Analysis of microstrip discontinuities using the method of integral equations for overlapping regions

A. B.Ya kovlev A.B. Gni len ko

Indexing terms: Microstrip discontinuities, Integral equations, Planar dispersive waveguides, Method of integral equations for overlapping regions

Abstract: The method of integral equations for overlapping regions (MIEOR) in conjunction with a planar dispersive waveguide model is proposed for the investigation of microstrip discontinuity problems. A general formulation for a microstrip multiport junction and the numerical solution for the scattering matrix of T- and right- angled bend junctions with and without matching stubs are presented. The proposed technique directly leads to Fredholm’s integral equation of the second kind or a coupled set of integral equations for the electric field components defined in overlapping subregions of the planar dispersive waveguide model. The Galerkin’s method is employed to find the unknown coefficients in the electric field series expansion which represent the S-parameters of the junctions. The mathematical substantiation of the algorithm is provided. Numerical results are presented for T- and right-angled microstrip junctions showing good agreement with results already published.

1 Introduction

To meet the requirements of modern integrated circuit (IC) fabrication technology, designers have to use sophisticated computer-aided design (CAD) tools that imply performing a three-dimensional (3D) hybrid mode analysis of various circuit discontinuities and composing circuit building blocks. The main part of this process is the simulation of electromagnetic field diffraction by microstrip junctions which serve as building blocks for a great number of printed circuit components. In recent years, full-wave integral equa- tion methods as well as numerical techniques have been developed for the accurate analysis of microstrip dis- continuity problems. A full-wave spectral-domain approach has been successfully applied for the investi- gation of open microstrip discontinuities of arbitrary 0 IEE, 1997 ZEE Proceedings online no. 19971404 Paper first received 3rd February and in revised form 16th May 1997 A 8. Yakovlev is with Ansoft Corporation, Four Station Square, Suite 660, Pittsburgh, PA 15219, USA A.B. Gnilenko is with the Department of Radiophysics, Dniepropetrovsk State University, Street Gagarin 72, Dniepropetrovsk-625, 320625, Ukraine

shape [l]. The current distributions on the microstrips and the scattering parameters of various junctions have been determined based on the rigorous method of moments solution. The full-wave analysis of shielded passive microstrip components on a two-layer substrate is provided in [2] using the method of moments proce- dure. The finite difference time domain (FDTD) approach has been presented in [3] for the calculation of the frequency-dependent characteristics of microstrip discontinuities. It is shown that the method can be applied in modelling various microwave components for MMIC CAD applications. The FDTD method has been also effectively applied for the characterisation of a right-angle microstrip bend and the mitred microstrip bend [4]. Good agreement between measured and com- puted S-parameters for a right-angle microstrip bend has been demonstrated. The transfinite element method in conjunction with the planar waveguide model has been proposed in [5] for modelling MMIC devices. The applications of the method have been also demon- strated for the analysis of arbitrary shape waveguide discontinuities.

Despite the advantage of accuracy, the 3D full-wave methods and numerical techniques discussed above still remain time consuming for direct usage in CAD pro- gram packages. In this paper, the method of integral equations for overlapping regions (MIEOR) in con- junction with the planar dispersive waveguide model is proposed for the analysis of microstrip discontinuities. The principles introduced in [6] for a waveguide model of the microstrip line have been used in this paper. It is proposed to convert a real microstrip line to a waveguide model with vertical magnetic walls, fre- quency-dependent effective geometrical and material parameters. The criterion of this transformation is the equality of impedances of an original open microstrip line and its planar waveguide model. Using sufficiently faithful empirical formulae [5, 7-91, effective geometri- cal and material parameters of a suitable dispersive waveguide model can be obtained from parameters of an open microstrip transmission line.

The idea of overlapping regions has been introduced in [lo] for the analysis of diffraction nonco-ordinate waveguide problems using a suitable ‘Zwischenme- diu” with known eigenfunctions. Various waveguide discontinuity problems, including junctions of circular and rectangular waveguides, junctions between canoni- cal horns and different types of waveguide bends have been investigated using the idea of a ‘Zwischenmedium’ and presented in [ll]. Schwarz’s iterative method of overlapping regions has been successfully applied in [12, 131 to waveguide discontinuity problems. This

449 IEE Proc.-Microw. Antennas Propag., Vol. 144, No. 6, December 1997

method involves dividing the geometry of a considered structure into several simple overlapping subregions wherein the wave equation may be solved by the sepa- ration of variables leading to the exact analytical solu- tion for the Green’s function [12, 131.

The method of integral equations for overlapping regions (MIEOR) has been introduced in [14] using a different integral formulation for overlapping regions and developed in a number of papers, for example [15, 161. This method exploits the same idea of com- plex structure dividing into simple overlapping subre- gions as the above technique. However, matching of field components on boundaries of common overlap- ping areas is used for uniting field integral formulations for overlapping subregions into the Fredholm’s integral equation of the second kind or a system of integral equations which may be algebraised applying Galer- kin’s method. Galerkin’s method results in obtaining an operator equation of the second kind with a com- pact operator. It is well known that the analytical Fredholm’s alternative theorem is satisfied under this condition and the existence of the bounded inverse operator can be proved. It is shown that the obtained infinite system of linear algebraic equations (SLAE) of the second kind represents the correct mathematical model of the initial boundary problem for the Helm- holtz’s equation which allows the application of numer- ical techniques yielding an accurate good convergent solution.

port 2

port 3

\

port N Fig. 1 Geometry of a generalised microstrip multiport junction

2 General integral equation formulation for a microstrip multiport junction

The geometry of a generalised H-plane multiport junc- tion of microstrip transmission lines is shown in Fig. 1. Assume that all ports are matched and the incident field, generated at infinity at port 1, represents the dominant mode of the microstrip line. Consider the lossless dielectric of the substrate with the relative die- lectric permittivity E,.. Strip width is denoted by a,, i = 1, 2, ..., N, and strip thickness is assumed negligibly small. Reference planes denoted by i- -i separate the region of a discontinuity from uniform semi-infinite microstrip lines. If the incident field is generated at port 1, then the total electric field in the first uniform liqe is de_termined by the incident and reflected fields (E,,, + Elr) , the total electric field in the ith uni5rm line with a matched load is the transmitted field (El t , i

450

= 2, ..., N) and the total electric field zD in the region of a discontinuity represents the dynamic sum of all transmitted and reflected fields concentrated in this region.

port 2

/ sM

/ - areas of overlapping Fig.2 microstrip multiport junction

Geometry of a dispersive waveguide model for a generalised

The geometry of the proposed dispersive waveguide model is presented in Fig. 2. The initial microstrip structure is changed to the multiport junction of waveguides with vertical magnetic walls, frequency- dependent effective dimensions, and effective dielectric permittivities. Using the previously mentioned empiri- cal formulae [5, 7-91, the following transformations have to be carried out for the strip width of the ith waveguide, (i = 1, 2, ..., N):

w! - a; 1 + flf2

w; = a; + where

h c w,o = 1207r- f,“ = 2-

2; a; & with

30.666h 7528

F, = 6 + (27r - 6)e-(?) and effective dielectric permittivity E:#

. 2

where

Here we denoted thickness of the substrate as h, veloc- ity of light as c and width of the ith modelled waveguide as w,.

IEE Proc.-Microw. Antennas Propag., Vol. 144, No. 6, December 1997

For solving the problem of electromagnetic field dif- fraction by a microstrip multiport junction, the rigor- ous numerical-analytical method of integral equations for overlapping regions is applied. The method involves dividing a complex region into simple overlapping sub- regions s, and SD bounded by surfaces S h U oj and Sa U o{, respectively, where S i and Sh are the sur- faces of magnetic walls in the region of discontinuity S D and regions of waveguides S,, as shown in Fig. 2. Coupling apertures between S, and SD are denoted as ~j and Oi.

The field distribution in each region S, (i = 1, 2, ..., N) satisfies a two-dimensional Helmholtz’s equation and Neiman boundary conditions on the boundaries Sh . The boundary-value problem is formulated for the E,-component of the electric field

wheref(x, z ) represents the incident field at port 1, SI, is the Kronecker delta, d/dn is the partial derivative with respect to the outward normal to the surface Si, ki = (2n/A)d~ifi. The boundary-value problem for the Ey)-component of the electric field in the region SD with boundary conditions on the surface of magnetic walls Sfi can be formulated similarly.

Suitably dividing the complex region into overlap- ping subregions allows a fundamental solution of eqn. 3 in the analytical form as the solution of the boundary-value problem for the Green’s function G(i)(x, z; x’, z’) to be obtained.

= -b(z - .’)S(% - % I ) ( 2 , z ; 2’, z’) E si (5)

The Green’s function G@)(x, z ; x’, z’) can be obtained as the solution of eqns. 5 and 6 for the region S, with Neiman boundary conditions on Sfi U 0;. It should be noted that the boundary conditions (eqn. 6) are for- mulated for the closed boundary Sh U of., meanwhile the electric field component E$!)(x, z ) is not determined in an explicit form on the coupling apertures of. .

Applying Green’s theorem to eqns. 3 and 5 and tak- ing into account the boundary conditions (eqns. 4 and 6), a set of integral representations expressing the elec- tric field in subregions S, and SD in terms of analyti- cally obtained Green’s functions can be written in the following form:

q y z , 2) = SlzEznc(J:, .)

(7 )

Assuming the equality of the electric fields associated with overlapping subregions in the common areas of overlapping and matching first partial derivatives of the electric field components on the boundaries of the overlapping areas

the set of integral representations (eqns. 7 and 8) can be coupled into a system of integra1 equations

(11) Eqns. 10 and 11 consists of N + 1 integral equations connected through coupling apertures o{ and oj. The incident field E,,,(x, z) represents the dominant mode of the dispersive waveguide with magnetic walls E,,,(x, z) = T Eoo(x, z). For the certain geometry, which allows dividing the complex region into a number of simple overlapping subregions, the Green’s functions can be determined in terms of the series expansion on a com- plete set of eigenfunctions of the Helmholtz’s operator.

Substituting eqn. 11 into eqn. 10, a set of the Fred- holm’s type integral equations of the second kind is obtained for the electric field component E$!)(x, z ) in the ith uniform region in terms of Green’s functions and the field distribution defined on the coupling aper- tures of and of..

E p ( 4 z ) = Eznc(2, 2 )

(12) where L(i)(x, z; x”, z”) are integral operators

L(i) ( 2 , z; d l , z”)

(13) As a result, the proposed approach leads to the second kind of Fredholm’s integral eqn. 12 with the kernel that involves a product of Green’s functions and/or their derivatives for overlapping regions. The algebraisation procedure of Galerkin’s method applied to eqn. 12 results in an infinite matrix equation of the second kind with a compact operator. Under this condition, a characteristic equation may be truncated and solved numerically yielding correct values of the scattering parameters.

3 T-junction of microstrip lines

The described technique has been applied for the analysis of the scattering parameters of a microstrip line T-junction. Geometry of a real unit and its waveguide model is shown in Fig. 3. It should be noted that a waveguide model demonstrated in Fig. 4 for a

45 1 IEE Proc.-Microw. Antennas Propag., Vol. 144, No. 6, Decemher 1997

microstrip T-junction with a matching stub will be considered later in this paper for the numerical applications. It is assumed that ports 1, 2 and 3 are matched and the dominant mode TEoo(x,z) is generated at the port 1 at z = -CO. Width of modelled waveguides is denoted as 2wl and 2w2, and effective dielectric permittivities are denoted as E$$ and E$), respectively. Dispersive expressions for geometrical parameters and effective dielectric permittivities of the waveguide model defined by eqns. 1 and 2 for a general geometry of a microstrip multiport junction can be easily transformed to the case of T-junction configuration. It should be noted that the representations (eqns. 1 and 2) are not required for the intersecting region. It will be shown that the integral equation formulation is obtained in terms of contour integrals over the coupling apertures. At the same time the coupling apertures represent the internal points of uniform waveguide regions, wherein the eqns. 1 and 2 are applicable.

matching stub -

X

l

I -magnetic walls

-area of overlapping

’ Einc

Geometries of a real microstrip T-junction and a dispersive port 1

Fig.3 waveguide model

X

I magnetic walls

c

area of overlapping

Einc

port 1 Fig.4 junction with a matching s t d

Geometry of a dis ersive waveguide model for a microstvp T-

It is well known that T-junction units play the role of impedance transformers between input and output ports. Therefore, the effective dielectric permittivity for the whole structure may be represented in the form

‘eff = (14)

452

The method of integral equations for overlapping regions allows dividing the complex field determination domain into simple overlapping subregions

s1 : -w1 5 x 5 w1 s2: - ~ < X < c C o < z < 2 w z

--03 < z 5 2w2

Eqns. 10 and 11 adapted to the geometry of the T-junc- tion waveguide model (for i = 1, 2; E$) E E,) form the

gral equations for the E,-component of the electric field in overlapping subregions SI and S,

( 2 , ~ ) = Eznc(s, 2 )

J -WI

Green’s functions Gz(x, z; x’, ‘) have been chosen in the form of a series expansion on a complete set of eigen- functions of the Helmholz’s operator

G 2 ( x , z; x’, z’)

(17) where @i)(x) and @L*)(z) are eigenfunctions of the Helmholz’s operator defined in regions SI and S,, respectively

fi)(z, z’) and fi*)(x, x’) are one-dimensional Green’s functions determined for semi-infinite and infinite regions

f%’ 4 eYc)(z’-2w2) cosh(yg)(z - 2 ~ 2 ) ) for z 2 z’ e ~ ~ ) ( z p 2 w n ) cosh(y2) (z’ - 2 ~ 2 ) ) for z < x’

(19)

with G,*) being the propagation constants in corre- sponding regions S , and S,

72) = 2/(”7r/2W# - k2

2n k = - J - - ‘eff

Solution of the integral equation system (eqns. 15 and 16) is represented in the form of a series expansion on a complete set of eigenfunctions. The total electric field El(x, z ) can be determined as a sum of the incident electric field ELnC(x, z) and series of reflected waves with unknown amplitudes Ak; the total electric field E2(x, z) is defined by a series of transmitted waves propagating

IEE Proc -Mtcrow Antennas Propag , Vol 144 No 6. December 1997

along the x-axis in opposite directions with amplitude coefficients B, and C,

cc ( 1 )

( 2 , ~ ) = &,,(x, z) + & e Y k (a-2wz) 4 k (‘I (4 k=O

Einc(z, 2 ) = 2 4 t ) ( x ) cosh(yF’(z - 2 ~ 2 ) ) (22) The unknown coefficients of the eqn. 22 A,, B, and C, stand for the reflection and transmission coefficients for the dominant and higher-order modes.

Taking into account the obtained representations for Green’s functions and fields in the subregions, the sys- tem of integral equations (eqns. 15 and 16) can be rewritten in the series form

m

k=O 0 0 0 0

m=O s=o

00 0 0 0 0

J -wi

w1 x f i 2 ) (x, ”’)OF) (z’)dx’

Using Galerkin’s method, the system of functional equations (eqns. 23 and 24) can be reduced to a cou- pled set of matrix equations with unknown coefficients Ak, Br and C,. The resulting system of linear algebraic equations of the second kind is obtained for the reflec- tion coefficients A,

s=o s=o k=O

where

x sinh(22~2y$~))yb~)w1

( ( -1 )k + (-l)p)e-Yj2’wl - (1 + ( - l )P+k)eYb2)w1 X ((2WlY!‘))2 + (kn)2)((2w2y, (1) ) 2 + (STl2)

x e-Fy!z)wr e -2w2Yk (1)

An infinite system of linear algebraic equations (eqn. 25) can be truncated and solved numerically to yield values of the reflection coefficients Ap. The trans- mission coefficients Bp and Cp can be found from the matrix form of the functional equation (eqn. 24) using the obtained reflection coefficients Ap.

cz-i. magnetic walls

c_

area of overlapping

t

0 w2 X

Fig.5 sive waveguide model

Geometries of a real mimustrip right-angled bend and its disper-

4 Microstrip right-angled bend

The geometry of a microstrip right-angled bend and a corresponding waveguide model is shown in Fig. 5. It is assumed that ports 1 and 2 are matched and the dominant mode of the waveguide model T Eoo(x, z ) is generated at the port 1 at x = 00. Assume also that the structure is filled with lossless dielectric having fre- quency-dependent effective dielectric permittivity As in the case of a microstrip T-junction, the integral equation formulation for the geometry of a right- angled bend makes it possible to avoid the considera- tion of a waveguide model for the intersecting region.

Following the idea of the method of overlapping regions, the complex structure is represented as the overlapping of two subregions having the common area Q: 0 i x 5 w2 , 0 5 z I wl. Simple overlapping subre- gions can be introduced geometrically as follows:

SI: o < x < m O < % < W l

s2: o < x < w 2 o < z < m The structure is uniform along the y-direction and, therefore, the discontinuity problem on the scattering matrix can be solved for the E,-component (Et ) = E,; i = 1, 2) of the electric field. Based on the proposed above technique, a system of integral equations in over- lapping subregions s, can be obtained in the form

El(%, z ) = G n c ( 2 , X )

IEE Proc.-Microw. Antennas Propag., Vol. 144, No. 6, December 1997

00

(27) Green’s functions GL(x, z; x’, z’) can be represented in the same manner as given in Section 3

Gz(x , z; x’, z’) 03 ( c $2) ( z )$2 ) (z’)f:) (z, x’) for i = 1

(28) where $ i ) ( z ) and q9A2)(x) are eigenfunctions of the Helmholz’s operator defined in regions SI and S,, respectively

fL:)(x, x’) and fi2)(z, z’) are one-dimensional Green’s functions determined for semi-infinite regions

where $1 and d2) are the propagation constants of the dominant and higher-order modes in regions SI and S,

7:) = d(rn7r/~1)~ - k2

72) = J ( n . i r / ~ a ) ~ - k2 (32) The solution of the system of integral equations (eqns. 26 and 27) can be found in terms of a modal expan- sion. The total electric field EI(x, z ) is determined by the incident electric field EinC(x, z) and a series of reflected waves with amplitudes A,; the total electric field E,(x, z ) is defined by a series of transmitted waves with amplitudes B,

M

k=O 00

s=o

The unknown coefficients Ak and B, represent the reflection and transmission coefficients for the domi- nant and higher-order modes.

The system of integral equations (eqns. 26 and 27) can be rewritten in a series form using expressions for the Green’s functions and the modal expansion repre- sentations determined above

M

k=O m=O s=o

(34) 454

s=o 03 0 0 0 3

n=O n = O k=O ( 3 5 )

where

J o Finally, the coupled set of functional equations (eqns. 34 and 35) has been reduced to a system of linear alge- braic equations of the second kind with the unknown reflection coefficients Ap

M

where

The above infinite system of linear algebraic equations can be solved numerically. The roots of the eqn. 36 are approximated by the roots of a truncated system of lin- ear algebraic equations. The transmission coefficients B, can be obtained from eqn. 35 using known reflection coefficients A,,.

5 Numerical results and discussion

A computation of infinite SLAEs obtained as a result of the application of Galerkin’s method can be carried out only after the truncation of infinite systems of equations. Actually, a procedure of truncation may lead to a ‘relative’ convergence or even to the absence of convergence for approximate solutions. This drawback is mainly caused by convolution-type equations involved in computational algorithms. That is why the regularisation methods, which allow the application of the general theory of operator equations for the substantiation of the truncation technique, have

IEE Puoc.-Microw. Antennas Propag., Vol. 144, No. 6, December 1997

to be used for the transformation of boundary problems for the Helmholtz equation to infinite SLAEs.

Infinite systems of linear algebraic equations (eqns. 25 and 36) obtained in Sections 3 and 4 may be written in the generalised matrix form

where I is the unit matrix, D = IIdpkj/pyjc=c=l is a matrix of coefficients of the characteristic eqns. 25 and 36, A = {ak}rzl is a vector of the unknown amplitude coeffi- cients in the field series expansions (reflection coeffi- cients), B = { b p } p i is a vector of right parts.

Using asymptotic evaluations of field components from [17], one can obtain for amplitude coefficients ak = O(k-5’3) and therefore A = { a k } c el, where the space of infinite sequences e’ is defined as follows:

(I -D)A z= B (37)

li=l

It should be noted that in the space of el, the finite energy condition in any limited volume is provided for the amplitude coefficients of field series expansions.

From eqns. 25 and 36, the evaluation for elements of the right part vector of eqn. 37 is derived in the follow- ing form:

Hence, for the vector of right parts, we obtain B = {b,};=! c c el+€, where E > 0 is an infinitesimal quantity.

The asymptotic behaviour of elements of the charac- teristic matrix D = lldpkll,S;cz1 for p , k >> 1 follows evaluation

M

with a, p > 0 being real parameters which depend on the geometry of the problem only.

1 t 1

1 0.3818

0.3810 0 10 20 30 40

N Fi .6 M a nitude of the reflection coefficient as a function o the number oj”%gheu-orc&’modes N at port I , for“dij$erent numbers oj&,,her-order modes M at ports 2 and 3 ___ M = 5 . . . . . . . . M = 10 - - - M = 1 5 - ~ M = 20 - . ~ . M = 25

In [16], it has been shown that the infinite matrix D - ( ( $ k / / p l defines w-complete continuous operator D: ti -+ e +&. The class of compact operators contains that of o-complete continuous operators. Hence, eqn. 37 is uniquely solvable in the given space of infinite

-

IEE Proc.-Microw. Antennas Propag., Vol. 144, No. 6 , December 1997

sequences in which the existence of the bounded inverse operator (I - D)-’ may be substantiated based on the Fredholm alternative theorem. Therefore, eqn. 37 is the correct mathematical model of the initial boundary problem (eqn. 3) and the scattering parameters of the microstrip junctions may be approximated by roots of the truncated equations (eqns. 25 and 36).

Correctness of the proposed mathematical algorithm has also been verified numerically studying convergence of the magnitude of the reflection coefficient for the example of a basic T-junction in the dominant mode propagation regime. Fig. 6 shows changes in the reflection coefficient magnitude against the number of terms N in truncated matrix blocks for different number of terms M of the internal summation. It should be noted that the number of terms N and M corresponds to the number of higher-order modes at port 1, and ports 2 and 3, respectively. We assume here that the dominant mode is generated at infinity at port 1. It is observed that for presented parameters of series truncation stability in the third decimal place can be obtained.

-2 t

0 1 2 10 frequency, Hz i x 10 1

Fig. 7 fie uency-dependent characteristics for the scattering matrix parameters of a basic microstrip T-junction Comparison with results by Wu et al. [IS], h = 0.635mm, 2 w l = 2wz = 0.6096mm, er = 9.9 - - - lSlll theory, __ ISzll theory, A IS1,/ Wu er al. [lS], + lSz11 Wu et al. [IX]

frequency,Hz (xlo’o) Fig. 8 Magnitude of the reflection and transmission coefficients against frequency jor a basic microstrip T-junction Comparison with results by Mehran [19], h = O.h35mm, 2w1 = 2w2 = 0.56mm,

Mehran [I91 E~ = 9.7 - - - lS,,I theory, __ Is211 theory, A I S H Mehran [W, + Is211

455

Numerical results for the dominant mode reflection and transmission coefficients of a basic T-junction have been compared with those obtained by a full-wave spectral-domain approach in conjunction with the moment method 1181 (Fig. 7) and the waveguide model approach used in [19] (Fig. 8). Excellent agreement is obtained in the low-frequency range, specifically up to lSGHz, with results in [l8] and 20GHz for data in [19]. Obviously, dispersive formulae (eqns. 1 and 2) for effective dimensions and effective dielectric permittivity used in the proposed waveguide model cannot ade- quately reflect the significant contribution of higher- order modes in a higher frequency range.

-15.0

m Q- -20.0

5 -25.0 E

(U U 7 Y .-

Y

-30.0

-35.0

0 1 2 3 10 frequency,Hz ( x 1 0 1

Frequency-dependent magnitude of the ref2ection coefficient for a

I.Omm, ~ - - - e = 1.5"

Fig. 9 microstrip T-junction with a matching stub C h = 0.635mm, 2 w l = 2w2 = 0.56mm, Er = 9.7 ~ e = 0.5mm, - - ~ e -

-

-

-

-

o.8 t

0 1 2 3 frequency , Hz ( x 7 O'oJ

Frequency-dependent magnitude of the transmission coeficcient Fig, 10 for a microstrip T-junction with a matching stub C h = 0.635mn1, 2w, = 2w2 = 0.56mm, E, = 9 7 __ e = OS", - - - e = l.Omm, ~ - - - e = 1.5"

Numerical results for the scattering matrix parame- ters (magnitude of the reflection and transmission coef- ficients) shown in Figs. 9 and 10 are obtained for a microstrip T-junction with a matching stub 4. The geometry of the corresponding waveguide model is depicted in Fig. 4. It is observed that the narrow-bend matching stub can be used to control the reflection and transmission coefficients. Particularly, an increase of the matching stub length results in obtaining the mini- muin of the reflection coefficient at lower frequencies.

456

Fig. 11 demonstrates results for the magnitude of the reflection and transmission coefficients for a T-j unction waveguide model with different widths of outgoing cir- cuits. It is shown that a change in strip width leads to a change of locations of minima and maxima of SI1 and S,, coefficients, reupectively. The minimum of the reflection coefficient at lower frequencies at port 1 can be obtained by increasing the width of outgoing ports 2 and 3.

0 1 2 3 L

Frequency-dependent S parameters for a nonuniform microstrip frequency, HZ ( x ldo )

Fig. 1 1 T-junction h = 0.65mm, 2wl = O.hOXnim, __ ISlIl ~ - - ISZll

= 10.1

40.01 . ' ' ' ' I . 8 , I 0 2 4 6 a 10

frequency, HZ ( x '10 '1 Fig. 12 Magnitude of the reflection coeff ient against frequency for the 90" microstrip bend Comparison with results by Harms and Mittra [20], h = 0.7874mm, w1 = w2 = 2 4mm, E, = 2 2 __theory 0-0 Harms and Mittra [20]

Comparative analysis of the presented technique with an equivalent circuit model [20] has been carried out for the example of a microstrip right-angled bend in the frequency range 1-9 GHz. The results are obtained for the magnitude of the reflection coefficient and shown in Fig. 12. Figs. 13 and 14 show dispersion behaviour of the reflection and transmission coeffi- cients for uniform and nonuniform 90" microstrip bends. It is observed that at low frequencies the mini- mum of the reflection coefficient can be obtained only for a uniform bend. The lower reflection coefficient in comparison with a uniform bend can be obtained at higher frequencies, decreasing strip width in the outgo- ing circuit.

IEE Proc -Microw Antennas Propug, Vol 144, No 6, December 1997

0

-10

%

5 -20

ai U 3

8 E T w- m

-30

-40 1 2 frequency,Hz (~10”)

Fig. 13

h = 0.1814mm, w1 = 1.2mm, 6, = 2.2 ~ w2 = 1.2mm, 0-0 w2 = 0.6mm, 0-0 w2 = 0.3”

Frequency-dependent magnitude o the re ection corficient for uniform and nonuniform microstrip right-ang ed ben 1.’

0.00 -

t -1.00 1

t \ I U

1 0 1 2 3

10 frequency,Hz [ x10 1 Fig. 14 Frequency-dkdent magnitude -of the transmission coefficient for uniform and nonuni orm microstrp right angled bends h = 0.7874mm, w, = 1.2mm, er = 2.2 ~ w2 = 1.2mm, 0-0 w2 = 0.6mm, 0-0 w2 = 0.3“

6 Conclusions

The method of integral equations for overlapping regions in conjunction with a dispersive waveguide model is presented as an efficient approach for the analysis of microstrip discontinuity problems. It is shown that the presented technique gives a correct mathematical model which leads to a matrix equation of the second kind with a compact operator. The existence of the bounded inverse operator and approximation of the solution by roots of the truncated matrix equation have been proved. Numerical results are shown for uniform and nonuniform microstrip T- junctions and right-angled bends and compared with results published by other authors. Advantages of the

presented technique can be emphasised as follows: (a) correctness of the obtained mathematical model proved by the use of concepts of functional analysis and numerically studied convergence of the algorithm (b) extremely low computational effort (c) applications to various types of microstrip disconti- nuity problems.

7 References

1 HORNG, T.-S., ALEXOPOULOS. N.G., WU, S.-C., and YANG, H.-Y.: ‘Fullwave spectral domain analysis for open microstrip discontinuities of arbitrary shape including radiation and surface-wave losses’, Int. J. of MIMICAE, 1992, 2, (4), pp. 224-240

2 HILL, A., and TRIPATHI, V.K.: ‘An efficient algorithm for the three-dimensional analysis of passive microstrip components and discontinuities for microwave and millimeter-wave integrated cir- cuits’, IEEE Trans., 1991, M‘IT-3, (l), pp. 83-91

3 ZHANG. X.. and MEI. K.K.: ‘Time-domain finite difference approach to the calculation of the frequency-dependent character- istics of microstrip discontinuities’, IEEE Trans., 1988, MlT-36, (12), pp. 1775-1787

4 FEIX. N.. LALANDE. M.. and JECKO. B.: ‘Harmonica1 char- acterization of a microstrip bend via the finite difference time domain method’, IEEE Trans., 1992, MTI“, (5), pp. 955-961 CENDES, Z.J., and LEE, J.F.: ‘The transfinite element method for modeling MMIC devices’, IEEE Trans., 1988, MTI-3, (12),

6 WOLFF, I., KOMPA, G., and MEHRAN, R.: ‘Calculation method for microstrip discontinuities and T-junctions’, Electron. Lett., 1972, 8, (7), pp. 177-179 MENZEL, W., and WOLFF, I.: ‘A method for calculating the frequency dependent properties of microstrip discontinuities’, IEEE Trans., 1977, MIT-2.5, pp. 107-1 12 YAMASHITA, E., ATSUKI, K., and UEDA, T.: ‘An approxi- mate dispersion formula of microstrip lines for computer-aided design of microwave integrated circuits’, IEEE Trans., 1979, MTT-27, pp. 1036-1038

9 GUPTA, K.C., GARG, R., and BAHL, I.J.: ‘Microstrip lines and slotlines’ (Artech House, Norwood, MA, 1979)

10 PIEFKE, G.: ‘The tridimensional ‘Zwischenmedium’ in the field theory’, Arch. Electr. Uebertrag., 1970, 24, pp. 523

11 PIEFKE, G.: ‘Feldtheorie’ (B. I.- Wissenschaftsverlag, 1973) 12 ISKANDER, M.F., and HAMID, M.A.K.: ‘Scattering coeffi-

cients at a waveguide-horn junction‘, Proc. IEE, 1976, pp. 123- 127

13 ISKANDER, M.F., and HAMID, M.A.K.: ‘Iterative solutions of waveguide discontinuity problems’, IEEE Trans., 1977, MlT-25,

14 PROKHODA, I.G., and CHUMACHENKO, V.P.: ‘The method of partial overlapping regions to investigation of the complicated form waveguide-resonator systems’, Izv. Vuzov. Radiophysics, 1973, 16, pp. 1578-1581 (in Russian)

15 PETRUSENKO, I.V., YAKOVLEV, A.B., and GNILENKO, A.B.: ‘Method of partial overlapping regions for the analysis of diffraction problems’, IEE Proc. Microw. Antennas Propag., June 1994, 141, (3), pp. 196-198

16 GNILENKO, A.B., YAKOVLEV, A.B., and PETRUSENKO, I.V.: ‘A generalized approach to the modeling of shielded printed- circuit transmission lines’, IEE Proc. Microw. Antennas Propug., April 1997, 144, (2), pp. 103-110

17 MITTRA, R., and LEE, S.W.: ‘Analytical techniques in the the- ory of guided waves’ (Macmillan, 19’71)

18 WU, S.-C., YANG, H.-Y., ALEXOPOULOS, N.G., and WOLFF, I.: ‘A rigorous dispersive characterization of microstrip cross and T junctions’, IEEE Trans., 1990, M’IT-3, pp. 1837- 1844

19 MEHRAN, R.: ‘The frequency-dependent scattering matrix of microstrip right-angle bends, T-junctions and crossings’, Arch. Electr. Uebertrag., 1975, 29, pp. 454-460

20 HARMS, P.H., and MITTRA, R.: ‘Equivalent circuits for multi- conductor microstrip bend discontinuities’, IEEE Trans., 1993, M T 4 1 , pp. 62-69

5

pp. 1639-1649

7

8

pp. 763-768

IEE Proc.-Microcv. Antennus Propug., Vol. 144, No. 6 , December 1997 451