A Novel ecouple Iterative Metho for eep-submicron MOSFET RF Circuit Simulation CHUAN-SHENG WANG an YIMING LI epartment of Mathematics, National Tsing Hua University, National Nano evice Laboratories, an Microelectronics an Information Systems Research Center, National Chiao Tung University, 1001 Ta-Hsueh R., Hsinchu city, Hsinchu 300, TAIWAN Abstract: - In this paper, we solve the MOSFET RF circuit orinary ifferential equations with the waveform relaxation metho, monotone iterative metho, an Runge-Kutta metho. With the monotone iterative metho, we prove each ecouple an transforme circuit equation converges monotonically. This metho provies an alternative in the time omain numerical solution of MOSFET RF circuit equations. Key Wors: - Nonlinear Circuit Moel, MOSFET evice, OE, Monotone Iterative Metho 1 Introuction Numerical methos for the raio frequency (RF) circuit provie an efficient alternative in the evelopment of integrate RF components, such as filter, low-noise amplifier, mixer, an power amplifier. The MOSFET evices use for esign such integrate RF circuit becomes a tenency because of the successful experience in igital circuit esign. ue to the unusually high linearity of MOSFETs at high frequencies, these active evices have been of great interests for RF an wireless applications in the recent years. The common approach to analyze the inter-moulation istortion an two-tone characteristics for an MOSFETs is to solve a set of equivalent circuit orinary ifferential equations (OEs) in the frequency omain. The harmonic balance metho is a stanar approach for solving such RF problems [1][2]. However, this frequency omain approach has its merits an limitations in stuying the physical properties of MOSFET with time variations. Another approach to the analysis of electrical characteristics for a MOSFET RF circuit is to solve a set of equivalent circuit OEs in time omain. The time omain results are then further calculate with fast Fourier transformation (FFT) for obtaining its spectrum. However, the iscretize OEs in circuit simulation are often solve with conventional Newton s iterative (NI) metho. The well-known HSPICE circuit simulator is right aopte the NI metho an its variants in its numerical kernel. It is well-known the NI metho is a local metho; in general, it has a quaratic convergence property in a sufficiently small neighborhoo of the exact solution, an hence it encounters convergence problem for practical engineering applications. These properties have their limitation an shoul be carefully verifie in the practical engineering application. In this work, we apply this metho to simulate MOSFET characteristics with exploiting the basic nonlinear property in the equivalent circuit moel. By consiering the Kirchhoff s current 1
law for each noe, the circuit governing equations are formulate in terms of the noal voltages (V G, V, V S, V, V S, V G ). The circuit moel is ecouple into several inepenent OEs with the waveform relaxation [3] ecoupling scheme. The basic iea of the ecouple metho for circuit simulation is similar to the well-known Gummel s ecoupling metho for evice simulation [5]; it is that the circuit equations are solve sequentially [4]-[5]. In the circuit moel, the given the pre- first equation is solve for V (g+1) G vious states, =, S, G, S, an, respectively. For the secon equation is solve for V (g+1) given, = G, S, G, S, an, respectively. We have the same proceure for other OEs. Each ecouple OE is transforme an solve with Runge-Kutta (RK) metho an the monotone iterative metho. We prove this approach converges monotonically for all ecouple circuit equations. It means that we can solve the circuit OEs with arbitrary initial guesses. Numerical results for various evices have been reporte to emonstrate the robustness of the metho in this paper. This paper is organize as follows. In Sec. 2, we state the OE moel. Sec. 3 is the monotone iterative metho for the ecouple OEs. For each ecouple equation, we prove the convergence property by using monotone iterative metho. A computational proceure is also introuce in this section. Sec. 4 shows some computational results an some iscussions. Sec. 5 raws the conclusions an suggests future works. 2 A MOSFET RF Circuit Moel As shown in Fig. 1, base on the noe current flow conservation (the well-known Kirchhoff s current law) an utilize the EKV large signal equivalent circuit moel (Fig. 2) for the MOSFET evice [6], the complete simulation moel can be formulate with noal equations. The system of noe equations for time epenent MOSFET circuit is a set of nonlinear cou- Figure 1 : A typical MOSFET RF circuit. ple OEs. At noes B,, S, an G we have the following equations (1)-(4), respectively [7]. (C gb + C gb0 )( V B (C gs + C gs0 )( V S (C g + C g0 )( V V G ) + V G ) + V G ) + V G V G R G = 0, I S I B + (C g + C g0 )( V G C b ( V B I S + (C gs + C gs0 )( V G C bs ( V B C G ( V in V ) + (1) V ) + V V R = 0, (2) V S ) + V S ) + V S V S R S = 0, (3) V G ) + V G V G + R G V IN V G = 0 (4) R IN 2
Figure 2 : The MOSFET EKV equivalent circuit. Similarly, at noes S an, we formulate the equations as follows: V V R V S V S R S + V V R + V S R S = 0 (5) + V R OUT = 0 (6) The Eqs. (1)-(4) are the OEs, an the Eqs. (5) an (6) are the algebraic equations. These equations are subject to proper initial values at time t = 0 for all unknowns to be solve. All currents I an capacitances C above are nonlinear functions of unknown variables. These nonlinear terms are There are 4 couple OEs with the nonlinear current an capacitance moels have to be solve an the unknowns to be calculate in the system of OEs are V G, V, V S, an V G, respectively. We note that the system consists of strongly couple nonlinear OEs, ue to the exponential epenence of current an capacitance moels. The moel parameters above for the EKV MOSFET moel can be foun in [6]. 3 Numerical Algorithms for the System of OEs We propose here a ecouple an globally convergent simulation technique to solve the system OEs in the large-scale time omain irectly [7]. Firstly, uner the steay state conition, we fin the C solution as the starting point to compute other time epenent solutions. For a specifie time perio T, to solve these nonlinear OEs in the time omain, the computational scheme consists of following steps: (i) Let an initial time step t be given. (ii) Use the ecoupling metho to ecouple all Eqs. (1)-(6). (iii) Each ecouple OE is solve sequentially with the MI an RK methos. (iv) Convergence test for each MI loop. (v) Convergence test for overall outer loop. (vi) If the specifie stopping criterion is reache for the outer loop, then go to step (vii), else upate the newer results an back to step (iii). (vii) If t < T, t = t + t an repeat the steps (iii)- (vi) until the time step meets the specifie time perio T. Firstly, all couple OEs must ecouple by the waveform relaxation (WR) metho [3]. We state here the WR metho in etails. Consier a general nonlinear system of OEs with associate initial conitions, V = f(v, t), V (0) = V 0, t [0, T ], (7) where T > 0, f : R [0, T ] R, V 0 = [V 1,0, V 2,0,, V,0 ] R is the initial vector of V, an V 0 = [V 1 (t), V 2 (t),, V (t)] R is the solution vector. The system can be written as follows, V 1 = f 1 (V 1, V 2,, V, t), V 1 (0) = V 1,0 V 2 = f 2 (V 1, V 2,, V, t), V 2 (0) = V 2,0. V = f (V 1, V 2,, V, t), V (0) = V,0 (8) The WR metho for solving (7) is a continuoustime iterative metho. Therefore, given a function which approximates the solution, it calculates a new approximation along the whole timeinterval of interest. Clearly, it iffers from most stanar iterative techniques in that its iterates are functions in time instea of scalar value. The iteration formula is chosen in such a way that one avois having to solve a large system OEs. A particularly simple, but often very effective iteration scheme is written below. It maps the ol 3
iterate V (n 1). V (n) 1 = f 1 (V (n) 1, V (n 1) 2,, t), V (n) 1 (0) = V 1,0 2 = f 2 (., V (n) 2, V (n 1) 3,, t), V (n) 2 (0) = V 2,0 V (n). V (n) = f (., V (n) 2,, V (n) (n), t), V (0) = V,0 (9) It is obvious similar to the Gauss-Seiel metho for iteratively solving linear an nonlinear systems of algebraic equations, so-call the Gauss- Seiel waveform relaxation scheme. It converts the task of solving a ifferential equation in variables into the task of solving a sequence of ifferential equations in a single variable. A closely relate iteration is the Jacobi waveform relation scheme, the iteration formula is given by, V (n) V (n) 1 = f 1 (V (n) 1, V (n 1) 2,, t), V (n) 1 (0) = V 1,0 2 = f 2 (., V (n) 2, V (n 1) 3,, t), V (n) 2 (0) = V 2,0. V (n) = f (V (n 1) 1,, V (n) (n), t), V (0) = V,0 (10) For a given specifie time step t an the previous calculate results, the ecoupling algorithm solves the circuit equations sequentially, for instance, the V G in Eq. (1) is solve for given the previous results (V, V S, V G, V, V S ). The V in Eq. (2) is solve for newer given V G an (V S, V G, V, V S ). The V S in Eq. (3) is solve for newer given (V G, V ) an (V G, V, V S ). We have similar proceure for other unknowns. Each ecouple OE is solve with the MI algorithm. To clarify the MI algorithm for the numerical solution of the ecouple nonlinear OEs, we write the above ecouple OEs as the following form = f(, t), (11) (g) (0) = V 0 where is the unknowns to be solve, g is the ecoupling inex g = 0, 1, 2,. We note that the f is the collection of the nonlinear functions an f C[I R, R] an I = [0, T ]. For a fixe inex g an, because the upper an lower solutions, an, exist in the cir-, we can prove the solution cuit an existence in the set Ω = {(t,, t I} for each ecouple circuit OE. ) Theorem 1 Let an are the upper an lower solutions of Eq. (11) in C 1 [I R, R] such that C[I R, R]. of Eq. interval I. in the time interval I an f Then there exists a solution (11) such that in the time Proof 1 It is a irect result with the continuous property of f, here the comparison theorem is applie [5]. Remark 1 We note that for each ecouple OE, the nonlinear function f is nonincreasing function of the unknown an the upper an lower solutions (0) an (0) of Eq. (11) in I can be foun. We can further prove there exists a unique solution of Eq. (11) in I an (0) (0). We see that the Theorem 1 provies an existence result of the problem, an we now escribe a monotone constructive metho for the computer simulation of the circuit OEs. The constructe sequences will converge to the solution of Eq. (11) for all ecouple OEs in the circuit simulation. In this conition, instea of original nonlinear OE to be solve, a transforme OE is solve with such as the RK metho. Now we state the main result for the solution of each ecouple MOSFET circuit OEs. Theorem 2 Let the f C[I R, R], (0) an (0) are the upper an lower solutions of Eq. (11) in I. Because f(t, (g) ) f(t, Ṽ ) λ( Ṽ (g) (g) ), V (0) Ṽ (g) (g) (0) an λ 0, sequences V an n unif. n unif. as n monotonically in I. Proof 2 The proof was one in [5]. 4
Theorem 3 For ecouple OEs. the nonlinear function f is nonincreasing in an f(t, 1 ) f(t, 2 ) λ( 1 2 ), 1 { } 2. Thus { } an n converge n=1 n n=1 uniformly an monotonically to the unique solution of Eq. (11). Proof 3 By using Theorem 2 an note the nonincreasing property of f, the result is followe irectly. 4 Results an iscussion As shown in Fig. 1, the input signal V IN is the C bias an the expression of the two-tone input signal V in is as follows: V in = V m sin(2πf 1 t) + V m sin(2πf 2 t), (12) V OUT (V) 1.52 1.50 1.48 1.46 1.44 1.42 1.40 1.38 0 2x 4x 6x 8x 10x 12x 14x Time (Sec.) 1.52 1.50 1.48 Figure 3 : Our time omain results. The input two-tone signal has an amplitue V m = 0.005 V, an funamental frequencies f 1 an f 2 are 1.71 an 1.89 GHz, respectively. Fig. 3-4 shows our simulation results an the HSPICE results in time omain, respectively. We also can fin the ifference in the Figs. 3 an 4 at the time between 2.5 ns an 3.5 ns, the HSPICE results show some non-smooth calculation, but our calculation oes not have this phenomenon. With the time omain results, we calculate the spectra of the output power by the FFT. Fig. 5 shows the corresponing spectra with our time omain results. We can fin the 3r-orer intermoulation (IM3) proucts at 2f 1 f 2 an 2f 2 f 1 clearly [7]. Fig. 6 shows the corresponing spectra with HSPCIE time omain result. However, it is ifficult to ientify the two IM3 proucts. Fig. 7 show the convergence property of our simulation kernel. Fig. 7(a) shows the inner loop iteration versus the maximum norm error at the beginning of the simulation. This figure shows the inner loop iteration has nlog(n) convergence where n is the number of iterations. The Fig. 7(b) shows the outer loop iteration convergence property in ifferent time perio. As we can see the spee of convergence is almost aroun n 2. V OUT (V) 1.46 1.44 1.42 1.40 1.38 0.0 2.0x 4.0x 6.0x 8.0x 10.0x 12.0x 14.0x Time (Sec.) Figure 4 : The HSPICE time omain outputs. Output power (W) 10-2 10-11 10-12 10-13 10-14 10-15 Funamental frequencies IM3 proucts 10-16 0 1x10 9 2x10 9 3x10 9 4x10 9 5x10 9 Frequency (Hz) Figure 5 : Our output power in frequency omain. 5
10-2 10 0 10 1 Output power (W) Funamental frequencies 0 1x10 9 2x10 9 3x10 9 4x10 9 5x10 9 Frequency (Hz) Figure 6 : The HSPICE output power in frequency omain. Maximun norm error 10-1 10-2 V V G V S 10 0 10-1 10-2 10-11 10-11 10-12 1 10 100 1000 Number of iterations t: 0 sec. t :13 f, 26 f, 13 p sec. 1 2 3 4 5 6 7 Number of iterations Figure 7 : (a) The maximum norm error vs. the number of inner iterations. (b) The maximum norm error vs. the number of outer iterations. 5 Conclusions A novel RF circuit simulation metho base on the waveform relaxation an monotone iterative methos has been propose. With monotone iterative technique, we have prove each ecouple circuit OE converges monotonically. The propose metho here is an alternative in the numerical solution of electric circuit equations. Computational results have been reporte in this work to emonstrate the robustness of the metho. This metho not only provies a computational technique for the time omain solution of circuit OEs but also can be generalize for circuit simulation incluing more transistors. Furthermore, this metho can be parallelize an systematically extene to simulate such as RF IC an system-on-a-chip large-scale VLSI circuit. ACKNOWLEGMENTS This work was sponsore in part by NSC 90-2112-M- 317-001 an PSOC 91-EC-17-A-07-S1-0011. References [1] B. Trpyanovsky, Zhiping Yu, Robert W. utton, Physics-base simulation of nonlinear istortion in semiconuctor evices using the harmonic balance metho, Computer Methos in Applie Mechanics an Engineering, vol. 181, 2000, pp. 467-482. [2] K. S. Kunert, J. K. White, an A. S.- Vinventelli, Steay-State Methos for Simulating Analog an Microwave Circuits, Kluwer Acaemic, 1990. [3] Vanewalle Stefan, Parallel multigri waveform relaxation for parabolic problems, Teubner, 1993. [4] Yiming Li, Kuen-Yu Huang, A novel numerical approach to heterojunction bipolar transistors circuit simulation, Computer Physics Communications, vol. 152, 2003, pp. 307-316. [5] Yiming Li, A monotone iterative metho for bipolar junction transistor circuit simulation, WSEAS Tranactions on Mathematics, vol. 1, no. 4, 2002, pp. 159-164. [6] C. Enz, F. Krummenacher, E. Vittoz, An analytical MOS transistor moel vali in all regions of Operation an eicate to low-voltage an low-current applications, Journal on Analog Integrate Circuits an Signal Processsing, Kluwer Acaemic Pub., 1995, pp.83-114. [7] Chuan-Sheng Wang, A novel ecouple iterative metho for eep-submicron MOSFET RF circuit simulation, M.S. Thesis, National Tsing- Hua University, 2003. 6