Analyzing oscillators using multitime PDEs

Oscillators are often difficult to analyze or simulate, because they generate waveforms that can span a range of widely separated time scales. We present a general oscillator formulation that separates slow and fast dynamics without approximations, and captures amplitude and frequency modulation in a natural and compact manner. To handle frequency-modulation effectively, we make use of a novel concept, warped time, within a multitime partial differential equation framework. The equations incorporate an explicit time-varying frequency variable that matches intuitive notions of changing frequency in a frequency-modulated signal. The formulation is useful for both hand analysis and numerical simulation.

[1]  Stephen A. Maas,et al.  Nonlinear microwave circuits , 1988 .

[2]  Miklós Farkas,et al.  Periodic Motions , 1994 .

[3]  R. Freund.  Reduced-Order Modeling Techniques Based on Krylov Subspaces and Their Use in Circuit Simulation , 1999 .

[4]  M. Crawford,et al.  Theory and methods , 1980 .

[5]  Balth van der Pol Jun. Doct.Sc..  LXXXV. On oscillation hysteresis in a triode generator with two degrees of freedom , 1922 .

[6]  Alberto L. Sangiovanni-Vincentelli,et al.  Steady-state methods for simulating analog and microwave circuits , 1990, The Kluwer international series in engineering and computer science.

[7]  Jiri Vlach,et al.  A piecewise harmonic balance technique for determination of periodic response of nonlinear systems , 1976 .

[8]  M.L. Liou,et al.  Computer-aided analysis of electronic circuits: Algorithms and computational techniques , 1977, Proceedings of the IEEE.

[9]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

[10]  George D. Vendelin.  Design of amplifiers and oscillators by the S-parameter method , 1982 .

[11]  David E. Long,et al.  Efficient frequency domain analysis of large nonlinear analog circuits , 1996, Proceedings of Custom Integrated Circuits Conference.

[12]  Leon O. Chua,et al.  Practical Numerical Algorithms for Chaotic Systems , 1989 .

[13]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[14]  Jacob K. White,et al.  Efficient Steady-State Analysis Based on Matrix-Free Krylov-Subspace Methods , 1995, 32nd Design Automation Conference.

[15]  J. Murdock.  Perturbations: Theory and Methods , 1987 .

[16]  Ulrich L. Rohde,et al.  Microwave and wireless synthesizers: theory and design , 1997 .

[17]  Ronald E. Mickens,et al.  Oscillations in Planar Dynamic Systems , 1996, Series on Advances in Mathematics for Applied Sciences.

[18]  J. Roychowdhury.  Analyzing circuits with widely separated time scales using numerical PDE methods , 2001 .

[19]  Michael B. Steer,et al.  Nonlinear circuit analysis using the method of harmonic balance—a review of the art. II. Advanced concepts , 1991 .

[20]  H. G. Brachtendorf,et al.  Numerical steady state analysis of electronic circuits driven by multi-tone signals , 1996 .

[21]  George Huitema,et al.  Quasi-Periodic Motions in Families of Dynamical Systems: Order amidst Chaos , 2002 .

[22]  Irving M. Gottlieb.  Practical Oscillator Handbook , 1997 .

[23]  Dafydd Gibbon,et al.  1 User’s guide , 1998 .

[24]  H. Weitzner,et al.  Perturbation Methods in Applied Mathematics , 1969 .

[25]  T. Aprille,et al.  Steady-state analysis of nonlinear circuits with periodic inputs , 1972 .

[26]  L. Chua.  The Genesis of Chua's circuit , 1992 .

[27]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[28]  Stig Skelboe,et al.  Computation of the periodic steady-state response of nonlinear networks by extrapolation methods , 1980 .

[29]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[30]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .

[31]  A. Neri,et al.  State of the art and present trends in nonlinear microwave CAD techniques , 1988 .

[32]  K. J. Antreich,et al.  Schnelle stationäre simulation nichlinearer Schaltungen im Frequenzbereich , 1992 .

[33]  Benjamin Parzen,et al.  Design of Crystal and Other Harmonic Oscillators , 1983 .