Correct Answer

Concept & PrincipleElectricalBasic Electrical

Option C

f0=1/2& $\pi$ ;√LC

Quick Summary: In an RLC series circuit, resonance occurs when the inductive reactance equals the capacitive reactance, i.e., $X_L = X_C$. At this frequency, the imaginary part of the impedance becomes zero, resulting in a purely resistive circuit where the current is in phase with the applied voltage.

💡 Explanation

In an RLC series circuit, resonance occurs when the inductive reactance equals the capacitive reactance, i.e., $X_L = X_C$ . At this frequency, the imaginary part of the impedance becomes zero, resulting in a purely resistive circuit where the current is in phase with the applied voltage.

🔢 Key Formulas

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

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

$f_0 = \frac{1}{2\pi\sqrt{LC}}$ — Resonant Frequency

⚙️ Working Principle

The condition for series resonance is $\omega L = \frac{1}{\omega C}$ , where $\omega = 2\pi f_0$ . Solving for $f_0$ gives $f_0°2 = \frac{1}{4\pi^2 LC}$ , which simplifies to $f_0 = \frac{1}{2\pi\sqrt{LC}}$ . At this point, the total impedance $Z = \sqrt{R^2 + (X_L - X_C)^2}$ reaches its minimum value $Z = R$ .

📌 Key Points

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At resonance, impedance is minimum and equal to resistance $R$ .
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Circuit current is maximum at the resonant frequency.
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The phase angle between voltage and current is zero degrees.
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The power factor of the circuit is unity (1.0).

✅ Advantages

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Used for voltage magnification in series circuits.
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Essential for tuning in radio and communication receivers.

❌ Disadvantages / Limitations

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Can lead to dangerously high voltages across L and C components.
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Highly dependent on the stability of component values over temperature.

🛠️ Applications / Uses

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Radio frequency tuning circuits.
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Band-pass filter designs in signal processing.

📄 Additional Information

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The quality factor $Q$ of the series resonant circuit is defined as $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$ .
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Option A is incorrect because it is missing the frequency inverse relationship; Option B and D are algebraically incorrect.

📊 Diagram / Illustration

✅

C is correct — The resonant frequency for a series RLC circuit is derived as $f_0 = \frac{1}{2\pi\sqrt{LC}}$ .

Core Concepts Used

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Inductive Reactance Capacitive Reactance Resonance Condition

💡 EXAM TIP

Always remember that at resonance, the energy stored in the magnetic field of the inductor is equal to the energy stored in the electric field of the capacitor, causing them to exchange energy back and forth.

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