Wave Propagation in Linear Media | re-examined

Straightforward integration yields for the pass band
ve;e
c =
and for the stop band
ve;e
c =
1+ 2 , , !p
!
,1 , 2 + , !p
!
2 + , !p
!
3Wave propagation in electromagnetic transmission lines
2 + , !p
!
2
2 1 , , !p
!
2
2 1, 2 , , !p
!
, !p
!
2 , 1
2 ,1+ 2 + , !p
!
2
2
r
sin 2L
!!p
r
2L
!!p
r
sinh 2L 1, !
!p r 2
2L 1, !
!p
2
,1
2
,1
2
(3.53)
: (3.54)
These velocities are depicted in g. 3.9 and g. 3.10, respectively. We see that if the line is
terminated with the characteristic impedance, = p 1 , (!p=!) 2 , the energy velocity in the
pass band is independent ofLand equal to the group velocity. Apart from very short lines,
the velocity is virtually independent of the length.
Remark (Velocity maximum) It is interesting to notice that for short lines, the
maximum velocity of energy propagation does not occur when the termination is optimal,
but for larger values of . For the example in g. 3.9, the value would be m 1:28 in
the limit L ! 0. It is not clear where this behaviour comes from.
A similar e ect appears for evanescence. Although a matched termination is not de ned
in this case, one might expect the velocity maximum to be related to the propagating
mode. In fact, for L ! 0 the value where the optimum is reached in g. 3.10 is m 1:6,
which has nothing p whatsoever to do with any other parameter. For large values of !p=!,
we nd m !p=!.
The result for the stop band ( g. 3.10) seems very plausible. Evanescence slows down the
energy transfer, and the energy velocity gradually diminishes. At any rate, energy transport
through an evanescent region is possible even in the strictly monochromatic case, when we
can guarantee that the spectrum of the signal is con ned to the stop band | unlike signals
with a broader spectral width, where we must always worry about components lying in the
pass band and thus distorting the result. On the other hand, no energy can be transferred if
the line is open ( ! 0) or shorted ( !1), which naturally re ects only our precondition
that the energy is to be dissipated at the termination resistance.
Remark (Formal equivalence to a wave guide) The results obtained for the transmission
line with a lossless plasma are formally identical with the H01 mode of a rectangular
wave guide with width b. The respective boundary conditions are that the excitation
at x = 0 is monomodal and that at x = l,
, Ez(l)
= R; (3.55)
Hy(l)
when x is the direction of propagation. The results from (3.46) onwards then apply with
the replacements !p 7! c=b and p L 0 =C 0 7! p =".
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