Chin. Phys. B, 2013, Vol. 22(3): 030506    DOI: 10.1088/1674-1056/22/3/030506
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# Transfer function modeling and analysis of the open-loop Buck converter using the fractional calculus

Wang Fa-Qiang (王发强), Ma Xi-Kui (马西奎)
State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical Engineering, Xi'an Jiaotong University, Xi'an 710049, China
Abstract  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 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 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.
Keywords:  Buck converter      small signal equivalent circuit model      fractional calculus      transfer function
Received:  20 May 2012      Revised:  08 October 2012      Accepted manuscript online:
 PACS: 05.45.-a (Nonlinear dynamics and chaos) 84.30.Jc (Power electronics; power supply circuits) 45.10.Hj (Perturbation and fractional calculus methods)
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 51007068), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100201120028), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2012JQ7026), the Fundamental Research Funds for the Central Universities of China (Grant No. 2012jdgz09), and the State Key Laboratory of Electrical Insulation and Power Equipment of China (Grant No. EIPE12303).
Corresponding Authors:  Wang Fa-Qiang     E-mail:  faqwang@mail.xjtu.edu.cn