Question

In series $\text{LR}$ circuit, $X_L=3R$. Now a capacitor with $X_C=R$ is added in series. Ratio of new to old power factor is

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

Answer 1

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

For $\text{LR}$ circuit:

$Z_i=\sqrt{R^2+X_L^2}$

$=\sqrt{R^2+9R^2}=R\sqrt{10}$

When capacitor is added, for the resulting $\text{LCR}$ circuit:

$Z_f=\sqrt{R^2+(X_L-X_C{)}^2}$

$=\sqrt{R^2+(3R-R{)}^2}=R\sqrt{5}$

Ratio of Power factors

$\frac{\cos {\varphi }_f}{\cos {\varphi }_i}=\frac{\frac{R}{Z_f}}{\frac{R}{Z_i}}=\frac{Z_i}{Z_f}$

$=\frac{\sqrt{10}}{\sqrt{5}}=\sqrt{2}$

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