Correct Answer

Concept & PrincipleElectricalBasic Electrical

Option A

Quick Summary: At series resonance in an RLC circuit, the inductive reactance ($X_L$) is exactly equal to the capacitive reactance ($X_C$), causing the net reactance to become zero. Consequently, the circuit behaves as a purely resistive load, resulting in a power factor of unity (1.0).

💡 Explanation

At series resonance in an RLC circuit, the inductive reactance ( $X_L$ ) is exactly equal to the capacitive reactance ( $X_C$ ), causing the net reactance to become zero. Consequently, the circuit behaves as a purely resistive load, resulting in a power factor of unity (1.0).

🔢 Key Formulas

$X_L = 2\pi f L$ — Inductive reactance

$X_C = \frac{1}{2\pi f C}$ — Capacitive reactance

$\cos(\phi) = \frac{R}{Z}$ — Power factor definition

⚙️ Working Principle

In a series RLC circuit, total impedance $Z = R + j(X_L - X_C)$ . Resonance occurs when $X_L = X_C$ , which forces the imaginary part to zero, leaving $Z = R$ . Since the phase angle $\theta = \tan^{-1}(\frac{X_L - X_C}{R}) = 0°$ , the power factor $\cos(\theta) = \cos(0°) = 1$ .

📌 Key Points

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At resonance, the current is in phase with the applied voltage.
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The impedance of the circuit is at its minimum value equal to R.
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The voltage across the inductor and capacitor are equal in magnitude but opposite in phase, canceling each other out.
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Circuit power factor is 1, implying a purely resistive nature.

✅ Advantages

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Maximum current flow for a given voltage source
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High selectivity in tuned circuits and communication systems

❌ Disadvantages / Limitations

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High voltage magnification (Q-factor) can damage insulation
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Potential for excessive current if resistance is very low

🛠️ Applications / Uses

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Tuning circuits in radio receivers
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Band-pass filter design
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Induction heating systems

📄 Additional Information

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Resonance frequency is given by $f_r = \frac{1}{2\pi\sqrt{LC}}$ .
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Option A (0) represents an ideal inductor or capacitor, which is not true for a resonant RLC circuit.
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Options B (0.5) and C (0.8) are typical power factors for inductive loads, but do not represent the resonance condition.

📊 Diagram / Illustration

✅

D is correct — At resonance, inductive and capacitive reactances cancel out, leaving the circuit purely resistive with a unity power factor.

Core Concepts Used

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Series Resonance Power Factor Phasor Relationships

💡 EXAM TIP

Always remember: in series resonance, Impedance is minimum (Z=R) and Current is maximum (I=V/R); in parallel resonance, Impedance is maximum and Current is minimum.

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