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Transcript

COMPANION MODELS AND BOUNDARY CONDITIONS FOR MULTI-RATE STEADY-STATE ANALYSIS OF INTEGRATED ANALOG CIRCUITS M. Iordache , Lucia Dumitriu Electrical Engineering Department, Politehnica University of Bucharest, Spl. Independentei 313, Bucharest, ZIP 060042, ROMANIA, e-mail [email protected] The RF-IC applications are, in general, strongly nonlinear and have carrier frequencies into the GHzrange, with modulated signals in the kHz-range. Due to these peculiarities, the integration time-step must be small enough to accurately capture the fast component and obtaining information about the slowly component needs a large number of time-steps. Shooting methods adjust the guess initial condition at the end of the period using a nonlinear solver (usually Newton-Raphson) and integrate by transient simulation. When signals with widely separated rates are involved, reaching steady-state needs a large number of time steps. Consequently, shooting cannot handle efficiently circuits driven by multi-tone signals, being definitely a single tone algorithm [5]. The finitedifference time-domain technique (FDTD) discretizes the differential equations over a period yielding a system of algebraic equations, which are solved simultaneously to find solutions in all time points of the discretization network. Another approach for the steady state simulation is harmonic balance method (HB) that operates in the frequency domain. Different methods, both in time domain and in frequency domain, were proposed for circuit simulation under modulated carrier excitation [2, 5, 6], and a mixed time-frequency method for the multi-tone steady state circuit simulation has been developed [3)]. A new timedomain approach based on multiple time scales has been developed in the last years for steady state analysis of nonlinear circuits with broad signal spectrum [8-12]. Using multivariate functions transforms the original ordinary differential equations (ODEs) or differentialalgebraic equations (DAEs), describing the circuit behavior, into partial differential equation form, in the general case resulting in multitime partial differential equations (MPDEs). In this new formulation the components with different rates of variation are decoupled, each disparate signal being represented by its own artificial time scale. Solving the MPDE numerically, by appropriate choice of boundary conditions [12], the quasi-periodic and envelope-modulated solutions are obtained for the circuits with combination of strong nonlinearities and multirate signals. In this paper, in order to preserve the easy formulation of the circuit equations, we use the semi-state method (SSM) equivalent with the modified nodal method in dynamic behaviour. For a lumped nonlinear analogue circuit the equations corresponding to SSM, have the following form [1, 5-7]: M xt x (t ) Gx (t ) F xt Bbt . (1) t where: x(t ) vnt 1, imt - is the independent variable vector, with x0 initial condition; M , G are square n 1 m n 1 m ; matrices F - is a nonlinear t t function of x; b j t , e - is the input vector; B is a selector matrix, with entries (1, 0 or 1), and the superscript “t” denotes the transpose. In the case when the circuit exhibits multirate behaviour, its variables can be represented efficiently using multiple time variables. Using p time-scales and denoting the multivariate forms of x (t) and b (t) by xˆ t1,...,t p and bˆ t1,...,t p , respectively, the MPDE corresponding to (1) becomes: xˆ xˆ (2) M xˆ ... Gxˆ F xˆ Bbˆ t1 ,..., t p . t p t1 Using equations (2) and replacing the dynamic elements by discrete resistive circuit models associated with an implicit numerical integration algorithm [13, 14] the transient analysis of the nonlinear circuit can be reduced to the dc analysis of a sequence of equivalent nonlinear resistive circuits, and efficiency in numerical computing of the associated MPDE is obtained. Considering the two-rate case, MPDE (2) becomes: xˆ xˆ ˆ (3) M xˆ Gxˆ F xˆ Bbt1 ,t 2 , t1 t 2 with the periodic boundary conditions (BCs) xˆt1 T1, t2 T2 xˆt1, t2 . We take a uniform grid { t i, j } of size n1 1 p2 1 on the rectangle [0, m1T1] x [0, T2] (Fig. 1), where t i , j t1i ,t 2 j , with: t 2 j i 2T2 j 1h2 . (4) meaning that at each integration step h1, p2 integration steps with step size h2 are performed, and t1i i 1h1 , i 1, n1 1, j 1, p2 1 ; (5) h1 m1T1 / n1 T1 / p1 , h2 T2 / p2 . Consider that the slow components of bt and xt depend on t1 and the fast components depend on t2. For the first periods T1 and T2 (corresponding to the grid size p1 1 p2 1 ), we assume that the BCs are: x̂1, j 0.0; j 1, p2 1 , (6,a) ˆ i ,1 0.0; i 1,2 , x ˆ i 1,1 x ˆ i , p2 1; i 2, p1 (6,b) x on the row t1 = 0, and on the column t2 = 0, respectively. We continue integration with the step h1 in respect of the slow time t1 from the point t p1 1, p2 1 to the point t p1 2,2 (see Fig. 1). Proceeding in this way for the other grids we shall integrate the MPDE until the point t n1 1, p 2 1 t1n11 ,t 2 p 21 , and t2 p 2 1 with t1n 1 1 m1T1 n1T2 . At each time moment t i, j we have to solve a nonlinear algebraic equation system. For this end we use the Newton-Raphson algorithm. The discrete resistive circuit equations, associated to the BDF (backward differential formula) of the first order, when the characteristics of the nonlinear elements are approximated by piecewise-linear continuous curves, at t i, j and at the (k+1)th iteration of the NewtonRaphson algorithm, corresponding to the SSM, have the following form: k Gdn1,n1 s k Bdn1,m sik, j v nk11i, j i sc ,i , j k 1 ki , j . (9) Rdm,m sik, j i mki ,1j e mk,i , j Adm,n1 si , j where: Gdn 1, n 1 sik, j is the incremental node-conductance matrix corresponding to n–1 independent nodes; Bdn 1, m sik, j is an (n-1)m matrix that contains the elements –1, 0, +1 and the current gains of the CCCSs; Adm, n 1 sik, j represents a m(n-1) matrix containing the elements –1, 0, +1 and voltage gains of the VCVSs; Rdm, m sik, j is a mm matrix having the elements made up of: the transfer resistances of the CCVSs, the incremental resistances of the discrete models of the current-controlled dynamic circuit elements and the incremental resistances of the current-controlled nonlinear resistors; v nk11i, j is the node-voltage vector corresponding to n–1 independent nodes at the (k+1)th iteration and at the time t i, j ; i mki,1j represents the current vector corresponding to the nonNA-compatible circuit branches, and i sck,n 1i , j , emki , j represent the contributions of the excitation sources (independent current and voltage sources), of the sources corresponding to the approximations of the nonlinear resistors, and the initial values of the inductor currents and of the capacitor voltages, which are determined from previous time steps t i 1, j of the slow time t1, and t i, j 1 of the fast time t2. The contribution of some dynamic circuit elements to the Eq. (9) is presented in APPENDIX. Figure 1. . A uniform grid { t i, j } of size n1 1 p2 1 . We start the integration process on the row 2 from the point t 2,2 t12 ,t 22 , with t12 h1 , t 22 h2 , in respect of the fast time t2 from the column 2 to the column p2+1, and so on until we arrive in the point t 2, p 2 1 ( t12 ,t 2 p 1 ) with t12 h1 , t 2 p 1 p 2 h2 . 2 2 After that we integrate one time step h1 in respect of the slow time t1 –assigning to x̂3,1 the value of x̂2, p2 1 – and then we start again the integration process on the row3 in respect of the fast time t2 from the column 2 to the column p2+1, and so on until we arrive in t p1 1, p 2 1 t1 p 1 ,t 2 p 1 , the point with t1 p 1 1 p1h1 T1 and t2 p 2 1 1 2 p1T2 . Remark 1. Before passing to the integration on the next grid (each grid having the size p1 1 p2 1 ), starting from the point t p1 2,2 t1 p 2 ,t 22 , with t1 p 1 2 p1 1h1 T1 h1 , and 1 t 22 p1T2 h2 , we must consider the following boundary conditions: xˆ p1 2,1 xˆ p1 1, p 2 1; xˆ p1 i ,1 xˆ p1 i 1, p 2 1 (7) for i 3, p1 1, on thecolumnt 2 0 , and x̂ p1 1, j , j 2, p2 1 i 2, p2 1 on the row t1 = T1. (8) [1] R. Achar, and M.S. Nakhla, Simulation of High-Speed Interconnects, Proceedings of the IEEE, 89, 5, 2000, pp. 693-728. [2] H. G. Brachtendorf, G. Welsch, and R. Laur, A novel timefrequency method for the simulation of the steady state of circuits driven by multi-tone signals, Proceedings of the ISCAS,1997, pp. 1508-1511. [3] Angela Hodge, and R. W. Newcomb, Semistate Theory and Analog VLSI Design, IEEE Circuit and Systems Magazine, 2, 2, 2002, pp. 30-49. [4] M. Iordache, Lucia Dumitriu, and L. Mandache, Time-Domain Modified Nodal Analysis for Large-Scale Analog Circuits, Revue Roum. Sci. Techn. - Électrotechn. et Énerg., 48, 2-3, 2003, pp. 257268.