LC resonance calculator

The frequency an inductor and a capacitor resonate at, or the L or C that puts the resonance where it is wanted, with the characteristic impedance, the reactance of each part at resonance, and the Q that decides how far an LC filter amplifies at f₀ before it attenuates. Resistance in series, a load across the capacitor and a damping resistor across the inductor are all in the model.

Fig 1 — f₀ = 11.3 kHz, Z₀ = 70.7 mΩ, Q = 7.44: the filter amplifies by 17.5 dB at 11.2 kHz before it attenuates.

Solve for

Inductance L (µH)

Capacitance C (µF)

Series resistance R (mΩ)

Load resistance R_L (0 for none) (Ω)

Damping resistor R_d across L (0 for none) (mΩ)

Resonant frequency f₀: 11.3 kHz
Angular frequency ω₀: 70.7 krad/s
Impedance Z₀ = √(L/C): 70.7 mΩ
X_L = X_C at f₀: 70.7 mΩ each
Period at f₀: 88.9 µs
Q from R alone, Z₀/R: 7.44
Q of the circuit: 7.44
Damping ratio ζ: 0.067
Gain at f₀: +17.4 dB
Peak gain: +17.5 dB at 11.2 kHz

Underdamped: noise near 11.2 kHz comes out 17.5 dB larger than it went in. SNVA801: "If LC filter is not well damped, the frequency response will peak near resonant frequency, which means the LC actually is amplifying the noise signal." A resistor of Z₀ = 70.7 mΩ across the inductor gives Q = 1 on its own.

How this is calculated

Standard: Würth ANP062 — LC Filter Design With MLCCs, Eq 1 (resonant frequency); TI SNVA801 — Analysis and Design of Input Filter for DC-DC Circuit, §4.1 Eq 17 and 21 (transfer function and damping ratio); TI SNVA871 — Output Noise Filtering for DC/DC Power Modules, §3.1 (second-stage filter and damping resistor); Würth ANP109 appendix and TI SLOA069 (reactance at resonance)

$f_0 = \frac{1}{2\pi\sqrt{L\,C}}, \qquad \omega_0 = 2\pi f_0 = \frac{1}{\sqrt{L\,C}}$: ANP062 Eq 1: "The cutoff frequency of a LC filter can be determined using the following formula." The note works it for 1.5 µH and 10 nF as 1.3 MHz. ANP109 gives the angular form for a capacitor and its own inductance.
$L = \frac{1}{(2\pi f_0)^2\, C}, \qquad C = \frac{1}{(2\pi f_0)^2\, L}$: Eq 1 solved for either part, for the solve modes.
$X_L = 2\pi f L, \qquad X_C = \frac{1}{2\pi f C}, \qquad X_L(f_0) = X_C(f_0) = Z_0 = \sqrt{L/C}$: X_C as ANP109 writes it. At resonance, in SLOA069's words, "the reactance from CO and LS are equal and opposite, yielding a net reactance of zero"; substituting f₀ into either reactance gives the characteristic impedance.
$H(s) = \frac{1 + s L G_d}{L C (1 + R G_d)\, s^2 + \left[ C R + L G_d + L G_L (1 + R G_d) \right] s + (1 + R G_L)}$: The circuit evaluated: series resistance R (source, DCR and ESR), L with an optional damping resistor across it (G_d = 1/R_d, SNVA871), and C with an optional load across it (G_L = 1/R_L). With G_d = 0 this is SNVA801 Eq 17, divided through by the load, with its converter resistance −R_in replaced by a passive load and R_ESR lumped into R. The figure and the peak are |H(j2πf)|, the peak found numerically.
$\zeta = \frac{R_{in} C (R_O + R_{DCR} + R_{ESR}) - L}{2\sqrt{L C R_{in} (R_{in} - R_O - R_{DCR})}}, \qquad Q = \frac{1}{2\zeta}$: SNVA801 Eq 21 as printed, for a converter load −R_in. The calculator takes ζ = a₁/(2√(a₀a₂)) from the denominator above. Replacing −R_in with a passive load R_L flips the sign of every coefficient of Eq 17 together, and this equation then gives the same magnitude. Each resistance alone gives the familiar limits Q = Z₀/R, Q = R_L/Z₀ and Q = R_d/Z₀.
$|H(f_0)| = Q, \qquad |H|_{max} = \frac{Q}{\sqrt{1 - 1/(4Q^2)}} \;\text{at}\; f_0\sqrt{1 - \frac{1}{2Q^2}}$: For the unloaded filter with no damping resistor, from the transfer function: at f₀ the s² and constant terms cancel and the gain is 1/(ω₀RC) = Q. A peak exists only above Q = 1/√2; at Q = ½ the gain at f₀ is −6 dB.

Assumptions

Ideal parts: the inductor has no self-resonance and the capacitor no series inductance. SNVA871 lists 68 MHz and 38 MHz self-resonances for its 1 µH and 2.2 µH inductors; above them the attenuation shown is not reached.
The capacitor's ESR is lumped with the series resistance. Unloaded that is exact; with a load, the ESR's true position across the output changes the result only in proportion to ESR/R_L.
Values at the operating point. ANP062 measures a 2.2 µF, 25 V ceramic losing 69 % of its capacitance at rated voltage and a 47 µF, 6.3 V one losing 80 %; enter the derated C, and the inductance at the DC current.
The load is a passive resistor. A switching regulator behind the filter presents a negative input resistance that reduces the damping instead (SNVA801); for that case the unloaded result is not the worst case.
Small-signal and linear. The filter's effect on a regulator's control loop, when it sits inside the feedback path, is outside the model.

What sets an LC circuit's resonant frequency

An inductor and a capacitor pass energy back and forth: the capacitor's charge drives a current through the inductor, the inductor's field collapses and recharges the capacitor the other way, and the exchange has a natural rate. The two reactances say where it is. An inductor's grows with frequency, a capacitor's, which Würth's ANP109 writes as X_C = 1/(2π·f·C), falls with it, and the two lines cross at exactly one frequency. There, in the words of TI's SLOA069, "the reactance from CO and LS are equal and opposite, yielding a net reactance of zero." That crossing is the resonant frequency f₀, and ANP062 gives it as its Equation 1, introducing it with "the cutoff frequency of a LC filter can be determined using the following formula".

Two things follow from the square root. First, f₀ depends only on the product L·C: 1 µH with 200 µF and 2 µH with 100 µF resonate at the same frequency. Second, moving f₀ is expensive. Halving it needs four times the product, and a capacitor that loses three quarters of its value to DC bias doubles it.

What the product does not fix is the ratio, and the ratio is the second number the calculator reports: the characteristic impedance Z₀ = √(L/C), which is also the reactance of each part at f₀. It is the yardstick every resistance in the circuit is measured against. A resistance in series with the inductor damps the resonance in proportion to R/Z₀; a resistance across the capacitor or the inductor damps it in proportion to Z₀/R. The quality factor Q collects the lot into one number, and for the low-pass filter the calculator draws it is also the gain at resonance: the unloaded filter's output at f₀ is Q times its input.

That is the point that matters for filtering. TI's SNVA801 puts it plainly: "If LC filter is not well damped, the frequency response will peak near resonant frequency, which means the LC actually is amplifying the noise signal." Würth's ANP045 says the same of an LC low-pass ahead of an ADC: "Instead of attenuating noise, the resonance peak is a reason that causes the noise to amplify." A filter built from a low-DCR inductor and ceramic capacitors has milliohms of resistance against a Z₀ of tens of milliohms, so a Q of ten or more is the normal outcome, not a corner case.

The Q expressions are the limits of SNVA801's Equation 21, the damping ratio of an LC filter with source resistance, DCR, ESR and a load. The calculator solves the whole second-order circuit rather than adding the separate Qs, because they do not add.

LC resonant frequency chart

f₀ for the inductors and capacitors a filter or a tank is built from, computed with the calculator's own function. The top left is RF; the bottom left, around a microhenry against tens of microfarads, is where the power-supply filters live.

C \ L	100 nH	1 µH	10 µH	100 µH	1 mH
10 pF	159 MHz	50.3 MHz	15.9 MHz	5.03 MHz	1.59 MHz
100 pF	50.3 MHz	15.9 MHz	5.03 MHz	1.59 MHz	503 kHz
1 nF	15.9 MHz	5.03 MHz	1.59 MHz	503 kHz	159 kHz
10 nF	5.03 MHz	1.59 MHz	503 kHz	159 kHz	50.3 kHz
100 nF	1.59 MHz	503 kHz	159 kHz	50.3 kHz	15.9 kHz
1 µF	503 kHz	159 kHz	50.3 kHz	15.9 kHz	5.03 kHz
10 µF	159 kHz	50.3 kHz	15.9 kHz	5.03 kHz	1.59 kHz
100 µF	50.3 kHz	15.9 kHz	5.03 kHz	1.59 kHz	503 Hz

Every decade step in either part moves f₀ by √10, a factor of 3.16, so two rows down is one decade lower in frequency, and so is two columns to the right. A value that is not in the table sits between its neighbours in the same ratio: 4.7 µF is √4.7 = 2.17 times lower in f₀ than 1 µF.

How resistance sets the peak

SNVA871's second-stage filter, 1 µH and 200 µF, run through the calculator with the resistances that damp it in practice. Z₀ is70.7 mΩ for every row; only the damping changes, and with it the height of the peak. f₀ itself does not move; the peak slides a little below it as the damping grows.

Damping	Q	Gain at f₀	Peak
1 mΩ, the ESR in SNVA871's simulation	70.7	+37.0 dB	+37.0 dB at 11.3 kHz
9.5 mΩ, adding the inductor's 8.5 mΩ DCR	7.44	+17.4 dB	+17.5 dB at 11.2 kHz
9.5 mΩ, with the 0.825 Ω load of 3.3 V at 4 A	4.57	+13.1 dB	+13.2 dB at 11.2 kHz
9.5 mΩ, with 500 mΩ across the inductor	3.66	+11.3 dB	+11.4 dB at 10.9 kHz
9.5 mΩ, with 100 mΩ across the inductor	1.24	+3.2 dB	+3.9 dB at 9.44 kHz
100 mΩ, √2 × Z₀: Q = 1/√2	0.71	−3.0 dB	none
141 mΩ, 2 × Z₀: Q = ½, critical damping	0.50	−6.0 dB	none

The same calculation reproduces SNVA871's own plots. Its Figure 7 simulates the filters with nothing but the capacitor's 1 mΩ of ESR and shows a spike that reaches a little over +40 dB; that is the 2.2 µH/100 µF filter, whose Z₀ of 148 mΩ against 1 mΩ gives Q = 148, 43.4 dB. Its Figure 8 adds a 100 mΩ damping resistor across the inductor and the two curves, read off the plot, top out near +5 dB and +2 dB; the calculator gives 4.9 dB and 1.9 dB for the same values. A resistor across the inductor has a cost its Q does not show: it lets high frequencies bypass L, so above R_d/(2πL) the roll-off falls from 40 dB per decade to 20, which is the shallower slope of that figure, and the peak sits a little higher than Q alone would put it.

Worked example: SNVA871's second-stage filter, 1 µH and 200 µF

The defaults are the filter SNVA871 simulates after a DC/DC power module, with the resistance a real build has: the 8.5 mΩ DCR the note lists for its 1 µH inductor, XAL5030-102ME, plus the 1 mΩ of ESR in its capacitor model.

f₀        = 1 / (2π × √(1 µH × 200 µF))          = 11.25 kHz   (SNVA871: "around 10 kHz")
ω₀        = 2π × 11.25 kHz                        = 70.7 krad/s
Z₀        = √(1 µH / 200 µF)                      = 70.7 mΩ
X_L       = 2π × 11.25 kHz × 1 µH                 = 70.7 mΩ = X_C
R         = 8.5 mΩ DCR + 1 mΩ ESR                 = 9.5 mΩ
Q         = 70.7 mΩ / 9.5 mΩ                      = 7.44
at f₀     = 20 × log₁₀ 7.44                       = +17.4 dB
peak      = 7.44 / √(1 − 1/(4 × 7.44²))          = 7.46, +17.5 dB at 11.20 kHz

The calculator shows f₀ = 11.3 kHz, Q = 7.44and a peak of +17.5 dB for these inputs. Noise near 11 kHz leaves this filter more than seven times larger than it arrived. Well above f₀ the attenuation climbs at 40 dB per decade, which is the job the filter was fitted for; the peak is the price, and SNVA871 names the case where it is paid: "Low frequency noise in the frequency range below this cutoff frequency can be amplified and pass through the LC filter", which matters in power savings mode, where "the frequency of the switching regulator folds back as load current decreases."

Solving backwards checks the arithmetic of a published example. ANP062 works its Equation 1 for a 1.5 µH ferrite and a 10 nF NP0 capacitor and gets 1.3 MHz; with the solve set to capacitance, 1.3 MHz and 1.5 µH return 9.99 nF. The same note gives the cutoff of the 2.2 µF version as 876 kHz. The formula gives 87.6 kHz, and the note's own list for the second board, 19 kHz for 47 µF down to 72 kHz for 3.3 µF, puts 2.2 µF just above 72 kHz, where 87.6 kHz sits. A slipped decimal point in a vendor note is a useful reminder to check the order of magnitude of any f₀ against a neighbour.

Where the LC resonance formula stops being valid

The parts have their own resonances. The formula assumes an inductor that is only inductance and a capacitor that is only capacitance. SNVA871 describes an inductor's self-resonant frequency as "the point when an inductor stops behaving like an inductor and instead behaves like a capacitor", and lists 68 MHz for its 1 µH part and 38 MHz for the 2.2 µH one. SLOA069 gives the capacitor's equivalent: a series self-resonance with its own lead and package inductance, above which "the impedance will be inductive". Above either frequency the filter's attenuation stops improving. SNVA871's Figure 7 shows both: its model gives the capacitor 1 nH and the inductor 2 pF, and the response levels off around −60 dB from about 1 MHz, then climbs back to about −20 dB by 1 GHz.

The capacitance is not the marked value. ANP062 is a whole note about this: a 2.2 µF, 25 V part in 0805 at its rated voltage lost 69 % of its capacitance, to 0.68 µF, and a 47 µF, 6.3 V X5R part fell to 10 µF. Since f₀ goes as 1/√C, the second case moves f₀ up by √4.7, 2.17 times. The note's Table 2 shows the resonance its measurement picks out for that filter moving from 950 kHz to 2.0 MHz, a ratio of 2.1: a different resonance from the 19 kHz of Equation 1, obeying the same square-root law. Temperature adds to it: X7R is allowed ±15 % over its range, on top of a delivery tolerance the note puts at typically ±10 %.

The inductance is not constant either. A ferrite bead or a gapless core loses inductance as DC current saturates it. ANP062 shows a bead's impedance falling with current and leaves the effect out of its analysis; the effect on f₀ is the mirror of DC bias, upward again.

The load is not a resistor. A switching regulator downstream draws constant power, so its input behaves as a negative resistance at low frequency. SNVA801 derives this and shows it reducing the damping: with the loop open, its filter's peak is "much less than 20 dB", and with the loop closed "the peak value is close to 20 dB". The calculator's load field is a passive resistor and damps the peak; behind a converter, treat the unloaded result as the optimistic case, not the pessimistic one.

The filter is inside a loop. SNVA871 warns that placing the LC filter inside the regulator's feedback path "may cause instability in the form of oscillations coming from the resonant frequency of the LC filter." That is a control-loop question, which this calculator does not answer.

Common LC resonance mistakes

Reading f₀ as a −3 dB corner. An LC low-pass has no fixed attenuation at f₀: the unloaded gain there is Q. ANP062 describes the filter as having "a fall of -6dB at its cutoff point", which holds only at Q = ½, critical damping; the defaults above are +17 dB at the same point.
Assuming ceramic capacitors damp anything. Their ESR is milliohms, and the peak is set by the ratio of Z₀ to the total series resistance. Adding capacitance lowers Z₀ and, with the same resistance, lowers Q; adding inductance raises both.
Damping with a series resistor that carries the load current. It works, and it drops I·R of DC and dissipates I²R. SNVA871 puts the resistor across the inductor instead, where the inductor's DCR, a small fraction of it, carries nearly all the DC past it; SNVA801 puts it in series with a separate capacitor across the output, with the capacitor "much larger than C", suggesting C_d ≥ 4C, so that no DC flows in it at all. Theferrite bead filter calculatoris the same problem with a lossy inductor.
Putting f₀ near a frequency that is present. A post-filter tuned to land near the regulator's light-load switching frequency, or near an audio band the downstream circuit cares about, amplifies exactly what it was fitted to remove.
Sizing from the nominal capacitance of a small, high-value ceramic. SNVA871's bench build used 300 µF where the simulation used 200 µF, raised "in consideration for the derating at 3.3 V output". Enter the derated value, and expect f₀ to sit higher than the nominal parts suggest.
Trusting a lossless simulation in either direction. Without the DCR, the same filter peaks at +37 dB instead of +17 dB; without the inductor's self-resonance, it attenuates forever. Both are modelling choices, not the circuit.