Stability Mechanism of the Load Converter Under the Switching Ripple Interaction

In a cascaded power converter, the switching ripple interaction between the source converter and the load converter reshapes the voltage ripple at the common coupling point (CCP). When the equivalent series resistance (ESR) of the output capacitor is large, the output voltage ripple of the source converter significantly affects the stability of the load converter. This study examines the effect of the switching ripple interaction on the stability of the load converter using the cascaded valley voltage-ripple (VVR) controlled Buck converter as an example. The operation principle of the VVR-controlled Buck converter is detailed in its schematic diagram and steady-state operation waveforms. When the source converter possesses a large output capacitor ESR, the voltage ripple at the CCP is determined by the voltage across the output capacitor ESR of the source converter. The characteristics of the voltage ripple at CCP are analyzed when the input current ripple of the load converter is fed forward to the output capacitor of the source converter. Based on the relationship between the duty cycles of the source converter and the load converter, the voltage ripple at CCP has three types of waveforms. Typical circuit parameters of the cascaded VVR-controlled Buck converter are selected for circuit simulation to demonstrate the stability effect of the voltage ripple at CCP on the load converter. Based on the typical circuit parameters, two sets of circuit simulation results are obtained. When the load converter is supplied by a constant input voltage (in standalone operation), it operates in a stable period-1 state, whereas when the load converter is supplied by the output voltage of the source converter (in cascaded operation), it operates in an unstable period-2 state. Both the inductance of the source and load converters (<italic>L</italic>₁ and <italic>L</italic>₂) can affect the characteristics of voltage ripple at CCP and are selected as bifurcation parameters to analyze the bifurcation behaviors of the load converter. The bifurcation diagrams with variations of <italic>L</italic>₁ and <italic>L</italic>₂ are presented in this study. As <italic>L</italic>₁ increases from 2 to 20 μH, the operation state of the load converter transitions from a chaotic state to an unstable period-2 state through a border-collision bifurcation at <italic>L</italic>₁ = 6.6 μH and then to a stable period-1 state via inverse period-doubling bifurcation at <italic>L</italic>₁ = 13.2 μH. As <italic>L</italic>₂ increases from 40 to 100 μH, the operation state of the load converter shifts from a stable period-1 state to an unstable period-2 state via period-doubling bifurcation at <italic>L</italic>₂ = 53 μH. The stability of the VVR-controlled Buck converter in standalone operation is independent of inductance; however, when the converter is in cascaded operation, its stability relates to the inductance of the source and load converters. When both the source and the load converters operate in continuous conduction mode (CCM), the cascaded VVR-controlled Buck converter has four switching states. A map between the state variables at the initial moment and the termination moment of the <italic>m</italic>th switch state is established by solving the state equation of the <italic>m</italic>th (<italic>m</italic> = 1, 2, 3, 4) switching state. Under the switching ripple interaction, the cascaded VVR-controlled Buck converter exhibits three switching state sequences in one switching cycle. With consideration of these three switching state sequences, an approximate discrete-time map model of the cascaded VVR-controlled Buck converter is developed. Based on this model, the corresponding Jacobian matrix around the fixed point and its eigenvalues are derived. The stability of the cascaded VVR-controlled Buck converter can be determined by monitoring the eigenvalues of the Jacobian matrix with variations in the chosen circuit parameters. As <italic>L</italic>₁ decreases or <italic>L</italic>₂ increases, the eigenvalue <italic>λ</italic>₁ exits the unit circle via −1 at <italic>L</italic>₁ = 13 μH or <italic>L</italic>₂ = 52 μH, while the other three eigenvalues, <italic>λ</italic>₂, <italic>λ</italic>₃, and <italic>λ</italic>₄, remain inside the unit circle, resulting in the occurrence of period-doubling bifurcation. Accordingly, typical eigenvalues for different <italic>L</italic>₁ and <italic>L</italic>₂ are listed in this study. The results of eigenvalue analysis are consistent with those of bifurcation analysis. The results showed that increasing the inductance of the source converter enhances the stability of the load converter, whereas increasing the inductance of the load converter diminishes its stability. Circuit simulation results for different <italic>L</italic>₁ and <italic>L</italic>₂ are presented to visually demonstrate the effects of <italic>L</italic>₁ and <italic>L</italic>₂ on the stability of the load converter. Finally, the experimental prototype of the cascaded VVR-controlled Buck converter is established. Based on the circuit simulation results, corresponding experimental results are provided. The experimental results confirm the accuracy of the simulation analysis and theoretical analysis. These results provide theoretical guidance for the parameter design of load converters in cascaded power converters. The research method can also be applied to load converters with other control methods, such as peak-current-mode control.