Post on 29-Mar-2018
1
ECE 391 supplemental notes - #9
150 Oregon State University ECE391– Transmission Lines Spring Term 2014
Terminated Transmission Line
V (z ') =V0+ e+γ z ' + Γe−γ z '( )
I(z ') = V0+
Z0e+γ z ' − Γe−γ z '( )
)0'()0'( === zIZzV L
'z 0
LZγ0Z
+
−
=Γ0
0
VV
L)'(zI
+ – )'(zV
ΓL = ΓL ejθL = ZL − Z0
ZL + Z0
Reflection Coefficient
1 -1
j
-j
Re(Γ)
Im(Γ) Γ
2
151 Oregon State University ECE391– Transmission Lines Spring Term 2014
Net Power Flow on Lossless Line
( ){ } [ ] const.12
)'()'(Re)'( 2
0
2
0*21
ave =Γ−==+
LZV
zIzVzP
( )''0)'( zj
Lzj eeVzV ββ −++ Γ+=
( )''
0
0)'( zjL
zj eeZVzI ββ −+
+
Γ−=
'z 0
LZβγ jZ =0
+
−
=Γ0
0
VV
L)'(zI
+–)'(zV
'z 0
LZβγ jZ =0
+
−
=Γ0
0
VV
L)'(zI )'(zI
+–)'(zV
+–)'(zV
reflected powerincident power
= ΓL2
152 Oregon State University ECE391– Transmission Lines Spring Term 2014
Transformation of Reflection Coefficient along Lossless Transmission Line
'0)'( zjeVzV β+++ =
'z 0
LZβγ jZ =0
1θ'1 zβθ +
Lθθ +1Lz θβθ +− '1
)'(zV +
)'(zV −
'0)'( zjeVzV β−−− =
3
153 Oregon State University ECE391– Transmission Lines Spring Term 2014
Transformation of Reflection Coefficient along Lossless Transmission Line
Γ(z') = V− (z ')
V + (z ')= V0
−e− jβz '
V0+e+ jβz '
= ΓLe− j2βz ' = ΓL e
+ j (θL−2βz ')
only phase change of twice the electrical length
'z 0
LZβγ jZ =0
+
−
=Γ0
0
VV
L)'()'()'(zVzVz +
−
=Γ
)'(zI
+ – )'(zV
)'(zI
154 Oregon State University ECE391– Transmission Lines Spring Term 2014
Development of Smith Chart )'2()(z' zj
LLe βθ −+Γ=Γxr)'()'(z
0
in jZzZz +==
r
x
0 0.5 1 1.5 2
0
0.5
-0.5
1
-1
r |x|
x>0
|x|
x<0
x=0
|Γ|=1
Γ=0
Γ−Γ+=+
11xr j
Γ z
z=
4
155 Oregon State University ECE391– Transmission Lines Spring Term 2014
Smith Chart
)'2()(z' zjL
Le βθ −+Γ=Γ
0
in )'()'(zZzZjxrz =+=
Γ−Γ+=+
11xr j
156 Oregon State University ECE391– Transmission Lines Spring Term 2014
Smith Chart
source
load
r = 1
OC SC
x = 0
Γ−Γ+=+
11xr j