Page 368 - Engineering Electromagnetics, 8th Edition

350 ENGINEERING ELECTROMAGNETICS

The voltage as a function of time at a given position in the line can now be
determined by adding the voltages in the waves as they intersect a vertical line drawn
at the desired location. This addition is performed starting at the bottom of the diagram
(t = 0) and progressing upward (in time). Whenever a voltage wave crosses the
vertical line, its value is added to the total at that time. For example, the voltage
at a location three-fourths the distance from the battery to the load is plotted in
Figure 10.21b.To obtain this plot, the line z = (3/4)l is drawn on the diagram.
Whenever a wave crosses this line, the voltage in the wave is added to the voltage that
has accumulated at z = (3/4)l over all earlier times. This general procedure enables
one to easily determine the voltage at any specific time and location. In doing so, the
terms in (117) that have occurred up to the chosen time are being added, but with
information on the time at which each term appears.
Line current can be found in a similar way through a current reflection diagram.
It is easiest to construct the current diagram directly from the voltage diagram by
determining a value for current that is associated with each voltage wave. In dealing
with current, it is important to keep track of the sign of the current because it relates to
the voltage waves and their polarities. Referring to Figures 10.19a and 10.20, we use
the convention in which current associated with a forward-z traveling voltage wave
of positive polarity is positive. This would result in current that flows in the clock-
wise direction, as shown in Figure 10.19a. Current associated with a backward-z
travelingvoltagewaveofpositivepolarity(thusflowingcounterclockwise)isnegative.
Such a case is illustrated in Figure 10.20. In our two-dimensional transmission-line
drawings, we assign positive polarity to voltage waves propagating in either direction
if the upper conductor carries a positive charge and the lower conductor a negative
charge. In Figures 10.19a and 10.20, both voltage waves are of positive polarity, so
their associated currents will be net positive for the forward wave and net negative
for the backward wave. In general, we write

V +
I + = (120)
Z 0
and

V −
I − =− (121)
Z 0
Finding the current associated with a backward-propagating voltage wave immedi-
ately requires a minus sign, as (121) indicates.
Figure 10.22a shows the current reflection diagram that is derived from the
voltage diagram of Figure 10.21a. Note that the current values are labeled in terms of
the voltage values, with the appropriate sign added as per (120) and (121). Once the
current diagram is constructed, current at a given location and time can be found in
exactly the same manner as voltage is found using the voltage diagram. Figure 10.22b
shows the current as a function of time at the z = (3/4)l position, determined by
summing the current wave values as they cross the vertical line drawn at that location.