Wave Propagation in Linear Media | re-examined

3.4 A simple thought experiment
I0
Z0
Figure 3.5: Transmission line charged with a short input current pulse.
3.4 A simple thought experiment
To judge whether (3.31) is indeed meaningful, we explore yet another example. Consider
the terminated transmission line like before, but with a pulse source rather than with a
monochromatic excitation ( g. 3.5). Let the source emit a current pulse with a duration
T l=c much shorter than the propagation time needed to reach the end of the line. Then
a total energy
W1 = I0 2 TZ0
l
R
(3.34)
will be transmitted to the load after a propagation delay = l=c. It will, however, only partly
be absorbed due to partial re ection. The re ected part will be entirely re ected in turn by
the ideal in nite internal impedance of the source. Therefore, the amount of energy that is
by and by absorbed by the load can be expressed by a series of contributions
W (t) =I0 2 T(1 + r) 2 R , (t , )+r 2 (t,3 )+r 4 (t,5 )+::: (3.35)
with the step function (t). The current (1 + r)I0 is the superposition of the incident and
re ected wave seen by the load resistance. For t !1,the in nite series in (3.35) of course
gives (3.34). The de nition of an energy velocity in this case is more or less arbitrary, but
may in a sensible way be given by
ve = l
e
; W ( e)
W1
1 , 1
; (3.36)
e
where e is the instance when a given portion of the total energy has been absorbed, the
remainder being determined by the base e of the natural logarithm. Thus the time-dependent
function of (3.35),
f(t) =
1X
i=0
(t , (2i +1) )r 2i ; (3.37)
is evaluated only to the index i = N where this condition is rst met. Since the summation
starts with index i = 0, we have at this point N +1 terms included. Knowing that
limt!1 f(t) =f(1)=1=(1 , r 2 ) and with the summation rule for nite geometric series, we
obtain
f( e)
f(1) = W ( e)
=1,r
W1
2(N+1) ; e =(2N+1) : (3.38)
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