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