An LC circuit, a coil and a capacitor connected together, has one special frequency where it naturally oscillates. At that resonant frequency the inductive and capacitive effects cancel, and the circuit passes or blocks signals most strongly. Radio tuners, filters, and oscillators all depend on this one number.
The formula
Resonant frequency f = 1 / (2 x pi x square root(L x C)). L is inductance in henries and C is capacitance in farads. The formula is sometimes called the Thomson formula. The result is in hertz.
Units are where most errors happen. Convert millihenries to henries (divide by 1,000) and microhenries to henries (divide by 1,000,000). Convert nanofarads to farads (divide by 1,000,000,000) and picofarads to farads (divide by 1,000,000,000,000). One missed conversion can shift the answer by a factor of 1,000.
Worked example
Take L = 10 mH and C = 100 nF. Convert: L = 0.01 H and C = 0.0000001 F. Multiply: L x C = 0.000000001. The square root is 0.00003162. Then f = 1 / (6.2832 x 0.00003162) = 1 / 0.00019868 = about 5,033 Hz, or 5.03 kHz.
Check the result by feel: bigger L or bigger C means a lower resonant frequency, because a heavier or slower circuit oscillates more slowly. If you double the capacitance, the frequency falls by the square root of 2, about 1.414 times, not by half. That square-root behaviour surprises many beginners.
What resonance means in practice
In a series LC circuit, impedance is lowest at resonance, so current is highest there. In a parallel LC circuit, impedance is highest at resonance, so the circuit blocks current there. Which behaviour you get depends on the wiring, but the resonant frequency formula is the same for both.
Real coils and capacitors have resistance and tolerance. A capacitor marked 100 nF might actually be 95 or 105 nF, which shifts the real resonant frequency a few percent from the calculated one. For a radio tuner that is why a variable capacitor is used: you tune until the circuit actually resonates.
Mistakes to avoid
Do not use the formula with L in mH and C in nF without converting; the answer will be wrong by a huge factor. Do not confuse resonant frequency with the frequency of your signal source; resonance is a property of the circuit, and the source may or may not match it. And remember the formula assumes ideal parts, so treat the answer as the design centre, then verify with a real measurement where it matters.
Working backwards: choose C for a target frequency
Often you know the frequency you want and one component, and must find the other. Rearrange the formula: C = 1 / ((2 x pi x f) squared x L). Suppose you want resonance at 10 kHz and already have a 1 mH inductor.
Compute: 2 x pi x 10,000 = 62,832. Square it: about 3,948,000,000. Multiply by L (0.001 H): about 3,948,000. Then C = 1 / 3,948,000 = 0.000000253 farads, or 253 nF. The nearest standard value, 250 nF, gives a resonant frequency very close to 10 kHz, and a small variable capacitor can trim the rest.
Try it faster with the tool
You can follow every step above by hand, but the free LC Circuit Resonance Frequency Calculator – Find Resonant Frequency Free does the arithmetic for you. Enter your numbers and it gives the answer right away, so you can check your hand calculation or skip straight to the result.
Quick tips
- Convert L to henries and C to farads first.
- Bigger L or C means a lower resonant frequency.
- Real part tolerances shift the true frequency a few percent.