Wave Propagation in Linear Media | re-examined

3.6 Inhomogeneous transmission line
0
1
0.75 75
ve/c
0.5 .5
0.25 25
0
100
2
X
4
10
6
8
1
ρ
0.1
0.01
Figure 3.11: Evaluation of the energy velocity (3.65) in the pass band of an exponential line depending
on the spatial coordinate X and the termination factor . The length of the line is L = 8, the signal
frequency is given by 1= =0:5. Note the partially logarithmic scale.
we nd for the special case of a matched termination the energy velocity
ve
c =
p
2 , 1
; (3.64)
which is exactly the group velocity we would have obtained from the dispersion relation.
If we can manage to terminate the transmission line with a real-valued multiple of the
characteristic impedance Z0, the energy velocity within a line of length l in the pass band,
2 > 1, becomes after a lengthy calculation
q
( 2 3
, 1)
ve
c =
2
2 (1 + 2 ) , 2 , ( , 1) 2 cos e X , ( 2 , 1) sin e X ; (3.65)
with the abbreviations = p 2 , 1 and e X = X , L. In the stop band, where 2 < 1,
a termination with the wave impedance makes no sense because no energy could then be
transmitted due to evanescence, so we assume a purely ohmic termination R. With the
settings = p 1 , 2 and r = R= p L0 0 =C0 0 ,we obtain the corresponding energy velocity
ve
c =
r , 1 , 2
(cosh e X + sinh e X , 2 )e L + r 2 (cosh e X , sinh e X , 2 )e ,L
47
: (3.66)