Charge-voltage relation for a universal capacitor

Abstract

Most capacitors of practical use deviate from the assumption of a constant capacitance. They exhibit memory and are often described by a time-varying capacitance. It is shown that a direct implementation of the classical relation, Q (t) = CV (t), that relates the charge, Q (t), with the constant capacitance, C, and the voltage, V (t), is not applicable when the capacitance is timevarying. The resulting equivalent circuit that emerges from the substitution of, C, by, C (t), is found to be inconsistent. Since, C (t), leads to a time-variant system, the current, Q, that is obtained from the product rule of the differentiation is not valid either. The search for a solution to this problem led to the expression for the charge, that is given by the convolution of the time-varying capacitance with the first-order derivative of the voltage, as, Q (t) = C (t) * V (t). Coincidentally, this equation also corresponds to the charge-voltage relation for a fractional-capacitor which is probably first reported in this Letter.