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Static Field Modeling of a Halbach Rotor
Introduction
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.
Application Library path: ACDC_Module/Magnetostatics/static_field_halbach_rotor_3d
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select AC/DC>Electromagnetic Fields>Magnetic Fields (mf).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies>Stationary.
6
Click  Done.
Geometry 1
Insert the geometry sequence from the static_field_halbach_rotor_3d_geom_sequence.mph file.
1
In the Geometry toolbar, click Insert Sequence and choose Insert Sequence.
2
Browse to the model’s Application Libraries folder and double-click the file static_field_halbach_rotor_3d_geom_sequence.mph.
3
In the Geometry toolbar, click  Build All.
4
Click the  Zoom Extents button in the Graphics toolbar.
5
Click the  Wireframe Rendering button in the Graphics toolbar.
Define a selection for the magnets.
Definitions
Magnets
1
In the Definitions toolbar, click  Explicit.
2
Select Domains 2–4 only.
3
Right-click Explicit 1 and choose Rename.
4
In the Rename Explicit dialog box, type Magnets in the New label text field.
5
Click OK.
Add a new cylindrical coordinate system. You will use this coordinate system to assign the magnetization of the permanent magnets.
Cylindrical System 2 (sys2)
1
In the Definitions toolbar, click  Coordinate Systems and choose Cylindrical System.
Define variables for the radial and azimuthal magnetic flux densities using the Vector Transform feature.
2
Click the  Show More Options button in the Model Builder toolbar.
3
In the Show More Options dialog box, in the tree, select the check box for the node General>Variable Utilities.
4
Click OK.
Vector Transform 1 (vectr1)
1
In the Definitions toolbar, click  Variable Utilities and choose Vector Transform.
2
In the Settings window for Vector Transform, type B_cyl in the Name text field.
3
Clear all domains.
4
Click the  Select All button in the Graphics toolbar.
5
Click Replace Expression in the upper-right corner of the Input section. From the menu, choose Component 1 (comp1)>Magnetic Fields>Magnetic>Magnetic flux density>mf.B_s - Magnetic flux density - T.
6
Locate the Output section. From the Coordinate system list, choose Cylindrical System 2 (sys2).
7
Locate the Transform Settings section. From the Transform as list, choose Flux vector.
View 1
Hide a few boundaries to view the results only in the inner part of the model domain.
Hide for Physics 1
1
In the Model Builder window, right-click View 1 and choose Hide for Physics.
2
In the Settings window for Hide for Physics, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 1, 2, and 4 only.
Magnetic Fields (mf)
Now, set up the Magnetic Fields physics. Model the permanent magnets using Ampère’s Law.
Magnet, Outward Magnetized
1
In the Model Builder window, under Component 1 (comp1) right-click Magnetic Fields (mf) and choose Ampère’s Law.
2
Select Domain 2 only.
3
In the Settings window for Ampère’s Law, locate the Coordinate System Selection section.
4
From the Coordinate system list, choose Cylindrical System 2 (sys2).
5
Locate the Constitutive Relation B-H section. From the Magnetization model list, choose Remanent flux density.
6
In the Label text field, type Magnet, Outward Magnetized.
Magnet, Inward Magnetized
1
In the Physics toolbar, click  Domains and choose Ampère’s Law.
2
Select Domain 4 only.
3
In the Settings window for Ampère’s Law, locate the Coordinate System Selection section.
4
From the Coordinate system list, choose Cylindrical System 2 (sys2).
5
Locate the Constitutive Relation B-H section. From the Magnetization model list, choose Remanent flux density.
6
Specify the e vector as
-1
r
0
phi
0
a
7
In the Label text field, type Magnet, Inward Magnetized.
Magnet, Counterclock-Wise Magnetized
1
In the Physics toolbar, click  Domains and choose Ampère’s Law.
2
Select Domain 3 only.
3
In the Settings window for Ampère’s Law, locate the Coordinate System Selection section.
4
From the Coordinate system list, choose Cylindrical System 2 (sys2).
5
Locate the Constitutive Relation B-H section. From the Magnetization model list, choose Remanent flux density.
6
Specify the e vector as
0
r
1
phi
0
a
7
In the Label text field, type Magnet, Counterclock-Wise Magnetized.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Air.
4
Right-click and choose Add to Component 1 (comp1).
5
In the tree, select AC/DC>Hard Magnetic Materials>Sintered NdFeB Grades (Chinese Standard)>N50 (Sintered NdFeB).
6
Right-click and choose Add to Component 1 (comp1).
7
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
N50 (Sintered NdFeB) (mat2)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Magnets.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Coarse.
Size 1
1
Right-click Component 1 (comp1)>Mesh 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Magnets.
5
Locate the Element Size section. From the Predefined list, choose Fine.
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
1
In the Model Builder window, right-click Mesh 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Edge.
4
Select Edges 6 and 31 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section. Select the Maximum element size check box.
7
In the associated text field, type 0.5.
Free Tetrahedral 1
1
In the Mesh toolbar, click  Free Tetrahedral.
2
In the Settings window for Free Tetrahedral, click  Build All.
Compare the mesh with the figure shown below.
Study 1
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, locate the Study Settings section.
3
Clear the Generate default plots check box.
4
In the Home toolbar, click  Compute.
Results
Use the Sector 3D dataset to produce a 3D dataset for the complete 3D model from the single-pole results.
Sector 3D 1
1
In the Model Builder window, expand the Results node.
2
Right-click Results>Datasets and choose More 3D Datasets>Sector 3D.
3
In the Settings window for Sector 3D, locate the Symmetry section.
4
In the Number of sectors text field, type 8.
5
From the Transformation list, choose Rotation and reflection.
6
Click to expand the Advanced section. Select the Define variables check box.
The Sector number will be used later on, to get the right expression for B_cyl.vphi 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
1
In the Results toolbar, click  More Datasets and choose Parameterized Curve 3D.
2
In the Settings window for Parameterized Curve 3D, locate the Data section.
3
From the Dataset list, choose Sector 3D 1.
4
Locate the Parameter section. In the Name text field, type phi.
5
In the Maximum text field, type 2*pi.
6
Locate the Expressions section. In the x text field, type 55*cos(phi).
7
In the y text field, type 55*sin(phi).
Parameterized Curve 3D 2
1
In the Results toolbar, click  More Datasets and choose Parameterized Curve 3D.
2
In the Settings window for Parameterized Curve 3D, locate the Data section.
3
From the Dataset list, choose Sector 3D 1.
4
Locate the Parameter section. In the Name text field, type phi.
5
In the Maximum text field, type 2*pi.
6
Locate the Expressions section. In the x text field, type 25*cos(phi).
7
In the y text field, type 25*sin(phi).
Use the following instructions to reproduce the plot shown in Figure 2.
B Field
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type B Field in the Label text field.
3
Locate the Data section. From the Dataset list, choose Sector 3D 1.
Slice 1
1
Right-click B Field and choose Slice.
2
In the Settings window for Slice, locate the Plane Data section.
3
From the Plane list, choose xy-planes.
4
In the Planes text field, type 1.
5
In the B Field toolbar, click  Plot.
Arrow Volume 1
1
In the Model Builder window, right-click B Field and choose Arrow Volume.
2
In the Settings window for Arrow Volume, locate the Arrow Positioning section.
3
Find the x grid points subsection. In the Points text field, type 60.
4
Find the y grid points subsection. In the Points text field, type 60.
5
Find the z grid points subsection. In the Points text field, type 1.
6
Locate the Coloring and Style section. From the Color list, choose Black.
7
In the B Field toolbar, click  Plot.
8
Click the  Go to XY View button in the Graphics toolbar.
Next, generate a plot for the radial magnetic flux density outside the Halbach rotor. Compare the result with Figure 3.
Br vs. phi
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Br vs. phi in the Label text field.
3
Locate the Data section. From the Dataset list, choose Parameterized Curve 3D 1.
4
Locate the Plot Settings section. Select the x-axis label check box.
5
In the associated text field, type Angle (rad).
Line Graph 1
1
Right-click Br vs. phi and choose Line Graph.
2
In the Settings window for Line Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Definitions>Vector Transform 1 (B_cyl)>Transformed vector - T>B_cyl.vr - Transformed vector, r component.
3
Locate the x-Axis Data section. From the Parameter list, choose Expression.
4
In the Expression text field, type phi.
5
Locate the y-Axis Data section. Select the Description check box.
6
In the associated text field, type Magnetic flux density, r component.
7
In the Br vs. phi toolbar, click  Plot.
Create the azimuthal magnetic flux density plot as shown in Figure 4.
Bphi vs. phi
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Bphi vs. phi in the Label text field.
3
Locate the Data section. From the Dataset list, choose Parameterized Curve 3D 1.
4
Locate the Plot Settings section. Select the x-axis label check box.
5
In the associated text field, type Angle (rad).
Line Graph 1
1
Right-click Bphi vs. phi and choose Line Graph.
2
In the Settings window for Line Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Definitions>Vector Transform 1 (B_cyl)>Transformed vector - T>B_cyl.vphi - Transformed vector, phi component.
3
Locate the y-Axis Data section. In the Expression text field, type B_cyl.vphi*(1-2*mod(sec1number,2)).
4
Select the Description check box.
5
In the associated text field, type Magnetic flux density, phi component.
6
Locate the x-Axis Data section. From the Parameter list, choose Expression.
7
In the Expression text field, type phi.
8
In the Bphi vs. phi toolbar, click  Plot.
Here, the expression B_cyl.vphi*(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_cyl.vphi 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 Arrow Volume 1 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. Line Graph 1 on the other hand, is in no way capable of relating its input to a certain "direction". It will therefore consider B_cyl.vphi 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.
normB vs. phi at r=55 mm
1
In the Home toolbar, click  Add Plot Group and choose Polar Plot Group.
2
In the Settings window for Polar Plot Group, type normB vs. phi at r=55 mm in the Label text field.
3
Locate the Data section. From the Dataset list, choose Parameterized Curve 3D 1.
4
Locate the Axis section. Select the Manual axis limits check box.
5
In the r maximum text field, type 0.56.
Line Graph 1
1
Right-click normB vs. phi at r=55 mm and choose Line Graph.
2
In the Settings window for Line Graph, locate the θ Angle Data section.
3
From the Parameter list, choose Expression.
4
In the Expression text field, type phi.
5
In the normB vs. phi at r=55 mm toolbar, click  Plot.
Finally, reproduce the plot for the magnetic flux density norm at a distance 25 mm from the rotor center.
normB vs. phi at r=25 mm
1
In the Home toolbar, click  Add Plot Group and choose Polar Plot Group.
2
In the Settings window for Polar Plot Group, type normB vs. phi at r=25 mm in the Label text field.
3
Locate the Data section. From the Dataset list, choose Parameterized Curve 3D 2.
4
Locate the Axis section. Select the Manual axis limits check box.
5
In the r maximum text field, type 0.12.
Line Graph 1
1
Right-click normB vs. phi at r=25 mm and choose Line Graph.
2
In the Settings window for Line Graph, locate the θ Angle Data section.
3
From the Parameter list, choose Expression.
4
In the Expression text field, type phi.
5
In the normB vs. phi at r=25 mm toolbar, click  Plot.
Compare this figure with that shown in Figure 6.