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 samarium and cobalt, 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 winding is also connected between the loops, interacting 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 imported 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 as the line integral of the electric field, E, along the winding. Because the winding sections are not connected in the 2D geometry, a proper line integral cannot be carried out. A simple approximation is to neglect the voltage contributions from the ends of the rotor, where the winding sections connect. The voltage is then obtained by taking the average z component of the E field for each winding cross-section, multiplying it by the axial length of the rotor, and taking the sum over all winding cross sections.

where L is the length of the generator in the third dimension, NN is the number of turns in the winding, and A is the total area of the winding cross-section.

Results and Discussion

The generated voltage in the rotor winding is a sinusoidal signal. At a rotation speed of 60 rpm the voltage has an amplitude slightly above 4 V for a single turn winding; see Figure 3.

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

The norm of the magnetic flux, |B|, and the field lines of the B field 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 brighter regions, which indicate the position of the permanent magnets in the rotor.

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 .

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
A	pi*(3[cm]/2)^2	7.0686E-4 m²	Area of single circular conductor coil

Geometry 1

Import 1 (imp1)

Import the geometry of the generator cross section from an external CAD file.

In the toolbar, click .

In the window for , locate the section.

Click .

Browse to the model’s Application Libraries folder and double-click the file generator_2d.mphbin.

Click .

Form Union (fin)

The geometry you imported is composed by two objects, an inner part (corresponding to the rotor) and an outer part (the stator). Create an assembly from these objects.

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click .

In this way, an identity pair connecting the shared boundaries has been automatically created.

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 .

Select Domains 2 and 28 only.

N50 (Sintered NdFeB) (mat3)

In the window, click .

Select Domains 20–27 only.

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
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
Recoil permeability	murec_iso ; murecii = murec_iso, murecij = 0	1	1	Remanent flux density
Remanent flux density norm	normBr	0.84[T]	T	Remanent flux density

In the text field, type Generic SmCo Magnet.

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.

Rotating Domain 1

In the toolbar, click and choose .

Select Domains 19–28 only.

In the window for , locate the section.

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.

Ampère’s Law 2

In the toolbar, click and choose .

Select Domains 20, 23, 24, and 27 only.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type Permanent Magnets Outward.

The other four magnets are reversed.

Ampère’s Law 3

In the toolbar, click and choose .

Select Domains 21, 22, 25, and 26 only.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Specify the e vector as


-1	r
0	phi
0	a

In the text field, type Permanent Magnets Inward.

Ampère’s Law 4

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Select Domains 2 and 28 only.

In the text field, type Iron.

Coil 1

In the toolbar, click and choose .

Select Domains 3–18 only.

In the window for , locate the section.

From the list, choose .

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

Locate the section. In the N text field, type 1.

In the acoil text field, type A.

Locate the section. Select the check box.

Reversed Current Direction 1

In the toolbar, click and choose .

Select Domains 3, 4, 9–12, 17, and 18 only.

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 ).

Continuity 1

In the toolbar, in the section, click and choose .

In the window for , locate the section.

Under , click .

In the dialog box, select in the list.

Click .

Study 1

Time Dependent

In the toolbar, click and choose .

In the window for , locate the section.

In the text field, type range(0,0.01,0.25).

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 window, under right-click and choose .

Size

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.

From the list, choose .

Find the subsection. From the list, choose .

In the window, under click .

In the window for , click to expand 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.

Magnetic Flux Density (rmm)

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Contour 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type 12.

In the toolbar, click .

The plot will show 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.

1D Plot Group 2

In the toolbar, click and choose .

In the window for , click to expand the section.

From the list, choose .

In the text area, type Induced voltage.

Locate the section. Select the check box.

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

Select the check box.

In the associated text field, type Voltage (V).

Global 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

In the toolbar, click .

The plot shows an induced EMF with an amplitude of about 4.2 V