Question

In an ac circuit, an inductor, a capacitor and a resistor are connected in series with \( X_L = R = X_C \). Impedance of this circuit is :

• A. Zero
• B. \( R\sqrt{2} \)
• C. \( R \)
• D. \( 2R^2 \)

Hint:

At resonance, the net reactance of the circuit is zero, and the circuit behaves as a purely resistive circuit with minimum impedance \( Z = R \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The impedance \( Z \) of a series RLC circuit is the combined opposition to current flow.
Step 2: Key Formula or Approach:
The formula for series impedance is:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
Step 3: Detailed Explanation:
Given the condition \( X_L = R = X_C \), we can specifically observe that:
\[ X_L = X_C \]
This is the condition for resonance.
Substitute these values into the impedance formula:
\[ Z = \sqrt{R^2 + (R - R)^2} \]
\[ Z = \sqrt{R^2 + 0^2} = R \]
Step 4: Final Answer:
The impedance is equal to \( R \).