Question

Problem 1 (12.5 marks) A uniform plane wave is normally incident as shown in the figure below. Assume that all regions are lossless. Prove that the input intrinsic impedance can be given by 13 cos(B21) + 1 n2 sin(21) Min = 12 n2 cos(B21) + j n, sin(B,1) 92 73 incident wave -1 nin 0 ho

Answer 1

Wave reflection from multiple interfaces: ng nin > Z 0 het ux assume that all regions are composed of lossless media and consider the two waves in region 2. take these as x-polarized, the ir electric fieldo add to of we Yield -jß z j B Z + + Ex523 ES I 20 αης Exzo, where B - wJesz/c, and where the amplitudes, Exio are complex. The yo polarized magnetic field is similarly written, using complex amplitudes; -j B2 z jBjZ Hy но е + Hyzo Tys2 We now note that the forward and backward electric field amplitude in region 2 are related through the reflection at the second interface, as where n -n₂ Tas 1, +1, Scanned with CamScanner

We thus have + Exz 123 Exco 220 We then wribe the mone bic field amplitudes inter my of electoric field amplitudes through + 1 Exzo s Hyzo ? 2 T and. Exo Hyzo Ex 7, ☆ [23 - jß ₂ z + + (Z) = We now define the wave impedence, nu as the z. dependent ratio of the total electric field to the total magnetic field Using Equations 0 + 9 jP, Z Exizo e Exzo e Exs2 v Bez th + Hy go şubstitute B, and in above Equation, -ja jß₂ Z 123 n (z) = jP2 z no jF, z Hijzo e + e کی z W Now Using sins (3) and Eulere identity lesz - cospz + jainpy] Leipz - Cosſz - j$inpz) (Me +ng) (Cospiz -j $ing, z) +(1,-1,) (cosmZ tj şin 2, 7) .: 7.(2) = ne (n+ n) (COSPZ - SMR2) (nn) (CSP,7 + j$in (2) Cscanned with CamScanner

This is easily Simplified to yield, nw (2) - کی I cosp, z - jy, sinpa z n cospz - jy sin ß, z. To find input impedence substitute 7--1 na co sp, I tj % Singal 2 nin 12 ", cosp₂l til, sinf I cscanned with CamScanner