Input intrinsic impedance via reflection

Input intrinsic impedance η_in(−l) is the ratio of the total electric field to the total magnetic field (E_x/H_y) at a distance l in front of the boundary, when there are both an incident and a reflected wave. It is the wave impedance "seen" looking toward the boundary.

Set-up

Region 1 (z < 0): lossless medium with η₁, β₁. Region 2 (z > 0): impedance η₂. The boundary is at z = 0. The reflection coefficient at the boundary is

Γ = (η₂ − η₁)/(η₂ + η₁)

Total fields in region 1 (E along a_x, H along a_y):

E_s(z) = E₀ ( e^{−jβ₁z} + Γ e^{+jβ₁z} )
H_s(z) = (E₀/η₁) ( e^{−jβ₁z} − Γ e^{+jβ₁z} )

Derivation

The wave impedance at any z in region 1:

η(z) = E_s/H_s = η₁ (e^{−jβ₁z} + Γ e^{jβ₁z}) / (e^{−jβ₁z} − Γ e^{jβ₁z})
     = η₁ (1 + Γ e^{j2β₁z}) / (1 − Γ e^{j2β₁z})

At distance l from the boundary, z = −l:

η_in = η₁ (1 + Γ e^{−j2β₁l}) / (1 − Γ e^{−j2β₁l})

Substitute Γ = (η₂ − η₁)/(η₂ + η₁) and multiply numerator and denominator by (η₂ + η₁):

numerator:   (η₂ + η₁) + (η₂ − η₁) e^{−j2β₁l}
denominator: (η₂ + η₁) − (η₂ − η₁) e^{−j2β₁l}

Multiply both by e^{jβ₁l} and use e^{±jβ₁l} = cos β₁l ± j sin β₁l:

numerator:   2η₂ cos β₁l + j2η₁ sin β₁l
denominator: 2η₁ cos β₁l + j2η₂ sin β₁l

Dividing both by 2 cos β₁l:

η_in = η₁ · (η₂ + jη₁ tan β₁l) / (η₁ + jη₂ tan β₁l)

Checks

• l = 0: η_in = η₂ (the load impedance).
• l = λ₁/2 (β₁l = π, tan = 0): η_in = η₂, so a half-wave layer repeats the load impedance.
• l = λ₁/4 (tan → ∞): η_in = η₁²/η₂, the quarter-wave transformer relation used for anti-reflection coatings (choose η₁ = √(η_a η₂) for matching).
• Perfect conductor (η₂ = 0): η_in = jη₁ tan β₁l, purely reactive.

This has exactly the same form as the input impedance of a transmission line, with η playing the role of Z₀.