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 determine the effect of increasing the frequency of an AC supply on the impedance of an L-C-R series circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Impedance**:
- Impedance (Z) in an L-C-R series circuit is defined as the total opposition to the flow of alternating current. It is given by the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where \( R \) is the resistance, \( X_L \) is the inductive reactance, and \( X_C \) is the capacitive reactance.

2. **Inductive and Capacitive Reactance**:
- The inductive reactance \( X_L \) is given by:
\[ X_L = 2\pi f L \]
- The capacitive reactance \( X_C \) is given by:
\[ X_C = \frac{1}{2\pi f C} \]
where \( f \) is the frequency, \( L \) is the inductance, and \( C \) is the capacitance.

3. **Behavior at Zero Frequency**:
- At \( f = 0 \):
- \( X_L = 0 \)
- \( X_C \) approaches infinity.
- Thus, impedance \( Z \) is infinite.

4. **Increasing Frequency**:
- As the frequency \( f \) increases:
- \( X_L \) increases (since it is directly proportional to \( f \)).
- \( X_C \) decreases (since it is inversely proportional to \( f \)).
- Initially, as frequency increases, \( Z \) decreases because \( X_C \) decreases faster than \( X_L \) increases.

5. **Resonance Condition**:
- There exists a frequency where \( X_L = X_C \). At this point:
\[ Z = R \]
- This is the minimum impedance of the circuit.

6. **Further Increase in Frequency**:
- Beyond the resonance frequency:
- \( X_L \) continues to increase.
- \( X_C \) continues to decrease.
- Eventually, \( Z \) starts to increase again because the increase in \( X_L \) outweighs the decrease in \( X_C \).

7. **Behavior at High Frequency**:
- As \( f \) approaches infinity:
- \( X_L \) approaches infinity.
- \( X_C \) approaches zero.
- Thus, impedance \( Z \) becomes infinite again.

8. **Conclusion**:
- The impedance of the L-C-R circuit first decreases with increasing frequency, reaches a minimum value at resonance, and then increases again as frequency continues to rise.

### Final Answer:
The impedance of an L-C-R series circuit decreases initially with increasing frequency, reaches a minimum value at resonance, and then increases again.