Power factor in series LCR circuit at reasonance is

Options

A. 1
B. `(1)/(sqrt2)`
C. zero
D. infinite

Answer

To find the power factor in a series LCR circuit at resonance, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Series LCR Circuit**:
- A series LCR circuit consists of an inductor (L), a capacitor (C), and a resistor (R) connected in series with an AC supply.

2. **Define Impedance (Z)**:
- The total resistance offered by the circuit is called impedance (Z). The formula for impedance in a series LCR circuit is:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
- Here, \(X_L\) is the inductive reactance and \(X_C\) is the capacitive reactance.

3. **Condition at Resonance**:
- At resonance, the inductive reactance equals the capacitive reactance:
\[ X_L = X_C \]
- This means that the difference \(X_L - X_C = 0\).

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

5. **Define Power Factor (PF)**:
- The power factor (PF) is defined as:
\[ PF = \cos \phi = \frac{R}{Z} \]
- Here, \(\phi\) is the phase angle between the current and voltage.

6. **Calculate Power Factor at Resonance**:
- Since at resonance \(Z = R\), we can substitute this into the power factor formula:
\[ PF = \frac{R}{R} = 1 \]

7. **Conclusion**:
- Therefore, the power factor in a series LCR circuit at resonance is:
\[ \text{Power Factor} = 1 \]

### Final Answer:
The power factor in a series LCR circuit at resonance is **1**.