~IEEE
TRAN SACTI 0 NS
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MICROWAVE THEORY AND TECHNIQUES A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY
OCTOBER 1994
VOLUME 42
NUMBER 10
IETMAB
(ISSN 0018-9480)
PAPERS AIGaAs/GaAs HBT Linearity Charcteristics .................................... .N.-L. Wang, W. J. Ho, and J. A. Higgins Analysis of MESFET Injection-Locked Oscillators in Fundamental Mode of Operation ...... C.-C. Huang and T.-H. Chu An Efficient Self-Oscillating Mixer for Communications .................................... .X. Zhou and A. S. Da7 oush A 94 GHz Planar Monopulse Tracking Receiver ............................................ C. C. Ling and G. M. Rebeiz Experimental Validation of Microstrip Bend Discontinuity Models from I 8 to 60 GHz ................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. J. Slobodnick, Jr., and R. T. Webster Spike Leakage of Thin Si PIN Limiters ................................................ A. L. Ward, R. J. Tan, and R. Kaul Dielectric Properties of Single Crystals of Al 2 0 3 , LaAlh, NdGal0 3 , SrTi0 3 , and MgO at Cryogenic Temperatures ..... . .. .. . . .. .. .. .. .. .. .. .. .. .. .. . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . J. Krupka, R. G. Geyer, M. Kuhn, and J. H. Hinken Power and Spatial Mode Measurements of Sideband Generated, Spatially Filtered, Submillimeter Radiation ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. R. ueller and J. Waldman Some Important Properties of Waveguide Junction Generalized Scattering Matrices in the Context of the Mode Matching Technique ......................................... G. E. Eleftheriades, A. S. Omar, L. P. B. Katehi, and G. M. Rebeiz Frequency Domain Analysis of RF and Microwave Circuits Using SPICE ................................... C. E. Smith Experimental Proof-of-Principle Results on a Mode-Selective Input Coupler for Gyrotron Applications ................. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. .. . .. .. .. J. P. Tate, H. Guo, M. Naiman, L. Chen, and V. L. Granastein Experimental Analysis of Millimeter Wave Coplanar Waveguide Slow Wave Structures on GaAs ...................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Spickerman and N. Dagli Leakage Characteristics of Groove Guide Having a Conductor Strip ............................ Z. Ma and E. Yamashita Infrared Surface Waves in Circular Hollow Waveguides with Small Core Diameters .................................... . . . . . . . . . . . . . , ............................... F. E. Vermeulen, A. M. Robinson, C. R. James, and J. N. McMullin Radiation Modes in Dielectric Circular Patch Antennas ........................................... H. How and C. Vittoria Study of Microstrip Step Discontinuities on Bianisotropic Substrates Using the Method of Lines and Transverse Resonance Technique ........................................................................... Y. Chen and B. Becker A Macroscopic Model of Nonlinear Constitutive Relations in Superconductors ...... J. J. Xia, J. A. Kong, and R. T. Shin Numerically Efficient Spectral Domain Approach to the Quasi-TEM Analysis of Supported Coplanar Waveguide Structures .......................................................................... K. K. M. Cheng and/. D. Robertson On the Use of Differential Equations of Nonentire Order to Generate Entire Domain Basis Functions with Edge Singularity .................................................................... Z. Altman, D. Renaud, and H. Baudrand Flexibility in the Choice of Green's Function for the Boundary Element Method ............ T.-N. Chang and Y. C. Sze Improved Wire Modeling in TLM ....................... A. P. Duffy, J. L. Herring, T. M. Benson, and C. Christopoulos (Continued on back cover)
1845 1851 1858 1863 1872 1879 1886 1891 1896 1904 1910 1918 1925 1932 1939 1945 1951 1958 1966 1973 1978
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TEC HNIQUES, VOL. 42, NO. IO, OCTOBER I994
I845
AlGaAs/GaAs HBT Linearity Characteristics Nan Lei Wang, Wu Jing Ho and J. A. Higgins, Senior Member, IEEE
Abstract-Communication systems require linear power amplifiers with high efficiency and very low intermodulation distortion. AIGaAs/GaAs heterojunction bipolar transistors (HBT's) were found to have very low intermodulation distortion in power operation. Two-tone tests were carried out on both commonemitter (CE) and common-base (CB) power HBT's. At 7 GHz, the CE HBT showed -20 dBc IM3 (third-order intermodulation ratio) and 12% power added efficiency (PAE) per tone at the 1 dB gain compression point; IM:1 dropped to -30 dBc at 1.5 dB output power backofT. The CE HBT has lower intermodulation distortion than CB HBT. Load pull data were collected to aid the understanding of the intermodulation. Parameters of the Gummel-Poon model (as used in SPICE) were derived for HBT's based on de data and small-signal S parameters at various bias points. The accuracy and validity of the model were confirmed by comparison to experimental two-tone results. SPICE predicts that the emitter and base resistances linearize the HBT and reduce the third-order intermodulation distortion. The excellent third-order intermodulation performance of the CE HBT makes it a very attractive choice for linear power amplifiers.
l. INTRODUCTION BTs have high efficiency in class AB saturated power operation at microwave frequencies [ l]. However the transmitter power amplifier (PA) of a communication system needs to have both high efficiency and good linearity. Nonlinearity creates intermodulation distortion and raises the bit error rate (BER), and is one of the key issues in microwave communication systems [2]. The nonlinearity is often measured by the two-tone intermodulation method. Traditionally the intermodulation distortion is reduced by backing off the output power of the PA, which reduces the efficiency and requires a higher output power PA (as rated by output power at l dB gain compression point) to achieve the same linear output power. AIGaAs/GaAs HBT's have very low intermodulation. The figure of merit is IP3/ Pbias [3]. The mode of operation described here is only suitable for receiver applications since neither power density nor efficiency are addressed. In transmitter applications, the intermodulation ratio (IM 3 ) must be balanced with the efficiency at various power levels as measured by the backoff from I dB compression point. AIGaAs/GaAs HBT's were found to have excellent performance in this mode. At the I dB gain compression point and operating at 7 GHz, IM 3 is approximately -20 dB and the power added efficiency (PAE) per tone is 12%. In this work, extensive two-tone testing was carried out on both CE HBT and CB) HBT. It was found that CE HBT
has lower third-order intermodulation distortion than CB HBT. Two different output matching conditions were explored: the best combination of gain and output power at IM 3 = - 40 dBc (IM 3 match), which is a common ly required specification for high linearity power transistors ; and the highest output power in the single tone power test (power match). Load pull data was also collected. SPICE simulation shows good agreement with measured values. The low intermodulation distortion of the CE HBT is attributed to the negative feedback and linearization effect from the emitter and base resistances. SPICE simulation also suggests that the nonlinearity in collector capacitance must be reduced to achieve low intermodulation distortion. Although the exponential 1-V relation in HBT's is thought to make the device very nonlinear, both our experimental results and analysis indicate otherwise. The HBT power added efficiency (PAE) per tone at IM 3 = -44 dBc is 6-73 , which is about twice the efficiency of a typical MESFET at the same IM 3 . The results reported in thi s paper indicate that HBT is not only suitable for saturated power application, but is also an excellent candidate for linear power amplification.
H
Manuscript rece ived March 4, 1991; revised November 26, 1993. The authors are with Rockwell International Science Center, 1049 Camino Dos Rios, Thousand Oaks, CA 91360 USA . IEEE Log Number 9404160.
IL Two-TONE TEST OF HBT The power HBT's used in this work were identical indesign to those reported previously [4]. At 10 GHz, the HBT delivers over 0.6 W saturated power in class AB operation. The HBT layer structure and electrical characteristics have been published in earlier work [5]. The device consists of nine HBT cells; each cell has 2 emitter fingers with dimensions of 2 µm x I0 µm. DC current gain is about 20; BY cbo is 20 V; BYceo is typically 13 V. One of the important processing features in the device fabricati on is the collector implant in the extrinsic base region [6]. The implant reduces the collector capacitance and helps maintain Cbc at a constant along the load line. This implant was found to be a key to maintaining the low intermodulation in the two-tone test (see analysis section). The two-tone test was carried out at 7 GHz with a 5 MHz separation between the two signals. The input was matched to provide the lowest return loss and the highest gain at a given output matching condition. The output was matched to provide the desired tradeoff of IM 3 , efficiency, and output power. With the matching condition established, the rf power level was swept to generate Pin versus Pout curves for both two-tone and single carrier operations. In all the experiments described below, higher-order intermodulation distortion is at least 5 to I0 dB below the third-order intermodulation at the I dB gain compression point. In the Pin versus Pout curve of two tone test, gain compression and a non 3: 1 slope for the third-order intermodulation distortion take place at power level approaching the saturated region. The linearity of the
0018-9480/94$04.00 © 1994 IEEE
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. IO, OCTOBER 1994
1846
20 25
0
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15
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ls
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Pin (dBm) per tone
Pout sci (dBm)
Fig. I. CB HBT two-tone test result for class A and class B operations at 7 GHz. out sci represent "single carrier output power," or output power for each tone in the two tone test. Class B has higher single tone output power and PAE, but the gain and the IM3 are worse than class A. The IM3 is nearly constant over the power range for class B operation , and is most likely a result of the current waveform clipping.
HBT for power applications is best displayed by examining the efficiency, IM3, and gain at various output power levels. Seven tests were conducted to characterize the HBT linear operation. The major findings are summarized below: 1) CB HBT class A operation has lower intermodulation distortion than class B operation. 2) CE HBT class A operation has even lower intermodulation distortion than CB HBT class A operation. 3) The output matching impedance has strong influence on the IM3, PAE, and output power. 4) The input matching impedance does not affect the IM 3 at the same output power.
Fig. 2. CE HBT single carrier and two-tone test results with IM3 match in class A operation at 7 GHz. The third-order intermodulation distortion increases faster than 3: I at high power level. The IP3 concept is not suitable describing the intermodulation distortion at power level approaching the I dB compression point.
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A. CB HBT in Class B and Class A Operations The CB HBT was tested in both class A and class B modes. In class A operation the device was biased at a constant emitter current, and in class B operation the device was biased at a constant emitter base tum on voltage (Vbe ) · In both modes Vcb was maintained at 6 V. In class B operation, the collector current increases with the rf driving power. Therefore, the heat dissipation is highest at the maximum power level. The situation in class A operation is the opposite: the heat dissipation is the highest with no rf driving power. At equivalent junction temperatures, class A operation provides half the output power of class B operation. This necessitates using a larger transistor in class A operation to achieve the same output power as in class B operation. Fig. 1 shows the two-tone testing results for both modes, with the output power per tone in the two-tone test as the horizontal axis. In class A operation lower output power with higher gain and lower power added efficiency were observed. The IM 3 was much lower at lower output power levels in the class A operation. Due to the lower output power capability of class A operation, the IM 3 is higher at high output power level as a result of overdrive into the saturation region. IM3 remains constant in class B operation, which suggests that collector current waveform clipping is the dominant cause of the intermodulation distortion in that class of operation.
Fig. 3. CE HBT and CB HBT two-tone resultsin class A operation. The PAE per tone is almost identical. CBHBT has higher gain, but also higher intermodulation distortion.Intermodulation distortion of CE HBT is IO to 15 dB lower at anypower level.
B. CE HBT in Class A Operation with "IM3 Match"
Two tone test results on a CE HBT at 7 GHz with the "IM3 match" condition in class A operation are illustrated in Fig. 2, together with the one-tone results. The CE HBT was biased at Vee = 7 V and Ic = 70 MA, which is very similar to the CB HBT in class A bias. The CE HBT one-tone test showed 23.8 dBm output power at 1 dB gain compression point, with PAE reaching 41.8%. In the two-tone test, 19.63 dBm output power per tone was achieved at 1 dB compression, 4.2 dB lower than the one-tone P_ 1 dB . The PAE for each tone reaches 12.43 at the 1 dB compression point. The third-order intermodulation distortion at low power levels follows a 3: 1 slope, and rises more rapidly at higher power levels. C. Comparison of CE HBT and CB HBT in Class A Operation
Fig. 3 shows the comparison of IM3, PAE and gain between CE HBT and CB HBT in class A operation. Both configurations have about the same PAE at all power levels, and the same P out at 1 dB gain compression. The CB HBT has higher gain; but CE HBT shows IO to 15 dB better IM3 throughout
1847
WA NG et al.: AIGaAs/GaAs HBT LI NEA RITY CHARACTER ISTICS
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Fig. 4. Two CE HBT' s identical in emitter junction area but different collector capacitance are compared for class A two-tone intermodulation perfo rmance.
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Fig. 5. Comparison of two-tone test results of the CE HBT with the "IM 3 match" at 7 and 10 GHz. Due to the di ffic ulty to optimize the "IM3 match," it is diffi cult to conclude the frequency dependency of the intermodulati on di stortion. Linear gain at 7 GHz is indeed higher.
the whole power range. This result clearly indicates that CE HBT is more suitable for the linear power application than CB HBT. -
-1 0
D. Comparison of CE HBT's with Different Cb c in Class A Operation
•,o -20
Two CE HBT's with different lateral layout (different Cbc values) were compared in Fig. 4. The two layouts provide the same emitter junction area, but one version (HBT-B) had a 20% higher collector capacitance than the standard one (HBTA) used in thi s experiment. The performance in PAE, IM 3, and I dB gain compression output power level was identical. The gain was different by only 1 dB (11.8 versus 10.8 dB). This suggests that the nonlinearity in collector capacitance of these HBT is weak and does not affect the IM 3. IM 3 = -30 dBc can be achieved with less than 1.5 dB backoff in output power from the 1 dB gain compression point, and the PAE for each carrier is about 9%. At the 4 dB output power backoff point, the IM 3 reached -44 dBc with PAE per tone around 6%. E. Comparison of CE HBT Two-Tone Results in Class A Mode at 7 and JO GHz
The CE HBT was tested at 10 GHz to explore the frequency dependency of the IM 3 ; the result is shown in Fig. 5. The gain at 7 GHz is higher than the gain at I 0 GHz, as expected. The PAE is about the same for the same output power level. The IM3 ratio is also very similar at high power level s; and is about 5 dB better at 7 GHz at low power levels. Load pull test indicate that the load resistance is 65 f2 at 7 GHz, and 69 f2 at 10 GHz. The reactive part is conjugately matched.
F. Effect of Output Matching Impedance of CE HBT in Class A Bias All the two-tone tests so far on CE HBT have the socalled "IM 3 match" output matching condition: the matching impedance provides the best combination of gain, output power and efficiency at IM 3 = - 40 dBc. Another matching condition is called "power match": the impedance provides the highest output power and gain in the single-tone test. Results
• ,O &, 6.
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2(GT aBT - BT aGT) aw aw +(
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(9)
where GT and BT are conductance and susceptance of YT. Based on (8) and (9) the locked state can be determined by
1853
HUANG AND CHU: ANALYSIS OF M ESFET INJECTION-LOCKED OSCILLATORS IN FUNDAMENTAL MODE OF OPERATION
verifying the stability properties of each possible value V 2 under the injection condition. Similarly a MESFET injection-locked oscillator is shown in Fig. 2(b) by adding an input signal to either port 1 (transmission-type ILO) or port 2 (reflection-type ILO). For the reflection-type ILO the circuit equation given in (4) can be written as
[YiFETport2 ( TT v2, W )
+ y;02 ]TTv2 e j (w t
+
- I inj e JWinj ·
2) -
(10)
In the fundamental mode operation, the dynamic frequency w becomes Winj· By separating (10) into real and imaginary parts, it becomes
10pF 50Q. microstrip line
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load
load
Fig. 3. Shunt feedback MESFET oscillator circuit with a 10 pF DC blocking capacitor.
( 11 ) [GFETport2(Vi , Winj) + Yo2]V2 = Iinj COS,. w / wo with Qext = 75.8.
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Fig. 6. Measured and predicted frequency FM no ise under 2nd and 3rd subharmonic injection locki ng as a function of the injection power ratio at 100 Hz off set from carrier.
-JO -2.0
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
Vgs (V)
Fig. 8. Measured and predicted frequency conversion gain of the circu it as a function of gate to sourcebias voltage.
The calculated FM noise level based on (6) is also shown in Fig. 6. These results indicate a good match with our analytical modeling. To verify that when the operating point of the devices is close to pinch-off a large subharmonic injection locking range is expected, we measured the 2nd and 3rd subharmonic injection locking range as a function of the gate bias. The results are depicted in Fig. 7, which indicates a considerable increase in the normalized subharmonic injection locking range at an operating point close to pinch-off, compared with that in class A. The other aspects of the measured characteristics of the oscillator are addressed in [9]. B. Mixer Performance
In the second set of measurements, we investigated the mixing performance of this self- oscillating mixer. A conversion gain as high as 13 dB is measured for the Vgs = -1.82 V, which corresponds to the LO output power of +2 dBm. The total power consumption of this self-oscillating mixer is about 60 mW. The predicted and the measured conversion gain as a function of the gate bias matched very well, as depicted in Fig. 8. Conversion gain as a function of the RF input power level for different operating points was also measured, as depicted in Fig. 9. The mixer conversion gain is flat up to PRF ~ -10
dBm. Noise characteristics of this mixer were measured as well, and a double sideband noise fig ure of 8 dB was measured for down conversion from 16.8 GHz to 5 GHz. This fig ure is good for our prototype design, because we didn't consider to short circuit the gate at the IF frequency to minimize noise amplification from the input source or bias circuit at the IF freq uency, recognized as a major source of noise in FET mixers [5] . However, the operating point of devices was set at close to pinch-off, the low and stable LO power as
1862
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
well as the maximum transconductance swing were achieved. These conditions are all appropriate for minimizing noise. If considering those conditions discussed in [5], it is possible to further improve noise performance of the proposed circuit. On the other hand, operating at this operation point, the push-pull self-oscillating mi xer has the minimum variation of oscillation freq uency vs. gate bias voltage, shown in Fig. 4. Which means a minimum AM to PM conversion. This is another advantage of this circuit. Furthermore, operating in this region, because a low DC power dissipated in the device, it is very good for the long term temperature stability of the oscillator. V. CONCLUSION
The first injection locked push-pull self-oscillating mixer has been analytically modeled in terms of subharmonic injection locking, FM noise performance under injection locking, and frequency down/up conversion by use of a power series. We also have successfully demonstrated its merits experimentally. The results shows that low power consumption, high subharmonic injection locking range, low FM noise and high frequency conversion gain and low noise figure all can be obtained through thi s proposed push-pull self-oscillating mixer topology . The modeled performance matched very well with the measurement results . Because of the efficient performance of this topology, thi s small size and low power consuming injection locked push-pull self-oscillating mixer is attractive for T/R level data mixing architecture [l] of the optically fed phased array antennas as well as for mobile communications. ACKNOWLEDGMENT
This work is supported in part by the NASA, Lewis Research Center. The authors are also grateful to Mr. D. Sturzebecher at Army Research Laboratory, Ft. Monmouth for his assistance in fabrication of this circuit. REFERENCES [l] A. S. Daryoush, A. P. S. Khanna, R. R. Kunath , and K. B. Bhasin, "Fiberoptic links for mi lli meter wave communication satellites," IEEE Int. Microwave Symp. Dig., New York, May 1988. [2] A. S. Daryoush, "Optical synchronization of millimeter-wave oscillator for distributed architecture," IEEE Trans. Microwave Theory Tech. , vol. 38, pp. 467-476, May 1990. [3] X. Zhou and A. S. Daryoush, "A push-pull self-osc ill ating mixer for optically fed phased array," IEEE Int. Microwave Symp. Dig., 1993 , pp. 321-324.
[4] D. J. Sturzebecher, X. Zhou, X. Zhang, and A. S. Daryoush, "Optically controlled oscillators for millimeter-wave phased array antennas," IEEE Trans. Microwave Theory Tech., vol. 41, pp. 998-1004, June/July 1993. [S] S. A. Mass, Microwave Mixer. Norwood, MA: Artech House, 1986. [6] A. S. Daryoush, T. Berceli, R. Saedi, P. R. Herczfeld, and A. Rosen, "Theory of subharmonic synchronization of nonlinear oscillators," IEEE Int. Microwave Symp. Dig., pp. 735-738, 1989. [7] X. Zhang, X. Zhou and A. S. Daryoush, "A theoretical and experimental study of the noise behavior of subharmonically locked local oscillators," IEEE Trans. Microwave Theory Tech., vol. 40, pp. 895-902, May 1992. [8] X. Zhou and A. S. Daryoush, "Power series nonlinear coefficient extraction of MESFET' s," submitted to IEEE Trans. Microwave Theory Tech. [9] X. Zhou and A. S. Daryoush, "An injection locked push-pull oscillator at Ku-band ," IEEE Microwave Guided Wave Lett., vol. 3, Aug. 1993.
Xue-Song Zhou received the B.S. degree from Sichuan University, Chengdu, China and the M.S. degree from the University of Electronic Science and Technology of China, Chengdu, China, both in electrical engineering, in 1982 and 1984, respectively. From 1984 to 1989, he was a Teaching Assistant and later a Lecturer in the Department of Atmosphere Sounding at Chengdu Institute of Meteorology. Currently, he is a Ph.D. student at Drexel University, Philadelphia, PA. Hi s research interests include analysis and modeling of nonlinear characteristics of various microwave and optoelectronic devices, and analysis and design of MMIC based circuits and optical links. He is also interested in the design of high efficient nonlinear circuit, such as oscillators, mixers, power amplifiers and frequency multipliers for wireless and satellite communications.
Afshin S. Daryoush (S'84-M' 87-SM'92) received the B.S. degree from Case Western Reserve University, Cleveland, OH, and the M.S. and Ph.D. degrees from Drexel University, Philadelphia, PA, all in electrical engineering in 1981, 1984, and 1986, respectively. In 1986, he joined the faculty of Drexel University as DuPont Assistant Professor of Electrical and Computer Engineering and was promoted to Associate Professor in 1990, where he taught undergraduate and graduate courses and conducted research in microwaves and photonics. He worked as ASEE Summer Faculty Fellow at NASA-Lewis Research Center, Cleveland, OH (1987-1988) on the use of fiber-optic links in communication satellites, and at Naval Air Development Center, Warminster, PA (1989-1990) on digital fiber-optic networks for computer backplanes. He has also been a consultant for Government laboratories and aerospace industry. He has authored or coauthored more than JOO technical papers in the area of optical interfaces to passive and active microwave devices, and its applications in microwave circuits, and subsystems. Dr. Daryoush earned the Microwave Prize in the 16th European Microwave Conference, Dublin, Ireland, he has also received the best paper award in the Impatt Session of the 1986 International Microwave Symposium, Baltimore, MD. He has been awarded fo ur U.S. patents in microwaves, antennas, and photonics. He served the Philadelphia Joint Chapter of the AP/MTT Societies as Vice-Chairman (1989-1990) and Chairman (1991-1992).
IEEE TRANSACTIONS ON MICROWAVE TH EORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
1863
A 94 GHz Planar Monopulse Tracking Receiver Curtis C. Ling, Member, IEEE, and Gabriel M. Rebeiz, Senior Member, IEEE
Abstract- This paper describes the design, fabrication and measurements of a 94 GHz integrated monopulse receiver with IF beam control. The receiver is integrated on a single chip, and is based on a 23 GHz local oscillator driving four separate phase-coherent 94 GHz subharmonic mixers. The resulting IF signals are takeoff-chip to a IF monopulse processor, which produces sum and difference monopulse patterns for the elevation and azimuth coordinates. Voltage-controlled phase-shifters in each of the IF channels allow the monopulse patterns to be electronically steered. All of the receiver circuits are realized using uniplanar coplanar-waveguide (CPW), slot lines and coplanar striplines (CPS). These features result in a compact, lowcost system suitable for tracking systems operating in poor visibility conditions, as well as in collision avoidance receivers for automotive applications. To our knowledge, this work represents the first demonstration of a fully integrated millimeter-wave subsystem to date.
I. INTRODUCTION
ILLIMETER-WAVE systems are dominated by wave guide-based designs which are built up using separately manufactured components. While these systems are efficient and can be assembled using off-the-shelf or customized components, they are also expensive and bulky. Recent advances in millimeter-waveplanar of nove !integrated subsystems which are compact, rugged, and exhibit performance comparable to waveguide systems, with much lower fabrication costs [I] , [2]. One such system, an integrated four-horn monopulse tracking receiver, has been designed and fabricated for operation at 94 GHz. The receiver consists of four antennas , each with an associated subharmonic mixer. The four subharmonic mixers are driven by a single local oscillator (LO) with a frequency which is one fourth the RF, approximately 23 GHz. This frequency is low enough to allow the LO power to be produced and distributed with low losses using planar transmission lines. The receiver is integrated on a single high-resistivity silicon chip and utilizes hybrid-mounted components in order to simplify fabrication. The antennas and mixersare arranged in a two-by-two array. Because the mixers are driven by the same LO source, their IF signals are phase-coherent. The idea central to thi s design is to use the four IF signals to produce the monopulse sum and difference patterns for both the elevation and azimuth coordinates, using inexpensive low-frequency signal processing components. The receiver is
M
Manuscript received September 16, 1993; revised December 2, 1993. This work is supported by the Army Research Office under contract DAAL03-9 1G--0116. C. C. Ling was with the University of Michigan Radiation Laboratory, Ann Arobr, Ml 48109 USA. He is now with Hong Kong University of Science and Technology. G. M. Rebei z is with the Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, MI 48109- 2122. IEEE Log Number 9404157.
Fig. 1. The 94 GHz integrated monopul se receiver, consisting of an active wafer sandwiched by horn side wall wafers stacked above and below.
composed of uniplanar CPW, slotline and CPS circuits. This eliminates the need for a backing ground-plane and precision via-hole placement when utilizing three-terminal devices in the circuit. The fabrication of the receiver is compatible with standard semiconductor processing techniques, making the design rugged and inexpensive to fabricate. II. ANTENNA AND MIXER LAYOUT
The receiver consists of an active wafer containing integrated antennas, mixers, and local oscillator (LO) circuits (Fig. 1). The antennas are based on integrated horn cavities. These are fabricated using anisotropic etching of silicon which results in a fixed flare angle of 70°. Several wafers are etched with different opening sizes inorder below the active wafer. The integrated horn antenna design was first developed in [6], and subsequently optimized in [5], [14]. It is important to note that the monopulse design presented here will work equally well with other types of planar antennas, such as dipole and slot antennas on dielectric lenses [3] , or microstrip patch antennas. In this paper, a novel method for coupling to the horn cavity is utilized. This technique replaces the membrane with a thin substrate with a thickness of about 0.1>.d [5]. When using silicon or GaAs as a substrate material at 94 GHz, thi s corresponds to approximately 90 µm . The antenna and
0018- 9480/94$04.00 © 1994 IEEE
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. IO, OCTOBER 1994
S upporting siliqon beam , /Via groove 90um-thick --- silicon layer
/ /
L.--, /
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S id e view Fig. 2. Etched 90 µ-thick substrate with via-trenches used to suppress substrate modes.
Fig. 4. Illustration of the monopulse array layout and LO distribution scheme. Dimensions are in micrometers. Detailed dimensions of the transitions, and an equivalent circuit diagram are shown in Fig. 5.
CPS Zcps=900 90"(94GHz)
To LO/ IF Circuits
Fig. 3. Antenna and mixer circuit dimensions. The dashed lines show the location of a polyimide insulating patch used in overlay capacitors. All dimensions are in micrometers. The cross section the horn corresponds to a distance of 0.36>. from the horn apex.
any semiconductor devices can be integrated on the thin substrate, and via trenches or grooves are etched around this area The via trenches are aligned with the stacked cavity side walls to complete the horn antenna. The substrate thickness is chosen so that the frequency of operation lies between resonant modes of the thin substrate. Work in [5, ch. 7] discusses the preliminary analysis of such a structure. A 90 µm-thick substrate is too thin for practical fabrication and handling. In order to overcome this practical difficulty, only the substrate directly underneath the antenna is etched away until only 90 µm remains (Fig. 2). Via grooves surround the 90 µm region, and thin silicon beams attach the thin substrate with the rest of the wafer. The structure shown in Fig. 2 was micromachined in silicon using a two-step etching process. By placing the mixing devices directly at the apex of the dipole, the mixer becomes an integral part of the antenna. The width and length of the dipole, and its position relative to the apex of the horn, can be adjusted to tune the input
impedance of the antenna. A coplanar strip feed is attached to the apex of the antenna in order to extract IF signals or inject LO power to the mixer (Fig. 3). The feed of the antenna includes a quarter-wave RF capacitive choke which isolates the RF signal received by the antenna from the rest of the circuit. The length of the choke, and the capacitance used, can be used as a tuning circuit for adjusting the embedding impedances seen by the mixer, though a simple quarter-wave choke was used in this design. The value for the RF choke capacitance is chosen to yield a susceptance of j50 n at the LO frequency, and j12 n at the RF-frequency (Fig. 3). Its distance from the dipole is 0.25>. 9 . The impedance and guided wavelength of the CPS on the thin silicon substrate are estimated to be about 90 n and 2040 µm, respectively. These were calculated using approximate analytical formulas developed by Gupta et al. [7]. The choke has an estimated return loss at RF of about 0.2 dB, neglecting conductor and dielectric losses. The antenna/mixer design is shown in Fig. 3, and the exact dimensions are determined from an optimization process described in the Section III. The four antennas are driven by the same LO source through a power distribution circuit shown in Fig. 4. The 23 GHz LO power is fed using a CPW transmission line and is produced by an oscillator and LO amplifier circuits presented in Section IV. The LO distribution circuit contains a CPW-to-slotline transition (balun), which also functions as power dividers. The design of the transition is straight forward, and is based on a quarter-wave choke [8], which is adapted so that it drives two equal loads (the subharmonic mixers for two antennas) via the slotlines. The layout and equivalent circuit are shown in Figs. 3 and 5. The design is a result of microwave modeling and optimization also discussed in Section III. The slotlines easily transition to CPS, which feeds the antennas I mixers. The IF is isolated from the LO circuit by a 20 µm-wide break
1865
LING AND REBEIZ: A 94 GHz PLANAR MONOPULSE TRACKING RECEIVER
TABLE l DIOOE PARAM ETERS USED IN MIXER DESIGN . THE VALUES LIE WITHI N THE MANUFACTURER-SPECIFIED RANGES FOR EACH PARAMETER
3l1
60fF
l.l
O.lnH
40fF
0.75V
Is l x l0 - 15 A
TABLE II MIXER EMBEDDING IMPEDANCES AT EACH LO HARMONIC
CP W Zc=
IF
•
85"
2x LO
3xLO
RF
72 - j2150
20 - j50!1
16!1
IF
600 6pF
LO 15 + j2ll1
6pF
Antenna/ .
• An l e n na/
Mixer
Mixe r
t ~&=o LO in
65.
z...
Fig. 5. The CPW-CPS transition and LO power splitter circuit. Dashed lines show the outline of insulating polyimide patches. All dimensions are in micrometers.
in the slotline. At LO frequencies, signals travel unimpeded across this break via a 0.25 pF capacitive overlay whose value is selected to present a very low impedance at the LO frequency of 23 GHz, and a very high impedance at the IF frequency of 200 MHz. Once the antenna and mixer designs are finalized, the input impedance of the four antenna/mixers driven in parallel must be matched to the output impedance of the LO drive circuits presented in Section IV. An antiparallel Schottky diode pair is hybrid-mounted at the feed of each antenna to form a 94 GHz subharmonic mixer. The subharmonic mixer (SHM) is a millimeter and submillimeter-wave frequencies to obtain noise temperatures and conversion loss which are only a few dB above the performance of fundamental mixers [9], [10]. III.
ANTENNA AND MlXER OPTIMIZATION
Full-wave analysis of the dielectric within the horn cavity is needed to obtain the antenna input impedance. At the time this work was performed, such analysis had not been yet completed, so antenna impedances were found by constructing a 2.7 GHz microwave scale model (corresponding to a second subharmonic LO frequency of 675 MHz) of the cavity structure. Stycast with Er = 12 was used to model the semiconductor substrate. A 125 mil-thick layer of stycast was used to model the 0.1>.d substrate. Strips of copper tape attached to the stycast are used to construct the dipole antenna and the feeding CPS transmission line. The dipole radiates preferentially into the substrate, and is placed facing the apex to allow radiation propagating through the dielectric to radiate
Fig. 6. Antenna/mixer embedding impedances obtained from a 2. 7 GHz microwave model. The 10% band widths at each LO harmonic are plotted.
toward the horn aperture . The microwave scale model was constructed to include the LO distribution circuits (the CPWslotline-CPS transition shown in Fig. 5) and the two antennas connected to the two outputs ofthe transition. The embedding impedances of the back-to-back diodes (located at the apex of the dipole) were then measured at the first four harmonics of the LO frequency . The LO distribution circuit had little effect on the RF antenna impedance at 2.7 GHz due to the presence of the RF choke containing the isolation capacitor, modeled as a 5 pF capacitor. The RF isolation capacitor also causes the LO embedding impedance to be fairly low (15-25 0 ). The commercially available diode pair with the highest operating frequency, the MNCOM 40422, was originally designed for operation in a K-band mixer, and is used here at 94 GHz. The embedding impedances are adjusted by changing the dimensions of the antenna and (to a lesser extent) the LO distribution circuit, and harmonic balance analysis (HBA [ 11]) is applied to determine how the diode impedances and conversion loss are affected. This process is iterated in order to optimize for conversion loss. The resulting embedding impedances are shown in Fig. 6 and Table II, and the resulting dimensions of the circuits are shown in Figs. 3-5. The antenna impedance is low ( 16 0) in order to match the low RF impedance of the diode pair due to their large junction areas (resulting in C 10 + Gp = 100 pF). The analysis predicts that 6-8 mW of LO power is needed to drive each mixer. Thus 24-30 mW is needed to drive the four subharmonic mixers . Computations also predict a typical conversion loss of 12-14 dB , and showed that the diode series resistance is
1866
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
I I I '
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Fig. 7. Microwave scale model measurements of the input impedance to the CPW-CPS transition (ZLo), marked on Fig. 5. Varac l or Bias
an especially important diode parameters in determining the performance limits of a second-subharmonic mixer. Once the antenna design was optimized, the input impedance ZLo at point A marked on Fig. 5, was measured in order to design the output matching of the LO-producing circuits which drive the mixers (Fig. 7). ZLO is combined in parallel with an identical impedance (corresponding to the other antenna/mixerpair) and is matched to the output of the LO amplifiers described in the next section. IV. LOCAL OSCILLATOR CIRCUITS
Fig. 8. Circuit diagram and layout of the VCO design. All CPW transmission line sections have an impedance of 40 n. ATTEN 1OdB
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A. Oscillator Design
The 23 GHz LO power is produced using a VCO utilizing a commercially available HEMT, the NEC NE32100. The transistor is hybrid-mounted in a uniplanar, CPW transmission line circuit and connected using 0.7 mil gold bond wire. The uniplanar HEMT oscillator is designed using the reflection amplifier technique presented in [13]. The transistor is placed in a one-port network, which is connected to a load. The oscillator design uses CPW transmission line tuning stubs attached to the source and gate of the transistor (Fig. 8). The CPW transmission line dimensions are calculated for a 360 µm thick silicon substrate (Er = 11. 7), using a commercially available software (EEsof/Linecalc [ 17]). The reflection coefficient looking into the drain of the transistor (Sn. marked on Fig. 8) is calculated using ideal transmission line models and the linear S-parameters of the transistor. Two varactors are attached to the CPW line connected to the gate of the transistor and allow the frequency of oscillation to be adjusted. The varactors, MSV-34064-C 11 chip varactors manufactured in [18], have a Cjo of 0.6 pF and a minimum reverse-bias junction ground plane of the transmission line to maintain the symmetry of the CPW line, there by avoiding excitation of the even transmission line mode. Small-signal analysis is used to obtain a design with an operating frequency range from 24.9 GHz to 25.3 GHz, which is about 103 higher than the chosen LO frequency of 23 GHz. The factor of 103 was empirically determined by constructing low-frequency
23 3326GHz
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23 GHz VCO output spectrum.
oscillators with similar geometries. In order to bias the gate and varactors, the transmission line attached to the gate of the HEMT is terminated in an overlay bypass capacitor, which act as a short-circuit at 23 GHz. Measurements on a prototype VCO based on the NE32100 indicate that the circuit produced stable oscillation at 23.3 GHz and an output power of 0.5-1 mW. Based on the measured transmission line losses and a varactor series resistance of 1 n, the external circuits at the gate and source have a calculated Q of 9 and 12, respectively. The output spectrum produced by the VCO is shown in Fig. 9, and can be used in the receiver without phase-locking, since the phase noise is minuscule (even when multiplied by a factor off our due to subharmonic mixer operation) compared to the wide bandwidth of available IF processing elements. Measurements of the VCO indicate that the output frequency is linear with varactor bias (0 to -15 V) with a tuning range of 250 MHz, and an output power between 0.5 and 1 mW.
1867
UNG AND REBEIZ: A 94 GHz PLANAR MONOPULSE TRACKING RECEIVER
2.5
Ill
Silicon
~ Ove rl ay capacitor
---- Polyim ide o utline
Output and Blas Choke
Amplifier Equiva lent Cir cu it
DC Bie.s
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Equivalen t Circ uit
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Fig. JO.
Layout of the single-transistor amplifier with the equivalent circuit.
V. LO AMPLIFIER CHAIN DESIGN AND MEASUREMENT
The four mixers located at the apexes of the four antennas require a total LO power level of at least 20 mW, not including circuit losses. Therefore the output of the VCO must be amplified. First, an estimate for circuit losses must be obtained in order to estimate the requirements of the LO amplifier chain. This is done by utilizing on-wafer probing techniques together with "through-reflect-line," or TRL calibration [20] . The test circuits were fabricated using 0.45 µm-thick gold evaporated on a 360 µm-thick silicon substrate with resistivity of 2000-3000 0-cm and a relative dielectric constant of 11.7. On-wafer measurements were first performed on a hybridmounted DC-isolation beam-lead capacitor and 50 0 sections of CPW transmission-line. The beam-lead capacitors (Metelics MBC50-10B 12 [ 18] were needed to allow seperate biasing of the oscillator and amplifier stages. Losses in the blocking capacitor were measured to be about 0.4 dB at 20-26 GHz. The 50 n CPW transmission line losses from 20-26 GHz were measured to be about 0.5 dB per guided wavelength (5200 µm), or about I dB per centimeter, and is comparable to figures reported in other work [21]. Therefore, assuming a three stage amplifier design, the estimate for the circuit losses in the system is 1.5 dB for the three DC-blocking capacitors, and 4.5 dB losses in the planar transmission lines. This results in a total estimated loss of 6 dB , and an amplifier with a gain of 20 dB is needed to deliver 6 mW of available LO power to each mixer. The LO three-stage amplifier is designed around a singletransistor amplifier, whose layout and circuit diagram are shown in Fig. 10. It is based on a K-band {medium-power} GaAs MESFET (Fujitsu FLR016XV [19]) which is hybrid mounted to the circuit. A quarter-wave bias choke with a center frequency of 23 GHz is used throughout the LO amplifier
Fig. 11. S -parameters for a single-stage amplifier based on the FLROI6XV 7 V, Vcs 0 V, I n s 60 mA. transistor biased at Vn s
=
=
=
circuit to provide gate and drain biases for the transistors. The equivalent circuit for the choke is shown in Fig. I 0, and results in an open circuit at 23 GHz. Wire-bond crossovers (marked on the layout) equalize the CPW ground planes, and are essential in suppressing the even modes excited by the T -junctions, which form the input and output matching circuits (Fig. 10). On-wafer measurements of the single-transistor amplifier yield very good agreement with small-signal simulations (Fig. 11), for the bias conditions specified on the manufacturer Sparameter data sheets. The S-parameter data for the transistor includes the parasitic inductance for 250 µm 0.7 mil gold bond wires attached to the gate and drain of the FET. For the amplifier S-parameter measurements shown, VDs is set at 7 V with Vcs set to zero, resulting in IDs = 60 mA. The close agreement between measurement and theory in Fig. 11 are obtained after small adjustments have been made to the transmission line lengths used in the input and output matching networks during the small-signal calculations. The single-transistor amplifier is used in a balanced amplifier configuration (Fig. 12), which results in better than 10 dB return loss throughout the 21-25 GHz range using small-signal simulations. The balanced amplifier approach also ensures that the amplifier is unconditionally stable. Sections are used to match each stage, and to allow the hybrid-mounted DC blocking beam-lead capacitors to be inserted. The FLR016XV has a ldB compression point of 20 dBm (100 mW) at 18 GHz, which is graphically extrapolated from the published data to 18 dBm (60 mW) at 23 GHz. This means that a single device is individually able to supply the required power to the four subharmonic mixers. Two additional amplifier stages consisting of single-transistor amplifiers are used to boost the gain of the design. For simplicity, each stage is identical to the one illustrated in Fig. 10, and the stages are connected by 40 n sections of CPW transmission line, which allow DC blocking capacitors to be attached while simultaneously acting to impedance match each stage. The amplifier is designed to
1868
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
lso\ aled Port (500 Loa d)
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Fig. 13. array.
E and H-plane patterns for an individual horn in the monopulse
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maximize the frequency range in which the gain is above 20 dB so that the receiver could be used across a wider range of LO frequencies. The amplifier exhibits a calculated gain of greater than 20 dB from 21 GHz to 24 GHz, but the total gain of the amplifier chain was not measured. The output of the final stage is then fed to the subharmonic mixers through the CPW T -junction and the CPW-to-slotline transitions (described in Section II).
u
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Fig. 15 .
Block diagram of the 200 MHz IF processor. 0 T A BLE lll D C DIODE P ARAMETERS EXTRACTED U SING 1-V C URVE F ITTI NG, LISTED IN P AIRS FOR EACH DIODE
Channel r,
(fl) n
(V) I, (fA)
¢bi
1 7,7 1.11 , 1.08 0.76, 0.75 14, 15
2 6,6 1.09,1.10 0.77,0.69 12, 32
3 8,9 1.05,1.09 0. 74,0.68 60,50
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co - 20 TABLE IV M EASU RED T OTAL CONVERSION Loss FOR THE FOUR M ONOPULSE R ECE IVER CHANNELS, D EFINED AS THE IF POWER DI VIDE D BY THE T OTAL P OWER I NCIDENT ON THE APERTURE
Chann P.] Conversion Loss
38. ldB
2 28.ldB
3 28 .9dB
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(a)
conversion loss of channel 1 (38.1 dB) is probably due to uneven distribution of the LO power, most likely caused by nonuniformity and defects in the polyimide used to produce the overlay capacitors at various points in the circuit, such as the LO-IF isolation capacitors, and the RF mixer choke. By using R s = 6 fl , GP = 35 fF , and C10 = 75 fF with the embedding impedances shown in Fig. 6 in the harmonic balance analysis, the calculated mixer conversion loss is 19.5 dB and is dominated by the effects of the high series resistance. When combined with the 3.5 dB antenna losses, this gives a total calculated conversion loss (PrF /Pinc) of 23 dB. It is clear that about 5-6 dB of loss remains unaccounted for. The discrepancy between calculated and measured conversion loss may be due to the following reasons: 1) An insufficient amount of LO power is being delivered to the antiparallel diodes for optimum conversion loss. 2) A higher series resistance exists in the diode due to skin effects at 94 GHz. 3) Antenna losses in the horn structure are higher than expected. 4) Physical differences between the modeled antenna and the fabricated antenna are influencing the impedance of the dipole. In order to see if the mixer performance was being opti-
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DISTANC E in mm
Fig. 7. Voltage standing wave pattern in unmitered single bend cavity. Nominal microstrip width = 358 µm.
no effect. This is illustrated in Fig. 7 which shows the standing wave pattern and bend placement corresponding to the 39.59 GHz resonance of Table IL Placement at a null is seen to correspond to very low sensitivity. ACKNOWLEDGMENT
Alumina with dielectric constant measured at approximately 14 GHz was obtained by Ion Beam Milling, Manchester, NH 03103 and used to fabricate microstrip cavities according to Rome Laboratory designs. Specifications included 99.6% alumina with 2 microinch surface finish or better and metalization consisting of 200-500 A titanium/tungsten base layer plus 150 microinches gold. Substrate thickness and microstrip dimensions were measured by View Engineering, Simi Valley, CA 93063. Technical assistance was provided by George A. Roberts of Rome Laboratory. REFERENCES [I] K. C. Gupta, R. Garg, and R. Chadha, Computer-Aided Design of Microwave Circuits. Dedham, MA: Artech House, 1981. [2] T. C. Edwards, Foundations for Microstrip Circuit Design. New York: Wiley, 1981. [3] R. K. Hoffmann , Handbook of Microwave Integrated Circuits. Norwood, MA: Artech House, 1987. [4] P. Silvester and P. Benedek, "Microstrip discontinuity capacitances for right-angle bends, T-junctions and crossings," IEEE Trans. Microwave Theory Tech., vol. MTT-21, pp. 341-346, May 1973. Correction, vol. 23, p. 456, May 1975) [5] R. Horton, "The electrical characterization of a right-angled bend in microstrip line," IEEE Trans. Microwave Theory Tech. , vol. MTT-21 , pp. 427-429 , June 1973. [6] A. Gopinath and B. Easter, "Moment method of calculating discontinuity inductance of microstrip right-angle bends," IEEE Trans. Microwave Theory Tech. vol. MTT-22, pp. 880-883, Oct. 1974. [7] A. F. Thomson and A. Gopinath, "Calculation of microstrip discontinuity inductances," IEEE Trans. Microwave Theory Tech., vol. MTT-23 , pp. 648-655, Aug. 1975 . [8] R. Mehran, "The frequency dependent scattering matrix of microstrip right-angle bends , T-junctions and crossings," AEU, vol. 29, pp. 454-460, 1975. [9] _ _ , "Frequency dependent equivalent circuits for microstrip rightangle bends , T-junctions and crossings," AEU, vol. 30, pp. 80-82, 1976. [IO] W. Menzel, "Frequency-dependent transmission properties of truncated microstrip right-angle bends," Elect. Lett., vol. 12, p. 641, Nov. 1976.
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[II] B. Easter, et. al., "Theoretical and experimental methods for evaluating discontinuities in microstrip," Radio Electronic Eng., vol. 48, pp. 73-84, Jan./Feb. 1978. [12] R. J. P. Douville and D. S. James, "Experimental study of symmetric microstrip bends and their compensation," IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 175-182, Mar. 1978. [13] K. C . Gupta, R. Garg, and I. J. Bahl, Microstrip lines and Slotlines . Dedham, MA: Artech House, 1979. [14] P. Anders and F. Arndt, "Microstrip discontinuity capacitances and inductances for double steps, mitered bends with arbitrary angle, and asymmetric right-angle bends," IEEE Trans. Microwave Theo ry Tech., vol. MTT-28, pp. 1213-1217, Nov. 1980. [15] M. Kirschning, R. H. Jansen, and N. H. L. Koster, "Measurement and computer-aided modeling of microstrip discontinuities by an improved resonator method," IEEE Int. Microwave Symp. Dig., 1983, pp. 495-497. [16] X. Zhang and K. K. Mei, "Time-domain finite difference approach to the calculation of the frequency-dependent characteristics of microstrip discontinuities," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 1775-1787, Dec. 1988. [17] A. D. Broumas, H. Ling, and T. Itoh, "Transmission properties of a rightangle microstrip bend with and without a miter," IEEE Trans. Microwave Theory Tech. , vol. 37, pp. 925-929, May 1989. [18] R. H. Jansen and L. Wiemer, "Full-wave theory based development of MM-wave circuit models for microstrip open end, gap, step, bend, and tee," IEEE MTT-S Int. Microwave Symp. Dig., 1989, pp. 779-782. [19] J. Moore and H. Ling, "Characterization of a 90° microstrip bend with arbitrary miter via the time-domain finite difference method," IEEE Trans. Microwave Theory Tech. , vol. 38, pp. 405-410, Apr. 1990. [20] D. C . Chang and J. X. Zheng, "Electromagnetic modeling of passive circuit elements in MMIC," IEEE Trans. Microwave Theory Tech., vol. 40, pp. 1741-1747, Sept. 1992. [21] P. H . Harms and R. Mittra, "Equivalent circuits for multiconductor microstrip bend discontinuities," IEEE Trans. Microwave Theory Tech., vol. 41 , pp. 62-69, Jan. 1993. [22] Touchstone®, a software simulator for linear microwave and RF circuits, is a product of EEsor®, Westlake Village, CA 91362. Versions 2.1, 3.0 and 4.0 are applicable to this paper. [23] Super-Compact®, a microwave computer aided design program for performing linear simulation, is a product of Compact® Software, 483 McLean Blvd, Patterson, NJ 07504. Super-Compact® PC, version 4.1 is applicable to this paper. [24] I. M. Stephenson and B. Easter, "Resonant techniques for establishing the equivalent circuits of small discontinuities in microstrip," Elect. Lett., vol. 7, pp. 582-584, Sept. 1971. [25] W. J. R. Hoefer and A. Chattopadhyay, "Evaluation of the equivalent circuit parameters of microstrip discontinuities through perturbation of a resonant ring," IEEE Trans. Microwave Theory Tech., vol. 23, pp. 1067-1071, Dec. 1975. [26] V. Rizzoli and A. Lipparini, "A resonance method for the broad-band characterization of general two-port microstrip discontinuities ," IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 655-660, July 1981. [27] T. Uwano, "Accurate characterization of microstrip resonator open end with new current expression in spectral-domain approach," IEEE Trans. Microwave Theory Tech., vol. 37, pp. 630-633 , Mar. 1989. [28] R. A. York and R. C. Comton, "Experimental evaluation of existing CAD models for microstrip dispersion," IEEE Trans. Microwave Theory Tech., vol. 38, pp. 327-328, Mar. 1990. [29] A. J. Slobodnik, Jr., R. T. Webster, and G. A. Roberts, " 18-42 GHz experimental verification of microstrip coupler and open end capacitance models," IEEE Trans. Microwave Theory Tech. , vol. 40, pp. 584-587, Mar. 1992. [30] J. C. Rautio, "Experimental validation of electromagnetic software," Int. J. Microwave and Millimeter-Wave Computer-Aided Eng., vol. 1, pp. 379-385, 1991. [31] L. P. Dunleavy and P. B. Katehi , "A generalized method for analyzing shielded thin microstrip discontinuities," IEEE Trans. Microwave Theory Tech. , vol. MTT-36, pp. 1758-1766, Dec. 1988. [32] _ _ , "Shielding effects in microstrip discontinuities," IEEE Trans. Microwave Theory Tech., vol. MTT- 36, pp. 1767- 1773, Dec. 1988. [33] I. Wolff, "From static approximations to full-wave analysis: the analysis of planar line discontinuities," Int. J. Microwave Millimeter-Wave Computer-Aided Engineering, vol. 1, pp. 117-142, 1991. [34] R. H. Jansen and M. Kirschning, "Arguments and an accurate model for the power-current formulation of microstrip characteristic impedance," AEU, vol. 37, pp. 108-112, 1983. [35] W. J. Getsinger, "Measurement and modeling of the apparant characteristic impedance of microstrip," IEEE Trans. Microwave Theory Tech., vol. MTT-31, pp. 624-632, Aug. 1983.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. IO, OCTOBER 1994
Andrew J. Slobodnik, Jr. (S '65-M' 68) received the B.S. and M.S. degrees in electrical engineering from the Massachusetts Institute of Technology in 1966. After an additional year of graduate study at the University of California, Berkeley, he began working for what is now the Electromagnetics and Reliability Directorate, Rome Laboratory, Hanscom AFB , MA, where he currently works. His research interest is in the area of monolithic millimeter-wave integrated circuit research and development.
Richard T. Webster (S'76-M'76) received the B.S. and M.E. degrees in electrical engineering from Rensselaer Polytechnic Institute, Troy, NY, in 1973 and 1976, respectively. In 1980, he joined what is now the Electromagnetics and Reliability Directorate, Rome Laboratory, Hanscom AFB, MA, where he currently works. His work in the Componenet Technology Branch has included research on surface acoustic waves (SAW's), the interaction of SAW's with semiconductors, and most recently , the development of monolithic microwave and millimeter wave integrated circuits. He is the author of a number of technical publications and presentations in those areas. Mr. Webster is a member of Sigma Xi.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. IO, OCTOBER 1994
Spike Leakage of Thin Si PIN Limiters Alford L. Ward, Robert J. Tan, and Roger Kaul, Senior Member, IEEE
Abstract-Thin PIN diode limiters (10 µm or less) are used to protect sensitive microwave components from fast-risetime microwave pulses having energies exceeding 1 to 10 µJ. This paper analyzes and experimentally confirms the performance of these PIN limiters. It is shown that spike leakage is a transittime effect that is controlled by the mobility of the carriers. A p-type background /-region should yield less spike leakage energy for a given thickness. It is proposed that the hysteresis effect observed when limiters are operated under cw conditions is due to space charge effects and stored charges remaining after the reverse-biased half cycle. Detailed agreement between the measured and calculated device voltage waveforms requires accurate modeling of the circuit parasitics because of the high rate-of-change currents arising from avalanching.
I. INTRODUCTION
NTENSE microwave pulses with energies on the order of 10 µJ damage sensitive electronic devices (e.g., metal semiconductor field-effect transistors (MESFET's) and high electron mobility transistors (HEMT's), in the front end of systems [1]. Today several options exist for protecting against these pulses, such as PIN limiters [2], monolithic microwave integrated-circuit (MMIC)-compatible FET limiters [3], gas plasma discharge devices [4], and several others [5]. The first choice for limiters, until the advent of the FET limiters, was the low-cost, low-loss PIN limiter. This device, which is available from several manufacturers, is suitable for insertion into wideband, 50-ohm coaxial systems to protect against "radar-like" pulses. Other designs, frequently combined with gas-discharge limiters, have been used in coax as radar receiver protectors [6]. The "radar-like" waveforms have pulse lengths to several microseconds and risetimes of tens of nanoseconds. This paper considers the theoretical and experimental performance of PIN limiters subjected to shorter, faster pulses with the same energy per pulse. Typical pulse lengths are 100 ns or less with pulse risetimes in the 1- to 2-ns range. The goal of our limiter designs is to minimize the spike leakage and maximize the isolation. To achieve this goal we have, in some cases, used dual-diode limiters [2], modified the doping of the PIN diode [7], and used the forward-biased de pulse characteristic [7] to parameterize the diode. Thin intrinsic region (typically less than 10 µm) PIN diodes are used in order to respond to the fast pulse risetime. As a result, the diffusion of the carriers from the heavily doped P and N regions has a dramatic effect on the diode's free carrier distribution. In fact, our analyses indicate that the performance of the thin diode could not be explained with the
I
Manuscript received February 10, 1992; revised November 5, 1993. A. L. Ward is deceased. R. J. Tan and R. Kaul are with RF Effects and Hardening Technology Branch, Army Research Laboratory, Adelphi, MD 20783 USA. IEEE Log Number 9404163.
Spike leakage
Flat leakage
Time
Fig. I. Spike leakage is the initial transient power passing limiter before it turns on. Shaded area indicates energy in the spike.
"textbook model" of abrupt junctions. The actual slope of the carrier density in the I-region was needed to accurately model the current-voltage characteristics so that they agree with the measured values when current densities of several thousand amperes per square centimeter are present. This paper presents analytical and experimental results related to spike leakage. These results are preceded by a definition of the terms in the paper, an overview of the computer program used in the analysis, and a description of the experimental setups used throughout. The paper concludes with a discussion of unresolved issues. Some de forwardbiased switching and forward-biased rf conductivity results are available elsewhere [7]. Initial damage threshold results have also been published [8]. There is a fairly extensive body of literature on PIN limiters. Reference [9] provides a full chapter on PIN diode limiters, including eight references. Other basic limiter properties are discussed in [10]. Reference [11] is still referenced extensively today. High power switching, with limiter implications, was discussed in [12]. Reference [13] provides a modem extension of this paper. There is much useful practical limiter information in the trade literature-for example, a paper on thermal properties of limiters [14]. To repeat, our work differs from these earlier works in two basic areas: 1) This work emphasizes thin I-region devices because we desire low spike leakage levels. 2) We include the charges in the P-I and the I-N junction regions since, for thin devices, these charges represent a significant percentage of the stored charge in the device.
II. DEFINITIONS Fig. 1 illustrates spike leakage. Although the energy in the PIN limiter spike is small compared to the gas plasma limiter, it is a risk factor to sensitive electronics, especially when wide (10 µm or more) intrinsic region diodes are used.
0018-9480/94$04.00 © 1994 IEEE
1880
IEEE TRANSACTIONS ON MI CROWAVE THEORY AN D TECH NIQUES, VOL. 42, NO. 10. OCTOB ER 1994
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about one decade for a three-decade increase of input power. The variation of spike energy with I-region thickness is much greater, as shown in Fig. 7. The measured spike leakage energy was also found to increase with frequency, as shown in Fig. 8. The risetime of the rf input pulse is 1.5 ns for the data in Figs. 5-8. B. Calculations
As discussed in Section III, the forward and reverse halfcycles of the rf input for a diode must be calculated in separate forward and reverse half-cycle runs. The forward half-cycle is similar to the initial portion of the video tum-on pulse and the reverse half-cycle is similar to the initial portion of the tum-off or recovery of the video pulse. Fig. 9(a) shows the composite curve of the diode voltage as a function of time of the two half-cycles fo r a 10-µm PIN diode with a 400-V rf signal applied. The frequency is 1.5 GHz and the diode area is 3 x 10- 4 cm- 2 . The diode current is shown in Fig. 9(b). The voltage and current rectification is only moderate for this diode width and frequency . The distribution of the carriers across the diode at selected times for the forward half-cycle of the waveform of Fig. 9
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Fig. 11 . Electric fie ld distributions corresponding to charge distributions of Fig. 10.
is shown in Fig. 10. Evidence of the lower hole mobility as compared to the electron mobility is noted during the initial portion of the half-cycle. The corresponding field distributions for the carrier distributions of Fig. 10 are shown in Fig. 11. The fields are high in the central intrinsic region of the diode. The final distributions of Fig. 10 are used as initial conditions for the reverse bias half-cycle calculation. The reverse halfcycle carrier distributions and fields are shown in Figs. 12 and 13, respectively. Note that the electrons are removed from the diode much more rapidly than the holes, resulting in high
1883
WARD et al.: SPIKE LEAKAGE OF THIN SI PIN LIMITERS
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fields on the P side of the diode. In Fig. 12, with reverse bias, the electrons move to the right and the holes to the left. The trailing edge of the electron distribution is extremely sharp because, as seen in Fig. 13, the field is higher at the trailing edge, and any backward diffusing electrons will move faster in the higher field and catch up. Further discussion is included in later sections, but note that the "text-book" distribution of charges for PIN diodes is not correct during the transient period. Also note the majority of the charge in the I-region is located adjacent to the junctions. It is impractical to try to calculate entire unbiased rf tum-on transitions, i.e., the full spike leakage transients of many tens or hundreds of cycles; th~ time and effort would be excessive. Instead, we have made calculations for only selected portions of the transient. These have been sufficient to increase our understanding of spike leakage. Fig. 14 shows one example of the fit obtained between a calculated and measured tumon voltage transient for a 2-µm diode. The applied 1.5-GHz power reached its maximum of 100 W in about five cycles. As seen in Fig. 14, the second cycle was fit adequately with a calculated sinusoidal of 25 V; and the third cycle, with 55 V. The calculated stored charge remaining at the end of the reverse portion of cycle 3 was inadequate to measurably increase the amount of stored charge in the following forward half-cycle. Two reasons for undercalculating the stored charge were boundary conditions and unrealistic recombination rates.
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The boundary conditions were not modeled exactly, because DIODE becomes unstable with negative electric fields. Negative fields exist at the P-I and I-N junctions at low bias levels to counter diffusion effects. This limitation establishes the maximum P and N region doping levels that can be simulated with DIODE at low applied bias. For high applied voltages, the maximum doping levels must be high enough to keep the boundary fields near zero; otherwise the calculated diode voltage is too high. Experience indicates that for valid calculations the maximum doping level must be several times the plasma density in the center of the I region. This problem is less severe with rf excitation because the low-field producing instabilities are moving. The stored charge had to be arbitrarily increased nearly an order of magnitude in order to fit the fifth reverse half-cycle. The spike-leakage transient measurement of Fig. 14 was repeated with a second diode of the same lot. This time the applied (input) voltage signal was also recorded and is included in Fig. 15 for a 5-W input pulse. Although the measured input power was 10 W, the maximum voltage excursion
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
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was measured to be only about 23 V. This corresponds to a power of slightly under 5 W. The oscilloscope calibration was checked and found to be about 13 percent below the true voltage. The reason for the remaining error has not been ascertained. A comparison of Figs. 14 and 15 shows that there is a wide variation of the amplitude and shape of the forward half-cycles in Fig. 14 but very small variation in those of Fig. 15. Calculations show little variation in the forward half-cycle voltage waveforms. The increase in spike leakage power with increased input power is readily calculated for the various diode widths. Howev.er, the spike duration may only be qualitatively seen to decrease with the increase in input power. This decrease in spike duration has been shown to result from the effect of increased stored charge in the forward half-cycle and the resulting longer time required to remove that charge during the reverse half-cycle. Space charge effects are dominant in slowing charge removal. This effect is also observed in de reverse switching and other charge collection calculations [19]. It was shown in Fig. 10 that the lower mobility of the holes results in a slower transit time across the intrinsic region than that for the electrons. The field in the intrinsic region remains high until the holes complete their transit and form a plasma with low space charge fields. Most PIN diodes are actually Pv-N diodes, with v indicating low density of donors (excess electrons) of about 10 14 cm- 3 or less. It was decided to make calculations for a P-7r-N diode, where 7r indicates a low density of acceptors (excess holes). Calculations with 7r = 1 x 10 14 cm - 3 did show a marginally faster forward tum on and a lower maximum voltage. Increasing p to 1 x 10 15 cm- 3 resulted in a marked improvement, as shown in the inset of Fig. 16. Unexpectedly, the P-7r-N diode showed a lower quasi-equilibrium impedance [7] than the P-v-N diode at low currents, as shown in Fig. 16. Experimental verification of this improvement will be required.
C. Spike Leakage Paradigm As a result of many calculations, the following paradigm has been proposed. At low powers, all the stored charges from the forward half-cycle are collected in the reverse half-cycle. At moderate input powers, a few stored (low mobility) holes remain at the end of the reverse half-cycle. These stored holes,
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Fig. 17. Comparison of measured short pulse (600 ns) and cw output power as a function of input power at 1.5 GHz for 10-µm diode. CW curves show a hysteresis effect. Also shown are correspond ing calculated results.
in tum, reduce the tum-on time and thus increase the charge stored in the next forward half-cycle. This process takes a large number of cycles before electrons also remain at the end of the reverse half-cycle. At this stage the effective diode width is reduced and the diode impedance rapidly drops. This can explain the long duration but rather sudden end of the spike leakage transient at moderate input power. At high input power, space charge effects are large enough to cause electrons to remain at the end of the reverse half-cycle and the transition is rapid. This paradigm is also a plausible explanation for a cw hysteresis effect first observed in this laboratory in 1986. We observed that as the cw rf power was increased, the limiter output voltage increased nearly linearly with the input voltage until a sudden drop occurred. Further increase of the input power showed only a slight increase in limiter output voltage. Even this drop is not included in the conventional limiter action. Moreover, once the limiter was turned on, the input power could be dropped below the transition point and the limiter output would remain low until it met the original curve. One hysteresis curve is shown in Fig. 17 for a 10-µm diode. Once the cw input power exceeds the transition level near 20 W, the lower input-output power curve may be traversed in either direction. But once the input power is reduced to the level corresponding to the upper curve ("stored-charge level"), only that curve is followed until the input power once again exceeds the transition level. It is postulated that the transition level for cw is the point where stored charges remain at the end of the reverse half-cycle. Once this stage of redistributed charges and electric field is reached, the stored charges would remain even as the input power is reduced. The cw transition power is close to the pulsed power where limiter action is first observed. No cw hysteresis is observed for thin intrinsic region diodes (below 2 µm), and spike leakage is also quite small. There are no hysteresis effects under pulsed conditions, but the maximum power in the spike is nearly continuous with the upper cw leg, and the flat leakage is approximately .the same as the cw lower leg above the transition.
..... iir
WARD et al. : SPIKE LEAKAGE OF THIN SI PIN LIMITERS
VI. CONCLUSION AND FuTURE WORK
A combined theoretical and experimental study of spike leakage has been made for thin silicon PIN limiters exposed to short, single-pulse excitation. The diodes ranged in intrinsic-level thickness from 0.5 to 10 µm thickness. The main experimental observations were as follows: 1) the spike leakage duration decreases with input power, 2) spike energy tends to increase slightly with input power, 3) the spike energy increases with intrinsic region width for a fixed input power, and 4) the spike energy increases with frequency. The computer studies showed that these results are caused by a complex interaction of transit-time effects, stored charges from forward conduction, and space charge effects during reverse conduction. Supporting our studies of spike leakage are studies of burnout [8], and de transients and the forward-biased rf conductivity of PIN diodes [7] published earlier. There are fewer limitations on the computer simulation of these latter studies, and comparisons with measurements are more complete. In all cases, the comparison of computed and measured voltage and current waveforms has contributed the most to our understanding of spike leakage and burnout. The waveform studies also point to the importance of the external circuit in the limiter's performance. In particular, the effect of a series inductance (derived from the diode leads, packaging, or probes) was simulated to provide good agreement with measured waveforms.
1885
Theory Tech., vol. MTT-30, pp. 87S-882, June 1982. [13) G. Hiller and R. H. Caverly, "The reverse bias requirement for pin diodes in high power switches and phase shifters," IEEE MTT-S Int. Microwave Symp. Dig., Dallas, TX, May 8-10, 1990, pp. 1321-24. [14] P. Sahjani and E. Higham, "Pin diode limiters handle high-power input signal s," Microwaves and RF, pp. 19S-99, Apr. 1990. [IS] A. L. Ward, Calculations of Second Breakdown in Silicon Diodes, Harry Diamond Labs., HDL-TR-1978, Aug. 1982. [16] A. L. Ward and S. L. Kaplan, User 's Manual for DIODE, Harry Diamond Labs, HDL-SR-91-1 , May 1991. [17] S. M. Sze, Physics of Semiconductor Devices. New York: Wiley, 1969. [ 18) R. Van Overstraeten and H. DeMan, "Measurement of the Ionization Rates in Diffused Silicon p-n Junctions," Solid-State Electron. vol. 13 , pp. S83-608, 1970. [ 19] Microwave Diode Research, prepared by Bell Telephone Labs., Rep. 17, Contract DA36-039 sc8902S , under contract to the U.S. Army Electronics Research and Development Laboratory and the U.S. Army Electronics Material Agency, 1964.
Alford L. Ward (deceased) was born August 14, 1919, in Rockville, MD. He received the Ph.D. from the University of Maryland, College Park, in 19S4. He spent his entire professional career with the United States Army Harry Diamond Laboratories (now part of the Army Research Laboratory). His activities included transport properties and second breakdown in semiconductors, avalanche and Gunn oscillators, breakdown in gases, and most recently , microwave limiting and switching in semiconductors. He authored more than 100 publications in engineering and science journals. Dr. Ward was a Fellow of the American Physical Society and the American Association for the Advancement of Science. He endowed the Alford Ward Chair in Semiconductor Science at the University of Maryland. He died March 3, 1992.
REFERENCES [I] J. H. McAdoo, W. M. Bollen, and R. V. Garver, "Single-pulse RF damage of GaAs FET amplifiers," IEEE MTT-S Int. Microwave Symp. Dig., New York, NY, May 2S-27, 1988, pp. 289-92. [2] R. J. Tan and R. Kaul, "Dual-diode limiter for high-power/low spike leakage applications," IEEE MTT-S Int. Microwave Symp. Dig., Dallas, TX, May 8- 10, 1990, pp. 7S7-60. [3] C. Vasile, E. Merenda, T. Marynoski, and R. Kaul, The GESD MMICCompatible Limiter, Annual Report of the Grumman Electronic System Division and Harry Diamond Laboratories Cooperative Research and Development Agreement, Aug. 30, 1991. [4] S. D. Patel, L. Dubrowsky, S. E. Saddow, R. Kaul, and R. V. Garver, "Microstrip Plasma Limiter," IEEE MTT-S Int. Microwave Symp. Digest, Long Beach, CA, June 10-14, 1989, pp. 879-882. [SJ R. A. Sparks and R. Dibiase, "Premature decline limiting in X-band YIG limiters," IEEE Microwave Theory Tech. Symp. Dig., Palo Alto, CA, 197S, pp. 243-4S, and S. Stitzer and H. Goldie, "A multi-octave frequency selective limiter," IEEE Microwave Theory Tech. Symp. Dig., Boston, MA, 1983, pp. 326-28. [6] S. D. Patel and H. Goldie, "A 100 kW solid-state coaxial limiter for L-Band," Microwave J., Part I, Dec. 1981 , p. 61 and Part II, Jan. 1982, p. 93. [7] R. J. Tan, A. L. Ward and R. Kaul, "Transient response of PIN limiter diodes," IEEE MTT-S Int. Microwave Symp. Digest, Long Beach, CA, June 13-IS, 1989, pp. 1303--06. [8) _ _ , "Calculated and Measured Silicon PIN Limiter Short-Pulse Damage Thresholds," IEEE MTT-S Int. Microwave Symp. Digest, Dallas, TX, May 8-10, 1990, pp. 761-64. [9] R. V. Garver, Microwave Diode Control Devices. Norwood, MA: Artech House, 1976. [!OJ H. A. Watson, Microwave Semiconductor Devices and their Circuit Applications. New York: McGraw-Hill, 1969. [I I] D. Leenov, "The silicon pin diode as a microwave radar protector at megawatt levels," IEEE Trans. Electron Devices, vol. ED-I I, pp. S3-61 Feb. 1964. [12) M. Caulton, A. Rosen, A. Stabile, and A. Gombar, "Pin diodes for low frequency high power switching applications," IEEE Trans. Microwave
Robert Tan was born March 20, 1962, in Baltimore, MD. He received the B.S . degree in electrical engineering from Drexel University, Philadelphia, PA, and the M.S. degree in electrical engineering from Catholic University, Washington D.C., in 198S and 1988, respectively.
Roger Kaul (S'60-M'62-SM'88) was born November 26, 1940, in Cleveland, OH. He received the Ph.D. in electrical engi neering and applied physics from Case Western Reserve University, Cleveland, OH, in 1969. From 1969 to 1974, he researched Gunn instabilities at the United Aircraft Research Laboratories. From 1974 to 1981, he performed space system studies at ORI, Inc. From 1981 to 1987, he conducted studies related to electronic warfare and millimeter-wave communication systems at Litton' s Amecom Division. He is currently responsible for developing protection circuitry for MMIC devices, as well as supervising a hardening technology team at the Army Research Laboratory. He is a part-time instructor at Johns Hopkins University, Baltimore, MD. Dr. Kaul is active in the Washinton DC/Northern Virginia MTI-S Chapter and a member of the MTT-16 Technical Program Committee on Microwave Systems.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. IO, OCTOBER I994
Dielectric Properties of Single Crystals of Al20 3, LaAl03, NdGa03, SrTi03, and MgO at Cryogenic Temperatures Jerzy Krupka, Richard G. Geyer, Matthias Kuhn, and Johann Heyen Hinken, Senior Member, IEEE
Abstract-A dielectric resonator technique has been used for measurements of the permittivity and dielectric loss tangent of single-crystal dielectric substrates in the temperature range 20-300 K at microwave frequencies. Application of superconducting films made it possible to determine dielectric loss tangents of about 5 x 10- 7 at 20 K. 1\vo permittivity tensor components for uniaxially anisotropic samples were measured. Generally, singlecrystal samples made of the same material by different manufacturers or by different processes have significantly different losses, although they have essentially the same permittivities. The permittivity of one crystalline ferroelectric substrate, SrTi03, strongly depends on temperature. This temperature dependence can affect the performance of ferroelectric thin-film microwave devices, such as electronically tunable phase shifters, mixers, delay lines and filters.
I.
superconducting films
dielectric sample
adjustable coupling loop
Cu I
f
~.,
e
~~1~.;. 0
e
e, µ,, b
[e0 0e 0]0 0 0 e,
p
a
Fig. I. Crosssection of resonant setup for measurements of permittivity and dielectric loss tangent at cryogenic temperatures .
INTRODUCTION
NIQUE performance parameters of microwave devices employing high temperature superconducting (HTS) films result from their small dissipation losses. Very often these parameters are limited by dielectric substrate losses rather than by the surface resistance value of the HTS film, especially at lower microwave frequency bands. Many attempts have been made to characterize dielectric properties of materials which are used for deposition of HTS films [1]-[4]. Those papers show similar values for the real part of the permittivity but different values for losses. In part, these differences are due to inaccuracies in the values of the loss tangent obtained using common measurement methods, especially for tan 8 less than 10- 4 . However, microscopic dissimilarities between materials which are produced by different manufacturers or by different techniques can also cause variations in dielectric loss. In this paper a very sensitive method is used to measure permittivity and dielectric loss tangent of cylindrical samples of dielectric substrates. This method makes it possible to measure accurately dielectric loss tangents below 10- 6 .
U
Manuscript received August 13 , 1993; November 23 , 1993 . This work was supported in part by the Polish-American Maria Sklodowska-Curie Foundation under contract MEN/NJST-88-92 and in part by the Deutscher Akademischer Austauschdienst (DAAD). J. Krupka is with Jnstytut Mikroelektroniki i Optoelektroniki Politechniki Warszawskiej , Koszykowa 75, 00-662 Warszawa, Poland. R. G. Geyer is with the National Institute of Standards and Technology, Electromagnetic Fields Division, Boulder, CO 80303, U.S.A. M. Kuhn and J. Hinken are with the Forschungsgesellschaft ftir lnformationstechnik mbH, Postfach 1147, D-31158, Bad Salzdetfurth, Germany. IEEE Log Number 9404161.
II. THEORY
The resonant system which was used to measure permittivity and dielectric loss tangent at cryogenic temperatures is shown in Fig. 1. The dielectric sample under test constitutes a cylindrical dielectric resonator situated between two parallel superconducting films. The height of the dielectric sample is slightly greater than the height of the copper cavity. Copperberyllium springs are used to hold the sample tightly between the cavity endplates. Provided that the sample faces are optically finished, there are essentially no air gaps between the sample and the endplate. The cylindrical samples may in general possess uniaxial anisotropy along the cylinder axis. Measurements on isotropic materials are performed using one of the TEonl modes (usually the TEon mode). The same circularly symmetric modes are also used for measurements of the permittivity tensor component (E) that is perpendicular to the anisotropy axis of uniaxially anisotropic substrates. For anisotropic materials, determination of the parallel permittivity tensor component (E z ) requires additional experiments with one of the hybrid or TM modes . We have used the HEn1 and the HE 211 modes since they can be easily excited. The permittivity perpendicular to the anisotropy axis has been evaluated by solving the characteristic equation for the TEonl mode family (see [5] and Appendix), while taking into account the geometric dimensions of the resonant structure and the resonant frequency for the appropriate mode. Since we know
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KRUPKA er al.: DIELECTRIC PROPERTIES OF SINGLE CRYSTALS
E and the resonant frequency for at least one hybrid mode, we can evaluate Ez by solving the characteristic equation for hybrid modes ((2) in Appendix). The dielectric loss tangent for isotropic materials has been evaluated using the formula
tan8 = Pe(l/Qu - l/Qc),
(1)
where Qu is the unloaded Q factor for the resonant system with dielectric sample, Q c is the Q factor depending on conductor losses, and Pe is the electric-energy filling factor. The electric-energy filling factor, whose value in most cases is close to I , can be evaluated with exact formulas which require prior knowledge of the permittivity. Evaluation of Qc requires additional knowledge of the surface resistances of the HTS films that are used and of copper. These surface resistances have been measured at 24.9 GHz using the sapphire resonator technique [5]. We can obtain surface resistances at any other frequency by assuming a f 2 surface resistance dependence for superconducting plates and a f 1/ 2 surface resistance dependence for copper. Evaluation of the dielectric loss tangent for the permittivity tensor component parallel to the anisotropy axis is also theoretically possible. However, in this case measurement error would be much greater due to the appearance of hybrid mode splitting. This splitting is caused by unavoidable imperfections in the axial symmetry of the resonant sample under test. The relative measurement error for E in our resonant system is approximately 2 times greater than that introduced by measurement errors of dielectric sample dimensions. Systematic errors in the determination of E are only weakly affected by imperfections of the dielectric sample faces, since for TE modes the electric field has only an azimuthal component which is tangential to and therefore contiguous with the whole surface of the sample. Measurement errors for Ez, on the other hand, would be strongly affected by endplate air-gap effects. Although these effects are significantly reduced by the construction of our resonant system, they still could exist unless the sample faces are parallel and polished with optical quality. These errors increase when Ez increases and when the dielectric sample aspect ratio (2b/L) decreases. Typical errors for well-machined samples of 3 mm diameter and 4 mm height are 0.3 percent for E and 1-2 percent for Ez (if Ez does not exceed 30-40). The error in measuring dielectric loss tangent depends predominantly on Q c, which is a function of many variables (such as sample dimensions, permittivity of sample under test, and surface resistance of the resonator endplates). We will discuss the details for only one example which is representative of the lanthanum alumina samples that were measured. These samples were 4.0 mm ± 0.025 mm in length and 3.1 mm ± 0.025 mm in diameter and were place between thallium-barium-calcium-copper oxide (TBCCO) superconducting films having a measured surface resistance of 0.57 mn at 18.5 GHz and 20 K. The theoretical QC factor (computed on the basis of the exact field distribution for this
TABLE I DIMENSIONS, THERMAL EXPANSION COEFFICIENTS AND RESONANT FREQUENCIES OF TE011, TE021 MODES AT 300 K, FOR SEVERAL SAMPLES f (GHz)
3.11
4.01
18.38
f (GHz) TE021 35.71
3.11
4.05
18.33
35.67
9.2
3.IO
4.05
18.37
35.78
9.2
3.27 6.97 3.928
4.05 4.13 3.945
18.49 17.46 24.77
9.3/5 .8* 12.8 7.7/8.3*
2.025 SrTi03 • parallel to c-axis
3.95
7.21
35.63 27.92 cavity mode 15.03
Material
2b (mm)
L (mm)
TEoll
LaAI03 (I) LaAI03 (2) LaAI03 (3) NdGa03 MgO Al203
TEC (J2J2mfK) 9.2
10.4
sample) was 805 x 103 . Since the measured unloaded Qu factor at the same temperature was 520 x 103 and Pe = 0.967, (1) shows that the dielectric losses constituted more than 30 percent of the total losses. In this case, tan 8 = 6.5 x 10- 7 and was still measurable. For lower operating frequencies the measurement sensitivity would be even greater due to lower surface resistance of the superconducting cavity endplates, provided all dimensions of the system are properly scaled. III. EXPERIMENTS
Experiments were performed on oriented cylindrical samples (with the c-axis along the cylinder axis) having dimensions given in the second and third columns of Table I. The resonant setup was cooled in a closed-cycle refrigerator system (Gifford-McMahon) similar to that described in [6] . For permittivity measurements at 300 K, the HTS films were replaced with copper plates. The diameter of the copper cavity was 11.00 mm, and its height was 3.93 mm. At each temperature, the resonant frequencies for HE 111 , TE011, HE 211 , and TEo 21 modes were determined. Two TE modes were employed in order to check possible permittivity changes versus frequency, while the two hybrid modes were used to evaluate possible uniaxial anisotropy for the materials under test. The unloaded Q factors were measured only for the TE 011 mode. Permittivities were calculated, taking into account thermal expansion corrections for the samples under test. The thermal expansion coefficients used are shown in the last column of Table I [7)-[9]. The results of these measurements are presented in Figs. 2-4. For every temperature and for all materials under test, permittivity values evaluated from TE 011 and TE02 1 resonant frequencies were the same. This means that for all those materials, permittivity is not frequency-dependent, at least within the frequency limits established by these modal resonances. Also, the permittivities of three LaAl0 3 samples were the same within 0.2 percent. No measurable anisotropy was found for LaAl0 3 and MgO samples. A small anisotropy with Ez about 2 percent greater than E was noticed for NdGa0 3 ; however, it is within experimental uncertainty. For SrTi0 3 , measurements of Ez were difficult to perform. This was partially due to the
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOB.ER 1994
LoAI03 {2 )
24
10 -·
LoAJ0 3
~ ~
_,
~
23
LoAJO, (1)
1 0 -·
'o c
--"CT
0 ....,
NdGo03 .....---' ~--
21
100
50
0
..--.-----~
150
Temperature
Fig. 2.
LoAI0 3
10 _,
22
200
250
~~ *- -""""
--
~
x
MgO
10 _,
40
30
20
300
50
60
70
80
90
Temperature ( K )
( K)
Fig. 5. Temperature variations of dielectric loss tangents for LaAl03 and MgO. Sample (1) is from different manufacturer than samples (2) and (3). Sample (2) is made of single, crystal grown by Vemeuil method, sample (3 ) by Czochralski method.
Te mperature vari ations of permitti vities for LaAl03 and NdGa03.
12 .0 11.5
t:, A120 3
11 .0
10
-
-J
"t ~
c..i
"-
'
\
•\
10.5 'O
....,0
MgO 9.5
~
~"" "-.__
A120 3
9.0
100
50
0
150
Temperature
Fig. 3.
250
200
...,,,
~
i.---
SrTi03
ir-
""'-..
-
t:
-' ----
' I'\
'\ \
c
10.0
"I
~
NdGo0 3
300
( K)
10 -·
Temperature variations of permittivities for Al20 3 and MgO.
0
50
100
150
Temperature
200
250
300
( K)
Fig. 6. Temperature variations ofdielectric loss tangents for SrTi03 and NdGa03.
"Iii
10
\
4
TEaz,
~
-
.P "\. J"' ;:;;.>f ~e spatial beam profile through conventional mode scans with an apertured incoherent detector. Spatial aperture techniques combined with careful optical adjustment yielded a reasonably pure spatial mode. The remainder of this paper will describe results in separate sections: II-Coupling the Laser to the Diode; III-Coherent Measurement of the Sideband Power; IV-Sideband Separation and Spatial Mode Performance; V-Summary and Conclusions.
II. COUPLING THE LASER TO THE DIODE In order to make the most efficient use of available laser power one must optimize the coupling of the laser mode to the antenna pattern of the comer-reflector-mounted diode. Several papers have been published on this topic [ 14]-[16] and scaled model experiments have been performed [7], [ 17]. Based on these results we decided, as have other groups [13], to have the bottom face tilted away thus resulting in a 4>. antenna mounted in a dihedral, instead of a "cats eye" comer-cube. Tilting of the bottom face of the mount by 19° resulted in a 9 dB decrease in backreflected laser radiation and no observed degradation in receiver noise temperature. The above results were measured with a bolometer and a beamsplitter in the former case, and by performing noise temperature measurements, with identical comer-reflectors with and without tilted bottom faces , in the latter case. In the present study, coupling experiments were performed in order to confirm the optimal coupling parameters predicted in [16], namely that the beam waist w 0 be 1.82>. at the diode. In our experiments the size of a collimated Gaussian
0018- 9480/94$04.00 © 1994 IEEE
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 10, OCTOBER 1994
LO Submillimeter Laser
rn
cDDDDQ 00000 00000 DODO
Spectrum Analyzer
Frequency Synthesizer
Fig. 1. Setup used for the sideband power measurements. Not included in the figure, is an amplifier placed between the receiver diode and the spectrum analyzer.
laser beam was varied and subsequently measured using a computer-controlled translation stage and an apertured LHecooled bolometer. The collimated beam then propagated to a 59.7 mm focal length off-axis parabolic reflector as the final optic before the diode. To determine which beam size was optimal, the Schottky's nonlinear video responsivity was measured over a range of incident power levels between I and 10 mW for each tested configuration. This was necessary since these Schottky diodes are known to be nonlinear video detectors above rvlOO µW (18] . Assuming the paraxial wave equation to be correct for systems employing optics this fast ( 19], and using the predicted waist size and location for an off-axis parabolic reflector focussing a Gaussian beam (20] (we cannot scan at the focus of the parabolic reflector due to spatial constraints), we have confirmed the optimal waist size at the diode Wo = 1.82.).,
(I)
for a 4.X antenna mounted 1.2.X from each face of a dihedral (16]. Further we have observed that this coupling is quite sensitive; a waist size of 2.X produces noticeably weaker diode response and waists smaller than that of condition ( 1) also result in significantly less response. As the predicted (20] waist size approaches .X, the correction terms to the paraxial equation become important, and thus our "assumed" waist at the diode is slightly too small. III. COHERENT MEASUREMENT OF THE SIDEBAND POWER The apparatus for the sideband power measurement is shown in Fig. 1. The submillimeter lasers are independent systems each pumped by a nominal 100 W, CW C02 laser. The LO and Drive lasers were running the 191.61960 µm laser line (21] in CH 3 0H (1564.5187 GHz) and were tuned to operate roughly 3 MHz apart. This separated the upper and lower sideband by 6 MHz and also served to separate the sidebands from any pick-up signal present at the microwave synthesizer frequency. Without this separation, signal at the synthesizer frequency picked-up by the receiver could be mistaken for sideband radiation. The synthesizer was operated at 10 GHz and provided ,.,_,5 dBm of microwave power at
the diode. The Si etalon will be described in detail in the next section. Essentially, it transmits nearly 100% of the laser radiation while reflecting 80% of the sideband radiation . The 1 mil Mylar® beamsplitter has a reflectivity of 50% for spolarized radiation at this wavelength [22]. The power of the LO was measured to be 5.5 mW and the Drive laser power was measured for each diode tested. The laser power data was obtained with a Keating Absolute Power Meter. 1 The receiver diode was a University of Virginia (UVa) model I Tl 7 and the SBG diode was one of the following: IT7 , ITl2 , ITl7 , 1Tl5. In order to accurately measure the sideband power, it was necessary to measure the conversion loss of the receiver diode. This was done in-situ by placing a cold load and chopper in the beam path between the two Schottkys and performing measurements of the system noise temperature for different values of IF temperature. This procedure was performed 5 times in order to assure accuracy . With this data the conversion loss and mixer temperature can be determined from [23] (2)
where Ts ys is the system noise temperature, T M is the mixer temperature, Le is the conversion loss, and Tw is the noise temperature of the IF electronics. Since the measurement was performed in-situ the conversion loss included the 3 dB contribution of the beamsplitter. The measured conversion loss was 17 .3 dB which corresponds to a diode conversion loss of 14.3 dB. This fairly poor value of Le is mostly due to the "'2.5 dB measured impedance mismatch at lO GHz (24] , and to a lessor extent, the fact that 2.8 mW of LO power is not quite enough to optimize this receiver at 1.6 THz. With the above information, the gain of the IF chain, and the fact that the etalon reflects 80% of the sideband, the power in the sideband can be determined by measuring the power at the IF. Namely, Psideband = Pw - Gw
+ Le + 1 dB (etalon loss)
(3)
where Psideband is the single sideband power in dBm, Pw is the IF power is dBm, Gw is the gain of the IF chain, and Le is the conversion loss of the receiver. The IF power was measured with the spectrum analyzer set to a resolution bandwidth of 100 kHz and the amplitude accuracy of the spectrum analyzer was checked against two calibrated power meters. The accuracy of the Pw readings is dominated by spectrum analyzer calibration error which is less than 0.5 dB. A typical IF spectrum is presented in Fig. 2. The resolution bandwidth was set to 30 kHz for this data to clearly resolve the mechanical-vibration-induced integrated-laser-linewidth. It is important to note that the noise floor shown in Fig. 2 is the noise floor of the spectrum analyzer and not that of the receiver. The fairly weak pick-up signal from the microwave synthesizer is clearly seen in the figure and smaller peaks from other laser modes are also observed. 1
Thomas Keating Ltd., Billingshurst, West Sussex, England.
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MUELLER AND WALDMAN : POWER AND SPATIAL MODE MEASUREMENTS
fr, ..,.
10 dB/Div. Sideba nds
10 GHz
~ck-Up
* t-- t--~
" ~ i'. / 4 at IOOO MHz).
[I I]
(all
TABLE IV PSPICE NETLIST FOR FILTER ANALYSIS
0 0 i----------------------------------.100Mh a VM (4) /VM (3)
+------------------------------------1. ---+ 06h
300Mh
Frequency
Fig. 6. IS11 1versus frequency for low-pass filter.
"TEST OF NETWORK ANALYZER FOR FILTER FROM WOLFF AND KAUL P.247 VS 21 0 AC 1.0 0.0 RS 21 1 50 X11234NA11 X2 5 6 NA12
·NETWORK ANALYZER INPUT PORT 1
:vs AND RS ARE SOURCE
!BO . Od
·DIRECTIONAL COUPLER SUBCIRCUIT CALL '. DIRECTIONAL COUPLER SUBCIRCUIT CALL '.NETWORK ANALYZER TR ANSMISSION PORT 5 :our BETWEEN PORTS 2 AND 5
• • • • • • • • • • •••••••"LOW-PASS FILTER TO BE ANALYZED••••••••••• ••••• •••••• Tl 2 0 9 0 Z0=130 F=lOOOMEG NL=0.25 T2 9 0 5 0 Z0=130 F=lOOOMEG NL=0.25 Tll 20120Z0=81 F=1000MEG NL=0.25 T19 9 0 29 0 Z0=81 F=lOOOMEG NL=0.25 Tl 5 5 0 15 0 Z0=81 F= lOOOMEG NL=0.25
Date/Time run: om§MF
;TRANSMISSION LINE ;TRANSMISSION LINE ;STUB ;STUB ;STUB
······················EQUATIONS FOR CALCULATIONS ······ ················ Sl 1 - REF.COEFF = VM(4JNMC31 PHASE OF S 11 = VP(4J - VPC31 S 12 - TRANS. COEFF = V M(6JNMC31 PHASE OF Sl 2 = VP(6J - VPC31
~~m~~lNAL YZER
FROM WOLFF AND KAUL P. f:~perature:
27 . 0
+-----------------------------------+--------------- ------ ---- --------------t '
'
:'
:'
l :
+!
:
!
!
100 . 0dt
."') -wo1
1 1
VSWR = (1 +VM(4JNMC3111(1 -VMC4JNMC3JJ
• ••••••••••••••••••••••••••ANALYSIS**• • •••••••••••••••••••••••••••• .LIB UWAVE .LIB .AC LIN 90 1OOM EG 1OOOMEG .PRINT AC VM(3) VM( 41 VM(6 ) VP(3) VP(4) VP(6 ) .PROBE .END
analysis using the basic network analyzer system configuration of Fig. 1, node voltages representing the incident, reflected, and transmitted signals are available for further computation.
IV. EXAMPLE: NETWORK ANALYZER SIMULATION The use of the proposed model for RF and microwave circuit analysi s can be demonstrated by simulating a network analyzer to measure both reflection and transmission coefficients. Two directional couplers are required for this simulation, which can be linked to a SPICE program with a library subcircuit call or entered into the netli st. The DUT to be analyzed in this example is a low-pass filter shown in Fig. 5 [11]. Note that the directional coupler used for the reflection measurement is NAu (a four-port) and the directional coupler used for transmission wave measurement is NA 12 (a two-port). Table IV presents the netlist for a PSPICE analysis of the filter circuit of Fig. 5 which includes equations used to compute reflection coefficient, transmission coefficient, VSWR, and other related qualities as given in comment lines.
-100 . oo!.------------------ -----------------+----- ---------- -- ----- -- ---------- ---!OOMh a VP (4) -VP {3)
300Mh
1. OGh
Frequency
Fig. 7. S11 versus frequency for low-pass filter. A simulated measurement of reflection and transmission coefficients where first made wi th a "thru" connection (DUT removed in Fig. 5). The magnitude of the reflection coefficient and the transmission coefficient was found to vary less than 0.05 db and the phase difference was less than 0.3 degrees over the test frequency range for a matched system. The RF circuit of Fig. 5 was analyzed using PSPICE and the results are presented in Figs. 6-10. Figs. 6 and 7 show the magnitude and phase reflection coefficient (Su) of the filter and Figs. 8 and 9 present the magnitude and phase of the transmission characteri stics, respectively. Fig. 10 displays the filter input VSWR versus frequency which demonstrates calcul ation capability. This filter was also analyzed using TOUCHSTONE, 1 and these results are included in Table V along with numerical results fro m the PSPICE analysis for comparison. A review of the results in Table V shows that the difference between the PSPICE and TOUCHSTONE magnitude data is in the range of 0.005 and the difference in phase data is on 1 TOUCHSTONE is a RF and Microwave Circuit Analysis program marketed by EEsof, Inc. that is one of the standards of industry.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. IO, OCTOBER 1994
*TEST OF NETWORK ANALYZER FROM WOLFF ANO KAUL P. 247 Date/ Time run: 07 / 20/94 14: 19: 42 Temperature: 1.0 :
-----------------------------------+--------------------
O.Br
27 . O
----------------1
r
0.6!
l
0.4r
r
0
'1
1
+-------------------------- ---------+------------------------------------300Mh 1. OGh
0.0 !OOMh c VM (6) /VM (3)
Frequency
IS12I versus
Fig. 8.
r---------________________________
*TEST OF NETWORK ANALYZER FROM WOLFF AND KAUL P . 247 Date / Time run: 07/20/ 94 15: 01: 23 Temperature: 27 .o 10 . 0 -+- ________ ________________ ___________ _
.·1 6.01
··1 2.0+
i---------------------------
1.0 --------+-------------- ----------------------- _..j. 100Mh 300Mh 1. OGh D (!+VM (4) /VM (3) I I (1-VM (41 /VM (3) I Frequency
Fig. 10. Input VSWR versus frequency for low-pass (VSWR = (1 + VM (4)/ VM (3))/(1 - VM (4)/ VM (3))) .
frequency for low-pass filter.
•TEST OF NETWORK ANALYZER FROM WOLFF AND KAUL P. 247 Date/Time run: 07 /20/94 14: 19: 42 Temperature: 27 . o !BO . Od -+--------------- - ----------------- ---
r----------------------------------
l
•oo ")
0
-•oo
-1
"!
1
'1
-1eo . ad+.------ -- --------- ---------------- - -+- ____ ---------- __ --------------- ____ ---+ !OOMh D
VP (6) -VP (3)
300Mh
1. OGh
Frequency
Fig. 9.
¢ S12 versus frequency for low-pass filter.
the order of a few tenths of a degree except at one frequency. At l GHz, the 8 12 data differs significantly; however, the magnitude of S 12 is essentially zero (2.8 E-07 as compared to 1.6 E- 22) and the phase is quite different (-96.80° as compared to -179.08). The phase difference is not really that significant because the magnitude is zero. However, further consideration of the phase at this frequency indicates that the phase is approaching -90° in the limit as frequency approaches 1 GHz from below. General usage of thi s network analysis technique suggests that the directional coupler length should be near )., j 4 at the center of the frequency range of interest. This approach has demonstrated remarkable accuracy well over a full decade of frequency range. Primarily, this is because the design has been selected so that the signal paths have the same length and have similar frequency characteristics. It is, however, anticipated that errors in the model will begin to increase as the swept bandwidth is increased well above a decade. Broadbanding of the directional coupler could be used to extend the range of the operation using the same techniques used for broadbanding of practical directional couplers [10]. Note that the frequency at
filter
TABLE V Low-PASS FILTER CHARACTERISTICS DETERMINED WITH PSPICE AND TOUCHSTONE CIRCUIT ANALYS IS PROGRAMS FREQ MHZ 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000
PS PI CE
ls,, I ¢,,,
0.196 0.286 0.365 0.431 0.482 0.520 0 .537 0 .521 0.447 0 .238 0 .272 0 .797 0.956 0 .990 0.999 0 .997 0.995 0 .998 1 .000
54.93 37.74 20.44 3.31 · 13.50 ·30.46 -48 .28 -67.45 ·89.48 - 119 .1 14.86 ·38.50 ·74.72 ·98 .92 ·118.2 · 135 . 1 · 150.5 · 165 . 2 · 180.0
TQUCHSTQN~
ls,, I ¢,,,
0 .197 0.286 0.365 0.432 0.484 0.521 0.537 0 .522 0.450 0.239 0.272 0.799 0.960 0.992 0.999 1.000 1.000 1.000 1.000
54.93 37.59 20 .45 3.450 · 13.50 -30.60 -48.28 ·67.3 1 ·89.48 · 119 .23 14.86 ·38.36 ·74.72 ·99.07 ·118.2 -135.0 -150 .5 · 165.4 180.0
TQUCHSTQNE
PS PI CE
Is,, I
0 .976 0.956 0.931 0 .900 0.871 0.852 0.844 0.851 0 .899 0.969 0.962 0.600 0.277 0 . 123 0.052 0 .019 5.2E·03 6.2E-04 2.BE·07
¢ , 12
-35.07 -52. 26 ·69 .56 ·86.69 · 103.5 ·120.5 -138.3 -157.5 -179.5 150.9 104.9 51.50 15 .28 ·8.93 -28 .21 ·45.10 ·60. 48 -75.23 ·9 6.80
Is,, I
0.980 0.958 0.931 0.902 0.875 0.854 0.844 0.853 0.893 0.971 0.962 0.601 0.278 0.123 0.052 0.019 5 .2E·03 6.2E-04 1.6E-22
¢ , 12
-35 .07 -52.41 -69 .56 ·86.55 ·103.50 -120.60 · 138 . 28 · 157 .31 · 179.48 150.77 104.8 6 51.64 15 .28 ·9.07 · 28 .22 ·44.96 ·60.48 -75 .38 · 179 .08
1
TOUCHSTONE is a RF and Microwave Circuit Analysis program marketed by EEsof, Inc . that is one of the standards of industry.
which the directional coupler is ).,j 4 in length must be changed in the subcircuit (or library) for a particular frequency range of operation. This can be easily accomplished with the find and replace option available in most editors or wordprocessors. V. CONCLUSION
Two directional coupler circuit models based on the MarxEastin MTL model [8] have been developed for RF and Microwave Circuit analysis using SPICE-type programs. These directional couplers can be employed to separate the incident, reflected, and transmitted signals from computed node voltages for a circuit. With these signals, S-parameters and related quantities can be calculated using the limited math capability of some of the post-processors avai lable with SPICE-type programs. The high-frequency results for a low-pass filter analyzed with PSPICE/PROBE using thi s developed system indicates excellent accuracy as compared to another widely used program, TOUCHSTONE. Errors are less than 0.005 in magnitude and a few tenths of a degree in phase.
SM ITH: FREQU ENCY DOMAI N ANALYSIS OF RF AND MICROWAV E CIRCUTS USING SPICE
Because SPICE-type programs are so widely available, this addition of models for RF and microwave analysis provides a useful , cost effective, and readily available computational tool for educational programs offering RF and microwave related courses and laboratories where network analysis and simulation is required. Because of the wide range of linear and non-linear models available for SPICE, the extension of the capability for RF and microwave circuit analysis is an important enhancement of this computational tool for both educators and practicing engineers. ACKNOWLEDGMENT
The author wishes to thank the students in hi s 1991 Spring course on Computer-Aided-Design in Microwaves at the University of Mi ssissi ppi for their assistance in providing early computational data related to thi s work. In particular, he wishes to thank Mr. Shi-Dong He for his work on this subject as a class project. Also, he owes thanks to Ms. Cindy Conner for typing many drafts of this paper during its development and to the reviewers for their timely reviews and useful comments on this paper. R EFERENCES [I ) L. W. Nage l, "SPICE2: A computer program to simul ate semiconductor circu its," Electron ics Res. Lab., Univ. Cali forni a, Berkeley, ERL Memo No. ERL-MS20, May 1975 . [2) J. R. Hines, "A compari son of Berkeley SPICES," Design Automat. , pp. 28-33, Mar., 1991. [3] S. Prigozy, "Novel applications of SPICE in engineering educati on," IEEE Trans. £ducat. , vol. E-32, pp. 35-38, Feb. 1989. [4] L. V. Hmurcik, M. Hettinger, K. S. Gottschalck, and F. C. Fitchen, "SPICE applications to an undergraduate electronics program," IEEE Trans. £ducal., vol. 33, pp. 183- 189, May 1990.
1909
[SJ W. Banzhaf, "Using SPICE .in the electronics curriculum," Int. J. Appl. Eng. Educ., vol. 3, p. 335, 1987 . [6] Y. K. Tripathi and J. B. Rettig, "A SPICE model fo r multiple coupled microstrips and other transmission li nes," IEEE Trans. Microwave Theory Tech. , vol. MTT-33, pp. 1513- 151 8, Dec. 1985 . [7] C. R. Paul , "A simple SPICE model for coupled tran smiss ion lines ," Proc. IEEE Int. Symp. Electromag. Compatibil., Seattle, WA , Aug . 2-4, 1988, pp. 327-333. [8] K. D. Marx and R. I. Eastin, "A configuration oriented SPICE model for multiconductor transmiss ion lines with homogeneous dielectrics," IEEE Trans. Microwa ve Theory Tech. , vo l. 38, No. 8, Aug. 1990. [9] D. Kajfez, Notes on Microwave Circuits, vol. 2. Univers ity, MS: Kajfez Consulting, 1986. [ IO] R. Hoffman, Handbook of Microwave Integrated Circuits. Norwood, MA: Artech House, 1988. [I I) E. A. Wolff and R. Kaul, Microwave Engineering and Systems Applications. New York: Wiley , 1988, p. 247.
Charles E. Smith (S'S9-M' 68-SM'83) was born in Clayton, AL. He received the B.E.E., M.S., and Ph.D. degrees from Auburn Un iversity , Auburn , AL, in 1959, 1963, and 1968, respectively . From 1959 to 1968, he was e mployed as a Research Ass istant with the Auburn Un iversity Research Fou ndation. He joined the Uni versity of Mississippi Uni versity, Oxford, in 1968 as an Assistant Professor of Electrical Engineering. He advanced to Assoc iate Professor in 1969 and was appoin ted Chairman of the Electrical Engineering Department in 1975 and he is currently Professor and Chair. Hi s r~searc h interests are related to the app lication of electromagnetic theory to microwave circuits and antennas . Hi s recent research has been on the app lication of numerical techniques to microstrip transmi ssion lines, antenna measurements in lossy media, measurement of electrical properties of material s, CAD in microwave circuits, and data acqu isition using network analyzers. Dr. Smith is a me mber of Phi Kappa Phi, Sigma Xi , Eta Kappa Nu, and Tau Beta Pi.
1910
IEEE TRANSACTIONS ON MICROWAVE THEORY ANO TECHNIQUES, VOL. 42, NO. 10, OCTOB ER 1994
Experimental Proof-of-Principle Results on a ModeSelective Input Coupler for Gyrotron Applications Jeffrey P . Tate, Hezhong Guo, Matthew Naiman, Leemian Chen, and Victor L. Granatstein, Fellow, IEEE
Abstract- Proof-of-principle results for a mode selective input coupler are presented. Transmission and reflection measurements for the TE02 cylindrical waveguide mode are given along with the output mode pattern. The results show good agreement for the cutoff frequency, mode pattern general behavior and variation with frequency for signals above cutoff. A maximum passband of 1.2 GHz ("' 7%) has been achieved. Comparisons with theory for overall frequency response (from 15 to 18 GHz) and mode pattern characteristics (at 17.5 GHz) are also presented. The design and concept are promising for harmonic gyrotron-traveling-wavetube amplifier and phase-locked gyrotron oscillator applications.
I. INTRODUCTION
T
HE research on gyrotron-traveling-wave-tube (gyroTWT) amplifiers has been presented with some important challenges [l]. Experimental and theoretical results in both fundamental and harmonic devices have shown the importance of suppressing spurious modes in order to optimize device performance [2]. These oscillations can be caused by absolute instabilities or by circuit feedback due to reflections from the output window or collector. The experimental work on gyro-TWT's was inspired by the results of Granatstein et al. , in which a 100 kW signal at XBand was amplified and exhibited a growth rate of I dB/cm [3]. Theoretical work by Ganguly et al. and Chu et al. has established a basi s for analyzing the operating mechanism [4] , [5]. This involves the azimuthal bunching of beam electrons by an injected signal which drives the electron cyclotron maser instability. A number of theoretical investigations have been conducted for both fundamental and harmonic devices [6]-[12]. Experimental research in fundamental gyro-TWT's has progressed significantly as the work of [ 13]-[16] indicates, but significant challenges remain for attainment of stable, efficient, wideband operation of harmonic gyrotron travelingwave amplifiers. For the stable operation of harmonic gyro-TWT' s, the problems caused by spurious mode oscillations are increased because of backward-wave interactions at the fundamental and higher electron cyclotron harmonics [l ]. Circuit feedback oscillations are also possible and must be eliminated as well. In a second harmonic gyro-TWT operating in the TE 02 mode for example, spurious backward-wave oscillations at the Manuscript received Jul y 23, 1993; revised December 10, 1993. This work supported in part by the DOD Vacuum Electronics Initiative and managed by the Air Force Office of Scientific Research under Grant AFOSR-91 -0390. The authors are with the Laboratory for Plasma Research , University of Mary land, College Park, MD 20742 US A. IEEE Log Number 940415 3.
TE 21 (fundamental), TE 22 (second harmonic) and TE 13 (third harmonic) modes are possible. Techniques such as restricting interaction length and using multi-stage severed configurations have been investigated experimentally and theoretically. In the work of Chu et al., a gyro-traveling-wave tube using a sever section for suppression of TE 21 oscillations achieved 27 kW (saturated gain of 35 dB), 7.5 3 bandwidth and 163 efficiency at 33.5 GHz [2] . To our knowledge this is the best overall performance for a Ka-band severed gyro-TWT to date and it has stimulated the work of many researchers. The theoretical work of Kou et al., also suggests that multistage configurations are promising but two concerns remain, viz. 1) reflections created at the severs and 2) the average power handling capability of the severs [12]. The sever concept arises from a technique used in conventional traveling-wave tubes of separating the helix into two or more distinct sections creating regions where the EM waves cannot propagate. The sever in gyro-TWT's is usually a high loss section which severely attenuates the propagating waves thus reducing spurious oscillations. To address these concerns, a technique for achieving high purity mode selective operation would be very useful. Such a method requires a circuit designed to support a particular mode, for example the TE 02 , while suppressing others [ 17]. This can be achieved using a mode selective input coupler which incorporates a novel helix/cage mode launcher and a cylindrical mode filtering section with radial vanes. These components acting together can preferentially excite the required mode. This is done by setting up an azimuthally directed current in the helical structure which excites the TE 01 coaxial and cylindrical waveguide modes. These couple to form the TE 02 mode in a transition region. The mode filter would then allow only the TE 02 mode to propagate. Non-azimuthally symmetric modes such as the TE 21, TE22 , and TE13 oscillations would be suppressed in such a system due to the filtering effect of the helix , cage and vaned mode filter structures. The mode selective input coupler concept developed in [18] has been used in a proof-of-principle experiment. The present paper will give an analysis of the input coupler operation, and will describe the results of the proof-of-principle experiment. Once developed a practical input coupler based on the concepts demonstrated here would be employed in conjunction with novel, mode selective, interaction circuits also developed in [ 19] for stable harmonic gyrotron amplifier operation. This would allow for gyro-TWT use in millimeter-wave communications and radar systems. This paper presents the experimental results and analysis in which the excitation of a TE 02 cylindrical waveguide
0018-9480/94$04.00 © 1994 IEEE
19 11
TATE et al.: EXPERIMENTAL PROOF-OF-PRINCIPLE RESULTS ON A MODE-SELECTIVE INPUT COUPLER
TEO! COAXIAL
IN THE HELIX REGION
IN THE MODE FILTER AND OUTPUT WAVEGUIDE
Fig. 2. The transformation of TEo1 coaxial and waveguide modes into the TE 0 2 waveguide mode. The drawing shows the radial variation of the E o fieldcomponent.
Fig. I. A diagram showi ng the various reg ions of the input coupler. ( 1) TEM coaxial-to-cyli ndrical waveguide transition , (2) heli x, (3) cage I mode coupling section, (4) TEom mode filter.and (5) coaxial waveguide with tunable reflector. Some basic dimensions (in millimeters) for the mode filter are also given.
mode is demonstrated. Section II gives a description of the experimental apparatus used. In Section III, a qualitative analysis of the structure, following the method outlined in [20) will illustrate how the preferred mode is created, and will seek to establish the basis for explaining the experimental transmission and reflection results. Section IV presents the experimental results. Section V provides some di scussion and concluding remarks. II. BASIC DEVICE DESCRIPTION
The Guo mode-selective input coupler investigated in this paper was designed to excite and propagate the TE 02 waveguide mode with a cutoff frequency of 16.64 GHz. Fig. 1 shows the complete structure. A coaxial waveguide section (region 2) is excited by the extended center conductor from a side-mounted 50 coaxial connector (region 1). The TEM signal from region I produces an rf current which flows in the helix of region 2. This helix is closely wound (helix pitch 40 rim the std > 0 .5 lines were not fabric ated. In reality all the curves converge at std = 0 and std = 1.
V. RESULTS ON LINES FABRICATED ON SEMI-INSULATING GAAS SUBSTRATES
In this section the measurement results are given by describing the effects of changing b, h and w on Vph , Zo , and a. These three measured line characteristics did not change at all for line length variation of I, 2, and 3 cm which verifies the accuracy of the measurement and analysis method. Lines are labeled as (b, h, w ), where b, h, and w are the parameters defined in Fig. I with units of µm. For example (5, 30, 20) is a line for which b = 5 µm, h = 30 µm, and w = 20 µm. Other values are kept constant at d = 50 µm, g = 206 µm , and l = 1 cm. In the figures data points are connected by straight lines. In many of the figures , the regions of 0 < s/d < 0.3 and 0. 7 < sld < 1.0 were explored in detail only for one particular b, h, and w combination. Therefore, plotted results for other b, h, and w combinations crossed over the results for the particular line investigated in detail due to the lack of data points in these regions, i.e., when data points sld = 0, 0.3, or sld = 0.7, 1.0 are connected by a straight line. This happened primarily in the vph graphs. In reality, all the plots have the same qualitative shape and do not cross. The result of varying the gap width b is shown in Fig. 3. One can see that for narrow gaps the slots have a greater effect on vph· For smaller gaps the field strength in the gaps is larger hence associated modes are perturbed more by the slots. At s/d = 0 the Vp h values coincide well , indicating that changing the gap on the smooth lines had little effect on v ph as expected [14]. For sld = 1 all the lines become identical, hence the results become identical at this point. The "bathtub" shape of the vp h graph indicates that nearly all of the Vph change possible for a given h is obtainable with sld ~ 0.3. Fig. 3 also shows that Z 0 decreases as the gap gets smaller as expected. Z 0 changes almost linearly as a function of sld for this value of h. Therefore, for small values of h ( < 40 µm ) Z o may be calculated by taking the weighted average of the impedances of the sld = 0 and sld = 1 lines according to their fractions in the unit cell. Fig. 3 also shows the variation of a as a function of s/d for different b values . As expected, a increases as b gets smaller indicating greater field concentration and current crowding for narrower gaps. One might expect that an increase in h would cause an increase in a due to an increase in R resulting from longer
§:
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.. 70 ,.. .............................. . 0
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0 .2
0.4
0.6
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Fig. 5. Behavior of Zo as a function of std with h as a parameter for b = 9 µm , w = 20 1.1 m. The values of h (in rim) are indicated on the curves. Gaps in the std < 0 .5 range of the curves are e ither due to miss ing data becau se of yield or for clarity. For h > 40 µm the std > 0. 5 lines were not fabri cated . In reality all the curves converge at std = 0 and std = 1.
current paths around the slots in the ground planes. While R is certainly increasing, Z o also increases and a which can be estimated as
R 2Zo
a ~ --
(8)
does not increase in direct proportion to R . Thi s result shows that one can achieve Vph reduction without being penalized with increased a due to the presence of slots. In fact, little variation in a with h up to 80 µ m was observed. Fig. 4 shows the Vph variation as a function of sld for a wide range of h. The measured Vp h variation is roughly symmetric with respect to s/d up to h = 40 µm and it is assumed that this symmetry holds for larger h values. For h > 40 µm only the range up to sld = 0. 5 was investigated. The v ph goal was reached with h = 80 µm at sld = 0 .35. Fig. 5 shows the variation of Z 0 for different h values. The data for h > 65 µm are not shown because for thi s range the accurate extraction of Z 0 becomes difficult. Thi s is due to the mode mismatch reflection at the pad structure/slow wave line interfaces. These interfaces are at the calibration planes shown in Fig. 2(a). For large h values thi s reflection becomes larger than the reflection cau sed by the impedance mismatch. This mode mismatch reflection is not accounted for in the calibration and results in a distortion of the extracted
1922
IEEE TRANSACTIONS ON MICROWAVE TH EORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
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Fig. 6. Variation of vph, Zo and a with center conductor width w for b = 9 µm, h = 30 µm . Note that only w = 20 µm was explored for 0 < std < 0.3 and 0.7 < std < 1.0. Data poi nts in these regions are missing in the other graphs, and that is why a cross over appears. In reality the 'Vp h graphs all coi ncide.
Fig. 8. Behavior of Zo and a as a functi on of frequency at std the (9,30,20) line.
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impedance value. This problem is expected to affect all the extracted line parameter values but it shows up first in the Zo results . This is because of the three line characteristics being studied, Z 0 is the most sensitive to the reflection S-parameters su and s22, and su and s 22 are the measurement parameters that are most severely affected by the mode mi smatch. Fig. 6 illustrates the effect of changing the center conductor width w. It shows the most striking property of these lines: changing the center conductor width has no effect on vph for any s/d. However, as also shown in Fig. 6, changi ng w does change Z 0 . This allows setting vph by appropriately choosing b and h and then adjusting Z 0 independently with w. As described later, this was in fac t what was done to arrive at the final design. Fig. 6 also shows a as a function of s/d for different w values. Increasing w decreases a by reducing the current crowding in the center conductor. Fig. 7 shows plots of vph versus frequency at sld = 0.3. The dispersion in Vp h is minimal for f > 5 GHz, especially for the high h values. The decrease in Vph for f < 5 GHz is reasonable because here R begin s to approach and ultimately dominate the frequency dependent terms as discussed earlier. Fig. 8 shows a typical Zo and a versus frequency behavior. For Zo, there is no dispersion and its value is equal to 50 n. All this shows is that the Z 0 of this line is the same as the Zo of the through line used in the calibration standard.
= 0.3 for
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However, as discussed earlier regarding (3), above 10 GHz what is measured is quite accurate. In this range a frequency independent Vph and Zo can be modeled using freq uency independent L and C values in standard transmission theory. The loss data shows that a increases approximately with the square root of frequency due to skin depth penetration. In the range of parameters used in the experiments several lines met the vph goal. But none of these lines also had a 50 n Z 0 . Their Z 0 were in the range of 55 D-70 n. We now describe how a line meeting the design criteria can be realized. Fig. 9 shows Z 0 as a function of h - b at sld = 0.3 and b = 9 µm for different w values . Data for different w values fa ll on lines with approximately the same slope. Thi s data and observation can be used to design for a spec ific Zo and Vph as follows. As stated earlier, one can adjust w to achieve the desired Z 0 independent of Vph · Furthermore, it was shown in Fig. 4 that at s!d = 0. 3, h = 80 µm slots give the desired Vph = 8.9 cm/nsec. A 50 line with this phase velocity in Fig. and h - b = 71 µm. 9 should have coordinates of Z 0 = 50 By plotting a line passing through this point with the same slope as the others we obtain another locus of data points corresponding to a different w value. A simple extrapolation yields a w value of 44 µm for this locus. Therefore a line with parameters b = 9 µm, h = 80 µm, w = 44 µm , and s/d = 0.3 should have Vph = 8.9 cm/nsec and Zo = 50 n. Using the data presented many other combinations of Zo and Vp h can be realized.
n
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1923
SPICKERMANN AND DAGLI : EXPERIMENTAL ANALYSIS OF MILLIM ETER WAV E COPLANAR WAVEGUIDE SLOW WAV E STRUCTURES
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VI. RESULTS ON LINES FABRICATED ON MBE GROWN EPITAXIAL LAYERS For device applications the slow wave electrodes would at least partially overlap with epitaxial layers. An epitaxial layer is essential to form the optical waveguide medium in a traveling wave electrooptic modulator. Therefore the behavior of slow wave transmission lines on epitaxial layers was investigated. The layer involved here were unintentionally doped and the background carrier concentration was believed to be p-type with a carrier concentration of less than 5 x 10 14 cm- 3 . Due to limited availability of this material only five different lines and two calibration standards were fabricated. The fabricated lines were subjected to the mechanical inspection previously outlined and found to be equivalent to the lines of the same dimensions fabricated on the semi-insulating wafer. The epitaxial layer structure used in the experiments is shown in the inset of Fig. 10. It was grown by molecular beam epitaxy. This structure is intended to form a dielectric slab waveguide at optical frequencies. The layer thicknesses were confirmed under scanning electron microscope (SEM) examination and Al compositions by room temperature photoluminescence measurements. Fig. 10 shows the results of these measurements superimposed on the results when the same line geometries were measured on semi-insulating material. The Vph increased by about 23 , and there is no appreciable change in Zo . No Zo change combined with a Vph increase implies that L and C decreased together proportionally. The C decrease can be explained by the smaller dielectric constant of the AIGaAs material used in the epitaxial structure. The decrease in L may be explained by modeling the slots as series slotline stubs loading a uniform coplanar line. The decrease in Vph due to the presence of AIGaAs material will decrease the electrical length of the slots. Therefore, the stub reactance, which is inductive, will be reduced. So a reduction in the inductance per unit length of the loaded line will result. A stub loaded coplanar line model developed with the tools described earlier gives results in very good agreement with the Vph increase observed. Finally, Fig. 10 shows that o: on these lines increased by 1.5 dB/cm compared to lines on semi-insulating GaAs. This is explained by the presence of mobile charge in the MBE
80
5
10
15 20 25 30 Frequency (GHz)
35
40
Fig. 11. Measured Vp h as a function of frequency on the MBE grown epitaxial layer for the (9,70,20) line at std = 0.4 superimposed with the results for the same line fabricated on semi-insulating GaAs.
layers due to unintentional background doping. This results in an increased conductivity for these layers which gives the observed loss increase. Another concern with the transmission lines that are fabricated on epitaxial layers with mobile carriers is the possibility of vph dispersion due to conductivity variation of the material with frequency. Fig. 11 shows the variation of Vph as a function of frequency for a particular line geometry fabricated on MBE grown epitaxial layers and semi-insulating GaAs substrates. As shown, vph increased uniformly over the measurement bandwidth and no anomalous di spersion effects were observed. VII. CONCLUSIONS In this work, a detailed experimental study of coplanar waveguide slow wave transmi ssion lines was undertaken . In the study both semi-insulating GaAs substrates and MBE grown epitaxial layers on semi- insu lating GaAs substrates were used. All the fabricated structures were very carefully examined and measured up to 40 GHz using coplanar microwave probes. The slow wave structure consi sts of a coplanar transmission line with periodic slots cut into the ground conductors. It is shown that with this approach significant slowing of the phase velocity is possible. In the experiments a phase velocity value of 8.9 cm/nsec was achieved which represents a 223 slowing compared to the phase velocity of a uniform coplanar line. It was found that the presence of the slots did not result in an increase of the loss coefficient. For a given slot depth almost all the phase velocity slowing is achieved when the slot width to slot period ratio is around 0.3. The periodic loading of the line due to the slots increases when the gaps between the center conductor and the ground planes are decreased. The width of the center conductor had no effect on the phase velocity although it did effect the characteristic impedance. These observations suggested a way of using the measured data to tailor a slow wave structure for a specific phase velocity and characteristic impedance. First this requires choosing the gap of the unperturbed line for an acceptable loss. Then for a given gap the slot depth is adjusted until the desired phase velocity is achieved. It was found that the velocity slowing varies linearly as a function of slot depth for small slot depths for a slot width to period ratio of 0.3. Finally the center
1924
IEEE TR ANSACTIONS ON M ICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. I O, OCTOBER 1994
conductor width is adjusted to get the desired characteristic impedance value. If more fine tuning is req uired one can vary the slot length to period ratio around the value of 0.3 chosen here. Dispersion in characteristic impedance and phase velocity is found to be minimal to 40 GHz both for devices on semi-insulating GaAs substrates and epitaxial layers.
REFERENCES [I] R. C . Alferness, "Waveguide electrooptic modulators," IEEE Trans. Microwave Theory Tech., vol. MTT-30, pp. 1121 - 1137, Aug. 1982. [2] R. G . Walker, "High speed ill-V electrooptic waveguide modulators," IEEE J. Quantum Electron., vol. 27, pp. 654-667 , Mar. 1991. [3] S. Y. Wang and S. H . Lin, "High-speed III-V e lectrooptic waveguide modulators at ,\ = 1.3 ~1m ," J. Lightwave Tech. , vol. 6, pp. 743-757 , June 1988. [4] W. H. Haydl, "Properties of meander coplanar transmission lines," IEEE Microwave Guided Wave Lett., vol. 2, pp. 439-441, Nov. 1992. [5] T. Wang and T. Itoh, "Confirmation of slow waves in a crosstie overlay coplanar waveguide and its applications to band-reject gratings and reflectors," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 18 11 - 18 18, Dec. 1988. [6] A. F. Harvey, "Periodic and gu iding structures at microwave frequencies," IRE Trans. Microwave Theory Tech., vol. MTT-8, pp. 30-6 1, Jan. 1960. [7] N. A. F. Jaeger and Z. K. F. Lee, "S low-wave electrode for use in compound semiconductor electrooptic modulators ," IEEE J. Quantum Electron., vol. 28, pp. 1778-1784, Aug. 1992. [8] R. E. Collin , Foundations for Microwave Engineering. New York: McGraw-Hill, 1966, pp. 366-369. [9] J. C. Yi , S. H. Kim, and S. S. Choi , "Finite-ele ment method for the impedance analysis of traveling-wave modulators," IEEE J. Lightwave Tech., vol. 8, pp. 817-822, Jun. 1990. [10] HP Product Note 8510-8, Network Analysis-Applying the HP 85 108 TRL Calibration for Non-Coaxial Measurements, Oct. 1987. [J l] G. L. Matthaei, G. C. Ch inn , C. H. Plott, and N. Dagli, "A simplified means for computation of interconnect distributed capacitances and inductances," IEEE Trans. Computer Aided Design, vol. 11 , pp. 5 13-523, Apr. 1992. (12] EESOF corporation, Linecalc, transmission line analysis and synthesis program, version 3.5.
[13] K. Kiziloglu , N. Dagli, G. L. Matthaei, and S. I. Long, "Experimental analy sis of transmission line parameters in high-speed GaAs digital circuit interconnects," IEEE Trans. Microwave Theory Tech. , vol. 39, pp. 136 1- 1366, Aug. 199 1. [14] R. K. Hoffman, Handbook of Microwave Integrated Circuits. Norwood, MA: Artech House, 1987, pp. 355.
Ralph Spickermann was born March 3, 1965 in Santa Clara, CA. He received the B.S. degree in electrical engineering and computer sc ience from the University of California at Berkeleyand the M.S. degree in electrical engineering from the University of California at Santa Barbara in 1986 and 1991 , respectively. He is currently pursuing the Ph.D. degree at the latter uni versity. From 1986 to 1989, he was employed at Teledyne MEC in Palo Alto, CA where he worked on the design and production of traveling wave tubes for radar and sate llite commun ications applications.
Nadir Dagli (S'77-M ' 86) was born in Ankara, Turkey. He received the B.S. and M.S . degrees in electrical engineering from Middle East Technical University , Ankara, Turkey in 1976. and 1979, respectively, and the Ph.D. degree, also in electrical eng ineering, from Massachusetts Institute of Technology, Cambridge, in 1986. After graduation , he joined the e lectrical and computer engineering department at Un iversity of California at Santa Barbara, where he is currentl y an Associate Professor. Hi s current interests are design , fabrication, and modeling of guided-wave components for optical integrated ci rcuits, so lid-state microwave and millimeter-wave devices, calculations on the optical properties of quantum wires, e lectron waveguides, and novel quantum interference devices based on electron waveguides. Dr. Dagli was awarded NATO sc ience and IBM predoctoral fellowships during his graduate studies. He is the rec ipient of 1990 UCSB Alumni Di stinguished Teaching Award and 1990 UC Regents Junior Facul ty Fellowship.
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IEEE TRA SACTIONS ON M ICROWAVE TH EORY AND TEC HNIQUES. VOL. 42 . NO. 10. OCTOBER 1994
Leakage Characteristics of Groove Guide Having a Conductor Strip Zhewang Ma and Eikichi Yamashita, Fellow, IEEE
Abstract- A full-wave characterization of the groove guide having a conductor strip is made based on the mode-matching procedure. All of the constituents of the structure, including a coupling strip, grooves and a radiating open end, are treated in a concise and rigorous way by using the generalized scattering matrix technique. Numerical results show that a set of channel guide leaky modes is also present in this structure in addition to the groove guide leaky mode. The complex propagation behaviors of both the groove guide leaky mode and the channel guide leaky mode are illustrated for various wa veguide geometric parameters, and some conclusions regarding the design of leaky wave antennas using this structure are drawn. I. INTRODUCTION
HE groove guide leaky wave antenna using an asymmetrically located metal strip, as shown in Fig. 1, was first proposed and analyzed in [ 1] and [2] , where the principal leakage properties of the antenna and detailed design considerations were provided . The basic principle of the antenna is that of adding a conductor strip into the groove guide in an asy mmetrical fashion so as to perturb the initi ally bound dominant mode to produce a leaky mode, which we call the groove guide leaky mode. The convenient closedform fo rmul as for the phase and the leakage constants of the groove guide leaky mode were developed in [ 1] by using a transverse equivalent network. However, as briefly di scussed in [3] , since approximations were introduced for the derivation of these formulas in the treatment of the asy mmetric coupling strip and the step junctions (the groove), and higher order mode couplings produced between these di scontinuities were neglected, thi s theory was valid and usefu l on ly for small values of the strip width [ 1]-[3] . This obv iously limits the application range of the method, because in practical usages of this structure as leaky wave antennas, one may wish to obtain greater leakage per unit length along the longitudinal direction, and therefore, wish to use wider strip width. Furthermore, recent computations by the authors, using a rigorous full-wave method, show that in additi on to the groove guide leaky mode, another set of leaky modes is also present in thi s leaky wave structure, and under appropri ate circum stances, it may seriously complicate the performance characteristics of leaky wave antennas using thi s structure. This set of leaky modes had been reported in the nonradiative dielectric waveguide with an ai r gap [4] and in the offset groove guide [5] , and was identified there as the
T
Manuscript rece ived Jan uary 14, 1993 ; revised November 22, 1993. The authors are with the University of Electro-communications, 1-5- 1 Chofugaoka, Chofu-shi , Tokyo, Japan 182. IEEE Log Number 9404 154.
channel guide leaky modes. Moreover, the basic phenomena associated with the coupling effects between the expected leaky mode and thi s set of channel guide leaky modes had been demon strated and explained the re for the fi rst time. For the present structure, however, no such information is provided in the references, and the network method [ 1], [2] may fai l to end thi s purpose because, as stated above, it can not reflect acc urately the propagation characteristics of leaky modes in the structure with relatively large coupling strip width. The updated results provided in [ 1]- [3], together with the detailed design consi derati ons g iven by [2] , are valid and useful onl y for antennas with narrow beamwidths, and are therefore inadequate fo r both the thorough understanding of leaky mode characteristics in the structure, and the des ign of leaky wave antennas using thi s structure for wider radiated beams. The main contribution of thi s paper is, therefore, to provide a complete knowledge of the guiding and leakage properties of both the groove guide leaky mode and the channel guide leaky modes in this structure under various geometric parameters by employing a rigorous method. In Section II , the full-wave theory, brie fl y outlined in [3] because of the restricted-length, is described in a more detailed manner for clear understanding. In Section III, first, the mode charts of leaky modes in the structure are given , showing the ex istence of both the groove guide leaky mode and the channe l guide leaky modes, as well as the ir mutual coupling. Then, a variety of figure s are provided to demonstrate the propagation behaviors of these leaky modes and their effects on the leaky wave anten na performance. Particul ar attentions are given to the comparison between the complex propagation behaviors of these two types of leaky modes, and some conclusions concerning the design of leaky wave antennas based on this structure are drawn . ll . TH EORY The groove guide leaky wave structure shown in Fig. 1 may be looked, in the tran sverse y-direction , as a combination of parallel plate waveguides linked by cascaded discontinuities, including an open end, an asymmetric metal strip, two step-junctions (the groove) and a short-circuited end . In our analysis, the electromagnetic fie ld s in a parallel plate waveguide are expressed as a superposition of the LSE and LSM modes with respect to the z-directi on (the TE and TM modes with respect to the y-direction). The LSE and LSM modes may be viewed as propagating in the transverse ydirection and coupling each other at d iscontinuities of vario us vertical planes. The complex modes of the structure are formed as a result of repeated reflecti ons of the LSE and LSM waves
00 18-9480/94$04.00 © 1994 IEEE
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IEEE TRANSACT IONS ON M ICROWAV E TH EORY AND TEC HNIQU ES. VOL 42. NO. 10, OCTOBER 1994
x h, h2 L,
L,
y
c
c'
Fig. 1. Cross secti on o f the groove gui de leaky wave structure w ith a n asy mmetrica lly located meta l strip o f width o.
Fig. 2 .
Step junc ti on of a paralle l pl ate wavegui de .
orthonormality relati ons hold: at the two ends and discontinu ities . With the choice o f the LS E and LSM modes as the constituent parts of the fi e lds, we can , as can be seen later, treat the radi ating open end simply but accurately by using the equivalent admittance of the open end on the propagating LSE0 mode, which is directl y available in [ I] and [8]. To obtain the eigenvalue equation fo r the phase and the leakage constants of the structure, first we deri ve the scattering matrix of a para lle l plate waveg uide step-junction, as shown in Fig. 2, for the LSE and LSM mode excitation. Then, we treat the asym metric metal strip and the groove as cascaded step discontin uities by using the generali zed scattering matrix technique to obtain an overall scattering matrix . Finall y we for mul ate the eigenvalue equation by considering the radi ating open end and the short-circuited end, and employ ing the transverse resonance condition. As part of the mathematical form ul ation in the analysis is simil ar to that of [6] , thi s process is stated in an abbreviated fo rm. The reader is referred to [6]. The step-j uncti on shown in Fig. 2 is located in the y = 0 plane. At z = 0, the transverse (with respect to they-direction) electric and mag neti c fie lds in the i-th guide (i = 1, 2, representi ng the left and right region of the structure on the two sides of the step-j uncti on, respecti vely. All the components in the left and right region of the junction are indexed with I and 2, respec ti ve ly) are expanded in infinite sums of the LSE- and LSM-mode components as E it(:i; , y)
L
=
2..:
L
y) =
c
h_e
• d .-
im . I y
X -
;·h;
-e
h ; - L ; ein X
he
• d .-
·i m . I y
X -
(j nm
(3b)
/''' . e(~ x
j h 1-L1
j
·li;
e X ein
h.Tm-iyd.x = /'" . e(~, x hfm · iydx = o(3c) J h,, - Li
h- ·ihm
. ·I y d:£
h 1- L i
=
1"'
-e X ein
h im " . ·I y dX
= 0(3 d)
h 1- L i
where Dnm is the Kro necker de lta (= I if n = m ; = 0 if n of. rn). Using the boundary conditions at y = 0 and z = 0 and matching the tangenti al fi e lds, E it and H it• lead to a pair of equati ons on the tangenti al components of electrom agnetic fi elds. Vector-multiplying the electric component vector equation successively by h.?m and hJ'w and the magnetic component vector equati on success ively by e ~m and e2m' using the orthonormality relation (3) and truncating both sides of these equati ons, we get a set of linear simul taneous equations in the fo llowing matrix form :
R"" [R eh
R"e] [A~+ B ~ ] = [A~+ B~] Ree A2+ B2 [R] A2+ B2 (4)
(5) (1 )
H it( -T ,
/ '"
J h ; - L , ein X
=[HJ[A A!"-B - B!"] .
n = 0 ,1 ,2 , ..
+
(3a)
(
A·h1n e- 1··k ",, ;,, ·Y
1
-
From (4) and (5), the scattering matrix S of the step-junction is deduced in a fo rm as shown below
I' e1: k "' ;,,Y • ) B.1n h 1n 'I (x) ·
n=0 ,1,2, ..
+
(6)
2..:
n = l ,2 ,3 , ..
(2)
ln (4)-(6), express ions fo r the matri ces, S.;1 (i , j = 1, 2) , and fo r the elements of matrices, R hh ,h e,e h ,ee and H hh ,h e ,eh ,ee ,
where the coefficients, and represent the amplitudes o f the incide nt and re fl ected waves in the i-th guide, and the space vectors e'.:;e(:i;) and h '.:;c(x ) are the transverse modefu nctions whose ex press ions were g iven in [6] . If a new set of tilded vector-mode function s, e '. ~ e and h.'.'.;c, are de fined by repl acing kz with -kz in the modefunctions, e'.~ e (.x) and h'. ~ e (x ), respecti vely, the followin g
were give n in [6] . Out of the uniform secti ons connecting the cascaded di scontinuities, we choose one hav ing a smaller verti cal dimension as the reference section. By employing the generali zed scattering matrix technique [6], [7] , we obtain the overall scattering matrices of the cascaded di scontinuities on the left and right sides of the reference section , and denote them by g ( L ) and
A'.:;c
Bi:;e,
1927
MA AND YAMAS HITA : LEAKAGE CHARACTER ISTICS OF GROOVE GUIDE HAV ING A CONDUCTOR STR IP
where
G =I - ( s ~f l E ( L J s i;) + s~; ) ) x n (Ml ( s i~J E ( flJ s ~~l Scanering matri x representation of the original groove guide leaky
wave structure.
g ( R) , respectively . Now the original leaky wave structure can be represented by the scattering matrix representation show n in Fig. 3. The column vectors, A ( L ) and B ( L ) , accounting for incident and reflected wave amplitudes, respectively, of matrix g (L ) in the left end region , and vectors A ( R ) and B ( R ) of matrix g (fl) in the right e nd regio n, are related through the left and ri ght end boundary conditions as follows:
A (LJ = R (Ll n (LJn CLJs (LJ,
A (RJ = R :::
B
0.7
a way simi lar to that for Fig . 5 from the point of view of mode power distributions. Also it is seen that the values of a/ k 0 of al l the lea ky modes vary over a wide range, with particular sensitivity to frequencies nea r cutoff, as can be expected. The variation s of f1/k0 and n/ko of the leaky modes with the normalized length b/a of the groove wall are shown in Fig . 7. As the power of the groove g uide leaky mode ( I ) lies mainly in the groove region, its values of /3/ ko and a/ko are considerably affected by the change of b/a. The null s in its n//.; 0 curve is caused, as explained in [21, by the nodes in the standing wave of the LSE 0 mode wh ich occur exactly at the location of the asymmetric strip, thereby shorting o ut the coupling between the LSE 0 mode and the higher order evanescent modes. On the other hand, the values of a/ k 0 of the channel guide leaky mode (2) and (3) vary s low ly, since their power distributions are not greatly affected by the variations of b/ a. As explained earlier, the states of the power distributions of the mode ( I) and (3) are closer, therefore, their behaviors of (3/ J,; 0 are more sim ilar, particul arl y at small values of b/a, near which the value of a/ k0 of the mode (3) also vari es more rapidly. Finally, we observe the variations of f3/ k 0 and a/ko of the leaky modes with the norm alized plate wall le ngth c/ a, as illustrated by Fig. 8. Also two cases of the strip widths are provided. It is easy to find that the curves of f3 / k 0 he re are quite similar to those in Fig. 4. T hi s is natural si nce both the c of Fig. 8 and c" of Fig. 4 represent the le ngths of the narrower parallel plate walls that are on the ri g ht and left sides of the groove, respectively . However, the behaviors of a/ko here are fairly different from those of Fig. 4. The values of a/ k 0 of the channel guide leaky modes here vary mo re slowly while that of
""'
Cl::l. 0.6 0.5 10-1
10-2 10-3 0
...>:::
10-•
""' 0.0175 quasi-resonant modes stop existi ng, and in thi s case the radiation modes become leaky modes. The measured resonant frequencies compare reaso nably well with calculations, even though the substrate sizes used were not considerably large (L/ R = 7.62). Fig. 6 shows the corresponding values of Q0 versus d/ R. It is seen in Fig. 6 that Q0 possesses higher values for leaky modes than for quasi-resonant modes. The measured and calculated Q0 compared reasonably well for quasi-resonant modes. There was qualitative agreement between theory and experiments for leaky modes. However, improved quantitative agreement may be obtained by allowing the substrate size, L, goes to infinity.
Fig. 8.
Critical substrate thickness as a funct ion of metal conductivi ty.
In fact by fabricating a larger device such that L / R = 21.6 instead of 7.62, we improved the comparison dramatically. This is further illustrated in Fig. 7 where Q0 is directly plotted as a function of L for fixed ratio of d/ R ( =0 .07874) . It is seen in Fig. 7 that the measured Q0 increases monotonically with L , approaching the calculated Q 0 value, which is shown as a dotted line, only when L goes to infinity. This proves our earlier assertion that leaky modes are global waves. Figs. 8 and 9 show the critical substrate thickness, de, as a function of 1/ CY and tan 8, respectively. It is seen in Fig. 8 and 9 that for a given value of E,., say Er = 2.2, both 1 /CY and tan 6 have the same effect such as to increase the value of de. One may visualize the transition of normal modes (local to global) in the following physical picture. The electromagnetic field s behave like some kind of fluid whose vi scosity is characterized by the sum of the conductor and dielectric losses. We now imagine that full amount of fluid is placed in the region bounded by the metal patch and the ground
1944
IEEE TRANSACTIONS ON MICROWAVE TH EORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
IV.
cr = 100 cr 0 0.2 ,----,-----,-----.------,--,
0.15
~ "Cl
0. 1
0.05
E, =
2.2
CONCLUSION
Normal modes in a circular patch antenna have been analyzed theoretically using the modified Green's function approach. The use of current potentials in this analysis renders itself an indispensable tool in simplifying the solutions. We found that the normal modes in most cases consist of leaky surface waves which extend far beyond the region containing the patch antenna. Localized quasi-resonant modes occur only if the surface wave loss is minimized. That is, for given values of metal conductivity and dielectric loss tangent the condition allowing localized volume resonance to occur is when the thickness of the dielectric substrate is very thin or when the dielectric constant is close to 1. Our theory compares very well with measurements. The authors wish to thank the support of the AFOSR.
10"3 REFERENCES
tan8 Fig. 9.
Critical substrate thickness as a fun ction of dielectric loss tangent.
plane. The fluid remains in the cavity region without spilling onl y if the fluid is viscous enough such that the viscosity of the fluid (dielectric loss) and the adhesive force between the fluid and the container wall (conductor loss) overcomes the tendency that sets the fluid to flow (the gravity force). In this case the fluid is confined locally under the metal patch and the normal modes are termed as quasi-resonant modes. However, when the thickness of the substrate increases, or the aperture of the leaky cavity increases, the viscosity of the fluid can no longer hold the fluid within the cavity region and the fluid will spread out all-over the substrate. In thi s case we term the normal modes as leaky modes. The critical thickness of the substrate, de, is defi ned to be the thickness at which the transition of normal modes occur. Therefore, de is a function of the fluid viscosity, 1/ u and tan 8, the larger the viscosity, the larger the value of de. It is apparent that de is also dependent on the dielectric constant of the substrate, Ed. This is due to the fact that there are more surface modes for larger value of Ed which form the driven force that sets the fluid in motion (the gravity force). This is specially true if Er is not close to 1. If Er is close to 1, say Er = 1.07, the quasi-resonant modes can be always stable for all values of d if 1/u and (or) ta n8 exceed some certain values. In this case the driven fo rce is too weak and localization of the fluid becomes always possible. This is illustrated in Figs. 8 and 9 as sharp vertical transition lines. For example, for Er = 1.07 and when u 0 ~ 2u (tan8 = 0.0001) or when tan 8 ~ 0.0007 (u = 100u0 ) quasi-resonant modes are always prevalent in the antenna structure for all values of d. As such, surface waves are quenched out. Thus, if crosstalk between radiation elements in an array is to be avoided, one may design material parameters such as to quench out surface waves.
[I] Y. T. Lo, D. Solomon and W. F. Richards, "Theory and experiment on microstrip antennas," IEEE Trans. Antenna Propagat. , vol. AP-27 , p. 137, 1979. [2] S. Yano and A. Ishimaru, " A theoretical study of the input impedance of a circular rnicrostrip disk antenna," IEEE Trans. Antenna. Propagat. , vol. 29 , p. 77, 1981. [3] L. C. Shen, " Resonant freq uency of circular disc, printed circuit antenna," IEEE Trans. Antenna Propagat., vo l. AP-25 , p. 596, 1977. [4] W . C. Chew and J. A. Kong, " Analysis of a circular microstrip disk antenna with a thick dielectric substrate," IEEE Trans. Antenna Propagat., vol. 29, p. 68, 198 1. [5] K. Araki and T. Itoh, " Hankel transform do main analysis of open circular rnicrostrip radiating structures," IEEE Trans. Antenna Propagat. , vol. AP-29, p. 84, 1981. [6] M. C. Bailey and M. D. Deshpande, "Integral equation formulation of microstrip antennas," IEEE Trans. Antenna propagat. , vol. AP-30, p. 65 1, 1982. [7] D. M. Pozar, " Input Impedance and mutual coupling of rectangular microstrip antennas," IEEE Trans. Antenna Propagat. , vol. AP-30, p. 1191 , 1982. [8] J. R. James and P. S. Hall , Ed. , "Handbook of microstrip antennas," in J. R. Mosig, R. C. Hall and F. E. Gardiol, Numerical Analysis of Micros/rip Patch Antennas. Lo ndon, U.K. , Peter Peregrinus, 1989, ch. 8. [9] H. How and C. Vittoria, "New formulation of dyadic green's function: Applied to a microstrip line," to be published. [IO] J. D. Jackson, Classical Electrodynamics. New York: Wiley, 1975 . [11] A. Sommerfeld, Partial Differential Equations. New York: Academic Press, 1962. [12] K. Ogusu, "Optical strip waveguide: A detailed analysis including leaky modes," J. Opt. Soc. Amer. , vol. 73, p. 353, 1983. [ 13] H. How, R. G. Seed, C. Vittoria, D. B. Chrisey, J. S. Horwitz, et al., "Microwave characteri stics of high Tc superconducting coplanar waveguide resonator," IEEE Trans. Microwave Theory Tech., vol. 40, p. 1668, 1992.
Hoton How, for a photograph and biography, see page 72 of the January issue of this TRANSACTIONS.
Carmine Vittoria, photograph and biography not available at time of publication.
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IEEE TRANSACTIONS ON MICROWAV E TH EORY AN D TECHNIQUES. VOL. 42, NO. 10, OCTOB ER 1994
Study of Microstrip Step Discontinuities on Bianisotropic Substrates Using the Method of Lines and Transverse Resonance Technique Yinchao Chen, Member, IEEE, and Benjamin Beker, Senior Member, IEEE
Abstract-A hybrid approach, combining the method of lines (MoL) and transverse resonance technique (TRT), is presented for the analysis of microstrip step discontinuities that are printed on uniaxial or biaxial bi-anisotropic substrates. The method of lines, formulated in terms of Kronecker products, is used to determine the characteristic equation for the resonant length. The transverse resonance technique is applied to obtain the S -parameters of the junction by casting the discontinuity problem as a microwave equivalent network. Good agreement is found between results of the MoL/TRT approach and those obtained by other methods. Effects of individual tensor elements of the substrate on the scattering parameters of the discontinuity are investigated at selected frequencies. The proposed MoL/TRT approach is found to converge very fast and does not require excessive computer memory, with all computations performed on a 486DX-SO MHz PC.
l b
j >.~ml) µyy x2
E:zhz 2 ([-\[vo] @[Uxl) Eyy z
[bz] = _1_ [Exx - µxx ] ([8z]t 0 [8S ). hxhz E:yy
µyy
(7)
Matrices [-\ ~~.ND ] , [8x,z ], and [T~·~, ND ] that are used above were defined in [12] and thus will not be repeated here. Since all matrices are diagonal, equation (6) can be easily solved by a technique described [9]. This results in two sinusoidal solutions within the planar anisotropic region in the transformed domain. The solutions for (Ex , Ez) within the isotropic region , can be obtained from the same equations when all tensor elements of [c] and [µ] are set to Er and µ r. respectively. The boundary conditions enforced at the air-substrate interface (y = D) E (2) = E (1) x,z x,z
(2) _ H (1) _ ( H x,z x,z -
+, - )Jz, x
(8)
lead to the explicit relation between tangential E-field components at the interface and current densities on the metal parts
(9)
CHEN AND BEKER: STUDY OF MICROSTRJP STEP DISCONTINUITIES ON Bl-ANISOTROPIC SUBSTRATES
the Z-parameters of the equivalent network can be extracted. Since S-parameters are more commonly used in the analysis of the junction, they can be determined from the following relation
Equivalent Network for the Junction
[SJ
[Z]
Fig. 3. Equivalent two-port network terminated with a short-circ uited and open-circuited load s.
where the transformed currents are given by
!ix)= ([Ttmr ® [Tl)Nii) IJx) liz) = ([T,\rv] ® [T~Dn IJz). 1
(10)
The coefficient matrices of (9) are defined in the Appendix. From (9) and ( JO) , one can easily construct the following impedance Green's function matrix in the transformed domain
Finally, when the boundary conditions are enforced on the components of the electric field in the space domain; namely, when the tangential E-field is set to zero on the metal, the characteristic equation for eigenvalue of the reduced matrix equation is obtained (12) where w,. is a resonant frequency, and l is the corresponding resonant length of the equivalent cavity.
B. Employing the Transverse Resonance Technique According to [4], the step discontinuity can be treated as an equivalent resonant microwave two-port network that is shown in Fig. 3. Thi s equivalent circuit is only valid when both microstrips operate below the cutoff frequency of the higher order modes, which were analyzed separately with results presented elsewhere [14]. Under such conditions, the two-port resonance in terms of impedance parameters can be expressed as (13) where all Z-matrix elements as well as load impedances z 1 and z 2 are normalized . In this formulation, the cavity is formed by terminating the guiding structure with an electric and magnetic wall a di stance li and l 2 from the junction, respectively. The normalized load impedances of the terminations are
z1 =jtan(JJ1li) z2
= -jcot( j32l2)
1947
(14)
where the j31 and /3 2 are the propagation constants of the two connected transmission lines . Once three different pairs of (li, 12) are calculated for a given resonant frequency Wr,
=
([z] + [J])- 1 ([z] - [I]),
(15)
where [J] is a 2 x 2 unit matrix. In order to accelerate the search for the resonant length of the cavity, the axial resonance condition of a uniform transmission line, whose j3 is an average of /31 and j32 and can be precomputed, may be used as an initial guess. The axial resonance at a di stance li from the electric wall and 12 from the magnetic wall leads to the approximate total length of the cavity
lres = l1+l2 =
(2n-l)7r 2/3
' n = 1, 2, ....
(16) A small step discontinuity is not expected to change the cavity 's first resonant length significantly. In this case, lres(n = 1) should be employed. For large discontinuities larger resonant lengths are needed to avoid evanescent wave effects; consequently, higher values of n should be used instead. Equation ( 16) was found to provide an accurate initial estimate of the total resonant length, which was used to start the search for lres in the eigenvalue relation (12). This approach proved to be numerically very stable in leading to accurate and fast solutions to equation (12). As pointed out above, to determine S-parameters of the junction three different pairs of ( li, l 2 ) are needed. This implies that a relationship between them is required to reduce the number of unknowns . Results of an extensive numerical study indicate that the following ratios of li to [z : 1.225 , 1.0, and 0 .825 appear to be quite satisfactory in computations for W 1 to W 2 ratios analyzed in this paper. Finally, it should be added that the TRT approach to the analysis of microstrip discontinuities in conjunction with the MoL is not the only possible technique to account for the axial dependence of the guided fields. The method of sources, which incorporates the source directl y into MoL formulation, has been implemented in the solution to the discontinuity problem as well (13] .
III.
NUMERICAL RESULTS
To verify the formulation and its numerical implementation, the numerical results of the MoL!fRT technique are validated against the published data. As the first example, the effective index of refraction is computed with the proposed approach for a uniform transmission line by setting W 1 = W 2 (see Fig. 1). In this case, the resonant length of the cavity can be related to the propagation constant in a similar way to that presented in [4]. Numerical results shown in Fig. 4 are compared to those computed with the spectral-domain approach (SDA) of [9] for a dielectrically anisotropic substrate. The comparison is also made between MoL!fRT and SDA when the substrate is also magnetically anisotropic. As can be seen from Fig. 4, the agreement is very good with relative error not exceeding 0.23 %.
1948
IEEE TRANSACTIONS ON MICROWAVE THEORY ANO TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
2.5.---------------------~
1. 2
2a = b = 12. 7, D = 0.5, Wl = 0.414 (all in mm)
2a = b = 12.7; Wl = W2=1.0, D = 0.5 (all in mm) 1. 0
-
2.4 0
( 2.89, 2.45, 2.95; 1.12, 1.22, 1. 35)
1s121
0. 8
~ 2.3 W2:
6 01
-a- 0.966;
__,.__ 2.071 (mm)
--- 1.519;
( 2.89, 2.45, 2.95; 1.0, 1.0, 1.0 )
• 2.2
o. 4
01
-
-
-
1s111 2.1
-
0. 2
•
SDA [9]
-
MOL& TRT
o
SDA
4
8 12 Frequency (GHz)
( 6.24, 6.64, 5.56; 1.12, 1.24, 1.18)
2.0+ - - - - - - . - - - - - - - - . - - - - - . - -- - - , - - - - - 4 0
-
16
o. 0
2
0
20
Fig. 4. Validation of MoLffRT approach for a uniform rnicrostrip printed on bianisotropic substrate.
4
6
8
10
12
Frequency (GHz)
Fig. 6.
S -parameters for a step discontinuity with different step widths.
0.30.-------------------~0.99
2a = b = 12. 7, D = 0.5, Wl = 0.414, W2 = 0.966 (all in mm)
1. 2,---------------------~
Frequency = 8 GHz
a= 2.0, b = 10.0, Wl = 0. 3175, W2 = D = 0.635 (all in mm) l.(t
0.28
n
,.
1s121
0.26
0.8 ~
~ 0.24
~
"
-
"g_
!:' 0.6
MoL & TRT
,.
Ref [5]
o
Ref [1 ]
a~
~;:~=:=:==:=:=;;;;;;;;~£;~;·;;;;·g=,=-----··='='~
=
0.97
~
=
"' 0.4
0.96
1s111
.
0.2
w
=: ( 4-7, 6.64, 5.56; 1.12, 1.24, 1.18 )
..
.
w
yy: ( 6.24, 4-7, 5.56; 1.12, 1.24, 1.18)
0
0.0·+----.----..,.----.---.----.--,----,--~-~___,
0
2
3
4 5 6 Frequency (GHz)
7
8
9
10
Fig. 5. Validation of MoLffRT approach for step di scontinuity printed on a dielectric substrate € ,. = 9.6.
In the second validation example, S-parameters are cal culated for a microstrip step discontinuity, whose substrate is assumed to be isotropic. Comparison with the MoLfTRT results and those obtained from [l], [5] is shown in Fig. 5. The results for S 12 match quite well. On the other hand, S 11 data match well with those of [l), but show some disagreement with those of [5]. · In all the following computations, the total number of discretization points was taken to be N = 11 and M = 19. The computations were performed on an IBM-compatible 486DX50 MHz PC, with the average program run-times ranging from 2.5 to 3 minutes per frequency point. The effect of changing the strip width ratio of Wif W2 on the scattering parameters was investigated. The computed data for the MIC with the bi-anisotropi"c substrate are presented in Fig. 6. The material in this case is characterized by permittivity and permeability tensors whose diagonal elements are (6.24, 6.63, 5.56) and (1.12, l.24, 1.18), respectively. Notice that the Sparameters remain nearly constant throughout the selected
zz: ( 6.24, 6.64, 4-7; 1.12, 1.24, 1.18 ) 0.18+----.-----,-----,-----..,.---..,.----+0.95 ~o ~5 50 55 60 65 70 Elements of Permittiuity
Fig. 7. tensor.
S-parameters as function s of scanned elements of the permittivity
frequency range, extending to 12 GHz. Moreover, as expected, the wider strip width ratio is associated with greeter reflection from and lower power transmission through the junction. It should be added that since the structure is symmetric and medium tensors are biaxial, the junction is reciprocal with S12 = S21 . To see the influence of anisotropy on guided wave propagation characteristics through the step discontinuity, a material parameter study is carried out next. The values of each substrate material element (permittivity or permeability) are scanned to determine which of them has the greatest effect of the S-parameters. One element is scanned at the time, while holding others fixed. The results for the permittivity scan are shown in Fig. 7 and, correspondingly, the permeability scans are di splayed in Fig. 8. Observe from Fig. 7 that when the change in Czz hardly affects S 11 or S 12 . However, for the lower values of cxx and cyy . the changes in both S 11 and S 12 are quite noticeable. This study indicates that the variation in cyy causes the most visible change in S 1 2, with €xx showing similar behavior.
1949
CHEN AND BEKER: STUDY OF MICROSTRIP STEP DISCONTINUITIES ON Bl-ANISOTROPIC SUBSTRATES
0.30 ,--- - - - - - - - - - , - - - - - - - - - - - r l . 0 0 2a = b = 12. 7, D = 0.5, WI = 0.414, W2 = 0.966 (all in mm) Frequency = 8 GHz
0.99
0.28
0.98
yy
:·· :::::... . .. . . ... {>
0.26
=-·.,:t~~~= === : : : : ==== === === ====== ===--xx
~ 0.24
0.97
"".....
C/:J
0.96 zz o.22f7"::::..._===t============= ] 0.95 yy xx : ( 6.24, 6.64, 5.56; 1-4, 1.24, 1.18) yy: ( 6.24, 6.64, 5.56; 1.12, 1-4, 1.18) zz: ( 6.24, 6.64, 5.56; 1.12, 1.24, 1-4)
0.20
0.94
0.18 +---~--~---~---,-----~---+o. 93
1.0 Fig. 8.
1.5
2.0 2.5 3.0 Elements of Permeability
3.5
4.0
S-parameters as fun ctions of scanned elements of the permeability tensor.
Next, Fig. 8 summarizes the computed results for .the permeability element scans, while the val ues of the permittivity are held fixed . In this case, it is evident that the µ xx element of the[µ] influences most significantly the values of Sn and 5 1 2. Just as in the permittivity element scans illustrated in Fig. 7, the effect of varying the zz-element of[µ] on the S-parameters is hardly noticeable. However, the effects of µy y are visible, though not as strong as those due to µ xx . Finally, it should be mentioned that the effects of medium parameters on 5 11 and 5 12 are larger than those due to the frequency , as shown in Fig. 5 and 6. It is apparent from the results of Figs. 7 and 8 that €xx and µyy are responsible for greatest observed changes in the S-parameters, and, hence, should be carefully taken into consideration in the MIC design . IV. CONCLUSION
TABLE I DERI VATIVES OF
(Ex, E , )
AND TH EIR CORRESPONDING M ATR IX FORMS [12]
Derivatives
Matrix Forms
oE, ox
.:.!. (( U,)®[D:o ]')I£,)
~
...!...([u.]®[D: J)JEJ
h,
ox o' E,
7
~
-! ([ U.} ®[ h,
P;N J)I£, ) 1 :; ([u.J ®[P;o))IEJ
ox' oE, oz
1 : , ([ D~D r
~
...!...([ D~0 ) ® [u, J)IEJ
oz o' E,
v
APPENDIX
The derivatives of (Ex, E z ) and their corresponding matrix form s are summarized in Tab le I, where [Ux,z ] are N x N and MxM unit matrices, with N and M being the total numbers of sampling points along the x - and z-directions, respectively .
®[u,J)JE,)
hi
-; ([P~N ) ® [U, j)JE,) h,
02 £
at o E, oxilz ~
oxoz
.
-; ([P~0 ) ® [u, J)IE,) h,
1
The method of lines and transverse resonance technique are applied to the analysis of microstrip step discontinuity problems. The Kronecker product is used extensively to set up diagonal linear equations in the transformed domain. The proposed approach is validated for both isotropic and anisotropic substrates. Numerous examples are presented, indicating that the amplitudes of S-pararrieters are most sensitive to changes in cy y and µ xx elements of the permittivity and permeability tensors. More importantly, it appears that the effects of anisotropy on the scattering parameters of the step junction are greater than those of the frequency . Computationally, it is found that the MoLffRT technique is numerically very fast and does not require excessive computer resources, allowing for the calculations to be performed on the PC.
0
h,
h,lh,
([D~0 )' ® [D: ]')iE,)
h,lh,
0
([D~0 ) ® [D:0 J)JE,)
The coefficient matrices that appear in (9) are given by
-1
[b2] = - [U] z oµ xx
1
+ Z Q(µ xx hz )2 ([ozii ® [Uxl) [sC 2
lr ([c5z] ® [Uxl) 1
(A-4)
1950
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. IO, OCTOBER 1994
[c1] =
-l [U] + Zo
1 h2 ([Uz] ® [ox]) [sC 1l]- 1 ([Uz] ® [ox]t) x
ZQ
(A-5)
[d1]
=
1
zo
h-lh ([Uz] ® [ox]) [3Cll]- ([oz] ® [Uxl) x
(A-6)
z
1
[cz] = --[U] zo µ zz ( Zo
1
h )2
µ zz x
1
([Uz] ® [ox]) [sC 2 l ] - ([Uz] ® [oxr) (A-7)
1
[d2]=---zo µxx µ zz hxhz
1
2 x ([Uz] ® [ox]) [sC l] - ([oz] ® [Uxl)
(A-8)
with
zo = jwµo
[sCll]
= :2 z
(A-9)
([.Xrm l ® [Uxl) + : 2 ([Uz] ® [.XNvl) - k6[U] x
(A-10)
[s 0 is required). In macroscopic theory, the energy for each electron consists of kinetic energy and potential energy, E = Ek + Ev =
0018-9480/94$04.00 © 1994 IEEE
IEEE TRANSACTIONS ON MICROWAVE TH EORY AND TECHNIQUES , VOL. 42, NO. 10, OCTOBER 1994
1952
We can relate the two macroscopic quantities, J and v A by ·+oo (5) J = _ qn(v)vdv = qnovA
1.5~-~-~-~-~-~-~--~~
j
00
at 6. V . The T dependence of n (v ) req uires that at T
= 0, (6)
n( v ) = n 0 8( v ),
a delta function [ 12]. For higher T, the spread of the distribution is wider.
0 .5
III. n 8 ( J) AND CRJTICAL CURRENT DENSITY Jc(T) O'----'--""""'"""'::__-'----'---'--""'-'-- ' - - _ _ , ·3
·2
·1
1
v I VA Fig. 1.
An assumed distribution of electron velocities for T = 20 K.
~ mv 2 + Ep, where m is the mass of an electron and v is its velocity. In quantum theory, v is the expectation value ('!/i lvl'l/J), where 'l/J is the wave function of the electron and v is the velocity operator. The electron energy E can be expressed as a function of temperature T, applied current density J, magnetic field H, and field frequency f, E = E(T, J , f , H) . Ee is the origin for critical values of Tc, Jc, f c and He. Consider a case where the electrons have one-dimensional velocities, for example, in a thin (radius a « A the penetration depth) wire within which the current flow s in only one direction. The electrons in the sample have different velocities due to thermal motion . The average velocity of all electrons is non zero along the current direction. The current density J is a macroscopic quantity. At a certain point r, J is related to the average velocity of the electrons in a small volume 6. V,
J = I:.~ qv.; 6. V
(2)
where N is the number of electrons in 6. V and q is the charge of an electron. Note that not every electron has the same velocity. The velocities Vi of the electrons obey a certain distribution . Here we assume that the number of electrons, 8N, which have the velocities between v and 8v obeys a one-dimensional distribution. If we define
8N n( v )= 8v6. V '
(3)
then [10], [11]
Since the electrons have different velocities, and therefore different energies at a certain applied current density J, the electrons will not all exceed the E e at the same time. The number density of superelectrons, n 8 , does not disappear abruptly when J exceeds the macroscopic critical current density Jc. Only when T = 0 and n(v) becomes no8( v ), n 8 (J) shows an abrupt drop to zero at a critical Jc(T = 0): for J < Jc(O ), n 8 = no ; and for J > Jc(O), ns = 0. In experiments, since absolute T = 0 K is not achievable, a completely sharp tran sition of n 8 is not observed. The l c(T ) is related to the average velocity VA and characteristic energy E e. First, we define a characteristic velocity Ve for a single e lectron corresponding to Ee. If the potential energy is included in the case of maximum kinetic energy, ~mv ~ = Ee. Hence, Ve = j2Ec/m. Since J = noqVA, we define
l c(T)
( T = qnovc = qnoyf7.04kBTc m 1 - l "'c
where VA is the average velocity of the e lectrons, m is the mass of an electron, and n 0 is the total number densi ty of electrons in 6. V. Fig. I shows a possible n( v) function where n( v) is assumed to be a continuous di stribution of a Gaussian form . The specific mathematical form of the distribution n(v ) may be obtained from the microscopic quantum theory. If the wave function of each electron 'l/J; is taken into account for a distribution n(v.;), where Vi = ('l/IJul'!/ii), such formula can be derived from statistical physics. The numerical res ul ts shown in thi s paper are obtained by using the distribution in Fig. 1.
12 (7)
as the critical current density. This Jc is different from the Ginzburg-Landau's de-pairing current density . Here Jc(T) is a quantity derived from the hypothetical characteristic energy Ee via the classical kinetic energy expression. For TI- 0, n 8 (J) is a smoothl y varying function and the sample is partially superconducting. The Jc(T) has a different characteristics from l e at T = 0. This smooth varying feature of the I - V curve has been widely observed in experiments [5] , [6] . Since VA = J / qn 0 , the number density of super electrons (here we cou nt the single electron density rather than the pair density) n 8 ( J) can be derived from the velocity distribution. The superelectrons are those whose energy are lower than Ee. In the one-dimensional case,
n 8 (J ) =
(4)
)°'
j
v c (T)
n(v)dv.
(8)
-vc(T )
In general, this integral can only be evaluated nµmeric ally . At T = Tc, Ve = 0, hence n 8 = 0 for all J 's. By using the electron velocity di stribution in Fig. 1, n s (J) is plotted in Fig. 2 for fo ur different temperatu res. Here J is normali zed by the Jc at T = 0. At T = 0, n 8 = no fo r J < Jc(O). This is what has been predicted by the two-fluid model. The temperature dependence of ns assumed in the two fluid model is that
1953
XIA er al. : A MACROSCOPIC MODEL OF NONLINEAR CONSTITUTIVE RELATIONS IN SUPERCONDUCTORS
10
T• O.SK
T:20K
0.8
?i' c
"' .:-
~
0.6
«
0.4
«
~
~
0.2
00
1.4
0.8
0.6
0.4
0.2
00
1.6
0.2
0.4
0.6
1.2
0.8
1.4
1.6
1.8
J I Jc(O)
JI l c(O)
Fig. 2. Superelectron number density ns(l)/no at different temperatures. In this example, a 3/2, and Tc 90 K. l c(O) is from (7).
Fig. 4. Penetration depth>. as a functi on of current density J at T = 88 K. Here Tc 90 K.
=
=
=
and A = m s/q;ns . The subscript s denotes that the quantity is of superelectrons. The nonlinearity will be included in A(J) = µo>- 2 , and ' -
0.8
~
0.6
: Two Fluid Model
'
>--(J)
ms q; ns(l)µo ·
( 11)
· ·-: This Model
'
0.4
Once ns(J) is known, >--(J) can be derived. Hence, the nonlinear constitutive relations are obtained. Nonlinear effects come in A and ns. Substituting (8) of ns( J) in A, we derive the >--( J ) function , which is plotted in Fig. 4. The penetration depth goes to infinity when J » l c(T) . Conductivities ..( J = 0) = 2. (a) current density J y ( x) distribution in a slab. (b) superconductivity u 8 ( x) distribution in a slab.
Fig. 8.
We first consider the solution of a linear superconductor for
>>
nnWT.
I cosh(x/ >.) ly(x) = 2 >. sinh( d/ 2 >.)
where A is the vector potential. In general , thi s equation can not be solved in a closed-form.
for lx l ~ d/2
(22)
forl x l ~ d/2 .
(23)
and
Hz(x) =
I sinh(x/>.)
-2 sinh(d/ 2>.)
An iterative scheme for solving the nonlinear problem is to initially use the linear solution ly (x) in (22) to calculate n 8 (J) in (8) and >.(x) in (11 ). Then >. in (22) is substituted with the >.(x) obtained to calculate ly(x). Then (8) is used to calculate n 8 [Jy(x) ] again. This procedure is repeated until ly(x) converges. Fig. 8 shows the results of J(x) and u 8 (x) obtained from the iterative method. The impedance of the slab is defined as
z (21)
-0.1
(b)
n(v)dvz
-Jv2-v2-v2
:8.5
x/d
J v2-v2-v2
dvx
0.35
0.3
ns
dvy
0.1
0.45
t)
(18)
c
0.5
0.5
dxO-(:r: )E(x)
dxO-(x)E(x)
v
0.4
0.55
JJ
where E(x) is the electric field . For a uniform current density, l y(x) = JA, J A =I/ D.zd where d is the thickness of the slab. For nonuniform ly( x), ly is bigger than J A at the edge of the slab and n 5 ( J ) is smaller at the edge. Therefore, for the same magnitude of I, inhomogeneous l y(x) will exceed l e at some x's even when JA is less than J c. If a three-dimensional velocity distribution is considered for a current flow in the y direction, ly = qnovA and
n 8 (Jy ) =
0.3
(a)
earlier in the R- I and Lk -I curves. The internal and external inductances Lin and Lex will also be nonlinear since ly (x) is determined by I,
JJ = J J =J J
0.2
x/d
A superconducti ng slab with a thickness d.
I=
0.1
1 =
d /2
(24)
J_d / 2 dxO-(x)
and resistance R = Re(Z) and kinetic inductance Lk = - Im (Z)/w. Fig. 9 shows the R - I and Lk - I relations from the final convergent l y(x).
1956
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TEC HNIQUES, VOL. 42, NO. 10, OCTOBER 1994
..-.. 0
2000
II J ___,,. Ul
0:::
-......._ ..-..
2 Ul 0:::
0
0
2
J/Jc(O) (a)
10 ..-.. 0
..-..
II
0
II
2 Ul
I
d/)..:112
I
_J
df)..:I
-......._
5
..-..
/
2 Ul _J
0
0
~~~~~~~~~~~~~~~
0.5
0
0
2
J/Jc(O) (b)
(b)
Fi g. 9. Ford/ >.(.J = 0) = 0. 5, 1 , 2, Tc = 90 K. (a) resistance of a slab, R, as a function of applied current inten sity I at T = 80 K; (b) kinetic inductance L k of a slab as a func tion of applied current intensi ty I at T = 80 K. Here T,. = 90 K.
Fig. 10. At T = 88 K, Tc = 90 K. (a) Surface resistance R s as a function of .J. (b) surface inductance L s as a function of .J. 0.04 0
0.035
For different thickness d, the nonlinear curve will be different. We have found that for smal ler d/ >.., R - I and Lk - I relations are more nonlinear. This is understandable since JY is bigger for smaller d for a given applied current I. Surface impedance of a superconductor is defined as
0.03
0.025
g ri
0.02
O.Q15
Zs =/¥
(25) 0.01
where 0.005
-
E =
ii7( J)
E +--.
w
(26)
Surface resistance is R s = Re(Zs ), and surface inductance is L s = - lm(Zs)/w. For WT= 10- 3 , R s and L s are plotted _in Fig. I 0. The nonlinear region appears near J c where the sample is partially superconducting. In the normal or superconducting states, the relations are linear. The L 8 is different at normal and superconducting regions. Experimental results [13) are compared in Fig. I I. As it shows, this model agrees with the trend of the experimental results. Since some parameters (a, T, etc.) are material dependent, by adjusting these parameters, a better fit between the theory and measurement may be found. Surface impedance Zs is a good description of superconductors since most of the currents and fields are confined within the penetration depth >.. from the surface.
~o·
10'
10'
10'
K (A /m)
Fig. I I. R s as a function of su.rface current density 1': at T = 77 K. The circles are the measured data from (13]. The sample isa YBCO at f = 10.4 K GHz with Tc = 92 K. The solid curve is from this model, where a = 3/2 , and wr = 2.8 x io - 2 . The surface current density is calculated from .J with a thickness of >.( .J).
Another case of interest is at very high frequencies and
O"n for T > Tc is not very big (which is true for the ceramic high Tc superconductors) so that E can not be neglected in (26) for JY near J c. Since gt -:/:- 0 in the Ma.xwell 's equations, a wave equation has to be considered. We will solve the guided wave case where the electric field Ey is decaying away outside the slab.
XIA et al.: A MACROSCOPIC MODEL OF NONLINEAR CONSTITUTIV E RELATIONS IN SUPERCONDUCTORS
The wave equation for E is 2
\7 E
+ k5 E~ E = 0
(27)
i CJ( ] ) 1 + - -.
(28)
where
Er =
1957
obtained. The geometry of a superconductor will introduce nonuniform distribution of the current density J. Therefore, non linearity will be enhanced at the edges and surfaces of superconductors. Using this macroscopic model , a solution scheme for electromagnetic properties of superconductors has been proposed.
W EO
If we replace E by 0- Jy in (27), (27) will look very similar to (21). We can also use an iterative scheme to solve for the nonlinear 0- case. First, we assume CJ is independent of J and solve for Jy(x) . Second, we calculate CJ (.7:) = CJ[Jy (x )]. Third, we solve the wave equation for the inhomogeneous medium problem and obtain Jy (x) . These steps will be repeated until Jy(x) and 0-(x) converge. This is a feed-back process. At the edge, increase in Jy will cause n 8 to decrease, which causes CJ 8 , and hence Jy to decrease. This will continue until the stable solution is obtained. Thi s procedure is similar to solving coupled two differential equations in the GL theory where the equation for A is essentially the same as the wave equation (27) for E or JY. The difference is the second equation for 'l/J, where 'if;(A ) (n 8 ( J)) is to be derived. We have derived n 8 ( J ) in this paper from the velocity distribution assumption. The results for the slab geometry can be used to study a microstrip geometry. If the dimension of z in Fig. 7 is reduced to d « >., the current will still be uniform in the z direction . Therefore, the same results can be applied. This nonlinear model can also be applied to study the dependence of R and L on the magnetic field H. Once the relation between the magnetic field H and the current J is determined, He and Jc can be related and the above discussion is directly applicable. Although the model presented is classical , the corresponding quantum statistical distribution can be used to derive the velocity (energy) distribution . Discrete distribution may be needed if the energy is quantized . VI.
CONCLUSION
A macroscopic model is proposed for nonlinear constitutive relations in superconductors. Distribution of electron velocities is used to derive the dependence of superelectron density n s on applied macroscopic current density J. Complex 0-( J ) is
REFER ENCES [I] P. England , et al. , "Granular superconductivity in R1Ba2C u30 7 _0 thin films," Phy. Rev. B, vol. 38, pp. 7 125- 7128 , Oct. 1988. [2] M. A. Dubson , et al. , " Non -Ohmic reg ime in the superco nducting tran siti on of Po ly-crysta lline Y 1 Ba2 Cu30 x," Phy. Rev. Lett. , vol. 60, pp . 1061 - 1064, Mar. 1988. [3] T . Van Duzer and C. W . Turner, PrinCiples of Superconductive Devices and Circuits. New Yo rk: Elsevier, 198 1. [4] G . J. Chen and M. R. Beasley, "Shock-wave generatio n and pul se sharpening on a series array Joseph son junctio n transmi ss io n line," IEEE Trans. Appl. Supcond., vo l. I, Sept. 199 1 [SJ D. H. Kim , et al. , " Possible ori gins o f res isti ve ta ils and criti cal currents in hi gh-temperature superconductors," Phys. Rev. B., vo l. 42, pp. 6249-6258 , 1990 . [6] X. Yu and M. Sayer, "Temperatu re de pendence of critical currents in YB aCuO ceramics," Phys. Rev. B., vo l. 44, no. 5, 199 1. [7] L. N. Shehata, "The wall energy and the critical current of an ani sotropic hi gh-temperature superconductor using modified Ginzburg-Landau Theory; ' J. Low Temperature Phys. , vol. 78, no . 1/2, 1990. [8] M. Tinkham, Introduction to Superconductivity. New York: McGraw Hill Book, 1975. [9] K. K. Mei and G. C. Liang, " Electromagnetics o f superconductors," IEEE Trans. Microwave Theory Tech. , vo l. 39, Se pt. 199 1. [IO] L. D. Landau and E. M. Lifshitz, Statistical Physics. New York : Addi son-Wesley, 1969. [ 11] G . H. Wannier, Statistical Physics . New York: Wiley, 1966. [ 12] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods fo r Scientists and Engineers. New York : McGraw- Hill , 1978. [1 3] Y. Kobayashi , T. Imai , and H. Kayano, " M icrowave measurements of te mperature and current de pendences of surface impedance for high-Tc superco nductors," IEEE Trans. Microwave Theory Tech. , vo l. 39, pp . 1530- 15 38, Se pt. 1991.
Jake J. Xia, photograph and biography not available at time of publication.
Jin A. Kong, photograph and biography not avail able at time of publicati on.
Robert T. Shin, photograph and bi ography no t availabl e at time of publication.
1958
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHN IQUES, VOL. 42, NO. IO, OCTOBER 1994
Numerically Efficient Spectral Domain Approach to the Quasi-TEM Analysis of Supported Coplanar Waveguide Structures Kwok K. M. Cheng, Member, IEEE, and Ian D. Robertson, Member, IEEE
Abstract- A numerically improved technique for the QuasiTEM analysis of supported coplanar waveguides is presented. The spectral domain method is combined with a special set of basis functions, to facilitate an accurate and efficient solution. The resulting integrals are evaluated using closed-form expressions instead of numerical integration scheme which leads to short CPU time. In this study, numerical results for the characteristic impedances of coplanar waveguides are presented. Comparisons are also made between the computed results and available ones. The charge density distributions on the center signal strip as well as on the ground plane are examined.
I.
,_,(x)
x
INTRODUCTION
C
OPLANAR waveguide [1] is often considered to have free space above and below the dielectric substrate. This configuration has not been found suitable for MMIC 's, where the substrate is typically thin and fragile. A solution is to mount the substrate directly on a conducting ground plane. In this case, the ground plane will support the fragile substrate, thus increasing both the mechanical strength as well as the power handling capability of the structure. However, it has been pointed out lately that the ground plane backing introduces some undesirable effects on the CPW behaviour of the structure due to the presence of the microstrip mode [2] . This mode can be suppressed by increasing the substrate thickness, but this is not always possible especially in MMIC's applications where semiconductor substrates are usually thin. An alternative is to mount the semiconductor substrate on a low-permittivity material such as quartz. A third solution is to grow a high quality GaAs layer on a Si substrate [3] and then to mount the entire assembly on a ground plane. In both cases, the thickness of the supporting dielectric material under the main substrate should be large enough so that the effect of microstrip mode may be ignored. Many papers have been devoted to the analysis of coplanar waveguide structure, and the frequency dependent solution is also available [4]-[6]. Although it has been pointed out that the quasi-TEM approximation is not valid at high frequencies, the dispersion characteristics presented in [7] suggests that I-percent accuracy in the effective dielectric constant can be maintained with this assumption up to 20 GHz. Several Manuscript received July 20, 1993; revi sed November 22, 1993. This work was supported by the Science and Engineering Research Council (SERC) UK. The authors are with the Communication Research Group, Department of Electronic and Electrical Engineering, King 's College London , Strand London, England WC2R 2LS. IEEE Log Number 9404140.
x bi Fig. 1. tions.
Cross-sectional view of supported coplanar waveguide configura-
approaches based on the quasi-TEM approximation have been reported, including the conformal mapping theory [8] and the finite difference method [9]. The well known spectral domain method [ 10]-[ 13] has also been successfully applied to solve a great number of planar configurations. Its main drawback is the long CPU time required for numerical integrations, especially when special functions [12], [13] are used. Owing to the increasing popularity of supported coplanar waveguides for the design of hybrid and monolithic integrated circuits, it becomes highly desirable to have an efficient and reliable method in obtaining the electrical parameters of these structures. In this paper, we present a new approach to improve the numerical efficiency of the quasi-TEM spectral domain computations for supported coplanar waveguides. We also extend this method to analyze the charge density distribution of the quasi-TEM mode for CPW's. The charge density distribution over all the CPW's conductor surfaces, together with their cumulative charge distributions, are examined. II. METHOD OF ANALYSIS The supported coplanar waveguide (SCPW) configuration under analysis is shown in Fig. 1, where the ground planes are assumed to be infinitely wide. All metallic conductors are
0018-9480/94$04.00 © 1994 IEEE
1959
CHENG AND ROBERTSON : NUMERICALLY EFFICIENT SPECTRAL DOMAIN APPROACH TO THE QUASI -TEM ANALYSIS
assumed to be infinitely thin and perfectly conducting. Three different configurations commonly found in MMIC's will be considered: these include the open SCPW, covered SCPW and the dielectric overlayed SCPW structures, where e 1, e2 , e3, and e 4 are the relative dielectric constant values of the various layers. The supported medium is assumed to extend to negative infinity in the y direction. Recent advances in GaAs growth on Si have resulted in high-quality and high-performance GaAs electronic and optoelectronic devices on Si substrates, and therefore, one must consider this composite structure as a substrate material for microwave and millimeter-wave monolithic integrated circuits. If we ass ume the quasi-TEM approx imation to be valid, our problem reduces to solving the Laplace 's equation in the x -y plane subject to the appropriate boundary conditions
c- 1 (a)°¢(a) = p(a)
where
cPa, i(x) = -1
2 _ _:.(_ x_-_a_i-_1_:.)__ _ 1 2~ a , ;(a; - a ;-1) .1: -
aN
~a ,i
=0
if 0
< x < ai-1
if ai-l
< x < a.;
< x < aN
if
ai
if
aN
< X < 00 (3 b) (3c)
cPb, i(x)
< x < bo if bo < x < bi-1 if 0
= 1
.x - bo
=l--~b , i
(b; - x) 2 2~b , .;(bi - bi-1)
(1)
< x < bi if b.; < x < 00 if bi-1
=0
(3d) where ¢(a), p( a) and G( a) are the transform of the potential distribution, charge density distribution and static Green's functions in the Fourier domain, respectively. These quantities are evaluated at the interface where the conductor strips are situated. The mathematical expression of c- 1 (a) for the SCPW structure is given by
~ , _ bi + bi-1 _ bo
a;=
0
kl
k _ 2 -
c4- c 3 c4+c3
=1
= 0
_ e1 - e2
----
e1
+ e2
{Dielectric overlayed} {shielded} {open.}
(2b)
(2c)
Of the spectral domain methods reported [12) , [13], the unknown potential distribution is usually expressed in terms of appropriate basis functions which incorporate the edge effect. High numerical accuracy has been demonstrated employing Bessel function and Chebyshev polynomial as the basis functions. The integrals in the resulting equations are usually determined by numerical integration schemes. To obtain higher efficiency in the computations, we propose to choose a set of basis functions that allows closed-form expressions to be used for the integrals. One possible representation in approximating the potential function ¢( x) at the dielectric interface is defined as N
M
¢(.x) = LKi¢a ,.;(x) + LLicPb , i(x) i=l
w
2' + (iS
< (1 < · · · < (N -1 < (N
= 1
(3f)
= fo < 6 < ·. · < ~M-1 < ~M = l.
(3g)
0 = (o
where
(3e)
2
b,i -
The graphical plots of expressions in (3b) and (3d) are depicted in Fig. I . The coefficients, Ki and L;, are the unknowns to be determined, (i and ~i are predefined constants depending on the method of discretization . If the slot is divided uniformly, the number of sections required becomes very large if a highly accurate solution is desired. Fortunately, by adopting a nonuniform discretization scheme, the numerical accuracy of the solution can be improved without the need to increase the number of basis function s. The dividing strategy is to use smaller sections over the region where the potential function is changing rapidly. Expressions for (i and ~i which have been very useful are shown below
~i = sin 3 ( 4 ~ 7r), (N- ·i
i
= 1,
=1-
f,;.
2, · · · , N - 1
(4a)
(4b)
By applying Galerkin 's procedure [13) to (1), the following system of linear equations are fou nd:
1960
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
The element of matrix .A_, denoted by 1, 2, · · ·, N + M), is defined by
Ai , j
=
100
A.; , j ( i, j
m/>c, i(a.)ef>d,j(a.) da.
2c2 oo 1·00 a.e-2nh°'ef>c, - i(a.)ef>d, - j(a.) da. + -+-L)~
c2
o
c3 n= l
00 100 a.e-2nt°'ef>c, - i(a.)ef>d, --1(a.) da.
2c3
+-+-I>~ c2
where Ca and Za are the characteristic capacitance and impedance, respectively , of the same guide structure when all dielectric material s are replaced by air. These parameters may be evaluated by conformal mapping theory or by the present method. The proposed method is therefore a very powerful analytical tool because one can get as close to the true value of capacitance as one desires by simply taking a sufficient number of terms of a finite expansion for the potential function .
o
c3 n=l
III. CHARGE D ENSITY DISTRIB UTION
(6) c
=a
< N else c = b
d
= a if j < N
if ·i
else d
=b
where the Fourier transform s of (3b) and (3d) are given by (7a) and (7b), as shown at the bottom of the page. Closed-form expressions are available (see the Appendix) for the integrals in (6) and the element Ai, j may therefore be rewritten as
Quantifying the charge density/field intensity at the conductor edges is of great important when determining parameters such as the insertion loss for a given structure. In determining the spatial waveforms of the quasi-static charge density distribution , it should be noted that the transverse electric field lines terminating on the surface of a perfect conductor must be perpendicular to that surface. As a result, there only exists an Ey component over the conductor surfaces of the CPW for the electric field. The expressions for the normal electric field distributions above and underneath the conductor surfaces, evaluated in the Fourier domain , can be shown as
Ebe low y
Eabove Y
=-
a.efJ(a.) { l+
2
f n=l
= a.efJ(a.) { l +
2
( c1 - c2 ) ne-2nh°' } c1 + c2
~
( c4 - c3 ) n e- 2nt01 } . L c4 + c3 n=l
(IOa) ,
(lOb)
(8)
u
= aN
if i
(a.)cos( xa.) da..
( 11 )
0
(9a)
Zo = Za y
re: C
ef>a,i (a.)
= 2 sin (ai-1a.) -
(9b)
The electric field intensi ty variations can also be found in a similar way but for the sake of brevity, only the charge density distribution function at the conductor surfaces will be derived . Hence, combining (2), (3) and (11), we have the charge density
ai-10.cos(aNa.) - sin (aia.) + aia.cos(aNa.) .6.a, ;(a; - a.; -1) 0. 3
(7a)
(7b)
1961
CHENG AND ROBERTSON: NUMERICALLY EFFICIENT SPECTRAL DOMAIN APPROAC H TO THE QUAS l-TEM ANALYS IS
distribution immediately over the conductors as p(x)
=
c:o ( c:2
+ c:3 )
N ~Ki
7r
{loo
w
s
-
i---~----1
a¢a , i (a)cos(.u x) da
0
y x
( p.z~~~~~~~-£3~~~-rt
f--~~~~~~~~~£2~~---l 1=h Replacing the integrals by the analytical formulae given in the Appendix, the above expression can then be put as
Fig. 2.
Graphical plots of basis functi ons.
3.0
· { Ho(aN, a;, ai-1, x)
N - Noo unifom1 discrctiw11011 U - Un ifonn discretization
W/h = I ConfonnaJ mappin g
+2
h~ ~knH ( aN ai 1
+2
+ C:3 n=l L
c2
n
a;-1 ,!'..._) 2h ' 2h ' 2h ' 2h
t~~knH(aN 2 c? -
+ c3 n= L l
n
a; a;-1 .:'.._ )}
2t ' 2t '
U-20
2t ' 2t
N-3
~~~~~§§~=:~:=::_u~-~so~ o.o-" 10
M
c:o(c:2
+
+ c:3) ~
7r
Li L 6. b ·(b · - bi -1) i =l 't 'l
S/h ratio Fig. 3. Plot of relative error in characteristic impedance calculations versus the ratio S/ h for different di scretization methods.
· { Ho(bo , b;, bi-1, x)
+
2h~ ~kn1 H (~ }!i_ c: 2 + C:3 ~ n 2h ' 2h '
+2
t~ ~knH 2 c2
+ C:3 n=l L
n
b;-1 ,!'. ._ ) 2h ' 2h
(bo b; b;-1 .:'.._ )} 2t' 2t' 2t ' 2t · ( 13)
The cumulative charge distribution function q(x) , which will be presented later on, is defined here as the integral of the charge density di stribution per unit length at the interface ( 14)
The proper choice of basis function clearly eliminates the need for tedious numerical integration to compute the integrals. At this point, it is very important to consider the
numerical efficiency in computing the summations in (8) and (13) . If a direct sum method is tried, the slow convergence and the oscillations of the sums make the computation time too long to be practical. Hence, to keep accuracy and computation times within reasonable bounds, it is necessary to accelerate the convergence of the sums. Two treatments are described in the appendix to reduce computation time. We have typically obtained a ratio between the time employed by direct sum and by these schemes of between 10 to 100.
IV. DISCUSS IONS
A computer program based on the form ul ae presented in thi s paper, has been developed to analyze the open SCPW configuration shown in Fig. 2. In our first example, numerical results are generated to examine the accuracy of the solution
1962
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
This Paper
s I µm w I µm
£1
=
Spatial charge density distribution (SCPW)
Spectral Domain
I 0 £ 1 = 3. 78 £ 1
=
£ 3 = 1.0
0.06
I0
c
£ 2 = I2.9
0
~
20
60
20 60 120 200 800
45.38 33 . 15 27.56 24.37 18.76
55.79 40 .79 33 .96 30. 12 23 .63
55.75 40.69 33.76 29.77 22.40
45 .85 33 .38 27 . 12 2450 18.83
56 .37 41 .08 34. 16 30 28 23 .72
56.33 40.98 33 96 29 .92 22 47
20 60 120 200 800
61 .49 45.60 37 .51 32.70 24.00
75.69 55.21 46 .37 40.58 30.49
75.46 55.86 45 .82 39.75 28.38
62 .01 45 .83 37 .67 32 .81 24.05
76 .32 56 .50 46.56 40.72 30 .56
7609 56 14 46.00 39 89 28 43
.2
0.0 4
£, = 3.78
0.02
S = 60 µm W = 60 µm
c
h = 100 µ111
.Q
E
E
"'
'6
.?:-
·u;
0.0
c ~ ~ -0.Q2 0
.c u
100
200
20 60 120 200 800
70 .02 52 .90 43 .73 38 07 27 .43
86.32 65.37 5423 47.44 35.09
85.77 64.64 53.22 4607 32.22
70 .60 53 18 43 .90 38 . 19 27 .48
87 .03 65.71 54.44 47 .60 35 . 16
86.48 64 97 53.42 46 22 32 25
20 60 120 200 800
82 .62 64.45 5409 47 .34 33 .63
I 02.42 80.21 67.67 59.61 43 .59
100.60 78.12 65. 18 56.66 38.95
83 24 64.80 54 31 47 .50 33.66
I 03.18 80 .66 67 .95 59 .80 43 .65
10 1.48 78 9"1 65 4"1 56 22 38 95
Fig. 4. Co mparison of characteri stic impedance values for the open SCPW with a rigoro us spectral domain technique and for different dielectric materials at low frequencies ( 1 GHz) .
based on different discretization methods. Fig. 3 shows the plot of the relative error in the characteristic impedance evaluation as a function of the ratio S/h (W/h = 1, c 2 = 10, and £1 = £3 = 1). Based on the uniform discretization principle, curves are produced for dividing number N = M = 10, 20, and 50. The reference points for the error calculations are taken as the impedance values generated by the present method with N = 200. This integer is chosen by increasing the number of subsections until the resulting impedance does not vary by more than 0.01 %. For purposes of comparison, values obtained by the conformal mapping theory and the non uniform discretization method (N = M = 3) are al so included in the diagram. It should be noted that the mapping theory produces quite large amount of error as the slot width is increased, as expected. The results also shown that in order to keep the "error" to below half a percent, the number of basis functions required is around 50. Upon examining the curves U - 20, U - 50, and N - 3, it is quite easy to see that the number of basis functions needed is reduced by a factor of about I 0, using the non uniform discretization formula given in (4). This is in fact a significant savings in terms of computational time and storage requirement since both of these parameters are roughly proportional to N 2 • The next example is to illustrate the accuracy of the method presented, by making a comprehensive comparisons with respect to the characteristic impedance of an open SCPW configuration (h = 200 µm) . As shown in Fig. 4, three supporting dielectric materials are considered here which are: GaAs (er = 12.9) supported by quartz (er = 3.78), GaAs (er = 12.9) supported by Alumina (er = 10) as well as a
- 0.04 -1- -1-----,
0
Fig. 5.
2
3
4 x/ b
5
6
7
8
Spatial charge density distribution , at the dielectric-air interface.
hypothetical substrate (er = 20) supported by Alumina (er = 10). In the second column of Fig. 4 shows the impedance values reported in [16], which were obtained by a rigorous spectral domain hybrid mode approach. Upon examining the two set of data, it may be concluded that the inaccuracy of the solution presented here is less than one percent for most of the applicable range of physical dimensions used in MMIC's and available dielectric materials (1 < E r < 20). It should also be pointed out that the numerical computations were run on a VAX Cluster 8700 machine, and the computing time for each impedance value is less than 80 ms with
N=M=3. Fig. 5 shows the charge density distribution over the conductor surfaces of the quasi-TEM mode for CPW's, where the central conductor, of width 2b (W), is placed between the two ground planes, of spacing 2a (W + 28), which are located on a main substrate of thickness h, with relative permittivity £ 2 . As we can see clearly from the diagram, the charge density distributions exhibit singular behaviour at the conductor edges. In practice, this gives a maximum electric field intensity over the conductor edges, as expected. The nonzero value of p( x ) in the gap very close to the edges of the conductors is caused by representing the highly discontinuous function with a series expansion of finite terms. It should also be noted that the set of basis. functions proposed in this paper are theoretically inferior to weighed Chebyshev polynomials [13] which take into the account the singular behaviour of the first derivative of the potential function at the conductor edges. From the practical point of view, however, this is not a serious problem. In Figs. 6 and 7 the spatial charge density distributions and the normalized cumulative charge distributions over the conductor surfaces, as a function of the CPW' s geometry, are shown. The normalized cumulative charge distribution function for the center conductor and ground plane are defined as q(x )/q(b) and (q(a)-q( x ))/q(a), respectively. These plots show the fractional electric charge as a function of distance
1963
CHENG AND ROBERTSON : NUMERICALLY EFFICIENT SPECTRAL DOMAIN APPROACH TO THE QUASI-TEM ANALYSIS
Spatial charge density distribution (center conductor)
Spatial cumulative charge distribution (center conductor)
0.5
1.0
c
c
0
£ 3 = 1.0
0
:;;
t; 0.4
£2 = 12.9
c
£1 = 3.78
.2
.2 0.8 c 0
:;;
0
'.E
:.sen
:J
5en 0.6
h = IOOµm S = 60 µm
o.3
'6
'6
Q)
"'5 0.4 .c
£0.2
en cQ) u
u
J
Q)
"'5 0.1 .c u
W= lOµm
u
W=IOµm l- - - - - - - - - - --
>
:g
-s 0.2
E
2oµm _ .. ----_,,
___,--
.---··:"::·::·::.·.:·.~:::::::C:-:-:.'.::::·:::::~·
Q)
60i.tm
:J
.. '1ooµm
u
......................--·:..·:::····
o.o-+-===r====~---,-----r---.
0.2
0.0
0.4
0.8
0.6
1.0
0.0
0.2
0.6
0.4
x/b (a)
IOOµm j
··-----····""-- .
1.0
(a)
Spatial cumulative charge distributi.on ( groundplane)
Spatial charge density distribution ( groundplane) 0.0
0.8
x/b
---
- - - ---·-·--·-·-
---------
1.0
100 µm _,..------··-··----- ···
c
0
:;;
c
u
t; -0.02
.2 0.8
2
c
0
c
60 µm
~--
----------·····
0
:;;
c
0
:J
5 0.6
W= JOµm
'.E-0.04
:.sen
'6
£-0.06
"'5 0.4 .c
en
'6
Q)
en cQ) u
u
Q)
>
:;;
Q)
~ 0.2
"'5 -0.08
E
.c u
:J
u -0 . 1 -+---~----.-----r-----,----,
0
2
3
5
(x-a)/b
0.0 0
2
3
4
5 6 (x - a)/ b
7
8
9
10
(b)
(b)
Fig. 6. Charge density distributions over the conductor surfaces as a function of W / h, (a) for the center conductor, (b) for the ground plane.
Fig. 7. Normalized cumulative charge distribution as a function of W lh , (a) for the center conductor, (b) for the ground plane.
at the conductors. And from Fig. 7(a), it can be seen that 50% of the total charge at the center conductor is within the outermost 25% of the center conductor. Furthermore, the level of charge crowding at the conductor edges decrease with W/h decreasing. This example is an extreme case where the effect of low impedance lines produces high E-field crowding at the center conductor edges. Therefore, with the gap width kept constant, insertion loss is reduced when the characteristic impedance of the line is increased. On the other hand, the rate of decay of electric charge density across the ground conductor as shown in Figs. 6(b) and 7(b), indicates the coupling effects that would exist between adjacent CPW transmission lines. One result of thi s observation is that coupling between parallel CPW lines will decrease with increasing strip width and hence, coupling effects are less severe between two impedance lines, as expected. At any rate, the presented
graphs lend themselves to the design of compact MMIC's where the measure of adjacent line coupling of CPW 's can be controlled.
V. CONCLUSIONS A numerically improved spectral domain approach has been described in obtaining the quasi-TEM parameters for a wide class of supported CPW configurations. Numerical results generated by these formulae have shown excellent agreements with other reported data. The graphical results presented agree with previous analytkal methods where the charge density distribution plots vary with CPW's structural dimensions, as predicted. The new technique is both easy to implement and reliable, thus making it an excellent choice for use in CADoriented design tool s.
1964
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
Method 1 Shanks ' Transformation (k 1 , k 2 < 0): It can be seen from the definitions in (2b) and (2c) that the absolute values of k 1 and k2 are always less than one for an unshielded SCPW configuration. In practice, when the dielectric constant values of the various layers of material s are taken into consid-
VI. APPENDIX
Fn(c, d, a, e,
f, b)
n4
e ration, the sign of k 1 and k2 are usually negative. Under this
circumstance, the terms of the infinite series are alternating and monotonically decreasing. It has been suggested that the convergence of such a series can be accelerated substantially by applying Shanks ' transform [14]. In principle, the limiting value of the partial sum defi ned by m
Sm= L )nFn(*) n =l
approaches the "true value" of the original summation as m tends to infinity. This "true" value can also be extracted from the partial sums of the first few terms by repeated application of Shanks ' transform [14]. An efficient algorithm for the computation of higher order Shanks' transform has been described in [15] . Method 2 Bound approximation (k 2 = l ): In this case, the general form of the sum may be written as 00
(Al)
It can be shown that the value of the above expression are bounded m-1
2
fi(f3) = {3 (1-
~)
L 2 In (1 + {3 )
+ h(f3)
h (f3 ) =
= (1 -
-1 ( f3 2
2
n=l
2( 1 - 3{3 3
{3 2) In (1
2
)
+ {3 2) -
arctan ({3 ) +
2{3 3
g
6
6
In (1
(A3)
+ f3 2 )
c -x) - I (a-n, n-b ,n-.'r )} .'r) = -n{ I (a -, -, 2 nn n
(B 1)
2
- 2 arctan (b - x)
1
00
Fn(*):::; f
n=l
n=l
Fn(*)
+
Fn(*) dn
m
and the term Fn( *) dn may be solved analytically. Note that both the upper and lower bounds may be made to approach the "true" value of the infinite sum simply by increasi ng m. A formula which has been found very useful in approximating this "true" value is given below
In both schemes, the series converge very rapidly. In practice, only 3 to 5 terms are necessary for a very good approximation (four to seven significant digits). R EFERENCES
- )I [l+(b+x) ] + (b x n 1 + (b - .'J; )2
+ x)
: :; f
(A4)
I (a b x)= bln{[l+ (a + x)2][1+ (a-x)2]} '' [l+(b+x) 2 ]2
- 2arctan (b
Fn( *) dn
m
J:
4{3 arctan ({3)
+ 2/3(1; {32 ) arctan ({3) + 7~4
Hn(a , b, c,
+
(A2)
4
- -/3 - -1 )
1
oo
Fn( *)
+ 4b. (B2)
To accelerate the convergence of the summations, two methods are described here:
[I] T. ltoh, "Overview of quasi-planar tran smission lines," IEEE Trans. Microwave Theory Tech. , vol. 37, pp. 275- 280, Feb. 1989. [2] H. Shi gesawa and M. Tsuji, "Conductor-backed slot line and coplanar waveguide: Dangers and full-wave analysis," IEEE MTT-S Dig., pp. 199-202, 1988. [3] M. I. Aksun and H. Morkoc, "GaAs on Si as a substrate fo r microwave and millimeter-wave monolithic integration," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 160-163, Jan . 1988. [4] J. B. Davies and D. Mirshekar-Syahkal , "Spectral domain solution of arbitrary coplanar transmi ssion line with multilayer substrate," IEEE Trans. Microwave Theory Tech., vol. MTT-25, pp. 143-146, Feb. 1977.
CHENG AND ROBERTSON: NUMERICALLY EFFICIENT SPECTRAL DOMAIN APPROACH TO THE QUASl-TEM ANALYSIS
[5] Y. Fukuoka, Y. C. Shih, and T. ltoh, "Analysis of slow-wave coplanar waveguide for monolithic integrated circuits," IEEE Trans. Microwave Theory Tech., vol. MTI-31 , pp. 567-573, July 1983. [6] J. B. Knorr and K. Kuchler, "Analysis of coupled slots and coplanar strips on dielectric substrate," IEEE Trans. Microwave Theory Tech. , vol. MTI-23, pp. 541-548, July 1975. [7] Y. C. Shih and T. Itoh, "Analysis of conductor-backed coplanar waveguide," Electron. Lett., vol. 18, no. 12, pp. 538-539, June IO, 1982. [8] G. Ghione and C. Nald i, "Coplanar waveguides for MM!C applications: effect of upper shielding, conductor backing, finite-extent ground planes, and line-to- line coupling," IEEE Trans. Microwave Theory Tech., vol. MTI-35 , pp. 260-267, Mar. 1987. [9] T. Hatsuda, "Computation of coplanar-type strip-line characteristics by relaxation method and its app lication to microwave circuits," IEEE Trans. Microwave Theory Tech. , vol. MTI-23, pp. 795-802, Oct. 1975 . [10] F. Medina and M. Homo, "Determination of Green ' s function matrix for multiconductor and an isotropic multidielectric planar transmission lines: A variational approach," IEEE Trans. Microwave Th eory Tech., vol. MTI-33, pp. 933-940, Oct. 1985. [I I] R. R. Boix and M. Homo, " Modal quasistatic parameters for coplanar multiconductor structures in multilayered substrates with arbitrary transverse dielectric anisotropy," Proc. IEE, vol. 136, pt. H, no. I, pp. 76-79, Feb. 1989. [12] K. Araki and Y. Naito, "Upper bound calcu lations on capacitance of microstrip line using variational method and spectral domain approach ," IEEE Trans. Microwave Theory Tech., vol. MTI-26, pp. 506-509, July 1978. [13] A. Sawicki and K. Sachse, "Lower and upper bound calculations on the capacitance of multi-conductor printed transmission line using the spectral-domain approach and variational method," IEEE Trans. Microwave Theory Tech. , vol. MTI-34, pp. 236-243, Feb. 1986. [14] D. Shanks, "Nonlinear transformations of divergent and slowly convergent sequences," J. Math. Phys., vol. 34, pp. 1-42, 1955. [15] P. Wynn , "On a device for computing the em(Sn) transformation ," Math. Tables Aids to Comp. , vol. IO, pp. 91-96, 1956. [16] S. S. Bedair and I. Wolff, "Fast, accurate and simple approximate analytic formulas for calculating the parameters of supported coplanar waveguides for (M)MIC's," IEEE Trans. Microwave Theory Tech. , vol. 40, pp. 41-48, Jan. 1992.
1965
Kwok K. M. Cheng (M'91) was born in Hong Kong. He gained a first class honour degree in e lectronic engineering in 1987 and a Ph.D. degree in 1993 , both from King' s College, University of London. From 1990 to 1992, he was employed as a Research Assistant at King 's College, and worked on osci ll ators, filters and MMIC design. He is currently a Post-Doctoral Research Assistant at King's . His main research interests include the computer-aided modeling of passive and active components and the analysis and design of microwave active filters , high-effic iency amplifiers and low-noise osci ll ators. Dr. Cheng has published over IO papers in leading international technical journals. He is an associate Member of the Institution of Electrical Engineers, England.
Ian D. Robertson (M ' 91 ) was born in London England in 1963. He received the B.Sc. and Ph.D. degrees from King's College, University of London, in 1984 and 1990, respectively. From 1984 to 1986, he was employed at Plessey Research (Caswell) in the MMIC Research Group, where he worked on MMIC mixers, on-wafer measurement techniques, and FET characterisation. In 1986, he returned to King 's as a Research Assistant, working on the T-SAT mobile commun ications payload. He is currently a Lecturer at King 's College and leader of the MMTC Research Team in the Communications Research Group. He has coauthored over I00 technical papers.
1966
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOB ER 1994
On the Use of Differential Equations of N onentire Order to Generate Entire Domain Basis Functions with Edge Singularity Zwi Altman, Daniel Renaud, and Henri Baudrand, Senior Member, IEEE
Abstract- Entire domain analytical basis functions with edge singularity are a very useful tool for analysing planar transmission lines and planar rectangular or circular circuits in a moment method solution. Analytical basis functions are limited to separable geometry. In this paper we introduce new entire domain basis functions including edge singularity at the edges of a domain with arbitrary shape. These basis functions are derived from a differential equation of nonentire order which includes a fractional derivative. The one dimensional case is considered first. In the two dimensional case the basis functions are constructed numerically using the boundary element method and the Galerkin method. The basis functions are applied in a moment method solution to a nalyse a shielded microstrip. The current and the electric field a re calculated and compared with the results obtained by analytical basis functions.
l. INTRODUCTION
HE.analysis of planar transmission lines like microstrips, finhnes and CPW has been carried out efficiently by the moment method using the spectral domain approach (SDA) [l] . Particularly good results are obtai ned when analytical basis functions (b.f.) with edge singularity are used [2]. The use of b.f. with edge singularity enables physical quantities, like current on planar conductors or fields on apertures, to be described with a very small number of functions. Later these b.f. were extended to rectangular and circular domains and today they are often used to analyse rectangular patch antennas [3] and other rectangular and circular structures. The diajor drawback of the analytical b.f. with edge singularity is that they can only be applied to separable geometry. The purpose of this paper is to introduce new entire domain b.f. including edge singularity, which are defined over a planar domain of arbitrary shape. At the edges of a metallic conductor the current and fields have si ngular components. This singular behaviour has been studied thoroughly in the literature [4]-[7]. The electromagnetic solution that involves a planar conductor with a complex shape is numerical. If the moment method is chosen [8], then sub-domain b.f. are used, like the "roof-top" functions [9]. The edge condition and a domain of complex shape impose
T
Manuscript received July 29, 1993; rev ised November 22, 1993. This work was supported in part by the DRET, French Ministry of Defence, under contract 91-441. Z. Altman and H. Baudrand are with the Laboratoire d' Electronique, E.N.S.E .E.I.H.T., 2, rue C. Camichel, 31071 Toulouse Cedex, France. D. Renaud is with ANTECH, 31 Av. J. F. Champollion, 31100 Toulouse, France. IEEE Log Number 940415 1.
a large number of sub-domain b.f. and so heavy calculations are involved. When entire domain b.f. with edge singularity are used, we may expect that the number of b.f. needed will be much smaller. The demand that the domain be of arbitrary shape imposes a numerical type of solution, and so a preliminary preparatory stage is required. Calculation of this additional stage for a given geometry will be worthwhile if the b.f. are used repeatedly, for example, when the frequency response of a planar circuit is required. The b.f. with edge singularity, which we call singular b.f., are constructed as follows: first a differential equation of nonentire order with Dirichlet boundary condition is introduced, and the solutions to this equation, which we call generating functions, are found numerically. Then the b.f. are obtained by deriving the generating functions in the one dimensional case, or by taking their gradient in two dimensions. The differential equation is obtained by changing the order of the derivative in the Helmholtz equation as will be explained presently. In the one dimensional case the generating functions resemble sine functions and the b.f. resemble cosines. The main difference is in the slope of the generating functions which tends to infinity at the edges. When the generating functions are derived, a singu lar behaviour at the edges is observed. The two dimensional case is given as a simple generalisation of the one dimensional case. The numerical construction of the generating functions and the b.f. resembles that of scalar potential and transverse electric field of TM modes in waveguides respectively. The differential equation of nonentire order is derived from the Helmholtz equation as follows: we consider the Laplacian in the Helmholtz equation as a linear operator. This operator is defined by its eigenvalues and eigenfunctions (sines and cosines). If we change the power of the Laplacian operator to a fractional power p/q , we introduce a fractional derivative, or a differential equation of nonentire order. The eigenvalues of the new operator are those of the Laplacian to the power of p/ q, while the eigenfunctions are left unchanged. Once we have the eigenvalues and eigenfunctions of the differential operator, we may define the Green's function for the differential equation and with it we construct the sol ution . The concept of a derivative of nonentire order is used in fractal theory, and some examples of such derivatives are illustrated in [IO]. In Sections II and III, the generating functions and the b.f. are constructed in one and two dimensions respectively. It is shown that the singul ar behaviour of the b.f. at the
0018-9480/94$04.00 © 1994 IEEE
ALTMAN er al. ; USE OF DIFFERENTIAL EQUATIONS OF NONENTIRE ORDER
1967
edges is determined by the parameter p/q of the differential equation. Away from the edges the generating functions and b.f. oscillate. In the last section we use the singular b.f. to calculate the current and fields for a shielded microstrip line. The results are compared with those obtained by analytical b.f..
D
S
a Fig. 1.
A domain S inside a domain D of length a.
II. SINGULAR BASIS FUNCTIONS IN ONE DIM ENS ION
In this section, we introduce the singular b.f. that are characterized by a singular behaviour on the edges of the domain of definition. The mathematical development is first introduced for the one dimensional case. In the next section the formulation is generalised to the two dimensional case for a domain of arbitrary shape. Consider a domain S defined by the segment [XA, XB] inside a domain D of length a (Fig. I). Define the scalar function ¢k> called a generating function, as a solution to the following differential equation of nonentire order and with Dirichlet boundary conditions:
a ) p/ q -k 2]cPk( x). -0 _ [( - 8x2 { cPk(XA) = cPk(XB) = 0 2
(1)
(- 8 2 /8x 2)P/q is a fractional derivative, or a derivative of nonentire order. We consider (-8 2 /8x 2 ) as a linear operator with a domain of definition D. This operator is defined by its eigenfunctions, and its eigenvalues, sin( mr /a) and its eigenvalues, ( mr /a )2 . When the power of the operator is changed from one to p/q, we leave the eigenfunctions unchanged and raise all eigenvalues to the power p/ q:
[
~] p/qSill . [n7r :r J -_ 8x 2
a
[n27r2] p/q . [n7r ·] a2
Sill
a
X .
(2)
The derivative is taken with minus sign in order to have real eigenvalues. For p/q = 1 we retrieve the Helmholtz equation with Dirichlet boundary condition. (1) may be solved with the aid of a Green 's function , defined by the following equation:
8 2 ) p/q [( - 8 x 2
] k 2 Gk(:r, .x 0 ) = -o(x , x 0 ).
-
(3)
The subscript " O" stands for the source point and x 0 can have one of the values XA or XB · Gk is constructed as an eigenfunction expansion of the operator (-8 2 /8x 2 )Pfq in the same way as the Green's function for the Helmholtz equation [ 11]:
Gk(x , xo) =
Ln
(n:;
(n7r ) (n7r )
2
1 sin - xo sin - x . / )Pq_k2 a a
(4) We may now write the solution to (1) as follows :
cp(xB) and cp(.xA) are two derivatives of nonentire order of ¢k which are found numerically for each solution of k. The
eigenvalues k of (1) are found by imposing the boundary conditions on cPk using (5): cPk (xA) = cPk( xB) = 0. This yields a homogeneous matrix equation of the form [A]
r-~-t-~',~-+-~---J x
Fig. 2.
~~/J
The first externa l solution in the region D-S, p/q = 3/4.
The infinite set of eigenvalues of (1) includes those values of k for which the determinant [A] vanishes. For each solution k, the determination of P is straightforward. (5) is obviously a solution to (1), as may be verified by substitution. In the case of the Helmholtz equation (p / q = 1) two kinds of solutions cPk are found : interior and exterior solutions. Interior solutions vanish in the exterior domain D-S, and exterior solutions vanish in the interior domain S. The property of a solution to vanish in the interior or exterior domain is well understood for the Helmholz equation [I I] . When p/q I- 1 this property no longer holds. An interior solution does not vanish in the exterior domain but gets small values there, and vice versa. In Fig. 2 the first exterior solution of the domain D-S is plotted for p/q = 3/4. We may distinguish between interior and exterior solutions by calculating a ratio Rk between the average absolute value of /k in the interior and exterior domains. For p / q = 3 / 4 we find the following empirical relation: Rk (interior solutions) lORk (exterior solutions ). The ratios corresponding to the interior and exterior solutions tend to bunch together into two di stinct values, and so their identification is simple. ¢k has the form of a sine (or cosine) series and so its calculation on a discrete set of points is performed efficiently using the fast si ne (or cosine) transform . The b.f. Fk(x) in one dimension is defined as the derivative of ¢k :
(6) From now on we consider the interior solutions only. That part of ¢k which passes over to the external domain D-S is truncated in the space domain using the fast sine transform , and then the sine series coefficients are obtained by taking the inverse transform. Fk (x) which is obtained from the truncated function / k vanishes in the exterior domain . Of special interest is the choice of p/q = 3/4. Equation (5) gives cPk in the form of a Fourier sine series. For large n the Fourier coefficients of ¢k behave as 1/n312 , and those of Fk as 1/n 112 . We notice that Fk becomes singular at the edges in the same order as 1/ VS where S is the distance from the border. (The proof is given in the Appendix for the two dimensional
1968
IEEE TRANSACTIONS ON MI CROWAVE TH EORY AND TECHNIQUES. VOL. 42, NO. 10, OCTOB ER 1994
lk----~-~x
c
\__) (a)
x
(b)
Fi g. 5. 1
fl
x l1
x
I
\,
(c)
Fig. 3.
The anal ogue to the constant functi on with edge singul arity.
(d)
a.
First four generating functi ons in one dimensio n. p / q = 3/4 .
c. Fig. 6. (a)
b.
p/q = 3/4
p/q = 213
d.
p/q = 5 /8
The first bas is fun ction plotted for different values of p/ q.
(b)
We note that (7) has the form of a sine or cosine Fourier series and so in certain applications may be used as a filter function in the spectral domain. In the one dimensional case (7) becomes
" I (c)
'!/Jc ( X ) =
x
1--
Fig. 4.
p/q = 1
Fk-----~--
\, (d)
First four bas is functi ons in one dimension . p / q = 3/4.
case). This singularity corresponds to the edge condition of current or electromagnetic fie ld at a 0° conducting wedge. In Fig. 3(a)-(d) the first four truncated interior solutions for cf>k with p/ q = 3/4 are plotted. We may notice the resemblance of the solutions to the first four sine functions. The important difference is found in the larger slope at the edges. In Fig. 4(a)-(d) the first four basis functions Fk with p/ q = 3/4 are plotted. The edge singularity is evident. In some applications it is interesting to add a constant b.f. which serves as a bias function . Thi s constant function can be considered as a particular solution of the Laplace equation, and the solution is characteri sed by a constant value on the boundary and a zero normal derivative there. In the two dimensional case we denote by I the contour of the domain, and by ii the outward normal. The constant function of unit amplitude is given by
'l/Jc (r ) = - { 8G~r , ro) dlo . 11 no
(7)
8G(x, XA) 8G(x , xa) - - ----'---8xo 8xo
(8)
where G in (7) and (8) is the Green ' s function of the Laplace equation. In the problem of a current on a metallic strip the first analytical b.f. [2] is always positive and contains the edge singularity. We may generate such a function by substituting the Green 's function given in (4) with k = 0 and p/ q = 3/4 into (8). For large n the Fourier coefficients of 'l/Jc (p/ q = 3/4) behave as 1/n 112 . Thi s is the same asymptotic behaviour as that of Fk(p/ q = 3/4) in (6), and that of the analytic b.f. in (18) . The filtered function 'tf;c (p/ q = 3/4) is shown in Fig. 5. It is the analogue of the constant solution of the Laplace equation, and will be used in section 4 as the first b.f. for the current on a microstrip. An interesting example is to show the dependence of the b.f. on the parameter p/ q. In Fig. 6(a)-(d) the first b.f. is plotted for four different values of p / q. It is noticed that the functions become more singular at the edges as the values of p/ q decreases. Thi s behaviour corresponds to the lowering of the convergence rate of Fk asp/ q decreases. In all four figures the number of terms in the series for Fk is 256. III. SINGULAR BASIS FUNCTIONS lN Two DIMENSIONS
In thi s section we generali se the formulation given in Sectio~ II to the two dimensional case. As the domain of
1969
ALTMAN et al. ; USE OF DIFFERENTIAL EQUATIONS OF NONENTIRE ORDER
definition is of arbitrary shape, a part of the calculation is numerical. In order to give some insight into the two dimensional formulation, we note that it is analogous to the construction of TM transverse electric fields in waveguides [12]: first a scalar potential is derived using a contour integral and a Green's function for a two dimensional rectangular resonator, and then the transverse electric field is obtained by taking the gradient of the potential. Consider a domain S with a contour l, inside a rectangular domain D with sides a and b (Fig. 7). We introduce the following differential equation of nonentire order:
[(-!J.)Pf q - k 2]¢k( .x, y) = 0 {
cPk(r )
= O; r
(9)
E1
(-!J.)P f q is the Laplacian with minus sign and the power of p/q:
d2
( -!J. )Pf q = [ - ( - 2 dx
+ -d22 )] p/q
(JO)
dy
As in the one dimensional case we consider (- !J. )Pf q as a linear operator which is defined by its eigenfunctions, sin(m7r/a)sin(m7r/b) and its eigenvalues, ((m7r/a) 2 +
(n1f /b)2)p fq: (-6.)pf qsin ( rr:7f x )sinC; m7f)2 = (( --;;:---
y)
-;;-.x) sin (n1f by ) .
n7r ) 2) p/qsin (m7f + (b
( 11)
!~
y
d
,. x ~
Fig. 7.
b
D
a
A domain S with a contour I inside a rectangular domain D.
are found, and may be selected in the same way as in the one dimensional case. Equation ( 12) is a solution to (9), as may be easi ly verified by substitution. In the case p/ q = 1 we may show the uniqueness of the solution in the sense that every solution to (9) may be written as (14). This is shown by using Green's theorem which is a special case of Gauss' theorem [11, pp. 803-806]. The proof of uniqueness of (14) in the general case where p/q -j. 1 is awkward because we do not have an analogous theorem to Gauss' theorem when fractional derivatives are involved. Some basic mathematical concepts must first be defined like the divergence of nonentire order, 'VP / q · F , the gradient and so on. This is beyond the scope of the present work. The functions { Aqq, Akk, A u are identity matrices because (3) does not contain the scalar potential , while A kp, Apki A qki Akq , At k, Akt , At q, A ql and 0 are all null matrices . The matrix elements shown above are obtained by some limiting procedures. For example, B~t is obtained by first assigning y = y0 = 0 in (4c), Then, U2(.T)/E2 =
J{~(2/mrE2)[coth(mrh2/b)
sin(mrxo/ b)]}
x sin (mrx/ b)[8U2/8n]dx (y =Yo = 0).
(7)
In (7), the integral path is along path CBAF which contains the conductor (with index q) and the non-conductor (with index k) region . After discretization, the matrix element is readily obtained. Other B matrix elements in region 2 can also be obtained by assigning y 0 and y the value 0 or -h 2 in Eq (4b) or (4c) depending on where the source and field points are located. In solving (5), the right hand side is considered as a known data file. If 1 volt is applied at the conducting plate, the elements of the column vectors UP and Uq are all equal to 1. The unknown contain 6 different column vectors. The elements of the column vectors U and G are the electric scalar potential and its positive normal derivative on the central point of each segment respectively. Again, the indexes p , q, k , and l represent specific regions. The boundary element method is itself very flexible in treating multi-media and multi-conductor systems. However, for several practical structures such as the suspended line and the coplanar waveguide, the conductor is on one interface plane only (no conductor along y = -h 2 in Fig. 2). Regions 2 and 3 can be treated as one homogeneous region with permittivity E2 by considering a combined Green's function. To achieve this end, 1.p 2 is expanded as
conditions are
dG1;,( y)/dy = 0 at y = 0 G1;,( y) = G;,( y) at y =Yo dG1;,( y)/dy - dG;,(y)/dy = [-l (bE2)] x sin(m rxo/ b) at y =Yo G;,( y) = G~ at y = -hz Ez [dG;,( y)/dy] = E3 [dG~(y)/ dy] at y G~( y ) = 0 at y = - (h 2 + h 3).
L a ;,( y) sin (mrx/ b) for Yo ::; y ::; 0
(9b)
(9c) (9d)
=-
h2
(9e) (9f)
In the above boundary conditions, (9a) is needed so that (3b) holds true; (9c) reflects the singularity at the source point (xo, yo). The other conditions are easily understood. For the present application, only (8a) is needed to establish the boundary integral equation along y = 0. Once G~ (y) is obtained, we then assign y = y 0 = 0 in G~ and get the required equation along the interface plane y = 0. i.e.
'Pz(Y =Yo = 0) =
L G1;, (y = Yo = 0) sin (mrx/ b) ,
( IO)
n=l
G1;, (y =Yo= 0) = (2 / m :) sin (mrxo/ b)](Pl / Ql )
(lOa)
Pl= cos h (mrhz/b) sin (mr h 3/ b)
+ (E3/E2) sinh (mrh2/ b) cosh (mrh3 / b) QI= E2 sinh(n7rh2/ b) sinh(mrh3/b)
+ E3 cosh(n7rh2 / b) cosh (n7rh3/b). This combined Green's function saves memory space as no node point is assigned on the dielectric interface plane y = -hz. The boundary elements are assigned on the plane y = 0 only. To facilitate the calculation of matrix elements (as in (6a)- (6e)), the geometric series method in [4] can be employed. (6a), for example, can be rewritten as NI
B;~
=L
(2/ n7r ) [ta nh (n 7rhi/b) sin(n7rx;/b)RLT(.T 1 )]
n=l 00
n=l Nl
- L(2/ n7r) sin(n7rx;/ b)RLT(x 1 )
00
'P2 =
(9a)
n= l
(8a) where NI is a number such that tanh ( n7r hi/ b) '.::::' l. The second series in the above expression can be simplified by the geometric series method [4]. We first note that
n=l
and 00
'P2
= L a ;, (y ) sin(mrx/ b) for
- h 2 ::; y ::; YO ·
(8b)
n= l
00
L (2/ n7r) sin (n7rx;) RLT(x1 ) n= l
In region 3, constructed as:
1.p 3
satisfies Laplace's equation and can be
L G~(y) sin(nn:/ b) for n=l
L(l /n 2) [siu (n:i:1) + siu (nx2) n= l
00
'P3 =
00
2
= (B / 7r )
- (h2 + h3) :S y :S - h 2 .
(8c) To derive the combined Green's function , a line source is assumed at (x 0 , y0 ) in region 2. The required boundary
where X 1 = 7r (x; - x-)/b, xz = 7r (X; + x-)/ b, x 3 7r (x; - x+)jb, and X4 = 7r (x; + x+)jb. x- = Xj - (w1/2), and x+ = Xj + (Wj / 2) as defined before. By the geometric
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. IO, OCTOBER 1994
1976
TABLE I COMPARISON OF CHARACTERISTIC IMPEDANCE (Zo) AND NORMALIZED GUIDED WAVELENGTH VALUES FOR MICROSTR!P LINE (DIMENSIONS IN MILS, b 20 h 2 + w,h1 lOh2,€1 €Q,€2 €r€o)
=
w 10 15 22 20
=
e_r 2.9 4.7 4.3 4.7
h2 67 14 19 31
=
y
I
=
Zo
Zo
>. gI >.o
>. 9 />.o
[10) 165.9 67.2 67.2 84.4
this work 163.5 67.4 67 .4 83.7
[10) 0.693 0.546 0.567 0.554
this work 0.695 0.547 0.565 0.568
ll
e_r b/w 2 5 9
=
=
1.0 this work 63.788 65.178 66.280
=
=
3.78 this work 52.950 55.147 56.571
[11) 63.0 64.0 64.0
=
52.6 54.8 55.0
l
=
case I
+ 2w)
2s' /((s
= 0.43, w/a = 0.14
s/a (I )
50.00 50.86 51.57 53.76
0.5 I
2 3
(2) 50.32 51. 13 51.80 52.38
[13) 46.10 46.50 46.60 46.60
=
=
case 2 sf a = 0.43, w/a = 0.07 (I) (2) [13) 40.48 40.92 36.10 41.78 42.17 36.80 42.56 43.00 37.20 43 .89 43 .77 37.40
TABLE IV COMPARISON OF COPL;ANAR WAVEGUIDE DATA BY THE PRESENT METHOD AND BY THE SPECTRUM DOMAIN TECHNIQUE h 1=h3 =4.5mm, h2 =l.Omm, w s =2.0mm, e1 €3 l.O eo , e2 €r€Q
=
= =
=
€r
2.55
9.35
20.0
[14] thi s work
1.22 1.1 952
1.78 1.7488
2.44 2.3454
series method, this series converges quickly. We list below the formula for [12] : 00
'.::::'.-[xi ln(xi) -
Xi -
5
(.Ti ) /72 - (xi ) /14400],
i = 1, 2, 3,and 4. As a result, Nl terms are enough to make (6a) convergent. For all examples studied in this paper, Nl is set to 20. Similar procedures can be employed to obtain fast converging forms for (6b) to (6e), (10), etc. For (6c), we need to choose Nl such that csch (mrh 2 /b) approaches zero. For (10), it is seen from (lOa) the Pl/Ql approaches 1 as n approaches NL
III.
A
A' F
x c,
r3
3
Sa
Cross section of a coplanar waveguide .
checked with [10], [11]. The data from [11] shown here were read from a graph. Good agreement was found. Tables III-IV show the calculated results for coplanar waveguide by boundary element method using the Green' s function in combined form and in original form ( 6a to 6e). Referring to Fig. 3, AB, BB' . B' C , DE, FA' and A' A were divided into 29, 2, 12, 6, 12, and 2 uniform segments (boundary elements) respectively. It is noted that no boundary elements are assigned on line DE when using the combined Green 's function . Thus, memory space is saved. Furthermore, as the number of segments increases, so does computation time. With the combined Green's function , typical computation time for examples shown here is about 10 seconds on a PC/486 computer. Computation time significantly increases (typically 25 minutes for examples shown here) if we do not adopt the geometric series method to facilitate computation. In comparing with data obtained in the literature [ 13] calculated by the finite difference method, some discrepancies were found. However, our results calculated by boundary element method are consistent using different approaches. Finally, we present results for coplanar waveguide obtained by the present method and compare with results obtained by the spectral domain technique [14]. Data from [14] are made at a frequency of 1 GHz in order to avoid the effect of frequency dispersion. Good agreement is found. IV.
n= l
3
B
1-s·-1-w-I- ~ -1-w+s·-1 Fig. 3.
[11) 31.5 40.7 41.0
TABLE Ill CALCULATED CHARACTERISTIC IMPEDANCE (ohm) FOR COPLANAR WAVEGUIDE (a = 4.3mm, h2 0.6 l mm , e1 e3 eo,e2 9 .4eo , (I): ORIGINAL GREEN'S FUNCTION, (2): COMBINED GREEN'S FUNCTION)
=
B'
=
25.0 this work 32.302 40.839 41.912
[II)
c
S,
1
a h2
TAB LE II CALCULATED CHARACTERISTIC IMPEDANCE FOR SUSPENDED LINE (h 1 h3 0.4a , h 2 0.2a ,w a,€1 €3 €0,€2 €r€O
=
c,
NUMERICAL RESULTS
Tables I and II show the calculated results for microstrip and suspended line respectively. Nl is set to 20 for all calculations . Uniform discretization was used for both cases, where AB, BC, DE, and FA were divided into 10, 10, 20, and 10 segments respectively. The data presented were
CONCLUSION
This paper is based on the work in [4] . The main purpose is to eliminate the disadvantage that our previous work is not directly applicable to multilayered structures. We therefore suggest several useful Green's functions . These new functions not only solve the aforementioned problem but also simplify the boundary integral equation. The integrand contains both the potential function and its derivative in most of the work based on the boundary element formulation . However, in the present work, the integrand contains only the positive normal component of the electric scalar potential. Therefore, half the work is saved in obtaining the matrix element. A combined Green 's function is also presented to further save memory space. This combined Green' s function is useful in treating suspended stripline and coplanar waveguide structure. To compute matrix elements, we use a geometric series method which simplifies the calculation. Normally, 20 terms
CHANG AND SZE: FLEXIBILITY IN THE CHOICE OF GREEN' S FUNCTION FOR THE BOUNDARY ELEMENT METHOD
are enough to get convergent results. Another advantage of the present work is that there is no singularity in the boundary integral equation. REFERENCES
I977
[ 13] T. Hatsuda, "Computation of coplanar-type strip line characteristics by relaxation method and its applications to microwave circuits," IEEE Trans. Microwave Theory Tech., vol. 23, pp. 795-802, 1975. [ 14] E. Yamashita and K. Atsuki , "Analysis of microstrip-like transmission lines by nonuniform discreti zation of integral equations," IEEE Trans. Microwave Theory Tech., vol. 24, pp. 195-200, 1976.
[I] T. Honma and I. Fukai, "An analysis for the equivalence of boxed
[2] [3]
[4]
[5]
[6]
[7] [8]
[9]
[ IOJ [ 11 ] [ 12]
and shielded strip lines by a boundary element method," Trans. IECE, Japan , vol. 165-B , pp. 497-498, 1982. M. Ikeuchi, " Boundary element analysis of shielded microstrip lines with dielectric layers," Trans. IECE, Japan, vol. J67-E, pp. 585-590, 1984. T. N. Chang and C.H. Tan, "Analysis of a shielded microstrip line with finite metallization thickness by the boundary element method," IEEE Trans. Microwave Theory Tech., vol. 38, pp. 1130-1132, Aug. 1990. T. N. Chang and Y. T. Lin, "Quasi-static analysis of shielded microstripline by a modified boundary element method," IEEE Trans. Microwave Theory Tech., vol. 41, pp. 729-731, Apr. 1993. T. N. Chang and Y. C. Sze, "Flexibility in the choice of Green's function for the boundary element method," IEEE MTT-S Int. Microwave Symp. Dig., 1993, pp. 921-922. B. H. Mcdonald, M. Friedman, M. Decreton, and A. Wexler, " Integral finite-element approach for solv ing the Laplace equation," Electron. Lett., vol. 9, no. 11 , pp. 242-244, 1973. B. Song and J. F, "Efficient analysis of finline structures by the boundary element method," IEE Proc., vol. 139, part H, Feb. 1992. N. Kish and T. Okoshi, "Proposal for boundary-integral method without using Green's function," IEEE Trans. Microwave Theory Tech., vol. 35, pp. 887-892, Oct. 1987. F. Tefiku and E. Yamashita, "An efficient methods for the determination of resonant frequencies of shielded circular disk and ring resonators," IEEE Trans. Microwave Theory and Tech., vol. 41 , pp. 343-346, Feb. 1993. T. Itoh, "A method for analyzing shielded microstrip lines," Trans. IECE, Japan, vol. J58-B, pp. 24-29, Jan., 1975. E. Yamashita and K. Atsuki , "S tripline with rectangular outer conductor and three dielectric layers," IEEE Trans. Microwave Theory Tech., vol. 18, pp. 238-243, May 1970. R. E. Collin, Field Theory of Guided Waves. New York: McGraw-Hill , 1960, pp. 52-54, 576-581.
The-Nan Chang was born in Tainan, Taiwan, in 1953. He received the B.S. degree in physics from National Taiwan University, Taipei, Taiwan, and the M.S. and Ph.D. degrees in electrical engineering from Tatung Institute of Technology, Taipei, Taiwan, in 1975, 1980, and 1987, respectively. From 1980 to 1981 , he was a Design Engineer in the Television Department at Tatung Company, Taipei, Taiwan. From 1981 to 1984, he joined the Antenna Department at the same company, where he was responsbile for antenna analysis and RF circuit design . In 1987, he joined the facu lty of Tatung Institute of Technology, where he became a Professor in 1990. His research interests include electromagnetic theory, antenna and planar circuit design .
Yung-Chang Sze was born in Kaohsiung, Taiwan in 1959. He received the B.S. degree in electronics from Tamkang University, Taipei, Taiwan and the M.S . degree in electrical engineering from Tatung Institute of Technology, Taipei, Taiwan, in 1987 and 1993, respectively. From 1988 to 1991, he was a Design Engineer in the Research and Development Department at Vidar-Sun Moon Star Company, Taipei, Taiwan, where he was responsible for setup and measurement of digital microwave communication systems. In 1993 , he returned to the same company, where he is now a Senior Engineer.
1978
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
Improved Wire Modelling 1n TLM Alistar P. Duffy, Jonathan L. Herring, Trevor M . Benson, and Christos Christopoulos
Abstract- An important aspect in the application of transmission-line modelling (TLM) to microwave circuit and electromagnetic compatibility (EMC) simulation studies is the accurate modelling of wires and wire-like structures. This paper outlines the cause of errors arising from simplified descriptions often used when modelling wires by integrated solution methods in TLM. It proposes a simple solution which is validated against experimental results for a number of test configurations.
a general rectangular Cartesian mesh. With such a simple description of a wire in a TLM model it is observed that the simulated electromagnetic resonances of the wire shift to lower frequencies by approximately 5-103 [3]. This will be referred to in this paper as "resonance error," and is discussed further in the next section . Another approach used to model wires is the separated solution [4], [5] method, where the field simulation and the wire simulation are decoupled. The process of implementing I. INTRODUCTION a separated solution method is to si mulate the structure under POWERFUL and flexible modeling method, such as study in order to extract information about the fields in the transmission-line modeling (TLM), can play an imporvicinity of the wire(s) assuming the wires are not present tant role in electromagnetic design, especially with regard to and then use that information to generate distributed current microwave circuits and electromagnetic compatibility (EMC) and voltage sources along the wires in a one-dimensional [l] . TLM can be applied at the outset, before prototyping model. Separated solutions can produce accurate results, free in order to assess the likely behavior of a system, or later, of resonance error effects, but they allow little interaction with to identify where improvements can be made to existing the environment, e.g. reradiation, to take place. designs. Conducting structures (for example: wires; wireA further class of solutions exist which are referred to looms; microstrip, and mechanical reradiating features , such as wire nodes [6]-[9]. These involve embedding a wire-like as support spars) are found in all systems . Thus the accurate structure within, or between, nodes. The wire nodes allow the and precise modeling of wire-like structures within TLM is modeling of wires with a smaller diameter than the node size of crucial importance if this technique is to be used for used in the simul ation . However, they add extra complexity to electromagnetic simulation. the model due to the additional node types required. The modeling of wires is often implemented by using short In [6], Naylor and Christopoulos devised the scattering circuit nodes . These are symmetrical condensed nodes [2] with matrix of a three-dimensional symmetrical condensed node the scatteri ng matrix S = -I where I is the identity matrix which had a wire running through it. This involved the with a value of 1 on the leading diagonal and a value of O addition of "pseudo-stubs" in order to allow propagation along elsewhere. S relates the incident voltages on a node at timea wire with a di ameter less than the node size. The model of step k (denoted by k V i) to the scattered, or reflected, voltages [7] adopts a slightly different approach in that the wire is (denoted by k v r) by: placed between adjacent symmetrical condensed nodes, rather than within a node. This approach allows the symmetrical condensed nodes to be unaltered, but requires additional Alternatively, wires may be modelled by placing short-circuits features to be included in the model. Finally, the model at the mid-points of the link-lines such that all the energy described in [8], [9] can almost be regarded as an hybridisation transmitted along a link line is returned with a 180° phase of the previous two, in that it adopts an approach similar to shift in the following time period. Thi s method, where the [7] but places the wire inside a symmetrical condensed node, wires are explicitly included in the model, is referred to as in the manner of [6], with good results. The integrated solution method offers generality, selfthe integrated solution method. Integrated solution methods consistency and ease of visualisation. It has the advantage over allow the modeller easy visualisation of the structure being the separated sol ution and wire node methods of requiring simulated, without recourse to a two-stage solution. In a neither a two stage-solution nor extra node types. To date, simple implementation of the model, errors are introduced the limitation on its usage has arisen from the resonance because computational resource limitations make it necessary error, described above. Thi s paper discusses the resonance to model the wire cross-section by only a single node within error and proposes a solution for the problem using standard nodes. Simulated transient responses obtained using the new Manuscript received April 27, 1993; revised November 12, 1993. This work method are compared against experimental results, for several was supported in part by DRA, Fort Halstead, U.K. configurations, with good agreement being obtained. The A. P. Duffy, T. M. Benson, and C. Chri stopoulos are with the University of Nottingham, University Park, Nottingham NG7 2RD, U.K. experimental validation of the new wire model suggests that J. L. Herring is with the University of Victoria, Victoria, Canada. the proposed approach is generally applicable. IEEE Log Number 9404166.
A
0018-9480/94$04.00 © 1994 IEEE
1979
DUFFY et al. IMPROVED WIRE MODELING IN TLM
Conductors Node boundaries
tJ . . 0 ... L!" ~
No direct connection of this node to the wire
•
Node boundaries Conductor
L_
I (b)
(a)
Fig. I. Description of wire cross-sections, (a) square cross-section and (b) stepped approximation to a cylindrical cross-section.
II. RESONANCE ERROR
TLM simulations of the simple structure, cons1stmg of a wire shorted at each end by two large metal plates, were undertaken using the integrated solution method with a single node modeling the cross-section of the wire and the resonant frequencies determined from the Fourier Transform of the time domain data. It was found that the resonances obtained were approximately 6% below those expected from the analytical solution (282 MHz compared with the expected value of 300 MHz). This is a manifestation of the problem which is found to occur whenever wires are modelled using the integrated method in rectangular Cartesian co-ordinates with a single node cross-section. The cross-section of the wire was increased to 4 nodes x 4 nodes as shown in Fig. l(a). It was found that this model gave more accurate results than when the cross-section of Fig. l(b) was used, which is a stepped approximation to a cylindrical cross-section. The square cross-section exhibited an error of approximately 1%, whereas the stepped cross-section exhibited an error of approximately 3%. The resonance error problem also manifests itself in twodimensional models, where other authors have referred to it as coarseness error [IO]. In a recent paper (11] the error was also attributed to the fact that nodes diagonally adjacent to an external comer have no direct contact with the wire. The solution proposed there was to introduce a stub to the 2D model such that direct interaction with the comer, by the adjacent comer node, was possible. Three-dimensional models of strip-line structures have also been the subject of previous investigations (12]. It was noted that the resonant frequency of the strip was approximately 10% below the expected value, and it was suggested that the cause of this error was poor interaction of the comer nodes diagonally adjacent to the conductor. The solution proposed was to model the stripline with two new node types-allowing accurate modeling of the conductor itself and of the edges-which embedded the conductor within the node, rather than modeling it by shorted link-Jines. It should be noted that, in the modeling of cavities, consisting of purely internal comers (13], no such resonance error is observed. In such problems, the resonances are predicted by TLM with a high degree of accuracy. It was also found that the number of nodes used to model the length of the wire had no significant effect on the magnitude of
Node link-lines
Fig. 2. Behavior of the corner node, showing no direct connection to the conductor.
the resonance error as long as at least ten nodes per wavelength were used. As mentioned earlier, the probable explanation for the error is that the communication of the signal around the comers is subject to additional delays. As there is no direct link between the nodes diagonally adjacent to the conducting surfaces, the circumferential signal communication takes longer than it should, with the (diagonal) comer node acting like a storage element. Fig. 2 illustrates this. This storage element would give rise to dispersion in the time domain signal, thus slowing down the propagating wave and, hence, reducing the resonant frequency. Increasing the number of nodes of the square crosssection representation would decrease the relative effect of the comers. This would explain the differences in behavior between the square cross-section and the piece-wise cylindrical representation, the latter containing a larger number of external comers. In order to investigate the effects of the comers further, the rod between parallel plates simulations were repeated using nodes which are elements of a cylindrical coordinate system. In this case, the curved surface of the conductor matched the coordinate system used to describe it as all nodes communicate directly with the wire. It was found that resonances were obtained which did not exhibit errors. This provides additional evidence as to the cause of the error. The purpose of this paper is to investigate further the origin of the resonance, or coarseness, error and to propose a remedy for it. Ill. A NEW WIRE MODEL In order to overcome the resonance error problem identified in the previous section, a method is proposed which corrects for the circumferential communication delay. This is achieved by correcting for the time delay introduced by the comer node, rather than modifying the structure of the comer node to make it interact directly with the wire comer. Only standard symmetrical condensed nodes are used. The communication delay can be introduced by introducing a lower nodal time step immediately around the wire compared with the rest of the work-space (14]. This is equivalent to reducing the background permittivity and permeability in a one-node thick layer around the wire. The arrangement is shown in Fig. 3(a). Tests were carried out in which the background properties were changed firstly only along the length of the wire, and then only in the transverse direction, this is
1980
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. IO, OCTOBER 1994
End of the conductor
Cross·section
A B
C
Actual path D
Desired path
H
E Wire
Wire
_ _ Node boundaries
•
D
(a)
Conductor
(b)
Fig. 4. Communication distance error calculation, (a) actual path and (b) the desired path.
Region of lower permittivity/permeability
(a)
"@'~~-~ correction
AB + BC distance is
+
FG
CD
+
+
GH
illustrated in Fig. 3(b). It was found that the resonance error was corrected with the transverse correction but not with the longitudinal correction. Further tests were undertaken in which the background properties were only changed in the external comer nodes, this also corrected the resonance error. These tests confirmed the circumferential communication delay of the external comer nodes as the root of the resonance error problem. Longitudinal only correction, if it had resulted in an improvement to the resonance error, would have indicated that the root of the problem was associated with the propagation of the signal along the wire rather than around it. The values for Er and µ r in the corrected circumferential velocity (CCV) region, the shaded region around the wire in Fig. 3(a), were chosen to be the same, thus avoiding the potential problems caused by the change in impedance as a signal passed between the region adjacent to the wire, and the rest of the work-space. The velocity LI, and the impedance Z thus have the values: 1 (~ ) µo
( ~) Eo = TJ
=we
HI = 6.l 2
The required
+ ~ 6.l + 6.l 2 2
2
26.l
(1
J( ~ )t:o( ~ )µo
= 46.1/2 = 26.1.
The error in distance is thus
Fig. 3. (a) Region of corrected circu mferential velocity around the wire. (b) Planes of correction .
Z=
+
correction
(b)
LI=
DE
(1)
(2)
where c is the velocity of light in free space (normalised velocity of waves travelling in a TLM mesh), ri is the impedance of free space and w is a velocity correction factor. It can be seen from the above equations that the local velocity in the CCV regions is increased by an amount w which results in a reduction in propagation time, as required. The time-step of the simulation is set to that appropriate to the CCV region. Stubs are added to the nodes in the rest of the work-space to compensate for the fact that, for those nodes, a slightly longer propagation time is required. Either stub loaded, or hybrid nodes can be used for this purpose. Fig. 4 shows communication distances travelled by pulses around a comer for a single node cross-section wire. The communication distance of a signal around the comer is
+ i) 6.l
= 1.1 =
w.
Hence, a value of l/w = 0.9 was used to adjust the relative medium properties around the wire in order to speed up the communication. This correction was applied isotropically on a single layer of nodes around the wire. Alternatively, a correction may be applied to the four comer nodes only. In this case, it is found that, using similar reasoning to that above, the required value is approximately l/w = 0.8. Similar results were obtained in both cases.
IV. METHODS FOR EXPERIMENTAL VERIFICATION An improved wire representation was presented in the previous section. In order to assess its applicability, the method must be verified against a set of results which together represent typical uses for the model. Either theoretical or experimental bench-marks could be used. This section identifies and documents a set of experimental comparisons which together cover a range of possible applications of the model. In constructing the model, the following requirements were kept in mind. 1) the model should simulate a wire accurately by using no more than one node cross-section. This requirement is based on the fact that computing resources are limited. If a single wire were to be modelled with several nodes per cross-section this would risk excessively long run-times and large storage. 2) the rest of the model, away from the wire, should be affected as little as possible by the wire model. 3) resonant effects on the wires should be modelled accurately. 4) the wire may, or may not, have its ends adjacent to metallic terminating planes (referred to here as terminated and open ended configurations respectively). 5) terminations, such as those used for connection to coaxialtype equipment should be modelled.
DUFFY et al. IMPROVED WIRE MODELING IN TLM
1981
Cross-section -I - 1-
····--·---··;.
:;;
_J -
- !...
_J -
I
/
Receiving Rod
~ "-··· '
Ce - ntrefed
I '~·· Fig. 5.
_______ ...... , . .·
~,
/
81
~
I
-
I- -I -
-:- -:- ~
I
"1 -.-
h> ~
~
- . - J.. -
-~ I
End of the conductor
I
I
l I I I - -T--r-,1 1 -1_ 1_ L I
I
I
I
I
-, -1-1- T _ L I __ _ I_ 1 I I
I
-
r-, -
_ _ Fine mesh nodes _ _ Coarse mesh nodes
Fig. 6.
•
Conductor
0
CCV region
Multigrid region enclosing the wire.
Screened room configuration (all dimensions are in cm).
The model can be tested against these requirements by studying the following configurations: I) electromagnetic coupling of linear dipole to a single conducting element within a screened room. 2) electromagnetic coupling of a linear dipole to multiple conducting elements within a screened room. 3) the modeling of a signal propagating along a wire in an enclosed cavity. Measurements were made of the coupling between wires placed in a screened room and of the response of a terminated wire placed in a closed rectangular conducting enclosure. The measurements were taken using a Hewlett-Packard network analyzer, model 8510B. In the screened room, ferrite beads were attached to the signal feed cables to eliminate the reradiative effects of the cables [15) . In the simulations, wires were modelled by a single node cross-section using internal boundaries. For the models simulating the screened room configurations, a multi-grid method [ 16) was used with a bulk work-space node size of 5 cm and a fine mesh region enclosing the wires of node size 1.67 cm (a 3:1 reduction). The enclosure si mulations used a regular mesh of size t::..l = l cm. Although physical dimensions in the model differ slightly from the actual ones, this has minimal effect on the resonances.
A. Screened Room Dipole-to-Rod Coupling The experimental configuration used is shown in Fig. 5. This study involves two I m long, l cm in diameter, wires . One of these wires is excited at its centre from a 50 connection (thi s is the transmitter, and henceforth referred to as the dipole) while the other is continuous and is placed above a conducting bench (this is the receiver and referred to as the "receivi ng rod"). Measurements were taken, using a current probe, of the current at a point halfway along the receiving rod. The modeling of the wires, in the multigrid scheme, is shown in Fig. 6. Both the dipole and receiving rod were modelled in the same way, and the 50 n feed was also embedded in the CCV region enveloping the dipole. The effects of the modified wire model can be clearly seen from Fig. 7. This figure compares the current in the receiving rod for measured and simulated cases with and without the modification proposed in this paper. It can be seen that the new method has further improved an already good simulation.
n
.8
.6 .4
.2
0
L-~J!;...--":..2"---~~......C.W!..._;::=-..=;:~
0.5x10 8
1.0x108
1.5x108
Frequency (Hz) Fig. 7. Comparison of currents in the single receiving rod. The solid line is the TLM sim ulati on with correction, the dotted line is the TLM simulation without correction and the dashed line is the experimental result.
B. Multi-Conductor Modeling This configuration, is similar to the configuration used for the above dipole-to-rod coupling study , except that a second receiving rod is brought into proximity with the original receiving rod, and is placed 30 cm further away from the dipole and parallel, at an equivalent height above the bench, to the first receiving rod. At this separation (30 cm ~ 0.15>.) receiving elements are near field coupled. The same modeling scheme was used for both wires, with the same value of w as described for the case of a single wire. Figs. 8 and 9 show the current in the rod closest to the dipole and the current in the rod furthest from the dipole, respectively. Each figure shows the experimental results, the CCV results and those for the simulation without any change in the material properties. A substantial improvement in agreement with experimental results is observed when using the improved model described in this paper.
C. Enclosed Cavity This configuration is shown in Fig. l 0. This is a system where the wire extends along the length of an enclosed rectangular box. Terminations for connection to the measurement system are available at both ends of the rod. The box was enclosed and thus the response obtained was the combination of the rod resonances and of the box resonances . The model used a regular node spacing of 1 cm with the 50 terminations modelled by single lossy nodes
n
1982
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 10, OCTOBER 1994
1.0
1.0
~
c
~
.8
c
~
~
:; ()
:;
'.
.6
()
' '
Q)
Q)
.!::!
.!::!
: ,:
.4 .2
0 z
.2
Fig. 8. Comparison of currents in the rod closest to the dipole for the two conductor configuration. The solid line is the TLM simulation with correction, the dotted line is the TLM simulation without correction and the dashed line is the experimental result.
.8
:;
.6
Fig. 11 . Comparison of currents for the enclosed cavity configuration. The solid line is the TLM simulation with correction, the dotted line is the TLM simulation without correction and the dashed line is the experimental result.
Fig. 11 shows the current in one 50 n termination with an excitation in the other one. It compares the CCV results with experiment and with and without correction. The use of the CCV correction shows a substantial improvement in the simulated response.
1.0
c
,,'
Frequency (Hz)
Frequency (Hz)
~
()
.4
0 0.4x10 9 0.6x1 o9 0.8x10 9 1.0x109 1.2x109
0 0.5x10 8
~
ca E
E
0 z
.6
-0
-0
ca
.8
"O
Q)
.!::!
ca
.4
0
.2
V. DISCUSSION AND CONCLUSION
E
z
0 0.5x10 8
1.0x108
1.5x108
2.ox108
Frequency (Hz)
Fig. 9. Comparison of currents in the rod farthest from the dipole for the two conductor configuration. The solid line is the TLM simulation with correction, the dotted line is the TLM simulation without correction and the dashed line is the experimental result.
E E
8co Rod with terminations
360mm 400mm Fig. 10. Enclosed cavity configuration.
between the ends of the wire and the box walls. The CCV region extended along the entire length of the wire to both the walls to which the wire was adjacent.
The results of the three comparisons made in the previous section confirm that, by introducing a CCV region in a one node thick layer around a TLM wire model, agreement between experimental and simulated results, to better than 13 can be obtained. The requirements of single node cross-section in a rectangular Cartesian mesh can be met. The configurations used represent a wide range of structures likely to be modelled and can be seen to have relevance to practical EMC problems. The nature of the tests used suggest that the CCV method may be generally applicable. It will be noted that the dipole current in the screened room and the enclosed box comparisons showed a significant contribution by the screened room and box resonances, which agreed very well with the experimental results. Further, no detrimental change in the Q-factor of the wires was observed. This paper has identified the nature of the error observed when modeling wires in TLM using the integrated solution method. An improved method of modeling wires in TLM has been introduced which appears to be generally applicable giving good comparisons against a set of experimental verification tests. This new method requires that the circumferential time-step around the wire is reduced. This involves changing the permittivity and permeability of the space immediately adjacent to the wire from their free-space values by a factor of 0.9. This may be performed isotropically for ease of data preparation, although circumferential only correction provides the same results . This approximate value provided results which showed a good agreement with measurements. The approach described in this paper has addressed the modeling of the wires of a single node cross-section, thus, the mesh size would need to be tailored to the problem. However, preliminary results for a further development of the technique
DUFFY et al. IMPROV ED WIRE MODELI NG IN TLM
to model wires with smaller di ameters than the node in which they are placed has been very encouraging. It is believed that the cause of the problem identified in thi s paper is common to many conducting structures other than wires, e.g. microstrip. Consequently, a sim il ar technique may be applied to these structures in order to reduce the resonance errors associated with them.
1983
Alistair P . Duffy was born in Yorkshire, United Kingdom, o n January 21 , 1966. He received the B.Eng. degree from University College, Cardi ff, United Kingdom and the M.Eng. degree in 1988 and 1989 respectively. He rece ived the Ph.D. degree fro m the University of Nottingham, United Kingdom, in 1993. He worked for Oyster Terminals Ltd between 1988 and 1990 before taking up hi s present positi on as a research assistant at the University of Nottingham. Hi s research interests include experimental and modeling techniques as applied to electromagnetic pro blems .
R E FER ENC ES [I] J. L. Herring, P. Nay lor, and C. C hri sto poulos, "Transmissio n-line modelling in electromagnetic compatibility studies," Int. J. Numerical Modelling , vol. 4, pp. 143- 152, 1991. [2) P. B. Johns, "A sy mmetrical condensed node for the TLM method," IEEE Trans. Microwave Theory Tech. , vol. 35, pp. 370-377, Apr. 1987. [3] P. Nayl or, "Coupling between e lectromagnetic waves and wires using transm ission-line modelling," Ph .D. dissertation, Univ . Nottin gham, 1986. [4) P. Naylor, C. Christopoulos and P. B. Johns, "Analysis of the coupling of electromagnetic radi ation into w ires using tran smiss io n-line modelling," !ERE Fifth Int. Conf Electromagnetic Compatibil. , vol. 71, pp. 129- 135 , Oct. 1-3 , 1986. [5] _ _ , "Coupling betwee n electrom agnetic fi elds and wires using transmission-line modelling," IEE Proc. , vol. 134, pt. A, no 8, pp. 679-686, 1987. [61 P. Naylo r and C. Christopoulos, " A new wire node for modelling thin wires in electromagnetic field problems solved by transmiss ion line modelling," IEEE Trans. Microwave Theory Tech. , vo l. 38, pp. 328-390, Mar. 1990. 17) A. J. Wlodarczyk and D. P. Johns, " New wire interface fo r graded 3-D TLM ," Electron. Leu., vol. 28 , no. 8, pp. 728-729, 1992. [8] J. A. Porti , J. A. Morente, M. Khall ad i, and A. Gallego, "Compari son of thin wire model s for TLM method," Electronics Leu., vol. 28 , no. 20, pp. 1910- 1911, 1992. [91 J. A. Morente, J. A. Porti , G. Gimenez, and A. Gallego, "Loaded-w ire node for TLM method," Electron. Leu. , vol. 29, no. 2, pp . 182- 183 , 1993. [10] W. J. R. Hoefer. "The tran smiss ion- line matrix method-theory and application'', IEEE Trans. Microwave Theory Tech. , vol. MTT-33 , pp. 882-892, Oct. 1985. [I I] U. Mueller, P. P. M. So, and W. J. R. Hoefer, "The compen sation of coarseness error in 2D TLM modelling of microwave structures," IEEE MTT-S Dig., 1992, pp. 373-376. (12] J. S. Nielsen and W . J. R. Hoefer, " New 3D TLM condensed node structures for improved simulatio n of conductor strips," IEEE MTT-S Dig. , 1992, pp. 122 1- 1223 .. [13] J. L. Herring, A. P. Duffy, T. M. Benson, and C. Chri stopoulos, "A transmission-line modelling study of screened room behavior," IEE Colloquium Radiated Emission Test Facilities, no. 1992/ 132, pp. 1/1 - 1/4, June 2, I 992. (14) A. P. Duffy, T. M. Benson, C. C hristopoulos, and J. L. Herring, " New method for accurate modelling of wires using TLM ," Electron. Lett. , vol. 29, no. 2, pp. 224-226, 1993. (15) A. P. Duffy, P. Nay lor, T. M. Benson, and C. C hristopo ul os, " Numerical simulatio n of e lectromagnetic coupling and comparison with experimental resu lts," IEEE Trans. Electromagnetic Compatibil., vol. 35 , no. I, pp. 46-54, 1993. [ 16) J. L. Herring and C. Christopoulos, " Multigrid transmiss ion-line modelling method for solving e lectromagnetic fi eld prob le ms," Electron. Lett., vol. 27 , no. 20, pp. 1794-1795, 1991.
Jonathan L. Herring was born in Middlesbrough, England, in 1966. He received the M.Eng. degree in Electrical and Electro nic Engineering and the Ph.D . degree in 1989 and 1993 respecti vely. From 1989 and 1993, he was employed as a Research Assi stant in the Department of Electrical and Electronic Engineering at the University of Nottingham, where he was working on the modelling of EMC problems. He is currently employed at the University of Victoria, Canada, where he is working on microwave device modelling.
Trevor M. Benson, born in 1958, received a First C lass Honours degree in Physics , and the Ph .D. in experimental phys ics, from the University of Sheffi eld in 1982. After spending over six years lecturing at University College, Cardiff, He joined the University of Nottingham as a Senior Lecturer in Electrical and Electronic Engineering in 1989. His current research interests include ex perimental and nume rical studies of e lectromagnetic fields and waves, with particular emphasis on electromagnetic compatibility and propagation in optical waveguides.
Christos Christopoulos was born in Patras, Greece, in 1946. He received the Diplo ma in electrical and mechani cal engineering from the Natio nal Technical Uni versity of Athens in 1969, and the M.Sc. and D.Phil. from the U niversity of. Sussex in 1970 and 1975, respectively. In 1974, he joined the Arc Research Project of the University of Liverpool and spent two years working on vacuum arcs and breakdown while on attachment to the UKAEA Cu lham Laboratories. In 1976, he joined the University of Durham as a Senior Demonstrator in Electrical Engineering Science. In October 1978, he joined the Department of Electrical and Electronic Engineering, University of Nottingham, where he is now Professor of Electrical Engineering. His research interests are in electrical di scharges and pl asmas , electromagnet ic compatibility, e lectromagnetics, and protection and si mulatio n of power networks.
1984
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. IO, OCTOBER 1994
Finite Element Computation of Electromagnetic Fields Scott T. Clegg, Katherine A . Murphy, William T. Joines, Member, IEEE, G. Rine, and Thaddeus V. Samulski
Abstract-A three dimensional finite element solution scheme is developed for numerically computing electromagnetically induced power depositions. The solution method is applicable to those problems for which it can be reasonably assumed that the magnetic permeability is homogeneous. The method employs an incident field/scattered field approach where the incident field is precalculated and used as the forcing function for the computation of the scattered field. A physically logical condition is used for the numerical boundary conditions to overcome the fact that electromagnetic problems are generally unbounded (i.e., the boundary condition is applied at infinity) but numerical models must have a boundary condition applied to some finite location. At that numerical boundary, an outgoing spherical wave is simulated. Finally, an alternate to a direct solution scheme is described. This alternate method, a preconditioned conjugate gradient solver, provides both a storage and CPU time advantage over direct solution methods. For example, a onethousand fold decrease in CPU time was achieved for simple test cases. Unlike most iterative methods, the preconditioned conjugate gradient technique used has the important property of guaranteed convergence. Solutions obtained from this finite element method are compared to analytic solutions demonstrating that the solution method is second-order accurate.
p(A ) 'lj;; c µo K
n
N r
NIT N co nt NIT ER
NA N D
Nno F
complex permitivity = C: r + j _.!!.__ w < ;;;N
0.15 0.1 0.05 0 -0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.075 -0.05 -0.025
X(m)
0
0.025
0.05
O.o75
0.1
X(m)
(a)
(a)
+------..··-·-...-............______ ..__...._...........______........................- .....+.........-........,.............. ;_............
z ::: ::
. ........_.................................................,.....................................,.......... ·········-· •·············-····t ·········--~ ~-- ·······---~·-······· ...
... . . . -.. . . . . .-.. . . . . . . . . . . ==~]L/. :=.t~~t: ' I
" ''
m-2 ~
....
g 0.1
;
,,.-·
Ul
-·-··
~
]
.,
0.1·- - - -·-·---··-""""'"""'"' ____
_,_~~~~~
/.
/ _ _ _ _,, ___................._,_,,,, __
/
/
/
/
'
,L~ J__--rt-,__ ·-· -____
!
i
-·+---+--· I 1
/ ! 0.01 + - - -- - - + - --1--- --1-- --1-- +---l--t--+-l
0.01 1
10
0.1
~
(b)
(b)
Fig. 2. (a) Plot of the analytical and numerical solutions for case I. The plot shows the magnitude of the electric field along the x -axis. The symbols indicate the FE solution values at node points on the x axis and the line shows the analytical solution. (b) Convergence study for case I. D represents the nondimensionalized grid size parameter and Error is the difference between the numerical solution and the analytical solution at the center of the spherical model. A linear fit with a slope of 2( m = 2) is shown on the plot to illustrate that the model exhibits a second order convergence rate.
Fig. 3. (a) Plot of the analytical and numerical solutions for case II. The plot shows the magnitude of the electric field along the x-axis. The symbols indicate the FE solution values at node points on the x -axis and the line shows the analytical solution. (b) Convergence study for case II. D represents the nondimensionalized grid size parameter and Error is the difference between the numerical solution and the analystic solution at the center of the spherical model. A linear fit with a slope of 2 ( m = 2) is shown on the plot to illustrate that the model exhibits a second order convergence rate.
direct solver. It should be noted that the bandwidth (NBAND) is based upon application of a GPS bandwidth minimization routine [28] .
it is faster and requires less storage. Results were presented comparing this FE method against known analytic solutions. It was shown that the algorithm is second-order accurate.
V . CONCLUSION
APPENDIX
A finite element formulation for the solution of the deposition of electromagnetic energy was derived. Since, physically, the problem is generally unbounded, special boundary conditions were applied to simulate an outgoing wave at the edge of the numerical model domain. An iterative solution method with guaranteed convergence was implemented. This solution scheme has a two-fold advantage over direct solution methods;
A preconditioned conjugate gradient solution method is used to solve for the unknown scattered field (Escat)· Given the system of equations resulting from ( 10)
[A]{ Escat}
= {b}
(A.l)
it was shown that standard iterative techniques will not converge. Thus, to insure convergence the system is premultiplied
1989
CLEGG et al.: FIN ITE ELEMENT COMPUTATION OF ELECTROMAGNETIC FIELDS
sparse nature of A resulting in slow convergence, eliminating one of the advantages for using an iterative technique. To accelerate the convergence, (A .2) is preconditioned. We will now derive the preconditioned conjugate gradient algorithm for (A.2). First some preliminary definitions. Let M be the preconditioning matrix which is real and symmetric. Now, define a matrix Q such that:
J.16 0.14 0.12
e
~ 8 ~
:><
ill
0.1 0.08 0.06
,,.
\
'
"'" .. '~
'
J
t/
'
0.04 0.02
0 -0.1 -0.08 -0.06 -0.04 -0.02
..,,. The preconditioned system is written as
~....t·1 Iv '
0
(A.3)
1
1
A *A' E'scat = A *b1
I
(A.4a)
where
O.Q2 0.04 0.06 0.08
0. I
A'= Q-lAQ-T
X (m) (a)
(A.4b)
E~cat = QT Escat b' = Q- 1 E.
and
(A.4c)
(A.4d)
Defining the residual error as: 1
rI
]
0.1 +··-·······--····-················-···········--·-+-·-············---·--·+--------i--- -i---- --/ +···-·--·-···-········----·-···----·-····;························----···t
~
I'
1 1 == A* A E scat
1
where E scat is the solution. Now let there be a quadratic function
-
E~'scat )
f (Escat)
(A.5)
where
::~ ~::::;-:~ :::.:t ::~ :;;k-:.....:. :=: . . .
···-··-···--·-·-·---~ ~1~i?~ 1-r,~~
J(E scat) = ~ (E~cat
-
1 ~1
E~cat) *A*' A' ( E~cat - E~cat)
- 2Es~at A*
/"'
~
I
(A.6)
A' Escat
Finally define the gradient of
O.Ql-+-------+--"--- +----+---+-- +--+--+--+--1 0.1
1
A* b1 == A* A '(E'scat
-
f
as:
g(Escat ) = Y' J = A*' A' ( E~cat -
~
E~cat)
(A.7)
(b)
Fi g. 4. (a) Plot of the analytical and numerical soluti ons for case 111. The plot shows the magnitude of the electric field along the .T-axis. The sy mbol s indicate the FE so lution values at node points on the x -axi s and the line shows the analyt ical solution. (b) Convergence study for case Ill. D represents the nondimensional ized grid size parameter and Error is the difference between the numerical solution and the analystic soluti on at the center of the spherical model. A linear fit with a slope of 2 ( m = 2) is shown on the plot to illustrate that the model exhibits a second order convergence rate.
To mm1m1ze f, search in the direction defined by the real number An and let dn be the exact line search. (Note that n is an iteration counter). Then
tn+l _ E'n E scat scat and
by the complex conjugate of A(A* is the complex conjugate of A. I). Thus, the system of equations is
AEscat =
b
(A.2a)
+\
1
and
b=
A*b
dn
r n+i =rm + An A*' A' dn
(A .Sa) (A.Sb)
Observe that the optimal value of An will make dn orthogonal to g1n+ 1 . Thus
where
A = A*A
An
(A.9) (A.2b) (A.2c)
The matrix A is Hermitian, which has the important property that the spectral radius is less than unity [25] and thus convergence is guaranteed. Unfortunately, the A matrix looses the
Hence, combining (A.S b) and (A.9),
(A. LO)
1990
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 10, OCTOBER 1994
Next define the search direction dn as
REFERENCES
(A.11) To obtain the parameter f3n, note that (A.12) Thus /Jn =
T*n+l A*' A' dn d*n A*' A' dn
(A.13)
To complete the derivative of this algorithm, the matrix M must be defined. It has already been indicated that M is based upon the finite element approximation of the discrete Laplacian. It can also be noted that evaluation of (A.10) and (A.13) require the inversion of the M matrix. Prior work [30] has established that a single SOR sweep giving a partial inverse of M is sufficient. Thus M is defined as
M =(Do+ wLo)D- 1 (Do
+ wUo)
Ao is the matrix obtained from the discrete approximation of the Laplacian (i.e., the \7 2 Escat term), Do is the diagonal of Ao, Lo is the lower diagonal matrix of A 0 , U0 is the upper diagonal matrix of Ao, and w is the over relaxation parameter (for convergence 1 < w < 2). Thus, the preconditioned conjugate algorithm proceeds as follows: Assume, E~cat = 0 TO=
9
0 =
-b ro
do= -To Mio= do get 1° from one SOR sweep, and let n = 0. Then begin the iterative procedure as follows: Step 1) An =
(LL) -y•n A-yn
n+l _En + \ n 2) E scat scat An I 3) Tn+l = Tn + AnA/n 4) gn+l = rn+1 5) Mµn+l = 9 n+l
6) get µn+l from SOR sweep µ• n +l 7) /Jn = ( -y•n AYn
n)
8) 9)
dn+l = -Tn+l + f3n dn M1n+l = dn+l get gn+l from SOR sweep
10) . T 11) Check for convergence, 1f rn Tn go to step 1. End of algorithm.
>c
ACKNOWLEDGMENT
A special acknowledgement is extended to Ms. Jeanne Forest for her valuable help in preparing this manuscript.
[l] S. B. Field and J. W. Hand, "Introduction," in An Introduction to the Practical Aspects of Clinical Hyperthermia. London: Taylor and Francis, 1990, pp. 1-9. [2] K. Hynynen, et al., "The effect of blood perfusion rate on the temperature distributions induced by multiple, scanned focused ultrasonic beams in dogs' kidneys in vivo," Int J Hyperthermia, vol. 5, no. 4, pp. 485-497, 1989. (3] R. B. Roemer, "Thermal dosimetry," Thermal Dosimetry and Treatment Planning. Berlin, Springer-Verlag, 1990, pp. 119-214. [4] J. W. Strohbehn, et al., "Optimization of the absorbed power distribution for an annular phased array hyperthermia system," Int. J. Radiation Oncology Biol. Phys., vol. 16, no. 3, pp. 589-599, 1989. [5] S.-I. Umemura and C. A. Cain, "Analysis of temperature responses to diffused ultrasound focal fields produced by a sector-vortex phased array," Int J Hyperthermia , vol. 6, no. 3, pp. 641-654, 1990. (6] D. Andreuccetti, et al., "Phantom characterization of applicators by liquid-crystal-plate dosimetry," Int. J . Hyperthermia, vol. 7, no. l , pp. 175-184, 1991. [7] R. Takemoto-Hambleton, et al., "A study of the sensitivity of SAR to geometry," Tenth Annual Meeting North American Hyperthermia Group, New Orleans, LA, Radiation Research Society, 1990. [8] C. Wang and 0 . P. Gandhi, "Numerical simul ation of annular phased arrays for anatomical based models using the FDTD method," IEEE Trans. Microwave Theory Tech., vol. 37, pp. 118- 126, Jan. 1989. [9] D. Sullivan, "Three-dimensional computer simulation in deep regional hyperthermia using the finite-difference time-domain method," IEEE Trans. Microwave Theory Tech., vol. 38, pp. 204-211, 1990. [JO] - - ·"Mathematical methods for treatment planning in deep regional hyperthermia," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 864-872, May 1991. [I I] K. D. Paulsen, "Calculation of power deposition patterns in hyperthermia," in Thermal Dosimetry and Treatment Planning. Berlin, SpringerVerlag, 1990, pp. 57-118. [I 2] K. D. Paulsen and M. P. Ross, "Comparison of numerical calculations with phantom experiments and clinical measurements," Int. J Hyperthermia, vol. 6, no. 2, pp. 333-349, 1990. (13] G. P. Rine, et al., "Feasibility of estimating the temperature distribution in a tumor heated by a waveguide applicator," Int. J. Rad. One. Biol. Phys. , submitted, 1991. (14] _ _ ,"Comparison of two-dimensional numerical approximation and measurement of SAR in a muscle equivalent phantom exposed to a 915 MHz slab-loaded waveguide," Int. J. Hyperthermia, vol. l , no. 6, pp. 213-226, 1990. (15] 0. C. Zienkiewicz, The Finite Element Method. London: McGraw Hill, 1977. (16] S. H. Wong and Z. J. Cendes, "Combined finite element-modal solutions of three-dimen sional eddy current problems," IEEE Trans. Magnetics, vol. 24, pp. 2685-2687, June 1988. (17] - - · "Numerically stable finite element methods for the Galerkin solution of eddy current problems," IEEE Trans. Magnetics, vol. 25, pp. 3019-3021 , Apr. 1989. [18] D. R. Lynch and K. D. Paulsen, "Origin of vector parasites in numerical Maxwell solutions," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 383-394, Mar. 1991. [ 19] W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism . New York: Addison Wesley, 1962. [20] G. P. Rine, Computer Based Dosimetry for Local Microwave-Induced Hyperthermia Using the Finite Element Method. Lawrence, KS : University of Kansas Press, 1988. (21] K. D. Paulsen and D.R. Lynch, "Elimination of vector parasites in finite element Maxwell solutions," IEEE Trans. Microwave Theory Tech. , vol. 39, pp. 395-404, Mar. 1991. . (22] K. D. Paulsen, et al., "Three-dimensional finite, boundary, and hybrid element solutions of the Max well equations for lossy dielectric media," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 682-693, Apr. 1988. (23] A. Bayliss, et al., "Boundary conditions for the numberical solution of elliptic equation inexterior regions," SIAM J. Appl. Math., vol. 42, no. 2, pp. 430-451 , 1982. (24] J. H. Ferziger, Numerical Methods for Engineering Application. New York, Wiley, 1981. (25] L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering. New York, Wiley, 1982. (26] G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods. New York, Oxford University Press, 1985.
CLEGG et al.: FINITE ELEMENT COMPUTATION OF ELECTROMAGNETIC FIELDS
[27) W. T. Joines, "Reception of microwaves by the brain," Medical Res. Eng., vol. 12, no. 3, pp. 8- 12, 1976. [28) B. G. Gibbs, et al., "An algorithm for reducing the bandwidth profile of a sparse matrix," SIAM J. Num. Anal, vol. 13, no. 2, pp. 236-250, 1976. [29] A. Bayliss, et al., "An iterative method for the Helmoltz equation," J. Comp. Phys., vol. 49, no. 3, pp. 443-457, 1983. [30] 0. Axelson, "Solution of linear systems of iterative methods," in Lexture Notes in Mathematics: Sparse Matrix Methods. Berlin: SpringerYerlag, 1977, pp. 1-5 1.
Scott T. Clegg was born in Sacramento, CA. He graduated with a B.S. in mechanical engineering from Tulane University, New Orleans, LA and the M.S. and Ph.D. degrees in mechanical eng ineering from the University of Arizona, Tucson, in 1985 and 1988, respectively. From 1978 to 1982, he worked for IBM, Tucson, AZ. He is currently on the facu lties of the Department of Radiation Oncology and the Department of Biomedical engineering at Duke University, Durham, NC. His research interests are in the development and application of numerical methods.
Katherine A. Murphy was born August 11, 1958, in Los Angeles, CA. She received the B.A. degree in mathematics from the State University of New York, Binghamton, and the Sc.M. and Ph.D. degrees in applied mathematics from Brown University, Providence, RI, in 1979, 1981, and 1983, respectively. From 1983 to 1984, she was a Visiting Assistant Professor in the Mathematics Department, Southern Methodist University, Dallas, TX. From 1984 to 1986, she was a Visiting Assistant Professor in the Divis ion of Applied Mathematics, Brown University. Since 1986, she has been with the Mathematics Department of the University of North Carolina, Chapel Hill , where she was an Assistant Professor until 1993 , when she became an Associate Professor. Her research interests include the theoretical and computational aspects of parameter estimation in differential equations with application to problems in biology and medicine.
1991
William T. Joines (M'61) was born in Granite Falls, NC. He received the B.S.E.E. degree from North Carolina State University, Raleigh, and the M.S. and Ph.D. degrees in electrical engineering from Duke University, Durham, NC, in 1959, 1961 and 1964, respectively. From 1959 to 1966, he was a member of the Technical Staff at Bell Laboratories, Winston-Salem, NC, where he was engaged in research and development work on microwave components and systems for military applications. He joined the faculty of Duke Univers ity in 1966, and is currently a Professor of Electrical Engineering. His research interests are in the area of electromagnetic wave interactions with material s
G. Rine, photograph and biography not available at time of publication.
Thaddeus V. Samulski was born in Augusta, GA. He received the B.S. degree in physics from the University of Notre Dame, and the Ph.D. degree in physics, in 1968 and 1976, respectively. From 1977 to 1978, he was a Postdoctoral Fellow in Medical Physics at the Mideast Center for Radiological Physics (MECRP) at the Allegheny-Singer Research Corporation, Pittsburgh, PA. From 1978 to 1982, he held joint staff positions at MECRP and the Department of Radiation Oncology at Allegheny General Hospital, Pittsburgh, PA. In 1982, he joined the Medical Physics staff at Stanford University Medical Center, Stanford, CA, to conduct research in hyperthermic oncology. In 1986, he joined the faculty in the Department of Radiology at Duke University Medical Center, Durham, NC, and is currently an Associate Professor in Radiation Oncology at Duke. He directs the technical research and development effort in an NCI Program Project Grant involving the study of hyperthermia in cancer therapy.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. IO, OCTOBER 1994
1992
Using Linear and Nonlinear Predictors to Improve the Computational Efficiency of the FD-TD Algorithm Ji Che n, Student Member, IEEE, Chen Wu, Member, IEEE, Titus K. Y. Lo, Member, IEEE, Ke-Li Wu, Member, IEEE, and John Litva, Senior Member, IEEE
Abstract- It is well known that the Finite-Difference TimeDomain (FD-TD) method requires long computation times for solving electromagnetic problems, especially for high-Q structures. The reason for this is because the algorithm is based on the leap-frog formula. In this paper, both linear and nonlinear predictors, which are widely used in signal processing, are introduced to reduce the computation time of the FD-TD algorithm. A short segment of an FD-TD record is used to train the predictor. As long as the predictor is set up properly, an accurate future realization can be obtained. We demonstrate, by means of numerical results, that the efficiency of the FD-TD method can be improved by up to 90 %. With this result, the FD-TD algorithm becomes a much more attractive technique for solving electromagnetic problems. I. INTRODUCTION
HE finite-difference time-domain (FD-TD) method has been used widely for solving electromagnetic problems [l], [2]. It gives the evolution of the fields in time, given a known excitation. This leads to a complete understanding of near fields and of transient effects. Also, it is easily applied to problems composed of complex structures, which may be difficult to solve using other numerical methods. However, a major draw back of the FD-TD method is its computation time. In order to obtain accurate frequency responses via a Fourier transform, a very long time-domain record is usually needed. For some structures, especially those with high Q-values, the computation time may require up to a few days. Premature termination in the time domain will result in inaccurate parameter extraction in the frequency domain. To enhance the FD-TD method for simulating microwave problems, one begins by combining it with signal processing techniques such as the MUSIC method [3], or the System Identification (SI) method [4]. Both of these techniques can give accurate spectra using only a short segment of the original FD-TD record. However, up to now, these methods have only been used to predict the resonant frequency or spectrum of resonant structures. Another successful example of combining the FD-TD method with a signal processing technique, in order to reduce the computational overhead, is to use the FD-TD method with Prony's technique [5], [6]. In this combination, Prony 's technique is used to predict the future realization of the time domain response by training a short segment of an FD-TD record and setting up coefficient-based
T
Manuscript received May 19, 1993; revised November 19, 1993. J. Chen, C. Wu, T. K. Y. Lo, and J. Litva are with the Communications
Research Laboratory, McMaster University , Hamilton, Ontario, Canada. K.-L. Wu is with COM DEV Ltd., Cambridge, Ontario, Canada. IEEE Log Number 9404148.
models, where the order is to be determined. Unfortunately, this method may require preknowledge to select the order of the coefficients for different structures [6]. Sometimes, it may be somewhat difficult to find the proper order. Even with these limitations, all of the above signal processing techniques can save computational time when using the FD-TD method to solve electromagnetic problems. It is very clear that the incorporation of signal processing techniques with the FD-TD method is an effective way to improve the efficiency of the FDTD method. Once the FD-TD technique is formulated to solve a particular electromagnetic problem, the algorithm can be treated as a system, whose execution can be carried out using the appropriate signal processing technique. The fundamental operation of the FD-TD algorithm is based on the leap-frog formula, which means that future realizations are based on calculations that occurred in the past. In this paper, linear and nonlinear predictors are introduced, which can take advantage of the leap-frog nature of the FD-TD method to predict the later response of the system in the time domain. The following section gives detailed descriptions of the linear and nonlinear predictors used in this study. In Section III, some numerical results are given to show how these techniques are used to improve the efficiency of the FD-TD method. Conclusions and future work will be given in Section IV. II. METHODOLOGY
A. General Description of the FD-TD Method
The FD-TD algorithm is a method in which the central difference scheme is used to discretize Maxwell ' s curl equations in both time and space. The central difference technique can contain the magnitude of the round-off errors so that second-order accuracy is achieved. To model the electrical and magnetic fields in space, Yee introduced the cell system in 1966 [7]. The physical basis for Yee' s cell system can be easily explained using Faraday's and Ampere's laws. After using Yee's cell system to describe the computation domain, which is bounded by electric, magnetic or application specific absorbing walls, Maxwell ' s equations are essentially replaced by a computer which calculates the fields at the grid points associated with the cells. In this paper, a Gaussian pulse, whose 3dB cut off frequency is designed to overlap with the frequency band of interest, is used as the excitation. The time step is given by
0018-9480/94$04.00 © 1994 IEEE
b..t
= 0.5b..h/ c
(I)
CHEN et al.: US ING LINEAR AND NONLINEAR PREDICTORS TO IMPROVE THE COMPUTATIONAL EFFICIENCY
where .6.h = min {.6..T, .6.y, .6. z}. .6.x , .6.y and .6.z are the space steps in X , Y, and Z directions. c is the free-space velocity. For a high dielectric constant material (cr = 9.9) , the waves that propagate along the microstrip line are strongly dispersive. It is appropriate, therefore, that Litva's [8] dispersive boundary condition be used here. Thi s boundary condition has been found to have very good performance with dispersive waves.
1993
solution of the equations,
Cxx (l,1) Cxx( l,2) Cxx (2, 2)
Cxx( l , p) Cxx( 2, p)
&(l)l rCxx (l,0)] &(:2) = _ Cxx (:2, 0)
rCxx (p , 1) Cxx(p,2)
Cxx(p, p)
&(p)
Cxx (2, 1)
(3)
where
~
Cxx(j, k) = N
N- l
p
L
x(n - j)x(n - k).
(4)
n=p
B. Linear Predictor Linear predictors or autoregressive (AR) models have found applications in many fields of research [9]. It is the most popular time series modeling approach being used in modem spectral estimation [10]. Thi s is because accurate estimation of the AR parameters can be derived by solving a set of linear equations. In contrast to Prony 's method, which uses a model consisting of deterministic exponentials to fit the data, the AR model seeks a random model , to the second-order, to fit a statistical data base. Also, compared with the autoregressive moving average (ARMA) method, AR modeling does not require the solution of a set of highly nonlinear equations. We say that the time series x(n), x(n - 1), . . . x(n - p) represents the realization of an autoregressive process of order p if it satisfies the equation,
Cxx(p, O)
The matrix is Hermitian and positive semidefinite. It can be solved by Cholesky decomposition. Under certain conditions [11], the solution of the AR parameters yields the following equations,
[x(~) ~g~
-
~g~ x(N
+ 1)
[~~: ~ ~~]
x(p) x (p ~ 1)
1[a11 ~2
x(p + N)
ap
·
(5)
x(p + N) Normally the value of N is chosen to be greater than 2p. Then the least-square algorithm is used to obtain the coefficients. C. Nonlinear Predictor
where c is an error term, and a 1 a 2 , .. . ap are constants called the AR parameters. From equation (2), we see that the present value of the process u( n) is equal to a finite linear combination of past values of the process, x(n -1 ), x(n -2) , .. . x(n -p) , plus the error term c(n) . Therefore, the present status can be predicted using a linear combination of previous observations. So, as long as the linear predictor is set up properly, through simple extrapolation, we can get the future realization. There are two issues that must be addressed when setting up an AR model. One is how to choose the order of the model. The other is how to estimate the coefficients. The selection of the model order in AR is a critical problem. Using an order value which is too low results in a high rate of attenuation when extrapolating into the future, while too large an order causes general numerical instability. Two commonly used model order estimators are the final prediction error (FPE) technique and the Akaike Information Criterion (AIC)[lO]. They are both based on the estimated predictor error power and are regarded as general guides for AR model selection. Although, the actual order which is selected in practice may be higher than the order given by these two techniques, at least, these two methods give a starting point for the selection of the order of the process. There are a large number of methods for estimating the AR parameters. In this paper, we choose the covariance method. The covariance for AR parameter estimation yields the
The development of the upcoming class of nonlinear autoregressive models is motivated by the observation that many processes in nature display random variations which are essentially non-Gaussian in character [12] . In real world applications, it is found that many random processes display essentially nonlinear behavior and hence some form of nonlinear time series modeling, is required to accurately capture the process [13]. These nonlinear models are usually classified under the name of amplitude-dependent exponential autoregressive models. The basic form for an EXPAR(exponential autoregressive) model of order p is p
x(t) = L[a1 + ,81 exp(- 8x 2 (t - j))]x(t - j) +ct j=l
(6) where ct is an error term. 8 is a constant. a and f3 are the EXPAR parameters to be trained. Note that the above model contains no "constant terms" because Ozaki seems to take the view that "if the vibration process starts from the zero initiation state, it stays at zero" [14]. From (6), we can set up equations for the parameter 8 as (7) shown at the bottom of the following page. The equations are then solved using the least squares method. Also, in (7) we require N to be greater than 2p. Experimental evidence suggests that there can be difficulties with the estimation of 8, which is a critical parameter [13]. Since 8 is essentially a scaling factor, it is reasonable to look
1994
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
at values of 8 in a range such that e -ox~_, is different from both zero and equal to one for most values of Xt-i [12] .
III. NUMER1CAL R ESULTS
In this section, three typical electromagnetic problems are studied. They consist of two microwave filters and a patch antenna. The usefulness of AR and EXPAR modeling for improving the computational efficiency of FD-TD will be demonstrated. Before one uses AR or EXPAR models to analyze FD-TD records, there are two issues that must be addressed. One is the selection of the segment of the FD-TD record to be used for training the AR or EXPAR models. The segment should cover a significant fraction of a FD-TD time record except for the very beginning of the FD-TD waveform. By avoiding the very beginning of the record, we ensure that the intricacies of the structure being analyzed are captured by the data used for training our model. For complex structures, we usually choose the beginning of the training segment to coincide with three times the round-trip time required for the launched pulse to be reflected from an extremity of the structure. The other requirement is that the original FD-TD records must undergo some decimation. To meet the FD-TD stability condition, the FD-TD algorithm usually oversamples the data compared to the needs of the AR and EXPAR models. If the oversampled FD-TD records were to be modeled without decimation, a very high order AR model would be required . This would lead to divergence between predicted and the true values. If we decimate the FD-TD records by a certain factor, the divergence of the predicted result is avoided. In real applications, the FD-TD records are usually decimated by a certain factor. As our first example, AR modeling is applied to an edge coupled bandpass filter [15] as shown in Fig. 1. In order to obtain an accurate estimation of the S parameters, over 30000 iterations are needed if the original FD-TD method is used. By using the AR approach, only 2000-3500 iterations of the FD-TD algorithm are required to generate the original data base for the AR process, denoted as {u}. Decimating the {u} by the factor 10, we get a time series {x }. Then {x } is used to develop an AR model of order 50. Fig. 2 gives a comparison between the result obtained by FD-TD plus AR model and the direct FD-TD computation alone. It should be added that once the AR parameters are set up, the AR model can accurately predict very long time-domain traces of transient waveforms. Fig. 3 shows the S parameter calculated by the FD-TD plus AR model, and the direct FD-TD using 3500 FD-TD time
D. Discussion
There are three reasons for our introducing signal processi ng based models here. They are as follows: l) As we know, the FD-TD algorithm for electromagnetic sim ulation is a leap-frog algorithm, which means that the present value is iterated from a previously known value. When we look at the AR and EXPAR models, we find that they also predict their future values based on past occurrences. Since they both use the past to predict the future, this similarity prompts us to incorporate the AR and EXPAR model s in the FD-TD algorithm, with a considerable saving in computation time. 2) It is well known in signal processing that a signal consisting of a number of si nusoids can be modeled using AR because of the recursive nature of this algorithm. When we use FD-TD to analyze high-Q structures, we rely on the fact that the time response can be approximated by a summation of sinusoidal waves. It follows that when high-Q structures are analyzed, an AR model can be used to provide an estimate of the future realization because of the similarity between the underlying basis functions for AR and high-Q structures. However, if there are some nonlinear effects in the system, we should add nonlinear terms to the model. Thus, for some cases, an EXPAR model is needed. 3) The classical problem in AR modeling is selecting of the correct order for the process. As mentioned earlier, there are two main tests that are used to determine order. However, in practice, they can't work perfectly all of the time. Normally we will choose a value for the order which is greater than that given by either PFE or AIC in order to be sure that the AR process adequately models the problem. Complex structures normally require high order models. It is found that on occasion the choice of a high order process leads to disastrous results in that the process fails to converge. We find that this conflict can be circumvented by using the EXPAR technique. Failure of convergence does not seem to be a problem when using this technique. In practice, when EXPAR is implemented, computational difficulties disappear.
li1 ti2
rx(x(2)l ) x(~V)
x(2) x(3)
x(p) x(p + 1)
exp( - c5x 2 (1)) exp( - c5x( 2) )
exp (- c5x 2 (2)) exp( - 8x 2 (3))
oxv ( _ ,,, (v IJ exp (- c5x 2 (p +l ))
2
2
1x
tip /31
:c(N + 1)
.'C(p + N)
2
exp( - 8x (N))
ex p(- 8x (N + 1))
exp(-c5x (p + N))
/32
r"(P + 1) 1
~ - :(:::)
/]p (7)
CHEN et al. : USING LINEAR AND NONLINEAR PREDICTORS TO IMPROVE THE COMPUTATIONAL EFFICIENCY
1995
h
w=t= l .272rnrn
;1 -10
- - - - - 5w - - - w w w
I Er=lO
t
.
I
""' i
a
-SO
(a)
-70
a
·---· l· ~ - i·--rAJWA.
Fitqucncy (Ghz)
Fig. 3.
Edged-coupled microstrip filter scattering parameter S 21 .
b
(b)
Fig. I. (a) Plane of edged- + ko EHcf> - -He/> - j-Hr
r2
r2
~~{OE [!_( r H¢) + jmHr] E r Or Or +
~; [jmHz + r :z Hc/> ]} =
0
2 1 2 \7 Hr+ ko EHr - 2 Hr
II .
FORMULATIONS
The structures of the dielectric resonator cavities to be analyzed are shown in Fig. 1 with one constant-¢ meridian plane. The perfect conducting metallic cylindrical cavity of radius b and height l is axisymmetrically loaded with a pillbox dielectric rod of radius a, height h, and dielectric constant Er (Fig. I (a)) or a dielectric ring of inner radius c and outer radius a (Fig. l(b)) . Mathematically, geometrical singularities at dielectric corners are difficult to be dealt with [29], [30]. Instead, each comer is replaced with a curvature of electrically small radius such that the tangential and the normal directions can be defined there and hence mathematical difficulty can be
. 2m + yr-2 He/> r _ ~OE ({)Hr_ {)H z )= O E OZ OZ Or
(2a)
\7 2H z
OE ( {)Hr + k2oEH z + ~Eur uz !:>
!:>
_ {)Hz )= O, !:> ur
(2b) (2c)
where an azimuthal dependence e-jmc/> is assumed for the axisymmetrical structure. It is readily seen that for the azimuthally invariant modes (m = 0), the azimuthal field He/> is uncoupled from the other two fields, Hr and Hz. Similar situation holds for the electric field. Thus, there exist TM and TE modes, of which the nonvanishing fields are (He/> , Er, E z) and (Ecf>, Hr. Hz), respectively. Those modes with m f. 0 are hybrid, where the six electromagnetic fields are nonvanishing in general.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
2000
Since the azimuthal field is uncoupled in the TE or the TM modes, it is beneficial to employ the formulation in terms of the azimuthal field only. For the hybrid modes, it is still not necessary to use the full H formulation. Instead, we propose to use the Hr-Hz formulation, since the field H does not appear in (2c) and that in (2b) can be expressed explicitly in terms of the other two magnetic fields, Hr and H z, via the divergence equation of magnetic field as 1 a
aHz J-H = --(rHr) + - . r r ar az .m
B. Hybrid Modes
Using the Hr-Hz formulation the governing equation for hybrid modes (m =/= 0) is two coupled equations which at a homogeneous region read as 3 a 1- m2 82 ) 2 a a2+ --+ + H +--Hz ( ar2 r ar r r az r2 az 2
= -k6 EH,.
a 1 a - -m2 + a2 a ) ( -arz +--a r r r z 2
2
(3)
In this investigation the associated partial differential equations
are applied at homogeneous region and those terms having derivatives of the permittivity disappear. Accordingly, the permittivity discontinuity is treated by enforcing boundary conditions derived later.
+
(4 )
where 'If; = rHq, for the TM and 'If; = rEq, for the TE modes, and E = 1 in air and E = Er in the dielectric. At permittivity discontinuity the continuity conditions of the tangential fields are imposed. Tangential fields at an air-dielectric interface (including the dielectric comers) with tangential direction i and normal direction n can be given via the curl equations of electric and magnetic fields as
- jwµoHt = ( :n
.mH Hr aHn a Ht ¢ =-+-- +r r an at . 1 ( aHn aHt) JWEoEq, = ~ an .
jw EoEt
for TE mode
= ~ ( :n + cos()~) H,
where () is the angle from the r axis to have made use of the relation:
[!]
= [-
(Sa)
(9b)
At permittivity discontinuity the terms Hr/r, 8Htf 8t, and aHn/8t are continuous automatically. By using this property, (9) yields the boundary conditions at permittivity discontinuity
as
aHn l 8n A aHt 1aHt -I --I A
E,.
an
D
aHnl an D (
l -
(!Oa)
~) aHn. Er
at
(!Ob)
The associated boundary conditions at the metallic walls are
for TM mode (Sb) at z
n.
(9a)
at -
an
+ cos ()~ )E
(8b)
H z =-ko EH z.
J-
For the TE or TM mode the governing equation at a homogeneous region is
a 2'lf; 2 az 2 = -ko E'lf;,
2
(8a)
The associated boundary conditions at permittivity discontinuity are the continuity of azimuthal fields E and Hq, , which can be expressed in terms of Hr and H z. For field H q, , the relation has been given in (3). The relation for field E can be derived from the curl equation of magnetic field . By using (6), these relations can be written as
A. TE and TM Modes
a 2'lf; 1 a'lf; ar2 - -:;: ar
2
= 0 or l
( I la)
In deriving (S), we
~~: ~ ~~~ ~ ] [ : ] .
Since at permittivity discontinuity the terms E , H , cos(), and r are continuous, the boundary conditions there can be written as
a'lf;I an A
a'lf;I an D
for TE mode
a'lf;I an A
E~ :~In'
for TM mode
at r
(6)
(7a) (7b)
where A and D denote the derivatives being performed on the air and dielectric sides, respectively . At both the z axis and the metallic walls the associated boundary conditions are the Dirichlet condition ('If; = 0) for the TE modes. While the boundary conditions for the TM modes are the Dirichlet condition at the z axis and the derivative condition (a'lf; /an = 0) at the metallic walls. On the middle section plane, the PEC (perfect electric conductor) or the PMC (perfect magnetic conductor) condition can be used, if the resonator structure is symmetric about that plane.
= b.
( 11 b)
The boundary conditions at the z axis (r = 0) can be obtained by multiplying r 2 to both sides of (8a) and (8b) (or r to those of (8a), for the case of m = 1). By so doing we obtain the boundary conditions at the z axis as
aHr ar {
=0
Hr= 0 Hz = 0
::: }
at r
= 0.
( I le)
III. FINITE-DIFFERENCE METHOD
The finite-difference method can be regarded as a special case of finite-element methods on rectangular elements. Algorithm based on the irregular-shaped elements needs extra computation time and memory space. Therefore, there is no need to use the finite-element method for the domain having rectangular boundaries. A finite-difference technique renders
SU AND GUAN: FINITE-DIFFERENCE ANALYSIS OF DIELECTRIC-LOADED CAVITIES USING THE POWER METHOD
the governing equation to a matrix eigenvalue problem. The dimension of the resulting matrix can be reduced if larger meshes are used near the metallic walls as usually done in the finite-element method [9]. However, a simple finite-difference scheme based on equally-spaced node points is employed for convenience in this investigation for the considered resonators in which the shielded metal enclosures are not far from the dielectric resonator. The grid must coincide with the metallic boundary and permittivity discontinuity to impose the required boundary conditions. That is, the cross section of the resonator is divided into a rectangular grid composed of m 1 x m2 identical meshes of size 6. by 6., where m 1 = b/ 6. and m2 = l/ 6.. Corresponding to the dielectric ring resonator in the cavity, we let mr6. = c, mo 6. = a, mB6. = h, and my6. = l2 + h, where subscripts I, o, B, and Y denote "inner," "outer," "bottom," and "top," respectively. If the resonator structure is symmetric about the middle section plane, the actual value of m 2 is divided by a factor of 2. In order to conform to the square mesh, the () angle at a dielectric comer is made to be ±45° or ±135°. The finite-difference scheme described below can be easily extended to the unequally-spaced rectangular grids which are more flexible with resonator dimensions. A. TE and TM Modes
By a use of the simple equally-spaced finite-difference method, (4) becomes the simultaneous algebraic equations:
E~j { ( 1 -
;i)
'1/Ji+l ,j
+ '1/Ji,j -1
- 4'1/Jii}
+ (1 + =-
i = 1, 2, .. ., (m 1 -1)
;i)
'1/Ji-1 ,j
+ '1/Ji, j+l
2 (ko6.) '1/Jij , j = 1, 2,. . ., (m2 -1)
(12)
where '1/Jij = 'lj; (i!:::,,, , j!:::,,,) and Eij = E(i!:::,,, , j!:::,,,). The algebraic equations corresponding to those node points at permittivity discontinuity are excluded from (12). In (12), the nodal fields at the metallic walls, the z axis, the symmetry plane, and permittivity discontinuity are either zero or can be evaluated in terms of neighboring nodal fields via the derivative condition or boundary condition (7), where the three-point forward or backward difference formula is used for the first derivatives involved. For example, the expressions of the nodal fields at the outer radius and comer of dielectric resonator for the TE modes are
2
'1/Jij
= 3 ('1/Ji+l ,j
i = mo, mB
+ '1/Ji - 1,j ) -
1
5('1/Ji+2,j
+ '1/Ji -2,j)
and 2 = 3 ('1/Ji+ l,j+l
1
- 6 c'l/Ji+2 ,j+2 + '1/Ji-2,j-2), respectively.
By the same method, (8) becomes the coupled simultaneous algebraic equations
E~j
{ (
1 + i i ) H ri+l,j
+ (1-
i i ) H ri-l ,j
(13b)
+ H ri,j+l
+ Hri,j- l + (
l~ 2m 4)
+~(Hz
H z1.,J._ 1 ) } = -(ko6. )2 H r1.J.
2
'/,
E~j
{ (
1+
·+ 1
i,J
-
-
;i)
H zi+l ,j
(~
+ (1-
H r;1
;i)
H zi-l ,j
+ H Zi,J+ l
2
+Hz;, 1 _ 1 j
-
+4)Hz; 1 }
=-
1, 2,. . ., (m 1 - 1) 1, 2,. . ., (m2 - 1)
2
(ko6. ) H z;1 , ( 14)
where Hr;1 Hr(i6. , j6.) and H z, 1 = H z(i6. , j6. ). Similarly, the algebraic equations corresponding to those node points at the dielectric discontinuity are excluded from (14). And, in (14) the fields at the metallic walls, the z axis, the symmetry plane, and permittivity discontinuity are either zero or can be evaluated in terms of neighboring nodal fields via the derivative condition or boundary condition (10) or (11). The two-point central difference formula is employed for the first derivative on the right-hand side of (lOb). And the three-point forward or backward difference formula is employed for the other first derivatives in the derivative condition and boundary conditions (10) and (11). Note that at permittivity discontinuity the field Ht are coupled to field H n. To evaluate these fields explicitly, the procedure used is described in what follows. The nodal fields Hn at the dielectric discontinuity are first evaluated via (lOa). Next, the H t fields at the dielectric comers are evaluated via (lOb), which together with the H n fields are used to evaluate the H r and H z fields at the comers via (6). Then, the other nodal fields H t at permittivity discontinuity can be evaluated explicitly via (lOb). The expression of fields Hr at the top surface of dielectric ring resonator, for example, is
( 15)
After the finite-difference scheme, we obtain a standard eigenvalue problem
Ax= >.x
+ '1/Ji-l,j-1)
i = m o, j =my
B. Hybrid Modes
mr < i < mo, j = m y.
< j < my (13a)
'1/Ji j
2001
(16)
for the TE, TM, or hybrid modes of a given azimuthal number m, where matrix A is real and nonsymmetric and constant >. = -( ko 6.) 2 . For the TE or TM modes in empty cavities, eigenvalues of matrix A are distributed somewhat uniformly between -8 and zero. While, in cavities loaded
2002
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL. 42, NO. 10, OCTOBER 1994
with dielectric resonator of large permittivity Er, it is expected from Gerschgorin's theorem that many of the eigenvalues will closely cluster near the zero end due to the factor 1/Er· Consequently, the convergence rate is very slow if the ordinary variants of the power method are directly used to find resonant frequencies of dominant modes. The separations of eigenvalues can be widened and hence the convergence rate can be accelerated if matrix A is preconditioned by the Chebyshev polynomial, as discussed in the following Section. IV. CHEBYSHEV ACCELERATION TECHNIQUE
The Chebyshev polynomial of degree n denoted by Tn ( x) is given by the recurrence form ula as
n = 2, 3, . ..
(17)
with To(x) = 1 and T 1 (x) = x. Like that of the power xn, the functional value of Tn( x ) is (18)
and exceeds unity as the argument x exceeds unity. However, it is the key point to note that the Chebyshev polynomial Tn(x) is a more rapid function of degree n than the power xn, when x exceeds unity, especially when xis close to unity . For example, T20(1. 01) = 8.469151 '.::::' 1.01 215 , T20(1.00l) = 1.427333 '.::::' 1.001 356 , and T100 (1.001) = 43. 76587 '.::::' 1.001 3781 . Suppose a is an eigenvalue and unity is the next one, where a = 1 + 8 and 8 is a small positive quantity. Then, by applying the Chebyshev acceleration to (simultaneous iteration based on) the power method, the convergence rate for finding the eigenvalue a can be increased by a factor of (, where Tn (a) = aCn. As n 2 8 « 1, the factor ( becomes as large as degree n . This is because that
where the identity Tn'(l) = n 2 has been made use of. As n --+ oo, the value of the Chebyshev polynomial is given as [27] , [31]
Thus, as n
--+
(a +~)/ ln (a)
'.: : ' .J2f8.
when 8 « 1.
B = __ 2_
C2 - C1
{A - C1 + C2 1} , 2
(21)
where c1 and c 2 are two arbitrary constants and I denotes the unity matrix. It is known that eigenvectors of matrix A are also eigenvectors of matrix B and of any polynomial of matrix B. If c1 and c 2 are eigenvalues of matrix A , then -1 and I will be the eigenvalues of matrix B corresponding to the same eigenvectors. By suitably choosing constants c1 and c2, the convergence rate of the composite matrix Tn( B ) will be much faster than that of s n. Suitable value of constant C1 can be given by applying the ordinary power method to find to the largest-magnitude of matrix A , which is close to -8 in the TE and TM cases, as seen from Gerschgorin 's theorem . Suitable value of c 2 is a small negative quantity. The smaller the magnitude of c2, the smaller the 8 and hence the greater the factor (. However, the maximum resonant frequency that can be calculated is reduced. V. SIMULTANEOUS ITERATION BASED ON THE POW ER METHOD
In practice, only the fundamental mode and a few higherorder modes are needed in each kind of modes. For this end, simultaneous iteration based on the power method [26], [32] can be applied to solve the eigenpairs of matrix B . When this method is used in conjunction with the Chebyshev acceleration technique, one actually solve the eigenpairs of matrix T , where the composite matrix T = Tn( B ) and matrix B is given in (21 ). To find the first q eigenvalues and the corresponding eigenvectors of matrix T of order M, the simultaneous iteration begins with p(?:. q) linearly independent starting vectors forming an M x p matrix X 0 . Then , eigenpairs can be found from the iteration formula k = 0, 1, ...
(22a)
(20a)
where ck is a p x p matrix chosen so that xkck are the optimum linear combinations in the vectors of Xk to represent the true eigenvectors. This reorientation matrix Ck in turn is determined from the transformed generalized eigensystem with matrices of p x p:
(20b)
(X kT TX k)Ck = (X kTXk)CkAk ,
oo, the factor ( remains finite and is given as ( oo =In
the Chebyshev-polynomial preconditioning can be used to accelerate the convergence rate in the power method. Since the wanted eigenvalues are larger in magnitude. To this end, one may scale matrix A to matrix B as
For example, ( 00 = 14.2 and 44. 74 for a = 1.01 and 1.001, respectively. That is, by using the Chebyshev acceleration, the factor ( is linearly proportional to the degree n at the beginning and finally saturates to a value close to y'2/8 (with 8 « 1), as the Chebyshev polynomial degree n is increasing. In an actual calculation, degree n is limited by the use of the simultaneous iteration discussed in Section V and is chosen according to the discretization size ~ such that the factor ( is close to ( 00 . If the eigenvalues of matrix A are shifted and scaled such that most of the unwanted eigenvalues are confined between -1 and 1, while all the wanted ones exceed unity,
(22b)
where superscript T means transpose, matrix Ak [= diag (1 1, 1 2, . . ., /p )], and matrix Ck correspond to the eigenpairs of the transformed problem. This small subspace system (p « M) in tum is solved by the QZ algorithm called from the EISPACK subroutine [33]. When k is sufficiently large, matrices A k and Xk (or XkCk)' correspond to the first p eigenvalues and eigenvectors of T, respectively. Since an iterative procedure is used, the efficiency of the proposed method depends on the convergence rate, which in tum depends on how fast the powers of the ratio rp+if rq go to zero [32, pp. 304-305]. It is seen that the use of a
2003
SU AND GUAN: FINITE-DIFFERENCE ANALYSIS OF DIELECTRIC-LOADED CAVITIES USING THE POWER METHOD
TABLE I COMPARISON OF THE RESONANT FREQUENCIES (GHz) FOR THE CYLINDRICAL CAVITY LOADED WITH A DIELECTRIC ROD . (€r 35.74, h 0 .3 IN, I 0.6 IN, a 0 .34 IN, b 0. 51 IN, 11 12 )
=
=
Present Mode Type Technique (FDM ) TE01 TE02 TE03 TEo, TEos TEoo TM01 TM02 TM03 TMo, TMos TM 06 HE" HE12 HE1 3 HE 14 HEIS HE16 HE21 HE22 HE23 HE24 HE25 HE, 6 HE31 HE,, HE33 HE3,. HE3s HE3r,
=
ModeMatching [6]
=
=
FEM [10]
FDTD [16]
=
Difference (F DM & [6])
s s A s s
3.429 5.4 12 5.908 7.497 8. 015 8.581
3.428 5.462 5.93
3.435 5.493
3.53
0.03 0.92 0.37
% % %
A A
4.542 6.361 7.254 7.641 9.093 9.942
4.551
4.601
4.53
0.20
%
0.08
%
0.4 5 0.37 1.36 0.27 0.46
% % % % %
5 .00 5 .33
0.16 0.36
% %
5.85 6.40
0.12 0.22
% %
A
s
A
s
A
s A
s
A A
s
A
s
A
s
A
s
A
s
A
s
A
s
4.205 4.310 5.662 5.924 6.33 1 7.213 4.992 5.311 6.943 7.285 7.355 8.208 5.843 6.386 7.90 1 8.098 8.453 9.365
7.26
4.224 4.326 5.74 5.94 6.36
greater p decreases the magnitude of ')'p+ 1 and hence improves the convergence rate, but increases the computation effort for each iteration as a compensation. In our calculation the subspace dimension p was chosen to be just q or be q + 1 to prevent possible near degeneracy. The QZ algorithm can solve the repeated eigenvalues and independent eigenvectors of the transformed generalized eigenproblem if some of the eigenvalues are degenerate. After the eigenpairs, /i and Xi, of matrix T are solved, the eigenvalues A; of matrix A are evaluated by the Rayleigh quotient as
4.271 4.373
3.90 4. 17
8.5
o.; ·
100 50 20
8
10
5
? s,.,
l
7.5
~
" J:
O"
6.5
..
5.5 .
A; = (xi, Axi).
(x;, X i )
i=l,2, ... , q.
(23) Number of Total Iterations
The resonant frequencies corresponding to each A; are i=l,2, ... , q
(24)
Fig. 2. Convergence dependences for the TE03 mode and subspace dimension p = 2 with Chebyshev polynomial degree n as a parameter (c 1 = - 8 and c2 = -2.83 x 10 - 3 ).
here c is the speed of light in free space. VI. NUMERICAL RESULTS
To begin with, we use the Hr-Hz formulation (8) and a 40 by 20 grid (by utilizing symmetry) to evaluate the resonant frequencies of azimuthally dependent modes of an empty cylindrical cavity with l = b, of which exact solutions are available. No spurious modes are found and our results are in
good agreement with the analytic solutions, especially for the lower-order modes. Most of the numerical results for lhe first ten modes of m = 1 for both types of symmetry are within 0.5%. Next, we deal with the dielectric-loaded cavity depicted in Fig. l{a) with Er = 35.74, h = 0.3 in, l = 0.6 in, a = 0.34 in, b = 0.51 in, and li = l 2 , which has been analyzed by
IEEE TRAN SACTIONS ON MICROWAVE TH EORY AND TEC HNIQUES, VOL. 42, NO. 10, OCTOBER 1994
2004
r
~z
(a)
r
~z
(b)
Fig. 3.
Field distributions of (a) H r and (b) H z (in relative magnitude) for the HE 11 mode of the rod dielectric resonator cavity considered in Table I.
other approaches [6] , [10], [16] . By utilizing symmetry in the z direction, a 51 by 30 grid was used to fit the boundaries of the conductor and the dielectric exactly. Values of constant c1 determined by applying the power method to matrix A are -8, -8.07, -10.32, and -15.12, form= 0, 1, 2, and 3, respectively. For most cases, the value of c2 is chosen to be -2.83 x 10- 3 , which corresponds to the upper limit of the calculated resonant frequencies being 10 GHz, as seen from (24). The calculated resonant frequencies of the TE, TM, and hybrid modes by the present method are presented in Table I. The mode designation we use in this investigation is the same as that in [IO] although the symmetry property is used in the computation. Note that fields Hr and Hz are of opposite symmetry in the TE or hybrid modes and the symmetry type in Table I is referred to field H z. For the TM modes, the symmetry type is referred to field E r. To compare the convergence rates, the calculated frequency of the TE 03 mode as a function of the number of iterations is plotted in Fig. 2, with the Chebyshev polynomial degree n as a parameter. (For each matrix-vector multiplication Tn(B)x ;, which is evaluated recursively according to ( 17), the number of iterations is counted as n. ) The line designated by 1
corresponds to the ordinary simultaneous iteration of the power method without the acceleration. Evidently, the convergence rate increases with increasing degree n . For n = 1 or 5, convergence is far from being reached even after 2500 iterations; while, for n = 50 or 100, the results converge to the third decimal after 1000 iterations. The modes presented in Table I were calculated with p = 4 and n = 100. The numbers of iterations required for convergence are about 1000. And the computation time for the first three hybrid modes is typically about 10 min for the 1000 iterations on a SUN Spare workstation. For the TE or TM modes, the computation time for each iteration can be reduced by half, approximately. The results by other methods are al so listed in Table I. The discrepancies between our results and those of the modematching method are listed in the last column for comparison . The data of the mode-matching method are read from the curves in [6] with a precision of 2 decimals and some of them are taken directly from [ 10] with 3 decimals. It is seen that all the differences are within one percent except the HE 13 mode. However, the measured data in [6] for this mode is about 5.70 GHz. The relative difference between our calculated result and thi s measured data is 0.67%.
SU AND GUAN: FINITE-DIFFERENCE ANALYSIS OF DIELECTRIC-LOADED CAVITIES USING THE POWER METHOD
2005
TABLE II COMPARISON OF THE R ESONANT FREQUENCIES
{GHz) FOR THE CYLI NDRI CAL 36.0, h 0.28 IN,
CAVITY LOADED WITH A DIELECTRIC RING. ( €r /1 = 0.205 IN, / 2 = 0.295 IN, a = 0.40 IN, c =
=
0.125
=
IN,
b = 0.60
IN )
~z r
(a)
(b)
Fi g. 4. Field distributions of (a) H r and (b) H , (in relative magnitude) for the HE21 mode of the rod dielectric resonator cavity considered in Table I.
The corresponding eigenfield distributions of resonant modes can be obtained directly by the present method. The modal fields Hr and Hz over the r- z plane depicted in Fig. 1(a) are illustrated in Figs. 3 and 4 respectively for the HE11 and the HE 21 modes, of which the fields are of opposite symmetry. It is seen that field Hz of the HE 11 mode and field Hr of the HE 21 mode are symmetric about the middle section plane and somewhat concentrate in the dielectric region. While, field Hr of the HE 11 mode and field Hz of the HE21 mode are antisymmetric and are found to concentrate along the air-dielectric interfaces parallel to the associated field . Field Hr of the HE 11 mode also has local peaks near the dielectric corners. The maximum-magnitude peak of Hr is much larger than that of Hz for the HE 21 mode. While, the peak of Hz is a little larger than that of Hr for the HE 11 mode. It is noted that the derivatives 8Hr/8z and 8Hz/8r are di scontinuous at the air-dielectric interfaces parallel to the associated field. This is due to the boundary condition (10b). The second dielectric-loaded cavity to be analyzed is the shielded dielectric ring resonator depicted in Fig. l(b) with Er = 36.0, h = 0.28 in , li = 0.205 in, lz = 0.295 in, a = 0.40 in, c = 0.125 in, and b = 0.60 in. A very fine grid of 120 by 156 is used and the calculated results with p = q = 2 and n = 300 are shown in Table II. Values of c1 determined from the power method are -8, -8.12, -10.37, and -15.17 for m = 0, 1, 2, and 3, respectively. Values of c2 are chosen such that the upper limit of frequency is about 7 GHz, which
corresponds to c2 = -3.47 x 10- 4. Also shown in Tabie II are the results of mode-matching method and measured data in [7]. It is seen that our results are in agreement with both the measured data and the mode-matching results. Except for the HE11 and HE 12 modes, the required numbers of iterations for convergence to the third decimal are about 3500 and the corresponding computation time of two hybrid modes of a given azimuthal number m is about 3.5 hr. The resonant frequencies of the HE 13 (5.334 GHz) and HE 14 (5.673 GHz) modes are somewhat close to that of the HEu and HE12 modes. Hence, more iterations are needed or the "guard" vectors (p > q in the preceding Section) are helpful in order to enhance the convergence. The relation between computation time and mesh size b. is discused in what follows. In the present two-dimensional problem, the total number of unknowns and the constant 8 in the Chebyshev acceleration are proportional to 1/b. 2 and b. 2, respectively. Thus, by a use of (20b), the computation time for converged results is proportional to 1/ b. 3 (rather than 1/ b. 4 ), if the Chebyshev polynomial degree n is large enough.
VII. CONCLUSION An efficient numerical procedure using the finite-difference technique and the simultaneous iteration of the power method in conjunction with the Chebyshev acceleration is proposed to analyze dielectric-loaded cavities. The efficiency is due to that the shifted eigenvalues corresponding to the dominant modes are greatest in magnitude and that the separations of eigenvalues are widened. Both the computation time for each iteration and the memory space are linearly proportional to M, the total number of unknowns. A matrix of order M as large as 37,000 has been handled on a workstation . Thereby, highly accurate results of resonant frequencies of the TE, TM, and hybrid modes in a dielectric-loaded cavity of two-dimensional problem have been obtained. REFERENCES [I] D. Kajfez and P. Guillon, Dielectric Resonators.
Norwood, MA: Artech House, 1986. [2] S. J. Fiedziuszko, " Dual-mode dielectric resonator loaded cavity filters " IEEE Trans. Microwave Theory Tech. , vol. MTT-30, pp. 1311-1316, Sept. 1982.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 10, OCTOBER 1994
[3) K. A. Zaki, C. Chen, and A. E. Atia, "Canonical and longitudinal dualmode dielectric resonator filters without iri s," IEEE Trans. Microwave Theory Tech., vol. MTI-35 , pp. 1130-1135 , Dec. 1987. [4] Y. Kobayashi and M. Mineg ishi , "A low-loss bandpass filter using electrically coupled high- Q TMol6 dielectric rod resonators," IEEE Trans. Microwave Theory Tech., vol. MTI-36, pp. 1727-1732, Dec. 1988. [5] V. Madrangeas, M . Aubourg, P. Guillon, S. Vigneron, and B. Theron, "Analysis and realization of L-band dielectric resonator microwave filters," IEEE Trans. Microwave Theory Tech. , vol. 40, pp. 120-127, Jan . 1992. [6) K. A. Zaki and C . Chen, "New results in dielectric-loaded resonators," IEEE Trans. Microwave Theory Tech. , vol. MTI-34, pp. 815-824, July 1986. [7] S. -W. Chen and K. A. Zaki , "Dielectric ring resonators loaded in waveguide and on substrate," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 2069-2076, Dec. 1991. [8] R. E. Collin and D. A. Ksienski, "Boundary element method for dielectric resonators and waveguides," Radio Sci., vol. 22, no. 7, pp. 1155-1167, Dec. 1987. [9] F. H. Gil and J. P. Martinez, "Analysis of dielectric resonators with tuning screw and supporting structure," IEEE Trans. Microwave Theory Tech. , vol. MTI-33, pp. 1453-1 457, Dec. 1985. [10] M. M. Taheri and D. Mirshekar-Syahkal, "Accurate determination of modes in dielectric-loaded cylindrical cavities using a one-dimensional fi ni te element method," IEEE Trans. Microwave Theory Tech., vol. MTI-37 , pp. 1536- 1541 , Oct. 1989. [I I] P. Guillon, J. P. Balabaud, and Y. Garault, "TMo 1 p tubu lar and cyli ndrical dielectric resonator mode," IEEE MTT-S Int. Microwave Symp., Dig., pp. 163- 166, 1981. [ 12] T. Weiland, "On the computation of resonant modes in cylindrically symmetric cavities," Nuc/. lnstrum. Methods, vol. 216, pp. 329-348, 1983 . [13] U. V. Rienen and T. Weiland, "Triangular discretization method for the evaluation of rf-fields in cylindrically symmetric cavities," IEEE Trans. Ma gn. , vol. MAG-21, pp. 231 7-2320, Nov. 1985 . [14] J. E. Lebaric and D. Kajfez, "Analysis of dielectric resonator cavities using the fi nite integration technique,'' IEEE Trans. Microwave Theory Tech. , vol. 37, pp. 1740-1748, Nov. 1989. [ 15] A. Navarro, M. J. Nunez, and E. Martin, "Study of TEo and TMo modes in dielectric resonators by a finite difference time-domain method coupled with the di screte Fourier transform," IEEE Trans. Microwave Theory Tech. , vol. 39, pp. 14-17, Jan. 1991. [16] P. H. Harms, J. F. Lee, and R. Mittra, "A study of the nonorthogonal FDTD method versus the conventional FDTD technique for computing resonant frequencies of cylindrical cavities," IEEE Trans. Microwave Theory Tech., vol. 40, pp. 741- 746, Apr. 1992. For corrections, see vol. MTI-40, pp. 2115-2116, Nov. 1992. [ 17] Z. Bi, Y. Shen, K. Wu , and J. Litva, "Fast finite-difference timedomain analysis of resonators using digital filtering and spectrum estimation techniques," IEEE Trans. Microwave Th eory Tech. , vol. 40, pp. 1611-1619, Aug . 1992. [ 18] J. A. Pereda, L. A. Vielva, A. Vegas, and A. Prieto, "Computation of resonant frequencies and quality factors of open dielectric resonators by a combination of the finite-differe nce time-domain (FDTD) and Prony 's methods," IEEE Microwave Guided Wave Lett., vol. 2, pp. 43 1-433, Nov . 1992. [ 19] X. Zhao, C. Liu, and L. C. Shen, "Numerical analysis of a TMo 10 cavity for dielectric measurement," IEEE Trans. Microwave Theory Tech. , vol. 40, pp. 1951 - 1959, Oct. 1992. [20] J. P. Webb, "Edge elements and what they can do for you ," IEEE Trans. Magn. , vol. 29, pp. 1460-1465 , Mar. 1993 . [2 1] J. B. Davies, "Finite e le ment analysis of waveguides and cavities-a review," IEEE Trans. Magn. , vol. 29, pp. 1578-1583, Mar. 1993. [22] J. F. Lee, D. K. Sun , and Z. J. Cendes, "Full-wave analysis of dielectric waveguides using tangential vector finite elements," IEEE Trans. Microwave Theory Tech., vol. MTI-39, pp. 1262-1271 , Aug. 1991.
[23] J. F. Lee and R. Mittra, "A note on the applicat ion of edge-e lements for modeling three-dimensional inhomogeneously-filled cavities," IEEE Trans. Microwave Theory Tech. , vol. 40, pp. 1767- 1773, Sept. 1992. [24] A. Chatterjee, J. M . Jin and J. L. Volakis, "Computation of cav ity resonances using edge-based finite e lements," IEEE Trans. Microwa ve Theory Tech., vol. 40, pp. 2106-2108, Nov. 1992. [25] C. C. Su, "A numerical procedure using the shifted power method fo r analyzing dielectric waveguide without inverting matrices," in Proc. URS/ Radio Sci. Meeting, p. 321 , 1990. [26] _ _ , "An efficient numerical procedure using the shi fted power method for analyzing dielectric waveguides without inverting matrices," IEEE Trans. Microwave Theory Tech. , vol. MTI-41 , pp. 539-542, Mar. 1993. [27] Y. Saad, "Chebyshev acceleration techniques for solving nonsym metric eigenvalue problems," Math. Comp., vol. 42, pp. 567-588, Apr. 1984. [28] A. T. Galick, T. Kerkhoven, and U. Ravaioli , "Iterative solution of the eigenvalue problem for a dielectric waveguide," IEEE Trans. Microwave Theory Tech. , vol. 40, pp. 699-705 , Apr. 1992. [29] R. E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 199 1, ch. I. [30] J. V. Blade!, Singular Electromagnetic Fields and Sou rces. New York: Oxford Press, 1991 , ch. 5. [31] D . Ho, F. Chatelin, and M. Bennani, "Arnoldi-Tchebyshev procedure for large scale nonsymmetric matrices," Math. Model. Numer. Anal., vol. 24, no. I, pp. 53-65, 1990. [32] A. Jennings, Matrix Computation for Engineers and Scientists. London: Wiley, 1977, ch. 10. [33) B. S. Garbow, J. M . Boyle, J. J. Dougarra and C. B. Moler, "Matrix eigensystem routines-EISPACK guide extension," in Lecture Notes in Computer Science, vol. 5 I . Heidelberg: Springer-Verlag, 1977.
Ching-Chuan Su was born October 2, 1955 in Taiwan. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from National Taiwan University in 1978, 1980, and 1985, respectively. From 1980 to 1982, he was employed at the Industrial Technology Research Institute, Hsinchu , Taiwan, where he was responsible for the development of several IC fabrication processes for MOS products. In 1985 he joined the fac ulty of National Tsinghua University, Hsinchu , Taiwan, where he is . an Associate Professor of Electrical Engineering. His research areas include electromagnetic theory, numerical solutions in scattering, waveguide, resonator, and MOS circuit, and fabrication of ferroelectric memory IC .
Jenn-Ming G uan was born October 17, 1967, in Taipei, Taiwan. He received the B.S. degree in electrical engineering from National Tsinghua University , Hsinchu , Taiwan , in 1989, and the M.S. degree in electrical engineering from National Tai wan University, Taipei , Taiwan, in 1991 . Since 199 1 he has been working toward the Ph.D. degree in the Department of Electrical Engineering at National Tsinghua University. His research interests include the numerical techniques for the waveguide and resonator problems.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
2007
Short Papers_ _ _ _ _ _ _ _ _ _ _ _ __ An Application of FD-TD Method for Studying the Effects of Packages on the Performance of Microwave and High Speed Digital Circuits
metal box microstrip line
Ke-Li Wu, Chen Wu, and John Litva
Abstract- The finite-difference time-domain (FD-TD) method is combined with an appropriate time-frequency discrete conversion technique to analyze packaging and time domain transition effects of microwave and high speed digital circuits. The output response of a given input pulse is obtained by linear convolution of the input signal with time domain system function , which is obtained through FD-TD simulation of the whole packaging system including coaxial to microstrip line transitions. As an example, a shielded microstrip line which is connected with coaxial lines, is analyzed and measured. The comparison between experimental and numerical results shows very a good agreement.
I. INTRODUCTION
With the development of high speed digital circuits and monolithic microwave integrated circuits (MMIC's), the detrimental effects of electronic packages and other transitions associated with housing these ci rcuits have become topics of considerable interest in the last few years. Of particular interest are the internal resonances of an enclosing structure, as well as the distortion in the shape of electrical pulses, which leads to a reduction in the transmission efficiency of signals propagating through these structures. It is obviously important to develop accurate and robust full-wave numerical models, which are capable of coping with the shielding and transition effects, for high performance circuit designs. The problems related to packaging side effects have been investigated in the spectral domain by a number of workers. One representative model is that developed by [I]. The reciprocity theorem is used to set up the model and the analysis is carried out using the moment method. Although this model is a significant step towards the goal that is being espoused by an increasing number of researchers, it is found that there are still some drawbacks in the model. For example, the hi gher order modes near the coaxial to microstrip transition have not been completely taken into account because the coaxial feed is described by a simple magnetic frill current which involves only the TEM mode in the coaxial line. As well, the relative convergence of the Green' s function makes it difficult to obtai n a reliable solution. The FD-TD method has been widely used to solve electromagnetic problems. In this short paper, the three-dimensional FD-TD method is applied to model a representative electronic package and associated transitions. In particular, the model developed earlier [2], [3] for the coax-to-microstrip transition is incorporated with the solution procedure. Since the size of the FD-TD' s lattice is fairly small compared with the wavelength of interest, higher order modes in the vicinity of the coaxial to microstrip transitions are easily incorporated into the model. Manuscript received November 9, 1992; revised December 6, 1993. K.-L. Wu is with the R&D Department, COM DEV LTD, 155 Sheldon Dr, Cambridge, Ontario, Canada NI R 7H6. C. Wu and J. Litva are with the Communications Research Laboratory, McMaster Un iversity, Hamilton, Ontario, Canada L8S 4KI. IEEE Log Number 9404149.
Fig. 1.
A shielded microstrip line connected with coaxial lines.
Since a nonphysical excitation plane issued in launching a numerical pulse at the input port, the difficulty in the investigation of a pulse distortion is one of launching pulse of a given shape. Therefore, we cannot directly simulate the output for a given input. However, we can reach the same objective indirectly by using the system function. The system function can be found by deriving the impulse response for certain bandwidth. To establish the system transfer function of a packaged circuit either in time or in frequency domain, the discrete Fourier transform (DFf) will be used to convert sequences of discrete data from time-domain into frequency-domain and vice versa. Once the discrete transfer function is found, the discrete linear convolution is used to determine the output from an input. As a typical example, a prototype of a microstrip line in a shielded metal box with coax-tomicrostrip transitions is made as shown in Fig. 1. The experimental results are in good agreement with numerical results. II. FD-TD SIMULATION It is well known that, in the application of the FD-TD method, the electromagnetic fields are directly discretized in time and in space. The curl operators in the Maxwell 's equations are implemented using the central difference scheme proposed in [4], which is referred to as Yee's lattice. The second order accuracy is achieved by the algorithm when using the first order central difference scheme. After setting up appropriate boundaries, consisting of electric conductor walls and absorbing boundaries, the analysis is initiated by introducing an impulse function at some appropriate location and updating the field quantities at the grid points in a time sequential manner. For the sake of simplicity, no further details of the FD-TD method will be discussed here. The algorithm has been described extensively in the literature. In this application, the coaxial line of a circular cross section is approximated by a stair casing boundary as shown in Fig. 2. Through a large number of numerical examples, it can be concluded that, although the inner and outer conductors of a coaxial line are approximated by staircasing, the accuracy is still retained as long as the characteristic impedance and the dimensions of the approximated line are close to that of the original one. Since the input and output planes are located in TEM coaxial lines, the first order absorbing boundary condition works very well in this study.
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Before implementing the linear convolution, we need to be aware of two conditions that must be satisfied when choosing the length of the sequences to be convolved, i.e.:
inner conductor I
. .v
~
/
I
,,v
/
I
;
\
\
~~
#'
~
~
~
I) The original input pulse must be sampled sufficiently often, > 213.fmax, where 13.fmax is the say at M points so that maximum frequency bandwidth of interest. 2) To avoid the spectrum overlap in carrying linear convolution through the periodic convolution, the sequences of Vi, Vo and the input pulse sequence with M points should be extended to the length of N, which is greater than or equal to M + Nr -1, by adding zeros.
L
'""' \ j
According to Symmetry Properties of DFT, the following conditions:
I/ ~
...
,
I
(3)
.....
~,,,,,,.
and
~
(4)
outer conduc tor\ Fig. 2. A stair casing approximation of a 50 Ohms coaxial line. The system transfer function for a packaged circuit can be derived from its impulse response. However, it is difficult to get an ideal impulse in a numerical process. To get the system function, a Gaussian pulse is used as the excitation, which is applied in the coaxial line of input port. This fictitious source at the excitation plane will become a TEM wave ·after propagating a few lattices away from the plane. After the Gaussian pulse has travelled through a package system, input and output pulses can be sampled at an appropriate reference plane. The discretized freq uency domain transfer function can then be determined using:
S 2 1 [f .] = DFT{V (n!3.t)}
(I)
0
k
= 1, 2, ... , Nr,
[!k] must satisfy
/~ ~
k
S21
n
DFT{V; (m!3.t)}
= 1, 2, . .. , N
0 ,
m
= 1, 2, .. . , N;
(2)
where Vi(m'3.t) and Vo(n!3.t) are the sampled input and output voltages, respectively. To be more specific, Vi refers to the N; data samples of the signal at the input port and Vo to the N 0 data samples of the signal at the output port, and 13.t is the time step used in FD-TD simulation. N r is usually equal to No or N ;, whichever is greater. The shorter of the two sequences must be extended by adding zeros until its length is equal to Nr. The fast Fourier transform (FFT) is used here to implement Discrete Fourier Transform (DFT), which converts the data from the time domain into the frequency domain. The frequency domain transfer function is of most interest to the microwave circuit designer. It is assumed here that the time step used in the FD-TD algorithm satisfies the stability condition and is therefore small enough so that the Nyquist rate is satisfied to cover the frequency bandwidth of interest. III. SIGNAL CONVERSION TECHNIQUES
In a high speed digital circuit design, one is interested to know the distortion of an input pulse after it passes through a packaging system. Such distortion is mainly caused by three factors , in addition to the properties of circuit itself: I) the dispersion down the printed transmission line, where the propagating wave is not a pure TEM wave; 2) mode coupling intransition discontinuities; and 3) internal resonances in an enclosing packaging metal box. In fact, all these factors have been included in the system transfer function. As a result, for a given input pulse, the output pulse can be obtained by convolving the input pulse with the system transfer function in the time domain.
where k = 1, 2, ... , ( N - 1)/2 when N is odd or k 1, 2, ... , (N /2) - 1 when N is even. These properties are important because when one converts an experimental frequency domain system function into the time domain, the function must be modified to meet the symmetry properties mentioned above. From modified time domain sequences of V0 and Vi , one can find the discrete frequency domain system function in a relatively straight forward manner. The product of the resulting frequency dom ai n sequence with the DFT of N point input pulse must also satisfy the symmetry properties. The inverse DFT of this product results in a real time-domain sequence, which is the output time domain response. IV. NUMERICAL AND EXPERIMENTAL RESULTS
To verify the proposed model, a prototype of the microstrip line in a metal box with coaxial to microstrip transitions is analyzed and measured. Fig. 3(a) and (b) shows the system transfer function obtained by experiment and the FD-TD method, where 13. z = 13. x = 13.y 0.3125 mrn, 13.t 0.5713.h/C and C is the speed of light. Details with regards to the discretization used at the coaxial to microstrip junction is given in Fig. 2. It can be seen that the numerical and experimental results are in good agreement both in magnitude and in phase. Because of a large mismatch between the coaxial line and microstrip line at high frequency (due to the dispersion in the microstrip line), the transmission properties will become unpredictable. In this example, it occurs when the frequency is higher than 12 GHz. Fig. 4 shows the input and the output of a rectangular pulse through the prototype, where the spectrum of the ideal rectangular shaped input pulse, with a rise time of 59.4 ps and pulse width of 154.5 ps, is truncated at 20 GHz. The output response are obtained by the linear convolution of the input pulse with its system transfer function shown in Fig. 3. The solid line is the result using the transfer function calculated by the FD-TD method and the dashed line is obtained by the measured transfer function. Due to the dispersive property of the microstrip line, the rectangular shaped pulse is greatly distorted after it passes through the system. The effects of the metal box and the transitions are better observed from the frequency domain transfer function . It should be mentioned that due to the errors in the errors in the dimen sions of the prototype, some degree of discrepancy between simulated and measured results can be observed.
=
=
V. CONCLUSION
A model of packaging and transition effects for microwave and high speed digital circuits is developed based on the 3D full-wave
2009
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
by this model for the purpose of verification. Digital signal processing techniques are used with the FD-TD algorithm in this analysis. The comparison of numerical results with experimental results is in a good agreement. The variable grid model will be adopted to improve the efficiency of the model in the next phase of the research.
- 10
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ACKNOWLEDGMENT
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The authors wish to thank their colleague Dr. Qiang Wu, for his helpful discussion during the course of this research. The authors also acknowledge Sharon R. Aspden of Rogers Corp. for providing the dielectric material used in this research under the auspices of their university program. The comments and suggestions from reviewers are also acknowledged. Funding was provided by the Telecommunication Research Institute of Ontario.
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REFERENCES
Frequency (Ghz)
[I) L. P. Dunleavy and L. P. B. Katehi, "A generalized method for analyzing shielded thin microstrip discontinuities," IEEE Trans. Microwave Theory Tech., vol. MTI-36, pp. 1758-1766, 1988. [2] C. Wu, K.-L. Wu, Z. Bi, and J. Litva, "Modelling of coaxial-fed rnicrostrip patch antenna by finite difference time domain method," Electronics Letters, vol. 27, pp. 1691-1692, 199 1. [3) __ , "Accurate characterization of planar printed antennas using finitedifference time-domain method," IEEE Trans. Antennas Propagat. , vol. 40, pp. 526-534, May 1992. [4] K. S. Yee, "Numerical solution of initial value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302-307, 1966.
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The Dominant Mode in a Parallel-Plate Chirowaveguide
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Liyang Zhang, Yongchang Jiao, and Changhong Liang
Frequency (Ghz)
(b)
Fig. 3. (a) The magnitude of the system transfer function for the packaged circuit shown in Fig. I. (b) The phase of the system transfer function for the packaged circuit shown in Fig. I. 1.4.-------,----.,...------r----~----,
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I. INTRODUCTION
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Abstract-It has been reported that the lowest cutoff frequency of the modes in a parallel-plate chirowaveguide is not zero. In this paper, we show that a dominant mode of trivial cutoff frequency may be supported by such a chirowaveguide. The mode exhibits the characteristics of a TEM mode when the chirality of the medium vanishes or the operating frequency is very low. An analogous mode also exists in a bianisotropic chiral coaxial line, i.e., the structure formed by more than one conductor.
The input and output pulses of the packaged circuit shown in Fig. 1.
numerical technique-the FD-TD method. The emphasis has been put on the combination of the FD-TD method with the time-frequency domain discrete conversion techniques. A typical problem is analyzed
Recently a number of papers on the theory or analysis of chirowaveguides has been published. Engheta et al. [l]-[2] studied the possible propagating modes in a cylindrical chirowaveguide and proposed the method for analyzing such structures using the theory of coupled modes. Investigation into the characteristics of the modes in a parallel-plate chirowaveguide was made and notable features of chirowaveguides, e.g. , bifurcation of the modes, which may have many potential applications, have been reported [l]-[2]. According to [l] , [2] no modes may propagate below nontrivial frequency, i.e., the lowest cutoff frequency in a chirowaveguide. However, the lowest cutoff frequency of the parallel-plate chirowaveguide was found to be nonzero [I ]-[2]. Moreover, this frequency does not Manuscript received July 29, 1993; revised November 15, 1993. The authors are with Department of Microwave and Telecommunication Engineering, 601, Xidian University, Xi'an 710071 , P.R. China. IEEE Log Number 9404145.
0018-9480/94$04.00 © 1994 IEEE
2010
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y
approach zero when the chirality of the medium vanishes. This means that the structures formed by two perfectly conducting plates but filled with a medium of weak chirality might prevent the transmission of low frequency current, which contradicts our conventional concept. In this note, the dominant mode of a parallel-plate chirowaveguide is found in the fast-slow-wave region of the Brillouin diagram having trivial cutoff frequency. Such a mode was determined by the characteristic equation given by [I] , [2] but neglected in their analysis. The field distribution of the dominant mode tends to that of the TEM mode in a conventional parallel-plate waveguide filled with non-chiral medium when the chirality of the medium disappears. Additionally, the longitudinal components of the dominant mode vanish when the operating frequency gets low enough. We speculate that such a mode may also exist in any chirowaveguide containing more than one separated conductor.
II.
Perfect Conductor
Filled with a Chiral Material (e,µ,¢ c) Fig. 1.
FORMULATION OF THE PROBLEM USING VECTOR WAVE FUNCTIONS
Considering the cross-sectional configuration of the parallel-plate waveguide shown in Fig. I, and assuming that no variations of the filed take place along the x -direction, we may choose the vector wave functions as
Fig. 1 shows a chirowaveguide formed by a pair of parallel plates of distance a and filled with chiral medium with parameters (c: , ft , Ee) and the constitutive relations
D = c:E + i EeB ,
( I)
H = i EeE + B / µ ,
(2)
+ • (' {cos(1+ Y) M 12 = x exp i(3z ) . ( )' ' Stn l+ Y
with c:, µ , Ee denoting the permittivity, permeability, and the chirality admittance of the chiral medium, respectively. Here we suppose the time dependence of the field to be exp (- iwt ). From the chiral constitutive relations (1) and (2), and the source-free Maxwell equations, we have [1] E - 2wµ Ee \7 x {E \7 x \7 x { H H - k 2{E H = 0,
A parallel -plate chirowaveguide.
N+ = _.!.._ex (i f3z) [ ' i(3 {cos(l+ y) + 1 2 k+ p Y sin b+ Y) •
z
I+
(8)
{ sin(l+ y) ] (9) -cosb+ Y) '
M -1 2 = x• exp ( i.(3 z ) {cosb. ( Y)) , ' Stn 1 - Y
(3)
N-
1• 2
where k = w .,f4i, with w being the operating frequency. We may use vector wave functions M , N to construct the general solution to (3), i.e.,
=_.!.._ex (i f3z) [ 'i(3 {cos (!_ y ) + . k_
Y
P
where 1± = Jk~
- (3
sin(!_ y )
( 10)
{sin(l_ y)
Z{-
]
-cos(!_ y ) ' ( 11 )
2.
The boundary conditions are
x·E
= 0 , at
y=
± a/ 2,
( 12)
z· E
= 0 , at
y=
± a/ 2.
( 13)
E = A(M{ + N {) + B (M t + N t) + C (M1 - N 1) + D (M 2 - N 2 ) ,
(4)
With the substitution of (4), and then (8)- ( 1 l ) in ( 12) and ( 13), we obtain a group of homogeneous linear equations, i.e., ( 14), (shown at the bottom of the page). Equation ( 14) can be separated as the following equations:
where A , B , C , D are expansion coefficients determined by boundary conditions, (j = 1, 2) are vector wave functions satisfying
MT, NT
(5)
± N1
1 = f;; \7
k+
± x M1,
cos ( 'Y 2a )
cos( ¥ ) [ - :1 sin ( 2±.:: )
(6)
and
]
[A ]
C
=O
*)
[- f!c~f
vk2
where k ± = ±wµEe + + (wµE c)2 . Then with the substitution of (4) in the source-free Maxwell equations, the magnetic field is determined as
~= sin( -y;a )
2
~~i~~s'Y(;'Y:)a ) ][~] = 0.
ZT/ e
cos ( ¥ ) cos( -yta ) [
f!- sin ( ¥ ) . ( -'-'--y.._a ) - -'Y+ Stn k+
2
sin( ¥ ) - sin( "ta ) - :1cos ( 2±.:: ) k+ 2 - :1 cos ( -r+a ) k+
2
( 16)
Equation ( 15) or ( 16) has a nontrivial solution if and only if the coefficient matrix is singular, i.e., its determinant equals to zero. This leads to the characteri stic equations
H = .;_[A (M { + N {) + B (M t + N t ) - C(M1 - N 1) - D (M 2 - N 2 )],
( 15)
:+ + '.- ) sin( '+ + I- a ) + ( : + - l - )sin( 1 + - i( k+ k_ 2 k+ k_ 2 = 0 , for even modes,
(7)
cos( ";a ) cos ( -y;a )
'Y - .in ("- ks - 2- a) k-y -_ cos (")' - 2_- a) -Y -
.
(
-y_ a)
k _ Stn - 2-
l
~~~~~~;)) [~] ~ - cos ( 'Y ; a)
D
= 0.
a) ( 17)
( 14)
2011
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
the field distribution of these modes is given as k_a
0/27T
2.0
E x = -iEo exp (i /3z ) [cos (";a) cos b +Y) - cos
c;a)
cos b-Y) ] ,
(21)
+ :_ cos(";a)cosb- Y)] ,
(22)
1.5
= Eo exp (i/3z) [ ~ cos(";a )cos(/'+ Y)
Ey 1.0
0.5
= - iEaexp (i/3z) [ Z:cos("; a )sin (/'+ Y)
E, dominant mode I /.
,. /
. (l'- Y)] , + I'k_ cos ( -l'+a) - sm 2
f3o
"
0.0
(23)
-4'-------~--~-~-~-~--~-~-~
0.0
4.0
8.0
12.0
16.0
20.0
Fig. 2. Brillouin diagram for propagating modes guided by a parallel-plate chirowaveguide filled with a chiral medium of parameters c, µ , /;c, € 8.854 x 10- 12F/m, I'= 47r X 10 - 7 H/m, /;c 0.001 mho.
=
=
Hx
= -T/~o exp(i /3z) [cos(";a)cos(l'+y) + cos(";a )cosb-Y)] ,
(24)
and I'+ + 'Y- )sin( 'Y+ + I'( k+ k_ 2
a) - ( I'+ k+
/'- )sin( /'+ - I'k_ 2 = 0, far odd modes,
a) (18)
which are exactly the same as (25) given by [2]. Following [2], let 6. 1 (/3 ) represent the function on the left hand of ( 17). It is obvious that 6., (/3 ) is a continuous real function provided L ~ /3 ~ k+. One may prove that by letting I'- = ir;,, then [6.1 (/3))* = 6.1 (/3 ). However, 6. 1 (k+)
=-
2J(k+ /L)2 - l s h[L a J (k+/k-)
2
-
1]
(19)
= 2J1 -
I'- cos ( -l'+a . b- Y) ] , - k_ - ) sm 2 by (4) and (7). Firstly, if ~c --+ 0, we readily get and /'+. I'- --+ 0, T/c --+ TJ. Therefore,
E
= 'f)2Eo exp(ik z ),
H
k_,
/3 , k+
= - x 2Eo exp (i k z), ~c
--+
0.
(26) --+
k,
(27)
1)
(20)
The condition in (20) can be satisfied if the operating frequency w is low enough. This causes 6.1 (/3) to have at least one zero in (k-, k+) at low operating frequency, therefore the cutoff frequency of the corresponding mode (the dominant mode) should be zero. Fig. 2 shows the dispersion diagram far propagating modes, plotted with the same parameters as used by [l], [2]. From the figure, we discover the dominant mode existing in ( k_ , k+) throughout the whole real frequency region and getting closer to k+ when the operating frequency goes higher. One should note that at each one of these frequencies satisfying k+a Jl - (k-/ k+)2 = n7r , one of the trajectories far even modes intersects the dashed straight line far k_ a in the Brillouin diagram. These corresponding frequencies were denoted with Os, by [2] . However it is more precise to adopt 11./27r (II. = wa ,f€ii,) to represent the title of the longitudinal ax is in [2, Fig. 4].
(25)
-Eo exp (./3 = ----:;;:i z ) [ 'Y+ k+ cos ( -'Y-a - ). sm ( I'+ y ) 2
Hz
(L/k+) 2 sin[k+a J1 - (L/k+)2 ]
> 0, if k+a Jl - (k - /k+)2 < 7T.
III.
/3 cos ( -l'+a - ) cos b-Y) ] , - k_ 2
< 0,
and 6. 1(L)
- iEo ./3z) [ k_ /3 cos ( -"(-- a) cos(/'+Y) Hy= ~exp(i 2
That means the dominant mode in a parallel-plate chirowaveguide approaches the TEM mode in a conventional parallel-plate waveguide when the chirality of the medium vanishes. Secondly, using sin(x) ~ x for x --+ 0 in (17) at low frequencies, we get an approximation to the propagation coefficient of the dominant mode
/3 ~ Jk+L
= k,
(28)
Therefore, with the operating frequency tends to zero, we get the field distribution of the dominant mode, i.e., E
H
= 'f)2EoJ1 +
= ( -x_ + 2 Zo..~, cT/Y_ )2Eo(l+TJ
2
TJ 2 ~2 exp (i kz) ,
~~ ) exp (.i k·z ) ,
T/
aJki - k'!._ «
1,
(29)
FEATURES OF TH E DOMI NANT MODE WITHWEAK CH!RALITY OF THE MEDIUM OR AT
Low
OPERATING FREQUENCY
With ( 15) and ( 16), we may choose A = - iEocos(l'_a/2), C = i Eo cos (l'+a/ 2) , and B = D = 0, far even modes, then
where the longitudinal components of both electric and magnetic fields tend to zero. However, it is clear electromagnetic power can be transferred in such a mode even the operating frequency goes to zero.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42 , NO. 10, OCTOBER 1994
IV.
CONCLUS IONS
In this note, it is verified that a mode with trivial cut off frequency exists in the parallel-plate chirowaveguide. The mode always gets its propagation coefficient in the fast-s low-wave region of the Brillouin diagram, and shows features close to those of the dominant mode in a conventional parallel-plate waveguide at low frequencies. The longitudinal components of the mode tend to vanish provided the operating frequency is low enough. It seems that electromagnetic power can be transferred in the mode at low frequencies. Also we have found an analogous mode supported by a circular coaxial line
filled with a bianisotropi c chiral medium. It is considered that such a mode may be supported by any chirowaveguide formed by more than one separated conductor.
REFERENCES
[I] N. Engheta and P. Pelet, "Modes in chirowaveguides," Opt. Leu., vol. 14, no. 11, pp. 593-595, 1989. [2) P. Pelet and N. Engheta, "The theory of chirowaveguides," IEEE Trans. Antennas Propagat., vol. 38, pp. 90-98, Jan. 1990.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
2013
Letters _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ Comments on "Analysis of the Effects of a Resistively Coated Upper Dielectric Layer on the Propagation Characteristics of Hybrid Modes in a Waveguide-Shielded Microstrip Using the Method of Lines" R. Pregla
In [ 1J, a long derivation is given to consider a thin resistive layer of a structure in the analysis procedure with the MoL. But all what is needed is given in [2J. With the definition of Jm there (line below (45)) we can write for a thin resistive layer of thickness T
Only Ohms law S = CT E has been used and S is the volume current density. It is clear that Sand E do not change in the vertical direction of the thin layer. Introducing the above equation into (46) in [I , (9)J of the paper of Chung and Wu is obtained. As shown the "extension" is given by one line only . REFERENCES
[ l] S.-J. Chung and L.-K. Wu, "Analysis of the effects ofa res istively coated upper dielectric layer on the propagation characteristics of hybrid modes in a waveguide-shielded microstrip using the Method of Lines,"/£££ Trans. Microwave Theory Tech., vol. 41 , pp. 1393- 1399, Aug. 1993. [2] R. Pregla and W. Pascher, "The method of lines," in T. Hoh, Ed. , Numerical Techniques for Microwave and Millimeter-Wave Passive Structures. New York: Wiley, 1989, ch. 6, pp. 38 1-446.
Comments on The Influence of Finite Conductor Thickness and Conductivity on Fundamental and Higher-Order Modes in Miniature Hybrid MIC's (MHMIC's) and MMIC's R. Pregla
This comment has been written • because some statements in the paper [AJ cannot remain uncontradicted. • to help researchers using the MoL in case of numerical difficulties (see point 5 below). Manu script received January 28, 1994. The author is with Allgemeine und Theoretische Elektrotechnik, Fern Universitat, 058084, Hagen, Germany. IEEE Log Number 9404142. Manu script received January 28, 1994. The author is with Allgemeine und Theoreti sche Elektrotechnik, Fern Universitat, 058084, Hagen, Germany. lEEE Log Number 9404144 ..
1) The authors of the paper [AJ states: "· · · we adopted the self-
consistent approach for the MoL [15J ([BJ) at approximately the same time as the authors in [16J ([CJ)." This is not true. The whole analysis procedure was given a long time before : The basic ideas and necessary formulas for the "self-consistent" analysis of composed dielectric waveguides (and the lossy metall strips or layers have to be analysed as dielectric too) with the method of lines was given in the paper [DJ ([2 1J in [AJ and [5J in [BJ-the year of the Romeo-Conference was not given in either of the papers). If a time comparison is made it must be done between these two papers: The paper [DJ was published three years before the paper [BJ and was known by one of the authors from the beginning on because he has also given a paper at the Rome-conference. This is important for the comparisons of the contents. The reader may do this by himself. The paper [DJ was cited by Wu and Vahldieck in the appendix of [AJ , but it was not stated that it is the basis of their whole analysis. (The reader may do the same work of comparison for the paper [AJ and the paper [DJ .) Instead the authors permit the impression that they describe their own independent algorithm (see quotation above). The algorithms in [AJ, [BJ are the same as in [DJ . A principal difference cannot be seen. Even in the case where there is some freedom the authors use the same procedure. Why do the above authors use the transformation in two steps of the difference matrices as in [DJ ? This is a not necessary procedure. The writer of this comment has used this only to obtain nicer formulas. From numerical point of view the one step transformation is better because an eigenproblem for a tridiagonal matrix with all advantages has to be solved whereas in the other case a full matrix is obtained. Therefore we use these advantages in our programs. Afterwards the matrices can be splitted because the transformation matrices for the homogeneous layers are given analytically. The writer of this comment means that it was also not necessary to repeat the analysis procedure especially because the whole procedure is described in detail with all necessary formulas in the well known book of Itoh [EJ , which was published one year before [BJ and four years before the above paper [AJ . Whereas the programming work can be done without extra analysis work using the details in [E], this is not the case with [A]. On the basis of [DJ the writer of this comment and his coworker have analysed lossy planar structures. The results were presented first at the U.R.S.I. conference in 1988 [FJ, that means two years before Wu and Vahldieck have gi ven the conference paper [BJ . In our MTT-paper [CJ, we have used nonequidistant discretization, too, which was published well before in some well known articles. Thus, this could not be a reason for a repeated description by the above authors. 2) Another statement is: "In contrast to [ l 6J ([CJ), the analysis presented here includes also higher-order modes .. .". Correct is that the authors have numerically calculated results for higher order modes. But the analysis procedure [CJ, [EJ always contains higher order modes. Thus, also this point is not a reason to repeat the algorithm.
0018-9480/94$04.00 © 1994 IEEE
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 10, OCTOBER 1994
3) "Now also half-open structures can be simulated . .. " Half-open structures were incorporated in the MoL algorithm from the beginning on. If the general solution for a layer is known (the results have a transm ission-line like form, see [E]), the limit of infinite thickness can be obtained by everyone in such a si mple manner that it is not worth mentioning as a motivation point for a report. Therefore even in the case of modelling open structures [G, HJ (two-dimensional case!) we have not reported the form ul a.-A generalization for cylindrical structures employing Bessel fu nctions has been presented in [IJ , however. 4) "The matrix elements in (19) can be easily derived analytically ... " The given result is not an end version and therefore not helpful for the user. Elaborated formulas are published before in [EJ. There the (30) or (35) gives the results for homogeneous and ( 138) for inhomogeneous layers (The matrix p in this can be replaced by 81 ). In [EJ a detailed recurrence algorithm for the admittance in a layer is given for the calculation of the field from one layer to the other. 5) The authors report that they have difficulties with numerical instability. Thi s point is much more problematic in the field of integrated optics because the wavelength is of the order of the dimensions. Such problems can always be solved by carefully rewriting the form ul as in a useful form. With the solutions mentioned in the foregoing point it is possible to overcome the problem under consideration by rewriting the admittance matrix for a layer as y ~k)
= y~k)(y\k)
[EJ R. Pregla and W. Pascher, "The method of lines," in T. ltoh, (Ed.), Nume rical Techniques fo r Microwave and Millim eter Wave Passive Structures. New York, Wiley, 1989. pp. 38 1-446. [FJ R. Pregl a and F. J. Schmtickle, "The method of lines for the analysis of lossy millimeter wave structures," Kleinheubacher Berichte Band vol. 32U. R.S.I. Conj, Klei nheubach, Germany, Oct. 1988, pp. 261-270. [G] A. Dreher and R. Pregla, "Analysis of microstrip structures with an inhomogeneous dielectric layer in an unbounded region," IEE Electron. Lett., vol. 28, no. 32, pp. 2 133-2 134, 1992. [HJ _ _ ,"Analysis of radiating planar resonators with the method of Jines," Proc. IEEE-MTT Symp., Albuquerque, NM , 1992, pp. 425-428. See also "Full-wave analysis of radiating plan ar resonators with the method of lines," IEEE Trans. Microwave Theory Tech., vol. 4 1, pp. 1363- 1368, 1993. [IJ W. Pascher and R. Preg la, "Vectorial analysis of bends in optical strip waveguides by the method of lines," Radio Sci., vol. 28, no. 6, pp. 1229-1233, 1993. [JJ U. Rogge, "The method of lines for the analysis of dielectric waveguides," Ph.D. di ssertation, Fem Un iv., Hagen, Germany, 1991. [KJ U. Rogge and R. Pregla, "The method of lines for the analysis of striploaded optical waveguides," J. Lightwave Tech., vol. 11 , pp. 20 15-2020, Dec. 1993. [LJ W. Pascher and R. Pregla, "Analysi s of curved optical waveguides by the vectorial method of lines," Proc. Int. Conf Integrated Optics and Optical Fibre Communicat., Pari s, pp. 237-240, 1991. [MJ U. Rogge and R. Pregla, "The method of lines for the analysis of strip- loaded optical waveguides," J. Opt. Soc. Am. B, vol. 8, no . 2, pp. 459-463, 1991.
- y ;k-1))-l y~k ) -y\ kl
where y 1 and y 2 are given in [E]. This form ul a has been used for a long time [JJ as it is self-evident that for programmin g a formula must be brought in a form that numerical difficulties do not arise. Therefore we have not published it until recentl y [KJ as we received several inquiries from people who had problems at this point. The same procedure has been used for the cylindrical case [I], [L]. Finally, outside the comment some statements should be given on the importance of the approach introduced in the MoL in 1987 [DJ. With this approach Maxwell 's divergence equation is automaticall y fu lfilled. This is one reason that the MoL does not give spurious solutions for inhomogeneous dielectric (in the general sense) waveguides. Because in [DJ the wave eq uation in the self-adjoint form was used, it was possible to incorporate the interface conditions on "Dielectric steps. Therefore the steps of the norm al field components are modelled correctly too [M] . With thi s approach even di ffused waveguides are modelled by using up to 80 or more layers [MJ . R EFERENCES
[AJ K. Wu, R. Vahldieck, J. Fikart, and K. Minkus, "The influence of finite conductor thickness and conductiv ity on fundamental and hi gher-order modes in mini ature hybrid MIC's (MHMJC' s) and MMJC's," IEEE Trans. Microwave Theory Tech., vol. 4 1, pp. 421-430, Mar. 1993. [BJ K. Wu and R. Vahldieck, "A self-consistent approach to determine loss properties in MIC/MMIC coplanar transmission lines," Proc. 3rd AsiaPac. Microwave Conf , Tokyo, Japan, Sept. 18-2 1, 1990, pp. 823-824. [CJ F. J. Schmtickle and R. Pregla, "The method of lines for the analysis of lossy wavegu ides," IEEE Trans. Microwave Theory Tech., vol. 38, pp. 1473-1479, Oct. 1990. [DJ R. Pregla, M. Koch, and W. Pascher, "Analysis of hybrid waveguide structures consisting of mi crostrips and dielectric waveguides," Proc. 17th Europ. Microwave Conj. , Rome, Ital y 1987, pp. 927-932.
Reply to Comments on "The Influence of Finite Cortductor Thickness and Conductivity on Fundamental and Higher-Order Modes in Miniature Hybrid MIC's (MHMIC's) and MMIC's" R. Vahldieck and K. Wu
Pregla's comments to papers [15] and the above implies that we have used the same analytical fo rmul ati on fo r the method of lines (MoL) as in his paper published at the 17th EuMc in 1987 ([2 1J in the above paper), and that we have referenced this work onl y in the appendix. This is not true and it appears that Pregla has not read the above paper carefully .
Response to I First of all, under "II. Method of Analysis" on p. 422 of the above paper, we have clearly pointed out the di fferences of our paper to the one in question ([21 J or [D] in hi s comments). We have further explained in the appendi x on p. 428 that because we have used nonequidi stant discreti zation (in contras t to [21 J in whi ch the transmi ssion line cross-section was discretized by equidistant lines), it is not possible to use the same expression as in [2 1] to calcul ate the value of fr to be inserted in[