Page 365 - Engineering Electromagnetics, 8th Edition

CHAPTER 10 Transmission Lines 347

and then does not reflect, as the load is matched. The transient phase is thus over, and
the load voltage is equal to the battery voltage. A plot of load voltage as a function
of time is shown in Figure 10.19b, indicating the propagation delay of t = l/ν.
Associated with the voltage wave V + is a current wave whose leading edge is
of value I . This wave is a propagating step function as well, whose value at all
+
points to the left of V + is I + = V /Z 0 ;at all points to the right, current is zero. A
+
plot of current through the load as a function of time will thus be identical in form
to the voltage plot of Figure 10.19b,except that the load current at t = l/ν will be
I L = V /Z 0 = V 0 /R L .
+
We next consider a more general case, in which the load of Figure 10.19a is again
a resistor but is not matched to the line (R L = Z 0 ). Reflections will thus occur at the
load, complicating the problem. At t = 0, the switch is closed as before and a voltage
wave, V 1 + = V 0 , propagates to the right. Upon reaching the load, however, the wave
will now reflect, producing a back-propagating wave, V . The relation between V 1 −
−
1
and V is through the reflection coefficient at the load:
+
1
V 1 − R L − Z 0 (115)
V 1 + = L = R L + Z 0

As V 1 − propagates back toward the battery, it leaves behind its leading edge a total
voltage of V 1 + + V .Voltage V 1 + exists everywhere ahead of the V 1 − wave until it
−
1
reaches the battery, whereupon the entire line now is charged to voltage V + V .
+
−
1
1
At the battery, the V 1 − wave reflects to produce a new forward wave, V . The ratio
+
2
of V and V is found through the reflection coefficient at the battery:
−
+
2
1
V 2 + Z g − Z 0 0 − Z 0 =−1 (116)
V 1 − = g = Z g + Z 0 = 0 + Z 0
where the impedance at the generator end, Z g ,is that of the battery, or zero.
V 2 + (equal to −V )now propagates to the load, where it reflects to produce
−
1
backward wave V 2 − = L V . This wave then returns to the battery, where it reflects
+
2
with g =−1, and the process repeats. Note that with each round trip the wave
voltage is reduced in magnitude because | L | < 1. Because of this the propagating
wave voltages will eventually approach zero, and steady state is reached.
The voltage across the load resistor can be found at any given time by summing
the voltage waves that have reached the load and have reflected from it up to that time.
After many round trips, the load voltage will be, in general,
+ − + − + −
V L = V + V + V + V + V + V + ···
3
1
2
1
2
3
3
2
2
2
2
= V 1 + ! 1 + L + g L + g + + +· · · "
L
g
g
L
L
With a simple factoring operation, the preceding equation becomes
+ ! 2 2 " (117)
V L = V (1 + L ) 1 + g L + +· · ·
1
L
g