SIMULATION OF A NONLINEAR GAAS MESFET MODEL FOR USE IN THE DESIGN OF NONLINEAR MICROWAVE CIRCUITS

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1 SIMULATION OF A NONLINEAR GAAS MESFET MODEL FOR USE IN THE DESIGN OF NONLINEAR MICROWAVE CIRCUITS Mohammed Abdo Tuko Department of Eletrial Engineering Addis Ababa University 29 ABSTRACT A omputer program has been developed that performs a large-signal simulation of a GaAs Metal-Semiondutor-Field-Effet-Transistor (MESFET) using the Curtie-Ettenberg model fl]. The model is then used to desig linear mirowave iruits suh as frequeny multipliers a11d power amplifiers. The simulation employs a two-stage numerial approximation. The so alled Hmultiple refletion" method, whih is a speial kind of "hamw11i balane" tehnique, is used between the FET and the li11ear embeddi11g network. The illtrinsi FET with its no11li11earities is a11alyzed with Newton's method in the time domain. Compariso11 of simulatio11 and experimental results of the desig11ed iruit show good agreemellt. INTRODUCTION In the past deades the aurate modeling of mirowave field effet transistors (FETs) has preoupied the attention of many iruit designers. The modelling of the linear behavior of the FET is now a resolved problem. What still hallenges the mirowave iruit designer is the aurate modelling of the nonlinear behavior of the FET. Many researhers ame up with different models; the Curtie-Ettenberg model [ l], the Materka Kaprzak mode [2], and the Raytheon model [3] are among the few. The purpose of this paper is to optimize one of these models to make them appliable in the design of optimum nonlinear mirowave iruits suh as frequeny multipliers, mixers, and power amplifiers. The Curtie Ettenberg model is used for the purpose beause it is less ompliated and requires less omputation time. The simulation is done by separating the whole iruit into nonlinear and linear subiruits and then analyzing it by the method of 'harmoni balane'. The idea of harmoni balane is to find a set of port voltage waveforms or harmoni voltage omponents that give the same urrents in both the linear network equations and the nonlinear network equations; when that set is found, it must be the solution. THE ANALYSIS METHOD The harmoni balane approah is most useful for strongly or weakly nonlinear iruits that have single-tone exitation. It is widely applied to analyze mirowave iruits suh as frequeny multipliers, power amplifiers and mixers subjeted to loal osillator drive. In these iruits the harmoni balane tehnique usually onsists of a time domain model of the GaAs FET oupled with the frequeny domain models of the input and output mathing networks. In other words, the harmoni balane method allows the use of multiport iruit theory [4) whih an be used to simplify part of the iruit by lumping all the linear reatanes, impedanes, transmission lines, and other linear elements into a single matrix of limited size. By desribing the linear part of the iruit as one multiport, it need be evaluated only one at eah harmoni, with the results stored in matries and no further evaluation is neessary. The model is illustrated in Fig. 1. In referene to Fig. 1, sine the urrents in the linear and nonlinear sub-iruits must be equal the matrix Eq.(l) must be satisfied. In Eq.(l) ln.k is a phasor quantity, the 'Jeh harmoni omponent of the port n, alulated via the port voltages and the f-matrix of the linear subiruit; ln,k is the urrent omponent alulated via the same port voltages and the nonlinear elements. The iruit in Fig. 1 is suessfully analyzed when either the steady-state voltage or urrent waveforms at eah port are known. Alternatively, knowledge of the frequeny omponents at all ports onstitutes a solution beause the frequeny omponents and time waveforms are related by the Fourier series. If, for example, the frequeny-domain port voltages

2 3 Mohammed Abdo are known, one an use the Y-parameter matrix for the linear subiruit to find the port urrents. The port urrents an also be found by inverse Fourier transforming the voltages, to obtain their timedomain waveforms, and determining the urrent waveforms from the nonlinear elements. 11.X 12, /2, X 12, 12,1 = (1) The linear suhiruit an be treated as a multiport and desribed by its Y- or S- parameters. The nonlinear elements are modelled by their global IN or QN harateristis and analyzed in the time domain (4). In our ase the FET onstitutes the nonlinear suhiruitry and hene is analyzed in the time domain in order to simulate it aurately. On the other band, the linear iruit response of the FET an be analyzed in the frequeny domain. Transformation between the time and frequeny domains is aomplished using a disrete Fourier Transform (DFT). It must be noted that the equations for the nonlinear suhiruit are known only in the time domain and annot be transformed to the frequeny domain. +.1 INPU'l' PORT + UNEAR SUBCTRCUTT NONUNEAR SUBCIRCUIT N OUTPUT PORT Figure I A Nonlinear Mirowave Ciruit, Divided into Linear and Nonlinear Suhiruits, with the Soure and Load Impedanes Absorbed into the Linear Suhiruit.

3 Thus, every voltage wave arriving at a port of the nonlinear subiruit is transformed to the time domain using the Inverse Fast Fourier Transform (IFFT). Design of Nonlinear Mirowave Ciruits 31 After the time domain analysis of the nonlinear iruit voltages and urrents, the results are transformed bak to the frequeny domain. It is to be noted that it is the intrinsi FET that makes up the nonlinear subiruit. In referene to Figs. 2 and 3 (with Z 1 = ~ = ZL, the intrinsi FET an be analyred using the following network equations. - i - dqgs - dq8'1 = 8fl dt dt Figure 2 The Nonlinear Transistor Model (2) (3) dq,, = + -- (4) dt where v 1 1 and v 2 1 are the inident voltages at ports l and 2, respetively. Qv and Q8'1 are the harges stored in the gate-soure and gate-drain juntions, respetively. For the atual omputation, Eqs. (3) and (4) are rearranged and ombined with Eq. (2) resulting in the following equations: 2v~ + 2v; - 2v.,, - v 48 + v 1 - v.,, - v 8fl = z, RI (5) The voltages v 1, Va., and v8'1 are taken as 1 1 independent variables, while v 1 and v 2 are obtained from the refletion algorithm and they hange at every iteration. The resulting system of 3n differential equations is disretized by approximating the time derivatives with seants through neighbouring samples as: v(t + J:lr) - v(t -.:it) 2.tl.t (6) This leads to a system of nonlinear equations with 3n independent variables where n indiates the n A intrinsi FET Figure 3 The Refletion Priniple n...\ number of time domain samples. This system is solved by means of Newton's method as desribed below. All derivatives an be expliitly alulated and therefore by easily evaluated. Beause the multiple refletion algorithm requires the equations to be solved many times with similar values, every iteration uses the previous solution as an initial guess. Only the very first iteration is supplied with a guess based on the DC omponents. NEWTON'S MEmOD The differential equations are disretired for all n samples forming the vetor of independent variables as given in Eq. (7). On the other band, the left sides of Eqs. (3), (4), and (5) are taken as omponents of a 3n-dimensional vetor error funtion F whose zero is found iteratively by the Newton's method. That is to say F an be expressed as shown in Eq. (8).

4 32 Mohammed Abdo V= vpn) v n v,,i(tl) v + 1 n (7) ijfi ijfi ijf av;- i)v2 av ijf2 ijf2 ijf2 ijf = av;- i)v2 i)v3 ijv ijf3 ijf3 ijf3 avl i)v3 av (1) v,itn) v:i.. v2(t1) v:i ' The Jaobian is then inverted using the Gaussian elimination algorithm [5]. THE MULTIPLE REFLECTION ALGORITHM 2v~ - v.u ZL - ids. dqgd + lgd + -- dt v, - v gs dqgd +-- F= Ri dt 2v: + 2v~ - 2v.u - v 41 + v 1 - v" - vgd ZL RI (8) Moreover, the updated voltage vetor v+ 1, is expressed as: ~.1 = ~ - ( ~~] ~1 F(~) (9) where p indiates the iteration number. Equation (6), whih is used for disretization purpose, produes a Jaobian ontaining many zero elements as shown in Eq. (1) below. As desribed earlier the harmoni balane method divides the network to be analyzed into linear and nonlinear subnetworks, whih are onneted at a number of ports. A solution is obtained when a set of port voltages auses a set of idential port urrents into both the linear and the nonlinear subnetwork. In the partiular ase of a frequeny doubler network, whih is onsidered in this paper, one is only interested in a steady-state solution whih an be obtained muh faster by this method rather than by the time domain integration. In the simulation the harmoni balane method is realized using the multiple refletion algorithm that requires no evaluation of derivatives and no initial guess. The method imitates the natural tum-on proess of a iruit. Ideal transmission lines, whih are a large integer multiple of the fundamental wave-length long, are assumed to exist between the two subnetworks (Fig. 3). On these lines the inident and refleted waves of the tum-on proess are alulated. The refleted wave, v/(w), travelling bak from the nonlinear subiruit is multiplied by the frequeny dependent refletion fator, f(w), of the linear subnetwork and is added to the initial inident wave, v 1 (w), to give [4]: v~ + l(w) "' v!(w) + f(w). v;(w) (11) where p indiates the iteration number. The refletion fators of the linear subiruit are given by: Journal, of EAEA, Vol. 11, 1994

5 Design of Nonlinear Mirowave Ciruits 33 r(w) (12) ZL is the harateristi impedane of the imaginary transmission lines between the subiruits. Its hoie is not so important, but fast onvergene an be ahieved ifzl is approximately taken to be equal to Z(w.,), i.e the port impedane of the iruit at the fundamental frequeny. For a rapid onvergene of the algorithm the refletion fators at both ends of the transmission lines must be kept small. The DC bias soures, however, have an internal impedane of nearly zero. In this ase a larger value of Z, (soure impedane) is assumed and the bias voltage v,.. is offset to ompensate for this artifiial inrease in the DC soure impedane. The flow diagram of the overall simulation is given in Fig. 4. To determine the parameters of the Curtie Ettenberg model one needs the DC as Wtlll as the S-parameter measurement results that desribe the transistor. Neither of the two measurement results is suffiient alone to determine all the equivalent Input: Model parameter, iruit parameter, <J estimate v 1, v 2 or take previous solution solving the differentja) equations of the intrinsi FET with Newtons ll'jf!th iruit elements shown in Fig. 2. This means both measurement results must be used and ompared with the voltages and urrents that are obtained during the simulation. Then, the model parameters are varied and with the help of an appropriate algorithm, the model is further optimized till a good agreement between simulation and measurement results is obtained. The algorithm used to optimize the model parameters of the FET is the so alled 'evolution priniple' and is well desribed in [6]. RESULTS The model parameter extration method desribed earlier has been applied to a GaAs MESFET type HLT dull-2. From the measured DC and S parameters at 15 different operating points onsisting of the ombination: V, {-15V, -.5V,.5V} and V.u {.5V, l.ov, l.5v, 3V, SY}, and measured over the frequeny range up to 4 GHz, the parameters of the Curtie-Ettenberg model have been extrated and given in Table 1. The relevant equations are as given in the Appendix setion. Note that at first the parameters a, a,, ai, a 3, (3 and 'Y are determined from the DC measurements. By using these values and by varying r,r,, C, 11 C~ and g,,.,,, the urve-fitting between the measured and simulated S-parameters is performed. Sine the influene of the individual model parameters in some ases is interdependent with one another, it may be neessary to do several swithing between the DC and S-parameter optimization yles and searh for optimum values and use different optimization variables at eah stage. In other words, it is important to note that a 1, as an example, influenes both the DC harateristis as well the small-signal transondutane gain and the paradox here is that the two optimization proedures do not always lead to the same value. For this reason both proedures must be interhangeably used to evaluate a 1 until a ompromise value is found. Figure 4 Flow Diagram of the Simulation

6 34 Mohammed Abdo 2 t db v-... CQ - ~ : «; o.2 "'... <IJ > : u I I /.A' /... I ""... ' ""',/ I""- ~ f lim{z{u>o))j -- Figure 5 Calulated Maximum Ahievable Conversion Gain of a 16 to 32 GHz Frequeny Doubler Ciruit as a Funtion of Seond Harmoni Load Reatane. Table 1: The extrated nonlinear model parameters of the HLT dull-2 GaAs FET. a =.3152 A a 3 =.549 AN 3 R; =.2742 vbi = o.8 v N = 1.5 Fe =.5 a, =.6463 A/V,,. = F vdro = 2.5 v T = I, = A 'Y = /V a 2 =.2868 A/V 2 C,.w = F VT =.26 v g""" = s f3 =.4 1/V.. - t txj 'O "«j.2 <11,_ Q) > u db 14 P ~ in Figure 6 Measured and Simulated Conversion Gain of a 16 to 32 GHz Frequeny Doubler Ciruit as a Funtion of the Input Fundamental Power

7 Design of Nonlinear Mirowave Ciruits 35 The transistor model whose parameter extration is shown above was used to design a 16 to 32 GHz frequeny doubler. Figure 5 shows the predited maximum ahievable onversion gain of the doubler as a funtion of the seond harmoni load reatane. The optimum load was found to be 12+j41. The measured and alulated onversion gain of the doubler as a funtion of the input power is given in Fig. 6. As an be seen from the figure there is good agreement between measurement and simulation results. doubler. The harmoni balane tehnique has been used for the simulation. The designed iruit has been tested experimentally and proved to be in good agreement with the simulation results, thus verifying the validity of the approah used. The 16 to 32 GHz frequeny doubler realized here has a maximum overison gain of -.5 db whih is quite a good result for suh a high frequeny of operation. CONCLUSIONS A simplified nonlinear GaAs MESFET model has been simulated and implemented for the design of nonlinear mirowave iruits suh as a frequeny APPENDIX The following equations for the nonlinear elements of the Curtie-Ettenberg mode [7] are implemented in the program: a) Condution urrent: l (a + a 1 v 1 + a2 v~ + a3 v ~) tanh(')'vdr), ' for v"' ;:::: for Vt.ls :::; (A. l ) where: (A-2) In the simulation, 7 is assumed to be zero. b) Diode Currents: /,(exp [ :~,] - 1) g..,, 1, -I, + gmmvj, (A.3)

8 36 Mohammed Abdo where, (A.4) and. I /,(exp NVr v gd - 1) + g<nun)v gd, [ I - 1, 1 + exp(- v "' Vr + v "') [ (A.5) where, v = v - v gd gs <U (A.6) Finally, the Shottky-ontat apaitanes are given by: = gs R. &'" j '"' Vt - Fe. [1 v. - FCV ] + J 2 V.,(l - Fe) for v 1 < F evl>i, for v. J ;;:::: FeVbi' (A.7) and = gd gdo H ' gd -. [I for v&<f < FeVw (A.8)

9 REFERENCES Design of Nonlinear Mirowave Ciruits 37 [1] W.R. Curtie, M. Ettenberg, "A nonlinear GaAs FET model for use in the design of output iruits for power amplifiers", IEEE Trans. Mirowave Theory Teh., vol. MTT-33, No. 12, pp , Deember (2] T.Kaprzak, A. Materka, "Compat DC model of GaAs FET's for large-signal omputer alulation, IEEE Journal of Solid-State Ciruits, vol. S-18, no.2, pp , April [3] Compat Software: Program Super Compat/Mirowave Harmonia, Version 2.1 and Manual, 199. [4] S.A. Maas, Nonlinear Mirowave Ciruits, Arteh House, Norwood, MA, (5) Bronstein, Semendjajew, Tashenbuh der Mathematik, Verlag Harri Deutsh, Thun and Frankfurt/Main (Germany), [6) H.H. Sledzik, "Ein Verfahren zur Modellierung der nihtlinearen Eigenshaften und zur Grobsignalsimulation von GaAs-MESFET und AIGaAs/GaAs HEFT", Ph.D. Dissertation (in German), University of Duisburg, 199. [7] EEsof In., EEsof Element Catalog 199.