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HW3Solution - ECE 201A Homework 3 solution 1 RegionI 0 0 i...

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ECE 201A Homework 3 solution 1. We can describe the problem with a transmission line equivalent circuit, which is The propagation constants of the equivalent transmission lines, 0 β and 1 β , are the components of the k vector in z direction, i.e., 0 0 0 2 cos z i k π β θ λ = = and 1 1 0 2 cos z t k n π β θ λ = = The characteristics impedances of the equivalent transmission lines, 0 Z and 1 Z , are the wave impedances in z direction, i.e., 0 0 0 cos z i Z Z η θ = = and 0 1 1 0 z Z Z n η λ = = since the wave is TE polarized. Since regions I and III are infinite in extend so are the equivalent transmission lines. Since we see the characteristic impedance looking into an infinitely long transmission line, the equivalent circuit becomes i θ t θ i H G i E G 0 ε 0 μ RegionI 1 ε 0 μ RegionII 0 ε 0 μ RegionIII d z x 1 0 r ε ε ε = r n ε = . . . 0 Z 0 Z 1 Z 0 β 0 β 1 β d
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To find the power reflected back to region I, let’s find the reflection coefficient Γ at the first junction. 0 0 in in Z Z Z Z Γ = + , where 0 0 1 1 1 0 1 0 1 tan tan in in in Z Z Z jZ d Z Z Z Z Z jZ d β β + Γ = = + + . Then ( ) ( ) 2 2 2 2 1 0 1 0 1 1 1 0 1 0 1 2 2 2 2 0 1 1 1 0 1 0 1 0 1 1 0 1 tan tan tan tan tan 2 tan j Z Z d Z Z jZ d Z Z jZ d Z Z jZ d Z Z jZ d Z Z j Z Z d β β β β β β + Γ = = + + + + + Power reflection coefficient is 2 Γ , so ( ) ( ) 2 2 2 2 2 1 0 1 2 2 2 2 2 2 0 1 1 0 1 tan 4 tan Z Z d Z Z Z Z d β β Γ = + + For total transmission into region III 2 0 Γ = . Because if there is no reflected power to region I, all the power must be transmitted to region III, since all the media are lossless. This requires either 0 1 Z Z = , which is the trivial solution or 1 d m β π = , which makes 1 tan 0 d β = . 1 0 2 cos t n d d m π β θ π λ = = , which yields 0 2 cos t m d n λ θ = . t θ is found using the Snell’s law, hence 1 sin sin sin sin i t t i n n θ θ θ θ = = 2 2 2 2 2 sin sin cos 1 sin 1 i i t t n n n θ θ θ θ = = = Therefore 0 0 2 2 2 2 sin 2 sin 2 i i m m d n n n n λ λ θ θ = = . . 0 Z 0 Z 1 Z 0 β 1 β d in Z
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