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# Re: [femm] inductance/flux calculations in 2D/3D

- To: femm@yahoogroups.com
- Subject: Re: [femm] inductance/flux calculations in 2D/3D
- From: Keith Gregory <k.gregory@lboro.ac.uk>
- Date: Tue, 10 Jul 2001 20:05:33 +0100
- In-reply-to: <25585.994768825@www46.gmx.net>

`Cornelius,`
`Dave is right, it really does depend on whether you have a problem including a high permeability material or not. I've done investigations into this stuff before and always found the same sort of results.`
`If you have a transformer or a machine with a core of permeability significantly greater than 1 then a 2D model works well. The flux linking the winding overhangs (i.e. that bit of the winding not inside the core) is mostly leakage, mostly in air, much smaller (several orders of magnitude) than the main flux component and makes very little difference to the overall result. However, you are right in thinking that ends of the winding make a contribution, they do but it's certainly not "one fourth".`
`The air-cored case though is significantly different. Every part of an air-cored coil contributes to the magnetizing force. But it is a bit simplistic to say that each side contributes a quarter of the total, that would be true only at the coil's centre.`
`I once did some simple experiments to compare inductances calculated from FE models with those determined using the analytic equations in Inductance Calculations - working formulas and tables by Federick W Grover (an old but still good book). For a long pair of parallel wires in a go and return arrangement the comparison was very good, within a few percent if I remember correctly, but for a square coil the differences were quite large, again if my memory serves more that 50% difference for the examples I tried. Circular coils modelled in FEMM axisymmetrically should produce a close agreement.`
`All this means that you have to be very careful how or even if you model what are essentially 3D problems using a 2D solver like FEMM. 2D models can help significantly in understanding a 3D problem but you have to question the absolute accuracy of derived things like inductance.`
`I would add one further point. The derivation of the simple equation you quote for the magnetizing force around a wire really implies that the current returns via some far off and undefined path (otherwise you are accumulating change at a point) so really it's a single turn coil not a wire in isolation.`
`Keith.`
`
At 14:40 10/07/01 +0200, you wrote:`

> `Dave meeker wrote:`
> `>The discrepancy in inductance between a 2-D and 3-D model is typically much`

>     >smaller than a factor of 2 error. It sounds like your rationale is that
>     >since you have a nearly square cross-section and the end-turns aren't taken

>     >into account, you'll have about a factor of 2 error in total flux. It
>     >doesn't work quite like that. Think about this terms of magnetic
>     >circuits--the only parts of the wire that actually "drive" flux are those
>     >that get encircled by the flux path. The part of the winding that actually

>     >drives the primary, desirable part of the flux in the transformer is the
>     >coils that passes through the "winding window" in the transformer, because

>     >the flux flowing through the core circles this part of the winding. In
>     >contrast, the only flux that actually encircles the end turns is leakage
>     >flux, traveling on a highly flux reluctant path, mostly through air.
>     >Because this end turn leakage path is highly reluctant relative to the
>     >primary flux path (hopefully), it doesn't add much inductance to the model.

> `>At least, this is the basic rationale in using a 2-D approximation...`

>     OK Dave, many thanks for the advice.
>     I tried out some configurations and it really seems to work
>     in practice and I am ready to accept it as the "basic rationale"

>     Although I still have difficulties understanding WHY it is like that.
>     A very simple hypothetical experiment:

> `- take a core with a quadratic cross section`

>     - wind a one turn coil around it. that is, this coil consists of
>     4 wires, each on every side of the core

>     - now every wire produces a H-Field around it H = I/(2*pi*r)
>     where r is the distance from the wire surface, I is current

> `- if the core is of linear material then B = mu * H`
> `- flux = integrate B over cross section`

>     THIS IS WHY I "FEEL" EVERY ONE OF THE 4 WIRES PRODUCES ONE FOURTH
>     OF THE INDUCTION OR FLUX IN THE CORE.

> `WHERE DOES "the only parts of the wire that actually "drive" flux are those`
> `that get encircled by the flux path" COME FROM???`
> `what am I missing ?`

>     regards,
>     Cornelius

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- References:
  - inductance/flux calculations in 2D/3D
    - From: corneliuspaul@gmx.de

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