LRC Resonance
Sweep frequency, find peak. Builds on RC Circuits and Inductance & RL Circuits.
You have an LRC series circuit: an inductor (L), resistor (R), and capacitor (C) connected in series, driven by an AC voltage source. You slowly sweep the driving frequency from very low to very high.
At resonance, when the driving frequency matches the natural frequency ω₀ = 1/√(LC), what is the total impedance of the circuit?
At resonance, the impedance is just R! The inductor’s reactance (ωL) and the capacitor’s reactance (1/ωC) are exactly equal and opposite. They cancel each other out, leaving only the resistance. This means the current peaks at its maximum value I = V/R — the circuit responds most strongly at its resonant frequency.
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Watch the frequency sweep from low to high. The current amplitude traces out a resonance curve with a sharp peak. At the peak, X_L = X_C and the impedance drops to just R. Below resonance, the capacitor dominates. Above, the inductor wins. The phasors show the cancellation in action.
ω₀ = 1/√(LC)
At resonance, X_L = X_C, impedance = R only, current is maximum. The Q factor (ω₀L/R) controls how sharp the peak is — high Q means a very selective circuit.
Every radio station broadcasts at a different frequency. The tuning dial adjusts C in an LRC circuit, shifting the resonant frequency to match the desired station. Only the resonant station’s signal gets through strongly.
A wine glass is a mechanical resonator. When a singer hits the glass’s natural frequency, energy builds up cycle after cycle until the glass shatters — the same principle as electrical resonance.
Magnetic resonance imaging exploits nuclear spin resonance. RF coils tuned to specific frequencies excite hydrogen atoms in your body. The resonant response is detected to create detailed images.
Metal detectors use LRC resonance. A metal object near the coil changes L, shifting the resonant frequency. The detector senses this shift and beeps. Different metals cause different shifts.
“Resonance is nature’s amplifier. At the right frequency, energy piles up cycle after cycle. In an LRC circuit, the reactive parts cancel, leaving only resistance — and current surges to its peak. Every radio, every MRI, every musical instrument exploits this beautiful phenomenon.”
LRC Resonance Lab
Explore the full frequency response of an LRC series circuit. Sweep the driving frequency and watch the resonance peak emerge. Adjust L, R, and C to control the resonant frequency and the sharpness of the peak.
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Start with a low R to see a sharp resonance peak (high Q). Now increase R — watch the peak broaden and flatten. Try changing L or C and see the resonant frequency shift. Mark ω₀ = 1/√(LC) and verify the peak lands right there. Compare Q = 5 vs Q = 50 — the difference in selectivity is dramatic. This is why radios need high-Q circuits to separate nearby stations.
Resonance occurs when energy sloshes perfectly between the inductor’s magnetic field and the capacitor’s electric field. At resonance, the reactive impedances cancel, only R remains, and the current peaks. The Q factor determines how selective the resonance is — high Q means the circuit responds to a very narrow range of frequencies.