Impedance Calculation for Layered Impedance Boundary

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 k^th layer:

• d_k is the layer thickness.
• Z_wk is the wave impedance:

• γ_k is the propagation constant:

• k₀=ω√_{μ0 ϵ0} is the free space wave number, where ω=2πf is the angular frequency.
• ϵ_rk the complex relative permittivity.
• μ_rk is the complex relative permeability.

ϵ_rk and μ_rk 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 k^th layer:

• σ_k is the bulk electrical conductivity.
• tanδ_ek is the electric loss tangent.
• tanδ_mk is the magnetic loss tangent.