Which increase in frequency of an AC supply , the impedance of an L-C-R series circuit

Options

A. remians constant
B. increases
C. decreases
D. decreases at first, becomes minimum and then increases

Answer

To solve the problem regarding the behavior of impedance in an L-C-R series circuit as the frequency of an AC supply increases, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Impedance in L-C-R Circuit**:
The impedance \( Z \) of a series L-C-R circuit is given by the formula:
\[ Z = \sqrt{(X_L - X_C)^2 + R^2} \]
where \( X_L \) is the inductive reactance, \( X_C \) is the capacitive reactance, and \( R \) is the resistance.

2. **Define Reactances**:
- The inductive reactance \( X_L \) is given by:
\[ X_L = \omega L = 2\pi f L \]
- The capacitive reactance \( X_C \) is given by:
\[ X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} \]

3. **Behavior at Low Frequency**:
- At \( f = 0 \) (low frequency), \( X_L = 0 \) and \( X_C \) approaches infinity. Thus:
\[ Z = \sqrt{(0 - \infty)^2 + R^2} \rightarrow \infty \]
- This indicates that the impedance is at its maximum value.

4. **Behavior as Frequency Increases**:
- As frequency \( f \) increases, \( X_L \) increases and \( X_C \) decreases.
- Therefore, the overall impedance \( Z \) starts to decrease because \( X_L \) increases at a slower rate compared to the rapid decrease of \( X_C \).

5. **At Resonance**:
- At the resonant frequency \( f_r \), \( X_L = X_C \). Thus:
\[ Z = \sqrt{(X_L - X_C)^2 + R^2} = R \]
- This is the minimum value of impedance.

6. **Behavior at High Frequency**:
- If the frequency continues to increase beyond the resonant frequency, \( X_L \) will dominate and tend to infinity while \( X_C \) approaches zero:
\[ Z = \sqrt{(\infty - 0)^2 + R^2} \rightarrow \infty \]
- This indicates that the impedance increases again.

7. **Conclusion**:
- From the analysis, we can summarize that the impedance \( Z \) first decreases to a minimum value at resonance and then increases again as the frequency continues to rise. Thus, the correct answer is that the impedance first decreases, becomes minimum, and then increases.

### Final Answer:
The impedance of an L-C-R series circuit **decreases at first, becomes minimum, and then increases**.