induction motor in 2d - comsol multiphysics · pdf fileinduction motor in 2d. 2 ... this is...

Created in COMSOL Multiphysics 5.2a

I n du c t i o n Mo t o r i n 2D

This model is licensed under the COMSOL Software License Agreement 5.2a.All trademarks are the property of their respective owners. See www.comsol.com/trademarks.

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Introduction

This is the model of a induction motor in which the eddy currents are induced in the rotor by the time harmonic currents on the stator windings and the rotation of the rotor. The rotational torque and the angular speed of the rotor when an alternating current is passed through the stator windings are calculated as a function of time. The effect of the step change in the load torque is also included in this model.

Air gap (white)Laminated stator steel Phase A (black)

Phase B (blue)

Phase C (red)

Rotor steel

Rotor aluminum

Figure 1: Model illustration of a three phase induction motor. The stator with windings carrying an alternating current and a rotor with aluminum and steel are shown.

Modeling in COMSOL Multiphysics

Setup the problem in a 2D modeling space using Rotating Machinery, Magnetics (rmm) physics interface. Import the model geometry in which the rotor and the stator are a separate parts. Finish the geometry using an assembly feature. This will automatically form an identity pair connecting the rotating rotor frame with the fixed stator frame between the rotor and the stator moving parts. The identity pair enforces continuity for the vector potential in the global fixed coordinate system (the stator frame). The stator and the inner part of the rotor are made of steel. The stator steel is laminated such that the conductivity is assumed to be zero. However, the rotor steel and aluminum have the conductivity of 1.6x106 S/m and 3.72x107 S/m respectively. The three phases of the motor are denoted

2 | I N D U C T I O N M O T O R I N 2 D

by three different colors as shown in Figure 1. Each stator winding phase spans 45 degrees and they are separated by 120 electrical degrees. The alternating current through the stator windings is applied using a Homogenized multi-turn coil feature with 2045 turns. The geometrical dimensions for this model are used from Ref. 1.

The rotational motion of the rotor given by Equation 1 and Equation 2 is modeled using a Global ODE and DAEs. The axial torque on the rotor can be calculated by adding the Force Calculation feature on the rotor domain. However, this method is sensitive to the mesh size. It is necessary to have a finer mesh to get the accurate torque calculation.

(1)

(2)

where Tm is the motor torque, TL is the load torque, ωm is the angular speed of the motor and φ is the rotor angle.

The alternative approach would be to use the Arkkio's method of torque calculation, a volume integration of the product of the magnetic flux densities. In this method, the electromagnetic torque in 2D models of electrical rotating machines can be calculated using Equation 3.

(3)

where ro is the outer radius, ri is the inner radius, and Sag is the cross sectional area of the air gap. The magnetic flux density in the radial and azimuthal directions is Br and Bφ , respectively.

Results and Discussion

A transient analysis is performed to calculate the torque and the speed of an induction motor with a conducting rotor. The magnetic flux density norm at t = 0.8 s is shown in Figure 2.

Figure 3 shows the current density norm and the contour plot of magnetic vector potential z-component at t = 0.8 s. The induced current is maximum in the rotor aluminum.

dωmdt

------------Tm TL–

J---------------------=

dφdt------- ωm=

Tm1

μ0 ro ri–( )--------------------------- rBrBφ Sd

Sag

=


Figure 4 illustrates the rotor torque and the load torque as a function of time. The rotor speed decreases slightly with the increase in load torque. The rotor torque is increased slowly to balance the load torque. The time delay for rotor torque to reach the load torque is because of the rotor inertia.

Figure 5 shows the rotor angular speed as a function of time. The rotor speed increases to the steady state speed of 60 Hz at about 0.4 s. This speed is equal to the stator electrical frequency.

Figure 6 displays the rotor power loss as a function of time.

Figure 2: The surface norm and the arrow plot of the magnetic flux density at t =0.8 s.


Figure 3: The current density norm and filed lines of magnetic flux density at t =0.8 s.

Figure 4: Rotor torque and a load torque as a function of time.


Figure 5: Rotor angular speed as a function of time.

Figure 6: Rotor losses as a function of time.


Reference

1. K. Davey, “Induction Motor Analysis: International TEAM Workshop Problem 30,” http://www.compumag.org/jsite/images/stories/TEAM/problem30a.pdf.

Application Library path: ACDC_Module/Motors_and_Actuators/induction_motor_2d

From the File menu, choose New.

N E W

In the New window, click Model Wizard.

M O D E L W I Z A R D

1 In the Model Wizard window, click 2D.

2 In the Select Physics tree, select AC/DC>Rotating Machinery, Magnetic (rmm).

3 Click Add.

4 In the Select Physics tree, select Mathematics>ODE and DAE Interfaces>

Global ODEs and DAEs (ge).

5 Click Add.

6 Click Study.

7 In the Select Study tree, select Preset Studies for Selected Physics Interfaces>

Time Dependent.

8 Click Done.

Define all the required parameters.

G L O B A L D E F I N I T I O N S

Parameters1 On the Home toolbar, click Parameters.

2 In the Settings window for Parameters, locate the Parameters section.


3 In the table, enter the following settings:

G E O M E T R Y 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 cm.

Import the geometry sequence of the motor cross section from a separate COMSOL file.

4 On the Geometry toolbar, click Insert Sequence.

5 Browse to the application’s Application Libraries folder and double-click the file induction_motor_2d_geometry_sequence.mph.

Form Assembly (fin)1 On the Geometry toolbar, click Build All.

The geometry sequence of the motor cross section you just inserted is composed by two unions, an inner part (corresponding to the rotor) and an outer part (the stator). The geometry is finalized using a Form Assembly. An identity pair is automatically created

between the common boundaries of the stator and the rotor.

Name Expression Value Description

f0 60[Hz] 60 Hz Supply frequency

w0 2*pi*f0 376.99 Hz Supply angular frequency

n0 2045 2045 Number of turns

L 1[m] 1 m Length of the motor


The geometry of the imported motor cross section is shown in the following figure.

D E F I N I T I O N S

Create a selection for the rotor solid parts and rotating domains.

Explicit 11 On the Definitions toolbar, click Explicit.

2 Select Domains 16 and 17 only.

3 In the Settings window for Explicit, type Rotor Parts in the Label text field.

Explicit 21 On the Definitions toolbar, click Explicit.

2 Select Domains 15–17 only.


3 In the Settings window for Explicit, type Rotating Domains in the Label text field.

Define the integration coupling variable in the rotor domain. This will be used to compute the rotor inertia.

Integration 1 (intop1)1 On the Definitions toolbar, click Component Couplings and choose Integration.

2 In the Settings window for Integration, type intop_inertia in the Operator name text field.

3 Locate the Source Selection section. From the Selection list, choose Rotor Parts.

Define the integration coupling variable in the air ring just outside of the rotor aluminum. This will be used to compute the axial torque in the rotor using the Arkkio’s method.

Integration 2 (intop2)1 On the Definitions toolbar, click Component Couplings and choose Integration.

2 Select Domain 15 only.

3 In the Settings window for Integration, type intop_torque in the Operator name text field.

Define the variables for the density of aluminum and steel in the rotor domain.

Variables 11 In the Model Builder window, right-click Definitions and choose Variables.

2 In the Settings window for Variables, locate the Geometric Entity Selection section.

3 From the Geometric entity level list, choose Domain.


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

Name Expression Unit Description

rho_r 2700[kg/m^3] kg/m³ Density of rotor aluminum


Variables 21 Right-click Definitions and choose Variables.

2 In the Settings window for Variables, locate the Geometric Entity Selection section.



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

Step 1 (step1)Create a step function to represent the step change in the load torque.

1 On the Definitions toolbar, click More Functions and choose Step.

2 In the Settings window for Step, locate the Parameters section.

3 In the Location text field, type 0.5.

4 In the To text field, type 4.

5 Click to expand the Smoothing section. In the Size of transition zone text field, type 0.05.

Define the variables to compute the rotor inertia, load torque, three phase currents, and the axial torque using Arkkio’s method.

Variables 31 Right-click Definitions and choose Variables.

2 In the Settings window for Variables, locate the Variables section.



rho_r 7850[kg/m^3] kg/m³ Density of rotor steel


I_rotor intop_inertia(rho_r*(X^2+Y^2)*L)

kg·m² Inertia of the rotor

T_Load step1(t[1/s]) Load torque

Ia 1[A]*sqrt(2)*cos(w0*t) A Current on phase A

Ib 1[A]*sqrt(2)*cos(w0*t+120[deg])

A Current on phase B

Ic 1[A]*sqrt(2)*cos(w0*t-120[deg])

A Current on phase C

r sqrt(X^2+Y^2) m Radial distance


R O T A T I N G M A C H I N E R Y, M A G N E T I C ( R M M )

Set up the physics for Rotating Machinery, Magnetic. Use the Homogenized multi-turn coil feature to model each of the stator windings. Specify the motor length at out-of-plane thickness.

1 In the Model Builder window, under Component 1 (comp1) click Rotating Machinery,

Magnetic (rmm).

2 In the Settings window for Rotating Machinery, Magnetic, locate the Thickness section.

3 In the d text field, type L.

For faster computation, use the linear mesh elements instead of the default quadratic mesh elements.

4 In the Model Builder window’s toolbar, click the Show button and select Discretization in the menu.

5 In the Settings window for Rotating Machinery, Magnetic, click to expand the Discretization section.

6 From the Magnetic vector potential list, choose Linear.

Coil 11 On the Physics toolbar, click Domains and choose Coil.

2 In the Settings window for Coil, type Coil 1: Phase A in the Label text field.

dr airgap/2 m Radial distance for torque integration

Bphi (-rmm.BX*Y+rmm.BY*X)/r T Azumuthal magnetic flux density

Br (rmm.BX*X+rmm.BY*Y)/r T Radial magnetic flux density

T_ark intop_torque(r*L*Br*Bphi/mu0_const/dr)

J Arkkio’s torque method




4 Locate the Coil section. From the Conductor model list, choose Homogenized multi-turn.

5 Select the Coil group check box.

6 In the Icoil text field, type Ia.

7 Locate the Homogenized Multi-Turn Conductor section. In the N text field, type n0.

Reversed Current Direction 11 On the Physics toolbar, click Attributes and choose Reversed Current Direction.



2 In the Settings window for Coil, type Coil 2: Phase B in the Label text field.


4 In the Settings window for Coil, locate the Coil section.

5 From the Conductor model list, choose Homogenized multi-turn.


7 In the Icoil text field, type Ib.






2 In the Settings window for Coil, type Coil 3: Phase C in the Label text field.


4 In the Settings window for Coil, locate the Coil section.

5 From the Conductor model list, choose Homogenized multi-turn.


7 In the Icoil text field, type Ic.




Prescribed Rotation 1Assign the rotational angle variable phi as a Prescribed Rotation. This variable will be solved in the Global ODEs and DAEs interface later on.

1 On the Physics toolbar, click Domains and choose Prescribed Rotation.

2 In the Settings window for Prescribed Rotation, locate the Domain Selection section.

3 From the Selection list, choose Rotating Domains.

4 Locate the Prescribed Rotation section. In the αrot text field, type phi.


The problem will be solved separately in the fixed frame of the stator and the rotating frame of the rotor. Apply a Continuity feature on the shared boundaries (that are connected by Identity Pair I).

Continuity 11 On the Physics toolbar, in the Boundary section, click Pairs and choose Continuity.

2 In the Settings window for Continuity, locate the Pair Selection section.

3 In the Pairs list, select Identity Boundary Pair 1 (ap1).

G L O B A L O D E S A N D D A E S ( G E )

Global Equations 1Model the rotational dynamics of the rotor using a Global Equation. Two separate equations will be solved for two state variables, the angular speed and the angle of the rotor.

1 In the Model Builder window, under Component 1 (comp1)>Global ODEs and DAEs (ge) click Global Equations 1.

2 In the Settings window for Global Equations, locate the Global Equations section.


4 Locate the Units section. Find the Dependent variable quantity subsection. From the list, choose Angular frequency (rad/s).

5 Find the Source term quantity subsection. From the list, choose Angular acceleration (rad/s^2).

6 In the Model Builder window, click Global ODEs and DAEs (ge).

Global Equations 21 On the Global ODEs and DAEs toolbar, click Global Equations.

2 In the Settings window for Global Equations, locate the Global Equations section.


Name f(u,ut,utt,t) (1) Description

W d(W,t)-(T_ark-T_Load)/I_rotor Angular speed of rotor

Name f(u,ut,utt,t) (1)

Initial value (u_0) (1)

Initial value (u_t0) (1/s)

Description

phi d(phi,t)-W 0 0 Angle of rotation


4 Locate the Units section. Find the Dependent variable quantity subsection. From the list, choose Plane angle (rad).

5 Find the Source term quantity subsection. From the list, choose Angular frequency (rad/

s).

M A T E R I A L S

Assign materials to the model. First, assign air in the model.

On the Home toolbar, click Windows and choose Add Material.

A D D M A T E R I A L

1 Go to the Add Material window.

2 In the tree, select Built-In>Air.

3 Click Add to Component in the window toolbar.


Next, specify aluminum for the rotor and steels for the stator as well as rotor.


2 In the tree, select Built-In>Aluminum.


M A T E R I A L S

Aluminum (mat2)1 In the Model Builder window, under Component 1 (comp1)>Materials click

Aluminum (mat2).


3 In the Settings window for Material, locate the Material Contents section.


Property Name Value Unit Property group

Relative permeability mur 1 1 Basic

Electrical conductivity sigma 3.72e7[S/m] S/m Basic

Relative permittivity epsilonr 1 1 Basic




2 In the tree, select Built-In>Steel AISI 4340.


M A T E R I A L S

Steel AISI 4340 (mat3)1 In the Model Builder window, under Component 1 (comp1)>Materials click

Steel AISI 4340 (mat3).

2 In the Settings window for Material, type Steel: Rotor in the Label text field.


4 Locate the Material Contents section. In the table, enter the following settings:


Steel AISI 4340 (mat4)1 In the Model Builder window, under Component 1 (comp1)>Materials click

Steel AISI 4340 (mat4).

2 In the Settings window for Material, type Steel: Stator in the Label text field.


4 Locate the Material Contents section. In the table, enter the following settings:

The stator steel is laminated, therefore, the conductivity is zero.

5 On the Home toolbar, click Add Material to close the Add Material window.



Electrical conductivity sigma 1.6e6[S/m] S/m Basic




Electrical conductivity sigma 0[S/m] S/m Basic



M E S H 1

1 In the Model Builder window, under Component 1 (comp1) click Mesh 1.

2 In the Settings window for Mesh, locate the Mesh Settings section.

3 From the Element size list, choose Extra coarse.

4 From the Sequence type list, choose User-controlled mesh.

Free Triangular 1Define a relatively smaller size mesh on the rotor boundaries compared with that of the stator side.

Size 11 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click

Free Triangular 1 and choose Size.

2 In the Settings window for Size, locate the Geometric Entity Selection section.


4 From the Selection list, choose Rotor Parts.

5 Locate the Element Size section. From the Predefined list, choose Finer.

Free Triangular 1Right-click Free Triangular 1 and choose Size.

Size 21 In the Settings window for Size, locate the Geometric Entity Selection section.




5 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 0.075.

Free Triangular 1Right-click Free Triangular 1 and choose Size.

Size 31 In the Settings window for Size, locate the Geometric Entity Selection section.





5 Click the Custom button.

6 Locate the Element Size Parameters section. Select the Maximum element size check box.


8 Click the Zoom Extents button on the Graphics toolbar.

9 Click Build Selected.

To resolve the induced eddy currents in the conducting aluminum and steel, add the Boundary Layer mesh.

Boundary Layers 11 In the Model Builder window, right-click Mesh 1 and choose Boundary Layers.

2 In the Settings window for Boundary Layers, locate the Domain Selection section.



Boundary Layer Properties1 In the Model Builder window, under Component 1 (comp1)>Mesh 1>Boundary Layers 1

click Boundary Layer Properties.

2 In the Settings window for Boundary Layer Properties, locate the Boundary Selection section.

3 Click Paste Selection.

4 In the Paste Selection dialog box, type 53-54, 57-58 in the Selection text field.

5 Click OK.

6 Right-click Component 1 (comp1)>Mesh 1>Boundary Layers 1>Boundary Layer Properties and choose Build Selected.

Boundary Layers 21 Right-click Mesh 1 and choose Boundary Layers.

2 In the Settings window for Boundary Layers, locate the Domain Selection section.



Boundary Layer Properties1 In the Model Builder window, under Component 1 (comp1)>Mesh 1>Boundary Layers 2

click Boundary Layer Properties.


2 In the Settings window for Boundary Layer Properties, locate the Boundary Selection section.

3 Click Paste Selection.

4 In the Paste Selection dialog box, type 51-52, 56, 59 in the Selection text field.

5 Click OK.

6 In the Settings window for Boundary Layer Properties, click Build All.

The final mesh should look like the image below.

Some adjustments to the solver settings are required in order to obtain a more stable solution.

S T U D Y 1

Step 1: Time Dependent1 In the Settings window for Time Dependent, locate the Study Settings section.

2 In the Times text field, type range(0,0.002,0.8).

Solution 1 (sol1)1 On the Study toolbar, click Show Default Solver.

2 In the Model Builder window, expand the Solution 1 (sol1) node.

3 In the Model Builder window, under Study 1>Solver Configurations>Solution 1 (sol1) click Time-Dependent Solver 1.


4 In the Settings window for Time-Dependent Solver, click to expand the Time stepping section.

5 Locate the Time Stepping section. From the Steps taken by solver list, choose Intermediate.

6 From the Maximum BDF order list, choose 2.

7 In the Model Builder window, under Study 1>Solver Configurations>Solution 1 (sol1)>

Time-Dependent Solver 1 click Direct.

8 In the Settings window for Direct, locate the General section.

9 From the Solver list, choose PARDISO.

10 In the Model Builder window, under Study 1>Solver Configurations>Solution 1 (sol1)>

Time-Dependent Solver 1 click Fully Coupled 1.

11 In the Settings window for Fully Coupled, click to expand the Method and termination section.

12 Locate the Method and Termination section. From the Jacobian update list, choose On every iteration.

13 In the Maximum number of iterations text field, type 8.

14 In the Tolerance factor text field, type 1e-3.

15 In the Model Builder window, click Study 1.

16 In the Settings window for Study, locate the Study Settings section.

17 Clear the Generate default plots check box, because you will add the desired plots manually.

18 On the Study toolbar, click Compute.

R E S U L T S

Now, plot the solution in the spatial frame (the stator’s fixed frame) at time t = 0.8 s.

1 In the Model Builder window, expand the Results node.

Study 1/Solution 1 (sol1)1 In the Model Builder window, expand the Results>Data Sets node, then click Study 1/

Solution 1 (sol1).

2 In the Settings window for Solution, locate the Solution section.

3 From the Frame list, choose Spatial (x, y, z).

2D Plot Group 11 On the Home toolbar, click Add Plot Group and choose 2D Plot Group.


2 In the Settings window for 2D Plot Group, type Magnetic Flux Density Norm in the Label text field.

3 Locate the Plot Settings section. From the Frame list, choose Spatial (x, y, z).

4 Right-click Magnetic Flux Density Norm and choose Surface.

Magnetic Flux Density NormIn the Model Builder window, under Results right-click Magnetic Flux Density Norm and choose Arrow Surface.

Arrow Surface 11 In the Settings window for Arrow Surface, locate the Arrow Positioning section.

2 Find the x grid points subsection. In the Points text field, type 50.

3 Find the y grid points subsection. In the Points text field, type 50.

4 Locate the Coloring and Style section. Select the Scale factor check box.


6 On the Magnetic Flux Density Norm toolbar, click Plot.

The plot shows a surface plot for magnetic flux density norm and arrow plot for a magnetic flux density.



2 In the Settings window for 2D Plot Group, type Current Density Norm in the Label text field.

3 Locate the Plot Settings section. From the Frame list, choose Spatial (x, y, z).

Surface 11 Right-click Current Density Norm and choose Surface.

2 In the Settings window for Surface, locate the Expression section.

3 In the Expression text field, type rmm.normJ.

Current Density NormIn the Model Builder window, under Results right-click Current Density Norm and choose Contour.

Contour 11 In the Settings window for Contour, click Replace Expression in the upper-right corner of

the Expression section. From the menu, choose Component 1>Rotating Machinery,

Magnetic (Magnetic Fields)>Magnetic>Magnetic vector potential (Spatial)>

rmm.Az - Magnetic vector potential, z component.

2 On the Current Density Norm toolbar, click Plot.


The plot shows a surface plot for current density norm and a contour plot for a magnetic vector potential z-component.


Now, plot the rotor axial torque and load torque together as a function of time for a comparison.

2 In the Settings window for 1D Plot Group, type Electromagnetic Torque in the Label text field.

3 Locate the Plot Settings section. Select the x-axis label check box.

4 In the associated text field, type Time (s).

5 Select the y-axis label check box.

6 In the associated text field, type Torque (Nm).

Global 11 On the Electromagnetic Torque toolbar, click Global.

2 In the Settings window for Global, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Definitions>Variables>

T_ark - Arkkio’s torque method.

3 Locate the y-Axis Data section. In the table, enter the following settings:

4 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Definitions>Variables>T_Load - Load torque.

5 Click to expand the Coloring and style section. Locate the Coloring and Style section. In the Width text field, type 2.

Expression Unit Description

T_ark Nm Arkkio’s torque method


6 On the Electromagnetic Torque toolbar, click Plot.

The electromagnetic torque exhibits an oscillatory behavior during the start-up and gradually reaches peak value at around t = 0.28 s. It decreases to zero when it reaches full synchronous speed at around t = 0.4 s. At t = 0.5 s, a step change in the load torque is applied. The motor gradually generates the equal amount of torque by sacrificing some speed.


Next, plot the rotor speed as a function of time.

2 In the Settings window for 1D Plot Group, type Rotor Speed in the Label text field.

3 Locate the Legend section. From the Position list, choose Lower right.

Global 11 On the Rotor Speed toolbar, click Global.

2 In the Settings window for Global, locate the y-Axis Data section.

3 Click Clear Table.

4 Click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Component 1>Global ODEs and DAEs>W - Angular speed of rotor.


5 Locate the Coloring and Style section. In the Width text field, type 2.

6 On the Rotor Speed toolbar, click Plot.

Surface Integration 11 On the Results toolbar, click More Derived Values and choose Integration>

Surface Integration.

2 In the Settings window for Surface Integration, locate the Selection section.

3 From the Selection list, choose Rotor Parts.

4 Click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Component 1>Rotating Machinery, Magnetic (Magnetic Fields)>

Heating and losses>rmm.Qh - Volumetric loss density, electromagnetic.

5 Click Evaluate.

TA B L E

1 Go to the Table window.

2 Click Table Graph in the window toolbar.


R E S U L T S

Table Graph 11 In the Model Builder window, under Results>1D Plot Group 5 click Table Graph 1.

2 In the Settings window for Table Graph, locate the Coloring and Style section.

3 In the Width text field, type 2.

1D Plot Group 51 In the Model Builder window, under Results click 1D Plot Group 5.

2 In the Settings window for 1D Plot Group, type Rotor Losses in the Label text field.


induction motor in 2d - comsol multiphysics · pdf fileinduction motor in 2d. 2 ... this is...

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