Which of the following curves represents the variation of impedance `(Z)` with frequency `f` in series `LCR` circuit?

Options

A.
B.
C.
D.

Explanation

To determine the variation of impedance \( Z \) with frequency \( f \) in a series \( LCR \) circuit, we can follow these steps:

### Step 1: Understand the Impedance Formula
The impedance \( Z \) of a series \( LCR \) circuit is given by the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where:
- \( R \) is the resistance,
- \( X_L = \omega L = 2\pi f L \) is the inductive reactance,
- \( X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} \) is the capacitive reactance.

### Step 2: Express Impedance in Terms of Frequency
Substituting the expressions for \( X_L \) and \( X_C \) into the impedance formula, we have:
\[ Z = \sqrt{R^2 + (2\pi f L - \frac{1}{2\pi f C})^2} \]

### Step 3: Analyze the Behavior at Different Frequencies
1. **At \( f = 0 \)**:
- \( X_L = 0 \)
- \( X_C \) approaches infinity.
- Therefore, \( Z \) approaches infinity.

2. **At Resonant Frequency \( f = f_R \)**:
- At resonance, \( X_L = X_C \) which implies \( 2\pi f_R L = \frac{1}{2\pi f_R C} \).
- Thus, \( Z = R \) (minimum impedance).

3. **At \( f \to \infty \)**:
- \( X_L \) approaches infinity.
- \( X_C \) approaches 0.
- Therefore, \( Z \) approaches infinity.

### Step 4: Sketch the Impedance vs Frequency Curve
- Starting from infinity at \( f = 0 \), the impedance decreases as frequency increases until it reaches a minimum value \( R \) at the resonant frequency \( f_R \).
- After the resonant frequency, as frequency continues to increase, the impedance increases again towards infinity.

### Step 5: Identify the Correct Curve
From the analysis, the curve that represents this behavior will start from infinity, decrease to a minimum at the resonant frequency, and then increase back to infinity. This corresponds to a U-shaped curve.

### Conclusion
The correct option that represents the variation of impedance \( Z \) with frequency \( f \) in a series \( LCR \) circuit is **option C**.

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