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.

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): The surface roughness of the metal specified as a RMS value in m.
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) $Z_{s^{'}} = \frac{1 - j}{2 π ϱ σ δ_{s k i n}} \frac{J_{0} [(1 - j) \frac{ϱ}{δ_{s k i n}}]}{J_{1} [(1 - j) \frac{ϱ}{δ_{s k i n}}]}$
where J₀ and J₁ are Bessel functions.
For a sheet of thickness, d, with properties $ε_{c}$ , $μ_{c}$ with $ε_{c} = ε + \frac{σ}{j ω}$ the wave propagation constant in the sheet is given by $β_{c} = ω \sqrt{μ_{c} ε_{c}}$ and the wave impedance in the sheet by $η_{c} = \sqrt{\frac{μ_{c}}{ε_{c}}}$ . The reflection coefficient at the interface between the sheet and the environment is defined as $Γ = \frac{E^{-}}{E^{+}} = \frac{η_{o} - η_{c}}{η_{o} + η_{c}}$ . The surface impedance is given by
(2) $Z_{s^{'}} = η_{c} \cdot \frac{1 + Γ e^{- j 2 β_{c} d}}{1 - Γ e^{- j 2 β_{c} d} + (Γ - 1) e^{- j β_{c} d}}$