Wllsinus_solutions - Solutions Set 4 Lossless Lines...

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ECE 303 - Sum2014 Solutions - Set 4: Lossless Lines - Sinusoids 1 4-3. Let a transmission line be used to connect a source and a load. The source produces a forward wave with amplitude A = 2 and the load produces a reflection coefficient of Γ L = 0. The source is at x = - 20 m, the load is at x = 0 m, the wavenumber β = 1 rad/m, and the operating frequency is f = 10 7 Hz. Do the following: (a) Compute ˜ v ( x,t ), the total phasor voltage on the transmission line. (b) Using the result in (a), compute ˜ v ( x,t ) for the three time values given by: (i) t = 0, (ii) t = 20 nsec, (iii) t = 40 nsec. (c) Compute the real-valued voltage waveform v ( x,t ) for the three time values given in (b). (d) One one graph, plot all three real-valued voltage waveforms v ( x,t ) for the three time values given in (b). Plot these over the range - 20 < x < 0. DETAILED SOLUTION: (a) Compute ˜ v ( x,t ) , the total phasor voltage on the transmission line. From eq. (15) in class Notes Set 4, the complex-valued total voltage wave can be written as ˜ v ( x,t ) = A bracketleftBig e j ( ω t - β x ) + Γ L e j ( ω t + β x ) bracketrightBig ( a ) From the problem statement we have A = 2, Γ L = 0, β = 1 and ω = 2 πf = 2 π 10 7 . Substituting these values into equation ( a ) then gives the desired solution to part (a): ˜ v = 2 e j (2 π 10 7 t - x ) ( b ) (b) Using the result from part (a), compute ˜ v ( x,t ) for t = 0 , t = 20 nsec , t = 40 nsec . (i) t = 0 : Substitute t = 0 into equation ( b ) to obtain the desired result : ˜ v = 2 e j (2 π 10 7 (0) - x ) = 2 e - j x ( c ) (ii) t = 20 nsec: Substitute t = 20(10) - 9 into ( b ) to obtain the desired result : ˜ v = 2 e j (2 π 10 7 × 20 × 10 - 9 - x ) = 2 e j (0 . 4 π - x ) ( d )
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ECE 303 - Sum2014 Solutions - Set 4: Lossless Lines - Sinusoids 2 Prob. 4-3 (cont.) (iii) t = 40 nsec:
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