Wave Propagation in Linear Media | re-examined

0
100000
2
10000
X
4
1000
6
r
8
100
3Wave propagation in electromagnetic transmission lines
10
1
0.2
0
0.4
ve/c
Figure 3.12: Evaluation of the energy velocity (3.66) in the stop band of an exponential line depending
on the spatial coordinate X and the termination factor r. The length of the line is L = 8, the signal
frequency is given by =0:5. Note the partially logarithmic scale.
The energy velocity in the pass band is depicted in g. 3.11. Its dependence on the position
along the line and the termination is very similar to that in a lossless plasma ( g. 3.7). Except
for matched termination, = 1, spatial harmonics appear in the local velocity. The situation
in the stop band, however, is a bit di erent, as can be seen by comparing g. 3.12 and g. 3.8.
While the local velocity still increases towards the end of the line, its maximum | at least in
a broad range of termination factors | is no longer located directly at x = l but signi cantly
ahead of this point. Note that in this case, r = 1 has nothing at all to do with matched
termination. Like in the previous example, we could eliminate the position dependence by
introducing the e ective velocity (3.51). However, this provides little additional insight and
has therefore been omitted.
3.7 Turn-on e ects in a lossless plasma
In the examples investigated so far we restricted ourselves predominantly to monochromatic
waves, which is an instructive special case and easy to analyse. On the other hand, we usually
apply waves to transmit signals, and as a strictly monochromatic wave cannot carry any
information, this case has no practical relevance. What we need instead, is a modulation of the
wave to form pulses we can identify with the information. A simple way to do so is to switch
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