中文摘要：

近年来，分数阶微积分在多个科学研究领域得到广泛的研究，但目前涉及磁能耦合、电能转换等方面的分数阶特性研究还较少。特别是随着无线电能传输的关注度越来越高，更有必要对通过磁能进行的电能转换的分数阶系统进行研究。因此，本文对互感电路及变压器模型在分数阶特性下的各种性能进行了分析。首先介绍了传统分数阶电容和电感串并联电路的阻抗分析，接着沿用该分析方法，研究了分数阶互感电路及变压器模型的等效电路阻抗矩阵参数的阻抗特性，得出其相位特性，并分析了不同阻抗特性下，阻抗频率、幅值与电感阶次的关系。从分析中可知，分数阶互感电路及变压器模型的电路参数增加了自感和互感的阶次，因此整个模型设计自由度更高，更具有一般性。本文的研究将有助于分数阶无线电能传输系统的发展。

英文摘要：

Fractional calculus has been applied in numerous scientific fields recently. But few researches have analyzed the fractional-order characteristics of magnetic coupling and electrical energy conversion. Hence in this paper, fractional-order characteristics of the mutual coupled circuit and transformer model are analyzed. First, impedance analysis of a fractional LβCα series circuit is introduced as an example. Then, the same analytical method is been used to analyze the impedance characteristics of the impedance matrix parameters of the fractional- order mutual coupled circuit and transformer model. As a result, phase characteristics as well as relationships between impedance frequencies and inductor orders are obtained. Compared with traditional circuits, the fractional-order mutual coupled circuit and transformer model has employed self-inductance and mutual-inductance of fractional orders, which have increased the design degree of freedom.