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Homopolar Generator
Introduction
A homopolar generator is composed of an electrically conductive rotating disc placed in a uniform magnetic field that is perpendicular to the plane of rotation. The motion of the conductor through the static magnetic field induces Lorentz currents in the disc. By connecting the outside rim of the disc to the center via a stationary conductor, a significant current can be generated. This example models the flow of current through the copper conductor and the rotating disc.
Figure 1: Model illustration of a homopolar generator. A spinning disc is placed in a uniform magnetic field.
Model Definition
The application is formulated in 3D and solved using a stationary formulation. The rotating conductive disc of radius 10 cm and height 3 cm is placed in a space with a uniform magnetic flux density of 1 T. Figure 1 shows the current direction, the rotational direction, and the direction of the magnetic field. The center of the disc is connected to its rim via a conductor, which forms a closed, solenoidal path for the circulation of electric current. This application also illustrates the proper use of a Lorentz term to model the rotation of the disc.
For a disc rotating about the z-axis, with an angular velocity ω, the velocity, v, at a point (x, y) is given by
The use of a Lorentz term to include the motion is valid in a situation when the moving domain does not contain any magnetic sources, such as currents or magnetization (fixed or induced), that move along with the material, and when the moving domains are unbounded and invariant in the direction of the motion. In this example, the induced current distribution is stationary and does not follow the rotation of the disc.
Results and Discussion
The model is solved using a stationary solver at an angular speed of 1200 rpm. A total current of about 45.159 kA flows through the conductor. Figure 2 shows the streamline plot of an induced eddy current density in the rotating disc and the conductor.
Figure 3 shows a volume plot of the magnetic flux density (total) and a streamline plot of the induced magnetic flux density.
Figure 4 illustrates the direction of the current density in the rotating disc and the conductor. In this figure, the current is flowing from the center of the disc to its rim (inside the disc). The current direction is reversed if either the rotational direction or the direction of the uniform magnetic field is changed.
Finally, Figure 5 displays a volume plot of the resistive loss in the copper domains.
Figure 2: Current density norm in the conducting domains.
Figure 3: The total magnetic flux density norm (surface plot) and the induced magnetic flux density (streamline plot).
Figure 4: The direction of current in the rotating disc and the conductor.
Figure 5: The resistive loss in the rotating disc and the conductor.
Application Library path: ACDC_Module/Motors_and_Actuators/homopolar_generator
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>Vector Formulations>Magnetic and Electric Fields (mef).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies>Stationary.
6
Click  Done.
Geometry 1
Define the rotational velocity of a copper disc as a global parameter.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
Bz0
1[Wb/m^2]
1 T
Constant magnetic flux density
RPM
1200[rpm]
20 1/s
Disc speed
Follow the instructions below to construct the model geometry. First, create the geometry of the rotating disc.
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose mm.
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 100.
4
In the Height text field, type 30.
5
Locate the Position section. In the z text field, type -15.
6
Click  Build Selected.
Cylinder 2 (cyl2)
1
Right-click Cylinder 1 (cyl1) and choose Duplicate.
2
In the Settings window for Cylinder, click  Build Selected.
Cylinder 3 (cyl3)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 15.
4
In the Height text field, type 30.
5
Locate the Position section. In the z text field, type 15.
6
Click  Build Selected.
Cylinder 4 (cyl4)
1
Right-click Cylinder 3 (cyl3) and choose Duplicate.
2
In the Settings window for Cylinder, click  Build Selected.
Add the conductor to connect the center of the disc to the rim.
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type 35.
4
Locate the Unite Objects section. Clear the Unite objects check box.
5
Click  Show Work Plane.
Work Plane 1 (wp1)>Plane Geometry
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 1 (wp1)>Rectangle 1 (r1)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 130.
4
In the Height text field, type 20.
5
Locate the Position section. In the xw text field, type -130.
6
In the yw text field, type -10.
7
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (mm)
Layer 1
60
8
Clear the Layers on bottom check box.
9
Select the Layers to the left check box.
10
Click  Build Selected.
Work Plane 1 (wp1)>Rectangle 2 (r2)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 13.
4
In the Height text field, type 20.
5
Locate the Position section. In the xw text field, type -130.
6
In the yw text field, type -10.
7
Click  Build Selected.
Extrude 1 (ext1)
1
In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Work Plane 1 (wp1) and choose Extrude.
2
Select the object wp1.r1 only.
3
In the Settings window for Extrude, click  Build Selected.
4
Locate the Distances section. In the table, enter the following settings:
Distances (mm)
10
Difference 1 (dif1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object ext1 only.
3
In the Settings window for Difference, locate the Difference section.
4
Find the Objects to subtract subsection. Click to select the  Activate Selection toggle button.
5
Select the object cyl3 only.
6
Click  Build Selected.
Extrude 2 (ext2)
1
In the Geometry toolbar, click  Extrude.
2
In the Settings window for Extrude, click  Build Selected.
3
Locate the Distances section. In the table, enter the following settings:
Distances (mm)
-50
Work Plane 2 (wp2)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
From the Plane type list, choose Face parallel.
4
Click the  Wireframe Rendering button in the Graphics toolbar.
5
On the object ext2, select Boundary 6 only.
6
Click  Show Work Plane.
Work Plane 2 (wp2)>Plane Geometry
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 2 (wp2)>Rectangle 1 (r1)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 20.
4
In the Height text field, type 30.
5
Locate the Position section. In the xw text field, type -10.
6
In the yw text field, type -5.
7
Click  Build Selected.
Extrude 3 (ext3)
1
In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Work Plane 2 (wp2) and choose Extrude.
2
In the Settings window for Extrude, click  Build Selected.
3
Locate the Distances section. In the table, enter the following settings:
Distances (mm)
30
Difference 2 (dif2)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object ext3 only.
3
In the Settings window for Difference, locate the Difference section.
4
Find the Objects to subtract subsection. Click to select the  Activate Selection toggle button.
5
Select the object cyl1 only.
6
Click  Build Selected.
Cylinder 5 (cyl5)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 300.
4
In the Height text field, type 400.
5
Locate the Position section. In the z text field, type -200.
6
Click  Build Selected.
Form Union (fin)
1
In the Geometry toolbar, click  Build All.
2
Click the  Go to Default View button in the Graphics toolbar.
3
Click the  Wireframe Rendering button in the Graphics toolbar.
The final geometry looks like as shown in the figure below.
Definitions
View 1
Hide some outer boundaries to view the rotating disc.
Hide for Physics 1
1
In the Model Builder window, expand the Component 1 (comp1)>Definitions node.
2
Right-click View 1 and choose Hide for Physics.
3
In the Settings window for Hide for Physics, locate the Geometric Entity Selection section.
4
From the Geometric entity level list, choose Boundary.
5
Select Boundaries 1 and 4 only.
Define boundary selections for the exterior boundaries of the outer cylinder. You will use these boundaries to define the uniform magnetic field.
Exterior Boundaries
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, locate the Input Entities section.
3
From the Geometric entity level list, choose Boundary.
4
Click  Paste Selection.
5
In the Paste Selection dialog box, type 1-4,38,43 in the Selection text field.
6
Click OK.
7
Right-click Explicit 1 and choose Rename.
8
In the Rename Explicit dialog box, type Exterior Boundaries in the New label text field.
9
Click OK.
Define the domain selections for the rotating disc.
Rotating Disc
1
In the Definitions toolbar, click  Explicit.
2
Select Domains 5 and 7 only.
3
Right-click Explicit 2 and choose Rename.
4
In the Rename Explicit dialog box, type Rotating Disc in the New label text field.
5
Click OK.
Next, assign a selection for the copper domains.
Copper Domains
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, locate the Input Entities section.
3
Click  Paste Selection.
4
In the Paste Selection dialog box, type 2-7 in the Selection text field.
5
Click OK.
6
Right-click Explicit 3 and choose Rename.
7
In the Rename Explicit dialog box, type Copper Domains in the New label text field.
8
Click OK.
Magnetic and Electric Fields (mef)
Set up the Magnetic and Electric Fields physics. Use Ampère’s Law for the air domain and the default Ampère’s Law and Current Conservation feature for the copper domains. Assign the rotational velocity of the rotating disc using a Velocity (Lorentz Term) feature.
Ampère’s Law 1
1
In the Model Builder window, under Component 1 (comp1) right-click Magnetic and Electric Fields (mef) and choose Ampère’s Law.
2
Select Domain 1 only.
Velocity (Lorentz Term) 1
1
In the Physics toolbar, click  Domains and choose Velocity (Lorentz Term).
2
In the Settings window for Velocity (Lorentz Term), locate the Domain Selection section.
3
From the Selection list, choose Rotating Disc.
4
Locate the Velocity (Lorentz Term) section. Specify the v vector as
-2*pi*RPM*y
x
2*pi*RPM*x
y
0
z
Use the Magnetic Potential boundary condition on the exterior boundaries to define the uniform magnetic fields in the z direction.
Magnetic Potential 1
1
In the Physics toolbar, click  Boundaries and choose Magnetic Potential.
2
In the Settings window for Magnetic Potential, locate the Boundary Selection section.
3
From the Selection list, choose Exterior Boundaries.
4
Locate the Magnetic Potential section. Specify the A0 vector as
0
x
Bz0*x
y
0
z
Materials
Assign materials to the model. First, assign air in the exterior region. Next, specify copper for the remaining domains.
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
Click Add to Component in the window toolbar.
5
In the tree, select AC/DC>Copper.
6
Click Add to Component in the window toolbar.
7
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Copper (mat2)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Copper Domains.
Mesh 1
The finite element mesh in the surrounding air domain can be relatively coarse, but the element size in the disc and the conductors must be small enough to resolve the current distribution in the domains.
Size 1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Size.
Size
1
In the Settings window for Size, locate the Element Size section.
2
From the Predefined list, choose Coarser.
Size 1
1
In the Model Builder window, click Size 1.
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 Copper Domains.
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 10.
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
In the Model Builder window, expand the Results node.
Study 1/Solution 1 (sol1)
Create a dataset for visualizing the results in the copper domain only.
Study 1/Solution 1 (2) (sol1)
1
In the Model Builder window, expand the Results>Datasets node.
2
Right-click Results>Datasets>Study 1/Solution 1 (sol1) and choose Duplicate.
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Copper Domains.
Use the following instructions to reproduce the plot for the current density shown in Figure 2.
3D Plot Group 1
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (2) (sol1).
Streamline 1
1
Right-click 3D Plot Group 1 and choose Streamline.
2
In the Settings window for Streamline, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Magnetic and Electric Fields>Currents and charge>mef.Jx,mef.Jy,mef.Jz - Current density.
3
Locate the Streamline Positioning section. In the Number text field, type 50.
4
Locate the Selection section. Click  Paste Selection.
5
In the Paste Selection dialog box, type 34 in the Selection text field.
6
Click OK.
7
In the Settings window for Streamline, locate the Coloring and Style section.
8
Find the Line style subsection. From the Type list, choose Tube.
9
In the Tube radius expression text field, type log10(mef.normJ)-5.
Color Expression 1
1
Right-click Streamline 1 and choose Color Expression.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type mef.normJ.
4
In the 3D Plot Group 1 toolbar, click  Plot.
3D Current Plot
1
In the Model Builder window, under Results right-click 3D Plot Group 1 and choose Rename.
2
In the Rename 3D Plot Group dialog box, type 3D Current Plot in the New label text field.
3
Click OK.
4
Click the  Zoom Extents button in the Graphics toolbar.
Next, generate a volume plot for the total magnetic flux density and a streamline plot for the induced magnetic flux density. Compare the plots with Figure 3.
3D Plot Group 2
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (2) (sol1).
Total B
1
Right-click 3D Plot Group 2 and choose Volume.
2
In the Settings window for Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Magnetic and Electric Fields>Magnetic>mef.normB - Magnetic flux density norm - T.
3
Right-click Volume 1 and choose Rename.
4
In the Rename Volume dialog box, type Total B in the New label text field.
5
Click OK.
Induced B
1
In the Model Builder window, right-click 3D Plot Group 2 and choose Streamline.
2
In the Settings window for Streamline, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (1) (sol1).
4
Locate the Expression section. In the z component text field, type mef.Bz-Bz0.
5
Locate the Streamline Positioning section. In the Number text field, type 100.
6
Locate the Selection section. Click  Paste Selection.
7
In the Paste Selection dialog box, type 24 in the Selection text field.
8
Click OK.
9
In the 3D Plot Group 2 toolbar, click  Plot.
10
Right-click Streamline 1 and choose Rename.
11
In the Rename Streamline dialog box, type Induced B in the New label text field.
12
Click OK.
Magnetic Flux Density (B)
1
In the Model Builder window, right-click 3D Plot Group 2 and choose Rename.
2
In the Rename 3D Plot Group dialog box, type Magnetic Flux Density (B) in the New label text field.
3
Click OK.
Reproduce Figure 4 using the following instructions.
3D Plot Group 3
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (2) (sol1).
Arrow Volume 1
1
Right-click 3D Plot Group 3 and choose Arrow Volume.
2
In the Settings window for Arrow Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Magnetic and Electric Fields>Currents and charge>mef.Jx,mef.Jy,mef.Jz - Current density.
3
Locate the Arrow Positioning section. Find the x grid points subsection. In the Points text field, type 40.
4
Find the y grid points subsection. In the Points text field, type 40.
5
Find the z grid points subsection. From the Entry method list, choose Coordinates.
6
In the Coordinates text field, type 0.
7
Locate the Coloring and Style section. From the Arrow length list, choose Logarithmic.
Color Expression 1
1
Right-click Arrow Volume 1 and choose Color Expression.
2
In the 3D Plot Group 3 toolbar, click  Plot.
Arrow Volume 2
1
In the Model Builder window, under Results>3D Plot Group 3 right-click Arrow Volume 1 and choose Duplicate.
2
In the Settings window for Arrow Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Magnetic and Electric Fields>Currents and charge>mef.Jx,mef.Jy,mef.Jz - Current density.
3
Locate the Arrow Positioning section. Find the x grid points subsection. In the Points text field, type 80.
4
Find the y grid points subsection. In the Points text field, type 1.
5
Find the z grid points subsection. From the Entry method list, choose Number of points.
6
In the Points text field, type 20.
7
Click to expand the Inherit Style section. From the Plot list, choose Arrow Volume 1.
8
In the 3D Plot Group 3 toolbar, click  Plot.
Create a volume plot for the resistive loss in the copper domains.
3D Plot Group 4
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (2) (sol1).
Volume 1
1
Right-click 3D Plot Group 4 and choose Volume.
2
In the Settings window for Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Magnetic and Electric Fields>Heating and losses>mef.Qh - Volumetric loss density, electromagnetic - W/m³.
3
Locate the Coloring and Style section. From the Color table list, choose Thermal.
4
In the 3D Plot Group 4 toolbar, click  Plot.
Compare this figure with that shown in Figure 5.
Finally, evaluate the total current through the conductor as the surface integral of the current density norm.
Surface Integration 1
1
In the Results toolbar, click  More Derived Values and choose Integration>Surface Integration.
2
Select Boundary 27 only.
3
In the Settings window for Surface Integration, click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Component 1 (comp1)>Magnetic and Electric Fields>Currents and charge>mef.normJ - Current density norm - A/m².
4
Click  Evaluate.
Table
1
Go to the Table window.
The total current through the conductor should be about 45 kA.