LC resonance calculator
Series and parallel tank circuits, solved in any direction: the resonant frequency from the parts you have, or the parts you need for a frequency and impedance you want — with the Q factor, the −3 dB bandwidth, the impedance at any frequency, and the resonance curve those numbers actually describe.
All the loss is modelled as resistance in series with the inductor, which is where it nearly always lives. See choosing real components.
| Frequency | Impedance | Response | Phase |
|---|
Response is the output relative to its own peak at resonance — the current through a series circuit, or the voltage across a parallel tank. Impedance is what the source sees looking in. The highlighted row is your test frequency; the rest are the −3 dB edges and the skirts either side.
How this is calculated
An inductor's reactance rises with frequency and a capacitor's falls. There is exactly one frequency where the two are equal and, being opposite in sign, cancel. That is resonance:
f0 = 1 / (2π·√(L·C))
At that frequency both reactances equal the characteristic impedance of the pair:
Z0 = √(L/C) = XL = XC at f0
Those two numbers describe the tank completely. f0 says where it resonates and Z0 says how much energy it swings while doing so. A 100 µH / 1 nF tank and a 1 mH / 100 pF tank share a resonant frequency and behave nothing alike, because their Z₀ differs by a factor of ten.
Nothing above mentions resistance, and that is why nothing above tells you how the circuit behaves near resonance. Real parts have loss — mostly the inductor's winding resistance and core loss, plus the capacitor's ESR. Put all of it in one series resistance R and you get the quality factor and the bandwidth:
Q = Z0 / R = (1/R)·√(L/C) BW−3 dB = f0 / Q
The response curve of every resonant circuit, series or parallel, is the same universal shape once you write it in terms of Q and the frequency ratio x = f/f0:
|H| = 1 / √(1 + Q²·(x − 1/x)²) φ = −arctan(Q·(x − 1/x))
The impedance column in the table is computed separately and exactly from the parts, not from that normalised curve: |Z| = √(R² + (XL − XC)²) for the series circuit, and |Z| = XC·√(R² + XL²) / √(R² + (XL − XC)²) for a parallel tank whose inductor carries the loss.
Series or parallel — they are opposites
Same two parts, same resonant frequency, opposite behaviour, and mixing them up is the most common mistake on this subject.
A series LC is a short circuit at resonance. The reactances cancel and only R is left, so the impedance hits a minimum and the current a maximum. Drive it from a voltage source and the voltage across the inductor alone reaches Q times the applied voltage — the reactance is still there, it is just cancelled by an equal and opposite one next to it. At Q = 100 a 1 V input puts 100 V across a capacitor that may well be rated for 50. Series resonance is used to pass one frequency to ground or to a load, and it is what an unintended series LC does to a power rail.
A parallel LC — a tank — is an open circuit at resonance. Its impedance peaks at the dynamic resistance Rdyn = L/(C·R) = Q·Z0 = Q²·R, which is where the surprising numbers come from: 2 Ω of coil resistance in a Q = 158 tank looks like 50 kΩ. Current circulates round the L–C loop at Q times the current the source supplies, and the source only has to make up the losses. Parallel resonance is used to reject one frequency (a trap or notch), to select one (a tuned amplifier's load), or to sustain an oscillation.
Both magnifications are the same physical fact seen from two directions: at resonance the tank stores Q/2π times as much energy per cycle as it loses, so something in it — voltage or current — has to be Q times larger than what you put in.
Reading Q and bandwidth
Q is the only number that matters once f₀ is set, and it means three things at once. It is the sharpness of the resonance: BW = f0/Q, so Q = 100 at 10 MHz gives a 100 kHz window. It is the magnification factor described above. And it is how long the circuit rings: struck once and left alone, a tank's amplitude decays to 4.3 % after Q cycles, and halves after 0.22·Q of them.
The two −3 dB edges are not symmetrical about f₀ on a linear scale. Their geometric mean is exactly f₀ — flo·fhi = f0² — so the upper edge always sits slightly further away than the lower one. Above Q ≈ 10 the difference is too small to matter and the usual approximation f0 ± BW/2 is fine; below that it is visible, and the table above uses the exact edges either way.
Q is also not a constant of the parts. It is Z0/R, and R itself climbs with frequency: skin effect raises the winding resistance roughly as √f, and core loss faster still. A coil quoted at Q = 80 has that Q at the one frequency in the datasheet, and usually less elsewhere. If you need a specific bandwidth, measure the assembled circuit rather than trusting a product of two datasheet numbers.
Choosing real components
For a given f₀ you may pick any L and C whose product is right, and that freedom is the actual design decision — it sets Z₀. A high-Z₀ tank (more inductance, less capacitance) reaches a higher Q for the same coil resistance and swings more voltage, but a few picofarads of stray board and probe capacitance become a real fraction of C, so the frequency drifts with everything nearby. A low-Z₀ tank is stable and immune to strays but demands a very low resistance to reach the same Q. In RF work Z₀ usually lands between about 50 Ω and 500 Ω for exactly this reason.
Then there is tolerance, and here resonance is unusually forgiving: because f₀ depends on √(L·C), a component error is halved on its way to the frequency. A 10 % capacitor moves f₀ by 5 %. Two 10 % parts at their worst-case extremes move it by 10 %, not 20 %. Put the precision in the capacitor — a C0G/NP0 or silvered-mica part at 1–5 % with a tiny temperature coefficient — because capacitance is far easier to buy accurately than inductance, and because a Class-2 ceramic (X7R, Y5V) drifts with temperature and applied voltage badly enough to move a tuned circuit out of band on its own.
The calculator gives the exact part and then the nearest E12 (10 % steps) and E24 (5 %) catalogue value, with the resonant frequency that value really produces. Where the maths asks for something between two steps, note that trimming is normal practice: a fixed capacitor near the target plus a small trimmer, or a slug-tuned coil, is how real tuned circuits are brought onto frequency.
Finally, every real part stops being what it says on the label at high enough frequency. A capacitor's own lead and plate inductance gives it a self-resonant frequency, above which it is an inductor; an inductor's inter-winding capacitance does the same in reverse. Operate a tank near either part's SRF and the calculated f₀ is simply wrong. Keep both SRFs well above f₀ — a factor of ten is a comfortable rule — and check the datasheet rather than assuming.
FAQ
My tank resonates at the right frequency but the Q is far below the calculation. Why?
Because Q is set by resistance and you have more of it than you entered. Candidates, in the order they usually turn out to be at fault: the inductor's AC resistance at your frequency rather than its DC resistance; core loss, if it is wound on ferrite or iron powder near its frequency limit; the source impedance driving a series circuit, which adds directly to R; and the load across a parallel tank, which is in parallel with Rdyn and can dominate it completely. That last one catches everyone — a 50 kΩ tank feeding a 10 kΩ input has a loaded Q barely a fifth of its unloaded one.
What is the difference between loaded and unloaded Q?
Unloaded Q is the resonator on its own, set by the parts' own losses — that is what this page calculates. Loaded Q includes whatever the circuit is connected to. For a parallel tank, put the source and load resistances in parallel with Rdyn and recompute; for a series circuit, add them to R. Loaded Q is always the lower of the two and is the one that determines the bandwidth you actually measure. Designing a filter means choosing the loading deliberately to get the bandwidth you want, and then the unloaded Q only has to be comfortably higher than that.
How do I convert a datasheet coil Q into a resistance?
Use R = 2π·f·L / Q at the frequency the datasheet specifies. A 100 µH coil quoted at Q = 80 at 500 kHz has an effective series resistance of about 3.9 Ω there — many times its DC resistance, which is why the DC figure flatters your bandwidth. The "Coil Q" input above does this conversion for you at f₀.
Does the resistor position matter — series with L, series with C, or across the tank?
For the resonant frequency and Q, barely; for the shape of the impedance curve away from resonance, yes. Loss in the inductor branch and loss in the capacitor branch produce the same Q to well within measurement error at any useful Q, so lumping both into one series R — as this page does — is the standard model. A resistor deliberately placed across a parallel tank is a different thing: it is a damping resistor, it appears in parallel with Rdyn, and it is how you set a bandwidth on purpose rather than accept the one the parts happen to give.
Why is my parallel tank's impedance peak slightly below f₀?
Because a tank with a lossy inductor is not quite the idealised parallel RLC. Its unity-power-factor resonance sits at f0·√(1 − 1/Q²), and the frequency of maximum impedance is a third, slightly different point. At Q = 10 that is a 0.5 % shift, at Q = 3 about 6 %, and above Q = 20 it is unmeasurable. This page reports the standard f0 = 1/(2π√(LC)) and the standard dynamic resistance L/(C·R), which is exact at that unity-power-factor point; below Q ≈ 7 it also shows the exact magnitude at f₀ so the two are never confused.
Can I use this for a speaker crossover or a notch in an audio circuit?
For a notch or a trap, yes — a parallel LC across a signal path blocks its resonant frequency, and a series LC to ground shunts it away, with the depth set by Q and the loading. For loudspeakers use the crossover calculator instead: a driver's impedance rises with its voice-coil inductance and swings wildly around its own resonance, so it is neither the constant resistance nor the ideal load these formulas assume, and crossover design starts from aligned second- and fourth-order topologies rather than from a single tank.
How does this relate to the RC filter calculator?
They sit either side of a real boundary. A first-order RC or RL filter has one reactive part, one pole, a 6 dB per octave slope and no resonance — it can never peak. An LC has two, and the pair can exchange energy, which is what makes a peak and a Q possible at all. Past resonance an LC's skirts fall at 12 dB per octave, twice as fast, which is the other reason to reach for one.
What does a negative or very low Q mean?
Q below about 0.5 means the circuit is overdamped: there is enough resistance that it no longer rings at all, the impedance curve has no useful peak or dip, and calling it a resonant circuit stops being meaningful. That is a legitimate design point for a damped supply filter, where ringing is the enemy — but if you wanted selectivity, it means R is far too large for the Z₀ you chose. Raise Z₀ (more L, less C) or lower the resistance.
What is the highest Q I can realistically build?
With ordinary parts, an air-cored or good ferrite coil and a decent capacitor, a few hundred is achievable and 100–200 is routine. Litz wire and careful layout push it further. Beyond about a thousand, LC stops being the right technology and the field moves to quartz crystals (Q in the tens of thousands to over a million), ceramic resonators, and cavity or dielectric resonators at microwave frequencies. If your design needs Q = 5,000, you need a different kind of resonator, not a better coil.