How do I determine the "recoil permeability" and "remanent flux density norm" of a permanent magnet?

Question

I am working on modeling a permanent magnet using finite element analysis.* The magnet in question is a neodymium cylinder 1 cm in diameter, and 5 cm in length.

The program uses the equation

$$ \vec{B} = \mu_0 \mu_{rec} \vec{H} + \vec{B}_r $$

to define the field caused by the permanent magnet, where $\mu_{rec}$ is the recoil permeability, and $\vec{B}_r$ is the remanent flux density.

It is my job to input values for $\mu_{rec}$ as well as the magnitude and direction of $\vec{B}_r$. I have a gaussmeter (i.e. a device that measures the $B$-field) and the magnet. Can I use this to determine $\mu_{rec}$ and $\vec{B}_r$, and if so how?

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Toward a solution, I know that the magnetic field may also be expressed as

$$ \vec{B} = \mu_0 \vec{H}+ \mu_0\vec{M} $$ where $\vec{M}$ is the magnetization vector. Thus,

$$ \mu_0 \vec{H}+ \mu_0\vec{M} = \mu_0 \mu_{rec} \vec{H} + \vec{B}_r. $$

This makes me think that $\vec{B}_r = \mu_0 \vec{M} $... but if so, I don't understand what the $\mu_{rec}$ is doing hanging out down there. Please help!

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*COMSOL Multiphysics 5.6, Magnetic Fields module, if you're curious. Here is a screenshot

Answer 1

In general, the relationship between $\vec{B}$ and $\vec{H}$ is nonlinear and history-dependent. The formula

$$ \vec{B} = \mu_0 \mu_{rec} \vec{H} + \vec{B}_r $$

is an expedient approximation that often works well for rare-earth permanent magnet applications. It refers to the ‘major hysteresis loop’ which is the path followed when $\vec{H}$ cycles between vary large positive and negative values, sufficient to fully saturate the material. By definition, $\vec{B}_r$ is the value of $\vec{B}$ when $\vec{H}=0$. The first term is simply a linear and isotropic approximation for how $\vec{B}$ varies around $\vec{H}=0$.

In principle, the values of $\mu_{rec}$ and $\vec{B}_r$ can be read off of measured hysteresis loops. Vendors of the highest quality NdFeB and SmCo provide such data. (See product literature from Shin Etsu, for example.) When adequate data is not available or it can’t be trusted, the only practical option is to map the exterior field of your magnets and then to fit the data with the help of finite-element calculations. Beware that the magnetic properties may not be spatially homogeneous, especially with poor quality material.

Answer 2

With some hints from a user of the COMSOL forums, I have been able to piece together an answer.

HOWEVER, if someone else has a description of how I could actually find the exact values for my magnet, I will happily accept your answer instead!

Per Permanent Magnet Materials and Their Application, by Peter Campbell, Cambridge Univ. Press, 1994 (p. 94-96), the equation

$$ B_m = \mu_0(\mu_{\text{rec}} H_m + M_r) $$ describes the relationship between the $H$ and $B$ fields within the magnet (hence the subscript $m$).

By comparing this to COMSOL's equation (in my original question), it is indeed clear that $\vec{B}_r = \mu_0 \vec{M}_r$, where $\vec{M}_r$ is the remanence. Furthermore, per An Overview of MnAl Permanent Magnets with a Study on Their Potential in Electrical Machines,

The recoil permeability would be $\mu_{\text{rec}}$ for an ideal [permanent magnet] with constant magnetization. The recoil permeability is near 1, or assumed to be 1 for many common materials such as NdFeB, SmCo and ferrites. However, some materials have a significant demagnetization, such as Alnico, with deviating recoil curves and higher recoil permeability. In those cases, it is more complicated to conduct a linear approximation.

The paper goes on to provide the following values for remanence and relative permeability for NdFeB, which is what I was looking for:

Finally:

• $|\vec{B}_r| = |\vec{M}_r| = \mu_0 \times (1.08 - 1.49)\mathrm{T}$
• $\mu_r \approx 1$