Correct Answer

Concept & PrincipleElectricalBasic Electrical

Option C

$R$

Quick Summary: At series resonance in an RLC circuit, the inductive reactance ($X_L = 2\pi fL$) and capacitive reactance ($X_C = \frac{1}{2\pi fC}$) are equal in magnitude but opposite in phase. Consequently, they cancel each other out, leaving only the ohmic resistance ($R$) as the net impedance of the circuit.

💡 Explanation

At series resonance in an RLC circuit, the inductive reactance ( $X_L = 2\pi fL$ ) and capacitive reactance ( $X_C = \frac{1}{2\pi fC}$ ) are equal in magnitude but opposite in phase. Consequently, they cancel each other out, leaving only the ohmic resistance ( $R$ ) as the net impedance of the circuit.

🔢 Key Formulas

$Z = \sqrt{R^2 + (X_L - X_C)^2}$ — General formula for impedance

$Z_{min} = R$ — Impedance at series resonance

⚙️ Working Principle

The total impedance of a series RLC circuit is given by $Z = \sqrt{R^2 + (X_L - X_C)^2}$ . At resonance, the resonant frequency $f_r = \frac{1}{2\pi\sqrt{LC}}$ is reached such that $X_L = X_C$ . Substituting this into the impedance formula yields $Z = \sqrt{R^2 + (0)^2} = R$ . This state results in minimum impedance and maximum current flow.

📌 Key Points

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At resonance, the phase angle of the circuit is 0°.
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The circuit behaves as a purely resistive load.
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Current is maximum at resonance: $I_{max} = \frac{V}{R}$ .
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Voltage across inductor and capacitor are equal in magnitude but 180° out of phase.

✅ Advantages

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Maximum power transfer efficiency at resonant frequency.
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Useful for frequency selection in tuning circuits.

❌ Disadvantages / Limitations

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High currents at resonance can potentially damage components.
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Voltage magnification can occur across L and C elements.

🛠️ Applications / Uses

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Radio frequency tuning circuits.
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Oscillators and filters.
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Band-pass filter design.

📄 Additional Information

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Option A (0) is incorrect because resistance R is never zero in a practical RLC circuit.
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Option D ( $\infty)$ is the condition for parallel resonance impedance in an ideal LC circuit, not series resonance.

📊 Diagram / Illustration

✅

C is correct — At series resonance, the inductive and capacitive reactances cancel out, reducing total impedance to the circuit's resistive component, R.

Core Concepts Used

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Series RLC Resonance Complex Impedance Phasor Cancellation

💡 EXAM TIP

Remember that in a series circuit, resonance minimizes impedance, while in a parallel (tank) circuit, resonance maximizes impedance.

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