Nonlinear charge-voltage relationship in constant phase element

The constant phase element (CPE) or fractional-order capacitor is an electrical device that has an impedance of the form Z(s)=1/Cαsα, where Cα is the CPE parameter and α is a fractional dispersion coefficient of values between 0 and 1. Here we show that in the time-domain the classical linear charge-voltage relationship of ideal capacitors, q=C·v, is not valid for CPEs. In fact the relationship is nonlinear and can be expressed as q=C(v;Cα,α)·v. We verify our findings using (i) circuit simulations of an integer-order emulator of a CPE, and (ii) experimental results from a commercial supercapacitor as well as an in-house fabricated device based on carbon film materials. © 2020 Elsevier GmbH