Numerical stability analysis of oscillating integrated circuits

Abstract

An important topic in modern circuit design is the development of mixed analogue-digital circuits. Analogue circuits usually contain oscillating elments. Self-excited oscillating circuits transfer a constant input signal into an oscillating periodic output signal. The stability properties of this class of circuits are of special interest. A loss of stability means bifurcation and bifurcation is one of the main reasons for the birth of an irregular behaviour of a dynamic system. The implementation of a tool for stability analysis in a simulation package usually requires a charge-oriented approach for the investigation of local stability. For the investigation of industrially relevant circuits using a commercial simulator, it is necessary to adapt that approach to differential-algebraic equations (DAEs). For that purpose a generalized definition of the monodromy matrix is introduced in the case of ordinary differential equations (ODEs). We show that — using this generalized definition — some basic results of classical ODE stability analysis can be transfered to the index-1 DAE case. The present approach uses for the computation of the monodromy matrix provisional results produced by harmonic balance, a nonlinear frequency analysis, that is used for the computation of stable and unstable periodic solutions. The Floquet multipliers help to characterize the stability and bifurcation behaviour of the periodic signals of an electronic circuit. Because in simulation packages stable as well as unstable periodic solutions can be computed, stability analysis is an indispensable tool for a proper interpretation of simulation results.