Coil Above Plate Sbs

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Induction Currents from Circular CoilsSOLVED WITH COMSOL MULTIPHYSICS 3.5a

coil_above_plate_sbs.book Page 1 Tuesday, November 25, 2008 2:17 PM

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I n du c t i o n Cu r r e n t s f r om C i r c u l a r Co i l s

Introduction

A time-varying current induces a varying magnetic field. This field induces currents in nmtro

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coil_above_plate_sbs.book Page 1 Tuesday, November 25, 2008 2:17 PMI N D U C T I O N C U R R E N T S F R O M C I R C U L A R C O

eighboring conductors. The induced currents are called eddy currents. The following odel illustrates this phenomenon by a time-harmonic field simulation as well as a ansient analysis, which provides a study of the eddy currents resulting from switching n the source.

wo current-carrying coils are placed above a copper plate. They are surrounded by r, and there is a small air gap between the coils and the metal plate. A potential ifference provides the external source. To obtain the total current density in the coils u must take the induced currents into account. The time-harmonic case shows the in effect, that is, that the current density is high close to the surface and decreases pidly inside the conductor.

odel Definition

Q U A T I O N

o solve the problem, use a quasi-static equation for the magnetic potential A:

At------- 0

1 r1 A( )+ Vloop2r-------------=

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Here 0 is the permeability of vacuum, r the relative permeability, the electric conductivity, and Vloop the voltage over one turn in the coil. In the time-harmonic case the equation reduces to

F O R C E S

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Fo

jA 01 r1 A( )+ Vloop2r-------------=


ou can compute the forces on the plate caused by the eddy currents by integrating e expression for the Lorentz force density, F (in N/m3):

or the time-harmonic case, the cycle average of the Lorentz force density components this axisymmetric model, Fr and Fz, are available in postprocessing through the riables FLtzavr_emqa and FLtzavz_emqa, respectively. In the time domain, the riables FLtzr_emqa and FLtzz_emqa store the corresponding time-dependent

uantities.

esults and Discussion

he applied voltage is constant across each wire, but the induced current makes the rrent density higher toward the end facing the copper plate. Figure 1 shows this fect for the time-harmonic solution and also displays the field lines from the magnetic ux.

imilarly, Figure 2 displays a snapshot of the induced current density and magnetic flux ensity for the transient solution in a combined surface and arrow plot.

inally, Figure 3 shows the total axial force between the coils and the plate as a function f time. For the chosen current direction, the force is repulsive.

F J B=

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Fp

F


igure 1: The component of the induced current density for the time-harmonic solution lotted together with the field lines of the magnetic flux.

igure 2: Induced current density and magnetic flux density for the transient solution.

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Fti

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1

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igure 3: Resultant repulsive axial Lorentz force (N) between the coils and the plate vs. me (s).

odel Library path: ACDC_Module/Electrical_Components/oil_above_plate

odeling Using the Graphical User InterfaceHarmonic Analysis

O D E L N A V I G A T O R

From the Space dimension list, select Axial symmetry (2D).

Select the AC/DC Module>Quasi-Statics, Magnetic>Azimuthal Induction Currents, Vector Potential>Time-harmonic analysis application mode.

Click OK.

P T I O N S A N D S E T T I N G S

From the Options menu, choose Axes/Grid Settings.

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2 Set axis and grid settings according to the following table. Before enter the settings on the Grid page, clear the Auto check box. When done, click OK.

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AXIS GRID

r min -0.05 r spacing 0.01

r max 0.15 Extra r 0.0125

z min -0.08 z spacing 0.01

z max 0.08 Extra z 0.0025 0.005


E O M E T R Y M O D E L I N G

Draw the surrounding rectangle R1 with opposite corners at (0, 0.05) and (0.1, 0.05).

Draw the rectangle representing the metal plate R2 with the corners at (0, 0.02) and (0.08, 0). Now change the position of R2 by clicking the Move button on the Draw toolbar or selecting it in the Draw menu under Modify. Set the z displacement to 0.001, then click OK.Draw a circle C1 representing the first current-carrying coil, centered at (0.0125, 0.0025) with radius 0.0025.

Select C1 and click the Array button. Set the Displacement in the r direction to 0.006 and the Array size in the r direction to 2.

H Y S I C S S E T T I N G S

calar VariablesFrom the Physics menu, select Scalar Variables.

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2 Set the frequency variable nu_emqa to 50, then click OK.Subdomain SettingsIn this model, use the COMSOL Multiphysics materials library to specify material parameters in the subdomains. Only the conductivity, the permeability, and the permittivity are determined by the material selected.

1 From the Physics menu, open the Subdomain Settings dialog box.

2 Select Subdomains 24.

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On the Electric Parameters page, click the Load button. In the dialog box that appears, select Copper. Click OK.

Enter subdomain settings according to the following table; when done, click OK.

oundary ConditionsFrom the Physics menu, open the Boundary Settings dialog box.

Specify boundary conditions according to the following table; when done, click OK.

E S H G E N E R A T I O N

Initialize the mesh by clicking the Initialize Mesh button on the Main toolbar.

Click the Refine Mesh button to refine the mesh.

O M P U T I N G T H E S O L U T I O N

lick the Solve button on the Main toolbar.

O S T P R O C E S S I N G A N D V I S U A L I Z A T I O N

he magnetic potential is the default visualization quantity. In this case it is of interest plot the eddy currents together with the magnetic flux density. These fields are used examine the magnetic forces that arise in current-carrying conductors.

Click the Plot Parameters button on the Main toolbar.

On the General page, select the Surface and Streamline check boxes.

Click the Surface tab. Select Induced current density, phi component from the Predefined quantities list on the Surface Data page.

ETTING SUBDOMAINS 1, 2 SUBDOMAINS 3, 4

loop 0 1e-4

ETTING BOUNDARIES 1, 3, 5 BOUNDARIES 2, 7, 9

oundary condition Axial symmetry Magnetic insulation

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4 Click the Streamline tab. Select Magnetic flux density from the Predefined quantities list on the Streamline Data page.

5 From the Streamline plot type list, select Magnitude controlled.

6 On the Line Color page, click first the Use expression option button and then the Color Expression button.

7 In the Streamline Color Expression dialog box, select Magnetic flux density, norm from the Predefined quantities list. Type uT in the Unit edit field and select Thermal from

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the Color table list. Click OK to close the Streamline Color Expression dialog box.

Click OK to close the Plot Parameters dialog box and generate the plot.

For a better view of the current-intensive region, use the Zoom Window button on the Main toolbar. The current density is highest near the edge of the plate.

he resulting plot should resemble that in Figure 1 on page 3.

ext, calculate the cycle average of the total axial force on the plate from the currents the coils:

From the Postprocessing menu, choose Subdomain Integration.

Select Subdomain 2 (the plate).

From the Predefined quantities list in the Expression to integrate area, select Lorentz force contribution cycle average, z component.

Select the Compute volume integral (for axisymmetric modes) check box. (Note that the default entry in the Unit of integral list changes from N/m to N, as appropriate.)

Click Apply. The value of the integral appears in the message log below the drawing area. The result should be approximately 9.6107 N, or 0.96 N. The minus sign indicates a downward force, corresponding to a repulsive force between the plate and the coils.

As a consistency check and to get an idea of the accuracy of the result, compute the axial force from the plate on the coils; from Newtons third law of motion, it follows that this force is equal in magnitude and opposite in direction to that from the coils on the plate.

Select Subdomains 3 and 4 only (the coils), then click OK. The result that appears in the message log should be approximately 9.6107 N.

odeling Using the Graphical User InterfaceTransient Analysis

he previous model was an example of harmonic field variations. To study a transient se where the current increases abruptly from zero to a constant value, you only need

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to make a few alterations. To adapt the application mode to a transient analysis, you must first change the analysis type.

C O M P U T I N G T H E S O L U T I O N

1 From the Solve menu, open the Solver Parameters dialog box and select Transient from the list of analysis types in the Analysis types area.

2 In the Times edit field, type 0 10^(range(-4,1/3,-1)). This gives solution output times t = 0 followed by 10 exponentially increasing values in the range 104 to 101.

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Set the Absolute tolerance to 1e-8, then click OK.

Click the Solve button on the Main toolbar.

O S T P R O C E S S I N G A N D V I S U A L I Z A T I O N

a way similar to the harmonic analysis, you can visualize the magnetic flux density, e induced currents, and the magnetic forces.

o reproduce the plot in Figure 2 on page 3, follow these steps:

Open the Plot Parameters dialog box.

In the Plot type area on the General page, select the Surface and Arrow check boxes and clear the Streamline check box. From the Solution at time list, select 0.002154.

Click the Arrow tab. On the Subdomain Data page, select Magnetic flux density from the Predefined quantities list. In the Arrow positioning area, set the Number of points to 25 in both directions. In the Arrow parameters area, set the Arrow length to Normalized and the Scale factor to 0.5.

Click OK to close the Plot Parameters dialog box and generate the plot.

Use the zoom features to see the eddy current distribution.

To see how the current distribution changes in time, return to the Plot Parameters dialog box and change the evaluation time on the General page or click Start Animation on the Animate page.

o reproduce the plot in Figure 3 on page 4 of the axial force on the coils as a function f time, follow these steps:

From the Postprocessing menu, choose Subdomain Integration.

Select the coil subdomains from the Subdomain selection list (Subdomains 3 and 4).

Keep the selections you made in the time-harmonic case, that is select Lorentz force contribution, z component from the Predefined quantities list and leave the Compute volume integral (for axisymmetric modes) check box selected.

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4 Click Plot, then click Cancel or OK to close the Subdomain Integration dialog box.

5 In the figure windows toolbar, click to select both the X log and Y log buttons.

6 Click the Edit Plot toolbar button and edit the plots title and axis labels as desired.

7 When done, click OK to finish the plot.


Induction Currents from Circular CoilsIntroductionModel DefinitionResults and DiscussionModeling Using the Graphical User Interface-Harmonic AnalysisModeling Using the Graphical User Interface-Transient Analysis

Coil Above Plate Sbs

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