Permanent Magnet

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Permanent Magnet

This example shows how to model the magnetic field surrounding a permanent magnet. It also computes the force with which it acts on a nearby iron rod. Thanks to the symmetry of the geometry and the antisymmetry of the magnetic field, only one fourth of the geometry needs to be modeled.

Figure 1: A full 3D view of the geometry. Left-right and top-down symmetry is used to minimize the problem size.

Model Definition

In a current free region, where

it is possible to define the scalar magnetic potential, Vm, from the relation

This is analogous to the definition of the electric potential for static electric fields.

Using the constitutive relation between the magnetic flux density and magnetic field

where Br is the remanent flux density, and together with the equation

you can derive an equation for Vm,

The model uses this equation by selecting the Magnetic Fields, No Currents interface from the AC/DC Module.

Boundary Conditions

The magnetic field is symmetric with respect to the xy-plane and antisymmetric with respect to the xz-plane. These planes therefore serve as exterior boundaries to the geometry.

On the symmetry plane, the magnetic field is tangential to the boundary. This is described by the Magnetic Insulation condition:

On the antisymmetry plane, the magnetic field is perpendicular to the boundary. This condition is represented by a constant magnetic scalar potential. The model uses the Zero Magnetic Scalar Potential condition.

If the air box is sufficiently large, the boundary condition used on its remaining exterior boundaries has little influence on the field in the vicinity of the magnet. Although an infinite element domain would give the very best results, this model uses the magnetic insulation condition for convenience.

Results and Discussion

The force on the rod is calculated internally as an integral of the surface stress tensor over all boundaries of the rod. The expression for the stress tensor reads

where n1 is the boundary normal pointing out from the rod and T2 the stress tensor of air. The integration gives 3.64 N, which corresponds to one quarter of the rod. The actual force on the rod is therefore four times this value, or 14.6 N.

ACDC_Module/Introductory_Magnetostatics/permanent_magnet

Modeling Instructions

From the menu, choose .

In the window, click

.

1

In the window, click

.

2

In the tree, select .

3

4

5

In the tree, select .

6

Import 1 (imp1)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

4

Browse to the model’s Application Libraries folder and double-click the file permanent_magnet.mphbin.

5

The imported geometry contains the permanent magnet and the rod that it is acting on. The following instructions show you how to create the air box and delete the part of the geometry that you do not want to include in the model.

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the text field, type 0.25.

4

In the text field, type 0.1.

5

In the text field, type 0.1.

6

Locate the section. In the text field, type -0.1.

7

Right-click and choose .

The air box now covers only the parts of the magnet and the rod that you want to include in the model. Perform a Boolean geometry operation to get rid of the superfluous parts.

Compose 1 (co1)

1

In the toolbar, click

and choose .

2

Click in the window and then press Ctrl+A to select both objects.

3

In the window for , locate the section.

4

In the text field, type blk1+imp1*blk1.

5

.

6

button in the toolbar.

The geometry now contains the air volume and one fourth of the imported objects.

1

In the window, under right-click and choose .

2

In the window for , type Iron in the text field.

3

Select Domains 2 and 4 only.

4

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	4000	1	Basic

1

Right-click and choose .

2

In the window for , type Air in the text field.

3

Select Domain 1 only.

4

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic

1

In the toolbar, click

to open the window.

2

Go to the window.

3

In the tree, select .

4

Click in the window toolbar.

5

In the toolbar, click

to close the window.

N54 (Sintered NdFeB) (mat3)

1

In the window for , locate the section.

2

.

3

In the dialog box, type 3 in the text field.

4

Magnetic Fields, No Currents (mfnc)

1

In the window, under right-click and choose .

2

Select Domain 3 only.

1

In the window, click .

2

Select Boundary 17 only.

1

In the window, click .

2

Select Boundary 12 only.

All exterior boundaries are magnetically insulated by default. Use the condition on those boundaries where antisymmetry holds.

Symmetry Plane 1

1

In the toolbar, click

and choose .

2

In the window, click .

3

Select Boundaries 2, 8, and 24 only.

4

In the window for , locate the section.

5

From the list, choose .

Next, add a force computation on the rod.

Force Calculation 1

1

In the toolbar, click

and choose .

2

Select Domain 2 only.

3

In the window for , locate the section.

4

In the text field, type rod.

Free Tetrahedral 1

To get an accurate force computation, you need a particularly fine mesh on the rod. It also makes sense to use a fine mesh in the magnet and its iron core, as this is where the magnetic field will be the strongest.

In the toolbar, click

.

1

In the window, click .

2

In the window for , locate the section.

3

From the list, choose .

1

In the window, right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Select Domains 2–4 only.

5

Locate the section. Click the button.

6

Locate the section.

7

Select the check box. In the associated text field, type 0.0025.

8

.

1

In the window, click .

2

In the window for , locate the section.

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Clear the check box.

4

In the toolbar, click

.

3D Plot Group 1

In the toolbar, click

and choose .

1

Right-click and choose .

2

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

3

Locate the section. From the list, choose .

4

From the list, choose .

5

In the text field, type 0.005.

6

Locate the section. Click

.

7

In the dialog box, select in the tree.

8

9

In the toolbar, click

.

The plot shows the magnitude of the flux density just above the symmetry plane. Add an arrow plot to see its direction.

1

In the window, right-click and choose .

2

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

3

Locate the section. Find the subsection. In the text field, type 100.

4

Find the subsection. In the text field, type 50.

5

Find the subsection. From the list, choose .

6

In the text field, type 0.0051.

7

In the toolbar, click

.

The plot shows only the quarter of the geometry used for the computation.

Introduce additional mirror datasets to plot the solution in the complete geometry.

Symmetry Condition

1

In the toolbar, click

and choose .

2

In the window for , type Symmetry Condition in the text field.

3

Locate the section. From the list, choose .

Antisymmetry Condition

1

In the toolbar, click

and choose .

2

In the window for , type Antisymmetry Condition in the text field.

3

Locate the section. From the list, choose .

4

Locate the section. From the list, choose .

5

Click to expand the section. From the list, choose .

3D Plot Group 1

1

In the window, under click .

2

In the window for , locate the section.

3

From the list, choose .

4

In the toolbar, click

.

1

In the window, click .

2

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

3

In the toolbar, click

.

3D Plot Group 1

Finally, use Global Evaluation to evaluate the force on the rod.

Global Evaluation 1

1

In the toolbar, click

.

2

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

The variable containing the force on a quarter of the rod will be added to the table. Multiply this value by 4 to compute the total force on the rod.

3

In the column, change the expression to mfnc.Forcex_rod*4.

4

In the column, write Total force on the rod.

5

The total force on the rod evaluates to 14.6 N.