Idea Transcript
~IEEE
TRAN SACTI 0 NS
ON
MICROWAVE THEORY AND TECHNIQUES A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY
AUGUST 1994
VOLUME 42
NUMBER 8
IETMAB
(ISSN 0018-9480)
PAPERS Analysis and Design of a Circular Disk 3dB Coupler .................................. M. E. Bialkowski and S. T. Je/lett Capacitance Computations in a Multi layered Dielectric Medium Using Closed-Form Spatial Green's Functions ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. S. Oh, D. Kuznetsm 1, and 1. E. Schutt-Aine Review of Printed Marchand and Double Y Baluns: Characteristics and Applications ...... V. Trifunovic and B. Jokanovic Accurate Measurement of Q-Factors of Isolated Dielectric Resonators ....................................... . ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. K. Mongia, C. L. Larose, S. R. Mishra, and P. Bhartia The Input Impedance of a Coaxial Line Fed Probe in a Cylindrical Waveguide ............ W. W. S. Lee and E. K. N. Yung Dyadic Green's Functions for Conductor-Backed Layered Structures Excited by Arbitrary Tridimensional Sources ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Bernardi and R. Cicchetti Rigorous Analysis of Shielded Cylindrical Dielectric Resonators by Dyadic Green's Functions ......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Sauviac, P. Guillot, and H. Baudrand Accurate Computation of the Electrostatic Charge Distribution on Shielding Plates of SAW-Transducers .............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. F. Molisch and F. 1. Seifert An Implementation of the Vectorial Finite Element Analysis of Anisotropic Waveguides Through a General-Purpose PDE Software ....... . ....... .. . . .. .. ............. .. ....................... S. R. Cvetkovic, A. P. Zhao, and M. Punjani Absorbing Boundary Conditions Applied to Model Wave Propagation in Microwave Integrated-Circuits .......... 1. Fang FD-TD Modeling of Digital Signal Propagation in 3-D Circuits with Passive and Active Loads .... ... ... .. ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Piker-May, A. Tafiove, and 1. Baron Powder Core Dielectric Channel Waveguide ...... . ............. . ......................... W. M. Bruno and W. B. Bridges Phase Shift and Loss Mechanism of Optically Excited £-Plane Electron-Hole Plasma ........... A. S. Rong and Z. L. Sun Full Wave Characterization of a Through Hole Via Using the Matrix-Penciled Moment Method ........................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.-G. Hsu and R.-B. Wu Coupling Between Different Leaky-Mode Types in Stub-Loaded Leaky Waveguides ................ ... ................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Shigesawa, M. Tsuji, P. Lampariello, F. Frezza, and A. A. Oliner A Modified Finite Element Method for Analysis of Finite Periodic Structures .............. . S.-1. Chung and 1.-L. Chen Multi mode Moment Method Formulation for Waveguide Discontinuities ...................... . ..... A. K. Bhattacharyya Microwave Imaging for a Dielectric Cylinder .................................................. H.-T. Lin and Y.-W. Kiang
1437 1443 1454 1463 1468 1474 1484 1494 1499 1506 1514 1524 1533 1540 1548 1561 1567 1572
SHORT PAPERS New Formulation of Dyadic Green's Function: Applied to a Microstrip Line ..... H. How, T.-M. Fang, and C. Vittoria Reducing Solution Time in Monochromatic FDTD Waveguide Simulations ..... . ....... D. T. Prescott and N. V. Shuley (Continued on back cover)
1580 1582
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10• 8 10 20
40
60
80
I 10
2
p (mm)
Fig. 5. Comparison of approximated and exact Green' s functions in the space domain.
J] 0
J] 0
the expressions for ·+(r I ro) and ·+(r I r 0 ) are given by (16a) and (16b), shown at the bottom of the page. Here, K:.:':,j denotes the asymptotic value of K:;,, ,n,j and we have assumed that K:, ,n,j is approximated by N+ . rn,n,3
K+ m,n,J.(rv) 1
=
~ c+ ,i .ea;'.:,·,',,,,-Y ~
J.
m,n,J
=
1 2 3 4 ' ' '
(17)
i=l
1
(18) Here, qk is the unknown coefficient to be determined, and Nr is the total number of cells used to discretize conductors. sk is the surface of the kth cell, and point matching is applied at the center of the ith cell (xi , Yi , zi). The similar equation can be also obtained for 2-D problems. Assuming that all possible ground planes are held with the same potential and the ith cell is in the jth conductor, Vi is the voltage difference between the jth conductor and the ground plane; in the case of no ground plane, any one of the existing conductors can be chosen to be the ground reference to define the voltage. For the capacitance computation, Vi takes the value of 1 or 0 depending on the excitation of the jth conductor. With manipulations in y , Yi · and the terms due to the exponential approximation, the surface integration in (18) can be put into the following forms:
f1
J(x - x')2
where N:,, ,n,j is the number of exponential functions used to approximate K°!:.,n ,j· The approximated Green's function is also compared in the space domain for the 2-D case. The same structure is used as in the previous one with c 2 = c 1 = 2co, and y and y0 were 0.6 mm and 1.6 mm, respectively. A maximum number of exponentials used in the approximation was five . The exact Green's function is obtained by applying the image principle in the space domain. Fig. 5 shows the comparison. The maximum relative error was 0.0033. Finally, we note that considering the forms of (15) and (16) it can be seen that the exponentials used to approximate the Green's function in the spectral domain correspond to the weighted images in the space domain. IV. SOLUTION METHOD FOR THE INTEGRAL EQUATION
Once we obtain the approximated Green's function, the integral (1) can now be solved by the moment method. First, discretizing the surface or the cross section of each conductor with polygonal-type elements, the unknown charge density can
G(xi,Yi, zilx, y, z) ds;i=l , 2, . .. ,Nr .
Sk
1
+(yd~ y')2 + (z - z' )2 = ff Ir ~ r'I =
J
jd;'. (19a)
Similarly, for 2-D problems we have
1 in( J(x - x') 2
+ (y - y') 2 )dl'=1 ln(IP - p'l)dl' =
1
ln(P)dl'. (19b)
The evaluations of (19a) and (19b) over an arbitrary polygonal patch and a line segment are well-known and the closed-form formulas are given in (19] . Now solving the system of linear equations obtained from (18) will give the unknown charge distribution in terms of its basis function. It is important to point out that without the closed-form expression of the Green's function, the integration (19) must be performed, theoretically, an infinite number of
f 13D,+(r Ir0) = K+, oo 1 m,n,l J(x - xo) 2 + (y +Yo - 2dn) 2 + (z - zo) 2
N!.n.1
+ ~ c+,i ~ m,n , 1 - r = = = = = = = 1= = = . ==== i=l (x - xo) 2 + (y +Yo - 2dn + a!;,~, 1 ) 2 + (z - zo) 2
J{ 0 ·+(p I Po)= K:,:r::,1 ·In ( J(x -
xo) 2 + (y +Yo - 2dn) 2)
N+
+
~·' c-:;.:~, 1
·
(16a)
ln ( J(x - xo) 2 + (y +Yo - 2dn
+ a;;:.',~, 1 ) 2 ).
(16b)
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
1448
TABLE I
,.
e,.= I
Computation
.,
350 µm
Delabare et al. [4]
[ 141.38 -21.493 --0.9924] -21.491 92.951 -17.844 (pF/m) --0.9023 -17.844 87.495
[ 142.09 -21.765 --0.8920] -2 1.733 93.529 - 18.098 (pF/m) --0.8900 -18.097 87.962
e,.=3.2
lOO µ m
e,.=4.3
200 µ m
150µm
Single Microstrip (wlh=l ,Er =4)
7. 70 10· 2 ,............,.....,....,..........,.-,....,.......,....,,....,.....,....,.....,....,.....,....,.--.,-.,.-,
Fig. 7. Three conductors in a layered medium. All dimensions of conductors and spacings are identical.
7.66 10· 2 1-..................... i ..............~C;....C...... .
E'
Li;
u"
1.64 10· 7.62 10·
2
t-·············· /'··/ ···········
2
~
+············· :::±==±::::::=.:·· 1
e,.=l
2
e,.=5 7.60 10· 2 7.58 10'
2
1
t-
······''······'·······················'······················-'·······················!············ -I
~
7.56 10· 2 ...___,_..........._._~--.J...~~....1....~~-L.._,_.........J
0
20
40
60
80
2 6
14-
4
~
~
12
9
1"'-----1 15
600
e,.=11
100
Number of Basis Functions
Fig. 6. Comparison of the present method with the spectral domain approach as a function of the number of basis functions. Fig. 8. Ten conductors in a layered medium. All dimensions of conductors and spacings are identical, and all units are in micrometer.
times. For most practical problems, the time to construct the matrix of the linear equation (moment matrix) takes the major portion of the computation time. We have significantly reduced this computation time by approximating the Green' s function with a finite number of exponentials.
TABLE II 307.13
-41.2
-11.34
-6.28
-5.351
-2 18.8
-4.966
-1.385
--0.814
-41.20
319.6
-28.04
-7.79
-4.987
-5 .025 -216.95
-3.54
--0.985
--0.666
-11.34 -28.04
310.5
-24.29
-8.587
-1.366
-3 .503
-218.4
-3.18
-1.148
-7.79
-24.29
303 .5
-24.73
--0.798
--0.946
-3.164
-219.4
-3.345
-5.35 1 -4.988
-8.588
-24.73
290.5
-0.708
-0.627
-1.12
-3.325
-221.3
-6.279
V. N UMERICAL EXAMPLES
A computer program was developed based on the above algorithm, and it is capable of handling an arbitrary number of dielectric layers and conductors and designed to read mesh data from a conventional mesh generator to allow computation of the complex geometries and meshes. The algorithm is tested for both 2-D and 3-D problems, and the equivalent crossover capacitance of two orthogonal microstrip Jines is computed based on the integral equation with both 2-D and 3-D Green's functions.
A. 2-D Problems First, a simple microstrip capacitance is computed to verify the method, and the results are compared with those for the spectral domain approach (SDA) in Fig. 6. The spectral domain method used for comparison solves the integral equation iteratively in conjunction with the minimization in the boundary condition error. A more complex geometry, a three-conductor system in a layered medium, shown in Fig. 7, is also computed. The number of basis functions used was 50 for each conductor. In both cases, the maximum number of exponentials used to approximate the coefficient function n i was 7. Comparison with the results in [4] is shown in Tabl~ I. In [4], the spectral-domain Green 's function is numerically integrated to obtain the corresponding spacedomain Green's function using a Gaussian quadrature formula in conjunction with analytical asymptotic extraction.
K!
--0.729
-5.01
-1.363
--0.796
--0.709
231.7
-2.074
--0.393
--0.182
--0.134
-4.954 -216.97
-3.494
--0.944
--0.628
-2.074
232.0
-1.19
--0.255
--0.135
-1.382 -3.532
-2 18.4
-3. 157
- 1.1 2
--0.393
-1.1 9
-0.8 13 --0.982
-3 .1 74
-219.5
-3.32
--0.182
--0.255
-0.875
231.6
--0.763
-0.729
-1.145
-3.34
-221.4
--0.134
--0.135
-0.242
--0.763
231.3
-2 18.8
--0.665
231.8 -0.8752
--0.242
A ten-conductor transm1ss1on line system above a thick dielectric substrate, shown in Fig. 8, is also considered. The total number of 300 basis functions was used to represent the unknown charges, and nine exponentials were used to approximate each coefficient function of the Green 's function . Table II shows the computed results. The same structure is considered in [3] using the free-space Green 's function with the basis functions which incorporate the edge singularities of the charge near the comers of the conductors. In [3], data are obtained using 160 and 190 basis functions for conductors and dielectric interfaces, respectively. The methods used in [2] and [4] are also employed to compute the same structure in [3]. According to [3], the methods used in [2] and [4] resulted in nonphysical values, for instance, negative selfcapacitance values. The method used in [4] took the CPU times of 89611.19 s on an IBM RS-6000 station with 300 basis functions, whereas, the method in [3] took 458.67 s. On the same machine, our method took only 85.7 s of CPU time. For the final 2-D example, a multilayered stripline case, shown in Fig. 9, is also analyzed. Fifty basis functions are used to discretize each conductor. We have obtained the values
1449
OH et al.: CAPACITANCE COMPUTATIONS IN A MULTILAYERED DIELECTRIC MEDIUM
TAB LE III
Imm
0.2rnm
Er=2.5
i
•,=JO
14----1
-I
~
0.2mm
3mm
Computation
Silvester et al. [20]
Gupta et al. [21]
c,, (IF)
65.3
72
67.65
Cnorma/ized
0.359
0.386
0.363
(Cex l(Cunif • h))
2mm
-.i wz 1. . -
Fig. 9. Two conductors in a layered medium with two ground planes. Dimensions of conductors are identical.
1 I
I I
Con~u::o~I :t~ 2 _ J_ --I --fWI
.
,
I
I
I1 I
I
I
I
I Conductor 2 I Center Lines
(a) Free Space
Fig. 10.
-::-J
Geometry of the test rnicrostrip bend.
C11 = C22 = 217.07 pF/m and C12 = C21 = -107. 76 pF/m. Data obtained from the Ansoft Maxwell software are C11 = C22 = 217.65 pF/m and C1 2 = C21 = -108.24 pF/m. B. Equivalent Capacitance for a Microstrip Bend In this numerical example, we will use the 3-D Green 's function to calculate the equivalent capacitances for a rightangle bend of a microstrip line. Although the equivalent capacitance or excess capacitance due to the right-angle bend discontinuities has been well-studied [20], [21], it is considered here again to verify the present method. An accurate characterization of a microstrip bend involves the semi-infinite rnicrostrip lines which, in tum, requires different expressions for the Green' s function. However, since the effects of the discontinuity are localized, it is expected that the equivalent capacitances can still be accurately calculated by truncating the semi-infinite lines with finite lengths. Referring to Fig. 10, the excess capacitance Cex can be defined by l1
(20)
Jim
1,2
-oo
l1 ,2- l~ .2
where CT is the total lumped capacitance and Cunif is the capacitance per unit length for the uniform microstrip line. Setting V to 1 in (18), (20) can be directly written in terms of charges: l1
Jim
1 ,2
-oo
[QT - Qunif(li
+ h)].
(21)
'1 .2 -'~ .2
Here, QT is the total charge on the conductor, and Qunif is the charge per unit length for the uniform line. The excess
Plane,,, ~Ground :>.."-.'\..'\.."-.'-'-'-'-"' ~'-'
Cil
~
30
30
~
,_ z
w
u-1111 u. u. w
third order (case a)
second order
0
u
z
-
z•. e
0
t;-2111 w ..J u.
R
R
Zo
w
0::
third order (case b)
Fig. 2.
fourth order
3111
6111
9111
Equivalent circuits of Marchand baluns. Fig. 3.
simple expressions using the general relations for distributed TEM networks [20].
A. Simple Expressions for the Design of Marchand Baluns All structures in Fig. 2 have been assumed to be loss-free networks. For the passband Chebyshev response the magnitude of reflection coefficient IPI is by [20]:
IPl 2 = In the passband Mchn(() , ()c) is
IP\
Mchn((),()c)
[c:Mchn(() , ()c)J2
1 + [c:Mchn(() , ()c)J2
.
oscillates between zero and
= T2(tan()()c )Tn-2 ( tan
(1)
\P\ max if
cos()( ))
cos c
- u2 (ta n()c ) un_ 2 ( cos() ) tan() cos () c
(2)
where
Tn-2(x) = cos((n - 2)arccos(x)) is the Chebyshev polynomial of the first kind and of degree ( n - 2) n = 2, 3, 4 for the balun of the second, third and fourth order respectively Un-2(x) = sin((n - 2)arccos(x)) is the Chebyshev polynomial of the second kind and of degree ( n - 2) () is electrical length of the transmission line () c is electrical length of the transmission line at the lower end of the passband and (7r - () c ) is electrical length at the upper end of the passband which follows from the symmetry of expression (1). All networks in Fig. 2 have equal-ripple characteristics in the passband (()c, 7r - ()c), with a maximum value of \P\max = (c: 2/ (c: 2 + 1)) 112 at the ends of passband. The number of nulls, where IPI vanishes, is equal to the order of Marchand balun, while the number of maxima is one less then the order of the balun. Also, the nulls and maxima of the passband characteristic are symmetrical to () 0 = 7r /2. It should be pointed out that these networks have multiple responses which are symmetrical to () = k7r /2, for k = 1, 3, 5, ... . A typical Chebyshev passband characteristic for a third order Marchand balun is given in Fig. 3. In the design of Marchand baluns, the passband is commonly specified through the lower (!1) and upper (Ju) frequencies. The relation between ()c and ft, f u is readily
12111 15111 8 [de9J
Chebyshev passband characteristic of the third order Marchand balun.
obtained, provided that the transmission lines are nondispersive (>.. 0 / ).. 9 (f) = const.) and bearing in mind that the physical lengths of the transmission lines which form the balun are I = >.g I 4 at the center of the passband, = 7r /2, so that
eo
()
-
c -
_7r_
(3)
B+ 1
where B = fu/ ft. The equivalent circuits of the Marchand baluns given in Fig. 2 can be reduced to the generali zed equivalent circuit of Fig. 4, which corresponds to a second order Marchand balun with complex load impedances Z; 1 and Z;4 , instead of pure real impedances Zo and R. Impedances Z; 1 = R;1 + .i X ;1 and Z;4 = R; 4 + .f X; 4 are equal to the transformed impedances Zo and R through the transmission lines with characteristic impedances Z1 and Z 4 . Therefore, for the second order balun r;1 = 1;
r;4
=
r;
X;1 =
X;4
= O;
(4)
for the third order balun (case a) r;1 = 1; X;1 = O; 2 2 1 + tan () r;4 = Z4T 2 2 2{) i z 4 + r tan z2 - r2 4 X;4 = Z4 tan() 2 2 2 (); z 4 + r tan for the third order balun (case b) 1 + tan 2 () zi - 1 r; 1 = z 12 2 (); x; 1 = z1 tan() 2 (); 2 z 1 + tan z 1 + tan 2 r;4 = r; X;4 = O;
(Sa)
(Sb)
(6a) (6b)
and for the fourth order balun 2 zi - 1 2 1 + tan () (7a) r;1 = z 1 2 () ; X;1 = z1 tan() 2 () ; 2 z 1 + tan z 1 + tan 2 1 + tan 2 () z42 - r 2 r; 4 = z~r 2 (); x; = Z4 tan() . (7b) 4 z 4 + r 2 tan 2 z 42 + r 2 tan 2 () In the above expressions we have used small letters for all impedances normalized with respect to Z 0 . The normalized input impedance of an equivalent circuit in Fig. 4 is rfn + xfn where e rin
r;4Z~b tan 2 () = r;2 + ( X;4 + Zab t an ())2 4
(8a)
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 8, AUGUST 1994
-
75
a,
•
~
65
-lil
\
m
\
55
\
45
Fig. 4.
Generalized equivalent circuitof Marchand baluns. 35
and
e
= rf42+ Xi4(Xi4 + Zab tan 2B) Zab tan B -
B
(8b)
25
and the input impedances of a series open circuited line and a parallel Short circuited line are given by - j Z b COt B and J Zab tan B respectively. The characteristic impedances of transmission lines (Z1 , Z4, Zb , Zab) can be expressed in closed form using equations (8a,b) and keeping in mind that for B = B~O' Where B~o is the i-th null of (1), there is no reflection in the circuit of the Marchand balun. This means that the left-hand side impedance is equal to the right-hand side impedance for every section plane of the circuit. Thus for the circuit in Fig. 4,
15
X in
ri4 + (xi4 + Zab tan B)
ril (B~o) = Xi 1(B~ 0 )
Zb
cot
2
Fig. 5.
= xfn(B~ 0 ).
(9)
It should be noticed that only the nulls in half of the passband (0, 7r /2), can be used to obtain an independent system of nonlinear equations, due to the symmetry of the passband characteristics. Marchand Balun of Second Order: The null of the passband characteristic for this balun is given by
Z4
B~ 0 = arctan( h tan Be)·
(I 0)
Substituting r i 1 and .Til from (4) into (9) gives the following system of nonlinear equations: 2 t 2 Bi r zab an 20 = 1 (1 la) 2 2 r + z~b tan B~ 0 and
r
Zab tan B~ 0
+ z~b tan
B~ 0
=
1 = 0. cot B20
(1 lb)
------~~-
J2(r - 1) tan ( 8 :
1)
= - . (12) Zab Marchand Balun of Third Order (Case A): The null s of (1) for this balun are Bj 0 and B50 = 7r /2. Applying the method of bisection, the null Bj 0 can be found numerically. Thus, only for the Marchand balun of second order the expressions given for computing the characteristic impedances are closed form. The dependence of Bj 0 on B is shown in Fig. 5. Zb
8
24. 679
10
20.605
~
-.............. -...
6 8 10 BANDWIDTH 8: I
=
Vr
(13)
r24 + X24i i (14) - (ri4 - 1) tan 2 Bj0 rf4 + Xi4(Xi4 + Zab tan B!o) 2 Bl ( Zb = 30· (15) 1 ) 2 Zab tan ri24 + Xi4 + Zab tan B30 Using (5b), ri 4 and Xi 4 are found as 1 + tan 2 Bj 0 (16a) r ; 4 = r 1 + r tan 2 Bj 0 and r::.
= V I (1 -
T)
tan Bj 0 Bl 1 + r tan 2 30
(16b)
Marchand Balun of Third Order (Case B ): The nulls of the reflection coefficient for the "case b" balun are the same as for "case a," so the unknown impedances can be readily obtained as follows: Z4
=
Vr
(17)
r
Zab = ~ sin2 Bl (18) 30 2 r + Zab tan Bj0 2 2 1 Zb = 2 tan 2 Bl Zab tan B30 - Xil tanB30· (19) r 2 + zab 30 Using (6a), x ; 1 is given by
r
r
30.826
b= vfr;
The solution is Zab
41 . 27 6
6
while the substitution Bj 0 in (9a) and (9b) gives 112 Zab = -b+ (b 2 + c) ;
X;4
for
2
63. 711
4
Substituting B50 in expression (9a) and (9b) gives
tan 20
r
4
2
Dependence 9! 0 versus B for the third order Marchand balun.
1 tan Be ) = 0 Mchn(B 20 , Be)= T2 ( --B1
2
"'""'
8~ 0 (deg)
c-
rfn(B~o)
-,-----=-----="-~ - Zb 2
\
8: I
Xil =
Jr(r - 1) tan Bj 0 r + tan 2 B1 30
(20)
Marchand Balun of Fourth Order: For the fourth order balun, the system of nonlinear equations derived from (9) can not be solved in closed form for Z1 , Zb , Zab and Z4. It is shown in the Appendix that for this network (21)
We introduce an intuitive approximation which comes from the similarity noted between a fourth order balun with Chebyshev response and the corresponding homogeneous quarter-wave
TRIFUNOVIC AND JOKANOVIC: REVIEW OF PRJNTED MACHAND AND DOUBLE Y BALUNS
1457
TABLE I COMPARISON BETWEEN THE CHARACTERISTIC IMPEDANCES WHICH ARE FOUND USING THE "CLOSED FORM " EXPRESSIONS
(23), (25)
AND THEIR EXACT VALUES
(I 0]
FOR A MARCHAND BALUN OF FOURTH ORDER AND
"closed form" expressions
R [fl]
Zab
54.65 58.15 64.43 76.04 99.12 152.16
[fl] 511.59 400.90 323.53 273.88 249.36 254.28
51.78 53 .06 55.24 58.97 65.51 77.84
[Cloete I0] lpl (dBJ -31.90 -27.30 -22.80 -18.43 -14.17 -9.97
transformer having the same order and response. Then we can easily derive expressions for the impedances Z 1 , Zb, Zab and
~
r
= - 4- r
\
45
(22a)
\
35
4
IPI Q max
81 1
\
55
and ZQl
[fl] 510.88 401.32 324.80 275.34 250.50 254.36
\
65
-~ m
z4 _...!_
25
(22b)
'
ZQ2
15
where the index Q is used to denote the quarter-wave transformer. Equating expressions (22a) and (22b) gives
2
Fig. 6.
(23) The closed form relations for the quarter-wave transformer are [21)
µo
2vz
(V2-1)µ6
= Slll ( ~ ~ ~ ~ ) .
(24)
The unknown impedances zb and Zab can be found by solving the system of nonlinear equations which are formed by substituting for ri1, Xi1 from (7a) and rf1, xf1 from (Sa) and (8b) into (9a) and (9b). Since Zi 1 and Zi 4 are known, the system should.be solved only for the null Bl0 which is nearest to Be. Null Bl0 can be found numerically or from the plot of Fig. 6, which shows the variation of Bl 0 with B. Thus 112 2 (25a) Zab = -b + (b - c) where
(25b)
6
29.228
8
23.376
10
19.540
!'-.... ~.....__
r--
6
8
1
= -
(26a)
30
1 1+tan 2 840 1 + z 2 t an 2 ulll
- r
r
'
39.395
2 2 1 + tan Blo ri1 = z1 z2 + tan2 (;ll
tan Bl 0
30
. '
1 - z 12 ?
(26b) 1 1 + Z 12 tan- 830 Although the "closed form" expressions (23) and (25) have been based on a rather rough ass umption, the characteristic impedances we have obtained from them are in good agreement with the exact solutions given in [10), as shown in Table I. The maximum absol ute error is less than l n which can be considered negligible in balun design. x .;4
- 1·
62. 183
4
and according to (7a) and (7b)
Z1
t2 =
2
10 8ANDWIOTH 81 1
4
.,4 -
2)2 - 1· ( J2 + 1)µ6 '
8!0 [degl
Dependence OJ 0 versus B for the fourth order Marchand balun.
r·
a'~ n,~,;)+ (( ~,~,;)' +~)1/2 t1 =
"'
1
ZQ1 _a· ' ZQ2
lpl [dB] -35.00 -30.00 -25.00 -20.00 -15.00 -10.00
Z ab
51 .58 52.75 54.79 58.38 64.9 1 77.79
°'•
The approximation made is that the fourth order Marchand balun and the corresponding quarter-wave transformer with identical resistances (Zo and R) loading both ends, have equal reflection-coefficient maxima given by =
Z1 [fl]
75
Z4.
IPlmax
B = 10 : 1.
B. Numerical and Experimental Results Using the "closed form" expressions which are derived in Section II-A. we can simply calculate the impedances of Marchand baluns of second, third and fourth order for different bandwidths B. Results are given in Table II. As mentioned above, the expressions for the second and third order Marchand baluns are exact, while those for the fourth order balun are very closely to exact relations, and are applicable in a practical design . The special case of a Marchand balun of second order, which is very popular in practice, has a load impedance equal to the impedance of the short circuited balanced line, i.e., Zab = r. In this case, the bandwidth B is in range 2.2 to to 70 respectively. 2.7 for load impedances ranging 60
n
n
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
TABLE II IMPEDANCES OF MARCHAND BALUNS OF SECOND, THIRD AND FOURTH ORDER CALCULATED BY "CLOSED FORM" EXPRESSIONS FOR DIFFERENT BANDWlTH B
=
Zo son; R impedances [fl] (see Fig. 2) zb Zab
B
Z1 2
122.47 44.83 112.05 70.53 51.37 32.98 50.54 42.78 34.05 25.62 35.98 31.01 25.74 20.88 28.70 24.37 20.76 17.60 24.11 20.07
70.71 59.57 4 70.71 60.89 6 70.71 62.74 8 70.71 64.41 10 70.71 65 .74
40.82 40.83 111.54 70.89 97.32 80.88 151.59 116.89 146.83 124.44 195.15 161.26 194.28 168.79 239.50 205.21 240.82 213.44 284.15 249.07
= f u j Ji.
= 10on IPlmax Z4 70.71 83 .93 70.71 82. 12
' 70.71 79.69 70.71 77.63 70.71 76.06
balun order n
[dB] -9.54 -29.21 -41.78 -48.77 -9.54 -17.2 1 -26.19 -26.49 -9.54 -14.27 -18.17 -19.41 -9.54 - 12.95 - 13.83 -15.99 -9.54 -12.2 1 - 11.51 -14.06
2 3a 3b 4 2 3a 3b 4 2 3a 3b 4 2 3a 3b 4 2 3a 3b 4
I I -12
f-----+--+----+---+-+-----;:H
-ie
-=-s.""s=t--~-----::-e.-=-e---c=""""""""1;41s.~e
I I FREQUENCY.
GHa
Fig. 7. Theoretical (dotted line) and experimental (solid line) results of back-to-back Marchand balun with CPWFc p-CPS lines.
The microstrip-slot line Marchand balun described in [5] has a 60 n slot impedance and a measured bandwidth of 2.7: 1 (2.6-7.1 GHz) with a VSWR less than 1.5. The theoretical value calculated for the bandwidth in this case is 2.2. The coplanar waveguide-slot line Marchand balun, [14] with slot impedance of 70 n shows a 1.5: 1 (3.0-4.6 GHz) bandwidth with a VSWR less than 1.5. The theoretical value of bandwidth calculated for this case is 2. 7: 1. We have realized a Marchand balun with CPWpap-CPS lines on 0.635 mm thick alumina substrate. The characteristic impedances of the CPWFGP and CPS lines were 50 n and 72 n respectively. Measured results of back-to-back baluns are given in Fig. 7. Although, lower and upper limits of bandwidth are in good agreement with theoretical results, there is a
Fig. 8. Double Y -junctions: Microstrip-slot CPWFcp-CPS and their equivalent circuit.
line,
CPW-slot
line,
considerable discrepancy in the passband. The main cause for that is radiation which appears particularly when the electrical length of open and short circuited lines becomes greater than 45°. Generally speaking, the coplanar Marchand baluns structures (CPW-slot line, CPWpGp-CPS) show narrower bandwidth when compared to the theoretical results because radiation cannot be neglected, particularly for CPWpap -CPS baluns. Among different realizations of Marchand baluns the best characteristics have been demonstrated with microstrip-slot line baluns, even though slot line is not a TEM line, leading to some discrepancy between theoretical and measured results, namely a wider frequency bandwidth.
TRIFUNOVIC AND JOKANOVIC: REVIEW OF PRINTED MACHAND AND DOUBLE Y BALUNS
Fig. 9.
1459
Measured back-to-back insertion loss and VSWR of CPWrcp -CPS double Y baluns versus lengths of open and short circuited stubs.
III. DOUBLE Y BALUNS Double Y baluns are based on the six port double Y junction, which consists of three balanced and three unbalanced lines placed alternately around the centre of the structure. Three different realizations of the double Y junction: microstrip-slot line, CPW-slot line and CPWpGp-CPS are shown in Fig. 8 together with their equivalent circuits. In this structure, each two opposite ports are uncoupled when the junction effects can be neglected and the other four ports are matched. The signal incident on the unbalanced port will be equally divided between the other four ports. Similarly, an input signal at the balanced port will also be equally divided between the four output ports, but one pair of opposite ports will be in phase opposition to the others. In order to make a structure which works as a balun with perfect transmission between an opposite balanced and unbalanced ports, opposed pairs of lines should have reflection coefficients with opposite phases, i.e., one pair of lines should be short circuited, the other one open circuited. The electrical length from the short and open circuits to the centre of the junction should be equal fJbaihal
= f3unbalfunbal·
(27)
Three different double Y baluns, based on the different double Y junctions, have been realized and shown in Fig. 1. According to the simple theory presented in [18], double Y baluns are all-pass networks, in contrast to Marchand baluns which are band pass networks. However, both microstrip-slot line [19] and CPW-slot line [17] double Y baluns do not demonstrate characteristics of all-pass network, because the slot line open circuit is realized as a circular slot line. Being a good open circuit only in a 2-3 octave frequency range, the circular slot line is mainly responsible for the bandwidth limitation in these baluns. To avoid this imperfection, an improved double Y balun CPWpGp -CPS is realized providing almost ideal open and short circuits. As a result, a wider frequency bandwidth
is obtained. The lower edge of the frequency bandwidth is practically DC, while the maximum upper edge of the bandwidth depends on transverse dimensions of the lines forming the double Y junction. The distance from the short and open circuits to the centre of the junction should be a few times greater than the radius of the junction to minimise junction parasitic effects. Also, this balun approaches the upper edge of the bandwidth, where the electrical length of this distance becomes 45° . Thus if we want to increase the upper edge of the frequency band by 2-3 times without changing the lower limit, it is necessary to reduce the width of the gaps 3-5 times. To illustrate this conclusion three CPWpGp -CPS double Y baluns having different junction radii have been realized on 0.635 mm thick alumina substrate. Characteristics of the 50 0 CPWFGP line and the CPS which form the double Y junction are calculated according to [25] , [26] respectively. Finite ground planes of CPWFGP lines are connected by air bridges in the vicinity of the junction. The measured back-toback insertion loss and VSWR are shown in Fig. 9. The baluns realized with narrowest gaps of 50 µm have the lengths of open and short circuited stubs of 2.7 mm (model 1) and 2.0 mm (model 2). Model 3 is realized with the narrowest gaps of 20 µm and lengths of the open and short circuited stubs are 1.0 mm. The baluns have a frequency range from DC to 6, 8 and 13.5 GHz respectively with VSWR less than 1.5 and insertion loss less than 1 dB for a single balun. IV. MULTI-OCTAVE UNIPLANAR MIXERS Recently developed uniplanar mixers [16], [23] use a combination of coplanar waveguides, slot lines, and bonding wires on one side of the substrate with no metallization on or connectors to the back side required. From a manufacturing standpoints these circuits present cost effective solution for broadband mixers and can be easily adopted to monolithic integrated circuits. Using the excellent features of uniplanar double Y baluns, two different configurations of double balanced mixers are
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
~
10.
er----+----+---+----i
9
w
RF
I. 5
5. 75
I. 5
s. 75
18. 11
If
FREQUENCY.
Fig. 1 I .
IF
Fig. 10.
Layout and equivalent circuit of DBDY mixer.
proposed. Both of them show superior characteristics to other uniplanar mixers published to date. A. Double Balanced Double Y Mixer
The double balanced double Y mixer (DBDY) [24], consists of a 6 port CPW-slot line double Y junction, where the diodes are mounted, and an improved uniplanar CPW-slot line balun [ 17]. Four matched diodes are placed at the open ended ports of the junction, whil st the other two ports serve as RF and LO ports. The broadband performance and extremely high LO/RF isolation of this mixer fo ll ow mainly from the characteristics of 6-port double Y j unction. If junction effects can be neglected and matched loads are placed at four ports, there are no bandwidth limitations concern ing the isolation between the other ports and matching conditions on them. Fig. 10 illustrates the mixer layout and its equivalent circuit. It is constructed on a 0.635 mm thick 25.4 x 25.4 mm alumina substrate. RF comes to the Schottky diodes directly through CPW while the LO signal is coupled to the diodes via a slot line, which is then transformed to CPW using a novel CPWslot line double junction. The fo ur wires connected to the series capacitors are DC decoupled fro m ground and provide the hot IF path, whereas the other four wires are connected to the LO side and provide the ground return. The measured conversion loss and LO/RF isolation with a fixed IF of 0.5 GHz and I 0 dB m LO power is shown in Fig 11. In the freq uency range 1.6-10.0 GHz conversion loss is between 6.0 and 8.5 dB , LO/RF isolation is greater than 30 dB and LO/IF isolation about 20 dB. The VSWR measured at RF
10.8 CH:ir
Measured conversion loss and LO/RF isolation of DBDY mixer.
and LO ports is less than 2:1 and less than 3:1, respectively. This design offers improved LO/RF isolation and a bandwidth greater than 1:6, whilst maintaining a conversion loss similar to that achieved with other double balanced mixers in the same frequency range.
B. A Coplanar Waveguide/Coplanar Strip Double Balanced Mixer Conventional double balanced mixers use ring quads connected between two back to back baluns. This diode arrangement requires two baluns to be aligned to each other on two sides of the substrate. Thi s causes difficulties in manufacturing as well as in mounting of the diode ring. A novel uniplanar double balanced mixer is realized using a diode crossover quad and CPWpcp -CPS double Y balun. In distinction from similar mixer configurations [16], this mixer does not require any additional circuit to minimize LO leakage to RF and IF ports because of coplanar strips which are used instead of slot lines. Fig. 12 illustrates the mixer layout realized on 0.635 mm thick 25.4 x 25.4 mm alumina substrate with unmetallized back side. The substrate metallization consists of three identical CPW pep-CPS double Y baluns on each mixer port. According to the measured back to back insertion and return losses of this balun, given in Fig. 10 (model 3), it can be used in frequency range from DC to 13.5 GHz with excellent properties. RF coplanar striplines are connected directly to Alpha diode quad DMF 3965--000, while LO signal reaches the diode chip through the series beam lead capacitors. These capacitors serve three purposes: LO matching, IF and DC blocking. It is necessary to prevent the IF current from flowing into the LO balun, which is only a few degrees long at IF frequencies, and would effectively shunt these currents to ground. Since this mixer circuit does not have inherent LO/IF and RF/IF isolation, inductances of four wire lines, which connect IF balun to diode chip, isolate the IF from the RF and LO. Measured conversion loss, LO/RF and LO/IF isolations are shown in Fig. 13. The average conversion loss is 6 dB in
TRIFUNOVIC AND JOKANOVIC : REVIEW OF PRINTED MACHAND AND DOUBLE Y BALUNS
146 1
R
Fig. 14. Equivalent circuit of Marchand balun of the fourth order which represents an antimetrical network.
Fig. 12. mixer.
Layout and equivalent circuit of CPWFGp-CPS double balanced
u; 11.1 ~ _, li ;;;
7. 5
~ 5.8
\
A.
~
~
~
......... /
fa..-
i§
u
•. 5
2.1
as a circular slot line. An improved CPW GFp -CPS double
Y baluns have an impressive frequency bandwidth of a few decades . To our knowledge, these baluns demonstrate the widest frequency bandwidth of all baluns published up to now. Presented results are not the highest achievable because of the technological limitations we have had to face (the narrowest gaps we where able to realize are 20 µm). Uniplanar double balanced mixers realized using double Y baluns illustrate a possible application of these baluns m modem MIC's and MMIC's.
IS. I
VI. APPENDCX ----LO-.,
-UHi'
41>-----+-------+-----t
2. 1
8. 5
15.11
FREQUENCY. CH:a
Fig. 13. Measured conversion loss, LO/RF and LO/IF isolations of CPWFc:p -CPS double balanced mixer.
midband, with rolloff to 9 dB at band edges for LO drive of I 0 dBm and IF frequency of 200 MHz. Both LO/RF and LO/IF isolations are greater than 20 dB across the entire band. V. CONCLUSION
The different realizations of two main groups of printed baluns are discussed: Marchand baluns with a "filter" structure and double Y baluns with a "bridge" structure. Well known synthesis of Marchand baluns given by Cloete is replaced in this work with simple "closed form" relations which can be easily incorporated within CAD packages. Among different realizations of Marchand baluns, the best characteristics have been demonstrated with microstrip-slot line baluns. The coplanar Marchand balun structures (CPWslot line and CPWFGp-CPS) show narrower bandwidth when compared to the theoretical results because radiation can not be neglected, particularly for CPWFGp-CPS baluns. Double Y baluns as all pass networks have potentially superior frequency bandwidth when compared to Marchand baluns. However, both microstrip-slot line and CPW-slot line double Y baluns do not demonstrate characteristics of all pass networks, because the slot line open circuit is realized
According to [22], when Chebyshev functions are used in the design of lossless networks with resistance loading at the both ends, the resulting networks will always be either symmetrical or anti metrical. Chebyshev functions of odd order will yield symmetrical networks, and those of even order will yield antimetrical networks . The equivalent circuit of a Marchand balun of the fourth order, which represents an antimetrical network, is shown in Fig. 14. A structure which is antimetrical about its middle should satisfy the following relation at all frequencies [21] : (28) where ZA and Z 8 are the impedances at the middle of antimetrical network and R~ is real positive constant. In [27] , this result was proved for quarter-wave impedance transformers . Using the notation of Fig. 14, we have
t:~ (}) -Xi4 + jri4 Zab tan 8 - - - - - - - - -
ZA = r i l
ZB
=
+ J( xil
(29)
-
r i4
(30)
+ j(xi4 + Zab tan B)
where the small letters have been used for the impedances normalized with respect to Z 0 . Substituting values ZA and z 8 from (29) and (30) into (28) gives ( r i1 -
2
R1i 7.2
i4
r;4
+
.2
)
+ .7. ( x;1
X.;4
+ J·( ZabR~ tane
- -Z b- )
tane
-
-
2
Rh 7.2
-
·i 4
0
Xi4
+
2
)
xi4
(31)
·
Equation (31) gives the following identities
R 2h
= Z b Za b = -
.Ti 1
r}4
+ XT4 = r il rl4 + XI4
.Ti4
r;4
(
32 )
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
Dividing the last two identities gives -Xiiri 4 /riiXi 4 = 1 and substituting for r i i , X i i , ri 4 and xi 4 from (7a, b) gives = 1.
(33)
Equation (33) is valid if (34) Application of middle of the substituting for of fourth order
(28), which is valid at all frequencies, at the passband (() = 7r /2) gives zf z~ = R~ or z 1 z 4 from (34) R~ = r. For a Marchand balun at all frequencies, we therefore found (35)
Through the use of this approach, the same expression ( zaZab = r) can be deduced for a Marchand balun of second order, which is also an antimetrical network. We have derived the identical relation (12b) in Section II-A by different procedure. ACKNOWLEDGMENT
The authors would like to acknowledge the valuable help of Dr. T. M. Benson of the Department of Electrical and Electronic Engineering, The University of Nottingham in language-editing of this paper. REFERENCES [I] S. B. Cohn, "Slot line on dielectric substrate," IEEE Trans. Microwave Theory Tech. , vol. MTT-17 , pp. 768-778, Oct. 1969. [2] E. A. Mariani, el al., "Slotline characteristics," IEEE Trans. Microwave Theory Tech., vol. MTT-17, pp. 1091-1096, 1969. [3] G. H. Robinson and J. L. Allen , "Slot line application to miniature ferrite devices," IEEE Trans. Microwave Theory Tech. , vol. MTT-17, pp. 1097-1101 , 1969. [4] J.B. Knorr, "Slot line transitions," IEEE Trans. Microwave Theory Tech. , vol. MTT-22, pp. 548-554, May 1974. [5] A. Podcameni and M. L. Coimbra, "Slotline-microstrip transition on ISO/anisotropic substrate a more accurate design," Eleclronics Leu., vol. 16, no. 20, pp. 780-781 , 1980. [6] B. Schuppert, "Microstrip/slotline transitions modeling and experimental investigation ," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 1272- 1282, Aug. 1988. [7] N. Marchand, "Transmission line conversion transformers," Electronics, vol. 17, pp. 142- 145, Dec. 1944. [8] W. K. Roberts, "A new wide-band balun," Proc. IRE, vol. 45, pp. 1628- 1631, Dec. 1957. [9] G. Oltman, "The compensated balun," IEEE Trans. Microwave Theory Tech., vol. MTT-14, pp. 112-119, Mar. 1966. [IO] J. H. Cloete, "Graphs of circuit elements for the marchand balun," Microwave J., pp. 125-128, May 1981. [I l] G. J. Laughlin, "A New impedance-matched wide band balun and magic tee," IEEE Trans. Microwa ve Theory Tech., vol. MTT-24, pp. 135-141 , Mar. 1976. [12] R. Bawer and J. J. Wolfe, "A printed circuit balun for use with spiral antennas ," IRE Trans. Microwave Theory Tech. , vol. MTT-8, pp. 319-325, May 1960. [ 13] A. Axelrod and D. Lipman, "Novel planar balun feeds octave band with dipole," Microwaves RF, pp. 91-92, Aug. 1986.
[14] V. Fouand Hanna and L. Rambroz, "Broadband planar coplanar waveguide-slotline transition," 12th Europ. Microwave Conj., 1982, pp. 628-631.. [15] H. Ogawa and A. Minagawa, "Uniplanar MIC balanced multiplier-A proposed new structure for MIC's," IEEE Trans. Microwave Theory Tech., vol. MTT-35 , pp. 1363-1368, Dec 1987. [ 16] D. Cahana, "A new coplanar waveguide/slotline double balanced mixer," 1989 IEEE MTT-S Dig., pp. 967-968. [17] V. Trifunovic and B. Jokanovic, "New uniplanar balun," Eleclronics Leu., vol. 27, no . 10, pp. 813-815, May 1991. [18] V. Trifunovic and B. Jokanovic, "Four decade bandwidth uniplanar balun," Eleclronics Leu., vol. 28, No. 6, 1992., pp. 534--535. (19] B. Schiek and J. Kohler, "An improved microstrip-to-microslot transition," IEEE Trans. Microwave Theory Tech., vol. MTT-24, pp. 231-233, 1976. (20] M. C. Horton and R. J. Wenzel , "General theory and design of optimum quarter-wave TEM filters," IEEE Trans. Microwave Theory Tech., vol. MTT-13 , pp. 316-327, May 1965. [21] G. L. Matthaei , L. Young, and E. M. T. Jones, Microwave Fillers, Impedance Matching Networks, and Coupling S1ruc1ures. New York: McGraw-Hill, 1964 pp. 38-42, 272-283. [22] E. A. Guillemin, Synthesis of Passive Networks. New York: Wiley, 1957' p. 600. [23] J. M. Goutoule and J. Burgarelli, "Standard planar biasable mixer for space telecommunications," Proc. 17th European Microwave Conj, pp. 838-840, 1987. [24] V. Trifunovic and B. Jokanovic, "A new uniplanar double balanced mixer with high port to port isolation," Proc. 21th European Microwave Conj , pp. 1539-1543, 1991. [25] G. Ghione and C. U. Naldi, "Coplanar waveguides for MMIC applications effect of upper shielding, conductor backing, finite-extent ground planes, and line-to-line coupling," IEEE Trans. Microwave Theory and Tech., vol. MTT-35 , pp. 260-267, Mar. 1987. [26] G. Ghione and C. Naldi , "Analytical Formulas for Coplanar Lines in Hybrid and Monolithic MICs," Eleclronics Leu., vol. 20, No 4, pp. 179-181, 1984. [27] H. J. Riblet, "General synthesis of quarter-wave impedance transformers," IRE Trans. Microwave Theory Tech., vol. MTT-5, pp. 36-43, Jan. 1957.
Velimir Trifunovic received the B.Sc. degree in electrical engineering from the Faculty of Electrical Engineering, University of Belgrade in 1978. Since 1980, he has been a design engineer in the Microwave Division of IMTEL (former the Institute of Applied Physics). His work in recent years has been concerned with the design of various printed antennas including two-dimensional dipole arrays, wideband antennas, phased arrays, feeding and beam forming networks.
Branka Jokanovic receives the B.Sc. degree in 1977 and the M.Sc. degree in 1988 from the Faculty of Electrical Engineering, University of Belgrade. In 1978, she joined Microwave Division, IMTEL as a member of Technical Staff where she was involved with design, fabrication and characterisation of two-dimensional slotted and printed antenna arrays. Her work in recent years deals mainly with the design of microwave/mm- wave components, especially mixers. She is currently Project Manager of mm-wave links.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
1463
Accurate Measurement of Q-Factors of Isolated Dielectric Resonators R. K. Mongia, Member, IEEE, C. L. Larose, Member, IEEE, S. R. Mishra, Member, IEEE, and P. Bhartia, Fellow, IEEE
Abstract-In this paper, a set-up used to accurately measure the resonant frequencies and Q-factors of isolated dielectric resonators is described. The measured resonant frequencies and Q-factors of first five lowest order modes of two cylindrical dielectric resonators of relative permittivity 38.0 and 79.7 respectively are reported. The measured values are compared with those of rigorous numerical methods available in the literature.
I. INTRODUCTION
HE knowledge of resonant characteristics of isolated dielectric bodies is of interest for remote sensing applications. "Isolated" dielectric resonators are also potentially useful as antenna elements [1]. A number of rigorous numerical methods have been reported in the last decade for evaluating the resonant frequencies and radiation Q-factors of isolated cylindrical dielectric resonators [2]-[5]. However, to verify the accuracy of these methods, very few results on the measurements of these quantities are available in the literature. To the best of our knowledge, the results reported in [2] are the only available experimental results which give measured data on the resonant frequencies and Q-factors of the first five lowest order modes of an isolated cylindrical dielectric resonator of a specific dielectric constant and aspect ratio. These results also suffer from some limitations. For example, in the set-up reported in [2], the HEM 110 mode, which is the lowest order hybrid mode was not observed. Further, the measured value of Q-factor of the HEM 210 mode is lower than the theoretically predicted values by about 40%. In this paper, we report the results of our measurements of the resonant frequencies and Q-factors on two dielectric resonator samples having values of er = 38.0 and 79.7 respectively. The quantities have been determined by measuring the radar cross section (RCS) of the dielectric resonator samples as a function of frequency. For the measurement of the Q-factor, an algorithm different from measurement of "half-power" frequencies has been used. The algorithm used in this paper is useful in determining if the measured resonator response corresponds to that of a "single" resonator. In our set-up, all the modes predicted by theory were easily observed. It was also found that interference may exist between the neighboring modes of isolated resonators, which may cause problems in determining the value of the Q-factor
T
Manuscript received July 19, 1993; revised October 12, 1993. R. K. Mongia and P. Bhartia are with the Department of Electrical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario, Canada. C. L. Larose and S. R. Mishra are with the David Florida Laboratory, Canadian Space Agency, 3701 Carl ing Avenue, Ottawa, Ontario, Canada. IEEE Log Number 9402936.
accurately. The measured values for the Q-factors are found to be in good agreement with theoretical values reported in the literature for all except one, for which the response was found to deviate significantly from that of a "single" resonator. II. EXPERIMENTAL SET-UP
The radar cross section (RCS) of dielectric samples was measured as a function of frequency in an anechoic chamber of inside dimensions 6 m x 5 m x 4 m. The sample was placed on a tall tapered Styrofoam cylinder and was excited by a transmitting horn placed at a distance of about 1.2 m from the sample. A receiving horn was placed by the side of the transmitting horn to receive the signal scattered by the resonator. The transmitting and receiving horns were connected to two ports of a Wiltron 360 network analyzer. More details about the set-up are described in [6]. A stepped synthesized CW signal was fed to the transmitting horn . In the usual RCS measurements, the received signal in the frequency domain is first transformed into the time domain (impulse response). The time domain response is then gated to eliminate the spurious signals scattered by bodies other than the target. In our measurements, no such time gating was used. It was found that when time gating was used, a considerable error was introduced in the measurement of RCS at and around the resonant frequencies, especially for high-Q modes. This is due to the fac t that a high- Q mode keeps "ringing" for a long time after an impulse is "incident" on it. Since the RCS of a target is directly proportional to the ratio of power picked by the receiver horn to that fed to the transmitting horn, the set-up used served as a "transmission cavity" method [7] of determining the resonant frequency and Q-factor. Further, since the transmitting and receiving horns are loosely coupled to the target, owing to the large distance between them and the target, the measured resonant freq uencies and Q-factors are essentially the unloaded quantities measured in an "isolated" environment. It may be noted that if the power is coupled to the resonator using probes kept close to the resonator as used in the set-up described in [2], the condition of "isolated" environment is altered considerably, especially for the measurement of radiation Q-factor. The emphasis in this paper is, therefore, on the more accurate measurement of the radiation Q-factor. III. ALGORITHM FOR MEASUREMENT OF Q-FACTOR
The most commonly used algorithm for the measurement of the Q-factor of a resonator is to measure the "half-power"
0018-9480/94$04.00 © 1994 IEEE
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IEEE TRANSACTIONS ON MICROWAVE TH EORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
measured response corresponds to that of a "single" resonator. By computing the standard deviation of M x N values, an estimate of the deviation of the measured response from that of a "single" resonator can be obtained.
0
i==
~ 1p0
3::
0 a..
p 2 p
D
1
UJ
IV. RESULTS
~
Measurements were made on two cylindrical dielectric samples A and B. The sample A was made of Trans-Tech 8623 material . For the sample A , the values of cr and Qd (Q-factor due to dielectric loss alone) as supplied by the manufacturer were equal to 79.7, and 3932 at 3 GHz respectively. The sample B was made of Trans-Tech 8812 material , and for this sample, the values of cr and Qd were specified as equal to 38.0, and 7809 at 4.5 GHz respectively. The aspect ratio (height/diameter) of both the samples were chosen as equal to 0.438. This choice was made because for this value of aspect ratio, numerical results of a few rigorous methods [2)-[5] and experimental results [2] for a value of cr = 38 are available. The measured values of resonant frequencies and average Qfactors are shown in Table I for the resonator sample A. The measured values of Q-factor are labeled as Q tot · The corresponding quantities for the resonator sample B are given in Table II. The number of frequency points chosen on either side of the peak, the standard deviation of the Q-factor and its coefficient of variation (coefficient of variation is defined as the value of standard deviation divided by the average value) are also shown in these tables. The values of and m~ as given by (2) and (3) were chosen bet\' een 2 dB and 4 dB . The peaks of the HEMn6. TMo 16 and HEM216 modes were observed when the electric field of the incident signal was polarized parallel to the axis of the resonator, while the peaks of the TE 016 and HEM 126 modes were observed when the incident signal was polarized perpendicular to the axis of the resonator. It is seen from Table I that the value of coefficient of variation of the Q-factor is quite low for all the modes for the resonator sample A, thus suggesting that the measured values are quite accurate. However, as seen from Table II, the values of coefficient of variation are generally higher for the modes of the sample B. This is attributed to the fact that for the sample B , the values of Q-factor are lower than the corresponding values for the sample A because of its lower cr value . It is thus expected that there will be a larger interference between neighboring modes of sample B than between the modes of sample A. In Fig. 2 the measured response of the TM016 and HEM 216 modes is plotted for the resonator sample B . It can be seen that for the TM 016 mode, the measured response deviates from that of a "single" resonator. This is reflected in the high value of the coefficient of variation (10.63 ) for this mode. It was found in this case that due to interference between modes, the value of RCS did not drop by more than 8 dB in the frequency range between the peaks of HEMn 6 (res. freq. = 5.18 GHz) and TM 01 5 (res. freq. = 6.133 GHz) modes. On the other hand, for the HEM 215 mode, the measured response seems to correspond closely to that of a "single" resonator as shown in Fig. 2. This is in turn reflected in the low value of coefficient of variation obtained for this mode (1.0%). The corresponding measured response of the TM01 6 and HEM21 6
II(/)
z
z' ,
(13)
and
I ·13mn -amn -J
vI'Y mn -
1
.
12
,m -- 1 ' 2 ' ... an d n = 1 ' 2 ' . .. '
k2.
(Sb) where x'mn is the nth root of 1:-n (x) = 0. Similarly, phase constants for TMmn modes, rmn, and amn, are Xmn
m = 0, 1, ... and n = 1, 2, . . . ,
'Ymn = - - ; a
ff
Ep(r , r') = _ _.!._
d:.rin (r')mJm(r:r,nP)
Ep m=l n=l
x cos(m¢)s inh[a~n(z + l)]
_ _j_ ~ ~ d
(6a)
( ')
L.., L.., mn r amn
wµE m=On=l
and
x cos(m¢)sinh[amn(z
amn = j f3mn = Jr'/nn - k 2 ;m = o, 1, .. . and n = 1, 2, .. . ' (6b) where .'Emn is the nth root of Jm(x) = 0. Due to the plunger placed at z = -l , Fz and Az for -l < z < z' are 00
00
E¢(r , r') =
~
m=l n=l x sin(m¢) sinh[a:.rin(z
ff
+ l)];
-l < z < z', (7)
p -l < z < z' ,
d:.rin(r') oJm~r'mnP) p
x sin(m¢)sinh[a~n(z 00
+ _J_
+ 1)];
(14)
m=l n=l .
OJm(rmnP) 0
+ l)]
00
L L dmn(r')mamnJm(rmnP)
wµEp m=On=l x sin(m¢)sinh[amn(z
+ 1)];
-l < z < z',
and 00
(15)
00
.
m=On=l x cos(m ¢)cosh[amn(z
Ez(r , r') = __J_
+ l)];
-l < z < z'. (8)
00
00
LL dmn(r')r!n Jm(rmn P)
wµE m=On=l x cos(m¢)cosh[amn(z
+ l)];
-l < z < z'.
In terms of the z-directed vector potentials, E and H are 2
E = p-1- o Az _ jwµ E OpOZ
_.!._ oFz Ep OcP
1 oFz , 1+--+ zE Op jWµE
(16)
2
+ ef>-1- o Az
Similarly, the components of H in the waveguide are
jwµ Ep OcPOZ
(k A
2
2
z
o Az ) +-oz 2 '
(9)
wµE m=l n=l X sin(m¢) e-et;,,nz
and
H =
2
2
p-1- o Fz + 2_ oAz + ef>-1- o Fz jwµ EOpOZ
_ ]_ oAz µ Op
µp OcP
1
jwµ Ep OcPOZ
X --
2
+ z-._1_ (k 2 Fz + o Fz ). 2 JWµE
oz
~ 1 ( ') t 8Jm(r'mnP) H p(r ' r ') = _j_ ~ L.., L.., cmn r am 0
(IO) X
00
p
00
LL Cmn(r')mJm(rmnP)
µp m=O n=l sin(m¢)e-et=nz ;
z
> z' ,
(17)
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
.
00
00
]_ LL
Hz(r , r' ) = __ c~n(r' h' ~n lmb'mnP) wµ f. m=ln=l X sin(m¢)e-°';.,.nz; z > z' , (19) and
00
oG; ] ( = _j_ o 5(p - p') 8( ¢ - ¢ 1 )8( ( - z'). o( z' +Cl Wf. P op
00
- -µp m=Dn=l L L dmn (r')mlm(/mn P) 00
LL
Hq,(r , r' ) = _ _ ]_ d~n(r')ma~nlm(r:,,nP) wµEp m=l n=l x cos(m¢) cosh[a~n(z + l)]
(21)
Hz(r , r')
00
LL d~n(r' )'y'~nlm(r:,,nP) wµf. m=l n=l + l)];
-l
(22)
As previously mentioned, the source is a radially directed impulse source; i.e. , (23)
The electric field intensity which satisfies ( 1) with the above source is the so-called electric type dyadic Green's function G e; while its magnetic counterpart is denoted by Gh. In general, a dyadic Green 's function (12] is associated with a subscript corresponding to the orientation of the source. The source under consideration is always radially directed; therefore this subscript is om itted for clarity. Since the zaxis is a base vector in a Cartesian coordinate system, the z-component of the Helmholtz equation is uncoupled from the transverse components. By expanding the \7 2 operator in the resultant scal ar equation in cyli ndrical coordinates, one obtains
0 20; o p2
i 0
i oc;
+ p op + p2
=
2
a;
o¢2
+
0 20; oz2
+
k2Ge z
_j_ o8(p - p') 8 (¢ _ ¢') o8(z - z' ).
Wf.P
op
oz
a e]
I
zz-
6
=
_j_ o5(p - p' ) 8(¢ wEp
(24)
op
¢ ' ).
(27)
Note that, a; in the above equation is not continuous. This anomaly to the Gauss ' Jaw is due to the doublet source which arises from differentiating impulse source. Substituting (13) and (16) into the above equation and by letting 8 --+ 0, Cmn and dmn are related 00
< z < z'.
J = p8(r - r') .
-
00
L L {Cmn(r' )e-°'"rnz' -
= __]_
x sin(m¢)sinh[a~n( z
I
z z +6
dmn( r') Olm~TmnP) µ m=On=l p x cos(m¢) cosh[amn (z + l)]; -l < z < z' , 00
to ( from z' to z' + 8. Then, combine the results of these integrations, one gets
a e]
- ~ ff .
(26)
z' - 8 to z' where 8 < b.. Similarly, integrate (26) with respect (20)
00
(25)
The next step is to integrate (25) with respect to ( from
x sin(m¢)cosh[amn(z + l)]; -l < z < z' , .
oG; ] ( = _j_ o 5(p - p') 8( ¢ - ¢ ')8( ( - z'). o( z' -Cl WEp Op
On the other hand, integrating (24) with respect to z backward from z' + b. to (; where b. --+ 0 and z' < ( < z' + b., one obtains
ff
H p(r , r' ) = _ _j_ d~n (r')a~n olm~r'mnP) wµ Em=l n=l p x sin(m¢) cosh[a~n ( z + l)] 1
In accordance with the Gauss ' laws, all field quantities are continuous in a space littered by finite sources, including an impulse source. However, for an impulse source, the derivatives of this field quantities are not necessarily continuous. For the radially directed impulse source placed at z = z', the differentiations with respect to p and ¢ are continuous, but the derivative in the z-direction is discontinuous. Based on the above observations, one integrates (24) with respect to z over an infinitesimal range; i.e. from z' - b. to (; where b. --+ 0, z' - b. < ( < z' and ( is a dummy variable, one gets
dmn(r')cosh[amn(z'
+ l)]}
m=On=l x r?nn l m(/mnP) cos(m ¢) =
_!!:. o8(p - p') 8(¢ P
- ¢ ' ).
(28)
op
Applying the orthogonal property of sinusoidal functions , one of the summations in the above equation can be eliminated; i.e., 00
L {Cmn( r' )e-°'=nz' -
dmn (r')cosh[amn(z'
+ l)]}
n=l X
f.Omr?nn lm(/mn P)
= _ _!!____ 08 (P - p') cos(m¢ ' ) , 27rp op
(29)
where l·
f.Q m
= { 1/2;
for m = 0 for m> 0 ·
(30)
Again, by invoking the orthogonal property of Bessel functions, a simple relationship between Cmn and dmn is obtained
Cmn(r') e-°'=nz' - dmn(r')cosh[amn(z' + l)] 1 µ OJm (/mn P ) ( ,1,1) cos m'+' , 2 0P 27r f.OmWmnTmn
(3 1)
LEE AND YUNG: THE INPUT IMPEDANCE OF A COAXIAL LINE FED PROBE IN A CYLINDRICAL WAVEGUIDE
where Wmn is the so-called Wronskian for TM modes
Similarly, for -1 (32)
'
dmn (p
,JJ
' '+' ' z
cos(m¢')
_ _.!!:__
-
)
e
e-a:;,."(z' +I) lm(r'mnP' ) cos(m¢') x --------'--"-'--'---'------'-------'-
p'
(33)
,1.,t
,Z
(34)
8Jm(rmnP aP 27f EomWmnlmn 1 X e-°'m" sinh[amn(z' + l)]. cos(m¢')
_.!!:__
' ) _
-
- O:mn(z' +1 )8Jm(rmnP ) ( ,1.,')· Bp cos m'+' ,
)
(35)
(36)
and
(
t ,1.,t
'
· Eme - a:'mn I S!Il · h[ I z' ) = JWµ Qmn ( ZI + l)] 2 1 12 / 7f Eom Wmn I mn amn lm(r'mnP') cos(m¢') (37) x p' '
E(r) =
(38)
Substituting (35) and (37), into (11), the radial component of the electric type dyadic Green' s function becomes
'
~~
27f L
-
t·E={t·Ea;
E
w'
,..,,2
a'
Om
x ---------'--"""'-'--'-----'---'-
p'
+-J-
00
00
LL
27rW E m=On=l
X
Omn 2 EomWmnfmn
J1
sinh [a'mn( z'
z > z'.
+ l)] BJm~;nP')
J1JL
J(r) · G(r, r' ) · J (r' )ds' ds.
(43) Based on the boundary conditions given in (42), the L.H.S. of (43) is reduced to
Jla
J(r) · E a(r )ds.
Vin
E _
e-°'mn(z+l) 8Jm(rmnP) cos(m¢)
a -
8p X
(42)
(44)
higher order modes in the coaxial-line excited due to the abrupt termination are negligible. It then follows that the fields in the aperture is equal to the incident TEM modes in the coaxial line plus the reflected waves; i.e.,
+ l)]Jmh'mnP') cos(m¢' ) --------------sinh [a'mn(z' .
(41 )
If the coaxial line opening is small, we may assume that all
p
x
otherwise ,
J(r) · E(r)ds =
mn 1 mn mn e-a:;,." (z+I) lm (r'mnP) cos( m¢) m =l n=l
on sa
O;
J1
m2
L
G(r , r') · J (r')ds,
where G is the matrix of dyadic Green 's functions. For an arbitrary source system, the surface s is such chosen that all sources are tangential to the surface. With this arrangement, a volume integral is reduced to a surface integral. Applying this technique to the probe in a waveguide, the surface chosen is indicated by a dotted line shown in Figure 2. Define a unit vector t which is tangential to the surface. The boundary conditions can be expressed as
J(r) · E(r)ds =
P
J1
where Sa is the aperture of the coaxial line. Since all sources are tangential to the surface, including J , dot multiplying E by J and integrating over the entire surface, one gets
where the Wronskian for TE modes, w'mn, is
ce( r r' ) - _jwµ
(40)
Based on [1], the electric field intensity in a source free region bounded by a surface s is
2
·wµ E m e- a:;,_n (z' +I) J d'mn (PI ''+',J.,I 'Z ') = -2/ 12 / 7f Eom Wmn I mn amn lm(r'mnP') cos(m¢') p'
mn P ''+'
< z < z'.
1
Following similar procedures, d'mn and c'mn are determined as
c'
+ l )]8Jm(rmnP) Bp cos(m¢) 1
X e
- l /
(
-O:mn (z'+I)
and
Cmn ( p , '+'
Om
w' p
x smh Omn z
1
m2
E
-----'---'---'------'~---'~--'-----'---'-
. [
8Jm(rmnP 2 aP 27f EomWmnfmn
') -
L
m=l n=l
Based on (31) and (33), dmn and Cmn are I
G~ is
a'mn mn ,..,,z 1 mn x sinh[a'mn(z + l)]Jm(r'mnP) cos(m¢) 27f L
-
To determine Cmn and dmn uniquely, another equation is required. As mentioned previously, both Ep and E q, must be continuous even at the source plane where the source is located. In terms of the eigenfunctions, this relationship is
= -dmn(r' )sinh[amn(z' + l)].
< z < z' ,
Ge(r r') - _jwµ ~ ~ P
Cmn(r' )e-,,.,.z'
1471
T
In (a 0 /ai)'
(45)
and J at the aperture is equal to cos (m¢'); (39)
J _ -
- I;n
27fT '
(46)
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 8, AUGUST 1994
where the T and () are the local coordinate system of the coaxial line as illustrated in Figure 3. Base on these approximations, the integrand on the R.H.S . of (44) can be expressed as
Jla
J(r) · Ea(r)ds =
=
-Vinlin
-Z;nl;~·
(47)
s
where Z;n is the input impedance of the coaxial cable. From (43), Z;n is Z;n =
~2~
J1J1
J(r) · Gp(r , r') · J(r')ds' ds.
I( ) = T sin[k(p - a+ h)] p m sin(kh) ·
00
m=l
n=l
00
00
m=On=l
(50) where R'mn + J X'mn are the input impedance due to a TEmn mode; while Rmn + J Xmn are due to a TMmn mode. Only TE 11 propagates. Thus, all R'mn and Rmn vanish except R~ 1 . After some lengthy algebraic manipulations, resistance and reactances given in (50) are (51)-(53), shown at the bottom of the page, and
X
Fig. 2.
The surface arrangement for a probe in a waveguide.
III. RESULTS AND CONCLUDING REMARKS
(49)
Substituting the dyadic Green 's function in (39) and (40), Z;n is broken down into a series of impedances corresponding to each mode
The formulas derived in the above section are used to find the optimal length of the probe in a cylindrical waveguide and its distance from the plunger. The waveguide of concern is of radius 11 mm. At 9 GHz, only TE 11 mode propagates and the radius is 0.33>... For TE 11 mode, the characteristic impedance is 997 0 and the guide wavelength >.. 9 is 67 mm. To excite TE 11 mode in a waveguide, a linear probe driven by a 50 n RG214/U coaxial cable is employed. The inner and outer radii of the cable are 0.65 mm and 2 mm, respectively, and its relative permittivity is 18.2. For perfect matching, Z;n = 500, and two formulas are obtained w µ sin 2 (,8~ 1 l)
50 =
I
71'W11
(l h) = -2wµ amn mn ' k 2 r;,n EOmWmn Sin 2 (kh)
x {
rkh a lo By Jmbmn(a - y/k)] sin(kh - y)dy x 1 1kh X
e-mni l sinil l
x cos [m tan X
1
k:~ ~:())]
(
:Y Jmbmn (a - y/k)] sin(kh - y)dydB. (54)
This completes the derivation of a formula for determining the input impedance of a radially directed coaxial line fed probe in a cylindrical waveguide.
wµsin 2 (,8~ 1 l)
R' (l h) = 11
I 71'W11
l
'
2 71'W I
11
,8'11 r 1211
rkh J1b~1(a-y/k)]sin(kh-y)dy}2 ' (55)
lo
ka -y
0=
00
00
00
m=l
n=l
00
LL X'mn(l, h) +LL Xmn(l, h).
There are two equations and two unknowns; therefore the unknowns can be uniquely determined. It is found that l = 13.5 mm (0.201>.. 9 ) and h = 7.5 mm (0.225>..). To further verify the formulas, input impedances of a 7.5 mm probe with various l's are measured by a HP8510 network analyzer. Comparison of the measured input impedances and theoretical results are sketched in Figure 4. There is a slight shift between
{1kh J1[r~ 1 (a O
2
-y/k)] sin(kh -y) d } ka - y y
(51) l
x
·
(56)
m=On=l
2 {1kh J 1 [r~ 1 (a -y/k)] sin(kh -y) d } ka - y y ' sm 2(kh) 2m wµ 1kh Jmb'mn(a - ky/k)] sin(kh - y) dY 12 1 · 2(kh)
wµsin(2,8~ 1 l)
X ' (l h) = 11
,8'11111 12 Slll ' 2(kh)
x
,8'11 T 1211 Slll . 2(kh)
and
2
""
Sa
'/
(48)
In the present study, narrow probes are used. Thus, the current on the probe can be better described by a line current I, which assumes a sinusoidal distribution; i.e.,
00
r···1
_________________ :: !t__________________________________________ _
(52)
0
2
X' (l h) = mn ' / Tmn Eom wmn Slll amn 2""
1o
1kh e-°':,.na; I
9 s in 1 cos
a- y
0
[m tan- 1 ( k~~'='~
9
)]
Jmb'mn(a - y/ k)] sin(kh - y)
~~~~~~-=-~~~-'--~~-'-='--~~~~~~~~~~dyd() ,
o
ka - y
(53)
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LEE AND YUNG: THE INPUT IMPEDANCE OF A COAXIAL LINE FED PROBE IN A CYLINDRJCAL WAVEGUIDE
latter process is not needed except in cases where more than two probes are placed in a waveguide and the coupling between them is strong. REFERENCES
z
Fig. 3.
The local coordinate system of the probe.
Input Impedance (o) 60 .--------------------~
x
30
x experiment•
20
- analytlcal
x ·10
..... .X
.. . . . . ... .
-20 ~---~~---~----~----~
11
12
13
14
15
l(mm) Fig. 4.
Comparison between measured data and analytical results.
the theoretical results and experimental data. The shift is approximately 1 mm. Given the constraints in measuring the separation between the center of the probe and the plunger inside a waveguide of radius 11 mm; 1 mm is well within the tolerance in measuring l within the waveguide. The agreement between theoretical results and the measurement is considered excellent. We have no intention to fine tune our techniques in measuring the separation because the aim of this experimental study is to verify the formulas. In conclusion, an accurate formula for determining the input impedance of a coaxial line fed linear probe in a cylindrical waveguide is derived. Surprisingly, there is no coupled equations to be solved. The solution is obtained by solving the Helmholtz equation with a doublet source. Although the probe is used to excite the dominant TEn mode, the formula is, in fact, valid for all higher modes. The formula is expressed in a closed form. It is lengthy; yet it is so straight forward that a personal computer is sufficient for its implementation. Note that, the current distribution on the probe is assumed to be sinusoidal. Should the current distribution be treated as an unknown, the dyadic Green's functions derived in the present study are applicable in determining the current distribution by the method of moments [3]. In general, the
(I] R. E. Collins, Field Theory of Guided Waves. New York: McGraw-Hill, 1960, pp. 258-271. [2] J.M. Rollins and J.M. Jarem, "The input impedance of a hollow-probefed, semi-infinite rectangular waveguide," IEEE Trans. Microwave Theory Tech., vol. 37, pp. 1144-1 146, July 1989. [3] J.M . Jarem, "A multifilament method-of-moments solution for the input impedance of a probe-excited semi-i nfinite waveguide," IEEE Trans. Microwave Theory Tech., vol Mtt-35, pp. 14-19, Jan . 1987. [4] A. Helaly and J. Jarem, "Input impedance of a probe excited semiinfinite rectangular waveguide with a tuning post," in Proc. 22nd Southeastern Symp. Syst. Theory Cookeville, TN, Mar. 11-13, 1990, pp. 161-167. [5] M. J. Al-Hakkak, "Experimental investigation of the input-impedance characteristics of an antenna in a rectangular waveguide," Electron. Lett. , vol. 5, no. 21, pp. 513-514, Oct. 16, 1969. [6] A. G. Williamson and D. V. Otto, "Coaxially fed hollow cylindrical monopole in a rectangular waveguide," Electron. Lett., vol. 9, no. IO, pp. 218-220, May 17, 1973. [7] A. G. Williamson, "Coaxially fed hollow probe in a rectangular waveguide," Proc. Inst. Elec. Eng., vol. 132, part H, pp. 273-285, 1985 . [8] A.G. Williamson and D. V. Otto, "Cylindrical antenna in a rectangular waveguide driven from a coaxial line," Electron. Lett., vol. 8, no. 22, pp. 545-547, Nov. 2, 1972. [9] A. G. Williamson, "Equivalent circuit for a radial-line/coaxial- line junction," Electron. Lett. , vol. 17, no. 8, pp. 300-301, Apr. 16, 1981. [IO] _ _ , "Analysis and modelling of a coaxial-line/ rectangularwaveguide junction," Proc. Inst. Elec. Eng. , vol. 129, Pt. H, no. 5, pp. 262-270, Oct. 1982. [I I] _ _ , "Radial line/coaxial-line junctions: Analysis and equivalent circuits," Int. J. Electron., vol. 58; pp. 91-104, 1985 . [12] C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory. Scranton, PA: Educational Pubs. , 1971.
Wilson W. S. Lee was born in Hong Kong, on January 28, 1964. He received the B.Sc. and M.Phil. degrees in electronic engineering from the City Polytechnic of Hong Kong, in 1989 and 1992, respectively. From 1989 to 1992, he was a Research Assistant with the Department of Electronic Engineering at the City Pol ytechnic of Hong Kong. His research interests include antenna design, antenna measurement, engineering electromagnetics, RF circuit design , and communication system.
Edward K. N. Yung (SM'85) was born in Hong Kong. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Mississippi, University, in 1972, 1974, and 1977, respectively. After working briefly in the Electromagnetic Laboratory, University of Illinois at Urbana-Champaign, he returned to Hong Kong. He is currently a Reader in the Department of Electronic Engineering, City Polytechnic of Hong Kong. He is the Head of the Telecommunications Research Center. His research interests are antenna theory, numerical analysis and microwave engineering. He is the author of more than 70 papers.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
Dyadic Green's Functions for ConductorBacked Layered Structures Excited by Arbitrary Tridimensional Sources Paolo Bernardi, Fellow, IEEE, and Renato Cicchetti, Member, IEEE
Abstract- A general procedure for the evaluation of the dyadic Green's functions for arbitrarily oriented electric- and magneticcurrent point sources embedded in a conductor-backed layered medium is presented. In the proposed approach a suitable integro-differential vectorial basis is used to split the inhomogeneous Maxwell equations, written in a transversely-invariant form, into the TE and TM wave equations. In the spectral domain (Fourier transformed domain) these equations are reduced to the usual transmission-line like equations which are solved by means of a set of auxiliary scalar Green's functions. These functions are used to express the dyadic Green's functions in the spectral as well as in the space domain. The expressions obtained contain explicit dyadic delta terms which are required to represent the electromagnetic field in the entire space including the source regions. The general theory is used to show that in the lowfrequency range the electromagnetic field can be approximated by means of a TM (with respect to the interfaces normal) wave. In this manner an asymptotic form for the dynamic characteristic impedance for a microstrip line is obtained. Finally, in order to show the feasibility of the method, some comparisons with previous results concerning the electrical characteristics of a microstrip line are carried out. I. INTRODUCTION
UMEROUS methods based on a full-wave type theory have been developed to determine the electromagnetic field produced by arbitrary distributions of planar surface currents embedded in si ngle- or multi-layered media [1]-[10], [12]-[20]. These methods can be used to analyze the electromagnetic behavior of microstrip lines, discontinuities, patch and phased array antennas, coupled striplines in high-density circuit boards, etc. Other methods have been developed for the study of layered structures with tridimensional sources [11], [21]-[27], which allow to model and analyze the electromagnetic radiation and scattering by objects of arbitrary shape embedded in the medium, pins which are used to interconnect metal patches printed on different interfaces, coaxial feeds of microstrip antennas, etc. One of the most common approaches adopted in the cited methods is that of expressing the electric dyadic Green's function (EDGF) in terms of Hertz's vector potentials [1]-[3], whereas another approach is that based on the scalar and vector potentials [4]-[11]. As discussed in [28], Hertz's potentials can produce spurious analytical solutions in the source regions.
N
Manuscript received June 29, 1992; revi sed October 19, 1993. The authors are with the University of Rome "La Sapienza," Dipartimento di Ingegneria Elettronica, Via Eudossiana, 18-00 184 Rome, Italy. IEEE Log Number 9402938.
Moreover, as pointed out in [26], some disadvantages arise from the mentioned methods when the EDGF has to be evaluated. In [26] it is also shown that in the case of tridimensional sources, analytical and numerical difficulties appear in the evaluation of the field in the source region. In [26), as well as in [29)- [31 ], dyadic delta functions have been introduced to obtain a complete representation of the field in this region. This paper presents a new formulation for the evaluation of the dyadic Green 's functions for arbitrarily oriented electric and magnetic currents embedded in a layered dielectric medium. In the proposed approach the dyadic Green 's functions are expressed in terms of the electromagnetic field decomposed into TE and TM waves. These waves form a complete set in terms of which an arbitrary field can be represented. Each of the dyadic Green's functions is expressed by means of three uncoupled auxiliary scalar Green's functions which satisfy transmission-line like differential equations. In the general case in which both electric and magnetic sources are present only six independent scalar Green's functions have to be calculated to obtain the electromagnetic field excited in the structure. It is important to note that because the boundary conditions at the source point can be split into separate conditions for the TE and TM waves an arbitrary orientation of the point source (electric and/or magnetic) does not produce coupling in the above mentioned scalar functions. Moreover, with the proposed methodology the scalar Green ' s functions depend only on the geometrical and electrical parameters of the structure, and not on the excitation. It is therefore possible to obtain an equivalent circuital representation of the field quantities and, in this manner, the physical behavior of the field is more comprehensible and easier to grasp. The format of this paper is as follows. In Section II, we make the statement of the problem and present the evaluation of the dyadic Green' s function in the spectral as well as in the space domain. In Section III, we present the spectral equivalent circuits referring to the TE and TM waves decomposition of the electromagnetic field . Finally, in Sections IV and V some analytical and numerical applications of the formulation are presented. II. THEORY A. Statement of the Problem Consider a general layered structure consisting of N + 1 dielectric layers separated by N planar interfaces parallel to
0018-9480/94$04.00 © 1994 IEEE
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BERNARDI AND CICCHETII: DYADIC GREEN'S FUNCTIONS FOR COUNDUCTOR-BACKED LAYERED STRUCTURES
B. Evaluation of the Dyadic Green's Functions in the Spectral Domain P ( r)
r
In order to evaluate the dyadic Green' s functions we employ an invariant transverse formulation of the inhomogeneous Maxwell's equations. Firstly, the vector operator 'V, the field, and the sources are split into their transverse and axial components with respect to the direction. Then, Maxwell' s equations are separated into two groups, the first containing only the transverse components of the field and the second expressing the z-components as a function of the transverse one. In the transverse plane we choose a suitable orthogonal system with unit vectors (see appendix) given by
I I
I
L-------
,'1 "
, ,""
/
" I- - - - - - -
p' ,
"",,"
/
/ L -
-
-
-
-
z
--.....- - - , - - - - - - - -
" "" I """"
/
,, "
I
""~------o~-------
//
"" //
/
-
-
-
-
I
':::::..~ -
- -
-
-
-
/
,, "
co· µ o
/
u,v
/ /
El' µI
/
Fig. I.
N -layers grounded structure.
(8)
the xy plane of a Cartesian coordinate system as depicted in Fig. 1. The permittivity c(z) and the permeability µ(z) are expressed as c((z)) = EoEri } { µ z = µoµri
for all z E Di, with i = 1, 2, .. . N (1)
where co and µo are the vacuum permittivity and permeability, respectively, and Di is the domain of the ith layer. Assuming a time variation e jwt , to obtain the space distribution of the electromagnetic field in the planar structure we have to solve the following set of dyadic differential equations: 'V x EGe(r Ir')= -jwµ(z)H G e(r Ir')
(2)
'V x
(3)
'V x
H H
G e(r Ir') = jwc(z)EG e(r Ir')+ 18(r - r')
Gm(r Ir') = jwc(z)EGm(r Ir')
(4)
'V x EGm(r Ir')= -jwµ(z)HGm(r Ir') -18(r - r') (5)
where EGe(r I r'), H G e(r I r'), H Gm(r I r'), and EGm(r I r') are the dyadic Green's functions, r and r' the position and source vectors, 1 the unit dyadic (or idemfactor), and 8(·) the Dirac delta distribution. As regards Green's functions, the left superscript indicates the field E or H, while the right superscript refers to the type of source (electric or magnetic). The linearity and time invariance of Maxwell's equations allow us to express the electromagnetic field in terms of the dyadic Green's functions and of the electric J(·) and magnetic J m ( ·) currents by means of the following integral representations
E(r) = +
H(r) = +
(9) where 'Vt is the transverse nabla operator and ~ is the magnitude of 'Vt· The generalized unit-vectors and satisfy the orthonormal condition and together with the unit-vector form a vectorial basis which is used to express the field equations in scalar form . Using the basis u, v, z, we can express the unit dyad 1 as
u
z
1 =
uu + vv + zz = l t + zz
z
a
.
az Eu(P, z ) = -Jwµ( z )Hv(P , z) - lmv(P, z)
a
1
2
2
Hz(p ,z ) =
~( )~Jmz (p,z) )Wµ Z
( 13)
f,
1,
H
J,
H
a
1
2
: z Hu(P , z ) = jwc(z )Ev(P , z) (7)
where the integrals are extended over the volume elements dV' within which the currents are non-vanishing.
2
-~( )~Jz (p,z)+lmu (p,z) ]WE Z
G e(r Ir')· J(r')dV' Gm(r Ir' ) · Jm(r')dV'.
( 12)
~( )~Eu(p,z)- ~ ( )Jmz( p,z) )Wµ Z ) Wµ Z
- Ev(P, z ) = --.-(-)('Vt+ K (z ))Hu(P, z ) 8Z )WE Z
(6)
(11)
- Hv(P , z ) = -.-(-)('Vt+ K (z ))Eu(P, z) - l u(P, z) 8Z ]Wµ Z
-TM wave
EGm(r Ir')· Jm(r')dV'
(10)
where lt is the transverse unit dyad. Equation (10) expresses the completeness relation. The scalar differential equations of the field in the space (u, v,z) can be derived by the left dot-product of u,v, and z with the vectorial equations written in the invariant transverse form. Subsequently, the scalar equations can be split into two groups (TE and TM with respect to the direction) -TE wave
1,
EG e(r Ir') · J(r')dV'
v
Ez(p, z ) =
+ l v(P , z )
(14) (15)
-~( )~Hu (p,z)- ~( ) Jz( p,z) . ]WE Z ]W E Z ( 16)
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 8, AUGUST 1994
In (11)-(16) p represents the transverse cylindrical coordinate (r = x x+yy+ zz = p+ zz), K( z ) = ~Jcr( z )µr ( z ) the wave number, and c the speed of light in vacuum. Notice that with the choice of and as basis vectors the v-components represent the irrotational (lamellar) parts and the u-components represent the solenoidal (rotational) parts of the corresponding vectorial fields. To solve the two sets of differential equations (11)-(13) and (14)-(16), we define the two-dimensional Fourier transform pair:
u
v
= ;f(a,(J,z )
f
+oo f+oo 1/; (x,y,z )ej(Oi x+/3 y)dx dy
- oo
-oo
1/; (x, y, z ) 1
f +oo f +oo ;f(a, (3, z )e-j(a x+/3 y) da d(J
= (27r )2 - oo
- oo
(18)
and transform the two sets of equations into the spectral domain. In this way the partial differential equations are reduced to total differential equations with respect to the z-coordinate. Let us consider, at first, the differential equations for the transverse components of the TE field
d dz Eu (a,(J ,z) = -jwµ (z) Hv(a,(J,z )- lmv(a ,(J,z) (19)
d -
-d Hv(a , (3, z ) = z
_,y2 (z) J-(-) Eu (a , (3, z ) - lu(a , (3, z ) wµ z
+ -wµ (-) lmz(a , (3, z ) z (
: z Ea:v(( ,z I z')
-
w2µ(z )c(z ) =
V';
(21)
is the spectral representation of the operator + K 2 ( z ). Since the set of differential equations (14)-(16) for the TM wave can be derived by (11 )-(13) using the duality principle, the differential equations fo r the transverse components of the TM field can be obtained by duality. In the following only the TE relevant expressions will be given. The two sets of differential equations for TE and TM waves have the canonical form of the inhomogeneous transmissionline equations. To solve the mentioned equations we use a set of auxiliary scalar Green's functions whose defining equations for the TE wave case are derived by (19)-(20) as
8(z - z') (24)
d Hc -m (C I Z') -d VV O
EG~y(~, z w->O
Z
2 rv
j f3
Gv(~,z I z')
(55)
W J
z')"" _{!_ dd W
Z
Gv(~, z I z').
(56)
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
Starting from the asymptotic approximations (51)-(56) it is possible, as an example, to derive the frequency behavior of the characteristic impedance Zc of a microstrip line. As it is known, different definitions for the characteristic impedance of a microstrip line have been given [33], [34]. These definitions correspond to different frequency behaviors that coincide only when w ---+ 0. Let us consider a microstrip line oriented along the y-axis. The voltage-current definition gives
Zc(w) =
f~dh E (O, z) · z e-1 f3v(w) Ydz +w/ 2 ~ .
J_w /2 J(x). y e-J f3v (w)Ydx
(57)
where d is the total thickness of the layered substrate, w and h the width of the strip and its distance from the interface, respectively, and
the boundary condition of zero tangential electric field on the strip. The following expansion for the surface current is used
J ;(x ) =
{t I;'.~(.7:)x+f;. I~,¥n (.7:)y
for all lxl < w/ 2
1
0,
Otherwise
(61) where I;'., I:X, are the unknown coefficients, and ¢~ ( x), ¢':r. (x) the corresponding basis functions that incorporate the appropriate edge singularity. As basis functions we have used those given in [19]. Using such functions, after some typical mathematical manipulations the EFIE is reduced to the following homogeneous matrix equation
(58)
(62)
the propagation constant. In (58), ceff (w) is the effective permittivity. Taking into account that in the limit of w ---+ OJx(x) ,..., 0 (57) becomes
from which the propagation constant {Jy of the fundamental mode is obtained. In (62) Z({Jy) is the matrix obtained through the application of the moment method and I is the vector corresponding to the unknown coefficients in (61). Once obtained the propagation constant {Jy we can analyze the case of an open end microstrip. For this purpose the surface current on the strip is written as
{Jy(w) = KoJce ff(w)
Using the asymptotic approximation (56) and the boundary condition Gv(a, -d I -h) = 0 at the ground plane, we obtain where
~ Jt~" Gv(a ""'2 , -hi - h)Jy(a)da Zc(w ) ,..., J ceff(w) ~~--+-; ______ _ w->O C J_ ; 12 ly (x )dx
=
~ Zc(O).
(63)
r
is the reflection coefficient at y = 0 (end of the line),
Ji (r) and Jr (r) are the incident and reflected surface currents of the fundamental mode, respectively, and J d(r ) is the current (60)
Jceff(O)
Equation (60) expresses the low-frequency behavior of the characteristic impedance (V-I definition) of an embedded microstrip line. According to (60) the characteristic impedance increases with the frequency . This behavior is well known, but it has been previously carried out by using numerical techniques based on full-wave type analyses [4], [19], and [21], or through approximated formulas based on simplified models [35]. V . NUMERICAL RESULTS
For the purpose of comparison and in order to show the feasibility of the proposed method, the electrical characteristics of a lossless microstrip line, working up to the millimeter frequency range, are computed in this section. The electrical characteristics considered are the effective dielectric permittivity and the characteristic impedance for an infinitely long microstrip line, and the voltage reflection coefficient I' for a semi-infinite microstrip line (open-circuited). As a consequence of the hybrid nature of the propagating field all the above parameters are frequency-dependent. At the beginning we consider the case of an infinitely long y oriented microstrip line. The unknown surface current-density J i(x) on the metal strip is obtained through the Galerkin ' s moment method in the Spectral Domain (SDA) applied to solve the electric field integral equation (EFIE) which enforces
excited near the open end discontinuity [36]. To express the currents J i( r) and Jr(r), the technique followed in [37] has been used. According to this technique the expressions of the incident and reflected waves are
J i,r(r) = J i,r(x) { U [ _
(y _ 2; ) ]
x cos {Jyyf:jU [-y] sin {Jyy }
(64)
where the upper/lower sign refers to the incident/reflected wave, U(-) is the Heaviside unit-step function, and Ji,r(x) represents the transverse behavior of the incident or reflected travelling waves . As regards the current Jd (r) it has been modeled by means of a suitable number of rooftop basis functions placed near the open end discontinuity . Using the expansion (63) and solving the corresponding EFIE by means of the SDA the reflection coefficient r has been obtained. A microstrip line with a substrate permittivity er = 9. 7, thickness d = 0.65 mm, and strip width w = 0.65 mm, has been considered for the computations. The numerical results, obtained up to the millimeter frequency range, are shown in Figs. 4-6. Fig. 4 shows the frequency behavior of the effective dielectric permittivity, obtained using three ¢~(x) and four ¢'!fn(x) basis functions to represent the transverse behavior of the surface current, compared with the numerical results computed using the dispersion formula given in [38].
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BERNARDI AND CICCHETTI: DYADIC GREEN'S FUNCTIONS FOR COUNDUCTOR-BACKED LAYERED STRUCTURES
1.0
0 -5
0.9
.IQ
0.8
· 15 -20 -25
0.7
-30
0.6
-35
t-
0.5
-40 -45
0.4
-50
0.3
-55 -60
0.2
-65
0. 1
-70 -75
0
0
40
20
60
"'
0
"' ~
-80 0
100
80
"O
::r
r,i
IO
20
30
40
50
60
Frequency (GHz)
Frequency (GHz)
Fig. 4. Microstrip effective permittivity vs. frequency (er = 9.7 , w = 0.65 mm, d = 0 .65 mm). The solid line corresponds to our results, the open squares correspond to the Kirschning and Jansen dispersion formula [36).
Fig. 6. Magnitude and phase of the reflection coefficient versus freque ncy for a semi-infinite microstrip line. The solid line corresponds to our results, the dashed line to the measurement presented in [40). Microstrip characteri stics: d 0 .65 mm, w /d 1,er 9.7
=
=
=
110
100
(1)
90
g 8
80
~
"O
15.
.§
70 60 (l)P-1 (2) V-I (3)P-V
50 40 0
20
40
60
80
100
Frequency (GHz)
Fig. 5. Microstrip characteristic impedance versus frequency (er = 9.7, w = 0 .65 mm, d = 0 .65 mm). The solid lines correspond to our results, the open circles correspond to P-I definition , the open squares to V-1 definition, and the filled squares to P-V definition, as computed in [37) .
The three curves in Fig. 5 have been computed considering the three different definitions usually adopted for the characteristic impedance [34) . On the same curves the values of the characteristic impedance obtained in [39) for the same structure are shown for comparison. Finally, Fig. 6 shows for the semi-infinite microstrip line the reflection coefficient r , in magnitude and phase, compared with the experimental results presented in [40). Our results were obtained using eleven rooftop basis functions to represent the longitudinal component of the current Jd(r) . It can be observed that all the figures show a very good agreement between previous and our results. Other examples of application of the proposed method to more complex geometries can be considered and are presently under examination [41) . VI.
CONCLUSION
A transversely-invariant representation of the dyadic Green's functions for arbitrarily oriented tridimensional electric and magnetic currents embedded in a planar layered
structure has been presented. Each of the four dyadic Green 's functions has been expressed by means of three auxiliary scalar Green's functions related to the TE and TM (with respect to the interface normal) waves decomposition of the electromagnetic field . It has been shown that in the general case in which both electric and magnetic sources are present only six independent scalar Green's functions have to be calculated to obtain the electromagnetic field excited in the structure. The expressions of the dyadic Green 's functions contain explicit dyadic delta terms which are required for completeness at the source points. The asymptotic behavior in the limit of zero frequency of the electric dyadic Green's function for planar sources shows that in the low-frequency range the field can be approximated by a TM wave. This approximation has been used to derive a low-frequency asymptotic expression for the characteristic impedance (VI definition) of a microstrip line in a layered structure. The spectral equivalent circuits for the transverse and longitudinal components of the electromagnetic field have been carried out and some of their applications have been outlined. As a numerical application of the proposed method the electrical characteristics of a lossless microstrip line working up to the millimeter frequency range have been calculated showing good agreement with previously published results. Other examples of application to more complex geometries can be considered and are presently under examination. In conclusion, the proposed approach has proved to be a useful and flexible tool for the analytical and numerical analysis of any kind of layered isotropic structures working in the microwave and millimeter frequency range. APPENDIX
Using the properties of the Fourier transform we define the differential and integro-differential operators used in this paper. The operator is defined by the following inverse Fourier integral representation
fit
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
~g(x, y, z) =
F- 1 {-j(q(a, ,B, z)}
= F-
1
{-jO * g(x, y, z)
1 = --2 . 3 *g(p,z) 7rJP
(Al)
where the convolution product * acts on the transverse cylindrical coordinate x, y. The integro-differential vector operator ii, and v have the following representations
To evaluate the term F- 1 { (xa + y,B / O in (A2) and the operators (A5)- (A8) the various Fourier integrals have been expressed in terms of the corresponding Fourier-Bessel transform and solved using the integral table appearing in [42].
ACKNOWLEDGMENT
The authors are indebted to Prof. N. G. Alexopoulos for his advise and stimulating discussions during the revision phase of this paper.
vg(x, y, z ) = \1\g(x, y, z )
vvt
+ y,B/~)g(a, ,B, z)} 1 F- { (xa + y,B)/0 * g(x , y, z) ~1 * g( p,z ) - 1-.p-
= F= =
~
v
{
REFERENCES
(xa/~
27rJ p
\lt x
ug(x, y, z) =
1
zg(x, y, z) =
m'I v t
1 ~1 - . ¢2'ff) p
* g(p, z)
(A2)
(A3)
where p and J; are the polar unit-vectors. Between the above mentioned operators the following relation exists: (A4) Starting from the operator ii and v, we can define the following integro-differential dyadic operators \lt\lt
~g(p,
~
z) = ppg(p, z)
t
- (pp -
¢¢) ~ P
r g(p', z )p' dp'
lo
(AS) (\lt x z)(\lt x z) \12
~~
~
~
1
g(p, z) = c/>c/>g(p, z) +(pp - ¢)2 p
t
x 1Pg(p',z)p'dp' _
- -
\ltX\ltX l t (
g p,z
\12
)
(A6)
t
\lt(\lt x z) \12
~~
g(p, z) = - p¢g(p, z)
t
~
~
1 p
+(pc/>+ ¢p2
1p g(p', z)p'dp' 0
(A7)
(\lt x z)\lt \12
~~
g(p, z) = c/>pg(p, z)
t
~
~
1
+ (pc/>+ c/>p) 2
p
1p g(p' ' z )p'dp' 0
(A8) Using (A6) and (A 7) we obtain
\1 t \1 t
lt = ~ t
\1 t x \1 t x l t \12
(A9)
t
which expresses the transverse unit dyad l t in its longitudinal and transverse components [28].
[l] E. J. Denlinger "A frequency dependent solution for microstrip transmission lines," IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp. 30-39, Jan. 1971. [2] I. E. Rana and N. G. Alexopoulos, "Current distribution and input impedance of printed dipoles," IEEE Trans. Antennas Propagat., vol. AP-29, pp. 99-105, Jan. 1981. [3] D. R. Jackson and N. G. Alexopoulos, "Analysis of planar strip geometries in a substrate-superstrate configuration," IEEE Trans. Antennas Propagat., vol. AP-34, pp. 1430-1438, Dec. 1986. [4] T. Itoh, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures. New York: Wiley, 1989. [5] _ _ , "Analysis of microstrip resonators," IEEE Trans. Microwave Theory Tech., vol. MTT-22, pp. 946-952, Nov. 1974. [6] J. R. Mosig and F. E. Gardiol, "Analytical and numerical techniques in the Green 's function treatment of microstrip antennas and scatterers," IEE Proc., vol. 130, Pt. H, no. 2, pp. 175-182, Mar. 1983. [7] J. R. Mosig and T. K. Sarkar, "Comparison of quasi-static and exact electromagnetic fields from a horizontal electric dipole above a lossy dielectric backed by an imperfect ground plane," IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 379-387, Apr. 1986. [8] N. K. Das and D. Pozar, "A generalized spectral-domain Green 's function for multilayer dielectric substrates with application to multilayer transmission lines," IEEE Trans. Microwave Theory Tech. , vol. MTT-35, pp. 326-335, Mar. 1987. [9] K. A. Michalski "On the scalar potential of a point charge associated with a time harmonic dipole in a layered medium," IEEE Trans. Antennas Propagat., vol. AP-35 , pp. 1299-1301 , Nov. 1987. [10] J. R. Mosig "Arbitrarily shaped microstrip structures and their analysis with a mixed potential integral equation," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 314-323, Feb. 1988. [11] K. A. Michalski and D. Zheng, "Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part I: Theory," IEEE Trans. Antennas Propagat., vol. AP-38, pp. 335-344, Mar. 1990. [12] T. Itoh, "Spectral domain immittance approach for dispersion characteristics of generalized printed transmission lines," IEEE Trans. Microwave Theory Tech., vol. MTT-28, pp. 733-736, July 1980. [13] T. Itoh and W. Menzel, "A full-wave method for microstrip structures," IEEE Trans. Antennas Propagat. , vol. AP-29, pp. 63-68, Jan. 1981. [14] F. L. Mesa, R. Marques, and M. Homo "A General algorithm for computing the bidimensional spectral Green's dyad in multilayered complex bianisotropic media: The equivalent boundary method," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 1640-1649, Sept. 1985. [15] N. G. Alexopoulos, "Integrated-circuit structures on anisotropic substrates," IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp. 847-881, Oct. 1985. [16] R. H. Jansen, "The spectral domain approach for microwave integrated circuits," IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp. 1043-1056, Oct. 1985. [17] N. Fache and D. De Zutter "Rigorous full-wave space-domain solution for dispersive microstrip lines," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 731-737, Apr. 1988. [18] M. Marin, S. Barkeshli, and P.H. Pathak, "Efficient analysis of planar microstrip geometries using a closed-form asymptotic representation of the grounded dielectric slab Green 's function," IEEE Trans. Microwave Theory Tech. , vol. 37, pp. 669-679, Apr. 1989. [19] K. Uchida, T. Noda, and T. Matsunaga, "New type of spectral-domain analysis of a microstrip line," IEEE Trans. Microwave Theory Tech., vol. 37, pp. 947-952, June 1989. [20] W. Schwab and W. Menzel, "On the design of planar microwave components using multilayered structures," IEEE Trans. Microwave Theory Tech., vol. 40, pp. 67-72, Jan. 1992.
BERNARDI AND CICCHETTI: DYADIC GREEN'S FUNCTIONS FOR COUNDUCTOR-BACKED LAYERED STRUCTURES
[21] C. L. Chi and N. G. Alexopoulos, "Radiation by a probe through a substrate," IEEE Trans. Antennas Propagat. , vol. AP-34, pp. 1080-1091, Sept. 1986. [22] S. Pinhas and S. Shtrikman "Vertical current in microstrip antennas," IEEE Trans. Antennas Propagat., vol. AP-35, pp. 1285-1289, Nov. 1987. [23] L. Vegni, R. Cicchetti, and P. Capece "Spectral dyadic Green's function for planar integrated structures," IEEE Trans. Antennas Propagat., vol. AP-36, pp. 1057-1065, Aug. 1988. [24] W. C. Chew, "Some observations on the spatial and eigenfunction representations of dyadic Green's functions ," IEEE Trans. Antennas Propagat., vol. 37, pp. 1322-1327, Oct. 1989. [25] J. F. Kiang, "Integral equation solution to skin effect problem in conductor strips of finite thickness," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 452-460, Mar. 1991. [26] S. Barkeshli and P. H. Pathak, "On the dyadic Green's function for a planar multilayered dielectric/magnetic media," IEEE Trans. Microwave Theory Tech., vol. 40, pp. 128-142, Jan. 1992. [27] N. Vandenberg and L. P. B. Katehi, "Broadband vertical interconnects using slot-coupled shielded microstrip lines," IEEE Trans. Microwave Theory Tech., vol. 40, pp. 81-88, Jan. 1992. [28] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, 1971. [29] J. H. Wang, Generalized Moment Method in Electromagnetics. New York: Wiley, 1991. [30] J. Van Blade!, Singular Electromagnetic Fields and Sources Oxford: Clarendon Press, 1991. [3 l] R. E. Collin, Field Theory of Guided Waves. New York: IEEE Press, 1991. [32] K. A. Michalski, "Missing boundary conditions of electromagnetics," Electron. Lett., vol. 22, pp. 921-922, Aug. 1986. [33] B. Bianco, L. Panini, M. Parodi, and S. Ridella, "Some considerations about the frequency dependence of the characteristic impedance of uniform microstrips," IEEE Trans. Microwave Theory Tech., vol. MTI26, pp. 182-185, Mar. 1978. [34] J. R. Brews, "Characteristic impedance of microstrip lines," IEEE Trans. Microwave Theory Tech., vol. MTI-35, pp. 30-34, Jan. 1987. [35] W. J. Getsinger, "Measurement and model ing of apparent characteristic impedance of microstrip," IEEE Trans. Microwave Theory Tech., vol. MTI-31, pp. 624-632, Aug. 1983. [36] R. W. Jackson and D. Pozar, "Full-wave analysis of microstrip openend and gap discontinuities," IEEE Trans. Microwave Theory Tech., vol. MTI-33, pp. 1036-1042, Oct. 1985. [37] N. G. Alexopoulos, T. Horng, S. Wu, and H. Yang, "Full-wave spectraldomain analysis for open microstrip discontinuities of arbitrary shape including radiation and surface-wave losses," Int. J. MIMICAE, vol. 2, No. 4 pp. 224-240, 1992. [38] M. Kirschning and R. H. Jansen, "Accurate model for effective dielectric constant of microstrip with validity up to millimeter-wave frequencies," Electron. Lett., vol. 18, pp. 272-273, Mar., 1982. [39] R. K. Hoffmann, Handbook of Microwave Integrated Circuits. Norwood, MA: Artech House, 1987. [40] G. Gronau and I. Wolff, "A simple broad-band device deembedding method using an automatic network analyzer with time-domain option," IEEE Trans. Microwave Theory Tech., vol. 37, pp. 479-483, Mar. 1989. [41] P. Bernardi, R. Cicchetti, and D. M. Svaluto, "Disturbance induced in interconnecting microstrip line excited in the near-field of an electric dipole," IEEE EMC Sympos., Dallas, Aug. 9-13, 1993, pp. 346-347.
1483
[42] G. N. Watson, A Treatise on the Theory of the Bessel Functions. Cambridge, U.K. : University Press, 1962.
Paolo Bernardi (M'66-SM'73-F'93) was born in Civitavecchia, Italy, in 1936. He received the degree in electrical engineering from the University of Rome in 1960, and the "Libera Docenza" in Microwaves in 1968. Since 1961, he has been with the Department of Electronics of the University of Rome "La Sapienza," where he became Full Professor in 1976. He was the Director of the Department from 1982 to 1988. His research work has dealt with the propagation of electromagnetic waves in ferrites, microwave components, biological effects of the EM waves, and electromagnetic compatibility. He is the author of more than 100 scientific papers and numerous invited presentations at internatioal workshops and conferences. He is the Chairman of the International Union of Radio Science (URSI) Commission K "Electromagnetics in Biology and Medicine," Chairman of a Commission of the Italian National Research Council (CNR) working at the "Progetto Finalizzato" on Electromagnetic Compatibility in Electrical and Electronic Systems, and Vice-Chairman of the European Community COST Project on Biomedical Effects of Electromagnetic Radiation. Mr. Bernardi is a member of the Editorial Board for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and for Microwave and Optical Technology Letters. He is a contributor to the Editorial Board of Alta Frequenza. He is "Socio fedele" of AEI. In 1979-80, he was the Chairman of the IEEE Middle and South Italy Section, and in 1984, he was recipient of the IEEE Centennial Medal.
Renato Cicchetti (S'83-M'83) was born in Rieti, Italy, in 1957. He received "cum Laude" the degree in electronic engineeering from the University of Roma "La Sapienza" in 1983. From 1983 to 1986, he worked as an Antenna Designer at Selenia Spazio S.p.A. (now Alenia Spazio S.p.A.), Roma, Italy, where he was involved in studies on theoretical and practical aspects of antennas for space application and scattering problems. Since 1986, he has been a Researcher at the Dipartimento di Elettronica of the University of Roma "La Sapienza." His research interests include electromagnetic field theory, electromagnetic compatibility, microwave and millimeter wave integrated circuits, and antennas. Dr. Cicchetti is a member of the Italian Electrical and Electronic Society (AEI).
1484
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
Rigorous Analysis of Shielded Cylindrical Dielectric Resonators by Dyadic Green's Functions Bruno Sauviac, Philippe Guillot, and Henri Baudrand, Senior Member, IEEE
Abstract- This paper presents a rigorous approach for the calculation of resonant frequencies of a metallic cavity loaded by a dielectric resonator. Tangential fields at the air-dielectric interface are derived from dyadic Green's functions and boundary conditions are applied. Dyadic Green's identity and the boundary element method are used to solve the numerical problem. In order to validate the method, resonant frequencies are calculated for a cylindrical cavity loaded with a dielectric cylinder and compared with available results in the literature. Then resonance is studied for dielectric cylinder in a rectangular cavity. In the case of multiple dielectric resonators in the cavity, the coupling coefficient is computed with an original method based on the use of symmetries. I. INTRODUCTION
ROBLEMS involving dielectric resonators have been treated in various studies [l]-[11]. Many applications are useful in telecommunications such as filters in the 1-5 Ghz frequency range. Together with the progress of microwave integrated circuits, dielectric resonators have found wide applications because of their small size compared with empty metallic cavities. Moreover, they present other interesting features; dielectric resonators are simple structures; they have a good quality factor and a good temperature stability; ceramic dielectrics with high relative permittivity are used to design microwave filters. Dielectric rod resonators shielded with circular waveguides have been studied by making use of field continuity and Galerkin method in [l]-[3] or by current continuity [4]. In this case, fields in the cavity can have an analytical form. An approximate method using the principle of the modematching technique was proposed in [5], [6] to study a dielectric post in a rectangular waveguide. In order to model discontinuity problems with dielectric resonators, [8]-[ 11] used the finite element method. In this work a rigorous method is proposed to study an arbitrarily-shaped resonator placed in a rectangular cavity. The method is based on the second dyadic Green's identity. The tangential components of the fields at the air-dielectric interface are expressed by the use of dyadic Green's functions. This theoretical approach is combined with the boundary element technique (12], (13], [16]-(18]. The method applied here is rigorous; it does not suffer from any approximation. Moreover, in contrast with the finite-
P
(I)
Er1
Fig. 1.
General structure.
element method, the numerical problem does not require the use of powerful computers and can be easily implemented on personal computers. First, the theory used to solve the problem is exposed. It is developed for an arbitrarily shaped resonator. Then, the method is validated by the study of cylindrical cavity loaded with a dielectric rod. Next, for a circular resonator in a rectangular cavity, the variations of resonant frequencies with parameters such as position or permittivity of the dielectric rod are studied. Finally, with a judicious choice of symmetries, coupling coefficient between parallel resonators and perpendicular resonators placed in the cavity are calculated. II. FORMULATION OF PROBLEM
The general structure analysed is shown in Fig. 1. It is composed of an arbitrarily-shaped resonator placed in a rectangular metallic cavity. Two opposite sides of the resonator are in contact with the cavity metallic walls. The application of the continuity of tangential electric and magnetic fields at the interface between regions (1) and (2) (Fig. 1) yields a system of homogeneous equations. This system has solutions only for the resonant frequencies. The continuity equations are the following:
ETl Hr1
= Er2 =Er
= Hr2 = Hr.
(l)
Regions l and 2 are homogeneous medias. The use of Green's functions gives a set of two equations involving tangential fields Er 1, Er 2, HTl , Hr2 (normal vector is oriented from region (l) toward (2)):
Manuscript received November 17, 1992; revised October 4, 1993. The authors are with the Laboratoire de Microndes E.N.S.E.E.I.H.T. 2, rue Charles Camichel-31071 Toulouse Cedex-France. IEEE Log Number 9402939. 0018-9480/94$04.00 © 1994 IEEE
{i!.Tl Hr2
= 6"1.ETl + 6~Jir1 = Gh2Er2 + 6~ 1 Hr2·
(2)
1485
SA UVIAC et al.: RIGOROUS ANALYSIS OF SHIELDED CYLINDRICAL DIELECTRIC RESONATORS
Subscripts 1 and 2 refer to regions (1) and (2) (Fig. 1) and Gh 1, G~ 1 , Gh 2, G~ 2 are dyadic Green's operators (see Appendix A) defined by
Gh 1· = jwc:1 { ·Ch1(r, ro)dSo
Gh2·
J (s)
= -jwc:2 f
·Gh2(r, ro)dSo
J (s)
6~ 1 - = f
·['Y'o x Gh1(r, ro)]dSo
l cs)
=J
(s)
·["9
0
6~ 2 -
xG1i2(r, ro) ]dso.
(3)
In these equations Ghl and G~il represent the dyadic Green's functions associated with magnetic field, respectively for region (1) and (2). Dyadic Green' s functions is given by the following relation [14], [15]:
an analytical form. For this reason they have been chosen for the calculation of the dyadic Green' s functions of region 2. This choice allows to simplify expressions and to reduce the size of the system to solve. By this method, an arbitrarily shaped resonator can be studied. If the shape of the resonator is simple (rectangular or circular), particular conditions for the calculation of Green' s functions can be used to reduce the size of the final system. Moreover, if the dielectric is uniform along its hight, equations involve only simple contour integrals. One can note that it is possible to place the dielectric resonator anywhere in the cavity. Ill. NUMERICAL IMPLEMENTATION
The unknowns (Ez, Et, Hz, Ht) are chosen as the sum of the trial functions. They can be expressed in the following forms:
L Htj 9t1 (r) Hz(r) = L Hzj9zj(r ) Ht(r) =
(4)
j
where eigenfunctions Hm must satisfy the eigenvalue equation:
j
Et(r) = 'L, Et1fo(r )
(5)
The expansion of Green ' s function must be set with eigenfunctions that verify div ii n = o and rot iin =!- o (for example TE and TM modes for a rectangular cavity) and longitudinal functions (with eigenvalue Am = kfic:r[19]) that verify div Hn =I- 0 and --+ rot Hn = 0. Eigenfunctions for each region are explicited in Appendix B. Using (2), tangential fields on the discontinuity surface ( s) can be expressed under (6), shown at the bottom of the page. In formulas (6), superscripts (1) and (2) of the eigenfunctions correspond to regions (1) and (2) respectively. The unknowns of the system are the tangential fields (Ez, Et , Hz, Ht) on the surface (s). The system is solved with Galerkin's method and application of the boundary element technique. The use of this method leads to a matrix system [A][x] = 0 to solve. The resonant frequency is determined by imposing that the determinant of the matrix should be zero. In the case of an arbitrarily-shaped resonator, dyadic Green's function of region (2) will be the two-dimensional free space dyadic Green s function. In the case of a circular resonator, that is the most common case, Green s functions with electric wall conditions can have
Ht(r) =
L m
H z(r)
Am
s
E z(r)
= L E zj f zj (r ).
(7)
j
gtj , gzj , ftj, f zj are trial func tions and Htj , Hzj, Et j , Ezj denote their amplitudes. Step functions have been used as trial functions, they are explained in Appendix C. The application of the Galerkin method to (6) allows to obtain the system to be solved. The complete system of the general case is presented in Appendix D. IV. NUMERICAL RESULTS A. Resonant Frequency
In this section we present some numerical results for a cylindrical dielectric resonator shielded by a circular cavity (Fig. 2) and by a rectangular cavity (Fig. 3). 1- Case of a Cylindrical Cavity Loaded With a Dielectric Cylinder In order to validate the method, the results are compared with those of [3] . The structure analyzed is presented in Fig. 2. It is composed with a cylindrical cavity axially loaded with a concentric dielectric cylinder. In this case dyadic Green's function for region 1 is calculated via formula (4),
E;;.l*(ro)Hz(ro)
+ H~l*(ro)Et(ro)H;;.£* (ro)Ez (ro)]dSo
= L j~~)1 H$,;l (r) J.[E~l*(ro)Ht(ro) - E;;.£*(ro)H z(ro) + H$,;l *(ro)Et(ro) - H;;.£*(ro)E z(ro)]dSo m
Ht (r)
j~~)1 H;;,f (r) J.[E~l*(ro)Ht(ro) -
j
Am
s
2
= L -j~~ H~£(r) J.[E~l*(ro)Ht(ro) - E~l*(ro)Hz(ro) + H~l*(ro)Et(ro)H~f* (ro)Ez (ro)]dSo m
Am
!.[ ( s
-jwc:2 (2) ( ) (2)*( ro ) H z ( ro ) H z (r ) -_"' 0 --( Em2z)*( ro ) Ht ( ro ) - E mt 2-)- Hmt r m Am s
(2) *( ro ) E z ( ro) ] dSo + Hm(2)*( z ro ) Et ( ro ) Hmt
(6)
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
1486
0,40
~
0::
::2>,
0,30
"" c:r ";:
0,20
c: :;;"'
0,10
.
ERR=3.5% +
a/R=b/R=IO +
+
u c:
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.£:
0 ~
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ID
Cl
E
s.o
ID 0
~
2.5
2.0
I
I \I
48
~~-'-'-~~~~~........__~_.__,_~~~~.._.__,
0
5
10
15
20
25
30
f (GHz)
1.5
~~~-'-'-~~~~~~~-'-'-~~~~
0
5
10
15
20
25
30
f (GHz)
Fig. 7. Characteristic impedance Zo (f) of the microstrip line computed by different absorbing boundary conditions (ABC). Curve I (dashed line): I st order ABC with P l = ./3.2 ; Curve 2: 2nd order ABC with p 1 = ./3.2 and P2 = J2.5 ; Curve 3: 2nd order ABC with Pl = ./3.2 and P2 = J3.3 ; Curve 4: 2nd ABC with P l = v'3.2 and P2 = ./4.0.
Fig. 8. Percentage error in the voltage when the far end boundary in the y direction is terminated by the first order boundary conditions with zero a 1 and different values of Pl. Curves I to 6 correspond to Pl = J4, J3, .,/2, JI, Jo.5 and v'Q.25, respectively. The dashed curve is for the electric wall boundary.
estimated propagation speed of outgoing waves. One may need to do some analyses or precalculations to find and narrow the range of the estimated propagation speed. The procedure of determining the parameter p;' s with precalculations can be as follows: first model the wave propagation in a relatively large computation domain with a relatively coarse finite-difference mesh; find the propagation speed of waves at places where absorbing boundary conditions are to be placed; distribute the values of p;' s to cover the whole range of the estimated propagation speed of outgoing waves. Depending on the spread of the wave propagation speed, the order of the absorbing boundary condition required to guarantee sufficiently small reflection usually varies from two to four. In the above tests of absorbing boundary conditions at the far end in the .7: direction, the absorbing boundary conditions applied at the far ends in the y and z directions, which will be discussed later, are sufficiently good so that reflection errors from the end boundaries in the y and z directions are negligible.
Consider the first order Higdon 's boundary condition. This boundary condition becomes the same as the first order Engquist & Majda' s boundary condition, and it can be shown that this boundary condition is also equivalent or comparable to the first order space-time extrapolation method, the impedance boundary condition, and the first order Liao et. al' s boundary condition [10] . The only parameter that can be adjusted in this boundary condition is p 1 . It has been discussed in the last section that Higdon' s boundary condition would totally reflect pure evanescent waves no matter what value of p 1 is chosen. Fig. 8 shows percentage errors caused by reflection waves in the voltage of the microstrip line for various values of p 1 and for the electrical wall boundary. Percentage errors in Fig. 8 are expressed as a function of frequency, which are obtained by taking Fourier transform of transient voltages. Results in Fig. 8 demonstrate that reflection errors from the boundary can not be substantially reduced no matter what value of p 1 is chosen. The corresponding effective dielectric constants Ereff(f) for the first order Higdon' s boundary condition with different p 1 are drawn in Fig. 9, which shows that no accurate solution of Ereff(f) can be achieved by using the first order Higdon ' s boundary condition. Percentage errors in the voltage for the second order Higdon's boundary conditions with p 1 = J4.0 and p2 chosen to be different values are displayed in Fig. 10, which shows that the second order Higdon 's boundary conditions can not provide much improvement on the first order ones. Next, consider the absorbing boundary condition proposed in this paper, which is expressed in (1). Let us begin with the test of the first order boundary condition, and let p 1 = 0, a 1 to be different values. Percentage errors in the voltage for different values of a 1 are shown in Fig. 11. Contrary to those shown in Fig. 8, reflection waves from the boundary can be substantially reduced by properly choosing parameter a 1 , as can be clearly seen from Fig. 11. According to (6), the parameter a;'s in an absorbing boundary condition should be chosen as the attenuation rate of fields in the y direction (ay) . However, exact value of ay on a boundary surface is usually not known. Moreover, ay varies
B. Boundary Conditions at the Far End in they-Direction. Let the computation domain in the y direction be terminated at y = 2500 µm. That is, the distance from the boundary to the center of the microstrip line is 2.5 times the width (W) of the metal strip, and the number of space-steps in the y direction of the computation domain is 20. Fields near thi s boundary mostly propagate in the direction parallel to the boundary surface, and evanesce in the direction normal to the boundary surface. Let us see how Higdon's boundary condition (with a;' s equal to zero in (1)) and the boundary condition of this paper perform for this type of fields. The numerical solution, used as the reference data to find reflection errors due to various absorbing boundary conditions applied at y = 2500 µm, is computed with the outer boundary in the y direction being moved to 210 space-steps (26.25 mm) away from the center of the microstrip and term inated with a fourth order absorbing boundary condition.
FANG: ABSORBING BOUNDARY CONDITIONS APPLIED TO MODEL WAVE PROPAGATION
151 I
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3.30
>
3.25
3.5
~
3.0
Q)
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Q)
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I
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I
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/
/
/
/
1.5 1.0 0.5
~~~~~~~~~~~~~~~
0
5
10
15
20
25
30
f (GHz)
5
Fig. 9. Effective dielectric constantof the microstrip line computed with the far end boundary in they direction being terminated by the first order boundary conditions with zero a1 and different values of Pl. Curves I to 6 correspond to P1 = V'.4, /3, .,/2, JI, v'lf.5 and J0.25, respectively. The dashed curve is for the electric wall boundary.
10
15
20
25
30
f (GHz)
Fig. 11 . Percentage error in the voltage when the far end boundary in the y direction is terminated by the first order boundary conditions with zerop 1 and different values of a1 . Curves I to 6 correspond to a 1dh = 0.02, 0.04, 0.06, 0.08, 0.1 0, and 0.12 respectively. The dashed curve is for the electric wall boundary.
4.0
3.5
> .!';
.,.,g c.,"' .,~
3.0
Cl
2.5
Cl.
2.0
5
10
15
20
25
30
f (GHz)
Fig. IO. Percentage error in the voltage when the far end boundary in the y direction is terminated by the second order boundary conditions with a1 a2 0, Pl V4 and different values of p2. Curves I to 6 correspond to P2 = V'.4, /3, .,/2, JI, v'Q.5 and v'Q.25, respectively. The dashed curve is for the first order boundary condition with 1 = 0 and Pl = V'.4.
=
=
=
with position and time on the boundary surface. Precalculations are sometimes desirable to get a good idea of how fields attenuate at places where absorbing boundary conditions are to be placed. The procedure of determining parameter ai' s can be as follows: first compute the fields in a computation domain with a relatively large dimension in the y direction ; find the attenuation rate of fields at places where absorbing boundary conditions are to be placed; then set values of a/s to be in the variation range of the attenuation rate of fields. For the example of the microstrip line considered here, Fig. 12 shows the attenuation rate per space-step (adh) of the electric field at y = 2500 µm, obtained from a computation domain with a larger dimension (y = 6250 µm) in the y direction . The a dh in Fig. 12 is defined as In [ Ez(xo,yo , z)/Ez(xo,Yo + dh,z) ], and plotted as a function of the vertical space index k (equivalent to the vertical space coordinate z ( dh = 125 µm)), where x 0 is at a position along the microstrip where the peak of the pulse just passes through, Yo + dh is the y coordinate at the boundary which is 2500 µm. The values of a/s can then be picked by considering the information
provided in Fig. 12. The parameter a / s can be chosen to be position-dependent according to the variation of attenuation rate of fields on the boundary surface; but numerical tests show that satisfactory results can be achieved by choosing a/ s to be constants, which makes the numerical implementation of absorbing boundary conditions much simpler. Actually one only needs to know an approximate, instead of a precise, value of adh to construct an absorbing boundary condition which can substantially reduce the reflection error. For example, if a 1 is selected to be 1/2ay, then the reflection coefficient R 1 ~(ay-a 1 )/(ay+a 1 )=1 / 3. That is, even the estimated attenuation rate is 50% off the actual value, two thirds of the reflection error can be removed by a first order boundary condition. Results in Fig. 11 demonstrate that a substantial reduction in reflection error can be achieved when a 1 dh is a constant on the boundary surface and in the range of 0.04 to 0.12. The reflection error can be further reduced by simply increasing the order of the absorbing boundary condition. Fig. 13 shows the percentage errors in the voltage with the first to fourth order boundary conditions, and the corresponding effective dielectric constants Ereff ( f) are shown in Fig. 14. As can be seen from Fisg. 13 and 14, as the reflection error being removed from the numerical solution, the rippled curve of Ereff(f) becomes a smoothed one. C. Boundary Conditions at the Far End in the z-Direction.
The situation at the far end boundary in the z direction is similar to that at the far end boundary in the y direction, except that the fields encountered here are more dominantly evanescent waves. Let the computation domain be terminated at z = 2500 µm. The percentage errors in the voltage for different absorbing boundary conditions are shown in Fig. 15, from which the same conclusion can be obtained as that from the tests on the far end boundary in the y direction. Absorbing boundary conditions with all a/ s being set to zero have little effect in reducing reflection waves as shown by curve 2 and 3 of Fig. 15, whereas boundary operators for evanescent waves play a very important role in reducing the reflection error. The
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 8, AUGUST 1994
0.20
3.35 3.30
0.15
.J::
".
3.25
s 0.10
~
3.20
,_
~
3.15 3.10
0.05 3.05
5
0.00 0
2
4
6
8
10
10
k
Fig. 12. Attenuation rate per space-step (a dh) of the electric fi eld component E , at the far end boundary in the y direction. The curve displays a dh as a function of the vertical space-index k (equivalent to the vertical space coordinate z (dh = 125 µm)).
a2 dh = 0.04, p3 =
J3,
a 4 dh = 0.12 .
reference data used to obtain the reflection error presented in Fig. 15 is from a computation domain of 110 space-steps in the z direction and terminated by a fourth order absorbing boundary condition.
D. Stability of High Order Absorbing Boundary Conditions When high order absorbing boundary conditions with zero a ;' s are used, the following phenomenon may occur. After a pulse passes through an outer computation boundary, the field values at the boundary do not fall back to zero, instead they gradually increase with time. The increase of field values at the boundary would in tum cause the increase of fields in the entire computation domain. The higher the order of the boundary condition, the faste r the fields would grow. For the case of the microstrip line, the instability is most likely to happen at the end boundaries in the x direction if high order boundary conditions of zero a ;' s are applied. As been di scussed in [13], this type of instability is caused by computer round-off error, and can be solved by following means. (I) Use double precision to reduce computer round-off error. (2) Add a small value of a ; in some of the boundary operators
20
25
30
Fig. 14. Effective dielectric constant Er e ff (f) computed with different boundary condition s applied at the far end boundary in the y direction . Curve I: electric wall boundary; Curve 2: 1st order ABC, a 1 dh = 0.06 ; Curve 3: 2nd order ABC, a 1dh 0.06 , a2 dh 0.04; Curve 4: 3rd order ABC, a 1dh = 0.06 , a2 dh = 0.04, p3 = J3 ; Curve 5: 4th order ABC,
=
a 1dh = 0.06,
a2 dh = 0.04, p3 =
10
Fig. 13. Percentage error in the vo ltage for different boundary conditions applied at the far end boundary in the y direction. Curve I: electric wall boundary; Curve 2: lst order ABC, a 1 dh = 0.06 ; Curve 3: 2nd order ABC, a1 dh = 0.06 , a2 dh = 0.04; Curve 4: 3rd order ABC, a 1 dh = 0.06, a2 dh = 0.04, p3 = J3 ; Curve 5: 4th order ABC, a 1 dh = 0.06 ,
15 f (GHz)
12
=
J3
a4 dh = 0.12.
15
20
25
30
f (GHz)
Fig. 15 . Percentage error in the voltage for different boundary conditi ons applied at the far end boundary in the z direction. Curve I: magnetic wall boundary; Curve 2: 1st order ABC, P l = JI ; Curve 3: 2nd order ABC, Pl = JI, P2 = J2 ; Curve 4: 1st order ABC, a 1dh = 0.08 ; Curve 5: 2nd order ABC, a 1dh 0.06 ,a2 dh 0.04; Curve 6: 3rd order ABC,
=
=
a 1dh = 0 .06 , a2 dh = 0.04 , a3 dh = 0.04 , p3 =
JI.
designated for propagating waves to suppress the increase of fields with time [13] . Fig. 16 displays the waveforms of the electric field Ez located at 3.75 mm away from the end boundary in the x direction where the third order absorbing boundary conditions are applied. As can be seen from Fig. 16, the field waveform obtained with the single precision and all a;'s to be zero diverges quickly. The divergence of the field can be suppressed by adding some small values of a ; 's (a;dh is chosen to be 0.02 in this test) to some of the boundary operators, as seen from curve 3 and 4 in Fig. 16. The effect of using double precision in numerical computations can also be clearly seen from Fig. 16. In the above tests of absorbing boundary conditions at the end boundaries in the x, y and z directions, all the computations are performed in double precision. Using double precision is usually the first, the simplest and the most effective option to solve the stability problem caused by absorbing boundary conditions. If using double precision is still not enough to suppress the rising of fields (which can happen
FANG: ABSORBING BOUNDARY CON DITIONS APPLIED TO MODEL WAV E PROPAGATION
-
0.6
Incident Gaussian Pulse
0.5 0.4 w " 0.3 "O
a;
0.2
u ·;:
0.1
;;:::
t5 Ql
0.0
[ij
-0.1 -0.2 -0.3 -0.4 -0.5 0
1000
500
1500
2000
2500
3000
Time step n
Fig. 16. Electric field Ez obtained with different boundary conditions applied at the far end boundary in the x direction. The boundary condiP2 = 3 .1, tions tested are all of the third order and pi = p3 2 .9 . Curve I: single precision, cr. 1 dh cr. 2 a3 dh 0 Curve 2: double prec ision, a 1 dh cr.2 a3 dh 0 Curve 3: single pre0 .0 2 , a 2dh a3 dh O; Curve 4: single precision, cision, a 1 dh a 1dh a2 dh 0.02 , a3 dh 0.
=
=
= =
=
= =
=
=
= = =
13.2,
=
for high order absorbing boundary conditions), small values of a;'s can be assigned to some of boundary condition operators. The larger the a;'s, the more stable the numerical solution. On the other hand, since non-zero a-;'s in an absorbing boundary condition deteriorate the absorption property of the boundary condition for propagating waves, as can be seen from (6), the value of a;'s should be kept as small as possible. The typical value of adh used to suppress the instability is in the range of 0.01 to 0.03 . IV.
151 3
[2] X. Zhang and K. K. Mei, "Time-domain fini te-di ffere nce approach to the calcul ation of the frequency-dependent characteristics of microstrip discontinuities," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 1775-1787, Dec. 1988. [3] A. Taflove and M . E. Brod win , "Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwa ve Theory Tech. , vo l. MTT-23 , pp. 623-630, Aug. 1975 . [4] S. Yoshida and I. Fukai, "Transient analysis of a stripline having a corner in three-dimensional space," IEEE Trans. Microwave Theory Tech., vol. MTT-32, pp . 491-498 , May 1984. [5] B. Engquist and A. Majda, " Absorbing boundary conditions for the numerical simulati on of waves," Maih. Comp., vol. 3 1, pp. 629-65 l , 1977. [6] Z. Li ao, H. L. Wong, B. Yang, and Y. Yuan, " A transmitting boundary fo r transient wave analysis," Scienlia Sinica, Series A , vol. XXVII , no . I0, pp. I062-1076, Oct. 1984. [7] R. L. Higdon, " Absorbing boundary conditions for difference approx imations to the multi-dimensional wave equation," Math. Comput., vol. 47, no . 176, pp. 437-459, Oct. 1986. [8] J. Fang and K. K. Mei, "A super-absorbing boundary algorithm fo r solving electromagnetic problems by time-domain finite-difference method," 1988 IEEE AP-S/URSI Inl. Symp., Syracuse, NY, June 6- 10, 1988 , pp . 472-475. [9] K. K. Me i and J. Fang, "Superabsorption-A method to improve absorbing boundary conditions," IEEE Trans. Anlennas Propagal., vo l. 40, pp. 1001-1010, Sept. 1992. [IO] J. Fang, "Time Domai n Finite Difference Computation of Maxwell's Equations," Ph.D. di ssertation, Dept. Elec . Eng. Computer Science, Univ . California at Berkeley, 1989. [l l] J. Fang and K. K. Mei, "Hi gh order absorbing boundary conditions without hi gh order derivatives," 1990 IEEE AP-S/URSI Int. Symp., pp. 1368- 137 1, Dallas, Texas, May 7-1 l , 1990. [12] K. C. Gupta, R. Garg and R. Chadha, Compuier-Aided Design of Microwave Circuits . Norwood, MA : Artech House, l 98 1, pp. 60-63. [13] J. Fang, "Investigation on the stability of absorbing boundary conditions for the time-domain finite-d ifference method," 1992 IEEE AP-SIURSI Int. Symp., Chicago, IL, July 18-25, 1992, pp . 548-551.
CONCLUSION
This paper shows that, in numerical modeling of wave propagation in microwave integrated-circuits, both propagating and evanescent waves at outer boundaries of computation domains need to be absorbed to obtain accurate numerical solutions. The absorbing boundary condition presented in thi s paper can effectively absorb both propagating and evanescent waves, and results in reduced requirement of computer resources and improved accuracy of numerical solutions. R EFERENCES [ l] X. Zhang, J. Fang, K. K. Mei, and Y. Liu, "Calculation of the dispersive characteristics of microstrips by the time-domain fi nitedifference method," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 263-267 , Feb. 1988.
Jiayuan Fang (S ' 85-M'90) received the B .S.E.E. in electrical engineering from Tsinghua Un iversity , Beijing, China, and the M.S. and Ph.D. degrees in electrical engineering from the Un iversity of Califo rnia at Berkeley, in 1987 and 1989 , respectively. Since 1990, he has been an Assistant Professor at the Department of Electrical Engi neering, State Uni versity of New York at Binghamton. His research interests are numerical methods, e lectronic packaging, microwave ci rcu it components modeling, antennas , and scattering. Dr. Fang received the National Science Foundation Research Initiation Award in 1991 and the National Sc ience Foundation Young Investigator Award in 1993 .
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
FD-TD Modeling of Digital Signal Propagation in 3-D Circuits With Passive and Active Loads Melinda Piket-May, Member, IEEE, Allen Taflove, Fellow, IEEE, and John Baron, Student Member, IEEE
Abstract- Most existing computer-aided circuit design tools are limited when digital clock speeds exceed several hundred MHz. These tools may not deal effectively with the physics of UHF and microwave electromagnetic wave energy transport along metal surfaces such as ground planes or in the air away from metal paths that are common at or above this frequency range. In this paper, we discuss full-wave modeling of electronic circuits in three dimensions using the finite-difference time-domain (FDTD) solution of Maxwell's equations. Parameters such as stripline complex line impedance, propagation constant, capacitance per unit length and inductance per unit length can be easily computed as a function of frequency. We also discuss FD-TD Maxwell's equations computational modeling of lumped-circuit loads and sources in 3-D, including resistors and resistive voltage sources, capacitors, inductors, diodes, and transistors. We believe that this approach will be useful in simulating the large-signal behavior of very high-speed nonlinear analog and digital devices in the context of the full-wave time-dependent electromagnetic field.
I.
INTRODUCTION
N the past, digital logic designers have not seriously considered the electromagnetic details of their systems. Clock speeds have been low enough and logic levels high enough that electromagnetic problems have been minor. But as voltage levels drop below one volt and clock speeds increase to I 00 MHz and above, key electromagnetics issues must be addressed. For example, most existing computer-aided circuit design tools (primarily SPICE) are limited at digital clock speeds exceeding 300 MHz. These tools do not deal with the physics of UHF/microwave electromagnetic wave energy transport along metal surfaces like ground planes, or in the air away from metal paths, that are common above this frequency. Effectively, electronic digital systems develop substantial analog wave effects when clock rates are high enough, with full-wave electromagnetic field phenomena markedly affecting the propagation, cross-talk, and radiation of electronic digital pulses in typical structures such as multilayer circuit boards and multichip modules. The general theme of this paper is that a synthesis of electromagnetic fields and waves and electronic
I
Manuscript teceived January 18, 1993; revised October 11 , 1993. This work was supported in part by Cray Research, Inc., Eagan, Minnesota, and in part by Los Alamos National Laboratory, Los Alamos, NM, under a Cooperative Research and Development Agreement. M. Piket-May is with the Department of Electrical and Computer Engineering, University of Colorado, Boulder, Colorado. A. Taftove is with the Department of Electrical Engineering and Computer Science, McCormick School of Engineering, Northwestern Univers ity , Evanston, IL 60208-3118 USA. J. Baron is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305-4055 USA. IEEE Log Number 9402931.
circuit devices will eventually become necessary to design very high-speed digital systems. The specific research summarized in this paper addresses numerical modeling of 3-D digital interconnects with passive and active loads directly from Maxwell's equations. We utilize the finite-difference time-domain (FD-TD) method, a simple, robust numerical technique for direct time integration of Maxwell's equations that has become the means to model a wide variety of electromagnetic wave problems [1] . Using the basic algorithm introduced by Yee [2], FD-TD implements the space derivatives of the curl operators via finite differences in regular interleaved (dual) Cartesian space meshes for the electric and magnetic fields . Simple leapfrog time integration is employed. The literature indicates that 3-D FD-TD modeling of metallic digital interconnects has commenced. Liang et al. [3] modeled picosecond pulse propagation along co-planar waveguides above a gallium arsenide substrate. Shibata and Sano [4] modeled propagation along metal-insulator-semiconductor lines. Lam et al. [5] modeled digital sig11al propagation and radiation for VLSI packaging, and Maeda et al. [6] modeled digital pulse propagation through vias in a three-layer circuit board. Work is also being done in FD-TD modeling of passive and active devices that can either source or terminate metallic transmission lines. Two groups reported incorporation of selfconsistent models of charge transport into a 3-D FD-TD solver to model picosecond pulse generation by an optically excited gallium arsenide device: Sano and Shibata [7] with a drift-diffusion charge-transport model, and El-Ghazaly [8] with a Monte Carlo model. A third group Sui et al. [9], developed lumped-circuit models of typical passive and active circuit elements (resistors, inductors, capacitors, diodes, and transistors) incorporated into a 2-D FD-TD solver for exploring electromagnetic modeling of nanosecond-regime circuits. The work of this paper follows directly from [6] and [9]. In Section III, we illustrate the power of FD-TD modeling to deduce a classical electromagnetic compatibility problem, ground loop current flow , in what may be the most complex 3D passive circuit model so far, a module consisting of a stack of multilayer circuit boards penetrated by scores of via pins. Using a uniform space resolution of 0.004 inch, each layer, via, and pin of every circuit board was modeled. 60-million vector field unknowns were solved per modeling run, a factor of perhaps 100 times larger than the capacity of the largest SPICE or finite-element CAD tool available.
0018-9480/94$04.00 © 1994 IEEE
PIKET-MAY et al. : FD-TD MODELING OF DIGITAL SIGNAL PROPAGATION IN 3-D CIRCUITS
In Section IV, we improve and extend the work of [9] to 3-D, 1introducing a semi-implicit algorithm that permits the large-signal behavior of nonlinear circuit devices to be simulated in a numerically stable manner in the context of the FD-TD Maxwell's equations solver. We report FDTD electromagnetic wave models of the resistor, resistive voltage source, capacitor, inductor, diode, and bipolar junction transistor.
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convenience is provided by denoting the Yee finite-difference analog to the curl of H observed at E z li,j,k as
n+l/2 \7 x H . . k i,1 ,
=
Hx / ~;~{~2,k +
H /n+l /2 _ H /n+l/2 Y i+l/2,j,k Y i-1 /2,j,k -~-----~--b.x
- Hx/~;~G2,k b.y (2b)
II. FD-TD Algorithm and Extension
Then, (lb) can be rewritten very si mply as
Ezln+l
A. Basic FD-TD Formulation
i,J, k
The derivation of the basic FD-TD equations used to solve Maxwell's equations has been documented extensively [1], [2], [10] so it will not be repeated. In all of the cases to be discussed, a uniform, 3-D Cartesian Yee mesh is used with a unit cell of dimensions b.x, b.y , andb.z. Numerical stability is maintained by bounding the time step [11]. Consider Maxwell's curl Hequation, suitable for timestepping the electric field
8D \7 x H =Jc + at
(la)
where the electric conduction current is ] c O' E and the electric displacement is D = cE. Using central differencing with the standard Yee subscript/superscript notation for space and time coordinates, respectively, (la) becomes
E n+l _ zli,j,k -
+~ ( 1-~) 2€;,j,k
l
E
+ (
l
+
fi,j,k Ui,j,k 6.t
,,,,k
2< · .
)
(
8D \7 x H =Jc + at
H x 1'.'+ l /2
"J- 1/2, k
+)
(lb)
-Hx ln+ l /2
6.y
,,,+ 1/2, k
in time relative to the stored electric field, E z
In. i ,J,
k,
(3a)
J
h
6.x
-
+ h.
We assume that a lumped element is located in free-space at the fieldEz i,J, k; is z-oriented in the grid; and provides a local current density that is related to the total element current, h, as
JL = uA xuy A '
(3b)
Here, h is possibly a time-derivative, time-integral, scalar multiple, or nonlinear function of the electric potential,V = E z i,j,kb.z, developed across the element. Note that the assumed positive direction of h is +z. Then, the following modified version of (2c) suffices to specify the presence of the lumped circuit element in the electromagnetic field grid J
An important observation is that all H quant1t1es on the right-hand side of (lb) are at time step n + 1/2, centered and the
newly updated electric field, Ez /~J.k . Further, the bracketed coefficients are derived assuming that J c is also evaluated at time step n + 1/2, taking
2 Jcl".1+!/ = O'i j kEz In+!/ 2 = O'i ,j,k (Ez In· k + Ez In+!) i ,J , ' ' t,J , 2 t,J, t,J, (2a) We denote this as the "semi-implicit formulation" for the conduction current in that this current relies in part upon the updated electric field to be determined as a result of the timestepping, and yet does not result in a system of simultaneous equations. This results in a numerically stable algorithm for arbitrary positive values of O' [1], [10]. For convenience in the discussions of this paper, we shall assume that all circuit components are located in a freespace region (E = 1: 0 , O' = 0, Jc = 0) . Additional notational 1
The electric field time-stepping algorithm is now modified to allow for the addition of lumped linear and nonlinear circuit elements. The basis of this formulation, reported in [9] for 2-D problems, showed that lumped circuit elements can be accounted for in Maxwell's equations by adding a Jumped electric current density term, h, to the conduction and displacement currents on the right hand side of (la). Equation (la) now becomes
n.
H ln+ l /2 -H ln+ l /2 y i+ l /2, j,k y i- 1/2,j, k
(2c)
B. Extended FD-TD Formulation
z li,1 ,k
2f i,j, k
_!±J_
= E z Jn. + b.t \7 X Hln+l/2. i,J,k Eo i,J,k
Note added in proof. See also Tsuei et al. [16] . for an independent expansion of the work of [9] to 3-D.
In the key departure from the work of [9], we have defined in (4) the lumped current as being evaluated at the mid time step, n + 1/2, just as we did in (2a) for the conduction current. Since the lumped current is a function of the electric field at the circuit element, this requires averaging the values of E z I"."+kl t,J, andEz ~. i,1, k• an operation that yields a numerically-stable semiimplicit time-stepping algorithm. Generalization of all of this to x and y orientations of a lumped element is straightforward by permuting the coordinate subscripts of the field quantities. J
III. INTERCONNECT GEOMETRIES WITHOUT LUMPED LOADS
A. Parallel Coplanar Microstrips Preliminary studies were performed to establish the validity of FD-TD results for characteristic impedance and delay of parallel co-planar microstrips . The first studies were done by
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 8, AUGUST 1994
JO
v
Even Excitation Center Exc itation Odd Excitation
70.00
-0 :J
11 11
~
§,60 .00
___ ___ ____ ____ ___
0
/
:::;?;
v
50.00
u
c
'1------------------
0
-04QOO
v
---------------------------- ,
Q_
E 30 .00
-
(a} Characteristic impedance vs . frequency
20.00 4-nrrnTTT~rrnTTT~rrnTTTTTrrrnTTTTTT'rrn"TTT"r 0.00 2.00 4.00 6.00 8.00 10.00
freque ncy (log) (a) ,,.--.
162.0
Even Excitation Center Excitation Odd Excitation
..c
u - ~ 159.0 '-.... u
(a)
v
~ 1 56.0
.S?
v
0
I
\J
153.0
0
c
Separation
''
>..
/
150.0
/
----------
:.;::; 0 g'147.0
(b) Propagation delay
Q_
vs . frequency
0
..... Cl.. 144.0 -+------~--~---~---~ 6.00
7.00
8.00
9.00
10.00
frequency (log) (b)
Fig. 1. FD-TD computed properties of three parallel coplanar microstrips: (a) characteristic impedance versus freq uency; (b) propagation delay versus frequency.
setting up FD-TD grids for single x -directed microstrips of negligible metalization thickness and a variety of widths over dielectric substrates in the order of 1 mil (0.001 inch) thick. Grid cell size 8 for these cases was in the order of 0.1 mil with t:.t in the order of 4.2 fs. In all cases, conductors were "on grid," i.e., located at planar loci of tangential E components in the FD-TD mesh that were set to zero for all time steps, and with zeroed tangential E components on the conductor edges. Mur absorbing boundary conditions [12] were set up at the outer lattice planes. To effectively decouple the Mur boundaries from the localized fields within and near the microstrip geometries, the FD-TD space lattice extended at least 20 free-space cells beyond the microstrip structures in all directions. Excitation of a microstrip was provided by specifying a Gaussian-pulse time history for a group of co-linear electric field components (usually E z ) bridging the gap between the ground plane and the strip conductor at the desired source location. The characteristic impedance, Z 0 , of the microstrip was then found by forming the ratio of the di screte Fourier transforms of the line voltage and current, V and I, obtained
22-layer board
Signal
Ground Return
(b)
Fig. 2. 3-D connector module: (a) stack of fo ur multilayer circuit boards perforated by scores of vertical via pins spaced on 0.1-inch centers; (b) generic sketch of digital signal and ground return paths in the top multilayer circuit board.
from the resu lting propagating E and H fields:
V(t, x, ) =
1
E(t , x, ) · dl,
I (t , .Ti )=
Cv
Zo(w,.Ti ) = F[V(t,xi )]/F[J(t,xi)].
j H(t, Xi )· dl f c1 (5)
Here, the contour path for V extends from the grou nd plane to the microstrip, while the contour path for I for extends around the strip conductor at its surface. Thi s method showed the FD-TD computed values of Z 0 to be virtually independent
PIKET-MAY et al.: FD-TD MODELING OF DIGITAL SIGNAL PROPAGATION IN 3-D CIRCUITS
of frequency up to 1.0 GHz, and in the order of 1 % agreement with textbook values [13]. Next, using similar FD-TD grid resolutions, we considered single microstrips with finite metalization thickness, possibly fully embedded within a dielectric layer. Here, FD-TD predictions were compared to measurements. In one example, a 1.1-mil-thick, 1.4-mil-wide metal strip was assumed to be suspended 1.1 mils above a large ground plane within a 3.1-mil-thick dielectric layer having c:,. = 3.2. The FD-TD numerical simulation predicted a flat characteristic impedance, Z0 , of 48 up to 1.0 GHz, agreeing with the experimental results to within the measurement uncertainty (about 0.2 D). The computed variation of Zo above 1 GHz was found to be ±2 n. Similar excellent agreement was found for the propagation delay, where the experimental value was 150.5 ± 1.5 psec/inch, while the FD-TD prediction was 149 .5 psec/inch. We then studied the impedance and propagation delay of three parallel, co-planar microstrips having finite metalization thickness. Each microstrip had the geometry discussed above and was separated by 3.6 mils from the adjacent line(s). Even-mode results were obtained with all three strips excited simultaneously with the same polarity, while odd-mode results were obtained with the two outer strips excited with the opposite polarity relative to the center strip. Results were also observed for the center strip excited with the two outer strips floating. Fig. 1 shows Zo and the propagation delay predicted by FD-TD for all three cases. These results were obtained with 8 = 0.1 mil , 6.t = 4.2 fs, and a lattice size of 200 x 134 x 42 cells. It is clear from the results that signal propagation on adjacent lines significantly influences the effective impedance of the line, and to a lesser degree affects the propagation delay. The FD-TD results were confirmed by laboratory studies which showed an approximate 7 n elevation of the characteristic impedance for the even-mode excitation, and an reduction of the impedance for the oddapproximate 7 mode excitation (both from de to about 1 GHz). At frequencies approaching I0 GHz, the evolution of non-transmission-line modes within the trio of parallel lines suffices to progressively disturb the observed Z 0 and propagation delay.
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n
n
B. Multilayered Interconnect Modeling Example This section illustrates the power of FD-TD modeling to investigate a classical electromagnetic compatibility problem, ground loop current flow. In what may be the most complex 3-D passive circuit modeled by any means so far, we constructed a high-resolution FD-TD model of sub-nanosecond digital pulse propagation and crosstalk behavior in a realworld computer module consisting of a stack of four multilayer printed circuit boards (each with > 10 metal -dielectric-metal layers) penetrated by scores of via pins forming a connector. A uniform space cell size of 8 = 4 mils was used with time step 6.t = 169 fs. The space cell was the thickness of a single layer in the circuit boards, and 1/3 the diameter of each 12-mil-diameter circular via. In this manner, each layer, via, and pin of every circuit board and every connector was modeled in the final simulation, resulting in almost 60-million
Fig. 3. Color plate showing the plan view of an outwardly propagating electromagnetic wave within the top multilayer circuit board generated by the passage of a subnanosecond pul se down a single vertical via pin. Color scale: yellow = maximum; green = moderate; dark blue = negligible.
vector field unknowns solved. Fig. 2(a) is a diagram of the full connector module, where each multilayer circuit board is shown as a cross-hatched horizontal slab. Vertical via pins located on 0.1-inch centers penetrate the stack of boards and connectors throughout the module. Fig. 2(b) is a generic sketch of a digital signal path and a ground-return path in the top multilayer circuit board. Single Multilayer Circuit Board, Early Time Response: The first step in the study was to model in detail the early-time response of the top multilayer circuit board to an impulse of current (90-ps Gaussian pulse, spectral width about 20 GHz) propagating down a single vertical via pin. The via pin (in the circular via hole) was excited by pulsing a vertical electric field component, Ez, in the FD-TD lattice just above the pin
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL. 42, NO. 8, AUGUST 1994
Fig. 4. Color pl ate of the early-time coupling of magnetic fields from the exc ited via pin to the adjacent unexcited via pins as seen in a vertical cut throu gh the top multilayer board and connector of the stack of Fig. 2. Color scale: red maximum ; yellow moderate; green low-level; dark blue negli gible.
=
and below a simulated ground strap connected to the outer via pins designated as ground returns. The short duration of the pulse was selected primarily to permit time resolution of layerto-layer pulse reflection effects within the circuit board. We utilized a 300 x 92 x 52 cell FD-TD grid having 1.44 million cell s (8 .6 million unknown s, 19 MW of memory) . Using a single processor of a Cray YMP-8, this grid size required 0.36 sec per time step. The power and signal metalizations of the circuit board were assumed to be infinitely thin. Fig. 3 is a color plate showing the plan view of IHI of the outwardly propagating electromagnetic wave within the board generated by the passage of the pulse down the via pin. Although the relatively intense magnetic field adjacent to the excited via (shown by the yellow color) is quite localized, moderate-level magnetic fields (shown by light blue) emanate throughout the entire transverse cross section of the board and link all of the adjacent via pins, shown as dark dots in a diamond pattern. The complete color video of the dynamics of this phenomenon shows repeated bursts of outward propagating waves linking all points within transverse cross sections of the board as the pulse passes vertically through the multiple metal-dielectric-metal layers of the board. The resulting pin-to-pin cross-talk is vividly illustrated in the color plate of Fig. 4, which depicts IHI in a vertical cut through the top multilayer board and connector of the stack of Fig. 2. The coupling of magnetic fields from the excited via pin to the adjacent unexcited via pins is clearly indicated. Complete Four-Board Connector Module, Late Time Response: The final step in the study was to model the response of the complete four-board connector module to the impulsive excitation of a single vertical via pin. Keeping the same space
=
=
=
resolution as for the single-board model , we enlarged the FDTD grid to 300 x 92 x 340 cells to contain the three additional multilayer circuit boards and connectors. The new grid totalled 9.4 million cells with 56 million vector field unknowns, and required 117 MW of memory. Using a single processor of a Cray YMP-8 , this grid required 2.13 sec per time step, or 71 minutes for a complete run of 2000 time steps. (Using all eight of the Cray YMP processors reduced the running time by almost 8: 1 to only 9 minutes.) Again, all power and signal metalizations of the circuit boards were assumed to be infinitely thin. Color videos of pulse propagation and crosstalk were constructed to illustrate these phenomena. Fig. 5 is a color plate showing the magnitude and direction of late-time currents flowing along the vertical cross section of the complete connector module of Fig. 2 for a subnanosecond pulse assumed to excite a single vertical via pin in the top multilayer board. The currents were calculated in a post-processing step by numerically evaluating the curl of the magnetic field obtained from the 3-D FD-TD model. The color red was selected to denote downward-directed current, while the color green was selected to denote upward-directed current. At the time of this visualization , current had proceeded down the excited via through all four boards and all three connectors. But, upwarddirected (green) current is seen to flow on the adjacent vias. This represents undesired ground-loop coupling to the circuits using these vias.
IV.
INTERCONNECT GEOMETRIES
WITH
LUMPED LOADS
In this section, we report 3-D FD-TD modeling of the connection of linear and nonlinear lumped loads to microstrip
1519
PIKET-MAY et al.: FD-TD MODELING OF DIGITAL SIGNAL PROPAGATION IN 3-D CIRCUITS
High·conductivity microstrip
11
Lumped
Source Thin-sheet ground plane 50-600 cells
4-10 cells I cell
Lumped Element
1-5 cells
'
,_j,
Fig. 6. Generic geometry used for 3-D FD-TD model s of a linear or nonlinear lumped element terminating a stripline.
A. The Resistor
Fig. 5. Color plate showi ng the magnitude and direction of late-time currents flowing along the vertical cross section of the complete multiboard module of Fig. 2 for a subnanosecond digital pulse assumed to exc ite a single vertical via pin in the top multilayer board. Color scale: red = net downward-directed current; green net upward-directed current; dark blue negli gible.
=
=
interconnects. Fig. 6 depicts the microstrip geometry considered. For convenience in the discussion, the microstrip is assumed to be oriented in the x -direction and the lumped load is assumed to be oriented in the z-direction. Extension of the theory to other Cartesian orientations in the FD-TD lattice is straightforward. Lattice resolution 8 in the range I to 8 mils was used in the simulations to study the effect of varying the number of cells spanning the strip conductor and the gap between it and the ground plane. For excitations with spectral components up to l GHz, the results were found to be only a very weak function of the grid space increment 8 in this range. In all cases, conductors were "on grid," i.e., located at planar loci of tangential E components in the FD-TD lattice that were set to zero for all time steps, and with zeroed tangential E components on the conductor edges. Mur absorbing boundaries were used as specified in Section III. A resistive source of the type discussed in Section B below was used in all of the FD-TD simulations of this section. The source resistance was matched to the line impedance so that retroreflections at the source were minimized.
When studying microstrips 1t 1s very useful to have the capability to termjnate the line in a resistive load, very likely matched to the characteristic impedance of the line. One method that we studied for terminating the line in an FD-TD grid is to insert a physical resistor block into the microstrip model using the relation R = pL/ A (where R is the desired resistance, p is the block 's resistivity, and L and A represent the length and cross sectional area of the block) to select the parameters of the insert. The second method is to insert a numerical lumped element in an extension to 3-D of that described in [9]. Here, assuming a z-directed resistor located in free-space at the field component Ez I 'l,..Ji k' the voltage-current characteristic that describes its behavior in a semi-implicit manner appropriate for stable operation of the FD-TD field solver is n+l/2
Iz l t,J, .. k
6. z ( n+l In ) = -R Ez i,J, .. k + Ez t,J, .. k , 2 1
J L -
f z l n+l/2 i,J,k
6. x 6.y
(6a)
where R is the value of the resi stance and 6..7.:, 6.y and 6. z are the grid increments in the x, y and z directions. The corresponding time-stepping relation for Ez I i,j,k is
(6b)
To compare the impedance match provided by the physical and numerical resistors in the FD-TD mesh, we modeled a terminated 50-D transmission line similar to that of Section III-A excited by a 90-ps Gaussian pulse (spectral width of 20 GHz). After performing a discrete Fourier transform of the reflected pulse for each case, we found that both the physical and numerical resistors provided reflection coefficients of less than I% up to l GHz. Fig. 7 shows the magnitude and phase of
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42. NO. 8. AUGUST 1994
1520
voltage source in a semi-implicit manner is ,.-...50.00
en
I ln+l/2
E
..c:
z ·i ,j,k
= _i..:._z (E A
2Rs
2,48.00
fz (lJ
:'::: 46.00
ln+l/2
2
0
::?; 44.00
· - -
(lJ
u
c
NUMERICAL RESISTOR IDEAL PHYSICAL RESISTOR
0
~ 42.00
Ez
I n+l i,3,k
a_ (a) Magnitude
E 40.00
=
1 ( l +
+-~~~~~-r-~rTTTTTT-~T""T~~
10.
10'
10.
10
(
+(
2.00 ,.-...
1.50
· - -
....
CJl (lJ
.6.t.6. z ) 2Rseo .6.x.6.y Ez Jn. .6.t.6. z ., ,J,k 2Rseo .6. x .6.y
+
10
Frequency
(lJ (lJ
(7a)
where V8n+I/ is the source voltage and Rs is the internal source resistance. The corresponding time-stepping relation for E zl t,J, . . k is
CJl
en
+
v:n+l/2 In ) _ s__ z i,j,k + Rs
/::,,x/::,,y
:::i
c
z i,j,k
E
i,3,k
J £=
-0
Jn+l
1+ l
.6.t ~
)
.6.t.6.z 2Rs eo .6.x .6.y
Rse~},x.6.y
+
NUMERICAL RESISTOR IDEAL PHYSICAL RESISTOR
)
.6.t.6. z 2Rseo .6. x .6.y
V
x
H n+ 1/2
Ii,j,k
v;n+l /2 S
·
(7b)
1.00
-0
"--" (lJ
en 0 ..c:
C. The Capacitor
0 .50
We next consider the insertion of a numerical lumped capacitor into the FD-TD grid in an extension to 3-D of that described in [9]. Again assuming a z -directed lumped element located in free-space at E z i, J , k' the voltage-current characteristic that describes the capacitor's behavior in a semi implicit manner is
0.00 -r---~'-'=-==-:
0... (lJ
u
-0.50
I· .
c
.g -1.00 (lJ
a_
E -i.5o - 2.00
.
(bl Phase
+-~-r-~~~-r--~~-~~~~
10'
10.
10.
10 ••
I ln+l /2 = C!::,,z ( E ln+l - E In /::,,l
z i,j,k
Frequency Fig. 7. Agreement of FD-TD computed effective load impedance (low frequency to I 0 GHz) for the numerical resistor and the physical resistor: (a) magnitude; (b) phase.
the effective load impedance versus the ideal (matched) load impedance for the two cases. Over the span de to I 0 GHz, the maximum deviation is (positive +20, -0.50) and (+ 1.75°, -1°). The fact that the physical and numerical resistor results agree so well implies that the numerical resistor model does an excellent job of picking up the parasitic capacitance or inductance that may be present between the terminals of the physical resistor. It is important to note that implementing (6b) at two adjacent electric field components between the microstrip signal line and ground plane simulates the presence of two parallel resistors terminating the line. The effective load resistance is thereby halved.
h
=
Izl
~t~ ''
z ·i, j ,k
z i,j,k
)
'
12
(8a)
/::,, x!::,,y
where C is the value of the capacitance. This formulation differs from that of [9] in that the electric field samples are separated here by one time step rather than two . In this manner, we are consistent with the electric field sampling used for the numerical resistor described previously. The corresponding time-stepping relation for E z Ii,J, · .k is
Ezl~J,~ = Ez l ~j,k + (i + .L)v x HJ ~j,~ 12 Eo .6. x .6.y
(8 b) For the parallel combination of a capacitor, C, and resistor, R, located at E z J .; ,j,k' the results of (6a), (6b), (8a) and (8b) can be readily combined to yield the following time-stepping relation:
B. The Resistive Voltage Source With the ability to model lumped elements in the context of FD-TD, it is a simple matter to model a nonreflecting (matched) sou rce as a resistive voltage source in an extension to 3-D of that described in [9]. Again assuming a z-directed lumped element located in free-space at E z I..,,11, · .k' the voltagecurrent characteristic that describes the behavior of a resistive
(8c) To test this FD-TD modeling approach, a variety of numerical capacitive loads were modeled at the end ofa long (but ::::::1 mil scale in the transverse plane) microstrip line subjected
son
PIK ET-M AY
e1
al.: FD-TD MODELI NG OF DIGITAL SIGNAL PROPAGATION IN 3- D CIRCU ITS
where q is the c harge of a n electron, V, 1 is the voltage across the diode, k is Boltzmann's constant, and Tis the temperature in degrees Kelvin. According to the 2-D study of [9], if one assumes a z -directed diode located in free-space at Ez i,j,k' the e lectric field time-stepping relation is given by
1.0
4 nF
-
-
I
FD - TD
00000
152 1
Theoretica l
Q)
1 Ez I 1+kl· =Ez 171· k
g1o6
1,·
,
0
>
,
' 1. ,
E0
\.._ 0 .4
+ 6.t \7 Ea
6.t Io 6. x 6.y
!
12 H I:''·+ 1 1
X
i
[e( - qE, I/; .,
D. z / "'T)
-
i] .
(I 1)
0
+-'
u
However, it has been determined that this expression yields a numerically unstable algorithm for diode voltages larger than 0.8 vo lts due to its exp licit formulation which employs the previously computed electric fi eld in the exponential. We have found that a numerically stable FD-TD algorithm for the lumped diode can be realized in 3-D by using the semi-implicit update strategy for the electric field
20 nF
0
°-0.2 0
()
0.0 _j__...__'-"""-----------~~ 90 1J O 170 50
Time ( psec ) Fig. 8. Agree ment of FD-TD and ex act solutions for the voltage across the capacitor termin ating the stripline for two values of the capacitor (stepwise incident pulse).
to a rectangular step-pu lse excitation I 000 time steps long. The FD-TD computed voltage response vs. time across each capacitor was then compared to the exact theoretical response. Results are shown in Fig. 8 for microstrips terminated with 4 nF and 20 nF capacitors. The theoretical and FD-TD curves are indistinguishable.
D. The Inductor We next consider the insertion of a numerical lumped inductor into the FD-TD grid in an extension to 3-D of that described in [9]. Again assuming a z-d irected lumped e lement located in free-space at Ez l.i ,j ,k' the voltage-current characteristic that describes the inductor's behavior in a manner appropriate for stable operation of the FD-TD field solver is
6. z6.l ~ E
n+ l / 2 _
I z l·i ..J, k
-
L
L_,
lrn.
z i,J,k'
m=l
h
I z 1 :1;~/2 ' '
=
(9a)
6. x 6.y
where L is the value of the inductance. This formulation differs from that of [9] in that the electric field samp les are summed only through time step n , which is consistent with the observation of the lumped current at time step n + 1/ 2 that we employ throughout this development. The corresponding time-stepping relation for Ez Ii..J, k is
Ez I ;~-;:! =Ez -
I~j,k + ~ot \7 x HI :'.J.!12 6. z (6.t)2
~
c0 L6.:i;6.y L_,
Ezl;n. k ·
(9b)
,.1 , ·
m=l
E. The Diode The current through a lumped-circuit diode is expressed by:
Id= Io
[ e( qVd/ kT )
-1]
(10)
( 12) In this manner, we obtain the following tran scendental equation:
Ez l '.1,1~, ! = Ezl'.1,1·, k + 6.l\7 Ea E0
X H I:'...+ ,, ! / 2
6.t Jo { e [- q(E, l"+ ,' .+E, l"1. 1.. ,) 6.z/2t
u;
..,'-'0
B- E B-E B- E B- E
0.2
-8 .0
0
0
-10.0 50
0
100
150
Time
-12.0 0
5
10
15
20
200
25U
(p sec )
Fig. 11. Agree ment of FD-TD and SPICE calculati ons for the transistor base-to-emitter vo ltage (stepw ise inc ident pul se).
Tim e (n s ) Fig. 9. Agree ment of FD-TD and SPICE calculations for the vo ltage across the dio determ in ating the stripline.
Substituting ( 16) and (17) into (14) and ( 15), we obtain:
l} - Io{ e- [%(Ez l~~ '+EzlG s)/kT] - 1} I~+l/2 =Io{ e [ 1(Ez 1~6 ' +Ez l8c )/ kT] - 1} I~+l/2
I cell
I cell
microstrip
Rs
vs
= O'. n}o { e[%(E z 1~6 ' +EJ8cl/ kT]
_
( 18)
_ O'.Fio{ e- [ %(Ez l~~ ' +Ez l;;; 8 )/kTj _ l }· ( !9)
1
Now, we obtain two coupled transcendental equations for the FD-TD e lectric field updates at the transi stor:
•cell
E
ground plane 25cells
E ln+l z EB
I cell--+-
= E z InEB + tJ.t\7 E0
X
Hln+l/2 ., ' ·l ' k
+
tJ.t
"
" .
E0 Ll .T Ll'.IJ
I n+l /2 E
Fig. 10. Generic geomet ry used for 3-D FD-TD model o f an NPN bipolar junctio n tran sisto r (BJT) termin ati ng a stripline in the co mmo n emitter confi g uration.
(21 )
Ebers-Moll transistor model described by the following circuit equations [14]:
IR = Io
[ e(q\!13 c/ kT ) -
J,
1
( 14) ( 15) Now, assuming a transi stor that is located in free-space and oriented in the z-direction in the FD-TD grid as shown in Fig. I 0, we can use a semi-implicit strategy to express the base-emitter voltage, VsE, in terms of Ez IEB, the FD-TD computed electric field in the one-cell gap between the ground plane and the end of the stripline
11,n+ l / 2 = _ tJ. z (E ln+l BE ZEB 2
+ E z InEB ) ·
+ E z InBC ) ·
n
( 16) V . CONCLUSION
A similar semi-implicit strategy is used to express the basecollector voltage, VBc, in terms of Ez IBc, the FD-TD computed electric field in the one-cell gap between the end of the stripline and the collector load:
11,n+ l/2 _ 6 (E 1n+ l BC 2 Z BC
The Newton-Raphson method may be used to solve these equations. To test this FD-TD modeling approach, a transistor at T = 300° K having I 0 = 10- 16 amp, O'.R = 0.5, and O'.F = 0.9901 was modeled at the end of a 50 microstrip line (of ~I-mil scale in the transverse plane) in the manner of Fig. I 0. The collector de supply was included in the electromagnetic simulation. Both the active (Re = 50 n) and saturated (Re = 10 f2) regions of operation were observed for a step function excitation of the stripline. A typical result is shown in Fig. 11 , where the FD-TD computed base-to-emitter voltage is compared to that obtained by a SPICE model. Very good agreement is observed.
( 17)
Analog coupling effects for passive metallic interconnects and packag ing of digital circuits operating at nanosecond clocks can be very complex. In fact, as clock speeds increase beyond 500 MHz, it may not be possible to design such systems and make them work in a timely and reli able manner without resorting to full-wave Maxwell 's equations solutions
PIKET-MAY et al.: FD-TD MODELING OF DIGITAL SIGNAL PROPAGATION IN 3-D CIRCU ITS
in 3-D. FD-TD numerical methods appear to provide sufficient accuracy and scaling ability for large interconnection and packaging problems to be of substantial engineering importance to the digital electronics comm unity. An emerging possibility is that FD-TD Maxwell's equations modeling can be linked directly to SPICE [ 15). Thi s would expand full-wave electromagnetic modeling of digital interconnects to include the voltage-current characteristics of the connected logic devices. It is conceivab le that eventually the logical operation of very complex, very high-speed digital electronic circuits can be directly modeled by FD-TD timestepping of Maxwe ll 's eq uations. A C KNOWLEDGM ENT
The authors wish to thank Cray Research , Inc ., with special thanks to Evans Harrigan for continuous support and encouragement. The authors also acknowledge the technical contributions of Dr. Mike Jones and Dr. Vince Thomas of Los Alamos National Laboratory, and Dr. Chris Reuter of Northwestern University. REFERENCES
III
121
!31
J4 1
JS!
j6 J
171
181
191
I 101
II II
I I2 1 11 31
A. Taflove, " Review of the formulation and applications of the finitedifference time-do main method for numerical modeling of electromagnetic wave interactions with arbitrary structures," Wave Motion, vol. 10, pp. 547- 582, Dec. 1988. K. S. Yee, "Numeri ca l so luti o n of initial bo undary va lue proble ms involv in g Maxwell 's equations in isotropic medi a," IEEE Trans. Antennas Propagat., vo l. 14, pp. 302-307, May 1966. G.-C. Liang, Y.-W. Liu, and K. K. Me i, "Full -wave anal ysis of copl anar wavegu ide and slotline using the time-domain finite-difference method," IEEE Trans. Microwave Theory Tech., vol. 37, pp. 1949- 1957, Dec. 1989. T. Shibata and E. Sano, "Characteri zati on of M JS structure copl anar transmi ss ion lines fo r investigation of sig nal propagation in integrated circuits," IEEE Trans. Microwave Theory Tech., vo l. 38, pp. 881-890, Jul y 1990. C. W. Lam, S. M. Ali, R. T. Shin, and J. A. Ko ng, " Radiatio n from discon tinuiti es in VLSI packaging structures," Proc. Prog ress in Electromagnetics Research Symp., Boston , MA , July 199 1, p. 567. S. Maeda, T. Kashi wa, and I. Fukai, " Full wave analys is of propagation characterist ics of a through ho le using the finite-d ifference time-do main method ," IEEE Trans. Microwave Theory Tech., vo l. 39, pp. 2 I 54-2 I59, Dec. 1991. E. Sano and T. Shibata, " Fu ll wave anal ysis of picosecond photoconductive sw itches," IEEE J. Quantum Electron., vol. 26, pp. 372-377, Feb. 1990. S. M. El-Ghazaly, R. P. Joshi and R. 0. Grondin , " Electromagneti c and tran sport considerations in subpicosecond photoconductive sw itch modeling," IEEE Trans. Microwave Th eory Tech., vo l. 38, pp . 629-637, May 1990. W. Sui , D. A. C hri stensen and C. H. Durney, "Ex tending the twodimensional FD-TD method to hybrid electromagneti c systems with active and passive Jumped elements," IEEE Trans. Microwave Theory Tech., vol. 40, pp. 724-730, Apr. 1992. A. Taflove, " Basis and Application of Finite-Difference Time-Domain (FD-TD) Techniques for Modeling Electromagnetic Wave Interacti o ns," (short course notes), 1992 JEEE Antennas and Propagation Soc. Int. Symp. and URS! Radio Sci. Meeting, Chicago, IL, Jul y I992. A. Taflove and M. E. Brodwin, " Numerical solution of steady-state e lectromag net ic scattering problems usin g the time-dependent Maxwe ll 's equations," IEEE Trans. Microwave Th eory Tech., vol. 23, pp. 623-630, Aug. 1975 . G. Mur, "Absorbi ng boundary condition s for the finit e-d ifference approximation of the time-domain e lectromag netic field eq uations," IEEE Trans. Electromagn. Compal., vol. 23 , pp . 377-382, Nov. I 98 I. S. Ramo, J. R. Whinnery and T. Van Du zer, Fields and Waves in Communications Electronics, 2nd ed. New York: Wil ey, 1984, p. 4 10.
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I 14] S. M. Sze, Physics of Semiconductor Devices. New York: Wiley, 198 I, p. 152. V. A. Thomas, M. E. Jones, M. J. Piket-May, A. Taflove and E. Harrigan , "The use of SPICE lumped circ uit s as sub-grid model s for FD-TD analy sis," IEEE Microwave Guided Wave Lett., vo l. 4, pp. 14 1- 143 , May 1994. I 16] Y.-S. Tsue i, A. C. Cangellaris, and J. L. Prince, "Rigorous electromag netic modeling o f chi p-to-pac kage (first- leve l) interco nnect ions,'' IEEE Trans. Components, Hybrids, and Manufacturin g Tech., vo l. 16, pp. 876-883, Dec. 1993.
I 151
Melinda Piket-May (S'89-M'92) received the B.S. in electrical engineering from the Uni versity of Illino is, Urban a-Champaign , IL, and the M.S. and Ph.D. degrees in elect ri ca l engineering from Northwestern Uni versity , Evanston, IL. During her latter studies , she interned for several acade mic quarters at Cray Research, Inc., Eagan, MN and C hi ppewa Fall s, WI. There, she collaborated w ith Cray software deve lopers and hardware engineers to deve lop beta so ftware for the finited il'ference time-domain (FD-TD) Maxwell ' s equati ons superco mputing simul atio n of rad ar cross section and high-speed d igital e ledctronic circuits. She has also co ll aborated w ith researchers at Los Alamos National Laboratory , Los Alamos, NM , in developing innovative means to mode l the o perati o n of nanosecond-regime no nlinear ana log and digital c ircuit s by applying FD-TD Maxwe ll 's eqauation s techniques. She cu rrent ly is an Assistant Professor in the Department of Electrical and Computer Eng ineering, Univers ity of Colorado, Boulder, CO.
Allen Taflove (M '75-SM '84-F'90 ) is a Professor in the Department of Elect rical Eng ineerin g and Computer Science, McCorm ick School of Engineering, Northwestern University, Evanston , IL. Hi s current research interests include firstprinciples (Maxwe ll ' s equati ons) supercomputin g simulation of nanosecond-regime electronic circuit s and sub-picosecond-regime nonlinear optica l phenomena and dev ices. Since the end of 1989, he has given 50 invited talks and lectures in the U.S. o n hori zons in superco mputing computatio nal e lectrom agnetics . He was named an IEEE Fel low for "Contributions to the development of the fi nite-difference timedomain (FD-TD) soluti on of Maxwe ll' s equations." In 1990-9 1, he was a Di stingui shed National Lecturer for the IEEE Antennas and Propagation Soc iety, and in 1992 was C hairman of the Technical Program of the IEEE Antennas and Propagati o n Soc iety Internat ional Symposium in Chicago, IL. He originated several inn ovative progra ms in the McCormick Schoo l, includin g the Honors Progra m in Undergraduate Research (a seven-year co mbined B.S./Ph.D. engineering deg ree progra m for extremely talented students), the Undergraduate Des ig n Competi ti on, and the Hi g h Schoo l Outreach Program. In 1991 , he was named McCormick Faculty Advi ser of the Year. He is a member of the Eta Kappa Nu , Tau Beta Pi , Sigma Xi, lnternti onal Union of Radio Science (URSI) Co mmi ss io ns B and K, the Electromagnetics Academy, AAAS, and New York Academy of Sc iences. Hi s biographical li stings include Wh o 's Who in Engineering, Who 's Who in America, Who's Who in Science and Engineering, and Who's Who in American Education.
John Baron (S'94) is a gradu age student in the De partme nt of Electrical Engineering , Stanford Univers ity, Stanford, CA. While an undergraduate electrical eng ineering student at No rth western University, Evanston , IL, he participated in a coll aboration w ith researchers at Los Alamos National Laboraory, Los A lamos, NM , in developing innovati ve means to model the operati on of nan osecond -reg ime non linear anal og and digital c ircuits by applying finit e-d ifference time-domain (FD-TD) Maxwe ll 's eq uations techniques. As part of thi s colla borati o n, he interned at Los Alamos during o ne summer qu arter. He is pursuing stud ies in space science.
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IEEE TRANSACTION S ON M ICROWAV E THEORY AND TECHN IQ UES, VOL. 42. NO. 8. AUGUST 1994
Powder Core Dielectric Channel Waveguide William M . Bruno and William B. Bridges, Fellow, IEEE
Abstract- A powder-filled rectangular groove in the surface of a plastic substrate has been demonstrated as a dielectric waveguide at 94 GHz. Propagation losses as low as 0.09 dB/cm were measured by direct transmission with nickel-aluminum titanate powder in a polypropylene substrate and with barium tetratitanate powder in a polytetraftuoroethylene (PTFE) substrate; values as low as 0.06 dB/cm were deduced from ring resonator measurements. Guide wavelengths measured for various combinations of guide dimensions, powders, and substrates agree within 10% with values predicted by the approximate theory of Marcatili for the mode. Effective loss tangents 1 for the powders at 94 GHz were calculated from waveguide attenuation measurements, using Marcatili's field solutions. Ring resonators fabricated by filling a groove in a polypropylene substrate with nickel-aluminum titanate powder exhibit Q 's as high as 2400 at 94 GHz in an 8 cm diameter ring. Coupling to the resonators was achieved with adjacent straight powder core channel guides as directional couplers.
Er
I. INTRODUCTION ie lectric waveguides have been widely studied for use in millimeter wave integrated circuits [ 1] since they generally exhibit much lower millimeter wave attenuation than structures which rely on metal surfaces for guiding, such as microstrip, slotline, and coplanar waveguide. (Hollow metal guide is low-loss, but not suitable for integrated circuits.) In add iti on, dielectric waveguides are particularly well-suited for coupled waveguide structures, because the evanescent fields of the guided mode extend outside the core. We have demo nstrated a powder-filled groove in the surface of a plastic substrate as a low- loss diel ectri c channe l waveguide at 94 GHz. Directional couplers and ring resonators with Q's as high as 2400 at 94 GHz have been implemented with thi s waveguide. Such powder core channel guides may be attractive for low cost passive integrated circuits such as feed structures for antennas. In the first half of thi s paper, we briefly survey theories applicable to rectangul ar dielectric channel waveguide, describe our powder core chan nel waveguides, and di scuss the measurements used to characterize the performance of straight segments. The second half of this paper treats the design and testing of ring resonato rs, and the desig n of the straight guides used to couple to the reson ators. In the appendi x, we derive an eq uation re lating the attenuation of the modes of rectangular chan nel waveguide to the loss tangents
D
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Manuscript received March 12, 1993; rev ised October 12, 1993. Thi s work was supported in part by an Army Research Fe ll owship under Grant DAAG2983 -G-OO 17 and in part by Anny Research under Grant DAAG29-84-K -O I00. W. M. Bruno is with TRW, One Space Park , Redondo Beach, CA 90278 USA. W. B. Bridges is with Cali fornia In stitute of Technology, Pasadena, CA 9 11 25 USA. IEEE Log Number 9402930.
of the core and its surroundings. Thi s eq uation is then used to calculate 94 GHz effective loss tangents fo r powders used in our experiments. II. POWD ER CORE CHANNEL WAVEGUIDE
A. Motivation The cost of convention al millimeter waveguide and waveguide components is substanti ally higher than that of their longer wavelength counterparts becau se of their proportion ally smaller dimensional to lerances. As a conseq uence, millimeter wave waveguide components are not mass-produced; rather, they are machined or asse mbled individually (or electroforming mandrels are mac hined individuall y) so that, in a sense, the finished part represents "stored preci sion machinist' s labor. " The situation is no different for the usual dielectric waveguide or im age gu ide component mach ined from teflon or si licon-in fact, the machining task may be even more difficult due to the ductile or brittle nature of dielectric material s. Metallic integrated circuits produced photolithographically on a dielectric su bstrate is one answer to thi s dilemma; however, MMIC is not an exact replacement fo r conventional waveguide components, but is more a complementary techno logy. One motivation fo r the present study was to see if millimeter waveguide and waveguide components co uld be reali zed (ultimately) by injection molding or emboss ing plastic substrates, so that the "stored mac hini st's labo r cost" resides in a precision form usable for thousands of components rather than being delivered with every piece. While, for convenience, we have used prec ision machining in our laboratory demonstrations, we have also demonstrated that waveguide sections at 94 G Hz can be for med by thermoplastic means (for example) with negligible change in characteristics [2]. The choice of placing the precision fabrication in the substrate rather than the core is somewhat arbitrary. We could have chosen to mo ld the waveguide cores from low-loss polymers and then surround them with foam or powdered material of lower dielectric constant (possibly powders of the core material , which automatically have lower dielectric constants). Our choice of a so lid , prec ision-formed substrate filled with a powder ex hibiting a higher dielectri c constant was ac tuall y chosen as a continuation of our studies on flexible dielectric millimeter waveguide using powdered cores [3].
B. Su rvey of Theoretical Approaches The rectangular di electric channel waveguide is shown in cross section in Fig. 1. No analytical solution ex ists fo r the propagating modes of thi s guide. However, there are several
001 8-9480/94$04.00 © 1994 IEEE
BR UNO AND BRIDGES : POWE R CORE DIELECTRIC CHANNEL WAVEGUIDE
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E, = I
diode detector
""\ B\\\\ ,,
~ ~
~E r
'\
Fig. 1.
flared metal
wavegu ide section
\
= 2.0 8 TEFLON
powder-filled
groove
2.2 5 POLYPROPYLENE
Cross section of rectangular dielectric channel waveguide.
numerical methods which yield approximate values of the propagation constants. Most of these techniques, such as those proposed by Yeh et al. (finite elements) in [4], [5] and (finite differences) [6}, require the use of large computers, Marcatili ' s approximate-mode method [7}, on the other hand, is fairly easy to use since the computations involved are much simpler. Marcatili ' s method makes an approximation based on the criterion that the refractive index of the core is similar to that of the cladding. That is, (l-nc1ad /n core « 1). However, selected cases analyzed with other methods agree reasonably well with guide wavelengths calculated from Marcatili's theory for the Ef.1 mode [6}, [7] even when the criterion (1- n c1ad/ncore « 1) is invalid. 1 Indeed, good agreement is obtained even for a rod with relative dielectric constant equal to 13 surrounded by air [6]. For these reasons, Marcatili's method was used here to predict the propagation constant of the Ef. 1 mode of powder core channel waveguide.
- lOdB direct ional coupler
Fig. 2.
Set-up for measuring attenuation and guide wavelength.
movable perturber - IO dB
source
waveguide cou ple r
-+---~~
C. Guide Wavelength and Attenuation Measurements
A rectangular groove was milled into the surface of a lowloss (PTFE or polypropylene) substrate and was filled with a powder having a high-dielectric constant to form the core of a dielectric waveguide (Fig. 1). With this configuration, the powder could be packed from the top to assure a sufficiently uniform density along the length of the groove. Rectangular grooves with cross-sectional dimensions varying less than ±0.001 inches or ±0.025 mm from the specified values (typically 1 mm x 1 mm) could be milled with relative ease. This degree of dimensional accuracy was found to be sufficient at 94 GHz to produce a guide wavelength uniform within our measurement accuracy. The guide wavelength and loss per unit length were measured for the fundamental vertically polarized (Ef.1 ) mode of various powder core channel waveguides using the set-up shown in Fig. 2. On each end of the substrate the dielectricfilled groove was extended with a thin-walled trough of substrate material. This trough fitted snugly into the end of a slightly flared section of WR- LO metal waveguide to couple to the dielectric guide. Lossy inserts made from Emerson and Cumming MF-110 absorber were placed at non-periodic intervals in the substrate 3 mm from the groove to attenuate any substrate modes (that is, propagation of energy through the substrate other than via the desired guided mode) that 1 We use Marcatili' s notation here, despite the difficulty that the magnetic field associ ated with the "E~q" mode is mainly x-directed and therefore cannot be designated "H~q" without confusion. Marcatili solves thi s problem by rarely referring to the magnetic field. See al so [6] for an alternative way of naming modes. ·
Fig. 3.
Guide wavelength measurement.
might have been excited at the coupling point and resulted in end-to-end coupling via the substrate. To measure the guide wavelength, a knife-edged metal shorting plane was held mechanically just above the surface of the powder, as shown schematically in Fig. 3. The perturber reflects a small fraction of the power traveling along the waveguide toward the feed, where it interferes with the reflection from the input coupler. The amplitude of this interference changes as the relative phase between these two signals changes. Thus, as the perturber was moved along the length of the groove, a sequence of maxima and minima in reflected power was sensed with a -10 dB directional coupler and a Schottky diode, as shown in Fig. 3. The guide wavelength is twice the distance the perturber is moved between successive minima. For various combinations of guide dimensions, dielectric powders, and substrate materials, the guide wavelengths were compared to the values predicted by Marcatili's approximate theory [7] for the fundamental vertically polarized mode. (No beats were observed in the measured patterns of reflected power versus perturber position, indicating that the waveguides were single-mode, as intended. Also, the polarization of the guided mode matched the vertical polarization of the excitation, as expected.) In order to use Marcatili' s theory, the dielectric constants of the powders were needed. The density of the powder in
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 8, AUGUST 1994
TABLE l COMPARISON OF M EASURED G UIDE W AV ELENGTH WITH PREDICTION OF MARACATILI
Powder Type
Substrate Width Depth Type of of Groove Groove (mm) (mm)
Powder Density (g/cm 3 )
Relative Dielectric constant
~g
(Meas.) (mm)
~g
(Marcatili) (mm)
±.02
D-30**
T
0.94
0.94
1.95 ± .07
5.78 ± .35
1.86
D-38**
T
1.12
1.12
1.77±.04
5.0 ±.4
2.06
MCT40•
T
1.83
1.04
1.33 ± .02
3.77 ± .07
2.07
2.11 ± .02
D-8512*
T
1.27
1.10
1.55 ± .03
3.85± .07
2.16
2.18± .02
D-8512* D-8512*
T p
1.27
1.10
1.47 ± .02
3.55± .22
1.10
1.55 ± .02
3.85± .08
2.22 2.04
2.26 ± .06
1.47
1.96 ± .08*** 1.9 ± .1
2.09± .02
Substrate T is PTFE Substrate P is polypropylene *** The uncertainty in the guide wavelength predicted by Marcatili's theory is estimated from the uncertainty in the dielectric constant of the powder. ** *
70% of particles between lOOµm and 43µm, 30% less than 43µm 100% of particles less than 43µm
the groove was determined by precision weight measurement, and previously-measured curves of dielectric constant versus density were used to find the effective dielectric constant of the powder packed into the groove. The dielectric constants of the powders had been measured at 10 GHz using the shortedwaveguide technique. These measurements were made at 10 GHz because of the difficulty of controlling the length of a powder sample in a shorted WR-10 metallic waveguide sufficiently accurately to measure its dielectric constant at 94 GHz. The effective dielectric constant of a powder composed of low-loss dielectric material should not vary much between 10 GHz and 94 GHz if the powder grains are small relative to wavelength at 94 GHz. (For further discussion, see [2].) The powders used were the same as for the flexible guide work described in [3] : Trans-Tech D-30 nickel-aluminum titanate, Trans-Tech D-38 barium tetratitanate, Trans-Tech MCT-40 magnesium calcium titanate, and Trans-Tech D-8512, an "improved" barium tetratitanate. For D-8512, MCT-40, and for one batch of D-30, all particles were less than 43 µm in size. For D-38 and for a second batch of D-30, 703 of the particles were between 100 µm and 43 µm and 303 were less than 43 µm. Trans-Tech gives f. 1 = 31 and tan 8 < .0002 for solid D-30 at 10 GHz, f. 1 = 37 and tan 8 < .0005 for solid D-38 at 6 GHz, f. 1 = 40 and tan 8 < .002 for solid MCT-40 at 6 GHz, and f. 1 = 38.6 and tan 8 < .0005 for solid D-8512 at 6 GHz; no data at 94 GHz is known, other than that in this report. To determine the loss-per-unit length of a channel waveguide, the power transmitted from end-to-end was measured by a detector connected to the flared section of metal waveguide surrounding the trough on the far end of the substrate (Fig. 2).
Shorted stub ("E/H") tuners were added to match the coupling sections. The power detected at the far end could not be significantly increased by adding the E/H tuners, so we assume that the couplers are reasonably well matched by themselves. In addition, removing the lossy substrate inserts did not affect the power received at the far end, indicating that little power is lost to substrate modes. A third detector connected to a small horn antenna was used as a movable probe to determine that the power radiated from the couplers and guide was small. Taken together, these observations indicate that almost all of the incident power was coupled into the dielectric waveguide, so that the difference between the incident power and the power detected at the far end mostly represents dielectric waveguide loss. The loss per unit length, including a small error due to coupling loss, is then this loss divided by the length of the dielectric waveguide. A comparison between the measured values of the guide wavelength with those predicted for the Ei_1 mode by Marcatili ' s approximate theory is given in Table I for various powders in plastic substrates at 94 GHz. Although Marcatili's theory assumes that the refractive index of the core is similar to those of the surrounding media and is thus only an approximation, the wavelengths it predicts for the Ei_1 mode (Table I) are in reasonable agreement with those measured. Hence, we conclude that the dielectric constants of the powders used were not too high for Marcatili' s theory to be useful in predicting guide wavelength. A selection of typical measured values of loss per unit length for straight powder-filled grooves in a plastic substrate are given in Table II. The measurements were made at 94 GHz. For given substrate material and channel dimensions, the
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TABLE TI AITENUATION OF P OWDER CORE CHANNEL WAVEGUIDES
Powder
Substrate Type
Type
**
Width of
Depth of
Groove
Groove
Density of Powder
(mm)
(mm)
(g/cm3 )
Loss (dB/cm)
D-30••
p
1.17
1.13
1.75 ± .03
0.34 ± .01
MCT 40•
T
1.50
1.05
1.26 ± .03
0.14 ± .01
D-8512•
T
1.27
1.10
1.47 ± .02
0.09 ± .01
D-30•
p
1.17
1.13
1.80 ± .03
0.17 ± .01
D-30•
p
1.17
1.13
1.68± .03
0.09± .01
70% of particles between lOOµm and 43µm, 30% less than 43µm 100% of particles less than 43µm
Substrate Tis PTFE, tan6 = .0002 @ 94 GHz (Ref. 15) Substrate P is polypropylene, tan6 = .0002 @ 94 GHz (Ref. 16)
transmission loss increased with powder density . This variation is consistent with a closer confinement of the mode energy to the core region, which has higher material loss, and with increasing powder loss tangent III. BENDING Loss AND RING RESONATORS Having demonstrated the realizability of straight waveguide sections with useful low loss and the applicability of an existing simple theory to describe the guide, a logical next step was to determine the bending loss of such guides; straight sections alone would not be useful in making complex circuits. Fortunately, Marcatili had also published a theory for curved waveguides [8], and we wished to see how well it applied to the values to be employed here. For other theories of propagation on curved dielectric waveguides, see [9]-[13]. For guides whose curvature radius is large compared to both wavelength and channel width, Marcatili treats curvature as a perturbation. The field distributions of his modes of curved guide are thus only slightly different from those of straight guide. Each mode of the curved guide is given the name of the associated mode of straight guide. Like Marcatili 's theory for straight channel guides, the theory of curved guides assumes that the differences in refractive index between the core and the surrounding media are small. For our powders and substrate material s, Marcatili 's theory predicts that, at 94 GHz, bending losses would be insignificant compared to our measured absorptive losses in straight guides for radii of curvature greater than about 1 cm. We checked this prediction by measuring the transmission losses of guides machined in 180° circular arcs, coupling in and out exactly as we had with the straight sections. We found that a 1 cm radius of curvature gave a measured loss much higher than that of a straight guide. On the other hand, the measured losses of 180° arc guides with 4 cm and 5 cm radius of curvature were
178mm
Fig. 4. Coupling scheme used to achieve adjustable distances between a ring resonator and two straight waveguides.
comparable to straight sections. Something seemed to be amiss between experiment and theory. Coupling to 180° arcs proved more difficult than coupling to the straight sections, particularly for the small radius arcs (1 cm). We were concerned that the transition from the straight "trough" section inserted in the flared metal waveguide to the curved section might produce a radiative loss. It occurred to us that a neat technique to eliminate this possible source of error was to use 360° arcs, or ring resonators. The guide loss could then be determined from the measured Q of the resonator under very loosely-coupled conditions. We decided to build one ring resonator with a 4 cm radius and another with a 5 cm radius. Coupling to the ring resonators was achieved by placing straight channel guides in proximity to the ring guide. We know of no theory to analyze such a coupling structure; Marcatili treats only the coupling between parallel straight guides [7] , predicting a coupling that is approximately an exponentially decreasing function of the spacing. We reason that the coupling for the ring-to-straight guide spacing will also vary exponentially with the guide spacing. Thus, we developed a technique that allowed this spacing to be adjusted easily. Straight channel guides were positioned on opposite sides of the ring to couple power in and out of the ring. The resonator and the guides used for coupling were each built on individual substrates. Material was cut away from two opposing edges of the resonator substrate until each edge was only 0.38 mm from the channel. The substrate of each straight guide had an edge that was cut at an angle to the channel. For both of these substrates, the distance between the channel and the edge was 0.46 mm at one end and 4.32 mm at the other. When the three substrates were placed together as shown in Fig. 4, the separations between the resonator and the coupling guides could be adjusted from about 0.9 mm to 4.7 mm by sliding the substrates with respect to one another. The substrate material was chosen to be polypropylene, since it is easier to machine than PTFE. The powder was chosen to be D-30 nickel-aluminum titanate, with all particles less than 43 µm in size. This powder gave the least lossy straight guides at 94 GHz, by measurement on straight powder core channel guides and on cylindrical dielectric guides made by filling hollow PTFE tubes with powder [2], [3] . Marcatili' s theory of straight channel waveguide was used to choose the channel dimensions of the guides used for coupling to the resonator so that only the E'f.1 mode would propagate,
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IEEE TRANSACTIONS ON MICROWAVE THEORY AN D TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
Source
Isolator
Cavity Wovemeter
Aluminum 1 - - - - - - - - - - - . Foil A
Precision Attenuator
Waveguide Coupler
Waveguide Coupler
0
f+-. 9 = 2.04 ± .02 mm for the Ef1 mode give Q = 1500 ± 200.) Thus, we are surprised that some of the actual measured Q values (Table III and Fig. 6) exceed 1700. These results suggest that measurement of the Q of a ring resonator may be a better method for determining waveguide dissipative and scattering losses than end-to-end transmission on a straight guide. The discrepancy in propagation loss values 2 A 94 GHz sweeper would obviously have been a better choice, but the klystron was the only source available.
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TABLE III M EASURED
Q V ALVES OF Radius of
RING RESONATORS
Frequency
Measured Q
Density of Powder (g/cm 3 )
Dielectric Constant
J.76 ± .02
4.28 ± .07
4.0
94.39
1100 ± 200
1.86 ± .02
4.65 ± .08
4.0
94.61
1300 ± 200
J.88 ± .02
4.73 ± .09
4.0
94.24
2400 ± 400
J.95 ± .02
5.06 ±.IO
4.0
94.61
1600 ± 200
2.10 ± .02
5.90 ± .13
4.0
94.31
1200 ± 200
J.67 ± .02
4.02 ± .06
5.0
94.86
810 ± 100
J.70 ± .02
4.10 ± .06
5.0
93.20
930 ± 150
J.78 ± .02
4.35 ± .07
5.0
94.32
1300 ±~00
J.83 ± .02
4.53 ± .08
5.0
94.28
1600 ± 200
1.88 ± .02
4.73 ± .09
5.0
94.47
1900 ± 200
1.89 ± .02
4.78 ± .09
5.0
94.44
1000 ± 200
+R = 4 .0cm
(GHz)
Curvature
0
(cm)
2000
[!'.
+
~z
R=5.0 cm
0
(j)
lr!
1000
++
t+ t
f ++
POWDER DIELECTR IC CONSTANT (E , )
Fig. 6. Measured Q values of ring resonators versus the dielectric constant of the powder (Trans-Tech D-30 nickel aluminum titanate).
'
Powder:
Nickel-aluminum titanate (Trans-Tech D-30) . All particles less than 43 µm .
Channel width: 1.83 mm Channel depth: 1.09 mm for R = 5.0 cm 1.05 mm for R = 4.0 cm
between straight guides and ring resonators is likely due to losses incurred in coupling to the straight guides. We tentatively attribute the initial increase in Q with increasing powder density to reduced bending loss concomitant with increased propagation constant. However, as powder density increases, dielectric loss increases as the fields become more confined to the (relatively) lossy channel and the effective loss tangent of the powder increases. Eventually this effect becomes dominant and the Q begins to decrease with increased powder dielectric constant. Propagation of a higher-order mode (or modes) may also have contributed to the eventual decrease in Q, particularly for the 5 cm radius ring for which the decrease in Q was abrupt. IV. SUMMARY Waveguides consisting of a low-loss, high-dielectric constant powder packed in a rectangular channel in the surface of a plastic substrate exhibit losses as low as 0.09 dB/cm at 94 GHz. These waveguides appear to be attractive for passive millimeter wave integrated circuits. Guide wavelengths calculated with Marcatili's approximate theory are in reasonable agreement with experimental measurements at 94 GHz. Powder loss tangents at 94 GHz were calculated from attenuation measurements. Ring resonators implemented with powder core channel waveguide exhibit Q's as high as 2400 at 94 GHz implying waveguide loss of approximately 0.06 dB/cm or 6 dB/m . Coupling to the resonators was achieved via adjacent straight powder core channel waveguides. V. APPENDIX COMPUTING WAVEGUIDE
Loss
USING MARCATILl'S THEORY
Marcatili ' s theory [7] applies to lossless rectangular dielectric waveguide. The difficulty in obtaining a closed-form
Fig. 7.
Cross section of dielectric waveguide analyzed by Marcatili.
solution for this waveguide is contained in the mixed boundary conditions in the "comer regions" shown shaded in Fig. 7. Marcatili observes that, for a guided mode, a very small fraction of the power propagates in the comer regions. Hence it is reasonable to ignore the fields in these regions. Fields are assumed to exist only in regions 1-5 and these are matched only at the boundaries of region 1. Marcatili simplifies the problem further by assuming that the refractive index difference between region 1 and any other region is small. Stated precisely, Marcatili assumes that 1 - n i /n 1 « 1, i = 2, 3, 4 , 5. (For our channel guides, the left-hand side of this inequality typically equaled 0.25 for i = 3, 4, 5 and 0.5 for i = 2; thus, strictly speaking, we do not satisfy Marcatili ' s approximation.) As a result of these assumptions, the modes found by Marcatili have almost purely transverse fields (TEM) and can be grouped into two sets, E~q and E;q. For both families of modes, the subscripts p and q indicate the number of extrema of the transverse field components in the x and y directions, respectively. The propagation constant of a mode, kz, is found by matching tangential field components at the edges of region 1 in two steps, equivalent to superposing two slab waveguides: n2 - n1 - n4 and n 3 - n1 - n 5. For E~q modes,
kz2 -- k21
-
k2x
-
k2y >
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
1530
where k 1 = n 1 k, k = 27f / >., and >. is the free space wavelength. The constant kx is found by solving the transcendental characteristic equation for the n 3 - n 1 - n 5 slab,
where
A = (wE~/tan 81P1 +WE~ tan 84(P3 + P4 + P5) +and W E~ tan 82P2)/(7r /kz)
in which
6
=
~5 =
((7r/A3)2 - k;)-1/2 ((7r/A5) 2 - k;)-1;2
and
The constant ky is found by solving the transcendental characteristic equation for the n 2 - n 1 - n 4 slab,
d 2 =cos ((kyb/2) + 'Y) exp (b/(2TJ2))
d1 =cos (kxa/2) exp (a/(26)) in which,
d4 =cos ((-kyb/2) + 'Y) exp (b/(2TJ4)) Ji= (a/2) +sin (kxa)/(2kx)
and
h = (b/2) +sin (kyb) cos (2'Y)/(2ky) 13 = (b/2) - sin (kyb) cos (2'Y)/(2ky)
A similar procedure is used to find the propagation constants of the E;q modes. Marcatili' s theory can be used to calculate waveguide loss as a function of material losses for low-loss rectangular dielectric channel waveguides in the same way the calculation is made for low-loss metal waveguide. Namely, we assume that the dielectric losses are so small in all regions that the fields are the same as for the lossless case [ 17]. Losses due to dimensional imperfections (e.g., waveguide wall roughness) and to scattering from material inhomogeneities are not included in this analysis. We also note that for our rectangular dielectric channel guide, regions 3-5 in Marcatili's analysis (Fig. 7) have identical material properties. Since the material losses are assumed to be small, they can be taken into account by multiplying each of Marcatili's field components by the factor exp (-az). Here 2a is given by the ratio of the average power dissipated per unit length, A, to the average power flowing along the guide, F. These quantities, in tum, are found from the field components given by Marcatili, assuming that the loss per unit volume is everywhere proportional to the square of the electric field. In the following, we take Marcatili's arbitrary field constant, M 1 , equal to unity . For the E$q modes, a can be expressed as 3
a= A/2F,
(1)
3 This expression for a could be written as a single large equation, but the form used below proved convenient for calculation.
/5 = (6/2) exp (-a/6)
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BRUNO AND BRIDGES: POWER CORE DIELECTRIC CHANNEL WAVEGUIDE
Note that W E~ tan 8i Pi/ (7f / kz) is the average power dissipated per unit length in region i. The variable F can be expressed as
where
Here, Wi represents the average power flowing in region i. In the equations above, E1 = Re ( E) and Eo is the permittivity of free space. Also, the subscripts 1, 2, 3, and 4 after the variables k, E, n, and tan 8 associate them with the corresponding regions shown in Fig. 7. In addition, we note that a depends implicitly on the mode indexes p and q through kx and ky. If the effective loss tangents of our powders had been known at 94 GHz, (1) could have been used to predict the transmission losses of our channel guides. However, since the 94 GHz effective powder loss tangents were unknown, (1) was used to calculate them from the measured transmission losses of the 1 mode, using literature values [15], [16] of the loss tangents of the substrate materials. (The dielectric properties of heterogeneous media, such as powders, are often called "effective" to distinguish them from the bulk properties of the constituent materials. In this paper, the term "effective" has been omitted when the properties being discussed are clearly those of powders.) Some of the results are given in Table IV. As shown there, the effective loss tangents increased with powder density, as expected [2] . This effect causes the waveguide loss to increase. However, the rise in waveguide loss accompanying increased powder density is not solely due to the increase in effective loss tangent for the powder. The increased effective dielectric constant of the powder also caused a greater fraction of the power of the guided mode to travel in the core, rather than in the lower loss substrate. As shown in Tables II and IV, waveguides made with 43 µm D-30 powder exhibited much lower losses than those using the larger grain D-30. As a result, the computed values of effective loss tangent for the 43 µm D-30 were smaller. The fact that higher losses were measured for the D-30 powder with larger particles suggests that the extra loss is due to scattering. However, we were not able to detect significant scattered radiation with the small horn-plus-Schottky-detector probe for any powder waveguide. In addition, theories of scattering by powders [18], [19] predict negligible scattering loss for any of these powder sizes. We do not know why the coarse D-30 powder is more absorptive.
E¥
REFERENCES [I] T. Itoh, "Dielectric waveguide-type millimeter-wave integrated circuits," Infrared and Millimeter Waves, vol. 4, pp. 199-273, Academic Press, 1981. [2] W. M. Bruno, "Powder core dielectric waveguides", Ph.D. dissertation, California Inst. Tech., pp. 151-154, 1986. [3 ] W. M. Bruno and W. B. Bridges, "Flexible dielectric waveguides with powder cores," IEEE Trans. Microwave Theory Tech. , vol. MTT-36, pp. 882-890, May 1988. [4] C. Yeh, S. B. Dong, and W. Oliver, "Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides," J. Appl. Phys., vol. 46, pp. 2125-2129, May 1975. [5] C. Yeh, S. B. Dong, and W. P. Brown, "S ingle-mode optical waveguides," Appl. Opt., vol. 18, pp. 1490-1504, May 1979. [6] E. Schweig and W. B. Bridges, "Computer analysis of dielectric waveguides: a finite-difference method," IEEE Trans. Microwave Theory Tech., vol. MTT-32, pp. 531-541, May 1984 .. [7] E. A. J. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Syst. Tech. J., vol. 48, pp. 2079-2132, Sept. 1969. [8] _ _ , "Bends in optical dielectric guides," Bell Syst. Tech. J. , vol. 48, pp. 2103-2132, Sept. 1969. [9] D. C. Chang and E. F. Kuester, "Radiation and propagation of a surfacewave mode on a curved open waveguide of arbitrary cross section," Radio Sci., vol. 11, pp. 449-457, May 1976. (10] E. F. Kuester and D. C. Chang, "Surface-wave radiation loss from curved dielectric slabs and fibers," IEEE J. Quantum Electron. , vol. QE-11, pp. 903-907, Nov. 1975. [I I] H. F. Tay lor, "Power loss at directional change in dielectric waveguides," Appl. Opt., vol. 13, pp. 642-647, Mar. 1974. [12] T. M. Benson, P. C. Kendall, and M. S. Stern, "Microwave simulation of optoelectronic ending loss in presence of dielectric discontinuity," IEE Proc., vol. 135, Part J, pp. 325-331, Aug. 1988. [13] L. Lewin, "Local form of the radiation condition: Application to curved dielectric structures," Electron. Lett. , vol. 9, pp. 468-469, Oct. 1973. [14] E. L. Ginzton, Microwave Measurements. New York: McGraw-Hill, 1957, pp. 403-405. [ 15] D. Jablonski, "Attenuation characteristics of circular dielectric waveguide at millimeter wavelengths," IEEE Trans. Microwave Theory Tech. , vol. MTT-26, pp. 667-671 , Sept. 1978. [16] G. W. Chantry, J. W. Fleming, and G. W. F. Pardoe, "Absorption spectra of polypropylene in the millimetre and submillimetre regions," Infrared Physics, vol. 11 , pp. 109-118, 1971. [17] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 2nd ed. New York: Wiley, 1984. [ 18] V. Twersky , "Propagation in pair-correlated distributions of smallspaced lossy scatterers," J. Opt. Soc. Am., vol. 69, pp. 1567-1572, Nov. 1969. [ 19] L. Tsang and J. A. Kong, "Effective propagation constant fo r coherent electromagnetic wave propagation in media embedded with dielectric scatterers," J. Appl. Phys., vol. 53, pp. 7162-71 73, Nov. 1982.
William M. Bruno was born in Ithaca, New York, in 1958. He received the B.S. in electrical engineering from the University of Californ ia at San Diego in 1981 and the M.S. and Ph.D. degrees in electrical engineering from the California Institute of Technology in 1983 and 1986, respectively. His graduate research at Caltech concerned millimeterwave dielectric waveguides. In 1985 he joined TRW as a Member of the Technical Staff, working in the area of optical communications. He investigated high-speed fiberoptic links and a line-of-sight laser transmitter designed to reduce sc intillation caused by atmospheric turbulence. Since 1989 he has been a Project Engineer at TRW working on optical coatings. His primary research activity has been the study of diffuse light scattering from optical coatings. Dr. Bruno is a member of Phi Beta Kappa.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
William B. Bridges (S'53-M ' 6 1-F'70) was born in Inglewood, CA, in 1934. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of California, Berkeley, in 1956, 1957, and 1962 respectively, and was an Associate in Electrical Engineering from 1957 to 1959, teaching courses in communication and circuits. Hi s graduate research dealt with noise in microwave tubes and electron-stream instabilities (wh ich later became the basis of the Vircator). Summer jobs at RCA and Varian provided stimulating experience with microwave radar systems, ammonia beam masers, and the early development of the ion vacuum pump. He joined the Hughes Research Laboratories in 1960 as a member of the Technical Staff and was a Senior Scientist from 1968 to 1977, with a brief tour as Manager of the Laser Department in 1969-1970. His research at Hughes involved gas lasers of all types and their application to optical commu nication, radar and imaging systems. He is the discoverer of laser oscillation in noble gas ion s and spent several years on the eng ineering development of practical high-power visible and ultraviolet ion lasers for military applications. He joined the faculty of the California Institute of Technology in 1977 as Professor of Electrical Engineering and Applied Physics. He served as Executive Officer for Electrical Engineering from 1979 to 1981. In 1983, he was appointed Carl F. Braun Professor of Engineering and conducts research in optical and millimeter wave devices and their applications. Current studies include the millimeter-wave modulation of light and innovation in waveguide gas lasers. Dr. Bridges is a member of Eta Kappa Nu, Tau Beta Pi, Phi Beta Kappa, and Sigma Xi, receiving Honorable Mention from Eta Kappa Nu as an "Outstanding Young Electrical Engineer" in 1966. He received the Distinguished Teaching Award in 1980 and 1982 from the Associated Students of Caltech, the Arthur L. Schawlow Medal from the Laser Institute of America in 1986, and the IEEE LEOS Quantum Electronics Award in 1988. He is a member of the National Academy of Engineering and the National Academy of Sciences, and a Fellow of the Optical Society of America and the Laser Institute of America. He was a Sherman Fairchild Distinguished Scholar at Caltech in 1974-1975, and a Visiti ng Professor at Chalmers Technical University, Giiteborg, Sweden in 1989. He is coauthor (with C. K. Birdsall) of Electron Dynamics of Diode Regions (New York: Academic Press, 1966.) He has served on various committees of both IEEE and OSA, and was formerly Associate Editor of the IEEE JOURNAL OF QUANTUM ELECTRONICS and the Journal of the Optical Society of America. He was the President of the Optical Society of America in 1988, and was a member of the U.S. Air Force Scientific Advisory Board I985-I989.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
Phase Shift and Loss Mechanism of Optically Excited E-Plane Electron-Hole Plasma Ao Sheng Rong and Zhong Liang Sun
Abstract-This paper describes the phase shift and loss mechanism of an optically excited E-plane electron-hole plasma. The formulation based on the integral equation with rapidly convergent spectral Green's function plus a closed form spatial representation is presented. It includes the possible physical effects, among which are the injected light power level, the nonuniform distribution of the plasma and the end discontinuity effect. For GaAs as the inserted semiconductor, it is shown that an optically sensitive regime occurs, where the phase shift is highly influenced by the illumination level and where a peak of the optically induced loss exists. The regime is changed by the distribution profile of the excess carriers. It is also observed that at the high injection light power level, the optically excited plasma behaves like a metallic strip does. The field distributions at the optically excited plasma section are also presented, which support the field-displacement effects of the plasma. I. INTRODUCTION
HE optical control of microwave and/or millimeter wave components is accomplished by passing millimeter wave signals through a medium whose properties (usually of complex dielectric constant) can be dynamically controlled by the optical illumination. The use of bulk optoelectronic effects in semiconductors in implementing optically controlled phase shifters [l], [2], modulators [3], [4], switches [5)-[9] and subpicosecond photoconducting dipole antennas [10) has been demonstrated. To design such devices to good purpose, an understanding of the physical properties of the optically excited electron-hole plasma is necessary. The paper describes the phase shift and the loss mechanism of an optically excited E -plane plasma. This structure has exhibited potential applications for the measurement of the optically induced complex permittivity of semiconductors [11). It has also been analyzed in [12], but their model assumed that the optically excited plasma is of uniform density in the inserted semiconductor and neglected the end discontinuity effects. The present formulation using the integral equation technique incorporates the possible physical effects, among which are the injection optical power level, the nonuniform distribution of the optically generated excess carriers and the end discontinuity effect. The formulation also involves the additional original feature that the integral kernel is expressed in terms of the integral of the fast convergent spectral Green's function plus its asymptotic terms with the analytical closedform representation in space domain. This facilitates the
T
Manuscript received March 22, I993; revised September 27, 1993. The authors are with State Key Laboratory of Millimeter Waves, Department of Radio Engineering, Southeast University, Nanjing 210018, Jiangsu, People's Republic of China. IEEE Log Number 9402943.
1
J_ ~-'--"'-~
l - semiconductor 2 - plasma
~h3 --j ~~ h1 ~
Fig. 1.
Configuration of the optically excited E-plane electron-hole plasma.
numerical computation of the scattering parameters of the optically excited E-plane plasma. In the following, the formulation will be summarized. II. FORMULATION
Fig. 1 shows the configuration under investigation. The semiconductor is longitudinally inserted in a rectangular waveguide and, at the section (0 < z < w), is laterally illuminated by laser radiation. The injected photons generate free carriers, or electron-hole plasma, over a thin layer directly under the surface. The free carriers are assumed to be uniform on the plane (0 < y < b, 0 < z < w ), but inhomogeneously distributed inside the semiconductor along the x direction due to the diffusion effect and the exponential absorption of the photon energy. For a more refined study, the diffusion of free carriers along z direction will be considered. For simplicity, the effects of optical injection holes are not addressed in this paper although their presence is unavoidable in practical applications. When the end discontinuity effect is taken into account, the problem of interest concentrates on calculating the scattering parameters of the optically excited E -plane plasma. The model above implies that the electric field in Fig. 1 has a y-component only, which is independent of the same variable. Suppose an incident field E~( x, z) of the dominant mode
Et(x, z ) = ¢(x)exp(-jf3z) ¢(x)
{
Asin(~x)
B sin 'T)( x - h 1 ) + Ccos 'T)(x - h 1 ) D sin~(a - x )
0018-9480/94$04.00 © 1994 IEEE
0 ::::; x < h1 h1 :::; x < h1 + h2 h1 + h2:::; x .. 9 from the input end and the output end of the plasma, respectively, in order that the higher order modes excited by the plasma have negligible amplitudes at the shifted reference planes. This arrangement ensures that S 11 and S2 1 can characterize the change of the dominant mode wave passing the plasma. The difference between the phase angles for sg1 and S~ 1 , without and with illumination, gives the phase shift due to the plasma ~ ¢, i.e., ~ = LS~1
+ 27r + f3 L .
The optically induced loss a can be derived from
a = 1-
ISf11 2 -
IS~11 2
(5)
Sf1 and S~1 (6)
Equation (1) is the Fredholm's integral equation of the 2nd kind, the kernel of which is slowly convergent and is rapidly oscillating for large kz. The direct implementation of (1) will be accompanied by poor numerical efficiency. Great improvement in numerical efficiency can be achieved by using the decomposition, or asymptotic extraction technique as follows : G(x Ix', kz) in (3) can be found in the form: -
1
G(x l x' , kz) = J.k x2 ( 1 - R 1 R 3e -2 J·kx2 h) 2 (e - jkx2X + R 3 ejkx2(x - 2h2)) ·(ejkx2x' + Ri e- jkx2 x' )(x > x') X ·(ejkx2x + Ri e- jkx2x ) { ·(e-jkx2 x' + R 3 ejkx2 (x'-2h2l )(x
Rewrite (7) in the form of
G(x
Ix', kz )
= (G( x
Ix', kz ) - Gas(XIx', kz)) + Gas(X Ix', kz). (9)
In (9), the first term exhibits rapid convergence. Compared with G(x Ix', kz ), it has values significantly different from zero only over a narrow range of the spectral variables. As a result, its infinite integral can be truncated in reasonably moderate value kzm• thus causing a dramatic reduction of CPU time. Careful attention to the second term of (9) reveals that Gas (x lx',kz ) corresponds exactly to the Green's function due to a linear distribution of charges with intensity Eo in a rectangular waveguide, or due to a point charge with intensity Eo between pairs of infinite plates. Gas(x Ix', kz) is the spectral expression of the solution to the following equation satisfying the homogeneous boundary conditions of the first kind at x = 0 and x = a:
82¢ 82¢ / / x + z = -8(x - x )8(z - z ) 8 2 8 2
lx=O
< x') (7)
= O;
(10)
lx=a = 0.
When (10) is solved by using the eigen-function expansion in terms of the x variable, an alternative representation
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RONG AND SUN: PHASE SHIFf AND LOSS MECHANISM OF OPTICALLY EXCITED E-PLANE ELECTRON-HOLE PLASMA
aera of the Ith cell • Al
Im ( kz)
observation point
A
B
(Xio ,Zio) +
c
P;
Pi
~~ oodpol~;
D'C
Pi 1 tho Ith odgo :
po•Wwlpolol
D
Re ( kz) B '
A'
Fig. 2. Meanings of the quantities defined in (15). Fig. 3.
G~ 5 (x,
Path of the spectral integral in ( 16).
z Ix' , z' ) of Gas (x, z Ix' , z' ) results
G~(x, z Ix' , z' ) =
oo
L nl'lr sin (na7r x ) sin (n: x') e- na~ lz- z'I n =l
(11 )
where G~5 (x, z Ix' , z' ) is exactly equal to Gas (x, z Ix' , z' ). It can be shown that G~(x, z Ix' , z' ) or Gas(x, z Ix' , z' ) has the analytical closed-form representation, i.e., , , 1 [cosh ~( z - z' ) - cos ~ (x + x' ) ] ( ) ( ) . Gas(x, z Ix , z ) = - ln 47r cosh ~ z - z' - cos ~ x - x ' (12)
When the observation point coincides with the source point, Gas(x , z Ix' , z' ) offers a logarithmic singularity that can be analytically extracted from Gas (x, z Ix' , z') via
Gas(x, z Ix' , z' ) = (Gas (x, z Ix' , z') - Go(x , z Ix' , z' )) +Go(x,z lx' ,z') (13) 1 7r ~---c--=--:------:-~ (14) 27r a The first term of (13) is regular from point to point. When D.. V is segmented into rectangular cells, its integral over the cells can be accomplished by using the two-dimensional Gaussian rule. On the other hand, the singular term Go( x, z Ix ', z' ) can be analytically integrated over the cells by using Gaussian integral theorem. For the ith cell, the integral of Go(x, z Ix' , z' ) is found [13]
Go(x, z Ix' , z') = --ln-y1(x - x') 2 + (z - z') 2 .
j liGo(x,z lx' ,z')dx'dz'
~ ~ P 0 [z+1nP+ -
- 47r
~
t
t
t
=
- 2~ Ailn(~)
z:-1np. t t
+pO t
i =l
. [tan- 1
zt - tan- 1
pio
where
z; ] - ~(z+ PP 2 '
z:-)] ,.
(15)
The quantities in (15) are defined in Fig. 2. Specifically when the observation point is located at the centrum of the cell, the second term in (15) will vanish. Expand Ey (x, z) in terms of the rectangular pulse functions. The Delta functions are chosen as testing functions . Following the standard procedure of the method of moments, one obtains from (1)
[Z][J]= [VJ
(16)
Zij = Dij - k6{
2~ 1-+: [i
x; [ (G(x Ix' , kz)
- Gas(X Ix', kz)) · i z; D..Erp(x', z') e-j k,(z-z')dz' ] dx' ] dkz
+
Jl
D..Erp(x',z')
X
(Gas(x,z lx' ,z')
J
- G0 (x, z lx' ,z'))dx 1 dz 1
+ D..Erpi j
l;
Go(x ,z lx' ,z')dx'dz' } ·
D..Erpi represents the average value of D..Erp over the ith cel l. Equation (16) is a determini stic equation, the right-hand terms of which are the known excitation vector. After solving the expansion coefficient vector [I] for Ey (x, z ), one can obtain the scattering parameters 8 11 and 82 1 from (4). It should be noticed that if the inserted semiconductor is lossless without illumination, the integrand involved in the above spectral integral contains the poles along real kz axis. These poles appear in pairs and correspond to the propagation constants of the propagating modes along the ±z directions. To include the contribution of these poles into the integral, the integral path is chosen as shown in Fig. 3, where the lengths of section OC' and of section OC should be larger than the maximum propagation constant /Jmax. III. NUMERICAL RESULTS AND D ISCUSSIONS
The formulation described above has been used to compute extensively the phase shift and the attenuation characteristics of the optically excited E-plane electron-hole plasma, where the inserted semiconductor is assumed to be GaAs. Its material and property parameters have been summarized in [2]. Figs. 4 and 5 show the amplitudes 1811 I and 1821 I of the scattering parameters as fu nctions of the diffusion length La and the optically excited excess carrier density Nvo at the surface of GaAs.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
ISlllc -SO
0.8
_,,_La
= 10 µm
1l
-o- La = 25 µm ..... La= 40µm
0.6
:::
tll
~
~
:
0.4
-150 -200
0.2
0
..-.La= lOpm -o-La = 25 pm ....-La= 40pm
-100
-250
L......................_.=..i......~.L..............J.~.....i...~..J
1012
1013
1014
101.'l
1016 3
NpO ( 1/cm
1017
1018
L.....~L.............i~....t~u...L~.....t..~....._~.....
1012
1019
1013
1014
101.'l
1016
NpO ( l/cm
)
Fig. 4. Amplitude ISi i I of the reflection coefficient versus the surface excess carrier density, with the diffusion length as a parameter. Where the dotted line designates S ill lc for the metallic strip case.
3
1017
1018
1019
)
Fig. 6. Phase shift versus the surface excess carrier density, with the diffusion length as a parameter.
1
--La= lOpm ~La "'
0.8
lOpm
-0-La = 25 pm
0.8
-o-La "' 2S pm ....-La"' 40pm
....-La= 40pm
0.6
0.6
.....
!I! 0.4
0.4
IS2llc 0.2
0.2 0
W....'-"'"'L.............i~....t~u...L~.....t..~....._~.....
1012
1013
1014
101.'l
1016 3
NpO ( 1/cm
1017
1018
1019
)
Fig. 5. Amplitude IS21 I of the transmission coefficient versus the surface excess carrier density, with the diffusion length as a parameter. Where the dotted line designates S12 i1c for the metallic strip case.
Where the waveguide dimensions are a x b = 2.54 x 1.27 mm 2 , h 1 = h 3 = 1.17 mm . For GaAs, the thickness h2 = 0.2 mm and the permittivity without illumination Er d = 13.1. The plasma length is maintained at w = 1 mm . The millimeter wave signal is generated by a source operating at a frequency f = 94 GHz. As indicated in Figs. 4 and 5, for all the given diffusion profiles of the plasma, ISn I and IS21 I exhibit a inverse dependence on the excess carrier density. The valley shape of IS21 I corresponds to the maximum optically induced loss. It can be concluded that, as the injection light power level becomes high, the millimeter wave signal wi ll propagate through the plasma at the expense of a large attenuation . Meanwhile, the reflection from the input end of the plasma will increase. At a sufficiently intense illumination (about Npo > 10 17 l/cm 3 ), both ISn l and IS21I exhibit an asymptotic behavior. In order to understand the physical background of these behaviors, the case of an E-plane metallic strip on the semiconductor substrate, with the same dimensions and operating conditions as the plasma, is also analyzed by using the program based on the generalized analysis by
0
L.......~'11E:!::::U:UwL.~=L~.....i.~........L_.......wloliE"(~
1012
1013
1014
1015
1016 3
NpO ( 1/cm
1017
1018
1019
)
Fig. 7. Optically induced loss versus the surface excess carrier density, with the diffusion length as a parameter.
one of these authors [14]. In Figs. 4 and 5, the horizontal dotted lines represent the amplitudes ISn lc and IS21 lc of the scattering parameters for the strip case. As observed, IS11 I and IS2 1I approach their limiting values ISnlc and IS21lc as the illumination becomes intense. This observation implies that when the illumination is intense enough an optically excited E-plane plasma behaves as an E-plane metallic strip on the semiconductor substrate. Figs. 6 and 7 illustrate the curves for the phase shift and the optically induced loss. It will be noted that an optically sensitive regime occurs where the phase shift is highly affected by the illumination and where a peak of the optically induced loss exists. It also be observed that the regime is changed by the diffusion length of the excess carriers. The dependency of a on Npo indicates that, under intense illumination, the plasma is mainly due to the reflection from the input end of the plasma rather than to energy dissipation. Figs. 8-11 show ISnl, IS21 I, the phase shift !1¢ and optically induced loss a versus the distance h1 between the
RONG AND SUN: PHASE SHIFT AND LOSS MECHANISM OF OPTICALLY EXCITED E-PLANE ELECTRON-HOLE PLASMA
1537
--hl = 0.1 mm 0.8
-bl= 0.1 mm
0.8
-o-hl = 0.3 mm
ohl =OJ mm ...-bl• 0.5 mm
...-bl= 0.5 mm 0.6
0.6
0.4
0.4
0.2
0.2
..... .....
"'
0
0
L........._j..........."".. Numerical results are obtained in the following examples by choosing L = 1.5>.. Fig. 5 shows the phase and magnitude of the reflection coefficient r for the via structures with ail a = o.75, b/a = 2, and h/ a 1 = 20 or 30. The solid curves are obtained by the present analysis. The phase is observed to be approximately -90° at the quasi-static limit and decreases almost linearly versus frequency. This implies that the via structure is dominantly capacitive. Additionally, the via structure exhibits a resonance phenomenon where if I reaches its maximun at the frequency of kh ~ 0.65, i.e., the via height h is roughly one tenth of a wavelength. The dashed curves also included in Fig. 5 for comparison are based on the quasi-static model shown in Fig. l(b). The model parameters, the excess capacitance Ce and inductance L e, are obtained from [2] and the reflection coefficient is given by
r
-jwCe[2 - Y02 L e/ Ce - w2 CeLe] = 2Y0 (1 - w 2 L eCe)+ ,jwCe [2 + Y} Le/Ce - w 2 LeCe] ( 18)
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
1544
1.0 1.0
.--==-----------~
IT I
0 .8
0 .8
L .. ...... 1.001' - - - - 1. 5 01' - - 1.751'
0. 6
e·; ;'!_;--..o
h/a 1 30 -- 2 0
,,, ,
--
1,
,, •
0.6
Lo ss
0.4
• 1 11 11 11 11 '1
lrl 0.4
0 .2
11 11 • 1 11 11 '1
0 .2 0.0
\ \
0.0
0 .4
I
1 1 II II II II
0 .8
k*h
1. 2
''
1.6
Fig. 4. Curves showing the convergence of the calculated propagation characteristics with truncation length L and number of unknowns N as the parameters.
where Yo is the characteristic admittance of the transmission line. As the frequency increases, (18) predicts that r would reach its resonance at Wres = J(2 - Y02 Le/Ce)/3LeCe and become reflectionless at w0 = V3wr es · The results by the quasi-static model and our full wave analysis are in good agreement at low frequencies and both present a similar resonance phenomenon except for a slight frequency drift. However, the quasi-static model deteriorates above the frequency Wres and fails to properly predict the magnitude of r . As the frequency becomes higher than Wo , the quasi-static model becomes so insufficient that not only the predicted magnitude but also the phase is entirely wrong. The reflectionless phenomenon does actually not occur, if the non-negligible radiation resistance at higher frequencies is counted. To be more quantitative, the results show that the quasi-static model is valid until kh = 0.4, which con-esponds to an operating frequency of 6.4 GHz if the via height is h = 3 mm. For further verification of this study, the finite-difference timedomain (FD-TD) technique is applied to obtain the scattering parameters [5]. Consider a through hole via structure with a 1 = a = 60µm , b = 180µm, and h = 1.8 mm. In the FDTD simulation, the circular wire, via, and hole are replaced by square ones with sizes 2a, 2a, and 2b, respectively. The solution region is truncated into a rectangular box and the first-order Mur's boundary condition [6] is employed. The truncated region is discretized using an 80 x 240 x 180 mesh, that is, 4.8 mm in width by 14.4 mm in length by 10.8 mm in height. The incident wave is assumed to be a Gaussian pulse having a quasi-TEM modal distribution along the transmission line. Fig. 6 compares the results obtained by the present approach and the FD-TD technique, which are denoted by the solid and dashed curves, respectively. The agreement between these two sets of curves is not satisfactory quantitively, since the structures considered in these two analyses are slightly different. Nonewtheless, both methods justify the high frequency radiation and predict similar progagation characteristices qualitatively. Fig. 7 shows the propagation characteristics versus the frequency for via structures with aif a= 1, b/a = 2, and h/ a as a parameter. At the low frequency limit, it is found tha lfl --+ 0, ITI --+ 1, and the radiation loss tends to zero, as expected intuitively. Although the reflection is smaller as
Full-wave ,.,1, - - - Quasi-static 11
-
'1J>'
0.0 0.0
0.4
0.8
1.2
1.6
k*h (a) 0
;· i''
h/a 1 30 -- 2 0
I
--
- 90
I
'O~,
'' I I I I I I I I I
Q.)
C/J
' '
""''
'""''
I I I I I I I
'-t;,:- .... 'Et" ~-:o
..c:(ij - 180 0.. -270 -
Full-wave - - - Quasi-static
-360 0.0
0 .4
0.8
1.2
1.6
k*h (b)
Fig. 5. Comparison between the full wave analysis and the quasi-static model which are denoted by solid curves and dashed curves, respectively, (a) amplitude, (b) phase. The via structures have parameters aif a = 0.7 5, b/a = 2 and h/a1 = 20 or 30.
LO
IT I 0.8
- - Full Wave - - - FD-TD (80X240X180)
0 .6
/
>'
/ / /
Loss
/
0.4
0.2
0.0 0 .0
0.4
0.8
1.2
1.6
k*h Fig. 6. Comparison between the results by the present approach and the FD-TD technique, which are denoted by solid and dashed curves respectively. The via structure has parameters ai = a = 60µm, b = l80µm, and h = 1.8 mm.
the frequency goes higher than the resonance frequency, the transmission coefficient always decreases and the radiation loss increase versus the frequency. The curve with h/a = 30 shows that the radiation loss may become as high as 0.4
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HSU AND WU: FULL WAVE CHARACTERIZATION OF A THROUGH HOLE VIA USING THE MATRIX-PENCILED MOMENT METHOD
(or 4 db) at kh = 1, which corresponds to an operating frequency of 16.7 GHz if the via height is h = 3 mm . This phenomenal situation clearly depicts that the via structure would cause severe electromagnetic interference (EMI) at ordinarily high frequencies. Fig. 8 shows the propagation characteristics versus the frequency for the via structures with h/a 1 = 30, b/a = 2, and ai/a as a parameter. It is observed that the reflection and radiation loss can be reduced by choosing a smaller via radius. Fig. 9 shows the results with ai/a = 1 while b/a is chosen as a parameter. The reflection is larger if b/a becomes smaller. Thi s occurrence can be attributed to a larger excess capacitance Ce since r ~ -jw(Ce/Yo - ~L e Yo), based on (18) at low frequencies. Additionally, radiation loss is notably almost independent of the via hole radius b. Figs.10 and 11 present the propagation characteristics versus the via height h while choosing various via radii a and hole radii b as the parameters. The operating frequency is taken to be 10 GHz and the radius of the transmission line is a 1 = 50µm . The reflection and radiation Joss generally become larger as the via height increases. For a fixed via height, Fig. I 0 depicts that the reflection and radiation loss decrease for smaller via radius a. As found from Fig. 11, the reflection can be reduced slightly by choosing a larger via hole, but the radiation loss remains almost invariant. In summary, better transmission characteristics can be obtained by choosing a shorter and thinner via and wider hole. Among the three parameters, the via height is dominant and the hole radius is the least relevant. Although this paper mainly deals with a via through a hole in a ground plane, the analysis procedure yields important information regarding the two constitutive problems as a by-product. For example, the associated short circuit problem shown in Fig. 2(c) is next considered. The normalized equivalent impedance Z sc of the grounded via is given by Zsc
= (1 + f sc)/(1- f sc) ·
(19)
Fig. 12 shows the real and imaginary parts of Zsc versus the frequency for the via structures with a 1 = a, b = 2a, and h/ a 1 as a parameter. As predicted in [2], the grounded via is basically inductive and can be modeled by a lumped inductance at low frequencies. Using the closed-form formula for the lumped inductance Le derived in [2] for the case of h/ a 1 = 40, the resultant dashed straight curve (Zs c = jwLeYo) is also shown for comparison sake. The agreement is good at low frequencies. However, the quasi-static assumption in [2] starts to fail at a frequency of about kh = 0.4, above which the radiation loss plays its role and the real part of Z sc is no longer negligible as compared to the imaginary part. Assuming the same via dimensions, Fig. 13 shows the input admittance Yant of the associated wire antenna as shown in Fig. 2(b). At low frequencies, the nearly linear behavior of the imaginary part of Yant depicts that the wire antenna is capacitive. The real part of Yant tends to become a constant at the quasi-static limit due to the presence of the infinite transmission line. The constant is observed to actually equal the characteristic admittance of the transmission line Yo~ 27r/(770 ln(2h/a 1 )). The difference between the real part of Yant and the characteristic admittance Yo can be defined as the radiation conductance Yrad, which is
1.0
:~·":·· '•••,,,ITI• •.·
0.8
- - 20 -------- 30 - - - - 40
0.6
0.4
0 .2
0.0 0.0
0.4
0.8
1. 2
1.6
k*h Fig. 7. The propagation characteristics versus the frequency for the via structures with aif a = 1, b/ a = 2 and h/ a 1 as a parameter.
1.0
0.8
a,/a -------- 0.75 I.DO - - 1.25
0.6
········'~'········y: .. -;·:·; .. -;·;"
0.4
/
Loss
,..:..;""'.,;·...:~·-=-= ... .,,........ •·;" .,~
0.2
0.0 0.0
0.4
0.8
1.2
1.6
k*h Fig. 8. The propagation characteristics versus the frequency for the via structu res with h/a1 = 30 , b/a = 2 , and aif a as a parameter.
proportional to the radiated power. As expected, the radiation loss becomes significant at high frequencies . It is interesting to compare the radiation conductance Yant of the wire antenna in Fig. 13 with the radiation resistance Re(Zsc) of the grounded via in Fig. 12. As the frequency increases, the radiation conductance Yrad is observed to increase and then slightly decreases. Comparatively, Re(Zsc) first increases gradually but then more drastically. Thi s in tum reveals the radiation mechani sm shown in Fig. 7 of our original through hole via structure. At low frequencies, a larger part of the radiation loss is a result of the associated wire antenna structure. However, at substantially higher frequencies , the capacitance tends to short circuit the hole such that the radiation loss can be attributed to the associated grounded via structure. V. CONCLUSIONS
A full wave approach was presented in this study for investigating the frequency dependent propagation characteristics of a via through a hole in a ground plane. The problem was first decomposed into a short circuit problem and a wire antenna problem, for each of which only the structure in a half space required consideration. The current distribution on the via and a truncated section of transmission line was solved by the
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 8, AUGUST 1994
1.0
3 .---------------~
0.8
b/a
h /a,
-------- 1.5 - - - - 2.0 - - 2 .5
0 .6
- - 20 ----- --- 30 - - - - 40
2
0.4
0.2
0.0 0.0
0.4
0.8
1.2
0 .0
1.6
0.4
k*h Fig. 9. The propagation characteristics versus the frequency for the via 30 , and b/ a as a parameter. structures with aif a 1, h/ a1
=
0.8
1.2
1.6
k*h
=
Fig. 12. The normalized impedance Z sc of short circuit problem in Fig.2(c) versus the frequency for the via structure with aif a 1, b/ a 2 and h/ a1 as a parameter.
=
=
1.0 - -- --~ ...... ....................... ...._....- a..-z...- s..- ,.,.-=--=-- ..... =--=--=--=----=--
--------------~
10 0.8
IT I
a ---- ---- 3 0 µm - - - - 40 µm - - 60 µm
0.6
h/a 1 - - 20 -------- 30 - - - - 40
8
6 0.4
Yant
lfl 0.2
4
~-=--=--=-- =--=-- -:- =--=--=--=--=--=--=--=------------- - ----- -
Loss
(mo) 2
Im(Y •• 1)
0 .0 1.0
1.5
2 .0
2 .5
h (mm)
0 .0
Fig. I0. The propagation characteristics at lO GHz versus h for the via 50µm, b 100 µm, and a 30, 40, or 60 µm. structures with a1
=
=
=
0.4
0 .8
1.2
1.6
k*h Fig. 13. The normalized input admittance Yan t of the wire antenna problem in Fig. 2(b) versus the frequency forthe via structure with aif a = 1, b/ a = 2 and h /a i as a parameter.
1.0
IT I
b
0 .8
- - 75 µm 100µ,m ---- 150µ,m
0.6
0.4
lrl 0.2
~-·-~ ·-·-·· ·· ·-···- ···---·-·-~·---·-···- ·-···--- ·- ·
Loss 0.0 1.0
1.5
2.0
2.5
h (mm) Fig. 11. The propagation characteristics at lO GHz versus h for the via structures wi th a 1 = a = 50µ m, and b = 75 , 100, or 150 µm.
moment mothed and the scattering parameters were extracted by the matrix pencil method. It is found that satisfactory convergence can be achieved in the present approach if the transmission line is truncated as least 1.5 >. away from the discontinuities. The present analysis was also compared
with the convensional quasi-static analysis. The correlation remained satisfactory up to the frequency that the via height is about one tenth of a wavelength. As the frequency becomes higher, the radiation effect is significant and the quasi-static model fails. Numerical results were presented for investigating the frequency dependence of the propagation characteristics and the effects due to various geometrical parameters of the via structure. Roughly speaking, the radiation loss may be as high as 0.4 ( 4 db) if the via height is larger than 0.16 >.. For reducing the radiation loss and also achieving more enhanced transmission characteristics, this study would recommend choosing a shorter and thinner via, while a wider hole would function in a minor role. REFERENCES
[I] T. Y. Wang, R. F. Harrington, and J. R. Mautz, "The equivalent circuit of a via," Trans. Soc. Comput. Simulation, vol. 4, pp. 97-123, Apr. 1987. [2] _ _ , "Quasi-static analysis of a microstrip via through a hole in a ground plane," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. l008-l013, June 1988. [3] _ _ , "The excess capacitance of a microstrip via in a dielectric substrate," IEEE Trans. Computer-Aided Design, vol. 9, pp. 48-56, June 1990.
HSU AND WU: FULL WAVE CHARACTERIZATION OF A THROUGH HOLE V IA USING THE MATRIX-PENC ILED MOMENT METHOD
[4] P. Kok, and D. De Zutter, "Capacitance of a circular symmetric model of a via hole including finite ground plane thickness," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 1229-1 234, July 1991. [5] S. Maeda, T. Kashiwa, and I. Fukai, "Full wave analysis of propagation characteristics of a through hole using the finite-difference time-domain method," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 2 154-2 159, Dec. 1991. [6] G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compal., vol. EMC-23, pp. 377-382, Nov. 198 1. [7) W. D. Becker, P. H. Harms, and R. Mittra, "Time-domain electromagnetic analysis of interconnects in a computer chip package," IEEE Trans. Microwave Theory Tech. , vol. 40, pp. 2155-2163, Dec. 1992. [8] D. G. Swanson, Jr. , "Grounding microstrip lines with vi a holes," IEEE Trans. Microwave Theory Tech. , vol. 40, pp. 1719-1721 , Aug. 1992. [9] R. Sorrentino, F. Alessandri, M. Mongiardo, G. Avitabile, and L. Roselli , "Full-wave modeling of via hole grounds in microstrip by threedimensional mode matching technique," IEEE Trans. Microwave Theory Tech. , vol. 40, pp. 2228-2234, Dec. 1992. [10] E. Zheng, R. F. Harrington, and J. R. Mautz, "Electromagnetic coupling through a wire-penetrated small aperture in an infinite conducting plane," IEEE Trans. Electromagn. Compal., vol. EMC-35, pp. 295-298, May 1993. [I I] P. B. Katehi and N. G. Alexopoulos, "Frequency-dependent characteristics of microstrip discontinuities in millimeter-wave integrated circuits," IEEE Trans. Microwave Theory Tech. , vol. MTT-33, pp. 1029- 1035, Oct. 1985. [12] T. Becks and I. Wolff, "Analysis of 3-D metallization structures by a full-wave spectral domain technique," IEEE Trans. Microwave Theory Tech., vol. 40, pp. 2219-2227, Dec. 1992. [13] Y. B. Hua and T. K. Sarkar, "Generalized pencil-of-function method for extracting poles of an EM system from its transient response," IEEE Trans. Antenna Propagat., vol. 37, pp. 229-234, Feb. 1989. [14] K. K. Mei, "On the integral equations of thin wire antennas," IEEE Trans. Antenna Propagat. , vol. AP-13, pp. 374-378, May 1965 .
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[15] R. Mittra Ed., Computer Techniques for Electromagnetics. New York: Pergamon, 1973, ch. 2. [ 16) G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: Johns Hopkins University Press, 1983, ch. 5. [ 17] J. P. Keener, Principles of Applied Mathematics. New York: AddisonWesley, 1988, pp. 38-43 .
Show-Gwo Hsu was born in Chunghwa, Taiwan, Republic of China, in 1966. He graduated from the Nation al Taipe i Institute of Technology, Taipei , Taiwan in 1986, and received his M.S. degree from the National Taiwan Un iversity, Taipei, Taiwan, in 1991. Currently, he is pursuing his Ph.D. degree in electrical engineering at the National Taiwan Univers ity. Hi s research interests include electromagnetic theory, antenna theory and measurement, and electronic packaging analysis.
Ruey-Beei Wu was born in Tainan, Taiwan, Republic of China, in 1957. He received the B.S.E.E. and Ph.D. degrees from National Taiwan University, Taipei, Taiwan, in 1979 and 1985 , respectively. In 1982, he joined the faculty of the Department of Electrical Engineering, National Taiwan University, where he is now a Professor. From March 1986 to February 1987, he was a Vi siting Scientist in IBM General Technology Division laboratory, East Fishkill Facili ty, Hopewell Junction , New York. Hi s areas of interest include computational electromagnetics, dielectric waveguides, slot antennas, transmission line discontinuities, and interconnection modeling for electronic packaging.
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IEEE TRA NSACTIONS ON MIC ROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
Coupling Between Different Leaky-Mode Types 1n Stub-Loaded Leaky Waveguides Hiroshi Shigesawa, Fellow, IEEE, Mikio Tsuji, Member, IEEE, Paolo Lampariello, Senior Member, IEEE, Fabrizio Frezza, Member, IEEE, and Arthur A. Oliner, Life Fellow, IEEE
Abstract-Several leaky-wave antennas have been described in recent years that possess excellent properties, and they all have in common a parallel-plate stub guide of finite height as part of the cross section. For taller stubs and for larger leakage rates (wider radiated beams), some interesting new exotic interactions occur due to coupling between the desired leaky mode and another leaky mode, which is a modification of the "channel-guide" mode. Although we have described such coupling behavior briefly previously, this basic new feature (the coupling between two leaky (complex) modes) is not generally known, and is discussed here in more detail and in a more general context. In addition to a broad qualitative discussion, numerical results are presented for structures based on NRD guide and on stub-loaded rectangular guide. We believe that these effects are universal to all leaky-wave structures that possess a finite stub height in the cross section, so that the discussion here serves as a model for what to expect for other structures.
I. INTRODUCTION
A
leaky-wave antenna is basically an open waveguide that supports a leaky mode (instead of a bound mode) and that therefore radiates all along its length, producing a travelingwave line-source radiator. Several leaky-wave antennas have been described in recent years that possess excellent properties, such as great versatility in the beamwidth, negligible cross polarization in the radiated beam, the ability to control the beam angle and the beamwidth essentially independently, etc. All these leaky-wave structures have in common a parallelplate stub guide of finite height as part of the cross section. For taller stubs and for larger leakage rates (resulting in wider radiated beams), we found some unexpected but interesting exotic behavior in the phase and attenuation characteristics as the stub height was varied. These initially puzzling effects are due to the simultaneous presence of another leaky mode, and to coupling between thi s additional leaky mode and the desired leaky mode. The additional mode is a modification of the "channel-guide" mode that was known about 30 years ago; its properties will be described later in this paper. We described some of these new effects, and why they occur, in two publications [l], [2], but since those descriptions appeared Manu script rece ived April 21 , 1993; revised October 11 , 1993. Thi s work was supported in part by Doshisha University 's Research Promotion Fund in Japan, in part by the Mini stero dell 'Un iversita e dell a Ricerca Scientifica e Tec nologica and the Consigli o Nazionale delle Ricerche in Ital y, and in part by the Air Force RADC Contract No. F l 9628-84K-0025 . H. Shigesawa and M. Tsuji are with the Department of Electronics, Doshi sha University, Kyoto, Japan. P. Lampariello and F. Frezza are with the Dipartimento di Ingeg neri a Elettronica, Uni versita "La Sapienza" di Roma, Rome, Italy. A. A. Oliner is with the Departme nt of Electrical Engineering and Weber Research Institute, Polytechnic Uni versity, Brooklyn, NY, USA. IE EE Log Number 9402945 .
only in digest papers, and only with other material in a larger context, the discussions were necessarily brief. Many more results were presented in a comprehensive report [3], but the report received only limited circulation. We believe that these effects are universal to all leakywave structures that possess a finite stub height in the cross section. For this reason , these coupling effects are described here in more detail and in a more general context, as a model for what to expect for other structures. As illustrative examples, we discuss in detail two specific structures, an asymmetric NRD (nonradiative dielectric) guide and a stubloaded rectangular guide, as extensions of the previously publi shed material [ 1], [2]. The interactions to be described involve the coupling between two leaky (complex) modes, so that the modes will not couple unless both their values of phase constant /3 and attenuation (or leakage) constant a are the same. That requirement has probably served to make such coupling relatively rare, and also serves to explain why these effects are not generally known. Since it is desirable to illustrate the coupling effects by reference to specific structures, we present in Section II the two specific leaky-wave structures and describe briefly their principles of operation. The two structures are shown in Figs. I and 2. In that context, we explain for each case the mechanism of radiation and the function of the parallel-plate stub guide. We also comment on the theoretical approaches used to obtain the numerical values of the propagation characteristics. For the asymmetric NRD guide, we employed the mode-matching procedure. For the stub-loaded rectangular guide two quite different methods of calculation were used : the mode-matching procedure and a novel transverse equivalent network. Excellent agreement was found between the numerical values obtained by the two different methods, and between those values and measurements taken at X -band and in the 40 GHz to 60 GHz range. Before presenting the details of the coupling behavior between the desired leaky mode and the channel-guide leaky modes (the dominant plus higher-order modes) for the two structures mentioned above, we discuss in Section III the nature of the channel-guide modes, why they arise, and how they differ from the desired leaky modes. We then examine in general terms the conditions necessary for coupling, and show qualitatively how these modes behave in very different ways when the leakage rate is small and when it is large, that is, before the two modes couple and then when they do couple. The coupling effects in the propagation characteristics are then described quantitatively in Section IV for the asymmetric
001 8-9480/94$04.00 © 1994 IEEE
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SHIGESAWA et al.: COUPLING BETWEEN DIFFERENT LEAKY-MODE TYPES IN STUB-LOADED LEAKY WAVEGUIDES
NRD guide and in Section V for the stub-loaded rectangular guide. The behavior of the phase constant (3 and the attenuation (or leakage) constant a, normalized in each case to the freespace wavenumber k0 , is plotted first as a function of the height c of the stub guide, normalized to the free-space wavelength Ao, for various degrees of coupling. In this context, we discuss how the antenna performance is affected in the regions of coupling, and how such effects can be minimjzed or avoided. We next present the dependences of (3 / ko and a/ k 0 as a function of different geometrical parameters, under various coupling conditions, and then, for the stub-loaded rectangular waveguide, these dependences as a function of frequency. Section VI contains a summary of the principal conclusions to be drawn from this study. II. Two LEAKY-WAVE ANTENNAS AND THEIR PRINCIPLES OF OPERATION
A leaky-wave antenna is basically a waveguide whose cross section is open in some fashion so that leakage of power can occur all along its length. The mode that leaks has a complex propagation wavenumber (3 - ja , with a phase constant (3 and a leakage constant a . A knowledge of these quantities (3 and a as a function of the geometric parameters is sufficient to permit the design of the antenna; for example, the beam angle follows from (3, the beamwidth depends on both (3 and a, but primarily on a, and information regarding their dependence on the geometric parameters allows one to taper the geometry longitudinally in order to control the sidelobe level and distribution. These considerations, together with many examples of leaky-wave antenna structures, are presented in many references, most recently in [4]. A large class of leaky-wave antennas, designed primarily but not necessarily for application at millimeter wavelengths, contains a parallel-plate stub guide of finite height as part of the guide cross section. Included in thi s class are the two structures to be described below: the asymmetric NRD guide and the asymmetric stub-loaded rectangular waveguide. Other examples include the asymmetric trough guide, groove guide with an asymmetrically located metal strip, L-shaped guide, offset groove guide, and asymmetric slitted ridge guide.
A. The Asymmetric NRD Guide Nonradiative dielectric (NRD) guide is a low-loss open waveguide for mjllimeter waves, first proposed and described in 1981 [5]. It is a modification of H guide where the spacing between the metal plates is Jess than -\ 0 /2 so that all junctions and discontinuities that maintain symmetry become purely reactive instead of possessing radiative content. Fig. l(a) shows the cross section of NRD guide when it is symmetrical and supports a dominant bound mode with the electric field orientation indicated. The dielectric material in the center portion confines the main part of the field, and the field decays exponentially in the vertical direction in the air regions away from the dielectric-air interfaces. When the vertical parallel metal plates (parallel-plate stub guides) are sufficiently long, the dominant mode field is completely bound, since the field has decayed to negligible values as it reaches the upper and lower open ends.
(a)
(b)
Fig. 1. (a) NRD guide, which is like H guide except that the spacing a is made less than A.o /2 to insure that discontinuitites are reactive. (b) Leaky structure obtained when the NRD guide is bisected and an air gap is introduced between the dielectric strip and one metal side wall. The modifications produced in the e lectric fields are also shown.
The cross section of the asymmetric NRD guide is presented in Fig. l (b); due to the asymmetry, the initially bound dominant mode becomes leaky and the structure can be employed as a leaky-wave antenna. In its operation, the structure is of course fed from one end, and the mode, characterized by (3 - ja, leaks power away at some angle along the length of the structure. The structure in Fig. 1(a) is first bisected horizontally to provide radiation from one end only; since the electric field is purely vertical in this midplane, the field structure is not altered by the bisection. An air gap is then introduced into the dielectric region, as shown, to produce asymmetry, and the value of a, the leakage constant, is increased by increasing the air-gap width t. As a result, a small amount of net horizantal electric field is created, which produces a mode in the parallel-plate air region that is basically a TEM mode. This mode then propagates at an angle between the parallel plates until it reaches the open end and leaks away . It is necessary to maintain the parallel plates in the air region sufficiently long that the vertical electric field component of the original mode has decayed to negligible values at the open end. Then the TEM mode at an angle, with its horizontal electric field, is the only field left, and the field polarization in the radiation is then essentially pure. Furthermore, the discontinuity at the open end does not introduce any crosspolarized field components. To be more precise, we should understand that the "TEM" mode is an inhomogeneous plane wave that propagates at an angle (instead of simply vertically), with magnetic field components in both the longitudinal (z) and vertical (y) directions, because the basic leaky mode propagates in the z direction . This "TEM" mode is part of the overall basic leaky mode, and it in fact produces the leakage. Nevertheless, the simplified phrasing in the previous paragraph contains the essence of the physical picture, and is useful in explaining in simple terms the basic principle of operation. We should add that the mode with vertical electric field also possesses a z compo11ent of electric field because it is also tilted, being a part of the overall longitudinally propagating leaky mode. Since this mode decays with y to negligible values at the radiating open end, the E, component, which gives rise
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
a'
d
Fig. 2. The asymmetric stub-loaded rectangular guide, shown in full view and in cross section with the dimensional parameters indicated. The leakage rate increases as the parallel-plate stub guide is shifted further off center.
to the cross polarization, is also negligible in the radiating aperture. Furthermore, since the discontinuity at the open end is completely uniform along the z direction, all modes excited by it are separable into those with Ez only and those with Hz only. Therefore, the TEM mode at an angle, which has an Hz component (and only Ex in its electric field), will not excite any z component of electric field in the radiating aperture. As a result, the geometry and the mode of operation described above permits the radiated beam to possess essentially pure horizontal polarization. The first theoretical treatment [6] for the asymmetric NRD guide considered parallel-plate stubs of infinite height only; the effects due to stubs of fi nite height, which include the exotic coupling effects discussed here, were reported in [1] and appeared in more detail in [3]. Both treatments employed mode matching at the air-dielectric interfaces, and numerical results were obtained as a functi on of the geometric parameters. The mode-matching procedure is time-consuming but provides accurate and reliable results when enough transverse modes are included; for most points, 100 TE and TM modes were used. Convergence in the mean-square sense was employed, and convergence was checked each time. There are many ways in which asymmetry can be introduced to create the leakage, but the section of parallel-plate stub guide would be needed in each case, to insure that the vertical electric field does not contribute to the radiation. The potential for the coupling effects between the desired leaky mode and the channel-guide mode would therefore be present in all of these structures.
B. The Stub-Loaded Rectangular Guide The stub-loaded rectangular guide is presented in Fig. 2 in full view and in cross section, and it is seen to be very simple in form. The exciting electric field in the main guide is vertical ; some horizontal component of electric field is also excited, however, where the main guide connects to the parallelplate stub guide on its top waJI , which is deliberately located off-center. In the paraJlel-plate stub guide, both vertical and hori zontal electrical field components are therefore present; the vertical component creates a TM 1 mode (viewed verticaJiy in the parallel-plate stub guide), which is below cutoff and decays
to negligible values at the upper end of the stub guide. The horizontal component produces a TEM mode propagating at an angle between the paraJlel plates (and therefore a TE 0 mode with respect to the y (vertical) direction); when it reaches the upper (open) end, this mode leaks power away at an angle all along the structure. The basic leakage behavior, and the role of the paraJJel-plate stub guide in suppressing the vertical electric field component so that the cross polarization is negligible, are seen to be very similar to what we noted for the asymmetric NRD guide, even though the exciting feed structure main guide region is quite different in each case. One of the virtues of this antenna is its versatility, which may be understood from the following considerations. When the stub guide is centered, the structure may be viewed as a slotted section cut in rectangular waveguide, which is nonradiating when the stub is taJJ enough vertically. When the stub is placed all the way to one side (d = 0), the result is an L-shaped structure, which was found previously to radiate very strongly [3]. Thus, for small off-center positions of the stub, the leakage rate will be small, yielding radiated beams of narrow width. When the offset is increased, a will increase, and the beamwidth will increase. We therefore have a relatively simple leaky-wave antenna, easily fed from a rectangular waveguide, that permits great versatility with respect to beamwidth by simply adjusting the position of the stub guide. It possesses other important virtues as well. A comprehensive treatment of this antenna type is being submitted for publication [7], [8]. The papers include the theoretical approach, performance characteristics, effects due to flanges and finite stub height, and comparisons with measurements, but they do not contain the complicated wavenumber behavior reported here due to coupling between the channelguide leaky mode and the desired leaky mode. Two very different theoretical approaches were used in the analysis of this structure; one was the mode-matching method, similar to the one employed for the asymmetric NRD guide, and the other was based on a novel transverse equivalent network for which simple closed-form expressions were derived for all the elements of the network. The network approach allows one to obtain numerical values quickly and inexpensively, but it is approximate in principle; in contrast, the mode-matching procedure is rigorous in principle and can be very accurate, but it is time-consuming and therefore costly. Numerical values obtained by these two independent methods were compared, and excellent agreement was found, thus confirming the accuracy of the network method. In addition, measurements were taken on structures at X -band and in the 40 GHz to 60 GHz range, and very good agreement was obtained with the theoretical values [8]. The structure shown in Fig. 2 was originally derived from groove guide, by first bisecting the structure horizontally to permit radiation from one end only, and then by offsetting the parallel-plate stub guide portion with respect to the central region of the groove guide. The resulting structure was termed "the offset groove guide," and numerical results were derived for several different cross-sectional aspect ratios. Because some of the best performance was found for an aspect ratio corresponding to that for rectangular waveguide, and because
SHIGESAWA et al.: COUPLING BETWEEN DIFFERENT LEAKY-MODE TYPES IN STUB-LOADED LEAKY WAVEGUIDES
the structure was easy to feed from rectangular waveguide, the name was changed to the present "stub-loaded rectangular guide". Since three presentations were made under the early name (offset groove guide), however, it is appropriate here to indicate those earlier publications. The first presentation [9] discussed the structure and showed numerical results for strip guides of infinite height, the second one [2] contained numerical values for stub guides of finite height, and included the early results on coupling, and the third paper [10] reported early measured values at millimeter wavelengths and their comparisons with theoretical calculations.
0.75
1551
~
p1k 0 0.70 >-
0.65 ~~-+-----+-----' 1.0 2.0 3.0
III. QUALITATIVE FEATURES OF THE COUPLING EFFECTS AND THE Two DIFFERENT LEAKY-MODE TYPES
A discussion of the principal qualitative features of the coupling effects is useful because it provides insight into what otherwise appears rather involved. The quantitative behavior is presented in Sections IV and V for the two specific leaky-wave structures described in Section II. Since we expect that similar coupling behavior will occur for all leaky-wave structures that contain a parallel-plate stub guide of finite height as part of the cross section (and that category should include many examples), the effects described here should serve as a model for what to expect for other structures. The discussion below begins with what one would expect when the stub height is finite, and then we proceed to what actually happens when the leakage rate is small and when it is large. The two very different behaviors that are found occur, respectively, before the two leaky modes couple and then when they do couple. We then describe in qualitative terms how the two modes, namely, the desired leaky mode and the extraneous channel-guide leaky mode, differ from each other and what they have in common. Finally, we return to the coupling effects and discuss in qualitative terms the nature of the various interactions. A. Qualitative Features of the Leakage when the Stub Height is Finite
As explained in Section II, the leaky-wave radiation is produced by perturbing the bound dominant mode by introducing asymmetry in some fashion . As a result, a mode akin to a TEM mode is created in the parallel-plate stub guide, and it propagates at an angle toward the open end and then leaks away. In practice, the stub guide must be finite in height, of course. The open end itself corresponds to a discontinuity on the transverse transmission line representative of this TEM mode at an angle, which in tum is a TE 0 mode on the transmission line. A simple rigorous equivalent network is actually available for this radiating open end [11], and it was used as part of the analysis for both of the two leakywave structures [1], [2] discussed in Section IL Although the expression in [11] for the discontinuity at the open end holds strictly for real wavenumbers, the expression can be analytically continued directly into the complex plane for the leaky-wave application, as is done customarily in such contexts. Further details relating to this network appear in [7, Part II], and in [3]. It is important to note that this network
Fig. 3. The variations of normalized phase constant f3 / ko and leakage constant a/ ko with normalized stub guide height c/ >.o, for asymmetric NRD guide when the leakage rate is very low (air-gap thickness t = 0.01>.o = 0.024a). No coupling effects are present, and a/ko exhibits the expected periodic variation.
is applicable to all leaky-wave structures that contain the parallel-plate stub guide as part of the cross section. The first effect caused by the finite stub height is a mild standing wave (in the vertical direction) that is produced because of the mismatch introduced by the radiating open end. The standing wave exists in the stub guide between the radiating open end and the junction between the stub guide and the main guide (which will be different for each leaky-wave structure). As the height of the stub guide is varied, therefore, the values of /3/ko and o:./ko would be expected to undergo a periodic variation. That is precisely what is found when the leakage rate is low, with f3 / ko being affected very little and sometimes hardly at all, and with o:./ ko enduring a cyclical variation in value. An illustration of what was found for the asymmetric NRD guide in the case of a very low leakage rate appears in Fig. 3; the normalized phase constant f3 / ko and leakage constant o:./ko are plotted as a function of the normalized stub guide height c/ >. 0 . When we obtained the computed results for much larger values of leakage, however, we found them unexpected and initially puzzling. An illustration of what we found in that category for the stub-loaded rectangular guide is shown in Fig. 4. The distinction between the dotted and the dashed regions is explained later. This form of curve is what one obtains no matter where one begins in the solution and then tracks the solution for lower and higher values of c/ >. 0 ; as seen, the expected periodic behavior is completely absent. Clearly, some new factor must be present to account for the completely different behavior. The initially puzzling behavior for larger values of leakage is due to the simultaneous presence of another leaky mode, and to the coupling between that mode and the desired leaky mode. The additional leaky mode is similar to the so-called channel-guide mode that was first described some thirty years ago in connection with a class of leaky-wave antennas based on rectangular waveguide with a completely open side wall. The field properties of the modified channel-guide modes that
1552
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST I994
1.0
J3ik 0 0.8
,.
f-
,· ···················· 0.6
-
E
t
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I
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2.5
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-
c
c
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c/A.o
b
b
,-,
.
(a)
·.
(b)
Fig. 5. Sketches of the electric field variations and orientations in the asy mmetric NRD guide for (a) the desired leaky mode, and (b) the lowest channel-guide leaky mode, illustrating important differences between them . 10·3 ~-~-~-~-~-~~ 0 0.5 1.0 1.5 2.0 2.5 3.0
c/A.o
Fig. 4. Computed solutions for f3 / ko and oc/ ko as a function of stub guide height c/ >.o for the asymmetric stub-loaded rectangular guide when the rate of leakage is large. The behavior is totally different from what one finds in Fig. 3, for low leakage rates. The difference is due to the presence of coupling, as explained in the text ; the distinction between the dashed and dotted regions is also di scussed.
occur here are discussed in the next subsection, and compared with those of the desired leaky mode. After that, we present a qualitative di scussion of the coupling effects. B. Properties of the Two Different Leaky-Mode Types
The two leaky-mode types that interact to produce the coupli ng effects are the desired leaky mode and the channelguide leaky mode. They are very different from each other in certain respects, but they have some field components in common so that coupling is possible in principle. However, the condition for coupling req uires that both the f3 / ko and the a/ko values of each mode be equal, which is a condition that cannot often be met, and that explains why such coupling phenomena are not commonly seen. In broad terms, the differences between the two mode types are the following . For the desired leaky mode, most of the field is concentrated in the main guide region, and then leaks away from that region at a relatively slow rate, resulting in a fairly low value of a/ ko in most cases. For the channel-guide leaky mode, most of the field appears in the parallel-plate stub guide region, so that the leakage fro m the open end is strong and the value of a /ko is relatively high. Furthermore, in the main guide region the dominant field components for each of these two modes are orthogonal to each other, with the electric field of the desired leaky mode being primarily vertical and that for the channel-guide mode being primarily horizontal. When geometrical changes are made in the main guide to alter the value of a/ ko for the desired leaky mode, the a/ko value for the channel-guide mode is influenced negligibly, in part because most of its field is in the stub guide region, and in part because the electric fields of the two modes are opposite in the main guide region. Since the stub guide is essentially a perturbation on the main guide for the desired leaky mode, a variation in the height c of the stub guide produces only the periodic variation seen in Fig. 3. On the other hand, since the stub guide is basically the "main guide" for the channel-guide
mode, we find that higher-order solutions of the channel-guide mode are created as height c is increased, corresponding to additional cycles of field variation in the vertical direction in the cross section. Some of these points are illustrated in Fig. 5, where some electric field variations are sketched in for the case of asymmetric NRD guide. The main guide for this structure corresponds, of course, to the partially dielectric-filled region of height b. For the desired (NRD guide) leaky mode, we see that the electric field in the main guide region is primarily vertical, and that this vertical component decreases exponentially in the stub guide region. The horizontal component, which is excited at the air-dielectric interface, increases slowly along the stub guide height in conformity with its leaky-wave properties. Depending on the guide wavelength (,\y) in the vertical direction along the stub guide, there can be a sinusoidal variation superimposed on the small exponential increase because of the standing wave mentioned earlier. In contrast, it is seen from Fig. 5 that the field of the channel-guide leaky mode (the lowest of the solutions) is weakest within the main guide (dielectric) region, and is strongest in the stub guide region nearest to the open end. Furthermore, the electric field is predominantly horizontal throughout. The principal electric field orientations in the stub-loaded rectangular waveguide for these two modes are similar to those shown in Fig. 5. The sketch of the field distribution for the channel-guide mode shown in Fig. 5 is valid for the lowest of the channel-guide modes. An actual vector electric field plot for the second-order channel-guide mode, which involves three half-cycles rather than only one, is presented in Fig. 6 for the stub-loaded rectangular guide. The dimensions employed in the calculation correspond to those used in measurements [3], [8]. In the light of the remarks above, the height of the stub guide has little influence on the desired leaky mode outside of the standing-wave effect, but it affects the channel-guide modes critically. In fact, the number of channel-guide modes present and above cutoff depends directly on the height c of the stub guide. Finally, we can see that the leakage rate of the desired mode may be large or small depending on the amount of stub guide offset (amount of asymmetry), but that the leakage rate for the channel-guide modes will always be large, because
SHIGESAWA et al.: COUPLING BETWEEN DIFFERENT LEAKY-MODE TYPES IN STUB-LOADED LEAKY WAVEGU IDES
a
= 4.6 (mm}
a'= 2.4 b = 2.4 c = 3.12 d = 0.9 -
.,.
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1553
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Fig. 6. Vector electric field plot for the second-order channel-guide leaky mode in the asymmetric stub-loaded rectangular guide, showing the field strengths and the field directions at each point in a grid of many points.
their fields are largest at the open end, and that changes in the stub guide offset will affect the channel-guide modes only very little. Figs. 14-16 in Section V bear out these statements. We may also observe that the two mode types do have field components in common, particularly at the open end, and that they can become coupled at the discontinuities at the open end and at the junction between the main and stub guides. Thus, when their individual f3 / ko and a./ ko values are respectively equal to each other the propagation wavenumbers will exhibit strong coupling effects. C. Qualitative Features of the Coupling Effects
The variations in the values of f3 / k0 and a./ ko for the channel-guide leaky modes as a function of stub guide height cf Ao are shown in Fig. 7 for the asymmetric NRD guide. The values of f3 / ko are seen to be particularly sensitive to the stub guide height c. We also may note that, as c increases, more and more channel-guide modes appear above cutoff. For the desired leaky mode, Fig. 7 represents the case of very small leakage. That behavior is seen to be exactly what was shown in Fig. 3. For the f3 / k 0 plot, the flat curve for the desired leaky mode is seen to cross every one of the curves for the channel-guide modes. On the other hand, the curves for a./ ko for the channel-guide modes are seen to lie substantially higher than the periodically varying one for the desired leaky mode. Since coupling can occur only when the values of both f3 /k 0 and a./ko of the two leaky-mode types are equal, we see that coupling does not occur here, even over the large range of stub guide heights shown in this figure . When the geometry of the leaky-wave structure is changed so that the leakage rate of the desired leaky mode is increased significantly, the value of f3 / k 0 changes a little bit but the whole periodic curve for a./ k0 rises substantially. The curves for the channel-guide modes change very little; however, for the reasons indicated in the previous subsection. Since the a./ ko curves for the desired leaky mode then rise substantially but those for the channel-guide modes remain essentially in the same location (in the plot vs. c/ Ao), the a./ k 0 curves for those two mode types may now cross each other. The corresponding curves for f3 / ko always cross each other because the curves of the channel-guide modes cover a large range of f3 / ko values.
5
5
NRD guide mode 1.0
2.0
c/"/...o
3.0
4.0
Fig. 7. Variations of (3/ >.o and a/ >.o with normalized height c/ >.o of metal parallel plates, for the asymmetric NRD guide shown in Fig. 1 when the parameter values are a/ >.o 0.423, bf >.o 0.250 , Er 2.56, and t/ >.o = 0.01. For this small air gap, no coupling occurs between the asymmetric NRD guide mode and the channel-guide mode solutions.
=
=
=
Thus, both the a./ ko and the (3/ko values can become equal, and coupling can occur. One may refer to Fig. 8 or 12, for the asymmetric NRD guide or the stub-loaded rectangular guide, respectively, as illustrations of how the coupling effects manifest themselves. Where the crossings occur, we observe that for the f3 / k 0 plot the crossings are replaced by gaps that are exactly of the form expected for classical forward directional coupling [12). We may note that in both Figs. 8 and 12 portions of the curves are shown with dashes instead of solid lines; in fact, the dashed curves in Fig. 12 are precisely those shown in Fig. 4. Referring to the f3 / k 0 curves in Fig. 12, let us first follow the n = 1 curve (which is the same as the one in Fig. 4), beginning at the lower values of c/ Ao. The vertical portion, shown dashed, is actually a portion of the lowest channelguide mode; in the absence of coupling, it would continue upward to join with the upper portion of the present n = 0 curve. After the first coupling region, the n = 1 curve flatten s out, and the dashes are replaced by dots because this section is actually a continuation of the curve for the desired leaky mode. For somewhat higher values of c/ Ao, a second coupling region is reached, and the n = 1 curve continues as a portion of the next channel-guide mode, and is shown dashed again. Although the n = 1 curve forms a continuous solution, it represents different physical behaviors in the dashed and the dotted regions, representative of different modal content. When one tracks the wavenumber values for any given one of
1554
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
channel guide modes 0.8
It= 0.041..a I
I I
NRD guide mode
@\
0.7
0.6
10
·1
5
5
1.0
2.0
c/'A.o
3.0
4.0
Fig. 8. Same as for Fig. 7, but fort/ >.o = 0.04. Coupling effects are present for larger values of c/ >.o; the dashed lines show how the modes actually interchange within one interaction region.
the curves labeled n, the result obtained has the curve form shown in Fig. 4, devoid of periodicity and initially confusing. The curves are understandable only in the larger context of coupling with another mode. Two practical questions arise in view of the presence of the extraneous channel-guide leaky modes: a) If we excite the desired leaky mode, how strongly will the channel-guide leaky mode be excited as a result? b) How can one avoid the complications introduced by the coupling region in an antenna application? With respect to the first question, we first recall that even for small values of leakage rate and/or small values of c/ Ao , for which the coupling effects are absent or minimal, both mode types are present in principle. Under those conditions, the channel-guide mode should be only weakly excited, since that mode is weakest in the main guide region where the exciting field is present. In addition, the field polarizations are opposite, to a primary extent, in that region. If some amount of the channel-guide leaky mode were to be excited, it would at worst result in a low-power spurious beam at some angle. In that context, measurements were performed at 50 GHz on a stub-loaded rectangular guide [3] , [8], and no evidence was found of any spurious beam. For the second question, relating to antenna applications, one must of course avoid the complications introduced by the coupling regions. Two options are available. The first is to avoid the coupling effects altogether by selecting a sufficiently small stub height c. The function of the stub is
basically to provide a length of below-cutoff guide in order to prevent the vertical electric field component from radiating, and thereby achieve essentially pure horizontally polarized radiation. Calculations made in [3], [7] demonstrate that one need not have c/ Ao greater than 0.4 or 0.5 in almost all cases, even for stronger leakage rates. In Fig. 12, we see that the first coupling region occurs when c/ Ao is slightly greater than 0.5. We should also realize that the decay rate for the vertical electric field can always be increased by simply making the stub width narrower, and that the stub height c can then be reduced further. The second option, when c/ Ao must, for some reason, be larger than that for the first coupling region, is to make sure that one operates inside the dotted region and avoids the coupling region itself, where the fields become quite complicated. Although the coupling behavior for f3 / ko follows a known pattern, that for a/ ko appears not to have been described previously. The crossings occur, as seen in Fig. 8 or 12, in a complicated-appearing fashion , but insight is again obtained by following the n = 1 curve (as found in Fig. 12 and in Fig. 4) and distinguishing between the dashed portions and the dotted one. When viewed in that way, we see that the portions corresponding to the desired leaky mode follow the original periodic variation in an approximate fashion, by taking the n = 0 curve up to the first crossing, then following the dotted curve up to the crossing with the n = 2 curve, then following the n = 2 curve down and up to the crossing with the n = 3 curve, and so on. We thereby reproduce the periodic variations, but with lower successive peaks as c/ Ao increases. The remainder of the curves will possess fields corresponding to the channel-guide modes, except in immediate neighborhood of the crossings, where the fields are in the center of the coupling regions and are therefore complicated. Further insight requires an examination of the wavenumber behavior in a quantitative fashion, as is provided in Secs. IV and V for the asymmetric NRD guide and the stub-loaded rectangular guide, respectively.
IV. COUPLING INTERACTIONS FOR THE ASYMMETRIC NRD GUIDE
The first structure for which quantitative numerical data are presented relating to the coupling effects is the asymmetric NRD guide . The structure itself was shown in Fig. 1, and the manner in which leakage occurs was explained in Section II, A. Referring to Fig. l(b), we should recognize that the best geometrical parameter to vary to control the leakage rate is the thickness t of the air gap, and that larger values oft correspond to larger values of a/ ko. The coupling interactions to be described below are due to the simultaneous presence of the desired leaky mode and a set of channel-guide modes. The properties of these modes and the way they interact in a qualitative fashion are discussed in some detail in Section III. Here, quantitative calculations are presented for the following dimensions: plate spacing a = 0.423Ao (the value chosen originally by Yoneyama and Nishida [5]), dielectric strip height b = 0.25A 0 , and dielectric
1555
SHIGESAWA et al.: COUPLING BETWEEN DIFFERENT LEAKY-MODE TYPES IN STUB-LOADED LEAKY WAVEGUIDES
constant er = 2.56. In the first set of wavenumber plots, different leakage rates for the desired leaky mode are obtained by varying the air-gap thickness t, and the normalized leakage constant a/ ko and phase constant /3 / ko are given as a function of the normalized height c/ >. 0 of the parallel-plate stub guide. After those plots, results are presented as a function oft/ .Ao . The first results are those for a small air gap, with t = 0.01>. 0 , appearing in Fig. 7. We have already discussed that figure in Section III, since it is a case for which the desired leaky mode and the channel-guide modes do not couple with each other. As a result, we can see how the /3/ko and a./ko curves for the two mode types appear separately. For the desired leaky mode, the /3 / ko curve is essentially fiat and the a./ ko curve shows the periodic variation expected in view of the standing wave introduced by the discontinuity at the open end. The curves for the successive channel-guide modes, corresponding to multiple half cycles in the field variation along the stub guide height, are labeled by numbers l through 5. More details regarding the properties of these modes are presented in Section III. For this value of t, however, these modes do not couple. The /3 / ko values always cross, but the values of a/ ko for the channel-guide modes are much larger than those for the asymmetric NRD guide mode. The situation is made clear by reference to points A in Fig. 7; we note that the /3 / k0 values coincide, but that at that value of c/ >. 0 the a/ k 0 values differ by more than an order of magnitude. For significant coupling to occur, both the /3 / ko and the a./ ko values must be very close to each other. Let us next examine the results for t = 0.04>. 0 , which seems to be a transition value; the curves appear in Fig. 8. Here again, there are always crossings in the /3 / k0 curves, so we must examine how closely the a/ko values approach each other. Interestingly, the a./ko values become closer as c/ >. 0 increases, and the corresponding values are indicated in Fig. 8 by the letters B, C and D. For points C, only the channel-guid~ mode curve seems to be influenced, and only slightly since the values are not that much alike yet. At point D the coupling is so strong that characteristic coupling behavior already occurs. We observe that curve splitting (directional coupling behavior) is produced in the /3 / ko plot, and that the a/ k0 curves cross and interchange with each other. The coupling behavior for a larger value of air-gap thickness t appears in Fig. 9. We again observe that the coupling becomes stronger as c/ >. 0 increases. As t/ >. 0 increases, so that the leakage for the asymmetric NRD guide mode increases, the a./ k0 values for the two modes become close to each other and the mode coupling and conversion effects grow stronger and occur for smaller values of c/ >. 0 . The behavior in the coupling region itself is actually rather complicated. To help clarify the behavior, one interaction region is shown dashed for both a/ko and {3/ko in Fig. 8. As one follows the dashed curves, one sees that the two modes actually fully interchange with each other. These dashed curves are similar to the ones in Fig. 4, and their significance has been discussed in some detail in Section III. In those regions in which coupling is absent, we saw (as in Figs. 7 and 8) that /3 / ko remains fairly flat as c/ >. 0 is changed,
channel guide modes
0.8
It= 0.05 l. I 0
0.7
0.6
channel guide modes
5
1.0
2.0
c/'J...o
4.0
3.0
Fig. 9. Same as for Fig. 7, but fort / >.o = 0.05. Strong coupl ing is seen to occur almost everywhere for thi s larger value of leakage.
10·1
a= 0.423 l. 0 Max
b = 0.25 l. 0
cx/k 0
€
r = 2.56
c = 00 ,/ /
,,,,...,,,....
Min
,,..,,,,,..,,,..,,,,.../ /
/
------
/ /
/
0
0.02
0.06
0.04
VA.a
Fig. 10. The value of o. /ko varies in periodic fashi on as c/ >.o is changed in those regions for which coupling is absent. The curves here indicate the range in that variation as a function of t/ >.o , when b/ >. 0 = 0 .25.
but that a/ ko undergoes a rather large variation, in periodic fashion, as c/ .Ao is increased. By appropriately selecting the value of c, therefore, we can adjust the leakage rate within a fairly wide range. That range changes with the value of air-gap thickness t, of course, and the curves in Fig. 10 tell us what that range is as a function of t/ >. 0 , for dielectric width b/ >. 0 = 0.25. From Fig. 10 we see that the difference between the maximum and minimum values (the crests and the troughs in the periodic variation of a./ ko) is somewhat less than an order of magnitude, and that it increases slowly as
1556
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 8, AUGUST 1994
f = 50 GHz a= 4.8 mm
a'= 2.4 mm
b = 2.4 mm
d = 1.0 mm
1.0
µ
0.8
kc 0.6
0.4 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
2.5
3.0
3.5
4.0
c/).o
10-·
0
0.5
1.0
1.5
2.0 c/).o
Fig. 11. Behavior of the normalized phase and leakage constants for the desired (stub-loaded rectangular guide) and channel-guide modes, for the case of small leakage, where coupling occurs only for large values of stub height.
t/ Ao increases. Of course, the median value increases strongly as the air gap is made wider. Also shown in that plot is the dashed curve representing the value for c = oo; the dashed curve possesses the same variation with t/ Ao, but it lies a little lower than the median value. As the air-gap thickness t increases, the curve of (J / k0 vs. c/ Ao remains less flat, and exhibits some fluctuation in a periodic fashion. The amplitude of that fluctuation increases rapidly with t/ Ao, but still remains quite small; for t/ Ao = 0.05 it is about 2%. Plots of this variation for b/ Ao = 0.25 and 0.30 are given in [3], as are curves similar to Fig. 10 for b/ Ao = 0.30. V . COUPLING INTERACTIONS FOR THE STUB-LOADED RECTANGULAR GUIDE
The structure of the leaky stub-loaded rectangular guide was shown in Fig. 2, and the way the leakage was produced was discussed in Section II, B. Referring to Fig. 2, one may increase the leakage rate, and therefore a/ k 0 , by increasing the amount of offset of the parallel-plate stub guide. The value of a/ k 0 can also be controlled by changing the stub guide width a', and the circumstances under which that may be preferable have been reported in [3], [7], [ 10]. In the set of plots presented here as a function of stub guide height c/ Ao, however, the leakage rate is changed by varying the offset value d. A smaller value of d indicates a greater amount of offset, as may be seen from Fig. 2. A. Variations With Stub Height
The preliminary comments given in Section IV for the coupling interactions for the asymmetric NRD guide are
applicable here as well, as are the many qualitative remarks presented in Section III. In line with those earlier discussions, we begin here with Fig. 11, for which we may observe no coupling, then the onset of coupling, and finally full coupling, all on the same plot, as the stub guide height c/ Ao is increased. For the smaller values of cf >. 0 , for which no coupling occurs, the n = 0 curve in each plot corresponds to the desired leaky mode, and the n = 1, 2 and 3 curves refer to the channelguide leaky modes, where the number n increases as c/ Ao increases because these modes resemble the modes that would be present in a rectangular waveguide of width c with an open side. As seen, coupling does not occur for low values of c/ Ao here because the a/ko values for the channel-guide modes are too high, and the condition for coupling requires that both the (J / ko and a/ k 0 values for both mode types must be equal. For channel-guide modes of higher n the value of a/ ko is smaller for the same value of /3 / k 0 . As a result, coupling will eventually occur for a sufficiently tall stub guide. This effect is observed in Fig. 11 , where for n = 4 the values of both a/ko and (J / ko become equal for the two modes. At that point we note a gap present instead of a simple crossing for (J / ko, and, for a/ ko, we see that the curves cross and then interchange with each other. This behavior is, of course, similar to what was found in Section IV. We also note that, when the values of a/ ko for the two mode types approach each other but do not yet cross, as for the n = 3 curve, the curves for a/ ko draw nearer to each other, again as we observed in Section IV. In Fig. 12, the stub guide is offset somewhat more, and the leakage rate of the desired mode is increased significantly. As a result, coupling occurs between the two mode types for all of the channel-guide modes, with the first coupling region occurring at about c/ Ao = 0.5. Note that the number n, after coupling occurs, no longer corresponds to the pure mode types but to curves that include both types, and switch from one to the other. The same situation occurs in Fig. 11 for the curves labeled n = 4 and n = 5. The physics of what occurs in the coupling region is illustrated by the curve labeled n = 1, and shown partly dotted and partly dashed. Let us follow the n = 1 curve as c/ Ao increases. It is first in the dashed region, where it is clearly a channel-guide mode. Then the curve reaches the first coupling region, where the character of the solution changes from that of the channel-guide mode to that of the desired mode; to reflect this change, the next portion of the curve is shown dotted. That changed character is retained until the next coupling region, at which a switch in character occurs back into the channel-guide mode, shown dashed again . If one knew nothing about the coupling process, and simply followed that solution from one end to the other, that person would not see any periodic character to the solution, and would not know that in the dotted region the field distribution in the solution would be very different from that in the dashed region. The coupling behavior in the /3 / ko plot corresponds to what one would expect. What is new and interesting is what one finds for the a/ ko plot. If one concentrates on only those portions that correspond to the desired mode, meaning those regions like the dotted one, the result is a continuous curve that is roughly periodic, though with decreasing amplitude,
1557
SHIGESAWA et al.: COUPLING BETWEEN DIFFERENT LEAKY-MODE TYPES IN STUB-LOADED LEAKY WAVEGUIDES
f = SO GHz
f = SOGHz 8 = 4.Smm
a' = 2.4 mm
b = 2.4 mm
a = 4.8 mm
d = 0.8 mm
a' = 2.4 mm
b "' 2.4 mm
d = 0.5 mm
1.0
µ
0.8
/, .......
k;;0.6
I
/
I I
I
I
0.4 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0
4.0
0.5
1.0
2.0
1.5
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2.5
3.0
3.5
4.0
2.5
3.0
3.5
4.0
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a
10- ·
ko 10- •
0
0.5
1.0
1.5
2.0 c/ ). 0
2.5
3.0
3.5
4.0
Fig. 12. Same as Fig. 11, except that the leakage rate for the desired mode is higher. The physical behavior associated with the coupling is illustrated by the dotted and dashed portions (see text).
0
0.5
1.0
2.0
1.5
c/ ).o
Fig. 13. Same as Fig. 11 , except that the leakage rate is now much higher, making it more difficult to identify separate regions. 1.0
and is also something resembling the desired mode curve in Fig. 11 before coupling occurs. For the f3 / ko plot, the portions corresponding to the desired mode are the flatter portions between the coupling regions, similar to the one shown dotted. It is therefore possible to identify the nature of the separate portions even though the plots are initially very confusing. In Fig. 13, the offset is made even greater, so that the leakage rate of the desired mode becomes very large. The lessons learned from Fig. 12 still apply, and one can still identify the separate regions, but it has become more difficult to do so since the boundaries have become less sharp in the f3 / ko plot.
0.8
J!_
0.6
1r:J
ko
04
0.2
0 0
a
0.05
0.10
0.15
0.20
0.25
0.15
0.20
0.25
d/a 0.30
B. Variations With Stub Offset In Fig. 14, we present the behavior of f3 / k 0 and a/ ko as a function of stub guide offset, measured in terms of d/ a, where d/a = 0.25 corresponds to the centered stub, for which there is no radiation. In this figure, c/ >. 0 , the normalized stub height, is the parameter. A set of values of c/ >. 0 have been chosen such that each value corresponds roughly to the middle of the desired mode portion of the curves. This assertion may be verified by an inspection of Figs. 11 to 13. The curves for f3 / ko in Fig. 14 are seen to be relatively flat and independent of the value of c/ >. 0 . This feature is consistent with the flatness of the (3 / ko curves in the desired mode portions found in Figs. 11 to 13. For the curves of a/ k 0 in Fig. 14, we observe that the leakage rates are higher for smaller values of c/ >. 0 , a feature that is especially pronounced at the larger values of leakage. The reason for this behavior lies in fact that the curves of a/ ko vs. c/ >. 0 slope downwards when the leakage rates are
0.05
0.10
d/ a
Fig. 14. Behavior of the normalized phase and leakage constants for the desired mode as a function of stub guide offset, for different values of stub height. f = 50 GHZ, a = 4.8mm, a' = 2.4mm, b = 2.4 mm.
higher, as seen in Fig. 13. (This feature would be clearer if the data in Fig. 13 were plotted on a linear scale.) It is also observed that all the curves go to zero when d/ a is increased. We shall see in a moment that this behavior does not hold for the channel-guide leaky modes.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 8, AUGUST 1994
1.0
1.0 n =2
n =O 0.8
p ~
n =1
0.6
p
'-n =1
\.n - o
0.15
0.20
0.25
0.20
0.25
n=3 n- 4
0.8
n•5
0.6
k;;' 0.4
0.4
0.2>-
0.2
0 0
0 0.10
0.05
0.15
0.20
0.25
0
0.05
0.10
d/a
d/ a 0.30
0.20 n =0
0.24
0.18 .!!._ x 10- 1 ko
0.16
n -3 ..fLx10- 1
ko
n =1
0.12
n- 4
0.08 n =2
0.06
0.04
n- 1 n-o
0 0
0.05
0.10
0.15
0.20
0 0
0.25
0.05
d/a
=
=
=
0.15
d/ a
Fig. 15. Behavior of the normalized phase and leakage constants for both the desired and the channel-guide modes as a function of stub guide offset. 1). f .50GHz, a 4.8 mm, a' 2 .4 mm, (The desired mode is n b 2.4mm, c/ >.o 0.7 5 .
=
0.10
=
=
In Fig. 15, we again present the behavior of (3 /ko and a /ko as a function of d/ a, but for all the mode solutions that exist at the stub height value c/ >. 0 = 0. 75. From Figs. l l to 13 we note that three modes are present, the n = 0, n = 1 and n = 2 modes; then = 2 mode is omitted from the a/ k0 plot because its leakage rate is so large at this value of c/ >. 0 as to be off scale. Let us also note from Figs. I 1 to 13 that the n = 1 mode for that value of c/ >. 0 possesses the character of the desired mode (see the dotted lines in Fig. 12), whereas the n = 0 and n = 2 modes are clearly of the channel-guide mode type. (A comment should be made here regarding the mode number that characterizes the desired mode. That mode number depends on when the coupling begins. If the coupling has not yet started for the value of c/ >. 0 , the mode number is still n = 0. If it started before that value of c/ >. 0 , n will be greater than zero. The n = 1 portion at c/ >.a = 0. 75 for d = 0.8 mm (Fig. 12) is the same as that labeled n = 0 for d = 1.0 mm (Fig. 11 ). For labeling purposes in Fig. 15 , however, the numbering used corresponds to that in Fig. 12.) We may first see from Fig. 15 that for f3 / k 0 all three modes are relatively flat as d/ a is varied, showing that the behavior of (3 /ko is not a good way of distinguishing between the two mode types. The behavior of a/ k 0 , on the other hand, is entirely different for the two mode types . The n = 1 mode, which for this value of c/ >. 0 has the nature of the desired mode, has its a/ k 0 value go to zero as the stub becomes symmetrically located, as we would expect of this mode type. (In fact, it is the same curve as the one in Fig. 14 for c/ >. 0 = 0.75). The n = 0 mode, however, which is
Fig. 16. Same as Fig. 15, except for a much larger stub height c/ >.o = 3.00, so that more modes are present simultaneously. (The desired mode is n = 4).
of the channel-guide mode type, does not go to zero, and in fact increases somewhat as the stub becomes centered. The channel-guide modes are not strongly dependent on d/a because of their electric field orientation, as shown in Figs. 5 and 6. A similar demonstration is made in Fig. 16, but for a much larger stub height, c/ >. 0 = 3.0, so that more modes are present simultaneously. There are actually 6 modes present, but only 5 are shown in the a/ko plot because the values for then= 5 mode are off scale. An inspection of Figs. 11 to 13 verifies that 6 modes are present, and that only one of them , the n = 4 mode, is the desired mode and the rest are channel-guide modes. The f3 / ko plot in Fig. 16 shows that all the modes are quite flat with d/ a, as was found in Fig. 15. For the a/ ko plot, we see clearly that only the n = 4 curve, which is of the desired mode nature for this value of c/ >. 0 , goes to zero as the stub becomes centered. The channel-guide modes are basically not affected when d/ a changes. The behavior of a/ ko when the stub offset is altered is thus a sensitive test of the character of the mode in question. C. Variations With Frequency
The curves in Figs. l 1 through 16 above present data at the mid-range frequency of 50 GHz. We next examine what occurs when we vary the frequency over a wide range. To illustrate more clearly the differences in behavior between the desired leaky mode and the channel-guide modes, we choose the case for which there is a small leakage rate (d = 1.0 mm); results for (3 /ko and a/ko over a large frequency range
SHIGESAWA et al.: COUPLING BETWEEN DIFFERENT LEAKY-MODE TYPES IN STUB-LOADED LEAKY WAVEGUIDES
0.8
J!_
0.6
ko 0.4
O'-""""--.J---'-'----'---'----'-----'--'---'-~
0
20
60 40 f (GHz)
80
100
10°
10-· L_J__.J.._.J--_J.__j.__J__J_--'---'--'
0
20
40 60 f (GHz)
80
100
Fi g. 17. Behavior of the normalized phase and leakage constants over a wide range of frequencies for both the desired and channel-guide modes for the case of small leakage. (The desired mode is n = 0). a = 4 .8mm, a' = 2.4mm, cf >.o 0. 75 , d 1.0mm.
=
=
are presented in Fig. 17. We note first that only three modes are present; that observation is consistent with the curves in Fig. 11 and the mode numbering used for the figure, since the dimensions correspond as well. In Fig. 11 , the n = 0 mode is clearly the desired mode, and the n = 1 and n = 2 modes are channel-guide modes. When we recall that the channel-guide modes have a much higher leakage rate than the desired mode, the behavior of the curves in Fig. 17 becomes clear. The n = 1 and n = 2 modes have much higher values of a/ko in the region above cutoff, and the /3 / k 0 behavior below cutoff shows that the channelguide modes have a much higher minimum and turn up much sooner. The behavior below cutoff is explained in [3], [7]. What is important to know here is that, when the a/ko value for the leaky mode is smaller, the /3 / k 0 value approaches zero more closely at cutoff. A final point of interest is that in the a/ko plot the n = 0 curve (for the desired mode) changes very rapidly at cutoff, thereby crossing the curve for one of the channel-guide modes.
VI. CONCLUSIONS A fairly large group of leaky-wave antennas, even though they are based on different waveguiding structures, such as rectangular guide, NRD guide, trough guide, groove guide, ridge guide, etc., all have in common a parallel-plate stub guide of finite height as part of the cross section. Sometimes this stub guide is present automatically as part of the cross section of
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the basic waveguide, as in NRD guide, trough guide or groove guide, but in other cases it is added deliberately, as is done for antennas based on rectangular guide or ridge guide. In all cases, however, the parallel-plate region serves to eliminate the vertical component of electric field, allowing the radiated beam to possess essentially pure horizontal polarization. The parallel-plate stub guide of finite height also serves to introduce an additional set of leaky modes, the channelguide modes, so that two different leaky-mode types are present simultaneously in the guiding structure: the desired leaky mode and the set of channel-guide leaky modes. As discussed in Section III, these two mode types are sufficiently different from each other that the channel-guide modes are only weakly excited. When both the phase constant /3 and the attenuation, or leakage, constant a of each of the two mode types become equal to each other, however, the mode types will couple to each other, and produce exotic-looking curves for the wavenumber behavior. This initially puzzling behavior is discussed and explained qualitatively in Section III, and demonstrated quantitatively for two different guiding structures in Secs. IV and V. The coupling behavior for /3 actually follows a known pattern, but that for a appears not to have been described previously. Since such coupling will occur only when both the /3 and a values are equal, it is not surprising that these effects are relatively rare and not generally known. An important concern involves how to get around these coupling effects so that antenna performance will not be affected adversely. As indicated in Section III, there are two ways. One is to recognize which regions of the wavenumber curves correspond to the desired leaky mode, and to design the antenna in one of those regions; the discussion in Section III describes those regions clearly. An easier way is make the stub-guide height short enough. An examination of Figs. 7-9 and 11-13 shows that the first coupling region seems to occur for c/ >. 0 just over 0.5, even when the leakage rate is high. Coupling problems can be avoided, therefore, by having c/ >. 0 < 0.5. The stub height c must be large enough to permit the n = 1 (cross-polarized) mode, which is below cutoff in the parallel-plate region, to decay to negligible values, but the decay rate of this mode obviously depends on the width of the parallel-plate region. For the stub-loaded rectangular guide, for the dimensions and wavelength chosen for Figs. 11-13, the reduction in the cross-polarized mode is about 29 dB for c/ >. 0 = 0.5 and about 23 dB for c/ >. 0 = 0.4. When the stub width is narrowed to a'/ a = 0.3 instead of 0.5, this reduction becomes 54 dB, 43 dB and 32 dB for c/ >. 0 = 0.5, 0.4 and 0.3 , respectively. These coupling effects can therefore be avoided by keeping the stub only as high as necessary and not any higher, and by reducing the stub width somewhat if necessary. The interesting coupling effects that arise because of the presence of the parallel-plate stub guide of finite height have therefore been explained as a coupling effect between two different types of leaky modes, and the additional set of modes has been identified as the channel-guide modes. A detailed qualitative discussion has been presented here because we expect that similar effects will occur for all leaky-wave
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
structures that possess such a stub guide in their cross sections. Furthermore, the wavenumber behavior is of fundamental interest in its own right. REFERENCES [I] H. Shigesawa, M. Tsuj i, and A. A. Oliner, "Effects of air gap and finite metal plate width on NRD guide," IEEE Int. Microwave Symp. Dig., Baltimore, MD, June 2-4, 1986, pp. 11 9- 122. [2] P. Lampariello, F. Frezza, H. Shigesawa, M. Tsuji, and A. A. Oliner, "Guidance and leakage properties of offset groove guide," IEEE Int. Microwave Symp. Dig., Las Vegas, NV, June 9-11 , 1987, pp. 731 -734. [3] A. A. Oliner, "Scannable millimeter wave arrays," Final Rep. RADC Contract No. Fl 9628-84-K-0025, Polytechnic Univ ., Brooklyn, NY, Sept. 30, 1988. [4] A. A. Oliner, "Leaky-wave antennas," in R. C. Johnson, Antenna Engineering Handbook, 3rd ed. New York: McGraw-H ill , 1993 . [5] T. Yoneyama and S. Nishida, "Nonradiative dielectric waveguide for millimeter-wave integrated c ircuits," IEEE Trans. Microwave Theory Tech. , vol. 29, pp. 11 88-1192, Nov. 198 1. [6] A. A. Oljner, S. T. Peng, and K. M. Sheng, "Leakage from a gap in NRD gu ide," 1985 IEEE Int. Microwave Symp. Dig. , St. Louis, MO, June 3-7, 1985, pp. 6 19-622. [7] P. Lampariello, F. Frezza, H. Shigesawa, M. Tsuji , and A. A. Oliner, "A versatile leaky-wave antenna based on stub-loaded rectangul ar waveguide, Part I: Theory; Part II : Effects of flanges and finite stub height," to be submitted to IEEE Trans. Antennas Propagat. [8] _ _ , "A versatile leaky-wave antenna based on stub-loaded rectangul ar waveguide, Part Ill: Measurements," submitted to IEEE Trans. Antennas Propagat. [9] A. A. Oliner and P. Lampariello, "A simple leaky wave antenna that permits flexibi lity in beam width," Dig. National Radio Science Meeting, Philadelphia, PA, June 9-13, 1986, p. 26. [IO] H. Shigesawa, M. Tsuji, and A. A. Oliner, "Theoretical and experimental study of an offset groove guide leaky wave antenna," IEEE Int. Symp. Antennas Propagation Dig. , Blacksburg, VA, June 15- 19, 1987, pp. 628-632 .. [I I] N. Marcuvitz, Waveguide Handbook. New York: McGraw-Hill , 1951, Sec. 4.6a, pp. 179-183 (Reprinted by IEE, IEEE, 1986). [12] J. R. Pierce, "Coupling of modes of propagation," J. Appl. Phys., vol. 25, pp. 179- 183, Feb. 1954.
Hiroshi Shigesawa (S'62-M'63-SM'85-F'94) was born January 5, 1939, in Hyogo, Japan. He received the B.E., M.E., and D.E. degrees from Doshi sha University, Kyoto, Japan, in 196 1, 1963 , and 1969, respective! y. Since 1963, he has been with Doshi sha University , where he is currentl y a Professor in the Faculty of Engineering. From 1979 to 1980, he was a Visiting Scho lar at the Microwave Research Institute, Polytechnic Institute of New York . Hi s present research involves microwave and millimeter-wave guiding structures and devices, and scattering problem s involving electromagnet ic waves. Dr. Srugesawa is a member of IEICE of Japan , the Institute of Electrical Eng ineers of Japan, and the Optical Soc iety of America.
Mikio Tsuji (S'77-M'82) was born September 10, 1953, in Kyoto, Japan . He rece ived the B.E., M.E., and D.E. degrees from Doshisha University , Kyoto, Japan, in 1976, 1978, and 1985, respectively. Since 198 1, he has been with Doshisha Un ivers ity, where he is now an Associate Professor. Hi s present research involves microwave and millimeter-wave guiding structures and devices, and scattering problems involving electromagnetic waves. Dr. Tsuji is a member of IEICE of Japan, and the Institute of Electrical Engineers of Japan .
Paolo Lampariello (M'73-SM '82) was born May 17, 1944, in Rome, Italy. He received the degree in electronic engineering at the University of Rome, Rome, Italy, in 1971. In 197 1, he joined the In stitute of Electronics, Uni versity of Rome, where he was an Assistant Professor of Electromagnetic Fields. Since 1976, he has been engaged in educational acti vities involving electromagnetic theory. He was made Professor in 1986. Since November 1988, he is Head of the Department of Electronic Engineering at the "La Sapienza" University of Rome. From 1980 to 198 1, he was a NATO Postdoctoral Research Fe llow at the Polytechnic Institute of New York, Brooklyn. His research interests inc lude electromagnetic and elastic wave propagation in anisotropic media, interactions with biological tissues, guiding structures, and antennas and resonators for millimeter waves.
Fabrizio Frezza (S'87-M '92) received the "Laurea" degree in Electronic Engineering and the Doctorate in applied electromagnetics from "La Sapienza" University of Rome, Rome, Italy, in 1986 and 199 l , respectively. In 1986, he joi ned the Department of Electronic Engineering, Uni versity of Rome, where he has been a Researcher since 1990. Hi s research interests include guiding structures, antennas and resonators for microwaves and millimeter waves .
Arthur A. Oliner (M'47-SM '52-F'61 -LF' 87) was born March 5, 1921 , in Shanghai , China. He received the B.A. degree from Brooklyn College, Brooklyn , NY, and the Ph.D. degree from Cornell University, Ithaca, NY, both in physics, in 1941 and 1946, respectively. While at Cornell University, he held a Graduate Teaching Assistantship in the Physics Department and also conducted research on a project of the Office of Scientific Research and Development. In 1946, he joined the Microwave Research In stitute of the Polytechnic Institute of Brooklyn, Brooklyn, NY. In 1957, he was made Professor. From 1966 to 197 1, he was Head of the Electrophysics Department and then became Head of the combined Department of Electrical Engineering and Electrophysics from 1971 to 1974. He was also the Director of the Microwave Research Institute from 1964 to 1981. Hi s research has included network representations of microwave structures, precision measurement methods, guided-wave theory with stress on surface waves and leaky waves, traveling-wave antennas, plasmas, periodic structure theory, and phased arrays, wavegu ides for surface acoustic waves and integrated optics, novel leaky-wave antennas for millimeter waves and leaky-mode effects in millimeter-wave integrated circuits. He is the author of about 200 papers and the coauthor or coeditor of three books. He served on the Editorial Boards of the journal Electronics Letlers and the volume series Advances in Microwaves (Academic Press). He is also a former Chai rman of a National Academy of Sciences Advisory Panel to the National Bureau of Standards (now NIST). Dr. Oliner is a Fellow of the AAAS and the British IEE, and is a member of the Nat ional Academy of Engineering. He has received the 1967 IEEE Microwave Prize and the 1964 Institution Premium, the highest award of the British IEE. He was named an Outstanding Educator of America in 1973 and one year later, he received a Sigma Xi C itat ion for Distinguished Research . He was a President of the IEEE MTT Society, its first National Lecturer, a member of the IEEE Publi cation Board, and General Chairman of three symposia. ln 1982, he received the IEEE Microwave Career Award, and he is one of six li ving Honorary Life Members of the IEEE Society. In 1984, he was a rec ipient of the IEEE Centenn ial Medal, and in 1988, a special retrospective session was held in his honor at the International Microwave Symposium. In 1993 , he became the first recipient of the MTT Society's Distringuidshed Educator Award. He is a member of several Commi ssions of the International Un ion of Radio Sc ience (URS !), a past U.S. Chairman of Commission I (now A) and D, and a member of the U.S . National Committee of URS!. In 1990, he received the URS ! van der Pol Gold Medal.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 7, JULY 1994
156 1
A Modified Finite Element Method for Analysis of Finite Periodic Structures Shyh-Jong Chung and Jiunn-Lang Chen
Abstract-A modified finite element method with new solving algorithm is proposed to analyze electromagnetic problems of finite periodic structures. Dielectric-loaded parallel-plate waveguides with rectangular and triangular dielectric gratings are tackled as an example of the present approach. Numerical results are checked by the self-convergence test and by comparing with those obtained by other methods. Finally, the dependence of the scattering parameters on the frequency, the period number, and the grating height is analyzed and compared. I. INTRODUCTION
N recent years, the finite element method (FEM) has been used to analyze wave guiding and scattering problems of microwave, millimeter-wave, and optical components [l]-[5]. When appropriately combined with other techniques (e.g., Green's function technique [2], [3], eigenfunction expansion method [4], absorbing boundary condition [5]), this method can, in general, tackle arbitrary geometries with inhomogeneous, anisotropic, and/or lossy mediums. Before applying the finite element method, one first establishes a variational equation for the problem to be handled. According to the geometry of the problem, the structure is then divided into several subregions, called elements. The fields in each element are expanded by the so-called nodal field values, (which are values of the fields in some particular points (nodes) of the element and are to be determined,) and the corresponding "shape functions" (bases) [6]. After using the Rayleigh-Ritz procedure, a submatrix of the element can be obtained. Finally, by assembling the submatrices of all the elements, one gets a system matrix equation from which the nodal field values are solved. In most cases, the system matrix is highly sparse, and thus the frontal solution technique [7] is usually used in the solving process of the equation to reduce the computation time and the core storage requirements. The technique includes two phases, i.e., assembly and elimination. Whenever a submatrix of a new element is gotten, it is assembled to the existing (working) matrix. The working matrix is then reduced by eliminating the variables that won ' t appear in the rest elements. As all the elements have been called in, the resulting working matrix becomes that corresponding to the variables of the boundary nodes. By this final matrix, the scattering parameters or the eigenvalues are calculated.
I
Manuscript received June I, I 993; revised October I 9, I 993. This work was supported by the National Science Council of the Republic of China under Grant NSC 82-0404-E-009-396. The authors are with the Institute of Communication Engineering, National Chiao Tung University, 1001 Ta Hsueh Rd., Hsinchu, Taiwan, R.0.C. IEEE Log Number 9402946.
Although the finite element method coupled with frontal solution technique is versatile for the analysis of electromagnetic problems, it still needs improvement to increase its efficiency, especially for the structures with high repetition, e.g., periodic structures, or with large areas of homogeneous regions. (The later can be treated as special cases of the former.) In these structures, although the field distributions are different from one period to another, the discretization of the structure in each period may be made the same. The resultant matrix obtained from the frontal solution technique for each period is thus the same. This means that one only needs to establish the matrix once but not for every period. Periodic structures appear in many devices, such as filters, gratings, and distributed feedback lasers. For the analysis of these problems, most researchers assumed infinite periods existing in the structure, and used Floquet's theorem [8] to focu s the problem into a single period. Only a few handled with the finite periodic structures [9]-[11], among which [9] used the network representation to connect the contribution of each period and [10] proposed a method based on the spectrumdomain analysis, combined with the sampling theorem, to treat finite periodic structures. Although available for arbitrary number of periods, these approaches could tackle only the periodic structures with step di scontinuities. In this paper, we propose a modified finite element method for the analysis of structures with high repetition. In the di scretization of the structure, we first divide the whole area into several blocks, called "super-elements". Due to the repetition property of the geometry, these super-elements belong only to a few patterns. (For example, in a finite periodic structure, each period is treated as a super-element, and the whole structure contains only one (geometric) pattern). For each pattern, we divide it into many ordinary elements, and expand the fields in each element by the nodal field values and the shape functions. Through the treatment of the Rayleigh-Ritz procedure and the frontal solution techniqure, one obtains a submatrix corresponding to the variables of boundary nodes of the pattern. To this end, the fro ntal solution technique is again used to assemble and eliminate the submatrices of the super-elements. The final matrix is then obtained after thi s process. The organization of this paper is as follows: Section II describes the theory of the method. Section III, as an example of the theory, deal s with a finite periodic planar dielectric structure with top and bottom covers. The numerical results are then presented in Section IV, fo llowed by conclusions in Section V.
0018-9480/94$04.00 © 1994 IEEE
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
I !
If'
I (;
11 ff' t
( f
PEC
" " :: I (": :I 1 I
(
I (
'f!CC'"
(" """'
If 1 {("'
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::. . ~. J.I. . . . . -
(a)
PEC
Fig. 1.
PATTERNS
Dielectric scatterers located in a parallel-plate waveguide.
I
~
II. THEORY As an illustration of the theory, consider a two-dimensional structure as depicted in Fig. 1, which shows four dielectric obstacles located in a parallel-plate waveguide. The spacing between the first (rectangular) and the second (triangular) dielectrics is the same as that between the third and fourth (rectangul ar) dielectrics. A TE mode of the parallel-plate waveguide is incident from the left-hand side (LHS), which will excite some modes reflected back to the LHS and some transmitted to the right-hand side (RHS). From a partial variational principle [12), one obtains a variational equation as follows:
where n, sandwiched by boundaries f1 and f2, denotes a finite region enclosing all the obstacles as shown in Fig. 2(a). f l, 2 means the inner face (inside 0) of f 1 , 2 . The variational operator 5a operates only on the terms with superscript "a" . To solve (1) by the finite element method, r2 is divided into N small triangular elements (as shown in Fig. 2(a)), so that the functional Ia can be written as N
l a = L)~ +If,.
L s = l ,3,4
8
I~ + LI~ + LI~ +If, , s =2
bJ D (b)
Fig. 2. (a) Finite-element discretization including super-elements and small triangular elements. (b) Three structure patterns of the super-elements.
where
I~
represents the functional of the s-th super-element, Ni
I~ = I.::r~e ·
s =5
(4)
e =l
The field in each triangular element is expanded by the field values at six nodes of the element, three on the vertices and three on the mid-pionts of the edges. By using the Rayleigh-Ritz procedure and the frontal solution technique, ~e variational operation on I~ would result in a submatrix As associated only with the field values of the boundary nodes of the s-th super-element. Note that, owing to the same dimensions and discretization, the super-elements of a pattern would have the same submatrix. Thus, one only needs to calculate three submatrices for the present example. (Each submatrix of a pattern is obtained by assembling and eliminating Ni sub-submatrices of the triangular elements.) By using _!!le frontal solution technique ~ain, the Ns submatrices (As, s=l,. ..,Ns) and the submatrix Ar (associated with If,) are then gathered in tum to get the final matrix equation,
A7f; = s,
Here I~ represents the func tional corresponding to the eth triangular element, and If, corresponding to the boundaries r 1 and f 2 . By the repetition nature of the structure, the N small elements are grouped into N s ( = 8) "super-elements", as indicated in Fig. 2(a). These super-elements belong to the three patterns shown in Fig. 2(b ): super-element 1, 3, 4 are of pattern I, super-element 2 is pattern Il, and super-elements 5, 6, 7, 8 pattern m. (Here super-element 6 is assumed to have the same dimensions as super-elements 5, 7, 8.) By appropriate division, the super-elements of the same pattern may have the identical discretization. Assuming pattern i, i=I, Il, fil, be comprosed of N i small elements, (2) is rewritten as
=
ill
(2)
e=l
la
II
(3)
(5)
with 1jj being the nodal field values of boundaries f1 and f2, and s the sources terms associated with the incident field. The scattering parameters are determined from the solution of (5). Before the end of this section, it is noticed that the eight super-elements shown in Fig. 2(a) can be considered as four new super-elements, that is, super-elements (1, 5), (2, 6), (3, 7), and (4, 8). These new super-elements belong to two patterns, thus only two submatrices are needed to be obtained. III. APPLICATION TO FINITE PERODIC STRUCTURES Fig. 3 shows a periodic dielectric waveguide with top and bottom PEC covers, where a Ncperiod structure is connected to two semi-infinite dielectric-loaded parallel-plate waveguides. The materials in each period may be arbitrary and may contain PEC's. Let the incident field be the TE dominant
CHUNG AND CHEN: A MODIFIED FlNITE ELEMENT METHOD FOR ANALYSIS OF FINITE PERJODIC STRUCTURES
1
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Fig. 3. Periodic dielectric waveguide with top and bottom covers. N e is the number of the periods in the structure.
Wlllalt
T '
;
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mode (slab-guide mode) of the dielectric-loaded parallel-plate waveguide. The fields exterior to the boundary I' 1 are expanded by the orthonormal modes of the dielectric-loaded parallel-plate waveguide, i.e.,
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;
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STRUCTURE B
M
Ey(I't)
= ei(x) + L
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Fig. 4. Periodic structures to be analyzed. For structures A and C, Er = 5, t = 1.Smm, h 1 = 0. For structure B, Er = 11.8 , t = 0.7mm, h 1 = h 2 = 1 mm.
and M
Hx(I't) = hi(x)
+L
R;h;(.r ),
(7)
·i =l
where e; (x ) is the modal function of the i-th mode, h; (x) = - f3i e; (x), with {3; being the propagation constant of the mode. wµ o R; is the reflection coefficient to be determined. By the continuities of the tangential fields and the orthonormality of the modes, one gets
R; = -8 1;
+ {
lr1
Ey(I'!)h;(x )dx,
(8)
Ey(I'!)h;( x')dx') h;(x ).
(9)
and thus
Hx(I' !) = Hx(I't) = 2h1(x )
-t, (fr
1
With a similar treatment to the boundary I' 2 , one obtains
T; =
{ Ey(I'2) h;(x )dx
lr2
I- p..;
STRUCTURE C
i =l
(10)
and
where T; is the (unknown) coefficient of the ith mode transmitted to the RHS waveguide. By casting (9) and (11) into (I), a variational equation containing only the electric field inside n is obtained, which is then solved by the method mentioned in the previous section. In the discretization of the structure, each period is treated as a super-element. Besides, two aditional uniform sections immediately adjacent to the periodic stucture are included in n and are considered as two super-elements (see Fig. 3). The inclusion of these two sections can reduce the number (M) of
the waveguide modes required to expand the fields on r 1 and I' 2 , due to the decays of the higher-order evanescent modes. Thus, there are totally Ne + 2 super-elements in D, which belong to two patterns, i.e., the pattern of the period and that of the uniform section. The essential boundary condition on the PEC covers (i.e. , tangential electric fields to be zero) is enforced in the process of obtaining the submatrices of the patterns. This makes the submatrix associated only with the nodal field values of the two side boundaries of the pattern. Thus, whenever a new super-element is called in, after assembly and elimination, the resulting working matrix always contains the variables of r 1 and those of the RHS boundary of that super-element, assuming that the super-elements are called in one by one from left to right. (The variables of r 1 can not be eliminated since they will appear in the last element, i.e., the last term of (3)). In other words, the size of the working matrix is kept constant, not getting larger with the increase of the number of treated super-elements. After the last element having been called in, the working matrix becomes the final (system) matrix and the source terms are introduced. The solutions of the resulting matrix equation are the electric fields on r 1 and I' 2 , i.e., Ey(I'~) and Ey(I'2) . The reflection coefficients R;'s and transmission coefficients T i's are then calculated from (8) and (10). IV. NUMERICAL RESULTS
We present in this section some numerical results for the finite periodic structures shown in Fig. 4. Some of the geometric parameters are indicated in the figure, others defined in Fig. 3. The number (M) of the modes, including propagation and evanescent modes, of the dielectric-loaded parallel-plate
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
TABLE I CONVERGENCE TEST OF THE SCATTERING COEFFICIENTS FOR STRUCTURE A SHOWN IN FIG. 4. Ne 2, f 40GHz, p l. 25 mm, h 2 2mm, Er 5.
=
=
=
=
=
- h : 0 .25t - -- - h : 0 .50 t
n set
Nx
Nz
IRILR
ITILT
6 8 10 12 14 18 12 12 12 14 18
4 4 4 4 4 4
0.5185L-137 .2°
6 8 10 10 10
0.5153L-140 .0° 0.5149L-140 .3° 0.5162L-140.3° 0.5158L-140.7° 0.5160L-140.7° 0.5161L-140 .8° 0.5159L-141.1° 0.5158L-141.2° 0.5157 L-141.3° 0.5157 L-141.3°
0.8551L0° 0.8570L0° 0.8573L0° 0.8565L0° 0.8564L0° 0.8567 L0° 0.8566L0° 0.8565L0° 0.8567 L0° 0.8568L0° 0.8567 L0°
24
12
0.5157 L-141.4°
0.8568L0°
.L
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
ow...............c...................,c................_,~......_,_~........._~-'-'-~....... ~ M ~ ~ 30 m ~ ~ FR EQUENCY (G Hz)
Fig. 6. Variation of the reflection coefficient as a function of the frequency for the structure A shown in Fig. 4. N e 20 ,p 3 .6mm, H 2 2t .
=
=
=
0.8 0
~
"'0~
~
-10
"'
-20
"' u [;; "" 0
-
"' 0.6
~
Q..
,
h : 0 .25t
--- -- h : 0 .50t
0.4
0.2
(J
z
3
;(J
"' "''"
5 -30 -
-'
"'
----- RESU LTS FRO M [II ]
-40
35
40
45
50
10
15
20
25
30
PERIOD NUMBER (Ne)
PRESENT AN ALYSIS
55
60
65
Fig. 7. Variations of the reflection (Pr ) and transmi ssion (P t) powers as a function of the period number (Ne ) for the structure A shown in Fig. 4. f 29.2 GHz, p 3 .6 mm, h 2 2t.
=
=
=
FREQU EN C Y (G l·h )
Fig. 5. Variation of the reflection coefficient as a function of the frequency fo r the structure B shown in Fig. 4. Ne = 20 , p = l.36mm, t p = 0. 03mm.
waveguide is set to be 10. Although not shown here, further increase of this number does not influence the calculated results. Table I shows a convergence test of the slab-guide reflection (R) and transmission (T) coefficients for the structure A with 2 periods (Ne = 2). Nx and Nz represent the division mesh numbers in the x and z directions, respectively, inside a period. It is seen that good convergence is obtained both for the magnitudes and phases of the coefficients. The power conservation law, i.e., the absolute squares of the scattering coefficients of all the propagation modes are added to be unity, is always obeyed for thi s and the following calculations. As a further test of the present method, Fig. S shows the magnitude of the reflection coefficient R (of the slab-guide mode) as a function of the frequency for the structure B with N e = 20. The results of the dash line are from [ 11], in which the obstacles (PEC' s) embedded in the uniform slab are replaced by highly dopped plasmas (u = 2.44 x 104 5/m). The infinite conductivity of the obstacles makes the curve of the present analysis (solid line) having higher values in the flat band and larger oscillations than that of [11]. Except for these differences, the two curves behave quite similarly in the whole frequency band.
Fig. 6 illustrates the frequency response of the reflection coefficient for the structure A (Ne = 20) with h = 0.25t (solid line) and h = 0.5t (dash line). It is seen that the increase of the height of the ridges has a little influence on the upper end of the reflection band, but has large on the lower end. The expense of this increased bandwidth is the raise of the side lobes in the lower frequency. There are three dielectric thicknesses associated with the stuctures of Fig. 6, i.e., t, I.2St (for h = 0.25t), and I.St (for h = 0.5t). For the given period length (p = 3.6mm) and dielectric constant (Er = 5.), to satisfy the Bragg reflection condition based on the first order perturbation theory [13], i.e., (J (f)p = ?r , the required frequencies for the slab waveguides with the three dielectric thicknesses are calculated to be 30.7S, 27.8S, and 2S.90 GHz, respectively. It is interesting to note that the mean value of the first two frequencies is 29.3 GHz, which is approximately equal to the maximum-reflection frequency (29.2 GHz) for the solid-line curve (which is the result for the periodic structure formed by two equally spaced dielectrics with thickness t and l.2St). Similarly, the mean value of the first and last frequencies, i.e., those for dielectric thicknesses t and l .St, is 28.33 GHz, which, again, is approximately equal to the maximum-reflection frequency (28.2 GHz) for the dash-line curve. Fig. 7 shows the variations of the reflection (Pr ) and transmission (Pt) powers (of the slab-guide mode) with the
CHUNG AND CHEN: A MODIFIED FINITE ELEMENT METHOD FOR ANALYSIS OF FINITE PERIODIC STRUCTURES
'
,,
' '
0.8
-
::"''0" 0.
---
,
t"i 'I
'
:' 'II
~
...z
0.8
."'
0.6
z 0 f:: ()
0.4
'
"'u
P,
;:
--- --Pt
0.6
1565
- Nc= 20
0
----Nc= 40
()
0.4
.."'"'_,
0.2
0.2
'"
0 10
15
25
20
22
30
PERIOD NUM BER (Ne)
26
30
28
34
32
36
FREQ UENCY (G Hz)
Fig. 8. Variation s of the reflection (so lid lines) and transmi ssion (dash lines) powers as a function of the period number (N e) for the structure A shown in Fig. 4. The arrows indicate the increase of h 2. f 29 .2GHz, p 3. 6 mm, h = 0.25 t.
=
24
=
Fig. 9. Variation of the reflection coeffic ient as a functi on of the frequency for the structure C shown in Fig. 4. p 3 .6 mm, h 0.25t, h 2 2t.
=
=
=
V. CONCLUSIONS
increase of the period number (Ne) for the structure A with h = 0.25t and h = O..St. Since the frequency is chosen to be 29.2 GHz, which is simultaneously inside the two reflection bands of Fig. 6, the reflection power Pr (transmission power Pt) increases (decreases) monotonically to unity (zero) both for h = 0.25t and h = 0.5t. For a given number of periods, the reflection power of h = 0.5t (dash line) is larger than that of h = 0.25t (solid line), due to the stronger reflection at each step-discontinuity. In other words, to get a given reflection power, the total length of the structure with h = 0.5t will be shorter than that of the structure with h = 0.25t, since less periods are needed for the former structure. Fig. 8 presents the reflection (solid lines) and transmission (dash lines) powers as a function of the period number for the structure A with several heights h 2 (defined in Fig. 3) of the upper cover. The curves for h 2 = 4t, 6t, lOt (t = l.8mm) are almost the same since their propagation constants of the slab-guide mode are nearly identical ((3 ~ 0.796 rad./mm). As the height of the upper cover is reduced to h 2 = 2t, the propagation constant is varied a little ( (3 = 0.788 rad./mm), but the curve of Pr (Pt) still increases (decreases) monotonjcally with the increase of the period number. This phenomenon is changed when the upper cover is further pressed down to h 2 = t. At this height the propagation constant is changed to 0.730 rad./mm due to the strong influence of the cover on the field of the slab-guide mode. This seriously destroys the Bragg reflection condition for the given length of the periods (p = 3.6mm). For this reason, the curves of Pr and Pt for h 2 = t oscillate and do not change monotonically with the increase of N e. The frequency response of the reflection coefficient of a triangular grating shown as the structure C of Fig. 4 is illustrated in Fig. 9. The solid line respresents the result for 20 periods, and the dash line for 40 periods. It is noted that the increase of the period number can raise the mainlobe level, while keep the main-lobe bandwidth and side-lobe levels unchanged. Also note that from our calculations, the magnitudes of the reflection coefficients for a TE dominant mode incident from RHS are the same as those for that coming from LHS , as is the consequence of the power conservation law [8].
We have improved the algorithm of the finite element method for solving scattering and guiding problems of highly repetitive structures. The whole structure is divided into several super-elements which can be grouped into a few patterns due to the repetition property of the structure. Each super-element is discretized to many small elements and the Rayleigh-Ritz procedure and the frontal solution technique are used to obtain a submatrix for the super-element. Only a few submatrices are needed to calculate since the super-elements belonging to a pattern would lead to the same matrix. By assembling the submatrices of all the super-elements, a final system matrix equation is obtained and then solved. The validity of the proposed method has been checked by the self-convergence test and by comparing the numerical results with those obtained by another method. Also the power conservation law has been fulfilled for all the calculations. Numerical results, including the frequency response and the dependence on the period number of the scattering parameters, for finite periodic structures in dielectric-loaded parallel-plate waveguides have been presented. It has been found that, to predict the maximum-reflection frequency of a rectangular grating, one may approximately take the mean value of the Bragg-reflection frequencies (for a given period length) of the two slab waveguides with thicknesses t and t + h (see the structure A of Fig. 4). The increase of the period number has little influence on the main-lobe bandwith and the sidelobe levels of the frequency spectrum, but it does increase the reflection power in the main lobe. The proposed method can be applied to any variational equation to be solved by the finite element method, although a partial variational equation has been used in this paper. Besides, the efficiency of the present method is more obvious when more complicated structures, such as semiperiodic structures arbitrarily formed by two or more structure patterns, are tackled.
REFERENCES [I] G. I. Costache, "Finite element method applied to skin-effect problems in strip transmission lines," IEEE Trans. Microwave Theory Tech., vol. 35, pp. 1009-1013, Nov. 1987 .
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
[2] J.-M. Jin and J. L. Volakis, "A finite element-boundary integralformulation for scattering by three-dimensional cavity-backed aperatures," IEEE Trans. Antennas Propagat., vol. 39, pp. 97-104, Jan. 1991. (3] S.-J. Chung and C.H. Chen, "A partial variational approach for arbitrary discontinuities in planar dielectric waveguides," IEEE Trans. Microwave Theory Tech., vol. 37, pp. 208-214, Jan. 1989. [41 R.-B . Wu and C.H. Chen, "Variational reaction formulation of scattering problem for ansiotropic dielectric cyinders," IEEE Trans. Antennas Propagat., vol. 34, pp. 640-645, May 1986. [5] J. P. Webb, "Absorbing boundary conditions for the finite-element analysis of planar devices," IEEE Trans. Microwave Theory Tech., vol. 38, pp. 1328-1332, Sept. 1990. [6] 0. C. Zienkiewicz, The Finite Element Method. New York: McGrawHill, 1977. [7] P. Hood, "Frontal solution program for unsymmetric matrices," Int. J. Num. Meth. Eng., vol. IO, pp. 379-399, 1976. [8] R. E. Collin, Field Theory of Guided Waves. New York: McGraw-Hill , 1960. [9] T. E. Rozzi and G. H. ln 'tveld, "Field and network analysis of interacting step discontinuities in plannar dielectric waveguides," IEEE Trans. Microwave Theory Tech., vol. 27, pp. 303-309, Apr. 1979. [I OJ K. Uchida, "Numerical analysis of surface-wave scattering by finite periodic notches in a ground plane," IEEE Trans. Microwave Theory Tech., vol. 35, pp. 481-486, May 1987. [I I] M. Matsumoto, M. Tsutsumi, and N. Kumagai, "Bragg reflection chacteristics of millimeter waves in a periodically plasma-induced semiconductor waveguide," IEEE Trans. Microwave Theory Tech. , vol. 34, pp. 406-411, Apr. 1986. [ 12] S.-J. Chung and C. H . Chen, "Partial variational principle for electromagnetic field problems: Theory and applications," IEEE Trans, Microwave Theory Tech., vol. 36, pp. 473-479, Mar. 1988. [13] D. Marcuse, Light Transmission Optics. New York: Van Nostrand, 1982.
Shyh-Jong Chung was born January 18, 1962 in Taipei, Taiwan. He received the B.S.E.E. and Ph.D. degrees from National Taiwan University, Taipei, Taiwan, in 1984 and 1988, respectively. Since 1988, he has been with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan, where he is currently an Associate Professor. Hi s research interests in-
clude waveguide discontinuities, wave propagation, and numerical techniques in electromagnetics.
Jiunn-Lang Chen was born December 26, 1967, in Taiwan. He received the M.S. degree from the National Chiao Tung University, Hsinchu, Taiwan in 1993. He has been engaged in the research on theoretical and numerical analysis of electromagnetic radiation, scattering problems.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNJQUES, VOL. 42, NO. 8, AUGUST 1994
1567
Multimode Moment Method Formulation for Waveguide Discontinuities Arun K. Bhattacharyya
Abstract- A computationally efficient method for analyzing waveguide discontinuities is presented. The method is based on Galerkin's moment method formulation. The generalized scattering matrix of a waveguide discontinuity is derived. A generalized equivalent circuit is constructed which gives physical insight to the problem. The method is found to be several times faster than the well-known mode matching technique. The accuracy of the proposed method is demonstrated by comparing the numerical result with the measured return loss of a multisection iris-coupled filter.
I. INTRODUCTION
A
number of analytical and numerical techniques are available in the literature to analyze waveguide discontinuity problems. They include the variational method [l], [2], the Mode Matching Technique (MMT) [3], [4] and the Method of Moments (MOM) [5], [6]. The variational method provides fairly accurate results for an isolated di scontinuity in a waveguide supporting only one propagating mode. The results become inaccurate if two consecutive discontinuities are very close to each other. This is due to the higher order mode interaction between two discontinuities which is usually ignored in the variational formulation. The MMT can incorporate the higher order mode interactions between two consecutive discontinuities. In this method, the generalized scattering matrix of a discontinuity is determined from the continuity conditions of the electromagnetic fields. The size of the scattering matrix is proportional to the total number of modes considered. If a large number of modes are taken, this method yields accurate results. However, due to large matrix size, the method requires large CPU time. The moment method analysis presented in [5] is computationally more efficient than the MMT. This is due to the introduction of appropriate basis functions, which provides a faster convergence. The MOM analysis in [5] is identical to the MMT if the modal functions are taken as basis functions. In this paper, we present a more efficient computational method for analyzing a waveguide discontinuity. The method takes care of the higher order mode interactions between two consecutive discontinuities. Moreover, it is based on the Galerkin MOM, so that the advantage of basis function expansion of the fields on the discontinuity surface is fully utilized. In the proposed method, we divide the modes that are generated from a discontinuity surface into two categories. They are i) interacting modes, which include all propagating Manuscript receive June 18, 1993; revised October I, 1993. The author is with Hughes Space and Communications, Los Angeles, CA 90009 USA. IEEE Log Number 9402932.
GUIDE 1
GUIDE Z
z=O
Fig. I.
A general waveguide discontinuity.
modes and few evanescent modes which do not decay down at a fast rate and ii) noninteracting modes, which are evanescent and decay before the appearance of the next discontinuity . A set of entire domain basis functions (in principle, subdomain basis functions could also be used) are used to represent the equivalent magnetic current on the discontinuity surface. The equivalent magnetic current will produce all possible modes. From the continuity condition of the magnetic field and applying Galerkin's technique, we find a set of equations relating the incident, reflected and transmitted mode amplitudes. The generalized scattering matrix is deduced from the equation set. The size of the scattering matrix is proportional to the number of interacting modes only. The effect of noninteracting modes appear through additional terms in the matrix elements. Therefore, the matrix size could be very small here, even though a large number of evanescent modes are considered in the analysis. This effectively increases the computational speed. It is found that for a similar problem with similar accuracy, the speed is about 8 to 30 times faster than MMT. The organization of the paper is as follows: Section II provides a formulation of the problem. The generalized scattering matrix is deduced in this section. Section III derives the generalized equivalent circuit of a waveguide discontinuity. The concept of the generalized transformer is introduced in this section. Section IV shows numerical results justifying the need for appropriate basis functions . Section IV also demonstrates the accuracy of the formulation by comparing the theoretical results with the measured result. The conclusion is outlined in Section V.
II. FORMULATION Fig. 1 shows the geometry of a general waveguide discontinuity. On the discontinuity surface (z = 0) an equivalent magnetic surface current is assumed to represent the fields in the waveguides. For the fields in waveguide 1 (z < 0), let the magnetic surface current be M. The equivalent magnetic surface current to represent the field in the waveguide 2 (z > 0) would be -M.
0018-9480/94$04.00 © 1994 IEEE
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
We represent the magnetic surface current in terms of a set of known vector basis functions as follows: N
M=
LAnf:(x , y) , n=i
(1)
where the vector function is oriented in the transverse direction, so that i · x, y) = 0. The magnetic current exists on the open region of the discontinuity. The magnetic current source will produce the electromagnetic fields in both regions. These fields are determined using dyadic Green 's functions. For the fields in the region z < 0, we should place a conductor behind = (i ) the magnetic surface current M. Let Ge, be the appropriate dyadic Green 's function which relates magnetic current to the transverse electric field in waveguide 1. Then the transverse electric field vectors in waveguide 1 would be
(8) Substituting the expression for M from (1), we obtain Eii in terms of the coefficients of the vector basis functions. N
1: (
-f 1 -
(2)
R
where R represents the discontinuity surface at z 0. The transverse electric field vector in the region z > 0 is
-f 1 -
E- (2) -
L
Anael(i, n)
(9)
n=i where
j L, j L(1: ·G~i))
ael(i , n) = }i)
· e}1lds dsi. (10)
it
Since f n is independent of Ri, we can interchange the order of integration, which results in . 1 ael(i,n)=e(i)
J r - [j r }Rfn·
=(i)
Jn,Ge
;: Mi, one writes N
(16)
n=l
In waveguide 2, the corresponding equations are N
Ri
(2) -eii
N
Et +EI;= LAnael(i,n)i :=:;Mi. n=i
EI;= LAnael(i, n) i >Mi.
e(i) . e(i) dsi t i
e(i ii) =
(14)
(5)
Et+ E2; = LAnae2(i, n) i :=:; M2 n=i
(17)
N
-(2) ds2, · ei
Rz
where Ri and R2 are the cross sections of waveguide 1 and 2, respectively. Substituting (2) in (6) one obtains:
E2; = LAn ae2(i , n) i > M2.
(18)
n=l
Equations (15)-(18) yield the continuity of the electric field across the discontinuous surface. The continuity condition is
BHATTACHARYYA: MULTIMODE MOMENT METHOD FORMULATION FOR WAVEGUIDE DISCONTINUITIES
satisfied since the same magnetic surface currents were used to obtain the electric fields in both sides. The magnetic field continuity condition is used to obtain another set of equations. The transverse magnetic fields on both sides are 00
jj(l)
=L
E;,;] hp)
[Eii -
( 19)
i=l 00
jj(2)
[Et - Ei;] h~2)
=-L
However, we can include a very large number of terms in (27) (perhaps several hundred terms) without increasing the size of the matrix inversion. It is the inclusion of these terms, without increasi ng the matrix size, that makes the proposed technique computationally efficient. Equations ( 15), ( 17), and (26) can be used to construct the generalized scatteri ng matrix of the junction. To that end, we can express the above equations in matrix notations as follows.
(20)
[EiJ + [E!] = [ael][A] [E;t-] + [E2] = [ae2][A] [hfl][EiJ - [hfl][E!] + [hf2][E;t-] - [hf2][E2] = [ah][A].
·i =l
where hp), hp) are the transverse magnetic modal vectors in waveguides I and 2, respectively. These quantities are related to e'.;(l) and eF) respectively by Maxwell's equations. The continuity of the transverse magnetic field components demands that the following equation holds 00
00
2
L[Eii - E;,;JhF) = - L[Et - Ei;]h~ ). i=l
(21)
i=l
We now take the vector product of (21) with .fn, retain the zcomponent and then integrate over the discontinuity surface. This yields 00
00
i=l
i=l
where
l Jh~l)
x ln. zds
. JR r 1-c2) -h Xfn·zds.
(24)
hf2(n ,i) =
1
Recall that E{i = 0 for i therefore (22) becomes
> M1
M1
+L
(Et - Ei;)hf2(n,i)
00
Ei;hf2(n , i) = 0.
L
i =M1 +l
(25)
i=M2+l
Using (16) and (18) in (25) one obtains,
L (E{i - E;,;)hfl(n , i)
+ L (Et - Ei;)hf2(n, i) i =l
i=l
L Amah(n,m)
=0
(26)
m=l
where 00
ah(n, m) =
ael(i, m)hfl(n, i)
L i=l+M1
00
+ L
ae2(i, m)hf2(n, i).
[Z1] = [ae1]{[ah] + [hfz][ae2]}- 1[hfi] [Z2] = [ae2]{[ah] + [hfl][ael]}- 1[hfz]
(33)
Ill. EQUIVALENT CIRCUIT
N
X
(32)
This completes the derivations of the generalized scattering parameters of the waveguide discontinuity.
M2
M1
511 = -{[J] + [Z1]}- 1 {[J] - [Z1]} 512 = 2{J + [Z1]} - 1[Z1] 521 = 2{[!] + [Z2]} - 1[Z2] 522 = -{[I]+ [Z2]}- 1 {[J] - [Z2]} with
00
E;,;hfl(n, i) -
L
(30)
where each element in the 5 matrix is a submatrix and is expressible in terms of known quantities. The expressions are derived as
= 0 for i > M2,
i=l
i =l
(29)
(31)
Mz
L (E{i - E;,;)hfl(n ,i) x
and Et
(28)
Note that [EiJ, [E!] are column matrices of order M 1 x 1, [E;t-], [E2] column matrices of order M 2 x 1, [A] is a column matrix of N x 1, [ael] and [ae2] have orders M 1 x N, and M2 x N, respectively. [hfl] and [hf2] have orders N x M 1 and N x M 2 , respectively. [ah] is a square matrix of order N x N. From (30) we express [A] in terms of [Ei], [E!] , [E;t-], [E2] and then substitute in (28) and (29). After a lengthy algebraic manipulation, we express the equations in the scattering matrix format
(22)
(23)
hfl(n, i) =
1569
(27)
An equivalent circuit of this discontinuity can be constructed from (28)-(30). The equivalent circuit is valid if mutually orthogonal modes are considered. 1 However, the analysis is presented in Section II is applicable even for nonorthogonal modes. To give a physical meaning to the equivalent circuit, we normalize the modal vectors. This is necessary to make the circuit reciprocal. Therefore, we select modal vectors, ~l) and
4_2 l such that elJl = 1 = e~zl. With this normalization we can
i=l+M2
define E{i, Et as modal voltages and [E{iY/1) ] and [EtY/2) ]
The infinite series in (27) gives the important influence of evanescent modes that decay in the close vicinity of the junction. In practice, this infinite series must be truncated.
1 It is to be pointed out that the TE., and TM., (a = x, y , z ) modes together form a complete set of mutually orthogonal modes. Howeve r, the TEx and TEy modes together form a complete set, but they are not mutually orthogonal.
1570
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
z=O
y
z =O
Li [o o l ) d I )
Fig. 2.
____.n.____
[ I )' [ae 2 ]
(a)
Equivalent circuit representation of the waveguide discontinuity. 8
y 8
.\ x
\
,,.,....,,...
(a)
...__.._ /
'-... ~
!
I
i::;..._
/\
8
,;
---------
~
v\
~
8
-""
'?-0 . 25
-0 . 15
-0 . 05
0 .05
0.15
0 . 25
\J I N INCHES
(b)
-
Fig. 4 (a) An iris di scontinuity in a waveguide. The waveguide dimensions are 2.54 cm x 5.08 cm. The aperture dimensions of the iris are 1.27 cm x 5.08 cm. Iris thickness= 0.0127 cm. D real, 0 imagi nary. (b) The electric field intensity on the iris aperture shown in Fig. 4(a). Frequency = 4.0 GHz. Orea! , 0 imaginary. 8 ,;-0 .25
-0 . 15
- 0.05
0.05
0 . 15
0 . 25
IV.
\J I N INCHES
(b)
Fig. 3. (a) A junction of two waveguides. The H-plane dimensions are equal. (b) The electric field intensity at the waveguide junction. The E-plane dimensions are 2.54 cm and 1.27 cm, respectively. The H-plane dimensions are 5.08 cm for both guides. Frequency = 4.0 GHz. D Real, 0 Imaginary.
as modal currents, where ~(l), ~< 2 l are mode admittances. From (23) and (24), it can be shown that for orthogonal modes,
n)}
hfl(n, i) = ~( l ) ael (i, hf2(n, i) = ~( 2 ) ae2 (i, n)
34 < )
so that (28)-(30) can be expressed as
[Vi] = [ael] [A] [Vi] = [ae2] [A] [ael] [Ii]+ [ae2] [h] =[ah] [A]
(35) (36) (37)
with [Vi ] = [Et+ E!], [Vi] = [Et°+ E2], [Ii] = [Y1][Et E!] and [h] = [Y2][Et°-E2 ]- [Y1] and [Y2] are characteristic admittance matrices whose off-diagonal elements are zeros. Equations (34), (35) and (36) yield an equivalent circuit of the discontinuity as shown in Fig. 2. In the equivalent circuit, [ael] and [ae2] act as the tum ratios of the transformers. The [ah] matrix is the admittance matrix responsible for the stored energy in the vicinity of the discontinuity. The scattering matrix of the discontinuity can be constructed from the equivalent circuit also. Note that the conventional equivalent circuit is a special case of the generalized equivalent circuit with only one incident mode.
RESULTS
AND
DISCUSSIONS
The entire formulation was computer programmed to compute the generalized scattering matrix of a waveguide discontinuity. The program was very general to accommodate different waveguide dimensions at the two sides of the discontinuity. To examine the nature of the field distribution, two different discontinuities were considered. Fig. 3(a) and (b) show a junction of two rectangular waveguides of different dimensions and the electric field di stribution on the junction plane. The Hplane dimensions were the same for both guides. The incident mode was the TE 10 mode. About seven basis functions were needed for obtaining a converged result. The field has a peak at the center in this case, and has a number of local maxima and minima. Fig. 4(a) and (b) show an iris discontinuity in a waveguide and the associated field distribution on the discontinuity plane. Unlike Fig. 3(b) the field has peak values at the two ends of the discontinuity plane. In this case about ten cosine basis functions yield converged results. From the nature of the field distribution, it is very apparent that a different type of basis function would be more appropriate for a faster convergence. In fact we also used the basis function with square-root singularity at the two ends as given in [2] , [7]. For a similar result, only six terms were sufficient. This justifies the use of alternate type of basis functions for the iris discontinuity instead of using the mode functions as basis functions . In order to validate the present formulation , the return loss of a four-section iris-coupled filter was computed and compared with the measured results. For numerical compu-
BHAITACHARYYA: MULTIMODE MOMENT METHOD FORMULATION FOR WAVEGUIDE DISCONTINUITI ES
g 0
0
~o
< ...:
0'
~ ~o
J
\
\ I f1
U1 U1 Do _Jo
z .;
"'' >-
0
:'.)
u.J
°'g
o
0
0
I
0
0
g
"'
11.10
0
l. 20
t.?>O
I.~
1.50
L. 60
1.70
FREa.ENCY IN Gil )( 10" 1
Fig. 5. Return loss of a four-section iris coupled filte.-theory, 0000 measurement.
tations, fourteen basis function s were used (N = 14) to characterize each discontinuity. The number of coupled modes were taken to be the same as N, that is M 1 = M 2 = 14. About 100 terms were taken in the Green 's function. Fig. 5 shows the comparison between the theoretical and measured return loss. Excellent agreement was found. To obtain a similar accuracy from a mode matching program one needs to consider about 100 modes . This necessitates 100 x 100 order of matrix manipulations including several times matrix inversions. However, in the present formulation the matrix size is only 14 x 14. The CPU time is much lower in this case as compared to a mode matching program. This proposed method was also used to predict the performance of several other microwave networks. They include iris coupled polarizer and rectangular corrugated horn. The computational speed is about 8 to 30 times faster than that of a mode matching formulation. V . CONCLUSION
A computationally efficient moment method formulation is presented for analyzing waveguide discontinuities. A generalized equivalent circuit is also developed. The advantages of thi s method over existing methods are: 1) The proposed method can isolate the modes that are responsible for interaction between two successive di scontinuities and the remaining higher order modes which are only responsible for energy stored in the vicinity of the discontinuity. The scattering matrix size is equal to the number of interacting modes on ly. In the well known mode matching formulation [3], [4], the matrix size is equal to the total number of modes considered. Therefore, for a simi lar accuracy, the matrix size is much smaller and the computational speed becomes greater. The earlier moment method [5] is very similar to the mode matching except the flexibility of usi ng different basis function instead of mode function s.
1571
2) With an appropriate choice of basis function, the solution converges at a faster rate. This flexibility is not available in the mode matching form ulation. In addition, the proposed method is applicable for arbitrary shaped aperture on the discontinuity plane, and offset waveguide junctions as in Fig. 1. 3) Depending on the problem, the number of coupling modes and basis functions can be adjusted to enhance the computational speed. For example, if two discontinuities are far apart, only very few coupli ng modes are sufficient. On the other hand, for a minor discontinuity only few basis functions may be sufficient to characterize the discontinuity plane. To sum up, the method is very flexible for adjusting the number of basis functions and the number of interacting and noninteracting modes. The matrix size involved in the computation is much smaller than that for the mode matching procedure, which improves the computational efficiency considerably. ACKNOWLEDGMENT
The author thanks Mr. Alan Keith for supplying the measured results of the iris-coupled filter. Thanks are also due to Dr. Mark Schalit for his valuable technical suggestions and to Mr. Frank Boldissar for his interest towards the research work. REFERENCES [ l] R. F. Harrington , Time-Harmonic Electromagnetic Fields. McGrawHill Book Co. , 1961 , ch. 8. [2] R. E. Coll in, Field Theory of Guided Waves, 2nd ed. IEEE Press, 1991. [3] G. L. James, "Analysis and design ofTEu to HEu corrugated cyli ndrical waveguide mode converters," IEEE Trans. Microwave Theory Tech. , vol. MTI-29, pp. 1059-1066, Oct. 1981. [4] _ _ ,"On the problem of applying mode-matching techniques in ana lyzing con ical waveguide discontinuities," IEEE Trans. Microwave Theory Tech. , vol. MTI-31 , pp. 718-723, Sept. 1983 . [5] H. Auda and R. F. Harrington, "A moment solution for waveguide junction problems," IEEE Trans. Microwave Theory Tech., vol. MTI-31 , July 1983. [6] M. Leong, P. S. Kooi , and P. Chandra, "A new class of basis functions for the solution of the E-plane waveguide discontinuity Problem," IEEE Trans., vol. MTI-35, no. 8, pp. 705-709, August 1987. [7] K. C. Gupta, R. Garg, and I. J. Bahl , Microstrip Lines and Slot Lines. Norwood, MA: Artech House.
Arun K. Bhattacharyya was born in India in 1958. He received the B.E. degree in electronics and telecommunications from Bengal Engineering College, University of Calcutta, and the M.Tech. and Ph.D. degrees from the Indian Institute of Technology, Kharagpur, in 1980, 1982, and 1985, respectively. From November 1985 to April 1987, he was with the Department of Electrical Engineering, University of Manitoba, Manitoba, Canada, as a Postdoctoral Fellow. From May to October 1987, he worked with Til-Tek Ltd. , Kemptville, Canada, as a Senior Antenna Engineer. In November 1987, he joined the Uni versity of Saskatchewan , Saskatoon, Canada, as an Assi stant Professor, then Associate Professor in 1990. In July 1991, he joined the Hughes Space and Communication Group. His research interests include printed antennas and circuits, modeling of planar and nonplanar microwave antennas and circuits. He is the author of a book, Electromagnetic Fields in Multilayered Structures: Theory and Applications (Artech House, 1994). He has several publications in his research fields .
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
Microwave Imaging for a Dielectric Cylinder Huang-Tien Lin and Yean-Woei Kiang, Member, IEEE
Abstract- The problem of reconstructing both the shape and the relative permittivity of a homogeneous dielectric cylinder from the measurement of scattered field is numerically simulated. The Newton-Kantorovitch algorithm and the moment method are used to solve a set of nonlinear integral equations. Numerical results show that, with multiple incident directions, good reconstruction is obtained. This algorithm can be applied at a single frequency without limitation on the value of dielectric constant. The effect of random noise on imaging reconstruction is also investigated. I. INTRODUCTION
T
HE inverse scattering problem is to recover information about some inaccessible region from the scattered wave fields measured outside. This problem has attracted increasing attention owing to its applications in remote sensing, medical imaging, nondestructive evaluation, etc. However, it is difficult to solve because of its ill-posedness and nonlinearity. For electromagnetic imaging of dielectric objects, several numerical methods have been proposed. By using the approximate approach, the recovering procedures may be simplified. Taking the conventional diffraction tomography technique [1] for example, good reconstruction can be obtained, while it is suitable only within the limitation of Born or Rytov approximation. The inverse problem can also be treated by solving relevant exact equations through numerical methods. Ney, Smith and Stuchly [2] calculated some two-dimensional inverse problems by the moment method and the pseudoinversion technique. It is an original method, but the employed data are limited to one incident wave. Besides, better reconstructed result may be obtained for the object of long and narrow cross section. Chew and Wang [3], [4] developed the Born iterative and distorted Born iterative methods. The latter converges faster than the former, while the distorted Born iterative method is not so robust to noise contamination in comparison with the Born iterative method. However, both of these two methods are very concise in principle. Colton and Monk [5], [6] transferred the inverse problem into an optimization one, and then offered another point of view theoretically on this problem. The "layer-stripping formula" was derived in [7] for solving the inverse problem, but further simulation was needed to verify the feasibility of the algorithm. Recently, Wang and Zhang [8] proposed an unrelated illumination method for this problem. It is novel to increase information by arranging various incident waves. Thus the difficulty of the ill-posedness is avoided and good reconstruction result can be obtained through simple matrix operations. Manuscript received July 29, 1993; revised October 11 , 1993. The authors are with Department of Electrical Engineering National Taiwan University Taipei, Taiwan, Republic of China IEEE Log Number 9402948.
In this paper, we confine the dielectric cylinder to be homogeneous. The object might be a strong scatterer of which the cross-section shape and relative permittivity are unknown. Based on the concept of equivalence principle, the scattered field can be expressed in terms of the equivalent surface electric and magnetic currents [9]. As the equivalent sources distribute on the object surface only, we need less basis functions for numerically solving the surface integral equations rather than for the volume integral equations (such as the Lippmann-Schwinger integral equation). Thus the number of unknowns can be reduced because it is unnecessary to divide the cross section of the object into cells for numerical computation. By applying the Newton-Kantorovitch algorithm [10] and the moment method [11], the inverse problem can be solved with iterative procedures. We need only approximately guessed values of the size and the relative permittivity of the dielectric object as the prior knowledge. Multi-incident waves are applied to get enough information for the reconstruction. It is worth noting that good reconstructed shape and relative permittivity can be obtained although no regularization is used. In Section II, the theoretical formulation is presented. In Section III, numerical results for both cases of lossless and lossy objects are given, and the effect of noise is also simulated. Finally, some conclusions are drawn in Section IV. II. THEORETICAL FORMULATION
A. Direct Problem
Consider a homogeneous dielectric cylinder located in free space as depicted in Fig. 1. The cross section of the object is of starlike shape, i.e., it can be described in polar coordinates in the xy-plane by the equation p = F( B) with respect to its reference center (x 0 , y0 ) . The permittivity and permeability of free space and the dielectric object are denoted by ( i: 0 , µ 0 ) and ( i: 2 , µ 2 ) respectively. Here i: 2 can be complex to take into account the loss effect. The dielectric object is illuminated by an incident plane wave whose electric field vector is parallel to the z-axis (TM polarization). Then the incident electric and magnetic fields can be given by Ei(X , y) = Eo e -jk 0 (xs in¢ -ycos¢) Z
H;(x , y) =
-Eo
(1)
~(cos. 0.50
'' ''' '
0.40
stepped
. ...
0.30 0.20
modified
'
'
'
''
RESULTS
A rectangular waveguide was modelled and excited with both the stepped and the modified excitations. The frequency of excitation was 1.25 times the cutoff frequency for the TE 10 mode. As can be seen from the results shown in Fig. 5, a vast improvement has been achieved in the settling time of the resulting transient, reaching a steady state much sooner without marked oscillation. These results are presented with the modified excitation having a rise period, T, of JO cycles of the excitation frequency . If the rise period is decreased, the amount of transient created by the modified excitation will increase, approaching that of the stepped excitation as the rise period is reduced to zero. This is because the frequency spectrum for the shorter rise time excitation is more spread out giving an increased spectral content below cutoff. Choice of the rise period, T, for the modified excitation must then be made in such a manner that both the rise time and the amount of transient produced are acceptable.
0.00 0
1000
2000
3000
Time (n.dt) Fig. 5.
Comparison of excitation envelopes
frequency increases above the cutoff frequency that this rise time can be decreased as long as we do not create too many of the below cutoff components. To reduce the transient we have chosen a new function to replace the stepped monochromatic excitation. Here we have based our new excitation on the Hanning window function [4]to alter the shape of the rise of the excitation. Modified monochromatic excitation (Hanning):
sin ( ~~) sin(2Ir f mt )u( t) - ~ cos(2f-))u(t)
ifT < t if 0:::; t:::; T. otherwise (3) This function has been chosen because it has a smooth shape and has an increased attenuation in the magnitude of the transients' spectral sidelobes. This is achieved at the expense of a slightly wider mainlobe. However, as time proceeds this mainlobe becomes insignificant as the spectrum approaches that of the single frequency. It should be noted at this point that these changes to the excitation function are only concerned with the time component and not the spatial component. Because we are enforcing the TE 10 mode across the whole of the exciation plane, higher order modes will not be generated by altering the time function as only TE 10 modes will exist.
Ey
= { ~in(~)sin(2Irfmt) ( ~
IV.
CONCLUSION
A simple technique has been presented which can be employed to decrease the transient time in a single frequency Finite-Difference Time-domain simulation. Results are presented which demonstrate both the methods' validity and effectiveness when applied to the waveguide structure. ACKNOWLEDGMENT
The authors would like to thank the University of Queensland Computer Centre for the use of their MasPar MP- I computer facilities. The authors would also like to acknowledge a similar approach reported in [5], published while this paper was in preparation. Their modified excitation function, although only mentioned briefly, makes use of a linear ramp function over several cycles to increase the electric field to its steady state value. REFERENCES
[l) Z. Q. Bi, K. L. Wu , and J. Litva, "Application of the FD-TD method to the analysis of H-plane waveguide di scontinuities," Electron. Lett., vol. 26, no. 22, pp. 1897-1898, Oct. 1990. [2) E. A. Navarro, Y. Such, B. Gimeno, and J. L. Cruz, "Analysis of H-plane waveguide discontinuities with an improved finite-difference time-domain algorithm," IEE Proc.-H, vol. 139, pp. 183- 185, Apr. 1992. [3] R. Bracewell, The Fourier Transform and its Applications. New York: McGraw Hill , 1965 . [4) A. Y. Oppenheim and R. W. Schafer, Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975. [5) J. H. Beggs and R. J. Luebbers, "Nu merical analysis of staircasing effects on cutoff frequencies for FDTD modeling of circular waveguides,", IEEE AP-S Int. Symp. Dig., June 1993, pp. 546-549.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES , VOL. 42, NO. 8, AUGUST 1994
1585
Letters _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ Comments on "Theoretical and Experimental Characteristics of Single V-Groove Guide for X-Band and 100 GHz Operation"
That result is in contradiction with both the physical meaning and the mathematical meaning we related above. In order that J( v may accord with them, it should be rewritten by
Hu Chi
Kv=
( 11 2ac 2 +
lale
2
-1; :-
)
h( x , y)cos
-a
- c
2 (
-7rX) 2c · cos
In the above paper [ l], the key question is to derive the scaling factor, h(x , y), which is necessary for the determination of the propagation characteristics of single V -groove guide. The authors use a conformal mapping technique to obtain it. Since the cross section of the V-groove guide is symmetrical, one quarter is considered only, which is shown in Fig. 15. According to the symmetrical principle of the conformal mapping, the transformed cross section is also one quarter of the parallel-plate waveguide which is filled with a nonisotropic and nonhomogeneous medium, and the scaling factor h( x, y) should be symmetrical about both x axis and y axis. Therefore, the transformed cross section which is plotted in the complex Z plane in Fig. 15 should be modified. In the Z plane, it should be plotted in the region where 0 :=:; x :=:; c and y 2: 0, and a series of conclusions (given in Appendix A, D and Section II-B-1) concerning it should also be modified respectively as follows [2], [3]:
Z =-Ki cos- 1 T T
= cos [- ~ ( Z -
h(x, y)
= Ibr I c
c)]
= -cos [ ~ ( x + j y)]
cos [~ (x + jy)] + ro [ cos [~ (x + JY)] + 1 ]
1/2-8/w
(3)
Last, according to the principles of electromagnetics, on the wall of a perfect conductor, the vertical components of a magnetic field should be zero. Hence, with respect to Fig. 5, the boundary condition (iii) should be modified as follows: (iii)
at x
= LBrsin (k,,sx) exp [-ky s(y -
a)]
y
2:
a
y :=:; -a
(5b-l)
(5b-2)
In addition, expression (6a) indicates a relation between the wavenumbers in region A shown in Fig. 5. In the expression (6a), h 2 (x, y) is replaced by its weighted average Kv in order to simplify the computation. In terms of the conformal mapping, if h( x, y) 1 in region A, it means that the parallel-plate waveguide is filled with an isotropic and homogeneous medium, and J( v = 1. However, according to the expression (6a), if h(x, y) = 1, we have
R EFERENCES
[I) Y. M. Choi , D. J. Harris, and K. F. Tsang, "Theoretical and experimental characteristics of single V-groove guide for X-band 100 GHz operation," IEEE Trans. Microwave Theory Tech. , vol. 36, pp. 715-723, Apr. 1988. [2] F. J. Tischer, "The groove guide, a low-loss waveguide for millimeter waves," IEEE Trans. Microwave Theory Tech., vol. 11 , pp. 291-296, Sept. 1963. [3] Hu Chi, "Comments on 'The groove gu ide, a low-loss waveguide for millimeter waves '," (submitted IEEE T-MTT).
Reply To Comments on "Theoretical and Experimental Characteristics of Single V-Groove Guide for X-Band and 100 GHz Operation"
In Hu's paper [l], it was pointed out that as the cross-section of the V-groove guide is symmetrical, one quarter of the section should be considered. This point exactly matches the authors ' idea in [2]. In fact, Hu's implementation of the transformation of the guide section from the U-plane to the Z-plane for the region 0 :=:; x :=:; c and y 2: 0 looks more straight forward. However, in the method discussed in [2], symmetry can also be acquired by translating the transformed coordinates in the Z-plane. Hu also pointed out that [2, (Sb)] should be given by
y
=
=
=±c
Y. M. Choi, K. F. Tsang, and D. J. Harris
r
,,. Av
(:~) dx dy.
+B
where K1 = c/7r and B = c. They are obtained by matching boundary values at T = -1 with Z = 0 and at T = 1 with Z = c. In Section II-B-2, it is difficult to understand that y 2: lal for a represents the half width of the groove. It must be that IYI 2: a . In the region B, magnetic fields attenuate as IYI is increased, so the field equation (5b) should be given by
H zs
2
1 [ c . (7ra)J [ 2a . (7rc )] c + -:;;:- sm 2a 2ac ( ~ + ~) a+ ; sm --;;-
Manuscript received January 14, 1994. The author is with the Department of Electronic Engineering, Southeast University, Nanjing, 210018, People 's Republic of China.
2: a.
A careful manipulation in the proposed expression reviews that it is always possible to absorb the term exp ( k ysa) into the amplitude Manuscript received March 16, 1994. Y. M. Choi and K. F. Tsang are with the Department of Electronic Engineering, City Polytechnic of Hong Kong, Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong. D. J. Harris is with the School of Electrical, Electronics and Communication Engineering, University of Wales College of Cardiff, Cardiff, Wales, United Kingdom.
0018-9480/94$04.00 © 1994 IEEE
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
coefficient, B r, that precedes. In fact, both forms of expression could be used to obtain the secular equation as shown in [2, (7)]. Nevertheless, the authors agree that there are typing mistakes in [2]. The amendments are made as follows:
1. Section II-B-2: "y ~ Jal should be written as 2. Expression (6a): J( v should be written as
Kv
= 2ac(l/21+ 1/r.)
laleh\x, -a
- c
"JyJ
~
a";
y) cos2 (r.x/2c) 2 · cos ( r.y/4a) d x dy
since the limits on x must be ± c and the limits on y must be
±a. 3. Section II-B-2: The boundary condition (iii) should be written as
oHzA /o x = oHzs/ox =
0
atx
only be made by the discretization of the partial differential equations and not by replacing the partial differential equations by different partial equations. This results in unphysical, spurious solutions: The discretization of the system of first order two-dimensional partial differential equations, ( 10), does not in general yield an algorithm conserving the energy (only in the special case of a= 1/4). This is in contradiction to the energy conservation embodied in Maxwell's equations. 2) For the discretization of the system of first order twodimensional partial differential equations, ( I 0), the authors are using Cw ave = Cm es h, where Cw a ve represents the wave propagation velocity and Cm es h the velocity of the voltage pulses on the mesh. This is in contradiction to the correct relation Cw a ve = Cm es h/2 for the TLM scheme based on the condensed symmetric TLM node [2], [3] .
= ±c
which corresponds to the requirement that the vertical component of the electric field (but not the magnetic field as described by [I]) should vanish on the metallic wall. In fact, if the boundary conditions (i) and (ii) are applied to the field equations (5a) and (5b ), the secular equation (7) can be obtained. Boundary condition (iii) is applicable only for higher-order solutions. 4. Table III: "100 MHz" should be written as " 100 GHz"
REFERENCES
[I] J. LoVetrii and N. R. S. Simons, "A class of symmetrical condensed node TLM methods derived directly from Maxwell's equations," IEEE Trans. Microwave Theory Tech., vol. 41 , pp. 1419-1428, Aug. 1993.
[2] P. B. Johns, "A symmetrical condensed node for the TLM-method," IEEE Trans. Microwave Theory Tech., vol. MTI-35, pp. 370-377, Apr. 1987. [3] J. S. Nielsen and W. J. R. Hoefer, "A complete dispersion analysis of the condensed node TLM mesh," IEEE Trans. Magnet., vol. 27, pp. 3982-3985, Sept. 1991.
REFERENCES
[I] H. Chi, "Comments on 'Theoretical and experimental characteristics of single Y-Groove guide for X-band and 100 GHz operation''', submitted to IEEE T-MTI. [2] Y. M. Choi, D. J. Harris, and K. F. Tsang, "Theoretical and experimental characteristics of single Y-groove guide for X-band and 100 GHz operation'', IEEE Trans. Microwave Theory Tech., vol. 36, pp. 715-723, Apr. 1988.
Reply to Comments to "A Class of Symmetrical Condensed Node TLM Methods Derived Directly from Maxwell's Equations" Joe LoVetri and Neil R. S. Simons
Comments on "A Class of Symmetrical Condensed Node TLM Methods Derived Directly from Maxwell's Equations" M. Krumpholz and P. Russer
The derivation presented in [I] is erroneous for the following reasons: I) No mathematical justification is given for the approximation of Maxwell's equations by a system of first order two-dimensional partial differential equations, (10). Adding the three (IO) yields
In our paper, [l] , we derive the three-dimensional symmetrical condensed node TLM algorithm usi ng a characteristic based field decomposition of Maxwell 's equations. We obtain identical scattering and transfer events as those originally presented in [2]. The goal and eventual result of our investigation was to present a mathematically sound method for deriving the TLM scattering and transfer events directly from Maxwell ' s equations (without recourse to the approximation of space by a mesh of transmission lines). The statement made by Krumpholz and Russer, that the derivation presented in [l] is erroneous, is not valid and the two specific points they raise will now be considered. First, Krumpholz and Russer suggest that, in [!], the use of the three two-dimensional partial differential equations, (10'),
(I)
Oiu + A c Oz U = 0
which is in contradiction to the correct (I). For the derivation of numerical methods simu lating the evolution of the electromagnetic field, an approximation of Maxwell's equations should
(I)
Manuscript received March 14, 1994. The authors are with Ferdinand-Braun-Insti tut ftir Hi:ichstfrequenztechnik, Rudower Chaussee 5, 12489 Berlin, Germany.
Manuscript received April 11 , 1994. J. LoVetri is with the Department of Electrical Engineering, University of Western Ontario, London, Ontario, Canada, N6A 589. N. R. S. Simons is with InfoMagnetics Technologies Corporation, 1329 Niakwa Road East, Winnipeg, Manitoba, Canada, R2J 3T4.
0018-9480/94$04.00 © 1994 IEEE
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 42, NO. 8, AUGUST 1994
to approximate (1') ~u+AE ~u+AF~u+A c ~u=O
(2)
is incorrect since (2) does not follow from (I) by summation (we use primed numbers to refer to equations appearing in [l]). This is obvious and it was never suggested that (2) could be derived from (I). Instead, as was described in [l] , the three equations of (1) are found by the approximation of assuming there to be no field variation in the spatial directions missing in each equation. This was explicitly described in the first paragraph of Section III of [l] entitled Numerical Approximations. Of course if one requires ( 1) to be exactly derivable from (2) then one could change (1) to Ot U
OtU
+ 3AF OyU = 0,
+ 3Ac O z U = 0
(3)
which is typical of dimensional splitting methods [3]. A finite difference method for a 1/3 time step could then be used for each of the three equations. This was not done so as to not obscure the physical interpretation of the approximation being made (i.e., of plane wave propagation along the rectangular arms of each cell). It should also be noted that (3) would produce the same Riemann invariants as in ( 1) but propagating at three times the speed. Starting from this approximation a class of TLM algorithms, based on a parameter a, were derived. The principle of energy conservation was then used to set a = 1/4. It was found that other values of a caused the numerical scheme to be dissipative and the only reason these other values of a were investigated was for the possibility of strategically using a small amount of artificial dissipation at different points in the mesh. This possibility has yet to be investigated. The second point raised by Krumpholz and Russer is the disparity between Cwave and CR J. (We refer to Cw a v e as the group velocity for wave propagation in the mesh as t::.1/>.. -+ 0, and CR J as the propagation velocity of the Riemann invariant or voltage pulse variables). This disparity has been associated with the TLM method since its introduction in 1971 [4] . Originally (in the two-dimensional formulation) Johns formulated the method with CR J equal to the velocity of light in free space Co . The resultant Cwav e is co / ./2, and therefore Johns referred to the "free space" medium modelled by the mesh as having a relative permittivity of 2 and a relative permeability of 1.0. To obtain results in a true free-space medium, frequency renormalization is required. For the three-dimensional symmetriccondensed TLM model, Cwave = CRJ /2. In the development of this model Johns no longer referred to CRJ as Co, and therefore the frequency renormalization is avoided, since it can be assumed that Cw ave = c0 • In general, when developing a . TLM model, the relationship between Cw a v e and CRJ should be determined by dispersion analysis as has been the case for the development of alternative two-dimensional TLM models [5]-[7]. The specification of a physical significance to CRJ is not required. In fact, CR J can be considered to represent a propagation velocity of information within the mesh. This information-propagation aspect of CR J was beneficial in the development of the two dimensional TLM models presented in [6] and [7] . In our derivation, we do not assume that the velocity of the physical wave being modelled will be equal to CR1. In fact, as was explained above, in the derivation one has the freedom to change the velocity of the voltage pulses relying on dispersion analysis to determine the correct speed. Unfortunately, we did not expand on this aspect of the
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derivation in our paper. The result is a sound and correct derivation (with the parameter a = 1/4) of the three-dimensional symmetric condensed node TLM algorithm. Any criticisms regarding spurious modes or other nonphysical behavior should be directed to the CEM community at large, and not to our specific derivation of the TLM method. REFERENCES
[I] J. LoVetri and N. R. S. Simons, "A class of symmetrical condensed node TLM methods derived directly from Maxwell 's equations," IEEE Trans. Microwave Theory Tech. , vol. 41 , pp. 1419-1428, Aug. 1993. [2] P. B. Johns, "A symmetrical condensed node for the TLM method," IEEE Trans. Microwave Theory Tech. , vol. MTT-35, pp. 370-377, Apr. 1987. [3] N. N. Yanenko, The Method of Fractional Steps, New York: SpringerVerlag, 1971. [4] P. B. Johns and R. L. Beurle, "Numerical solution of two-dimensional scattering problems using a transmission-line matrix," Proc. IEE, vol. 118, pp. 1203-1208, 1971. [5] N. R. S. Simons and A. Sebak, "New transmission line matrix node for two-dimensional electromagnetic field problems," Canadian J. Physics, vol. 69, no. 11 , pp. 1388-1398, 199 1. [6] __ , "Spatially-weighted numerical models for the two-dimensional wave equation: FD algorithm and synthesis of equivalent TLM model ," Int. J. Num. Mod., vol. 6, pp. 47-65, 1993. [7] __ , "A fourth-order-accurate in space and second-order-accurate in time TLM model," accepted for publication.
Comments on "Improvement in Calculation of Some Surface Integrals: Application to Junction Characterization in Cavity Filter Design" G. G. Gentili
In ( l], the authors claim to have " . . .developed a rigorous method. . ., allowing the reduction of a surface integral to a contour integrar' with application to mode-matching analysis of waveguide junctions. The method have the advantage of a 50% time reduction in the evaluation of the coupling integrals in mode-matching techniques. The authors have applied the method to analyze the transition from circular waveguide to rectangular waveguide. This same method has been derived in [2] in its general form . In [3] some preliminary results on the discontinuity between ridged and rectangular waveguide have been presented. The only original contribution recognized in [l] is therefore the application of method [2] to the analysis of coupling between circular cavities through rectangular irises. REFERENCES
[I] P. Guillon, P. Couffignal, H. Baudrand, and B. Theron "Improvement in calculation of some surface integrals: Application to junction characManuscript received March 28, 1994. The author is wi'th the Politecnico di Milano, Dipartimento di Elettronica e Informzaione, via Ponzio 34/5, 20133 Milano, Italy. IEEE Log Number 9402925.
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terization in cavity filter design," IEEE Trans. Microwave Theory Tech. , vol. 4 1, pp. 2156-2160, Dec. 1993. [2] G. G. Gentili, "Properties of TEn'M mode-matching techniques," IEEE Trans. Microwave Theory Tech., vol. 39, pp .. 1669-1672, Sept. 1991. [3] G. G. Gentili, M. Politi, R. Maffi a, G. Macchiarella, and A. Melloni, :'Analysis of the discontinuity between a ridged and a rectangular waveguide by an efficient mode-matching technique," Proc. Europ. Microwave Conference, Stuttgart, 199 1.
fectly conducting walls are considered. In the particular case, where the waveguides on both sides of the discontinuity are homogeneous, the expressions that we have presented for the different couplings of TE and TM modes [2] are in accordance with those obtained in [I] . However, we would like to emphasize an important feature of the following contour integral developed in [2] (21)
I
=
1{ ~ A
µ (E V' e + H \7 h)
+ Reply to Comments on "Improvement in Calculation of Some Surface Integrals: Application to Junction Characterization in Cavity Filter Design" P. Guillot and H. Baudrand
+µ~
1(
A (e \7 E
e \7 H
+ h \7
+ E \7 h)
H)}· di
x ii · di.
ii
(2 1)
In order to obtain this expression, no assumption has been made on the TE or TM character of the existing modes. As a consequence, it can be equally applied to the matching of hybrid modes in inhomogeneous structures. In fact, this point constitutes the main originality of the work presented in [2]. It could be applied, for example, to the study of dielectric rod-antennas [3] . REFERENCES
We would like to thank Dr. Gentili for caJling our attention to the work described in [I]. His paper focuses on some general properties of TE-TM field expansions when homogeneous waveguides with perManuscript received April 18, 1994. The authors are with the Laboratoire d' Electronique, ENSEEIHT, 31071 , Toulouse, France. IEEE Log Number 9402926.
[I] G. G. Gentili, "Properties of TErrM mode-matching techniques," IEEE Trans. Microwave Theory and Tech. , vol. 39, pp. 1669-1672, Sept. 1991. [2] Ph. Guillot, P. Couffignal, H. Baudrand, and B. Theron, "Improvement in calculation of some surface integrals: Application to junction characterization in cavity filter design," IEEE Trans. Microwave Theory Tech., vol. 41 , pp. 2156-2160, Dec. 1993. [3] M . Aubrion , A. Larminat, B. Chan, and H. Baudrand, "Design of a dual dielectric rod-antenna system," Microwave Guided Wave Letters, vol. 3, pp. 276-277, Aug. 1993.
0018-9480/94$04.00 © 1994 IEEE
MTT-S Technical Committee Chairmen Coordinating Chairman: J. RAUE Computer-Aided Design
Microwave Systems
Microwave High-Power Techniques
Microwave and Millimeter-Wave Integrated Circuits
K. C. GUPTA I. WOLFF
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Manufacturing Technology C. HUANG
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Microwave low Noise Techniq11es J. J. WHELEHAN, JR.
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(Continued from front cover)
LETTERS Comments on 'Theoretical and Experimental Charoacteristics of Single V-Groove Guide for X-Band and 100 GHz Operation" ........................................................................................................ . H. Chi Reply to Comments on "Theoretical and Experimental Characteristics of Single V-Groove Guide for X-Band and 100 GHz Operation" ............................................................. Y. M. Choi, K. F. Tsang, and D. J. Harris Comments on "A Class of Symmetrical Condensed Node TLM Methods Derived Directly from Maxwell's Equations" ............................................................................................ M. Krumplwlz. and P. Russer Reply to Comments on "A Class of Symmetrical Condensed Node TLM Methods Derived Directly from Maxwell's Equations" ....... .. ..................................................................... J. Lo Vetri and N. R. S. Simons Comments on "Improvement in Calculation of Some Surface Integrals: Application to Junction Characterization in Cavity Filter Design" ...................................................................................... G. G. Gentili Reply to Comments on "Improvement in Calculation of Some Surface Integrals: Application to Junction Characterization in Cavity Filter Design" ................................................................... P. Guillot and H. Baudrand
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