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

580 Chapter 19 Ordinary differential equations I
This equation can alsobe written intermsof the reflection coefficient
i L
= v 1 e−jβl
[1 − ρ(l)]
Z 0
The voltage and currentatthe load arerelated by the simple equation
Z L
= v L
i L
and hence
Z L
= v 1 e−jβl [1 + ρ(l)]
v 1
e −jβl
[1 − ρ(l)]
Z 0
=Z 0
[1 + ρ(l)]
[1 − ρ(l)]
Itcan be shown by further manipulation that
ρ(l) = Z L −Z 0
Z L
+Z 0
Thereflectioncoefficientattheloadisthereforedependentonlyonthecharacteristic
impedance of the line, which is usually known, and the load impedance. Some RF
measuringinstrumentssuchasnetworkanalyserscanmeasuretheamountofforward
and backward waves. From thisthey areable todetermine the load impedance.
Engineeringapplication19.11
Standingwavesontransmissionlines
Thedifferentialequationsthatmodelvoltageandcurrentwavesontransmissionlines
alsodescribethepresenceof standingwaves.Astandingwave issocalledbecause
it appears to remain stationary in space. Standing waves are an effect caused by the
interference pattern created when two waves propagate in opposite directions in the
sametransmissionmedium.
Thepreviousexamplesuggestedthatminimizingbackwardwavesisoftendesired.
In this case the standing wave component on the line is also minimized. In order to
studythisinmoredetailwewishtoplotthevoltagestandingwavepatternforagiven
loadimpedance,Z L
.
Recall from Equation (19.23) (with α = 0 for a lossless line) that the voltage at
any pointzon the lineisgiven by
v(z) = v 1
e −jβz +v 2
e jβz
This voltage is the sum of the forward and reflected waves. The modulus of this
function can be plotted withzas the independent variable. Once the amplitude v 1
is
known, then the amplitude v 2
can be determined using the reflection coefficient at
the load
ρ(l) = v 2
v 1
e j2βl
from which v 2
= v 1
ρ(l)e −j2βl so that
v(z) = v 1
e −jβz + v 1
ρ(l)e −j2βl e jβz