IEEE MTT-V034-I11 (1986-11)

158 Pages·1986·14.908 MB·English

by  FANTOMASPING

~IEEE TRANSACTIONS ON
MICROWAVE THE.ORY
AND TECHNIQUES
NOVEMBER 1986 VOLUME MTT-34 NUMBER 11' (ISSN 0018-9480)
A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY
PAPERS
Scattering at Circular-to-Rectangular Waveguide Junctions ............................. J. D. Wade and R.H. MacPhie 1085
Green's Function Treatment of Edge Singularities in the Quasi-TEM Analysis of Microstrip ................. V. Postoyalko 1092
Quasi-TEM Analysis of Microwave Transmission Lines by the Finite-Element Method ............. Z. Pantie and R. Mittra 1096
Finite-Difference Analysis of Rectangular Dielectric Waveguide Structures ........... K. Bierwirth, N. Schulz, and F. Arndt 1104
A SPICE Model for Enhancement-and Depletion-Mode GaAs FET's ... S. E. Sussman-Fort, J.C. Hantgan, and F. L. Huang 1115
Vectorial Finite-Element Method Without Any Spurious Solutions for Dielectric Waveguiding Problems Using Transverse
Magnetic-Field Component ................................... . K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki 1120
Improved Approximations for the Fringing and Shielded Slab-Line Capacitances ............................ H.J. Riblet 1125
A Procedure for Calculating Fields Inside Arbitrarily Shaped, Inhomogeneous Dielectric Bodies Using Linear Basis Functions
with the Moment Method ............................. C.-T. Tsai, H. Massoudi, C.H. Durney, and M. F. Iskander 1131
A Combined Method for Dielectric Waveguides Using the Finite-Element Technique and the Surface Integral Equations
Method ............ : ........................................................................... C.-C. Su 1140
Systematic Characterization of the Spectrum of Unilateral Finline .......................... C. A. 01/ey and T. E. Rozzi 1147
Printed Circuit Transmission-Line Characteristic Impedance by Transverse Modal Analysis ........... H.-Y. Yee and K. Wu 1157
Vortex Formation Near an Iris in a Rectangular Waveguide ........................... R. W. Ziolkowski and J.B. Grant 1164
Transit-Time Effects in the Noise of Schottky-Barrier Diodes ................... M. Trippe, G. Bosman, and A. van der Ziel 1183
Broad-Band Noise Mechanisms and Noise Measurements of Metal-Semiconductor Junctions ........................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Jelenski, E. L. Kol/berg, and H. H. G. Zirath 1193
Reflecting Characteristics of Anisotropic Rubber Sheets and Measurement of Complex Permittivity Tensor ............... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0. Hashimoto and Y. Shimizu 1202
Low-Frequency Characteristic Modes for Aperture Coupling Problems .................................... Y. Leviatan 1208
Unloaded Q-Factor of Stepped-Impedance Resonators ................................... G. B. Stracca and A. Panzeri 1214
SHORT PAPERS
Measurements of Time-Varying Millimeter-Wave Vectors by a Dynamic Bridge Method .. . A. M. Yurek, M. G. Li, and C.H. Lee 1220
An X-Band Four-Diode Power Combiner Using Gunn Diodes ..................................................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Bhattacharyya, S. K. De, G. Ghosh, P. C. Rakshit, P. K. Saha, and B. R. Nag 1223
LETTERS
Comments on "Nonuniform Layer Model of a Millimeter-Wave Phase Shifter" ...................................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Ogusu, I. Tanaka, J. K. Butler, T. F. Wu, and M. W. Scott 1226
Comments on "Self-Adjoint Vector Variational Formulation for Lossy Anisotropic Dielectric Waveguide" ................ .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Hoffmann, S. R. Cvetkovic, and J. B. Davies 1227
Comments on "Computer-Aided Design Models for Millimeter-Wave Finlines and Suspended-Substrate Microstrip Lines" ....
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. K. Piotrowski, P. Pramanick, and B. Bhartia 1228
Corrections to "Normal Modes in an Overmoded Circular Waveguide Coated with Lossy Material" ...................... .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. S. Lee, S. W. Lee, and S. L. Chuang 1229
PATENT ABSTRACTS ....................•.....•..•.......••...........................•..........•. J. J. Daly ll230
ANNOUNCEMENT
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO. 11, NOVEMBER1986 1085
Scattering at Circular-to-Rectangular
Waveguide Junctions
JOHN DOUGLAS WADE AND ROBERT H. MACPHIE, SENIOR MEMBER,IEEE
Abstract —A formafly exact solution is givenfor the problem of scatter- Modal analysis, coupled with the technique of matching
ing at a circukw-to-rectarrgnbu waveguide junction and at a thick di- the tangential electric and magnetic fields at waveguide
aphragm, with a centered circular aperture, in a rectangular wavegnide.
junction discontinuities, has been used to solve avariety of
The method uses normal ‘II? and TM mode expansions of the wavegnide
problems [3]–[6]. Modal analysis suffers from the relative
fields and trarfitionaf mode matcfring of the transverse electric and mag-
netic fields at the jnnction boundary. Exact closed-form expressions are convergence problem, but this can be eliminated by a
obtained for the electric field mode-matching coefficients which conple the judicious choice for the ratio of the number of modes ursed
‘II?,(TM) modes in the reetangnlar gnide to the TE(TM) andTM(TE) for the aperture field expansion to the number of modes
modes in the circnfar gnide. Numerical results are presented for the case of
used for the waveguide field expansion [7], [8].
TEIo mode propagation in the larger rectangnfar guide with all other
In this paper, modal analysis and the matching of the E
modes cutoff. Convergent numerical results for the equivalent shunt sus-
ceptaoces of snch junctions are obtained when abont 12 modes (eight ‘II and H fields at the junction of a circular and a rectangular
and four TM) are retained in the circular waveguide or in the circular waveguide lead to a closed-form analytical solution for the
aperture of the diaphragm. The resnlts are graphically compared with scattering matrix [S] of the junction. It is demonstrated
formulas and curves due to the quasi-static theory of Bethe and the
that convergent numerical results for the dominant-mode
variational theory given in the Waceguidc Handbook [2].
(TEII or TEIO) reflection and transmission coefficients can
be obtained by retaining about 12 modes (eight TE and
I. INTRODUCTION four TM) in the smaller, circular waveguide (see Fig. 1).
The case of a thick diaphragm, with a centered circular
ELECTROMAGNETIC DIFFRACTION by a circular
hole, in a rectangular waveguide is then treated as a
aperture in a conducting screen is important in micro-
cascaded connection of rectangular–circular–rectangular
“wave engineering. Waveguide diaphragms with circular
guides. The generalized scattering matrix technique [8], [9]
apertures can be used as matching elements in microwave
is used to deduce the scattering matrix [Sd] of the di-
circuits or in the construction of cavity filters. Waveguide-
aphragm. In both cases (simple circular-to-rectangular
to-cavity coupling is often accomplished with a circular
junction and the rectanguku-circular-rectartgular di-
aperture.
aphragm junction), the numerical results are presented in
The theory of diffraction by small holes was pioneered
terms of the inductive shunting susceptances across equiv-
by Bethe [1]. He showed that a small aperture in a con-
alent transmission lines. The results are compared with the
ducting screen is approximately equivalent to an electric
formulas and curves due to Bethe [1], [10] and those in the
dipole normal to the screen with a strength proportional to
Waveguide Handbook.
the normal component of the exciting field, and a mag-
netic dipole in the plane of the screen with a strength
proportional to the exciting tangential magnetic field. II. ELECTRIC FIELD MODE MATCHING
The most complete variational solution of scattering ATTHEJUNCTION
from a diaphragm with centered circular aperture in a As illustrated in Fig. 1, the circular waveguide of radius
rectangular waveguide is given in Marcuvitz [2, pp. R (guide 1) forms ajunction at z = Owith a larger rectan-
238–240]. The equivalent circuit for the aperture consists gular waveguide of lateral dimensions a and b (guide 2),
of a susceptance shunted between two wires of a transmis- with b > 2R. Note that the z axis is the axis of symmetry
sion line. Full mode expansions of the waveguide fields are for both guides. The more general problem of a junction
used but the aperture E field is approximated with a static, with laterally displaced axes appears to have a solution
small-hole distribution. Therefore, these formulas will be based on the techniques given in this paper but, at present,
less accurate for larger holes. the details have not been worked out.
In the circular waveguide (guide 1 for which z < O), the
Manuscript receivedJanuary 28, 1986; revised June 27, 1986. This tangential electric field just to the left of the junction
work was supported in part by the Natural Science and Engineering
(z= O)can be given as a superposition of TE (h-type) and
Research Council, Ottawa, Canada, under Grant A-2176.
J. D. Wade is with the National Research Council’s Herzberg Institute TM (e-type) modal fields:
for Astrophysics in Ottawa, Canada KIA 0R6.
R. H. MacPhie is with the Electrical Engineering Department, Univer-
sity of Waterloo, Waterloo, Ontario, Canada N2L 3G1.
IEEE Log Number 8610510. qr
0018-9480/86/1100-1085 $01.00 01986 IEEE
1086 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO. 11, NOVEMBER 1986
case of greatest practical interest, the tangential E field
just to the right of the junction (z= O+ ) can be expanded
as follows:
72(X, y) = ~ ~ [b~~;zj:~n(x, y)+ b$jzj:~n(x, y)] (7)
inn
with m=l,3,5, ... and n = 0,2,4, ....
Moreover, to obtain E fields with even symmetry with
Fig. 1. A circular-to-rectangular waveguide junction. Gtudes have a respect to the x and y axes in the circular waveguide, the
common axis of symmetry (z axis).
series expansion (1) is for odd q (q= 1,3,5,”““).
In (7), the modal fields are
Traditionally, the modal fields in (1) are written as (see
Collin [10, p. 110])
@l!ln(x7 Y) = Nmn[~8,nsin(Bxmx)s~(BYnY)
+ Jflxmcos (Bxmx) Cos(BynY)] (8)
“sin(q$) + @& Y/( P&.P) cos(q+)] (2)
Z4,%(X, y) = N~. [%~sin(&~x)sin(& .y)
- JPyncos (Bxmx) Cos(PynY)] (9)
.sin(q+) +$(q/P)Jq(D~JP) cos(q@)] (3)
where
where
r N~ti = 2 ~ab[l?~n +2&’~/c~] (lo)
2 /
,–
is the rectangular waveguide modal normalization factor,
in which
&~a=m~, &.b=n~and c.=lifn=O, e.=2ifn#0.
and However, we now find it convenient in this circular-to-
rectangular waveguide junction problem to transform the
/77
coordinates (x, y) in (8) and (9) into their circular cylin-
drical equivalents. First, one can show that the double sine
product occurring in (8) and (9) can be written as
sin (PXnX) sin (&Y)
are normalization constants in which ~&R and f?$R are,
respectively, the rth roots of l;(x) and J~(x).
= sin[/3~. cos@~.pcos@] sin[~~. sin+~.p sine]
However, for the circular-to-rectangular junction, it is
where
convenient to replace the unit uectors ~ and ~ in (2) and
(3) by their Cartesian (2, j) equivalents. It is not difficult
P2=X2+Y’
to show that (2) and (3) can be reexpressed as follows: B:n = I?:m+ B:n,
+ Jq+I(B;rP)~q+@)] (4)
Then, by means of standard trigonometric identities and a
well-known Bessel–Fourier expansion [11, p. 361], we find
4,’;,(P> 4) = %[J,.O;P)U$’) that
– J,+l(P;;P)’%+A4’)] (5) sin (P.mx) sin (&Y)
= - ii ( - l) PJ2,(/3rnnP) siII(QP%7m)siTI(zP@) (11)
~=1
are @-dependent unit vectors.
Moreover, in the same way, one can verify that
In the rectangular waveguide (guide 2 for which z > O),
with the z axis passing along its center and not along its
lower left corner (as is normally assumed), the modes for COS(&mX)U@ynY) = ~ CP(–1)PJ2P(%2P)
~=o
which the E fields are even functions of both x and y
would be those scattered by a TEIO mode field incident
from guide 2 or a TEII mode incident from guide 1- In this .cos(2pC$mn) cos(2p@). (1’2)
WADE AND MACPHIE: CIRCULAR-TO-RECTANGULAR WAVEGUIDE JUNCTIONS 1087
Use of (11) and (12) in (8) and (9) leads to (11.3.29)], one can show that
H =A ~n,qNmnN;:)cos( q@mn)
mn, qr
“[Bynsmn,p(B+x)fm+cmn,p(@)j]
‘%:22.(A4=)%.PEOJ2P(LO) “[B& ,q_l,, – B“~n,q+l,r] (21a)
where
(14)
Amn,q= 2W(–l P-1)’2L% (22)
wherein
%.,,($) = -(-l) ps~(2p@mn)sin(2p@)
cmn,p(@) =~p(-l)pco5(2p@mn) cos(2p@).
@j
We are now in a position to enforce the electromagnetic ‘~;,~q+z(P:,R)~q+~(8~.R)
boundary conditions on the transverse electric field at the q I
circular-to-rectangular waveguide junction. The field must with t= (or ‘f.
be continuous (matched) in the circular aperture O< p < R However, by making use of the fact that .lq(/l;;R) = O,
and vanish everywhere else in the region 21x1< a,21yl <b; we can, with recursion formulas for Bessel functions, show
we tacitly assume that the conductivity of all the metal that (21a) vanishes, i.e.,
surfaces is infinite. Thus, the boundary conditions are such K o. (21b)
mn, qr=
that
This curious phenomenon also occurs for the rectangular-
~l(P>@),O<p GR,
Z*(p, @)’= o to-rectangular waveguide junction [9, p. 2061].
{ 9 p> Rand 21xl<a,21yl <b. If we now scalar multiply (18) by Z~.e~~(p,+) and in-
(17) tegrate over the rectangular cross section at z = O, we
Scalar multiplication of (18) by ~j,~n(p, +) and integra- Again, closed-form expressions for these coupling coeffi-
tion over the complete cross section of guide 2 yields, due cients (TM–TE and TM–TM) can be deduced
to the orthogonality of the guide 2 modes,
Qmn, qr= –A ~n,qNm~N$!’)sin(q@m~)
where
The latter can be shown to reduce to
and
are, respectively, the TE–TE and TE–TM E-field mode-
osin (q@mn)Jq(BmnR)Jq- l(B~jR). (Zbb)
coupling coefficients for the junction.
These coefficient integrals can be evaluated analytically Equations (19) and (24) can be cast into matrix form as
if we use (4)–(6) and (13), (15), and (16) in (20) and (21). follows:
Then, in view of orthogonality of the Fourier harmonics
[sin(q + 1)$, cos(q & 1)+] and a Bessel function integral
provided by Abramowitz and Stegun [11, p. 484, eq.
1088 IEEE TRANSACTIONS ONMICROWAVE THEORYAND TECHNIQUES, VOL. MTT-34, NO. 11, NOVEMBER 1986
where a(m) and b(m), with m = h or e, being the modal and (30), that
weighting coefficient vectors in guides 1 and 2, respec-
a-= (([ Y1]+[YL1])-l([Y1 ]-[yL1]))tz+ (34)
tively, and [H], [0] = [K], [Q] and [E] being the sub-
matrices of the overall E-field mode-matching matrix [M], where
with
[YL,] = [M]=[Y2][M] (35)
b= [M]a (28)
is the “load” admittance matrix of guide 2 as “seen” by
and
guide 1. In view of the fact that b+ = O,it follows from
~~= [a(~)~, ~(e)~ ] b~= [b@)~, b(e)~] (29) that
with T indicating the transpose operator. [s,l] = ([Y,]+ [YLJ)-l([Y1]- [YLJ). (36)
The other submatrices in (29) are then deduced by
III. THE SCATTERING MATRIX OFTHE
simple matrix algebra
CIRCULAR–I?XCTANGULAR JUNCTION
[s21] = [M]([SIJ+ [I]) (37)
We define the II-field modal coefficient scattering ma-
trix [S] of the circular-to-rectangular waveguide junction
[S,2]=2([Y,]+[YL1 ])-’[M]T[Y2] (38)
to be such that
[s22] = [M][S12] -[1] (39)
where [1] is the identity matrix and T indicates the trans-
pose operation. These results may also be obtained by
where, as is traditional, the + and – superscripts denote, means of the conservation of complex power technique [9],
respectively, incident and scattered waves. [12].
In the case of the lossless structure, one can use H-field
IV. TRANSVERSEDIAPHRAGM WITH CENTERED
mode matching in the circular junction aperture to deduce
CIRCULAR HOLE IN RECTANGULAR WAVEGUIDE
the following matrix equation:
Fig. 2 illustrates the geometry of a diaphragm (perfectly
[kf]T[Y,](b-- b+) = [Yl](a+ -a-). (30) conducting) of nonzero thickness 1with a centered circular
hole of radius R. This structure can be regarded as a
H-field mode matching, analogous to the E-field mode rectangular–circular–rectangular cascaded connection and
matching described in Section II, is well known [3], [5] and the generalized scattering matrix technique [8, pp. 207-217]
will not be treated in detail in this paper. In (30) [~], for may be used to determine the overall scattering matrix
i = 1 and 2, is the modal admittance matrix for the ith [Sd] of the diaphragm
guide.
[s;l] = [s:3] = [s, J+[s121[~l[s221
“{[11 -[ W,21[MS221[W221 }-1[W2J (40)
[s:3] = [s$,]
where the two submatrices in (31) are diagonal. In particu-
lar, for the circular guide, the diagonal elements are
= [s,21{[Hm s221[Im2,1}-’[Ll[ s,J. (41)
Here, the transmission-line matrix [L] of the central cir-
cular guide is a diagonal matrix such that
and for the rectangular guide they are
(42)
with submatrix diagonal elements given by
L$?q,=e.p(-/Hz)
/(%77%”
“;’” ’33) L(e) (43)
qr>qr=exP( -/ml).
In (43), the subscript 2 denotes the circular guide’ (see Fig.
2).
In (32) and (33), p~is the permeability and El and cz are
V. NUMERICAL RESULTS
the permittivities of the dielectrics filling guides 1 and 2,
respectively. We begin with the case of a circular–rectangular wave-
Then, if we assume that incidence is from guide 1 only, guide junction. In our numerical computations, which con-
so that b+ = O, it is straightforward to show, using (28) sider only air-filled guides, we selected a frequency range
WADE AND MACPHIE : CIRCULAR-TO-RECTANGULAR WAVEGUIDE JUNCTIONS 1089
TABLE I
NUMBER OFRECTANGULAR MODES FORAGIVEN NUMBER OF
CIRCULAR MODES
Number .f Number of ReCCaWUhr Modes (TE, TM)
circular
Modes R-b12 R-3b18 R-b14 R-b18
(TE, Tu)
7
2,1 13,6 20,10 60,23 1S5,95
4,2 18,10 35,20 65,50 275,20
8,4 35,20 65,40 130,90 500,350
Fig. 2. A thick diaphragm, with centered circular hole, in a rectangular
waveguide. TABLE II
CONVERGENCE OFBJ AND ED AT8GHz
b18
for which the only propagating mode in the larger rectan-
Nun. of
gular guide was the TEIO mode. Consequently, there were Modes XJ iD FJ XD iJ iD i, ill
TE, TM I I I
no propagating modes in the smaller circular guide. Thus, 2,1 9.62 6.85 23.7 17.4 82.1 61.8 670 514
the reflection coefficient p10 of the TEIO mode in the 4,2 9.18 6.15 22.6 15.7 78.0 55.7 632 466
8,4 9.11 6.o1 22.4 15.4 77.2 54,6 628 454
rectangular guide was of primary interest 1
(44) a = 2.25b = 2.286 cm.
Plo = ~22,10,10.
From plo one can determine the normalized load admit-
tance (for TEIO rectangular modes) of the junction TABLE III
CONVERGENCE OFBJ AND ~D AT14 GEl
r
~=–jE’=— 1 – Plo (45) 9. b12 3b/8 b14 - b,, ~
1+ plo Nun. of
Modes iJ iD i, iD FJ iD i, iD
TE, TM I I I I
which is pure imaginary since the circular guide is cut off. 2,1 2.56 2.08 7.39 5.111 28.0 20.4 242 184
Moreover, the minus sign indicates that the junction sus- 4,2 2.k5 1.87 7.04 4.83 26.6 18.2 230 I<6
8,4 2.44 1.86 6.99 4.76 26.3 17.9 227 162
ceptance is inductive.
In the numerical work, the number of modes assumed in a = 2.25b = 2.286 cm.
each waveguide strongly depends on the size of the circular
guide relative to the rectangular. Table I provides the
30t\ R
relevant information. Normally, half as many TM modes
as TE modes are used in the circular guide. Moreover, as a . 2.2s b
R/b diminishes and with a = (9/4)b, the number of rect-
angular guide modes increases dramatically. Fortunately,
2.0
this large number of modes need only be used in the
Im-
calculation of the load admittance matrix [YLI] as given by
~=
(35); therefore no inuersion of a large ma@x is required. ~
Tables 11and III show the convergence of B7 together with
ED for the thin diaphragm with a circular hole as a !.0–
function of the number of modes in the circular guide for
various values of R/b and at two frequencies; the rectan-
gular guide is assumed to be standard X-band guide with l MARCUVITZ
a = 2.286 cm. It is seen in Tables II and III that for about “o~
1.5
a dozen modes (eight TE and four TM) in the circular
a/A
guide, the numerical results have converged quite well for
Fig. 3. Susceptance of rectangnfar-to-circular waveguide junction.
all R/b ratios and at both the low and high frequencies.
In Fig. 3, the susceptance B1 is plotted as a function of
a/A with R/b as a parameter. Twelve modes were used in where D = 2R and ~~ is the wavelength of the propagat-
the circular guide in all cases. Not surprisingly, the normal- ing mode in the rectangular waveguide. The values for ED
ized susceptance increases as the circular waveguide radius can be obtained from a graph elsewhere in the handbook
decreases and diverges as a/A -+ 0.5, since the rectangular [2, p. 240]. We note that the agreement between the
TEIO mode’s admittance vanishes at this point. quasi-static variational solution (46) and the more rigorous
Also plotted at discrete values of a/A are the load scattering matrix solution (solid lines in Fig. 3) is quite
susceptances given in the Waveguide Handbook [2, p. 327]. good, even for large values of R/b.
For small irises, one has Numerical results for the more interesting case of a thin
(1= O) diaphragm with centered circular aperture in a
rectangular guide are presented in Fig. 4. Again, it is
convenient to represent the zero-thickness diaphragm as a
shunting susceptance ~~ = B~/ Ylo normalized with re-
1090 IEEE TRANSACTIONS ON MZCROWAVE THEORY AND TECHNIQUES, VOL. MTr-34, NO. 11, NOVEMBER 1986
30 3.0,
R
\
\
20
&_
‘;’O
—.
.::.:’”-- Im
—g
.
‘\ z
\ -. -,.
:\ ‘. --... ----- -- LbL8_ Lo
0 ---,---
----
0 MARCUVITZ b12
o
0.5 07 0.9 1.1 I.3 1,5
o/i
oI I I I I
Fig. 4. Susceptance of a thin diaphragm with circular aperture. Solid 0.5 0.6 0.7 0.8 0.9 [0
lines are calculated using the present method while broken lines are o1A
from the small-hole formula.
Fig. 5. Susce@mce of a thin diaphragm, with acentered circular aper-
ture of radius eR, in arectangular waveguide (a= 2.25b).
spect to the rectangular TEIO mode’s admittance. ED is
computed for four aperture radii, and the convergence
behavior is indicated in Tables II and III. For twelve
modes in the circular guide, it is estimated that the error in
ED is 2 percent or less. Also given in Fig. 4 are curves
based on the well-established formula
RECTANGULAR
3abA~ wAvEGUIOE B.
zD’— R<< Ag (47)
1677R3‘
Yc Bp Bp Yc
.-— H
derived in Collin [10, pp. 190–194] and originating with I ——
EQUIVALENT TRANSMISSION
the quasi-static theory of Bethe [1]. As expected, (47) LINE ANO ?TNETWORK
agrees well with the present results for small holes, R < Fig. 6. Thick diaphragm with centered circular hole in rectangular
b/8. For larger apertures, the simple formula overesti- waveguide and the equivalent transmission line with H network.
mates ED and hence underestimates the transmitted field.
Bethe shows that for small apertures the fields scattered in can be shown that
the forward direction are proportional to R3/A2, whereas — (1- d2-
for large Kirchhoff-type apertures they vary as R2/A. do
jBP = (48)
Accordingly, as R/A increases (but in our case still re- (1+ PIO)2- &
mains less than unity), the small-aperture (Bethe) theory
predicts a forward scattered field that is too weak and 2‘rIo
j~~ = (49)
hence a ED that is too large. This is confirmed by the
(1+ PIO)2- ~l?o
results presented in Fig. 4.
The diaphragm’s susceptance is also compared with the where
variational calculus susceptance provided by the Waoe- 7~~= S31,10,10 (50)
guide Handbook [2, p. 240]. The latter gives lower suscep-
is the transmission coefficient of the TEIO mode and is
tances than the small-hole expression, but for large holes
such that
(R= b/2) gives values that differ from ours by about 10
percent. (51a)
IP1012+ l~lo12 ‘1
Fig. 5 provides curves of FD for seven different ratios of
arg(plo)–arg(rlo) = T/2. (51b)
R/b. The range of a/A is only over that in which the
TEIO mode alone can propagate, i.e., the range of greatest Clearly ~P and ~~ can be deduced from a knowledge of
practical interest. ~lo alone. We choose I~loI and arg (~10) for our graphical
We next turn to the case of a diaphragm of nonzero results since ~~ -+ co when 1~ O.The amplitude of ~lo is
thickness (1# O). When the normalized load admittance is plotted in Fig. 7(a) for four aperture radii and a series of
calculated, the real part is no longer unity. A more diaphs-agm thicknesses ranging in increments of Al/a=
sophisticated circuit representation is thus required. We 0.02 from I/a = Oto l/a= 0.08. As is expected, the trans-
have chosen the r-equivalent circuit shown in Fig. 6. It mitted wave’s amplitude decreases with increasing di-
1091
WADE AND MACPHIE: CIRCULAR-TO-RECTANCIH,AR WAVEGUIDE JUNCTIONS
[3] A. Wexler, “Solution of waveguide discontinuities by modal analy-
sis;’ IEEE Trans. Microwave Theory Tech,, VOL MIT-l 5, pp.
I
90” 508-517, Sept. 1967.
[4] P. H. Masterrnan and P. J. B. Clarricoats, “Computer field-match-
.3.0- ing solution of waveguide transverse discontinuities,” Proc. Inst.
80” Elec. Eng., vol. 118, pp. 51-63, Jan. 1971.
[5] R. E. Collin, Field Theory of Guided Waves. New York:
McGraw-Hill, 1960, ch. 8.
E -20 - 57 [6] R. Vahldieck and J. Bornermann, “A modified mode-matching
-Q
: technique and its application to aclass of quasi-planar transmission
s
lines< IEEE Trans. Microwave Theoiy Tech., vol. MTT-33, pp.
916-926, Oct.1985.
-1.0- 60 [7] R. Mittra, T. Itoh, andT. S.Li, “Analytical andnumericafstudies
of the relative convergence phenomenon arising in the solution of
an integral equation by the moment method,” IEEE Trans. Micro-
5 wave Theory Tech., vol. MTT-20, pp. 96–104,Feb.1972.
OL I 1 1 I —1 1 t r 1 00 I [8] R. Mittra and S.W. Lee, Analytical Techniques in the Theoiy of
05 0.6 0.7 0.s 0.9 1.0 0.5 06 0.7 0.6 0.9 1.0 Guided Waves. New York: Macmillan, 1971.
OiA d A
[9] R. Safavi-Nairri and R. H. MacPhie, “Scattering at rectangular-to-
Fig. 7. Amplitude and phase of the transmission coefficient (TEIO rectangular waveguide junctions: IEEE Trans. Microwave Theory
mode) for a thick diaphragm, with centered circular hole, in rectangular Tech., vol. MTT-30, pp. 2060-2063, Nov. 1982.
wavegnide. The thinnest diaphragm (1= O)has the largest 171. [10] R. E. Collin, Foundations for Microwave Engineering. New York:
McGraw-Hill, 1966, pp. 190-194.
[11] M. Abramowitz and I. A. Stegun, Handbook of Mathematical
aphragm thickness. Moreover, the effect is greater for Functions. New ,York: Dover, 1965.
small irises since in a guide of smaller radius the modes [12] R. R. Mansour and R. H. MacPhie, “Scattering at an N-furcated
parallel-plate wavegnide junction~’ IEEE Trans. Microwave Theory
that try to” tunnel” across the diaphragm are more strongly Tech., vol. MT”I’-33,pp.830-835,Sept.1985.
attenuated. [13] R. Levy, “Improved singleandmultiaperture waveguidecoupling
The phase, of the transmission coefficient, arg (~lo), is theory,including explanationofmutualinteractions< LEEE Trans.
Microwave Theo~ Tech., vol. MTT-28, pp. 331-338, Apr. 1980.
plotted in Fig. 7(b). Here the thicknesses are l/a= 0,0.04, [14] N. A. McDonald, “Electric and magnetic coupling through small
and 0.08, except for the case of R = b/8, where the phase apertures in shield walls of any thickness,” IEEE Trans. Microwave
Theory Tech., vol. MTT-20, pp. 689-695, Oct. 1972.
is almost invariant with thickness. The greatest phase
[15] J. D. Wade, “The conservation of complex power technique and
change occurs for the largest hole (R = b/2). Not surpris- scattering from circular apertures in rectangular waveguides,”
ingly, the phase is ahnost + 90° for small holes, since in M.A.SC. thesis, University of Waterloo, Waterloo, Ontario, Canada,
1984.
such a case p10= – 1 and (51b) must be satisfied.
E+
VI. DISCUSSIONAND CONCLUSIONS
Jofm Douglas Wade was born in London,
This paper has provided an exact modal solution (in Ontario, Canada, on March 27, 1953. He re-
principle) to the problem of scattering at circular-to-rect- ceived the B.SC.degree in physics from the Uni-
versity of Toronto in 1976, the M.Sc. degree in
angular waveguide junctions. In practice, numerical results
astronomy from the University of Western
for dominant-mode reflection and transmission coeffi- Ontario in 1979, and the M.A.SC. degree in elec-
cients accurate to 1 or 2 percent are possible when 12 trical engineering from the University of
Waterloo, Ontario, in 1984.
modes are considered in the smaller circular guide. In the
He is presently with the National Research
case of diaphragms with centered circular holes, the effect Council of Canada in Ottawa, where he is in-
volved with the development of centimeter- and
of the thickness of the diaphragm has been shown to be
millimeter-wave receivers for the Algonquin R~dio Observatory.
always significant (see Fig. 7). The effect of sidewall thick-
ness in single and multiapertWe waveguide couplers has
previously been taken into account by Levy [13], who used
the earlier small-aperture work of McDonald [14]. ix
Although this paper has postulated throughout that the
waveguides are perfectly conducting, the effect of a large
Robert H. MacPhie (S’57-M63-SM79) was
but finite conductivity can be accommodated by a gener- born in Weston, Ontario, Canada, on Se~tember
alization of the conservation of complex power technique 20, 1934. He received the B.A.SC.’ degree-in elec-
trical engineering from the University of Toronto
[15]. Moreover, the analysis of cavity resonators and filters
in 1957 and the M.S. and Ph.D. degrees from the
formed by transverse diaphragms with centered circular University of Illinois, Urbana, in 1959 and 1963,
holes is a quite simple extension of the present work, even respectively.
In 1963, hejoined the University of Waterloo,
if conductivity losses are included [15].
Waterloo, Ontario, Canada, as an Assistant Pro-
fessor of electrical engineering and at present he
is Professor of electrical engineering at Waterloo.
ReferenCeS
His researchinterests currentlv. focus on din.ole‘antennas. wavemide
[1] H. A. Bethe, “Theory of diffraction by small holes:’ Phys. R.eu., scattering theory, scattering from prolate spheroid systems, and &cro-
vol. 66, pp. 163–182, Oct.1944. strip structures. During 1984–1985, he was on sabbatical leave as a
[2] N. Marcuvitz, Ed., Waveguide Handbook. New York: McGraw- Professeur Associ6 at the Universit6 de Aix-Marseifle I, France, working
Hill, 1951. at the Department de Radio41ectricit.&
1092 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO. 11, NOVEMBER 1986
Green’s Function Treatment of Edge
Singularities in the Quasi-TEM
Analysis of Microstrip
VASIL POSTOYALKO, MEMBER, IEEE
Abstract —A new Green’s function approach to the quasi-TEM amdysis known from the study of Motz expansions [11] that at the
of microstrip is presented. By expressing the charge density on the strip edges of such a mathematically idealized strip both the
conductor as the sum of asingular term, derived from the consideration of
charge density and the electric field are singular. In the
a Motz expansiou, and a continuous term, the integrat equation defining
application of numerical techniques to problems involving
this charge density is transformed into an integral equation for the
continuous term. An accurate numerical solution to this new integral edge singularities, special consideration of the singularities
equation can he obtained by approximating the continuous term by a can often greatly speed the rate of convergence [2], [3], [5],
low-order unit-pulse expansion. It is seen that the numerical scheme [6], [12], [13]. By employing an analytic approximation for
developed in this work is both easy to implement and rapidly convergent,
the charge density near the edge of the strip conductor, a
thus making it an excellent choice for use in microwave CAD packages.
new Green’s function approach to the numerical analysis
I. INTRODUCTION of microstrip is developed in this paper. It is seen that this
u new approach is both rapidly convergent and easy to
NDER THE ASSUMPTIONS that loss is negligible
implement, thus making it an excellent choice for use in
and that the mode of propagation is quasi-TEM, the
microwave CAD packages.
characteristic impedance 20 of microstrip is given by [1]
1
20= (1) II. THE GREEN’S FUNCTION TECHNIQUE
c(cca)l’2
For an open microstrip of zero thickness (see Fig. 1), the
where c is the velocity of light in free space, C is the capacitance C can be expressed as
electrostatic capacitance per unit length of microstrip, and
C=/”’* u(x)dx (2)
C. is the electrostatic capacitance per unit length of the —w/2
structure obtained from microstrip by replacing the dielec-
where the charge density U(x) is the solution of the follow-
tric substrate with air. A wide variety of methods have
ing Fredholm integral equation of the first kind:
been used for the numerical evaluation of C and C= for
both open and shielded microstrip (see, for example, [1]-[8]
/“’2 G(~-~’)u(x’)~~’=1, +<;. (3)
and the references cited therein). If sufficient computing —w/2
resources are available, any of the methods cited above can
Here, the kernel G(x – x’) is given by [14]
be used to generate accurate design tables or design curves
‘:,:)
which relate 20 to the substrate dielectric constant and
G(x-xf)=
line dimensions. In view of the current interest in the
computer-aided design (CAD) of microwave integrated
1
circuits (both hybrid and monolithic) [9], attention is (x-x’) 2+4n%’
. ~ ~n-1 in (4)
focusing on numerical methods which are not only capable
~=1 (x - x’)2+4(n -1)’hz
[
of the accurate one-off calculation of 20 but are also well
suited for use as part of a fast and flexible microwave where K, referred to as the partial image coefficient, is
CAD package. Green’s function techniques have been given by
successfully used in microstrip analysis programs [10] and 1–6,
thus appear to be good choices for use in general-purpose K=— (5)
l+cr”
microwave CAD packages.
In the mathematical modeling of microstrip, the strip By exploiting the symmetry of the microstrip structure
conductor is often taken to be of zero thickness. It is well about the line x = O(o(x) = U(– x)), eq. (3) reduces to
4‘(“(2
G x–x’)+G(x +x’)) IJ(x’)dx’ =1,
Manuscript received January 27, 1986; revised May 21, 1986.
The author is with the Department of Electrical and Electronic En-
gineering, University of Leeds, Leeds LS2 9JT England.
IEEE Log Number 8610281.
0018-9480/86/1100 -1092$01 .00 01986 IEEE