Under a particular geometrical arrangements of impedances of the type resistors and capacitors for the modeling of a transmission line, the voltage and current along the line are known to follow the standard partial differential equation of diffusion. In this work we propose a generalization of this circuit network by considering the non-ideal fractional capacitive element, also known as constant phase element (CPE), as the energy storage component. The CPE’s impedance is given by Zc(s)=1/(Cαsα), where Cα>0 and 0<α⩽1, and offers an extra degree of freedom compared to the ideal capacitor of impedance Z=1/(Cs). This leads to an anomalous time-fractional diffusion equation that we solve considering the Caputo fractional derivative definition for the case of one-dimensional, semi-infinite propagation under a constant voltage excitation at x=0. The voltage and current responses are found analytically in terms of the Fox’s H-function. We discuss the implications of the dispersive nature of the CPE on the time-domain response along the transmission line system, as well as on the frequency-domain input impedance.

中文翻译：

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在半无限分布式电阻恒相元件传输线上


在用于传输线建模的电阻器和电容器类型的阻抗的特定几何布置下，已知沿线路的电压和电流遵循标准的偏微分扩散方程。在这项工作中，我们通过将非理想的分数电容元件（也称为恒相元件 （CPE））视为储能元件，提出了该电路网络的推广。CPE 的阻抗由 Zc（s）=1/（Cαsα） 给出，其中 Cα>0 和 0<α⩽1，与阻抗 Z=1/（Cs） 的理想电容器相比，它提供了额外的自由度。这导致了一个异常的时间分数扩散方程，我们考虑了 Caputo 分数阶导数定义，在 x=0 的恒定电压激励下进行一维半无限传播的情况。电压和电流响应是根据 Fox 的 H 函数进行分析的。我们讨论了 CPE 的色散特性对传输线系统的时域响应以及频域输入阻抗的影响。