082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

19.6 Constant coefficient equations 579
A definition used regularly when analysing transmission lines is the voltage
reflection coefficient. Usually this is denoted by ρ and can be defined either at a
specific position on a lineor as a function ofdistance
ρ(z) = v 2 ejβz
v 1
e −jβz = v 2
v 1
e 2jβz
Itisadimensionlessquantityandistheratioofthereversetoforwardwavecomponents.
Inmanysystemsencounteredinradio-frequency(RF)engineeringitisdesirable
to minimize the relative amplitude of the backward wave component and hence the
reflection coefficient. An example of this can be found in the transmit circuit of a
mobilehandsetwhereatransmissionlinecarryingthesignalisattachedtoanantenna.
Duringtransmission,forwardwavespropagatetowardstheantennaterminals.Atthis
point they are either radiated from or dissipated within the antenna, or they reflect
back.Ifreflectedbacktheymayreturntotheamplifiercircuitwhichgeneratedthem,
and are wasted in the form of heat energy. As a consequence, battery life can be
reduced due to wasted power. Hence the minimization of the reflection coefficient,
usually by carefully designing the antenna, at the working frequency of the handset
isan important activity.
Fortheantennaitisdesirablefor ρ(z)tobeassmallaspossibleandfor v 1
≫ v 2
.
Note that in a system such as this |ρ(z)| < 1 and that ρ(z) is in general a complex
number.
We now consider a transmission line of length l with characteristic impedance
Z 0
terminated at z = l with a load having impedance Z L
. The setup is shown in
Figure 19.10.
z = 0
y 2 e jbz
Z 0
y 1 e –jbz
z = l
Z L
z
Figure19.10
Lossless transmissionline with
termination.
The total voltage at the load end of the line, v L
, is the sum of the forward and
backward wave componentsatz = l, thatis
v L
= v 1
e −jβl +v 2
e jβl
Using the definition ofthe reflection coefficient atz = l,
ρ(l) = v 2
v 1
e 2jβl
we can rewrite the voltage atthe load as
v L
= v 1
e −jβl e jβl
+ρ(l)v 1
e = v j2βl 1 e−jβl + ρ(l)v 1
e −jβl = v 1
e −jβl [1 + ρ(l)]
Thecurrentattheload,i L
,canbefoundfromfirstprinciplesbyasimilaranalysisto
the voltage, butwill be statedhere for simplicity
i L
= 1 Z 0
(v 1
e −jβl − v 2
e jβl )
➔