Improved Sampled-Data Modeling Method and Stability Analysis of Digitally Controlled DC-DC Converters
Li Ziyang1,2, Hang Lijun1,2, Zheng Xiang1,2, He Zhen1,2, Zeng Pingliang1,2
1. Regional Energy Internet Technology Zhejiang Engineering Laboratory Hangzhou Dianzi University Hangzhou 310018 China; 2. College of Automation Hangzhou Dianzi University Hangzhou 310018 China
Abstract：The discrete-time modeling method is a tool that can accurately describe the characteristics of the digitally controlled converter. By using the discrete-time modeling approach, the state variables are not averaged, so the precise state transition characteristics can be obtained at the sampling point. However, the derivation process of the traditional discrete-time model is usually computationally complex and the analysis of the model usually relies on numerical calculation instead of an equivalent circuit so the physical meaning is not clear, which is not conducive to the design of the controller. So far, the intuitive and accurate large-signal equivalent circuit models are rarely reported in the research of digital-pulse-width-modulated (DPWM) DC-DC converters. Based on the reasonable approximation of the sampled-data model, this paper derives the second-order global equivalent circuit for the DC-DC converters with leading-edge and trailing-edge modulation, which resolves the contradiction between the intuitiveness and accuracy of the traditional model. By improving the conventional sampled-data modeling method, both the mathematical model and equivalent circuit are built by merging the two switching states of a cycle into one state in the detailed switching circuit, which can overcome the contradiction between the accuracy and intuitiveness of the existing model. The equivalent circuit model proposed by this paper can accurately track the operating waveform of the detailed switching circuit. To verify the results, the detailed switching circuit and the equivalent circuit were built on the PLECS platform. The simulation phenomenon shows that when the system enters instability from stability, the closed-loop system loses its steady-state operating point. Under this condition, the waveform of the detailed switching circuit can still be tracked and represented by that of the equivalent circuit model proposed by this paper, which further proves the accuracy of the proposed model. On this basis, the control delay tcntrl and the modulation delay tDPWM of digital control are analyzed, and the small-signal model of z-domain is derived. Furthermore, the magnitude and phase responses respectively are obtained by the small-signal model from the proposed method and the experiment are compared to verify the accuracy of the model. In addition, the method for analyzing the stability of the system by using the equivalent circuit is given, and the theoretical range of parameters of proportional (P) control that can stabilize the digitally controlled Buck converter is further derived. And also, the case that the stability boundary varies with the duty cycle under two different modulations is studied, and the nature of the effect of the modulation on the system is revealed. Then, the influence of certain circuit component parameters on system stability is explored, including the input voltage, the load, the output capacitor, the inductor, and the parasitic parameters. Comparing the stability boundaries of these parameters under different models, the simplicity and accuracy of the equivalent circuit proposed in this paper are proved. Finally, the prototype of the DC-DC converters were constructed to verify the effectiveness of the model and theoretical analysis.
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2023, Vol. 38 
