Question

A series LCR circuit with \( L = \frac{100}{\pi} \) mH, \( C = \frac{10^{-3}}{\pi} \) F and \( R = 10 \, \Omega \) is connected across an AC source of 220 V, 50 Hz supply. The power factor of the circuit would be _____.

Answer 1

Correct Answer: 1

The power factor of the LCR circuit is calculated using the provided values. Initially, the inductive reactance \( X_L \) and capacitive reactance \( X_C \) are determined via the formulas:

• \( X_L = 2\pi f L \)
• \( X_C = \frac{1}{2\pi f C} \)

Given \( f = 50 \) Hz, \( L = \frac{100}{\pi} \) mH, and \( C = \frac{10^{-3}}{\pi} \) F, the reactances are:

• \( X_L = 2\pi \times 50 \times \frac{100}{\pi} \times 10^{-3} = 10 \, \Omega \)
• \( X_C = \frac{1}{2\pi \times 50 \times \frac{10^{-3}}{\pi}} = 10 \, \Omega \)

Subsequently, the circuit's impedance \( Z \) is computed using \( Z = \sqrt{R^2 + (X_L - X_C)^2} \). As \( X_L = X_C \), the impedance simplifies to:

• \( Z = R = 10 \, \Omega \)

The power factor \( \cos\phi \) is the ratio of resistance to impedance:

• \( \cos\phi = \frac{R}{Z} = \frac{10}{10} = 1 \)

Consequently, the circuit's power factor is 1, which is within the acceptable range.