Consider a general LCR AC circuit with the instantaneous voltage (of the source) and the instantaneous current expressed as

`E = E_0 sin(omega t)` and `i=i_0sin(omega t+phi)` where `i_0 =E_0/Z ,` respectively.

The instantaneous power supplied by the AC voltage source is defined to be

`P = Ei = E_0i_0sin(omega t) sin (omega t+phi)`

Therefore the average power delivered over a full cycle is given by

`vecP =E_0i_0[(int_0^T(omega t)sin(omega t+phi))/T]` , where `T = (2 pi)/omega`

`vecP =(E_0i_0)/T int_0^T sin(omega t)[sin(omega t)cos phi + cos(omega t)sin phi]dt`

`vecP =(E_0i_0)/T int_0^T [sin^2(omega t)cos phi + sin(omega t)cos(omega t)sin phi]dt`

`vecP =(E_0i_0)/T[ cos phi int_0^Tsin^2(omega t)dt+sin phi int_0^T sin(omega t)cos(omega t)d t]`

Average value of the function `sin^2(omega t)` is equal to `(1/2)` as seen before, whereas the average value of the function `Sin(omega t)cos(omega t)` is zero, therefore, average power

`=> =(E_0i_0)/2 cos phi = E_(rms) i_(rms) cos phi`

`Cos phi = R/Z` from the phasor diagram is called the Power factor for the given circuit and as `-pi/2 < phi < pi/2` , Power factor can have values ranging from zero (purely capacitive or purely inductive circuits) to 1 (purely resistive circuits, or a circuit operating at it's resonant frequency `f_0 =1/(2pisqrt(LC))` when a given circuit is driven by an AC source operating at this resonant frequency, the power drawn from the source is maximum, since

`Cos phi =1` and `Z` is minimum or peak current is maximum

Note that the average power over one complete cycle is also equal to the average power for a long time.