The power factor of a series LCR circuit is

Options

A. Z/R
B. R/Z
C. R/X
D. X/R

Answer

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

### Step-by-Step Solution:

1. **Understand the Components of the Circuit**:
- A series LCR circuit consists of an inductor (L), a capacitor (C), and a resistor (R) connected in series to an AC source.
- The inductive reactance (XL) and capacitive reactance (XC) are defined as:
- \( X_L = 2\pi f L \)
- \( X_C = \frac{1}{2\pi f C} \)

2. **Calculate the Impedance (Z)**:
- The total impedance (Z) of the circuit can be calculated using the formula:
\[ Z = \sqrt{(X_L - X_C)^2 + R^2} \]
- Here, \( X_L \) is the inductive reactance and \( X_C \) is the capacitive reactance.

3. **Determine the Phase Angle (φ)**:
- The phase angle (φ) between the total voltage and the total current in the circuit can be calculated using the relationship:
\[ \tan(\phi) = \frac{X_L - X_C}{R} \]
- This angle indicates the phase difference due to the reactance and resistance in the circuit.

4. **Calculate the Power Factor (PF)**:
- The power factor (PF) is defined as the cosine of the phase angle (φ):
\[ \text{Power Factor} = \cos(\phi) \]
- From the right triangle formed by R, \( X_L - X_C \), and Z, we can express the power factor as:
\[ \cos(\phi) = \frac{R}{Z} \]

5. **Conclusion**:
- Therefore, the power factor of a series LCR circuit is given by:
\[ \text{Power Factor} = \frac{R}{Z} \]

### Final Answer:
The power factor of a series LCR circuit is \( \frac{R}{Z} \).