Micros Rip Line Analysis

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,

Variational Method forVOL. MTT-16, NO. 4, APRIL 1968

the Analysis ofMicrostrip Lines

EIKICHI YAMASHITA, MEMBER, IEEE, AND RAJ MITTRA, MEMBER, IEEEAbstractThis paper reports a method for computing the l ine capaci-

tance of a microstrip line based on the application of Fonrier transformand variational techniques. The characteristic impedance, guide wave-length, and the surface potential distribution in the microstrip line areobtained for a range of structnre parameters and the dielectric constant.The resnlts calculated from the expressions developed in the paper arecompared with the theoretical results presently available in the literatureand good agreement is found. Comparison with available experimentalresolts i s also made where feasible. Possible applications and limitationsof the method are discussed.

I. INTRODUCTION

\vITH THE RAPID development of the transistorsand other semiconductor devices in the microwavefrequency regions, the microstrip line structure is

finding increasing use in the microwave integrated circuit.This has been evidenced by a number of recent publicationson the subject, e.g., Hyltin, [l] Vincent, [z] Guckel and Bren-nan, [3] Webster, [41 and Sobol. [51Although the basic properties of the microstrip line have

been well known for some time, [61there still appears to be aneed for rigorous theoretical formulas which are useful overa wide range of design parameters. Among the number ofresearchers who have studied the problem, Dukes[ 71 re-ported an experimental method using an electrolytic tank forstudying the microstrip line. WU[81 proposed a set of dualintegral equations for the two current components on thestrip and presented an iterative solution to the integral equa-tions. Wheeler[g] studied the microstrip line based on con-formal mapping and Guckel and Brennan[31 applied a widestrip approximation to analyze a microstrip line structureplaced on a dielectric and a conducting layer.In this paper an attempt is made to develop an analytical

method which is general and useful for calculating the linecapacitance for a rather wide range of geometrical and di-electric parameters of the microstrip line. Furthermore, thetechnique developed here has the potential for application toa wider class of geometries, such as a multilayered dielectricinsert in the stripline. The analytical approach in the presentpaper is based on the Fourier transform and variationaltechniques. The knowledge of the line capacitance is em-ployed to calculate the characteristic impedance, guide wave-

Manuscript received May 5, 1967; revised August 18, 1967. Thework reported here was supported by U. S. Army Research GrantDA-G-646.E. Yamashita was with the Antenna Laboratory, University ofIllinois, Urbana. He is now with the University of Electro-Communica-t ions, Tokyo, Japan.R. Mittra is with the Antenna Laboratory, University of Illinois,Urbana, I ll .

251

length, and the surface potential on the dielectric sheet of themicrostrip line. The above results are compared where pos-sible with the available experimental data and other theo-retical results.

II. OUTLINE OF THEORETICAL PROCEDUREThe static potential distribution r#J(x, Y) in the microstrip

line structure satisfies Poissons equation

v(l)(z, y) = ~ P(%!/) (1)eand the boundary conditions on the surface of the dielectricmaterial as well as the conductor. Here p(x, y) is the chargedistribution on the surface of the conducting strip.The line capacitance can be evaluated by inserting the

solution of (1) in the variational expression[ll

(2)where

Q=s P(X, y)dz (3)8

and the integral in (2) and (3) are to be taken on all the sur-faces over which the charge P(X, y) is distributed.The characteristic impedance of a TEM transmission line

in free space is well known and is given byzo=& (4)

where COis the line capacitance of the stl ucture and c is thevelocity of light.Let the structure be modified by inserting a uniform dielec-

tric sheet along the line as shown in Fig. 1. The line capaci-tance of this new structure C is larger than CO. The newguide wavelength A is smaller than the free space wavelengthxO. Hence, the characteristic impedance of the new line isobtained by modifying (4) as follows:2=9+20()Similarly, the new guide wavelength is given by

(5)

(6)

Thus, all of the basic properties of the stripline, whether di-electric loaded or unloaded, are derivable from the knowl-

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252 IEEEY

Lt


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YAMASI-IITA AND MITTRA: VARIATIONAL METHOD FOR ANALYSIS OF MICROSTRIP LINES 2s3

Fig. 3.

I t= /1I

1 1,00 I I162 10- I 10

W/hLine capacitance versus strip width and strip height.

z(n)

THIS THEORY250 MODIFIEO

CON FORMALMAPPING [91

200 -

I50 - C* = 11.7

100

50 -

0 ! 1 ! I I , ,,1 ! , I I ,,1 I , ! ,,10-2 10- I 10W/h,

Fig. 4. Calculated results for characteristic impedanceand comparison wi th other theory.

choice of a trial function is also supported by the current dis-tribution in plane parallel transmission lines. 1121Accord-ingly, we choose the trial function

[(i -++-f(x) = ~ )(otherwise.) (15)From (15), one has by Fourier transforming

~ Si. !2!() ~ ()]w 2m 2 sin 4 (16)Q= flw -@w,

z(.(2)


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254 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, APR~ 196$

200

Z.(L-1)

100

I/h =0000020040060080,10

o~162 d 10

W/h

Fig. 7. Effect of strip thickness on characteristicimpedance with 6*= 1.00.

IV. STRIP OF FINITE THICKNESSThe calculations presented were based on the assumption

that the thickness of the strip t is negligible compared to thewidth w. It becomes necessary to modify the formulas in theevent that the strip has a reasonable thickness. The modifi-cation required is described in the following.One observes that the potential function $(B, y) in the

region above y= h has an exponential behavior exp ( Ip I y).Hence, the potential at y = h+ tsatisfies

4(8, h + O = e-lfi[d(il, h). (17)Considering the two layers of charge at y = h and y = ii+ t,one obtains a modified formula for the line capacitancewhich reads

Using (18), the characteristic impedance is calculated for thecase C*= 1 and plotted in Fig. 7. Since adequate results werenot available from other sources for the thick strip case, itwas felt appropriate to make an approximate comparisonwith the impedance of a circular wire above a ground plane.The impedance of the circular wire is also plotted in Fig. 7.If the cross-sectional area of the strip and the circular wireare equivalent, the two characteristic impedances should ap-proximately agree with each other. It is seen that forw/h = 0.1, the wave impedance of a circular wire line is222 ohms and the wave impedance of the stripline with thesame area (w/h = 0.1, t/h= 0.786) is 220 ohms. Hence, the

(:)

Fig. 8.

1 c = 11 ,7 roils THEORY100 1/h=OOO . . . . . .\ EXPERIMENT [!]50

o~16= d 10

W/hTheoretical and experimental results for the effect of stripthickness on character ist ic impedance with C*= 11.7.

I h

0 10 20X /h

Fig. 9. Potential distribution on surface of dielectric sheet.

expression (18) is believed to be a good approximation ofthe thick strip case.The effect of strip thickness on the characteristic im-

pedance of a dielectric loaded line is shown in Fig. 8 forE*= I I . 7. The present theory is compared with the experi-mental data of Hyltin[ll for a silicon slab with t/h= 0.01(t= 0.1 mil and h= 10 roils). The experimental valuest1 areslightly lower than the ones calculated by this theory.The potential distribution on the surface of the dielectric

sheet is evaluated by computing the Fourier transform[lll

s(Z,h) = ~ _@(@,z)e-~RZ/3. (19).An example for the case w/h= 0.1 is shown in Fig. 9. Theknowledge of the potential distribution is useful for esti-mating the coupling of the strip to adjacent elements in amultielement circuit.

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YAMASHITA AND MITTRA : VARIATIONAL METHOD FOR ANALYSIS OF MICROSTRIP LINES 255

V. POSSIBLE APPLICATIONS AND LIMITATIONSOF THE THEORETICAL PROCEDURE

In addition to the problem studied, a variety of transmis-sion systems comprising different arrangements of conduct-ing strips, a ground plane, and dielectric layers could be ana-lyzed by the present method. One of these configurations isillustrated in Fig. 10.2 It is obvious that with an increase ofthe number of boundary and continuity conditions the poten-tial distribution function would take more and more compli-cated forms. However, the calculations would still be alge-braic and straightforward. The resulting formulas are ex-pressible in closed forms which are convenient for calcula-tion on a digital computer.Another application of the method would be the calcula-

tion of the dc resistance per unit length between a conduct-ing strip and a ground plane filled with resistive media. Thecase of a single resistive medium was discussed by Smythe. [131Obviously, this procedure can be extended to the case ofmultiple resistive media by knowing the line capacitance formultiple dielectric media.Suppose one has a system of conductors with two resistive

media as shown in Fig. 11. Let resistivity of each medium bespecified by RI and Rz. As a first step, one calculates the linecapacitance C of a microstrip line in Fig. 1, taking the dielec-tric constant C*= RJR1. The total dc resistance R per unitlength can then be obtained by using

(20)

To illustrate this type of calculation, numerical resultswere obtained for Rz= 20R1 and Rz= RI, respectively. Theseare plotted in Fig. 11.There are some theoretical limitations which should be

considered before applying this method.1)

2)

3)4)

The dielectric material should be of low loss. Quantita-tively, this condition is stated by

weRd >> ~where Rd is the resistivity of the dielectric material.The present method assumes a TEM mode and ne-glects the radiation effects. These assumptions are im-posing a condition

kO >> h.The strip is assumed to be thin. Therefore, t


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256 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, APRIL 1968REFERENCES

P] T. M. Hyltin, Microstrip transmission on semiconductor dielec-trics, IEEE Trans. Microwave Theory and Techniques, vol. MTT-13,pp. 777-781, November 1965.

[z] B. T. Vincent, Jr., Ceramic microstrip for microwave hybridintegrated circuitry, presented at the Internat1 Microwave Symp.,Palo Alto, Calif., May 1966, Session V-2.[?]H . Guckel and P. A. Brennan, Picosecond pulse response ofinterconnections in a common substrate monol ithic system, presentedat the Internat1 Solid State Circuits Conf,, Philadelphia, Pa., February1967, Session XI.[4] R, R. Webster, Integrated microwave oscil lators, ampl if iers,

switches and converters, presented at the Internat1 Solid State Cir-cuits Conf., Philadelphia, Pa., February 1967, Session IV.[s]H. Sobol, Extending IC technology to microwave equipments,

Electronics, vol. 40, pp. 112-124, March 1967.[b]M. Arditi, Characteristics and applications of microstrip for

microwave wiring, IRE Trans. Microwave Theory and Techniques, vol.MTT-3, pp. 31-56, March 1955.[T]J. M. C. Dukes, An investigation into some fundamental proper-ties of strip transmission lines with the aid of an electrolytic tank;Proc. IEE (London), vol. 103B, pp. 319-333, May 1956.[81T. T. WU, Theory of the rnicrostrip, J. AppL ~h~s., VO1. 28,pp. 299-302, March 1957.[91H. A. wheeler, Transmission-line properties of parallel stripsseparated by a dielectric sheet, IEEE Trans. Microwave Theory andTechniques, vol. MTT-13, pp. 172-185, March 1965.[IO] R. E. Collin, Field Theory of Guided Waues. New York:McGraw-Hill, 1960, p. 162.[H] A. Papotdis, The Fourier Integral and Its Applications. NewYork: McGraw-Hill, 1962.W] I. Palocz, The integral equation approach to currents and fieldsin plane parallel transmission lines, IBM Rept. RC 1220, June 1964.(This literature was suggested by a reviewer.)11$1W. R. Smythe, Static and Dynamic Electricity. New York:McGraw-Hill, 1950, p. 234.

CorremondenceILThe Theoretical Design of Broadband3-Port Waveguide Circulators

AbstractAn extension of earlier fieldanalysis has made possible the prediction,purely by theory, of broadband 3-port H-planecirculator designs. Using a configuration con-sisting of concentric rings of ferrite and dielec-tric material surrounding a conducting post inthe center of the junction, circulators with iso-lation bandwidths up to 42 percent have beenpredicted. In practice, bandwidths np to 30 per-cent have been obtained in X band and band-widths of about 20 percent in C and Q band.

INTRODUCTIONThe design, by purely theoretical means, of

a broadband waveguide 3-port ci rculator hasbeen briefly reported. [1 The purpose of thiscorrespondence is to give some details of thetheory which is an extension of the earlierfield analysis,[il together with recent experi-mental details of the resulting circulators.Broadband waveguide circulators havepreviously been designed on an empiricalbasis.However, h has beenfound possible toeliminate much laborious and time-consum-ing measurement by the combination of fieldanalysis with the digital computer.The original field analysis[1wasconcernedwith the full-height ferrite post mounted cen-trally in the waveguide junction. For thisgeometry, the applied magnetic field and theferrite post diameter are the only physical pa-rameters that can be varied continuously.Hence, in designing a 3-port circulator where

Manuscript received October 31, 1966; revisedAugust 4, 1967.The work describedhere wascarriedout at Mullard ResesrchLabs., Redhill, Surrey,Eng-land.

Fig. 1.

two real equations have to be satisfied for ci r-culation at a given frequency, we would haveno more than the necessary two physical pa-rameters. One would expect to be able toachieve circulation at any (reasonable) fre-quency with any (reasonable) ferri te mater ialby suitable choice of ferrite-post diameterand applied magnetic field. However, onewould also expect the result to be a narrow-band circulator as was found in the 3 and 5percent bandwidths originally computed. [1In order to achieve broadband circulation,more degrees of freedom must be available.The earlier field analysis has therefore beenextended to the structure of Figs. 1 and 2where, instead of a ferrite post, we have aconducting post surrounded by a ferrite tubewhich in turn is surrounded by a dielectrictube. (This arrangement had earlier provedsuccessful in high-power circulators .)[al Asdielectric materials are available over a widerange of permittivity in fine steps, this ex-tended struc ture has three additional degreesof freedom.With the above field analysis, it takes less

Conducting DielectricFerrite v in material

/SectionAA I 1IStatic magk.tic fieldFig. 2. Three.port circulator ferrite configuration.

than a second to compute circulator per-formance at any particular frequency. It istherefore now feasible to use linear program-ming on the computer to optimize (and sodesign) broadband circulators. A computerprogram has been written that chooses thevalues of the five parameters (conductor, fer-rite and dielectric diameters, applied mag-netic field, and permittivity ) to give the great-est percentage frequency bandwidth overwhich the reflection coefficient remains lessthan 0.1, and hence the isolation is greaterthan 20 dB.In this analysis only two significant ap-proximations are made. First, although theoriginal field analysis was exact, the same ap-

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Micros Rip Line Analysis