In the above question size element will raise the power factor to unity?

Options

A. an inductor should be placed in series
B. a capacitor should be placed in series
C. a resistance should be placed in series
D. an inductor or a resistance should be placed in series

Answer

To solve the question regarding how to raise the power factor to unity in an LCR circuit, 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 current in an AC circuit. It is mathematically 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**:
The impedance \( Z \) in 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 for Unity Power Factor**:
For the power factor to be unity (PF = 1), the cosine of the phase angle must equal 1, which implies:
\[ \cos \phi = 1 \implies R = Z \]
This occurs when the reactive components (inductive and capacitive) cancel each other out, i.e., \( X_L = X_C \).

4. **Reducing Impedance**:
To achieve a unity power factor, we need to reduce the impedance \( Z \). This can be done by manipulating the reactance:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
To decrease \( Z \), we need to make \( |X_L - X_C| \) smaller. This can be achieved by:
- Adding a capacitor (to increase \( X_C \))
- Adding an inductor (to increase \( X_L \))

5. **Adding Elements**:
- If we add a capacitor in parallel, it will increase the capacitive reactance \( X_C \).
- If we add an inductor in series, it will increase the inductive reactance \( X_L \).
- The best approach to achieve unity power factor is to add a resistance in series with the circuit, as it will help in balancing the reactance.

6. **Conclusion**:
To raise the power factor to unity, we can add a resistor or an inductor in series with the circuit. This will help in reducing the overall impedance and achieving the condition \( R = Z \).

### Final Answer:
To raise the power factor to unity, add a resistor or an inductor in series with the circuit.

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