329sp09hw12sol - ECE-329 Spring 2009 Homework 12 — Solution 1 Let us consider two transmission lines with characteristic impedances Z 1 and Z 2

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Unformatted text preview: ECE-329 Spring 2009 Homework 12 — Solution April 22, 2009 1. Let us consider two transmission lines with characteristic impedances Z 1 and Z 2 joined at a junction that includes a "shunt" resistance R . ECE 329 Homework 12 Due: Wed, Apr 2 1. Two T.L.’s with characteristic impedances Z 1 and Z 2 are joined at a junction that “shunt” resistance R as shown in the diagram below. R Z 1 Z 2 z v ++ v +- v- Z 1 v ++ /Z 2 v + + v- +- +- Assume that a voltage wave v + ( t- z v p 1 ) from the left produces reflected and tran v- ( t + z v p 1 ) and v ++ ( t- z v p 2 ) on lines 1 and 2 traveling to the left and right, respective of the junction. Using an abbreviated notation, Kirchhoff’s voltage and current laws (K applied at the junction can be expressed as v + + v- = v ++ and v + Z 1- v- Z 1 = v ++ Z eq , where Z eq ≡ RZ 2 R + Z 2 is the parallel combination of R and Z 2 . a) Verify the KVL and KCL equations given above, explaining carefully how they a b) Solve the KVL and KCL equations above to obtain the reflection and transmiss Γ 12 ≡ v- v + and τ 12 = v ++ v + for the junction. c) Calculate Γ 12 ≡ v- v + and τ 12 = v ++ v + for Z 1 = 50 Ω , Z 2 = 100 Ω , and R = 100 Ω . 2. Two T.L.’s with characteristic impedances Z 1 and Z 2 are joined at a junction that series resistance R as shown in the diagram below. R Z 1 Z 2 z v ++ v +- v- Z 1 v ++ /Z 2 v + + v- +- +- Write the pertinent KVL and KCL equations at the junction that relate v + ( t- z v p 1 ) , v ++ ( t- z v p 2 ) as in Problem 1, and then repeat 1b and c for this revised T.L. ckt. 3. A lossless transmission line is terminated with an “short” at z = 0 , and in the region z < a forward propagating monochromatic voltage waveform v + ( z, t ) = Re { 10 e- jβz e jωt } 2 π × 10 8 rad/s is the wave frequency. Also, the propagation speed on the line is v p m/s, wavenumber β = ω v p , and the characteristic impedance of the line is 100 Ω . Taking into account the short termination at z = 0 — where the voltage v must vanis — determine, for z < , a) The complete expression for the voltage waveform, v ( z, t ) = v + ( z, t ) + v- ( z, t select v- ( z, t ) such that it represents a monochromatic wave propagating in- then enforce the v + (0 , t ) + v- (0 , t ) = 0 condition....
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This note was uploaded on 11/16/2009 for the course ECE 329 taught by Professor Franke during the Spring '08 term at University of Illinois at Urbana–Champaign.

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329sp09hw12sol - ECE-329 Spring 2009 Homework 12 — Solution 1 Let us consider two transmission lines with characteristic impedances Z 1 and Z 2

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