Power factor is one for

Options

A. pure inductor
B. pure capacitor
C. pure resistor
D. Either an inductor or a capacitor

Answer

To determine when the power factor is equal to 1, we need to analyze the relationship between resistance (R) and impedance (Z) in an AC circuit. The power factor (PF) is defined as:

\[ \text{Power Factor (PF)} = \frac{R}{Z} \]

Given that the power factor is 1, we can set up the equation:

\[ PF = 1 \implies \frac{R}{Z} = 1 \implies R = Z \]

### Step 1: Understanding Impedance
The impedance (Z) of an LCR circuit is given by:

\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]

where:
- \(X_L\) is the inductive reactance,
- \(X_C\) is the capacitive reactance.

### Step 2: Setting Up the Equation
From the power factor condition \(R = Z\), we can substitute for Z:

\[ R = \sqrt{R^2 + (X_L - X_C)^2} \]

### Step 3: Squaring Both Sides
To eliminate the square root, we square both sides:

\[ R^2 = R^2 + (X_L - X_C)^2 \]

### Step 4: Simplifying the Equation
Subtract \(R^2\) from both sides:

\[ 0 = (X_L - X_C)^2 \]

### Step 5: Analyzing the Result
The equation \((X_L - X_C)^2 = 0\) implies:

\[ X_L - X_C = 0 \implies X_L = X_C \]

This means that the inductive reactance equals the capacitive reactance, which occurs in a resonant circuit where both inductor and capacitor are present.

### Step 6: Considering the Other Condition
Alternatively, if both the inductor and capacitor are absent, then:

\[ X_L = 0 \quad \text{and} \quad X_C = 0 \]

In this case, the circuit is purely resistive, and the power factor will also be 1.

### Conclusion
Thus, the power factor is 1 in two scenarios:
1. When both inductor and capacitor are present and in resonance (i.e., \(X_L = X_C\)).
2. When both inductor and capacitor are absent (i.e., the circuit is purely resistive).

Given the options:
1. Pure inductor
2. Pure capacitor
3. Pure resistor
4. Either an inductor or a capacitor

The correct answer is **Option 3: Pure Resistor**.

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