COMSOL 6.4 - Iron Sphere in a Magnetic Field

This tutorial is part of a series on modeling an iron sphere in a background magnetic field within the Introduction to Electromagnetics tutorial group. This tutorial focuses on the case of a magnetically permeable iron sphere in a spatially uniform magnetic field where the magnetic field sinusoidally varies in time at a frequency of 60 Hz. At this frequency, the skin depth of the iron used in the model is ~ 0.3 mm, which is larger than the sphere itself.

A factor that requires consideration in this tutorial scenario is the conductivity of air, σair. In practice σair is negligible but setting it to zero leads to numerical difficulties. This tutorial covers two methods of approaching this by using a stabilization conductivity in the feature or using . To assess the performance of these methods, we will look at the magnetic dissipation. Using a stabilization conductivity in will produce artificial magnetic dissipation in that domain. This will be calculated for the different methods and compared to the larger dissipation in the iron sphere itself.

Model Definition

Figure 1: A magnetically permeable iron sphere in a spatially uniform background magnetic field that sinusoidally varies in time with angular frequency, w. The sphere at the center is surrounded by air and enclosed in a region of Infinite Elements.

Each model in this tutorial series uses the same basic structure illustrated in Figure 1. It consists of a 0.25 mm diameter iron sphere, with a relative permeability of, μr = 4000, placed in a spatially uniform background magnetic field of strength B0 = 1 mT. In this case, that magnetic field oscillates at a frequency of 60 Hz.

As discussed in the introduction, the surrounding air normally has an infinite skin depth, which leads to numerical difficulties when solving. We can overcome this in two ways:

Use a stabilization conductivity corresponding to a ratio of 1000:1 between the largest and smallest skin depths in the model. In this case, this means using a value of σstab = 5000 S/m. This will add artificial magnetic dissipation in the air domains in the model.

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. However, this approach does not require a stabilization conductivity, resulting in negligible artificial magnetic dissipation in the air domain.

Results and Discussion

Figure 2: Cross-section of the iron sphere showing the induced current in the sphere and the magnetic flux calculated using gauge fixing.

Figure 2 plots the magnetic field and the induced current density for the model with gauge fixing. Without gauge fixing, a stabilization 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 3.8×10−16 W. When using gauge fixing, the artificial conductivity is not needed and set to 0 S/m. This corresponds to an infinite skin depth in the surrounding air and negligible dissipation. In both cases, the total dissipation in the iron sphere is 9.1×10−14 W. This is the dominating value in this model.

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. It’s important to note, however, that Gauge fixing is sometimes necessary for convergence. For example when solving a stationary problem in the magnetic fields interface using a direct solver¹. 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.

ACDC_Module/Introductory_Electromagnetics/iron_sphere_bfield_02_60hz

Modeling Instructions

This tutorial will demonstrate the physics of an iron sphere in a spatially uniform magnetic field sinusoidally varying at 60 Hz. The instructions on the following pages will help you to build, configure, solve, and analyze the model. If anything seems out of order, please retrace your steps. The finalized model — available in the model’s Application Libraries folder — can help you out. You can compare it directly to your current model by means of the option in the toolbar.

Root

The geometry, materials, and selections have been prepared in the Introduction tutorial (chapter 1). They have been saved in the file iron_sphere_bfield_00_introduction.mph. You can start by opening this file and saving it under a new name.

Hint: if you are new to COMSOL Multiphysics, it is worthwhile to check out the Introduction tutorial first.

From the menu, choose .

Browse to the model’s Application Libraries folder and double-click the file iron_sphere_bfield_00_introduction.mph.

From the menu, choose .

Browse to a suitable folder and type the filename iron_sphere_bfield_02_60hz.mph.

Magnetic Fields (mf)

Free Space 1

Next, set up the stabilization conductivity of free space using the parameter sigma_air. This allows you to set the desired value in the studies later in this tutorial.

In the window, expand the > node, then click .

In the window for , locate the section.