Problem Description

Solution

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 included in the list of predefined quantities in the Plot Parameters dialog box, 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. Click to enlarge.

Unresolved skin depth	Well-resolved skin depth

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 boundary condition exists in most application modes in the AC/DC and RF Modules. To use it, you will first need to deactivate the subdomain that you would otherwise have used for your metal. Then, select the Impedance boundary condition for the boundaries of the inactive metal domain, and enter the properties of the metal on the Material Properties tab of the Boundary Settings dialog box. 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 3D.  

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. If this makes you run out of memory in a 3D model, an iterative solver such as GMRES with the Geometric Multigrid preconditioner might help you. This preconditioner starts operating on a coarser mesh (down to 2 linear elements per wavelength) and works its way to the final mesh. For information, see "Solving Large 3D Problems" under RF Modeling in the RF Module User's Guide.

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