Power definition in sinusoidal steady-state conditions
Fig.1
A voltage generator connected to a generic network supplying an instantaneous voltage:
(1)
is conveniently described in harmonic regime by the phasor:
(2)
and the relationship between phasor and instantaneous scalar quantity is:
(3)
The instantaneous power erogated by the generator is:
(4)
where the first term represents the fraction of power that is varying (sinusoidally) with twice as the fundamental frequency and whose average is zero over a period.
The second term represents the fraction of power that is constant over a period i.e. the average value of the erogated power.
Therefore if the complex power is defined as:
(5)
where from the observation above
(6)
is the time average erogated power