Closed-form Model of W/h- Dependent Equivalent Isotropic Relative Permittivity of Microstrip on Multilayer Anisotropic Substrate

Transcription

1 107 VOL.4, NO., MARCH 009 Closed-form Model of W/- Dependent Equivalent Isotropic Relative Permittivity of Microstrip on Multilayer Antropic Substrate A.K.Verma 1, Y.K. Aasti *, Himansu Sing 3, Microave Researc Laboratory, Department of Electronic Science, University of Deli, Ne-Deli, INDIA bol.net.in,. 3. Abstract-Te equivalent tropic relative permittivity of te multilayer microstrip line on te tropic substrate for 0 < / 10, tropic ratio 0.5 n 3 for bot lo and ig permittivity substrates is reported. Model as accuracy 0.5% against te full-ave metod. It computes effective relative permittivity and caracteristic impedance of microstrip on composite tropic substrates it deviations 5.3% and 1.78% respectively against te EM- softare Empire. It is used to obtain dispersion in multilayer tropic substrate microstrip up to mm ave range it ig accuracy against te results of HFSS. Index Terms- Antropic medium, Caracteristic Impedance, Equivalent effective permittivity, Isotropic medium, Microstrip transmission line etc. I. INTRODUCTION Te tropy in dielectric substrates affects te propagation parameter and caracteristic impedance of a microstrip line. Te EMsimulators are available to take in to account te tropy in a substrate. Hoever, te closedform expressions tat could be incorporated in te Circuit- Simulators are not available. Suc closed-form expressions are also useful for te development of te CAD oriented syntesis algoritm needed for designing te planar circuits on tropic substrates. Tese closed - form expressions are furter useful to analyze distortion in te pulses propagating on te tropic microstrip interconnects. Effort as been made by several investigators to account for te effect of tropy on te effective relative permittivity and caracteristic impedance of a microstrip line [1]-[7]. Te closed-form expressions are reported tat convert te tropic substrate to te equivalent tropic substrate it equivalent substrate tickness. Hoever, tese expressions do not consider te idt/ eigt (/) ratio of an tropic microstrip line ile obtaining te equivalent tropic relative permittivity [1, 5, 7]. As tis popular metod is not accurate, e must take into account te effect of /-ratio of te microstrip for accurate determination of te equivalent tropic relative permittivity. Edard et al. [4, 5] ave used te finite-difference metod it an analytical formula to acieve tis end; ereas Fritsc and Wolff ave used te full-ave metod [6]. Edard et al. ave also given a closed-form expression to get an equivalent tropic relative permittivity. Hoever, it is applicable only to te Sappire substrate [4]. Tese researcers ave not considered microstrip on multilayer tropic substrates. Te present ork reports te /-ratio dependent closed form model to compute te equivalent tropic relative permittivity of te tropic microstrip line itout using te finite difference or te full-ave metod. Te model is valid over te ide range of line parameters. Te accuracy of te model is tested against te available experimental and numerical results and also against te results obtained from te EMsoftare Empire [8]. Te model is furter extended to te microstrip on multilayer tropic substrates. II. THE MODEL JMOT ISRAMT

2 108 VOL.4, NO., MARCH 009 Te process of getting / dependent equivalent / tropic relative permittivity ( ) of te tropic substrate is son in Fig.1. It involves to steps. (i) To get te equivalent tropic substrate it tropic relative permittivity ( r,eq ) and equivalent substrate tickness ( eq ) itout consideration of /-ratio. (ii) To get te / dependent equivalent tropic / relative permittivity it original substrate tickness (). We ave used te metod of partial capacitance to develop te model. Fig.1: Process of obtaining / dependent equivalent / req, tropic substrate ( ) Step #1: Te folloing standard relations convert te tropic substrate of a microstrip to an equivalent tropic substrate r, eq and equivalent substrate tickness eq itout accounting for te affect of / ratio [1, 7] req eq Were, W Antropic Substrate Original tropic Substrate r xx r xx ( r xy r xy eq ), + Δ Equivalent Isotropic Substrate r, eq Step #1 W Equivalent Isotropic Substrate ( / ) req, Step # (1) r xx r xy r ς rς cos r yx ( rς sin θ + rς θ + r, rη r, Antropic ratio n rη sin rη ) sin rη cos rxx ryy θ, θ cosθ, θ () Were (x,y) are te pysical axes of te microstrip and ( η, ζ ) are te crystal axes of te tropic substrate. Te parallel and normal components of te uniaxial tropic substrate and r r respectively are provided by a manufacturer. Te relative permittivity components along te X- axis and Y axis are r and xx r respectively. Te angle θ is te yy misalignment angle beteen te substrate s crystal axis and pysical axis. Te component provides us te main central parallel plate capacitance p.u.l. of te microstrip. Te tropic ratio is defined by n. Step #: Te metod of partial capacitance provides te / dependent equivalent tropic / relative permittivity ( req, ) of te substrate it original idt and tickness as son in Fig.1. Te idt () of a microstrip is extended to + Δ due to te fringe field. Te fringe electric field creates te fringe capacitance (C f ). We assume tat te main central parallel plate capacitance (C p ) is caused by te component. We furter assume tat te fringe capacitance of te original microstrip on tropic substrate is same as te fringe capacitance of microstrip of te step #1on te equivalent tropic substrate relative permittivity r,eq and substrate tickness eq. Tus te total line capacitance p.u.l. on te / dependent tropic substrate could be ritten as a combination of te fringe capacitance and te central capacitance JMOT ISRAMT

3 109 VOL.4, NO., MARCH 009 / 0 ) / ( + Δ 0 r, eqδ 0 r, yy + Δ r, eq + r, yy + Δ + Δ (3) Te line capacitance p.u.l. is proportional to te idt of te strip conductor. Tus te fringe capacitance C f Δ, te central parallel plate capacitance C p and te total line capacitance C t ( + Δ). In vie of tese expressions, above equation gives expression for te / dependent equivalent tropic relative permittivity / r, eq C f + r, yy C p Ct r, eq Ct + ( r, yy r, eq) C p Ct (4) Were, C t C p + C f. Te total line capacitance Ct p.u.l. and te fringe capacitance C f p.u.l. of microstrip on te equivalent tropic substrate are obtained from te step #1 Te central parallel plate line capacitance p.u.l. is obtained from te component of te tropic substrate. ryy r eff (, eq, req ) Ct v Z0 (, eq, req 1) C p ryy 0 ( a) ( b) (5) Were, v is velocity of EM-ave in te freespace. Te caracteristic impedance Z0 (, eq, req 1) on air substrate and te effective relative permittivity r eff (, eq, req) on te equivalent tropic substrate it relative permittivity req and substrate tickness eq are obtained from te closed-form expression of Hammerstad-Jensen [8]. We can use te W/ dependent equivalent / tropic relative permittivity req, it substrate tickness and strip idt to compute te correct static effective relative permittivity and caracteristic impedance of te Equivalent Isotropic Relativ e Permittivity original microstrip on te tropic substrate by using again te closed-form expression of Hammerstad-Jensen [8]. Any dispersion model / can be used on te substrate it to get te frequency dependent effective relative permittivity and caracteristic impedance of te tropic substrate microstrip [7.9]. III. VALIDATION OF MODEL Fig. compares te present model against te finite difference metod results on Sappire substrate [5]. Te present model as maximum deviations 1.3% for te narro idt microstrip line. Te table-1 furter compares te model against te full-ave results of Fritsc and Wolff [6]. Te maximum deviation is about 0.5%. Table-1: Deviation in equivalent tropic relative permittivity of present model rx ry Wolff [ 6] Present Model Finite- Difference Metod [5] Present Model % Deviation / Fig.: Comparn of present model against te finite difference metod JMOT ISRAMT

4 110 VOL.4, NO., MARCH 009 Table-: Average % deviation in models against Empire ( 0.1 / 10.0) ( 0.5 n 3.0) % Dev. in Static Dev. in Static Z 0 r,eff Alexo Present Alexo Present ry [] Model [] Model Using te present model, e ave computed te effective relative permittivity and caracteristic impedance of microstrip over ide range of tropic ratio (n) on substrate it ryy.55, 4.5, 9.9 against te results obtained from te EM-softare- Empire [13]. We ave also compared te results of Alexopoulos and Krone [] obtained by numerical metods against te results of Empire. Fig.3 and Fig.4 so tese comparns for te effective relative permittivity and caracteristic impedance for W/0.1,1.0,10. It is apparent tat te present model sos better agreement it te results of Empire as compared to te numerical metod of Alexopoulos and Krone. Te table- as summarized te comparn of te average deviations on tree tropic substrates it ide range of tropic ratio (n). Te numerical metod of Alexopoulos and Krone as average deviation up to 13.58% for te effective relative permittivity and 9.11% for te caracteristic impedance; ereas te present model as tese deviations only 5.3% and 1.78%. We note tat te computed results of effective relative permittivity by several EM-softare also ave large variations among temselves. Te table-3 sos % deviation in computed results of te effective relative permittivity obtained by Empire and HFSS against te results of Sonnet. Te maximum deviation in Empire and HFSS are 10.9 % and 7.08% respectively. Empire as maximum deviation for te narro line; ereas HFSS as maximum deviation for te ide line. Te average deviation for te Empire and HFSS are 5 % and 4% respectively it respect to Sonnet. In vie of tese comparns, te present model appears to be accurate and acceptable for te design ork. Effective Relative Permittivity Static Caracteristic Impedance Finally te present model as been used along it te dispersion model of Jensen [9] to compute te dispersion in microstrip line on Sappire substrate for 0.4 mm and 0.4 mm, 0.6 mm up to 30 GHz. Fig.5 compares te present model against te experimental results of Fritsc and Wolff [6]. Te maximum deviation in te present model is itin 1% tat is itin te accuracy limit of Jensen dispersion model for te tropic substrate /10.0, Alexopoulos[] /10.0, Present Model /10.0, Empire /1.0, Alexopoulos[] /1.0, Present Model /1.0, Empire /0.1, Alexopoulos[] /0.1, Present Model /0.1, Empire Fig. 3: Comparn of Static Effective Relative Permittivity against Empire ( 9.9) Ratio of te substrate indices of refraction (n) ryy /10.0, Alxopoulos[] /10.0, Present Model /10.0, Empire /1.0, Alxopoulos[] /1.0,Present Model /1.0, Empire /0.1, Alxopoulos[] /0.1, Present Model /0.1, Empire Ratio of te substrate indices of refraction (n) JMOT ISRAMT

5 111 VOL.4, NO., MARCH 009 Fig. 4: Comparn of Static Caracteristic Impedance against Empire ( 9.9) Effective Relative Permittivity y ryy Present Model,0.4mm,0.4mm Measured Data[6],0.4,0.4mm Present Model,0.6,0.4mm Measured Data[6],0.6,0.4mm Frequency (GHz) Fig. 5: Comparn of dispersion using present model against experimental results [6] on Sappire. IV. MULTILAYER ANISOTROPIC SUBSTRATE Te present model can be extended to te microstrip line on te multilayer tropic substrate son in Fig. 6. For te step #1, on using equation- 1 and equation-, e convert eac tropic layer of tickness m (m1, N) it relative permittivity ( rm, xx, rm, yy ) to te equivalent relative permittivity rm, eq (m1,...n) and equivalent substrate tickness eq,m (m1,...n). Te step # provides te / dependent equivalent tropic relative / H permittivity for single layer substrate of tickness H N. Fig. 6: Multilayer tropic substrate microstrip line. We convert te multilayer tropic substrate to an equivalent single layer tropic substrate r,eq by using te single layer reduction (SLR) formulation [10]. At first e convert eac tropic layer to te equivalent tropic layer mentioned above. Next, e compute te effective relative permittivity r, eff ( / H eq, eq ) of te multilayer microstrip line on te equivalent tropic substrate by using te variational metod it TTL tecnique [11, 1]. Tus te effective relative permittivity of te multilayer tropic substrate is r, eff ( / H, C (, eq1,... C (, H, 0 eq eq eqn, eq1 ) eq1, eqn eqn ) 1) (7) Were layer ise equivalent tropic relative permittivity (m1, N) and equivalent eqm eqm substrate tickness (m1, N) for eac layer are obtained from equations (1). Te line capacitance p.u.l. of te microstrip line on te multilayer substrate is obtained by te variational metod [10, 11, & 1] C (, 1 π eq1, eq, ~ [ f ( β ) / Q] Y β eqn, 1 dβ eq1, eq... eqn ) (8) r, xxn r, yyn r, xx m r, yy m N m r, xx 1 r, yy 1 1 Te line capacitance on te air substrate C0 (, H, eq1... eqn 1) is also obtained from above expression. Y is admittance function of te TTL metod signifying te Green s ~ function, f ( β ) is te assumed trial carge distribution function in Fourier domain and Q is te total carge on microstrip line for te trial carge distribution. In te single layer reduction (SLR) formulation te multilayer microstrip line is reduced to an JMOT ISRAMT

6 VOL.4, NO., MARCH equivalent single layer substrate of tickness H ( N ) and equivalent tropic relative permittivity r, eq( / Heq, eq).te equivalent tropic relative permittivity is obtained from te Weeler s transformation [10] r, eq ( / H eq, eq ) r, eff ( / H eq, eq ) q( / H ) (9) Were, H eq 1 eq + eq Neq and te filling factor q( / H ) is obtained from 1 q (1 + p), 1H 1 + p 1H 1 + 1/ 1/ H 1 H, H < 1 Te total line capacitance of te multilayer microstrip line is computed from Ct V r, eff ( / H eq, eq ) * * Z 0 ( / H, r 1... rn 1) (10) (11) Were, V is te velocity of EM-ave in te freespace. Te caracteristic impedance of microstrip * * on air substrate, Z ( / H,... 1) 0 r1 rn is obtained from te closed-form expression of Hammerstad Jensen [8]. Te central capacitance of te multilayer tropic substrate microstrip is obtained from te yycomponent of te tropic relative permittivity o ryy, eq C p H H ryy, eq N i i ryy, i ( a) ( b) (1) Finally te /H dependent equivalent single layer / H tropic substrate of relative permittivity req, and tickness H is obtained from equation- (4) in ic substrate r,eq is taken from equation- (9), r, yy is replaced by ryy, eq from equation (1b), C t and C p are taken from equations- (11) and (1a) respectively. Te variational metod or Hammerstad-Jensen formula [8] is applied on tis equivalent tropic substrate to obtain te effective relative permittivity reff of te multilayer tropic microstrip line. Te static caracteristic impedance Z O of te multilayer tropic microstrip line can also be computed on tis equivalent tropic substrate by using te Hammerstad Jensen expression. Te unified dispersion model [10] can be used to compute te dispersion in te multilayer tropic microstrip line. V. RESULTS OF MULTILAYER ANISOTROPIOC SUBSTRATE MICROSTRIP LINE We ave computed te static effective relative permittivity and caracteristic impedance Z O reff of te microstrip for to cases of composite layer tropic substrate- case I: first layer alumina ( r, xx 8.607, r, yy ) and second layer sappire ( r, xx 9.4, r, yy ); case II: first layer boron nitride ( r, xx 3.4, r, yy 5. 1 ) and second layer sappire ( r, xx 9.4, r, yy ); 1. Fig.7 compares te static effective relative permittivity computed by te present model against te reff results obtained from te EM- softare HFSS [14]. Te average deviation of results is about 5%. We ave noted in section-3 tat tere is a significant variation in results obtained from various EM- softare. In vie of it, 5% average deviation of te present model against te results of HFSS is reasonable. Fig.8 sos te caracteristic impedance Z O for bot cases of te composite tropic substrate microstrip line. JMOT ISRAMT

7 113 VOL.4, NO., MARCH 009 Table-3: % Deviation in reff computed by commercial softare s against Sonnet r 9.9 r 4.5 r.55 / Empire HFSS Empire HFSS Empire HFSS Static Effective Relative Permittivity A B Alumina&Sapire Present model Alumina&Sapire HFSS BoronNitride&Sapire Present Model BoronNitride& Sapire HFSS W/ reff Fig.7: Effective relative permittivity of composite layer tropic substrate microstrip line. A B Once W/H dependent equivalent single layer substrate is obtained, e can use te unified dispersion model [10] tat as adopted Kriscnining Jansen [9] dispersion model to compute dispersion in te multilayer tropic microstrip line. Fig. 9a and Fig.9b so comparns of te dispersion computed by te present model and HFSS. Te microstrip lines are taken for to cases- sappire on alumina and boron nitride on sappire substrates; for W/H 0.101, 1.6, Bot te cases so good agreement of results up to 30 GHz. Even at muc iger frequency good agreement as been observed. Te maximum 5% deviation is only for te static case tat decreases it increase in frequency. Static caracteristic Impedance p Alumina & Sapire Boron Nitride & Sapire W/ Effective Relative Permittivity y HFSS W/ 9.14 Present Model W/ 9.14 HFSS W/ 1.6 Present Model W/ 1.6 HFSS W/ present Model W/ Frequency (GHz) F (GH ) Fig.8: Caracteristic impedance Z O of composite layer tropic substrate microstrip line. Fig. 9a: Dispersion of microstrip for sappire on alumina substrate JMOT ISRAMT

8 VOL.4, NO., MARCH Effective Relative Permittivity y HFSS W/ 9.14 Present M odel W/ 9.14 HFSS W/ 1.6 Present M odel W/ 1.6 HFSS W/ Present Model W/ Frequency (GHz) Fig. 9b: Dispersion of microstrip for boron nitride on sappire substrate. VI. CONCLUSION We ave developed closed-form model to obtain te equivalent tropic relative permittivity of te microstrip line on te tropic substrate for 0 < / 10, tropic ratio 0.5 n 3 for bot lo and ig permittivity substrates. Te model as accuracy 0.5% against te full-ave metod. Te model is used to compute effective relative permittivity and caracteristic impedance of microstrip on tropic substrate it deviations 5.3% and 1.78% respectively against te EM- softare Empire; ereas te numerical metod of Alexopoulos and Krone as average deviation up to 13.58% for te effective relative permittivity and 9.11% for te caracteristic impedance. Te model is extended to te multilayer tropic substrate microstrip line it deviation 5%. Te model computes dispersion in multilayer tropic substrate microstrip it good accuracy. Te model is useful for te development of accurate syntesis programs for te single layer and multilayer tropic microstrip lines needed for te design of circuits and components. Te model can also be used for analyzing te effect of tropy on te pulse propagation on te multilayer tropic microstrip line. [1] N.G.Alexopoulos, Integrated-Circuit Structures on Antropic Substrates, IEEE Trans. on Microave Teory and Tecniques, Vol.MTT- 33, pp , Oct [] N.G.Alexopoulos and C.M.Krone, Caractristics of Single and Coupled Microstrips on Antropic Substrates, IEEE Trans. on Microave Teory and Tecniques, Vol.MTT-6, pp , June [3] Jon L. Tsalamengas, N.K.Uzunoglu and N.G. Alexopoulos, Propagation Caracteristics of a Microstrip Line Printed on a General Antropic Substrate, IEEE Trans. on Microave Teory and Tecniques, Vol.MTT-33, pp , Oct [4] Terence C. Edards and Roger P. Oens, -18- GHz Dispersion Measurements on Ω Microstrip lines on Sappire, MTT-4, Aug 1976 pp [5] R.P.Oens, J.E.Aitken and T.C. Edards, Quasistatic Caracteristics of Microstrip on an Antropic Sappire substrate, IEEE transaction on MTT, Vol.MTT-4, NO. -8, August [6] U.Fritsc and I. Wolff, Caracterization of Antropic Substrate Materials for Microave Applications, IEEE.MTT-S Digest, pp , 199. [7] R.K.Hoffmann, Handbook of Microave Integrated Circuits, Capter -8, pp.13-4, Artec House, USA, [8] E. Hammerstad., O. Jensen: Accurate models for Microstrip Computer Aided Design, IEEE MTT- S, Int. Microave Symposium Digest, pp , [9] M., Kriscnining, R.H. Jansen, Accurate model for effective dielectric constant it validity up to millimetre-ave frequency, Electronics Letters, Vol. 18, pp. 7-73, Jan [10] A.K.Verma, G. Hassani Sadr, Unified Dispersion Model for Multilayer Microstrip Line, IEEE Trans. on Microave Teory and Tecniques, Vol-40, pp , July 199. [11] E.Yamasita, Variational Metod for te Analysis of Microstrip like Transmission Lines, IEEE Trans. on Microave Teory and Tecniques, Vol.MTT-16, No.4 pp , Aug [1] R. Crampagne, M.Amadpana and J.L.Guiraud, A Simple Metod for Determining te Green s Function for a Large Class of MIC Lines aving multilayered Dielectric Structures, IEEE Trans. on Microave Teory and Tecniques, Vol.MTT- 6, No., pp.8-87, Feb [13] IMST-Empire v.4.0, GmbH Germany. [14] Ansoft HFSS v Ansoft Corporation. Pittsburg, USA. [15] Sonnet-Lite v11.55.,syracuse, NY, USA. REFERENCES JMOT ISRAMT