Reactance calculator

The reactance of a capacitor, X_C = 1/(2πfC), and of an inductor, X_L = 2πfL, at any frequency, or the frequency at which either reaches a reactance you need. Add the capacitor's ESR and ESL, or the inductor's DC resistance and winding capacitance, to see where the real part departs from the ideal one and where it self-resonates.

Fig 1 — |X| against frequency, both axes logarithmic; at 1.00 kHz, X_C of 470 nF = 339 Ω; X_L of 10.0 µH = 62.8 mΩ. The lines cross at f₀ = 73.4 kHz, where both are 4.61 Ω. Thin: |Z| of the real part with its parasitics.

Solve for

Frequency f (kHz)

Capacitance C (0 to leave out) (nF)

Inductance L (0 to leave out) (µH)

Voltage across the part V (0 to skip) (V rms)

Capacitor ESR (0 to skip) (mΩ)

Capacitor ESL (0 to skip) (nH)

Inductor DCR (0 to skip) (mΩ)

Inductor winding capacitance C_P (0 to skip) (pF)

Capacitive reactance X_C of 470 nF at 1.00 kHz: 339 Ω · −90° · 2.95 mA at 1.00 V
Inductive reactance X_L of 10.0 µH at 1.00 kHz: 62.8 mΩ · +90° · 15.9 A at 1.00 V
Where X_L = X_C, the LC resonance: 73.4 kHz · both 4.61 Ω, √(L/C)
|Z| of the real capacitor, ESR 40.0 mΩ and ESL 14.3 nH: 339 Ω at −89.993° · DF 0.0118 %
Capacitor self-resonance, where X_ESL = X_C: 1.94 MHz · |Z| = ESR there, inductive above

How this is calculated

Standard: Würth Elektronik ANP109 — Impedance Spectra of Different Capacitor Technologies (rev b); ADI MT-101 — Decoupling Techniques; TI SLOA069 — How (Not) to Decouple High-Speed Operational Amplifiers; Würth Elektronik ANP045 — High Frequency SMT Chip Bead Ferrites

$X_C = \frac{1}{2\pi f C}, \qquad X_L = 2\pi f L$: ANP109 states the capacitive term as "Capacitive Reactance: Xc = 1/(2 π f Cs)"; its equivalent circuit (A1) carries the inductive term as iωL with ω = 2πf. MT-101 names both: "the reactance of the capacitor (1/ωC)" and "the reactance of the ESL (ωESL)". The current through a capacitor leads the voltage by 90°, through an inductor it lags by 90°, and its magnitude is V/X.
$f = \frac{1}{2\pi X C}, \qquad f = \frac{X}{2\pi L}$: The same two relations solved for the frequency at which a part reaches a target reactance.
$f_0 = \frac{1}{2\pi\sqrt{L C}}, \qquad X_0 = \sqrt{\frac{L}{C}}$: Where X_L = X_C. ANP109 Eq 6 gives it as the characteristic frequency of the L–C unit, MT-101 Eq 1 as the self-resonance, and SLOA069 as F_SR: "the reactance from CO and LS are equal and opposite, yielding a net reactance of zero." MT-101: "0.1 µF and 1 nH will resonate at 16 MHz."
$|Z_C| = \sqrt{ESR^2 + \left(2\pi f\, ESL - \frac{1}{2\pi f C}\right)^2}, \qquad DF = \frac{ESR}{X_C} = 2\pi f C\, ESR$: The capacitor as ANP109's series model, leakage neglected. ANP045 Eq 1 writes the magnitude as |Z| = √(R² + (X_L − X_C)²). The dissipation factor is ANP109's, stated "to improve comparability with datasheets"; at self-resonance |Z| falls to the ESR alone.
$Z_L = R_{DC} + \frac{j 2\pi f L}{1 - (2\pi f)^2 L C_P}, \qquad Q = \frac{2\pi f L}{R_{DC}}$: The inductor as ANP045's equivalent circuit less its parallel loss resistance: a series resistance ahead of L shunted by the winding capacitance C_P, which resonates with L at 1/(2π√(LC_P)) and makes the part capacitive above it. Q is MT-101 Eq 3, "a measure of its reactance to its resistance".

Assumptions

A single sine wave in steady state. A square wave or an edge contains many frequencies, and each sees a different reactance.
C, L and the parasitics are constants, which ANP109 calls "sufficiently accurate for electrical engineering". A class II ceramic under DC bias, or an inductor near saturation, is not at its marked value; enter the value at the operating point.
The capacitor's leakage resistance is neglected, as ANP109 says it usually can be. Its effect shows only at very low frequencies, below about 1 Hz.
The ESR is one number. ANP109 notes that measured ESR varies with frequency and is most trustworthy around the self-resonance.
Only the series self-resonance of a capacitor is modelled. SLOA069 describes parallel resonances above it, from plate-to-plate and plate-to-plane capacitance, which this model does not include.
The inductor model has no core loss. A ferrite bead is dominated by exactly that loss near its resonance; the ferrite filter calculator models it.

What sets capacitive and inductive reactance

Reactance is the part of an impedance that stores energy instead of dissipating it: a capacitor and an inductor hand the energy back each half cycle, and how much current they pass for a given voltage depends on how fast that voltage changes. Würth's ANP109 puts both into one equivalent circuit and states the capacitive term plainly: "Capacitive Reactance: Xc = 1/(2 π f Cs)". The inductive term sits in the same note's impedance expression as iωL, with ω = 2πf. ADI's MT-101 names the pair in a single sentence, "the reactance of the capacitor (1/ωC)" and "the reactance of the ESL (ωESL)".

The two move in opposite directions. X_C is inversely proportional to frequency and capacitance: a capacitor blocks DC and becomes a short at high frequency, which is why it decouples a supply pin and sets the corner of an RC filter. X_L is directly proportional to both: an inductor passes DC and chokes high frequencies, which the bead in aferrite filter relies on. On the figure's log-log axes each is a straight line of one decade per decade.

Reactance also carries a phase: the current through an ideal capacitor leads the voltage by 90°, through an ideal inductor it lags by 90°. ANP109: |Z| "represents the ratio of the voltage amplitude to the current amplitude, while ϕ gives the phase difference between voltage and current at a given frequency." The calculator reports the current an RMS voltage drives, I = V/X; being 90° out of phase, it does no net work.

Where the two lines cross, X_L = X_C at 1/(2π√(LC)), and both equal √(L/C). The same crossing exists inside every real capacitor, between its capacitance and its own series inductance, and past it the capacitor is an inductor. TheLC resonance calculator works the crossing in detail.

Capacitive and inductive reactance chart

X_C and X_L for the values in a parts drawer, computed by the calculator above. These are the ideal part's numbers; a 100 µF capacitor or a 1 mH inductor at 100 MHz is well past its self-resonance, and the sections below show where that starts.

Capacitive reactance X_C

C \ f	50 Hz	1 kHz	100 kHz	1 MHz	100 MHz
1 nF	3.18 MΩ	159 kΩ	1.59 kΩ	159 Ω	1.59 Ω
10 nF	318 kΩ	15.9 kΩ	159 Ω	15.9 Ω	159 mΩ
100 nF	31.8 kΩ	1.59 kΩ	15.9 Ω	1.59 Ω	15.9 mΩ
1 µF	3.18 kΩ	159 Ω	1.59 Ω	159 mΩ	1.59 mΩ
10 µF	318 Ω	15.9 Ω	159 mΩ	15.9 mΩ	159 µΩ
100 µF	31.8 Ω	1.59 Ω	15.9 mΩ	1.59 mΩ	15.9 µΩ

Inductive reactance X_L

L \ f	50 Hz	1 kHz	100 kHz	1 MHz	100 MHz
10 nH	3.14 µΩ	62.8 µΩ	6.28 mΩ	62.8 mΩ	6.28 Ω
100 nH	31.4 µΩ	628 µΩ	62.8 mΩ	628 mΩ	62.8 Ω
1 µH	314 µΩ	6.28 mΩ	628 mΩ	6.28 Ω	628 Ω
10 µH	3.14 mΩ	62.8 mΩ	6.28 Ω	62.8 Ω	6.28 kΩ
100 µH	31.4 mΩ	628 mΩ	62.8 Ω	628 Ω	62.8 kΩ
1 mH	314 mΩ	6.28 Ω	628 Ω	6.28 kΩ	628 kΩ

At 50 Hz even 100 µF is 31.8 Ω, which is why mains filtering needs bulk capacitance. And 100 nF is 15.9 mΩ at 100 MHz only on paper: by then the part's own inductance has taken over.

Worked example: ANP109's 470 nF film capacitor

The calculator's default is the film capacitor ANP109 measures in its section 2.3. The note's Table 2 lists the imaginary part of its impedance at three frequencies, 338.63 Ω at 1 kHz, 33.86 Ω at 10 kHz and 0.75 Ω at 450 kHz, which are exactly X_C of 470 nF. (The table's column head reads 4.7 nF; the values are those of the 470 nF part the section measures.) The same section gives the resonance, "a sharp minimum at fLC = 1.94 MHz", and about 0.04 Ω of ESR there.

X_C at 1 kHz    = 1 / (2π × 1 kHz × 470 nF)          = 338.6 Ω   (ANP109: 338.63 Ω)
X_C at 10 kHz   = 1 / (2π × 10 kHz × 470 nF)         = 33.86 Ω   (ANP109: 33.86 Ω)
X_C at 450 kHz  = 1 / (2π × 450 kHz × 470 nF)        = 0.753 Ω   (ANP109: 0.75 Ω)
I at 1 V rms    = 1 V / 338.6 Ω                      = 2.95 mA, leading by 90°
ESL             = 1 / ((2π × 1.94 MHz)² × 470 nF)    = 14.3 nH   (derived, not stated)
|Z| at 1 kHz    = √(0.04² + (90 µΩ − 338.6 Ω)²)      = 338.6 Ω at −89.993°
DF at 1 kHz     = 2π × 1 kHz × 470 nF × 0.04 Ω       = 0.0118 %

At 1 kHz the parasitics are invisible. The note measured 2.2 Ω of ESR at 1 kHz (its stated DF of 0.68 %), but treats the low-frequency rise as very likely a measurement artefact and says "the values around or at fLC are most trustworthy"; the calculator uses the 40 mΩ from the resonance.

The 10 µH inductor is a hand-worked case, as no source in the library gives a numerical X_L: 62.8 mΩ at 1 kHz, so 1 V would drive 15.9 A, limited in practice only by its DC resistance and the source. The lines cross at 1/(2π√(10 µH × 470 nF)) = 73.4 kHz, where both are 4.61 Ω. Solving for a 1 Ω target instead gives 339 kHz for the capacitor and 15.9 kHz for the inductor, and 73.4 kHz is their geometric mean.

Where the reactance formulas stop being valid

At the capacitor's self-resonance. ANP109: "Below this frequency the capacitor acts as capacitor, i.e. can be charged. Above this frequency, the capacitor acts as inductor." TI's SLOA069 describes the same point from the other side: "the reactance from CO and LS are equal and opposite, yielding a net reactance of zero", leaving only the ESR. The table shows where that falls. Half a nanohenry is what ANP109's 22 nF MLCC resonance implies; 1 nH is the figure MT-101 uses in its "0.1 µF and 1 nH will resonate at 16 MHz"; SLOA069 puts common PCB traces at "between 6 nH and 12 nH per centimeter", so a centimetre of trace to a via lands a part between the last two columns, whatever its package.

C \ ESL	500 pH	1 nH	2 nH	5 nH	20 nH
1 nF	225 MHz	159 MHz	113 MHz	71.2 MHz	35.6 MHz
10 nF	71.2 MHz	50.3 MHz	35.6 MHz	22.5 MHz	11.3 MHz
100 nF	22.5 MHz	15.9 MHz	11.3 MHz	7.12 MHz	3.56 MHz
1 µF	7.12 MHz	5.03 MHz	3.56 MHz	2.25 MHz	1.13 MHz
10 µF	2.25 MHz	1.59 MHz	1.13 MHz	712 kHz	356 kHz
100 µF	712 kHz	503 kHz	356 kHz	225 kHz	113 kHz

ANP109 measured three technologies and gives each one's capacitance, resonance and ESR. The ESL each implies is not in the note; it follows from L = 1/((2πf)²C), computed here from the measured capacitance.

Part	C measured	f_LC	ESR at f_LC	ESL implied
22 nF MLCC (WCAP-CSGP)	23 nF	45.8 MHz	60 mΩ	525 pH
470 nF film (WCAP-FTBE)	495 nF	1.94 MHz	40 mΩ	13.6 nH
270 µF aluminium electrolytic (WCAP-AIG8)	265 µF	68.5 kHz	40 mΩ	20.4 nH

The electrolytic resonates thirty times lower than the film capacitor on a similar ESL, because its capacitance is five hundred times larger. SLOA069 says as much of electrolytics: "Their self-resonant frequency is limited to a range between 100 kHz and 1 MHz."

At the inductor's self-resonance. Adjacent turns of a winding form a capacitance across the part. Würth's ANP045 models a chip bead with that parallel capacitance and shows that past the self-resonance "the ferrite bead becomes capacitive", its reactance falling instead of rising. A power inductor's datasheet gives the same thing as an SRF.

Off the marked value, or off a sine wave. A class II ceramic loses capacitance under DC bias and an inductor loses inductance as it saturates. And reactance is defined for one frequency in steady state; for a step, thecapacitor charge time calculator is the right view.

Common reactance calculation mistakes

Using ω where f belongs, or the other way round. X_C = 1/(2πfC) with f in hertz; writing 1/(fC) is out by a factor of 6.28, and so is entering a simulator's radian frequency as hertz.
Losing a prefix. A nanofarad entered as a microfarad is a factor of a thousand. The capacitance converter handles pF, nF, µF and the three-digit codes on the part.
Trusting the ideal X_C above self-resonance. The chart gives 100 µF at 100 MHz as micro-ohms; with the ESL ANP109's electrolytic implies, the real part is 12.8 Ω there. SLOA069's "Misconception number 2" follows: when high-frequency decoupling is poor, a larger capacitor does not help.
Adding reactances as magnitudes. In series, X_L and X_C subtract, which is how the net reaches zero at resonance; a resistance adds in quadrature, so 40 mΩ beside 339 Ω changes nothing visible.
Designing to the resonance itself. ANP109 notes that the parasitic inductance "may change as function of the length of circuit path or temperature", and the resonance moves with it. Thedecoupling calculator holds a target impedance across a whole band instead.