Surface Impedance Boundary Condition¶

You apply a surface impedance boundary condition at the boundaries of well-conducting materials. Their conductivity is high but cannot be considered infinite. In such cases the electromagnetic field goes into the boundary material to a skin depth that you cannot ignore. The condition is valid only when the skin depth is small compared with the conductor dimensions. The amplitude of electric field \(\tilde{\mathbf{E}}\) at the boundary with such a material must satisfy the following equation:

\[ \hat{\mathbf{n}} \times \left(\frac{1}{\mu} \nabla \times \tilde{\mathbf{E}}\right) = j \frac{\omega}{Z_s} \hat{\mathbf{n}} \times \left( \tilde{\mathbf{E}} \times \hat{\mathbf{n}} \right), \]

where \(Z_s = \sqrt{\mu_s/(\varepsilon_s - j\sigma_s/\omega)}\) is the surface impedance with \(\mu_s\), \(\varepsilon_s\), and \(\sigma_s\) being the permeability, permittivity, and conductivity of the boundary material.

To create a SurfaceImpedanceCondition, give a name and the boundary Marker. You must also supply the magnetic permeability \(\mu_s\), electric permittivity \(\varepsilon_s\), and electric conductivity \(\sigma_s\) Coefficients:

mu0 = 1.25663706127e-6  # [N/A**2] vacuum permeability
eps0 = 8.8541878188e-12  # [F/m] vacuum permittivity

cff_magnetic_permeability = CffConstantScalar(value=mu0)
cff_electric_permittivity = CffConstantScalar(value=eps0)
cff_electric_conductivity = CffConstantScalar(value=4.7e7)  # [S/m]

condition = SurfaceImpedanceCondition(
    name = "My Surface Impedance Boundary Condition",
    marker = my_marker,
    magnetic_permeability = cff_magnetic_permeability,
    electric_permittivity = cff_electric_permittivity,
    electric_conductivity = cff_electric_conductivity,
)

If the boundary material parameters are constant, you can use the SurfaceImpedanceCondition.Constant constructor with the relative permeability and relative permittivity values:

condition = SurfaceImpedanceCondition.Constant(
    name = "My Surface Impedance Boundary Condition",
    marker = my_marker,
    relative_magnetic_permeability = 1,
    relative_electric_permittivity = 1,
    electric_conductivity = 4.7e7,  # [S/m]
)