Small-Signal Analysis of Naturally-Sampled Single-Edge PWM Control Loops

This paper presents a simple method to analyze the behavior of feedback loops that contain a naturally-sampled single-edge pulse-width modulator. A small-signal model is derived by means of simple geometric arguments. It is shown how this small-signal model can be used to analyze the stability of the continuous-time pulse-width modulated feedback loop by using standard $z$ -domain techniques. The strategy relies on familiar concepts like transfer functions and small-signal gains and does not require any in-depth knowledge of nonlinear systems. A simple design process, where the continuous-time compensator is designed directly in the $z$ -domain, is developed and detailed design equations are derived for a PI current regulator. It is shown how the proposed strategy can accurately predict instability that cannot be explained by means of the well-known average model of the pulse-width modulator. The theoretical analysis is confirmed by means of detailed time-domain simulations. The mechanisms that lead to instability are discussed and an equation for the critical loop gain is derived.