Technical memorandumHYBRID-MODE ANALYSIS OF SHIELDED ANDCOUPLED SLOTLINES ON A SUSPENDEDSUBSTRATE

mode indicates even Ey,Ez, and Hx with respect to x = ain Fig. 1. The structure is assumed to be uniform andinfinitely extended in the axial direction, i.e. the y direc-tion in Fig. 1.

Indexing terms: Waveguides and waveguide components, Transmissionlines, Microwave passive components

Abstract: A hybrid mode analysis is presented forshielded single and coupled slotlines on a sus-pended substrate. The method is based on thetransverse modal analysis (TMA) technique.Numerical results are consistent with other theo-ries.

Fig. 1 Cross-section of shielded coupled slots on a suspended substrateand analytical model

a Cross-section b Analytical model

Introduction

A slotline is a transmission structure proposed for use inMICs by Cohn [1] in 1968. Coupled slotlines (oftencalled coplanar waveguides) was proposed by Wen [2] in1969. One advantage of such structures lies in the factthat the mounting of lumped components in a shunt ismuch easier.

Various studies have already been carried out. Gener-ally these include the quasi-TEM approximation [3] andthe spectrum domain analysis [4-6]. In this paper, thetransverse modal technique of Yee [7] is applied to give afrequency-dependent analysis of an enclosed structure,which can take account of the effect of sidewalls. Theemphasis is on the potential of the TMA method.

As the structure under study is laterally as well as ver-tically shielded, a (truncated) infinite summation ratherthan a (truncated, discretised) infinite integration isrequired to compute the elements of the transfer matrix.Each term in the summation series may be put in aclosed form in the TMA technique. Thus, the numericalwork and the computation time is greatly reduced.

In the TMA formulation, a rectangular cavity isformed by placing two electric walls transverse to thetransmission-line system with half guided wavelengthspacing. As the wave propagation is transverse to thetransmission line, the rectangular cavity can be viewed asthree rectangular waveguide sections joined by the dis-continuities. The transverse fields in each region areexpressed in terms of the normalised waveguide modalfunctions. Then, applying boundary conditions on thesurface of the inhomogeneity, a matrix equation isformed. The dispersion characteristics are obtained fromthat matrix.

Formulation

The cross section of the shielded coupled slotlines isshown in Fig. la. By symmetry, only half of the structureneeds to be analysed. The analytical model is shown inFig. Ib. The same model is suitable for the single slotlineby choosing w equal to half the width of the slotline. Anelectric wall is assumed at x = a for the single line,whereas an electric or magnetic wall is needed for theodd or even mode for the coupled slotlines. The even

The transverse fields in the three regions shown in Fig.1 are expressed in terms of waveguide modal functions.For region 1, 0 < z < ax:

2 MIT X"1 \"*

t /_| Z-lp= 1 m

2 M

p= 1 m

For region 2, gj

2 M

£r= Z Zp= 1 m

2 M

p= 1 m

X <£pm

For region 3, g

2 M

p= 1 m

2

H, = -j Ip = l

A,

t <

i <

M

1'm

, s i n

Z <

n sin

,[/ , .

z <

, sin

BprnJ

, c o s ( ^ m z ) z x <f>pm

92-

(P'm z) + Rpm COS (/?^ Z ) ] 0 p m

,cos(^z)-i?pmsin(^z)]z

h:

>" c o s [_f$m(h — zj\z x 0 m

(1)

(2)

(3)

(4)

(5)

(6)

where p is an index indicating TEZ (p = 1) or TMZ (p = 2)modes and m specify the waveguide modal values of the xco-ordinate. <f>pm is the orthonormalised waveguide modalfunction which satisfies

/ = 1, 2 ; = 1, (7a)

and

<i>Pm • <f>pn dx dy = dmn (p = 1, 2 m,n are the

modal indices in the x direction) (1b)

In eqns. 1-6, the total number of modes M' (M' = (M/2) + 1 for odd modes; M' = (M + l)/2 for even modes)needed is determined by the requirement of convergence.For a single slotline, only the odd mode is treated. Thedefinitions of ypm, y'pm, y"pm, 0m, fl'mt $"m and 4>pm are listedin Table 1. Note that m are even (odd) integers for odd

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 3, JUNE 1987 327

Table 1 : Expressions of waveguide modal functions

Region 1

Region 2

Region 3

Pm=[Klzr- (m%l2a)2(nlb)2V12

Yym = f}m/uj/j; w is the angular frequency

Vim = WlPmPm = [K2t'r - (mn/2a)2 - (n/b)2]"2

7- (rm:/2a)2(n/b)2V12

e = e, e 0 , e' = t'r e 0 , e" = e" e0

* i » = Nm[{Vb')E,m * - (m/2a)Eym 9]

<t>2m = Nm[(m/2a)E,m* + (1/b)Eym 9)

Nm = {26Jab[(m/2a)2 + (1/6)2]}1 '2 , Ko

EMm = cos (m7tx/2a) sin (TIK/6)

fKm = sin (mnx/2a) cos

(even) modes. The electric field intensity on the slot aper-ture is represented by

(8a)

where Cq are unknown constants to be determined, andQ is the total number of expansion terms. Qq are theexpansion functions given by

Qq = {cos [qn(x — cyw'] sin (ny/b)}x

+ {sin [qn(x — c)/W] cos (ny/b)}y (8b)

Applying the boundary conditions and using Galerkin'smethod, the following matrix equation is derived:

where

= 0

2 M

-zzp= 1 m

Wqpm)GGpm

&t • <t>pm dS•aperture

-I * <ApmJaperture

GGpm = y" cot - 0a)] + y'pm(A)

Gpm-G'pmtan(pmgi)tan(Pmg2)

Gpm = ypm cot W'm9l) + y'pm tan

cot - ypm cot

(9)

(10a)

(10b)

(10c)

(11)

(12)

(13)

(14)

The elements of the transfer matrix Mtq can be put in aclosed form because Qt and 0pm are both triangular func-tions resulting in algebraic expressions for Wtpm andvVqpm. (See eqns. 10b and c).

To obtain a set of nontrivial coefficients C'q, the deter-minant of the transfer matrix [Mtq] must vanish. Thisrequirement determines the resonant frequency / 0 of thecavity formed by placing electrical walls at both y = 0and y = b, where b is equal to half of the guided wave-length along the line direction. Note that the cavity fieldsmay be regarded as the wave propagating in the +ydirection. The effective dielectric constant is obtained by

seff = (v/2bf0)2

where v is the wave velocity in vacuum.

(15)

Numerical results

It has been found that numerical convergence dependslargely on the ratio of the truncated waveguide to slotlinemodal expansion terms, i.e. depends on M ' : Q'(Q' = Q — 1, not including q = 0). We have studied manycases and we conclude that the ratio should be chosen as2a : w. This ratio is also available in calculating the char-acteristic impedances. The calculation for impedanceswill be reported elsewhere. Note that the ratio chosenin Yee's paper is a : w.

Fig. 2 shows the even and odd mode propagation con-stants of coupled slotlines. The data agree well with thefull wave analysis.

0.530.52

0.50

0.48

0.46

0.44

0.42

0.40

6 9 12 15 18 21

frequency, GHz

Fig. 2 Even and odd mode dispersion characteristics for coupled slots

a = 10.0 mm, c = 9.2335 mm, w = 0.25 mm, gv =4.5 mm, g2 = 5.5 mm,h = 10.0 mm, er = 1.0, e'r = 11.0, e" = 1.0)

Reference 5Reference 6

x x x TMA

For the single slotline, the model shown on Fig. \b canstill be applied by taking w equal to half of the line widthand locating the slot at z = g2, a — w < x < a. An elec-tric wall is assumed at x = a for the dominant mode. Dis-persion characteristics of the single shielded slotline areshown in Fig. 3.

2 5 10 20

frequency, GHz

50

Fig. 3 Dispersion characteristic of shielded slot on a suspended sub-strate

a = 10 mm (30 mm), c = 9 mm (29 mm), w (half width of the slotline) = 1 mm,gl = 4.5 mm, g2 = 5.5 mm, h = 10 mm, er = 1 oe'r = 9.35, e" = 1.0

Reference 4x x x TMA

328 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 3, JUNE 1987

Conclusion

The TMA method has been extended for the determi-nation of frequency dependent characteristics of singleand coupled slotlines. Numerical results have been com-pared with published papers and are found in goodagreement with them. The potential of the TMA methodis proved to be applicable for similar structures. Theadvantages of TMA are its simple formulation and com-putation for various transmission lines.

18th July 1986

Department of Electrical EngineeringTatung Institute of Technology40 Chung-Shan North Road 3rd SectionTaipeiTaiwanRepublic of China

T. N. CHANG

References

1 COHN, S.B.: 'Slotline on a dielectric substrate', IEEE Trans., 1969,MTT-17, pp. 768-778

2 WEN, C.P.: 'Coplanar waveguide: A surface strip transmission linesuitable for non-reciprocal gyromagnetic device application', ibid.,1969, MTT-17, pp. 1087-1090

3 HATSUDA, T.: 'Comoutation of coplanar-type strip line character-istics by relaxation method and its applications to microwave cir-cuits', ibid., 1975, MTT-23, pp. 795-802

4 YAMASHITA, E., and ATSUKI, K.: 'Analysis of microstrip-liketransmission lines by nonuniform discretization of integral equations',ibid., 1976, MTT-24, pp. 195-200

5 KNORR, J.B., and KUCHLER, K.D.: 'Analysis of coupled slots andcoplanar strip on dielectric substrate', ibid., 1975, MTT-23, pp.541-547

6 DA VIES, J.B., and MIRSHEKAR-SYAHKAL, D.: 'Spectral domainsolution of arbitrary coplanar transmission line with multilayer sub-strate', ibid., 1977, MTT-25, pp. 143-146

7 YEE, H.Y.: 'Transverse modal analysis for printed circuit transmis-sion lines', ibid., 1985, 33, pp. 806-816

5325H

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