Question

Define the terms (I) resonant frequency, and (II) power factor of a series LCR circuit. For what value of the power factor will the power dissipated in the circuit be maximum?

Hint:

In LCR circuits: \[ Z=\sqrt{R^2+(X_L-X_C)^2} \] At resonance: \[ X_L=X_C,\quad Z=R,\quad \cos\phi=1 \]

Answer 1

Resonant Frequency

The resonant frequency is the frequency at which the inductive reactance becomes equal to the capacitive reactance in a series LCR circuit.

\[ X_L = X_C \]

At resonance:

• The impedance of the circuit is minimum.
• The current in the circuit is maximum.
• The circuit behaves like a purely resistive circuit.

The resonant frequency is given by

\[ f_r=\frac{1}{2\pi\sqrt{LC}} \]

---

Power Factor

The power factor of a series LCR circuit is defined as the cosine of the phase angle between the applied voltage and the current.

\[ \boxed{\text{Power Factor}=\cos\phi=\frac{R}{Z}} \]

where,

• \(R\) = Resistance of the circuit
• \(Z\) = Impedance of the circuit
• \(\phi\) = Phase difference between voltage and current

The average power consumed in the circuit is

\[ P=VI\cos\phi \]

---

Condition for Maximum Power Dissipation

Power dissipated in the circuit is

\[ P=VI\cos\phi \]

For fixed values of \(V\) and \(I\), power is maximum when

\[ \cos\phi=1 \]

This occurs at resonance, where

\[ X_L=X_C \]

Hence,

\[ \boxed{\text{Power factor}=1} \]

Therefore, the power dissipated in a series LCR circuit is maximum when the power factor is unity.