LC Resonance Frequency: How f = 1/(2π√(LC)) Tunes Every Radio and Filter

Put an inductor and a capacitor in a loop and you have built the most important two-component circuit in all of radio. Tap it once and it rings, the way a struck bell rings, at one specific frequency that depends only on the two part values. That frequency is the resonant frequency of the LC tank, and it sits underneath every AM dial, every FM front-end, every band-pass filter, and every Colpitts oscillator you have ever met.

The whole behavior collapses into one tidy equation:

f = 1 / (2π·√(L·C))

with L in henry and C in farad. Everything else in this article is just learning to read that line and trust the number it hands you.

What resonance actually means in an LC tank

An inductor resists changes in current; a capacitor resists changes in voltage. Energy sloshes between them, magnetic field in the coil to electric field in the cap and back again, like a pendulum trading speed for height. The coil's reactance climbs with frequency (X_L = ω·L) while the cap's reactance falls (X_C = 1/(ω·C)). There is exactly one frequency where the two are equal and opposite, and they cancel. That cancellation point is resonance.

At resonance the tank looks purely resistive to the outside world, the circulating current peaks, and the circuit happily oscillates with almost no prompting. Set the equality X_L = X_C and solve, and the 2π and the henries and farads rearrange themselves straight into f = 1/(2π·√(L·C)). The angular form, ω = 2π·f = 1/√(L·C), is the same fact written in radians per second instead of cycles per second. Engineers writing the differential equations reach for ω; anyone reading a tuning dial reads f.

The formula, slowly, with one worked example

Let me walk a clean example end to end, because the arithmetic is where most people slip. Take an inductor of L = 100 µH and a capacitor of C = 100 pF, a pairing you would genuinely see in a high-frequency tank.

First convert to SI: 100 µH = 1.0 × 10⁻⁴ H, and 100 pF = 1.0 × 10⁻¹⁰ F.

Multiply them: L·C = 1.0 × 10⁻⁴ × 1.0 × 10⁻¹⁰ = 1.0 × 10⁻¹⁴.

Take the square root: √(1.0 × 10⁻¹⁴) = 1.0 × 10⁻⁷.

Multiply by 2π: 2π × 1.0 × 10⁻⁷ ≈ 6.283 × 10⁻⁷.

Invert: f = 1 / (6.283 × 10⁻⁷) ≈ 1.59 × 10⁶ Hz, or about 1.59 MHz.

So 100 µH with 100 pF resonates just above the top of the AM broadcast band. Notice how the answer rides on the square root of the product: to double the frequency you do not halve a component, you cut it to a quarter. Drop the cap from 100 pF to 25 pF and the frequency doubles to roughly 3.18 MHz. That square-root relationship is exactly why variable tuning capacitors have such a wide mechanical sweep for what feels like a modest frequency span.

If hand-cranking exponents is not your idea of fun, the LC Resonance Frequency Calculator does this conversion-and-invert dance for you, autoscaling the output so 100 µH and 250 pF read cleanly as about 1.007 MHz instead of a wall of zeros. It also runs the equation backward, which is where it earns its keep on the bench.

Solving backward: the capacitor or inductor you actually need

Most real design does not start with two parts and ask for the frequency. It starts with a target frequency and one part you already own, and asks for the other. Rearrange the formula and you get two mirror-image results:

Capacitor for a target frequency: C = 1 / ((2π·f)²·L)
Inductor for a target frequency: L = 1 / ((2π·f)²·C)

Suppose you want a 1 MHz tuned circuit and you have a 100 µH coil already wound. Then (2π × 10⁶)² ≈ 3.948 × 10¹³, multiply by 100 × 10⁻⁶ to get 3.948 × 10⁹, and C ≈ 1 / 3.948 × 10⁹ ≈ 253 pF. Round to the nearest stock 250 pF part, allow a trimmer for the last few percent, and you are done.

The same move sizes a coil when the capacitor is fixed. For a classic 455 kHz intermediate-frequency stage with a 250 pF cap, L works out to about 0.49 H. Switching modes on the calculator instead of re-deriving the algebra each time is the difference between a five-second check and a fresh chance to drop a factor of 2π.

Where LC resonance shows up: radio, filters, oscillators

Three families of circuit lean on this one equation:

Radio tuning. When you turn an AM dial you are physically rotating a variable capacitor across the LC tank, dragging its resonance from 540 kHz to 1600 kHz so the receiver amplifies one station and rejects the rest. A 100 µH loop antenna paired with a roughly 90–800 pF variable cap covers the whole band. The tank's sharpness, its Q, decides how cleanly adjacent stations separate.

Filters. A passive LC band-pass or notch filter is built around a resonance placed exactly at the frequency you want to pass or kill. Plug the table values for L and C into the formula, confirm the center frequency lands in the passband, and only then commit to a board. If it is off, the inverse formula tells you which component to nudge and by how much.

Oscillators. Colpitts, Hartley, and Clapp oscillators all set their output frequency with an LC tank. You fix the inductor you have wound and solve for the capacitor that lands the output on spec, exactly the backward solve above.

A note from the bench

The first time the formula bit me, I was building a regenerative receiver and could not understand why my carefully calculated 1.0 MHz tank insisted on peaking nearer 940 kHz. The math was right; the breadboard was not lying either. The gap was stray capacitance, maybe twelve picofarads of it, hiding in the wiring and the coil's own self-resonance, quietly adding to my 250 pF cap. Once I accepted that the calculated number is the design center and not the bench truth, the fix was obvious: trim the variable cap down a touch and the peak walked right onto frequency. The ideal LC value gets you within a few percent every time, which on a real build is exactly close enough to start trimming rather than guessing.

That is the honest framing for this number. The formula assumes a pure inductor and a pure capacitor with no resistance, no leakage, no Q. A damped RLC circuit resonates a hair lower and rings with finite Q, so treat f = 1/(2π·√(L·C)) as the place to begin, then adjust for tolerance and stray capacitance on the bench.

Quick reference and a sibling tool

Three things to keep straight every time:

Watch your prefixes. A 100 µH coil is 1 × 10⁻⁴ H, not 1 × 10⁻⁶; a 250 pF cap is 2.5 × 10⁻¹⁰ F. One wrong dropdown shifts the frequency by orders of magnitude.
Never drop the 2π. f = 1/(2π·√(L·C)). Leaving out the 2π gives you the angular frequency ω, which is about 6.28× too large to report as f.
The square root rules everything. Frequency tracks 1/√(L·C), so component changes move it less than your intuition expects.

When you need to decode the cap value printed on a part before you feed it in, the capacitor code calculator turns a three-digit marking like "101" into 100 pF so you are entering the real value, not a guess. Pair it with the resonance calculator and the path from a bag of parts to a tuned circuit is a couple of clicks.

LC resonance is one of those rare ideas where a single short equation genuinely runs the show. Learn to read f = 1/(2π·√(L·C)), keep your units honest, and remember the bench always trims the last percent.

Made by Toolora · Updated 2026-06-13