FINDING THE PERIODIC STEADY STATE OF STRONGLY NONLINEAR CIRCUITS BY SHOOTING WITH IMPROVED EXPONENTIAL EXTRAPOLATON F. Constantinescu,, M. Nitescu, C. V. Marin, Department of Electrical Engineering, "Politehnica" Universify, Bucharest' Romania I. INTRODUCTION .\ very important step in the design of radio-frequency integrated circuits (RF-IC) is circuit simulation. Because the prototypingir very costly [1] the validity of a RF-IC design is testedonly by simulation. A typical RF-IC^application is strongly nonlinear and has carrier frequencies in the GHz-range with modulating signals in the KHz-range. Due to the broad signal spectrum (about six orders oi ,nugnitude) finding of the steady state by the "brute - force" method (time domain integration ofiircuit equations until all transients decay)is very time consuming. There are three classes of methods which can solve this problem for circuits with non-modulated signalsusing a reasonable amount of CPU time: shootingmethods,finite difference methods,and eipansion methods. The best known from the third class is the harmonic balance method which usesthe Fourier seriesexpansionof the circuit response; generallythis method is the most tirne- consuming and its efficiency is proved only for almost sinusoidalresponses. The fastestare the shooting methods which are effective for arbitrary response waveforms; using the integration of circuit equations, the result may be influencedby the existence of some local unstable modes.The finite difference methods have almost the same efficiency as shooting methods, the result is not influenced tr.ri the iocal unstable modes, butthe equation s)'Stem to be solvedis verv large i3J" The shootingmethods have been developed in two directions:shootingwith Newton-Raphson ana shootingtry ixtrapolation. In the first case the circuit state at the beginningof the period is correctect iierativel)' b), tlie Neg,ton-Raphson method untii the pedociicresponse is ohtained [3.|" The lirreal extrapoiation method [aj assumes that ihe state vector r"" at the beginrling of the period is an ::ffin: function on the state vector at the beginning of the previous period. It ftrliorrys that afier tlre sweeping of p excitation periods 1p being at most the dirnensionof x""1 and a minimization procedure some p coefficients defirning the "solutionrr *(0' are for,rnd. 'i'he sequence0f these "solutions" convetrges to x(u)corresponding to the periodic steady state. For circuits with modulated signais as RF-IC's simulation by time domain methods for fast computation of the steady-state (finite difference methods [3,5], shooting with Newton-Raphson [6], shooting by linear extrapolation [4], shootingby exponential extrapoiation [7]) is not efficient. The harmonic balance method [8] may be used only for circuits with weakly or miid nonlinearities. Recently, a new approach has been proposed [9,10.1i] for efficient finding of the steadystarein nonlinear circuits *itn broad signal spectrum. The circuit equations are formuiated in terms of r**o time variables corresponding to slow and fast signal components. Using this approach- the differential-algebraic equations (DAE) of the circuit becomepartial differential-aigebraic equations (PDAE). For examplea period of s(r) = sin 2r(t l4 )sin Zn(r l?i ) where Ti:lms. T2 : trns. can he described by 10".i00 sampleslperiod :lO8sampies. Using the two-rate representaticn s(tr^tr) = sin zr(trl ! )sin )ft(t2rr, ) only 100samplesrperiod x 100 sampiesrperiod:lU' sampreE arr necessarv. in this case an excitation period is sweptby a numerical rnethod using 108 time stepsin the classical tin:re Comainanall,sis and only 104stepsin the fwo-ratetime dornainanaivsis. -Jhi' approach has been usedfor the multivariate finite difference time domain nrethod anii iiieraiihi::ei shooting method [10.11]. In [9] the circuit PDAEs are solved Lrya variational techi-ristie usinq 17