Question

In an LCR series AC circuit at resonance, the value of power factor will be …….

Hint:

At {resonance} in an LCR circuit, the impedance equals the resistance (\( Z = R \)), making the {power factor} equal to \( 1 \). This means all supplied power is converted into useful work.

Answer 1

Step 1: Understanding Power Factor 
The power factor (\(\cos \phi\)) in an AC circuit is given by: \[ \cos \phi = \frac{R}{Z} \] where:
- \( R \) is the resistance,
- \( Z \) is the impedance of the circuit. 
Step 2: Condition at Resonance 
- The impedance (\( Z \)) in an LCR circuit is given by: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] where:
- \( X_L = 2\pi f L \) is the inductive reactance,
- \( X_C = \frac{1}{2\pi f C} \) is the capacitive reactance. - At resonance, \( X_L = X_C \), which simplifies the impedance to: \[ Z = R \] 
Step 3: Calculating Power Factor 
\[ \cos \phi = \frac{R}{Z} = \frac{R}{R} = 1 \] Thus, the power factor of an LCR circuit at resonance is \( 1 \).