Page 374 - Engineering Electromagnetics, 8th Edition

356 ENGINEERING ELECTROMAGNETICS

Figure 10.27 Voltage across the resistor as a function of time, as
determined from the reflection diagram of Figure 10.26, in which
R g = Z 0 ( = 0).

A special case of practical importance is that in which the resistor is matched
to the line, or R g = Z 0 .In this case, Eq. (122) gives V 1 + =−V 0 /2. The line fully
discharges in one round trip of V and produces a voltage across the resistor of value
+
1
V R = V 0 /2, which persists for time T = 2l/ν. The resistor voltage as a function
of time is shown in Figure 10.27. The transmission line in this application is known
as a pulse-forming line; pulses that are generated in this way are well formed and
of low noise, provided the switch is sufficiently fast. Commercial units are available
that are capable of generating high-voltage pulses of widths on the order of a few
nanoseconds, using thyratron-based switches.
When the resistor is not matched to the line, full discharge still occurs, but does
so over several reflections, leading to a complicated pulse shape.

EXAMPLE 10.12
In the charged line of Figure 10.25, the characteristic impedance is Z 0 = 100 , and
R g = 100/3 . The line is charged to an initial voltage, V 0 = 160 V, and the switch is
closed at time t = 0. Determine and plot the voltage and current through the resistor
for time 0 < t < 8l/ν (four round trips).
Solution. With the given values of R g and Z 0 , Eq. (47) gives g =−1/2. Then,
with L = 1, and using (122), we find

V 1 + = V 1 − =−3/4V 0 =−120 V
V 2 + = V 2 − = g V 1 − =+ 60 V
V 3 + = V 3 − = g V 2 − =−30 V
V 4 + = V 4 − = g V 3 − =+15 V
Using these values on the voltage reflection diagram, we evaluate the voltage in time
at the resistor location by moving up the left-hand vertical axis, adding voltages as
we progress, and beginning with V 0 + V at t = 0. Note that when we add voltages
+
1
along the vertical axis, we are encountering the intersection points between incident