analysis of slot-coupled transitions from microstrip-to-microstrip and microstrip-to-waveguides

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 7, JULY 1997 1127

Short Papers

Analysis of Slot-Coupled Transitions fromMicrostrip-to-Microstrip and Microstrip-to-Waveguides

Amjad A. Omar and Nihad I. Dib

Abstract— This paper provides an accurate, versatile, andcomputationally efficient method for the analysis of slot-coupledtransitions from microstrip-to-microstrip, and microstrip-to-rectangularand parallel-plate waveguides. The accuracy of this method isensured by satisfying all the boundary conditions through a mixedelectric–magnetic current integral equation formulation, combined withthe moment method. Computational efficiency is achieved by limitingthe discretization to only the strips and apertures and by using theaccurate and rapidly convergent complex images. To verify the accuracyof this method, the transitions are analyzed using the finite-differencetime-domain (FDTD) method. Experimental results are also obtained forsome structures. Close agreement is found between the complex imageresults, the FDTD results, and experiment over a wide frequency range.

Index Terms—Complex images, integral equation, microstrip, slot,waveguides.

I. INTRODUCTION

There has been a recent trend toward building circuits whichhave more than one dielectric layer and more than one level ofmetallization to reduce the size of microwave and millimeter-wavecircuits. Some of the metallization layers are used to isolate the activefrom the passive elements to reduce the parasitic coupling betweenthem. To this end, it is important to study the transition betweenmicrostrip and microstrip through a slot in the common ground plane,as shown in Fig. 1.

Two other important classes of transitions are the microstrip-to-parallel-plate waveguide (MS–PWG) and microstrip-to-rectangularwaveguide (MS–RW) transitions via a slot in the microstrip groundplane (Figs. 2 and 3) [1], [2]. These transitions are important inhigh-Q filters, high-Q oscillators, and nonreciprocal components. Theability to analyze these transitions becomes very important, especiallyin the millimeter-wave frequency range.

To achieve optimum design of the different transitions, it isimportant that an accurate and computationally efficient method bedeveloped which can be easily interfaced with available optimizationdesign software. As far as is known, there have been little or nopublications on the theoretical analysis of slot-coupled microstripline(MS)-to-waveguide transitions [1]. However, several methods havebeen presented in the literature to analyze the transition between twomicrostriplines (MS–MS) through a slot, shown in Fig. 1. Some ofthese methods, like the finite-difference time-domain (FDTD) methodand transmission-line matrix (TLM) method, require the discretizationof the entire volume of the circuit. They are, therefore, slowlyconvergent and require excessive memory. Also, the application of theelectric-field integral-equation technique (EFIE) requires segmentingthe entire conducting surface of the transition, including the infi-

Manuscript received February 22, 1996; revised March 24, 1997.A. Omar is with the Royal Scientific Society, Princess Sumayya University

College of Technology, Amman, Jordan.N. I. Dib is with the Department of Electrical Engineering, Jordan Univer-

sity of Science and Technology, Irbid, Jordan.Publisher Item Identifier S 0018-9480(97)04465-7.

Fig. 1. A three-dimensional (3-D) view of the microstrip-to-microstrip tran-sition via an aperture.

Fig. 2. The MS–PWG transition.

Fig. 3. The MS–RW transition.

nite microstrip ground plane.1 Therefore, the EFIE is numericallyinefficient for solving the aperture coupling problem.

1em, Sonnet Software Inc., Liverpool, NY, 13090.

0018–9480/97$10.00 1997 IEEE

1128 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 7, JULY 1997

(a)

(b)

Fig. 4. Splitting the MS–MS transition into two sub-problems equivalent tothe upper and lower half spaces, respectively. (a) Lower half space. (b) Upperhalf space.

A more efficient form of the integral equation technique wasapplied in [3] for the analysis of the MS–MS transition. Unlike theEFIE, which uses only electric currents, this technique uses bothelectric and magnetic currents, which flow on the strips and apertures,respectively. This limits the discretization to only the strips andapertures, and, thus, saves a lot of memory and computation time.However, the time-consuming numerical integrations of Sommerfeldintegrals are evaluated in [3] to obtain the spatial Green’s functionsof the structure. Therefore, there is still a need for a technique whichis faster and, hence, more suitable for use with CAD programs.

In this paper, a mixed electric–magnetic current integral equationformulation similar to that in [3] is used. However, this paper’sGreen’s functions for the coupling between the electric currents ofthe MS’s, magnetic current of the aperture, and for the cross couplingbetween the electric and magnetic currents, are all calculated usingthe accurate and rapidly convergent complex image technique [4],[5]. The only case where complex images are not used is when theGreen’s functions of the rectangular waveguide (RW) are evaluated.In this case, the real waveguide images are used instead. The compleximage technique avoids the time-consuming numerical integrationof Sommerfeld integrals and yields at least a ten-fold reductionin computation time with less than 1% error, as compared to thenumerical integration. These complex images include the effect ofthe surface waves and leaky waves and, therefore, have no restrictionon the substrate thickness.

To verify the accuracy of the results obtained using the compleximage technique, the same transitions are analyzed using the FDTDmethod. The FDTD formulation is simple and well known [6]–[8]and, thus, will not be presented here. Instead, the emphasis will beon the formulation of the mixed electric–magnetic current integralequation and the use of the computationally fast complex imagetechnique to evaluate all the Green’s functions.

II. THEORY

A. MS–MS Transition

To solve the MS–MS problem, the structure of Fig. 1 is split intotwo sub-problems, equivalent to the lower and upper microstrips,as shown in Fig. 4(a) and 4(b), respectively. Equivalent electric and

magnetic currents flow on the strips and apertures, respectively, ofboth sub-problems to satisfy the boundary conditions, as follows.

1) The continuity of the tangential electric field on the aperture isenforced by choosing the magnetic current on the aperture ofthe lower sub-problem, which flows along they-direction [I(1)my

of Fig. 4(a)], to be equal and opposite to that on the apertureof the upper sub-problem [I(2)my of Fig. 4(b)].

2) The continuity of the tangential magnetic field on the apertureis satisfied by the following equation:

H(1)

y I(1)

ex +H(1)

y I(1)

my = H(2)

y I(2)

ex +H(2)

y I(2)

my (1)

whereI(1)ex andI(2)ex are thex-directed electric currents flowingon the lower and upper microstrips, respectively, whileI

(1)

my

and I(2)my are they-directed magnetic currents flowing on theaperture in the lower and upper sub-problems, respectively.H

(i)y (I

(i)ex ) andH(i)

y (I(i)my) denote they-directed magnetic field

on the aperture of theith sub-problem caused by anx-directedelectric current on the MS and ay-directed magnetic currenton the aperture, respectively.The fields in (1) can be expressed in terms of the mixedpotentials of electric and magnetic currents [9], as follows:

H(i)y I

(i)my = �j!

Aper:

G(i)

F (r=r0

)J(i)my(r

0

)dS0

+1

j!Aper:

@

@yG(i)qm(r=r

0

)@J

(i)my

@y0dS

0 (2)

H(i)y I

(i)ex = H

(i)y;xx I

(i)ex +H

(i)y;zx I

(i)ex (3)

where

H(i)y;xx I

(i)ex = ~r�

M

G(i)

A (r=r0

)J(i)ex (r

0

)x dS0

� y (4)

H(i)y;zx I

(i)ex = ~r�

M

G(i)

A (r=r0

)J(i)ex (r

0

)z dS0

� y: (5)

In (2), use has been made of the continuity equation formagnetic current and charge [9]. In (2),G(i)

F is the Green’sfunction for they-directed electric vector potential due to ay-directed magnetic current in theith sub-problem [10], whileG(i)qm is the scalar potential due to a magnetic charge [10].

G(i)

A in (4) is the Green’s function for thex-directed magneticvector potential due to anx-directed electric current in theith sub-problem, whileG(i)

A in (5) is the Green’s functionfor the z-directed magnetic vector potential due to anx-directed electric current, with both source and field locatedinside a microstrip substrate. These Green’s functions are givenin Appendix A. The areaMi, over which the integrals areperformed, is the microstrip of theith sub-problem.

3) The vanishing of the tangential electric field on the MS’s ofthe upper and lower sub-problems is satisfied by the followingequation:

E(i)x I

(i)ex + E

(i)x I

(i)my = �E

ext

x (6)

where i = 1 and 2 corresponds to the lower and upper sub-problems, respectively.E(i)

x and Eext

x are the scattered andimpressed electric fields, respectively.

The electric and magnetic current densities are expanded as fol-lows:

J(i)ex (r

0

) =

N

n=1

I(i)ex;nf

en(x

0

)�en(y

0

) (7)

Jmy(r0

) =

N

n=1

Imy;nfmn (y

0

)�mn (x

0

) (8)


whereN (i)e andNm are the number of segments on the microstrip

of the ith sub-problem and aperture, respectively.fen and fmn arepulse basis functions, which are unity along thenth electric ormagnetic current segments, respectively, and zero elsewhere.� en and�mn account for the edge condition on the strips and apertures [11],respectively, and are given by

�en(y

0) =1

�W

2

2

� (y0)2

�mn (x0) =

1

� W

2

2� (x0)2

(9)

whereW (i)e andWm are the widths of the microstrip of theith sub-

problem and the width of the aperture, respectively. The choice ofthe basis functions in (7) and (8) was first validated by Glisson andWilton [12], who showed that the use of pulse basis functions insteadof rooftop basis functions has very little effect on the moment-methodsolution. This choice, however, significantly reduces the computations[13].

The Galerkin moment method [14] is then applied to (1) and(6), using the expansion functions of (7) and (8) also as weightingfunctions. This results are shown in the following matrix equation:

V(1)ext

00

=

Z(1)ee Z

(1)em [0]

Z(1)me Z

(1)mm + Z

(2)mm � Z

(2)me

[0] Z(2)em Z

(2)ee

I(1)exI(1)myI(2)ex

(10)

whereZ(1)ee andZ(2)

ee are the lower and upper microstrip impedancematrixes, respectively [15].Z(1)

mm and Z(2)mm are the lower and up-

per magnetic–magnetic (aperture) impedance matrixes, respectively.They are the dual of the electric–electric impedance matrixes in [4].Z(1)me and Z

(2)me are the matrixes for the magnetic voltages on the

aperture due to unit electric currents on the microstrips of the lowerand upper sub-problems, respectively. Since these two matrixes havesimilar forms, only the expression ofZ(1)

me is given in AppendixB. Z(1)

em andZ(2)em are the matrixes for the electric voltages on the

microstrips of the lower and upper sub-problems, respectively, dueto unit magnetic currents on the aperture. These matrixes were derivedfollowing the same procedure explained in Appendix B for obtainingZ(1)me. The details of the calculations of the self terms of the matrixes

Zee andZmm in (10) can be found in [15].It is important to mention that the Green’s functions in this

paper are evaluated using the complex images technique [4], [5],as explained in Appendix A. This technique avoids the numericalintegration of Sommerfeld integrals used to perform the inverseFourier transform of the spectral functions. Instead, the spectralfunctions are approximated as finite sums of complex exponentialsusing Prony’s method [16] and the Sommerfeld integral is performedanalytically using Sommerfeld’s identity, resulting in a small numberof images with complex amplitudes and complex locations. Theauthors’ numerical tests have revealed that the complex imagesprovide at least a ten-fold reduction in computation time with less than1% error as compared to the numerical integration of Sommerfeldintegrals.

B. MS–PWG and MS–RW Transition

To solve the MS–PWG or MS–RW transitions shown in Figs. 2and 3, respectively, (10) needs to be slightly modified, as follows:

V(1)ext

0=

Z(1)ee Z

(1)em

Z(1)me Z

(1)mm + Z

(2)mm

I(1)exI(1)my

: (11)

All the matrixes in (11) remain the same as in (10), except forthe magnetic–magnetic coupling matrix of the upper sub-problem

[Z(2)mm], which results from a different set of Green’s functions. In the

case of MS–PWG transition, the complex image Green’s functionscorrespond to a magnetic current between two parallel conductingplates. The expressions for these Green’s functions can be found in[17]. In the case of MS–RW transition, the Green’s functions neededare those for a magnetic current inside a rectangular waveguide. Toavoid the complexity of the complex image solution of a waveguide,only the classical real images are used. The computational effortinvolved is explained in the following section.

III. N UMERICAL RESULTS

A. MS–MS Transition

Fig. 5 compares theS-parameters of the MS–MS transition ob-tained using the complex image technique, the numerical integrationof Sommerfeld integrals in [3], and FDTD. Fig. 5(a) shows that thedifference injS11j between the complex image results and the resultsof [3] is less than 1 dB over the range of frequencies between 2and 4 GHz. Fig. 5(b) shows a maximum difference injS21j of 0.15dB, while Fig. 5(c) shows a maximum difference injS31j of 1.2dB. However, there is practically no difference between the compleximage results and the FDTD results.

To demonstrate the accuracy of the complex image technique athigher frequencies, Fig. 6 shows a comparison with experimentalresults and FDTD theoretical results, over the range of frequenciesbetween 0.5 and 11 GHz for an MS–MS transition. The maximumdifference between the complex image results and the experiment isless than 2 dB over the whole frequency range investigated. Thisdifference may be attributed to the fact that the authors’ modelassumes infinitely thin perfect conductors and infinitely wide groundplanes. Fig. 7 showsjS21j andjS31j as compared to FDTD results forthe same frequency range and dimensions of Fig. 6. The maximumdifference injS21j and jS31j is less than 2 dB.

The above results verify the accuracy of the complex imagemethod as compared with the FDTD method and experiments. Asfor computational efficiency, the authors’ method requires less than4 min per frequency point on an 80 386 40 MHz PC. On the otherhand, the FDTD code had to be executed on a DEC ALPHA machineand the entire volume of the structure had to be discretized.

B. MS–Parallel Plate and MS–Rectangular Waveguide Transitions

Fig. 8 showsjS11j of an MS–PWG transition computed usingboth the complex image technique, described above, and the FDTDmethod. Clearly, the two sets of results agree very well. It should bementioned that there has been no attempt to optimize the structureto get better coupling.

As for the MS–RW transition, Fig. 9 comparesjS11j, calculatedusing the complex image method, with that calculated using theFDTD method. The maximum difference is only 3%. The numberof real images used to calculate each of the Green’s functions insidea RW is 30� 30.

The results for the MS–MS, MS–PWG, and MS–RW transitionsclearly show the accuracy as well as the versatility of the compleximage solution of the transitions in terms of the ability to solve avariety of transitions. Currently, work is being done to extend thistechnique to other multilayered transitions (e.g., coplanar waveguide(CPW) patch, CPW–RW).

IV. CONCLUSION

In this paper, an accurate and computationally efficient methodfor solving the problems of aperture coupling from microstrip-to-microstrip, microstrip-to-parallel plate, and MS–RW transitions has


(a)

(b)

(c)

Fig. 5. Comparison between the authors’ results, FDTD results, and theresults of [3] for theS-parameters of two aperture coupled microstrips. (a)jS11j, (b) jS21j, (c) jS31j. (Wu = WL = 0:254 cm, "u = "L = 2:22,hu = hL = 0:0762 cm, L = 1:5 cm, W = 0:11 cm.)

Fig. 6. Comparison between experimental results, complex image re-sults, and FDTD results forjS11j of two aperture-coupled microstrips.(Wu = WL = 0:2028 cm, "u = "L = 2:94, hu = hL = 0:0762 cm,L = 0:96 cm, W = 0:068 cm.)

been presented. This method uses a mixed electric–magnetic currentintegral equation formulation with moment-method and compleximages. The results were compared with the FDTD method andexperiments to demonstrate the accuracy of the proposed method.

Unlike the other methods presented in the literature to solvethe transitions, the efficient application of the moment method (ofdiscretizing only the strips and apertures) combined with the accurate

Fig. 7. Comparison between complex image results and FDTD results forjS21j, jS31j of two aperture-coupled microstrips. (Dimensions are given inFig. 6.)

Fig. 8. The return loss versus frequency for MS–PWG transition(W = h = 4 mm, "r = 2, Ls = 10 mm, aa = 10 mm, ba = 1

mm, D = 24 mm.)

Fig. 9. The return loss versus frequency for MS–RW transition (W = 2

mm, h = 0:8 mm, "r = 2:3, Ls = 4 mm, aa = 5 mm, ba = 1 mm,a = 23 mm, b = 10 mm.)

and the rapidly convergent complex-image Green’s functions, makesthis method more suitable for use with optimization software toachieve optimum designs of the different transitions analyzed.

APPENDIX A:COMPLEX IMAGE SPATIAL GREEN’s FUNCTIONS

(GA AND GA ) IN A MICROSTRIP SUBSTRATE

The Green’s functions (GA andGA ) in the space domain arederived using the complex image technique [4], [5]. The Green’sfunctions given below are for the lower substrate where the groundplane is located atz = 0, the source is located at a general location(x, y, z) and the field is located at a general location (x0, y0, z0),where�h � z, z0

� 0 (see Fig. 1). The corresponding Green’sfunctions of the upper substrate can be easily obtained by replacingz with �z and z0 with �z0.

A. Green’s Function forGA

Let k� be the spectral variable, andkz , kz be related tok� asfollows:

k2

z + k2

� = k2

0"r; k2

z + k2

� = k2

0 : (12)


Then the Green’s function forGA in the spectral domain is givenby

~GA =�0

j2kze�jk jz�z j

� e�jk jz+z j

+

TEpoles

j4Respk�pkz

k2� � k2�p+G(kp) S1(kz ; z; z

0) (13)

where

S1(kz ; z; z0) = �e

�jk (z�z +2h)� e

�jk (�z+z +2h)

+ e�jk (z+z +2h)

+ ejk (z+z �2h) (14)

G(k�) = RTE �

TEpoles

j4Respk�pkz

k2� � k2�p(15)

RTE =rTE10

DTE

rTE10 =

kz � kz

kz + kz

DTE = 1� rTE10 e

�j2k h: (16)

In (15), k�p andResp are, respectively, the TE pole and its residuegiven by

Resp = limk !k

(kp � k�p)RTE

j2kz: (17)

Using Prony’s method [16],G(k�) can be approximated as follows:

G(k�) =

N

i=1

axx

i e�k b (18)

whereaxxi and bxxi are complex coefficients.Performing the inverse Fourier transform of (13) and making use of

the Sommerfeld identity and residue theorem [16], the spatial Green’sfunction for GA is given by

GA (�; z; z0) = GA +GA +GA (19)

where � is the radial distance between the source and field. ThetermGA in (19) represents the contribution of the quasi-dynamicimages, which are dominant in the near field, while the termGA

represents the contribution of TE surface waves dominant in the farfield, and the termGA represents the contribution of a set ofimages with complex amplitude and complex location, which aredominant in the intermediate field. These three contributions are givenby

GA =�0

4�

e�jk r

r0�

e�jk r

r00(20)

GA =

TEpoles

�0

4�(�2�j)RespS1(kz ; z; z

0) (21)

GA =�0

4�

N

i=1

axx

i �

e�jk r

r1�

e�jk r

r2

+e�jk r

r3+

e�jk r

r4(22)

where

k1 = k0p"r

r0 = �2 + (z � z0)2

r0

0 = �2 + (z + z0)2

r1 = �2 + z � z0 + 2h� jbxxi

2

r2 = �2 + �z + z0 + 2h� jbxxi

2

r3 = �2 + z + z0 + 2h� jbxxi

2

r4 = �2 + �z � z0 + 2h� jbxxi

2: (23)

B. Green’s Function forGA

GA =�0

4�

1

�1

�jkx(j2kz )jkz

RqS2H(2)0 (k��)k� dk� (24)

whereH(2)0 is the Hankel function of the second kind of order zero,

and

S2(kz ) = 2e�jk (3h) � 2e

�jk (h) (25)

Rq =2("r � 1)k2z

(kz + kz )(kz + "rkz )DTEDTM(26)

DTM = 1� rTM10 e

�j2k h

rTM10 =

kz � "rkz0

kz + "rkz(27)

and DTE is given in (16).Next,�jkx is replaced in (24) with@

@x, and the following Prony

approximation are then performed:

Rq

jkzS2 =

N

i=1

azx

i e�k b (28)

whereazxi and bzxi are complex coefficients.Substituting (28) in (24) and performing the integration using the

Sommerfeld identity results in

GA =�0

4�

N

i=1

azx

i

@

@x

e�jk r

ri(29)

where

ri = �2 + �jbzxi

2:

APPENDIX B:COUPLING MATRIXES BETWEEN THE MAGNETIC CURRENT OF THE

APERTURE AND THEELECTRIC CURRENT ON THE MICROSTRIPLINE

Applying the Galerkin moment-method procedure and using theGauss–Chebyshev integration rule to take care of the transversecurrent distribution functions�en and �mn in (7) and (8) gives thefollowing expression for the coupling matrixZ(1)

me of the lowersub-problem:

Z(1)me = Z

xx

me + Zzx

me (30)

V0(x; y; x

0; y0) =

�e�jk r �k21(x� x0)2 � jk1ri � 1 r2i + 3(x� x0)2(jk1ri + 1)

r5i

(37)


where

Zxxme(m;n) =

�1

4�NM

N

k=1

M

L=1

Ixxm;n;k;L (31)

Ixxm;n;k;L =

l l

V (xL; y; x0

; yk) dydx0 (32)

whereN andM represent the number of Gaussian quadrature nodeson the magnetic and electric segments, respectively.xL and ykare the Gaussian quadrature nodes on the aperture and microstrip,respectively, andlen;k and lmm;L are the lengths of the electric andmagnetic segments, respectively. Also

V (x; y; x0

; y0) = 2h

(�jk1r0 � 1)

r30

e�jk r (33)

where

r0 = (x� x0)2 + (y � y0)2 + h2: (34)

On the other hand

Zzxme(m;n) =

�1

4�NM

N

i=1

azxi

N

k=1

M

L=1

Izxm;n;i;k;L (35)

whereNzx andazxi are, respectively, the number and amplitude ofthe complex images ofGA . Thus

Izxm;n;i;k;L =

l l

V0(xL; y; x

0

; yk) dy dx0 (36)

V 0(x; y; x0; y0) is shown in (37) at the bottom of the previous page,and

ri = �2 + �jbzxi2

: (38)

ACKNOWLEDGMENT

The authors would like to thank the Communications researchCentre in Ottawa, Ont., Canada for providing the experimental resultsof Fig. 6.

REFERENCES

[1] W. Grabherr and W. Menzel, “A new transition from microstrip line torectangular waveguide,” inProc. of 22nd EuMC, Espoo, Finland, 1992,pp. 1170–1175.

[2] L. Hyvonen and A. Hujanen, “A compact MMIC-compatible microstripto waveguide transition,” inIEEE MTT-S Dig., San Franciso, CA, June1996, pp. 875–878.

[3] N. Herscovici and D. Pozar, “Full-wave analysis of aperture-coupledmicrostrip lines,” IEEE Trans. Microwave Theory Tech., vol. 39, pp.1108–1114, July 1991.

[4] A. A. Omar and Y. L. Chow, “A solution of coplanar waveguidewith air-bridges using complex images,”IEEE Trans. Microwave TheoryTech., vol. 40, pp. 2070–2077, Nov. 1992.

[5] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed formspatial Green’s functions for thick microstrip substrates,”IEEE Trans.Microwave Theory Tech., vol. 39, pp. 588–593, Mar. 1991.

[6] K. Kunz and R. Luebbers,The Finite Difference Time Domain Methodfor Electromagnetics. Boca Raton, FL: CRC Press, 1993.

[7] J. Cheng, N. I. Dib, and P. B. Katehi, “Theoretical modeling ofcavity-backed patch antennas using a hybrid technique,”IEEE Trans.Microwave Theory Tech., vol. 43, pp. 1003–1013, Sept. 1995.

[8] J. Yook, N. I. Dib, and P. B. Katehi, “Characterization of high frequencyinterconnects using finite difference time domain and finite elementmethods,”IEEE Trans. Microwave Theory Tech., vol. 42, pp. 1727–1736,Sept. 1994.

[9] R. S. Elliott, Antenna Theory and Design. Englewood Cliffs, NJ:Prentice-Hall, 1981.

[10] D. G. Fang, J. J. Yang, and K. Sha, “The exact images of a horizontalmagnetic dipole above or within a microstrip substrate,” inProc. Sino-Japanese Joint Meeting Optical Fiber Sci. and EM Theory, Nanjing,China, 1987, pp. 693–698.

[11] R. E. Collin,Field Theory of Guided Waves. New York: McGraw-Hill,1991, pp. 23–26.

[12] A. W. Glisson and D. R. Wilton, “Simple and efficient numericalmethods for problems of electromagnetic radiation and scattering fromsurfaces,”IEEE Trans. Antennas Propagat., vol. AP-29, pp. 593–603,Sept. 1980.

[13] G. E. Howard, “Analysis of passive and active microwave integrated cir-cuits by the field approach,” M.A.Sc. thesis, Univ. Waterloo, Waterloo,Ont., Canada, 1988.

[14] R. F. Harrington,Field Computation by Moment Methods. Malabar,FL: R. E. Krieger, 1968, pp. 62–81.

[15] D. A. Huber, “A moment method analysis of stripline circuits throughmulti-pipe field modeling,” M.A.Sc. thesis, Univ. Waterloo, Waterloo,Ont., Canada, 1991.

[16] R. W. Hamming, Numerical Methods for Scientists and Engineers.New York: Dover, 1973, pp. 620–622.

[17] A. A. Omar and Y. L. Chow, “Coplanar waveguide with top and bottomshields in place of air-bridges,”IEEE Trans. Microwave Theory Tech.,vol. 41, pp. 1559–1563, Sept. 1993.

Effect of Conductor Backing on the Line-to-LineCoupling Between Parallel Coplanar Lines

Kwok-Keung M. Cheng

Abstract—A good estimate of the coupling effect between parallelcoplanar waveguide (CPW) lines is important, especially for monolithicmicrowave integrated circuit (MMIC) applications where unnecessarycrosstalk between conductors could be a serious problem. This papershows how these coupling parameters may be analytically obtained inthe presence of the back-face metallization. Closed-form formulas aredeveloped for evaluating the quasi-TEM characteristic parameters basedupon the conformal-mapping method (CMM). Very good agreement isobserved between the values produced by these formulas and by aspectral-domain method (SDM).

Index Terms—Coplanar waveguide, coupled lines.

I. INTRODUCTION

Coplanar waveguide is often considered to have free space aboveand below the dielectric substrate. However, this configuration hasbeen found unsuitable for monolithic microwave integrated circuits(MMIC’s), where the substrate is typically thin and fragile. Prac-tical realizations of coplanar waveguides (CPW’s) usually have anadditional ground plane beneath the substrate. The main advan-tages of this back-face metallization are principally to increase themechanical strength as well as to improve heat dissipation. Thestandard CPW, plus this additional conducting ground plane, is oftencalled conductor-backed CPW (CBCPW). Various approaches havebeen reported on the characterization of coplanar transmission linessuch as the finite-difference method [1], the spectral-domain method

Manuscript received July 7, 1996; revised March 24, 1997.The author is with the Department of Electronic Engineering, The Chinese

University of Hong Kong, Shatin, Hong Kong.Publisher Item Identifier S 0018-9480(97)04466-9.

0018–9480/97$10.00 1997 IEEE

analysis of slot-coupled transitions from microstrip-to-microstrip and microstrip-to-waveguides

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