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329sp09hw14sol(2)

329sp09hw14sol(2) - ECE-329 Spring 2009 Homework 14...

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ECE-329 Spring 2009 Homework 14 — Solution May 8, 2009 1. Let us analyze the following T.L. circuit. Γ ( l ) Γ b Γ a Γ (0) a) Using the Smith chart and working in terms of impedances. Reflection coefficient at the load Γ L = - 1 Reflection coefficient A at shunt connection Γ a = - 1 e - j 4 π · 0 . 1 = 1 e j 0 . 6 π Normalized input impedance A shunt connection z a = 1+Γ a 1 - Γ a = j cot ( 3 π 10 )= j 0 . 727 Normalized input impedance B shunt connection z b = 1 50 50 + 1 za = 0 . 345 + j 0 . 476 Reflection coefficient B at shunt connection Γ b = z b - 1 z b +1 = 0 . 567 e j 2 . 174 Reflection coefficient at the generator Γ( l ) = Γ b e - j 4 π · 0 . 1 = 0 . 567 e j 0 . 917 Normalized impedance at the generator z ( l ) = 1+Γ( l ) 1 - Γ( l ) = 1 . 074 + j 1 . 425 Impedance at the generator Z ( l ) = 50 · z ( l ) = 53 . 71 + j 71 . 23 Ω 0.2 0.5 1.0 2.0 5.0 +j0.2 -j0.2 +j0.5 -j0.5 +j1.0 -j1.0 +j2.0 -j2.0 +j5.0 -j5.0 0.0 z (0) z a z b z ( l ) 0 . 1 λ 0 . 1 λ 1

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ECE-329 Spring 2009 b) Using the Smith chart and working in terms of impedances. Reflection coefficient at the load Γ L = - 1 Reflection coefficient A at shunt connection Γ a = - 1 e - j 4 π · 0 . 1 = 1 e j 0 . 6 π Normalized input admittance A at shunt connection y a = 1 - Γ a 1+Γ a = - j tan ( 3 π 10 )= - j 1 . 376 Normalized input admittance B at shunt connection y b = 1 + y a = 1 - j tan( 3 π 10 ) = 1 . 0 - j 1 . 376 Reflection coefficient B at shunt connection Γ b = 1 - y b 1+ y b = 0 . 567 e j 2 . 174 Reflection coefficient at the generator Γ( l ) = Γ b e - j 4 π · 0 . 1 = 0 . 567 e j 0 . 917 Normalized admittance at the generator y ( l ) = 1 - Γ( l ) 1+Γ( l ) = 0 . 337 - j 0 . 448 Normalized impedance at the generator z ( l ) = 1 y ( l ) = 1 . 074 + j 1 . 425 Impedance at the generator Z ( l ) = 50 · z ( l ) = 53 . 71 + j 71 . 23 Ω 0.2 0.5 1.0 2.0 5.0 +j0.2 -j0.2 +j0.5 -j0.5 +j1.0 -j1.0 +j2.0 -j2.0 +j5.0 -j5.0 0.0 z (0) y (0) y a y b y ( l ) z ( l ) 0 . 1 λ 0 . 1 λ 2. The voltage in a T.L. ( Z o = 50 Ω ) has a standing wave pattern with V max = 8 . 4 V and V min = 2 . 1 V .
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