A mathematical analysis of a series circuit containing a nonlinear capacitor

This paper presents a mathematical analysis of a general series circuit composed of an inductance and a resistance in series with a nonlinear capacitor. The analysis is based on the assumption that the saturation curve of the dielectric material of the capacitor may be represented by a hyperbolic sine function. A 1-term approximation of the forced steady-state oscillation of the circuit produced by the application of a periodic electromotive force and bias potential is obtained. This one term solution gives the best “least squares” approximation to the amplitude and phase of the steady-state charge and current oscillation of the system and reduces to the usual expression for the steady-state current, if the capacitor is linear. The method of analysis is a general one and is applicable to many problems of forced oscillations of nonlinear dynamical systems, both electric and mechanical. It depends on a procedure by which the “mean square error” involved in the 1-term approximation is minimized. Several special cases are considered, and an estimate of the natural frequency of the free oscillations of the circuit is obtained.