COMSOL 6.4 - Permanent Magnet Motor in Steady State

This tutorial covers the fundamentals of modeling an electrical machine with the Magnetic Machinery, Rotating, Time Periodic (MMTP) interface. MMTP solves for the steady-state operation directly while fully including the effects of nonlinear materials and induced currents. Concepts for determining the minimum temporal and spatial periodicities are discussed.

The geometry of a distributed wound interior permanent magnet machine is imported and configured with materials and winding layout before finding the current angle which gives peak torque production. Following this, overall results such as shaft power, loss per component and efficiency are evaluated together with plots of magnetic flux density, torque, voltages, and electromagnetic loss distribution.

Figure 1: Complete geometry of motor design.

Model Definition

The Magnetic Machinery, Rotating, Time Periodic (MMTP) interface solves for a time period with the condition that field variations are periodic, or will repeat in subsequent time periods. The time period is the reciprocal of the which is specified in the MMTP interface. The handle for time in this framework is the phase angle which always spans from 0 to 2π.

When modeling an electrical machine it can be beneficial to determine some aspects of the design a priori to conserve computational resources. Evaluation of a few expressions gives an understanding of the minimum spatial and temporal extents necessary for a reduced model to produce the same result as a full model.

Spatial Periodicity

The motor topology is a 10 pole 60 slot distributed wound interior permanent magnet synchronous machine. For most motor designs the spatial periodicity of the electromagnetic field can be found by evaluating Nsec = gcd(Np, Ns), where Np and Ns are the number of poles and stator slots, and where gcd() finds the greatest common divisor of these integers. In this case, Nsec = 10 and hence it is sufficient to represent one tenth of the geometry to capture all spatial field variations. Field variations in neighboring sectors are identical although perhaps with an opposite polarity depending on the periodicity type.

Figure 2: Sector symmetric geometry of motor design.

As a rule of thumb when there is an odd number of poles inside the modeled sector the periodicity type is antiperiodic, meaning the field variations in adjacent sectors have opposite sign. With an even number of poles in the sector the periodicity is symmetric. In this case there are two magnets arranged in a v-shape making out a single pole and hence the periodicity is antiperiodic.

Having created a sector symmetric geometry and specified the number of poles, the periodicity type and necessary rotational boundary features can be automatically configured by MMTP.

Temporal Periodicity

The MMTP interface solves directly for the steady-state operation without resolving any startup transients. For most synchronous machines the time period in which fields are periodic is equivalent to the electrical excitation period.

It is not always however the rotor eddy currents will have the same time periodicity as the excitation even for a synchronous machine. A rather safe assumption is that the eddy currents are periodic with the spatial periodicity, meaning they will at least repeat after rotation has completed an angle of 360°/Nsec.

Generally, the MMTP time period should be the least common multiple of the periods of all fields being modeled. In this case the temporal periodicity of any induced currents will be at least twice that of the excitation period, since 360°/Nsec · Np/2 = 180°E (electrical degrees). Hence, setting the time periodic frequency equal to the excitation frequency will capture all field oscillations correctly.

Temporal Resolution

The cogging torque or torque due to reluctance variations as the rotor magnets are passing by stator teeth, will in this design occur on the 12'th harmonic of the fundamental electrical frequency according to the formula;

To obtain data for consistent analysis of the torque output it is a good idea to ensure the cogging torque is sufficiently resolved. With, for example, six frames per cogging period this will result in 72 time frames per excitation period. The software will calculate the solution to all time frames simultaneously while including effects of time derivatives.

Initial Electrical Angle

The default behavior of the MMTP interface assumes rotational synchronization between the and the field excited by the . To obtain peak torque production however a specific angle offset between the rotating magnetic field and the rotor has to be found.

The optimal angle offset, or in MMTP, depends on several factors such as geometry, electrical steel grade or magnet strength, any of which will alter the magnetic reluctance of the design. While there are many motor control strategies deducing this offset analytically, the initial electrical angle can also be determined rather easily by letting the rotor remain stationary while stator field is rotating. The resulting torque output will resemble a sine curve where the maximum corresponds to peak torque in motoring mode, and the minimum to peak torque in generating mode.

Torque Accuracy

The same result yielding initial angle for peak torque production can also be used to evaluate the accuracy of torque calculation. If no induced currents are present, the net torque experienced by a locked rotor when stator field is revolved exactly one electrical period should be zero. Hence calculating the average torque from this result and comparing it with the peak torque gives an idea of the torque accuracy in the model.

Results and Discussion

The first simulation is run with stationary rotor and no induced currents to determine the initial electrical angle giving peak torque as shown in Figure 3.

Figure 3: Torque curve with locked rotor.

Evaluating the average torque and dividing it by the peak value shows that the error in torque calculation is less than 0.1%.

Having updated the initial electrical angle, a second simulation with rotation produces steady-state torque as shown in Figure 4 and magnetic flux density distributions as shown in Figure 5 and Figure 6.

Figure 4: Steady-state torque.

Figure 5:

Magnetic flux density distribution at phase angle 0°E.

Figure 6: Magnetic flux density distribution at phase angle 90°E.

While the temporal variation of electromagnetic loss typically is not relevant in a thermal time scale, the spatial distribution can be of significance. Figure 7 shows this distribution across all motor components.

Figure 7: Distribution of time averaged electromagnetic loss.

To verify assumptions on the temporal periodicity of induced currents the plot in Figure 8 can be examined. Here the induced current density evaluated at a point in the magnet is scaled up and plotted together with the excitation current density of phase 1. The two vertical lines indicate half an electrical period, which corresponds with the spatial periodicity.

Figure 8: Induced current at a point in magnet and current excitation in phase 1.

By inspection it is seen that the induced currents are periodic with half the electrical period and completes six cycles in the span of the full period.

The overall results — such as shaft power, average torque, total losses, and electromagnetic efficiency — are calculated and tabulated in an at the end of the Modeling Instructions. They can also be inspected by opening the model from the Application Library and selecting the node under the branch in the .

ACDC_Module/Devices,_Motors_and_Generators/pmm_steady_state

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

Geometry 1

Insert the geometry sequence from the pmm_steady_state_geom_sequence.mph file.

In the toolbar, click and choose .

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

In the toolbar, click

Global Definitions

Parameters 1 - Main

Move all geometry parameters to a new node to separate them from the main parameters.

In the window, under click .

In the window for , locate the section.

Click on the first row, then Shift-click on the last nonempty row to select all rows in the table.

Click .

In the window, click .

In the text field, type Parameters 1 - Main.

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


Name	Expression	Value	Description
speed	5000[rpm]	83.333 1/s	Shaft speed
f_el	speed*Np/2	416.67 1/s	Electrical excitation frequency
Ipk	50[A]	50 A	Phase current peak
init_ang	0[rad]	0 rad	Initial current angle
n_cog	Ns/gcd(Np,Ns)*2	12	Cogging torque harmonic order
Nframes	n_cog*6	72	Number of frames
L	250[mm]	0.25 m	Motor axial length

Parameters 2 - Geometry

In the window, click .

In the window for , type Parameters 2 - Geometry in the text field.

Add Material

In the toolbar, click

to open the window.

Go to the window.

Add magnet and electrical steel materials only. The default node specifies the magnetic properties of air for remaining domains.

In the tree, select > > > .

Click the button in the window toolbar.

In the tree, select > > .

Click the button in the window toolbar.

Materials

Silicon Steel NGO 35PN270 (mat2)

In the toolbar, click

to close the window.

In the window for , locate the section.

From the list, choose .

N42 (Sintered NdFeB) (mat1)

In the window, click .

In the window for , locate the section.

From the list, choose .

Magnetic Machinery, Rotating, Time Periodic (mmtp)

In the settings of enter key parameters which defines the temporal periodicity and resolution, and which allows for automatic configuration of the spatial periodicity.

In the window, under click .

In the window for , locate the section.

In the d text field, type L.