Question

An RCP wave in free space with

Eˢᵢ =(^aₓ - j^aᵧ)e⁻ʲᵏᵒᶻ

is normally incident upon a semi-infinite lossless medium with εᵣ and μᵣ.The reflected and transmitted field phasors are given by

Eˢᵣ =(Γₓ^aₓ - jΓᵧ^aᵧ)eʲᵏᵒᶜ²

and

Eˢₜ =(Tₓ^aₓ - jTᵧ^aᵧ) e⁻ʲᵏᶻ

Given that the boundary is z = 0, obtain Γₓ, Γᵧ, Tₓ, and Tᵧ.

Answer 1

To obtain the values of Γₓ, Γᵧ, Tₓ, and Tᵧ, we can use the boundary conditions at z = 0. The reflection coefficients are given by Γₓ = (1 - |Γₓ|) / (1 + |Γₓ|) and Γᵧ = (1 - |Γᵧ|) / (1 + |Γᵧ|). The transmission coefficients are given by Tₓ = (2|Tₓ|) / (1 + |Γₓ|) and Tᵧ = (2|Tᵧ|) / (1 + |Γᵧ|).

Step-by-step explanation:

To obtain the values of Γₓ, Γᵧ, Tₓ, and Tᵧ, we can use the boundary conditions at z = 0. At the boundary, the tangential components of the electric and magnetic fields must be continuous. Using these conditions, we can set up a system of equations and solve for the unknowns. Let's denote k₀ = √(ε₀μ₀ω² - γ²) and k = √(εᵣμᵣω² - γ²), where ω is the angular frequency and γ is the attenuation constant. Then the reflection coefficients can be found using the equations:

Γₓ = (Eˢᵢₓ - Eˢᵣₓ) / (Eˢᵢₓ + Eˢᵣₓ) = (aₓ - Γₓaₓ) / (aₓ + Γₓaₓ) = (1 - |Γₓ|) / (1 + |Γₓ|)

Γᵧ = (Eˢᵢᵧ - Eˢᵣᵧ) / (Eˢᵢᵧ + Eˢᵣᵧ) = (aᵧ - Γᵧaᵧ) / (aᵧ + Γᵧaᵧ) = (1 - |Γᵧ|) / (1 + |Γᵧ|)

Similarly, the transmission coefficients can be found using the equations:

Tₓ = (Eˢₜₓ) / (Eˢᵢₓ + Eˢᵣₓ) = (Tₓaₓ) / (aₓ + Γₓaₓ) = (2|Tₓ|) / (1 + |Γₓ|)

Tᵧ = (Eˢₜᵧ) / (Eˢᵢᵧ + Eˢᵣᵧ) = (Tᵧaᵧ) / (aᵧ + Γᵧaᵧ) = (2|Tᵧ|) / (1 + |Γᵧ|)