Power in AC Circuits

Power can be given by the following relationship

P =VI Watts

Thus the instantaneous power can be given by  be the current fiowing through the impedance, Z, then voltage across the impedance will be

 

The 1st term is a time-varying sinusoidal waveform with twice the frequency of v (t) or i (t) and an average value of zero. Conversely, the 2nd term is a constant quantity and is called the DC level or average value of the power signal, p (t) or average power delivered to the load that is
 

where φ =α - β is the phase angle between v (t) and i (t) .

The waveform p (t) is plotted in Figure drawn below for an arbitrary value of φ where the negative shaded portion of the waveform is the power returned to the source.

The waveform p t is plotted in Figure drawn below for an arbitrary value of φ where the negative shaded portion of the waveform is power returned to source.
 

Figure-Power supplied to generic load impedance.

A purely resistive load (φ= 0°), Equation stated above becomes
 

and the resulting power signal, p (t) is depicted in the Figure (a). In this case, there is no negative portion of the waveform and thus all the power is dissipated in the load.

 

Figure--Power supplied to the purely resistive and reactive loads.

For the purely reactive load, that is Z =±jX, φ =90° and P_av=0 . The power signal waveform is drawn in Figure (b) showing that the power oscillates between source and electric or magnetic field of the load. Therefore the power dissipated in load is zero for the purely reactive load.

In the terms of RMS voltage and current, the average power can be given by

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