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TH2E-6A Conducting Sheet Model for Efficient Wide Band FDTD Analysis of PlanarWaveguides and Circuits

A. Lauer and I.WolffInstitute of Mo bile and Satellite Com mun ication Techn iques,D-4 74 75 Kam p-Lintfort, Germany, e-mail lauer@imst.de

Abstract-A simp le wide band equ ivalent circuit for the surface asimpedance of cond ucting sheets is introduced into the three di-skin effect, thus the frequency dependence of losses and of the ES ( l + j ) a sented as w ell as numerical results for the attenuation coefficientof microstrip and coplanar waveguides in comparison to refer-ence data.

tanh((1 +j)&) Hsmensional Yee FDTD schem e. The model is based on the plane Yo =2Ys =2- =G(1 )inner indu ctivity is included. Stability considerations are p re- 0 8 1C l = - - , WO=--, G o o = - = K t .WO pot2 R0,o

Herein t is the sheets thickness and CJ is its conductivity.A second order rational approximation2 for real value argu-ments x yieldsI . INTRODUCTION

Metallic losses in m icrowave waveguides and circuits cannotbe treated efficiently using straight forward FDTD simula-tions, since the skin effect is much smaller and faster thanthe wave propagation itself. Thus the simulation time wouldbe increased by magnitudes. the ap proxim ation interval was [0,80], the maximum relativeTherefo re thin conducting sheet approx imations have been in- error was estimated to be 0.076% .troduced into the Yee [ l] FDTD scheme, which work well in For the needed complex3 =mono-frequent and narrow band applications [2]-[4]. (1 + )2i2, the relative errorThis contribution presents a conducting sheet approximation I - (x)l/lf(x)l is less thansuitable for wide band FDTD simulations, which properly 3 % for Cl

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Equation (3) can be solved analytically, resulting in withE v = 1 (l+E++E3-+E-E+)&,0: =- 1 +

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V. NUM ER I C ALALIDATION

Fig. 4.Modified equivalent circuit for a tange ntial electrical field no de in theconducting sheet.At the modified nodes, @,,I and O V , 2 are subtracted from theright side of (8). The newly introduced nodes are treated as

For the discretization in time, one of the well known steppingschemes for lossy FDTD cells can be used.At the edges of the con ducting sheets, G, has to be halved aswell as L,J, Lv,2 and & , I , Rv,2 are to be doubled for properoperation.

IV . STABILITYONSIDERATIONSAs shown in [8] a stability criterion for the unmodified Yeescheme can be w ritten as

If the left hand side of (12 ) is interpreted as an estima te for themaximum angular resonance frequency of the FDTD schem e,an equation for C, of the modified cells can be obtained,

The use of (13) to calculate C, has two advantages:1. The time step At remains unchanged.2. In practical applications, oC, is in the range from

AN, hereas IYmJ>10AN, .g. theparasitic capa citor does not influenc e the accuracy ofthe simulations.AN o

It is also possible to eliminate the c apacitors from the mod-ified nodes, but in this case the time step possibly has to beadjuste d to ensure stability.

For the empirical validation of the approximation, a mi-crostrip and a coplanar waveguide, for which reference datais available, have been sim ulated to obtain attenu ation coeffi-cients.Both are on a 250 pm A1203 substrate, the conducting Au(0=41 MA Nm ) films are 5pm thick. The microstrip lineswidth is 2 25 pm, yielding a characteristic impedance of ap-prox. 53.2 SZ. The coplanar waveguide has an inner conduc-tors width of w = 125pm, the distance between the innersides of the ou ter conducto rs is d=225 pm, the characteristicimpedance is approx.48.6 R.A 1=10 mm long part of the waveguides was used for thesimulation.Since a very high accuracy is needed for the simulation oflosses

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sheets conductance as well as its the inner inductivity, de-pending on the frequency due to the skin effect, are prop-erly modeled. Excellent results have been obtained simulatinglossy microstrip and coplanar waveguides. Because the ap-proximation proposed here has almost no impact on compu-tation time and computer me mory consum ption, it is suitablefor the simulation for most p assive planar microwave circuits.

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T-v-REFERENCES - 0.1I

[I ] K. S . Yee, Numerical solution of initial boundary value problems in-volving Maxwells equ ations in iso tropic media, IE EE T r m .AntennusPrupugut., vol. AP-14, pp . 302-307, 1966.[2] J. G.Maloney and G. . Smith, The efficient modeling of thin mate-rial sheets in the finite -difference ime-domain (FDT D) method: IEEETruns. Antennus P ropugut., vol. AP-40, pp. 323-330, 1992.[3] L.-K. Wu and L.-T. Han, Implementation and application of resistivesheet boundary condition in th e finite-difference time-domain method:IEEE Truns. Antennus Propugur., vol. AP-40, pp. 628-633, 1992.[4] J. G. aloney and G. S . Smith, A comparison of methods for modelingelectrically hon dielectric and conducting sheets in the finite-difference fig. 6. Attenuation -IS21 I of a 1 =10mm long microstrip waveguide (E, =time-domain (FDTD)method, IEEE Trans. Microwme Theory Tech., 9.8, h =25 0 pm, w =225 pm , t =5 pm , (J =41 MA Nm ) in dependence ofvol. MTT-41, pp. 690 494 , 1993. the frequency f . The thick solid line represents the FDTD results, the thin[5] G. Kibuka, R.BertenbuS, M. Naghed, and I . Wolff , C op la n~umped solid line is measured using open circuits of two different lengths, the otherelements and their application in filters on ceramic and gallium w- ine is obtained using an approximative ormula [6]. The FDTD results do notsen ide substrates: in 19th Europeun Microwave Cunference Pmceed- start at a frequency of zero, because a modulated Gaussian pulse was usedings, pp. 656-661, 1989.[6] E. Hammerstad and F. Bekkadal, Micrustrip Hundbook, vol. S T F 44 A74169 of E U B rep or t.171 I. J. Craddock, C. J. Railton, and J. P. McGeehan, Derivation and np-plication of a passive equivalent circuit for the Finite Difference TimeDomain algorithm, IEEE Micruwuve und Guided Wave Lett.. vol. 6,pp. 404 2 , 1996.[8] A. Laue r and I. Wolff, Stable and efficient ABCs for graded mesh fdtdsimulations, in IEEE MTT-S Int. Micruwuve Symp. Digest, pp. 461-

464. 998.

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f(GHz) + Fig. 7. Attenuation -IS211 of a 1 =10 mm long coplanar waveguide (E, =9.8, h =250 pm , w =125 p m , d =225 pm,t =5 pm, o =41 M N V m ) inFig. 5. Real and imaginary Part of the admittance Yo n dependence of the dependence of the frequency .f. The thick solid line represents the FDTDfrequency f . The solid lines represents he approximation obtained from volt- results, the thin solid line is measured using open circuits of two differentage and current FDT D results in a microstrip waveguides backside metalim - lengths, the oth er line is obtained using a s tatic Finite Differen ce methodtion ( t =5 p m , o =41 MN Vm ), the thin line shows the exact results from in conjunction with an ohmic loss approximation. The FD TD results do notthe plane skin effect analysis. The FDT D results do not start at a frequency start at a frequency of zero, because a modulated Gaussian pulse was usedof zero, because a modukuted Gaus sian pulse was used for excitation. for excitation.

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