How this instrument works
An LC circuit is two ways of storing the same energy trading places. Charge a capacitor and it holds energy in the electric field between its plates; push current through an inductor and it holds energy in the magnetic field around its windings. Left alone, with no resistance draining anything away, the two trade that energy back and forth — capacitor empties into inductor, the inductor's collapsing field refills the capacitor with the opposite polarity, and the cycle repeats. That trade happens at one specific rate set entirely by L and C, and that rate is what f₀ = 1 ⁄ (2π√(LC)) reports.
The formula is not an empirical fit; it falls straight out of Kirchhoff's voltage law. Writing the loop equation for a capacitor's charge q gives L(d²q/dt²) + q ⁄ C = 0 — the same shape as Newton's second law for a mass on a spring, m(d²x/dt²) + kx = 0, with inductance standing in for mass and the reciprocal of capacitance standing in for stiffness. A spring's angular frequency is √(k/m); swap in the electrical analogues and out comes ω₀ = 1 ⁄ √(LC), the natural rate in radians per second. Dividing by 2π converts that rotation rate into complete cycles per second, hertz — the only difference between the two numbers this page returns.
Radio tuning is the classic application: a receiver's front end is built around an inductor and a variable capacitor sized so f₀ sweeps across a broadcast band, and turning the dial shifts C until the circuit resonates with — and pulls out — one station's carrier while rejecting the rest. The formula itself assumes an ideal, lossless loop, though. Every real coil has wire resistance and every real capacitor has some equivalent series resistance, so an actual tank circuit's response peaks a hair off f₀ and its stored energy slowly bleeds away as heat, a behavior captured by a separate figure called Q. It is also easy to report ω₀ where f₀ was wanted, or the other way round — they differ by exactly 2π, roughly 6.28, and that factor is worth checking any time a resonance figure looks off by about that much.
- Enter the coil's value into Inductance, H. There is no unit menu on this field — it takes henries directly, so a 1 µH coil is typed as 0.000001 or 1e-6.
- Enter the part's value into Capacitance, then pick pF, nF or µF from the unit menu to match the marking on the component.
- Read Resonant frequency for f₀, switching the unit menu to kHz or MHz once the coil and capacitor are RF-scale.
- Read Angular resonant frequency for ω₀ in rad/s — the same physical rate, ready to drop straight into a phasor or impedance equation.
- To land on a target station or filter frequency, hold one component fixed and adjust the other until Resonant frequency reaches it.
Worked example — a 1 µH, 100 pF RF tank circuit
A ham-radio builder is winding a small air-core tank circuit for a shortwave antenna tuner: a 1 µH inductor, a handful of turns around a ceramic form, paired with a 100 pF variable capacitor set near its midpoint. Entering 0.000001 into Inductance, H and 100 into Capacitance (pF) gives f₀ = 1 ⁄ (2π√(1×10⁻⁶ × 1×10⁻¹⁰)) = 1 ⁄ (2π × 1×10⁻⁸) = 15,915,494.3092 Hz, read out as roughly 15.915 MHz — squarely inside the shortwave broadcast range, which is exactly what a coil and capacitor of this size are built to select.
Angular resonant frequency reads ω₀ = 1 ⁄ √(1×10⁻⁶ × 1×10⁻¹⁰) = 1 ⁄ (1×10⁻⁸) = 100,000,000 rad/s — exactly 2π times 15,915,494.3092 Hz, since one full cycle is 2π radians. Quadruple either component alone, swapping in a 4 µH coil or a 400 pF capacitor, and resonance falls to 7,957,747.15459 Hz, about 7.958 MHz: L and C enter the formula as a symmetric product under a square root, so halving the frequency takes a fourfold change in either one, never a doubling.
Questions
What actually happens physically at resonance in an LC circuit?
Energy stops flowing in any net direction and simply trades between the inductor's magnetic field and the capacitor's electric field once every cycle. At f₀ the two components' reactances are equal in size and opposite in sign, so in a series loop they cancel and only whatever resistance is present limits the current; in a parallel tank the two branch currents cancel instead and the impedance seen from outside peaks.
Why does the calculator give two different outputs, f₀ and ω₀?
Because a resonant rate can be reported two ways: cycles per second or radians per second. ω₀ = 1 ⁄ √(LC) is the natural angular rate the loop equation actually produces; f₀ = ω₀ ⁄ 2π restates the same physical rate in hertz, a count of full 360-degree turns per second. They describe one resonance, not two — divide ω₀ by 2π, about 6.283, and f₀ comes out every time.
What happens to the resonant frequency if I double the inductance?
It falls, but not by half. Because f₀ depends on the square root of the LC product, doubling only L multiplies the frequency by 1 ⁄ √2, about 0.707. The worked case here, 1 µH with 100 pF resonating at 15,915,494.3092 Hz, needs a fourfold change in L, to 4 µH, before the frequency actually halves, landing at 7,957,747.15459 Hz.
Does it matter whether L and C sit in series or in parallel?
Not for the frequency itself. f₀ = 1 ⁄ (2π√(LC)) gives the same natural rate whether the inductor and capacitor are wired in series or in parallel, since the value depends only on the LC product. What differs is behavior at that frequency: a series LC loop's impedance drops to a minimum at f₀, while a parallel LC tank's impedance rises to a maximum there, which is why the two arrangements get used for opposite jobs, trapping a frequency versus rejecting one.
Why does a real circuit resonate slightly differently from this figure?
Because the formula assumes zero resistance, and no real coil or capacitor is perfectly lossless. Wire resistance in the inductor and equivalent series resistance in the capacitor damp the oscillation, which in a driven RLC circuit nudges the peak-response frequency just below f₀ and broadens the response around it. The shift is tiny for a high-Q RF tank, often well under a tenth of a percent, but grows noticeable in lossier circuits such as large electrolytic filter stages.
What is this formula not valid for?
Circuits with more than one inductor or capacitor, or a source driven far from resonance. Combine multiple parts into one effective L or C first — this site's series and parallel inductor and capacitor instruments do that — before applying f₀ = 1 ⁄ (2π√(LC)). Real inductors also stop behaving as pure inductance above their own self-resonant frequency, which caps how high this formula can be trusted for a given part.