The Interdependence of Frequency Variation and Harmonic Content, and the Problem of Constant-Frequency Oscillators

TLDR: In this article, the symbolic calculus has been used throughout in a complete and exact manner, by employing it for the fundamental frequency as well as for all harmonic frequencies which appear in the system.

Abstract: The investigations of the frequency variations in oscillating systems with negative nonlinear resistance were based until now on the fundamental differential equation of the simple oscillatory circuit. This method, although very exact, is not always sufficiently simple and even not always possible, especially if it concerns more complicated schemes. The symbolic calculus, although very simple in use, if applied to the nonlinear systems, gives only approximate results. The following paper represents the operation of nonlinear systems from a different point of view. Here the symbolic calculus has been used throughout in a complete and exact manner, by employing it for the fundamental frequency as well as for all harmonic frequencies which appear in the system. This could be done owing to the investigation of the negative resistance operation from the energy point of view. It was taken into consideration, that in the negative resistance characteristics i = f(v), i must be the univocal function of v, and therefore the area described by the instantaneous point of work during one cycle of the fundamental oscillation must be zero or, ⨖ idv = 0. On the other hand i and v can be considered, with regard to the external circuit connected to the negative resistance, as the sum of harmonic currents and voltages. In this way we obtain the formulas which allow the interdependence of the frequency variation and the content of harmonics to be determined.