Multi-Time PDEs for Dynamical System Analysis

  • Jaijeet Roychowdhury
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 629)


Signals with widely separated time scales often appear in dynamical systems, especially those from electronic communication circuits. This paper reviews recent techniques that reformulate such problems as PDEs with artificial time variables, and describes their uses in circuit simulation.


Fourier Coefficient Instantaneous Frequency Harmonic Balance Differential Algebraic Equation Quasiperiodic Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Aliaga, D. Boley, R.W. Freund, and V. Hernandez. A Lanczos-type algorithm for multiple starting vectors, 1996. Numerical Analysis Manuscript No. 96-18, Bell Laboratories.Google Scholar
  2. [2]
    T.J. Aprille and T.N. Trick. Steady-state analysis of nonlinear circuits with periodic inputs. Proc. IEEE, 60(1):108–114, January 1972.MathSciNetCrossRefGoogle Scholar
  3. [3]
    D.L. Boley. Krylov space methods on state-space control models. IEEE Trans. Ckts. Syst. — II: Sig. Proc., 13(6):733–758, 1994.MathSciNetMATHGoogle Scholar
  4. [4]
    H.G. Brachtendorf and R. Laur. Transient simulation of oscillators. Technical Report ITD-98-34096K, Bell Laboratories, 1998.Google Scholar
  5. [5]
    H.G. Brachtendorf, G. Welsch, R. Laur, and A. Bunse-Gerstner. Numerical steady state analysis of electronic circuits driven by multi-tone signals. Electrical Engineering (Springer-Verlag), 79:103–112, 1996.CrossRefGoogle Scholar
  6. [6]
    M. Celik and A.C. Cangellaris. Simulation of dispersive multiconductor transmission Lines by Padé approximation by the Lanczos process. IEEE Trans. MTT, (44):2525-2535, 1996.CrossRefGoogle Scholar
  7. [7]
    M. Celik and A.C. Cangellaris. Simulation of multiconductor transmission Lines using Krylov subspace order-reduction techniques. IEEE Trans. CAD, (16):485–496, 1997.Google Scholar
  8. [8]
    E. Chiprout and M.S. Nakhla. Asymptotic Waveform Evaluation. Kluwer, Norwell, MA, 1994.MATHCrossRefGoogle Scholar
  9. [9]
    L.O. Chua and P-M. Lin. Computer-aided analysis of electronic circuits: algorithms and computational techniques. Prentice-Hall, Engle-wood Cliffs, N.J., 1975.MATHGoogle Scholar
  10. [10]
    L.O. Chua and A. Ushida. Algorithms for computing almost periodic steady-State response of nonlinear systems to multiple input frequencies. IEEE Trans. Ckts. Syst., CAS-28(10):953–971, Oct 1981.MathSciNetCrossRefGoogle Scholar
  11. [11]
    D. Long, R.C. Melville, et al. Full chip harmonic balance. In Proc. IEEE CICC, May 1997.Google Scholar
  12. [12]
    A. Demir, A. Mehrotra, and J. Roychowdhury. Phase Noise in Oscillators: A Unifying Theory and Numerical Methods for Characterization. In Proc. IEEE DAC, pages 26–31, June 1998.Google Scholar
  13. [13] A. Demir, A. Mehrotra, and J. Roychowdhury. Phase noise in oscillators: a unifying theory and numerical methods for characterization. IEEE Trans. Ckts. Syst. — I: Fund. Th. Appl., March 1999. Accepted for publication.Google Scholar
  14. [14]
    I. Elfadel and D. Ling. A block rational Arnoldi algorithm for multipoint passive model-order reduction of multiport RLC networks. In Proc. ICCAD, pages 66–71, November 1997.Google Scholar
  15. [15]
    P. Feldmann and R.W. Freund. Efficient linear circuit analysis by Padé approximation via the Lanczos process. IEEE Trans. CAD, 14(5):639–649, May 1995.Google Scholar
  16. [16]
    P. Feldmann and R.W. Freund. Reduced-order modeling of large linear subcircuits via a block Lanczos algorithm. In Proc. IEEE DAC, pages 474–479, 1995.Google Scholar
  17. [17]
    P. Feldmann and R.W. Freund. Circuit noise evaluation by Padé approximation based model-reduction techniques. In Proc. ICCAD, pages 132–138, November 1997.Google Scholar
  18. [18]
    P. Feldmann, R.C. Melville, and D. Long. Efficient frequency domain analysis of large nonlinear analog circuits. In Proc. IEEE CICC, May 1996.Google Scholar
  19. [19]
    P. Feldmann and J. Roychowdhury. Computation of circuit waveform envelopes using an efficient, matrix-decomposed harmonic balance algorithm. In Proc. ICCAD, November 1996.Google Scholar
  20. [20]
    R.W. Freund. Reduced-order modeling techniques based on Krylov subspaces and their use in circuit simulation. Technical Report 11273-980217-02TM, Bell Laboratories, 1998.Google Scholar
  21. [21]
    R.W. Freund. Reduced-order modeling techniques based on Krylov subspaces and their use in circuit simulation. Applied and Computational Control, Signals, and Circuits, Volume 1, pages 435–498, 1999.MathSciNetCrossRefGoogle Scholar
  22. [22]
    R.W. Freund, G.H. Golub, and N.M. Nachtigal. Iterative solution of linear systems. Acta Numerica, pages 57–100, 1991.Google Scholar
  23. [23]
    R.J. Gilmore and M.B. Steer. Nonlinear circuit analysis using the method of harmonic balance — a review of the art. Part I. Introductory concepts. Int. J. on Microwave and Millimeter Wave CAE, 1(1), 1991.Google Scholar
  24. [24]
    S.A. Haas. Nonlinear Microwave Circuits. Artech House, Norwood, MA, 1988.Google Scholar
  25. [25]
    X. Huang, V. Raghavan, and R.A. Rohrer. AWEsim: A program for the efficient analysis of linear(ized) circuits. In Proc. ICCAD, pages 534–537, November 1990.Google Scholar
  26. [26]
    I.M. Jaimoukha. A general minimal residual Krylov subspace method for large-scale model reduction. IEEE Trans. Automat. Control, (42):1422–1427, 1997.MathSciNetMATHCrossRefGoogle Scholar
  27. [27] J. Kevorkian. Partial Differential Equations: Analytical Solution Techniques. Chapman and Hall, 1990.Google Scholar
  28. [28] J. Kevorkian and J.D. Cole. Perturbation methods in Applied Mathematics. Springer-Verlag, 1981.Google Scholar
  29. [29]
    K. Kundert, J. White, and A. Sangiovanni-Vincentelli. A mixed frequency-time approach for distortion analysis of switching filter circuits. IEEE J. Solid-State Ckts., 24(2):443–451, April 1989.CrossRefGoogle Scholar
  30. [30] Markus Rösch. Schnell Simulation des stationären Verhaltens nichtlinearer Schaltungen. PhD thesis, Technischen Universität München, 1992.Google Scholar
  31. [31]
    R.C. Melville, P. Feldmann, and J. Roychowdhury. Efficient multi-tone distortion analysis of analog integrated circuits. In Proc. IEEE CICC, pages 241–244, May 1995.Google Scholar
  32. [32] R. Mickens. Oscillations in Planar Dynamic Systems. World Scientific, 1995.Google Scholar
  33. [33]
    M.S. Nakhla and J. Vlach. A piecewise harmonic balance technique for determination of periodic responses of nonlinear systems. IEEE Trans. Ckts. Syst., CAS-23:85, 1976.CrossRefGoogle Scholar
  34. [34]
    O. Narayan and J. Roychowdhury. Multi-time simulation of voltage-controlled oscillators. In Proc. IEEE DAC, New Orleans, LA, June 1999.Google Scholar
  35. [35]
    S. Narayanan. Transistor distortion analysis using Volterra series representation. Bell System Technical Journal, May–June 1967.Google Scholar
  36. [36]
    A. Nayfeh and B. Balachandran. Applied Nonlinear Dynamics. Wiley, 1995.Google Scholar
  37. [37]
    E. Ngoya and R. Larchevèque. Envelop transient analysis: a new method for the transient and steady state analysis of microwave communication circuits and systems. In Proc. IEEE MTT Symp., 1996.Google Scholar
  38. [38]
    A. Odabasioglu, M. Celik, and L.T. Pileggi. PRIMA: passive reduced-order interconnect macromodelling algorithm. In Proc. ICCAD, pages 58–65, November 1997.Google Scholar
  39. [39]
    P. Oswald. Personal communication, 1998.Google Scholar
  40. [40]
    L. Petzold. An efficient numerical method for highly oscillatory ordinary differential equations. SIAM J. Numer. Anal., 18(3), June 1981.Google Scholar
  41. [41]
    J. Phillips. Personal communication, June 1999.Google Scholar
  42. [42]
    L.T. Pillage and R.A. Rohrer. Asymptotic waveform evaluation for timing analysis. IEEE Trans. CAD, 9:352–366, April 1990.Google Scholar
  43. [43]
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipes — The Art of Scientific Computing. Cambridge University Press, 1989.Google Scholar
  44. [44]
    V. Rizzoli and A. Neri. State of the art and present trends in nonlinear microwave CAD techniques. IEEE Trans. MTT, 36(2):343–365, February 1988.CrossRefGoogle Scholar
  45. [45]
    M. Rösch and K.J. Antreich. Schnell stationäre simulation nichtlinearer Schaltungen im frequenzbereich. AEÜ, 46(3):168–176, 1992.Google Scholar
  46. [46]
    J. Roychowdhury. Efficient methods for simulating highly nonlinear multi-rate circuits. In Proc. IEEE DAC, 1997.Google Scholar
  47. [47]
    J. Roychowdhury. General partial differential forms for autonomous systems, 1998. Bell Laboratories Internal Document.Google Scholar
  48. [48]
    J. Roychowdhury. MPDE methods for efficient analysis of wireless systems. In Proc. IEEE CICC, May 1998.Google Scholar
  49. [49]
    J. Roychowdhury. Reduced-order modelling of linear time-varying systems. In Proc. ICCAD, November 1998.Google Scholar
  50. [50]
    J. Roychowdhury. Reduced-order modelling of time-varying systems. IEEE Trans. Ckts. Syst. — II: Sig. Proc., 46(10), November 1999.Google Scholar
  51. [51]
    J. Roychowdhury. Analysing circuits with widely-separated time scales using numerical PDE methods. IEEE Trans. Ckts. Syst. — I: Fund. Th. Appl., May 2001.Google Scholar
  52. [52]
    J. Roychowdhury, D. Long, and P. Feldmann. Cyclostationary noise analysis of large RF circuits with multitone excitations. IEEE J. Solid-State Ckts., 33(2):324–336, Mar 1998.CrossRefGoogle Scholar
  53. [53]
    J.S. Roychowdhury. SPICE3 Distortion Analysis. Master’s thesis, EECS Dept., Univ. Calif. Berkeley, Elec. Res. Lab., April 1989. Memorandum no. UCB/ERL M89/48.Google Scholar
  54. [54]
    Y. Saad. Iterative methods for sparse linear systems. PWS, Boston, 1996.MATHGoogle Scholar
  55. [55]
    D. Sharrit. New Method of Analysis of Communication Systems. MTTS WMFA: Nonlinear CAD Workshop, June 1996.Google Scholar
  56. [56]
    S. Skelboe. Computation of the periodic steady-state response of nonlinear networks by extrapolation methods. IEEE Trans. Ckts. Syst., CAS-27(3):161–175, March 1980.MathSciNetCrossRefGoogle Scholar
  57. [57]
    R. Telichevesky, K. Kundert, and J. White. Efficient steady-state analysis based on matrix-free krylov subspace methods. In Proc. IEEE DAC, pages 480–484, 1995.Google Scholar
  58. [58]
    V. Volterra. Theory of Functionals and of Integral and Integro-Differential Equations. Dover, 1959.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Jaijeet Roychowdhury

There are no affiliations available

Personalised recommendations