A method for analysing the transient and the steady-state oscillations in third-order oscillators with shifting bias

We provide an asymptotic method for systematically analysing the transient and the steady-state oscillations in third-order oscillators with shifting bias. The method allows us to construct the general solution of the weakly non-linear differential equation describing these oscillators through an iteration procedure of successive approximations typical of perturbation methods. The approximation to first order is obtained solving a system of two first-order non-linear differential equations in the leading terms of solution (dc component and fundamental harmonic), whereby the dominant dynamics, the stationary states and their stability can be easily analysed. Unlike existing approaches, our method also enables us to determine the higher harmonics as well as the frequency shift from the system's natural frequency in the exact solution through analytical formulae. In addition, formulae for higher-order approximations of the above quantities are determined. The proposed method is applied to a practical circuit to show its usefulness in both analysis and design problems. Copyright (C) 2001 John Wiley & Sons, Ltd.