Question

An inductor of inductive reactance \( R \), a capacitor of capacitive reactance \( 2R \), and a resistor of resistance \( R \) are connected in series to an AC source. The power factor of the series LCR circuit is?

• A. \( \frac{1}{\sqrt{2}} \)
• B. \( \frac{1}{\sqrt{3}} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{2} \) \

Hint:

The power factor of an LCR circuit is given by \( \cos \phi = \frac{R}{Z} \). To compute impedance, use \( Z = \sqrt{R^2 + (X_L - X_C)^2} \).

Answer 1

The Correct Option is A

Step 1: Power Factor Formula for an LCR Circuit
The power factor (\( \cos \phi \)) of 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: Compute the Impedance \( Z \)
The impedance of a series LCR circuit is: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] Given: - Inductive reactance: \( X_L = R \), - Capacitive reactance: \( X_C = 2R \). The net reactance: \[ X = X_L - X_C = R - 2R = -R \] \[ Z = \sqrt{R^2 + (-R)^2} \] \[ Z = \sqrt{R^2 + R^2} = \sqrt{2R^2} = R\sqrt{2} \] Step 3: Compute the Power Factor
\[ \cos \phi = \frac{R}{Z} = \frac{R}{R\sqrt{2}} \] \[ = \frac{1}{\sqrt{2}} \] Thus, the power factor of the circuit is \( \frac{1}{\sqrt{2}} \). \bigskip