Bifurcation of power electronic circuits

Abstract

This paper studies various bifurcations of periodic orbits in power electronic circuits: cyclic fold bifurcations, period-doubling bifurcations, and bifurcations due to Poincaré map discontinuities. We focus on circuits operating under closed-loop control and/or containing nonlinear reactive components. Section III contains an exploration of cyclic fold bifurcations and the associated resonant jump phenomenon in circuits containing saturable reactors. Section IV gives a comprehensive overview of period-doubling phenomena in closed-loop DC-DC conversion circuits. We study circuits with homeomorphic and unimodal Poincaré maps, those that period-double a single time and those that period-double repeatedly in a cascade to chaos. This section ends with a result relating non-genericity of a period-doubling bifurcation to halfwave orbital symmetry. An interesting feature of power electronic circuits is that they may have Poincaré maps that are continuous but not everywhere differentiable, or discontinuous. In Section V we study, in detail, bifurcation behavior in a thyristor controlled VAR compensator, understood in terms of Poincaré map discontinuities. We show that Poincaré map discontinuities are due to jumps in circuit switching times. We show how map discontinuities lead to steady state jump phenomena, and distinguish between transient behavior related to switch time jumps and steady state bifurcations. The paper ends with an Appendix, in which concepts underlying cyclic fold bifurcations for the case of a continuous but not everywhere differentiable map are developed.