Technical memorandumTRANSVERSE MODAL ANALYSIS OFMULTILAYER MICROSTRIP TRANSMISSIONLINES

Indexing terms: Microstrip and stripline, Transmission lines

Abstract: A hybrid mode analysis is presented fora multilayer dielectric within a rectangular con-ducting box. Specific results are obtained for a3-layer structure and compared with those in theliterature.

Introduction

With the increased use of suspended substrate lines,various studies have been made of multilayer microstriptransmission lines [1]. Similar structures have beentreated in References 1-4. In this paper, the transversemodal analysis [5] is applied to give a frequency depen-dent solution of an enclosed multilayer structure. Generalprocedures are shown and examples of results for singlemicrostrip on a suspended substrate are presented.

Formulation

Consider a multilayer structure shown in Fig. 1. It isassumed that there is an infinitely long thin strip on the

9N

e

£

e

e

(N*1)

w

(N)

(i*1)

( i )

e<2)

e<i>

Fig. 1 Cross-section of a multilayer microstrip-like transmission line

top of the Nth layer. To obtain the dispersion character-istics of this type of structure, the TMA method is used.Two electric walls are placed at y = 0 and y = b, where bis equal to half of the guided wavelength of the structureconsidered. The transverse fields in each region areexpressed in terms of the normalised waveguide modal

324

functions. For region 1 (0 < z < g^):

2 M

E == / /̂ F./4 sin (B zY\d)p = l m = 0

2 M

/ / , j = j ^ V [v4p^y_^ cos (P^zJ](z x ^p = l m = 0

for region i(i # 1, i # JV + 1, #,•_! < z < #,):

2 M

p= 1 m = 0

2 M

H = i Y V F/(I) cos /?(l)zp = l m = 0

— Rp^, sin f$\}}z']yp'll(z x < p̂m)

for region N + 1 (gN < z < h):

2 M

Et, N+l = Lu 2-,p= 1 m = 0

2 M

sin

(1)

(2)

m (3)

(4)

M (5)

Z Z ^p= 1 m = 0

x cos [^N + l\h - x <f>pm) (6)

where p is an index indicating TEZ or TMr modes, mspecified the waveguide modal numbers in the x direc-tion. <j>pm are the normalised waveguide modal functionsas given in Reference 5. The wave admittances ypm

(i = 1 , . . . , N + 1; p = 1, 2) are related to propagationconstants fj$ in the z direction by

(7)

(8)

where w is the angular frequency and

(9)

k0 is the wave number in free space and 8{rl)(i = 1, . . . ,

N + 1) are the relative dielectric constants in each region.Boundary conditions at the interface of the ith and

(i+ l ) t h ( i# N) layer are:

Ettl = Etti+x (10)

Hti = Hti+y (11)

Multiplying both sides of eqns. 10 and 11 by <f>pm a n dintegrating over the waveguide cross-section (via the nor-malisation conditions), then yielding a relation between/(') D(0 /( '+!) J?('+1)

pin' ptn 5 ntn ) pm )1S

The [/4BCD] matrix can be called 'chain matrix' as coef-ficients of the fields of any layer can be expressed in termsof any other one. After chaining the subsequent matrix,we finally have

1 pm

RiN2 e am (13)

or /J2 =0

(P= 1,2; note that= 0 in region 1).

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The boundary conditions at z = gN are

\EuN+x _ on the strip (14a)

[EttN + l elsewhere (146)

x {Hu jy — HtN+1) —0

on the strip

elsewhere(15a)(156)

The strip surface current density is represented by J = jS?= oCq&q> &q *s t n e vectorial basis functions. For most

cases, trigonometric basis functions may be used toobtain accurate results [5].

To enforce boundary conditions at z = gN, we takethe scalar product of eqn. 14 with <f>pm, then integrateover the waveguide cross-section and take the scalarproduct of the same equation with Qq, and integrate overthe strip area. It yields

- 9N) = <2 fl£sin sin

cos2 M

p= 1 m = 0

Wtpm = Qt ds

(16)

(17)

(18)

We may note that both I[p2 and R{

p2 can be expressed interms of Upm by eqns. 13 and 16.

Substituting eqns. 4 and 6 into eqn. 15 and taking thescalar product of eqn. (15) with <j)pm, then integratingover the waveguide cross-section, we obtain

<j = o

GG = vl,1+1) cot

(19)

(20)

- C'/{/4' + [ C cot (j8£JV)]} (21)

Substituting eqn. 19 into eqn. 17, we obtain a matrixequation given by

[M]\C\ = 0 (22)

where, the elements of [M], Mtq are given by

2 M

p = l m = 0

and [C] = [C1? C 2 , . . . , C9]T.After solving the determinant equation for the cavity

resonant frequency, say f0, the effective dielectric con-stant is obtained using the following definition:

eeff = (v/2bf0)2, (24)

where v is the speed of light in vacuumThis general procedure can be applied to various types

of structures. One of them is the suspended stripline. Inthis case, it is easy to obtain that

0i) (25)

(26)

h) + y(P

2m tan ( j ^ i ) (27)

h ~ y(pi c o t ifim9\) (28)A matrix similar to eqn. 22 with different values of ele-ments is obtained and dispersion characteristics of thesuspended stripline can be solved from that matrix.

Numerical result

To test the theory, the dispersion characteristics of a sus-pended stripline are investigated. Fig. 2 shows our results

A" = Gpm cot

C = G'pm

Gpm = 3 $ cot

G'pm = y™ cot

against a comparison with results of Yamashita andAtsuki [7], and they agree quite well. Dispersive naturesin terms of the thickness of the dielectric substrate (i.e.

3 . U

3.0

2.0

1.0

e(r2)=20.0,

= 9.35

5 10

f.GHz

20 30

Fig. 2 Dispersion characteristics of a suspended striplineN = 2, £<" = £<3) = 1.0, ei2) = 2.55; 9.35; 20.0, a = 20 mm, w = 2.0 mm, g,4.5 mm, g2 = 5.5 mm, h = 10.0 mm

• • TMAx Reference 7

(a2 — gi)/h) are shown in Fig. 3. It is expected that thethicker the dielectric substrate is, the more the dispersiveit will be.

1.3

1.2

1.1

1.0

(g2-g,)/h=l

10 15f .GHz

20 25 30

Fig. 3 Dispersive nature for suspended stripline as a function of(g2 — g^/hfor the same configuration of Fig. 2 with e(

r2) = 2.55

In the calculation, M'( = M — 1 = 40, not includingm = 0) terms of waveguide modes and Q'( = Q — 1 = 2,not including q = 0) terms of strip modes are chosen suchthat M': Q' = 2a : w. This choice will speed up thenumerical convergence, as has already been seen in Refer-ence 6. Trigonometric basis functions as in Reference 5are used in this paper.

Conclusion

The work presented here is an extension of the transversemodal analysis. The 'chain matrix' technique is used toanalyse a particular case, the multilayer single microstrip.Specifically, the dispersion characteristics on suspendedstripline have been calculated and general procedures fortreating multilayer microstrip-like transmission lines are

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 3, JUNE 1987 325

presented. Good agreement with the literature verifies theaccuracy of this method.

The theory can easily be extended to coplanar strip-lines/slotlines.

8th September 1986 T.N. CHANG

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

References

1 DA VIES, J.B., and MIRSHEKAR-SYAHKAL, D.: 'Spectrumdomain solution of arbitrary coplanar transmission line with multi-

layer substrate', IEEE Trans., 1977, MTT-25, pp. 143-1462 FARRAR, A., and ADAMS, A.T.: 'Multilayer microstrip transmis-

sion lines', ibid., 1974, MTT-22, pp. 898-8913 JANSEN, R.: 'Zur numerischen Berechnung geschirmter

Streifenleitungs-Strukturen', Arch. Elektron. und Ubertragungstech.,1975, 29, pp. 241-247

4 JANSEN, R.: 'Computer analysis of edge-coupled planar structures',Electron. Lett., 1974,10, (24), pp. 520-521

5 YEE, H.Y.: 'Transverse modal analysis for printed circuit transmis-sion lines', IEEE Trans., 1985, MTT-33, pp. 806-816

6 CHANG, T.N., and WU, K.Y.: 'Transverse modal analysis of edge-coupled microstrip resonators', Electron. Lett., 1986, 22, (11), pp.608-609

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

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