Magnet (unidirectional): demagnetization curve (Hc, Br)

Presentation

This model ( Nonlinear magnet described by H_c and the B_r module ) defines a nonlinear B(H) dependence with taking into account of demagnetization, wherever the curve knee is.

Main characteristics:

the mathematical model and the direction of magnetization are dissociated
a single material for description of several regions

Mathematical model

In the direction of magnetization the model is a combination of a straight line and an arc tangent curve.

The corresponding mathematical formula is written:

with:

where:

μ₀ is the permeability of vacuum, μ₀ = 4 π 10^-7 (H/m)
μ_rmax is the maximal relative permeability of material
B_r is the remanent flux density (T)
J_s is the saturation magnetic polarization (T)

The shape of the B(H) dependence in the direction of magnetization is given in the below figure:

In transversal directions one can write:

B_⊥(H)= μ₀μ_r⊥H_⊥

where μ_r⊥ is the transverse relative permeability

Direction of magnetization

The various possibilities provided to the user are the same ones as those presented in § Magnet (unidirectional): linear approximation.

Demagnetization during solving

With this non-linear model, it is now possible consider the demagnetization during solving (i.e. the irreversible degradation of a magnet's remanent magnetic flux density due to its interaction with a demagnetizing field) by checking the specific option that is provided during the material creation. This demagnetization model is based on a static Preisach approach, and may apply in all quadrants of the magnet B(H) characteristic. The following additional remarks apply:

Available for 2D and 3D in transient magnetic application
In projects containing a magnet material with this option active, it is advisable to set the application to perform an initialization through a static calculation (Application > Transient initialization).
This model does not take in account temperature variations

To use this model in a solved project:

Delete the results;
Go in Application > Transient initialization and select: Initialization by static calculation;
Create a new material Nonlinear magnet described by Hc and Br module;
Check the option Taking in account demagnetization during solving;
Assign the material to the regions representing the magnets in the project;
Go to Physics > Face regions (in 2D) or Volume regions (in 3D) > Orient material for face / volume regions to define the direction of magnetization in the magnet;
Solve the scenario.

After solving, to display the current remanent magnetic flux density of a magnet at a given time step:

Select the desired time step;
Create a new Arrows plot, then select the magnet as support;
Then, in the formula editor, click on button Br Magnet (available in the Magnetic Volume region tab) or type BrMagnet (without a space) in the formula field.
Note: In the formula editor, a related button Br is available and the user could also type the corresponding formula Br in the formula field. Remark that these recover the original remanent magnetic flux density stored in the material description, and not the remanent magnetic flux density modified by demagnetization effects at a given time step represented by BrMagnet.

In Flux, hard magnetic materials may be imported, exported and reused with the help of dedicated data exchange tools. These tools are notably useful when used in conjunction with magnets described by the B(H) property of type Nonlinear magnet described by H_c and the B_r module, with the option for consideration of demagnetization during resolution. For a complete discussion on this topic, refer to the following documentation chapter: Import, export and reuse magnets.

Example of results

Let's consider the permanent magnet motor below, which was represented in a Flux 3D project:

For the control angle values ψ = [0 ; π/6; π/4 ; π/3], we can see that there is a local effect on the remanent flux density, but the repartition depends on ψ, as shown in the picture below:

Figure 2. Arrow plots of the remanent magnetic flux density of one of the machine's magnet as a function of the control angle ψ. This localized effect modifies the machine's performance (e.g., torque and back EMF), as illustrated in the picture below.