Question

In an electrical circuit R, L, C and an a.c.voltage source are all connected in series.When L is removed from the circuit, the phase difference between the voltage and the currentin the circuit is $\pi /3$. If instead, C is removed from the circuit the phase difference is again $\pi /3$. The power factor of the circuit is

• A. 1
• B. $\sqrt{3}/2$
• C. 1/2
• D. $\frac{1}{\sqrt{2}}$

Answer 1

The correct option is A 1

The Power factor of a circuit is defined as:

$cos\varphi =\frac{Z}{R}$

Here, $Z$ is the impedance of the circuit and $R$ is the resistance connected.

In the first case, the inductor is removed from the circuit. Therefore, the phase difference in the voltage and current due to resistance and capacitor will be:

$tan{\varphi }_1=\frac{X_C}{R}$

$\Rightarrow tan\frac{\pi }{3}=\frac{X_C}{R}$

In the second case, the Capacitor is removed. So, the phase difference between the current and voltage is:

$tan{\varphi }_2=\frac{X_L}{R}$

$\Rightarrow tan\frac{\pi }{3}=\frac{X_L}{R}$

Since the phase difference is the same in both cases, the impedance of the capacitor and inductor will be the same.

$X_L=X_C$

Therefore, the total impedance of the circuit will be:

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

The power factor of the whole circuit is:

$cos\varphi =\frac{R}{R}⟹cos\varphi =1$

Option $\left(A\right)$ is correct.