The power factor of an RL circuit is \( \displaystyle\frac{1}{2} \). If the frequency of a.c. is doubled, the power factor will be:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{1}{\sqrt{13}} \)
The power factor of a series RL circuit is \( \cos\phi = R/Z \) with \( Z = \sqrt{R^{2} + X_L^{2}} \), and the inductive reactance \( X_L = 2\pi fL \) is proportional to the frequency.
Initially \( \cos\phi = \frac{1}{2} \):
\( \displaystyle \frac{R}{\sqrt{R^{2} + X_L^{2}}} = \frac{1}{2} \Rightarrow R^{2} + X_L^{2} = 4R^{2} \Rightarrow X_L = \sqrt{3}\,R \)
When the frequency is doubled, \( X_L' = 2X_L = 2\sqrt{3}\,R \), so \( X_L'^{2} = 12R^{2} \).
\( \displaystyle \cos\phi' = \frac{R}{\sqrt{R^{2} + 12R^{2}}} = \frac{1}{\sqrt{13}} \)