Question

For series L-C-R circuit, resonant frequency is ${\omega }_0$. At ${\omega }_1<{\omega }_0$, impedance of the circuit is $R\sqrt{2}$, then which of the following is correct about the phase difference between $i$ and $V$ at ${\omega }_1$?

• A. $i$ lags $V$ by $\frac{\pi }{2}$
• B. $i$ leads $V$ by $\frac{\pi }{4}$
• C. $i$ lags $V$ by $\frac{\pi }{4}$
• D. $i$ leads $V$ by $\frac{\pi }{2}$

Answer 1

The correct option is B $i$ leads $V$ by $\frac{\pi }{4}$

At ${\omega }_0$, the reactance of the circuit is zero. At ${\omega }_1$, which is less than ${\omega }_0$, the reactance of the capacitor dominates that of the inductor. If the net impedance is $R\sqrt{2}$, and the resistance is $R$, then $\cos \varphi =\frac{R}{Z}=\frac{R}{R\sqrt{2}}=\frac{1}{\sqrt{2}}$
$\Rightarrow \varphi =+{45}^{\circ }or-{45}^{\circ }$.
Since, reactance of the capacitor is dominating that of the inductor, $\varphi =+{45}^{\circ }$