Created in COMSOL Multiphysics 6.

Magnetic Field of a Helmholtz Coil

This model is licensed under the COMSOL Software License Agreement 6.3.
All trademarks are the property of their respective owners. See www.comsol.com/trademarks.
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 the application.

Applications include canceling 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.

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Model Definition
The application shows how to compute the magnetic field with two different approaches,
one using the Magnetic Fields interface and the other the Magnetic Fields, Currents Only
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:

–1
      A  = Je

where  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

B = A
–1
H =  B

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This model uses the permeability of vacuum, that is,   4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.

SPATIAL DERIVATIVE OF MAGNETIC FIELD

Computing the spatial derivative of the magnetic field or magnetic flux density is useful in
areas such as radiology, magnetophoresis, particle accelerators, and geophysics. One of the
most important use cases is the design of magnetic resonance imaging (MRI) machines,
where it is necessary to analyze not only the field strength but also the spatial variation of
the field. This application demonstrates how to compute the spatial derivative of the
magnetic flux density in the postprocessing step.

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 norm. The arrows indicate the
magnetic field (H) strength and direction.

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Figure 4 and Figure 5 compare the results from using the two different physics interfaces.

Figure 4: Comparison of the y component of the B field along the centerline of the Helmholtz
coil using two different approaches.

Figure 5: Comparison of the gradient (with respect to the y direction) of the y component of the
B field along the centerline of the Helmholtz coil.

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Application Library path: ACDC_Module/Devices,_Inductive/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 In the Select Physics tree, select AC/DC > Electromagnetic Fields > Vector Formulations >
Magnetic Fields, Currents Only (mfco).
5 Click Add.
6 Click Study.
7 In the Select Study tree, select General Studies > Stationary.
8 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)

1 In the Geometry toolbar, click Work Plane.

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2 In the Settings window for Work Plane, click Go to Plane Geometry.

Work Plane 1 (wp1) > Plane Geometry

In the Model Builder window, click Plane Geometry.

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.

Revolve 1 (rev1)
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click
Work Plane 1 (wp1) and choose Revolve.

Sphere 1 (sph1)
1 In the Geometry toolbar, click Sphere.
2 In the Settings window for Sphere, locate the Size section.
3 In the Radius text field, type 1.3.
4 Click to expand the Layers section. In the table, enter the following settings:

Layer name Thickness (m)

Layer 1 0.3

5 Click Build All Objects.

6 Click the Zoom Extents button in the Graphics toolbar.

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7 The geometry is now complete. To see its interior, click the Wireframe Rendering button
in the Graphics toolbar.

Line Segment 1 (ls1)

1 In the Geometry toolbar, click More Primitives and choose Line Segment.
2 On the object sph1, select Point 4 only.
3 In the Settings window for Line Segment, locate the Endpoint section.
4 Click to select the Activate Selection toggle button for End vertex.
5 On the object sph1, select Point 9 only.

DEFINITIONS
Next, define the Infinite Element Domain.

Infinite Element Domain 1 (ie1)

1 In the Definitions toolbar, click Infinite Element Domain.
2 In the Settings window for Infinite Element Domain, locate the Geometry section.
3 From the Type list, choose Spherical.
4 Select Domains 1–4 and 8–11 only.

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Hide for Physics 1
1 In the Model Builder window, right-click View 1 and choose Hide for Physics.
2 In the Settings window for Hide for Physics, locate the Geometric Entity Selection section.
3 From the Geometric entity level list, choose Boundary.
4 Select Boundaries 6 and 10 only.

MAGNETIC FIELDS (MF)

Coil 1
1 In the Physics toolbar, click Domains and choose Coil.
2 Select Domain 6 only.
3 In the Settings window for Coil, locate the Coil section.
4 From the Conductor model list, choose Homogenized multiturn.
5 From the Coil type list, choose Circular.
6 In the Icoil text field, type I0.
7 Locate the Homogenized Conductor section. From the list, choose User defined.
8 Find the High-frequency effective loss subsection. Clear the Include harmonic loss
checkbox.

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, click Coil Geometry 1.
2 In the Settings window for Coil Geometry, locate the Edge Selection section.
3 Click Clear Selection.

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4 Select Edges 25, 26, 46, and 49 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 7 only.
3 In the Settings window for Coil, locate the Coil section.
4 From the Conductor model list, choose Homogenized multiturn.
5 From the Coil type list, choose Circular.
6 In the Icoil text field, type I0.
7 Locate the Homogenized Conductor section. From the list, choose User defined.
8 Find the High-frequency effective loss subsection. Clear the Include harmonic loss
checkbox.

Coil Geometry 1
1 In the Model Builder window, click Coil Geometry 1.
2 In the Settings window for Coil Geometry, locate the Edge Selection section.
3 Click Clear Selection.

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4 Select Edges 30, 31, 72, and 75 only.

MAGNETIC FIELDS, CURRENTS ONLY (MFCO)

In the Model Builder window, under Component 1 (comp1) click Magnetic Fields,
Currents Only (mfco).

Conductor 1
1 In the Physics toolbar, click Domains and choose Conductor.
2 Select Domain 6 only.

Terminal 1
1 In the Model Builder window, expand the Conductor 1 node, then click Terminal 1.
2 Select Boundary 13 only.
3 In the Settings window for Terminal, locate the Terminal section.
4 In the I0 text field, type 10*I0.

Conductor 2
1 In the Physics toolbar, click Domains and choose Conductor.
2 Select Domain 7 only.

Terminal 1
1 In the Model Builder window, expand the Conductor 2 node, then click Terminal 1.
2 Select Boundary 20 only.
3 In the Settings window for Terminal, locate the Terminal section.
4 In the I0 text field, type 10*I0.

MATERIALS
Define the insulating material for the coils.

Coil Insulator
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 6 and 7 only.

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4 Locate the Material Contents section. In the table, enter the following settings:

Property Variable Value Unit Property

group
Relative permeability mur_iso ; murii 1 1 Basic
= mur_iso,
murij = 0
Relative permittivity epsilonr_iso ; 1 1 Basic
epsilonrii =
epsilonr_iso,
epsilonrij = 0
Electric conductivity sigma_iso ; 6e7[S/m] S/m Basic
sigmaii =
sigma_iso,
sigmaij = 0

MESH 1
Next, set up the mesh.

Edge 1
1 In the Mesh toolbar, click More Generators and choose Edge.
2 Select Edge 40 only.

Distribution 1
1 Right-click Edge 1 and choose Distribution.
2 In the Settings window for Distribution, locate the Distribution section.
3 In the Number of elements text field, type 50.

Free Tetrahedral 1
1 In the Mesh toolbar, click Free Tetrahedral.
2 In the Settings window for Free Tetrahedral, locate the Domain Selection section.
3 From the Geometric entity level list, choose Domain.
4 Select Domains 5–7 only.

Size 1
1 Right-click Free Tetrahedral 1 and choose Size.
2 Select Domains 6 and 7 only.
3 In the Settings window for Size, locate the Element Size section.
4 Click the Custom button.

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5 Locate the Element Size Parameters section.
6 Select the Maximum element size checkbox. In the associated text field, type 0.05.

Swept 1
In the Mesh toolbar, click Swept.

Distribution 1
1 Right-click Swept 1 and choose Distribution.
2 Right-click Distribution 1 and choose Build All.

STUDY 1

Step 2: Stationary 2
In the Study toolbar, click Study Steps and choose Stationary > Stationary.

Step 1: Stationary
1 In the Model Builder window, click Step 1: Stationary.
2 In the Settings window for Stationary, locate the Physics and Variables Selection section.
3 In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for
Magnetic Fields, Currents Only (mfco).

Step 2: Stationary 2
1 In the Model Builder window, click Step 2: Stationary 2.
2 In the Settings window for Stationary, locate the Physics and Variables Selection section.
3 In the Solve for column of the table, under Component 1 (comp1), clear the checkbox for
Magnetic Fields (mf).
4 In the Model Builder window, click Study 1.
5 In the Settings window for Study, locate the Study Settings section.
6 Clear the Generate default plots checkbox.
7 In the Study toolbar, click Compute.

Add a selection to the computed dataset to exclude the outer boundaries.

DEFINITIONS

Coils
1 In the Definitions toolbar, click Explicit.
2 Select Domains 6 and 7 only.
3 In the Settings window for Explicit, locate the Output Entities section.

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4 From the Output entities list, choose Adjacent boundaries.
5 In the Label text field, type Coils.

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.

Magnetic Flux Density, MF

1 In the Results toolbar, click 3D Plot Group.
2 In the Settings window for 3D Plot Group, type Magnetic Flux Density, MF in the
Label text field.

Slice 1
1 Right-click Magnetic Flux Density, MF 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 (comp1) > Magnetic Fields > Magnetic > mf.normB -
Magnetic flux density norm - T.
6 In the Magnetic Flux Density, MF toolbar, click Plot.

Arrow Volume 1
1 In the Model Builder window, right-click Magnetic Flux Density, MF 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 (comp1) >
Magnetic Fields > Magnetic > mf.Hx,mf.Hy,mf.Hz - Magnetic field.

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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.
7 Select the Scale factor checkbox. In the associated text field, type 25.
8 In the Magnetic Flux Density, MF toolbar, click Plot.

To make the coil look like a solid object, you can add a surface plot on its boundaries.

Surface 1
1 Right-click Magnetic Flux Density, MF 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.

Next, compare the results of By and Byy calculated from the two interfaces.

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Comparison of By
1 In the Results toolbar, click 1D Plot Group.
2 In the Settings window for 1D Plot Group, type Comparison of By in the Label text
field.

Line Graph 1
1 Right-click Comparison of By and choose Line Graph.
2 Select Edge 40 only.
3 In the Settings window for Line Graph, locate the y-Axis Data section.
4 In the Expression text field, type mf.By.
5 Locate the x-Axis Data section. From the Parameter list, choose Expression.
6 In the Expression text field, type y.
7 Click to expand the Coloring and Style section. Find the Line markers subsection. From
the Marker list, choose Cycle.
8 From the Positioning list, choose Interpolated.
9 Click to expand the Legends section. Select the Show legends checkbox.
10 From the Legends list, choose Manual.
11 In the table, enter the following settings:

Legends
Magnetic Fields interface

Line Graph 2
1 Right-click Line Graph 1 and choose Duplicate.
2 In the Settings window for Line Graph, locate the y-Axis Data section.
3 In the Expression text field, type mfco.By.
4 Click to expand the Title section. From the Title type list, choose None.
5 Locate the Legends section. In the table, enter the following settings:

Legends
Magnetic Fields, Currents Only interface

Comparison of By
1 In the Model Builder window, click Comparison of By.
2 In the Settings window for 1D Plot Group, locate the Legend section.
3 From the Position list, choose Upper middle.

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4 In the Comparison of By toolbar, click Plot.

Comparison of Byy
1 Right-click Comparison of By and choose Duplicate.
2 In the Model Builder window, click Comparison of By 1.
3 In the Settings window for 1D Plot Group, type Comparison of Byy in the Label text
field.
4 Locate the Legend section. From the Position list, choose Lower right.

Line Graph 1
1 In the Model Builder window, click Line Graph 1.
2 In the Settings window for Line Graph, locate the y-Axis Data section.
3 In the Expression text field, type d(laginterp(2,mf.By),y).

The mf interface is using Curl shape functions and the higher order spatial derivative is not
available in postprocessing. In this case, use the laginterp operator.

Line Graph 2
1 In the Model Builder window, click Line Graph 2.
2 In the Settings window for Line Graph, locate the y-Axis Data section.

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3 In the Expression text field, type d(mfco.By,y).
The mfco interface is using secondary order Lagrange shape functions and the second
derivative is available. The curves of Byy can be improved by using cubic elements.
4 In the Comparison of Byy toolbar, click Plot.

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