Power factor is maximum in an `LCR` circuit when

Options

A. `X_(L)=X_(C)`
B. `R=0`
C. `X_(L)=0`
D. `X_(C)=0`

Explanation

To determine when the power factor is maximum in an LCR circuit, we can follow these steps:

### Step 1: Understand the Power Factor
The power factor (PF) in an LCR circuit is defined as:
\[ PF = \cos \phi = \frac{R}{Z} \]
where \( R \) is the resistance and \( Z \) is the impedance of the circuit.

### Step 2: Identify the Range of Power Factor
The power factor can vary between 0 and 1:
- \( PF = 0 \) indicates a purely reactive circuit (either inductive or capacitive).
- \( PF = 1 \) indicates a purely resistive circuit.

### Step 3: Determine Conditions for Maximum Power Factor
The power factor is maximum (equal to 1) when:
\[ \frac{R}{Z} = 1 \]
This implies that:
\[ R = Z \]

### Step 4: Relate Impedance to Circuit Elements
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.

### Step 5: Set Up the Equation for Maximum Power Factor
For maximum power factor, we set:
\[ R = \sqrt{R^2 + (X_L - X_C)^2} \]

### Step 6: Square Both Sides
Squaring both sides gives:
\[ R^2 = R^2 + (X_L - X_C)^2 \]
This simplifies to:
\[ 0 = (X_L - X_C)^2 \]

### Step 7: Solve the Equation
The equation \( (X_L - X_C)^2 = 0 \) implies:
\[ X_L - X_C = 0 \quad \Rightarrow \quad X_L = X_C \]
This means that the inductive reactance equals the capacitive reactance.

### Conclusion
The power factor is maximum in an LCR circuit when the inductive reactance \( X_L \) is equal to the capacitive reactance \( X_C \).

### Final Answer
The power factor is maximum in an LCR circuit when \( X_L = X_C \).

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