Iron Sphere in a 60 Hz Magnetic Field

An iron sphere is exposed to a spatially uniform, sinusoidally time-varying, background magnetic field. The frequency of the field is low enough that the skin depth is larger than the radius of the sphere. This application uses a reduced field formulation to impose the background field and demonstrates two approaches for solving the problem. The application computes the induced currents in the sphere and the perturbation to the background field.

Figure 1: An iron sphere is exposed to a spatially uniform, sinusoidally time-varying, background magnetic field.

Model Definition

Figure 1 shows the setup, with an iron sphere placed in a spatially uniform time-harmonic background magnetic field. The background field is applied using the Reduced field formulation available in the Magnetic Fields interface. The model space is truncated by an Infinite Elements region, a domain condition approximating a domain that extends to infinity. When using Infinite Element Domain features, the boundary condition on the outside of the modeling domain only marginally affects the solution, since it is placed at a large physical distance.

The iron sphere has a relative permittivity of εr = 1, a relative permeability of μr = 4000, and an electric conductivity of σ = 1.12·107 S/m. The explicit assumption of modeling in the frequency domain is that all material properties are independent of the field strength. At the applied field strength of 1 mT, the permeability can be assumed to be constant — saturation effects in the iron are negligible.

For all models with time-varying magnetic fields, it is important to first consider the skin depth, δ, which is given by:

At the operating frequency of 60 Hz the skin depth of iron is δ ~ 0.3 mm. The surrounding air has εr = 1, μr = 1, and σ = 0 S/m, thus the ratio of the largest to the smallest skin depth is infinite, and this leads to numerical difficulties when solving the problem.

It is possible to avoid this numerical difficulty by adding an artificial conductivity to the air domain. The basic concept behind this approach is to consider the skin depth in all the domains in the model and, in domains where the skin depth is very large or infinite, the conductivity should be increased. This artificial conductivity should be large enough so that the ratio of the largest to smallest skin depth be around 1000:1. The greater the artificial conductivity, the less accurate the results, but a too small artificial conductivity negatively affects convergence.

An alternative approach, that does not require increasing the artificial conductivity as much, is to use gauge fixing. This adds an additional equation to the system of equations being solved, and as a consequence significantly increases the computational effort needed to solve the model.

Results and Discussion

Figure 2 plots the magnetic field and the induced current density for the model without gauge fixing, while Figure 3 shows the same results for the model with gauge fixing. The results agree well between the two approaches.

When using gauge fixing, the artificial conductivity of 5 S/m leads to a skin depth in the air of ~29 m and a total dissipation in the air domain of 4.53×10-11 nW. Without gauge fixing an artificial conductivity of 5000 S/m is used, leading to a skin depth in the air of ~0.9 m and a total dissipation in the air of 4.53×10-8 nW. For both approaches, the total dissipation in the sphere is 0.41×10-5 nW, two orders of magnitude higher.

Using gauge fixing increases the solution time and memory needed to solve the problem, and generally only slightly improves the solution. Therefore, it should be used sparingly. In any case, it is always recommended to carefully study the effects of artificial conductivity on the relative skin depths in the model, and to keep in mind that this is a function of the operating frequency.

Figure 2: The induced currents and the magnetic field for the model without gauge fixing.

Figure 3: The induced currents and the magnetic field for the model with gauge fixing.

ACDC_Module/Introductory_Electromagnetics/iron_sphere_60hz_bfield

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
B0	1[mT]	0.001 T	Background magnetic fields
r0	0.125[mm]	1.25E-4 m	Radius, iron sphere

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Sphere 1 (sph1)

Create a sphere with two layers plus an inner core. The outermost layer represents the exterior air region, scaled using the Infinite Element Domain, the middle layer is the unscaled air domain, and the core represents the iron sphere.

In the toolbar, click

In the window for , locate the section.

In the text field, type 3*r0.

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


Layer name	Thickness (mm)
Layer 1	r0
Layer 2	r0

Click

Click the

button in the toolbar.

Definitions

Create a set of selections before setting up the physics. First, create a selection for the surface of the iron sphere.

Core

In the toolbar, click

Select Domain 9 only.

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

Add a selection for the Infinite Element Domain feature.

Infinite Element domains

In the toolbar, click

Select Domains 1–4, 10, 11, 14, and 17 only.

In the window for , type Infinite Element domains in the text field.

Add a selection for the domain in which to plot the magnetic flux density norm. It is the complement of the selection.

Analysis domain

In the toolbar, click

In the window for , locate the section.

Under , click

In the dialog box, select in the list.

Click .

In the window for , type Analysis domain in the text field.

Add an Infinite Element Domain node. Use the selection .

Infinite Element Domain 1 (ie1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Magnetic Fields (mf)

Set up the physics applying a uniform background magnetic fields. In the physics, the background field must be specified in terms of its vector potential field.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Specify the Ab vector as


0	x
0	y
B0*y	z

Materials

Then, assign material properties. First, use air for all domains.

Air

In the window, under right-click and choose .

In order to improve the convergence rate of the solver, use an artificial conductivity of 5000 S/m.

In the window for , locate the 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	5000	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

In the text field, type Air.

Override the core sphere with iron.

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 toolbar, click

to close the window.

Materials

Iron (mat2)

In the window for , locate the section.

From the list, choose .

Mesh 1

The interface’s creates a swept mesh for the Infinite Element Domain.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Click

Plot the meshed structure to review the quality of the mesh.

In the toolbar, click

Results

Mesh 1

By default, the boundary mesh is plotted, so only the triangular elements on the outer boundaries are visible. Perform the following operations to inspect the tetrahedral elements in the interior of the geometry.

In the window for , locate the section.

From the list, choose .

Click to expand the section. Select the check box.

In the text field, type z<0 to plot a section of the mesh.

In the toolbar, click

Click the

button in the toolbar.

Study 1

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type 60[Hz].

In the toolbar, click

Results

Magnetic Flux Density Norm (mf)

The default plot shows the magnetic flux density norm. Suppress the Infinite Element Domain for the result analysis and add an arrow plot for the current density.

Study 1/Solution 1 (sol1)

In the window, expand the node, then click .

Selection

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Arrow Volume 1

In the window, right-click and choose .

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

Locate the section. Find the subsection. In the text field, type 1.

Find the subsection. In the text field, type 21.

Locate the section. From the list, choose .

Compare the reproduced plot with Figure 2.

Add a plot for the skin depth.

Skin Depth (mf)

In the toolbar, click

In the window for , type Skin Depth (mf) in the text field.

Slice 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 1.

Click in the upper-right corner of the section. From the menu, choose .

In the toolbar, click

The second study uses gauge fixing to improve convergence. The artificial conductivity in the air can therefore be reduced.

Materials

Air (mat1)

In the window, under click .

In the window for , locate the 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	5	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic