How this instrument works
An LC circuit rings at a single frequency because the inductor and capacitor keep swapping the same packet of energy: the capacitor discharges into the coil, building a magnetic field, and that collapsing field recharges the capacitor with reversed polarity, over and over. How fast that trade happens depends only on L and C — how much energy each part can store per unit of current or voltage — which is exactly what f = 1 ⁄ (2π√(LC)) computes. The relation is sometimes called Thomson's formula, after William Thomson (Lord Kelvin), who worked out the oscillation period of a capacitor discharging through a coil in 1853, decades before radio existed to put it to use.
The formula falls out of setting two reactances equal. Inductive reactance, X_L = 2πfL, grows with frequency; capacitive reactance, X_C = 1 ⁄ (2πfC), shrinks with it. There is exactly one frequency where the two match in size, and because they push current in opposite phase, they cancel there rather than add — leaving a circuit that either passes one narrow band freely or blocks it hard, depending on whether L and C sit in series or in parallel with the load. That is the real distinction between an LC filter and a plain RC one: an RC stage only rolls off gradually, since it has a single reactive element fighting a resistor, while an LC pair genuinely resonates, producing a sharp peak or a deep notch.
The formula assumes an ideal, lossless pair, which is never quite true. Real inductors carry winding resistance, and that resistance sets the circuit's quality factor, Q = (1 ⁄ R)√(L ⁄ C) for a series arrangement — a low Q broadens and flattens the peak this formula predicts as razor-sharp. Radio engineers lean on that sharpness to pull one station out of a crowded band; power-supply designers place the resonance well below a converter's switching frequency so ripple gets attenuated instead of amplified; loudspeaker crossovers use the same math to hand bass and treble to the right driver. In every case, the number this instrument returns is the centre of the action, not the whole story.
- Enter the coil's value in Inductance, H — the field expects henries directly, so a typical 1 mH choke is entered as 0.001.
- Enter the capacitor's rating in Capacitance, then set its unit menu to nF, µF, or mF to match what is printed on the part.
- Read the result in Resonant frequency; switch between Hz, kHz, and MHz depending on whether the circuit sits at mains-ripple or radio frequencies.
- If the page flags an error, check that Inductance, H is above zero — the formula divides by the square root of L times C, so a zero or negative inductance has no defined answer.
Worked example — a 1 mH choke with a 10 µF capacitor
Take a 1 mH inductor, a common choke value, paired with a 10 µF capacitor — a combination that turns up in switching-power-supply output filters. The instrument works in henries and farads, so L = 0.001 and C = 0.00001 (10 µF) go straight in. The product LC is 1 × 10⁻⁸; its square root is 1 × 10⁻⁴; and 1 divided by 2π × 10⁻⁴ comes out to 1591.54943092 Hz, which the results panel rounds to 1.5915 kHz.
That number matters to whoever picked those parts. A designer sizing a post-regulator filter for a buck converter switching near 100 kHz wants this resonance sitting comfortably below the switching frequency and above the audio band, so ripple gets knocked down rather than reinforced — 1591.5 Hz clears both marks easily. Because the formula assumes zero resistance, it predicts an infinitely sharp peak at that frequency; the actual inductor's winding resistance will round that peak off and set how much ripple truly survives.
Questions
Why does quadrupling the capacitance only halve the frequency?
Because f depends on the inverse square root of LC, not on LC directly. Multiply C by four and √(LC) doubles, so f is cut in half — raising this circuit's 10 µF to 40 µF while holding L at 1 mH drops the resonance from 1,591.5 Hz to 795.8 Hz, not to a quarter of the original value.
How is an LC filter different from a simple RC filter?
An RC filter has only one reactive part, so it just rolls off gradually past its cutoff. An LC filter has two reactances that cancel exactly at one frequency, so it genuinely resonates — producing a sharp peak in a parallel arrangement or a deep notch in a series one, rather than a smooth slope.
Does this formula account for the inductor's winding resistance?
No — it assumes an ideal, lossless L and C, which is why it predicts an infinitely sharp resonance. Real coils carry resistance that sets the circuit's quality factor, Q = (1 ⁄ R)√(L ⁄ C) for a series RLC; a lower Q rounds off and broadens the peak this instrument computes as a single point.
What happens if I set Inductance, H to zero?
The instrument flags it as invalid. The formula divides by the square root of L times C, and a zero inductance makes that square root — and the whole result — undefined, so the field requires a value greater than zero before it will compute a frequency.
Where does f = 1 ⁄ (2π√(LC)) actually come from?
From setting the two reactances equal: inductive reactance 2πfL grows with frequency, capacitive reactance 1 ⁄ (2πfC) shrinks with it, and they match in magnitude at exactly one frequency. Solving that equality for f gives the formula — a result William Thomson (Lord Kelvin) worked out in 1853 for a capacitor discharging through a coil.
What units does the Capacitance field expect?
Whatever is printed on the part. The field carries its own menu for nF, µF, and mF, so a 10 µF capacitor can be entered as 10 with µF selected rather than converted to farads by hand; the instrument performs that conversion internally before applying the formula.