The energy is in the gap

An inductor made on a ferrite core has a material in it chosen for one property: its permeability, which is a couple of thousand times that of air and which is the whole reason the core is there. Manufacturers then supply the same core cut into two halves with a deliberate gap between them, and a designer picks the gap.

That looks like undoing the choice, and it is not. It is a way of putting the energy somewhere that behaves better than the material does.

Reluctance in series #

The magnetic circuit is written the same way an electric one is, and this site’s habit of building things from primitives applies to it directly.

A path of length l, area Aₑ and permeability µ has a reluctance

$$\mathcal{R} = \frac{l}{\mu A_e}$$

which plays the part resistance plays: the flux is the magnetomotive force divided by it, exactly as a current is a voltage divided by a resistance. Reluctances in series add.

A gapped core is two reluctances in series: the ferrite’s lₑ/(µ₀µᵣAₑ) and the gap’s $l_g/(\mu_0 A_e)$. Note what the gap’s expression is missing — µᵣ, because the gap is air. So a gap two hundred microns long in a path sixty millimetres long, on a core whose relative permeability is 2000, contributes

$$\frac{\mathcal{R}_{\mathrm{gap}}}{\mathcal{R}_{\mathrm{core}}} = \frac{l_g\,\mu_r}{l_e} = \frac{200\times10^{-6}\times2000}{60\times10^{-3}} = 6.67$$

— nearly seven times the ferrite’s own reluctance, from a gap that is a three-hundredth of the path. The permeability is exactly what makes a short gap dominate: it divides the core’s contribution and does nothing to the gap’s.

The inductance is N²/ℛ, so it falls by the same factor: the core’s ungapped 41.89 mH becomes 5.46 mH at two hundred microns.

Where the energy is #

The stored energy divides between the two reluctances in proportion to them, so the fraction in the gap is

$$\frac{l_g}{l_g + l_e/\mu_r}$$

which is 87.0% for the numbers above.

Two things about that expression are worth reading carefully, and the gate holds both.

It contains no turns. Doubling the winding quadruples the inductance and quadruples the stored energy at a given current, and moves not one per cent of it out of the gap. The gate checks this by computing the share at 100 turns and at 400 and requiring them equal to 10⁻¹²; they agree to four decimal places of a percentage.

It contains no area. A bigger core stores more and stores it in the same proportion.

So the share is a ratio of two lengths and a permeability, and nothing about the winding or the size of the part enters. That is unusual enough on this site to be worth remarking on: almost every other result here depends on at least two component values, and this one depends on a geometric ratio.

Why put the energy in the air #

Because air has properties the ferrite does not, and every one of them is a property the ferrite lacks in a direction that matters.

Air does not saturate. The ferrite’s flux density is capped at about 0.35 T, and past it the permeability collapses — which is the next essay’s subject and is the hardest limit in this field. A gapped core reaches a given inductance with more current before the core reaches its cap, because the gap is carrying most of the magnetomotive force.

Air’s permeability does not drift. A ferrite’s µᵣ moves with temperature by tens of per cent and with the flux level, and varies from part to part by a quoted −20% to +30% on ordinary grades. A gapped core’s inductance is dominated by the gap, so it inherits the gap’s tolerance — a mechanical dimension, held to a few per cent — rather than the material’s.

Air has no loss. Hysteresis and eddy currents are properties of the material and scale with the flux in it. Moving the energy into the gap does not remove the flux from the core, but it does mean that a given stored energy corresponds to a smaller flux density there.

That third one has a limit, and the honest version is worth stating: the flux still passes through the ferrite, the loss still occurs, and the gap’s benefit is that a given energy is reached at a lower flux density rather than that the loss disappears. A figure claiming otherwise would be claiming a magnetic circuit can store energy without a field in it.

The same result read as an energy density #

There is a second way to arrive at the 87% and it is worth a paragraph, because it explains the number rather than computing it.

The energy density in a magnetic field is B²/2µ. The flux density B is continuous across the gap — it has to be, since the same flux passes through both — so the only thing that differs between the ferrite and the air is µ, and the air’s is two thousand times smaller. Therefore the energy density in the gap is two thousand times the energy density in the core, at the same instant, in a component whose two regions differ in length by a factor of three hundred.

Two thousand against three hundred is 6.67, which is the ratio computed above. The reluctance calculation and the energy-density calculation are the same statement, and the second one makes it obvious why a short gap wins: it is holding a vastly denser field over a slightly shorter distance.

That is also the cleanest way to see why the share contains no turns and no area. Both regions’ energies scale with the same B² and the same cross-section, so both scale together and the ratio does not move.

What the gap costs #

The trade is visible on the figure as the two curves crossing, and it is entirely the inductance.

At a millimetre the ferrite is holding 2.9% of the energy of a component it is 99.998% of by volume, and the inductance has fallen by a factor of thirty-four. Recovering that inductance means more turns, which is more copper, more winding resistance and more winding capacitance — and the winding resistance rises with frequency while the capacitance moves the self-resonance down.

So the gap is bought with copper and with bandwidth, and the exchange rate is steep. That is why gapped cores are the norm in energy-storage applications — a flyback transformer, a boost inductor, anything that deliberately stores energy in a magnetic field between one instant and another — and are absent from applications that only need coupling, where the whole point is to store as little as possible.

What the permeability slider shows #

The figure’s slider is the relative permeability, and moving it makes the essay’s central expression visible as a statement about the product $\mu_r l_g$ rather than about the gap alone.

At µᵣ = 200 the same two-hundred-micron gap holds only 40% of the energy, because the ferrite’s own reluctance is no longer negligible. At 15,000 it holds 98%.

So the higher the permeability, the more completely a gap of a given size takes over — which reads as a paradox and is not one. A high-permeability material has very little reluctance of its own, so almost any interruption of the path dominates it. The material’s job in a gapped design is to be absent magnetically over most of the path, and a better material does that job better.

The practical version, which is how the two levers interact: a designer who wants a particular inductance and a particular energy capacity picks the gap for the energy and then picks the permeability high enough that the gap is what decides. Once the ratio $\mu_r l_g/l_e$ is comfortably above one, the inductance depends on the gap and not on the grade of ferrite — and inherits a mechanical tolerance instead of a material one, which was the second reason for the gap in the first place.

The distinction the field turns on #

There is a difference between two uses of the same object that this essay’s arithmetic makes sharp, and it is worth stating because the word “transformer” covers both.

A transformer that transfers. Both windings conduct at once, the flux is whatever is needed to support the difference between the two ampere-turns, and the ideal is that the magnetising current — the energy stored — is as small as possible. Here a gap is a nuisance: it lowers the magnetising inductance, raises the magnetising current, and moves the lower band edge up.

A transformer that stores. One winding conducts, then the other. The energy is deliberately put into the field during one interval and taken out during the next, and the component is an inductor with two windings rather than a transformer at all. Here a gap is essential, because the ungapped core saturates long before it has stored anything useful.

The two are wound the same way and drawn the same way, and the arithmetic that distinguishes them is this essay’s: how much energy is in the field, and where.

What this model does not contain #

Three simplifications, all of them stated rather than discovered.

The permeability is a constant. It is not: it falls as the flux approaches saturation, so a real core’s inductance is a function of its current. That non-linearity is exactly what saturation is, and this essay’s linear reluctance model is a small-signal statement about a core well below it — the same relationship the small-signal essay has with an exponential device, and with the same requirement that the boundary be quoted.

The gap’s flux is confined to the gap. It is not: flux bulges outward at a gap — fringing — which lowers the effective reluctance below the simple $l_g/\mu_0 A_e$ and raises the inductance above what this model gives, by a few per cent at a small gap and by tens of per cent at a large one. Handling it properly needs a field solution rather than a circuit one, which is a different subject from this site’s.

The path length is one number. A real core has a cross-section that varies along it and a path that is not equally travelled, and the “effective” length and area a manufacturer quotes are themselves the results of a field calculation. They are taken here as given, which is what a data sheet is for.

None of the three changes the essay’s result, because all three affect the two reluctances in ways that are small compared with the factor of 6.67 between them. What they would change is the third significant figure of the inductance, and that is stated rather than quietly claimed to be exact.

What a designer actually chooses #

The figure’s two curves are what a designer trades between, and the choice is usually made from the other end — from how much energy has to be stored — so it is worth running the arithmetic in that direction once.

An inductor storing energy W at a peak current I needs L = 2W/I². The core must not saturate, which caps the flux and therefore caps N I; and the inductance is N²/ℛ. Putting the three together, the gap is determined by the energy and the core’s dimensions, and the turns follow.

The result is a rule of thumb worth knowing: the energy a gapped core can store is set by the gap volume, because that is where the energy is and its density is capped by the material’s saturation flux. A core with twice the gap stores about twice the energy at the same peak flux, with half the inductance and √2 times the current — which is the same energy per unit of gap volume either way.

That is the practical content of the 87%. It says that sizing an energy-storage inductor is sizing an air gap, and that the ferrite around it is there to complete the magnetic path cheaply rather than to hold anything. A designer who thinks of the core as the storage medium will pick a high-permeability material and be surprised that it saturates immediately; one who thinks of the gap as the storage medium will pick the gap first.

The site’s own habit applies to that rule as much as to anything else here: it is an approximation whose range is the range over which this essay’s linear model holds, which is flux densities well below saturation. What happens at the cap is the next essay’s, and it is not a gradual departure.

The magnetic circuit as a circuit, and where the analogy stops #

This essay has used an analogy throughout — reluctance for resistance, flux for current, magnetomotive force for voltage — and the site’s standing rule is that a model is drawn with the place it stops being true. So it is worth ending on where this one does.

It holds for the series arithmetic, which is all this essay uses. Reluctances in series add, the flux is common to both, and the division of the magnetomotive force between them is exactly the division of a voltage between two resistors. Everything above is that one statement.

It fails on energy. A resistance carrying a current dissipates; a reluctance carrying a flux stores. The analogy maps a lossy element onto a lossless one, so every energy statement in the electrical picture has to be re-derived rather than translated, and the “energy divides in proportion to the reluctance” used here is the magnetic fact rather than the electrical one. The correct electrical analogue of a reluctance is a capacitance, not a resistance — and then the analogy breaks the other way, because reluctances in series add while capacitances in series do not.

It fails on non-linearity. A resistor’s resistance does not depend on the current through it, within the ranges this site works in. A core’s reluctance depends heavily on the flux through it once the flux is large, which is saturation, and no linear circuit analogy contains it.

That last one is the boundary this essay stands on. Everything above is a small-signal statement about a magnetic circuit, in exactly the sense the semiconductor field’s small-signal model is a small-signal statement about an exponential — accurate below a boundary, silent about it, and useless above it. A boundary in volt-seconds measures the boundary, and it is not a frequency and not a current: $N\cdot A_e\cdot B_{sat}$, 3.500 mWb-turn for the core measured there, with everything a data sheet says about a frequency rating being that one number divided by $2\pi f$ once.

What else the same ratio of two lengths does not contain #

$l_g/(l_g + l_e/\mu_r)$ containing neither the turns nor the area is the kind of result that is easy to over-read, so it is worth naming what it is silent about — and the silences are where the rest of this field lives.

It is silent about where the gap is. The gap that is three gaps shows one gap of 1.2 millimetres and three of 0.4 giving the same inductance, the same saturation current and the same energy in the air, while the winding beside the single gap dissipates 6.52 times its direct-current loss against 3.00 beside three.

It is silent about how big the gap really is. The gap that is bigger than it is finds the reluctance computed as $g/\mu_0 A$ always low, with the standard correction supplying half of what is missing at a 0.3 mm gap and 85 per cent at 1.7 — on a correction worth 10 and 33 per cent of the inductance respectively.

And it is silent about what the fringing does to the copper. The turns nearest the gap finds the worst turn of an eight-turn winding at 37.5 times its direct-current loss four tenths of a millimetre from a one-millimetre gap, over a governing distance of 0.60 millimetres that is not the gap length and does not scale with it.

Three silences, all about position, from a result that is a ratio of two lengths — which is the clearest statement available of what a magnetic circuit is and is not.