Question

If for the given circuit capacitive reactance is two times of inductive reactance, and resistance $R$ is equal to inductive reactance then, the power factor of the circuit is,

• A. $1$
• B. $0$
• C. $\frac{1}{2}$
• D. $\frac{2}{\sqrt{5}}$

Answer 1

The correct option is D $\frac{2}{\sqrt{5}}$

Given:

$X_C=2X_L$ and $R=X_L$

$\therefore X_C=2X_L=2R$

The impedance of the circuit is given by:

$Z=\frac{1}{\sqrt{\frac{1}{R^2}+{(\frac{1}{X_L}-\frac{1}{X_C})}^2}}$

$=\frac{1}{\sqrt{\frac{1}{R^2}+{(\frac{1}{R}-\frac{1}{2R})}^2}}=\frac{2R}{\sqrt{5}}$

Therefore, the power factor of the circuit is given by,

$\cos \varphi =\frac{Z}{R}=\frac{2R}{\sqrt{5}}\times \frac{1}{R}=\frac{2}{\sqrt{5}}$

Hence, option $\left(D\right)$ is correct.