Fractional Order Circuit Elements Derived from Electromagnetism

In this paper, derivations of fractional-order (FO) circuit-element equations from electromagnetism are presented. Whilst many papers are devoted to FO modelling of electrical circuits, there are no strong foundations for such an approach. Therefore, we investigate relations between the FO electromagnetism and the FO circuit theory. Our derivations start from quasi-static (QS) approximations of Maxwell's equations in media with FO constitutive relations. Hence, FO lumped-element equations are derived which are supported by an approximate applicability condition. If the FO capacitor/inductor satisfies the derived condition, then the energy of the magnetic/electric field is negligible in comparison to the energy of the electric/magnetic field in the considered FO element. Then, we demonstrate that FO lumped-element parameters can be derived assuming the equality of electromagnetic power flowing into the circuit element and calculated as the current and voltage product. The obtained results support researchers and engineers employing FO modelling in electrical and electronics engineering.