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

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 $v(t)=\lambda t^2$. 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.