LC resonant frequency explained

Connect a coil and a capacitor and energy swings between them, from the capacitor's electric field to the coil's magnetic field and back, at one natural rate: the resonant frequency, f₀ = 1 ÷ (2π√(LC)). It's the frequency where the coil's reactance and the capacitor's are equal. Bigger parts swing more slowly.

Why it rings

Charge a capacitor and connect it across a coil. Current flows, building the coil's field. As the capacitor empties, the coil keeps the current going, charging the capacitor the other way, and then it all flows back. That's a tank circuit: like a swing, its energy moves between two forms, and like a swing, it responds hugely to pushes in time with its own rhythm.

Live experiment

Push the swing in time

Dot pushes Pip on a swing with a steady rhythm, and the swing has its own. Push in time and every push adds up until Pip goes sky-high; push off the beat and it stays small. The bars show energy swapping between height and speed.

Try it yourself: free lesson

“LC resonance” is lesson 6 of 6 in Oscillation Coast, Electronics World 4. Free with an account. Lessons unlock in order, so you’ll do the 26 before it first.

The formula

The coil's reactance rises with frequency, X_L = 2π f L, and the capacitor's falls, X_C = 1 ÷ (2π f C). Set them equal and solve:

Example: a wireless phone charger's coil of 10 µH with 100 nF. √(10⁻⁵ × 10⁻⁷) = 10⁻⁶ s, so

That's inside the 110–205 kHz band phone chargers use. To tune for a frequency, turn it round:

A 100 µH coil tuned to an AM station at 1,070 kHz needs about 221 pF.

Series and parallel LC

In a parallel LC, the tank, the coil's and capacitor's currents are equal and opposite at f₀, so the tank's impedance soars. Feed it through a resistor and it passes f₀ while shorting other frequencies to ground. That's how an old radio's dial picks one station: turning it changes C.

A series LC is the mirror image: at f₀ its reactances cancel, its impedance falls to just the wire's resistance, and current peaks. Off resonance, one reactance or the other blocks it.

How sharp: Q

Resistance slowly drains a ringing tank. The quality factor, Q, says how sharp the peak is: the band it passes is about f₀ ÷ Q wide. A high-Q tank rings for many cycles and picks one frequency out of its neighbours.

Resonance is everywhere. Contactless bank cards resonate at 13.56 MHz, metal detectors listen for their tank's frequency shifting, and the quartz crystal in a watch is a tiny, very sharp resonator at 32,768 Hz.