On the semi-infinite distributed resistor-constant phase element transmission line

Abstract

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 $Z_c\left(s\right)=1/\left(C_{\alpha }{s}^{\alpha }\right)$$Z_c\left(s\right)=1/\left(C_{\alpha }{s}^{\alpha }\right)$, where $C_{\alpha }>0$$C_{\alpha }>0$ and $0<\alpha ⩽1$$0<\alpha ⩽1$, and offers an extra degree of freedom compared to the ideal capacitor of impedance $Z=1/\left(Cs\right)$$Z=1/\left(Cs\right)$. 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$$x=0$. The voltage and current responses are found analytically in terms of the Fox’s $H$$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.