A New Approach for Accurate Prediction of Subharmonic Oscillation in Switching Regulators—Part I: Mathematical Derivations

This part of the paper takes a new look at the stability of the fundamental periodic behavior and the associated subharmonic oscillation boundary in dc-dc switching regulators with fixed-frequency pulse width modulation. After revisiting different approaches applied to a unified reduced-order model of switching regulators, the work presents a novel way of obtaining such a boundary without any simplification nor order reduction. A new simple closed-form condition is then obtained using the new approach, which is valid for different strategies with both trailing edge and leading edge modulation schemes. The critical condition is obtained from the steady-state response using an asymptotic approach without resorting to frequency-domain Fourier analysis or using the monodromy or the Jacobian matrix of the discrete-time model. Unlike the method based on the Fourier series expansion, the proposed method does not require the use of any transform and, most importantly, can be applied to bilinear switching regulators. As a byproduct of the proposed method, a singularity problem that is encountered in steady state, when the system involves integrators in the feedback loop, is addressed and a solution is developed to achieve a closed-form expression in the presence of such integrating feedback loops.