Question

In an LCR series circuit, at resonance

• A. the current and voltage are in phase
• B. the impedance is maximum
• C. the current is minimum
• D. the quality factor is independent of R
• E. the current leads the voltage by \( \frac{\pi}{2} \)

Hint:

At resonance, an LCR circuit behaves as a purely resistive circuit. This is why the power factor (\( \cos \phi \)) is equal to 1.

Answer 1

The Correct Option is A

Concept: Resonance in a series LCR circuit occurs when the inductive reactance (\( X_L \)) and capacitive reactance (\( X_C \)) are equal in magnitude but opposite in phase, effectively canceling each other out.
• Condition: \( X_L = X_C \implies \omega L = \frac{1}{\omega C} \).
• Impedance (\( Z \)): \( Z = \sqrt{R^2 + (X_L - X_C)^2} \). At resonance, \( Z = R \), which is its minimum value.

Step 1: Analyze the phase relationship.
The phase angle \( \phi \) between voltage and current is given by \( \tan \phi = \frac{X_L - X_C}{R} \). Since \( X_L = X_C \) at resonance, \( \tan \phi = 0 \), meaning \( \phi = 0 \). Thus, current and voltage are in phase.

Step 2: Evaluate the other options.

• Impedance and Current: Since \( Z \) is minimum, the current (\( I = V/Z \)) is maximum, contradicting options B and C.
• Quality Factor (\( Q \)): Defined as \( Q = \frac{1}{R} \sqrt{\frac{L}{C}} \), which clearly depends on \( R \), contradicting option D.