EVALUATION OF END EFFECTS AND LUMPED CAPACITANCE OF MICROSTRIP WITHANISOTROPIC SUBSTRATES - TRANSVERSE TRANSMISSION LINE TECHNIQUE

Shiban K. Koul* and Bharathi Bhat*

ABSTRACT

In this paper transverse transmission line technique is used to analysethe lumped capacitance and end effects of microstrip line with anisotropicsubstrate. In the resulting capacitance expression, the only parameterdependent on the transmission media is the admittance at the charge planewhich can be easily determined using a two wire transmission line equiva-lent. Extensive design data on the lumped capacitance and end effects ofmicrostrip with anisotropic substrates, sapphire and pyrolytic boronnitride, are generated. This data should be very useful for the design offilters and directional couplers in microstrip configuration with aniso-tropic dielectrics.

INTRODUCTION

A great deal of work has been published on the propagation character-istics, namely, characteristic impedance and phase velocity of microstriptransmission line with anisotropic substrates [11 - [5]. Practical micro-wave integrated circuits, however, use a variety of discontinuity structuresin the stripconductpr, such as, an abruptly ended stripconductor, a patchof stripconductor and so on. The lumped capacitance of a patch of strip-conductor and end effects of an abruptly ended stripconductor in microstripwith isotropic substrates have been analysed by several authors [6]- [8]..However, to date, analysis and design information on the end effects andlumped capacitances of conductor patches in microstrip with anisotropicsubstrates are not available in open literature.

This paper presents the analysis and design data on the end effectsand lumped capacitance of conductor patch in microstrip with anisotropicsubstrates. The analytical technique presented here uses the variationalmethod in conjunction with the transverse transmission line technique. Itis identified that the only parameter dependent on the transmission mediain the capacitance expression is the admittance at the charge plane. Thetechnique can be extended to analyse the end effects and lumped capacitancesof a variety of microstrip-like transmission line with anisotropic substra-tes by simply determining the admittance parameter.

ANALYSIS

The open-circuit end effect AZ of a general TEk transmission linecan be written as

CeEQ Ce ~~~~~~~~(1)u

where ce= Lim 0.5 [CQ - SCu] (2)e £4-co

Centre for Applied Research in Electronics, Indian Institute of Technology,Hauz Khas, New Delhi-110016, INDIA

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(a) (b) ~FFig.1 (a) Finite length stripconductor microstrip with anisotropic

dielectric substrate.(b) Two wire transmission line equivalent. Y is the charac-

teristic admittance. y is the propagatiSn const_ant.Y =y + Y_

In these expressions, CR, is the lumped capacitance of a section of strip-conductor of length Z and width w, C is the edge- capacitance of anabruptly ended stripconductor of widEh w and C is the line capacitanceper unit length of a uniform line of the same width. For the purpose ofanalysis, we consider the microstrip structure shown in Fig. 1(a). Thelumped capacitance C can be obtained by using the variational expre-ssion in the Fourier ransform domain given by

1 1 ~~~Co 00

Ci 7- 2r ¢(a, Y ,3) g(a,

(Y) da d$ (3)Q (27rQ) 0

where g (ac, O) is the Fourier transformed charge distribution on thestripconductor, Q is the total charge on the stripconductor and ¢(c.zyoj,)is the transformed potential distribution function at the charge planey= y0. Using the transverse transmission line technique, the expressionfor ° (o,y, () turns out to be

(a,yO, ) = g (c,)/[ Y(c2+ B3) Yj (4)

where, Y is the admittance at the charge plane obtained from Fig. l(b)using standard transmission line formulas, The expression for yis givenby by~~~O [ 1+ c coth{./ E } (5)

0 VII coti-i &¾-t Cx/Cy.blITo evaluate (3) we need to specify the charge distribution g(x,z) on thestripconductor which can be further written as

g(x,z) = g(x) g(z) (6)For g(x) and g(z) , the following trial functions are used

g(x) = J l+J412 3 ; -w/2 < x < w/2

L 0 ; otherwise (7)

149

.

1 ; -(Q/2-p) < z s (Q/2-p)

g(z) = l+(A/p) [izl- 2 P)] ; -Q/2 z < -(Q/2- p) (8)

(Q/2 -p) <z < 1/2

0 ; Otherwise

A and p are constants which have to be numerically optimized. The lumpedcapacitance C, can be obtained by substituting (4) and Fourier transformof the charge distribution g (x,z) in (3). The expression for C isalready available in an earlier paper by the authors [4] . Usingu(l) and(2), the end effects of an abruptly ended stripconductor in microstripwith anisotropic substrate can be numerically computed.

NUMERICAL RESULTS

Numerical computations of lumped capacitance C and end effects Atare carried out for microstrip having anisotropic &ielectric. Fordetermining the lumped capacitance of square and rectangular conductorpatches, a uniform charge distribution in the z-direction (A=0 in(8) )is assumed. The results obtained using this simple distribution are quiteaccurate for most applications. For computing the end effects, the factorA is set equal to 1. Figure2(a)shows the variations of normalised lumpedcapacitance of square and rectangular conductor patches in microstripconfiguration. The substrate considered is sapphire with principal axesrotated by 90 degrees with respect to the x-y coordinate system. Similarvariations with sapphire substrate having principal axes aligned with thex-y coordinate axes are shown in Fig. 2(b). The variations of normalisedlumped capacitance of square and rectangular conductor patches in micro-strip with pyrolytic boron nitride substrate are shown in Fig.3. Figure 4shows the variations of edge- capacitance and end effects of abruptlyended strip conductor in microstrip configuration. The substrateconsidered is sapphire.

CONCLUSIONS

A simple technique to conpute the lumped capacitance of a patch ofstripcondtuctor and end effects of abruptly ended stripconductor in micro-strip with anisotropic substrates is presented. The technique can beeasily extended to analyse the end effects and lumped capacitance offinite length stripconductor in various microstrip-like transmission lineswith anisotropic substrates. It is only necessary to determine a singleparameter, namely, the admittance at the charge plane. Numerical dataon lumped capacitance and open circuit end effects are presented formicrostrip with sapphire and pyrolytic boron nitride substrates.

REFERENCES

1. R.P. Owens, J.E. Aitken and T.C.Edwards,"Quasi-static characteristicof microstrip on an anisotropic sapphire substrate", IEEE Trans.Microwave Theory Tech.,Vol. MTT-24, pp. 499-505, Aug. 1976.

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0.1 0*2 0.5 1 2 5 10b /w

Fig.3 Variation of normalised lumped capacitance ofsquare and rectangular conductor patch in micro-strip with pyrolytic boron nitride substrateEX = 5.12, y = 3.4

2. C.M. Krowne," Microstrip transmission lines on pyrolytic boronnitridell, Electron. Letters, Vol. 12, pp.642-643, Nov. 1976.

3. N.G. Alexopoulos and S.A.- Maas, " Characteristics of microstripdirectional couplers on anisotropic substrates", IEEE Trans.Microwave Theory Tech., Vol. MTT-30, pp. 1267-1270, Aug. 1982.

4. S.K. Koul and B. Bhat, " Inverted microstrip and suspended micro-strip with anisotropic substrates", Proc. IEEE, Vol. 70, pp.1230-1231, Oct. 1982.

5. S.K. Koul and B.Bhat, "Generalised analysis of microstrip-like trans-mission lines and coplanar strips with anisotropic substrates forMIC, Electro-optic modulator and SAW application", IEEE Trans.Microwave Theory Tech., Vol. MTT-31, PP.1051-1059, Dec. 1983.

6. P. Silvester and P. Benedek, "Equivalent capacitances of microstripopen circuit", IEEE Trans. Microwave Theory Tech., Vol. MTT-20,pp 511-516, Aug. 1972.

7. T.Itoh, R. Mittra and R.D. Ward, "A method for computing edge capa-citance of finite and semi-infinite microstrip lines", IEEE Trans.Microwave Theory Tech., Vol. MTT-20, pp. 847-849, Dec. 1972.

152

8. B.Bhat and S.K. Koul,I" Lumped capacitance open- circuit end effectsand edge-capacitance of microstrip-like transmission lines formicrowave and millimeter-wave applications" IEEE Trans. MicrowaveTheory Tech, Vol. MTT-32, April 1984.

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