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Magnetic Field of a Helmholtz Coil
Introduction
A Helmholtz coil is a parallel pair of identical circular coils spaced one radius apart and wound so that the current flows through both coils in the same direction. This winding results in a uniform magnetic field between the coils with the primary component parallel to the axis of the two coils. The uniform field is the result of the sum of the two field components parallel to the axis of the coils and the difference between the components perpendicular to the same axis.
The purpose of the device is to allow scientists and engineers to perform experiments and tests that require a known ambient magnetic field. Helmholtz field generation can be static, time-varying DC, or AC, depending on application.
Applications include canceling the Earth’s magnetic field for certain experiments; generating magnetic fields for determining magnetic shielding effectiveness or susceptibility of electronic equipment to magnetic fields; calibration of magnetometers and navigational equipment; and biomagnetic studies.
Figure 1: The Helmholtz coil consists of two coaxial circular coils, one radius apart along the axial direction. The coils carry parallel currents of equal magnitude.
Model Definition
The application is built using the 3D Magnetic Fields interface. The model geometry is shown in Figure 2.
Figure 2: The model geometry.
Domain Equations
Assuming static currents and fields, the magnetic vector potential A must satisfy the following equation:
where μ is the permeability, and Je denotes the externally applied current density.
The relations between the magnetic field H, the magnetic flux density B and the potential are given by
This model uses the permeability of vacuum, that is, μ = 4π ×10-7 H/m. The external current density is computed using a homogenized model for the coils, each one made by 10 wire turns and excited by a current of 0.25 mA. The currents are specified to be parallel for the two coils.
Results and Discussion
Figure 3 shows the magnetic flux density between the coils. The flux is relatively uniform in the region between the coils. This uniformity is the main property and often the sought feature of a Helmholtz coil.
Figure 3: The slice plot shows the magnetic flux density. The arrows indicate the magnetic field (H) strength and direction.
Application Library path: ACDC_Module/Inductive_Devices_and_Coils/helmholtz_coil
Modeling Instructions
From the File menu, choose New.
New
In the New window, click Model Wizard.
Model Wizard
1
In the Model Wizard window, click 3D.
2
In the Select Physics tree, select AC/DC>Electromagnetic Fields>Magnetic Fields (mf).
3
Click Add.
4
Click Study.
5
In the Select Study tree, select General Studies>Stationary.
6
Click Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
I0
0.25[mA]
2.5E-4 A
Coil current
Geometry 1
Work Plane 1 (wp1)
In the Geometry toolbar, click Work Plane.
Work Plane 1 (wp1)>Plane Geometry
Right-click Work Plane 1 (wp1) and choose Show Work Plane.
Work Plane 1 (wp1)>Square 1 (sq1)
1
In the Work Plane toolbar, click Square.
2
In the Settings window for Square, locate the Size section.
3
In the Side length text field, type 0.05.
4
Locate the Position section. From the Base list, choose Center.
5
In the xw text field, type -0.4.
6
In the yw text field, type 0.2.
Work Plane 1 (wp1)>Square 2 (sq2)
1
In the Work Plane toolbar, click Square.
2
In the Settings window for Square, locate the Size section.
3
In the Side length text field, type 0.05.
4
Locate the Position section. From the Base list, choose Center.
5
In the xw text field, type -0.4.
6
In the yw text field, type -0.2.
Work Plane 1 (wp1)
In the Model Builder window, click Work Plane 1 (wp1).
Revolve 1 (rev1)
In the Geometry toolbar, click Revolve.
Sphere 1 (sph1)
1
In the Geometry toolbar, click Sphere.
2
Right-click Sphere 1 (sph1) and choose Build All Objects.
3
Click the Zoom Extents button in the Graphics toolbar.
4
The geometry is now complete. To see its interior, click the Wireframe Rendering button in the Graphics toolbar.
Materials
Define the materials for the model.
Add Material
1
In the Home toolbar, click Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Air.
4
Click Add to Component in the window toolbar.
5
In the Home toolbar, click Add Material to close the Add Material window.
By default, the first material added is applied on all domains.
Add another material for the coil domains. Since the coil will use a homogenized model for the metallic wires, only the relative permittivity and permeability are required from the material.
Materials
Material 2 (mat2)
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Coil Insulator in the Label text field.
3
Select Domains 2 and 3 only.
4
Locate the Material Contents 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
 
S/m
Basic
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
1
1
Basic
Magnetic Fields (mf)
Coil 1
1
In the Model Builder window, under Component 1 (comp1) right-click Magnetic Fields (mf) and choose the domain setting Coil.
2
Select Domain 2 only.
3
In the Settings window for Coil, locate the Coil section.
4
From the Conductor model list, choose Homogenized multi-turn.
5
From the Coil type list, choose Circular.
6
In the Icoil text field, type I0.
In order to specify the direction of the wires in the circular coil, use the Coil Geometry subfeature to select a group of edges forming a circle. The path of the wires will be automatically computed from the geometry of the selected edges. For the best results, the radius of the circular edges selected should be close to the average radius of the coil.
Coil Geometry 1
1
In the Model Builder window, expand the Coil 1 node, then click Coil Geometry 1.
2
In the Settings window for Coil Geometry, locate the Edge Selection section.
3
Click Clear Selection.
4
Select Edges 20, 21, 36, and 39 only.
Now set up the second coil in the same way.
Coil 2
1
In the Physics toolbar, click Domains and choose Coil.
2
Select Domain 3 only.
3
In the Settings window for Coil, locate the Coil section.
4
From the Conductor model list, choose Homogenized multi-turn.
5
From the Coil type list, choose Circular.
6
In the Icoil text field, type I0.
Coil Geometry 1
1
In the Model Builder window, expand the Coil 2 node, then click Coil Geometry 1.
2
In the Settings window for Coil Geometry, locate the Edge Selection section.
3
Click Clear Selection.
4
Select Edges 25, 26, 56, and 59 only.
Create a finer mesh in the coils.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Coarse.
Free Tetrahedral 1
Right-click Component 1 (comp1)>Mesh 1 and choose Free Tetrahedral.
Size 1
1
In the Model Builder window, right-click Free Tetrahedral 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 2 and 3 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section. Select the Maximum element size check box.
7
In the associated text field, type 0.05.
8
Click Build All.
Study 1
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, locate the Study Settings section.
3
Clear the Generate default plots check box.
4
In the Home toolbar, click Compute.
Add a selection to the computed data set to exclude the outer boundaries.
Explicit 1
1
In the Definitions toolbar, click Explicit.
2
Select Domains 2 and 3 only.
3
In the Settings window for Explicit, locate the Output Entities section.
4
From the Output entities list, choose Adjacent boundaries.
5
Right-click Explicit 1 and choose Rename.
6
In the Rename Explicit dialog box, type Coils in the New label text field.
7
Click OK.
Now add the plots.
Results
In the Model Builder window, expand the Results node.
Study 1/Solution 1 (sol1)
In the Model Builder window, expand the Results>Datasets node, then click Study 1/Solution 1 (sol1).
Selection
1
In the Results toolbar, click Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Coils.
3D Plot Group 1
1
In the Results toolbar, click 3D Plot Group.
2
In the Settings window for 3D Plot Group, type Magnetic Flux Density in the Label text field.
Slice 1
1
Right-click Magnetic Flux Density and choose Slice.
2
In the Settings window for Slice, locate the Plane Data section.
3
From the Plane list, choose XY-planes.
4
In the Planes text field, type 1.
5
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Magnetic Fields>Magnetic>mf.normB - Magnetic flux density norm - T.
6
In the Magnetic Flux Density toolbar, click Plot.
Arrow Volume 1
1
In the Model Builder window, right-click Magnetic Flux Density and choose Arrow Volume.
2
In the Settings window for Arrow Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Magnetic Fields>Magnetic>mf.Hx,mf.Hy,mf.Hz - Magnetic field.
3
Locate the Arrow Positioning section. Find the X grid points subsection. In the Points text field, type 24.
4
Find the Y grid points subsection. In the Points text field, type 10.
5
Find the Z grid points subsection. In the Points text field, type 1.
6
Locate the Coloring and Style section. Select the Scale factor check box.
7
In the associated text field, type 25.
8
In the Magnetic Flux Density toolbar, click Plot.
To make the coil look like a solid object, can add a surface plot on its boundaries.
Surface 1
1
Right-click Magnetic Flux Density and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type 1.
4
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
5
From the Color list, choose White.
To verify that the current path is computed correctly, plot the Coil direction variable for each coil.
3D Plot Group 2
1
In the Home toolbar, click Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, type Coil Direction in the Label text field.
Streamline 1
1
Right-click Coil Direction and choose Streamline.
2
Select Boundary 5 only.
3
In the Settings window for Streamline, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Magnetic Fields>Coil parameters>mf.coil1.eCoilx,...,mf.coil1.eCoilz - Coil direction.
Streamline 2
1
In the Model Builder window, right-click Coil Direction and choose Streamline.
2
Select Boundary 12 only.
3
In the Settings window for Streamline, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Magnetic Fields>Coil parameters>mf.coil2.eCoilx,...,mf.coil2.eCoilz - Coil direction.
4
Locate the Coloring and Style section. Find the Point style subsection. From the Color list, choose Blue.
5
In the Coil Direction toolbar, click Plot.
The streamlines show the computed path of the coil currents.