Surface Impedance Boundary Condition¶

A surface impedance boundary condition is applied at the boundaries of well-conducting materials, whose conductivity, although high, cannot be considered infinite. In such cases, it is assumed that the electromagnetic field penetrates the boundary material to a non-negligible skin depth. It is important to note that the surface impedance boundary condition remains valid only if this skin depth is small relative to the dimensions of the conductor to which the boundary condition is applied. 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, the user must specify a custom name for the condition and provide the boundary Marker. Additionally, the user must 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, the SurfaceImpedanceCondition.Constant constructor with the relative permeability and relative permittivity values can be utilized:

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]
)