Conservation of reactive EM energy in reactive time
The complex Poynting theorem (CPT) is extended to a canonical time-scale domain (t, s). Time-harmonic phasors are replaced by the positive-frequency parts of general fields, which extend analytically to complex time t + is, with s > 0 interpreted as a time resolution scale. The real part of the extended CPT gives conservation in t of a time-averaged field energy, and its imaginary part gives conservation in s of a time-averaged reactive energy. In both cases, the averaging windows are determined by a Cauchy kernel of duration Δt ~ ±s. This completes the time-harmonic CPT, whose imaginary part is vaguely supposed to be “related to” reactive energy without giving a conservation law, or even an expression, for the latter. The interpretation of s as reactive time, tracking the leads and lags associated with stored capacitative and inductive energy, gives a simple explanation of the volt-ampere reactive (var) unit measuring reactive power: a var is simply one Joule per reactive second. The related complex radiation impedance density is introduced to represent the field's local reluctance to radiate.