Motor 001 - Permanent magnet synchronous electric motor

Model definition

A 3D permanent magnet synchronous electric motor is considered under magnetostatic conditions using $\phi$-formulation. The symmetry cell (1/8) of the motor having the periodic face aligned with the x-axis is illustrated below.

Component	Dimensions
Stator steel outer radius	0.100 m
Stator steel inner radius	0.059 m
Rotor steel outer radius	0.055 m
Rotor steel inner radius	0.025 m
Permanent magnet outer radius	0.058 m
Air gap thickness	0.001 m

Output Results:

• Torque (Nm).

Material Data

• Linear steel

Property	Value
Magnetic permeability	2000 $\times \mu_0$ = 2000 $\times$ 4$\pi \times$10$^{-7}$ H/m

Boundary conditions

Name	Type	Value
left and right y-planes	periodicity: rotation	antiperiodic
rotor-stator interface	continuity	potential equality on both sides

Step-by-step guide

Here you’ll find a step-by-step tutorial on how to simulate this in Quanscient Allsolve.

Step 1 - Create geometry

1. Start with a new project.  

2. Import the geometry by uploading a step file.  

3. Click on + icon next to Geometry elements and click on Translate.  

Note

The translation is done to separate the rotor and stator for meshing, and so to have a duplicate mesh on the rotor-stator interface.

4. In the geo operations settings, choose the rotating region target as shown in the image below and provide the translation in meters: X=0; Y=0; Z=0.02.  

5. Click on Apply and then Confirm model changes.

Position of geometry

In modelling a symmetry cell of a motor having periodic boundaries, the position of the geometry needs to be such that the rotation happens around the z-axis, and one of the periodic boundaries needs to be aligned with the x-axis.

Step 2 - Define regions and materials

1. Proceed to the Properties section to define regions and materials.  

2. Click on the + icon next to Materials, select Air from the list, and click Confirm.  

3. To apply this material, Click on Add volume in Materials settings and select the permanent magnet domain, and all air domains as shown in the image below. Click Apply.  

4. Similarly add a new material and select Carbon steel AISI 1020. Apply this material on the steel domain.

Nonlinear material

Notice that when Carbon steel AISI 1020 material was added, a new shared expression BHsteel1020 gets automatically added to the Shared expressions.

 

5. In this example, only linear steel is considered. To apply linear permeability for the steel, Click Properties under the Target tags in Materials settings and replace the non-linear permeability BHsteel1020(norm(H) + 1e-10) / (norm(H) + 1e-10) with 2000 * mu0. Click Apply.  

6. Now define a rotor volume as a shared region so that it can be referred when shifting the rotor back to its original position and when applying rotation. To do so, click + icon next to Shared regions and choose volume on the list.  

7. Name the volume to rotor and pick all rotating regions including half of the airgap as shown in the image below.  

8. Repeat the steps 6 and 7 to add the airgap volume. The air gap volume is referred to when calculating the torque.  

9. Now define a shared expression for current amplitude, Click on the + icon next the Shared expressions. Name the expression to I, give the description, and the value of 300 Amperes. Click `Apply’.  

10. Repeat the steps 9 and 10 to add a shared expression for the mechanical angle alpha and for the electrical angle phase having the values of 27 degrees and 0.0 degrees, respectively.  

Step 3 - Define the physics and apply boundary conditions

1. Proceed to the Physics section to define physics and interactions.  

2. Click on the + icon to add a new physics. Select Magnetism $\phi$.  

3. To apply this physics on the geometry, Click on Add volume, select all the volumes except the windings and click Apply.

Quick selection of volumes

1. Select all the volumes: click on the + icon under the targets and click Select all button.
2. Deselect the windings: click the winding volumes in the model view window to deselect them.

 

4. To add interactions, click on the + icon next to Magnetism$\phi$ and select Constraint. Add a point region, select a point, and give the value of 0. This is the gauge condition for the scalar potential $\phi$ and will quarantee the uniqueness of the solution. Click Apply.

Choosing point for gauge condition

Choose the point on a Magnetism $\phi$ region so that it is not on the periodic boundaries nor on the sliding interface. E.g. the corner point of the permanent magnet in air gap is a proper one.

 

5. Introduce the permanent magnet. Add a Remanence interaction on the permanent magnet volume. Address a 0.5 T magnetization radially by giving X = 0.5*x / sqrt(x*x + y*y), Y = 0.5*y / sqrt(x*x + y*y), and Z=0. Click Apply.  

6. Add Periodicity interaction. Click Add surface under the Peridocity target 1 and select the periodic boundary alinged with the x-axis. Similarly, select the opposite surfaces for the Periodicty target 2 (second image). Under Parameters, select the periodicity type to be rotation, give the center of rotation: X=0, Y=0, Z=0, and the rotation angle in degrees: X=0, Y=0, Z=45, and select Antiperiodicity. Click Apply.

Caution

When applying rotational periodicity, the first periodic surface needs to be aligned with the X-axis.

7. Add Continuity interaction. Click on Add surface under Continuity target 1 and select the interface on the stator side. Similarly, select the interface on the rotor side for the Continuity target 2. Click Apply.  

8. Introduce a current source. Add Lump I/V cut interaction. Click on Add curve and select a closed loop around the winding closest to the x-axis. Set value of I*sin((phase + 4.0 * alpha - 0.0) * pi/180.0) for the current under the parameters.

Note

Here, the rotor has 8 poles. Therefore the electrical angle is 8/2 times the mechanical angle at synchronous speed.

 

9. Repeat the step 8 for the winding in the middle and for the winding on the left with the current values of I*sin((phase + 4.0 * alpha - 60.0) * pi/180.0) and I*sin((phase + 4.0 * alpha - 120.0) * pi/180.0), respectively.  

Step 4 - Apply simulation settings

1. Proceed to the Simulations sections and add a new mesh by clicking the + icon next to Meshes.  

2. Under the Mesh quality choose Expert settings. Set the maximum size to 0.001 and enable the Curved mesh. Click on Add mesh refinement and select Volume in the list.  

3. Click on Refinement . Volume, select the air gap volume and set Max size to 0.0005. Click on Apply & mesh.  

4. Click on Show preview to preview the generated mesh.  

5. Add a new simulation. Select Steady state as the Analysis type.  

6. Set the previously generated mesh for the simulation by clicking on Mesh under Simulation 1 and by choosing Mesh 1.  

7. For visualizing the magnetic flux density after simulation, click on + icon next to the Outputs under the Simulation 1 and select Magnetix flux density.  

8. Click on Add volume and select all volumes except the windings. Click Apply.  

9. In the scripting, the following three modifications need to be done:

• a). moving the rotor back to its original position and applying the rotation.
• b). Apply antioperiodicity and rotation for the continuity condition.
• c). Calculating the torque.

9. To enable the scripting mode, click on Script under Simulation 1, click on Scripting mode toggle button, and click Yes on the prompt window.  

10. To shift the rotor back to its original position and to apply rotation, Add the following lines to the script in the scope as shown in the image below:

mesh.mesh.shift(reg.rotor, 0.0, 0.0, -0.02)
mesh.mesh.rotate(reg.rotor, 0.0, 0.0, expr.alpha)

 

11. To address the antiperiodicity regarding the rotated position for the continuity condition, Add the following arguments for continuitycondition() function:

qs.continuitycondition(..., [0,0,0], expr.alpha, 45.0, -1.0)

 

12. To calculate the torque via Arkkio’s method[1] and via Maxwell’s Stress Tensor, add the following lines to the script

# Calculating torque:

# The torque is scaled (8*5) to the actual z length of the motor 0.05 m and to take into account full 360 degrees of the motor.

# 1) Using Arkkio's method
rs = 0.059 # outer radius of airgap
rr = 0.058 # inner radius of airgap
radius = qs.sqrt( qs.pow(qs.getx(),2) + qs.pow(qs.gety(),2) )
ephi = qs.array3x1(-qs.gety()/radius, qs.getx()/radius, 0.0)
er = qs.array3x1(qs.getx()/radius, qs.gety()/radius, 0.0)
Bphi = df.B*ephi
Br = df.B*er
dr = (rs-rr)
magforcedensity1 = (8*5*radius*Br*Bphi/qs.getmu0()/dr)
torque1 = magforcedensity1.allintegrate(reg.airgap,5)

# 2) Using Maxwell stress tensor
leverarm = qs.array3x1(qs.getx(), qs.gety(), 0.0)
T = 1/qs.getmu0() * ( df.B*qs.transpose(df.B) - 0.5*df.B*df.B * qs.eye(3) )
magforcedensity2 = qs.on(reg.airgap, T)*qs.normal(reg.airgap)
torque2 = 8*5*qs.compz(qs.crossproduct(leverarm, magforcedensity2)).allintegrate(reg.continuity_target_2, 5)

qs.setoutputvalue("Mechanical angle", qs.evaluate(expr.alpha)[0])
qs.setoutputvalue("Torque (Arkkio)", qs.evaluate(torque1))
qs.setoutputvalue("Torque (Maxwell's Stress Tensor)", qs.evaluate(torque2))

 

Step 5 - Running the simulation and checking the results

1. Click on Not run next to Simulation 1 to start the simulation.  

2. To follow the simulation in progress, Click on Logs under the Results. Wait until the simulation status changes to Success.  

3. To visualize the magnetic flux density, Click on + icon next to Visualizations, click create filter icon, and select B on the list.  

4. Add glyphs by clicking + icon next to B, and selecting Glyph on the list.  

5. Click on Glyphs, select Data for the scaling mode, and Click on Activate current visualization on top of the geometry.  

6. Glyphs become visible.  

7. To see the value of torque in [Nm] at the applied mechanical angle in [deg], Click on Summary under the Simulation 1 --> Results  

References

[1] Arkkio’s Method for Torque Computation