Resonance in RLC Circuits

At one special frequency, an RLC circuit's two reactances cancel and the circuit looks purely resistive. That's resonance — and it's the phenomenon that lets a radio pick out one station from a sky full of broadcasts, a tuning fork stay in tune, and a power-supply output filter kill one specific ripple harmonic without touching the rest.

The resonant frequency

In a series RLC circuit the reactances are $X_L = \omega L$ and $X_C = 1/(\omega C)$. They are equal when:

The same formula holds for a parallel RLC tank (as long as we're treating both as ideal elements). At $f_0$:

• Series RLC: net reactance is zero, so $|Z| = R$ (minimum). Current $I = V/R$ is maximum — a sharp peak.
• Parallel RLC (tank): the inductor and capacitor trade energy back and forth; external current draw is minimum, impedance is maximum.

Underneath the peak is a lovely energy story. At resonance the capacitor and the inductor pass energy back and forth through a quarter-cycle rhythm — each reaches maximum while the other is zero. The source only has to top up whatever $R$ dissipates per cycle. In a perfect lossless circuit ($R = 0$) the oscillation would continue forever with no external help. Real components have losses; the source keeps them oscillating.

Peak current in series RLC

Three different component selections that share the same $f_0$ but have different $Q$ (quality factor). All three current curves have their peak at the same frequency, but they widen or narrow around it. The width is what matters for selectivity.

Q factor — “how peaky is the peak?”

The quality factor $Q$ is the ratio of energy stored per radian to energy dissipated. For a series RLC:

Those three forms are all equivalent at $\omega_0$. Intuition:

• Big L, small C, small R → high Q — lots of energy sloshing back and forth, little lost each cycle. Sharp peak.
• Small L, big C, big R → low Q — less energy stored, more dissipated. Broad peak.

In practice:

• An AM-radio front-end tuned circuit runs $Q \approx 100$ — sharp enough to split 10 kHz channels on the broadcast band.
• A typical audio band-pass (say a guitar wah pedal) runs $Q \approx 2–5$ — broad enough to feel musical.
• A power-factor correction capacitor on a motor feed behaves like a very low-Q resonator — its job is to cancel a single mains-frequency reactance, not to filter.

Bandwidth = f₀ / Q

The two frequencies $f_1$ and $f_2$ where the response crosses $1/\sqrt{2}$ of peak bracket thepassband. Their difference is the bandwidth:

The trade-off is inescapable: high Q means narrow bandwidth (great for channel selectivity, bad for wide- band signals); low Q means wide bandwidth (great for passing a full audio band, bad for picking one tone out of many).

Common applications

• Radio tuning. Every AM / FM set has a variable capacitor or varactor diode that slides $f_0$ across the broadcast band to pick a station.
• Crystal oscillators. A quartz crystal is a mechanical resonator with $Q \approx 10^4$ to $10^6$ — absurdly sharp — which is why your wristwatch keeps accurate time and your microcontroller's clock is stable to parts per million.
• Tank circuits in amplifiers. A parallel LC in the collector of an RF transistor lets you pull huge voltage swings at $f_0$ without wasting DC power — the inductor looks like a short at DC and high-impedance at the resonant frequency.
• Notch filters. A series RLC shunted across a signal path sinks all the energy at $f_0$ to ground — used to kill 50/60 Hz mains hum in audio gear, or to remove a specific harmonic in power-quality applications.
• Inductive wireless charging. The transmitter and receiver coils are tuned to the same $f_0$ so power transfers efficiently even across a small air gap. Off-resonance the gap is too lossy; at resonance the tank impedance matches and energy couples across.