Power factor in a series R-L-C resonant circuit is

Options

A. 0.5
B. 0.707
C. 1
D. 0

Answer

To find the power factor in a series R-L-C resonant circuit, we can follow these steps:

### Step 1: Understand the Components of the Circuit
In a series R-L-C circuit, we have:
- Resistance (R)
- Inductance (L)
- Capacitance (C)

### Step 2: Define Power Factor
The power factor (PF) is defined as the cosine of the phase angle (φ) between the voltage across the circuit and the current flowing through it. Mathematically, it is given by:
\[ \text{Power Factor} = \cos(\phi) \]

### Step 3: Relate Power Factor to Impedance
The power factor can also be expressed in terms of the circuit's impedance (Z):
\[ \text{Power Factor} = \frac{R}{Z} \]

### Step 4: Calculate Impedance in a Series R-L-C Circuit
The impedance (Z) in a series R-L-C circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where:
- \( X_L = \omega L \) (inductive reactance)
- \( X_C = \frac{1}{\omega C} \) (capacitive reactance)

### Step 5: Condition for Resonance
At resonance, the inductive reactance equals the capacitive reactance:
\[ X_L = X_C \]
This means:
\[ \omega L = \frac{1}{\omega C} \]

### Step 6: Impedance at Resonance
When \( X_L = X_C \), the difference \( (X_L - X_C) \) becomes zero. Therefore, the impedance simplifies to:
\[ Z = \sqrt{R^2 + 0^2} = R \]

### Step 7: Calculate Power Factor at Resonance
Substituting the value of Z back into the power factor equation:
\[ \text{Power Factor} = \frac{R}{Z} = \frac{R}{R} = 1 \]

### Conclusion
Thus, the power factor in a series R-L-C resonant circuit is 1.

### Final Answer
The power factor in a series R-L-C resonant circuit is **1**.

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