Static Field Modeling of a Halbach Rotor

This example presents the static-field modeling of an outward-flux-focusing magnetic rotor using permanent magnets, a magnetic rotor also known as a Halbach rotor. The use of permanent magnets in rotatory devices such as motors, generators, and magnetic gears is increasing due to their no-contact, frictionless operation. This model illustrates how to calculate the magnetic field of a 4-pole pair rotor in 3D by modeling only a single pole of the rotor using symmetry.

Figure 1: Illustration of a 16-segments, 4-pole pair Halbach rotor. The symmetries of the problem allow restricting the model to a single pole of the rotor.

Model Definition

Set up the problem in a 3D modeling space. Due to symmetry, it is sufficient to model a single pole of the rotor. Figure 1 shows a 3D view of the complete rotor with the magnetization direction of the magnets indicated. The black arrows show the radial and axial magnetization directions of the permanent magnets in the rotor. The permanent magnets are arranged in such a way that the magnetic flux density is minimized inside the rotor and maximized outside the rotor. The model consists of 16 permanent magnet pieces arranged to form a 4-pole pair rotor. The inner and outer rotor radii are 30 mm and 50 mm, respectively. The axial length of the rotor is 30 mm.

Results and Discussion

A steady-state study analysis is performed to calculate the magnetic fields of the Halbach rotor. The magnetic flux density is shown in Figure 2.

Figure 3 and Figure 4 illustrate the variations of the radial and azimuthal magnetic flux density as functions of rotor angle. The magnetic flux density norm is evaluated outside the Halbach rotor at a radial distance of 55 mm from the center.

Finally, Figure 5 and Figure 6 show the polar plots of the magnetic flux density norm at radial distances from the rotor center of 55 mm and 25 mm, respectively.

Figure 2: Magnetic flux density norm at the cross section of the Halbach rotor (at z = 0 mm).

Figure 3: The radial magnetic flux density as a function of rotor angle measured at a radial distance of 55 mm from the rotor center.

Figure 4: The azimuthal magnetic flux density as a function of rotor angle measured at a radial distance of 25 mm from the rotor center.

Figure 5: Polar plot of the magnetic flux density norm at a radial distance of 55 mm from the rotor center.

Figure 6: Polar plot of the magnetic flux density norm at a radial distance of 25 mm from the rotor center.

ACDC_Module/Magnetostatics/static_field_halbach_rotor_3d

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

Insert the geometry sequence from the static_field_halbach_rotor_3d_geom_sequence.mph file.

In the toolbar, click

Browse to the model’s Application Libraries folder and double-click the file static_field_halbach_rotor_3d_geom_sequence.mph.

In the toolbar, click

Click the

button in the toolbar.

Click the

button in the toolbar.

Define variables for the radial and azimuthal magnetic flux densities.

Definitions

Variables 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
R	sqrt(x^2+y^2)	m	Radial distance
B_r	(mf.Bxx+mf.Byy)/R	T	Radial magnetic flux density
B_phi	(-mf.Bxy+mf.Byx)/R	T	Azimuthal magnetic flux density

Define a selection for the magnets.

Magnets

In the toolbar, click

Select Domains 2–4 only.

Right-click and choose .

In the dialog box, type Magnets in the text field.

Click .

Add a new cylindrical coordinate system. You will use this coordinate system to assign the magnetization of the permanent magnets.

Cylindrical System 2 (sys2)

In the toolbar, click

and choose .

Hide a few boundaries to view the results only in the inner part of the model domain.

Hide for Physics 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 1, 2, and 4 only.

Magnetic Fields (mf)

Now, set up the physics. Model the permanent magnets using .

Magnet, outward magnetized

In the window, under right-click and choose .

Select Domain 2 only.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type Magnet, outward magnetized.

Magnet, inward magnetized

In the toolbar, click

and choose .

Select Domain 4 only.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Specify the e vector as


-1	r
0	phi
0	a

In the text field, type Magnet, inward magnetized.

Magnet, counterclock-wise magnetized

In the toolbar, click

and choose .

Select Domain 3 only.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Specify the e vector as


0	r
1	phi
0	a

In the text field, type Magnet, counterclock-wise magnetized.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click

In the tree, select .

Click

In the toolbar, click

to close the window.

Materials

N50 (Sintered NdFeB) (mat2)

In the window for , locate the section.

From the list, choose .

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Size 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Specify a very fine mesh on the curves where the magnetic flux density is to be evaluated. This helps to obtain a smooth curve for magnetic flux density.

Size 2

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Edges 6 and 31 only.

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type 0.5.

Free Tetrahedral 1

In the toolbar, click

In the window for , click

Compare the mesh with the figure shown below.

Study 1

In the window, click .

In the window for , locate the section.

Clear the check box.

In the toolbar, click

Results

Use the Sector 3D dataset to produce a 3D dataset for the complete 3D model from the single-pole results.

Sector 3D 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the text field, type 8.

From the list, choose .

Click to expand the section. Select the check box.

The will be used later on, to get the right expression for B_phi in Figure 4.

Next, construct circles to visualize the magnetic flux density on the inside and the outside of the Halbach rotor.

Parameterized Curve 3D 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type phi.

In the text field, type 2*pi.

Locate the section. In the text field, type 55*cos(phi).

In the text field, type 55*sin(phi).

Parameterized Curve 3D 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type phi.

In the text field, type 2*pi.

Locate the section. In the text field, type 25*cos(phi).

In the text field, type 25*sin(phi).

Use the following instructions to reproduce the plot shown in Figure 2.

3D Plot Group 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Slice 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 1.

In the toolbar, click

Arrow Volume 1

In the window, right-click and choose .

In the window for , locate the section.

Find the subsection. In the text field, type 60.

Find the subsection. In the text field, type 1.

Locate the section. From the list, choose .

In the toolbar, click

Click the

button in the toolbar.

Next, generate a plot for the radial magnetic flux density outside the Halbach rotor. Compare the result with Figure 3.

1D Plot Group 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Select the check box.

In the associated text field, type Angle (rad).

Line Graph 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

In the text field, type phi.

In the toolbar, click

Create the azimuthal magnetic flux density plot as shown in Figure 4.

1D Plot Group 3

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Select the check box.

In the associated text field, type Angle (rad).

Line Graph 1

Right-click and choose .

In the window for , locate the section.

In the text field, type B_phi*(1-2*mod(sec1number,2)).

Select the check box.

In the associated text field, type Azimuthal magnetic flux density.

Locate the section. From the list, choose .

In the text field, type phi.

In the toolbar, click

Here, the expression B_phi*(1-2*mod(sec1number,2)) perhaps requires some additional clarification: The even sectors of the rotor are mirrored in the phi direction, with respect to the odd ones. The original dataset contains the values of B_phi for one odd sector only. The added correction term uses the modulus operator; it will flip between +1 and -1 every other sector.

The reason did not need such data manipulation, is because it considers its input (mf.Bx, mf.By, mf.Bz) a vector field and is able to apply the transformation itself. on the other hand, is in no way capable of relating its input to a certain "direction". It will therefore consider B_phi a scalar.

Next, generate the polar plot of the magnetic flux density norm at a distance 55 mm away from the center of the rotor.

Polar Plot Group 4

In the toolbar, click