The paper considers a discrete state-space model for transients in a three-level flying capacitor DC–DC converter. A transition matrix is obtained for a pulse width modulation (PWM) period. The matrix elements are expanded into a power series using a selected small parameter. The matrix eigenvalues that determine the natural balancing dynamics transients are presented in the form of power series as well. Four separate transients are constructed based on four possible PWM period initial states (topologies). Inductor current and capacitor voltage transients are found for the voltage source power-up as the arithmetic average of the four separate transients. The discrete solutions are replaced by continuous ones. The resulting transients that are elementary functions of the circuit parameters, PWM period, and voltage reference demonstrate good agreement with the simulation results.

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A 4-level flying capacitor converter (FCC) operation is considered on a base of discrete state-space model. A transition matrix is obtained for a pulse width modulation (PWM) period for small normalised voltage command values [0, 1/3]. The transition matrix elements are expanded into power series by small parameters. The matrix eigenvalues are presented in the form of power series as well. Six separate transients are constructed for six possible initial FCC states on a PWM period. Inductor current and capacitors’ voltage transients are found for the voltage source power-up as the arithmetic average of the six separate transients. Finally, the discrete solutions are replaced by equivalent continuous ones. Simple and accurate formulas for inductor current and capacitors’ voltage transients demonstrate good agreement with simulation results.