# Skin Effect and Ohmic Losses

Specify the skin effect through the high frequency, static or exact expression.

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.

## High frequency approximation skin effect

The required parameters are ${\mu }_{r}$ , $\mathrm{tan}{\delta }_{u}$ and $\sigma$ (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 $\sigma \gg \omega {\epsilon }_{0}$ .
• For wires the skin depth must satisfy the condition ${\delta }_{\text{skin}}<\varrho$ where $\varrho$ is the wire radius. The surface impedance is given by ${\text{Z}}_{s}^{\prime }=\frac{1}{2\pi \varrho }\sqrt{\frac{j\omega \mu }{\sigma }}$ .
• For metallic surfaces the skin depth must satisfy the condition ${\delta }_{\text{skin}}<\frac{d}{2}$ . The surface impedance is given by ${Z}_{s}=\frac{1}{2}\sqrt{\frac{j\omega \mu }{\sigma }}$ .

## Static approximation skin effect

The required parameters are ${\mu }_{r}$ , $\mathrm{tan}{\delta }_{u}$ and $\sigma$ (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 $\sigma \gg \omega {\epsilon }_{0}$ .
• For wires the skin depth must satisfy the condition ${\delta }_{\text{skin}}>\varrho$ where $\varrho$ is the wire radius. The surface impedance is given by ${\text{Z}}_{s}^{\prime }=\frac{1}{\pi {\varrho }^{2}\sigma }$
• For metallic surfaces the skin depth must satisfy the condition ${\delta }_{\text{skin}}>\frac{\text{d}}{2}$ . The surface impedance is given by ${Z}_{s}=\frac{1}{\sigma d}$ .

## Exact expression for the skin effect

The required parameters are ${\mu }_{r}$ , $\mathrm{tan}{\delta }_{u}$ and $\sigma$ (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 $\sigma \gg \omega {\epsilon }_{0}$ .
• For wires with wire radius $\varrho$ the surface impedance is given by
(1) ${Z}_{{s}^{\prime }}=\frac{1-j}{2\pi \varrho \sigma {\delta }_{skin}}\frac{{J}_{0}\left[\left(1-j\right)\frac{\varrho }{{\delta }_{skin}}\right]}{{J}_{1}\left[\left(1-j\right)\frac{\varrho }{{\delta }_{skin}}\right]}$
where J0 and J1 are Bessel functions.
• For a sheet of thickness, d, with properties ${\epsilon }_{c}$ , ${\mu }_{c}$ with ${\epsilon }_{c}=\epsilon +\frac{\sigma }{j\omega }$ the wave propagation constant in the sheet is given by ${\beta }_{c}=\omega \sqrt{{\mu }_{c}{\epsilon }_{c}}$ and the wave impedance in the sheet by ${\eta }_{c}=\sqrt{\frac{{\mu }_{c}}{{\epsilon }_{c}}}$ . The reflection coefficient at the interface between the sheet and the environment is defined as $\Gamma =\frac{{E}^{-}}{{E}^{+}}=\frac{{\eta }_{o}-{\eta }_{c}}{{\eta }_{o}+{\eta }_{c}}$ . The surface impedance is given by
(2) ${Z}_{{s}^{\prime }}={\eta }_{c}\cdot \frac{1+\Gamma {e}^{-j2{\beta }_{c}d}}{1-\Gamma {e}^{-j2{\beta }_{c}d}+\left(\Gamma -1\right){e}^{-j{\beta }_{c}d}}$