7.7 Power in AC Circuit: The Power Factor

We have seen that a voltage \( v = v_{m} sin \omega t \) applied to a series RLC circuit drives a current in the circuit given by \( i = i_{m} sin(\omega t + \phi) \) where

\[ i_{m} = \frac{v_{m}}{Z} and \phi = tan^{-1} \left( \frac{X_{C} - X_{L}}{R} \right) \]

Therefore, the instantaneous power p supplied by the source is

\[ p = vi = (v_{m}sin \omega t) \times [ i_{m} sin(\omega t + \phi)] \] \[ = \frac{v_{m} i_{m}}{2} [cos \phi - cos (2 \omega t + \phi)] \]

The average power over a cycle is given by the average of the two terms in R.H.S. of Eq. (7.37). It is only the second term which is time-dependent. Its average is zero (the positive half of the cosine cancels the negative half). Therefore,

\[ p = \frac{v_{m} i_{m}}{2} cos \phi = \frac{v_{m}}{\sqrt 2} \frac{i_{m}}{\sqrt{2}} cos \phi \] \[ = VI cos \phi \] This can also be written as, \[ P = I^2 Z cos \phi \]

So, the average power dissipated depends not only on the voltage and current but also on the cosine of the phase angle \(\phi \) between them. The quantity \(cos \phi \) is called the power factor. Let us discuss the following cases:

Case (i) Resistive circuit: If the circuit contains only pure R, it is called resistive. In that case \(\phi = 0\), \(cos \phi = 1\). There is maximum power dissipation.

Case (ii) Purely inductive or capacitive circuit: If the circuit contains only an inductor or capacitor, we know that the phase difference between voltage and current is \(\pi /2\). Therefore, \(cos \phi = 0\), and no power is dissipated even though a current is flowing in the circuit. This current is sometimes referred to as wattless current.

Case (iii) LCR series circuit: In an LCR series circuit, power dissipated is given by Eq. (7.38) where \( \phi = tan^{-1} (X_{C} - X_{L})/R. \) So, \( \phi \) may be non-zero in a RL or RC or RCL circuit. Even in such cases, power is dissipated only in the resistor.

Case (iv) Power dissipated at resonance in LCR circuit: At resonance \(X_{c}\) – \(X_{L}\) = 0, and \(\phi \) = 0. Therefore, cos \(\phi \) = 1 and P = \(I^{2}\) Z = \(I^{2}\) R. That is, maximum power is dissipated in a circuit (through R) at resonance.