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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 Transmissionline +
Vg Z0  ZL
z z=0 z=l (a) Transmissionline circuit
Rg
+
Vg  I1+
+
V1+ Z0 (b) Equivalent circuit at t=0+ Figure 31:233 transmission line immediately after turnon.
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 234
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.
 Fall '12
 MartinSiderious
 Electromagnet

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