AN EXACT METHOD FOR ANALYZING BOXED MICROSTRIPS ON LOSSY SUBSTRATES Ricardo Marques and Manuel Horno Department of Electronic and Electromagnetism Faculty of Physics University of Seville Av Reina Mercedes s / n 41 01 2-Seville Spain

KEY TERMS Microstrip lines, lossy substrares, Wiener- Hopf techniques, modtjed residue calculus techniques

ABSTRACT A n exuct method for the hybrid mode analysis of shielded microstrip lines on anisotropic and lossy dielectric substrates is presented. The method is based on the MRCT and other modified Wiener-Hopf techniques. Results are given in tabular and graphic form.

1. INTRODUCTION

The modified residue calculus technique (MRCT) [l] and other modified Wiener-Hopf techniques have been applied successfully to the analysis of microstrip lines via a quasi- static approach [2] or by a full wave analysis [3, 41. Neverthe- less, in these papers, the authors consider lossless substrates, propagating modes only, and structures without lateral side walls.

In this paper we present a modified Wiener-Hopf analysis of shielded microstrip lines with lateral side walls. Complex permittivities and propagation constants are considered from the beginning and complex modes and dielectric losses can be treated in a straightforward way.

II. METHOD OF ANALYSIS

Consider a shielded microstrip line as is shown in Figure 1. The substrate is an isotropic lossy dielectric or a uniaxial anisotropic one, with the optical axis perpendicular to the interface. For simplicity, we consider only even moves, but the odd ones can be analyzed in the same way.

It is useful, as a first step, to introduce the transformation of fields and currents:

- - Q = ( -8, where EZ( a), ( a) are the Fourier trans- forms of fields E z ( x ) , E,(x) and currents J . ( x ) , J , (x ) at the interface, and p is the complex propagation constant along the line.

This transformation leads to two uncoupled relations be- tween the transformed fields and the currents at the interface

( a ) , x( a), and

[3-5]:

C ( a ) = g , ( P ; a)J ,Y(a ) , i = 1 , 2 ( 2)

Fields with W, = 0 and W, f 0 are LSE, and fields with W, # 0 and W, = 0 are LSM [5] .

Following the standard MRCT analysis and considering the symmetry constraints, we can write the total waves gener- ated by the transformed currents 4 on the strip in the form

V ( a ) = K:, , (a) k K,O(-a)e-'aw ( 3)

where fl,o represent the waves generated at x = 0 and the plus sign holds for the LSM ( i = 1) fields and the minus sign holds for the LSE (i = 2) fields. For the waves generated at x = 0 we can also write

where P,, A , , n , and B,, are coefficients to be determined, and v,, are the poles and zeros with negative imaginary part of the functions g,(p; a).

In the first factor of (4b), we can recognize the Wiener-Hopf solution to the semiinfinite bifurcation prob- lem. Terms with A, , represent perturbations to that solution caused by the finite width of the strip, and terms with B,, are perturbations caused by the reflected waves at the later side walls. The coefficients A,, and B,, must be related because the reflection coefficient of waves at lateral side walls is known.

On the other hand, because the F; are essentially the Fourier transforms of functions defined on a finite range of x, they can have no poles, and therefore the zeroes of (3) must coincide with the poles v , , ~ of g,(p; a) . Introducing this condition in (3) with (4), we obtain a system of equations to determine A,, and B,, N .

Following this method we obtain two infinite sets of linear equations to determine A,, and B,, n . Because terms with A, . and B,,n are perturbations of the solution with w -+ co and a 00, the infinite number of equations can be reduced to a finite one, typically less than 10, putting the remaining A, . n , B,, equal to zero, with a good numerical accuracy.

Once A, ,n and B f , n have been determined, P, must be determined to couple LSE and LSM fields. Reversing trans- formation (1) we can see that there appear spurious poles for kz and kx at a = &jP unless

and a,,

0 = u,(JP>u,(-"lP> + Y(- jP)u*(JpP) (5)

Solving Equation (5) for p, we obtain the complex propa- gation constant for a given frequency.

111. RESULTS

In Table I we present the complex propagation constants of a boxed microstrip (Figure 1) for different values of the conduc- tivity at 20 GHz. Results for the lossless structure are also compared with the ones provided in Ref. 6, which are shown in the last column.

In Figure 2 we present the propagation constant [Re@)] and the ratio between the atenuation constant and the'propa- gation constant [Im( P)/Re( p) ] for the fundamental mode of the same structure and for different conductivities of the

MICROWAVE A N D OPTICAL TECHNOLOGY LETTERS / Vol 2, No, 2, February 1989 39

T d

i

1 .0

0.5

/ /

T h

* -"A X

Figure 1 Shielded microstrip line

Re( P ) mm-'

//

/i/

I

I m( ,

\U = l:U/m

0.1 0.2 0.3 0.4 / I,

(mm-')

Figure 2 Propagation constant and attenuation-propagation constants ratio for the fundamental mode of the structure of Table I, for various values of the conductivity of the substrate ( u in mho/meter)

40 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 2, No. 2, February 1989

TABLE I Complex Propagation Constants for the Seven First Modes in the Structure of Figure 1 for Different Conductivities of the Substrate (a = 12.7 mm, w = 1.27 mm, h = 1.27 mm, d = 12.7 mm, &/go = 8.875, freq. = 20 GHz, and u in mho/mm)

u = 0 mho/mm" 0.001" 0.002a 0.003a 0.004a 0.005" Oh

P (mm-') ~ ~~~

1.1352 1.138 1.142 1.148 1.157 1.167 1.1355 -j0.061 -j0.021 -jO.l80 -j0.239 -j0.296

0.46222 0.4559 0.4372 0.4088 0.3731 0.3293 0.46231 -j0.0361 -~0.0678 -j0.0908 -j0.1037 - i0.1050

0.30394 0.3040 0.3045 0.3060 0.3092 0.3165 0.29551 -j0.0065 -j0.0129 -j0.0194 -j0.0263 -;0.0329

0.24926 0.2462 0.2403 0.2344 0.2292 0.2247 0.24902 -j0.0103 -j0.0165 -j0.0199 -j0.0214 -j0.0218

0.0268 0.0540 0.0821 0.1118 0.1414 - j0.23 13 5 -j0.2331 -j0.2384 -j0.2479 j0.2641 - j0.2926 -10.23168

0.06185 0.151 0.2366 0.3100 0.3821 0.4563 0.06434 -j0.31570 -j0.3212 -j0.3854 -j0.4670 -j0.5588 -10.6448 -10.32344

- 0.06185 - 0.0123 0.0123 0.0267 0.0325 0.0325 0.06434 -j0.31570 -J0.3366 -j0.3521 J0.3687 -J0.3840 -j0.3961 -j0.32344

aThis method. bHuang and Itoh.

substrate. The same quantities are shown in dotted lines for the parallel plate waveguide between the strip and the ground plane. For low losses the attenuation-propagation constants ratio for the microstrip and for the waveguide are very close (indistinguishable in a graphic).

A quantity of interest is the LSE/LSM ratio, defined as the quotient between P2 and PI in (4a). This quantity is shown in Figure 3 for the modes in Figure 2. As is expected, fields are predominantly LSE at low frequencies and become more LSM for higher frequencies

In Figure 4, the complex propagation constant for the 10 first modes in a shielded microstrip on a sapphire substrate is plotted against frequency. One can see the presence of com- plex modes under the cutoff frequency.

IV. CONCLUSIONS

As a general conclusion, we can say that the method presented here is suitable for analyzing shielded microstrip lines on lossy anisotropic substrates, without any previous assumption on

LSE/LSM

u = OD/m

u = 5 D/m

1.3

ko 0.3 0.4 (mm-1) 0.1 0.2

I I I

Figure 3 LSE/LSM ratio of the modes analyzed in Figure 2

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol 2, No. 2, February 1989 41

; 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 k o (mm-’)

Figure 4 Complex propagation constants for the 10 first modes of a shielded microstrip line on a sapphire substrate ( u = 12.7 mm, w = 1.27 rnm, h = 1.27 mm, d = 12.7 mm, q/ql = 11.6, and c l / c l l = 9.4)

the form of fields and/or currents, and with a high degree of accuracy.

Losses in the substrate and complex modes in lossless substrates are analyzed in a straightforward way, because the method i s formulated directly in the complex p plane.

On the other hand, the method is extendable to analysis of multilayer structures because the effects of substrate are fully contained in the g,(p; a) functions, which can be easily calculated in the spectral domain. Fin-line structures also can be analyzed because modes in fin-lines are identical to odd modes in suspended microstrips. So we expect that the method presented here will be useful in the analysis of microstrips and fin-lines on multilayered dielectric and semiconductor sub- strates.

ACKNOWLEDGMENT

This work was supported by the C.I.C.Y.T., Project number pb86-0144.

REFERENCES 1. R. Mittra, Computer Techniques for Electromagnetics, Pergamon

Press, 1973. 2. R. Mittra and T. Itoh, “Charge and Potential Distributions in

Shielded Striplines,” IEEE Trans. Microwave Theov Techniques, Vol. MTl-18, 1970, pp. 149-156.

3. A-M. A. El-Sherbiny, “Exact Analysis of Shielded Microstrip Lines and Bilateral Fin-Lines,” IEEE Trans. Microwaae Theorb, Techniques, Vol. M’IT-29, 1981, pp. 669-675.

42 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 2, No 2, February 1989

4. A-M. A. El-Sherbiny, “Hybrid Mode Analysis of Microstrip Lines on Anisotropic Substrates,” ZEEE Trans. Microwave Theory Tech- niques, Vol. MTT-29, 1981, pp. 1261-1265.

5. R. Marques and M. Horno, “Dyadic Green’s Function for Mi- crostrip-Like Transmission Lines on a Large Class of Anisotropic Substrates,” IEE Proc., Pt. H, Vol. 133, 1986, pp. 450-454.

6 . W-X. Huang and T. Itoh, “Complex Modes in Lossless Shielded Microstrip Lines,” ZEEE Trans. Microwave Theory Techniques, Vol. MTT-36, 1988, pp. 163-165.

Received 10-1 8-88

Microwave and Optical Technology Letters, 2/2, 39-43 0 1989 John Wiley & Sons, Inc. CCC 0895-2477/89/$4.00

INTERPOLATION SOLUTIONS FOR THE PROBLEM OF SYNTHESIS OF

REFLECTOR ANTENNAS DUAL-SHAPED OFFSET

Joseph A. Jewase and Raj Mittra Electromagnetic Communication Laboratory University of Illinois Urbanna, Illinois

Victor Galindo-Israel and W. lmbriale Jet Propulsion Laboratory California Institute of Technology Pasadena, California

KEY TERMS Rejector antenna, numerical method, antenna feed

1. INTRODUCTION

Synthesis of dual-shaped offset reflector antennas to control the exit aperture distribution of amplitude and phase has received considerable attention in recent years because the problem is both practically important and theoretically chal- lenging [l, 21. Figure 1 shows the geometry of the dual-reflec- tor system. For a given feed illumination and desired aperture field distribution, an exact formulation of the problem of simultaneously synthesizing the shapes of the sub and main reflectors was presented recently by Galindo-Israel et al. [l] in terms of a set of nonlinear first-order differential equations. In this paper, we discuss a numerical approach to solving these equations that circumvents some the difficulties encountered in [l], particularly for small values of 0.

2. NUMERICAL APPROACH

Given the feed distribution function, the desired aperture field, the system parameters, viz. B,, p M , Q , and a (see Figure l), and the optical path length K , the numerical synthesis of the reflector surfaces is carried out by using the procedure outlined below. First, we solve for r on the periphery of the subreflector by using the following steps.

(a) Assume a functional relationship between 11, and + on the peripheries.

(b) Assign values to r! (initial value for r at (p = -a/2) and N+ (number of points on the peripheries).

(c) Solve for r+ using any of the available IMSL subrou- tines, e.g., DIVPAG, for the solution of ordinary dif- ferential equations.

The corresponding values for z are deduced using the path length equation. With the values of the variables on the peripheries known, we proceed next to generate the rest of the reflector by decreasing 8 in small increments and computing the values of r , p, 11, as functions of 9, such that the condi- tions on the path length, the conservation of energy, and the total derivative are simultaneously satisfied. This calls for simultaneous solution of a system of differential equations, which has been discussed in [l]. However, the numerical approach to deriving the partial derivatives p+ and 11,+ is different in the present paper from that employed in [l], and is found to obviate some of the difficulties encountered in the previous work. The derivatives are computed by using the following procedure:

(a) Generate the knot sequence for p and 11, using, for

(b) Compute their corresponding B-spline representation

(c) Use IMSL subroutine DBSDER to evaluate p+ and ++. Having obtained values for p+ and ++, we substitute these

in the equations for the total derivative and the conservation of energy, and solve for pe and 11,e. The system of differential equations comprising re, pe, and $0 is then solved simultane- ously using any of the standard numerical techniques for the solution of an initial-value problem for ordinary differential equations such as the Adams-Moulton or Gear method em- bodied in IMSL subroutine DIVPAG.

For small values of 0, e.g., 0 < 5 O , a different approach is followed because the aperture field begins to deviate apprecia- bly from the prescribed value in this angular region. This may occur, for example, because of a nonzero nodal value B, [l], or it may occur due tc increasing numerical errors common to initial value problems. With reference to the condition of conservation of energy (Eq. (4.2) of [l]), which is given by V,V(p, 11,)p[pe11,+ - ~+11,e] = Z(B,+)sinO, it is noted that for B = 0, two possibilities exist: These are either p = 0 or the Jacobian of the transformation, i.e., [PO#+ - P + # ~ ] = 0. We choose the condition p = 0 when 0 = 0 for our interpolation process. This implies that the center of the subreflector maps into the center of the main reflector which excludes the existence of nodes. With this knowledge, we then proceed to synthesize the rest of the reflectors following the method described below.

Instead of generating p as before, we interpolate it instead using the B-spline functions between 0 = 0, where we set p = 0, and the last value of 0 for which the simultaneous differential equations were solved numerically rigorously. The accompanying 11, values are derived via extrapolation, again using the same B-spline functions. This results in sets of values for p and 11, as functions of B and +. Substituting these values in the differential equation re, we obtain r and finally use the path length condition to evaluate z. The values for p and 11, may also be used to obtain PO, p+, $0, and #+ by means of B-spline functions. The corresponding illumination function may then be deduced by substitution in the energy equation.

The reflector surfaces generated in the manner described above are much smoother than those that would be obtained via the differential equation approach, if it were employed all the way to B = 0. We point out that even while using interpo- lation, the path length and the total derivative conditions are exactly satisfied, and only the energy condition is relaxed. Numerical results show that the departure from the prescribed

example, IMSL subroutine DBSNAK.

using subroutine DBSLSQ.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 2, No. 2, February 1989 43