Generator in 2D

This example shows how the circular motion of a rotor with permanent magnets generates an induced EMF in a stator winding. The generated voltage is calculated as a function of time during the rotation. The model also shows the influence on the voltage from material parameters, rotation velocity, and number of turns in the winding.

The center of the rotor consists of annealed medium carbon steel, which is a material with a high relative permeability. The center is surrounded with several blocks of a permanent magnet made of sintered NdFeB, creating a strong magnetic field. The stator is made of the same permeable material as the center of the rotor, confining the field in closed loops through the winding. The winding is wound around the stator poles. Figure 1 shows the generator with part of the stator sliced in order to show the winding and the rotor.

Figure 1: Drawing of a generator showing how the rotor, stator, and stator winding are constructed. The coils are connected in series in order to give the highest possible voltage.

Modeling in COMSOL Multiphysics

The COMSOL Multiphysics model of the generator is a time-dependent 2D problem on a cross section through the generator. This is a true time-dependent model where the motion of the magnetic sources in the rotor is accounted for in the boundary condition between the stator and rotor geometries. Thus, there is no Lorentz term in the equation, resulting in the PDE

where the magnetic vector potential only has a z component.

Rotation is modeled using a ready-made physics interface for rotating machinery. The center part of the geometry, containing the rotor and part of the air-gap, rotates relative to the coordinate system of the stator. The rotor and the stator are built as two separate geometry objects, so it is possible to use an assembly (see Finalizing the Geometry in the COMSOL Multiphysics Reference Manual for details). This has several advantages: the coupling between the rotor and the stator is done automatically, the parts are meshed independently, and it allows for a discontinuity in the vector potential at the interface between the two geometry objects (called slits). The rotor problem is solved in a rotating coordinate system where the rotor is fixed (the rotor frame), whereas the stator problem is solved in a coordinate system that is fixed with respect to the stator (the stator frame). An identity pair connecting the rotating rotor frame with the fixed stator frame is created between the rotor and the stator. The identity pair enforces continuity for the vector potential in the global fixed coordinate system (the stator frame).

The stator and center of the rotor are made of annealed medium-carbon steel (soft iron), which is implemented in COMSOL Multiphysics as an interpolation function of the B-H curve of the material; see Figure 2.

Figure 2: The norm of the magnetic flux, |B|, versus the norm of the magnetic field, |H|, for the rotor and stator materials.

The generated voltage is computed automatically with the use of the Coil feature. In this simplified example all coils are series connected making out a single stator winding. The series connection of coils is realized in COMSOL Multiphysics by adding coil domains to a Coil group. In this way, the output voltage is the sum of voltages generated by each coils even though they are not geometrically connected in the 2D model.

The resistive and iron losses are computed with the use of a Loss Calculation subfeature and a Time to Frequency Losses study. In this 2D model, the magnets are modeled by using the Coil feature, which allows for the calculation of induced currents and resistive losses. The settings impose a zero net perpendicular current in each of the magnets by activating the Coil group checkbox. For the iron part of the generator, the Bertotti loss model is selected to compute the cycle averaged loss power density. A surface integration is then made to compute the total loss power of the generator. The generated voltage and the losses are compared with the 3D counterpart of this model, that is, Modeling of an Electric Generator in 3D.

Results and Discussion

The generated voltage in the rotor winding is close to a sinusoidal signal. At a rotation speed of 60 rpm the voltage has an amplitude of 700 V in total; see Figure 3.

Figure 3: The generated voltage over one quarter of a revolution. This simulation used a 100-turn winding.

The norm of the magnetic flux, |B|, and the B field lines are shown below in Figure 4 at time 0.20 s.

Figure 4: The norm and the field lines of the magnetic flux after 0.2 s of rotation. Note the darker regions, which indicate the position of the permanent magnets in the rotor.

The cycle averaged losses power density is shown below in Figure 5.

Figure 5: The volumetric loss density of the generator.

ACDC_Module/Motors_and_Actuators/generator_2d

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
L	0.4[m]	0.4 m	Length of generator
rpm	60[1/min]	1 1/s	Rotational speed of rotor
d_wire	3[mm]	0.003 m	Diameter of wire in the winding
N	100	100	Number of turns in the winding

Next, build the generator using rotor and stator parts from the geometry part library. Initialize the parts, and tick the selections that are pre-defined to make it convenient to assign material properties and magnetization direction.

Part Libraries

In the toolbar, click

and choose .

In the window, select in the tree.

Click

Part Libraries

In the toolbar, click

and choose .

In the window, under click .

In the window, select in the tree.

Click

Geometry 1

In the toolbar, click

Click the

button in the toolbar.

Internal Rotor – Surface Mounted Magnets 1 (pi1)

In the window, under click – .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
number_of_poles	8	8	Number of magnetic poles in rotor
number_of_modeled_poles	8	8	Number of magnetic poles included in the geometry
shaft_diam	10[cm]	0.1 m	Diameter of the shaft
rotor_diam	30[cm]+2*6.5[cm]	0.43 m	Diameter of the rotor
cont_diam	0.45[m]	0.45 m	Diameter of the stator-rotor continuity interface
magnet_h	6.5[cm]	0.065 m	Height of the magnets
magnet_w	10 [cm]	0.1 m	Width of the magnets (set to 0 to use all available space)
magnet_fillet_size	0.6[cm]	0.006 m	Radius of magnet fillet

Click to expand the section. In the table, enter the following settings:


Name	Keep	Physics	Contribute to
Shaft	√	√	None
Rotor iron	√	√	None
Odd magnets	√	√	None
Even magnets	√	√	None
Rotor magnets	√	√	None
Rotor solid domains	√	√	None
Rotor air	√	√	None
All	√	√	None

External Stator – Slotted 1 (pi2)

In the window, click – .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
number_of_slots	8	8	Number of slots in stator
number_of_modeled_slots	8	8	Number of slots included in the geometry
backiron_th	6 [cm]	0.06 m	Thickness of back-iron
stator_diam	80 [cm]	0.8 m	Diameter of the stator
external_air_size	0[cm]	0 m	Size of air external to stator
cont_diam	45[cm]	0.45 m	Diameter of the stator-rotor continuity interface
shoe_h	1.75[cm]	0.0175 m	Height of the shoe
shoe_w	11.5[cm]	0.115 m	Width of the shoe
tooth_h	10.5[cm]	0.105 m	Height of the tooth
tooth_w	9[cm]	0.09 m	Width of the tooth
shoe_fillet_size	0.4[cm]	0.004 m	Radius of the shoe fillet
slot_outer_fillet_size	0.4[cm]	0.004 m	Radius of the outer slot fillet
slot_inner_fillet_size	0.4[cm]	0.004 m	Radius of the inner slot fillet
slot_winding_type	2	2	Slot winding type: 1-No partition, 2-Radial partition, 3-Azimuthal partition, 4-Radial and azimuthal partition.

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


Name	Keep	Physics	Contribute to
Stator iron	√	√	None
Stator slots	√	√	None
Stator air	√	√	None
All	√	√	None

Definitions

Iron

In the toolbar, click

In the window for , locate the section.

Under , click

In the dialog box, in the list, choose – , – , and – .

Click .

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

Iron and Magnets

In the toolbar, click

In the window for , locate the section.

Under , click

In the dialog box, in the list, choose and – .

Click .

In the window for , type Iron and Magnets in the text field.

Create an assembly from the two geometry objects, connected by a pair boundary.

Geometry 1

Form Union (fin)

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Click the

button in the toolbar.

Definitions

Cylindrical System 2 (sys2)

In the toolbar, click

and choose .

The cylindrical coordinate system you just added will be used to define the field of the permanent magnets.

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 tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Soft Iron (Without Losses) (mat2)

In the window, under click .

In the window for , locate the section.

From the list, choose .

N50 (Sintered NdFeB) (mat3)

In the window, click .

In the window for , locate the section.

From the list, choose – .

Rotating Machinery, Magnetic (rmm)

In the window, under click .

In the window for , locate the section.

In the d text field, type L.

Next, use a Rotating Domain feature to specify the rotor’s rotational velocity.

In the toolbar, click

and choose .

Moving Mesh

Rotating Domain 1

In the window for , locate the section.

From the list, choose – .

Locate the section. From the list, choose .

From the list, choose .

In the f text field, type rpm.

Rotating Machinery, Magnetic (rmm)

The permanent magnets on the rotor have a radial field. Use the cylindrical coordinate system specified earlier to define it.

Permanent Magnets Outward

In the toolbar, click

and choose .

In the window for , type Permanent Magnets Outward in the text field.

Locate the section. From the list, choose – .

Locate the section. From the list, choose .

Locate the section. Select the check box.

In the Icoil text field, type 0[A].

Locate the section. From the list, choose .

Loss Calculation 1

In the toolbar, click

and choose .

Permanent Magnets Inward

In the window, right-click and choose .

In the window for , type Permanent Magnets Inward in the text field.

Locate the section. From the list, choose – .

Locate the section. Specify the e vector as


-1	r
0	phi
0	a

Iron

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type Iron.

Add the subfeature to compute iron losses.

Loss Calculation 1

In the toolbar, click

and choose .

Set loss model to .

In the window for , locate the section.

From the list, choose .

Coil 3

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose – .

Locate the section. From the list, choose .

In the Icoil text field, type 0[A].

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

From the list, choose .

In the dcoil text field, type d_wire.

Locate the section. Select the check box.

Reversed Current Direction 1

In the toolbar, click

and choose .

Select Domains 1, 4, 8, 10, 11, 13, 16, and 18 only.

Study 1

Time Dependent

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type range(0,0.005,0.25).

The problem is set and it would be essentially ready to solve. In the next section some adjustments to physics, mesh, and solver settings are presented. These offer a more stable solution which is also found in a shorter time.

Suggestion to make solution faster and more stable

There are several settings which, when applied to model like this, make solution faster and more stable. Even though not all of these are required for the presented simple generator model, all of these may turn to be useful for more complex problems. In particular, these settings are being applied:

- use of elements for as this is offering a better alternative in models with magnetic saturation;

- use in the condition applied where the pair with sliding meshes is active; this offers an higher precision for a given mesh;

- use, in the identity pair, a destination mesh finer than the source as this simplifies continuity constraint handling;

- use tighter tolerances than default in stationary preliminary step as this will simplify the starting of the subsequent transient which is in general more demanding;

- use of physical variables in time dependent study step; eventually set also scaling of rotation angle variable; such settings will make the time dependent solver able to better choose the time step;

- set to as this makes each of the time dependent solver step more stable;

- set in error estimate as this will enable time stepper to better balance equation among the actively inductive contributions and those corresponding to instantaneously adapting fields;

- set to or as this will offer a better value also for quantities which contains time derivatives as the electric potentials on the.

Definitions

Identity Boundary Pair 1 (ap1)

In the window, under click .

In the window for , locate the section.

Click

In the dialog box, type destination in the text field.

Click .

Mesh 1

Free Triangular 1

In the toolbar, click

Size

In the window, click .

In the window for , locate the section.

From the list, choose .

Click the button.

Locate the section. In the text field, type 0.015.

Size 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Click

Rotating Machinery, Magnetic (rmm)

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Click the

button in the toolbar.

In the dialog box, in the tree, select the check box for the node .

Click .

Continuity 1

In the window, under click .

In the window for , click to expand the section.

Select the check box.

Study 1

Solution 1 (sol1)

In the toolbar, click

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type 1e-5.

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the window, under click .

In the window for , click to expand the section.

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Postprocessing the results

Now, plot the solution in the spatial frame (the stator’s fixed frame) at time t = 0.2 s.

In the window for , locate the section.

From the list, choose .

In the toolbar, click

As shown in Figure 4, it shows the rotor position at t = 0.2 s, the magnetic flux density norm and magnetic flux lines. Next, plot the induced EMF in a quarter of the cycle.

Induced Coil Voltage

In the toolbar, click