Wave Propagation in Linear Media | re-examined

I 0
3Wave propagation in electromagnetic transmission lines
Figure 3.4: Transmission line with real-valued termination
With these results we can calculate the averaged propagated and stored energies
P =Re UI
2
; W = UU
4
dB 0
d!
l
+ II
4
which after some simple manipulations yields the velocity of energy transport
ve =
x
dX 0
d!
R
; (3.28)
2k
, 2 sin k(l , x)+ 2cos2 k(l , x) B0 dX0 d! + , cos2 k(l , x)+ 2sin2 k(l , x) X0 dB0 d!
(3.29)
with the termination factor = Z0=R. This complicated expression is reduced dramatically
if we evaluate it for the special case of a dispersionless line with
Then (3.29) becomes, with 1=c = p L 0 C 0 ,
B 0 = !C 0 ; X 0 = !L 0 : (3.30)
ve
c
= 2 Z0
R
1+ Z0
R
If we introduce the re ection factor of the current
r =
we can rewrite (3.31) in a more plausible way,
ve
c
Z0
R
Z0
R
, 1
: (3.31)
+1 ; (3.32)
1 , r2
= : (3.33)
1+r2 This invokes the picture of an overall transport velocity weighted by the power the waves
carry: the incident wave transports a certain amount of energy per unit time, P0, whereas the
re ected wave carries a part r 2 P0 of it back. Apart from this interpretation, (3.33) mirrors
exactly the basic idea of the energy velocity de nition: a net propagated energy per unit
length divided by the total energy content of that unit length. We notice that the energy
velocity isnot equal to the group velocity vg = c. In the limit of a matched termination r =0,
however, we haveve=vg. Expression (3.33) also holds for an unterminated or short-circuited
transmission line R = 1 or R = 0, thus r 2 = 1, where ve = 0 indicates that an energy
transport is impossible at all.
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