Engineering Department, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran

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1 Published in IET Microwaves Antennas & Propagation Received on 7th December 7 In Special Issue on Asia Pacific Microwave onference 7 ISSN Nonlinear and fully distributed field effect transistor modelling procedure using time-domain method K. Afrooz A. Abdipour A. Tavaoli M. Movahhedi Microwave/mm-Wave & Wireless ommunication Research Lab Radio ommunication enter of Excellence Electrical Engineering Department Amirabir University of Technology 44 Hafez Ave. Tehran Iran Electrical Engineering Department Shahid ahonar University of Kerman Kerman Iran ambiz.afroozaut.ac.ir Abstract: An accurate and efficient modelling approach for field effect transistors (FET) as nonlinear active transmission lines is presented. The nonlinear active multiconductor transmission line (NAMTL) equations are obtained by considering the transistor as three active coupled lines operating in a nonlinear regime. This modelling procedure accurately spots the effect of wave propagation along the device electrodes. This modelling approach is applied to an FET by solving the NAMTL equations using a finite-difference timedomain technique. The results of this model are compared with the semi-distributed (slice) model. This method produces more accurate results than the slice model especially at high frequencies. Introduction The demand for millimetre-wave monolithic microwave integrated circuits (MMIs) has been increasing rapidly because of the need for higher performance and lower cost. Also the increasing demand of processing and transmitting more information at a faster rate leads the analogue and digital electronic systems to operate at higher frequencies or higher cloc speeds []. Therefore the MMIs circuits are used in more and more highfrequency applications. A field effect transistor (FET) is one of the most important devices used in developing MMIs for systems such as an automobile radar and wireless local area networ which operate at high frequencies. To design the MMIs precisely an accurate FET model is indispensable. As the operating frequency of a microwave FET increases to the millimetre-wave range the dimensions of the electrodes become comparable with the wavelength l g []. When the device width becomes comparable with the wavelength the reactance of the gate electrode (input active transmission line) becomes different from the drain electrode (output active transmission line) [3]. Thus the electrodes of the device exhibit different phase velocities for input and output signals. y increasing the frequency or device dimension the phase cancellation because of the mismatching of the phase velocities will affect the overall performance of the device especially in the nonlinear regime. In a nonlinear circuit such as an oscillator mixer high-power amplifier frequency multiplier and so on several harmonics of the excitation source will be generated in the circuit. Hence the effect of wave propagation along the device electrodes has a more significant influence on the circuit performance as compared with a linear circuit. In such cases for accurate modelling the wave propagation effect needs to be considered in the device modelling. The full-wave analysis and global modelling approach can be used to consider the wave propagation effects along the device structure accurately. However the computation cost of this type of approach is very expensive [4]. Although some efficient numerical methods have been recently proposed for reduction in simulation time [5 ] these approaches seem to need more attention to be appropriate for implementing in simulation softwares. On the other hand device behaviour at high frequencies can be well 886 IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

2 described using a semi-distributed (slice) model that can be easily implemented in computer aided design (AD) routines of simulators. In semi-distributed modelling with the assumption of a quasi transverse electromagnetic (TEM) approximation the device is divided into finite cascade cells. Each cell contains the coupled electrode transmission lines resistance and internal inductance of the electrodes and an intrinsic FET equivalent circuit [ ]. The semidistributed model with both linear and nonlinear lumped elements for the active part has been investigated in several papers. Abdipour and Pacaud [] applied the semidistributed model to the FET in a linear regime based on a three-line structure and confirmed the results with measurement data. Also Ongareau et al. [] considered a semi-distributed model to analyse a FET in a nonlinear regime with the assumption of a two-line structure based on a harmonic balance technique. y increasing the frequency the semi-distributed model cannot precisely model the wave propagation effect and phase cancellation phenomena on the electrical performance. Therefore to achieve more accurate results in high-frequency applications one needs to develop the semi-distributed model. A fully distributed model is a modified version of the semi-distributed model in which the number of slices goes to infinity. Thus the fully distributed model can consider the effect of wave propagation along the device electrodes more accurately than the semi-distributed model although the PU time of this model is a little greater than the slice model. In this modelling approach the (TEM) wave propagation is inspected on the electrodes of the device. A fully distributed model of an FET with three active coupled lines in a linear regime is embodied in the active multiconductor transmission lines (AMTL) equation [3]. The AMTL equations are coupled linear and first-order differential equations that can give a good prediction of FET behaviour in linear and high-frequency applications [3]. These equations are based on the three active coupled lines and hence the distributed effect of the source electrode is considered at the AMTL equations for accurate modelling of devices in high-frequency applications [4]. The transistor at the AMTL equation is assumed to operate at the linear region as many microwave circuits such as mixers power amplifiers and so on operate under nonlinear conditions. Thus in this paper we introduce a nonlinear active multiconductor transmission line (NAMTL) equation using the fully distributed model by considering the distributed effect of the source electrode. To obtain these equations first the device width is divided into infinity segments. Each segment is considered as a system with a combination of three coupled lines and a conventional equivalent circuit of a FET in the nonlinear regime. The intrinsic FET equivalent circuit in a nonlinear regime modelled on the urtice cubic model [5]. The urtice cubic model parameters are obtained from D and low-frequency measurements. The transmission line theory is applied to a segment of the transistor to obtain the wave equation in a FET structure. Now this system of nonlinear differential equations (NAMTL equation) must be solved. Since an analytical solution does not exist for this system this problem is to be solved using a numerical technique. The finite-difference time-domain (FDTD) method is widely used in solving various inds of electromagnetic problems in lossy nonlinear inhomogeneous media and transient problems can be considered [6]. The FDTD technique is used to solve the NAMTL equations. The results achieved from this model (fully distributed) are compared with the slice model at several bias points. It is shown at the low frequencies the results of the semidistributed and fully distributed models are the same. However by increasing the frequency the results of two models are not in a good agreement. The fully distributed model is a modified version of the semi-distributed model when the number of slices increases to the infinity and consider the wave propagation effects. We expect the proposed method becomes more accurate than the slice model. Derivation of the NAMTL equations Fig. shows a typical millimetre-wave FET. The device consists of three coupled electrodes fabricated on a thin layer of GaAs supported by a semi-insulated GaAs substrate. y increasing the frequency the dimensions of the device become comparable with the wavelength. In this condition the distributed effect on the device electrodes should be considered in the device modelling. At a low enough frequency or for a device with dimensions much smaller than the wavelength the magnitude of longitudinal electromagnetic field is negligible compared with that of the transverse field. Hence in these cases the quasi-tem mode which is a dominant mode propagated along the device electrode can be approximated. The fully distributed model is one of the accurate models applied to consider the distributed and wave propagation effects on device behaviour. Thus the quasi- TEM approximation can be considered and the NAMTL equations are generalised using a fully distributed modelling approach. Figure Schematic showing the structure of a typical millimetre-wave FET IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

3 Figure Different parts of a differential slice in the fully distributed model We plan to extract a set of equations that show the transistor s behaviour. A differential section of the device is shown in Fig.. Each differential section combines a passive part and an active one. The passive and active parts describe the electromagnetic interaction between the coupled lines and the behaviour of the intrinsic device in the nonlinear regime respectively. All parameters are per unit length. y applying Kirchhoff s current law to the left loop of the circuit shown in Fig. and by considering the limiting case Dz! we obtain the following three equations z I d (zt) þ ~ t V d (z t) ~ t V g (z t) 3 t V s (z t) þ G ds (V d (z t) V s (z t)) þ ~I ds ¼ () z I g(z t) ~ t V d(z t) þ ~ t V g(z t) 3 t V s(z t) þ ~ gs V t g (z t) ¼ () z I s(z t) 3 t V d(z t) 3 t V g(z t) þ 3 t V s(z t) ~ gs t V g (z t) G ds (V d (z t) V s (z t)) ~I ds ¼ (3) ~ ¼ dp þ ds þ dsp þ ~ gd þ gdp ~ ¼ gp þ gsp þ ~ gd þ gdp 3 ¼ sp þ ds þ dsp þ gsp ~ ¼ ~ gd þ dgp 3 ¼ ds þ dsp 3 ¼ gsp Similarly by applying Kirchhoff s voltage law to the main node of the circuit and considering the limiting case as Dz! in Fig. gives z V d(z t) þ R d I d (z t) þ L d t V d(z t) þ M dg t V g(z t) þ M ds t V s(z t) ¼ (4) z V g(z t) þ R g I g (z t) þ L g t V g(z t) þ M dg t V d(z t) þ M gs t V s(z t) ¼ (5) z V s(z t) þ R s I s (z t) þ L s t V s(z t) þ M ds t V d(z t) þ M gs t V g(z t) ¼ (6) 888 IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

4 We can also write another equation as follows V g (z t) þ V s (z t) þ R i ~ gs t V g (z t) V g (z t) ¼ (7) The urtice cubic nonlinear model is selected to model the active part (intrinsic device) [5]. In this model the drain source current (~I ds ) gate source junction ( ~Q gs ) and gate drain junction ( ~Q gd ) are rewritten as the following with ~I ds ¼ I dso tanh (gv ds ) (8) I dso ¼ A o þ A V þ A V þ A 3 V 3 þ V ds V dsc R dso (9) V ¼ V gs [ þ b(v dso V ds )] V ds. V V gd [ þ b(v dso þ V ds )] V ds V () For values of V below the internally computed maximum pinch-off voltage (V pmax ) ~I ds is replaced with the following expression I dso ¼ A o þ A V pmax þ A V pmax þ A 3 V 3 pmax þ V ds V dsc R dso () if the value of I dso for V ds. V is negative the drain source current becomes set to zero. The drain source and gate drain charge and capacitance are formulated in the urtice cubic as follows 8 " sffiffiffiffiffiffiffiffiffiffiffiffiffiffi V bi o V # V V F c V bi bi p >< V bi o [ ffiffiffiffiffiffiffiffiffiffiffiffiffi F c ] "! Q ¼ o þ ( F c ) 3= 3F c V. F c V bi V (F c V bi )!# >: (V F c V bi ) and ¼ Q V ¼ 8 >< o 4V bi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V F c V bi ðv =V bi Þ o ( F c ) 3= >: 3F c þ V V bi V. F c V bi () (3) ¼ d c gco ¼ gso and V gc ¼ V gs.also gso and gdo are the zero-bias gate and drain capacitances respectively g the current saturation b acoefficientfora pinch-off change with respect to V ds F c the forward-bias depletion capacitance coefficient (diode model) and V bi the built-in gate potential. It is obvious that the value of each element in each segment is per unit length. Hence some of the urtice cubic parameters should be converted to per unit parameters. Equations () (3) and (4) (7) can be simplified into two matrix equations as the following I d ~ ~ 3 V d I g z I s A þ ~ ~ 3 ~ gs V g t ~ gs A V s A R i ~ V gs g G ds G ds V d ~I ds V g þ G ds G ds A V s A þ ~I ds A ¼ V g (4) V d V z g A þ L d M dg M ds I d M dg L g M gs A I t g A V s M ds M gs L s I s R d I d þ R g A I g A ¼ (5) R s I s I d V d I g V g and I s V s are the drain gate and source currents and voltages respectively; V g is the gate source capacitance voltage; and I ds is the drain source current. 3 FDTD solution of the NAMTL equations The nonlinear fully distributed model of an FET is embodied in the following NAMTL equations I(z z t) þ ~(V d V g V s V g ) V t (z t) þ G V (z t) þ ~I NL (V d V g V s ) ¼ (6) z V (z t) þ L I(z t) þ RI(z t) ¼ (7) t IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

5 V d (z t) I d (z t) V (z t) ¼ V g (z t) A I(z t) ¼ I g (z t) A V s (z t) I s (z t) ~I ds (z t) V d (z t) ~I NL (z t) ¼ ~I ds (z t) A V g (z t) V (z t) ¼ V s (z t) A V g (z t) I d (z t) I and I(z t) ¼ g (z t) I s (z t) A The NAMTL equations are coupled nonlinear and firstorder partial differential equations. A numerical technique is needed to solve the NAMTL equations. One of the best methods which can be employed to solve the NAMTL equations is the FDTD technique. The FDTD technique sees to approximate the derivatives with regard to discrete solution points defined by the spatial and temporal cells [7]. An explicit time-space centred finite-difference scheme is chosen to discretise the NAMTL equations [4]. Thus each voltage and adjacent current solution point is separated by Dz=. In addition the time points are also interlaced and each voltage time point and adjacent current time point are separated by Dt= as illustrated in Fig. 3. To ensure the stability of the discretisation and the second-order accuracy we interlace the N z þ voltage points V V... V Nz V Nz þ and the N z current points I I... I Nz as shown in Fig. 3 [7 9]. Discretising the derivatives in the NAMTL equations using the proposed algorithm gives nþð=þ nþð=þ I I Dz V þ G nþ þ V n with ¼... N z þ V nþ þ V nþ Dz þ R I nþ(3=) with ¼... N z. þ ~( V nþ V n V ) nþ V n Dt þ ~ I NL ( V nþ V n ) ¼ (8) þ L I nþ(3=) þ I nþð=þ I nþð=þ Dt ¼ (9) Figure 3 Relation between the spatial and temporal discretisation and terminal voltages and currents a Relation between the spatial and temporal discretisation to achieve second-order accuracy in the discretisation of the derivatives b Discretisation of the terminal voltages and currents! ~I ds ¼ I dso tanh g V d n þ Vd nþ Vs n Vs nþ and () I dso ¼ A o þ A V þ A V þ A 3 V 3 þ V n d þ V nþ d V n nþ () s Vs V dsc R dso 8 V n þ V nþ V n s V nþ s "!# þ b V dso V n d þ V nþ d V n s V nþ s >< V ¼ V n þ V nþ V n d V nþ d "!# þ b V dso þ V n d þ V nþ d V n nþ s Vs >: We denote V j i ; V ((i )Dz jdt) V j i ; V ((i )Dz jdt) I j i ; I((i =)Dz jdt) I j i ; I((i ð=þ rdz jdt) ~I NL ¼ [ ~ I ds ~ I ds ] T Vd n þ Vd nþ. Vs n þ Vs nþ V n d þ V nþ d V n s þ V nþ s () 89 IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

6 For values of V below the internal computed maximum pinch-off voltage (V pmax ) ~ I ds is replaced with the following expression I dso ¼ A o þ A V pmax þ A V pmax þ A 3 V 3 pmax þ V d n þ Vd nþ If the I dso value for Vd n þ Vd nþ Vs n Vs nþ V dsc (3) R dso. Vs n þ Vs nþ is negative the drain source current becomes set to zero. The drain source and gate drain charges and capacitances are modelled in the urtice cubic model as follows 8 gmo q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððv n þ V nþ Vm n nþ Vm Þ=V bi Þ >< gm ¼ V n þ V nþ 3 gmo ( F c ) 3= 3F c þ V n m V nþ m V bi 5 >: V n þ V nþ V n nþ m Vm F c V bi V n þ V nþ Vm n Vm nþ. F c V bi (4) m ¼ d c gco ¼ gso and V n gs þ V nþ gs V nþ V n nþ c Vc. Simplifying (8) and (9) we obtain F NL (V nþ m I nþ(3=) ¼ V n þ Vm n V nþ V n I nþ(=) m ) ¼ m ¼ d s g (5) ¼ L Dt þ R ( L Dt R ) I nþ(=) V nþ þ V nþ Dz (6) ecause of the simplicity and accuracy the leap-frog algorithm for alternately computing the voltage and the current is used to solve the NAMTL equations [6]. In this algorithm first the solutions start with an initially relaxed line having zero voltage and current values. Then voltages along the electrode of the transistor are solved for a fixed time from (5) in terms of the previous solutions and then currents are solved from (6) in terms of these and previous values. 3. Solution of the nonlinear equation F NL ¼ Equation (5) should be solved to calculate the voltage along the electrodes of the transistor. One of the most useful and best nown algorithms to solve nonlinear systems of equations is the Newton Raphson method that converges faster than the bisection and false position methods []. Exactly (5) is a set of four algebraic nonlinear equations with four unnown parameters as the following F NL ¼ [F NL F NL F 3NL F 4NL ] T (7) Our purpose is to obtain the four unnown parameters (Vd nþ V nþ Vs nþ nþ V ). In this technique first we start with the initial values for unnown parameters and then the Jacobian matrix is calculated upshot. The value of the unnown parameters are calculated in the next stage as follows ( V nþ ) mþ ¼ ( V nþ ) m ( JA(( V nþ ) m )) F NL (( V nþ ) m ) (8) this iterative algorithm is executed until jf NL ((V nþ ) m )j e. The Jacobian matrix is given as JA ¼ F NL V V nþ ¼ F NL Vd nþ F NL Vd nþ F 3NL Vd nþ F 4NL Vd nþ F NL V nþ F NL V nþ F 3NL V nþ F 4NL V nþ F NL V nþ s F NL V nþ s F 3NL V nþ s F 4NL V nþ s F NL V nþ F NL V nþ F 3NL V nþ F 4NL A V nþ (9) This procedure gives us the voltages along the electrodes of the transistor. Also the voltage of the gate source capacitance is obtained. Then using the voltage at the new time step we can obtain the current at the new step using (6). 4 oundary conditions In many applications in microwave circuits transistors are used in the common source configuration because of high stability and high gain of this configuration. In this configuration the source electrode is grounded at the beginning and at the end of electrode. Also the transistors have been excited at the beginning of the gate electrode and loaded at the end of drain electrode for the consideration of the wave effects [3 8]. Thus we need to investigate this case of loading and exciting configuration as the other configuration can be simply investigated using the NAMTL equations. Fig. 4 shows the assumed structure of the transistor with the biasing and loading circuits. With this consideration of biasing circuits () () and (7) for z ¼ and z ¼ W become IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

7 Figure 4 iasing and loading circuits considered for the transistor (V s (z t) ¼ j z¼z¼w ) respectively z I d(z t) þ ~ t V d(z t) ~ t V g(z t) þ G ds V d (z t) þ ~I ds ¼ j z¼z¼w (3) z I g(z t) ~ t V d(z t) þ ~ t V g(z t) þ ~ gs t V g (z t) ¼ j z¼z¼w (3) V g (z t) þ R i ~ gs V t g (z t) V g (z t) ¼ j z¼z¼w (3) Also two equations are given as KVL and KL in the biasing circuit at the beginning of the gate electrode as follows (Fig. 4) V V in (t) þ R s cs (t) b þ V t cs (t) þ V g ( t) ¼ (33) I V g dc L Ls (t) b þ V t g ( t) ¼ (34) Equations (3) (34) can be simplified as F NL ¼ I(z t) þ ~ z t X (z t) þ GX(z t) þ ~ I NL þ V in (t) ¼ j z¼ (35) I d (z t) V d (z t) I g (z t) V g (z t) I(z t) ¼ X (z t) ¼ V g (z t) A V cs (t) A I Ls (t) ~I ds (z t) I NL ¼ V in (t) ¼ A V in (t) A V g dc G ds ~G ¼ A ~ ~ ~ ~ ~ gs ~ ¼ R i ~ gs R s b A L b Therefore the NAMTL equations at z ¼ become (35). Applying the FDTD technique to (35) and considering Fig. 3 (this equation requires that we replace Dz with Dz=) gives F NL ¼ nþ(=) I nþ(=) I I Dz= ¼ [I nþ(=) d V n csþ=dtþ ð(i n Ls þ I nþ nþ(=) þ G X nþ þ X n þ ~ X nþ X n Dt þ V in nþ þ V n in þ ~ I NL (X nþ X n ) ¼ (36) I nþ(=) g Ls )=Þ ] T. ] T ¼ [ b ððv nþ Exactly (36) is a set of five algebraic nonlinear equations with five unnown parameters as the following F NL ¼ [F NL F NL F 3NL F 4NL F 5NL ] T (37) Our purpose is to obtain the five unnown parameters (Vd nþ Vg nþ nþ V g V nþ cl and ILs nþ ). Also two equations are given as KVL and KL in the biasing circuit at the end of the drain electrode as follows V R L cl (t) b þ V t cl (t) V d (W t) ¼ (38) I V d dc L LL (t) b þ V t d (W t) ¼ (39) cs 89 IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

8 Equations (3) (3) and (38) and (39) can be simplified as For ¼ FL NL ¼ I(z t) þ ~ X z t (z t) þ G X (z t) (4) þ ~ I NL þ V in (t) ¼ j z¼w (X nþ ) mþ ¼ ( X nþ ) m ( JA ((X nþ ) m )) F NL ((X nþ ) m ) (43) V d (z t) V g (z t) X (z t) ¼ V g (z t) V in (t) ¼ V cl (t) A A I LL (t) V d dc ~ ~ ~ ~ ~ gs ~ ¼ R i ~ gs R L b A L b G ds and ~G ¼ A Applying the FDTD technique to (4) and considering Fig. 3 (this equation requires that we replace Dz with Dz=) gives FL NL ¼ nþ(=) I N z þ þ G nþ(=) I N z þ ~ Dz= X nþ N z þ þ X n N z þ X nþ N z þ X n N z þ Dt V þ nþ in þ V n in þ ~ I NL ( X n N z þ X nþ N z þ ) ¼ (4) nþ(=) I N z þ ¼ [I nþ(=) dn z þ I nþ(=) gn z þ ]T ¼ [ b (VcL nþ VcL)=DtÞþ(I n LL n þ ILL nþ )= ] T Exactly (4) is a set of five algebraic nonlinear equations with five unnown parameters as the following FL NL ¼ [FL NL FL NL FL 3NL FL 4NL FL 5NL ] T (4) Our purpose is to obtain the five unnown parameters (V nþ dn z þ V nþ gn z þ V nþ gn z þ V nþ cl and I nþ LL ). Thus the finite difference approximation of the NAMTL equations can be written as follows and ¼ [Vd nþ Vg nþ V nþ Vcs nþ ILs nþ ] T (44) X nþ JA ¼ F NL X nþ X F NL F NL Vd nþ Vg nþ F NL F NL Vd nþ Vg nþ F 3NL F 3NL ¼ Vd nþ V g nþ F 4NL F 4NL Vd nþ Vg nþ F 5NL F 5NL Vd nþ Vg nþ For ¼ 3... Nz ( V nþ ) mþ ¼ ( V nþ JA ¼ F NL V F NL F NL ¼ F 3NL F 4NL V nþ d V nþ d V nþ d V nþ d For ¼ N z þ g F NL V nþ g F NL V nþ g F 3NL V nþ g F 4NL V nþ g F 5NL V nþ g F NL V nþ cs F NL V nþ cs F 3NL V nþ cs F 4NL V nþ cs F 5NL V nþ cs F NL I nþ Ls F NL I nþ Ls F 3NL I nþ Ls F 4NL I nþ Ls F 5NL I nþ Ls ) m ( JA(( V nþ ) m )) A (45) F NL (( V nþ ) m ) (46) V nþ F NL V nþ F NL V nþ F 3NL V nþ F 4NL V nþ F NL V nþ s F NL Vs nþ F 3NL Vs nþ F 4NL Vs nþ F NL V nþ F NL V nþ F 3NL V nþ F 4NL V nþ A ( X nþ N z þ) mþ ¼ ( X nþ N z þ) m JA L (( X nþ N z þ) m ) (47) FL NL (( X nþ N z þ) m ) (48) IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

9 X nþ N z þ ¼ [V nþ dn z þ V nþ gn z þ V nþ gn z þ V nþ cl I nþ LL ] T (49) And JA L ¼ FL NL X F NL VdN nþ z þ F NL VdN nþ z þ F 3NL ¼ VdN nþ z þ F 4NL VdN nþ z þ F 5NL VdN nþ z þ X nþ N z þ F NL V nþ gn z þ F NL V nþ gn z þ F 3NL V nþ gn z þ F 4NL V nþ gn z þ F 5NL V nþ gn z þ F NL V nþ gn z þ F NL V nþ gn z þ F 3NL V nþ gn z þ F 4NL V nþ gn z þ F 5NL V nþ gn z þ F NL VcL nþ F NL VcL nþ F 3NL VcL nþ F 4NL VcL nþ F 5NL VcL nþ F NL ILL nþ F NL ILL nþ F 3NL ILL nþ F 4NL ILL nþ F 5NL A ILL nþ (5) and for ¼ 3... Nz I nþ(3=) ¼ L Dt þ R L V nþ þ V nþ Dz Dt R I nþ(=) ) (5) The voltages and currents are solved by iterating for a fixed time step and then iterating the time. Fig. 5 shows a flowchart that describes the solution of the problem. 5 Numerical results Here the proposed approach is used for modelling a sub-micrometre-gate GaAs transistor. The device has a. mm mm gate. The input and output nodes are connected to the beginning and the end of the gate and drain electrodes respectively. The source electrode is grounded at the beginning and the end of the electrode. The structure of the considered transistor and its biasing and loading circuits are shown in Fig. 4. The values of elements used in the distributed model are shown in Table and parameters of the urtice cubic model are listed in Table. This transistor is simulated at the several bias points using the fully distributed model and its results are compared with the slice model to confirm the results of this study. First the transistor is biased at V gs ¼ VV ds ¼ 5 V and the device excited by a GHz sinusoidal excitation source with a.5 V amplitude. Figs. 6 and 7 show the voltage and current at the end of drain electrode using the fully distributed and slice models under this bias condition respectively. As these figures show Figure 5 Flowchart of leap-frog algorithm for solution of the NAMTL equations Table Numerical values of the lumped distributed model elements L d L s L g M gd M gs M ds gp dp sp gdp gsp dsp The distributed model elements Numerical values (per unit length) 78 nh/m 78 nh/m 6 nh/m 36 nh/m 36 nh/m 4 nh/m.6 pf/m 87 pf/m 48 pf/m 9 pf/m 9 pf/m 6 pf/m 894 IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

10 Table urtice cubic parameter values A o ¼.58 gso ¼.486 pf R dso ¼ V A ¼.34 gdo ¼.8 pf A ¼.94 A 3 ¼.48 V bi ¼.76 V b ¼.95 g ¼.3457 V dso ¼ 5V F c ¼.5 R i ¼ V V To ¼. V V dsc ¼ 5V there is a good agreement between the fully distributed and slice models. The drain voltage spectrum at the end of the drain electrode is presented in Fig. 8 which describes the nonlinear characteristic of the transistor under this condition very well. The load cycle for this transistor at V gs ¼.5 V V ds ¼ 5 V excited by a source with a V Figure 8 Drain voltage spectrum at the end of drain electrode when the device is biased at V ds ¼ 5V V gs ¼ V and excited by a GHz sinusoidal excitation source with.5 V amplitude Figure 6 Voltage at the end of drain electrode when the device is biased at V ds ¼ 5VV gs ¼ V and excited by a GHz sinusoidal excitation source with a.5 V amplitude amplitude and f ¼ 3 GHz when the load is a 5 V resistance parallel with a 5 ff capacitance is shown in Fig. 9. The result of the fully distributed model is compared with that of the slice model. Fig. depicts the results of the fully distributed and slice models at V gs ¼.5 V V ds ¼ 4 V and with an excitation source with a V amplitude but at a frequency of 6 GHz. In this figure a fractional difference exists between these two models because of the high-frequency effects. Here the device dimension becomes comparable with the wavelength. Moreover the slice model cannot consider the effect of wave propagation along the device electrodes and accurately analyse Figure 7 urrent at the end of drain electrode when the device is biased at V ds ¼ 5VV gs ¼ V and excited by a GHz sinusoidal excitation source with.5 V amplitude Figure 9 Load cycle at V ds ¼ 5 V and V gs ¼.5 V and a source with V amplitude and f ¼ 3 GHz when load is a 5 V resistance parallel with a ff capacitance Fully distributed model (þþþ) and slice model () IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

11 Figure Voltage at the end of drain electrode when the device is biased at V ds ¼ 4V V gs ¼.5 V having a source with.5 V amplitude and f ¼ 6 GHz it. Since the fully distributed model is a modified version of the slice model when the number of slices becomes infinite so the results of this model is more accurate especially at the highfrequency applications. In a nonlinear circuit because of higher-order harmonics these effects are more antiseptic than the linear circuits. In most of the papers the source electrode is assumed grounded at overall the electrode as the source electrode is grounded at the beginning and the end. In this wor the distributed effect of the source electrode is considered to obtain accurate results. The voltage at the middle of the source electrode using the fully distributed and slice models when the transistor is biased at V gs ¼ :5VV ds ¼ 4 V with a GHz sinusoidal excitation source and a.5 V amplitude is shown in Fig.. This figure shows that the distributed effect of the source electrode must be accounted for the analysis of high-frequency devices and circuits. Finally the device being biased at V ds ¼ 3.5 V and V gs ¼ V and the frequency of the Figure Voltage at the end of drain electrode when the device is biased at V ds ¼ 3.5 V V gs ¼ V and excited by a 4 GHz sinusoidal excitation source with different amplitudes excitation source being 4 GHz the voltages of the end of the drain electrode are plotted in Fig. for the fully distributed and slice models for several amplitudes of the excitation source. 6 onclusion The NAMTL equations have been introduced using a fully distributed model based on a three-line structure. These equations can accurately predict the behaviour of microwave/ millimetre wave transistors in a nonlinear regime. The FDTD technique is used to solve these nonlinear equations in the time domain. This modelling approach is applied to an FET and the results for example the voltage and current waveforms are compared with those of the slice model. At a low frequency the results of the fully distributed model have a good agreement with those of the slice model. However by increasing the frequency a fractional difference exists between two models. Since the device dimension is comparable with the wavelength the wave propagation effects obtained with the fully distributed model are more accurate than those with the slice model. Therefore in high-frequency applications the use of the fully distributed model is recommended. In addition in devices with a width comparable to the wavelength the distributed effect of the source electrode must be considered. Another advantage of this model is the ease of its integration into AD optimisers. 7 References Figure Voltage at the middle of source electrode at V ds ¼ 4V V gs ¼.5V and excited by a GHz sinusoidal excitation source with.5 V amplitude [] IMTIAZ S.M.S. GHAZALY S.M.: Global modeling of millimeter-wave circuits: electromagnetic simulation of amplifiers IEEE Trans. Microw. Theory Tech () pp. 8 6 [] ADIPOUR A. PAAUD A.: omplete sliced model of microwave FET s and comparison with lumped model and 896 IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8

12 experimental results IEEE Trans. Microw. Theory Tech () pp. 4 9 [3] GHAZALY S.M. ITOH.T.: Traveling-wave inverted-gate fieldeffect transistor: concept analysis and potential IEEE Trans. Microw. Theory Tech pp. 7 3 [4] ALSUNAIDI M.A. IMTIAZ S.M.S. GHAZALY.S.M.: Electromagnetic wave effects on microwave transistors using a full-wave time-domain model IEEE Trans. Microw. Theory Tech pp [5] GHAZALY S.M. ITOH T.: Inverted-gate field-effect transistors: novel high frequency structures IEEE Trans. Electron. devices pp [6] GOASGUEN S. TOMEH M. GHAZALY S.M.: Electromagnetic and semiconductor device simulation using interpolating wavelets IEEE Trans. Microw. Theory Tech. 49 () pp [7] HUSSEIN Y.A. EL.GHAZALY S.M.: Modeling and optimization of microwave devices and circuits using genetic algorithms IEEE Trans. Microw. Theory Tech. 4 5 () pp [8] MOVAHHEDI M. ADIPOUR A.: Efficient numerical methods for simulation of high-frequency active devics IEEE Trans. Microw. Theory Tech (6) pp [9] MOVAHHEDI M. ADIPOUR A.: Improvement of active microwave device modeling using filter-ban transforms. Proc. 35th Euro. Microwave onf. Paris France 5 pp. 3 7 [] MOVAHHEDI M. ADIPOUR A.: Accelerating the transient simulation of semiconductor devices using filter-ban transforms Int. J. Numer. Model 6 9 pp [] ONGAREAU E. OSISIO R.G. AUOURG M. OREGON J. GAYRAL M.: A non-linear and distributed modeling procedure of FETs Int. J. Numer. Model pp [] WALIULLAH M. GHAZALY S.M. GOODNIK S.: Largesignal circuit-based time domain analysis of high frequency devices including distributed effects. Microwave Symp. Digest IEEE MTT-S Int. vol. 3 pp [3] AFROOZ K. ADIPOUR A. TAVAKOLI A. MOVAHHEDI M.: Timedomain analysis of active transmission line using FDTD techniques (application to microwave/mm-wave transistors) Prog. Electromagn. Res. 7 PIER 77 pp [4] AFROOZ K. ADIPOUR A. TAVAKOLI A. MOVAHHEDI M.: FDTD analysis of small signal model for GaAs MESFETs based on three line structure. Asia-Pacific Microwave onf. (APM 7) ano Thailand December 7 [5] URTIE W.R. ETTENERG M.: A nonlinear GaAs FET model for use in the design of output circuits for power amplifiers IEEE Trans. Microw. Theory Tech () pp [6] TAFLOVE A.: omputational electrodynamics the finitedifference time-domain method (Artech House Norwood MA 996) [7] PAUL..R.: Incorporation of terminal constraints in the FDTD analysis of transmission lines IEEE Trans. Electromag. ompat pp [8] ORLANDI A. PAUL.R.: FDTD analysis of lossy multiconductor transmission lines terminated in arbitary loads IEEE Trans. Electromagn. ompat pp [9] RODEN A.J. PAUL.R. SMITH W.T. GEDNEY D.S.: Finitedifference time-domain analysis of lossy transmission lines IEEE Trans. Electromagn. ompat pp. 5 4 [] YPMA J.T.: Historical development of the Newton Raphson method Soc. Ind. Appl. Math () pp IET Microw. Antennas Propag. 8 Vol. No. 8 pp & The Institution of Engineering and Technology 8