Iron Sphere in a 13.56 MHz Magnetic Field

An iron sphere is exposed to a spatially uniform, sinusoidally time-varying, background magnetic field. The frequency of the field is so high that the skin depth in the sphere is much smaller than the radius. At such high frequencies it is possible to model only the fields and induced currents on the surface of the sphere, thus avoiding the need for solving for the fields within the volume of the sphere, resulting in significantly reduced model size.

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.

At 13.56 MHz the skin depth of iron is ~0.65 μm. The surrounding air has εr = 1, μr = 1, and σ = 0 S/m, which implies an infinite skin depth. To improve the stability of the solver, the application uses a small but nonzero artificial conductivity in the air (σ = 0.1 S/m).

Because the skin depth in the iron sphere is much smaller than the sphere radius, it is possible to assume that the induced currents flow only in a thin surface layer with negligible thickness. This phenomenon can be modeled using the Impedance Boundary Condition on the iron sphere surface. The inside of the iron sphere is not modeled at all, since it is assumed that there are no significant currents, and negligible fields, within the sphere.

The interior of the sphere is explicitly removed from the geometry by means of a Delete operation. An alternative approach is to modify the selection of the Magnetic Fields interface to exclude the interior of the sphere, without deleting the corresponding geometrical domain. This second solution can be useful for example in multiphysics applications when another physics interface (such as Heat Transfer) must be solved in the core as well.

Results and Discussion

Figure 2 plots the magnetic field and the induced current density along with a visualization of the mesh at the sphere surface.

Figure 2: The induced currents on the surface of the iron sphere and the magnetic field in the surrounding space.

ACDC_Module/Tutorials/iron_sphere_13mhz_bfield

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

Define a few parameters to be used in the model setup.

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

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 (m)
Layer 1	r0
Layer 2	r0

Remove the core domain from the model domain.

Delete Entities 1 (del1)

In the window, right-click and choose .

Click the button in the toolbar.

In the window for , locate the section.

From the list, choose .

On the object , select Domain 9 only.

Click .

The surface of the iron sphere is now an exterior boundary and it is now possible to apply the Impedance boundary condition.

Definitions

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

Explicit 1

In the toolbar, click .

In the window for , locate the section.

From the list, choose .

Select Boundaries 17–20, 31, 32, 39, and 42 only.

The simplest way to do this is to select the check box, then click on any boundary on the surface of the iron sphere.

Right-click and choose .

In the dialog box, type Iron surface in the text field.

Click .

Add an Infinite Element Domain on the outermost layer of the model domain.

Infinite Element Domain 1 (ie1)

In the toolbar, click .

Select Domains 1–4, 9, 10, 13, and 16 only.

In the window for , locate the section.

From the list, choose .

Magnetic Fields (mf)

Set up the physics interface to apply a uniform background magnetic fields. In the interface, the background field must be specified in terms of a 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

Impedance Boundary Condition 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Materials

Assign the material properties. First, use air for all domains.

Add Material

In the toolbar, click to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

Materials

Air (mat1)

Specify the conductivity of the air to a small value in order to improve the convergence rate.

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
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0.1[S/m]	S/m	Basic