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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.

V2controlled buck converter with ramp compensation is studied, its two border voltages are derived and one-dimensional discrete-time model is established, upon which the dynamical effect of V2control technique on stability control and mode shift of V2controlled buck converter with ramp compensation is studied by using the bifurcation diagram, Lyapunov exponent spectrum, and parameter space map. The research results verified by simulation and experimental measurements indicate that by utilizing ramp compensation, V2controlled buck converter can not only shift its operation mode from discontinuous conduction mode to continuous conduction mode, but also effectively stabilize its operation state from unstable subharmonic oscillation to stable periodic state.