LC resonance: the frequency where an inductor and a capacitor agree
An inductor's reactance rises with frequency, a capacitor's falls, so for any pair there is one frequency where the two are equal. There the energy swaps back and forth between the coil's magnetic field and the capacitor's electric field with nothing to stop it: the circuit resonates. That frequency, f₀ = 1 / (2π√LC), is what radio tuners, oscillators, filters and wireless chargers are built around. This calculator finds it, draws both reactances crossing, gives the Q and bandwidth for a known loss resistance, and works out the capacitor or inductor needed to hit a frequency you want.
How to use the LC resonance calculator
- Type the inductance (10m,
100u, 4.7mH) and the capacitance (100n, 100p, 1uF). f₀ and the reactance at resonance appear at once, and the graph shows the two reactance lines crossing. - Optional: a series resistance — mostly the coil's winding resistance — gives the Q, the bandwidth and the −3 dB limits, and draws the impedance dip on the graph.
- Optional: a target frequency works out the capacitor that hits it with your inductor (or the inductor with your capacitor); the buttons put that value straight into the calculator.
- Standard capacitors lists E12 values near yours with the f₀ each gives; Calculation writes the maths out; click the coil or the capacitor in the drawing for what changing it does.
The formula
f₀ = 1 / (2π √(LC)) Z₀ = √(L / C)
Set the two reactances equal, 2πfL = 1 / (2πfC), and solve for f. The product LC in henry-farads has the units of seconds squared, so its square root is a time and the reciprocal a frequency. At f₀ both reactances equal Z₀ = √(L/C), the characteristic impedance of the pair, which sets how large the circulating voltages and currents are compared with what drives them.
Your values f₀ = 1 / (2π √(10 mH × 100 nF)) = 5.033 kHz, where XL = XC = 316.2 Ω.
Series and parallel resonance
The same f₀ does opposite things depending on how the parts are connected. In series, the voltage across the coil leads the current by 90° and the voltage across the capacitor lags by 90°, so at f₀ they are equal and opposite and cancel: the pair looks like a short circuit (only the loss resistance remains) and the current peaks — an acceptor circuit, used for notch filters and as the low-impedance path in crystal-like oscillators. In parallel, it is the two currents that cancel: the pair looks like an open circuit and the voltage across it peaks — a rejector or tank circuit, the heart of radio tuning and LC oscillators. The drawing here is the series case.
Q and bandwidth
Q = Z₀ / R BW = f₀ / Q
A perfect LC pair would ring for ever; real coils have resistance that drains the energy each cycle. The quality factor Q compares the reactance at resonance with that loss: Q = Z₀ / R for a series circuit. It says two things at once: the voltage across L or C at resonance is Q times the driving voltage, and the response is within 3 dB of its peak over a band f₀ / Q wide. A tuned radio stage wants Q of 50–200 to separate stations; a filter that must pass a whole band wants a lower Q, which is why a resistor is sometimes added deliberately.
| R | Q | Bandwidth | Character |
|---|---|---|---|
| 1 Ω | 316.23 | 15.9 Hz | sharp: a good tuned circuit |
| 10 Ω | 31.62 | 159 Hz | moderate: a usable filter |
| 100 Ω | 3.16 | 1.59 kHz | broad: barely selective |
| 1 kΩ | 0.32 | 15.9 kHz | overdamped: no real resonance |
Designing for a frequency
C = 1 / ((2πf)² L) L = 1 / ((2πf)² C)
Usually one part is fixed by what you have or can make, and the other is chosen to land on the frequency. Because f₀ depends on the square root of the product, the frequency is forgiving: a 10% error in C moves f₀ by only 5%. The ratio L/C is the other design choice: for the same f₀ a larger L with a smaller C gives a higher Z₀ and so a higher Q for the same loss, but big coils have more resistance and stray capacitance; a smaller L with a larger C is lower impedance and less sensitive to the stray capacitance of wiring and tracks. Standard capacitors come in the E12 series (10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 and their decades), so the last bit of tuning is done with a trimmer.
Your values, step by step
- Resonance: √(10 mH × 100 nF) = 31.62 µs; 1 / (2π × that) = 5.033 kHz.
- Reactance there: √(L / C) = 316.2 Ω for both parts; period 1 / f₀ = 198.7 µs.
Worked example: 10 mH and 100 nF
LC = 0.010 × 100 × 10⁻⁹ = 10⁻⁹ s², so √(LC) = 31.62 µs and f₀ = 1 / (2π × that) = 5.033 kHz. At that frequency each part has a reactance of √(0.010 / 10⁻⁷) = 316.2 Ω. If the coil has 10 Ω of resistance, Q = 31.62 and the bandwidth is 159.2 Hz. For an AM radio instead: 240 µH with a 100 pF trimmer gives 1.027 MHz, and opening the variable capacitor to 365 pF brings it down to 537.7 kHz, the bottom of the band.
Questions
Does the resistance change the resonant frequency?
Hardly. A series resistance leaves f₀ exactly where it is and only broadens the peak. A resistance in parallel with the capacitor shifts it very slightly, and only matters when Q is below about 2.
Why does a crystal behave like an LC circuit?
A quartz crystal is mechanically resonant, and electrically it looks like a series LC with an enormous L, a tiny C and very little loss: Q in the tens of thousands, which no coil and capacitor can match. That is why crystals set the frequency in clocks and microcontrollers.
What limits the frequency range of an LC circuit?
At low frequencies the inductors get large and lossy; at high frequencies the coil's own stray capacitance and the capacitor's lead inductance take over, so above a few hundred megahertz the parts are replaced by transmission-line sections and cavities.
Can I resonate a transformer or motor winding?
Yes, and sometimes by accident: any inductance with a capacitor across it has an f₀. Power-factor-correction capacitors on motors and the capacitors across a transformer's windings can resonate with harmonics on the mains, which is a known cause of overheating.