Series LCR CircuitJEE Advanced

Impedance, phase and resonance — interactive Physics simulation for IIT-JEE.

Concept

In a series LCR circuit driven by AC, the resistor, inductor and capacitor each oppose current differently. Their combined opposition is the impedance $Z$ . At one special resonant frequency the inductive and capacitive effects cancel, $Z$ drops to just $R$ , and the current peaks.

Key formula

Z = \sqrt{R^2 + (X_L - X_C)^2}, \quad X_L = \omega L, \; X_C = \frac1{\omega C}, \quad f_0 = \frac1{2\pi\sqrt{LC}}

Derivation

Voltages across $L$ and $C$ are $90°$ out of phase with the resistor's, and $180°$ from each other. Adding them as phasors gives $V = \sqrt{V_R^2 + (V_L - V_C)^2}$ , so dividing by current, $Z = \sqrt{R^2 + (X_L - X_C)^2}$ .

The phase angle is $\tan\varphi = (X_L - X_C)/R$ . At resonance $X_L = X_C$ , so $\varphi = 0$ , $Z = R$ (minimum), and $I$ is maximum.

Scenarios to explore

Series LCR Circuit — Impedance phasor, resonance and power factor.

Real-world applications

Radio & TV tuning — selecting one station's frequency.
Band-pass and band-stop filters.
Metal detectors and induction heating.

JEE exam tips

At resonance: $Z = R$ , current maximal, power factor $= 1$ , circuit is purely resistive.
Quality factor $Q = \frac1R\sqrt{L/C}$ measures how sharp the resonance is.

Common mistakes

Adding $X_L$ and $X_C$ instead of subtracting (they oppose).
Forgetting to convert mH→H and μF→F before computing reactances.
Thinking $V_L$ or $V_C$ alone can't exceed the supply voltage — at resonance they can.

Exam traps to avoid

Below $f_0$ the circuit is capacitive (current leads); above $f_0$ it is inductive (current lags).
Average power $= V_{rms}I_{rms}\cos\varphi$ — the power factor matters, not just $VI$ .