What expression is HFSS using to compute the frequency dependent impedance with the layered impedance boundary?

A layered impedance boundary is used to model multiple thin layers in a structure as one impedance surface.

The impedance of the layered structure is calculated recursively. First the impedance of the lowest layer is calculated, and then the impedance of the layer above is computed from transmission line theory as follows:

The process is completed when the uppermost layer is reached.
Here, for the kth layer:

• dk is the layer thickness.
• Zwk is the wave impedance:

• γk is the propagation constant:

• k0=ω√μ0 ϵ0 is the free space wave number, where ω=2πf is the angular frequency.
• ϵrk the complex relative permittivity.
• μrk is the complex relative permeability.

ϵrk and μrk are given by the following formulas:

The real parts ϵ'rk and μ'rk are entered directly. The imaginary parts of the complex relative permeability and permittivity may be entered directly, or derived from the conductivity and loss tangents of the materials as follows:

Here, for the kth layer:

• σk is the bulk electrical conductivity.
• tanδek is the electric loss tangent.
• tanδmk is the magnetic loss tangent.