Small-signal analysis of naturally-sampled single-edge PWM control loops

This paper presents a simple method to analyse the behaviour 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 analyse 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 non-linear 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 timedomain simulations. The mechanisms that lead to instability are discussed and an equation for the critical loop gain is derived