On the perturbation analysis of the limit cycle in oscillators with shifting bias

Abstract

We propose a method of perturbation analysis of nearly sinusoidal oscillators with shifting bias, obtained by generalizing a method recently discussed in the literature [1] (Buonomo A, Di Bello C. IEEE Transactions on Circuits and Systems, 1996; CAS-43:953–963). The problem of periodic oscillations is formulated as a regular perturbation problem, Pϵ(x) =0, whose peculiarity is that the limiting linear problem, P0(x)=0, obtained when the perturbation parameter ϵ tends to zero, has a non-purely harmonic solution x0=B0+A0 cos ϑ. We give a simple condition for the existence of a periodic oscillation and an analytical method for constructing it in the form of a power series in ϵ. Unlike the existing perturbation methods, the method here proposed, which remains in the spirit of the bifurcation process of Poincaré, allows us to obtain the coefficients of the series solution, to an order in ϵ as great as we want, using recurrence formulae. The results of the analysis of a typical LC oscillator are given to show that these formulae are very useful as a practical method for determining all of the characteristics of the periodic oscillation, such as the harmonic content and the frequency correction due to the non-linear effect. Copyright © 2000 John Wiley & Sons, Ltd.