916 I E F t RANSACTIONS ON UICROWAVL THEORY ANI) K C H N I Q U k S , LO1 36. NO 6, JUNE 1988

Frequency-Dependent Analysis of a Shielded Microstrip Step Discontinuity Using an

Efficient Mode- Ma t ching Technique

Ahsfrucf -The frequency-dependent characteristics of microstrip step discontinuities are analyzed by employing a mode-matching technique. The fields on both sides of a discontinuity are expanded in terms of the normal hybrid modes of the shielded microstrip line. The properties of these hybrid modes are determined by applying a previously developed analytical approach using singular integral equation techniques. In addition to propa- gating modes, higher order modes are also taken into account. The higher order modes are evanescent-type waves. The propagation constants of the evanescent waves in general are found to be complex numbers. A mode- matching procedure is developed to determine the reflection and transmis- sion coefficients of the discontinuity. The use of two types of products to treat the boundary conditions for the continuity of the tangential electric and magnetic fields results in a highly efficient and numerically stable solution. Numerical results are computed for several step discontinuities and the results are compared with previously published data.

I. INTRODUCTION COMMONLY encountered discontinuity structure A in microstrip lines is the abrupt change of strip line

width. This type of discontinuity is widely employed in low-pass filters, multisection quarter-wavelength trans- formers, stepped coupled line directional couplers, and many other microwave circuits. Therefore it is important to develop analytical techniques to compute accurately the characteristics of step discontinuities. These solutions can be incorporated into computer-aided design software packages of conventional and monolithc microwave in- tegrated circuits. Furthermore, solutions developed for step discontinuity can serve as a starting point for more general treatments of discontinuity structures in planar microwave and millimeter-wave networks.

Several approaches have been proposed to treat the step discontinuities in microstrip lines. Presently there are several comprehensive reviews on t h s matter, such as the books by Hoffmann [1], Gupta et al. [ 2 ] , [3], Edwards [4], and Mehran [5]. There is also a detailed revised description and a presentation of a rigorous frequency-dependent technique by Koster and Jansen [6].

A discrete element T-type two-port network consisting of two series inductive elements and a single shunt capaci- tive element has been proposed to model the step discon- tinuity approximately [7], [8]. Another approximate tech-

Manuscript received July 15, 1987; revised January 22. 1988. The authors are with the Department of Electrical Engineering, Na-

IEEE Log Number 8821071. tional Technical University of Athens, Athens 10682, Greece.

nique used by several authors is to treat the microstrip lines on both sides of the discontinuity with an equivalent magnetic waveguide model [9], [lo]. Then the rather simple normal modes of the equivalent microstrip line are em- ployed in formulating a mode-matching procedure. A sys- tem comprising an infinite set of equations involving the unknown expansion coefficients is obtained. This system could be solved numerically, by analytic matrix inversion, or by applying residue calculus techniques. The inherent limitations of these approximations and the necessity of seeking rigorous solutions have been discussed recently by Koster and Jansen [6]. The rigorous frequency-dependent method developed by Jansen [ll], [12] to analyze step discontinuities, microstrip, and slot end effects is based on a spectral-domain approach using hybrid-mode analysis. The boundary value problem is reduced to the planar structure surface by incorporating initially into the field solutions the boundary conditions on the shielding walls. Then a Galerkin approach in conjunction with a spectral- domain Green’s function interpolation technique with especially suited expansion functions is employed to solve the problem [ll].

In this paper the very basic concepts of the mode-match- ing technique are employed to formulate the boundary condition problem associated with the microstrip step dis- continuity problem. The fields on both sides of the discon- tinuity are expanded in terms of hybrid modes. The char- acteristics of these modes are determined by utilizing the analysis developed by Mittra and ltoh [13] in determining the dispersion characteristics of microstrip lines. The highly analytical approach used in computing the mode proper- ties allows the development of an efficient mode-matching procedure. Furthermore, a fast convergence with number of modes is obtained by using products involving both microstrip line orthogonal mode functions. This feature is the key point of the proposed technique and is in agree- ment with the conclusions of Chu, Itoh, and Shih [14] in solving the step discontinuity problem using a magnetic wall equivalent waveguide microstrip model. The method presented in this paper seems to have some resemblance to techniques developed by Schmidt [15] and Chang [16], although the modal characteristics are determined by an entirely different method and the mode-matching tech- nique used is completely different.

0018-9480/8S/0600-0976$01 .OO 01988 IEEE

UZUNOGLU et d.: FREQUENCY-DEPENDENT ANALYSIS 917

The necessity of computing exactly the spectrum of ph ~

higher order modes of microstrip lines leads us to the determination of evanescent waves with complex propa- gation constants having properties similar to those ob- served in finlines, as recently reported by Omar and Schiinemann [19]. To the authors’ knowledge, this is the first report of complex propagation constant waves in shielded microstrip lines.

In the following analysis the time dependence of field quantities is assumed to be exp( + jut) and is suppressed throughout the analysis. The free-space propagation con- stant is shown with k , = o / c where c = 3 X lo8 m/s is the velocity of electromagnetic waves in vacuum.

I ’ 7

4 dj. ~

A,

Fig 1 Microstrip step discontinuity geometry

11. FORMULATION OF THE BOUNDARY VALUE PROBLEM

A . Computation of Mode Characteristics of Microstrip

The geometry pertaining to the microstrip discontinuity is given in Fig. 1, where the step discontinuity is located at the z = 0 plane. The shielding box height and width are denoted by h and (2L), respectively. The substrate dielec- tric constant and thickness are denoted by c r and d , respectively.

It is well known that microstrip line supports a domi- nant mode whose characteristics at sufficiently low fre- quencies can be determined by employing quasi-static TEM mode analysis. However, because of the partial di- electric filling, only hybrid modes can be guided. In char- acterizing microstrip lines the primary quantity to be known is the dominant mode propagation constant. Be- cause of this, the literature on microstrip lines mostly concerns the computation of propagation constants with either quasi-static or frequency-dependent characteristics. There have been only few reports on the properties of higher order modes [17], [B]. The basic approach em- ployed in the present analysis is to use the analytical technique developed by Mittra and Itoh [13] to determine the properties of higher order modes. In the following, the same notation as in [13] is adopted.

The mode characteristics are determined by computing the nontrivial solutions of the system [13],

eqs. (22), (23), (24), (25) , (48), (49), (52) , and (53) respec- tively]. Furthermore, in order to compute the Dnm, Pnlq, and K , terms up to an arbitrary order of solution, new algorithms have been developed while in [13] only first- order results are presented. In the Appendix details of the procedures used in computing the values of D,,,, P,,, and K , are presented.

The mode propagation constants p (which in general can take complex values) and their associated mode expan- sion coefficients x m ( e ) and Xih) are computed numeri- cally by truncating the infinite system of (1) and (2) into a finite order system. To this end, it has been demonstrated by Mittra and Itoh [13] that the system in (l), ( 2 ) possesses highly convergent properties in terms of the truncation orderi This is especially valid when only the value of the propagation constant is desired. It has been shown that taking only the first-order terms (Le., terms with m = 1 and n = 1) and solving a 2 x 2 system, fairly good accuracy is obtained [13]. Assuming the numerical values of the p propagation constants that give a zero determinant for the system comprising (1) and (2 ) are known, the correspond- ing x:) and xih’ coefficients can be determined by arbi- trarily setting xie) = 1. Then the mode field distribution can be determined by first computing the potential func- tions,

sinh ( aL1)y) sinh ( ai1)d )

X?’

for 0 < y < d I f o r d < y < h

- sinh( ai2)( h - y)) sinh ( a$( h - d ))

B,“’

+ m m

( LpapmamDprnMmkp) 2:) +‘e ’= cos(L,x) m = l n = l

+ m

- ( b , D p , N , K p ) ~ ~ h ’ = O , p = 1 , 2 . . . (1) n = l

for 0 < y < d - cosh h - y ))

sinh(ai2)(h-d)) Bn(h)- I for d < y < h

n =1 + W

+ ( h ) = s in( i ,x ) n = l

where i = (2n - l)?r/(2L), 8p,,, is the Kronecker symbol, and 227, Tihh’ are the normalized mode expansion coeffi- cients [13, eqs. (12) and (13)]. The definitions of the a,, b,, c,, d , , M,, N,, X,, and Y, terms are given in [13, (4)

978 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. VOL. 36. NO. 6, JUNE 1988

where

f i = p / k o , p o = 477 x l o p 7 (H/m).

Then the electric and the magnetic field for a specific mode with p = p,,, can be computed by using the relations

TABLE I NUMERICAL RESULTS FOR THE MODE PROPAGATION CONSTANTS

AT 10 GHz AND 20 GHz

F r e q u e n c y = 1 0 G H z

w = 0 . 9 5 3 mm w = 4 . 5 7 2 mm

m 0, ( m n - l ) E- ( m . - l I

1 0 . 2 8 8 4 + j O . O

2 0 .0 -JO , 2 4 8 3

3 0 . 0 - ] 0 . 5 5 4 7

4 0 . O - ] O .5886

5 0 . 0 - j 0 . 9 6 5 2

6 0 . 0 - 1 1 . 0 2 0 9

0 . 3 0 2 1 t j O .O

0 . 0 - ~ 0 . 2 4 6 9

0 . O - 1 0 . 5 5 0 8

0 . 0 - 1 0 . 6 0 2 7

0 . 0 3 0 . 9 6 4 0

0 . O & ] O , 9 7 7 3

7 0 . 0 0 4 4 9 - ] 1 .OB49 0 . 0 - 1 1 . 0 7 3 1

8 - 0 , 0 0 4 4 9 - j 1 .Of349 0 . 0 - 1 1 . 1 0 2 3

9 0 . 0 - ~ 1 . 1 2 3 6 0 . 0 - ~ 1 . 1 0 7 5

where F r e q u e n c y = 2 0 GHz

1 0 . 5 8 2 5 + ] 0 .O

2 0 . 2 7 7 8 + ] 0 . 0

3 0 . 0 - ~ 0 . 4 1 2 7

4 0 . 0 - ~ 0 . 4 5 5 8

5 0 . O - ] O .8880

and the transversal field components are

6 0 . 0 - j l .0450

e,,,(x, y ) =VI*(') - 5; x VI*( ' ) (7) P m

0.6: 3 2 + ] 0 .O

0 . 2 7 5 1 + ~ 0 . 0

0 . 0 - 1 0 . 4 0 3 4

0 . O - j 0 . 4 8 0 2

0 . 0 - ~ 0 . 8 6 2 6

0 . 0 - 1 1 . 0 6 3 1

z, = 2.32, L = 4.76 mm, and h = 6.35 mm (see Fig. 1). The two microstrip widths are 0.953 mm and 4.572 mm.

with a a

V,=,?- + 9 - . ax ay In (5)-(8) the subscript m indicates the mode number.

The microstrip line being an inhomogeneously dielectric loaded waveguide, the mode power orthogonality is satis- fied [19]:

where A is the cross-sectional area of the microstrip and the mode power coefficients C,,, are computed by substitut- ing (3), (4), (7), and (8) into (9) and performing the integrations for the x and y variables. The expression for the C,,, coefficient is given in the Appendix.

The closed wall nature of the shelded microstrip line ensures the existence of only discrete eigenwaves, exclud- ing the possibility of having a continuous spectrum of modes observed in open microstrip line. Then, the roots p, of the determinant of the system comprising (1) and (2) should be determined carefully and the modes should be treated as an ordered set. In practice only a single mode is allowed to propagate on microstrip line. There are an

in some cases, it is possible to have complex /3 roots of the determinant of the system (1) and (2). These complex roots are traced by applying the procedure described in [19]. The properties of the complex microstrip modes will be dis- cussed elsewhere. In Table I mode propagation constants of two sample cases are given at 10 GHz and 20 GHz.

In order to determine the numerical values of the ,8, propagation constants in the first place, the simplified 2 X 2 version of (1) and ( 2 ) is solved. Then, the values of p,,, and the expansion coefficients 7:) and XL'), ( zie' = 1) are determined accurately by using ( 2 M ) number of equations in the system (l), (2). To t h s end, a Regula-Falsi al- gorithm has been adopted to compute the roots of the determinant equation. In most cases only a single propa- gating ( p = p, = real) wave is encountered, although for sufficiently large c r and k,d values a second propagating mode could exist.

The convergence patterns of the modal field expansion coefficients A:) for a sample microstrip line are presented in Table 11, where a very good numerical stability and convergence are observed. Notice that the low-order coeffi- cients ( A i e ) , Ac,'); see the Appendix for the definition of A:)) in the case of evanescent modes are very insensitive to the order of solution M .

infinite number -of evanescent waves. Numerical computa- tions showed that usually the evanescent waves have purely imaginary propagation constants and therefore they are strongly attenuated along the propagation axis. However,

B.

In order to determine the frequency-dependent char- acteristics of the step discontinuity, an incident propagat-

Technique

UZUNOGLU et ul.: FREQUENCY-DEPENDENT ANALYSIS

TABLE I1 MODAL FIELD EXPANSION COEFFICIENT CONVERGENCE

979

,In 1 2 3 4 5 6 7

B(mm-1) 0.3000tjO.O 0.3017tjO.O 0.3020tjO.O 0.3021tjO.O 0.3021tjO.O 0.302ltj0.0 0.3021tjO.O

5.0223tj0.0 4.9958tjO.O 4.9910tjO.O 4.9905+jO.O 4.9902tjO.O 4.9902+jO.O 4.9901tjO.O -2.7243tjO.O -2.6457tjO.O -2.6624tjO.O -2.6636tjO.O -2.6647tjO.O -2.6649tjO.O

0.4844+jO.O 0.5160tjO.O 0.5202tjO.O 0.5214tjO.O 0.5218+jO.O 0.2677+jO. 0 0.2611tjO. 0 0.261O+jO. 0 0.2608+jO. 0

-0.121OtjO.O -0.122OtjO.O -0.1221+jO.O -0.0521tjO.O -0.0517tjO.O

0.0310tjO.O B ( mm-1) 0.0- jO.2470 0.0-j 0.2469 0.0-j0.2469 0.0- j0.2469 0.0-j 0.2469 0.0- j0.2469 0.0-j 0.2469

0.0- j6.8 313 O.O-j6.8038 O.O-j6.8041 0.0-j6.8041 0.0- j6.8041 0.0- j6.804 1 O.O-j6.804 1 -3.0447tjO.O -3.0285tjO. 0 -3.032 5 +$I. 0 -3.0332tjO. 0 -3.0334tjO. 0 -3.0334tjO. 0

0.6759tjO.O 0.6828tjO.O 0.6836tjO.O 0.6838tjO.O 0.6838tjO.O

A p ) 0.1865tjO.O 0.1854tjO.O 0.1854tjO.O 0.1854+jO.O -0.1171tjO.O -0.1172tjO.O -0.1172tjO.O

-0.0376tjO.O -0.0375+jO.O 0.0279+jO.O

O(mm-1) 0.0- jO.6020 0.0- jO.6026 0.0- j0.6027 0.0- jO.6027 0.0- jO.6027 0.0- jO.6027 0.0-j 0.6027

0.0-j2.7057 0.0-j2.7030 O.O-j2.7028 O.O-j2.7028 O.O-j2.7028 0.0-j2.7028 O.O-j2.7028 -3.9191tjO.O -3.8977tjO.O -3.9019tjO.O -3.9022tjO.O -3.9021+jO.O -3.9020tjO.O

A p ’ 0.7390tjO. 0 0.7444tj 0.0 0.7449t j 0.0 0.7448tj 0.0 0.7447tj 0.0 0.1941+jO.O 0.1935tjO.O 0.1935+jO.O 0.1936tjO.O

-0.1218+jO.O -0.1218+jO.O -0.1218tjO.O -O.O381+jO. 0 -0.0382+jO. 0

0.0286tj0.0 f= lOGHz, c ,=2 .32 , L=4.76mm, h=6.35mm(seeFig . 1),and w=4.572mm.

ing wave ( p = pl) from z = - 00 toward z = 0 (see Fig. 1) is taken. Furthermore, assuming the mode properties of the microstrips on both sides of the discontinuity are known, inside the z < 0 half-space the transversal electric and magnetic field components can be expressed as a superposition of the incident and the sum of all the reflected waves:

E , ( x , y , z ) = e , ( x , y)e-JPlz+

H , ( x , y , z ) = h , ( x , y)e -JP lz -

+ m

A,e,(x, y)e+”+ (IO)

~ , h , ( x , y)e+@n’

n = 1 + m

n = 1

( Z < O > (11) where A , ( n = 1,2, - ) are unknown coefficients to be determined. The corresponding transversal field compo- nents inside the z > 0 semi-infinite space can be written as follows:

+ m

~ ; ( x , y , z ) = B,,,eA(x, y)e-JPhr (12)

H , ’ ( x , y , z ) = ~ , , , h h ( x , y)e-@L’ (13) ( Z > O >

m = l + m

m = l

where the prime symbol is used to distinguish the two different microstrip line modal field distributions and

propagation constants. Again B,,, ( m = 1,2, . . . ) are un- known coefficients to be determined.

Applying the boundary conditions on the z = 0 plane for the continuity of the transversal field components,

+ m + m

e l ( x , Y ) + C Anen(x, Y ) = C BmeL(x, Y )

hl(x, Y ) - C A n h n ( x , Y ) = C BmhL(x, Y )

(14)

(15)

n = l m = l t m + m ( z = o )

n = l m =1

are obtained. Then, in order to determine the unknown A , and B, expansion coefficients, (12) and (13) should be transformed by some means into an infinite system of equations and the dependence on the x and y coordinates should be removed. To this end, it is possible to pursue several strategies in computing the A , and Bn coefficients. A quite similar problem of choosing the best mode-match- ing approach when the magnetic wall microstrip model is employed has been addressed recently by Chu, Itoh, and Shih [ 141. Furthermore, the trial of several approaches showed that there is also an optimal strategy in terms of convergence behavior in the present hybrid mode analysis. To this end it is found that the optimal way of solving (14) and (15) is to take the vector products of them with e m ( x , y ) and h h ( x , y ) , respectively. Following the ortho- gonality properties of the e,,,,h,,, and e ; , h h eigenwaves

980 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 36, NO. 6, JUNE 1988

0.3 - 1 2 0 G H z 20 GHz

10 GHz

/._----- -.___.___.-_ -. .>.-./*--*-.

0.1

4 8 12 M Fig. 2. Convergence of lSlll reflection coefficient with M (truncation order of (1) and (2)) and N (number of modes)

taken into account on both sides of discontinuity for w2/wI = 2, wl/d = 3, and E , = 2.32.

given in (9), it is found that + m

m = l + m

n = l

where

are coupling integrals and Cn,C,l have already been de- fined in (9).

Eliminating the Bm coefficient in (16) and (17), a new set of infinite system of equations is obtained:

n = 1 , 2 , 3 , . . * (19)

T h s system constitutes the basis of the analytical ap- proach of t h s paper. In order to determine the unknown expansion coefficients A, , A , ; * a , the infinite summations appearing in (19) should be truncated into finite order by taking N terms (modes). Then, convergence of the com- puted results is examined to verify the accuracy. This subject will be treated in the next section.

The coupling integrals Cmn are computed by substituting the field expressions given in (7) and (8) into (18) and then performing the integrations for the x and y variables. The result of tlm analysis is given in the Appendix. After determining the A , reflected wave coefficients, the trans- mitted wave expansion coefficients Bn are computed easily by using (17), where of course, the same Nth-order trunca- tion of the infinite sums is used. In practice the interested quantities are the doniinant mode reflection S,, and trans- mission S,, coefficients. Then S,, = A , and according to the definition of S parameters

s,, = B , E. (20)

111. NUMERICAL RESULTS Numerical computations have been performed by apply-

ing the theory developed in Section 11. For each pair of microstrip lines the spectrum of propagation and evanes- cent waves is determined up to sufficient order by takmg 2 M equations in the system (l), (2). A perfect agreement with the /? values given in [13] is observed. Then, (19) is solved numerically by keeping N terms in the infinite summation. Both M and N truncations affect the accu- racy of the obtained results for the S,, and S,, parameters. In each case, extensive convergence tests have been per- formed to verify the accuracy of the obtained results. In Figs. 2 and 3 sample convergence patterns are presented. In general, the phase quantities and /s21 are much more sensitive than the corresponding amplitudes IS,, I and &,I. The truncation order M of the system of (l), (2) seems to have the primary role in the convergence. The numerical results presented in this paper have been com- puted by using M = 8 and N = 8 truncations.

The “ relative convergence” aspect [20] by taking a non- equal number of modes in (19) in the two microstrip lines has been investigated. It has been shown that some im- provement in the convergence speed can be achieved by taking the ratio of mode numbers equal to the ratio of microstrip widths. However, because of the smooth con- vergence of the results presented in this paper no use is made of this property.

Several independent checks have been performed nu- merically, such as the validity of the power conservation, reciprocity (S,, = S,,), and the boundary conditions on the y = d plane (see Fig. 1) by the mode field distributions. Furthermore, the orthogonality conditions given in (9) have been verified by direct numerical computation.

Step discontinuities on low E , = 2.32 (polyethylene) and high E , = 10 (alumina) dielectric constant substrates have been considered. In all the computed results the shielding box dimensions are taken to be

2 L = 0.53 mm h = 6.35 mm.

UZUNOGLU et p i . : FREQUENCY-DEPENDENT ANALYSIS

I

9x1

I

~ r 1 2 . 3 2

4 8 12

180°-

1 70°-

\ '\

6 8 N 2 4

Fig. 3. Convergence of & with M and N truncation orders for the same discontinuity of Fig. 2.

I 0.5

I SllI

0.3

0.1

I I I ' d 0.02 0.04 -

i o

Fig. 4. parameters with d / A , ( A , being the free-spqce wavelength) for several discontinuity dimensions and c r = 2.32.

Variation of ISl,[ and

/&I 0

0 -

0 -10 -

I I 1 d 0.02 0.03 0.04 7

I I I d 0.02 0.03 0.04 -

ho Fig. 5 . Variation of & and /s21 phase parameters with d / X , for

several discontinuity dimensions and c r = 2.32.

1.0

0.6

0.;

W1 2

3

3

I I 1 d a02 0 0 3 004 -

A 0 Fig. 6 . Variation of lSlll and & l l parameters with d / A , for several

discontinuity dimensions, c r = 10, and w1 / d = 1.5.

9x2

1 8 0"-

160"-

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 36, NO. 6, JUNE 1988

E r ' l O

- w2 w1 0"- w 2 -

0 -10 -

The scattering parameters S,, and S,, are presented in terms of their amplitudes and phases versus the normal- ized substrate thickness d/A, , where A, is the free-space wavelength (A, = 2 r / k 0 ) .

In Figs. 4 and 5 results are given for the c r = 2.32 substrate, several microstrip width w1/w2 ratios, where w1 and w2 are defined in Fig. 1, and two w l / d ratios. The results obtained are compared with those given by Koster and Jansen [6], which are also drawn on the same figures for w , /d=3 and w 1 / w 2 = 2 and 3. Good agreement is observed for the JS,,I, 1S2J amplitude quantities, while the comparison of and /szl phase quantities is also reasonably good.

Discontinuities in microstrip lines with c, = 10 alumina substrate have been computed and are presented in Figs. 6 and 7. A similar behavior is observed with the c r = 2.32 substrate case.

IV. CONCLUSIONS A frequency-dependent analysis has been presented for

the microstrip discontinuity problem. The proposed method is shown to be efficient in terms of the required numerical labor and is easy to program. The evanescent mode spectrum of microstrip lines has been investigated and the existence of modes with complex propagation constants has been verified. Numerical results have been presented for several discontinuity structures and a com- parison with previously published data has been per- formed. In principle the same method could be used to treat other types of discontinuity problems in microstrip lines and the junctions between different types of wave- guides and microstrip lines.

APPENDIX

A . Computation of Pmq, D,, and K , Terms

Following a standard trigonometric analysis, the Pmq terms defined in [13, eq. (68)] after algebraic manipula- tions, are found to be

I q = 2 u

I q = 2 u - l

where a, and a , are defined in [13, eq. (61)] and the A , , and Bnq are given by the recursive relations

U Z O n - 1

A , , = 2 n - l q = o

2" ,cl Bn4( R ) = - n ( n - l ) ( n + / - I ) , U Z O 4 = 11 ( 2 u ) ! / = I

n - 1

B =l. "4 q = O

UZUNOGLU et ul.: FREQUENCY-DEPENDENT ANALYSIS 9x3

Following an analogous procedure, the Dnm and K n terms, defined in [13, eqs. (73) and (72) respectively], are given as follows:

where

when k - q +1= even and q -1 5 k I \ o

1 2k - I 0

' k q =

and

elsewhere

k - q + l 2 when k - q + l = e v e n a n d q + l s k elsewhere

and 11, 12, 13,14 are the following integrals:

11 = idsinh ( a f ) y ) . (sinh ( a f ' y ) ) * dy

12 = i dcosh ( a i l ) y ) . (cosh ( ail )y))* dy

1 3 = Lh sinh ( ai2)( h - y )) . (sinh ( ai2)( h - y ))) * dy

14 = [cash( ai2)( h - y))-(cosh(ai2)( h - y ) ) ) * d y .

C. Computation of the Cmn Terms (m(n) = 1,2,3, . . ) The coefficients Cmn defined in (18) are given as follows:

. ( o B ( e ) a ( 2 ) yn k n k n + Bih,)[,i * . I4

where

a( l ) k m = /- a i 2 = /-

a(l) k n = \i'm ai: = /-

and 11, 12,13,14 are the following integrals:

11 = J o d s i n h ( a i ~ y ) . ( s i n h ( a i ~ y ) ) * ~ y

12 = i dcosh ( a i ; y ) . (cosh ( a f i y ) ) " dy

1 3 = (sinh (ai?( h - y ) ) .(sinh (ai:( h - y )))* d y

14 = Lhcosh (ai: ( h - y )) . (cosh ( ai:l ( h - y ))) * dy .

- - B;e) B p

BAe) = Bib) = [3] sinh ( a!2)( h - d ) ) sinh ( .I,"( - )> Microuuoe Circuirs. Dedham, MA: Artech House. 19x1. sec. 6.2.4

K. C. Gupta, R. Garg, and R. Chadha, Computer Aided Desigii of

REFERENCES [ l ] R. K. Hoffman, Iniegrierie Mikrowelleiishculiungeii. Berlin:

Springer-Verlag, 1983. [2] K. C. Gupta, R. Garg, and I. J. Bahl, Microstrip Lines und Slotlines

Dedham, MA: Artech House, 1979, sec. 3.4.3.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 36, NO. 6, JUNE 1988

T. C. Edwards, Foundations for Microstrip Circuit Design. New York: Wiley, 1981, secs. 5.8, 5.13. R. Mehran, Grundelemente des rechnergestiitzten Entwurfs von Mikrostreifenleitungs - Schaltungen . Aachen: Verlag H. Wolff, 1983. N. H. L. Koster and R. H. Jansen, “The microstrip step discontinu- ity: A revised description,” IEEE Trans. Microwave Theory Tech ., vol. MTT-34, pp. 213-222, 1986. P. Benedek and P. Silvester, “Equivalent capacitances for micro- strip gaps and steps,” IEEE Trans. Microwave Theory Tech., vol.

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Nikolaos K. Uzunoglu (M82) received the B Sc. degree in electronics engineering from the Istanbul Technical Umversity, Turkey, in 1973 He obtained the M.Sc and Ph D degrees from the University of Essex, England, in 1974 and 1976, respectively.

He worked for the Hellenic Navy Research and Technology Development Office from 1977 to 1984. During thls period, he also worked on a part-hme basis at the National T e c h c a l Um- versity on electromagnetic theory In 1984, he

was elected Associate Professor at the National Techmcal Institute Um- versity of Athens, the position that he holds presently. His research 11 terests are microwave applications, fiber optics, and electromagnetic ‘heory.

8

I Christos N. Capsalis was born in Nafplion, Greece, on September 25, 1956 He received the Diploma of E.E. and M E from the National Techmcal University of Athens (NTUA), Greece, in 1979 and the bachelor’s degree in economics from the Umversity of Athens, Greece, in 1983 He also received the Ph.D degree in electrical engineering from NTUA in 1985.

Since January 1982, he has been a Research Associate in the Department of Electncal En- gineenng at NTUA. In November 1986 he was

elected Lecturer at NTUA, the position that he currently holds. His main research interests are in the electromagnetic field area, with emphasis on scattenng and propagation at millimeter and optical wavelengths

Constantinos P. Chronopoulos was born in Tnp- olis, Greece, in 1964 He received the Diploma in electncal enpeering from the National Techm- cal University of Athens (NTUA) in 1986 Since then, he has been working at NTUA toward the Ph D. degree with a scholarship from the Bodosalus Foundation His thesis deals with mi- crostnp discontinuities His m a n field of interest includes fiber optics and millimeter-wave apph- cations

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