Analysis of a Fundamental Oscillatory Process in Three Phases and Some of Its Applications

Oscillatory processes in two dimensions are well known, and most of the electronic oscillators exhibit a dynamic behavior, which is related to them. The aim of the present paper is to demonstrate and analyze an oscillatory process in three dimensions. The mathematical model considered possesses, as solutions in steady state, three stable sinusoidal waveforms arranged symmetrically as is usual in a three-phase supply system. The dynamic behavior of the model is thoroughly investigated and it is shown that it is related to the dynamic behavior of a well-known symmetrical quadrature oscillator in two phases. This relationship suggests that a practical system realized according to the model equations should possess the properties of a high-quality quickly responding voltage-controlled oscillator (VCO). This useful feature has been demonstrated experimentally by investigating the performance of a corresponding instrument. Beside the applications of the model in instrumentation and power electronics, it should also serve in developing simplified models for representing synchronous machines and other power systems elements by computers.