Capacitor theory

A new linear capacitor model is proposed. It is based on Curie´s empirical law of 1889 which states that the current through a capacitor is i(t)=U₀/(h₁tⁿ), where h₁ and n are constants, U₀ is the dc voltage applied at t=0, and 0<n<1. It implies that the insulation resistance is R_i(t)=h₁tⁿ, that is, it increases almost in proportion to time since n nearly equals 1.0. For a general input voltage u(t) the current is i(t)=Cdⁿu(t)/dtⁿ where use is made of the fractional derivative, defined by means of its Laplace transform. The model gives rise to a capacitor impedance Z(iω=1/[(iω)ⁿC], with a loss tangent that is independent of frequency. The model has other properties: the capacitor `remembers´ voltages it has been subjected to earlier, dielectric absorption is an example of this. Capacitor problems require solving integral equations. The model is dynamic, i.e. electrostatic processes are simply slow dynamic processes. The model is applied to several problems that cannot be treated with conventional theory