# Does a parallel air gap increase or decrease maximum energy stored in magnetic field of an inductor?

I understand that a serial air gap increases the amount of energy that can be stored in an inductor.

The reasoning is as follows: according to Gauss's law for magnetism, divergence of B-field is zero. Thus, if there's a serial air gap, the B-field is continuous within that air gap because the B-field at the ends of the air gap is perpendicular to the surface of a cube drawn partially outside and partially inside the air gap. The B-field going into the cube must be the same as the B-field going out of the cube (since there are no magnetic monopoles), so B-field in the air gap is the same as B-field out of the air gap.

The H-field, on the other hand, is B/(µ0*µr), so if µr of air gap is 1, but µr of core material is let's say 1000, the H-field in the air gap is 1000 times bigger. Thus, the energy density in the air gap, which is proportional to product of H and B fields, is 1000 times bigger. Thus, the air gap stores most of the energy, increasing stored energy at the value of B field which would saturate the core, while still having some of the inductance-increasing properties of the ferrite core. (Of course, you could create a purely air-cored inductor with theoretically infinite saturation energy, but that would require so many turns that resistance would be huge, and the field would leak way too much outside of the core, creating RF interference.)

However, in my parts bin I happen to have a different kind of inductor with an air gap. In this case, the core is not a toroid but cylinder-like. A hole is drilled right through the middle of the cylinder, making it a tube.

I don't understand how this kind of parallel (as opposed to serial) air gap could be beneficial.

If we modify the core slightly to make it a hollow toroid to make calculations simpler, the H-field inside is N*I/(2*pi*r) where r is toroid radius, I is current and N is number of turns, according to Ampere's circuital law. This should be true no matter whether we make the path integral at the hole or at the core material. So, H-field is continuous inside the hollow core.

We also have B = µ0*µr*H. So at a certain value of H, we have a weaker B-field inside the parallel air gap. Thus, the weaker B-field and the same H field mean that the parallel air gap stores less energy than the material around it. Thus, a parallel air gap should decrease energy stored when core becomes saturating, as opposed to a serial air gap which should increase energy stored when core becomes saturating.

Is my analysis correct? Does a parallel air gap decrease saturation energy as opposed to a serial air gap which increases saturation energy? If so, what's the benefit of this parallel air gap?

Edit: here's a picture:

You would see right through the inductor via the hole if the camera was positioned correctly.

Inductance about 200 µH. Resistance smaller than 0.1 Ohm so I can't measure it with my multimeter. Length 2.5 cm, core outside diameter 4mm, inside hole diameter 1.5mm, whole inductor diameter 1cm with all the wires around the core.

• I think this question boils down to what use is there for the inductor core in my parts bin. Post a picture. What is a hollow toroid btw? All inductors have a parallel air-gap btw. Oct 22, 2022 at 11:21
• That 'air gap' doesn't sound electromagnetic, but mechanical. It's a tube. OK, it would have more active cross-section if it was a cylinder, but they happen to have made it a tube. What's the active cross-section of the tube? You might as well worry abut the air around the core that's 'in parallel' with the core. What's the active cross-section of the rest of the magnetic circuit? Is there a 'rest of the magnetic circuit' even? Show us a picture of what you have. Oct 22, 2022 at 11:25

Does a parallel air gap decrease saturation energy as opposed to a serial air gap which increases saturation energy? If so, what's the benefit of this parallel air gap?

The hole down the middle of the core will have virtually no effect whatsoever. Your coil (as seen in the picture) has a massive series air-gap that dominates everything: -

See the lines of flux coming out of one end and returning back to the other end; that is your "series" air-gap that I'm talking about. The hole down the middle of the core brings nothing to the magnetism party.

Image from here.

• Ok, so could it be a cost reduction measure? Practically no effect on magnetism properties, but less ferrite material used? Oct 22, 2022 at 12:30
• It might be there so that some ferrite slug can be inserted to tune the inductance slightly or, as a means for securing it more rigidly. I have no great thoughts about cost. Oct 22, 2022 at 12:35
• It's also possible that this is meant to run a wire through it for measurements. Oct 22, 2022 at 12:55
• @Janka what sort of measurements do you envisage? I mean, the direction of the wire would not cut any lines of flux hence there would be no induced voltage. But, if you wanted to wind a vertical inductor with radial connections, the hole would be useful for the return wire from the top end of the solenoid. Oct 22, 2022 at 13:08
• You can heal that by doing one turn. Oct 22, 2022 at 13:10