Gapped Core Inductance Calculator
Gapped Core Inductance Calculator
Effective permeability, Aₗ and inductance for an inductor wound on a gapped ferrite core — with the peak flux density against the material’s saturation limit, which is the number that actually decides the design, the stored energy, and McLyman’s fringing correction, which this page checks against a manufacturer’s own published gapped Aₗ figures.
Gap, Aₗ, inductance and the flux density that decides it
an ETD 39/20/13 in N87 — Aₑ 125 mm², lₑ 92.2 mm, µᵣ 2,200 — with a 1.00 mm gap, 40 turns and 4.5 A peak
Two reluctances in series, and the flux that leaks round the gap
AL = µ₀ · Ae ÷ (lg + le/µᵣ) L = AL · N²
B̂ = L · Î ÷ (N · Ae) = µ₀ · N · Î ÷ (lg + le/µᵣ)
F = 1 + (lg ÷ √Ae) · ln(2G ÷ lg) (McLyman), and AL is multiplied by it
E = ½ L β , all of it in the gap once the gap dominates
- mu e
- the effective permeability the winding sees. With a 1 mm gap in a 92 mm path it is about 88, not 2,200 — the gap has taken over
- AL
- inductance per turn squared, usually quoted in nH. A gapped core is normally sold by its Aₗ rather than by its gap
- F
- McLyman’s fringing factor. G is the winding length beside the gap: the longer the winding, the further the flux can bulge and the more inductance you get
- B peak
- the number that decides the design. Note it does NOT fall when you add turns at a fixed current — it rises
Worked example
an ETD 39/20/13 in N87 — Aₑ 125 mm², lₑ 92.2 mm, µᵣ 2,200 — with a 1.00 mm gap, 40 turns and 4.5 A peak
The gap is 1.00 mm and the core path contributes lₑ/µᵣ = 0.0419 mm, so the gap is 96.0% of the reluctance and µₑ has fallen from 2,200 to 88.5
That gives Aₗ = 150.8 nH/turn² on the simple model; McLyman's fringing factor is 1.3524, so the real figure is 203.9 nH/turn² — and TDK publishes 196 nH for this exact core at this exact gap, which the corrected model is 4.0% above and the uncorrected one 23.1% below
40 turns therefore give 326.2 µH — 241.2 µH if you ignore fringing
At 4.5 A peak the flux density is L·I ÷ (N·Aₑ) = 293.6 mT, which is 75.3% of the 390 mT hot saturation figure; 5.98 A would take it all the way there
The winding stores 3.303 mJ, and 96.0% of that is in the air gap, not in the ferrite — which is the whole reason a gap is there
This page’s model against TDK’s published gapped Aₗ for the ETD 39/20/13 in N87
| Gap | TDK’s published Aₗ | Simple reluctance model | With McLyman fringing |
|---|---|---|---|
| 0.50 mm | 326 nH | 289.9 nH (-11.1%) | 349.9 nH (+7.3%) |
| 1.00 mm | 196 nH | 150.8 nH (-23.1%) | 203.9 nH (+4.0%) |
| 2.00 mm | 115 nH | 76.9 nH (-33.1%) | 121.6 nH (+5.7%) |
What a gap does, on the default core
| Gap | µₑ | Aₗ with fringing | Turns for 326 µH | Peak current to saturate |
|---|---|---|---|---|
| 0.10 mm | 650 | 1,169 nH | 17 | 2.54 A |
| 0.25 mm | 316 | 602 nH | 24 | 3.59 A |
| 0.50 mm | 170 | 350 nH | 31 | 4.63 A |
| 1.00 mm | 88 | 204 nH | 40 | 5.98 A |
| 2.00 mm | 45 | 122 nH | 52 | 7.77 A |
| 4.00 mm | 23 | 74 nH | 67 | 10.01 A |
The gap is the inductor; the ferrite is just the return path
An ungapped ferrite core makes a wonderful transformer and a hopeless inductor. Its permeability is enormous, so a handful of turns gives a huge inductance — and the flux density that goes with it saturates the core at a fraction of an amp. Cut a gap in the magnetic circuit and everything changes. The gap’s reluctance is in series with the ferrite’s, and because air is 2,200 times worse at carrying flux, even a tenth of a millimetre of gap in a 92 mm path dominates the sum. The inductance falls, the turns go up, and the current it takes to saturate goes up with them.
Three things follow. First, the inductance stops depending on the permeability. A gapped design is insensitive to a µᵣ that is specified ±25% and that moves with temperature and drive level, because the gap is doing almost all the work — the page prints what share. Second, the energy goes into the gap. Integrate B²/2µ over the volumes and almost all of ½LI² turns out to be sitting in the air, which is exactly why an inductor needs a gap and a transformer does not: an inductor’s job is to store energy, and ferrite is bad at storing it. Third, and least intuitively, adding turns does not reduce the flux density. At a fixed current, B = µ₀NI ÷ (lg + le/µᵣ) rises with N. More turns get you more inductance and less headroom, not more.
Fringing. The simple model pretends the flux crosses the gap in a neat column of area Ae. It does not: it bulges outwards, the effective area is larger than Ae, the gap’s reluctance is lower than the model says, and the real inductance is higher. On the default core, checked against TDK’s own published gapped Aₗ values, the uncorrected model is 11% low at a 0.5 mm gap and 33% low at 2 mm. McLyman’s correction — one plus the gap divided by the square root of the core area, times the natural log of twice the winding length over the gap — brings all three within 8%. That is good enough to design with, and it is a very long way better than ignoring it.
Fringing has a second consequence the inductance does not capture. The bulging flux passes through whatever is beside the gap, which is usually copper, and it induces eddy currents in it. A winding that sits right across a large gap can lose more to that than to everything else put together — which is why a gap is often split into several smaller ones, why the winding is kept back from the gap, and why a distributed-gap powdered core is worth considering when the gap gets long.
This page is the complement of the toroid inductor turns calculator, which starts from a data sheet Aₗ and does not model the gap at all: use that one when the core is sold to you with an Aₗ, and this one when you are choosing the gap yourself or checking the manufacturer’s figure. For the loss in the core see the Steinmetz core loss calculator, for the loss in the winding the litz wire calculator, and for the converter that needs the inductor in the first place the buck converter designer or the flyback converter calculator.
Frequently asked questions
Why does an inductor need an air gap?
Because its job is to store energy, and ferrite stores almost none. Energy density is B²/2µ, so a material with a permeability of 2,200 stores 2,200 times less energy per cubic metre at the same flux density than air does. The gap is where the energy actually lives; the ferrite is only there to carry the flux back round to the other side of it.
Does adding turns stop an inductor saturating?
No — it makes it worse. At a fixed current, flux density goes as µ₀NI ÷ (gap + core path ÷ µᵣ), so every turn you add raises B. What lowers B is a longer gap, a larger core area, or less current. Turns raise the inductance, and the inductance is not what saturates.
How much does fringing flux increase the inductance?
Typically 10 to 35% at practical gaps, and more as the gap gets longer relative to the core’s cross-section. On an ETD 39/20/13 checked against TDK’s own published figures it is 21% at a 0.5 mm gap, 35% at 1 mm and 58% at 2 mm. Ignoring it is the single biggest error in a hand-calculated gapped inductor.
Is a 0.5 mm spacer the same as a 0.5 mm gap?
Usually not. A spacer between two ungapped core halves puts a gap in every leg, so the flux crosses two of them on its way round: a 0.5 mm spacer is a 1 mm total gap in the magnetic circuit. A centre-leg-ground core has the whole gap in one place and the outer legs closed. Enter the total the flux actually crosses.
Which saturation flux density should I use?
The hot one. N87 is 490 mT at 25 °C and 390 mT at 100 °C; 3C90 is 470 and 380 mT. A design that uses the cold figure has about 25% less margin than it looks like it has, and it will pass on the bench and fail in the box.
Why doesn’t this page ask for the material’s Aₗ?
Because it is working the other way round — from the geometry and the permeability to the Aₗ, so you can choose a gap or check a manufacturer’s number. If your core already comes with an Aₗ, the toroid inductor turns calculator on this site starts from that instead, and the two are meant to be used together.
Related calculators
References
- TDK, Ferrites and accessories — ETD 39/20/13. The core data used throughout: Σ(l/A) = 0.74 mm⁻¹, lₑ = 92.2 mm, Aₑ = 125 mm², Aₘᵢₙ = 123 mm², Vₑ = 11,500 mm³, 60 g per set, and the published gapped Aₗ values in N87 this page’s model is checked against — 326 nH at 0.50 mm, 196 nH at 1.00 mm and 115 nH at 2.00 mm, against 2,700 nH ungapped.
- Ferroxcube, ETD39/20/13 data sheet. The same core’s coil former data, from which the 25.7 mm minimum winding width used as G in the fringing factor and the 69 mm average length of turn are taken.
- McLyman C W T. Transformer and Inductor Design Handbook, 4th ed., chapter 8 (DC inductor design using gapped cores). The fringing flux factor F = 1 + (l_g/√A_c)·ln(2G/l_g) and the revised turns calculation that uses it.
- TDK, SIFERRIT material N87. µᵢ 2,200 ±25%, Bs 490 mT at 25 °C and 390 mT at 100 °C, Curie temperature above 210 °C — the permeability and saturation figures used as defaults here.