Field Precision title

Iron in magnetic-field calculations (2)

Our programs PerMag (2D) and Magnum (3D) can handle saturation effects in nonlinear magnetic materials. One of the challenges in calculations with steel is tracking down data for the particular material you are using. In another article, I discussed how to represent steel at low values of B. Here, the term low means that the magnetic flux density inside the material is small compared to the saturation flux density Bs. For iron and steel, Bs is typically in the range 2-3 tesla. In this regime, I pointed out that if the relative magnetic permeability is large (μr > 1), then the exact value has little effect on the magnetic field distribution. One might object that the choice of μr has a significant effect on the inductance. My reply is that the concept of inductance is useful primarily for linear circuits. It is important to avoid saturation in practical inductors and transformers. To calculate inductance, you should assign a fixed value of μr equal to the small signal relative magnetic permeability.

In this article, I will shift to the other end of the scale and look at the variation of μr at high field levels (B > Bs). Figure 1 illustrates the field regimes. The red dots show values of μr(B) taken from familiar soft iron table supplied with the Poisson codes from Los Alamos National Laboratory. We shall concentrate on the portion of the curve marked saturated. There are two important practical questions:

  1. How can we determine the value of Bs from the curve?
  2. How can we extend the curve to higher values of B?

The second issue is important for numerical calculations with nonlinear materials. It often happens that the field values in some locations of the material may exceed the maximum value in the available table, preventing solution convergence.


Figure 1. Variation of the relative magnetic permeability with B is soft iron.

To start, we need some definitions:

  • I use the symbol B0 to denote the applied flux density, created by drive coils. The quantity B0 equals the flux density in the absence of steel.
  • The material contribution to the flux density is denoted Bm.
  • The total flux density is B = Bm + B0.
  • For the numerical solutions in PerMag and Magnum, the relative magnetic permeability at a point in the material is defined as μr = B/B0.

First, let's discuss PerMag solutions. Because of the solution technique used, we input material characteristics with a table of values that gives μr as a function of B. This is the type of data shown in the figure below. The term saturation means that the applied field is so strong that all magnetic domains in the steel are aligned. In this case, the contribution of the material to the flux density reaches the fixed value Bm = Bs. Therefore, the total magnetic flux density is B = Bs + B0. In saturation, the relative magnetic permeability is:

B/(B-Bs). [1]

For known μr and B, we can find Bs from:

Bs = B - B/μr. [2]

Using the average of the last six values [B,μr] from the soft iron table shown in the figure, Eq.[2] implies that Bs = 2.15 tesla. Inserting the value in Eq.[1] gives the blue line shown in the figure. The fit is very good for B > 2.2 tesla, and we can reliably extend the curve to any higher value of B.

Magnum uses a different solution technique, so that permeability tables should be entered in the form [B0,μr]. In this case, the functional form for μr at high field is

μr ≅ 1 + Bs/B0. [3]

We can derive the saturation flux density from

Bs = (μr - 1)*B0.

To conclude, here's a useful shortcut for calculating fields when there are highly-saturated steel parts. Suppose you had a high-field (B > 2-3 tesla) superconducting magnet and you wanted to add some small 3D steel inserts to tune the field distribution. You could use the full nonlinear capabilities of Magnum with a saturation curve for the steel. The approach would be time-consuming, especially if you wanted to try several different geometries. There is quite a lot of wasted effort. You would be using the code to tell you that the steel inserts are completely saturated, something that you already know. Here's better way. A steel part with perfect domain alignment is equivalent to an ideal permanent magnet with Br = Bs. You can quickly find the contributions of the steel inserts with a simple, linear Magnum calculation where the inserts are treated as permanent magnets with the easy axis along the local direction of applied field. In this way, you could investigate several configurations and then do the complete nonlinear calculation to determine the total fields and the small effect of the inserts on the magnet circuit reluctance.

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