In series LCR circuit, the phase angle between supply voltage and current is

Options

A. `tan phi = (X_(L)-X_(C ))/(R )`
B. `tan phi = (R )/(X_(L)-X_(C ))`
C. `tan phi = (R )/(X_(L)+X_(C ))`
D. `tan phi = (X_(L)+X_(C ))/(R )`

Answer

To determine the phase angle between the supply voltage and current in a series LCR circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Circuit**:
- In a series LCR circuit, we have a resistor (R), an inductor (L), and a capacitor (C) connected in series. The total voltage across the circuit is the sum of the voltages across each component.

2. **Phasor Representation**:
- The current (I) is the same through all components. The voltage across the resistor (V_R) is in phase with the current, while the voltage across the inductor (V_L) leads the current by 90 degrees, and the voltage across the capacitor (V_C) lags the current by 90 degrees.

3. **Voltage Relationships**:
- The voltages can be represented as:
- \( V_R = I \cdot R \)
- \( V_L = I \cdot X_L \) (where \( X_L = \omega L \))
- \( V_C = I \cdot X_C \) (where \( X_C = \frac{1}{\omega C} \))

4. **Resultant Voltage**:
- The total voltage across the circuit can be expressed in terms of the voltages across the inductor and capacitor:
- \( V_{total} = V_R + V_L - V_C \)
- The effective voltage across the inductor and capacitor can be combined as:
- \( V_{L} - V_{C} = I \cdot (X_L - X_C) \)

5. **Using the Parallelogram Law**:
- To find the phase angle \( \phi \), we can use the tangent function:
- \( \tan(\phi) = \frac{V_L - V_C}{V_R} \)

6. **Substituting Values**:
- Substituting the expressions for the voltages:
- \( \tan(\phi) = \frac{I \cdot (X_L - X_C)}{I \cdot R} \)
- The current \( I \) cancels out:
- \( \tan(\phi) = \frac{X_L - X_C}{R} \)

7. **Final Expression**:
- Therefore, the phase angle \( \phi \) can be expressed as:
- \( \tan(\phi) = \frac{X_L - X_C}{R} \)

### Conclusion:
The phase angle \( \phi \) between the supply voltage and current in a series LCR circuit is given by:
\[ \tan(\phi) = \frac{X_L - X_C}{R} \]