Exact Expression for the Skin Effect (Including Surface Roughness)

This option adds an exact skin effect (which includes the effects of surface roughness) to metallic surfaces and/or wires.



Figure 1. The SK - Add a skin effect (finite conductivity) dialog.
Note: The material parameters for the skin effect are defined with the DI card. The SK card then uses the label defined at the DI card.

Parameters:

Thickness of elements
The thickness d of the surface elements in metres (if an SF card is present, this is always scaled).
Surface roughness (RMS value in m)
In many striplines, PCB or similar planar devices, when the frequency increases, the surface roughness of the employed metal (for example, copper) becomes larger than the skin depth. To model a conductor accurately, specify the surface roughness as an RMS value in m. Common values are typically in the range 50 nm to 100 μm. Specifying the surface roughness will increase the losses as the frequency increases.
Note: Setting the surface roughness will not impact the total runtime or memory used.
Material label
Label of the material which will be used (as specified in the DI card).

The required parameters are μ r , tan δ u and σ (defined with the DI card). If applied to surfaces then also the thickness d is required.

The following restrictions apply:
  • A good conductivity is required, satisfying the condition σ≫ω ε 0 .
  • For wires with wire radius ϱ the surface impedance is given by
    (1) Z s ′ = 1−j 2πϱσ δ skin J 0 [(1−j) ϱ δ skin ] J 1 [(1−j) ϱ δ skin ]
    where J0 and J1 are Bessel functions.
  • For a sheet of thickness, d, with properties ε c , μ c with ε c =ε+ σ jω the wave propagation constant in the sheet is given by β c =ω μ c ε c and the wave impedance in the sheet by η c = μ c ε c . The reflection coefficient at the interface between the sheet and the environment is defined as Γ= E − E + = η o − η c η o + η c . The surface impedance is given by
    (2) Z s ′ = η c ⋅ 1+Γ e −j2 β c d 1−Γ e −j2 β c d +(Γ−1) e −j β c d