The power factor of LCR circuit at resonance is-

Options

A. `0.707`
B. 1
C. Zero
D. `0.5`

Answer

To determine the power factor of an LCR circuit at resonance, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Power Factor**:
The power factor (PF) is defined as the cosine of the phase angle (φ) between the voltage and the current in an AC circuit. Mathematically, it is expressed as:
\[ \text{Power Factor} = \cos \phi = \frac{R}{Z} \]
where \( R \) is the resistance and \( Z \) is the impedance of the circuit.

2. **Expression for Impedance (Z)**:
The impedance \( Z \) of an LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where \( X_L \) is the inductive reactance and \( X_C \) is the capacitive reactance.

3. **Condition at Resonance**:
At resonance in an LCR circuit, the inductive reactance \( X_L \) equals the capacitive reactance \( X_C \):
\[ X_L = X_C \]
This means that the difference \( (X_L - X_C) \) becomes zero.

4. **Simplifying Impedance at Resonance**:
Substituting \( X_L = X_C \) into the impedance formula, we get:
\[ Z = \sqrt{R^2 + 0^2} = \sqrt{R^2} = R \]

5. **Calculating Power Factor at Resonance**:
Now, substituting \( Z = R \) back into the power factor equation:
\[ \text{Power Factor} = \cos \phi = \frac{R}{Z} = \frac{R}{R} = 1 \]

6. **Conclusion**:
Therefore, the power factor of an LCR circuit at resonance is:
\[ \text{Power Factor} = 1 \]

### Final Answer:
The power factor of an LCR circuit at resonance is **1**.

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