Which of the following is not the characteristic of resonance in an LCR series circuit?

Options

A. `X_(L)=X_(C )`
B. `omegaL=(1)/(omegaC)`
C. `2pi fL = 2pi fC`
D. `f_(0) =(1)/(2pi) sqrt((1)/(LC))`

Explanation

To determine which of the following is not a characteristic of resonance in an LCR series circuit, we need to analyze the properties of resonance in such circuits.

### Step-by-Step Solution:

1. **Understanding Resonance in LCR Circuits**:
- In an LCR series circuit, resonance occurs when the inductive reactance (XL) equals the capacitive reactance (XC). This condition leads to maximum current flow in the circuit.

2. **Condition for Resonance**:
- The condition for resonance can be mathematically expressed as:
\[ XL = XC \]
- Where:
- \( XL = \omega L \) (Inductive reactance)
- \( XC = \frac{1}{\omega C} \) (Capacitive reactance)

3. **Impedance at Resonance**:
- The impedance (Z) of the circuit at resonance is minimized. The formula for impedance in an LCR circuit is:
\[ Z = \sqrt{R^2 + (XL - XC)^2} \]
- At resonance, since \( XL = XC \), the impedance simplifies to:
\[ Z = R \]
- This indicates that the circuit behaves purely resistively at resonance.

4. **Resonance Frequency**:
- The resonance frequency (f₀) can be derived from the relationship between inductance (L) and capacitance (C):
\[ f_0 = \frac{1}{2\pi\sqrt{LC}} \]

5. **Evaluating the Options**:
- Now, we need to evaluate the given options to identify which one does not represent a characteristic of resonance. The characteristics typically include:
- Maximum current at resonance.
- Minimum impedance at resonance.
- The relationship between inductance and capacitance leading to a specific frequency.
- If any option contradicts these established characteristics, it would be the correct answer.

6. **Conclusion**:
- After evaluating the options based on the characteristics of resonance, we find that one of the options does not align with the established principles. This option is the answer to the question.