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 $i=i_{m}sin(\omega t+\varphi )$ where

$$i_m=\frac{v_m}{Z}and\varphi =ta{n}^{-1}\left(\frac{X_C-X_L}{R}\right)$$

Therefore, the instantaneous power p supplied by the source is

$$p=vi=(v_{m}sin\omega t)\times [i_{m}sin(\omega t+\varphi )]$$ $$=\frac{v_m{i}_m}{2}[cos\varphi -cos(2\omega t+\varphi )]$$

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,

$$p=\frac{v_m{i}_m}{2}cos\varphi =\frac{v_m}{\sqrt{2}}\frac{i_m}{\sqrt{2}}cos\varphi$$ $$=VIcos\varphi$$ This can also be written as, $$P=I^{2}Zcos\varphi$$

So, the average power dissipated depends not only on the voltage and current but also on the cosine of the phase angle $\varphi$ between them. The quantity $cos\varphi$ 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 $\varphi =0$, $cos\varphi =1$. 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, $cos\varphi =0$, 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 $\varphi =ta{n}^{-1}(X_C-X_L)/R.$ So, $\varphi$ 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 $\varphi$ = 0. Therefore, cos $\varphi$ = 1 and P = $I^2$ Z = $I^2$ R. That is, maximum power is dissipated in a circuit (through R) at resonance.