Transfer function modeling and analysis of the open-loop Buck converter using the fractional calculus∗

Based on the fact that the real inductor and the real capacitor are fractional order in nature and the fractional calculus, the transfer function modeling and analysis of the open-loop Buck converter in a continuous conduction mode (CCM) operation are carried out in this paper. The fractional order small signal model and the corresponding equivalent circuit of the open-loop Buck converter in a CCM operation are presented. The transfer functions from the input voltage to the output voltage, from the input voltage to the inductor current, from the duty cycle to the output voltage, from the duty cycle to the inductor current, and the output impedance of the open-loop Buck converter in CCM operation are derived, and their bode diagrams and step responses are calculated, respectively. It is found that all the derived fractional order transfer functions of the system are influenced by the fractional orders of the inductor and the capacitor. Finally, the realization of the fractional order inductor and the fractional order capacitor is designed, and the corresponding PSIM circuit simulation results of the open-loop Buck converter in CCM operation are given to confirm the correctness of the derivations and the theoretical analysis.