Question

For a series RLC circuit $R=X_L=2X_C$. The impedance of the circuit and phase difference (between) $V$ and $i$ will be

• A. $\frac{\sqrt{5}R}{2},{\tan }^{-1}\left(2\right)$
• B. $\sqrt{5}R,{\tan }^{-1}\left(\frac{1}{2}\right)$
• C. $\frac{\sqrt{5}R}{2},{\tan }^{-1}\left(\frac{1}{2}\right)$
• D. $\sqrt{5}{X}_C,{\tan }^{-1}\left(2\right)$

Answer 1

The correct option is C $\frac{\sqrt{5}R}{2},{\tan }^{-1}\left(\frac{1}{2}\right)$

Given:
$X_L=R,X_C=\frac{R}{2}$

$\therefore \tan \varphi =\frac{X_L-X_C}{R}=\frac{R-\frac{R}{2}}{R}=\frac{1}{2}$

$\Rightarrow \varphi ={\tan }^{-1}\left(\frac{1}{2}\right)$

Also $Z=\sqrt{R^2+(X_L-X_C{)}^2}=\sqrt{R^2+\frac{R^2}{4}}=\frac{\sqrt{5}}{2}R$

Hence option $\left(B\right)$ is correct.