We analyze the stability of a boost power-factor-correction (PFC) circuit using a hybrid model. We consider two multi-loop controllers to control the power stage. For each closed-loop system, we treat two separate cases: one for which the switching frequency is approaching infinity and the other for which it is finite but large. Unlike all previous analyses, the analysis in this paper investigates the stability of the converter in the saturated and unsaturated regions of operation. Using concepts of discontinuous systems, we show that the global existence of a smooth hypersurface for the boost PFC circuit is not possible. Subsequently, we develop conditions for the local existence of each of the closed-loop systems using a Lyapunov function. In other words, we derive the conditions for which a trajectory will reach a smooth hypersurface. If the trajectories do not reach the sliding surface, then the system saturates. As such, the stability of the period-one orbit is lost. Using the conditions for existence and the concept of equivalent control, we show why, for the second closed-loop system, the onset of the fast-scale instability occurs when the inductor current approaches zero. For this system, we show that the onset of the fast-scale instability near zero-inductor current occurs for a lower line voltage. Besides, when the peak of the line voltage approaches the bus voltage, the fast-scale instability may occur not only at the peak but also when the inductor current approaches zero. We develop a condition which ensures that the saturated region does not have any stable orbits. As such, a solution that leaves the sliding surface (if existence fails) cannot stabilize in the saturated region. Finally, we extend the analysis to the case in which the converter operates with a finite but large switching frequency. As such, the system has two fundamental frequencies: the switching and line frequencies. Hence, the dynamics of the system evolve on a torus. We show two different approaches to obtaining a solution for the closed-loop system. For the second closed-loop system, using the controller gain for the current loop as a bifurcation parameter, we show (using a Poincaré map) the mechanism of the torus breakdown. If the mechanism of the torus breakdown is known, then, depending on the post-instability dynamics, a designer can optimize the design of the closed-loop converter.
A single stage ZVS Power factor correction converter
