Resistance is 50 ohm in series resonance, what is value of impedance?

Question

Examoogle · Accepted Answer

50

Explanation: In an RLC series circuit, resonance occurs when the inductive reactance equals the capacitive reactance, i.e., $X_L = X_C$. At this condition, the net reactive component becomes zero, and the total impedance of the circuit is purely resistive and equal to the resistance R.

Working Principle: The total impedance $Z$ in an RLC series circuit is given by $Z = \sqrt{R^2 + (X_L - X_C)^2}$. At resonance, the frequency of the source is such that the inductive reactance $X_L = 2\pi fL$ and capacitive reactance $X_C = \frac{1}{2\pi fC}$ cancel each other out ($X_L - X_C = 0$). Consequently, the impedance reaches its minimum value, which is equal to $R$.

Key Formulas: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ — General impedance formula for RLC series circuit; $X_L = X_C \implies Z = R$ — Condition and impedance at resonance

Key Points: At resonance, the phase angle between voltage and current is zero.; The power factor of the circuit at resonance is unity (1.0).; The current in an RLC series circuit is maximum at the resonant frequency.

Advantages: Maximum power transfer; Selectivity in radio tuning circuits

Disadvantages: High current can damage components if not controlled; Voltage magnification across L and C components

Applications: Radio and television tuning circuits; Oscillators and filter design

Additional Info: In this problem, since R = 50 \Omega and resonance implies Z = R, Z must be 50 \Omega.. Options A, C, and D are incorrect as they deviate from the condition where reactive components cancel out at resonance.

Answer: B is correct — At resonance in an RLC series circuit, the total impedance is purely resistive and equal to the value of R.

Exam Tip: Remember that at resonance, the circuit behaves as a purely resistive load, meaning the voltage and current are in phase.

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