Energy-based interpretation of the dispersion coefficient of the constant phase element

TL;DR

The paper tackles the unclear physical meaning of the CPE dispersion coefficient $α$ in impedance analysis. It introduces an energy-based interpretation by mapping the CPE to an infinite RC-network representation and formulating a detailed energy balance among input, stored, and dissipated energies. Across step, ramp, and quadratic excitations, the authors derive expressions showing that energy ratios depend only on $α$, enabling $α$ to be recovered from experimental energy measurements and interpreted as an energy division parameter. This thermodynamic perspective provides a practical framework for analyzing CPE-like behavior in supercapacitors, batteries, and related electrochemical devices, bridging impedance spectroscopy with energy conservation principles and facilitating improved device modeling.

Abstract

The dispersion coefficient of the constant phase element (CPE) is typically treated as an empirical fitting parameter in the analysis of impedance spectroscopy data, with no clear physical meaning. Here we seek to establish a energy-based interpretation for this coefficient by linking it to the ratio of the dissipated or stored energy in the CPE relative to that supplied by the input source. Using the $RC$ network equivalency of a CPE, we decompose the total input energy into a contribution stored in the capacitive modes and another dissipated in the resistive modes. Analytical expressions are derived for three test examples: (i) a constant voltage, (ii) a voltage ramp, and (iii) a quadratic input of the form $v(t)=λt^2$. In all cases we found that the ratios of any two of these energy quantities reduce to pure functions of the dispersion coefficient of the CPE, independent of excitation amplitude or material parameters. This result provides a new perspective of the CPE's dispersion coefficient from a thermodynamic/energetic basis, with direct implications for supercapacitor characterization, battery modeling, as well as for the analysis of other electrochemical systems and devices exhibiting the CPE behavior.

Figures (4)

• Figure 1: Schematic diagram representing the different energy metrics in a constant phase element (CPE); the source input energy at a time $\tau$ ($E_{{in}}(\tau)$) is equal to the sum of the stored energy in the capacitive part of the CPE ($E_{s} (\tau)$) and the dissipated energy in its resistive part ($E_{d}(\tau)$). The inset shows an $RC$ equivalent circuit for a CPE consisting an infinite number of series $R_{\omega} C_{\omega}$ branches connected in parallel, wherein each branch contributes with an elemental or modal stored energy $e_{\omega}(t)$ (in the capacitor $C_{\omega}$), and generates an elemental dissipated power $p_{\omega}(t)$ (due to the resistor $R_{\omega}$)
• Figure 2: Energy metrics in a CPE of different dispersion coefficient $\alpha$ under a constant voltage step excitation; the solid line plots are for Eqs \ref{['eq:EsEin']}-\ref{['eq:EsEd']} and the dotted plots for the CPE emulator based on the distribution function of relaxation times given by Eq. \ref{['eq:DFRT']}
• Figure 3: Energy metrics in a CPE of different dispersion coefficient $\alpha$ under a voltage ramp excitation; the solid line plots are for Eqs. \ref{['eq:EsEinRampV']}-\ref{['eq:EsEdRampV']} and the dotted plots for the CPE emulator based on the distribution function of relaxation times given by Eq. \ref{['eq:DFRT']}
• Figure 4: Energy metrics in a CPE of different dispersion coefficient $\alpha$ under a voltage excitation of the form $v(t)=\lambda t^2$; the solid line plots are for Eqs. \ref{['eq:EsEinV2']}-\ref{['eq:EsEdV2']} and the dotted plots for the CPE emulator based on the distribution function of relaxation times given by Eq. \ref{['eq:DFRT']}