# Formula for inductance with a rectangular non-air core coil

Does anyone have a formula for calculating the inductance for a coil wrapped around a rectangular non-air core? Seems trivial but this is the closest I've come in searching, and it is for air-core only.

What I am trying to determine is the amount of inductance I will have with a 25.4 x 6.36 mm core currently made up of micrometal mix 40 (60 ur) and N turns of wire.

While I may be able to simply multiply the value from the above site by 60 for the low frequencies I am using (< 5 KHz if it were to be driven continuously), without knowing the formula used I'm not certain of that.

I do not yet know the number of turns of the coil (I will use as many is required), or the pulsed amperage (as that depends on the total inductance and resistance, which I am trying to minimize to reduce the total current required). My best WAG for current is around 100 A at 6 V for < 0.1 ms. Minimizing the width of the coils is important to keep the magnetic flux lines as short as possible to the keeper.

Physically the core is arranged in a pattern as shown below, similar to a U for a transformer. A permanent magnet is inserted between the vertical cores (6.36 mm width as shown, 25.4 mm depth). One "coil pair" is on each side of the U. These coils will be quickly pulsed (maybe 3T) once to create a large magnetic force to change the flux path of the permanent magnet from going across the keeper to going across the gap at the bottom (with a piece of steel in place as shown on the bottom). The coil polarity can then be reversed and quickly pulsed to drive the flux path though the keeper to release the steel.

I plan to drive the coils with a MOSFET H-bridge.

• My first inclination is to ask you to develop a detailed drawing. Every little detail matters. Also, in air cores there is very significant fringing and that site says that they handle this for short coils, which is pretty amazing but also says they use rather detailed formulas, as well. When you insert a length of micrometal mix, you do at least two things: shorten the magnetic path length (increasing the inductance) and changing the fringing behavior at the ends, too. So you are probably right that blind use of a constant won't be accurate. How much so may depend on how long, etc.
– jonk
Commented Sep 5, 2021 at 22:18
• Give us a dimensional diagram, please. I also gather your thought about "low frequencies." But knowing the frequency range may trigger other thoughts. So include it, as well. And the peak current expected. And anything else you can think of that may in any way be relevant (which may include the applied voltage, though that often isn't as important to know unless it is high.)
– jonk
Commented Sep 5, 2021 at 22:19
• I still want that diagram. As an aside I forgot to earlier mention, the $\mu_r=60$ is not an excuse to multiply an air core calculation by!!! That's not even in the same universe of right. If all you are doing in inserting a length of material about the same length as the coil, then all you've done is shorten the magnetic path length by the length of that material. You do adjust the physical length and use $L^{'}=L\left(1-\frac1{\mu_r}\right)$ when working out how much shorter it is. But that's by no means a multiplier!! (If you fill the entire magnetic path, then maybe.)
– jonk
Commented Sep 5, 2021 at 22:32
• I updated the initial question - hopefully that answers at least some questions. I realize I am a complete novice in terms of magnetics, so forgive me if what I'm asking is incorrect. Commented Sep 5, 2021 at 23:04
• That drawing is hugely important. Much of what I was earlier thinking is changed, now. But of course it now makes me wonder what material you are using for the keeper and I'm no longer thinking about the coil, anymore. Do I gather that you are using this as a bistable locking mechanism? Because, if so, your keeper material is pretty much everything important about the design. That, and the gaps involved, of course. I've stopped caring about the coil question.
– jonk
Commented Sep 5, 2021 at 23:25

$$\L=N^2*A_L\$$ where $$\A_L\$$ is listed in catalog core area and permeability in nH/N^2 for N turns.

$$\A_L\$$ depends on geometry, size and permeability. Usually E cores or Torroids are used, not cubic Rectangles.

It is a misnomer to define the core as non-metallic, as it is a ferrite mix of magnetic, dielectric and metallic particles. This is a special blend that is NP0/C0G at 25' and NTC outside this range in a parabolic curve.

Imax is usually rated at 90% L nominal.

NP0 means 0 negative temperature coefficient (NTC) and 0 PTC but with a tolerance +/- xx ppm, often pronounced NP-Oh.

You must cite the catalog or datasheet and actual 3D geometry, wire diameter, and frequency to compute L within certain tolerances. No assumptions ought to be made except following the OEM's guidelines per the above formula.

• Tony, from what I can tell Al is listed for transformer cores as something such as 32/30. Of course this is not a transformer, so it does not use that shape of a core and although I understand what you are trying to say, I don't follow how that would translate into a usable formula that I could plug into excel to get a ballpark figure from the information on this datasheet for this material, or anything similar to it. If you can explain how you would calculate that using this data, or point me to a source that explains this further, I would be grateful. Commented Sep 6, 2021 at 16:19
• micrometals.com/products/materials/-40 Commented Sep 6, 2021 at 16:19
• Not enough details on 3D dimensions and surface roughness for air gaps, which makes a difference. What is DCR ? 6V/100= 60 mOhm or much less with L/DCR < 100 us? Below saturation? But if L/DCR= 40uH*N/4mohm then T=10ms . So what is your goal for Bmax? And Tau? Commented Sep 6, 2021 at 16:51
• I would guess they used AlNiCo magnetics at B >=1T , so to accomplish complete flux transfer you need to create -1T but partial transfer is adequate to exceed the holding force for some multiple electro-magnets.mCan this ferrite achieve 1T , No but almost of a value of AL for inductance with only 1A after 1/2 s on the boots dI/dt=9V/L so L= >= 0.5/9 H yet you want L/R around <0.0001s . Hmm that's tricky with 100A and a big arc on release of battery. Nice comet trail. Commented Sep 6, 2021 at 17:15
• Thanks for your comments, very helpful. Hopefully I can get away with less than 1T holding force release, there just isn't much information on flux transfer that I can find to determine what is really required. That said, I have ordered an LCR meter to get a beter idea of what the actual materials create with their interactions to find out how many windings and what actual power levels are required to make this work. A flyback suppression setup should take care of the comet trail from the coil de-energizing - hopefully at well under 100A. Commented Sep 6, 2021 at 17:30