Wave Propagation in Linear Media | re-examined

whereas the energy velocity in the evanescent region is
ve
c =
,1 , 2 + , !p
!
2 + , !p
!
If we chose a matched termination
then (3.46) would reduce to
3Wave propagation in electromagnetic transmission lines
2
, !p
!
2 ,1+ 2 + , !p
!
R = Z0 =
ve
c =
r L 0
s
C 0
2 , 1
2 cosh
1
q 1 , , !p
!
1 , !p
!
2
2
2(X , L)
r
1 , !
!p
! : (3.47)
2
; (3.48)
; (3.49)
which is exactly the group velocity vg = d!=dk referred to the speed of light. Numerical
examples of the energy velocity in dependence on the position and the termination ratio are
given in g. 3.7 for the pass band as well as in g. 3.8 for the evanescent region. It is worth
noticing that (3.46) and (3.47) are continuous at the plasma frequency. They do not exhibit
a zero, but the nite value
ve(!p)
c
= 2
2+
2 : (3.50)
The gures clearly show that the energy velocity depends on the position along the transmission
line. As could have been expected, the local velocity in the pass band oscillates
uniformly. In the stop band, however, the velocity nearly vanishes almost everywhere, but
increases sharply towards the end of the line within a characteristic distance that is practically
independent of the signal frequency. This behaviour stems from the fact that the stored
energy per unit length in the transmission line decays exponentially, whereas the transmitted
energy remains constant due to the absence of losses. Consequently, the energy velocity must
rise as the wave approaches the end of the line.
From a macroscopic point of view, the local energy velocity is of only minor importance |
an overall energy propagation time through the transmission line is much more interesting.
This can be de ned via
e =
Z l
and we obtain an `e ective' velocity like in (3.36),
0
1
ve
ve;e = l
42
e
dx ; (3.51)
: (3.52)