Question

In an a.c. circuit the resistance (R), inductance (L) and capacitance (C) are connected in series. The impedance (Z) of the circuit would be:

• A. \(Z = \sqrt{R^2 + X_L^2 - X_C^2}\)
• B. \(Z = \sqrt{R^2 + (X_L - X_C)^2}\)
• C. \(Z = \sqrt{X_L^2 + (R - X_C)^2}\)
• D. \(Z = R + (X_L - X_C)\)

Hint:

Series RLC impedance: \(Z = \sqrt{R^2 + (X_L - X_C)^2}\).
Net reactance is the difference.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Impedance in a series RLC circuit is calculated using a specific formula.
It combines resistance and reactance.

Step 2: Key Formula or Approach:
\(Z = \sqrt{R^2 + (X_L - X_C)^2}\).

Step 3: Detailed Explanation:
For a series RLC circuit:
- Resistance = R
- Inductive reactance = X\(_L\) = \(\omega L\)
- Capacitive reactance = X\(_C\) = \(1/\omega C\)
The net reactance is (X\(_L\) - X\(_C\)).
Impedance = \(\sqrt{R^2 + (X_L - X_C)^2}\).
Thus, the correct formula is option (B).

Step 4: Final Answer:
The impedance is \(Z = \sqrt{R^2 + (X_L - X_C)^2}\).
Hence, the correct option is (B).