Conservation of reactive electromagnetic energy in reactive time

Abstract

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 width $\Delta t\sim \pm s$. This completes the time-harmonic CPT, whose imaginary part is generally supposed to be vaguely `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.