Chapter2_Notes

Chapter2_Notes

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Unformatted text preview: switches. A typical setup is presented in Fig. 31: DC source, switch, transmission line and load. Here, ZL = R ⇒ all impedances are real. • Close the switch at t = 0. What’s the impedance the source “sees?” • The voltage V1 is now an initial condition, and is given in Fig. 31. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines Rg t=0 116 Transmission-line + Vg Z0 - ZL z z=0 z=l (a) Transmission-line circuit Rg + Vg - I1+ + V1+ Z0 (b) Equivalent circuit at t=0+ Figure 31:2-33 transmission line immediately after turn-on. Figure The Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 117 • How do we get the current? + I1 = Vg Rg + Z0 (209) V1+ = V g Z0 Rg + Z0 (210) + • I1 , V1+ are waves that start traveling down the transmission line. At what velocity? Note: + sign indicates travel in the positive z direction. Note also that we have switched from the previous convention of z = 0 at the load to here, z = 0 at the generator (more convenient this way). • A “snapshot” of the voltage and current along transmission line, at three different times is shown in Fig. 32. Note: Rg = 4Z0 and ZL = 2Z0 . • At first V1+ traverses transmission line, then it “hits” the load at t = T (T = l/up ). What happens? If ZL = Z0 we have a Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 118 V(z, 3T/2) V(z, T/2) V(z, 5T/2) + - + (V1 +V1 ) + (V1 ) V + V1 V1 - + (V1 +V1 +V2 ) + (V1 ) V + - (V1 +V1 ) V + V1 - + + V2 = Γg V1 - + V1 = ΓL V1 z 0 l/2 z l 0 (a) V(z) at t = T/2 z l 0 (b) V(z) at t = 3T/2 I(z, T/2) + (I1 ) + + I1 I + l/2 l (c) V(z) at t = 5T/2 I(z, 3T/2) (I1 ) I1 I l/2 I(z, 5T/2) I1 + (I1 +I1 ) = -ΓL I1 + + - + I2 = -Γg I1 z 0 l/2 (d) I(z) at t = T/2 l + - (I1 +I1 ) + (I1 +I1 +I2 ) + I1 I - z 0 l/2 (e) I(z) at t = 3T/2 l z 0 l/2 l (f) I(z) at t = 5T/2 Figure 32: Time evolution of voltage and current on transmission line. Figure 2-34 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 119 reflection. V1− = ΓL V1+ and ΓL = ZL − Z0 1 = ZL + Z0 3 (211) • Once we hit the load, we get the negative z traveling wave and we sum the forward and backward traveling waves. V1− traverses the transmission line from the load to the generator (source) and when it is half way there — take another snapshot. • What happens once V1− reaches the generator end (at what time?)? If Zg = Z0 ⇒ another reflection! V2+ = Γg V1− = Γg ΓL V1+ , where Γg = Rg − Z 0 = 0.6 (212) Rg + Z 0 • V2+ starts its travel from the source to the load and the total voltage is a sum of three components. Take snapshot half way Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 120 (t = 5T...
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This note was uploaded on 09/25/2013 for the course ECE 331 taught by Professor Martinsiderious during the Fall '12 term at Portland State.

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