PROPOSAL TO THE NATIONAL SCIENCE FOUNDATION Attn: Dr. Lawrence Goldberg for continuation of COMPUTER MODELING OF MICROSTRIP INTERCONNECTS IN MILLIMETER WAVES NSF Grant ECS-8657951 Submitted by P.B. Katehi, Principal Investigator Radiation laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 Period of Study: May 1, 1989-April 30, 1990 Cost: $25,000 Date of Submission: May 20, 1989.

PROGRESS REPORT The research conducted during the second year of this study concentrated on the following two problems; a) Characterization of a via-hole b) Switches in coplanar waveguide. This second task replaced the one on the characterization of superconducting interconnects that we had proposed last year, due to the uncertainty in the validity of the existing models which is intesified by the lack of experimental verification. The progress in each one of the two tasks is described below. a) Characterization of a Via-Hole. The theoretical formulation that we developed last year was applied to the problem of the via but it exhibited numerical instabilities which were inherent in the solution due to the dramatic change in size between the via and the surrounding cavity. As a result, the method which originally was a combination of a modal expansion and a finite element method had to be modified in order to improve the convergence of the solution. At first, the shape of the via was changed to a hollow rectangular conducting post as shown in figure 1 in order to make the application and as a result the testing of the method easier. Second, the original boundary value problem was divided into two simpler ones; the boundary value problem i) inside of the conducting via ( volume V1), and ii) outside the via (surrounding volume V2). Due to the properties of the electromagnetic field in the conducting via as oppose to the surrounding volume the two problems can be successfully decoupled. In fact, the fields inside the conductor of the via can be solved through an appropriate integral equation and their values on the surface of the via provide the outside region with appropriate boundary conditions. Furthermore, the fields in volume V2 may be found by solving a second integral equation subject to the appropriate boundary conditions. During the past year we completed the formulation of the method and we applied it to a simpler two dimenstional problem in order to check its convergence and verify its accuracy. The problem that we solved was that of wave propagation along the printed line of

figure 2a. The application of this method was successfull and some preliminary results are presented on figure 2b. During the third year, this method will be applied to the via-hole of figure 1 in order to derive frequency dependent equivalent circuits or scattering parameters including conductor o0 dielectric losses. b) Switches in Coplanar Waveguide. For the complete characterization of this circuit element, the diode's on and off states have to be considered separately and their frequency dependent equivalent circuits have to be derived accurately. Due t the planar nature of these switches, an integral equation method can be applied successfully for their analytical modeling. During this year, results will be derived for the simple developed, at first, for the case of a simple coplanar discontinuity such as an open end. The details of this development are presented in [1]. During this coming year, results will derived for this simple discontinuity and will be compared to available experimental data.After successfull verification of the validity of the developed programs we will proceed with the study of the switches The theoretical data will be tested by comparing to measurements. The experiments will be performed at NASA Louis Reasearch Center,Ohio. ii

1) Characterization of the via hole of Figure 1 using the developed method of the combined integral equations. Results will be given in the form of equivalent circuits or sca ttering parameters. 2) Characterization of open end discontinuities in coplanar waveguide using a surface integral equation. The method will be tested by comparing to available experimental data. 3) Extension of the method to characterize switches in coplanar waveguide. Results will be given in the form of equivalent circuits or scattering parameters for the on and off state of the diodes. REFERENCES [1] N.I.Dib and P.B. Katehi, "Theoretical Analysis of Coplanar Waveguide Open Circuit Discontinuity", Radiation Lab Report, The University of Michigan, NSF-024601-2-T, May 1989. ini

Figure 1: Simplified Geometry of a Via-Hole v

-w - Microstrip line Passivation /Silicon nitride.;...,..... Polyimide dielectric GaAs Ground plane (2a) Conductor Losses vs. Strip Width 0.6 _ 0r*^~~~~ \ h~h Silicon -5 lm o.s- \h Polyimide = 95 gm EQ]?~~~~ \h GaAs =3.9 mm cw~a~~ I' ~ \ ~h Air = 6 mm 0.4- f= GHz o b = 2 a = 20 mm co 0.3 U 0.2 0 1 2 3 4 w (mm) (2b ) Figure 2: Conductor Losses in Microstrip Lines with Passivation Layer vi

NSF: ECS-8657951 024601-\-T "Theoretical Analysis of Coplanar Waveguide Open Circuit Discontinuity" N.I. Dib P.B. Katehi May 1989

TABLE OF CONTENTS Page TABLE OF CONTENTS....................................................................................i LIST OF FIGURES........................................................................................ A bstract....................................................................................................... 1 1. INTRODUCTION.................................................................................2 2. ANALYSIS........................................................................................4 2.1 Introduction. 4 2.2 Derivation of Green's Functions........................................................4 2.2.1 Green's function in region (1) for a z-directed magnetic current................................................................. 6 2.2.2 Green's function in region (1) for a y-directed magnetic current............................................................... 16 2.2.3 Green's function in region (2)................................................20 2.3 Application of Method of Moments...................................................23 3. SUMMARY....................................................................................... 33 Appendix A................................................................... 34 4. REFERENCES............................ 36 i

LIST OF FIGURES Page Figure 1. A bridge coplanar waveguide............................................................3 Figure 2. A cutview of a coplanar waveguide open circuit discontinuity inside a cavity............................................................................5 Figure 3. Four subportions to be solved. Other than the impedance boundary side, the sides are assumed to be perfect conductors (cavity walls).......................7 Figure 4. The magnetic source raised to apply the boundary conditions..................... 10 Figure 5. The magnetic current source raised to apply boundary conditions................ 17 Figure 6. Structures to be solved to obtain Green's function in region (2)............................................. 21 Figure 7. Geometry for use in basis function expansion of magnetic current............... 27 ii

THEORETICAL ANALYSIS OF COPLANAR WAVEGUIDE OPEN CIRCUIT DISCONTINUITY N. Dib, P. Katehi Radiation Laboratory, University of Michigan, Ann Arbor, MI Abstract The theoretical analysis of a coplanar waveguide open circuit discontinuity inside a rectangular cavity is presented. First, the dyadic Green's function of a y- and z- directed dirac delta magnetic currents inside a cavity will be derived. Then the method of moments will be used to solve the integral equation for the unknown magnetic current distribution in the slots. The scattering parameters of such discontinuity could be determined from the knowledge of the magnetic current distribution. 1

1 Introduction The widespread use of microwave integrated circuits (MIC's) in recent years has caused rapid progress in its theory and technology. The first transmission line used in MIC's was, indeed, microstrip laid on dielectric substrate, and then other transmission lines such as slot lines, coplanar lines, finlines, and so on, were introduced and improved. Initially the analysis for this class of transmission lines was invariably a quasi-TEM approximation which can yield satisfactory results at low frequencies. However, at high frequencies its weakness becomes apparent. To feature the frequency dependence of these lines, a full wave analysis must be employed. Recently, new uniplanar circuit configurations for monolithic MIC's were proposed [1]. The fundamental components in these uniplanar MIC's are the coplanar waveguides (CPW), slot lines and air bridges (Fig. 1). Coplanar waveguides (CPW) offer several advantages over conventional microstrip line: there is no need for via holes which simplifies mounting of active and passive devices and they have low radiation loss. These as well as other advantages make CPW ideally suited for MIC's [2]. This report presents a full wave analysis of one type of coplanar discontinuity, namely the CPW open circuit. The ultimate goal of this study is to characterize various coplanar discontinuities up to the terahertz region. This is intended to be a step towards characterizing the coplanar air bridge discontinuity and other discontinuities. 2

conductor dielectric substrate Fig. 1 A bridged coplanar waveguide 3

2 Analysis 2.1 Introduction A CPW open circuit is shown in Fig. 2. The CPW lies inside a rectangular cavity with a multidielectric structure. The main steps in the formulation of the problem are as follows: 1. Derive the fields in the two regions directly above and below the conductor strip. 2. Formulate the integral equation. 3. Solve this equation using the method of moments. In the formulation, a few simplifying assumptions are made to reduce the complexity of the problem: 1. The width of the slots is small compared to the coplanar line wavelength Ag. This will facilitate the assumption of undirectional magnetic currents in the slots with negligible loss in accuracy. 2. The dielectric layers are lossless and the conductors are perfect. However, the analysis can be easily extended to take losses into consideration. 3. The time dependence is of the form ejit which will be suppressed throughout the analysis. 4. The input is a travelling wave with variation ej3 where, is the propagation constant of an infinite coplanar line [3]. 2.2 Derivation of Green's Functions In this section, the tensor Green's function [G] will be derived for the fields in regions (1) and (2) (see Fig. 2). The transmission line theory will be used to transform the surrounding layers into an impedance boundary. Throughout the analysis, LSE(TE-2) and LSM(TM-x) modes are used to derive the Green's function. The dyadic Green's function denotes the fields of a point source. Hence, the electric field can be computed from 4

Fig.2 A cutview of a coplanar waveguide open circuit discontinuity inside a cavity. 5

E =J J. GdS'd + f M * (1) where the integration is done over the surface of the source. In rectangular coordinates [G]e, for example, becomes GE = Gz^X + Gasy + Gzzaz + Gvyyx + Gvyyy + GZyyz + Gz,,z + Gziz + Gzzz (2) where Gij is the jth component of the electric field due to a unit idirected electric current element. In the same manner, the magnetic field can be derived as a I J. GdS'+. M- G(dS' (3) In our problem, the two slots are assumed to have magnetic currents. In order to obtain the scattering parameters, the distribution of this magnetic current must be determined. Using the equivalence principle, our original problem is divided into four subproblems, Fig. 3. We have to solve for the Green's functions in both regions due to magnetic currents in the y and z directions. After that has been accomplished the continuity of the tangential fields at the interface will be used to arrive at the integral equation. 2.2.1 Green's function in region (1) for a z-directed magnetic current The fields due to an infinitesimal z-directed magnetic current inside a cavity will be derived. Fig. 4 shows the structure with the magnetic current alleviated from the ground of the cavity. The magnetic current is assumed to be M = aS6(z - z')6(y - y')6(z - z') (4) Notice that at the end of the analysis x' will be substituted by zero. As mentioned before, a hybrid mode analysis (LSE and LSM) will be considered [4]. The following vector magnetic potential A and electric vector potential F for the LSM and LSE modes respectively are assumed 6

d1 d2 —..... —-------------—... —.. —... —-—........ impedncboundary side impedance boundary side X d1 d d2 (2) 1 /o impednc undry (btom idimpedance boundary (bottom side) impedance boundary (bottom side) Fig.3 Four subproblems to be solved. Other than the impedance boundary side, the sides are assumed to be perfect conductors (cavity walls) 7

A = aF, F- a. (5) Through the manipulation of Maxwell's equations V x E= -jwfHf (6) x H = jwE (7) along with - 1- H -= x A (8) 1E = - x F (9) one can obtain the field components in terms of (5) as Ex 1 027k 10 (o ~E, = w ——.[~ a~ (10) 1 92, 1 Or)f Ey. - -- (11) jwtte aOxy e az 1 9! 19Or) Ez = + - (12) jwwIE8x9z eE y Hz = --- - 2-a 2] (13) jwt~E Oy2 az2 1O'^b 1 aO2 Hy = i —+ jw1 - 9 (14) A az )jwE Edxay 1 9- + 1 02 (1) t ay jWJLE OxOz Both vector potentials should satisfy the homogeneous wave equation (away from the source) V2k + +k = 0 (16) V2' + kf2,, = 0 (17) 8

where k2 = w2lE1As shown in Fig. 4, it is assumed that the current source divides the cavity into two regions I and II. Applying the method of separation of variables to solve (16) and (17) with the following boundary conditions Exy = 0 at z = 0, (18) E.-=O at z=O,a (19) EI = 0 at x = (20) y,z one can obtain [A - d ) ) n= m O[AmnSin(,( - di)) + Bmncos(k=(x - dl))]cos(-y)cos(- z) (21) a I = SEn=o_ oCmnsin(kz,)cos( y)cos( Tz) (22) =o =o [Kmnein(k{( x - d1)) + Nmncos(k(x - dl))]sin(m-y)sin( z) (23) a I E 00 - r n70II = __ o Dncos(kcx )sin( — y)sin( -z) (24) n M ma T In the above equation, the following equation is satisfied 2 2 2 2 kx + k +y kz k where MW n7r 2 2 ky = --, k - and k1 - w 2=l a 1 To simplify the notation, one can consider Oq(x,y, z) = 2nZm~(x)cos( kyy)cos5(kzz) (25) 4p(x, y,z) = SJn (),m)(x)sin(kyy)sin(k,z) (26) 9

LSE LSM T or Z,or d I y Fig.4 The magnetic source raised to apply the boundary conditions 10

where O(x) = / o /J ((x,y,z)cos(kyy)cos(kz)dydz (27) la 2O O 4,1 ra Ot(x) = - j (x,y, z)sin(kyy)sin(kzz)dydz (28) q(x) and ~(x) can be considered as the coefficients of the double Fourier series of q and p. In other words, one can consider them as q$ and 4' in the Fourier domain. Substituting (25) and (26) in (10) - (15) one can obtain the fields in the Fourier domain as E~ = - [k2- k 2] (29) JW/LE1 Ey = -kz + I k (30) e1 jW/et1 Oa E, = k —ky +- kz (31) el jwlAel 02 H = — [k2 - kc] (32) jwILE1 - 1 04) 1 - Hy = -- kyc + -k (33) JW-1 k, p iz - kz -- k 1- (34) jWJLE1 aTX where E.= EnE,,sin(kyy)sin( k,z) (35) Ey= Em,EnEcos( kyy)sin( kz) (36) E, = EmEnEzin(kyy)cos( kz) (37) H.= EEnHcos(kyy)cos( kz ) (38) Hy = E EnHsin( kyk)cos( kcz ) (39) H, = EmEfHcos( ky )sin( kz ) (40) 11

Expressions for 5 and b are obtained from (21) - (24) as (I = Amnsin(k(z - dl)) + Bmnco(k,(x - dl)) (41) /II = Cmsin( kc ) (42) 1/I = Kmnsin( k( x-dl)) + NmnCco (k,( -ddl)) (43) tit = Dmnco( kcz) (44) Up to this point, one has to solve for the constants A, B, C, D, K and N, where the subscript mn will be suppressed for simplicity. The following six boundary conditions will be employed to solve for the six unknowns, Ef= EII at x=x' (45) H =H"y at x x (46) Hf =HII at x =x' (47) EI LSE = ZLSE at = dl (48) Hz EI ( ) LSM ZLSM at x = d (49) HI 1 E -~ E = 6(z - x')^(y - y' )(z - z') (50) In equation (48) and (49), ZLSE and ZLSM are the LSE and LSM impedances looking up at x = dl. These can be computed using transmission line theory. That is, each layer is simply considered a transmission line with a characteristic impedance (ZosE or ZLSM) and an eigenvalue ki where 12

.ki + k2 + k2 = W2fi -- k) and (-o - w —''(LSE) = WI' and (ZLSM)i X Equations (48) and (49) should be satisfied for each LSE and LSM mode, respectively. Equation (50) signifies the discontinuity in the y-component of the electric field due to the magnetic source. In the transform domain, the boundary conditions (45) - (50) become EI = EZt at x = x' (51) Ht=Hi, at x=x' (52) Hi H = HI at x=x' (53) ( )LSE ZLSE at x = dl (54) — 2 ( )LSM =ZLSM at x d (55) (EY' - Ey) = esin(kz')cos(kky')65( - x') (56) where Em = 1 m=O Equation (56) can be obtained by substituting (36) in (50) and using the orthogonality properties of the sin's and cos's. If (30) and (34) are substituted in (54) and (55), the following can be obtained B = jk SEA (57) Wit and 13

K = -jw- ZLMN (58) This will reduce the problem to solving 4 equations for 4 unknowns. Taking into consideration the other 4 boundary conditions (51) - (53) and (56) and after simplification, one can obtain C = A (59) D = jw-zkyadA (60) k^k: b, LS zLM b k K - EZLM k A (61) ky aA N = jwuri tk A (62) kw kZ b where 9 l k A = -am * z -ksin(kzz')co0(kyy) (63) al a 2E^ - k2 where a zero was substituted for x' and a -j -k — ZLSEcos(kdl ) - sin(kdl ) (64) WL& b = sin(k2dl)-jk ZL cos(k-di) (65) In this manner, expressions for the six unknowns have been derived. Substituting (57) - (65) in (41) and (43), one can obtain complete expressions for'I and iI from which ( x, y, z) and p( x, y, z) can be derived. Using (29) - (40), the Green's functions (the fields due to a unit magnetic current element in the z-direction) for this subproblem are as follows (GMl =2, Em 1 1 ky vZZ~~ x C L jwjA al b k. * [sin(k(ax - dl))] ZWLSM 14

+ jwpcos(k_(x - dl))] * sin( kxz')cos( kyy')sin(kyy)sin( kz) (66) where (GEM)1 is E, in region 1 due to MIz (GE,MA) _ -em el kz k2& 1 *[(k, — ).-.sin(k2(x-dl)) k, b E1 jkmk ZLSE jw V zLSM ) 0 (( -a ) )] + WL1 -, j7) - cos(k_(x -di))] w/cIE kw kz b * sin(kz')cos( ky')sin(kz)cos(kyy) (67) E(GM) = EZZ *2m E^ kb (ofM)1 TI: al a'k2 -k2 - m k, ZLsMaw2 El + k2k ZLSEb - s( in(kk(z d1))]) bjw sin( kzz')cos( kyy')sin( kyy)cos(kz ) (68) (a,M.)= 1 E -2Em kz 1 (Gzx E E." I I Tw n m al a jwjis jkL * [in( k,( x - d )) + -ZLcos( k,( k - d))] w/z *sin(kz')cos(kyy')sin(kyy)cos(kzz) (69),U1:_ El ~ * ^'* kl- k_ n m 1 [cos(k,(x - d)) ( j.a - k b 1jw5CE 15

k 2k nw el _ a^_$ + sin(k,(x - di)) ( 2 ZLSE + — k-ZL )] W2/A2E k2b 1 sin( kz' )cos( kuy')sin( kyy)cos( k:z) (70) (G f,M)1 2em l1 2k n al a k2 k2 [cos(k(x - di)). ( — j JWU/CE1 kckz b k2k 2 l2 + sin(k,(x - dl)) ( k2 ZLSE _W ky a LSM ] * sin( kz')cos(kyy')sin( kz )cos( kyy) (71) 2.2.2 Green's function in region (1) for a y-directed magnetic current Fig. 5 shows the structure under consideration with the magnetic current alleviated from the ground of the cavity. The magnetic current is assumed to be M =.ay6(x - z')5(y - y')6(z - z') (72) The same method used in 2.2.1 will be applied here. Equations (18) - (44) are applicable also to the structure of Fig. 3. Moreover, the boundary conditions (51) - (55) still hold in addition to (E~ - E ) = 6(x -')(y - y')(z - z') ( 73) which- can be written as E- _ 1 = _encos( kz')sin(kyy')6( -') (74) al where n = 1 n = O 2 n#O Simplifying the boundary conditions equations, the following expressions are derived 16

LSE LSM T r- Zor Z1 I X i, - - a... 1 II Fig.5 The magnetic current source raised to apply boundary conditions 17

A = ~En -E ^ in(kyy')os( k z') (75) l a - k21 k c k B = jk LSEA (76) W/I C= cA (77) D = -jwi b k A (78) k, a(79) K = -W2 k2k.ZLMA (79) kx a N = -jwo k 7k7A (80) where a and b are given by (64) and (65) and kL C = cos(k.,dl) + j ZLSEsin(kdl) (81) wtz wE zSMi d = cos(kmdl) + j -ZLMsin(kdl) (82) Finally, the dyadic Green's function is (GEM)xl n m a l z [-sin((k(x - d)) jWEiZLSM k, 1 + cos(k(a - di))] sin( kyy')cos( kzz')sin( kyy)sin( kzz) (83) 18

GE;M 1al'Ma k1k -- kk2 n m ~ a ~ * [sin(kx( z-dl ))* ( — ) C1 E1 b ZIsm z zLSE)] + cos(k,(z -dl)). ( ZLSM+j k ) k, b eIW sin(k y' )cos( kzz')sin( Ikz)ws(kyy) (84) E1 ky l b + co(k(l k-d)). (kZZLE. Z W/Ami f+)kk 1 * sin( kyy' )co8( kzz' )sin( kyy )cos( k*z) (85) (GH)M1 = E - k - 1 _,n m a m *[sin( k( - dl)) + - Zcos - d + cos(k(z - dl))(- _ j+ jw WIL1 kx ky b * sin( k y')cos( kz')sin( kyy)cos( kz) (85) H ~,Ml ) — 2 ky 1 n m al a 3we tt [sin(kx(x - d )) + j ZLSEcos(k=(x - d ))] wI, sin( kyy')cos( kz')cos( kyy)cos(kz) (86) ( a, k 2e - k n m [ain(k,(x - di)). 2w,2,1 - k 1,k.... + cos( k=( -- dI))'( j kWt —- ky k )] sin(kyy')cos(kzz')sin(kyy)cos(k~z) (87) 19

1GH,M )= 2 9,en el Iy \yz'~I v CC al a k2 k2 n m I *[si,(k~( - /. ( k Z-LSE + W k azLSM (sin(k.(xa - di)) ( 2 Z Z + kzZ M W2 1211 E b kk, wk, a + cos(k x -d))(j 1+ j- )] wtel k, b sin(kyy')cos(kzz')sin( kz )cos(kyy) (88) Up to this point, the dyadic Green's function GE = (GiM) y + (G )~y + (GM)1( )yz — "ye yy \ + (G EM)Z + (GM)ly) + (G;M)liz (89) 1 = ( GH,M ) (,M),M GH (Gym G yz + (G HM)i + (G M)l + (G' M) li (90) have been obtained. 2.2.3 Green's function in region (2) The fields in region (2) (see Fig. 3) due to unit magnetic currents in the y and z direction are to be derived. Fig. 6 shows both structures to be solved. The scalar potentials in the Fourier domain can be written as fI = Asin(k(z - d2)) + Bcos(k(x - d2)) (91) <qS" = Csin(kxz) (92) /'^ = Ksin(k(x - d2 ) + Ncos(k(x - d2))] (93) Il^ = Dcos(k.x) (94) The boundary conditions that apply for both structures in Fig. 6 are 20

xT xl I A sI M II| M II Y X - - - t —1x' --' F 1 d2, 1 d2 LSE LSM I LSE LSM Z 2or Z2 L Z2or Z2 Fig.6a Fig.6b Fig.6 Structures to be solved to obtain Greens function in region (2). 21

E = EI at x x' (95) HI = H' I at x=x (96) Hz' = Hl at = x' (97) ( ) LSE= _ZLSE at zx d2 (98) z ( H )= _ZLSM at =d2 (99) where ZLSE and ZSEa are the impedances at x = d2 looking in the negative x-direction. For Fig. 6a, a discontinuity in Ey exists such that E -y~ Et = - 8min( kz )C0o( kyy')6(x - ) (100) " al While a discontinuity in Ez exists for Fig. 6b E[I _ E' = ZE^C0o( kz'in(ky)kyy6 - z') (101) One can notice the similarity between (91) - (94) and (41) -(44). In addition, the boundary conditions (95) - (101) are the same as the ones applied in 2.2.1 and 2.2.2 except that ZLSE(LSM) replaces ZSE(LSM), d2 replaces dl, (-em) replaces (Em) and (-en) replaces En. So, in general, the Green's function in region (2) are similar to those in region (1) with the following changes ZLSE, zLSE 1 2 zLSM z. LSM dl -- d2 en -> -en 22

Em > -Em 1' >'2 kl2, k2 1 2 In summary, the Green's function for the open circuit coplanar line discontinuity (inside a cavity) has been determined in this section. This was accomplished by working with Maxwell's equations and by representation of our source as dirac delta functions. Then, boundary conditions were applied to solve for the fields 2.3 Application of Method of Moments In order to obtain the fields inside the cavity, one should integrate over the source coordinates (i.e. the slots). E1 J=.M.GE)d8' (102) E2 = - M - G(E d (103) f1 = J M. G)ds' (104) fH2 = / |/X M. G ds' (105) The choice of the same magnetic current M to compute the fields in both regions reflects the continuity of the tangential electric field in the slot region. The negative sign that appears in (103) and (105) is due to the fact that the assumption M()-E (1) x a- =f (106) leads to SM(2) = g(2) x (-a, ) = -M (1(7) The remaining boundary condition to be used in order to arrive at the integral equation is the continuity of the tangential magnetic field 23

(tang. - tang. (108) in the slots regions. Equation (108) may be written as Hv(') = H(2) (109) Y y Hf1) = H(2) (110) If the magnetic current is assumed to be M = -ayMy + zM, (111) the following equations for the magnetic field in both regions can be obtained H(1)= J/ [M(GyHM + Mz(GjHM)']ds' (112) H = J [M, (GHM)l + Mz(G;HM)l]ds' (113) 2) = - ff [MGy( ^+ MIz(GM) ]ds (114), _~ + bI~( ~v] z s Hz2) = -/ J [My(G,M) + M,(G M )2]ds' (115) where (G$HM )' is the Hn component due to Mm component in the ith region. Substituting in (109) and (110), one can obtain the following integral equations /f My(G() + G(2)) + Mz(G() + G(2)d1' = 0 (116) f My(G( + G)) + Mz(G( + G2))ds' = 0 (117) where the superscript H, M is suppressed for simplicity. The integral equations (116) and (117) are to be solved for the unknown magnetic current distribution using the method of moments. The method of moments is a numerical technique used for solving functional equation for which closed form solutions cannot be obtained [5]. By reducing the functional relation to a matrix equation, known 24

methods can be used to solve for the unknown current distribution. The general steps involved in the moment method for the computation of surface currents can be summarized as follows: 1. The integral equation for the electric or magnetic field in terms of the unknown surface electric and/or magnetic currents is formulated. The resulting integral equation can be put in the form OP( J190) = 3(B) (118) where Lop is an integral operator on J, and/or M,, and 3 is a vector function of either E and/or H. 2. The unknown currents are expanded in terms of known, basis functions as N1 J, = Eai1i (119) i=1 N2 M, = E bjj (120) j=1 where the a's and bYs are complex coefficients and N1 and N2 are the number of basis functions for J, and M, respectively. 3. A suitable inner product is defined and a set of test (or weighting) functions W is chosen. If (119) and (120) are substituted in (118) and the inner products with the weighting functions are performed, the results may be expressed as N1 N2 ai, < Wq, Lop(i) > +E bj < Wq, Lop(j) >=< Wq,g > i=1 j=l (121) where the inner product is defined as < a,b >= a. ds (.122) In Galerkin's procedure, which will be adopted here, the test functions are chosen to be the same with the basis functions. 25

4. A matrix equation is formed after the integrals (122) are computed. The unknowns in the matrix equation are the current amplitudes ai and bj which can be solved for by matrix inversion. One can notice that the method is computationally intensive, but with the advent of faster computers the moment method has become feasible. In our problem, equations (116) and (117) represent the general integral equation (118). Now, applying step (2), the y-component of the magnetic current will be expanded as M My - bpp(y, z) (123) p=1 The z- component of the magnetic current will be assumed to be composed of 5 components, incident and reflected travelling waves in each slot (y, < y < yi + WT1,y + WI + s <y< y + W1 + W2 + s) up to some point z = zl and the sum of basis functions for z > zl (see Fig. 7). That is M, = [(Alijz' +Blejz)(u(y- y)-u(y - yl - W1)) + (A2i0' + B2e'ioz(u(y - y. - W, - s) - u(y - yl - W, - W2))] * (uzZ)-U(Z- zi)) N + [Eanfn(y,z)]u(z-z) (124) n=l where f is the propagation constant in the coplanar waveguide and u(. ) is the unit step function. Substituting (123) and (124) in the integral equation (116) and (117). The following expressions can be obtained - LJA2ijz' (G6() + G(2) )ds' -A2eiz'( + G(2) )d 1 f Bej"( G(1) + G() )ds' + j B2ei (G(l) + G(2 )ds 26

z Y1,.!. I I.!.' > It I l l l Z N+2 N+ 1 w2t 1 1 1 1M M+2 I fN Fig. 7 Geometry for use in basis function expansion of magnetic current. 27

M + Ebpj | p(y',z')(G() + G(2))d8' N + E an | fn(Y',z')(G(^) + G(2 )ds' n=l - n a (125) and - j/ Alejz' (G(1) + G(2))d' 12 J A2eJ3z'(G(1) + G(2j)ds' = JB /ieJiz'(G(1) + G(2))d + j JB2ej Gz) + G(2))d' + bJ J~y ~/ p + E bp p(Y'/, z)(G() + G(2z))d + E an fn(y', z')(G() + G()d8 (126) (126) where s1 denotes the area for which yl < y < yl + W1 and o < z < z1, and s2 denotes the area for which yl + W1 + s < y < + W1 + s + W and o < z < zl. Sp and Sn are the area over which Op and f, are defined respectively. Galerkin's procedure will be applied where the test function are the same as the basis functions. The inner product of (125) with Ak(y, z), k = 1, * * m, is performed which will result in M equations, each one of the following form - J JS Ale-j'(G{(l) + G )k(Y) z)ds'dS A2ej3z (G(G) + G(2) ) )d'ds zy zy = L J BieJiz'(G () + G(2) )k(y,z)ds'ds + j B2,ej (G() + G(2)))k y,z)ds'ds ~k w2 S28 28

+ ~ bp j p(y',z')(G") +G2))Ak:(y z)ds'ds N + E, an f,(y', z)( Gzy + G ))k(, z )ds'ds (127) n==l n In the same manner, performing the inner product of (126) with fk(y, z), k 1,..N, N equations are obtained as - 15 |L Aiejz (Gl) + GZ2) )fk(y,z)ds'd - 1 1 A2e-iz'(G(l) + G(2))fk(y,z)ds'ds = j Bie'I3(G! J + G })fk(y,z)ds'ds + I 1|2 B,2el3z'(G2) + G2))fk(y,z)dsd - k 2 M + ~ bpj j|qp(yz',)(G() + G+ z)f(yz )d'd N + Ebp fn(y'',')(G() + G(2)fk(yz)ds'ds (128) where Sk in (127) is the area over which k(y, z) is defined, while in (128) denotes the area over which fk(y, z) is defined. Notice that the Green's functions are in terms of the source coordinates (y' and z') and the observation point coordinates (y and z). The Green's functions are obtained from (69) - (71) and (86) - (88). Finally, the inner product equations (127) and (128) are solved to form the matrix equation. The matrix equation obtained will be of the following form 29

B1 B2 [Y].. =[I] (129) a, aN where [Y] is an (M + N + 2) x (M + N + 2) matrix and [L] is a vector of (M + N + 2) elements where Ij = -/J Ae (G(v) + G2) ) j( y,^ z )d9'd9 - J 1 A2dz' (G/() + Gzy)(2)q,(y, z)ds'ds (130) for 1 <j < M and Ij = - // AIEjaz'(Gl)) +G GZ)fjM(y,z)ds'd8 - IJS A2eJiz'(Glz) +G z)f-JM(yz)ds'ds (131) for M + 1 < j < M + N. The elements of [Y] are obtained from (127) and (128) giving the following expressions. For 1 < i < M Y(i, 1) = j s eijz'(G(1) + G iy, z)ds'ds Y(i,2) = s eJOz (G( y+G!) G )i(yz)ds'ds (132) For 3 < j < M + 2 Y(itj) = j j ^-2( y',z' )( G,) + GJ(2))4i(y, z)ds'd8 (133) For M + 3 <j< M+N+ 30

Y(ij) = j fj M-2(y',z')(G(?) + G(2))i(y,z)ds'ds (134) Yii~ji S J S' M2 Z 7dd 13 For M + 1 < i < M +N Y(i, 1) = |i eij'(G(2) + G2 )fi_M(y,z)ds'ds (135) Y(i,2) = j f ejz'( G(!) + G2) )fi-M(y,z )ds'ds (136) 2 For 3 < j < M + 2 Y(ij)= / f/,j2(y',z')(G + G2 )fiM y)ds'ds (137) S Is Is' L v ) f -M(y Yz ()d For M+.3 < j < + N +2 Y(i,j)= j js fj-M-2(y',z')(G( + G )fiM(y, z)ds'ds (138) It can be observed that two more equations are needed to solve for the (M + N + 2) unknowns. The basis functions are chosen to be piecewise sinusoidal functions as shown in Fig. 7 such that P,, = sin(k*(y - y,)) Vv~Y) =. Y < yp+i sin( k*(yp+l -p Y)) sin(k*(yp+2 - y)) sin( k*(yp+2 - y+l )) = 0 elsewhere (139) where k* = w v/,f and ecff is the effective permitivity of the coplanar waveguide defined as Ee!f = ( )2 (14)) where /, is the free space propagation constant. The z- variation of qp is assumed to be unity over the slot. 31

The basis functions for Mll are sin(k*( z2 - z)) fl(z) = Sin(k*(z,,-Z < z < z sin(k*(zp+l - zp)) sin(k*(z - Zp+l)) = * i~i —- -~, zv+1 _ z < zp (141) sin(k*(Zp - pl ) ) for 2 <p < 2 and yl < y < yi + W1 sin(k*(zp+3 - z)) fp(Z) = - Z+3 < Z < Zp+2 sin(k*(Zp+3 - Zp+2)) sin(k*(z - zp+l)) = in(k*(z +2 - zp+)) + < Z < Zp+2 (142) in(kc*( Zp+2 - Zp+l)) for + 1 < p < N - 1 and yx + W1 + s < y < yi + W< + s + W2 sin(k*(z - ZN+1)) fN ( ) = i - -ZN+) ZN+2 < Z < ZN+l (143) sink'(ZN+2 - ZN+1 ) ) So, the other two equations needed can be obtained by imposing the continuity condition of the magnetic current M, such that Ale-j3zl +,/ ejzl = al (144) and A2e-jiz1 + B2ej3l = aN (145) So, one can write Y(M+ N+1,1) = -ej2"l (146) Y(M+N+ 1,M+3) = ej'Z (147) Y(M + N + 2,2) = -ej2l (148) Y(M+N+2,M+N+2) = eiZ' (149) I(M+N+1) = A1 (150) I(M+N+2) = A2 (151) 32

Finally, the integrals involved in the elements of [Y] and [I] can be performed analytically. In fact, one can find that seven integrals only have to be performed to get the elements of the matrices. Appendix A shows the derivation of these integrals. Once the element of [Y] and [I] are determined, (129) can be solved for the unknown current amplitudes by inverting [Y]. Using the derived current distribution in the slots, one can determine the scattering parameters characterizing the open and coplanar waveguide discontinuity. 3 Summary The open circuit CPW discontinuity has been analyzed theoretically in this report. The dyadic Green's functions for y and z directed dirac delta magnetic currents, placed in a rectangular cavity, were obtained. The fields were assumed to be a superposition of LSE and LSM modes. Then, the continuity of the tangential electric and magnetic fields in the slot region was used to arrive at the integral equation. Finally, the integral equation was solved using the method of moments to obtain the unknown magnetic current distribution from which the scattering parameters can be evaluated. This study is intended to be a step towards characterizing various CPW discontinuities including the CPW air bridge. A computer program, that solves the CPW open circuit discontinuity, is in the process of writing. 33

Appendix A ryi + W Iy 1 = cos(kyy)dy [in( ky{ yl + W1) - sin( ckyl)] ky - O0 ky W1 ky- =O rY +WI +S+W2 Y1/2 l cos(kyy)dy - [sin(k,( yl + wl + s + w2)) - sin(ky(y1 + w1 + 9))] ky 0 ny W2 k = 0 Iz -= j e Jzsin(kzz)dz 1o - -32 [kz - kzeiJ3l cos( kzz ) + j/3ej"z sin( kzl )] kC22 0 2Z zl +ld+g Iz2 = J+ 9 cos( kzz)dz JZ1 +Id 1 1 k [sin(kz(z, + sin( k )sin(kz(z, - z))] kz = sin(k*(zz - zl)), Jin(kzz)sin(k* z))z sin( k(zz )cs k2 2 - i1)) k + ki( z ( + k*sin( kzl) cos( k'(z - z)) + kcos( k, zl)sin(k'(z2 - z))] 34

1 r+ If- =. sin( kz )sin(k*(z - ZN+ ))dz N sin(k*(zN+2 ZN+1)) Z+2 1 1 ~=' 2~ —-- * *- ^'[-k sin(kzzN+l) sin( k*(ZN+2 - ZN+1 )) kz- k*2 + k sin(kzzN+2)cos(k'"(zN+2 - ZN+2)) + kzcos(kAzzN+2).sin(k*(ZN+l - ZN+2))] y'+l. sin( k*(y y- yi)) I(yi ) = * sin(kyy) ) dy Yi2. sin(k*(yi+ - yi)) + +i ( ) sin(k*(Yi+2 - Yi+)) _ _ * 1 11n(k*(yi2 -yi+l)) b - k*2 sin(k*(yi+l -yi)) [-sin(kyyi+z)sin(k*(yi+2 - Yi)) - sin( kyyi )sin(k*(Yi+2 - Yi+l)) + sin(kyyi+l )sin(k*(yi+2 - yi))] 35

4 References 1. T. Hirota, Y. Tarusawa and H. Ogawa, "Uniplanar MMIC Hybrids. A Proposed New MMIC Structure," IEEE Tms. Microwave Theory Tech., Vol. MTT-35, pp. 576-581, June 1987. 2. R. Simons and G. Ponchak, "Modeling of Some Coplanar Waveguide Discontinuities," IEEE Trans. Microwave Theory Tech., Vol. MTT-36, pp. 1796-1803, Dec. 1988. 3. G. Hasnain, A. Dienes and J. Whinnery, "Dispersion of Picosecond Pulses in Coplanar Transmission Lines," IEEE Trans. Microwave Theory Tech., Vol. MTT-34, pp. 738-741, June 1986. 4. R. Harrington, Time-Harmonic Electromagnetic Fields, New York: McGraw Hill, 1961, p. 152. 5. R. Mittra, Ed., Computer Techniques for Electromagnetics, Pergamon Press, 1973. 36