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

19.6 Constant coefficient equations 581
which simplifies to
v(z) = v 1
e −jβz[ 1 + ρ(l)e j2β(z−l)]
The modulus of v(z) is now plotted against z for several different values of the
reflection coefficient at the load ρ(l). Consider a transmission line of total length
l = 10 m. Another quantity weneed toknow inorder toplot the voltage on the line
is β, which is the phase constant. For a transmission line of given construction and
ataparticular frequency, β isconstantand represents thephase change per metreof
transmissionline. Herewe take β = π/2 rad m −1 .
Case1: ρ(l)= −1
Consider the case when the transmission line is terminated in a short circuit. The
reflection coefficient atthe load is
ρ(l) = Z L −Z 0
Z L
+Z 0
= 0−Z 0
0+Z 0
= −1
Taking the amplitude v 1
= 1, which represents a 1 V peak sine wave, and plotting
the modulus of v(z), gives the resultshown inFigure 19.11.
y(z)
2
1
0
2 4 6 8 10 z
Figure19.11
Voltage standing wave pattern forashort-circuitloadwith a 1 Vinput wave.
Noticethatthevoltagepeakonthelineis2Vwhereasthevoltageinputwasonly
1V.Thisisduetotheforward-going1Vinputwavereachingtheendofthelineand
reflectingbackuponitself.Atsomevaluesofzitconstructivelyinterfereswithitself
giving double the input; atothers itdestructively interferesgiving zero volts.
Case2: ρ(l)=1
Asimilareffectisseenforanopen-circuitload.HerebyconsideringZ L
→ ∞,ρ(l)
can be shown to equal +1, giving rise to the standing wave pattern shown in
Figure 19.12. Note here that the voltage maximum is at the load, unlike the case
ofthe shortcircuitwhere the minimum wasatthe load.
y(z)
2
1
0
2 4 6 8 10 z
Figure19.12
Voltage standing wave pattern foran open-circuit loadwith a1Vinput wave.
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