Constant phase element (CPE)

A constant phase element is the circuit element used in place of an ideal capacitor to model real electrode interfaces, whose impedance magnitude falls with frequency as a non-integer power. Its impedance is Z = 1 / (Q (jw)^a), where a between 0 and 1 sets the phase angle (a = 1 is an ideal capacitor, a = 0 a resistor) and Q is a magnitude coefficient. Empirically most microelectrodes show a around 0.8 to 0.95, meaning the phase is nearly but not exactly capacitive across a broad band.

The CPE is usually attributed to non-idealities: surface roughness, a distribution of relaxation times, porous coatings, and inhomogeneous current density across the contact. Because a is non-integer, a CPE cannot be reduced to a finite network of ideal R and C elements, which is why fitting electrode impedance spectra with Randles-plus-CPE models yields the compact descriptors (Q, a) reported when comparing electrode materials and tracking chronic changes.