Internal Resonance in Circuits Containing Nonlinear Resistance

The simultaneous presence of oscillations locked in synchronism is investigated for circuits containing a nonlinear element. Under these conditions the circuit is called internally resonant, the two frequencies of oscillation being related by an integral ratio r/s. The discussion is confined to self-excitation in a highly oscillatory circuit connected across a nonlinear element. Nonlinear oscillations are generally characterized by a certain Fourier spectrum of the component frequencies; as a first approximation, it is reasonable to assume that only the fundamental components need be considered. The nonlinear element is described in terms of its current voltage characteristic and an equivalent linearized parameter defined by considering only the two fundamental frequencies. The variation-of-parameters method is applied to the resultant "linearized" equations and the equilibrium conditions determined. The conditions for stable equilibrium are then investigated by assuming a small departure from the equilibrium and studying the motion of the system following this displacement.