Limit capacitance of the constant phase element

The constant-phase element (CPE) is a universal electrical model widely used to describe the intricate nature of a multitude of materials and processes under real-world conditions. The physical interpretation of the corresponding anomalous phenomenology is a challenging task, which traditionally relies on calculating an effective capacitance in the sense of a classical charge accumulation. However, a picture of this electrical element is not yet complete for cases of practical interest, and many questions remain open in relation to the intrinsic characteristics that makes it “unphysical” at long time scales. In this work, we derive mathematical formulas for estimating the limit capacitance of the CPE associated with surface and normal time-constant distributions. For this purpose, we obtain the transient responses, in term of multi-exponential relaxation patterns, attributable to the charge processes of micro-capacitances that constitute the “macroscopic CPE” with a dynamical behavior described in terms of the Mittag-Leffler function. As both transient dynamics can be considered negligible in practice from a certain time instant, we subsequently find the limit capacitance from a direct comparison of both steady-state times in the style of CPE reference works. Simulations are used to show that the obtained limit capacitance yields reasonable values for cases of multidisciplinary interest. Our study contributes to the advanced understanding of the pervasive presence of the CPE in natural and engineering contexts, shedding light on the problem of infinite charge and energy in complex systems.