On the Distributed Resistor-Constant Phase Element Transmission Line in a Reflective Bounded Domain

In this work we derive and study the analytical solution of the voltage and current diffusion equation for the case of a finite-length resistor-constant phase element (CPE) transmission line (TL) network that can represent a model for porous electrodes in the absence of any Faradic processes. The energy storage component is considered to be an elemental CPE per unit length of impedance z c ( s ) = 1/( c α s α ) with constant parameters ( c α , α ) instead of the ideal capacitor of impedance z ( s ) = 1/( c s ) usually assumed in TL modeling. The problem becomes a time-fractional diffusion equation for the voltage that we solve under galvanostatic charging, and derive from it a reduced impedance function of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mi>coth</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:math> , where s n = jω n is a normalized frequency. We also derive the system’s step response, and the distribution function of relaxation times associated with it. The analysis can be viewed and used as a support for the fractal finite-length Warburg model.