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Unformatted text preview: ECE 3030: Electromagnetic Fields and Waves Fall 2009 DEMO 8 INSTRUCTOR NOTES: Transmission Lines Reminder: Prelim 2 is next Tuesday night. Date: Tuesday 10/20 1. Since Prelim 2 is next Tuesday, there is no Homework 8 nor Demo 9. 2. Problem 13.27 Sliding-probe measurements made on a Z = 50 Ω transmission line indicate a voltage standing-wave ratio (VSWR) of 4, with a voltage minimum at a distance of λ/ 5 from the load. a. What is the magnitude of the reflection coefficient | Γ | ? b. What is the complex value of Γ at a point at which the voltage is minimum? c. Show that the impedance on the line ( V Total /I Total ) at a point 3 λ/ 8 toward the gen- erator from the voltage minimum is 23 . 5- j 44 . 1 Ω. d. At what distance from the voltage minimum should one insert a series-reactive ele- ment in order to match the line? e. What value of reactance would be required? f. If, instead of the reactive element of part (c), a shorted stub of the same transmis- sion line is to be connected in parallel across the line for matching, where should it be connected and how long should it be? Solution: I assumed a value of 5 volts for V + to make the plot shown in Figure 1 using the | Γ | found in part (a) below and the phase found in part (b). (The plot is really not necessary to do this problem, but it helps in understanding it.) a. The magnitude of the reflection coefficient is easily found from the SWR: | Γ | = SWR- 1 SWR + 1 = 4- 1 4 + 1 = 0 . 6-2.5-2-1.5-1-0.5 1 2 3 4 5 6 7 8 9 10 z [Fractional Wavelength] Maximum Voltage [V] Figure 1: Standing wave pattern with V + = 5 V (an arbitrary value). b. The minimum in the standard wave pattern occurs where the complex value of the reflection coefficient is Γ( z =- λ/ 5) =-| Γ | =- . 6 (a real number). Exploring this a bit further, since Γ L = | Γ L | e jφ , we can write Γ( z =- λ/ 5) = Γ L e j 2 kz fl fl z =- λ/ 5 = | Γ L | e j ( φ- 2 kλ/ 5) = | Γ L | e j ( φ- 4 π/ 5) =-| Γ L | Then the phase factor must be such that e j ( φ- 4 π/ 5) =- 1 = ⇒ φ = 4 π/ 5 ± π = 1 . 8 π or- . 2 π CORNELL UNIVERSITY c WES SWARTZ (09/9/16) 8–1 ECE 3030: Electromagnetic Fields and Waves Fall 2009 DEMO 8 INSTRUCTOR NOTES: Transmission Lines Now we can write the reflection coefficient at the load as Γ(0) = Γ L = 0 . 6e jφ = 0 . 6e- j . 2 π = 0 . 485- j . 353 from which can determine the load impedance: Z L = Z 1 + Γ 1- Γ = 82 . 22- j 90 . 62 Ω c. We can use the above technique to find the impedance 3...
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- Electromagnet, Transmission line, Impedance matching, Smith chart, J0, WES SWARTZ