Discrete-Time Modeling and Symmetrical Dynamics of V²-Controlled Buck Converters With Trailing-Edge and Leading-Edge Modulations

Through analyzing the operating principles of
$V^{2}$
controlled buck converters, 2-D discrete-time models and accurate switched models of
$V^{2}$
controlled buck converters with trailing-edge and leading-edge modulations are established, respectively. The discrete-time models demonstrate the advantages of simple modeling and less computational complexity. Based on this, the dynamical behaviors of
$V^{2}$
controlled buck converters are revealed through bifurcation diagrams and the boundary equations are derived by using the Jacobian matrix. Meanwhile, the operating regions with varying output capacitor time-constants are divided effectively by the boundary equations. The time-domain waveforms and phase portraits are also obtained through time-domain simulation. The research results indicate that trailing-edge- and leading-edge-modulated buck converters have symmetrical dynamical behaviors, such as symmetrical bifurcation, symmetrical time-domain waveforms, symmetrical phase portraits, and symmetrical period-doubling bifurcation boundaries. A symmetrical axis (SA) or a symmetrical point (SP) can be found in bifurcation diagrams, time-domain waveforms, phase portraits, and period-doubling bifurcation boundaries with varying output capacitor time-constants and duty ratios. By ramp compensation, the operating states of
$V^{2}$
controlled buck converters with trailing-edge and leading-edge modulations could be effectively stabilized from unstable to stable periodic state. Experimental results are further given to verify the theoretical analysis and simulation results.