Analytic solutions of limit cycles in a feedback-regulated converter system with hysteresis

A mathematical model is derived for the voltage step-down dc-to-dc converter in which a hysteretic bistable trigger circuit is used to regulate the output voltage. Normalized second-order differential equations are derived for the output-voltage error, or output-voltage ripple, measured with respect to a constant reference. The method of successor functions is applied to the piecewise analytic phase plane trajectory for the errors and the conditions leading to a limit cycle are initially formulated in two transcendental equations. With the objective of obtaining an analytic solution for the period of the limit cycles these two equations are then replaced by approximate algebraic equations which are solvable in general terms. The approximations are carefully based on properties that are typical of all converters of this type, and lead to quite simple but accurate expressions for the period of the limit cycle in terms of arbitrary system parameters. Expressions for the amplitude of the limit cycle are also given, and its stability is tested. A numerical example based on an actual representive system is given. Certain unusual characteristics of the limit cycle as a function of certain system parameters are pointed out.