Capacitance and Inductance Matrices for Multistrip Structures in Multilayered Anisotropic Dielectrics

1002 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 11, NOVEMBER 1987 Capacitance and Inductance Matrices for Multistrip Structures in Multilayered Anisotropic Dielectrics FRANCISCO MEDINA AND MANUEL HORNO, MEMBER, IEEE Ab.vtract —In this paper we present aunified variational approach to determine the capacitance and inductance matices of generalized multistrip systems embedded in amultilayered iso/anisotropic dielectric lossless medium. The analysis is earned out in the spectraf domain to take advantage of previously obtained recurrence relations which calculate the Green’s function in the spectral domain. The method leads to alow-order system of linear equations, which is shown explicitly. Examples and comparison with previously published results have been included. I. INTRODUCTION T HE PHYSICAL behavior of uniform multiconductor transmission lines when they are used in the design of high-frequency electrical filters and couplers is well understood [1]. It is also well known that at the low-frequency end of the spectrum, aquasi-TEM approximation can be assumed even when inhomogeneous and/or anisotropic dielectrics are involved [2, 3]. Under this assumption, the problem reduces to the determination of the Maxwell capacitance matrix of the system. Particular interest has been focused on the planar structures used in MIC technology. Numerous papers dealing with quasi-static propagation in twoor three-conductor strip systems can be found in the literature [4–8]. The more general problem of determining the Maxwell capacitance matrix for ageneralized multistrip system has been treated for homogeneous [9] and nonhomogeneous [10] media. However, for planar structures embedded in amultilayered dielectric medium, the spectral-domain analysis seems to be especially suitable [11, 12]. Recently an efficient recurrence algorithm to obtain the Green’s function in the spectral domain associated with an anisotropic multilayered dielectric medium has been used to analyze several coplanar [13] and noncoplanar [14] structures. Similar work dealing with isotropic dielectrics has also been recently published [15]. The aim of the present paper is to apply variational analysis in the spectral domain in order to obtain the capacitance and inductance matrices that characterize amultistrip system of the type shown in Fig. 1. The method leads to a low-order system of linear equations whose coefficients and independent terms are explicitly shown in the work. Manuscript recewed April 20, 1987; revised July 2, 1987. This work was supported by the Comisinr Asesora de Investigaci6n Clentifica y T4cnica (Prey. 3028/83), Spain. The authors are with the Departamento de Electncldad yElectronics, Facultad de Fisica, Uruversidad de Sevilla, 41012, Sevilla, Spain. IEEE Log Number 8716592, @ ~——––— ~–———= I%%1 I ig H, I LO_______ ———______L ________— d 63 (a) @ ~= —————— —–r ——– ———–––– ——– ——— ——–~ ,E% H% I JL .“ . ‘f .. ]q‘f HI L—— ———_J_______________________J IbI (b) Fig. 1. (a) Cross section of generic multilayered coplanar strip lines. (b) Cross section of ageneral noncoplanar multistrip system. Uand B denote electric wall, magnetic wall, or o~en boundarv: L and R denote electric or magnet;c wall. ‘J The choice of basis functions is discussed and several examples are introduced to show the method’s strength. II. STATEMENT OF THE PROBLEM: SPECTRAL ANALYSIS Consider asystem of strip lines in astratified lossless dielectric region enclosed in aset of rectangular boundary conditions. Conductors can be coplanar (Fig. l(a)) or 0018-9480/87/1100-1002$01.00 01987 IEEE Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 21,2020 at 15:35:50 UTC from IEEE Xplore. Restrictions apply. MEDINA AND HORNO; CAPACITANCE AND INDUCTANCE MATRICES non-coplanar (Fig. l(b)) strips. The top and bottom shields can be considered electric walls, magnetic walls, or open boundaries. So, open geometries can be simulated by choosing an open boundary in the top shield and taking “b“ in Fig. 1large enough. The permittivity of the jth dielectric layer is adiagonal tensor ~j. (It is important to take into account the dielectric anisotropy from apractical and theoretical point of view [16].) If nonmagnetic materials (p, =p~)are assumed, the problem will be reduced to, the determination of the capacitance matrix for the structures under consideration with ([ C,j]) and without ([C:]) substrates [3]. These two-dimensional electrostatic problems can be readily formulated in the spectral domain instead of working in the space domain [13] –[15]. The first step in the solution process is to determine Green’s function in the spectral domain associated with the structure. When several dielectric layers are involved, this task may become very tedious. Nevertheless, very simple recurrence expressions to determine Green’s scalar function for coplanar structures [13] (Fig. l(a)) and Green’s function matrix for noncoplanar ones [14] (Fig. l(b)) have been published. In this way, the Fourier transforms of the potential function ~, and the surface charge ~, on the interfaces where the strips are located are related via ~(rz)=G.(lZ) -Fs(rz) (coplanar strips) (la) [~(n)] =[G,.(n)] [p,(n)] (noncoplanar strips). (lb) These relations will be used in the following paragraphs to derive the inductance and capacitance matrices. HI. VARIATIONAL ANALYSIS In amulticonductor system such as the one described in the previous section, charges and potentials on the strips are related by means of the matrices [C] and [P] in the following way: [Q,]=[c,, ][L] [K] =[~,,][Q,] [8,] -1=[c,,]. (2) The i, jth element of the [C] matrix is the free charge per unit length on the ith conductor when all conductors except the jth one are grounded and the jth conductor is charged to apotential of 1V. Hence, the elements of [C] can be determined by relating the charge on the conductors to their potentials. However, in our work, we will take adifferent approach, one based on energy calculations. In this way, we can take advantage of the variational nature of the energy expressions. For an arbitrary distribution of charges on the strips, the electrostatic energy stored in the system per unit length can be expressed as u=+[Q,][~,J][Qj] =+i[Izl[ct,][TJl (3) Note that [P] can be expressed in terms of the electrostatic energy for different situations in the following way: P,, =2U’’/Q: P,, =(UiJ– U“- UjJ)/Q,Q,, i+j (4) 1003 where U‘J is the electrostatic energy stored in the system per unit length when all conductors except the ith and jth ones are isolated and discharged and the ith and jth conductors support charges Q, and Qj, respectively. Now, () our problem is the evaluation of the energy for the N+“c different distributions of charges necessary to compute ill the elements of the [P] matrix (and consequently the [C] matrix). The electric energy can be expressed in the Fourier domain as follows: u=: g’[F+’z)]’[ti$r(n)][fir(n)] n—1 (5) M=number of interfaces with conducting strips (M= 1for coplanar strip5) s,r=l,. ... M where P,(n) is the Fourier transform of the surface charge density on the jth interface (the expressions (1) have been used in (5)). Aknowledge of these distributions of charges is only possible in very limited cases. However, the stationary nature of (5) allows us to apply Ritz’s minimization procedure to obtain avery accurate estimation of U. Let n~be the number of conducting strips lying on the sth interface. The surface charge density can be expanded in aset of basis functions: (0 elsewhere (6) where b J() p:. x.U!x=l and b ~() P:p x.dx=O. The variational coefficients a~P are obtained by minimization of the electrostatic energy in (5) (Ritz’s procedure). This process leads to the following system of linear equations: Mn, # where the coefficient matrix is expressed in terms of the Fourier transforms of the basis functions in the following way: m and the independent terms column is related to the particular distribution of charged and isolated strips to be considered for each energy calculation. To determine the inverse of the capacitance matrix from (4), it is clear that we must only consider cases with one or two strips charged Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 21,2020 at 15:35:50 UTC from IEEE Xplore. Restrictions apply. 1004 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 11, NOVEMBER 1987 I“s ~ss —II .’ 1( ‘~... ––—— b1 I1 Fig, 2. Detail of the sth interface of the structures in Fig. 1. and the rest isolated and without charge. Therefore, if the lth strip of the uth interface and the mth strip of the uth interface are charged, the independent terms of the equation system will be given by (9) n=l Once (7) is solved, the electric energy per unit length can be computed from the following expression: Expression (10) is evaluated for all possible values of ~,~=l,... ,Mand 1,m=1,. . . . n,; the [P] coefficients can then be obtained by using (4). This process must be carried out for the structure with and without dielectric layers to obtain [Ci,] and [C;]. In this way, the capacitance and inductance matrices are obtained and the system is completely characterized under aquasi-TEM operation. IV. TRIAL FUNCTIONS Before generating numerical results, a good choice of trial functions is necessary. Different sets of basis functions have been considered in this work. From this study, we conclude that an adequate choice of trial functions must take into account the singularities of the charge distributions at the edges of the strips. Avery simple set of functions is to consider aconstant term covering the total charge on the strip and two terms for the singularities at the edges (see Fig. 2): [1 h(x) =; FL(x)= 1-2 “j) sT ~— -&i 3 w: ,2(+=(A-k’ (11) This choice is good enough to yield very accurate results for coplanar strip structures. This fact has been concluded from systematic comparisons with previously published results and with the data generated by using amore complete set of basis functions such as the following one (Fig. 2): {1 1/2 2 p:o(x)=— vW$’ [1 ~_ (XL) 2 (Wy;) Hpv (-L)) P:p(x) =;P:o(x)” Cos —’ x x : -COS(;)JO(;)). (12) In ageneral case involving broadside couplings, the charge distribution is more complicated and one must use the expansion in (12) to obtain good results. The convergence of the Fourier series is slower with these functions, but the accuracy in certain cases is improved substantially. In order to improve the convergence, the asymptotic performance of the series appearing in (8) and (9) has been considered in the computer programs. In this way, the number of Fourier terms necessary to obtain the desired accuracy is drastically reduced and both choices of trial functions are suitable for quick calculations. CPU time is less than one second per conductor strip for multistrip structures on a VAX1 1/780 computer. V. NUMERICAL RESULTS The theory presented in this paper was used to write a computer program which provides the capacitance and inductance matrices for asystem of coupled strips embedded in amultilayered isotropic or anisotropic medium. From these matrices, mode impedances and phase velocities are readily obtained [3]. In order to check the results, we analyzed several particular structures previously studied by other authors. In the following paragraphs, we show these comparisons and some new results. In order to show the effect of a good choice of basis functions we compare our results with the ones reported by Koul and Bhat [12]. (These authors provide ‘a useful method to analyze awide variety of symmetrical striplike structures on anisotropic substrates.) In Fig. 3we compare the results obtained in [12] for the interelectrode capacitance of acoplanar structure (electro-optic modulator) with the ones calculated by means of the method in this paper. We show two curves by using the trial functions in (12) with n=Oand n=5. In both cases we found avery good agreement for narrow strips. However, asignificant discrepancy is observed for wide ones. In our opinion, ttis discrepancy is due to the fact that the trial function used in [12] cannot conveniently represent the charge distribution on wide strips. These curves were also computed by using (11) and the discrepancy was less than 0.5 percent for all Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 21,2020 at 15:35:50 UTC from IEEE Xplore. Restrictions apply. MSDINA AND HORNO: CAPACITANCE AND INDUCTANCE MATRICES w(}D) 1005 Lo 80 120 160 . .. . ... S.mn ~“’””4 ..”, -- ‘oot .7 ,Ooc 40 80 120 160 20 s(pm) Fig. 3. Interelectrode capacitances of electro-optic modulator as a function of the width of the strips and their separation. ---- and — are the results reported in [12]. –.– .– represent our computations with one basis function, and . . . . represent our computations with five basis functions. dimensions. This fact confirms that (11) is a good set of basis functions for coplanar strips. The structures shown in Fig. 4(a) and (b) were analyzed by Kitazawa and Mittra [7] and by Kitazawa and Hayashi [8]. The effective dielectric constants for Cand n modes of these structures are represented in Figs. 5 and 6. As we can see, the results in [7] and [8] are in good agreement with our data. Table Icompares our data with those obtained by Wei et al. [10] for the case of two coupled microstrip lines between two ground planes in ahomogeneous and isotropic medium. Table II shows the results obtained for a structure with three strips embedded in athree-layered dielectric medium between two ground planes. This structure was also studied by Wei et al. in [10]. We compare both results in Table II. Significant differences are detected, but these are within the margin of error given in [10]. More exact calculations were previously reported by Kammler on multiconductor structures in ahomogeneous and isotropic medium [9]. In Table III, we show the results obtained in the analysis of apair of asymmetrical coupled strips between two ground planes. An excellent agreement (within +0.002 in all cases) was found. (It must be emphasized that the Kammler results are exact to within ~0.001.) These data were generated with the trial functions in (12). Atypical discrepancy of 1percent to 5 percent was observed by using (11). However, the set of I (a) I)’ h Zd h i (b) Fig. 4. (a) Cross-sectionaf view of asymmetrical coupled strip lines. (b) Cross-sectionaJ view of broadside-coupled strip lines of unequaf width. 12.0 , n ----- ------- --------- --------- --------- —--..-= P...-.... 10.0 COUPLED STRIPS WITH OVERLAY L—C-mode present work 8.0 ‘--’~’/ +! ~b--- \ u ,0 /cOu’’’:_::::::::>>>~##=#=”z”==z”== ------------- --------------- ------------ --------- ---.” -_ 4.0 .-.. ......... ............ ~ ICOUPLED SUSPENDED STRIPS 2.0 L..—dL_LL_LL—L—_J 0.1 1.0 4.0 Fig. 5. Quasi-static characteristics versus W2\WI, (The dimensions and dielectric constants have been takcnfmnr[7j:) basis functions in (11) allows us to obtain very good results in the case of coplanar structures. In this way, Table IV compares our results for the case of several multiconductor, coplanar structures with those reported in [9]. The Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 21,2020 at 15:35:50 UTC from IEEE Xplore. Restrictions apply. 1006 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 11, NOVEMBER 1987 8.0, TABLE II VALUES OF THE CAPACITANCE AND INDUCTANCE COEFFICIENTS FOR THREE COUPLED STRIP LINES IN ATHREE-LAYERED DIELECTRIC BETWEEN GROUND PLANES AND COMPARISON WITH THE DATA REPORTED IN [10] h 6.0 l\ —Present work ~eff, ll .Ref. [81 o2 4 6 8 SI!J ~ !- .6 ~.15 J’s Ej. 4.2 E. ‘-. 2~ ~ 1—.6< .2 I4x (a) Ref. (10 ) 0.4900 x10 -9 4.5737 x10 -12 -0.6457 x10-10 0.2459 x10 -0.6138 x10::0 0.2865 x10 -9 0.7773 x10-’0 -0.1036 x10 -12 -0.7193 x10-” 0.5212 x10-10 -0.9766 x10 -11 0.3676 x10-’0 Present win-k Difference 140r h‘cl -9 0.5115 x10 -12 –0.5929 x10 -0.6972 x. 10-10 0.2572 X10-9 –0.6659 x10 –lo 0.2977 x10 -9 0.8110 x10-10 -0.1075 x10-’2 -0.7812 x,0-’1 -10 0.5445 x10 -0.1C158 x10-10 0.4040 x10-10 4.3 % 3.3 % 7.7 % 4.5 % 8.1 % 3.8 % c11 c12 c13 c22 C23 c33 c011 c012 c013 c022 c023 c033 L11 L12 L13 ’22 L23 L33 c 60 SINGLE SIJSPENDED sTRIPLINES . — W=W1 s — —— w~= W2 4.2 % 3.7 % 6.2 % 4.4 % 8.9 % 4.1 % 04 6 e S/wl (b) Fig. 6. (a) Effective dielectric constant versus S/ W1. (b) Characteristic impedances versus ,S/ WI. (Dimensions and dielectric constants have been taken from [8].) -6 0.1456 x10 -8 0.5630 x10 0.2s44 x10-7 0.2240 x10 * 0.5762 X10 -7 -6 0.3065 x10 -6 0.1403 x10 o.58a7 xlC 4 -7 0.2862 X10 0.2157 X10+ -7 0.5%15 x10 0.2963 x10 -6 3.9 % 4.5 % 0.6 % 3.8 % 1.0 % 3.4 % TABLE I VALUES OF THE CAPACITANCE AND INDUCTANCE COEFFICIENTS FOR Two COUPLED MICROSTRIP LINES BETWEEN Two GROUND PLANES IN ATABLE III ASYMMETRICAL BROADSIDE-COUPLED STRSPSBETWEEN GROUND PLANES: COMPAtUSON WITH THE CAPACITANCE COEFFICIENTS REPORTED BY IQMLER [9] HOMOGENEOUS MEDIUM AND COMPABJSON WITH THE DATA REPORTED IN [10]. 1 t-3 +’+ ,’:;3 1.5 Ref. ~10~ Present work –9 -9 Cll 0.5356 X10 0.5320 x10 c-0.9250 x10-11 -0.1008 x10 -1( 12 c0.7834 X10 -9 0.7790 x10 -9 22 c0.5466 x10-10 0.5600 x10-1( 011 c-0.9439 r. 10-12 -0.1062 x10-’” 012 c022 0.7994 x10-’0 0.8200 x10-1[ L0.2033 X10-6 0.1987 X10-6 11 -8 -8 ’12 0.2401 X10 0.2570 X10 -6 -6 L22 0.1390 x10 0.1357 x10 3 ) ) 1 3 ‘2 1.0 0.8 0.6 0.5 0.4 0,3 0.2 0.1 I 0.05 –4 C1l/.o 9.136 8.862 8.434 8.179 7.904 7.611 7.300 6.956 6.746 c12/ro -5.355 -4.743 -3.953 -3.518 –3.064 -2,592 -2.100 -1.561 -1.234 C2240 9.136 7.760 6.301 5.558 4,807 4.048 3.268 2.424 1.916 . c/. 11 ~ ‘1 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 I *I● c12/e oC22]’0 I –5.352 9.133 -4.742 7.759 –3.953 6.301 –3.518 5.557 -3.063 4.807 –2.592 4.046 -2.099 3.264 -1.561 2.423 -1.235 1.915 9.133 8.860 8.434 8.179 7.903 7.610 7.299 6.956 6.746 w~erence ~9] Present work Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 21,2020 at 15:35:50 UTC from IEEE Xplore. Restrictions apply. MEDINA AND HORNO: CAPACITANCE AND INDUCTANCE MATMCES 1007 TABLE IV CAPACITANCE COEFFICIENTS COMPUTED FOR SEVERAL COPLANAR CONFIGURATIONS AND COMPARISON WITH RESULTS IN KAWLER [9] —. — — — — Present work (Ref. [$ cll/s O=2.4617 (2.4618) C1lAO =2.8878 (2.8!388) c12/c ~=-1 .0372 (-1 .0379; cll/Fo =2.8903, (2.8914) C22/E ~=3.2908 (3.2915) C121E ~=-1 .0060 (-1 .0064) C13/E ~=-0.0834 (-0.0841) C1l/FO =2. E!904 (2.8914) C22/C0 =3.2921 (3.2938) CJFO =-1 .0057 (-1 .0061) C231F0 =-0.9766 (-0.9767) C131C0 =-0.0788 (-0.0795) C141E0 =-0.0124 (-0.0125) C1l/sO =2.8904 (2.8914) C22/E0 =3:2921 (3,2939) c33/co =3.2943 (3.2961) CIZ/CO =-1 .0057 (-1 .0061) C2J=0 :-0.9763 (-0.9764) c13/Eo =-0.0789 (-0.0794) c24/F ~=-0.0745 (-0.0751) c14/co =-0.0117 (-0.0117) c15/Eo .-0.0020 (.0.0020) difference is less than 0.05 percent for the worst of the cases studied. Finally, another configuration was analyzed with our method, one consisting of athree-line symmetrical coupler with atwo-layered anisotropic substrate: sapphire (h z) and pyrolitic boron nitride (P.B.N.) (h J. Special coupler structures such as this one are often required in communication systems and other microwave applications. In these structures, the quasi-TEM modes (A, B, C) can be propagated [4]–[6]. Fig. 7represents the dependence of the mode characteristics on the ratio hi/h. It can be noticed that there are two values of hi/h that equalize the mode phase velocities. This interesting result is aconsequence of the combined effect of the geometry of the structure and the use of anisotropic substrates [17]. VI. CONCLUSIONS In this paper the authors have discussed the analysis of shielded multiconductor strip lines embedded in amultilayered anisotropic medium by employing avariational technique in Fourier’s discrete domain. The calculation of 8.0 7.0 6.0 5.0 4.0 3.0 l\ E=5.12; E= xl 3.40 (P. B.N yl mode AE== 9.40 ;EY2 .11-60 (sapphire) .2 .d .6 .8 hi/h Fig. 7. A, B, and Cmode effective dielectric constants for three strips on two anisotropic dielectric layers as afunction of the relative thickness of each-layer. the capacitance and inductance matrices for lossless configurations is achieved by computing the electric energy per unit length of the structures. The Rayleigh-Ritz procedure has been applied to optimize the solution using adequate trial functions. The method is numerically very efficient and can be easily implemented in acomputer program. The number of anisotropic dielectric layers and conductor strips is no longer adifficulty because the Green’s function matrix is evaluated by means of avery simple recurrence algorithm. Some examples have’ been included to illustrate the strength of the method and its accuracy. The propagation modes of three lines with two anisotropic layers have been studied as aparticular application. REFERENCE5 [1] [2] [3] [4] K. D. Marx, “Propagation modes, equivalent circuits, and characteristic terminations for multiconductor transmission lines with inhomogeneous dielectrics,” IEEE Trans. Microwave Theory Tech., vol. MTT-21, pp. 450–457, July 1973. I. V, Linden, “On the quasi-TEM modes in inhomogeneous muRiconductor transmission linesj’ IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 812-817, Aug. 1981. R. Marqw$s and M. Homo, “Propagation of quasi-static modes in anisotropic transmission lines: Application to MIC lines,” IEEE Trans Mzcrowaue Theo~ Tech., vol. MTT33, pp. 927-932, Oct. 1985. D. Pavlidis and H. L. Hartnagel, “The design and performance of three-hne microstrlp couplers,” IEEE Trans. Microwave Theosy Tech., vol. MTT-24, pp. 631-640, Oct. 1976. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 21,2020 at 15:35:50 UTC from IEEE Xplore. Restrictions apply. 1008 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 11, NOVEMBER 1987 V. K. Tripathi, “On the analysis of symmetrical three-line microstrip circuits,” IEEE Trans. Microwave Theory Tech., vol. MTT-25, pp. 726-729, Sept. 1977. V, K, Tripathi, “The scattering parameters and directional coupler analysis of characteristically terminated three-line structures in an inhomogeneous medium,” IEEE Trans. Microwave Theoty Tech., vol. MTT-29, f)f). 22–26, Jan. 1981. T. Kitazawa and R. Mittra, “Anafysis of asymmetric coupled striplinesj” IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp. 643-646, July 1985. T. Kitazawa and Y. Hayashi, “Analysis of unsymmetrical broadside-coupled striplines with anisotropic substrates,” IEEE Trans. Micrwwaue Theory Tech., vol. MTT-34, pp. 188-191, Jan. 1986. D. 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Homo, “Upper and lower bounds on mode capacitances for alarge class of anisotropic multilayered microstrip-like transmission lines,” Proc. Insl. Elec. Eng., pt. H, vol. 132, no. 3, pp. 157–163, June 1985. F. Medina and M. Homo, “Determination of Green’s function matrix for multiconductor and anisotropic multidielectric planar transmission lines,” IEEE Trans. Microwave Theoiy Tech., vol. MTT-33, pp, 933-940, tlct. 1985. A. Sawicki and K. Sachse, “Lower and upper bound calculations on the capacitance of multiconductor printed transmission line using the spectral domain approach and variational method,” IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 236-244, Feb. 1986. N. G. Alexopoulos, ‘<Integrated-circuit structures on anisotropic substrates,” IEEE Trans. Microwave Theo~ Tech., vol. MTT-33, Pp. 847–881, C)ct. 1985. [17] N. G. Alexopoulos, S. Kerner, and C. M. Krowne, “Dispersionless coupled microstrip over fused silica-like anisotropic substrates,” Electron. Letrer, vol. 12, no. 22, pp. 579-580, Oct. 28, 1976. Francisco Medina was born in Puerto Reaf, Spain, on November 9, 1960. He received the Licenciatura degree in physics from the University of Seville, Spain, in 1983. He is currently Assistant Professor of Electricity and Magnetism in the Department of Electricity and Electronics, University of Seville, where he is studying for the Ph.D. degree. His current interest is in multiconductor planar transmission lines and MIC design. * Manuel Homo (M75) was born in Terre del Campo (Ja+n), Spsrin, on June 29, 1947. He received the degree of Licenciado in physics in 1969 and the degree of Doctor en Ciencias in Physics in 1972, both from the University of Sevilla, Spain. Since October 1969 he has been with the Department of Electricity and Electronics at the University of Sevilla, where he became an Assistant Professor in 1970, Associate Professor in 1975, and Professor in 1986. His main fields of interest include boundary vafue problems in electromagnetic theory, wave propagation through anisotropic media, and microwave integrated circuits He is presently engaged in the analysis of planar transmission lines embedded in anisotropic materiafs, multiconductor transmission lines, and planar slow-wave structures. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on July 21,2020 at 15:35:50 UTC from IEEE Xplore. Restrictions apply.