A General Approach to Sampled-Data Modeling for Ripple-Based Control—Part II: Constant ON-Time and Constant OFF-Time Control
Na Yan, Xinbo Ruan, Xin Li
Abstract
This part is the second part of this article and mainly presents the sampled-data modeling method for the dc–dc converters with the constant <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> -time (COT) and constant <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> -time (COFT) controls. Compared with the constant frequency control in the first part, the duty cycle perturbation signal in the COT/COFT control is found to have two sets of narrow pulse signals in each switching period. Then, by establishing the relationship between the two sets of signals and using Shannon's sampling theorem, the transfer functions of the modulator and the loop gain are derived. Based on the theoretical results, the phenomenon that a current-mode constant <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> / <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> time controlled buck converter is stable at any duty cycle, and the instability of a voltage-mode constant <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> / <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> -time controlled buck converter with the smaller time constant of the output capacitor could be predicted. Moreover, the sampled-data modeling method is further extended to the digital control by taking a digitally current-mode controlled buck converter as an example. Simulation results are provided to confirm the validity of the proposed modeling method.