/~\ ..—L ‘ECH--’’‘1o{c2ElLT’{ ‘%$2‘4E’EL~f~cpoi{-’+‘E’”IEoR“>’}-’I;2 ‘5 The “iiicro}l aye Th :or; an d Tech-,)q~es,SocieI> is 2fi o:g.lniz~t:o:.. ,!:!’ h.... .-.....,. .,n:, _,....LL.l-. ...= IEEE. o.” v.e-.:~.oe:s ,1.::F, ~:~q:i~~~ ~:~f’~j~:l~?,~l [p.ter~st i* the fieid of micro\\ iie :hcor] and twnnlques, All lmembers d me IEEE are ei~io;t for :7.er710er Sfi;p lr. lfie >Oc:e:\ ?.nc $$:1. rxe:, e :Eis TRJ,\5.iCTIO>-S LLFOrIpaymen~ of the annual Society membership fee of $8,00, Affiliate membership isavailable upon pajment OFthe annua afi-iliate fee of S20.00. plus the Society fee of $8.00. For information onjoining write to tbe IEEE at tbe address belov. ADMIINISTRATI}7E COMMITTEE C. T, fltJCKER, Pwsiden~ H. G. ~LTW4K, JR., Vice ?residenl >“, W COx, .$ecreta{y- Treasurer ~. ~. ADAM’: H. ~OVv’E> JR. Y, KOXISHI J. M ROE J. E. XGEISFORD. JR. T. ~TOH H. J. ~-orth Jersey: R. SNYDER Syracuse: B. MITCHELL Connecticut: M, GILDEN ‘ Orange ~ounty ~. ~. .~CiKLAISD Tok~o: T. OKOSHi ~a]~as: ~. 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Pc&mastec Send Address changes to IEEE, 445 Hoes Lane, I%wataway, NJ 08854. IEEE TRANSACTIONSON MICROWAVE THEORYAND TECHNIQUES, VOL. MIT-31, NO. 6,JUNE 1983 429 Propagation Characteristics of Striplines with Multilayered Anisotropic Media TOSHIHIDE KITAZAWA AND YOSHIO HAYASHI Abstract —Various types of striplines with auisotropic media are anatyzed. The anafytfcat approach used in this paper is based on the network analytical method of electromagnetic fields, and the formulation iiCe process is straightforward for complicated stroctnres. Some numericaf REGION(1) d, I +%+ results are presented and comparison is made with the resnlts available in ~x the literature. REGION(2) ’32t E2C0 I. INTRODUCTION REGION(3) d3 33C0 I T HE NETWORK analytical method of electromagnetic r- fields has been successfully applied to analyze the Fig. 1. Generalstructureof coupledstripshaving multilayered aniso- tropicmedia. propagation characteristics of planar transmission lines. The hybrid-mode analysis of single and coupled slots was presented by employing this method [1], [2]. Recently, the As a first step we express the transverse fields in each dispersion characteristics of single rnicrostrip on an aniso- region by the following Fourier integral: [t3ro].pic substrate have been obtained using this approach @ )=;J“{ J’p(a;z)fl(a; x) )dae-j~oY 7 Single and coupled striplines on an anisotropic substrate qo) [=1 –~ Ip(a;z)zoxj(rx; x) have been analyzed by several investigators [3]-[7], but i=l,2,3 (2) hybrid-mode analysis is available only for the single micro- where strip case [3], [7], [8]. The purpose of this paper is to outline a new approach ““JG__ Koe -j.., f2=flxzo which is an extension of the treatment used in [1]–[3] and is capable of giving the propagation characteristics of various types of striplines with anisotropic media inclu- Ko=: sively. In what follows, the formulation process is il- yo~,o K = Xorx+ K=IKI (3) lustrated using the general structure with multilayered uniaxially anisotropic media. Two methods of solution are where POis the propagation constant in they-direction, X., presented. One is based on the quasi-static approximation Yo, and Z. are the x-, y-, and z-directed unit vectors, and it derives the transfo~ation from the case with aniso- respectively, and 1= 1and 1= 2 represent E waves (Hz= O) tropic layers to the case with equivalent isotropic layers. and H waves (E, = O), respectively. Equation (2) shows The other is based on the hybrid-mode formulation and it that the field components are a superposition of inhomoge- gives the frequency dependent solutions. The numerical neous waves whose spatial variation is exp{ – j(ax + Doy)}. results will be presented for single and coupled microstrips, Substituting the above expression into Maxwell’s field coupled suspended strips, and coupled strips with overlay. equation, we obtain the following transmission-line equation in each region: II. THE NETWORK ANALYTICAL METHOD OF ELECTROMAGNETIC FIELDS Fig. 1 shows the cross section of coupled strips having multilayered uniaxially anisotropic media; whose permittiv- ity tensors are where Eil 00 ~[0 = F== “)=- {i= o EiL () , i=l,2,3. (1) o 0 Cill [1 1)= Z! ManuscriptreceivedJune3, 1982;revisedJanuary18,1983. T. Kitazawaiswith theDepartmentof ElectricalEngineering,Univer- sity of Illinois, Urbana, IL, on leave from the JQtami Institute of Technology,Kitami, Japan. Y}’) = (5) Y. Hayashiiswith theKitami Institute of Technology,Kitami, Japan. 0018 -9480/83/0600-0429$01,00 01983 IEEE 430 IEEE TRANSACTIONS ON MZCROWAVETHEORY AND TECHNIQUES, VOL. MTT-31, NO. 6, JUNE 1983 SHORTCIRCUIT gion (1) can be obtained from the transverse fields accord- -, ,=,, ing to _._.!.- ~:v . v.(lfpxzo). (13) j(l.)q$,,l Substituting (2) and (12) into (13) and applying PO+O, E(l) can be obtained as z “.(/m aT{l)(a; z)tX(x’).e–~~(x–x’) dx’da. (14) ~ z=-(d,+d, ) —m SHORTCIRCUIT Fig. 2. Equivalent transmission-linecircuits for transversesection of Performing the integration by parts, using the equation of coupledstrips. continuity Notice that ~(i) and Kf) are the propagation constants in – j(.ou(x’) = *2X(X’) (15) the z-direction for E waves and H waves, respectively, and and applying the zero, frequency approximation u ~ O to Z(i) and Z2(i) are the characteristic impedance for these (14), we get waves. The boundary conditions to be satisfied are expressed as cosh{pl(z – dl)14} E$l)(~>z) = ‘2~7~rcm0 -~ ‘(a)~, sin~(p,d,]al) follows: ~(’)(dl)=O (6) .a(x~)e–l~(x–x’)~a~x’ (16) T“y(+o) = vp(-o) (7a) where 1~1)(+ O)-- I}z)( –O) = i, (7b) F(a) = 1 (17) ~l=coth(pldllal) +czeL ~(2)(–d2+O)=qt3)(–d2–0) (8a) I/2J(–d2+O) =lf3)(–d2–0) (8b) 1+ ~ tanh( p2d,lal) tanh( p~dqlal) L= (18) ~(3)(--d2–d3)=0 (9) tanh( pzdzlal) + ~ tanh( P3d~lal) i,= - f~ f~(a;x’)i(x’)dx’ (lo) J—W where the asterisk signifies the complex conjugate func- Pl=~ ~ze=g (19) tions, and i(x’) is the current density on the strip conductors at z = O“and may be expressed as and U(X’) is the charge distribution on the strip conduc- e = xOiX(x’)+ yoiY(x’). (11) tors. The potential distribution at z = Obecomes Considering the transmission-line equation (4) together V(X) ‘J%Z(X, Z) dz with the boundary conditions (6)–(9), we can obtain the equivalent circuits in the z-direction (Fig. 2). By conven- ‘;~~G(a;xlx’)u(x’) dad~’ (20) tional circuit theory, the mode voltages ~(i) and currents aO II’) in each region can be expressed in terms of il as where 2 F(a) fi(i)(a; z)= ZjZ)(a; z)il(a) G(a; xlx’)=m. — cosax cosax’ (for even modes) [al Ip(cK;z) =Zp(a;z)i/( a). (12) 2 F(a) —— — smax sin ax’ (for odd modes). The electromagnetic fields in each region can be obtained ~ “ Ial by substituting (12) into (2). (21) III. VARIATIONAL EXPRESSION FOR THE LINE On the strip conductor a < x < b, V(x) is equal to a CAPACITANCE constant VO, that is, the potential difference between the In the quasi-static approximation, the characteristic im- strip and the ground conductors pedance and the normalized propagation constant can be obtained from the line capacitance per unit length. We will V(X) =~o =~b/~G(a; XIX’) U(X’) dadx’, .0 derive a variational expression of the line capacitance of a.0 .,> ‘.0 1[ / Fig. 7. (a) Dispersion characteristics of coupled microstrips on sapphire. (b) Characteristic impedance of coupled microstrips onsapphire. (w/h ,0 0,,05 =1, a/h = 0.25; —even mode (hybrid-mode); ———odd mode (hy- / / brid-mode); —-—even mode (quasi-static); —--—odd mode (quasi- ,, ~~ / (//./’ static).) ,, .—. 002 ,.0’ ,0, 00, 0,0 h,, ,0.0, I Fig. 5. Dispersion characteristics of the first higher order mode of single microstrip on sapphire. (— this theory; —-—TM. mode of a sapphire-coated conductor; ———E1-Sherbiny’s [4].) 1~ /.-” m --- --- :, ---- ---- : -- ./,.,. ;. .0 ---- 2.0 ----~ I +---- ‘.0 I 0 0’ ,., ,,2 .,, h,, Fig. 8. Dispersion characteristics of coupled suspended strips. ([ ~ = 9.4, c,, = 11.6, W/h = 1, a/h = 0.25; —even mode (hybrid-mode); ———odd mode (hybrid-mode); —-— even mode (quasi-static); —-- Fig. 6. Characteristic impedance of single microstrip on sapphire — odd mode (quasi-static).) ,,0 Fig. 4 shows the dispersion characteristics, the frequency ,.2==7 dependence of the effective dielectric constant C,ff = ,/,., ~~/ti2cOp0, of single microstrip on sapphire substrates, -0.0i where Cefffor the dominant mode is reported and com- pared with the results of E1-Sherbiny [8]. The agreement is ~

IEEE MTT-V031-I06 (1983-06)

Similar IEEE MTT-V031-I06 (1983-06)