Generalised approach to modelling shielded printed-circuit transmission lines

Generalised approach to modelling shielded printed- circuit transmission lines

A.B. G nilen ko A.B. Ya kovlev I.V. Petrusenko

Indexing terms: Planar waveguides, Numerical tecluziques, Printed-circuit tvansini.wion lines

Abstract: A new generalised approach for the full-wave analysis of shielded printed-circuit transmission lines with finite metallisation thickness is presented. A rigorous formulation for the electric field vector is employed, representing the hybrid electromagnetic field in layers of multidielectric media. The dyadic Green's function of the electric kind is derived as a kernel of the vector integral representations for the electric field. Based on the method of overlapping regions, the approach can be applied to the characterisation of a large class of planar waveguiding structures to yield a set of linear algebraic equations of the second kind with a compact matrix operator. Numerical examples are shown for single and coupled shielded planar transmission lines illustrating application of the method. The technique described allows obtaining accurate and mathematically correct solutions of planar waveguide eigenvalue problems.

1 Introduction

Modern CAD tools have become increasingly important for the design of high-performance MMICs which serve as key components in compact and low- cost communication systems [l, 21. In the development of CAD programs, accurate knowledge of the propagation characteristics of printed transmission lines plays a vital role. The current trends in MMIC fabrication technology toward higher frequencies, higher component density, and increasing component complexity place very stringent demands on circuit design tools. This requires development of versatile techniques which can handle a large class of multilayered and multiconductor planar waveguiding structures taking into account electromagnetic coupling between conductors of finite thickness. As a result, attention has been directed in recent years to

0 IEE, 1997 Eb' Proceeding.r online no. 19971036 Paper first rcceived 1st July 1996 A.B. Cnilenko and 1.V. Pctruscnko are with Dniepropetrovsk State Uni- versity, Department of' Radiophysics, Street Gagarin 72, Dnicpropetro- vsk-625, 320625, Ukraine A.B. Yakovlev is with the University of' Wisconsin ~ Milwaukee, Depdrt- ment of Electrical Engineering and Coniputer Science, P.O. Box 784, Mil- waukee, Wisconsin 53201. USA

generalised hybrid mode approaches for the rigorous analysis of complicated printed-circuit configurations.

Various full-wave techniques have been developed for the rigorous analysis of multidielectric and multiconductor transmission line configurations. The spectral domain approach (SDA) [3] and the singular integral equation (SIE) method [4] are techniques which give efficient and accurate algorithms for characterising printed transmission lines embedded in a multilayered dielectric medium by relatively small-order matrices. Unlike the SDA and the SIE method dealing with zero-thickness conductors, the mode-matching technique (MMT) [5] is more versatile and can treat multilayered/multiconductor planar waveguiding structures with finite metallisation thickness as well as mounting grooves. However, this method operates with relatively large characteristic matrices and requires more computational effort.

It is well known that the mode-matching technique results in obtaining infinite sets of linear algebraic equations (SLAE) of the first or second kinds. The numerical solution of the infinite SLAE of the first kind by the truncation of infinite series leads to 'relative' convergence or to the absence of convergence of approximate solutions. The choice of a certain ratio between an order of truncation for the infinite SLAE and series forming matrix coefficients makes it possible to obtain the co-ordinate convergence. Despite the popularity of the MMT, the applications of this method are empirical rather than based on strict theory because the accuracy of previous analyses has been checked only by looking at the convergence of com- puted transmission line characteristics. The study of resulting operator properties from the standpoint of functional analysis substantiating the correctness of the transformation from an infinite problem to a finite one has not been investigated.

These drawbacks of the mode-matching technique are caused by convolution-type matrix operators of the first- or second-kind SLAEs. Thus, regularisation of the convolution-type equations is required lo obtain the correct solution of a problem.

To improve the accuracy and efficiency of the MMT, suitable basis functions (Chebyshev polynomials) with the singularity of field behaviour at the edges of zero- thickness conductors, which are eigenfunctions of the Cauchy operator, are commonly applied in Galerkin's method. However, this choice involves special functions such as Bessel functions in the algorithm and therefore increases computational effort. Other ways to regular- ise the incorrect first-kind equations, based on addi-

103 IEE Proc.-Microw. An~ennus Propug, Vol. 144, No. 2, Aprrl 1997

tional analytical preprocessing, have been proposed using the semi-inversion method [6] and the modified residue calculus technique [7] for some specific configurations. Unfortunately, these powerful techniques are not versatile enough to be applied for the analysis of complicated multilayeredimulticonductor waveguiding structures.

Recently, the method of overlapping regions (MOR) has been presented for the analysis of diffraction problems [8]. This method generally involves dividing a complex domain of the field determination of an investigated structure into a number of simple overlapping subregions for which a fundamental solution of Helm- holtz's equation exists as a solution of the Green's function problem. Existence of common regions allows formulation of the Fredholm equation of the second kind, or a coupled set of integral equations, which may be algebraised using Galerkin's method. The advantage of the MOR is in the matrix operator of the characteristic SLAE, a principal part of which is the idem-factor and an additional part is a compact operator in a suitable pair of spaces. Based on the analytical Fredholm alternative theorem, the existence of the bounded inverse operator and application of the truncation method to solving the characteristic equation can be demonstrated. This method directly leads to the second-kind SLAE with a compact operator. Thus, the method of overlapping regions may be considered as an efficient regularisation of the mode-matching technique equations.

The objective of this paper is to present a generalised approach for the full-wave analysis of multilayered and multiconductor printed-circuit transmission line structures based on the method of overlapping regions. By considering a rectangular multilayered region as a key structure of the approach, the integral representations for the electric field vector are constructed. Components of the dyadic Green's function of the electric type are derived in the most general form. This formulation is quite versatile and handles a wide variety of planar waveguiding structures with finite thickness conductors. The shielded microstrip line and coupled strips on two-layered substrate are considered. Based on the properties of the characteristic matrix operators, the strong convergence of solutions and application of the truncation method are substantiated. Calculated results are compared with data available in the literature.

-~ .

Fig. 1 Generalisedpluncir M-uvcpiding structure

104

2 Formulation of eigenvalue problem

Fig. 1 shows the cross section of a generalised waveguiding structure which consists of a number of finite thickness strip conductors embedded in layered medium. Using the method of overlapping regions, a field determination domain of the structure may be divided into a number of simple rectangular subregions: homogeneous and filled with more than one layer. A generalised multilayered region can be obtained by dividing a complicated transmission line configuration into subregions (Fig. 2). The region con- sisting of y1 isotropic lossless layers is assumed to be regular in the z-direction and bounded by a piecewise smooth perfectly conducting surface S. The region's boundary S has a number of open parts ~ apertures D,. (1 . = 1, ..., N,; N , is the number of apertures for the ith layer; i = 1, ..., n) in gaps between strip conductors. Dependence of the electromagnetic field on d( 'N~+~'z ) is assumed, with y = ja - @(Zwl[y] 2 0) being the propagation constant along the z-direction.

D n 2 Sn

. . . . . ---1 V" E n P n D,N,

L' I

The boundary value problem in the region is formu- lated for the full electric field vector which must satisfy

homogeneous vector wave equations

TxTxE ' ( r ) -k~E ' ( r )=O r E K i = l , . . . , n (1)

Dirichlet boundary conditions on the conducting surfaces Si

n x Ez(r) = 0 r E 5'; (2) continuity conditions for tangential components of

the electric and magnetic fields on interface surfaces between the layers Li n L x E'(r) = n L x Ezsl(r) r t L, 1 1

- n L x (V x E'(r)) = -nL x (V x E"'(.)) ( 3 ) I*.% Pz+l

condition at the edge in the form of belonging solutions to the class of Holder functions having proper behaviour in the vicinity of the conductor edges, and

radiation condition at infinity for waveguides. The corresponding boundary value problem for the

IEE Proc.-Microw. Antennas Propag., Vol. 144, No. 2, April 1997

dyadic Green’s function of the electric type is based on the formulation given in [9] for the Green’s dyadic of two different isotropic media and extended here to the case of a multilayered structure. According to this technique, the dyadic Green’s function of the electric type must satisfy: * dyadic differential equations

c x D x Gh.(rlr’) - k:ck( r i r ’ ) = h , k l b ( r - r’)

r g K r ’ g V k 7 , k = 1 . . . . , 72 (4) boundary conditions for the Green’s dyadic of the

first kind on the conducting surfaces SI and surfaces of the apertures D,,

(5) n x G‘k(rlr’) = o r E s,. u,=~ y, D,, * mixed continuity conditions for the Green’s dyadic of the third kind on interface surfaces between the layers L, n L x ck(r l r ’ ) = nl, x G+’ ‘(rlr’) r E L ,

radiation condition at infinity for waveguides. In eqns. 4-6, 6, is the Kronecker delta; d(rir’) is the three-dimensional Dirac’s delta function; _I is the unit dyadic; r and r’ are the observation and source position vectors; superscripts i and k serve to denote, respectively, the ith and kth layers where the observation and source points are located.

3 Construction of integral representations

Generally, the mathematical modelling of wave propagation in waveguiding structures assumes obtaining solutions to Maxwell’s equations or the wave equations (Helmholtz’s equations) under certain boundary conditions. There is a large group of approaches based on the equivalent transformation of initial boundary value problems to an integral equation or a coupled set of integral equations, which may be effectively used for the development of computational algorithms. This transformation usually employs integral representations of fields inside a considered structure in terms of field quantities on certain boundaries.

To construct the integral representations of the electric field vector in the key structure introduced in Sec- tion 2, we apply the superposition principle to eqns. 1 and 4 for the ith layer along with the dyadic form of the Green’s theorem and take into account boundary conditions eqns. 2, 5 and the radiation conditions. As a result, the following expression is obtained:

v, (nu x E’(r))(V x G’(r1r’))dS 5 L I . !

S , k E Z ( Y ” ) = -

- / { ( n i x E’( r ) ) (V x C’(r1r’))

(11; x G”’(rlr’))(C x E Z ( ~ ) ) } d~

. l,, -1

/ { ( n i x Ei ( r ) ) (V x Gk(rlr’))

- (n i x ~ ” c ( r l r ’ ) ) ( ~ x ~7(r))} d~

. I>

(7) IEE Proc..-Microw. Antmnua Pf’O~Jclg., Vol. 144, No. 2, April 1997

where nD is an outward normal to the surfaces of the apertures D,,, nlf= yo, nL = -yo.

Summing eqn. 7 for each ith layer and making use of the continuity conditions eqns. 3, 6 on interface surfaces between the layers, a set of integral representations of the electric field vector in the layers can be obtained

n V,

(8) Introducing Fourier transformations of the field and the dyadic Green’s function with respect to the z-co- ordinate

Ez((z. y. Z ) = - ~~ lm Ez((z. y, ?‘)eJYtZdy‘ 1

G k ( % , y . Z l X r , y r , Z ’ )

(9) G z k (., yJT ’ , y’. y ” ) e ” f ( ” - Z ’ ) d 3 ’ ’

the problem may be reduced to one on the cross-sectional plane of the region convidered

. ((VkU - j , z , ) x gyd. y’lz, y , -y ) )d l ’ (10)

where nd is an outward normal to aperture lines dkkr on the cross-sectional plane; i, k = 1 , ..., n. The set of electric field integral representations (eqn. 10) expresses the full electric field vector inside each layer in terms of field values at aperture lines in the boundary of the generalised region.

Applying the same technique, the electric field integral representation can be obtained for a homogeneous region in the following form:

l\r

r=l

G A ( x $ ylx’. yr. 7) (n, x E(z’, yr . y ) ) d ~

(11) i,-

where GA is the potential-type dyadic Green’s function of thesecond kind and ng is an outward normal to aperture lines g, in the boundary of the homogeneous region.

4 Derivation of electric-type Green‘s dyadic

Numerical efficiency of integral equation methods has to be paid by extensive analytical preprocessing which includes finding the Green’s functions. To apply the integral representations of eqn. 10 to the hybrid mode analysis of planar waveguiding structures, one must derive the electric-type dyadic Green’s function which is the kernel of an integral operator moving the electric field vector from the region boundary to any point inside. The dyadic Green’s function as a solution of the dyadic analogue for the vector wave equation is a 3 x 3 matrix with nine nonzero components.

Components of the dyadic Green’s function of the electric type are sought in the form of series expansion

105

in the x-direction with coefficients that are unknown functions of the y and y‘ co-ordinates

5 Application to transmission line modelling

Using the vector integral representations constructed in

m=O

Eigenfunctions ~,??~,(x) satisfy the Helmholtz’s equation and, taking into account the boundary conditions of eqn. 5 by x = t , a, may be written as

Substituting expressions for the Green’s dyadic components (eqn. 12) into eqn. 4 projected on the co-ordinate system axes, we reduce the boundary problem of eqns. 4-6 to solving the following differential equations for one-dimensional functions (y, y ’ , -y):

Solutions of eqn. 14 are found as a sum of the general solution of corresponding homogeneous equations and the partial solution of inhomogeneous equations, and may be associated with a wave travelling away from the source point plus waves reflected by interfaces between the layers

where

Upper signs in the expressions correspond to the case of y > y’ and the lower ones to that of y < y r .

Dirichlet boundary conditions (eqn. 5) at the shield case by y = lo, 1, and continuity conditions (eqn. 6) at interface lines between the layers allow construction of a set of linear algebraic equations for the determination of unknown coefficients in eqn. 15. To calculate an& and /3& a computer subroutine has been developed whose input parameters are permittivities and thick- nesses of the layers. The technique presented for the derivation of the Green’s dyadic makes it possible to obtain values for all components at any location of the observation and source points in the dielectric layers. The increase of the number of layers does not create obstacles for the analysis.

IO6

Section 3 along with the presented form of the dyadic Green’s function, and following the scheme of the method of overlapping regions [SI, any kind of shielded planar waveguiding structures can be investigated in a rigorous manner. To illustrate and validate the approach, two planar transmission line configurations, namely the shielded microstrip line and broadside-coupled microstrip lines with finite thickness conductors, have been analysed. The electric and magnetic walls are placed in the plane of symmetry of the structures at x = 0, as depicted in Figs. 3 and 4, for characterising odd and even modes of the eigenmode spectrum, respectively. The geometry of the structures under consideration can be divided into simple rectangular subregions which overlap in the shaded areas (see Figs. 3 and 4)

1st layer E:(x,y,y) t I z I a 0 5 1~ I I1

3rd layer E f ( x , y , ? ) t I x 5 a Z2 I y I l 3

2nd layer E;(x,y,y) t I z 5 a ZI L y I 12

E2(x,w17) 0 5 5 I a 0 I y I Sl Es(z,y,y) 0 I I a Cl I y I sa E4(.,Y,Y) 0 Ix I a e 2 I y I b

2nd region 3rd region 4th region

Y l

e lect r ic and

magnet IC

wal ls

0 t 0 X Fig. 3 ililicrostrip line con3guration

e iec t r i c and

magnetic w a l l s

b

c 2

52

C 1

51

0 0 t a X

Fig. 4 Broadside-coupled microstrip line configuration

IEE Proc.-Microw. Antennus Propug., Vol. 144, No. 2, April 1997

(4th region appears for the broadside-coupled microstrip lines only). Unknown fields inside each region may be expressed in terms of field values at apertures using the integral representations of eqn. 10 for the electric field vector in the multilayered (two-layered for the microstrip line and three-layered for the coupled microstrip lines) regions

and of eqn. I 1 for the homogeneous regions N ? L ,

(17) where subscript i + 1 denotes the (i + 1)th homogeneous region; superscripts i and k denotes the ith and kth layers of the layered regions; i, k = 1, ..., n; n = 2, N 2 3 = 1 for the microstrip line; n = 3, N2,4 = 1, N3 = 2 for the coupled microstrip lines; integration is taken along the aperture lines d, = {x‘ = t , y’ E [0, sl]}, d2 = {x’ = t , Y’ E [c,, Jzl}, 4 = {x’ = t , y‘ E [c2, 611, s 2 1 = CX’ E 14 a], y‘ = X I } , g31 = {x’ E [t , a], y’ = y’ = sZ}, g41 = {x’ E [ t , U ] , ,v’ = ~ 2 ) .

g32 = {x’ E a],

The assumption about field equality in the areas of overlapping makes it possible to match the tangential field components across the apertures, which are open boundary lines of the overlapping areas

n x E,+l (d, y’? y) = n x E: (z’, y’, y)

x/, Y’ E &, UTI1 g2+1 1. = 1, . . . ? n (18) N,+1

and substitute the integral representations for the multilayered regions into ones for the homogeneous regions. As a result, the set of integral representations may be coupled in a system of Fredholm integral equations of the second kind for the electric field vector in the homogeneous regions

n ,.

where K’icL(x’, y’lx, y , -y) are integral operators - k? K (x”; Y”lZ? Y? -7)

1v = - C ( V Z Y + j Y 0 ) x

. { n, x ((vi, - jyzo) x ~ : ‘ ( x ’ / ? y ” ~ z / , yf, -7))) dz’

Using Galerkin’s method, the set of integral equations is reduced to an infinite SLAE of the second kind which can be written in matrix form as

G,+I ( 2 , YIZ/ , d ; 7) 7.=1 L - *

( I - L ) X = 0 (20) whcre I is the identity matrix, L is a hypermatrix, and X is a vector of amplitude coefficients in series expan- sions of field components.

Consider the properties of the operator eqn. 20.

JEE Proc.-i!4iicroiv. Antennas Propug, Vol. 144, No. 2, April 1997

Applying the technique of [7], one may obtain X C I’ I M

to ensure the field’s condition at sharp edges of the strip conductors. Asymptotic behaviour of elements of the diagonal matrix blocks A included in the hypermatrix L

where 8, q > 0 are real constants for the given geometry of the problem, allows factorisation in the form

A = B C ( 2 3 ) Here, the similar factor-matrices B and C are defined by their elements d,, = b,i2(c,,2)

(24)

Using conditions of continuity and w-complete continuity for a matrix operator D:I1 - 1” [IO]

00

00

respectively, and taking into account that

1 00 n = -coth 7r (7) - -

n2 + < 2 m 2 2 c 2n m = l

const m=l 2 ( n 2 + < 2 m , 2 ) 5 7

n>>l t > O continuity of D = B(C):I’ - I’ and w-complete continuity of D = B(C): 1’ - I1+& can be substantiated. Hence, the diagonal matrices are w-complete continuous operators A:/’ -+ I’ + & ( E > 0) as products of the continuous operators B(C):I’ - I’ and the w- complete continuous operators C(B): 1’ - I1+&.

Nondiagonal matrix blocks of the hypermatrix L give asymptotic estimates which allowed us to prove the o-complete continuity of the corresponding operators in the pair of spaces I’ - I’ in the same way.

Thus, the infinite matrix equation of the second kind (eqn. 20) generates the Fredholm meromorphic operator-function in C with discrete spectrum. Based on the properties of the resulting operator, existence of a bounded inverse operator and the possibility of spectrum’s approximation by roots of the truncated eigenvalue equation det[Z - LIN = 0 may be proved.

6 Numerical results

To check the efficiency of the method, it is used in the computation of a relative error caused by truncation of the infinite eigenvalue equation: ax, = (y/kO), - (y/ko)n-l. The results shown in Fig. 5 reveal the error of truncation decreases as the size of the resulting matrix IZ increases following the evaluation x, = O(nP) with z > 2 for the various number of expansion terms m in matrix elements.

107

Dispersion characteristics for the dominant, &-even- H,-odd, and E,-odd- H,-even higher-order modes are obtained for ridge, traditional, and buried single microstrip line shown in Fig. 3. Dispersion curves for the dominant mode of different microstrip lines are shown in Fig. 6. The results are obtained for the following parameters: dielectric substrate thickness is I , = 1.27mm, strip width is 2t = 1.27mm, shield is 2a x s? = 12.7 x 12.7mm, = E ~ , ,ul = ,u2 = ,U,. An increase of strip thickness in a buried microstrip line leads to an increase of electromagnetic field concentration in the substrate. The propagation constant of the dominant mode approaches the propagation constant of the TEM-wave in free space loaded with dielectric E ~ : ylk, - d ~ , . The field distribution changes in going from buried to ridge microstrip line. Extension in dielectric E~ is consistent with decreasing propagation constant ylk,. However, the effect of strip thickness for ridge microstrip line becomes less significant at higher frequencies, which is explained by concentration of the electromagnetic field in the substrate.

= 8.875&,,

1 0 1.5 2 0 ' h ( n )

Fig.5 Relative error oj truncation uguinst size of mutrh blocks fix,u,, = O(n-3: n >> I; z = 2.59(m = 3); z = 2.28(m = 5 ) : z = 2 1 l(m = IO): T = 2.02(m = 15) *-* n z = 3 A-A in = 5 0-0 n? = 10 x-x m = IS

3.01 I 1

0 Y \ >

2 . L I 10 15 20 25 30

frequency, G H z Fig. 6

~

-~ ~ ~

~~

~~

Donfinant mode ?fridge, buried, and traditional nzicrostrip line s, = O.lmm, cI = 1.271" s , = 0.3175mm, c , = 1.271mm si = 0.9525nim, c , = 1271" s I = 1.2699nim, cl = 1.271mm s I = 1.2699mm, c, = 3.81" sI = 1.2699mm, cI = 8.89mm

. . .

lox

t f . 15 25

frequency, G H z Fig. 7 trip line

s I = 1.2699mni. cI = 2.54min s I = 1.2699". c , = 7.621nm

Ez-ew-HZ-odd dominant and higher order mode.7 of iidge micros-

1 5

n l ? . :'"I 0 5

z - . i *! 01

10 15 20 25 f requency, G H z

Fig. 8 0 s I = O.63Smn1, c1 = 1.271mm A sI = 0 3175mm. c , = 1.271mm * s1 = 0.1". c , = 1.271" 0 coupling regions

Ez-odd-Hz-even higher order modes o j buried microstrip line

3

2

0 Y >

1

C

l*.A..~* .............. &..A ...... ..........

* A

I 1. A A ..... .... A :

A . 1 .

1 2 3 L 5 striD hal f -width. mm

Fig. 9 line

ISGHz A 20GHz

Ez-odd-Hz-even higher order nzodes of negligibb thin microstrip

Dispersion characteristics for the dominant and E,- even-H,-odd higher order modes of ridge microstrip line are presented in Fig. 7. An increase in strip thickness causes large changes in higher order modes. In

IEE P i x - M i c r o w . Antennas Propag., Vol. 144, No. 2, April 1997

particular, the cut-off frequency for the first higher mode is significantly decreased. Redistribution of cou- piing regions and an increase of mode coupling is obscrved. An increase of strip thickness in burried microstrip line leads to a loss of mode coupling. Dis- persion characteristics and coupling regions for ,?,-odd- H,-even higher order modes are shown in Fig. 8. An increase of strip width results in significant changes of Ez-odd-Hz-even modes. Fig. 9 demonstrates the propagation constant behaviour for negligibly thin microstrip line against strip width. Results, obtained for negligibly thin shielded microstrip line, have been compared with data of other authors and good agreement has been observed, for example [I I , 121.

As a result, positive changes of strip thickness in ridge microstrip lines lead to an increase of higher- mode coupling, especially for E,-even-HZ-odd modes. Strip deepening in buried structures causes loss of higher-mode coupling, specifically for E,-odd-H,-even modes. An increase of strip width predominantly changes characteristics of Ez-odd-Hz-even modes in comparison with the E,-even-H,-odd mode set.

c 0 Ln c c

8 1 . 6 - U

L U W

- Y

L

.a, 1 . L -

3

U W

Y U

f 1 . 2 - W

1 .01 I

0 0.5 1 . o 1 . 5 2.0 2 t / b

l2 ~ I , = I 52": ( I 2 - /])/I? = 0.4; ( i ) Odd mode 0 ~ -0 Bahl el ai. I131 A--A Kawano 1141 '< t h k theory

= 1.0: c , ~ = 2.32: j = 1GHz (11) Evcn rnode

, . , . , . I .

5 10 15 20 25 frequencv. GHz

IEE Proc -Microw. Antennas Propug, Vol. 144, No. 2, April 1997

For the coupled microstrip line structure (Fig. 4), numerical results have been compared with those obtained in the literature for broadside-coupled suspended substrate microstrip lines. Fig. 10 shows the effective dielectric constant Eeii of c- and x- dominant modes (even and odd dominant modes for broadside- coupled suspended substrate microstrip lines) as a function of the relative strip width. These results have been compared with data of [13, 141 shown also in Fig. 10. Dispersion characteristics of E.il for dominant modes of the coupled microstrip lines reveal good agreement with those of [I41 for even and odd modes of broadside-coupled suspended substrate microstrip lines as demonstrated in Fig. 11.

53 t i

1 . o 0.2 0.5 1.0 1.5 20

~ r 2 ' E ~ I Fig. 12 2a = 3 5mm: 21 = 0.68": h = 3.0": 1, = 0.5": l2 ~ I, = 0.25mm; c1 I 12,, = 12,! ~ sL I = 0.01Xmm; E " , = 5.0 (i) z-mode (ii) c-mode

ivormnlised~~r[~u~ution constunt u<quinst S,~/E,.,

*-* 1GHs o-n 2 0 ~ ~ 2 x ~ - ~ x 50GHi A- A 100GHz

3 5 I

frequency, GHz Fi 13 Norvzulised propuption constmi of' domirzunt and f iw higher or%r niocle,~,fi~r broudside-coupled microstril, line,c us,fimction of,fi.equeiicy 2~2 = 3.5nini: h = 3.0mm; 21 = 0.68inm; 1, = 0.5": i2 ~ 1, = 0.25mm; c ~ , ~ ~ 12,1 = lZ,! (i) x-mode (ii) c-modc

A ? , ] = 0.01Xmm: c,, = 5.0; E , ~ = 10.0

cvcn modes odd modes -~~~

Fig. 12 shows the dependence of the normalised propagation constants ylkO of the dominant c- and x- modes on the ratio of relative permittivities of dielectric

109

layers E , ~ / F , . , for a fixed E , ~ . The variation of E , . ~ / E , ,

results in the transformation of the dominant modes in the region of E , . ~ = E , ~ : c-mode transforms to x-mode and vice versa. The propagation constant of the c- mode, which mainly concentrates in the lower ‘supporting’ layer, changes slightly because of the fixed E , ~ and tends to the value of d ~ , , as frequency increases. The propagation constant of the x-mode, which mainly concentrates in the layer between the strip conductors, increases following a rise in the value of F , ~ . Dispersion characteristics of the dominant and several higher order modes are illustrated in Fig. 13. It is easy to see that higher order modes may be divided into two sets, the propagation constants of which approach those of the dominant modes as frequency increases. This reflects the fact of corresponding mode propagation principally in a certain layer of the dielectric substrate.

7 Conclusion

A novel efficient technique based on the idea of integral representations for overlapping regions has been presented for rigorous investigation of a wide variety of planar transmission line eigenvalue problems. The approach can be used for creation of modern powerful CAD programs taking into account multilayer/multiconductor complexity of waveguiding structures. Appli- cations of the method to single and coupled shielded transmission lines have been demonstrated. The approach presented has been validated by analytically evaluating the properties of the resulting matrix operator, checking the convergence of numerical solutions. and coinparing the results obtained with data available in the literature.

8 Acknowledgments

The authors thank Dr. I.G. Prokhoda for encouraging and stimulating suggestions.

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11 YAMASHITA, E., and ATSUKI, K.: ‘Analysis of microstrip-like transmission lines by nonuniform discretization of integral equations’, IEEE Trnns., 1976, MTT-24, (4), pp. 195-200

12 HASSAN. E.E.: ‘Field solution, polarization and eigenmodes of shielded microstrip transmission line’, IEEE Trans., 1986, MTT- 34, (8), pp. 845-852

13 BAHL, I.J., and BHARTIA, P.: ‘Characteristics of inhomogeneous broadside-coupled striplines’, IEEE Trans., 1980, MTT-28, (6), pp. 529-535

14 KAWANO, K.: ‘Hybrid-mode analysis of a broadside-coupled microstrip line‘; IEE Proc. H, Microw. Antennas Propag., 1984, 131, ( I ) , pp. 21-24

110 IEE Proc.-Micvow. Antennas Pvopug., Vol. 144, No. 2, April 1997

Documents

Generalised approach to modelling shielded printed-circuit transmission lines