereas it is bulky that complicates the design, fabrication, and assembly. In this paper, a broadband input coupler with mode selectivity is proposed for HE04 mode confocal gyro-TWA operating at 0.1 THz. The experimental results show excellent agreement with the simulation one for a 0.1-THz input coupler operated at HE04 mode. The wave is coupled into the confocal waveguide via two nonstandard rectangular apertures to enhance the coupling efficiency. The mode-converting section is enclosed with mode-selective grooves to suppress the unwanted modes and thus achieve a high efficiency over a broad frequency band. The proposed input coupler exhibits an excellent performance compared to the reported configurations for confocal gyro-TWAs, moreover, the achieved performance is close to that of the state-of-the-art HOM input couplers for conventional gyro-TWAs. The merits of the novel coupler are listed as follows: 1) symmetrically dual-fed technology to suppress asymmetrical modes; 2) enclosed structure to decrease diffractive losses

0018-9480 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

TABLE I O PTIMIZED PARAMETERS L ISTS

Fig. 1. (a) 3-D configuration of the input coupler and main dimensional parameters. (b) E-field pattern.

during mode converting process; 3) grooves to suppress parasitic modes and increase the bandwidth; and 4) compact and simple structure to be easily fabricated. II. P RINCIPLES AND D ESIGN The input coupler consists of two parts, namely, the power dividing section and the mode converting section. The mode converting section is numerically investigated and optimized by computer simulation technology simulation codes, then a corresponding power divider is designed. The coupler is based on the dual-fed inserted-waveguide configuration, two signals with equal amplitude and 180° out of phase simultaneously excite the HE04 mode. Some unwanted modes can be suppressed by utilizing the symmetrically dual-fed technology; as a result, high coupling efficiency as well as high mode purity can be realized. Moreover, nonstandard coupling aperture is applied instead of standard WR10 rectangular waveguide to maximize the converting efficiency. Fig. 1(a) shows the 3-D configuration and main dimensional parameters of the proposed input coupler. Generally, the input coupler is placed between the electron optical system and the interaction circuit. The injected power should propagate toward the interaction circuit while be sufficiently cutoff in the reverse direction to maintain the stable operation of the electron optical system. The transition section is made of two pairs of mirrors to keep the wave from propagating toward the reverse direction. Fig. 1(b) shows the electric field (E-field) pattern in the coupler, it indicates that a HE04 mode with high mode purity is achieved. The two coupling apertures (port 1 and port 2) are symmetrically placed. Assume that the output port and the cutoff port are port 3 and port 4, respectively, the equivalent transmission of the coupler simultaneously excited at port 1 and port 2 with equal amplitude and 180° out of phase can be roughly approximated by the S-parameters regarding the coupler as a four-port component, that is Seq (dB) ≈ S31 (dB) + 3 dB.

(1)

Note that (1) is satisfied once the conversion to parasitic modes, the isolation coefficients (S41 , S42 ) and reflection coefficients (S11 , S22 ) are small enough. The performance of the coupler is jointly determined by the cutoff section, transition section, and coupling apertures.

Fig. 2. S-parameters of the 3-dB power divider for TE10 mode and main unwanted modes. The 2-D E-field vector distribution in E-plane is shown.

To achieve the optimal performance, global optimization algorithms such as the genetic algorithm and particle swarm optimization are applied to our design, and then followed by local optimization algorithms such as the interpolated quasinewton algorithm. Table I shows the optimized parameters list. A. Network Design The Y-type power divider configuration proposed by the National Tsing Hua University is chosen due to its merits of broad bandwidth, compact, and easy fabrication [20], [21]. According to the optimized results in Table I, the optimal coupling aperture height b2 is slightly smaller than its width that equals the WR10 waveguide width. As a result, some unwanted modes are easy to be excited in the overmoded aperture, which makes the design of the power divider more complicated. One possible solution is to design a Y-type divider with an appropriate output aperture and then change it into the optimal aperture by step-matching technology. Fig. 2 shows the refection of the input port and transmission of several parasitic modes. The reflection is well below −17.7 dB throughout the frequency band of interest.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. WANG et al.: BROADBAND HIGH-EFFICIENCY INPUT COUPLER

3

Fig. 5. 3-D configuration of the input coupler with grooves for mode selectivity.

Fig. 3. Frequency response of the input coupler with the 3-dB power divider.

Fig. 6. (a) Different groove widths Q 1 on the conversion to the PMA. (b) Different groove widths Q 1 on the conversion to the PMB.

Fig. 4. E-field vector distribution at the cross section of the waveguide for the operating mode and main parasitic modes, the rectangular frame denotes the position of grooves for mode selectivity.

The conversion of injected power to TE10 mode and the unwanted mode (TE11 and TM11 ) are better than −3.1 dB and below −25 dB, respectively, within the frequency band. In addition, the E-field vector distribution in E-plane is also shown. The input power is divided into two signals of equal amplitude and out of phase through a Y-type power divider. The field strength is expressed as a grayscale image and the length of arrows. In summary, the proposed dividing network achieves wide bandwidth and high mode purity, which is very suitable since it is compact and easy to be fabricated. Fig. 3 shows the performance of the input coupler using the optimal parameters in Table I with the power divider designed in part B. The conversion to main parasitic modes, i.e., the parasitic mode A (PMA) and the parasitic mode B (PMB) which will be discussed in detail later, are shown in Fig. 3. The maximum transmission of HE04 mode is −1.25 dB at 93.5 GHz. The transmission of the HE04 mode is better than −3 dB over 91.8–110.4 GHz, achieving a 3-dB bandwidth of 18.6 GHz. However, the conversion to the PMB is larger than −10 dB when the operating frequency is higher than 96.5 GHz, and the conversion increases with the increasing of frequency, which deteriorates the mode purity and conversion efficiency of the coupler. B. Mode-Selective Technology Fig. 4 shows E-field vector distribution of the HE04 mode and main parasitic modes at the output port of the coupler.

Fig. 7. (a) Different groove depths da5 on the transmission of the PMA. (b) Different groove depths da5 on the transmission of the PMB.

It indicates that the E-field strength of HE04 mode is weak near the closed boundary of the confocal waveguide; however, it is very strong for the PMA and PMB. As a result, two symmetrical grooves are introduced to disturb the E-field distribution of the parasitic modes. The rectangular frames in Fig. 4 denote the position of grooves in the cross section of the waveguide. Fig. 5 shows the 3-D configuration of the coupler with grooves and power dividing network. The main dimensions of the grooves are numerically analyzed and optimized. Fig. 6 shows the conversion to the two parasitic modes with different groove widths Q 1 . It indicates that the groove width Q 1 has a large influence on the conversion to the PMB but a negligible influence on the PMA. Moreover, the conversion to the PMB is not monotonic with the changing of the groove width, thus, there exists a groove width Q 1 which can minimize the conversion to the PMB. Fig. 7 shows the conversion to the two parasitic modes with different groove depths da5 . The groove depth has a large

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 8. Frequency response of transmission of the desired mode for with and without the mode-selective grooves, the filled marker, and hollow marker represents the structure with and without mode-selective grooves, respectively; a marker shape represents a mode.

influence on both the conversion to the PMA and PMB, especially the conversion to the PMA at the central frequency band and the conversion to the PMB at the upper frequency band. Specifically, the conversion to the PMA changes a little but dramatically increase with the increasing of the groove depth da5 when da5 ≤ 1 mm and da5 > 1 mm, respectively. In a contrast, the conversion to the PMB generally improves with the increasing of the groove depth, especially when da5 ≤ 1 mm. Therefore, there is a compromise for the groove depth to keep a low level of the PMA and PMB. Fig. 8 shows the comparison of the performance of the coupler with and without the optimized grooves. The frequency response of the conversion to the operating mode and main parasitic modes as well as the reflection coefficients at the input port are shown in Fig. 8. The filled markers and hollow markers represent the coupler with and without grooves, respectively. The conversion to the PMB is enormously reduced, while the reflection at the input port gets a little improvement with the grooves almost over the entire frequency band. As a result, the conversion to the HE04 mode has improved although there is some deterioration of the conversion to the PMA. Specifically, the achieved maximum conversion to HE04 mode is −1.03 dB at 99.5 GHz. The conversion to the HE04 mode is better than −3 dB over 91.6–112.9 GHz, achieving a 3-dB bandwidth of 21.3 GHz. Moreover, the conversion to the HE04 mode is better than −2 dB over 92–110 GHz, achieving a broad 2-dB bandwidth of 18 GHz. C. Open Boundary The mode-converting section is enclosed, with mode selectivity grooves. The closed boundary can reduce diffraction loss in the process of mode conversion. Besides, the closed confocal waveguide is convenient to add the grooves for mode suppressing, as a result, the converting efficiency to the desired mode can get further improved. Thus, high converting efficiency can be achieved, which can ease the pressure of the input signal source as a consequence.

Fig. 9. (a) 3-D configuration of the coupler with open boundary. (b) E-field pattern in yz plane.

A half wide mirror width a = 3.6 mm is chosen due to the desired mode has a quite low diffraction loss ( 0 (B.5) 2B3 + 3B2



2

B3 b 0 B2 B3 a 0 2 2B2 +2B3 2 − − ≥ 0. (B.6) b0 2B3 +3B2 2B3 +3B2 2B2 + 3B3 b0

Condition (B.4) and (B.6) can be rewritten as follows: (3B2 + 2B3 )3 (B2 + 2B3 ) ≥ Y02 (2B2 + 3B3 )2 (3X 2 + 2X 3 )3 (X 2 + 2X 3 ) ≥ Z 02 (2X 2 + 3X 3 )2

v ds (θ ) = a0 + a1 cos θ + a2 (2 cos2 θ − 1) + a3 (4 cos3 θ − 3 cos θ ) + c1 sin θ + c2 · 2 sin θ · cos θ + c3 sin θ (4 cos2 θ − 1) = (a0 − a2 ) + (a1 − 3a3 ) cos θ + 2a2 cos θ + 4a3 cos θ + sin θ (c1 − c3 + 2c2 cos θ + 4c3 cos2 θ ) 8B2 + 8B3 12B2 + 16B3 = a0 + a0 cos θ 8B2 + 12B3 8B2 + 12B3 8B3 4B2 + a0 cos2 θ − a0 cos3 θ 8B2 + 12B3 8B2 + 12B3 −b0 + sin θ · (4 + 8 cos θ + 4 cos2 θ ) 8B3 + 12B2 a0 = [8B3 + 4B2 (2 − cos θ )](1 + cos θ )2 8B2 + 12B3 4b0 − sin θ (1 + cos θ )2 8B + 12B 3 2

2B3 + 2B2 B2 a 0 b0 = a0 − cos θ − sin θ 2B2 + 3B3 2B2 + 3B3 2B3 + 3B2 2 · (1 + cos θ ) (B.1) i ds (θ ) 2

−B2 B3 a0 (4 − 8 cos θ + 4 cos2 θ ) 8B2 + 12B3

3

= b0 + b1 cos θ + b2 (2 cos2 θ − 1) + b3 (4 cos3 θ − 3 cos θ ) + d1 sin θ + d2 · 2 sin θ · cos θ + d3 sin θ (4 cos2 θ − 1) = (b0 − b2 ) + (b1 − 3b3 ) cos θ + 2b2 cos2 θ + 4b3 cos3 θ + sin θ (d1 − d3 + 2d2 cos θ + 4d3 cos2 θ ) 8B3 + 8B2 12B3 + 16B2 = b0 − b0 cos θ 8B3 + 12B2 8B3 + 12B2 8B2 4B3 + b0 cos2 θ + b0 cos3 θ 8B3 + 12B2 8B3 + 12B2

(B.7) (B.8)

where b0 a0 a0 Z0 = b0 Y0 =

X 2,3 = −

1 . B2,3

(B.9) (B.10) (B.11)

By observation, conditions (B.7) and (B.8) are sufficient conditions for condition (B.3) and (B.5). As a conclusion, according to these conditions, the design space for the general continuous mode can be obtained. R EFERENCES [1] A. Grebennikov and N. O. Sokal, Switchmode RF Power Amplifiers. Newton, MA, USA: Newnes, 2007. [2] S.-A. El-Hamamsy, “Design of high-efficiency RF class-D power amplifier,” IEEE Trans. Power Electron., vol. 9, no. 3, pp. 297–308, May 1994. [3] A. J. Wilkinson and J. K. A. Everard, “Transmission-line load-network topology for class-E power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 6, pp. 1202–1210, Jun. 2001. [4] M. Thian and V. F. Fusco, “Transmission-line class-e power amplifier with extended maximum operating frequency,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 58, no. 4, pp. 195–199, Apr. 2011. [5] A. Grebennikov, “High-efficiency class-E power amplifier with shunt capacitance and shunt filter,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 63, no. 1, pp. 12–22, Jan. 2016. [6] T. Mury and V. F. Fusco, “Inverse class-E amplifier with transmission line harmonic suppression,” IEEE Trans. Circuit Syst. I, Reg. Papers, vol. 54, no. 7, pp. 1555–1561, Jul. 2007.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 12

[7] P. Chen and S. He, “Analysis of inverse class-E power amplifier at subnominal condition with 50% duty ratio,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 62, no. 4, pp. 342–346, Apr. 2015. [8] F. H. Raab, “Class-F power amplifiers with maximally flat waveforms,” IEEE Trans. Microw. Theory Techn., vol. 45, no. 11, pp. 2007–2012, Nov. 1997. [9] F. H. Raab, “Maximum efficiency and output of class-F power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 6, pp. 1162–1166, Jun. 2001. [10] Y. Xu, J. Wang, and X. Zhu, “Analysis and implementation of inverse class-F power amplifier for 3.5 GHz transmitters,” in Proc. Asia–Pacific Microw. Conf., Dec. 2010, pp. 410–413. [11] J. H. Kim, G. Do Jo, J. H. Oh, Y. H. Kim, K. C. Lee, and J. H. Jung, “Modeling and design methodology of high-efficiency class-F and class-F−1 power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 1, pp. 153–165, Jan. 2011. [12] J. Moon, S. Jee, J. Kim, J. Kim, and B. Kim, “Behaviors of class-F and class-F−1 amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 6, pp. 1937–1951, Jun. 2012. [13] M. Sakalas, S. Preis, D. Gruner, and G. Boeck, “Iterative design of a harmonically tuned multi-octave broadband power amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2014, pp. 1–4. [14] N. Kumar, C. Prakash, A. Grebennikov, and A. Mediano, “Highefficiency broadband parallel-circuit class E RF power amplifier with reactance-compensation technique,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 3, pp. 604–612, Mar. 2008. [15] S. C. Cripps, P. J. Tasker, A. L. Clarke, J. Lees, and J. Benedikt, “On the continuity of high efficiency modes in linear RF power amplifiers,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 10, pp. 665–667, Oct. 2009. [16] P. Wright, J. Lees, J. Benedicta, P. J. Tasker, and S. C. Cripps, “A methodology for realizing high efficiency class-J in a linear and broadband PA,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 12, pp. 3196–3204, Dec. 2009. [17] S. Rezaei, L. Belostotski, F. M. Ghannouchi, and P. Aflaki, “Integrated design of a class-J power amplifier,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 4, pp. 1639–1648, Apr. 2013. [18] J. Moon, J. Kim, and B. Kim, “Investigation of a class-J power amplifier with a nonlinear Cout for optimized operation,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 11, pp. 2800–2811, Nov. 2010. [19] V. Carrubba et al., “The continuous class-F mode power amplifier,” in Proc. Eur. Microw. Conf. (EuMC), Sep. 2010, pp. 1674–1677. [20] V. Carrubba et al., “On the extension of the continuous class-F mode power amplifier,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 5, pp. 1294–1303, May 2011. [21] N. Tuffy, L. Guan, A. Zhu, and T. J. Brazil, “A simplified broadband design methodology for linearized high-efficiency continuous class-F power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 6, pp. 1952–1963, Jun. 2012. [22] M. Yang, J. Xia, Y. Guo, and A. D. Zhu, “Highly efficient broadband continuous inverse class-F power amplifier design using modified elliptic low-pass filtering matching network,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 5, pp. 1515–1525, May 2016. [23] V. Carrubba et al., “Inverse class-FJ: Experimental validation of a new PA voltage waveform family,” in Proc. Asia–Pacific Microw. Conf. (APMC), Dec. 2011, pp. 1254–1257. [24] V. Carrubba et al., “Exploring the design space for broadband pas using the novel ‘continuous inverse class-F mode,”’ in Proc. 41st Eur. Microw. Conf., Oct. 2011, pp. 333–336. [25] Y. Sun and X. Zhu, “Broadband continuous class-F−1 amplifier with modified harmonic-controlled network for advanced long term evolution application,” IEEE Microw. Wireless Compon. Lett., vol. 25, no. 4, pp. 250–252, Apr. 2015. [26] K. Chen and D. Peroulis, “Design of broadband highly efficient harmonic-tuned power amplifier using in-band continuous class-F−1 /F mode transferring,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 12, pp. 4107–4116, Dec. 2012. [27] S. D. Kee, I. Aoki, A. Hajimiri, and D. Rutledge, “The class-E/F family of ZVS switching amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 51, no. 6, pp. 1677–1690, Jun. 2003. [28] Z. Kaczmarczyk, “High-efficiency class E, EF2 and E/F3 inverters,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp. 1584–1593, Oct. 2006. [29] M. Ozen, R. Jos, and C. Fager, “Continuous class-E power amplifier modes,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 59, no. 11, pp. 731–735, Nov. 2012.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

[30] V. Carrubba et al., “Continuous-class F3 power amplifier mode varying simultaneously first 3 harmonic impedances,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2012, pp. 1–3. [31] T. Canning, P. J. Tasker, and S. C. Cripps, “Continuous mode power amplifier design using harmonic clipping contours: Theory and practice,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 1, pp. 100–110, Jan. 2014. [32] V. Carrubba et al., “The continuous inverse class-F mode with resistive second-harmonic impedance,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 6, pp. 1928–1936, Jun. 2012. [33] T. Sharma, R. Darraji, F. Ghannouchi, and N. Dawar, “Generalized continuous class-F harmonic tuned power amplifiers,” IEEE Microw. Wireless Compton. Lett., vol. 26, no. 3, pp. 213–215, Mar. 2016. [34] M. Roberg and Z. Popovic, “Analysis of high-efficiency power amplifiers with arbitrary output harmonic terminations,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 8, pp. 2037–2048, Aug. 2011. [35] P. Colantonio, F. Giannini, G. Leuzzi, and E. Limiti, “Multiharmonic manipulation for highly efficient microwave power amplifiers,” Int. J. RF Microw. Comput.-Aided Eng., vol. 11, no. 6, pp. 366–384, 2001.

Xiang Li (S’13–M’17) received the B.Sc. and M.Sc. degrees in electronic engineering from Tsinghua University, Beijing, China, in 2009 and 2012, respectively. He is currently pursuing the Ph.D. degree at the University of Calgary, Calgary, AB, Canada. His current research interests include multiband/wideband power amplifier design, outphasing transmitter design, and monolithic microwave integrated circuit power amplifiers for wireless and satellite communication. Mr. Li was a recipient of the Student Paper Award of 2010 Asia-Pacific Microwave Conference.

Mohamed Helaoui (S’06–M’09–SM’17) received the M.Sc. and Ph.D. degrees in communications and information technology from the École Supérieure des Communications de Tunis, Tunisia, in 2003 and 2006, respectively, and the Ph.D. degree in electrical engineering from the University of Calgary, Calgary, AB, Canada, in 2008. He is currently an Associate Professor with the Department of Electrical and Computer Engineering, University of Calgary. He is also the Chair of the IEEE local COM/MTT/AP joint chapter in the Southern Alberta Section. His has authored or co-authored over 150 publications, 1 book, 3 book chapters, and 11 patents applications (7 of them have been granted). His current research interests include digital signal processing, power efficiency enhancement for wireless transmitters, efficient and broadband power amplifiers, monolithic microwave integrated circuit power amplifiers for wireless and satellite communications, six-port receivers, and advanced transceiver design for software-defined radio and millimeterwave applications.

Xuekun Du (S’14) received the M.Sc. degree in communication and information system from Liaoning Technical University, Fuxin, China, in 2014. He is currently pursuing the Ph.D. degree in communication and information system at the University of Electronic Science and Technology of China, Chengdu, China. Since 2016, he has been a Visiting Ph.D. Student with the Intelligent RF Radio Laboratory (iRadio Lab), Department of electrical and computer engineering, University of Calgary, Calgary, AB, Canada. His current research interests include passive circuit design, high efficient wideband power amplifier design, and monolithic microwave integrated circuit modeling and design.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Postmatching Doherty Power Amplifier With Extended Back-Off Range Based on Self-Generated Harmonic Injection Xin Yu Zhou , Student Member, IEEE, Shao Yong Zheng , Member, IEEE, Wing Shing Chan, Member, IEEE, Xiaohu Fang, Member, IEEE, and Derek Ho, Member, IEEE

Abstract— In this paper, an asymmetric drain biased postmatching Doherty power amplifier (DPA) using harmonic injection is proposed for further back-off extension. The injected harmonic components are generated by the two active devices. Aided by the proposed harmonic injection network, drain waveform amplitude modulation for both devices can be achieved at saturation, which results in enhanced saturated power for both carrier and peaking devices, while carrier back-off power remains unchanged. As a consequence, the back-off region is extended. Moreover, the power utilization factor of the asymmetric drain biased DPA is also improved. A DPA for wideband code-division multiple access systems was designed and fabricated based on commercially available gallium nitride HEMT (Cree CGH 40010F) devices to validate the proposed technique. Measured results of the proposed DPA demonstrated that operation at 10-dB back off is possible between 1.6 and 1.9 GHz, with efficiency better than 46%. Index Terms— Doherty power amplifier (DPA), extended back-off range, harmonic injection, high efficiency.

I. I NTRODUCTION

T

HE classic Doherty power amplifier (DPA) provides high efficiency from 6-dB output back-off power to saturation [1]. It consists of two identical active devices with symmetric biasing to achieve the required profiles of load modulation. However, the more efficient digital modulation schemes with even higher peak to average power ratios (PAPRs) have been adopted in modern and future wireless communication systems. For example, wideband codedivision multiple access (WCDMA) has a PAPR of 9.6 dB. Power amplifiers (PAs) in base stations are therefore required

Manuscript received September 11, 2017; revised November 10, 2017; accepted November 29, 2017. This work was supported in part by the City University of Hong Kong under a City University of Hong Kong Applied Research Grant under Project 9667123, in part by the National Natural Science Foundation of China under Grant 61401523, in part by the Fundamental Research Funds for the Central Universities under Grant 16lgzd04, and in part by the UGC-funded English Language Collaboration Project under Project 6361003. (Corresponding author: Wing Shing Chan.) X. Y. Zhou, W. S. Chan, and X. Fang are with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong (e-mail: [email protected]). S. Y. Zheng is with the School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou, China (e-mail: [email protected]). D. Ho is with the Department of Materials Science and Engineering, City University of Hong Kong, Hong Kong (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2784811

to maintain high efficiency over a larger dynamic range from extended back off to saturation. The analysis of the principle behind the extended back off for Doherty topology was shown in [2] and surveyed well in [3]. The output back off (OBO) in decibel was expressed as OBO(dB) = 10 log[(1 + α)βd ] α = Psat, p /Psat,c βd = Psat,c /Pback-off,c

(1) (2) (3)

where Psat,c and Psat, p refer to the saturated carrier and peaking fundamental power and Pback-off,c is the back-off carrier fundamental power. The multiway Doherty configuration was proposed for larger α in [4]–[6]. Golestaneh et al. [4] presented a comprehensive methodology that addresses the practical design challenges resulting from transistor nonidealities. The circuit parameters for back-off extension are identified with enhanced bandwidth. However, the increased cost, limited bandwidth, and lower power utilization factor (PUF) [7] remain a major issue when compared with the two way symmetrical and asymmetrical DPAs with extended back-off region range [8]–[13]. In [9], an electronically reconfigurable Doherty amplifier capable of maintaining high efficiency was expounded. Its design procedure involved adjustment of the DPA optimal load impedance and the necessary reconfiguration required for the Doherty output matching networks to compensate for the load-impedance change. Pang et al. [10] used two equal-cell active devices with asymmetrical DPA (ADPA) based on postmatching topology for broadband high-efficiency range extension. However, further extension of the high-efficiency range was limited by the poor gain compression at back off. In [12], a novel symmetrical DPA was developed that provided high efficiency over a large dynamic range (>6 dB). The use of smaller class-C cells compared to ADPAs brings the advantage of higher gain and improved PAE performance. In [14], an efficiency enhanced active second-harmonic injection PA was proposed. This relied on a frequency doubler and a separate second-harmonic amplifier for injection. However, this method only focused on the performance enhancement at saturation, which restricted its use in modern wireless communication systems. Inspired by [14], with two transistors already present in the classic Doherty topology, this secondharmonic amplifier [14] may not be necessary. It is proposed

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

that the second harmonics generated by these two transistors can be injected into the drain of the other device by a harmonic injection network (HIN) for the ADPA postmatching topology. This results in enhancements of both Psat,c and Psat, p together with a small variation of Pback-off,c . Correspondingly, a higher βd can be obtained while α remains the same. This leads to a DPA with extended Doherty range, PUF enhancement, and high saturated efficiency, which are suitable for real wireless communication system. The rest of this paper is organized as follows. Section II describes the theoretical analysis of the proposed PMDPA. Section III discusses the design guidelines of the fabricated DPA. Section IV presents the experimental results of the proposed DPA. Finally, Section V concludes this paper.

Fig. 1.

Topology of the proposed DPA.

Fig. 2.

Geometry of the proposed DPA.

Fig. 3.

Operating principle of the proposed DPA at back off.

II. T HEORETICAL A NALYSIS In conventional DPAs, the value of α and βd can be theoretically expressed as [3] βd = 1 + α.

(4)

Hence, the OBO range can be further expressed as OBO(dB) = 10 log[(1 + α)2 ].

(5)

Obviously, the back-off range is solely determined by the saturated power ratio between two transistors and consequently a higher OBO necessitate the underutilization of the carrier device (e.g., reduced drain bias). However, with the decrease of carrier drain bias which is used for a wider back-off range, the carrier device will suffer from being over-driven at back off, which results in the deterioration of gain compression. Considering that the DPA is widely recognized as a linear PA, meaningful back-off extension should be achieved with a good balance between efficiency and linearity. Hence, the conventional asymmetrical drain biased DPA has limitations. It can be concluded that there are two major targets that needs to be met if further back-off extension of asymmetrical drain biased DPA is to be achieved. 1) First Target: The value of βd should be controlled independently, which can be achieved with higher Psat,c while maintaining Pback-off,c . 2) Second Target: The value of α in ADPA should be maintained, as the enhancement of Psat,c in the first target, a higher Psat, p is required to stop α reducing. To achieve these, the topology of the DPA using asymmetrical drain biased configuration is shown in Fig. 1. The input side consists of an input power splitter (IPS), one offset line [15], input matching networks (IMNs), stabilization networks (SNs), and biasing networks. At the output side, there are two fundamental-impedance inverters (FIIs), an HIN, one offset line, and postmatching network (PMN). The asymmetrical drain biased carrier (20 V) and peaking (28 V) devices can be considered as two current sources in phase quadrature. Hence, the peaking device behaves as a voltage controlled current source providing the current to modulate the load impedance of the carrier device for Doherty behavior. As shown in Fig. 2, the paths for the two fundamental signals are combined at the combining node. This combined

signal is transferred to the load (50 ) through the PMN, while the second-harmonic components of carrier and peaking devices are both injected into the drain of each other through the HIN (180°). A. Operating Principle of Asymmetrical Drain Biased Doherty Amplifier The operating principle of the asymmetrical drain biased Doherty amplifier can be separated into back-off point and saturation, respectively. In Fig. 3, Rc,back-off refers to the output impedance of carrier branch at back-off point. R p,back-off refers to the output impedance toward the peaking branch at back-off point. Z c,back-off represents the drain impedance seen by the carrier device at back-off point. RHEC refers to impedance toward the peaking branch at harmonic extracting point. In Fig. 4, Rc,sat and R p,sat refer to the output impedance of carrier and peaking branch at saturation, respectively. Z c,sat and Z p,sat represent the drain impedance seen by the carrier and peaking device at saturation, respectively, while

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ZHOU et al.: POSTMATCHING DPA

3

For the second-harmonic component generated by the peaking device (180°), it is injected into the drain of carrier device through the HIN with 0°. The carrier drain waveforms with injected second harmonic from the peaking device can be written as v D (ωt) = VDD +v 1c cos(ωt) + v 2c cos(2ωt) + ainj_ p cos(2ωt) = VDD +v 1c cos(ωt) + (v 2c + ainj_ p ) cos(2ωt)

(13)

i D (ωt) = IDD − i 1c cos(ωt) − i 2c cos(2ωt)+binj_ p cos(2ωt) (14) = IDD − i 1c cos(ωt) + (binj_ p − i 2c ) cos(2ωt) Fig. 4.

Operating principle of the proposed DPA at saturation.

RHEp refers to the impedance seen toward the carrier branch at the harmonic extracting point. RC.N. refers to the impedance at the signal combining point. Collectively, this is the overall load impedance of the entire load modulation network. Before the back-off point, only the carrier device is conducting with a resistive load impedance of R L at Rc,back-off . With the high fundamental-impedance characteristic of HIN and offset line 1, the peaking branch and HIN appears as an open circuit to the carrier branch. The values of RC.N. and Rc,back-off are effectively the same. The load conditions are as follows: Rc,back-off at f 0 = R L  Rp,back−off at f 0 = RHEC at f 0 = ∞ RC.N at f 0 = R L .

(6) (7) (8)

At saturation, both carrier and peaking devices are conducting. Offset line 1 compensates for the phase difference between the two fundamental signals. In this region at the fundamental frequency, RC.N. should remain the same as R L . Correspondingly, Rc,sat at f 0 is (1 + α) times that of R L . As well as, R p,sat at f 0 should be equal to (1 + 1/α) times R L . Moreover, RHEp should be infinite to prevent the peaking fundamental signal leaking into the HIN and the carrier branch. The ideal load conditions are as follows: Rc,sat at f0 = (1 + α)R L Rp,sat at f0 = (1 + 1/α)

(9) (10)

RHEp at f0 = ∞

(11)

RC.N at f0 = R L .

(12)

where VDD and IDD are the dc components of time-domain signals, v nc and i nc (n = 1, 2) are the voltage and current components of the fundamental, second harmonics of the carrier device, and ainj_ p and binj_ p are the amplitudes of the voltage and current of the injected second-harmonic component from the peaking device, respectively. In terms of voltage waveform, the peak amplitude can be obtained when ωt = 0. Its value is v 1c + v 2c + ainj_ p , which achieves an increase in amplitude for the voltage peak. While for the current waveform, due to the presence of self-generated second-harmonic current of the carrier device, the current waveform in amplitude is i 1c − i 2c . When the peaking secondharmonic current component (binj_ p ) is injected, the peak current amplitude is modulated to (i 1c − i 2c + binj_ p ), which results in a higher value. Enhancements can be seen in the carrier saturated amplitude of both voltage and current. The carrier saturated output power (Psat,c ) can be expressed as Psat,c = 1/2 × [(v 1c + v 2c +ainj_ p )(i 1c −i 2c +binj_ p )].

(15)

More importantly, there are nearly no second-harmonic components which can be provided to the carrier device at the back-off point because the peaking device is just at “turn ON.” Hence, the back-off voltage is v 1c + v 2c due to the voltage saturation characteristic of DPA, and the back-off current is 1/(1 + α) times i 1c − i 2c because of the load condition mentioned in (10). Hence, Pback-off,c remains unchanged from the original ADPA Pback-off,c = 1/2 × [(v 1c + v 2c )((i 1c − i 2c )/(1 + α))].

(16)

B. Operating Principle of Harmonic Injection Network

Dividing (15) by (16), βd can be calculated as    ainj_ p binj_ p βd = 1 + 1+ (1 + α). v 1c + v 2c i 1c − i 2c

When the two active devices reach saturation, both devices generate large second-harmonic components, which are used to advantage as no additional components are necessary. To achieve this, an HIN is integrated into the postmatching Doherty topology. From [16], we know that the best performance for output power and efficiency can be obtained when the phase of HIN is 180° at 2 f 0 . Since the HIN is designed to match the impedance characteristics with the FET’s output impedances at second harmonic, the secondharmonic components of the carrier and peaking devices only flow into the HIN.

It can be observed that the value in the first bracket is always larger than 1. Similarly, i 1c larger than i 2c in the carrier amplifier, and hence, the second bracket is also greater than 1. Compared with βd = 1 + α of the asymmetrical drain biased DPA without harmonic injection, the βd of proposed DPA is enhanced by the injected peaking second-harmonic components. Namely, the first target can be satisfied by the proposed harmonic injection concept. While for the second-harmonic component generated by the carrier device (0°), it is injected into the drain of the peaking device through HIN. The peaking drain waveforms

(17)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

with second-harmonic injection from the carrier device can be expressed as v D (ωt + 90°) = VDD + v 1 p cos(ωt + 90°) + v 2 p cos(2(ωt + 90°)) + ainj_c cos(2ωt − 180°) = VDD + v 1 p cos(ωt + 90°) (18) + (v 2 p + ainj_c ) cos(2(ωt + 90°)) i D (ωt + 90°) = IDD − i 1 p cos(ωt + 90°) − i 2 p cos(2(ωt + 90°)) + binj_c cos(2ωt − 180°) = IDD − i 1 p cos(ωt + 90°) (19) + (binj_c − i 2 p ) cos(2(ωt + 90°)) where VDD and IDD are the dc components of time-domain signals, v np and i np (n = 1, 2) are the voltage and current amplitudes of fundamental, second harmonics of peaking device, and ainj_c and binj_c are the amplitudes of the voltage and current of the injected second-harmonic component from carrier device, respectively. Similarly, for the carrier device, the enhanced Psat, p can be written as Psat, p = 1/2 ∗ [(v 1 p + v 2 p +ainj_c )(i 1 p −i 2 p +binj_c )].

(20)

Dividing (20) by (15), α can be calculated as

Fig. 5. Simulated drain impedances at f 0 of carrier and peaking devices at fundamental ( f 0 ), second (2 f 0 ), and third harmonic (3 f 0 ) for real and imaginary parts at saturation. (a) Carrier device. (b) Peaking device.

(v 1 p + v 2 p + ainj_c )(i 1 p − i 2 p + binj_c ) (v 1c + v 2c + ainj p )(i 1c − i 2c + binj_ p )    v 1 p + v 2 p + ainj_c i 1 p − i 2 p + binj_c = . (21) v 1c + v 2c + ainj_ p i 1c − i 2c + binj_ p

α=

Considering that the value of ainj_c and ainj_ p are relatively small compared with (v 1 p + v 2 p ) and (v 1c + v 2c ), the first bracket has small variations in value no matter what both the injected power are. Similarly, the value of binj_c and binj_ p are also relatively small compared with (i 1 p − i 2 p ) and (i 1c − i 2c ), therefore the second bracket has no significant change in value with or without both injected power. Hence, α of the proposed ADPA is not influenced by both harmonic injection. This achieves the second target. In [17], the electrical-impedance synthesis for arbitrary impedance, by controlling the relative power and phase between two excitation sources was demonstrated. When the relative phase is 180° and the injected power is larger than the source power, the negative real part of input impedance can be obtained and vice versa. According to (13), (14), (18), and (19), the sign of Z c at 2 f 0 and Z p at 2 f 0 depend on the value of injected current component (binj_ p and binj_c ) and selfgenerated current component (i 2c and i 2 p ). Due to the use of asymmetric drain biasing, the 2 f 0 power of peaking device is larger than that of the carrier device. Correspondingly, binj_ p > i 2c and binjc < i 2 p . Therefore, the negative real part of Z c at 2 f 0 can be achieved, which leads to better offset between carrier drain voltage and current waveform. On the other side, the positive sign of Z p at 2 f 0 can also be obtained. The corresponding drain-impedance conditions can be expressed as Real part of Z c at 2 f0 < 0, binj_ p > i 2c

(22)

Real part of Z p at 2 f0 > 0, binj_c < i 2 p .

(23)

Fig. 6. Simulated real part of second-harmonic drain impedance of carrier and peaking devices versus frequency.

Their simulated saturated drain impedance at center frequency for carrier and peaking devices at f0 , 2 f 0 , and 3 f 0 , for real and imaginary parts based on circuit dimensions given in Section III are shown in Fig. 5, to verify the theory. The carrier second-harmonic drain impedance has a negative real part, whereas for the peaking second-harmonic drain impedance it is positive. To further demonstrate the effect of second-harmonic injection, the simulated real part of Z c at 2 f 0 and Z p at 2 f 0 versus frequency is shown in Fig. 6. The real part of carrier device is negative, whereas for the peaking device is positive over the entire operating frequency range. C. Theoretical Analysis of Harmonic Injection Network The HIN is composed of a harmonic extracting circuit (HEC) and a second-harmonic injection offset line, which is shown in Fig. 7. Geometry and operating principle of the HEC for low and high frequencies are shown in Fig. 8. The topology is composed of two transmission lines (TL1 and TL2 ) in series

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ZHOU et al.: POSTMATCHING DPA

Fig. 7.

5

Geometry of HIN. Fig. 9.

Fig. 8.

Frequency response of HEC. (a) S-parameter. (b) Smith chart.

Geometry of HEC for low and high frequencies.

with a capacitor (C0 ) that is in parallel with a coupled line that is shorted to ground. TL1 and TL2 are feed lines for the input and output port. The shorted-coupled line is defined as a twonode connection network, where the inner side of Node A and Node B are connected to capacitor (C0 ). On the other side, Node A and Node B are also connected to the input–output feed lines, respectively. The odd-mode ABCD matrix of a shorted-coupled line can be expressed as     1 0 A B = . (24) − j Y1o cot θ1 1 C D odd The even-mode ABCD matrix of this shorted-coupled line can be expressed as     1 0 A B = (25) − j Y1e cot θ1 1 C D even where θ1 is the electrical length of end short-circuited coupled line. At low frequency, the impedance between Node A and Node B is relatively large because of the capacitor’s stopband characteristics. Under this condition, the corresponding even/odd mode S-parameters of the defined two-port network are j cot θ1 j cot θ1 S11o = (26) S11e = 2 − j cot θ1 2 − j cot θ1 2 2 S21e = S21o = . (27) 2 − j cot θ1 2 − j cot θ1 The S-parameters can be calculated as 1 j cot θ1 S11= (S11e + S11o ) = 2 2 − j cot θ1

1 S21 = (S21e − S21o ) = 0 2 (28)

Fig. 10.

Dimensions of input side network.

which shows the HEC has a stopband frequency response at the low frequency. At high frequency, the impedance between Node A and Node B decreases to a relatively small value, which result in the signal being shorted by the path between Node A and Node B. Consequently, the high-frequency signal transfers directly from TL1 to TL2 through C0 . To verify the theoretical analysis of HEC, the full-wave simulated S-parameter of the HEC based on the dimension in Fig. 9 in a 50- system was performed. This is shown in Fig. 9(a) as a magnitude, and on a smith chart shown in Fig. 9(b). Next, the second-harmonic injection offset line (offset line 2) is cascaded to satisfy the 180° phase requirement of the whole HIN. III. DPA FABRICATION AND R EFERENCE C ASE D ESIGN The proposed DPA consists of the input side network, FII and offset line, HIN, and PMN. Four steps design procedures are given as a guideline for DPA systematic realization. The circuit is designed using Rogers substrate 4003C (εr = 3.38 and h = 0.813 mm). A. Input Side Network (Step 1) The input side of the proposed DPA is composed of the IPS, OL, IMNs, drain bias, and stabilization networks. Their dimensions are shown in Fig. 10. The input network is similar to the conventional input networks used for Doherty PAs. Similarly the LC SN should be tuned properly to realize stable operating conditions and

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

Fig. 11.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 13.

Phase difference between the carrier and peaking device.

Simulated phase shift of the HIN.

Therefore, the real part of Z c,sat at ω0 has a relationship of R L and α R L (1 + α) . (33) 2+α Based on the above, the value of C can be determined by (32) and L should be tuned to an appropriate value based on load-pull data. For the peaking L-type FII, it will suffer from the impedance transformation ratio (TR) of R p,sat /Z p,sat . Thereafter, offset line 1 should be introduced to connect two FIIs for signal combining. Theoretically, the electrical length of offset line 1 should be 90° at f 0 to compensate for the phase difference between the two signal paths. While the impedance of offset line 1 (Z offset1) should be equal to R p,sat in order to avoid power loss. According to (10), Z offset1 can be expressed as Re{Z c,sat } =

Fig. 12.

Equivalent circuit of the L-type FII.

flatness of small signal gain within the operating frequency. With the aid of offset line before the IMN of the carrier device, the gate excitation signal between carrier and peaking is at phase quadrature. Also, a 180° phase difference at the second harmonic (shown in Fig. 11) is required for the harmonic injection technique. B. Fundamental Impedance Inverter and Offset Line (Step 2) Before the design of FII, the optimal drain impedance of carrier (Z c,sat ) and peaking (Z p,sat ) device should be obtained using load pull which was performed using Keysight ADS simulation software. From the variational trend of the optimal impedances required by the carrier PA in [18], the Doherty region requires a kind of impedance transformation which keeps the real part invariant while making the imaginary part decrease as seen within the package. To achieve this, the carrier FII adopts the L-type, which can be considered as an LC tank equivalent circuit, shown in Fig. 12. The Z c is expressed as 1 Z c = j ωL + . (29) j ωC + R1 This expression can be expanded to its real and imaginary parts as   R ωC + j ωL − . (30) Zc = 1 + R 2 ω2 C 2 ω2 C 2 + 12 R

Considering the load modulation leads to Rc changing from R L (back off) to (1 + α)R L (saturation) and that it is required that the real part of Z c does not change at ω0 , we have RL (1 + α)R L = . (31) 1 + R 2L ω02 C 2 1 + ((1 + α)R L )2 ω02 C 2 From (31), ω0 is calculated as  ω0 =

1 . (1 + α)C 2 R 2L

(32)

Z offset1 = R p,sat = (1 + 1/α)R L .

(34)

C. Harmonic Injection Network (Step 3) Since detailed analysis of the HIN was carried out in Section II-C, some practical considerations are presented here. First, the phase requirement for the HIN is theoretically 180°. However, due to internal parasitic effects of the two active devices, the HIN phase shift deviates from this in practice. Second, the capacitor (C0 ) should be selected with a high-Q factor to minimize power loss, such as the muRata GQM 18 series. Third, the impedance of offset line 2 should be chosen carefully. That is because the larger insertion loss of second harmonic is obtained with high impedance, while the undesirable fundamental leakage is caused by low impedance. Eventually, the length l should be further tuned when HIN is integrated into the post-matching DPA topology for stopping the fundamental components. The simulated phase shift of the HIN in a 50- system is shown in Fig. 13. Due to the HIN adopts a “low-pass filter with offset line” type configuration, this results in the large phase slope variation, which limits the effective 180° phase condition to a narrowband and consequently, that of the proposed DPA. With the aid of step 2 and step 3, the entire load modulation network has been fabricated and its dimensions are shown in Fig. 14. The detailed equivalent circuit of the proposed load modulation network is shown in Fig. 15. The carrier- and

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ZHOU et al.: POSTMATCHING DPA

Fig. 14.

Fig. 15.

7

Dimensions of the entire load modulation network. Fig. 17.

Photograph of the fabricated DPA.

Fig. 18.

Photograph of the test bench.

Equivalent circuit of the proposed load modulation network.

E. Reference Case Fig. 16.

Dimensions of the PMN.

peaking-impedance inverter can be considered to be an LC circuit (L 1 = 2 nH, C1 = 2.7 pF, L 2 = 2.4 nH, and C2 = 3 pF, respectively). Whereas, the fundamental and second-harmonic offset lines (offset line 1, offset line 2, and offset line 2 ) are equivalent with the inductors as 7, 4, and 0.3 nH, respectively. For the HEC, the equivalent even/odd mode impedance and electrical length of shorted-coupled line are 84.7 , 49.7 , and 38.1°, which is paralleled with a capacitor (C0 = 1 pF). D. Postmatching Network (Step 4) A conventional high-order low-pass topology is adopted for the PMN (shown in Fig. 16). Its main function is for maximum power transfer of the combined signal to the 50- system load (R0 ). TR of the PMN can be expressed as TR = R0 /RC.N .

(35)

After all the four steps design procedures are completed, the proposed postmatching DPA with extended back-off region can be achieved. Photographs of the fabricated DPA and test bench are shown in Figs. 17 and 18, respectively.

For comparison, a postmatching DPA with the same asymmetrical drain bias configuration but without a HIN was designed and simulated for use as reference. The reference case should have RC.N. to be the same as the proposed DPA’s simulation result. Both load conditions of the reference DPA at back-off point and saturation should be equal to (6)–(8) and (9)–(12). Namely, the reference case should have the same back-off efficiency and back-off output power with the ones of the proposed DPA. Then, their back-off value, PUF, and drain waveforms can be correctly compared. The reference case uses an identical input side and PMN as that of the proposed DPA. Simulated carrier and peaking output power between the reference DPA and proposed DPA are shown in Fig. 19 for a more indicative view at increased output power. For the proposed DPA (with HIN; carrier: 20 V, peaking: 28 V), the Psat, p and Psat,c are 13 and 9.6 W, and Pback-off,c is 2.25 W, leading to an α value of 1.354 and a βd value of 4.174, while for the reference DPA, the Psat, p and Psat,c are 8.2 and 6 W and Pback-off,c is 1.8 W, resulting in an α value of 1.36 and a βd value of 3.33. To better clarify the contributions of this paper and what has been achieved by each of these techniques independently, an ideal 6-dB DPA is also introduced. In Table I,

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

TABLE I PARAMETERS OF OBO R ANGE IN F IG . 19

Fig. 20.

Simulated DE versus output power at f 0 for three cases.

Fig. 21.

Equivalent circuit of the device adopted for the de-embedding.

Fig. 19. Simulated carrier and peaking output power with and without HIN ( f 0 ) versus input power.

the OBO enhancement of the proposed DPA can be observed (9.92 dB versus 8.95 dB versus 6 dB). The reason for this substantial back-off improvement can be viewed by the larger value of βd (4.174 versus 3.33 versus 2). This is because the carrier device (under 20-V drain bias) can generate a higher Psat,c compared with the reference case, while Pback-off,c is almost unchanged. Meanwhile, the proposed work’s α value remains unchanged when compared with the typical asymmetrical drain biased DPA (1.354 versus 1.36 versus 1). This benefits from both power enhancement of carrier and peaking device (13/9.6 versus 8.2/6). In short, the asymmetrical drain bias configuration extends the back-off region from 6 to 8.95 dB by increased α. After that, the proposed HIN further enlarges the value of βd while maintain the same value of α. As a result, further back-off extension from 8.95 to 9.92 dB can be achieved. The simulated drain efficiency (DE) versus output power at f 0 for the three cases is shown in Fig. 20. With the dual-harmonic injection technique, the back-off region of the proposed DPA is extended to 10 dB and an increase in efficiency at saturation is achieved. Based on the circuit dimensions mentioned above (proposed case and reference case), the de-embedding of a commercial large-signal model of CGH40010F [19] (shown in Fig. 21) is utilized to obtain the intrinsic drain waveforms of carrier and

Fig. 22. De-embedding drain waveform of carrier and peaking devices at saturation with and without harmonic injection. (a) Carrier device. (b) Peaking device.

peaking devices at saturation. Intrinsic drain waveform of the reference DPA and proposed DPA is shown in Fig. 22. For the reference case, the voltage amplitude of the carrier device does not exceed 40 V because of its lower drain bias (20 V). While the carrier current amplitude has

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ZHOU et al.: POSTMATCHING DPA

9

Fig. 23. Simulated saturated output power of symmetrical and asymmetrical with and without harmonic injection versus frequency.

a rectangle shape due to operation in deep saturation. Whereas the peaking device (28 V) exhibits a nonlinear current bifurcation due to its class-C bias. For the proposed case, the phase offset between carrier drain current and voltage waveform is separated because of the negative Re{Z c,sat at 2 f 0 }. Namely, the carrier device operates like a “class-J” PA [20] at saturation. Consequently, the overlap between voltage and current is decreased which enhances efficiency. The enhanced voltage and current amplitude generates an additional fundamental power compared with the references DPA. Also, the peaking voltage and current amplitude can be modulated at a higher level and the current bifurcation remains because of its class-C bias. Hence, the fundamental output power will be increased. As can be expected, the saturated efficiency of the proposed DPA will be higher than the conventional AB-C DPA [21] because of this enhanced higher efficiency. To demonstrate the improvement of the proposed DPA, simulated saturated power of the reference DPA and proposed DPA versus frequency is shown compared in Fig. 23. Compared with the reference DPA, output power of the proposed work is enhanced. It can be observed that the proposed DPA sacrifices bandwidth. Considering that this paper focuses mainly on the extension of back-off region and designed for WCDMA systems (narrowband system), a sacrifice in bandwidth is acceptable. To verify the theoretical analysis mentioned in (6)–(12), simulated load conditions of the proposed DPA at back-off point and saturation within the entire operating frequency are shown in Fig. 24, respectively. The Rc,back-off , Rc,sat , and R p,sat follow their relationship about α. Meanwhile, RHEC , R pback-off and RHEp are always located near the open circuit as seen on the Smith chart to stop the fundamental components. In Fig. 25, the simulated second-harmonic carrier and peaking output power versus input power is presented. These results show that the second harmonic of the peaking device is much larger than that of the carrier as input power increases. Finally, carrier and peaking intrinsic drain impedance of the proposed DPA trajectories (fundamental and second harmonic) versus input power level is shown in Fig. 26. As the proposed HIN introduces a mismatch at the fundamental frequency, the load modulation behaviors cannot be satisfied perfectly over the entire operating frequency range. However, the intrinsic drain impedance behaviors can still be controlled to within

Fig. 24. Simulated load conditions at low power and Doherty region over entire operating frequency range.

Fig. 25. power.

Simulated carrier and peaking output power (2 f 0 ) versus input

an acceptable range. In terms of the second harmonic, with increasing input power, the real part of Z c at 2 f 0 goes into the negative impedance region and remains there even up the saturation. While for Z p at 2 f 0 , the impedance always has a positive real part. IV. DPA M EASURED R ESULTS To demonstrate the feasibility of the proposed topology, a DPA using self-generated second-harmonic injection is implemented. The carrier (IDQ = 60 mA) and peaking devices (biased at −5.8 V) use identical 10-W GaN HEMT CGH40010F from Cree. Asymmetric drain bias is 20 V for carrier device and 28 V for peaking device, respectively. Simulated and measured small signal performances of the proposed DPA are shown in Fig. 27. A. Continuous Wave Signal Measurement Measured DEs versus output power over the entire operating frequency range are shown in Fig. 28. At 10-dB back off, efficiencies of at least 46% can be obtained from 1.6 to 1.9 GHz, which corresponds to a bandwidth of 17.1%. While the saturated DE ranges from 78.3% to 83%. As shown in Fig. 28, the first efficiency peak in the proposed DPA is not

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 28. Measured DE versus output power within entire operating frequency of the proposed DPA.

Fig. 29. Measured gain versus output power within entire operating frequency of the proposed DPA. Fig. 26. Carrier and peaking intrinsic drain impedance of proposed DPA trajectories versus input power level. (a) Fundamental. (b) Second harmonic.

Fig. 27. Simulated and measured small-signal performance of proposed DPA.

so obvious. This is because the saturated efficiency of the proposed DPA exceeds 80%, which results in the diminished visibility of Doherty behavior. Measured gain with respect to the output power is shown in Fig. 29. The gain decreases from 11 to 9.9 dB with varying output power with the compression point at saturation is less than 2.6 dB over the operating frequency band. A variation in gain can be observed, which is mainly due to two reasons. The first being that a lower carrier drain bias leads to an overdriven carrier device at the back-off point. Inevitably, the gain is compressed. While the second reason is that both harmonic injection only occurs from back-off point to saturation, which results in a faster growth in output power. Correspondingly, the gain increases sharply after the back-off point.

Fig. 30. Simulated and measured saturated output power versus frequency of the proposed DPA.

Due to the limited accuracy of the peaking amplifier simulation, the power from the peaking device is lower than the expected result. Moreover, the measured small signal gain (shown in Fig. 27) is slightly lower than simulation. This is because the transistor simulation model cannot provide the best accuracy with the drain bias used (20 V) which is different from the specified drain voltage (28 V). Consequently, the carrier device power at saturation is lower than that obtained from simulation. Although the measured saturated output power is lower than the simulation result, it is still within an acceptable range over the entire operating frequency. The simulated and measured saturated output power versus frequency is shown in Fig. 30.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ZHOU et al.: POSTMATCHING DPA

11

TABLE II P ERFORMANCE C OMPARISON W ITH S TATE - OF - THE -A RT W IDEBAND DPAs

Fig. 31. Measured ACPR (lower) and average modulated efficiency versus output power within entire operating frequency of the proposed DPA.

B. Modulated Signal Measurement

Fig. 32. Normalized power spectrum density of the proposed DPA (1.75 GHz with an average output power of 32.2 dBm) with and without DPD.

To test the linearity of the proposed DPA, measurements were done using a WCDMA test signal with PAPR of 9.6 dB. Adjacent channel power ratio (ACPR) was measured with a channel integration bandwidth 3.84 MHz at 5 MHz offset using the Agilent CXA Signal Analyzer N9000A. As the ACPRs in both upper and lower bands are similar, only one of them (lower) is shown in Fig. 31. ACPR performance of the proposed DPA was better than −32 dBc at back off and −20 dBc at saturation from 1.6 to 1.9 GHz. The average efficiency is shown in Fig. 31. The average DE at 32.2 dBm (10-dB back-off point of CW measurement) is better than 46.5%, which is the same as results obtained with CW excitation. Average DE at 40 dBm (saturation) is better than 66%, which is similar to that obtained under CW excitation.

Fig. 32 shows the power spectral density of the proposed DPA at a center frequency of 1.75 GHz. The proposed DPA amplifier was linearized by using digital predistortion (DPD) technique based on the dynamic deviation reduction Volterra series [22]. In particular, the static nonlinearity order is set to 9, kernel order is setting to 3 with memory depth of 5, and dynamic nonlinearity order set to 1. For the test, the linearization is done at back-off output power (32.2 dBm). After linearization, the DPA gave better than −53.3 dBc at back off, which is an improvement of at least 21.3 dB. A comparison between the performance of the proposed DPA and recently published DPAs are summarized in Table II. The designed prototype exhibits a wider back-off range compared with most other structures. It can be observed that 46%

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 12

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

efficiency at 10-dB back off can be achieved over 1.6–1.9 GHz with the proposed DPA, which is higher than that in [10], [13], and [22]. Furthermore, the proposed DPA has the highest saturated efficiency. The proposed DPA also exhibits the best ACPR performance at 10-dB back-off no matter what DPD technique is applied. It is important to emphasize that due to the work in [11] and [12] which are strong candidates for back-off extension DPA, an additional comparison is necessary to demonstrate the significance of our work. In [11], the Doherty behavior is evident but at the expense of gain compression within the Doherty region. Compared with this paper, the proposed DPA has advantages of back off, PUF, saturated gain, and efficiency. In [12], at least 47% DE can be achieved at 9-dB back off over the 1.9- to 2.2-GHz frequency range. However, an ACPR performance of only −24.4 dBc could be achieved at the back-off point. Compared with this case, our proposed DPA has similar back-off DE (46%) but with at a wider back-off region (10 dB) and operating bandwidth (17.1%). Meanwhile, the obvious advantage is the ACPR performance at back off (−32 dBc) which can only be achieved in the proposed DPA. Since the work done in [23] also focuses on the secondharmonic components, it is selected as the reference for PUF comparison. Even though [23] uses the normal 6-dB OBO symmetrical DPA, PUF of the proposed ADPA still has a comparable performance. Compared with [4] and [11], the PUF of the proposed ADPA has an obvious improvement for both multiway and ADPA. Overall, the proposed DPA exhibits a good combined performance in terms of back-off range, PUF and ACPR performance no matter what DPD technique is used. V. C ONCLUSION In this paper, for the first time, the asymmetrical drainbiased postmatching DPA based on the self-generated harmonic injection is proposed. By integrating an HIN between the carrier and peaking branch, the self-generated second harmonic can be injected into the drain of the other device. Waveform amplitude modulation can be achieved in both carrier and peaking devices, which results in the enhancement of saturated power for both carrier and peaking devices, while the carrier back-off power remains unchanged. Consequently, the higher value of βd (power ratio between the carrier saturated output power and carrier back-off power) can be achieved while maintaining the same α (saturated power ratio between the peaking and carrier device). This results in an extended back-off range (10 dB), higher PUF, and higher saturated efficiency for the proposed DPA. To validate the theory, a GaN demonstration circuit with proposed HIN was designed. The measurements reported a DE larger than 46% at 10-dB back-off region, achieved in the frequency range from 1.6 to 1.9 GHz. Meanwhile, good linearity at back off is also achieved, the ACPR performance without and with DPD technique obtained is −32 and −53.3 dBc, respectively. With the help of a large back-off region and a good balance between efficiency and linearity,

the proposed DPA is a good candidate for high efficient wireless transmitters. R EFERENCES [1] W. H. Doherty, “A new high efficiency power amplifier for modulated waves,” Proc. IRE, vol. 24, no. 9, pp. 1163–1182, Sep. 1936. [2] M. Iwamoto, A. Williams, P.-F. Chen, A. G. Metzger, L. E. Larson, and P. M. Asbeck, “An extended Doherty amplifier with high efficiency over a wide power range,” IEEE Trans. Microw. Theory Techn., vol. 49, no. 12, pp. 2472–2479, Dec. 2001. [3] X. H. Fang and K.-K. M. Cheng, “Extension of high-efficiency range of Doherty amplifier by using complex combining load,” IEEE Trans. Microw. Theory Techn., vol. 62, no. 9, pp. 2038–2047, Sep. 2014. [4] H. Golestaneh, F. A. Malekzadeh, and S. Boumaiza, “An extendedbandwidth three-way doherty power amplifier,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 9, pp. 3318–3328, Sep. 2013. [5] Y. Yang, J. Cha, B. Shin, and B. Kim, “A fully matched N-way Doherty amplifier with optimized linearity,” IEEE Trans. Microw. Theory Techn., vol. 51, no. 3, pp. 986–993, Mar. 2003. [6] S. Chen, P. Qiao, G. Wang, Z. Cheng, and Q. Xue, “A broadband threedevice Doherty power amplifier based on a modified load modulation network,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, USA, May 2016, pp. 1–4. [7] S. C. Cripps, RF Power Amplifiers for Wireless Communication. Norwood, MA, USA: Artech House, 1999. [8] V. Camarchia, M. Pirola, R. Quaglia, S. Jee, Y. Cho, and B. Kim, “The Doherty power amplifier: Review of recent solutions and trends,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 2, pp. 559–571, Feb. 2015. [9] A. M. M. Mohamed, S. Boumaiza, and R. R. Mansour, “Doherty power amplifier with enhanced efficiency at extended operating average power levels,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 12, pp. 4179–4187, Dec. 2013. [10] J. Pang, S. He, Z. Dai, C. Huang, J. Peng, and F. You, “Design of a post-matching asymmetric Doherty power amplifier for broadband applications,” IEEE Microw. Wireless Compon. Lett., vol. 26, no. 1, pp. 52–54, Jan. 2016. [11] D. Gustafsson, C. M. Andersson, and C. Fager, “A modified Doherty power amplifier with extended bandwidth and reconfigurable efficiency,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 533–542, Jan. 2013. [12] M. Özen, K. Andersson, and C. Fager, “Symmetrical Doherty power amplifier with extended efficiency range,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 4, pp. 1273–1284, Apr. 2016. [13] J. Xia, X. Zhu, L. Zhang, J. Zhai, and Y. Sun, “High-efficiency GaN Doherty power amplifier for 100-MHz LTE-advanced application based on modified load modulation network,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 8, pp. 2911–2921, Aug. 2013. [14] A. AlMuhaisen, P. Wright, J. Lees, P. J. Tasker, S. C. Cripps, and J. Benedikt, “Novel wide band high-efficiency active harmonic injection power amplifier concept,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2010, pp. 664–667. [15] R. Quaglia, M. Pirola, and C. Ramella, “Offset lines in Doherty power amplifiers: Analytical demonstration and design,” IEEE Microw. Wireless Compon. Lett., vol. 23, no. 2, pp. 93–95, Feb. 2013. [16] T. Nojima and S. Nishiki, “High efficiency microwave harmonic reaction amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., New York, NY, USA, May 1988, pp. 1007–1010. [17] M. D. Roberg, “Analysis and design of non-linear amplifiers for efficient microwave transmitters,” Ph.D. dissertation, Dept. Elect. Eng., Colorado Univ., Denver, CO, USA, 2012. [18] J. Pang, S. He, C. Huang, Z. Dai, J. Peng, and F. You, “A post-matching Doherty power amplifier employing low-order impedance inverters for broadband applications,” IEEE Trans. Microw. Theory Techn., vol. 63, no. 12, pp. 4061–4071, Dec. 2015. [19] X. Chen, W. Chen, F. M. Ghannouchi, Z. Feng, and Y. Liu, “A broadband Doherty power amplifier based on continuous-mode technology,” IEEE Trans. Microw. Theory Techn., vol. 64, no. 12, pp. 4505–4517, Dec. 2016. [20] A. Alizadeh, M. Yaghoobi, and A. Medi, “Class-J2 power amplifiers,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 64, no. 8, pp. 1989–2002, Aug. 2017. [21] P. Colantonio, F. Giannini, R. Giofre, and L. Piazzon, “Theory and experimental results of a class F AB-C Doherty power amplifier,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 8, pp. 1936–1947, Aug. 2009.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ZHOU et al.: POSTMATCHING DPA

[22] A. Zhu, J. C. Pedro, and T. J. Brazil, “Dynamic deviation reductionbased Volterra behavioral modeling of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 12, pp. 4323–4332, Dec. 2006. [23] X. Y. Zhou, S. Y. Zheng, W. S. Chan, S. Chen, and D. Ho, “Broadband efficiency-enhanced mutually coupled harmonic postmatching Doherty power amplifier,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 64, no. 7, pp. 1758–1771, Jul. 2017.

Xin Yu Zhou (S’15) was born in Shandong, China. He received the M.Sc. degree in electronic engineering from the City University of Hong Kong, Hong Kong, in 2014, where he is currently pursuing the Ph.D. degree in electronic engineering. From 2014 to 2015, he was a Research Assistant with the SYSU-CMU Shunde International Joint Research Institute, Shunde, China. His current research interests include broadband, high efficient power amplifiers, and microwave passive circuits. Mr. Zhou was a recipient of the First Place Award of the High Efficiency Power Amplifier Student Design Competition and the IEEE Microwave Theory and Techniques Society International Microwave Symposium in 2017.

Shao Yong Zheng (S’07–M’11) was born in Fujian, China. He received the B.S. degree in electronic engineering from Xiamen University, Xiamen, Fujian, in 2003, and the M.Sc., M.Phil., and Ph.D. degrees in electronic engineering from the City University of Hong Kong, Hong Kong, in 2006, 2008, and 2011, respectively. From 2011 to 2012, he was a Research Fellow with the Department of Electronic Engineering, City University of Hong Kong. He is currently an Associate Professor with the Department of Electronics and Communication Engineering, Sun Yat-sen University, Guangzhou, China. His current research interests include microwave/millimeter-wave circuits and evolutionary algorithms.

13

Wing Shing Chan (M’94) received the B.Sc. (engineering) degree in electronic engineering from Queen Mary College, University of London, London, U.K., in 1982, and the Ph.D. degree from the City University of Hong Kong, Hong Kong, in 1995. From 1982 to 1984, he was an Engineer with the Solid-State Techniques Department, Plessey RADAR, London, as part of a team that produced the world’s first solid-state RADAR transmitter in s-band. From 1984 to 1988, he was a Senior Design Engineer of RF/microwave amplifiers with Microwave Engineering Designs Limited. In 1988, he joined the Department of Electronic Engineering, City University of Hong Kong, as a Lecturer, where he is currently an Associate Professor. Dr. Chan is a Chartered Engineer of the Engineering Council, U.K. He was the past Chairman of the IEEE AP/MTT Chapter, HK Section. He served as a member of the Radio Spectrum Advisory Committee in the Office of the Telecommunications Authority. Xiaohu Fang (S’12–M’16) received the B.Eng. and M.Eng. degrees in electronic science and technology from the Huazhong University of Science and Technology, Hubei, China, in 2008 and 2011, respectively, and the Ph.D. degree in electronic engineering from the Chinese University of Hong Kong, Hong Kong, in 2015. He is currently a Post-Doctoral Fellow with the Department of Electronic Engineering, City University of Hong Kong. His current research interests include the design of advanced high-efficiency power amplifiers and the associated linearization methods. Derek Ho (M’15) received the B.A.Sc. degree (first class) in computer engineering and M.A.Sc. degree in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 2005 and 2007, respectively, and the Ph.D. degree in electrical and computer engineering from the University of Toronto, Toronto, ON, Canada, in 2013. He has held several engineering internships with PMC-Sierra, Vancouver, BC, Canada, where he was involved in the development of communication integrated circuits. He is currently an Assistant Professor with the Department of Physics and Materials Science, City University of Hong Kong, Hong Kong. His current research interests include energyconstrained systems that incorporate nanostructured sensors and mixed analogdigital integrated circuits with an emphasis on chemical and biomedical sensing applications.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

Broadband Nonreciprocal Phase Shifter Design Technique Mirko Palomba , Diego Palombini, Sergio Colangeli , Walter Ciccognani , and Ernesto Limiti, Member, IEEE

Abstract— In this contribution, a novel method to design broadband nonreciprocal phase shifters (NRPS) is illustrated: this new topology is based on the combination of two directivedistributed amplifiers with a four-port phase-shifting network. The presented approach includes explicit design formulae allowing for arbitrary differential-phase implementation. Extensive design description of a 180° NRPS, acting as broadband differential-phase inverter, is also provided. The 3.5 × 3.4 mm2 demonstrator, realized in 0.5 µm GaN HEMT technology, has an average zero insertion loss up to 25 dBm of input power, 180° ± 2° differential-phase shift, and good port matching over the 3–7-GHz operating bandwidth. Index Terms— Broadband, circulator, distributed amplifier, GaN, microwave monolithic integrated circuit (MMIC), phase shifter.

I. I NTRODUCTION

G

ALLIUM arsenide has been historically used for severalintegrated circuit (IC) applications in the microwave and millimeter-wave domain. Examples include high-power and low-noise amplifiers, oscillators and mixers, and signalrouting and signal-conditioning circuits; recently, GaAs logic families for mixed-signal microwave monolithic IC (MMIC) have presented too. The maturity of this technology process permits to get reliable and precise models, which provide good agreement between simulations and measurements. Thanks to its versatility GaAs represents, in most cases, the best choice for a high-volume and cost-effective MMICs production; however, when extreme robustness or high linearity is requested, different technologies have to be selected. GaN technology, first appeared in the 90s, soon exhibited its superiority in terms of high-power handling [1] to the high electronic bandgap, new system capabilities were achieved in a number of different fields such as RADAR, measurements instrumentation, space communications, and electronic warfare. Nevertheless, the growing system complexity for the Manuscript received May 17, 2017; revised October 18, 2017; accepted November 29, 2017. (Corresponding author: Mirko Palomba.) M. Palomba was with the Electronics Engineering Department, University of Roma Tor Vergata, 00133 Rome, Italy. He is now with Leonardo S.p.A., 00156 Rome, Italy (e-mail: [email protected]). D. Palombini was with the Electronics Engineering Department, University of Roma Tor Vergata, 00133 Rome, Italy. He is now with Elettronica S.p.A., 1329, 00131 Rome, Italy (e-mail: [email protected]). S. Colangeli, W. Ciccognani, and E. Limiti are with the Electronics Engineering Department, University of Roma Tor Vergata, 00133 Rome, Italy. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2017.2785816

above-mentioned applications [2], [3], a challenging roadmap for GaN technology evolution. In order to save cost and area occupation, IC downsizing is becoming more and more crucial: the introduction of multifunctional ICs, capable of implementing different functionalities, allows reducing chipset complexity and minimizing unwanted connections and hybrid circuitry. Another aspect, following the integration, is the best agreement obtained between simulations and measurements: the design of a single MMIC makes the modeling phase an easier task compared to the case in which different chips and technologies have to be interconnected. For these reasons, the scientific world is striving to integrate different functionalities into a single chip. There are several examples of GaN MMICs implementing high-power amplifier, low-noise amplifier (LNA), and switch circuitry as standalone entities [4]–[14], and there are also some examples of their integration [15]–[17]. Unfortunately, the lack in signal routing and conditioning circuitry realized with this technology has prevented for a long time a higher grade of integration: GaAs, CMOS, and other cheaper technologies cannot be easily interfaced with GaN circuits as a consequence of the inherent mismatch in power levels. Such low-bandgap technologies cannot handle signals coming from GaN ICs because breakdown voltages, or maximum current limits are easily reached. Therefore, the realization of signal-conditioning circuits in GaN technology could represent the real breakthrough toward system integration and costs reduction. Among all the signal-routing circuits, a particularly useful functionality consists in circulators, especially for those systems in which a full-duplex link is necessary. Circulators are used essentially in communication, RADAR, and measurement systems. The first—and probably still the most used— type of circulators was based on passive structures exploiting ferrite materials. This approach leads to cumbersome, scantly integrable, narrowband devices, and requiring an ofteninconvenient magnetic bias [18]–[23]. A very interesting improvement could be represented by active circulators. In this case, there are a dc power consumption and a lower power handling as drawbacks, compared to the ferrite-based solution, but a better performance in terms of integration level and frequency bandwidth is expected thanks to intrinsic technology advantages (it is possible to use the same technology for different functionalities) and dedicated design techniques. A former active, wideband approach was based on a distributed structure exhibiting a wideband behavior (6/18 GHz) but a very low isolation between ports, key performance for

0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

front-end circuits [24]. A wider bandwidth circuit was proposed in [25], but also in this case the insertion loss is critical, especially if the circulator is placed before an LNA. Another distributed approach making use of distributed amplifiers connected in a ring topology exhibits a 40% bandwidth [26] with an improved insertion loss. By the connection of an active splitter and an active combiner, it is possible to implement a two-branch active quasi-circulator [27]–[29] in which the circulation of signals is guaranteed by a phase-cancellation operation. Such an operation does not exceed the 12% operating bandwidth. Another phase-cancellation approach is reported in [30], leading to an even narrower bandwidth. A 20% operating bandwidth is possible by the combination of the Wilkinson power dividers and amplifiers [31]. A 3:1 bandwidth is achievable by using slow wave directional coupler and wideband amplifiers, as reported in [32]. Another interesting architecture makes use of the combination of particular two-port subcircuits able to vary differently the phase of signals flowing in different directions through them [33], [34]. Such special circuits are called nonreciprocal phase shifters (NRPSs). By means of NRPSs, it is possible to build a three-port architecture in which, by properly summing or subtracting signals at the ports, the circulation functionality can be obtained. The reported circulators do not exceed bandwidths of 10%/15%, but this is strictly correlated with the bandwidth of the NRPSs. From this point of view, an effort was spent to widen the operating bandwidth by proposing a new NRPS architecture based on Wilkinson power dividers, classical phase shifters and amplifiers (which introduce the nonreciprocal behavior thanks to the unilaterality of active devices), achieving a 28% bandwidth [35], [36]. Such NRPS topology provided the basis for a novel circulator architecture, achieving a measured 26% operating bandwidth in hybrid technology for L-band applications; even better results were obtained in the X-band with the design of a GaAs MMIC demonstrator [37], [38], which exhibited a simulated 36% band. With the aim of extending the operating bandwidth of the NRPSs, a new active topology [39] has been proposed, based on a distributed amplification concept. In this contribution, a complete theoretical analysis is provided together with a step-by-step circuit design flow description. The present contribution is structured as follows. In Section II, the circuit architecture is introduced. In Sections III and IV, the detailed circuit analysis and test circuit design are, respectively, presented. In Section V, the experimental results are analyzed and compared with simulations. Finally, Section VI gives the main conclusion. II. C IRCUIT A RCHITECTURE Ideally, an NRPS is an active two-port network able to produce two different phase delays depending on the direction of the incoming signal. The difference between these delays, namely, the phase shift, should be kept as constant as possible on a wide frequency range to enable broadband circuit operation. Furthermore, the NRPS should not affect signal amplitude, providing negligible insertion loss in both signal directions. Such a complex analog operation can only

Fig. 1.

Schematic representation of a quasi-circulator.

Fig. 2.

NRPS block diagram.

be achieved by partitioning the process in different elementary operations which, in turn, can be addressed separately: in more detail, signal routing and signal delay functionalities are implemented by means of a quasi-circulating three-port network and a four-port filtering network, respectively. A quasi-circulator is a three-port network that allows the signals to flow only from port A toward port B and from port C toward port A, as schematically depicted in Fig. 1. On the other hand, the four-port filtering network consists of two separated paths, each featuring a different signal delay: obviously, by properly choosing such delays, different phaseshifting performances will be achieved. NRPS operating principle can be easily understood by referring to the simplified block diagram proposed in Fig. 2: the signal at port 1 flows along the allowed path B-C, passing through the filtering (phase shifting) section φ1 and the allowed path A–B, and finally emerges at port 2; on the other hand, a signal at port 2 flows along the allowed path B-C, passing through the filtering (phase shifting) section φ2 and the allowed path A-B, and finally emerges at port 1. A signal flowing in the structure therefore undergoes different phase delays, depending on its direction: this phase difference, namely, differential-phase shift, is kept as constant as possible over the entire operating bandwidth. In principle, every differential-phase shift can be implemented by the proper choice of the filtering section: depending on this value, different circulators or isolator architectures are possible (see Fig. 3). In more detail, connecting three NRPSs (featuring 60° each) in a delta topology, a broadband circulator is possible [33]; on the other hand, by connecting a single NRPS (featuring 180°) with a 0° delay line or two 90° delay lines, a broadband isolator or circulator are, respectively, achieved [40]; again, using a more complicated topology with two NRPSs (featuring 90° each), a 90° coupler and a 180° coupler, a broadband circulator featured by two isolated ports

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PALOMBA et al.: BROADBAND NRPS DESIGN TECHNIQUE

3

Fig. 3. Circulator and isolators achievable by means of (a) 60°, (b) 180°, and (c) 90° NRPSs.

Fig. 4.

Distributed amplifier as a quasi-circulator.

can be realized [41]. Furthermore, implementing a 180° NRPS, would theoretically enable the realization of a broadband microwave gyrator, providing that the insertion phase is at the same time minimized. III. D ESIGN M ETHOD As already stated, NRPS implementation can be successfully accomplished by addressing separately the signal routing and signal delay functionalities. In the following, two distinct sections are, respectively, dedicated to the directive-distributed amplifier and to the phase-shifting section designs. A. Directive-Distributed Amplifier Distributed amplifier represents a fairly old and well-known concept and is usually employed in a variety of applications where ultrabroadband performance represents a fundamental prerogative. Despite its inherent versatility, due to the presence of four ports, this topology is typically used as a singleinput/-single-output network except for very rare applications like meta-distributed amplifiers [42] or quasi-circulating structures [43], [44]; in the latter case, the distributed amplifier actually behaves as an active three-port network, as shown in Fig. 4. Using the same port reference of Fig. 1, the related scattering parameter matrix should be the following: ⎞ ⎛ SAA SAB SAC ⎟ ⎜ S = ⎝ SBA SBB SBC ⎠ SCA SCB SCC ⎛ ⎞ 0 0 0 ⎜ ⎟ = ⎝ G fwd 0 1 ⎠ ⎛

G rev

1

0

0

⎜ = ⎝1 0

0 0 1

0 ⎞

1⎟ ⎠. 0

(1)

Fig. 5.

Equivalent circuit of a symmetrically tapered distributed amplifier.

Looking at the matrix in (1), Sii parameters (on the main diagonal) are 0 if the circuit shows perfect matching at the corresponding port; furthermore, SAB and SAC terms can be approximated to 0 as an effect of the intrinsic nonreciprocity of the active devices. Incidentally, it is worth pointing out that the term SBC (ideally 0 in a quasi-circulator) maintains the same value of SCB , because of the amplifier drain line’s reciprocity; however, this is not a problem considering that C port is inaccessible in the presented topology. On the other hand, SAB and SAC terms deserve a separate discussion since they must be carefully optimized by design: in particular, by tapering transconductances of the active devices, reverse gain G rev can be minimized over the entire operating bandwidth while maintaining a forward gain G fwd near to unity. Referring to the equivalent model of an N-cell symmetrically tapered distributed amplifier (see Fig. 5), the following expressions for the forward and reverse gains can be, respectively, obtained:  2 M     G fwd =  Z 0 gmi  (2)   i=0  2 M     G rev =  Z 0 gmi cos[(N − 2i)θ (ω)] = |Z 0 F(ω)|2 (3)   i=0

where M = N/2 for N even and M = (N − 1)/2 for N odd. Directivity D(ω) can be defined as the ratio between the above-mentioned gains: note that the expression in (2) is completely frequency independent, whereas array factor F(ω) in (3) depends on frequency by means of the electrical length θ (ω). Controlling the array factor permits to obtain the desired directivity 2  

 M  F(0) 2 gmi  G fwd  i=0  . = (4) D(ω) =  =   F(ω)  G rev F(ω)  In particular, the maximum product between directivity and bandwidth can be achieved only if the array factor is designed to be a Chebyschev polynomial Tn (x). Following the analysis in [18], an n-order polynomial has to be modified by a proper variable substitution which allows to restrict the ±1 ripple within the range [θm , π − θm ]: cos(θ (ω)) (5) x = cos(θm )    cos θ cos θ cos θ cos θ =2 − Tn−2 . Tn Tn−1 cos θm cos θm cos θm cos θm (6)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

Fig. 6.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

Fig. 7.

Phase-shifting section schematic (APN/BPN).

Fig. 8.

Port reflection components.

Directive-distributed amplifier design chart.

The parameter θm is the electrical length at the beginning of the operating band and is related to the achievable fractional bandwidth by the following expression: π θ2 − θ1 1 π − B% , where B% = 2 . (7) θm = 2 2 θ2 + θ1 Now the array factor can be reformulated as     cos θ  . F(ω) = K Tn cos θm 

(8)

Considering last equality in (4), the K factor should be determined by imposing the desired value of forward gain   −1  G fwd  1  . T K = (9) n  Z0 cos θm  Finally, the directivity can be rewritten as follows:     Tn cos1θm  Dm  =   D(ω) =     cos θ cos θ  Tn cos Tn cos θm  θm 

(10)

where Dm is the minimum directivity, i.e., the directivity at extreme electrical lengths (frequencies) θm and π − θm when the polynomial Tn approaches unity. A useful design chart can be constructed by plotting, on the same graph, the minimum directivity (in log units) as a function of the parameter θm for different cell numbers (different polynomial orders). As can be noticed from Fig. 6, fixing a number of cells and decreasing parameter θm (i.e., increasing fractional bandwidth B%), lower values of minimum directivity are possible; on the other hand, fixing the parameter θm (i.e., fixing fractional bandwidth B%) and increasing the number of cells, higher values of minimum directivity can be achieved. B. Phase-Shifting Section The proper choice of this delay element allows for any value of differential-phase shift: in principle, any switched network topology can be exploited and rearranged to work as a four-port network featured by two isolated paths. Even limiting the attention to 180° differential-phase shift, as the most difficult to be implemented on a wide band, several

distributed and lumped realization can be found: in [45], a distributed implementation is illustrated featuring pretty good phase shift and magnitude unbalance but implying an excessive amount of area occupation; in [46] and [47], two different lumped implementations are proposed, respectively, based on all-pass/all-pass network (APN/APN) and all-pass/bandpass network (APN/BPN). The latter solution (APN/BPN), reported in Fig. 7, seems to guarantee a significantly better level of phase deviation in the bandwidth together with a circuit topology more suitable for active device biasing purposes. Circuit elements can be easily determined by means of the following formulas (where ω0 is the center frequency and r is a dimensionless parameter setting the frequency slope of the passband response): Z0 1 C1 = ω0 ω0 r Z 0 Z0 1 L2 = C2 = . ω0 ω0 Z 0 L1 = r

(11)

IV. T EST C IRCUIT A monolithic test circuit has been designed to verify the feasibility of the proposed topology and to evaluate its performance. Design starts fixing the number of distributed amplifier cells and associated directivity: in an ideal case of lossless and perfectly matched structure, G fwd should be designed to be unity, while G rev is numerically equal to the reflection coefficient  at both ports (see Fig. 8). Unfortunately, drain line loss Ad imposes a forward gain greater than unity which in turn worsen port reflections by the same factor, since the minimum directivity has already been chosen (see again Fig. 8). Drain line loss Ad mainly depends on drain–source admittance G dsi of the active devices, through the device size Wi , and the normalized drain–source admittance G dsw. Simplifying the analysis [48] in the low-frequency approximation

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PALOMBA et al.: BROADBAND NRPS DESIGN TECHNIQUE

5

TABLE I C IRCUIT E LEMENTS OF P HASE -S HIFTING S ECTION

Fig. 9.

Schematic of the elementary cell.

(far below line cutoff frequency), drain line loss expression becomes Ad =

N Z0 Z0 WTOT G dsw. Wi G dsw = 2 2

(12)

i=0

The choice of the actual periphery should be carried on the basis of the desired output power level (PTOT ), having device bias condition and relative output power density (PW ) already been determined WTOT

PTOT = . PW

(13)

Once the desired return losses and drain line insertion loss have been defined, a simple linear system allows setting the correct value for the forward and reverse gain (minimum directivity) ⎧ ⎨|dB = G fwd |dB + G rev |dB − Ad |dB (14) ⎩G fwd |dB − Ad |dB = 0. Solving the linear set of equations (14), the required value of minimum directivity can be determined Dm |dB = |dB + Ad |dB .

(15)

The broadband NRPS has been designed using the 0.5-μm GaN/SiC HEMT process developed by SELEX ES (now Leonardo S.p.A.), featuring IDSS = 600 mA/mm at VDD = 25 V. The target design covers more than one octave, i.e., 3-7 GHz, with a saturated output power (PTOT ) of at least 30 dBm. Analyzing the device bias condition (VDD = 25 V and I D = 50% IDSS ) lets the output power density and the normalized drain–source admittance to be assessed: considering 15% of drain efficiency (realistic for a distributed topology), a PW of 1.12 W/mm is inferred; on the other hand, S-parameter measurements of FET devices suggest 13 mS/mm to be a consistent value for G dsw . From (13), WTOT results to be around 900 μm which, substituted into (12) gives an Ad approximately equal to −3 dB. Considering expression in (15) and imposing a return loss of 12 dB, a Dm of 15 dB is mandatory. Looking at the design chart in Fig. 6 and applying (7), minimum directivity should be achieved for θm /π = 0.3: desired value can be met by means of three cells. Total active periphery can be thus partitioned into three active devices

featured by 4 × 75 μm periphery each. Applying a secondorder polynomial, as reported in (6), the following expressions for the normalized transconductances can be derived:   2 2 1 1 g¯ m1 = − 1. (16) g¯ m0 = g¯ m2 = cos(θm ) cos(θm ) Furthermore, substituting the second of (14) into (9) allows calculating K coefficient which, substituted in (8), returns the array factor (i.e., actual transconductance values) F(ω) = 16 + 24 cos(2θ )

(17)

where gm0 = gm2 = 24 mS and gm1 = 16 mS. Now, these values have to be implemented by inserting a proper capacitance in series to the FET gate, whose expression, being gmw the normalized transconductance and Cgsi the i th intrinsic gate capacitance, would be Csi =

G dsw Cgsi . gmw − G dsw

(18)

Another capacitance should be placed in parallel to the drain node, in order to equalize gate and drain lines phase velocity, according to the following expression: Caddi =

G gsi G si − Cdsi . G gsi + G si

(19)

Finally, for amplifier proper operation, it is mandatory to set 90° as the electrical length of the elementary cell (see Fig. 9) at center frequency, that is, the maximum phase shift achievable by a single T-section made up of lumped elements [49]. To improve amplifier frequency response, three elementary cells, made up by an active T-section (including the active device) and a dummy T-section (including capacitance), were employed. Last step in MMIC design is the definition of the APN/BPN phase-shifting section: applying formulae in (11) for f 0 = 5 GHz and r = 1.45 (for best broadband performance [47]), the values of circuit elements are easily obtained (see Table I). The photograph of the designed broadband NRPS is presented in Fig. 10. Chip size is 3.5 × 3.4 mm2 . As effect of gmi modulation, achieved by series capacitors insertion, the use of bypass resistances becomes necessary for biasing FETs. According to (19), drain additive capacitances Caddi resulted to be so small that they were omitted, and phase velocities were equalized by simply tuning the inductor values. Looking at Fig. 10, port 1 and port 2 are visible in the upper left and right borders, respectively. Bias pads are placed on the upper and lower sides of the MMIC. The four-port phase-shifting network can be seen between the two distributed

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

TABLE II C OMPARISON OF R EPORTED NRPSs

Fig. 10.

Fig. 11.

Port matching.

Fig. 12.

Insertion gain.

Photograph of the fabricated MMIC.

amplifiers: the APN and the BPF are on the upper and lower sides, respectively. The classical APN structure was modified to bias the drain of the left distributed amplifier; the BPF circuit topology on the contrary was exploited in order to bias the gate of the left distributed amplifier and the drain of the other one. The remaining gate of the right distributed amplifier was biased by the termination resistor directly connected to the bias pad. V. E XPERIMENTAL R ESULTS The presented MMIC was successfully tested by means of linear S-parameter and gain compression measurements, proving the effectiveness of the approach. As an effect of the design algorithm proposed in Section IV, both port matchings are below the desired level of 12 dB, except for a small portion of the band between 4 and 5 GHz (see Fig. 11); furthermore, a pretty good accordance between simulation and measurements can be noted. The same accordance between simulated and experimental data can be appreciated about the insertion gain, roughly equal to 0 dB all over the operating band (see Fig. 12).

The most important figure of merit, i.e., the differentialphase shift between the two ways, is reported in Fig. 13. The proposed topology provides an almost perfect phase inversion between signals traveling in opposite directions: 180° differential-phase shift with negligible errors as small as 2°, except for the really low portion of the band, again in good accordance with predictions. Finally, large-signal measurements of insertion gain show a Pout,1 dBc around 25 dBm, with a saturated output power in line with the required level of 30 dBm (see Fig. 14). Unfortunately, the absence of a nonlinear model for the concerned FET periphery does not allow for an accurate measurement versus simulation comparison. A comparative table illustrates the differences among the present work and other NRPS implementation both as stand-alone topologies or as a part of isolator/circulator circuits

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PALOMBA et al.: BROADBAND NRPS DESIGN TECHNIQUE

7

R EFERENCES

Fig. 13.

Differential-phase shift.

Fig. 14.

Power and gain compression at center frequency (5 GHz).

(see Table II). Different implementations, herein considered, span across several technological domains from MMIC/HMIC to ferrite circuits. As can be noticed, this paper exhibits the best tradeoff between bandwidth and insertion loss; powerhandling capability is definitely lower with respect to the ferromagnetic solutions which, in turn, are less suitable for integration and need to be provided with cumbersome magnetic bias circuitry. VI. C ONCLUSION A novel technique for the realization of broadband NRPSs has been proposed: the design approach is based on the integration of directive-distributed amplifiers, employed as quasi-circulators, and a four-port filtering network as a signal delay element. A complete description of the circuit synthesis algorithm has been provided and accompanied with explicit design formulae and a design chart. A monolithic test circuit in the 3–7-GHz band has been designed and measured to validate the effectiveness of the method: a 180° NRPS, acting as broadband differential-phase inverter, has been selected as the most challenging phase shift value to be implemented. The realized GaN MMIC features a quite compact layout (3.5 × 3.4 mm2 ) and achieves the target 180° with a typical error of ±2°, showing an insertion loss of about 1 dB, a port matching better than 10 dB, and a saturated output power near to 1 W.

[1] D. W. Runton, B. Trabert, J. B. Shealy, and R. Vetury, “History of GaN: High-power RF gallium nitride (GaN) from infancy to manufacturable process and beyond,” IEEE Microw. Mag., vol. 14, no. 3, pp. 82–93, May 2013. [2] N. J. Kolias and M. T. Borkowski, “The development of T/R modules for radar applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Montreal, QC, Canada, Jun. 2012, pp. 1–3. [3] A. Bentini, W. Ciccognani, M. Palomba, D. Palombini, and E. Limiti, “High-density mixed signal RF front-end electronics for T-R modules,” in Proc. IEEE-AESS Eur. Conf. Satellite Telecommun., Rome, Italy, Oct. 2012, pp. 1–6. [4] P. Colantonio et al., “A C-band high-efficiency second-harmonic-tuned hybrid power amplifier in GaN technology,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 6, pp. 2713–2722, Jun. 2006. [5] R. Giofrè, L. Piazzon, P. Colantonio, E. Cipriani, and F. Giannini, “GaN power amplifiers: Results and prospective,” Int. J. Microw. Opt. Technol., vol. 9, no. 1, pp. 13–19, 2014. [6] E. Cipriani, P. Colantonio, and F. Giannini, “The effect of 2nd harmonic control on power amplifiers performances,” in Proc. Eur. Microw. Integr. Circuit Conf., Amsterdam, The Netherlands, Oct. 2012, p. 345 348. [7] S. Miwa et al., “A 67% PAE, 100 W GaN power amplifier with on-chip harmonic tuning circuits for C-band space applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Baltimore, MD, USA, Jun. 2011, pp. 1–4. [8] P. Colantonio, F. Giannini, R. Giofrè, and L. Piazzon, “Evaluation of GaN technology in power amplifier design,” Microw. Opt. Technol. Lett., vol. 51, no. 1, pp. 42–44, Jan. 2009. [9] H. Xu, C. Sanabria, A. Chini, S. Keller, U. K. Mishra, and R. A. York, “A C-band high-dynamic range GaN HEMT low-noise amplifier,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 262–264, Jun. 2004. [10] K. W. Kobayashi et al., “A cool, sub-0.2 dB noise figure GaN HEMT power amplifier with 2-watt output power,” IEEE J. Solid-State Circuits, vol. 44, no. 10, pp. 2648–2654, Oct. 2009. [11] W. Ciccognani, S. Colangeli, E. Limiti, and P. Longhi, “Noise measurebased design methodology for simultaneously matched multi-stage lownoise amplifiers,” IET Circuits, Devices Syst., vol. 6, no. 1, pp. 63–70, Jan. 2012. [12] A. Barigelli et al., “Development of GaN based MMIC for next generation X-Band space SAR T/R module,” in Proc. Eur. Microw. Integr. Circuit Conf., Amsterdam, The Netherland, Oct. 2012, pp. 369–372. [13] B. Y. Ma, K. S. Boutros, J. B. Hacker, and G. Nagy, “High power AlGaN/GaN Ku-band MMIC SPDT switch and design consideration,” in IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA, USA, Jun. 2008, pp. 1473–1476. [14] A. Bettidi et al., “High power GaN-HEMT SPDT switches for microwave applications,” Int. J. RF Microw. Comput.-Aided Eng., vol. 19, no. 5, pp. 598–606, Jun. 2009. [15] E. Limiti et al., “T/R modules front-end integration in GaN technology,” in Proc. IEEE Wireless Microw. Technol. Conf., Cocoa Beach, FL, USA, Apr. 2015, pp. 1–6. [16] P. Schuh et al., “GaN MMIC based T/R-module front-end for X-band applications,” in Proc. Eur. Microw. Integr. Circuit Conf., Amsterdam, The Netherland, Oct. 2008, pp. 274–277. [17] D. Palombini, A. Bentini, and D. Rampazzo, “A 6–18 GHz integrated HPA-DPDT GaN MMIC,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, USA, May 2016, pp. 1–4. [18] R. E. Collin, Foundation For Microwave Engineering (Series on Electromagnetic Wave Theory), 2nd ed. Piscataway, NJ, USA: IEEE Press, 2000. [19] A. M. T. Abuelma’atti, A. A. P. Gibson, and B. M. Dillon, “Analysis of a 10 kW-2.45 GHz ferrite phase shifter,” in Proc. Eur. Radar Conf., Manchester, U.K., Sep. 2006, pp. 261–264. [20] “Circulators and isolators, unique passive devices,” Philips Semiconductors, Appl. Note AN98035, 1998. [21] S. D. Ewing and J. A. Weiss, “Ring circulator theory, design, and performance,” IEEE Trans. Microw. Theory Techn., vol. MTT-15, no. 11, pp. 623–628, Nov. 1967. [22] J. A. Weiss, “Circulator synthesis,” IEEE Trans. Microw. Theory Techn., vol. MTT-13, no. 1, pp. 38–44, Jan. 1965. [23] A. E. Booth, “Dual balanced reciprocal waveguide phase shifter,” U.S. Patent 3 833 868, Sep. 3, 1974. [24] P. Katzin, Y. Ayasli, L. D. Reynolds, Jr., and B. E. Bedard, “6 to 18 GHz MMIC circulators,” Microw. J., vol. 35, no. 5, pp. 248–256, 1992.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

[25] I. D. Robertson and A. H. Aghvami, “Novel monolithic ultra-wideband unilateral 4-port junction using distributed amplification techniques,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 2. Albuquerque, NM, USA, Jun. 1992, pp. 1051–1054. [26] D. Palombini, M. Palomba, W. Ciccognani, and E. Limiti, “Novel active distributed circulator,” Int. J. Microw. Opt. Technol., vol. 9, no. 1, pp. 44–48, 2014. [27] D. Kother, B. Hopf, T. Sporkmann, I. Wolff, and S. Kosslowski, “New types of MMIC circulators,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 2. Orlando, FL, USA, May 1995, pp. 877–880. [28] A. Moussa, Y. Khalaf, and A. Khalil, “Design of a novel circulator for wireless transceivers,” in Proc. Int. Conf. Microelectron., Cairo, Egypt, Dec. 2007, pp. 267–270. [29] A. Gasmi, B. Huyart, E. Bergeault, and L. Jallet, “Noise and power optimization of a MMIC quasi-circulator,” IEEE Trans. Microw. Theory Techn., vol. 45, no. 9, pp. 1572–1577, Sep. 1997. [30] M. J. Cryan and P. S. Hall, “An integrated active circulator antenna,” IEEE Microw. Guided Wave Lett., vol. 7, no. 7, pp. 190–191, Jul. 1997. [31] G. E. Stratakos, “CAD design of an active MMIC circulator at 21–26 GHz frequency band,” in Proc. Gallium Arsenide Appl. Symp., 1998, pp. 355–360. [32] C. E. Saavedra and Y. Zheng, “Active quasi-circulator realisation with gain elements and slow-wave couplers,” IET Microw., Antennas Propag., vol. 1, no. 5, pp. 1020–1023, Oct. 2007. [33] Y. Ayasli, “Field effect transistor circulators,” IEEE Trans. Magn., vol. 25, no. 5, pp. 3242–3247, Sep. 1989. [34] A. Ohlsson, V. Gonzalez-Posadas, and D. Segovia-Vargas, “Active integrated circulating antenna based on non-reciprocal active phase shifters,” in Proc. Eur. Conf. Antennas Propag., Edinburgh, U.K., Nov. 2007, pp. 1–5. [35] M. Palomba, W. Ciccognani, E. Limiti, and L. Scucchia, “An active nonreciprocal phase shifter topology,” Microw. Opt. Technol. Lett., vol. 54, no. 7, pp. 1659–1661, Apr. 2012. [36] M. Palomba, A. Bentini, M. Ferrari, E. Limiti, and D. Palombini, “Novel active non-reciprocal phase shifter,” in Proc. Int. Symp. Microw. Opt. Technol., Prague, Czech Republic, 2011, pp. 381–384. [37] M. Palomba, A. Bentini, D. Palombini, W. Ciccognani, and E. Limiti, “A novel hybrid active quasi-circulator for L-band applications,” in Proc. Int. Conf. Microw., Radar Wireless Commun., Warsaw, Poland, May 2012, pp. 41–44. [38] M. Palomba and E. Limiti, “Microwave signal conditioning through nonreciprocal phase shifting,” IET Microw., Antennas Propag., vol. 7, no. 10, pp. 809–818, Jul. 16 2013. [39] D. Palombini, M. Palomba, S. Colangeli, and E. Limiti, “Novel broadband nonreciprocal 180° phase-shifter,” in Proc. Int. Conf. Microw., Radar Wireless Commun., Warsaw, Poland, May 2012, pp. 31–34. [40] I. J. Bahl, “The design of a 6-port active circulator,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 2. New York, NY, USA, May 1988, pp. 1011–1014. [41] Y. Naito, M. Iwakuni, K. Araki, and A. Ikeda, “A new-type of electronic circulator at 800 MHz band,” in Proc. Eur. Microw. Conf., Warsaw, Poland, Sep. 1980, pp. 502–506. [42] J. Mata-Contreras, D. Palombini, T. M. Martín-Guerrero, A. Bentini, C. Camacho-Peñalosa, and E. Limiti, “Active GaN MMIC diplexer based on distributed amplification concept,” Microw. Opt. Technol. Lett., vol. 55, pp. 1041–1045, May 2013. [43] S. N. Prasad and Z. M. Li, “Optimal design of low crosstalk, wideband, bidirectional distributed amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 2. San Francisco, CA, USA, Jun. 1996, pp. 847–850. [44] O. P. Leisten, R. J. Collier, and R. N. Bates, “Distributed amplifiers as duplexer/low crosstalk bidirectional elements in S-band,” Electron. Lett., vol. 24, no. 5, pp. 264–265, Mar. 1988. [45] S.-Y. Eom, “Broadband 180° bit phase shifter using a λ/2 coupled line and parallel λ/8 stubs,” IEEE Microw. Compon. Lett., vol. 14, no. 5, pp. 228–230, May 2004. [46] D. Adler and R. Popovich, “Broadband switched-bit phase shifter using all-pass networks,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 1. Boston, MA, USA, Jul. 1991, pp. 265–268. [47] X. Tang and K. Mouthaan, “Design considerations for octave-band phase shifters using discrete components,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 12, pp. 3459–3466, Dec. 2010. [48] J. B. Beyer, S. N. Prasad, R. C. Becker, J. E. Nordman, and G. K. Hohenwarter, “MESFET distributed amplifier design guidelines,” IEEE Trans. Microw. Theory Techn., vol. MTT-32, no. 3, pp. 268–275, Mar. 1984.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

[49] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks and Coupling Structures, 2nd ed. Norwood, MA, USA: Artech House, 1980. [50] G. Hamoir, J. D. L. T. Medina, L. Piraux, and I. Huynen, “Self-biased nonreciprocal microstrip phase shifter on magnetic nanowired substrate suitable for gyrator applications,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 7, pp. 2152–2157, Jul. 2012. [51] S. Adhikari, A. Ghiotto, S. Hemour, and K. Wu, “Tunable non-reciprocal ferrite loaded SIW phase shifter,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, USA, Jun. 2013, pp. 1–3.

Mirko Palomba received the M.S. degree in electronic engineering and Ph.D. degree in telecommunication and microelectronics engineering from the University of Roma Tor Vergata, Rome, Italy, in 2010 and 2014, respectively. He was a Researcher at a Consortium born within the University of Roma Tor Vergata until 2017, when he has joined Leonardo S.p.A., Rome. He has treated nonlinear topics concerning circuitry useful to frequency conversion and high-performing circuits for new-generation front-end circuits, especially those involving GaN technology. He was involved in national and European projects as a Designer responsible for programs, working for known space agencies (European Space Agency, Italian Space Agency), aerospace companies (Thales Alenia Space, Rheinmetall, Space Engineering), and European foundries (Leonardo, UMS, OMMIC). He has authored or co-authored over 40 scientific publications. His current research interests include microwave/millimeter-wave circuit design, linear circuits, in particular those circuits used for signal conditioning functionalities useful to multifunction chips, core chips, and transmit–receive modules.

Diego Palombini was born in Rome, Italy, in 1983. He received the M.S. degree in electronic engineering and Ph.D. degree in telecommunication engineering and microelectronics from the University of Roma Tor Vergata, Rome, in 2008 and 2012, respectively. From 2012 to 2014, he was a Microwave Engineer with Wave Advanced Technology Application s.r.l, Rome. Since 2014, he has been a Microwave Engineer with the Microwaves and Antenna Department, Elettronica S.p.A, Rome, where he is involved in the design of advanced GaN MMICs for EW applications. He has authored or co-authored over 30 scientific publications, mainly in the field of ultrawideband microwave circuits development, with particular interest in distributed and meta-distributed structures for signal amplification, conditioning, and routing applications. His current research interests include wideband power amplifiers, high linearity switches, and microwave mixers.

Sergio Colangeli received the M.S. degree in electronic engineering and Ph.D. degree in telecommunications and microelectronics from the University of Roma Tor Vergata, Rome, Italy, in 2008 and 2013, respectively. He is currently a Researcher with the University of Roma Tor Vergata. He has designed a number of low-noise amplifiers in different commercial GaAs and GaN HEMT technologies from C- to Ka-band. He is familiar with typical characterization techniques, in particular S-parameters and Y-factor, but also with less common ones, such as cold-source noise measurements, especially as applied to noise source–pull. He has carried out several measurement campaigns at cryogenic temperatures. He has authored or co-authored over 50 scientific papers. His current research interests include low-noise design methodologies for microwave applications, low-noise amplification and smallsignal and noise characterization, modeling of microwave active devices, and stability analysis of microwave circuits, mainly at small signal.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PALOMBA et al.: BROADBAND NRPS DESIGN TECHNIQUE

Walter Ciccognani received the degree in electronic engineering and Ph.D. degree in telecommunications and microelectronics from the University of Roma Tor Vergata, Rome, Italy, in 2002 and 2007, respectively. He has been an Assistant Professor with the University of Roma Tor Vergata, since 2012, where he has been teaching a course in microwave measurements since 2013. His current research interests include linear microwave circuit design methodologies, linear and noise analysis/measurement techniques, and small-signal and noise modeling of microwave active devices.

9

Ernesto Limiti (M’93) was a Research and Teaching Assistant from 1991 to 1997 and an Associate Professor from 1998 to 2001, and is currently a Full Professor of electronics with the Electrical Engineering Department, University of Roma Tor Vergata, Rome, Italy. He is currently the Head of the Electrical Engineering Department, University of Roma Tor Vergata. He has authored or co-authored over 350 publications on refereed international journals and presentations within international conferences. His current research interests include three main lines, all of them belonging to the microwave and millimeter-wave electronics area. The first one is related to characterization and modeling for active and passive microwave and millimeter-wave devices. Regarding active devices, the research line is oriented to small-signal, noise, and large-signal modeling. For active devices, novel methodologies have been developed for the noise characterization and the subsequent modeling, and equivalent-circuit modeling strategies have been implemented both for small- and large-signal operating regimes for GaAs, GaN, SiC, Si, and InP MESFET/HEMT devices. The second line is related to design methodologies and characterization methods for low-noise devices and circuits. The main focus is on cryogenic amplifiers and devices. Finally, the third line is in the analysis and design methodologies for linear and nonlinear microwave circuits. Prof. Limiti is a member of the Steering Committee of international conferences and workshops. He is a Referee for international journals of the microwave and millimeter-wave electronics sector. He has been the President of the Laurea and Laurea Magistrale degrees in electronic engineering of the University of Roma Tor Vergata, and is a member of the committee of the Ph.D. program in telecommunications and microelectronics, tutoring an average of four doctoral candidates per year.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

1

A W-Band LNA/Phase Shifter With 5-dB NF and 24-mW Power Consumption in 32-nm CMOS SOI Mustafa Sayginer , Member, IEEE, and Gabriel M. Rebeiz, Fellow, IEEE

Abstract— This paper presents a W-band phased array receive front end in 32-nm CMOS silicon-on-insulator technology. The architecture is based on cascode low-noise amplifiers and passive switched LC 5-bit phase-shifters and with root-mean-square (rms) phase error of