Frequency Domain Study of Three-Phase Motor

This verification application is based on the first case detailed in the TEAM benchmark problem 30, “Induction Motor Analyses” (Ref. 1). The application performs a frequency domain analysis of a simple three-phase induction motor in 2D, with a circularly-symmetric rotor made of aluminum and iron. The solution is compared with analytic results; in particular, the torque, coil voltage, and losses are evaluated as functions of the rotational speed.

Model Definition

The model geometry is two-dimensional and is represented in Figure 1. This simplified geometry was chosen in Ref. 1 to allow computing the analytic solution to use as a reference for the numerical results.

Figure 1: Two-dimensional geometry of the three-phase induction motor.

The rotor consists of an iron core surrounded by an aluminum layer. The air gap and the region containing the six coils (windings), including the coil themselves, are nonmagnetic and nonconductive. The motor is surrounded by a layer of laminated iron and the outer region is air.

The rotating part is rotationally symmetric and it does not have magnetic sources (such as permanent magnets), so the rotation can be modeled by adding an artificial term (Lorentz’s term) to the constitutive relation for the conduction current:

(1)

where v is the local velocity of the material. The properties of the rotor and the constant rotational velocity ensure that all the fields and currents in the model are time-harmonic; so a frequency-domain analysis can be performed.

Using Lorentz’s term and a frequency-domain analysis allows setting up a numerical problem without the need for moving meshes, as it is normally done with the interface. The Magnetic Fields interface can be used instead. The Lorentz’s term is added by using a feature.

The three-phase coils (windings) are modeled with three features using the conductor model, which provide a homogenized model of bundles of tiny conducting wires. The three Coils use the functionality to model the two series-connected phase groups with current flowing in opposite direction.

The quantities of interests, for which analytic data is provided in Ref, are:

•

the torque at different rotational speeds, computed using a feature,

•

the coil voltage, computed automatically by the Coil features,

•

the losses in the rotor, computed by integrating the power loss density.

Results and Discussion

Figure 2 and Figure 3 show the axial current density and the magnetic flux density lines for two different angular velocities, Ω = 200 rad/s and Ω = 800 rad/s, respectively, below and above the rotational velocity of the stator field (about 377 rad/s or 3600 RPM).

Figure 2: Axial current density and magnetic flux density lines for Ω = 200 rad/s.

Figure 3: Axial induced current density and magnetic flux density lines for Ω = 800 rad/s.

Since the analysis performed is in the frequency domain, the solutions plotted in Figure 2 and Figure 3 are the real part of the current density and magnetic flux density phasors. It is possible to visualize the evolution of the fields during a cycle by changing the value of the phase in the dataset or by creating an animation as demonstrated in the application.

Figure 4, Figure 5, and Figure 6 show the computed torque, the coil voltage, and the rotor losses as a function of the rotational velocity. The plots show the expected results for an induction motor; in particular, the torque is zero, the voltage is maximum, and the losses are at a minimum at the synchronous speed (Ω = 377 rad/s).

The computed solution shows very good agreement with the analytic data (green markers).

Figure 4: Computed (blue line) and analytic (green markers) torque as a function of the angular velocity of the rotor.

Figure 5: Computed (blue line) and analytic (green markers) coil voltage per turn as a function of the angular velocity of the rotor.

Figure 6: Computed (blue line) and analytic (green markers) rotor losses as a function of the angular velocity of the rotor.

Reference

1. http://www.compumag.org/jsite/team.html

ACDC_Module/Verification_Examples/three_phase_motor_frequency

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
f0	60[Hz]	60 Hz	Supply frequency
w0	2pif0	376.99 Hz	Supply angular frequency
n0	1000	1000	Number of turns
L	1[m]	1 m	Length of the motor
Omega	200[rad/s]	200 rad/s	Angular speed of the rotor
coil_wire_current	2045.175[A]*sqrt(2)/n0	2.8923 A	Current amplitude in coil wire

The geometry sequence is available in a separate file.

Geometry 1

In the toolbar, click

Browse to the model’s Application Libraries folder and double-click the file three_phase_motor_frequency_geom_sequence.mph.

In the toolbar, click

Click the

button in the toolbar.

Definitions

Integration, Steel

In the toolbar, click

and choose .

Select Domain 19 only.

In the window for , type Integration, Steel in the text field.

In the text field, type int_steel.

Integration, Aluminum

In the toolbar, click

and choose .

Select Domain 18 only.

In the window for , type Integration, Aluminum in the text field.

In the text field, type int_al.

Cylindrical System 2 (sys2)

Add a cylindrical coordinate system to simplify the definition of the rotational velocity.

In the toolbar, click

and choose .

Infinite Element Domain 1 (ie1)

In the original specification, the exterior region extends to infinity. Apply an Infinite Element Domain scaling system on the outer layer for this purpose.

In the toolbar, click

Select Domains 1 and 2 only.

In the window for , locate the section.

From the list, choose .

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Aluminum

In the window, under right-click and choose .

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

Select Domain 18 only(the outer layer of the rotor).

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
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	3.72e7[S/m]	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

Rotor Steel

Right-click and choose .

In the window for , type Rotor Steel in the text field.

Select Domain 19 only(the inner core of the rotor).

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	30	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	1.6e6[S/m]	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

Stator Steel

Right-click and choose .

In the window for , type Stator Steel in the text field.

Select Domain 15 only(the outer layer of the stator, before the infinite element region).

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	30	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

Magnetic Fields (mf)

Create three features using the functionality and apply each of them on pairs of windings on opposite sides of the rotor.

Coil, Phase A

In the window, under right-click and choose the domain setting .

Select Domains 4 and 13 only.

In the window for , type Coil, Phase A in the text field.

Locate the section. From the list, choose .

Select the check box.

In the Icoil text field, type coil_wire_current.

Locate the section. In the N text field, type n0.

The current flows in opposite directions in the two domains. Use a dedicated subfeature to specify this.

Reversed Current Direction 1

In the toolbar, click

and choose .

Select Domain 13 only.

Coil, Phase B

In the window, right-click and choose .

In the window for , type Coil, Phase B in the text field.

Locate the section. Click

Select Domains 7 and 9 only.

Reversed Current Direction 1

In the window, expand the node, then click .

In the window for , locate the section.

Click

Select Domain 7 only.

Coil, Phase B

Specify the second coil current with a 120° phase shift with respect to the first. In frequency domain, this corresponds to a multiplication by a complex phase factor.

In the window, click .

In the window for , locate the section.

In the Icoil text field, type coil_wire_current*exp(j*2*pi/3).

Coil, Phase C

Right-click and choose .

In the window for , type Coil, Phase C in the text field.

Locate the section. Click

Select Domains 5 and 11 only.

Specify the third coil current with a -120° phase shift.

Locate the section. In the Icoil text field, type coil_wire_current*exp(-j*2*pi/3).

Reversed Current Direction 1

In the window, expand the node, then click .

In the window for , locate the section.

Click

Select Domain 5 only.

Velocity (Lorentz Term) 1

In the toolbar, click

and choose .

Select Domains 18 and 19 only.

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the v vector as


0	r
Omega*sys2.r	phi

Here, Omega is the model parameter corresponding to the angular velocity, while sys2.r is the radial coordinate in the cylindrical coordinate system (with tag sys2).

Force Calculation 1

In the toolbar, click