Resolving the Skin Depth

Problem Description

Why do the currents in my metal look strange? How can I get a realistic solution at high frequencies?

Solution

Not resolving the skin depth is one of the more common modeling mistakes in AC and RF applications. Luckily, in most cases where you cannot afford to mesh fine enough, you can use an impedance boundary condition instead.

The skin depth is defined as the depth at which the electromagnetic field in a conducting material has decreased to 1/e of its value just outside the material. It is useful to check that you can resolve the skin depth by at least a few mesh elements before you add a highly conductive or permeable object to your model. For a good conductor, the skin depth is given by the expression

where ω is the angular frequency, μ the permeability, and σ the conductivity of the material. For example, at 50 Hz, the skin depth in copper (μ_r = 1, σ = 6·10⁷ S/m) is around 9 mm and that in iron (μ_r = 4000, σ = 1.12·10⁷ S/m) is 0.34 mm. Since most of the current in the metal will be contained within one skin depth from the surface, you need to resolve this region with at least a few mesh elements. In case you would want COMSOL Multiphysics to calculate the skin depth for you, it is available as a plot expression in all time-harmonic electromagnetics application modes.

The pictures below show what the local resistive heating near the surface of a metal might look like with an insufficiently resolved skin depth and a well resolved one. If you click to enlarge the pictures, you can see that the insufficiently resolved mesh gives a resistive heating that is distributed over a greater area and has a lower peak value.

Figure 1: Unresolved and well resolved field penetrations, respectively.

If you need to resolve the skin depth while maintaining a relatively small total number of mesh elements, a Boundary Layer mesh can be helpful. Use the Boundary Layers feature available from the Mesh node.

Impedance Boundary Condition 

For high frequencies and for highly conductive or permeable materials, resolving the skin depth is not always a realistic option. In such cases, the Impedance Boundary Condition may come in handy. This condition exists in most physics interfaces in the AC/DC and RF Modules. To use it, you will first need to inactivate your physics interface in the domains which you would otherwise have used for your metal. This is done from the Settings tab, by removing them from the Domains list. Then, select the Impedance boundary condition for the boundaries of the inactive metal domain. With the impedance boundary condition, all currents will run on the surface of your metal, eliminating the need to resolve the skin depth. When using the impedance boundary condition, the skin depth will be available as a predefined expression on the Boundary tab of the Plot Parameters dialog box.

The impedance boundary condition is a valid approximation when the thickness of the material is much greater than the skin depth. For most practical purposes, a factor of 3 - 4 is sufficient. This signifies that there is often a slight overlap between the frequencies where you can resolve the skin depth and those where you can use the impedance boundary condition.

The AC/DC Module contains an example model showing both how to resolve the skin depth with a fine mesh and how to switch to the impedance boundary condition when this is no longer possible. You can find it in the Model Library under AC/DC Module > General Industrial Applications > eddy currents.  

The Transition Boundary Condition resembles the impedance condition and is used for approximating thin metal sheets that are interior to the model.

Resolving the wavelength 

Keep in mind that when solving electromagnetic (and other) wave equations, you also need to resolve the wavelength. As a rule of thumb, use at least 10 linear or 5 second order elements per wavelength. Second order elements are the default in the relevant physics interfaces.

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