Why supercapacitor follows complex time-dependent power law (t∝) and does not obey normal exponential (e-t/(RC)) rule?

It is shown that the self-discharge characteristics of the supercapacitor cannot be explained using the existing fractional-order model since there is no self-leakage path of the said model. The basis of this fractional-order model was to explain empirically why the supercapacitor discharges using a complex time-dependent power law (t(alpha)) and does not obey the usual discharge method, like e (-t/(RC)). The conventional capacitor models have a series resistance and a leakage-parallel resistance. The fractional-order models have very little physical reasoning behind it. In this paper, we have calculated an expression using a free-electron "Drude Model" and showed that parallel leakage resistance of the supercapacitor depends on the value of the current going through it. This concept causes complex time-dependent power-law behavior. We have taken the data from the literature, where researchers came across the idea of fractional-order model and applied our new concept of the "Drude Model" on those data from literature and found that the dependence of leakage resistance of the supercapacitor on the current passing through that leakage-parallel resistor. We have also done measurements in our laboratory at the centre of Innovation, using NIPPON supercapacitors and found the leakage parallel resistance is also current dependent, which leads to this interesting complex time-dependent power law.