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In some cases it is necessary that core of inductor should have a gap, unlike with transformer core. I understand the reason with the voltage transformer core; there is nothing to worry about core saturation and we want to keep the winding inductance as high as possible.

The formula for the inductance is:

$$ L = N^2A_L = N^2\dfrac{1}{R} = \dfrac{N^2I}{\dfrac{\ell_c}{\mu_cA_c} + \dfrac{\ell}{\mu_0A_c}} = \dfrac{N^2IA_c}{\dfrac{\ell_c}{\mu_c} + \dfrac{\ell}{\mu_0}} $$

And, the formula for the magnetic flux density:

$$ B = \dfrac{\mu N I}{\ell} = \dfrac{N I}{\dfrac{\ell}{\mu}} = \dfrac{N I}{\dfrac{\ell_c}{\mu_c} + \dfrac{\ell_g}{\mu_0}} $$

Where,

\$N\$: Number of turns
\$R\$: Total core reluctance
\$A_L\$: The \$A_L\$ factor
\$I\$: Current through the wire
\$\mu_c\$: Permeability of the core
\$\ell_c\$: Mean magnetic path of the core
\$\ell_g\$: Length of the gap
\$A_c\$: Cross-section area of the core
\$L\$: Inductance
\$B\$: Magnetic flux density

What I understand from these two formulas is, the length of the gap affects both the magnetic flux density and inductance with the same proportion. When designing inductor, we would like to keep magnetic flux density low, so that the core wouldn't saturate and core loss stay low. People say that they leave the gap in order to keep the reluctance high, so that there are less flux flowing in the core, and the core stays away from the saturation region. However, doing so will reduce the inductance as well. By leaving the gap, we reduce magnetic flux density and inductance with the same coefficient. Then, instead of leaving the gap, we can also decrease the number of turns in the winding as well.

The only reason to leave gap that makes sense to is to increase the number of design parameters to obtain a closer resulting inductance value at the end. I can't find any other reason to leave gap.

What makes leaving the gap an inevitable action while designing an inductor?

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    \$\begingroup\$ On a project I was working on, I had identified an inductor design that needed a gap, and there's some justification on this question: electronics.stackexchange.com/questions/210640/… . \$\endgroup\$ – W5VO Jan 18 '18 at 11:42
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    \$\begingroup\$ I think this webstie is ideal for the answer you're looking for, sorry dont have time to put in an answer form, info.ee.surrey.ac.uk/Workshop/advice/coils/gap/index.html \$\endgroup\$ – Pop24 Jan 18 '18 at 11:43
  • \$\begingroup\$ @W5V0 question edited to make it more accurate and universally applicable. \$\endgroup\$ – RoyC Jan 18 '18 at 14:17
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Why do we want gap in the core material while designing inductor?

And...

The only reason to leave gap that makes sense to is to increase the number of design parameters to obtain a closer resulting inductance value at the end. I can't find any other reason to leave gap.

There is a major reason and it's clear from the formulas you quote: -

What saturates an inductor is too much current and too many turns for a given core geometry and core material. However, by adding a gap we might halve the permeability of the core and this means that we could double the amps (or double the turns) to obtain the same level of saturation we had before but, the inductance will have halved when we halved the permeability.

Fortunately, when we halve the core permeability, in order to restore the original value of inductance, we only need to increase the number of turns by \$\sqrt2\$ so, if we have halved the permeability with a gap, the potential for avoiding saturation has improved by \$\frac{2}{\sqrt2}\$ = \$\sqrt2\$.

This means that you get the same inductance but now you can have an operating current that is \$\sqrt2\$ higher for the same level of core saturation when the core was not gapped.

What I understand from these two formulas is, the length of the gap affects both the magnetic flux density and inductance with the same proportion

And...

By leaving the gap, we reduce magnetic flux density and inductance with the same coefficient

No; look at your 1st formula - it tells you inductance is proportional to turns squared whilst in your 2nd formula, flux is proportional to turns (no square term) so no, they don't alter with the same proportion or coefficient.

If a gap causes permability to halve, flux density also halves for the same operating current but, to return inductance to what it was previously, turns must increase by \$\sqrt2\$ hence the bottom line is that flux density has gone down by \$\sqrt2\$ for the same operating current. This is a benefit and a big one.

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    \$\begingroup\$ I prefer this kind of answer (quantitative, with added qualitative) over Neil's (essentially qualitative analogy), if I have to make a choice between them. Nice. \$\endgroup\$ – jonk Jan 18 '18 at 18:18
  • \$\begingroup\$ Where I struggled with my answer Andy, and I notice you don't address it either, is what the optimum size of airgap is, why not make it larger or smaller? Obviously if we do the magnetic sums, let's say for an inductor of constant volume, and differentiate, then we'll find a max stored energy at some gap, for pure (rather than distributed gap) core materials, but that's not very intuitive. Or we could do the physicist thang of both zero gap and all gap are bad, and 'somewhere between' is better, intuitive but not very quantitative. Thoughts? \$\endgroup\$ – Neil_UK Jan 19 '18 at 6:54
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    \$\begingroup\$ @Neil_UK I didn’t see it as requiring to be answered but, it depends on how much hysteresis loss versus copper loss a particular application could handle. Plus how much leakage to other circuits is acceptable. \$\endgroup\$ – Andy aka Jan 19 '18 at 9:08
  • \$\begingroup\$ Thinking about optimum size of air-gap, I came up with another answer, which addresses the specific permeability we want to acheive. It's horrible and rambling though, not particularly happy with it. Got any suggestions for improvement, while keeping it intuitive and formula-free? \$\endgroup\$ – Neil_UK Jan 21 '18 at 8:17
  • \$\begingroup\$ @Neil_UK I think I would begin by not mentioning a gap. I would want to make the argument about turns and permeability trade offs but keep in mind the specific aim of a fixed inductance as goal 1 and higher current capability as goal 2. Goal 3 is probably field confinement. At the end bring in gapping versus distributed gaps. \$\endgroup\$ – Andy aka Jan 21 '18 at 10:28
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Saturation is always an issue in both transformer and inductor design. If we're going to spend money on a heavy and expensive iron core, then we want to work it as near to saturation as we can.

The reason inductors are gapped, and transformers are not, is that they are trying to do different things.

The purpose of an inductor is to store energy. This means that to get the core close to the saturation B field should take as much H field, that is ampere turns, as possible. This needs a high reluctance magnetic path.

The purpose of a transformer is to transmit energy, with as little stored in the transformer as possible. In fact, energy storage in a transformer is a Bad Thing, needing snubbers to protect inverter drives. This needs a low reluctance path, so no air gap, as as high a permeability as possible.

Here's an analogy I like to use, and it's a bit odd, so I'm cool if not too many people grok it, is mechanical energy. In this analogy, stress is the equivalent of B field, so the saturation level is equivalent to the breaking strain of a material. Strain, the elongation, the change in length, is equivalent to the H field, the ampere turns. The stiffness is equivalent therefore to the permeability. An air-gap is a rubber rope, that takes a lot of change in length to get up to a decent stress. An iron core is a polypropylene rope, which takes very little strain to get it up to stress.

Now, which rope would you use for a pulley system? Obviously the non-stretchy one. You don't want to store energy in the rope between the pulleys, you just want input to become output.

Which rope would you use to store energy? The rubber one. If both the poly rope and the rubber rope had the same breaking strain, you could store 100x the energy using the rubber rope, if it stretched 100x more than the poly rope.

Bonus marks. Why do we use iron at all in an inductor? It's to do with the magnitudes of the permeability, copper losses etc. It so happens that it's not easy for the current to 'get hold of' the air round a conductor. It's a long way round the conductor, the H field is very low for any given current. It needs a lot of current to get a decent field. That's equivalent to our rubber rope being very long and thin, so we need to use some poly rope to 'gear it down' to the sort of distances and forces that are more in keeping with the rest of our system. The iron core concentrates the H field down to the small air-gap.

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    \$\begingroup\$ Brilliant analogy +1. \$\endgroup\$ – RoyC Jan 18 '18 at 14:19
  • \$\begingroup\$ There are gap requirements in some ferrite transformer designs, usually E cores and potted cores, for just the reasons you mentioned. +1. \$\endgroup\$ – Sparky256 Jan 19 '18 at 0:02
  • \$\begingroup\$ Your rope analogy works well also for using inductors to dampen noise. (along with a hanging counterweight - a capacitor) \$\endgroup\$ – Stian Yttervik Jan 19 '18 at 9:13
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You are correct that the maximum inductance is achieved with no gap, but core materials have varying permeability with changes in magnetic field strength. See the chart below:

enter image description here

There is also a change in permeability with temperature.

You can see that with no gap, the value of inductance would vary greatly as the current through your inductor changed. However, the permeability of free space (μ0) is constant. Even with a small gap length, the value of ℓg/μ0 can be much larger than ℓc/μc, so the contribution of the gap geometry in your equation can dominate the variability of the core material. This makes it possible to construct an inductor with a fairly constant value of inductance across a wide range of currents and temperatures.

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Because almost all of the magnetic energy is stored in the air gap !

The energy density is BxH. B is the same in air and iron but H is a factor 1/mu_r larger in the air gap, so that counts. Instead of an air gap you can also choose a ferrite with a low mu_r value, what I think of as an "airy" core.

Only if you don't need to store magnetic energy, like in case of a transformer where the power passes through without being stored, should you use a core without an air gap.

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  • \$\begingroup\$ ... for a small gapped core, B in the gap is the same as B in the iron core. Maybe rephrase it like that? \$\endgroup\$ – Andy aka Jan 19 '18 at 15:46
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Andy aka's answer was enlightening for me. Indeed, when we add a gap and decrease the overall effective permeability \$\left(\mu_e=\dfrac{\mu_0 \mu_c (\ell_c + \ell_g)}{\mu_0 \ell_c + \mu_c \ell_g}\right)\$, we reduce the flux density and gain more saturation margin. Therefore, we become able to add more turns into the winding. And since inductance increase with the square of number of turns, we increase the maximum obtainable inductance without saturating the core. In the extreme case, if we completely remove the core material, the maximum inductance without core saturation becomes infinite.

The formulas for inductance and magnetic flux density are:

$$ L = \dfrac{N^2IA_c}{\dfrac{\ell_c}{\mu_c} + \dfrac{\ell}{\mu_0}}, \quad B = \dfrac{N I}{\dfrac{\ell_c}{\mu_c} + \dfrac{\ell_g}{\mu_0}} $$

If we want to keep the flux density fixed without changing the requested amount of current, we must keep the following ratio fixed to a coefficient, say \$k\$.

$$ \dfrac{N}{\dfrac{\ell_c}{\mu_c} + \dfrac{\ell_g}{\mu_0}} = k $$

Rearranging the terms:

$$ \ell_g = \dfrac{\mu_0}{k}N - \dfrac{\mu_0}{\mu_c}\ell_c $$

Summarizing it, We leave the gap in order to increase inductance without saturating the core. This is achieved by the fact that \$B\propto N\$ and \$L\propto N^2\$ despite that \$B\propto\mu_e\$ and \$L\propto\mu_e\$.

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Why do we want gap in the core material while designing inductor?

Because we don't have the ideal materials readily available, to make a good inductor.

OK, so what's a good inductor?

We are going to be using expensive materials, so for any limited quantity of them, we want the most inductance, the highest energy storage, out of some fixed quantity of them. Different materials limit the energy storage in different ways.

Tell me more about these limits

Copper limits the current we can push through an inductor, because of heating. If we make an air-core inductor, this is invariably the thing that limits the maximum energy storage. If we wanted to run a higher current, we could do it briefly before the coil overheated.

Ferromganetic materials like iron or ferrite limit the B-field in the core. Once we've hit saturation, the permeability drops, and we get no further benefit from the core. The benefit is that it gives us a lot of B-field for our ampere-Turns (H-field). The permeability of these materials is in the 1000 range, meaning that very little current is needed to saturate them. As energy stored is the product of H and B field, we would like to increase the H field without a corresponding B field increase.

Why are the limits important for good inductor design?

A good inductor is equally limited by both the copper, and the magnetic material.

With a low permeability magnetic material like air, the current is limited by coil heating. We could store more energy with more magnetic field, so would ideally like to increase the permeability to get more B-field for our current. Unfortunately, with the resistivity of copper, the permeability of air, and the typical geometries of coil/core that are possible, the ideal permeability turns out to be in the 10s to very low 100s.

High permeability materials, ferrite and iron have figures in the 1000 and 1000s range respectively, tend to reach saturation at a lower coil current than the coil can handle for heating. We need to find a way to use more current. What we need is a lower permeability core so that more current will increase the H-field without increasing the B-field. A series air-gap reduces the effective permeability down from the 1000 range to the 10-100 range.

Are there other materials we could use instead of a core with an air-gap?

Yes. We can synthesise materials with an effective bulk permeability in the 10s to 100 range by using a resin-bound magnetic powder. This gives us the so-called distributed air-gap materials. When you see a reference to an 'iron-powder' core, or ferrite toroids with a permeability in the 10s, this is what's going on. A solid core with an air-gap is cheaper, and more flexible to manufacture.

Remember, the copper was just as important in setting the ideal permeability, through its losses. If we had a conductor without losses, then we could make use of a lower permeability core, because we could use a much higher current. This is what happens in superconducting solenoids, as used in MRI machines and the LHC. Fields in these run to many Tesla, above the saturation of both ferrite and iron.

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