On the Distributed Resistor-Constant Phase Element Transmission Line in a Reflective Bounded Domain
Anis Allagui, Enrique H. Balaguera, Chunlei Wang
Abstract
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.