Series LCR Circuit: Power Factor at Resonance

In a series LCR circuit, the power factor is defined as the cosine of the phase angle (\(\phi\)) between the voltage and current, or equivalently, the ratio of resistance (R) to impedance (Z).

Mathematically, the power factor is expressed as:

\(\text{Power Factor (PF)} = \cos(\phi) = \frac{R}{Z}\)

Understanding Resonance in LCR Circuits

Resonance occurs in a series LCR circuit when the inductive reactance (\(X_L\)) equals the capacitive reactance (\(X_C\)).

• Inductive Reactance: \(X_L = \omega L\)
• Capacitive Reactance: \(X_C = \frac{1}{\omega C}\)
• Resonance Condition: \(X_L = X_C\)

Calculating Impedance at Resonance

The total impedance (Z) of a series LCR circuit is given by:

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

At the resonance condition (\(X_L = X_C\)), the term \((X_L - X_C)\) becomes zero:

\(Z_{\text{resonance}} = \sqrt{R^2 + (0)^2} = \sqrt{R^2} = R\)

Therefore, at resonance, the impedance of the circuit is minimal and equal to its resistance.

Determining Power Factor at Resonance

Using the power factor formula with the impedance at resonance (\(Z=R\)):

\(\text{PF}_{\text{resonance}} = \frac{R}{Z_{\text{resonance}}} = \frac{R}{R} = 1\)

When the power factor is 1, the phase angle (\(\phi\)) is \(0^\circ\), indicating that the voltage and current are in phase. This is the condition for maximum power transfer in the circuit.

Conclusion

The power factor of a series LCR circuit at resonance is always 1.