Charge-voltage relation for a universal capacitor

TL;DR

This paper introduces a new charge-voltage relation for capacitors with time-varying capacitance, addressing inconsistencies in classical models and linking it to fractional calculus and dielectric response.

Contribution

It proposes a convolution-based charge-voltage relation for universal capacitors, extending the understanding of dielectric behavior and fractional capacitors.

Findings

Classical relation $Q=CV$ is invalid for time-varying capacitance.

The new relation uses convolution of capacitance with voltage derivative.

Connects universal capacitor behavior with fractional calculus and dielectric laws.

Abstract

Most capacitors do not satisfy the conventional assumption of a constant capacitance. They exhibit memory which is often described by a time-varying capacitance. It is shown that the classical relation, $Q\left(t\right)=CV\left(t\right)$, that relates the charge, $Q$, with the capacitance, $C$, and the voltage, $V$, is not applicable for capacitors with a time-varying capacitance. The expression for the current, $dQ/dt$, that is subsequently obtained following the substitution of $C$ by $C\left(t\right)$ in the classical relation corresponds to an inconsistent circuit. In order to address the inconsistency, I propose a charge-voltage relation according to which the charge on a capacitor is expressed by the convolution of its time-varying capacitance with the first-order time-derivative of the applied voltage, i.e., $Q\left(t\right)=C\left(t\right)\ast dV/dt$. This relation corresponds to the universal capacitor which is also known as the fractional capacitor among the fractional calculus community. Since the fractional capacitor has an inherent connection with the universal dielectric response that is expressed by the century old Curie-von Schweidler law, the finding extends to the study of dielectrics as well.