Dynamic Modeling of Peak Current Mode Controlled Two-phase PWM Converters with Coupled Inductors

Abstract

This paper presents simple and accurate large-signal and small-signal averaged models for basic two-phase coupled-inductor PWM converters operating in continuous conduction mode with peak current-mode (PCM) control. A large-signal model is developed, which captures the effects of inductor-current slopes using equivalent, duty-cycle-dependent inductances. A small-signal averaged model is then derived, which includes sampled-data effects. The small-signal model is used to explain differences compared to well-known PCM control models for single-phase or uncoupled multi-phase converters. It is shown how subharmonic instability regions depend on the coupling coefficient and the duty cycle. A continuous-time transfer function is derived for the control-to-inductor current response, with the high-frequency resonant pole <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(F_{res})$</tex> that occurs in the range <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$[0.5F_{s},\ F_{s}]$</tex> and varies with the coupling coefficient. A new closed-form expression for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$F_{res}$</tex> is then derived, which allows simplification to a design-oriented <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$3^{rd}$</tex> order transfer function. The proposed small-signal model can also predict audio-susceptibility, input, and output impedances accurately. Finally, the model is validated using simulations and experiments on a two-phase buck converter prototype with inversely coupled inductors.