Iron Sphere in a 20 kHz Magnetic Field

An iron sphere is exposed to a spatially uniform, sinusoidally time-varying, background magnetic field. The frequency of the field is such that the skin depth is smaller than the sphere radius. The application computes the induced currents in the sphere and the perturbation to the background field. In addition, it addresses the question of how to properly mesh domains with significant skin effect.

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 implicit 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.

At the operating frequency of 20 kHz, the skin depth in the iron is 16.8 μm. The surrounding air has εr = 1, μr = 1, and σ = 0 S/m. This implies an infinite skin depth, which causes difficulties for the default solver. Instead, solve the model using the artificial conductivity approach by modifying the air conductivity to be σ = 50 S/m.

Due to the small skin depth, the solution can be assumed to have steep gradients normal to the boundary of the sphere. Such cases are well suited for boundary layer meshing, an operation which creates short triangular prismatic elements along the direction normal to the surface. The thickness of these prismatic elements should be equal to, or smaller than, the skin depth. In this way, the steep gradients normal to the boundary are better resolved by the mesh.

Results and Discussion

Figure 2 plots the magnetic field and the induced current density, while Figure 3 shows the mesh. The mesh resolves the skin effect well.

Figure 2: The induced currents and the magnetic field in the iron sphere.

Figure 3: The mesh in the iron sphere and surrounding air.

ACDC_Module/Tutorials/iron_sphere_20khz_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

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

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.

Right-click and choose .

In the dialog box, type Core in the text field.

Click .

Add a selection for the Infinite Element Domain feature.

Infinite Element domain

In the toolbar, click

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

Right-click and choose .

In the dialog box, type Infinite Element domain in the text field.

Click .

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 .

Right-click and choose .

In the dialog box, type Analysis domain in the text field.

Click .

Add an Infinite Element Domain. 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 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

Materials

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

Air

In the window, under right-click and choose .

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
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	50	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic