Capacitor Self-Resonance Calculator
Capacitor Self-Resonance Calculator
Every capacitor is a series R-L-C, and above its self-resonant frequency it is an inductor. This gives the resonance, the impedance at the frequency you care about and whether the part is capacitive, resistive or inductive there — and then the anti-resonance between two paralleled capacitors, where the combination is worse than either one alone.
Self-resonance, impedance and the anti-resonance between two parts
a 10 µF bulk capacitor with 10 nH of ESL and 5 mΩ of ESR, with a 100 nF part (2 nH, 10 mΩ) beside it, examined at 2 MHz
A capacitor, and then two of them
fanti = 1 ÷ (2π√((L₁+L₂)·C₁C₂÷(C₁+C₂)))
|Z|peak = √((R₁R₂ + X²)² + X²(R₂−R₁)²) ÷ (R₁+R₂) ≈ X² ÷ (R₁+R₂)
where X is branch 1’s reactance at fanti, and branch 2’s is −X
- ESL
- the part’s terminal inductance plus the loop you mounted it in. The mounting normally dominates, which is why two identical capacitors on different footprints have different resonances
- f SRF
- below it a capacitor, at it a resistor of exactly its ESR, above it an inductor. A 10 µF part with 10 nH resonates near 500 kHz, which is well inside the conducted-emissions band
- f anti
- always between the two self-resonances. The larger part is inductive there and the smaller one is still capacitive, and they resonate with each other
- R1 + R2
- the only thing limiting the peak. Lower losses make a taller peak, which is the one context in which a low-ESR part is the wrong choice
Worked example
a 10 µF bulk capacitor with 10 nH of ESL and 5 mΩ of ESR, with a 100 nF part (2 nH, 10 mΩ) beside it, examined at 2 MHz
The bulk capacitor self-resonates at 1 ÷ (2π√(10 nH × 10 µF)) = 503.3 kHz, and at that frequency its impedance is exactly its ESR, 5 mΩ. Above it the part is an inductor — so across most of the 10 kHz to 10 MHz conducted-emissions band this “10 µF” is not a capacitor
The small part resonates much higher, at 11.25 MHz, because both its capacitance and its mounting inductance are smaller
At 2 MHz the bulk part presents 117.8 mΩ (inductive) and the small one 770.7 mΩ (capacitive), and in parallel they give 139 mΩ
Between the two resonances there is an anti-resonance: the bulk part's inductance against the small part's capacitance, at 1 ÷ (2π√((L₁+L₂)·C_series)) = 4.617 MHz. There the two reactances are equal and opposite, ±286.7 mΩ
And the impedance there is not small, it is 5.483 Ω — against 286.7 mΩ for either capacitor on its own, which is 25.6 dB WORSE than fitting just one of them. That is why adding a 100 nF next to a 10 µF sometimes makes a rail worse, and the fix is damping or several parts of the same value, not more capacitance
Where a capacitor stops being a capacitor
| Part and mounting | ESL | 10 µF | 1 µF | 100 nF | 10 nF |
|---|---|---|---|---|---|
| Axial electrolytic, long leads | 200 nH | 113 kHz | 356 kHz | 1.13 MHz | 3.56 MHz |
| Radial electrolytic, short leads | 20 nH | 356 kHz | 1.13 MHz | 3.56 MHz | 11.3 MHz |
| MLCC on pads with a long return loop | 10 nH | 503 kHz | 1.59 MHz | 5.03 MHz | 15.9 MHz |
| MLCC, tight pads, via beside each pad | 2 nH | 1.13 MHz | 3.56 MHz | 11.3 MHz | 35.6 MHz |
| MLCC, vias in the pads, thin dielectric | 700 pH | 1.9 MHz | 6.02 MHz | 19 MHz | 60.2 MHz |
It is a series R-L-C, and above resonance it is an inductor
Draw a capacitor as a capacitor and the impedance falls forever: 1/2πfC, on and on down. Real parts do not do this. Every capacitor has series resistance and series inductance — its own terminations, and the loop formed by the pads, the vias and the return path you mounted it in — so its impedance is ESR + j(2πf·ESL − 1/2πfC). That falls, reaches a minimum of exactly the ESR at the self-resonant frequency 1/(2π√(ESL·C)), and then rises again, because above resonance the part is an inductor.
The numbers are worse than people expect. A 10 µF bulk capacitor with 10 nH of total loop inductance self-resonates around 500 kHz. CE102 runs from 10 kHz to 10 MHz. That capacitor is therefore behaving as an inductor across most of the band it was fitted to fix, and no amount of extra bulk capacitance will change that — the extra part brings its own inductance with it. The inductance is usually set by the mounting rather than the component, which is why two identical capacitors on two different footprints resonate at different frequencies and why “add more vias” is real advice.
Then the part that earns the page. Put two capacitors of different values in parallel — the standard 10 µF plus 100 nF — and somewhere between their two self-resonances the larger part is inductive while the smaller one is still capacitive. They resonate with each other. That anti-resonance is a PARALLEL resonance, so the impedance does not dip, it peaks, and it peaks above what either capacitor would have presented on its own. The anti-resonant frequency is 1/(2π√((L₁+L₂)·C₁C₂/(C₁+C₂))) and it always sits between the two resonances; the peak is approximately X²/(ESR₁+ESR₂), where X is the branch reactance there. With low-ESR ceramics on both branches that peak can be tens of decibels above either part alone.
Which means the losses are the fix, not the capacitance. The peak height is inversely proportional to the sum of the two ESRs, so anything that raises them flattens it: a deliberately lossier small capacitor, a few tens of milliohms in series with the small branch, or a controlled-ESR part. Past a certain point that stops helping, because the added resistance becomes the impedance — there is an optimum, and this page lets you find it by changing the second ESR and watching the peak. The other answer, and usually the better one, is to parallel several capacitors of the SAME value rather than a decade-spaced set: identical parts have identical resonances, so there is nothing for them to anti-resonate against, and N of them give N times the capacitance with ESL/N and ESR/N.
The measurement point matters as much as the part. If you are chasing a conducted-emissions problem, the limit is measured on the power leads entering the unit, through a LISN — not on an internal rail. A reader looking at a noisy 3.3 V rail usually needs the filter at the 28 V or mains input, where the measurement actually happens. Decoupling an internal rail better is worth doing for the circuit’s sake and may do nothing at all for the emissions measurement. Above roughly 10 to 30 MHz the answer stops being component values and becomes layout: the loop the capacitor is mounted in, where the heatsink is bonded, how the harness is routed. A component-value prediction carried to 100 MHz without that caveat is wrong.
What this page is not. It is not the ripple and life calculation: for ESR heating, the three components of ripple voltage and the Arrhenius life model, see the capacitor ripple and ESR calculator, which covers those and does not do impedance against frequency or anti-resonance. For what the capacitance really is once DC bias has had its share, the MLCC DC bias derating calculator — worth running first, because the resonance moves with the real capacitance and not the marked one. For the resonance of an L and a C you chose deliberately, the LC resonant frequency calculator.
Frequently asked questions
What is the self-resonant frequency of a capacitor?
1/(2π√(ESL·C)), where ESL is the part’s series inductance plus the loop you mounted it in. Below it the part is a capacitor, at it the impedance is exactly the ESR — the lowest it will ever be — and above it the part is an inductor whose impedance rises with frequency. A 10 µF part with 10 nH resonates near 500 kHz.
Why does adding a 100 nF next to a 10 µF sometimes make things worse?
Anti-resonance. Between the two parts’ self-resonances the 10 µF is inductive and the 100 nF is still capacitive, so they form a parallel resonant circuit, and a parallel resonance is a peak in impedance, not a dip. At that frequency the pair can be tens of decibels worse than either capacitor on its own. This page computes where the peak is and how high.
How do I damp an anti-resonant peak?
Raise the losses in one branch. The peak height goes roughly as X²/(ESR₁+ESR₂), so a deliberately lossier small capacitor, or a few tens of milliohms of real resistance in series with it, flattens it — at the cost of a slightly higher impedance away from the peak. Beyond a point it stops helping because the added resistance becomes the impedance. Change the second ESR above and watch the peak to find the compromise.
Is it better to parallel several identical capacitors or a spread of values?
For a power rail, usually several identical ones. N identical parts give N times the capacitance, ESL/N and ESR/N, and because they all resonate at the same frequency there is nothing for them to anti-resonate against. A decade-spaced set buys a wider frequency coverage at the cost of an anti-resonant peak between every adjacent pair.
Does ESL come from the capacitor or from the board?
Usually mostly from the board. The part’s own terminal inductance for a small MLCC is well under a nanohenry; the loop formed by the pads, the vias and the return path through the planes is typically several. That is why the same part on two footprints resonates at different frequencies, and why reverse-geometry and via-in-pad constructions exist.
My 10 µF capacitor is not fixing my conducted emissions — why?
Two likely reasons, and this page and its companion cover both. Above about 500 kHz a 10 µF part with an ordinary mounting is an inductor, so it is not shunting anything. And the capacitance may not be 10 µF: a Class II ceramic under DC bias can retain a small fraction of its marked value. Also check where you are measuring — the conducted limit applies at the unit’s power input through a LISN, not on an internal rail.
Related calculators
References
- Cornell Dubilier, Aluminum Electrolytic Capacitor Application Guide. The ESL figures used in the table: 10–30 nH for radial-leaded parts, 20–50 nH for screw terminals and up to 200 nH for axial, with the note that the winding itself is usually under 2 nH — so the inductance is the terminations and the loop, not the capacitor.
- Thiadmer Riemersma, CompuPhase, Parallel Capacitors and the effect of Antiresonance (2022). Vector network analyser measurements of a 100 nF and a 100 pF in parallel showing “two resonant frequencies, with a peak halfway clearly demonstrating antiresonance”, and the conclusion that “mounting multiple capacitors of the same type in parallel will generally improve decoupling, whereas mixing capacitor types for decoupling may well be counter-productive” — which is the measured confirmation of what the closed forms on this page predict.
- The anti-resonant frequency and peak impedance printed here were checked, when this page was built, against a 200,001-point numerical search for the true maximum of the parallel impedance magnitude of two series R-L-C branches. The closed form for the frequency agrees to within 0.02% and for the peak to within 0.004%, and the damping claim was verified the same way by sweeping the second branch’s ESR.
- MIL-STD-461G, Requirements for the Control of Electromagnetic Interference Characteristics of Subsystems and Equipment, 11 December 2015, superseding MIL-STD-461F. A US Department of Defense interface standard and a work of the US Government, distributed without charge. Read from the document: paragraph 5.5.1 makes CE102 applicable from 10 kHz to 10 MHz on all power leads including returns that take power from a source outside the equipment, and paragraph 5.5.3 measures it at the signal output port of the LISN of Figure 6. That is the band across which a bulk capacitor with an ordinary mounting is an inductor. The limit itself, Figure CE102-1, is implemented on the conducted emissions margin page.