A Derivation of Stored Electromagnetic Field Energies in an Arbitrary Medium

The stored energies of time-harmonic field in a medium are traditionally analyzed by using narrow-band (NB) approximation. The analysis, however, suffers from two problems. Firstly, it involves a specific constitutive relation and is thus only applicable to a specified medium. Secondly, it is incompatible with the time-harmonic field theory for the stored energy expressions derived from it include time-dependent complex envelops rather than phasors. This article aims to solve the problems with the traditional NB analysis. The expressions of the stored energies in an arbitrary medium have been derived by introducing a new NB technique, which is based on the understanding that a pure sinusoidal field may be viewed as the limit of a NB field as its bandwidth shrinks to zero. To facilitate the derivation, an explicit form of the complex envelope is first deduced by expanding the Fourier frequency spectrum of the NB field around the mid-band frequency. The inflow power from the external source can be divided into the sum of the rate of dissipated energy and the rate of increase of stored energy. After introducing the time-domain fields derived from the new NB approach, the rates of dissipated energies and the stored energies can be determined.