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:

Always remember that power is ONLY dissipated across the resistor in an LCR circuit.
Ideal inductors and capacitors simply store and release energy back and forth without consuming any real power, which is why maximum power requires the reactive parts to cancel out completely.

Answer 1

Concept:
• In a series LCR (Inductor, Capacitor, Resistor) circuit, the overall opposition to AC current is called impedance ($Z$).
• Resonance occurs under specific conditions where the reactive components perfectly cancel each other's effects out.
• Power dissipation heavily depends on the phase alignment between the applied voltage and the resulting current, described by the power factor.

Step 1: Define Resonant Frequency
(I) Resonant Frequency:
In an alternating current series LCR circuit, the resonant frequency is strictly defined as the specific driving frequency of the AC source at which the inductive reactance ($X_L = \omega L$) becomes exactly equal in magnitude to the capacitive reactance ($X_C = \frac{1}{\omega C}$).
Because these two reactances are exactly $180^\circ$ out of phase, they completely cancel each other out ($X_L - X_C = 0$).
Consequently, the total impedance of the circuit drops to its absolute minimum possible value, which is simply the pure ohmic resistance ($Z = R$).
At this specific frequency, the circuit permits the maximum possible amplitude of current to flow.
Mathematically, the angular resonant frequency is expressed as $\omega_r = \frac{1}{\sqrt{LC}}$, and the linear resonant frequency is $f_r = \frac{1}{2\pi\sqrt{LC}}$.

Step 2: Define Power Factor
(II) Power Factor:
The power factor of an AC circuit is fundamentally defined as the cosine of the phase angle ($\phi$) that exists between the total applied voltage and the resulting circuit current.
It serves as a crucial indicator of what fraction of the total apparent power is actually converted into useful work (true power dissipation).
Mathematically, it is written as $\cos\phi = \frac{R}{Z}$, representing the simple ratio of the true resistance $R$ to the total impedance $Z$ of the circuit.
A higher power factor means the circuit is more purely resistive and highly efficient at dissipating energy.

Step 3: Determine condition for maximum power dissipation
The average power dissipated in an AC circuit over a complete cycle is governed by the formula:
\[ P_{avg} = V_{rms} \cdot I_{rms} \cdot \cos\phi \]
To strictly maximize this average power $P_{avg}$, the multiplicative power factor term $\cos\phi$ must reach its maximum mathematical value.
The cosine function reaches its absolute maximum value of $1$ when the phase angle is precisely zero ($\phi = 0^\circ$).
Therefore, the power dissipated in the circuit will be maximum strictly when the power factor equals $1$.
This idealized condition naturally occurs at electrical resonance, where the entire circuit behaves exactly like a purely resistive circuit.