Symbolic time-varying root-locus analysis for oscillator design

The small-signal analysis of an oscillator relative to a periodic steady-state (PSS) would generate periodic time-varying characteristic poles. Analyzing periodic root-loci can provide useful design information, which is not available from the existing circuit simulation tools. Although the numerical QZ algorithm can be used to generate periodic root-loci, this paper proposes an alternative symbolic computation method for repeated pole computation. It is demonstrated that the Muller algorithm can be used for finding the dominant periodic roots of a characteristic polynomial with periodic coefficients, whose efficiency is superior to the matrix-based numerical QZ method. Other advantages of symbolic root-locus analysis also are explored by applying the proposed method to the analysis of two oscillator circuits.