full wave solution of dielectric loaded waveguides and shielded microstrips utilizing finite...

Full Wave Solution of Dielectric Loaded Waveguides and Shielded Microstrips Utilizing Finite Difference and the Conjugate Gradient Method

V. Narayanan, M. Manela, R. K. Lade, and T. K. Sarkar Department of Electrical and Computer Engineering, Syracuse University, Syracuse, New York 13244-1 240 Received April 3, 1990, retmrsed February 18, 1991

ABSTRACT

Dynamic analysis of waveguide structures containing dielectric and metal strips is presented. The analysis utilizes a finite difference frequency domain procedure to reduce the problem to a symmetric matrix eigenvalue problem. Since the matrix is also sparse, the eigenvalue problem can be solved quickly and efficiently using the conjugate gradient method resulting in considerable savings in computer storage and time. Comparison is made with the analytical solution for the loaded dielectric waveguide case. For the microstrip case, we get both waveguide modes and quasi-TEM modes. The quasi-TEM modes in the limit of zero frequency are checked with the static analysis which also uses finite difference. Some of the quasi-TEM modes are spurious. This article describes their origin and discusses how to eliminate them. Numerical results are presented to illustrate the principles.

1. INTRODUCTION

Dielectric substrates, dielectric loaded waveguides, and microstrip structures have become an essential part of modern technology for a vast band of frequencies. from dc to millimeter waves. A comprehensive analysis of such complex structures is very important. This article is intended as a step toward such an analysis.

In our experience, one may incorporate, in the standard finite difference procedure [ 1-41. some features which increase the flexibility of the method and make it efficient and convenient.

We apply the method to a variety of systems in order to compute the propagation constants and the fields of the propagating modes. In this process, we encounter the problem of spurious modes, which occurs for both finite difference and finite elements techniques. We find that these

modes are inherent in the way the boundary conditions in the two dielectric surfaces are imple- mented. Consistently, for homogeneous systems, we do not get spurious modes. Knowing the origin of those nonphysical modes, one can now devise a procedure to identify and discard them. The procedure is described in section 2.

In ref. [4], for example, it has been pointed out that, for a inhomogeneously filled waveguide system, taking the longitudinal fields as the unknowns gives rise to spurious modes, whereas, utilizing the transverse components do not give rise to spurious modes, and numerical results have been presented to demonstrate this point. N o theoretical proof is available why it should be so.

In this article, we choose the node points for the finite difference method such that they do not coincide with the metal waveguide boundary as shown in Figure 1. With such a mesh definition,

International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering. Vol. 1, No. 4. 346-357 (1991) 0 1991 John Wiley & Sons. Inc. CCC 1050- 1827/91/040346-12$04.00

346

Full Wave Solution 347

9 and for QTE we have

Figure 1.

we do not numerically obtain any spurious solutions utilizing the longitudinal components of the fields as unknowns. A systematic theoretical study is in order to determine the origin of spurious modes. However, we do obtain spurious solutions when metal conductors are introduced in the waveguides. Unfortunately, this situation has not been dealt with in ref. [4]. There appears to be a lack of sound theoretical proof on the origin and elimination of spurious modes in the published literature-this paper is no exception. Here, a heuristic method, based on the continuity of the boundary conditions of the various field components, is utilized to eliminate spurious modes.

2. FORMULATION OF THE PROBLEM

We are interested in guided waves along the z - direction. So we assume field behavior of exp{j

In a homogeneous source-free region, the fields can be expressed as the sum of TM and TE to z fields [5]. These two fields are derived from two scalar potentials satisfying two-dimensional (2D) Helmholtz equation.

(of - Pz)).

(V: + k2 - p’)Q I T M = 0 (1) TE

for +TM we have [3]:

a* ax

Hy = -- (2)

1 a2* H, = T - a* aY z axaz

E = --

1 a’* v ax 2 ayaz

H , = -- a* E = -

where

P = ~ O E and Z = jok

( 3 )

(4)

When the system contains only one dielectric, these two potentials, JITM and +TE, are not coupled. When we have a surface between dielectrics, the boundary conditions provide coupling between them, unless there is no variation along the surface or when P = 0.

So, it is advisable to work with two potentials having the same units, and we define:

6% 9 = - E E , P

in addition

(7)

where k, is the free space wave number and then

i-

Both the potentials have units of A l m , and satisfy eq. (1). Defining

K j = E,p.,k~ - P2 (10)

(i runs on all the dielectrics involved)

we transfer the problem into finding the eigen- spectrum of the vf operator.

348 Naravanan et a / .

Since, in every medium. we have different eigenvalues, we multiply the equation of every medium by T, which is defined as

and obtain

where

Mathematically. we are looking for the eigen- spectrum of the operator --,V2.

We represent the Laplacian by the five points difference formula which is accurate to h', when h is the distance between mesh points. To calculate the Laplacian at a point P . we use Taylor

I .6

1.2

-

0.E

0 4

a = 0 9 b = 2 0 d = 0 2 ,' = 1 0 0

= 2 4 5 b 1 1

r 2

c , 3 = l o o

expansion around the symmetric points to get a set of linear equations for the different order de- rivatives. Figure I describes the set-up.

For the metal boundaries, we use a grid which does not coincide with the boundaries, but is shifted by 4h from them. The points outside the region are considered to be a mirror image of the interior points. so there is equality for the + and negative equality for 8. For every operator, using the needed number of imaginary points outside the boundary yields solutions with appropriate accuracy. This can be shown using Taylor expansion. The shifted mesh serves two objectives. It slightly increases the accuracy of the boundary conditions and, mainly. it enables us to avoid the problem of singularities of the fields at inner corners and edges.

In order to get a solution, we must impose the boundary conditions. We have six components of fields. They all can be generated by the two potentials, plus two more boundary conditions, de-

[ 31 x Results from Horrington

a/Xo Figure 2.


fined by

(where B is the magnetic flux density, and the directions x and y are shown in Fig. 1) at a dielectric interface. By utilizing eqs. (15), (2), and (3) we obtain

and from eqs. (16), (2), and (3)

Concentrating on the nonmagnetic case, and using the idea of virtual points introduced by Col- lins and Daly 141, one can translate the boundary conditions into second-order finite difference form as

where

The superscript, V , stands for a virtual point; n , s, w , and e , are defined in Figure 1. Following ref. [4], we combine the boundary conditions and

0 0.2

a . 2 0 dl = 0 4 5 b i. 0 9 d2 z 0 2 3 1 5

63 i 01115

l b l

0.4 0.6 0.8 1.0 a 1x0

Figure 3.

350 Narayanan et a / .

0 4

0 -

wave equations on the surface to form a new version of the discrete wave equation as:

1 1 2 A+,, = R(a i - a,)8, - - R(al - a?)O,,

+ d)p(l + R2)(?] + 7')

- (C)

-

1 I 1 I I I I I I I

and

+ (71 t 7:)R+, + (2(1 + R?)a, (23)

where

A = k $ ht (24)

Using a mesh of N points, we get a vector of 2N points for the two potentials.

Some of the solutions are false, since they satisfy the sum of eqs. (1). (22 ) , and (23 ) , but fail to satisfy each of the three equations individually. So, we identify the spurious modes by individually checking eqs. (22 ) and (23) on the boundaries between the dielectrics. For example, in ref. [4]. i t is claimed that spurious modes exist when the longitudinal field components are taken as unknowns. However, if we choose the shifted mesh on the external metal boundaries, then we do not have any spurious solutions. However, we do obtain spurious solutions when metal conductors are inserted in the waveguide. This case has not been dealt with in ref. [4].

The discrete wave equations are applied to nodes. If the nodes are adjacent to a metallic wall, then different boundary equations have to be con-


sidered. This results in 2N linear equations. In matrix form, we then have a matrix eigenvalue problem:

A X = AX (25)

where A is a 2N x 2N (generally nonsymmetric) matrix, and X is a vector of length 2N where the first N terms are . . . +N, and the last N terms are -ql . . . , -qN. Although the matrix A is nonsymmetric, the conjugate gradient method is not eliminated as a solution technique. This is because some modification of A can be done so that a new matrix, P , which is symmetric, is developed. The eigenvalues and eigenvectors of A can then be determined from those of P as follows.

Because of certain properties of the equations used to develop the matrix A , a diagonal positive definite matrix B can always be found such that S = BA is symmetric. Therefore, premultiplying both sides of eq. (25) by B gives:

BAX = SX = ABX (26)

It can be shown by simple matrix manipulations that the matrix:

where

B = D2 (28)

has the same eigenvalues as the original matrix A . Also, the eigenvectors of A , X I , are related to the eigenvectors of P , Z,,

Xi = D-*Z, (i = 1, 2, . . . , 2 N ) (29)

Note that D is invertable (D-’ exists) because B is positive definite. Again, because of properties in the finite difference equations, the matrix P is always symmetric. Therefore, the conjugate gradient method can be applied to find the eigenvalues and eigenvectors of P and, subsequently, A .

- 8 1 w 8 2 w 8 1

a = 2.0 81 = 0.5 I I 0.047 b I 0.9 82 I 0.6 d = 2.45 w I 0.2

( d )

I I 1 1 I I I I I I 0 0.2 0.4 0.6 0.8 1.0

0 1x0

Figure 5.

352 Naravanan er d.

0 .

3. THE CONJUGATE GRADIENT METHOD

c = 0 4 5 I = 0 0 4 7 ( e J

-

1 I 1, I 1 I 1 I 1 I

In this section, the method of conjugate gradi- ents as applied in this application is discussed. The conjugate gradient method (CGM) is used to iteratively find the minimum eigenvalues and corresponding eigenvectors of an operator equation

A X = AX (30)

where, in the general case. A is a positive semi- definite operator. In the application presented here. the condition that A be a symmetric matrix ( A = A T ) is sufficient to allow application of the CGM. In order to solve for the minimum eigenvalue of the system (25). the functional

F( X ) = ( A X ; X ) = X'AX (31)

is formed subject to the constraint

( X ; x) = X'X = 1. (32)

Letting Y denote the normalized form of X as

Y = "x; x) = X / . \ / ( X X ) . (33 )

The minimum of F( Y ) is the minimum eigenvalue, i.e.,

F(Y)rn,n = ( A Y ; Y)rn ,n

= ( A X ; X ) l ( X ; x)irn,,

= (AX; x ) I ( X ; X )

- X,,"

(34)

-

To minimize F ( Y ) , start with an initial guess X ' (0) and then orthogonalize it to all the eigen-

I I r3 I 2

a /A,

Figure 6.

functions (E , , i = 1, 2, . . . , n ) which have pre- viously been found, i.e.,

n

X ( 0 ) = X‘(0) - 2 (Ei, X’(O))E, (35) i = 1

0

or n

X ( 0 ) = X’(0) - 2 {X’(0)TEi}Ei (36)

The reasoning here is that since each of the eigenvectors are orthogonal, keeping the current values of Xorthogonal to all previous eigenvectors keeps the process from converging to eigenvalues which have already been found.

Next the first approximation of the minimum eigenvalues is found as

i = I

d l = 0 4 5 W = 0 2 I = 0 0 4 7

-

1 I I 1 I I 1 I 1 1

h(0) = F[Y(0)] = (AY(0); Y(0)) (37)

= X ( 0 ) ‘AX(0) / X ( 0 ) ‘X(0)


and the residual is calculated as

R(0) = h(O)X(O) - A X ( 0 ) . (38)

By selecting the initial direction vector as

P(0) = R(0) (39)

the successive approximation to Xis developed as

X ( k + 1) = X ( k ) + t ( k ) P ( k ) (40)

Here, t ( k ) is chosen to approach the minimum of F [ X ( k + l)] in the direction of P ( k ) by setting the derivative of F [ X ( k + l)] with respect to t ( k ) equal to zero and solving for t ( k ) . The result is given by

t ( k ) = { - B + V(B2 - 4 C D ) } / 2 D (41)

354 NaraJlanan et aI.

It can be shown that {B2 - 4CD) L 0, which shows that eq. (50) does indeed hold, and that the functional F is minimized at each iteration.

This new approximation X ( k + 1) is again or- thogonalized to all previous eigenvectors, and then is used to calculate to the next approximation to the minimum eigenvalue as

X(k + 1) = (AX(k + 1); (51)

X ( k + l ) ) / ( X ( k + 1); X ( k + 1)).

Next, the updated residual is calculated as

R ' ( k + 1) = h(k + 1) (52)

X ( k + 1) - A X ( k + 1)

and if the two convergence conditions

F [ Y ( k + l ) ] - F [ Y ( k ) ] = - f ( k ) W { B ' [h(k + 1) - h ( k ) ] / h ( k + 1) (53)

5 tolerance 1 (50)

- 4 C D } / ( X ( k f 1); X ( k + I ) ) 5 0

a = 0 9 b = 2 0 d = 0 2 w = 0 2 s 2 0 5 I = O ?

I = 1 0 0

I - 2 4 5

I - 1 0 0 13- ::- f l

a 1x0 Figure 8.


and

IIR’(k + 1)11 5 tolerance 2 (54)

are not met, the iterations are continued. From experience, these two conditions ensure an accurate result if the tolerances are chosen well.

Next, a new search direction P ( k + 1) is selected. It is given by

P ( k + 1) = R ( k + 1) + q(k)P(k) (55 )

where R(k + 1) is the orthogonaiized version of R‘(k + l), i.e.,

R(k + 1) = R ’ ( k + 1)

I .6

I .2

- P 0.8

0.4

0

r-----l

~

r 3

a = 0 9 b = 2 0

w = 0 2 _i_ I = 0 5 I = 0.2

b T I , ,= 100

I d : 0 2

Again, this othogonaiization is performed to keep X ( k + 1) orthogonal to all previous eigenvectors. In eq. ( 5 5 ) , q ( k ) is selected such that P ( k + 1) is conjugate to P ( k ) with respect to some weight- ing matrix H such that

( P ( k + 1); H P ( k ) ) = 0. (57)

Next, one obtains

R ( k + 1) = R ( k ) ( 5 8 )

- i i [ X ( k ) - X ( k + l)]

Combining eqs. (74), (76), and (77) results in

. . ‘r; 2 4 5

-

-

1 I I 1 1 0.2 0.4 0.6 0.8 I .o

a 1x0 Figure 9.

where

r ( k ) = R ( k ) / ( R ( k ) : R ( k ) ) . (60)

With this new search direction, defined by eq. (59). eq. (44) is used to update the eigenvalue and eigenvector. When conditions (57) and (58) have been met, the iterations are stopped and the current eigenvalue and current eigenvector X are the des I re d so I u t i o n 5 .

4. NUMERICAL RESULTS

We now present some selected numerical results. All the dimensions are hereby expressed in cen- timeters. Spurious modes were sorted out by individually checking each of the boundary conditions and are not presented. There are a few spurious modes for every case. Most of them are characterized by being very “flat .*’ almost straight lines, parallel to the frequency axis.

For the first example, we consider a waveguide with a centered dielectric slab. We solve it using the finite difference method. and analytically. following ref. [ 3 ] . We use this case as a measure for our accuracy. Figure 2 gives the configuration and the dispersion curves. Square mesh has been utilized with 17 = 0.1, t i = 90. The analytical solution has been marked by points, whereas the numerical solution is plotted. As the frequency (alh, ,) and the order of mode increase, the accuracy deteri- orates. For the first mode. the difference is less than 0 .2% at cut otf and reaches 1% for a l h - 0.6.

The second example is a waveguide with three dielectrics. We use rectangular mesh. r = 9/19. tz = 190, for obtaining the results of Figure 3.

Next. we insert a metal strip at the middle of a dielectric loaded waveguide. The strip modifies the waveguide modes and generates one “quasi- TEM“ mode. Figure 4 shows the mode structures with square mesh. h = 0.1. IZ = 90. In the limit of zero frequency. the “quasi-TEM” mode coin- cides with static analysis. We solve the Laplace equation and get p = 1.498 for the static solution. The dynamic analysis yields. as the frequency goes to zero. p = 1.500.

The next step was to insert two metal strips side by side. Figure 5 shows the configuration. the first two waveguide modes, and two “quasi-TEM” modes, even and odd. Both “quasi-TEM” modes start at the same point, p = 2.292. The even and

odd modes can be distinguished from the field configuration. That complies with the static analysis which makes p = 2.291. We use rectangular mesh with R = 9/19. n = 190.

We later arrange the strips one above the other. The results are given in Figure 6. The mesh is the same as for the previous case. We get the static velocities and p. The static analysis yields p, = - 1.339. p? = 1.738, - and the dynamic analysis gives

We next consider a waveguide with three metal strips in two levels. The static analysis give p, = 1.345. B2 = 1.355, and p3 = 1.725. For the dynamic analysis, we have p, = 1.343, = 1.361, and p1 = 1.726. The dispersion relations and con- figurations are given in Figure 7. The mesh is rectangular R = E, IZ = 380.

Finally. we consider the unilateral and bilateral lines of Figures 8 and 9. Comparison with published results show reasonable agreement.

fiI = 1.351, f i 2 = 1.741.

5. CONCLUSIONS

We have presented a dynamic finite difference approach for analysis of dielectric loaded waveguides and shielded microstrips. The main contribution of this work has been the detection and elimination of spurious modes. We explain their origin and accordingly establish a simple procedure which clearly differentiates between a physical and spurious mode. This procedure is supported by the results of a static analysis, also carried out by finite difference.

ACKNOWLEDGMENTS

This work was supported in part by E. I . duPont de Neniours & Company.

REFERENCES

1.

7 -.

H. J . Beaubien and A. Wexler, “Unequal arm finite difference operators,” IEEE Trans. Microwave The- ory Tech., Vol. MTT-18, December 1970. D. G. Corr and J . B. Davies, “Computer analysis of the fundamental and higher order modes in single and coupled microstrips, IEEE Trans. Microwave Theory Tech., Vol. MTT-20, October 1972.

Fir11 Wave Solution 357

3. J. B . Collins and P. Daly, “Calculations for guided electromagnetic waves using finite difference meth-

380.

ference analysis of rectangular dielectric waveguide structures,” IEEE Trans. MTT, Vol. 34, November

5. R. F. Harrington, Time Harrnonic Electromagnetic ods.” J . Electronic Control, Vol. 14, 1963, pp. 361- 1986, pp, 1104-1114.

4.- K. Bierwirth, N. Schulz, and F. Arndt . “Finite dif- Fields, McGraw-Hill, New York, 1962.

BIOGRAPHY

Viswanathan Narayanan was born in Ban- galore, India on December 14, 1965. He received the BE degree in Electronics and Communications from B.M.S. College of Engineering, Bangalore, in 1988.

He joined the Department of Electri- cal Engineering at Syracuse University for his graduate studies in 1989 where he is currently a research assistant. His research

interests are in microwave measurements, numerical electromagnetics, and signal processing.

Tapan K. Sarkar (Sf69-M’76-SM’X1) was born in Calcutta. India, on August 2, 1948. He received the BTech degree from the Indian Institute of Technology, Kharag- pur, India, in 1969, the MScE degree from the University of New Brunswick, Fred- ericton, Canada, in 1971. and the MS and PhD degrees from Syracuse University. Syracuse, NY, in 1975.

From 1975-1976 he was with the TACO Division of the General Instruments Corporation. He was with the Rochester Institute of Technology (Rochester, NY) from 1976-1985. He was a Research Fellow at the Gordon Mckay Laboratory, Harvard University, Cambridge, MA, from 1977 to 1978. He is now a Professor in the Department of Electrical and Com-

puter Engineering, Syracuse University. His current research interests deal with numerical solutions of operator equations arising in electromagnetics and signal processing with application to system design. He obtained one of the “best solution” awards in May 1977 at the Rome Air Development Center (RADC) Spectral Estimation Workshop. He has authored or coauthored more than 154 journal articles and conference pa- pers, and has written chapters in eight books.

Dr. Sarkar is a registered professional engineer in the state of New York. He received the Best Paper Award of the IEEE Transactions on Electromagnetic Compatibility in 1979. He was an Associate Editor for feature articles of the l E E E A n - tennas arid Propagation Sociefy Newsletter, and was the Tech- nical Program Chairman for the 1988 IEEE Antennas and Propagation Society International Symposium and URSI Ra- dio Science Meeting and an Associate Editor of the IEEE Transactions of Electromagnetic Compatibility. He is an As- sociate Editor of the Journal of Electromagnetic W a w s and Applications. He has been appointed U.S. Research Council Representative to many URSI general assemblies. He is the Chairman of the Intercommission Working Group of Inter- national URSI on Time Domain Metrology. He is a member of Sigma Xi and International Union of Radio Science Com- missions A and B.

Biographies and photos are not available for M. Manela and R. K. Lade.

full wave solution of dielectric loaded waveguides and shielded microstrips utilizing finite...

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