modeling of nonlinear active regions with the fdtd method

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IEEE MCROWAVE AM) UIDED WAVE LETTERS, VOL. 3, NO. 9, SEPTEMBER 1993 333

Modeling of Nonlinear Active Regionswith the FDTD Method

B. Toland, Member, ZEEE, B . Houshmand, Member, ZEEE, and T. Itoh, Fellow, ZEEE

Abstract-The FDTD method is extended to include nonlinearactive regions embedded in distributed circuits. The proceduresnecessary to produce a stable algorithm are described, and asingle device cavi ty oscillator is simulated with this method.I. INTRODUCTION

ECENTLY, the two-dimensionalFDTD method was ex-R ended to include active, passive and possibly nonlinearlumped circuit elements [l]. In that work the incorporation ofthe lumped elements into the FDTD lgorithm is described,and transmission lines with various lumped element loadswere simulated. Only one active load was modeled, whichwas linear, and this was modeled in the same fashion as thepassive lumped elements. However, we have found that toaccurately model active nonlinear regions requires som e extrasteps and precautions which are not necessarily needed forpassive element modeling. Some of these precautions havebeen noted in [2], in which active nonlinear circuit modelswere incorporated into the TLM method.For instance, in [2] the TLM method was used to modelactive nonlinear subregions of distributed circuits, and itwas noted that regions of negative conductivity may causespurious oscillations at the TL M mesh cut-off frequencies.To circumvent this problem, the TLM mesh cut-off is chosento be well above the active device cut-off frequency. It istherefore not too surprising that to avoid spurious oscillationsat the FDTD mesh cut-off frequency, the FDTD mesh cut-off frequency must be well above the active device cut-offfrequency. In addition, we have found that in many cases theFDTD algorithm will become unstable unless some care istaken in incorporating the active device model into the FDTDmethod. This is because a realistic active device can produceextremely large local currents which produce large fields thatare fed back in to the device mo del that can create a "runaway"effect.The purpose of this letter is to describe the steps we haveimplemented to produce a stable algorithm which simulatesrealistic devices with both active and nonlinear regions. Weuse this algorithm to simulate a single device cavity oscillator.The device is both nonlinear and active, and the nonlinearactive region is distributed over more than one cell.

Manuscipt received May 25, 1993. This work was supported by JSEPContract F49620-92-C-0055 and the Office of Naval Research Contract"14-91-J-1651 and A rmy Research Office Contract DAAHO4-93-G-0068.Editorial decisio n made by S. Maas.The authors are with the departmentof Electrical Engineering, Universityof Califomia, Lo s Angeles, 66-147A Engineering IV, 05 Hilgard Avenue,Lo s Angeles, CA 90024 1594 .IEEE Log Number 9211895.

11. INCORPORATIONOF THE ACTIVE EVICEMODEL NTO THE FDTD ALGORITHMThe active device which we will model is a resonanttunneling diode (RTD) hich was fabricated at UCLA. Theequivalent circuit and IV characteristic are shown in Fig. 1 . The junction capacitance C, series resistance R and nonlinearcurrent source F (K ) are converted into distributed parametervalues E , and F ( V , ) which will be used for each mesh cellin the packaged device volume. This is done so that each cellin the active region is a voltage source, with the aggregateeffect of all cells in the active region representative of thephysical device. Note also that we subtract out the dc biaspoint from the voltages and currents, i.e., V , an d F ( V , ) arewith respect to th e bias point. Voltage or curren t sources can beincluded into the FDTD algorithm as described in [11, but withthe follow ing additional steps to ensure a stable algorithm . Forthe device shown in Fig. 1, the voltage sou rce is a solution of

To solve this equation, we use forward time average differ-encing [3]. This requires that we expand the nonlinear currentsource in a Taylor series [4]:dF(V =V,")[v,"+l-:]. (2)(V,"+l)=F(V,")+ dV

We then obtain the finite difference form of ( l) , which isKn+' =AiV," - A2F(V,")- A3A?/(E;+'+ E ; ] , (3 )

where2 R A t A t,A 2 =-RC - A t [ l - R&(V,")] , A 3 = 7

$4)PA1 =

with P =2R C +A t [ l+RF(V,")]and & ( V r )=dF(l;y.This can be put back into the FDTD algorithm which isof the forward time average form as in [ l] . Since we areconsidering a y-directed source, we have the following forthe E y component:E,"+'

[&- & ( l - A B ) ][ z+k ( 1 - A B ) ]=E,"H E + 1 ' 2 ( i , j +1 / 2 , k +1 / 2 ) - HE+1'2(i,j+1 / 2 , k - /2 )

AZ +& ( l - A S ) ]+ [ A t1051-8207/93$03.00 0 1993 IEEE

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Fig. 1 . Circuit model of resonant tunnelling diode and IV characteristic.

Fig. 2. The layout of the waveguide oscillator and the steady-state electricfield distribution at one instant in tim e.

- H : + q +1 / 2 , j +1 /2 , k) - H,+/2( i - / 2 , j +1/2 , k)

Ev s evaluated at the point ( i , j + 1/2, C )( Yees notation [ 5 ] ) .By using (3) in conjunction with (5) , instabilities have beeneliminated from all tunnel diode cases that we have considered.

UI. MODELING F A SINGLE DEVICE AVITY OSCILLATORIn this section, we describe the FDTD simulation of areduced height WR28 waveguide cavity oscillator shown inFig. 2.Th e RTD parameter values of C =0.05 pF , R =9.7 Rand an extrapolated polynomial fit for F(Vbias+Vs) (seeFig. 1) were used [ 6 ] . At the bias point of vbia = 1.25

ao U) M bo a0 10.0(4 x 10-10

Fig. 3. The time development of the electric field E , at anobservationpointoutside the resonant cavity.

a10

0 36 72 1 0FWU-Y (GW

Fig. 4. The frequ ency spectrum of the steady-state electric field.

V, Ibias = 11.86 mA the small signal conductance G =k(vbia)= -32.5 m R- and the diode cut-off frequencyis 152 GHz. The mesh sizes were Ax = AZ = 0.17775mm. The cavity width is 40Ax, the distance between thebackshort and inductive iris is 22Az, and the diode covers 6cells and is cen tered in the cavity. The symm etrical, perfectlyconducting iris has zero thickness, an aperture 12Ax wide,with the waveguide 5.0 mm high.In Fig. 2, we show the layout of the oscillator and thesteady state electric field distribution Ev at one instant intime. Fig. 3 is a plot of the electric field vs. time at anobservation point outside the cavity. A small amount ofnoise is introduced into the cavity, and the oscillations buildup with time until saturation occurs. In Fig. 4, we show thecorresponding spectrum of the oscillator in the steady state(a fifth-order polynomial was used to model the ND R regionof the RTD). Both spurious oscillations at the mesh cut-offfrequency and algorithm instabilities have been suppressedwith this algorithm and by modeling the correct physicalbehavior of the active device.


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TOLAND er al.: MODELING OF NONLINEAR ACTIVE REGIONS WITH THE FDTD METHOD 335

IV . CONCLUSION T.-W. Huang, R. Chan, and M. Jensen for their advice andA FDTD lgorithm that simulates distributed circuits con-taining nonlinear active elements has been described, and a sin-gle device cavity oscillator has been modeled. The additionalsteps necessary to obtain a stable algorithm that simulates arealistic active device embedded in a distributed circuit havebeen described. An application of the algorithm described in

this paper can be applied to a number of circuits which havebeen characterized by the harmon ic balance method. However,the present method has the adv antage in that the passive planarstructures, including discontinuities, can be handled withoutresorting to the equivalent circuits which are used in theharmonic balance method.ACKNOWLEDGMENT

The authors are grateful to 0. Boric for for providing uswith the RTD parameters. The authors would also like to thank

encouragement.

REFERENCES[ l ] W. Sui, D.A. Christensen, and C.H. Dumey, Extending the two-

dimensional FDTD method to hybrid electromagnetic systems withactive and passive lumped elements, IEEE Trans. Microwave TheoryTech., vol. 40, o. 4, pp. 724-730, Apr. 1992.[2] P.Russer, P.M. So, and W. J. R.Hoefer, Modeling of nonlinear activeregions in TLM, IEEE Microwave Guided WaveLen., vol. 1, no. 1, pp.10-13, Jan. 1991.[3] L. Lapidus and G. F. Pinder, Numerical Solution of Partial DirerentialEquations in Science and Engineering. New York John Wiley, 1982.[4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling,Numerical Recipes.[5] K. .Yee, Numerical solution of initial boundary value problems in-volving Maxwells equations in isotropic media, IEEE Trans . AntennasPropagar., vol. Ap-14, no. 3, pp. 302-307, May 1966.

New York Cambridge UNv. Press, 1986.

[6] 0. Boric-Lubeeke, private communication.

modeling of nonlinear active regions with the fdtd method

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