# [Getdp] numerical saturation effect - 2

m.fenner at gmx.net m.fenner at gmx.net
Thu Sep 9 14:20:47 CEST 2004

```Dear Christophe,

first of all thanks for your comments on the wiki examples. I am going to test your suggestions asap.

But here is another problem: a while ago I asked about the saturation of the magnetic flux in the core of a coil. I investigated the problem a little further:

1. The physics says that the magnetisation M is dependent on the shape of the body of the magnetisable material. M is then given by:

M = chi / (1 + N * chi ) * H_ext

where chi is the magn. susceptibility and N is called demagnetisation factor, with 0 < N < 1. Exact values exist for simple geometries like a sphere: N = 1/3.

With chi -> infinity obviously M saturates. For the sphere you get M = 3 * H_ext.

The total flux is then B =  mu_0 (H_ext + M) = mu_0 * 4 * H_ext. Compared to the vacuum case the flux gain is 4.

2. When I solve the problem of a small sphere in a long coil for varying mu_r I find that the value of {d a} (which should be the magnetic flux) has a gain of ~3!

This suggests that {d a} represents only the magnetisation. But from the formulation with

nu [ Sphere ]  = 1. / (murSphere * mu0);

I would not expect that. Do you have an explanation? (see also the attached files)

3. One experiment to measure chi employs a torus with a coil wound around it to get rid of the demagnetisation factor. I wanted to model this problem. Though my .geo and .pro files seem to be okay (The current is wound around a torus anyway) I get a

Warning   : Null right hand side in linear system
....
Solver    : Generalized Minimum RESidual (GMRES)
1  0.0000000e+00  NaN

I do get a solution in the very same geometry when I define a current density with a z-component (i.e. phi-component in zylindrical coordinates) in the coil region. Do you have any idea what goes wrong here? (again files are attached)

Many thanks in advance and best regards

Matthias

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```