The Constant Phase Element is not a generalized capacitor

The constant phase element (CPE) is an ideal circuit element used for modelling e.g. supercapacitors in energy storage systems. Its charge-voltage relation is a convolution in the time domain and a multiplication in the frequency domain. This is the opposite of those for an ordinary time-varying capacitor. A CPE is therefore not a generalization of a capacitor. The current response of the CPE is the classical empirically determined Curie-von Schweidler relation. An ordinary capacitor can be viewed as a Debye dielectric in the limit when time is much larger than the relaxation time. Similarly the CPE can be considered to consist of e.g. a Cole-Cole dielectric, but in the opposite case of time much smaller than the relaxation time.