Conservation of reactive electromagnetic energy in reactive time

TL;DR

This paper extends the complex Poynting theorem to a combined time-scale domain, providing a conservation law for reactive energy in reactive time, and offers a new interpretation of reactive power units.

Contribution

It introduces a complex time-scale extension of the CPT, establishing conservation laws for reactive energy and clarifying the physical meaning of reactive power and reactive time.

Findings

Conservation laws for reactive energy in reactive time are derived.

Reactive power units are interpreted as Joules per reactive second.

A new 'complex radiation impedance density' concept is introduced.

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.