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Analysing Forced Oscillators with Multiple Time ScalesOnuttom Narayan Jaijeet RoychowdhuryUC Santa cruzBell Labs, Murray HillUSAUSAAbstractWe present a novel formulation, called the WaMPDE. for solvmg systems withforced autonomous components. An nnponant feature of the WaMPDE is itsability to capture frequency modulation (FM) in a natural and compact man-ner. This is made possible by a key new concept: that of warped rime. relatedto normal time through separate time scales. Using warped time, we obtain acompletely general formulation that captures complex dynamics in autonomousnonlinear systems of arbitrary size or complexity. We present computation-ally efficient numerical methods for solving large pracucnl problems using theWaMPDE. Our approach exphcitly calculates a time-varying local frequencythat matches intuitive expectauons. Applications to voltage-controlled oscilla-tors demonstrate speedups of two orders of magnitude.1 IntroductionOscillarory behaviour is crucial to the operation of many electronicsystems, such as voltage-controlled oscillators (VCOs), phase-lockedloops (PLLs), frequency dividers, ZA modulators, etc.. It is, how-ever, difficult to predict the response of such systems in a satisfactoryand reliable manner. In this paper. we present the WaMPDE (WarpedMultirate Partial Differential Equation), a new approach for analysingforced and unforced oscillatory systems. The WaMPDE provides aunified framework for treating phenomena like quasiperiodicity (par-ticularly frequency modulation (FM)), mode-locking and period multi-plication. (Conventional methods, discussed in Section 2, are typicallyerror-prone and computationally intensive for oscillators in general,and especially for forced ones exhibiting FM-quasiperiodicity.) Thekey to the WaMPDE is a compact representation of FM signals us-ing functions of several time variables, some of which are “warped”,i.e., stretched or squeezed by different amounts at different times inorder to make the density of the signal undulations uniform. The vari-ation of this stretching is much slower than the undulations themselves,hence a multiple time approach is used to separate the time scales. Aparticularly important feature of the WaMPDE is that, unlike previousmethods. it automatically and explicitly determines the local frequencyas it changes with time. Our approach also eliminates the problem ofgrowing phase error that limits previous numerical techmques for os-cillators.Numerical computations for the WaMPDE can be performed us-ing time-domain or frequency-domain methods, or combinations. Inparticular, existing codes for previous methods like the MPDE andharmonic balance (see Section 2) can be modified easily to performWaMPDE-based calculations. The use of iterative linear techniques(e.g., [Saa96, RLF98]) enables large systems to be handled efficiently.The remainder of this paper is organized as follows. Section 2 con-tains a brief review of previous work. Section 3 is a tutorial-style ex-position of the main concepts of the WaMPDE formulation, the math-ematical details of which are presented in Section 4. In Section 5.the new methods are applied to practical VCO circuits and comparedagainst existing techniques.2 Previous WorkMost previous analyses of oscillators have typically apply purely Im-ear concepts (e.g., [Ven82, Got97]) to obtain simple design formu-lae. Nonlinear analytical studies have largely been of polynomially-perturbed linear oscillators (e.g.. [Far94]). For real oscillators, nu-merical simulation has been the predominant means of predicting de-tailed responses. A fundamental problem, however, is the intrin-sic phase-instability of oscillators, leadmg to unbounded increase inphase error during simulation. Boundary-value methods like shooting(e.g., [NB95, TKW95]) and harmonic balance (e.g., [NB95, MFR95])can be applied to unforced oscillators in steady-state. but not to forcedoscillators with FM-quasiperiodic responses, which require an imprac-tically large number of time-steps or variables (see Section 3). In prac-O-7695-0013-7/99 $10.00 0 1999 IEEEFigure I: Example 2-tone quasi-periodic signal ?.(I)x 1”11 (fast me)0. 012 (slow me)Figure 2: Corresponding 2-periodic bivariate form f(rl ,I?)tice, the separation of the time scales is often reduced artificially tomake the problem tractable. As illustrated in Section 5. such ad-hocapproaches can lead to qualitatively misleading results.The warped-time approach presented in this paper is a general-ization of a recent multi-time approach (the MultIrate Partial Dif-ferential Equation (MPDE) [BWLBG96, Roy97, Roy98]) for non-autonomous systems with widely separated time scales. Earlier ef-forts to generalize the MPDE to autonomous systems [BL98] usednon-rectangular boundaries to capture frequency vanatlon. It has beenshown [Roy98], however, that this approach is limited to oscillationsthat eventually become periodic, and cannot. for instance, accommo-date FM-quasiperiodicity.3 Essential conceptsIn this section, we introduce several concepts at the core of this work.We first review why it is advantageous to use two or more timescales for analysing quasiperiodic signals, usmg amplitude-modulated(AM) signals for illustration. Then we show that although frequency-modulated (FM) signals can be quasiperiodic. the multl-time ap-proaches that work for AM do not confer the same advantages. Next,we introduce the concept of warped time and show how it can be usedto remedy the situation for FM.‘Finally. we outlme the basic featuresof the WaMPDE.Consider the waveform x(t) shown in Figure 1. a simple two-tonequasiperiodic signal given by:y(,i=sin(FI) sin(Ei),Tl = 0.02s.Tz = Is(1)The two tones are at frequencies ft = $; = 50Hz and fi = k =lHz, i.e., there are 50 fast-varying smusoids of period TI = 0.02s mod-ulated by a slowly-varying smusoid of period Tz = 1 s. When such sig-nals result from differential-algebraic equation (DAE) systems beingsolved by numerical integration (~.e., transient simulation). the time-62112”’ Inrerncuionnl Corzference on VLSI Design ~ .I~~nurr~ I999Figure 3: it : simplistic bivariate representation of FM signalsteps taken need to be spaced closely enough that each rapid undula-tion of b(t) is sampled accurately. If each fast sinusoid is sampled at npoints, the total number of time-steps needed for one period of the slowmodulation is n %, To generate Figure I, I.5 points were used per