EMSP 27jun2022 with solutions

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Electromagnetics and Signal Processing for Spaceborne Applications June 27th, 2022 1 2 3 4 do not write above Problem 1 A plane sinusoidal EM wave (f = 9 GHz) propagates from a medium with electric permittivity r1 = 3 into free space (assume r = 1 for both media). The expression of the incident electric field is: 𝐸⃗ 𝑖(𝑧,𝑦)= 𝐸0 (√3 2 𝜇 𝑦 + 1 2𝜇 𝑧 + 𝑗 2𝜇 𝑥)𝑒−𝑗𝛽√3 2 𝑧𝑒𝑗𝛽 2𝑦 V/m 1) 1 Determine the incidence angle . 2) 2 Determine the polarization of the incident field 𝐸⃗ 𝑖 (it is sufficient to state whether the polarization is elliptical, circular or linear; additional details, such as rotation direction or tilt angle, are not required). 3) 3 Determine the polarization of the reflected field 𝐸⃗ 𝑟. 4) 4 Calculate the absolute value of E0 by knowing that the power density of the electric field in A(x = -1 m, y = -1 m, z = -1 m) is ST = 0.5 W/m 2 (consider only the contribution of the reflected field). Solution 1) The incidence angle can be derived, for example, from the y component of : 𝛽𝑦 = 𝛽sin(𝜃)= 𝛽/2  sin(𝜃)= 1/2  𝜃 = 30° SURNAME AND NAME ____________________________________________________ ID NUMBER ___________________________________________________________ SIGNATURE ___________________________________________________________ y z 2 1  A 2) The wave has two components: a TE one, along x, and a TM one, given by the combinations of the two fields along y and z. The absolute value of the components is E0 V/m (TM) and E0/2 V/m (TE), and the differential phase shift is /2 3) Considering that the wave also has a TM component, it is worth checking the Brewster angle: 𝜃𝐵 = tg−1 (√ 𝜀𝑟2 𝜀𝑟1 ) = 30° As  = B  the TM component is totally transmitted into the second medium. As a result, the polarization of the reflected wave will be linear (along x). 4) To determine E0, it…