  The complex Poynting theorem is extended canonically to a time-scale domain
$(t, s)$ by replacing the phasors of time-harmonic fields by the analytic
signals $X(r, t+is)$ of fields $X(r,t)$ with general time dependence. The
imaginary time $s>0$ is shown to play the role of a time resolution scale, and
the extended Poynting theorem splits into two conservation laws: its real part
gives the conservation in $t$ of the scale-averaged active energy at fixed $s$,
and its imaginary part gives the conservation in $s$ of the scale-averaged
reactive energy at fixed $t$. At coarse scales (large $s$, slow time), where
the system reduces to the circuit level, this may have applications to the
theory of electric power transmission and conditioning. At fine scales (small
$s$, fast time) it describes reactive energy dynamics in radiating systems.
