Origin of the Curie-von Schweidler law and the fractional capacitor from time-varying capacitance

Abstract

Most dielectrics of practical purpose exhibit memory and are described by the century old Curievon Schweidler law. Interestingly, the Curie-von Schweidler law is the motivation behind an unconventional circuit component called fractional capacitor which due to its power-law properties is extensively used in the modeling of complex dielectric media. But the empirical nature of the Curie-von Schweidler law also plagues the use of the fractional capacitor. Here, I derive the Curie-von Schweidler law from a series combination of a resistor and a capacitor with a linearly time-varying capacitance. This may possibly be its first derivation from physical principles. However, this required a modification of the traditional charge-voltage relation of a capacitor to account for the time-varying capacitance, that is also supported with suitable arguments. Consequently, the parameters of the Curie-von Schweidler law and the fractional capacitor gain physical interpretation. The Debye response of dielectrics is shown to emerge naturally from the limiting case of the power-law response at short time scales. The obtained results are validated by matching them with the published experimental reports.

Charge-voltage relation for a universal capacitor

“…But that did not yield any physical interpretation of its parameters, τ and α, which was however provided recently in Ref. [18]. In addition to that interpretation, the finding of the charge-voltage relation for a fractional capacitor reported here, should further pave way in the emerging field of fractional-order circuits and systems [26].…”

“…Furthermore, in appropriate limiting conditions, fractional derivatives asymptotically converge to the integer-order derivatives. In recent years, a connection between the fractional derivatives and the physics of complex media has also been established [14,[16][17][18]. It is worth noting that the expressions for, Q C and I C , in Eqs.…”

Comment on "Nonlinear charge--voltage relationship in constant phase element" [AEU-Int. J. Electron. Commun. 117, 153104 (2020)]

“…Second, it has been already established in Ref. [8] that the Eq. (3) when interpreted in the light of fractional calculus describes the century-old dielectric relaxation law, the Curie-von Schweidler law [9][10][11], with an experimentally verifiable interpretation which also extends to the fractional-order that appears in the expression for current of a fractional capacitor.…”