The internal resonance relations in the pause states of a nonlinear LCR circuit

Abstract

Electrical current i( t) ( t is time) responses, having pauses, and named pause states, of an LCR circuit with a nonlinear (ferro- electric) capacitor C, whose voltage/charge characteristic v c( q) is not strictly prescribed in the analysis, to a periodic rectangular- wave input voltage wave v( t) are considered. The pauses of the current function are the intervals where the function oscillates, but are relatively small. There is a resonance relation between the amplitudes of the pause and nonpause oscillations in the pause states which are, in the usual sense, nonresonant states, since the extreme value of the nonpause oscillations is nonresonant with respect to the input. The pause oscillations may be considered as a local time, resonantly suppressed response which is an extension of the resonance situations which are traditionally considered. There is a specific distribution of the “potential” and “kinetic” energies of the oscillations, between the pause and nonpause intervals. We have here both a simple feature of a nonlinear response, which is rare, and a qualitatively interesting oscillatory state which is relevant, in many details, to both the nonlinear and linear versions of the system. (The relevant analysis is missing in the classical linear system theory which is almost completely devoted to the topics of spectrum and harmonic superposition or state-space equations which are not the most suitable tools for the analysis of complicated waveforms.) The system-theoretic interest in the study of the states is not limited, however, by the topic of the resonance relation, as is finally discussed. The purpose of the work is to introduce the concept of the pause state as a concept in system theory.