transmission lines - feup
TRANSCRIPT
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Faculdade de Engenharia
Transmission Lines
ELECTROMAGNETIC ENGINEERINGMAP – TELE 2008/2009
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EE 0809Lines 2
Faculdade de EngenhariaTransmission Lines
transmission lines à waveguides supporting TEM waves
parallel-plate waveguides
coaxial waveguides
two-wire waveguides
most common types
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Faculdade de EngenhariaTransmission Lines
general transmission line equations
time-harmonic solutions
finite transmission lines
voltage, current and impedance along the line
transmission lines in circuits
Smith chart
impedance matching
λ/4 transformer
reactive elements
single-stub
double-stub
transients
today
next week
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Faculdade de EngenhariaTEM waves in parallel-plate waveguides
b
y
z
x
W
xE
H
yEE
ˆ
ˆ
00
00
η−=
=r
r
βγ j= xeE
H
yeEE
zj
zj
ˆ
ˆ
0
0
β
β
η−
−
−=
=r
r
inside the guide:
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Faculdade de EngenhariaVoltage between the plates
b
y
z
x
W
voltage between the plates: zjebE β−−= 0
∫ ⋅−=−2
1
12
P
P
ldEVVrr
xeE
H
yeEE
zj
zj
ˆ
ˆ
0
0
β
β
η−
−
−=
=r
rinside the guide:
zjy eEE β−= 0
( ) ∫−=b
ydyEzV0
voltage à
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Faculdade de EngenhariaCurrent density on the plates
b
y
z
x
W
xeE
H
yeEE
zj
zj
ˆ
ˆ
0
0
β
β
η−
−
−=
=r
r
1
2
na
current density on the plates:
upper plate:
yan ˆˆ −=
inside the guide:
lower plate:
yan ˆˆ = 1
2
na
( )21ˆ HHaJ ns
rrr−×=
02 =Hr
xeEH zj ˆ01
β
η−−=
r( ) zeEbyJ zj
s ˆ0 β
η−−==
r
02 =Hr
xeEH zj ˆ01
β
η−−=
r ( ) zeEyJ zjs ˆ0 0 β
η−==
r
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Faculdade de EngenhariaCurrent on the plates
b
y
z
x
W
upper plate current:
∫ ⋅=A
sdJIrr
zjeEW β
η−−= 0
xeE
H
yeEE
zj
zj
ˆ
ˆ
0
0
β
β
η−
−
−=
=r
rinside the guide:
( ) zeEbyJ zjs ˆ0 β
η−−==
r
currentà
( ) ∫ ⋅=W
s zdxJzI ˆr
( ) zeE
byJ zjs ˆ0 β
η−−==
r
( ) zeE
yJ zjs ˆ0 0 β
η−==
r
current density:
lower plate current: zjeEW β
η−+= 0( ) ∫ ⋅=
Ws zdxJzI ˆ
r
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Faculdade de EngenhariaLossless transmission line equations
b
y
z
x
W
( ) zjebEzV β−−= 0
( ) zjeEWzI β
η−−= 0 zj
zj
eE
WjdzdI
ebEjdzdV
β
β
ηβ
β
−
−
=
=
0
0
εµη
εµωβ
=
=
VbW
jdzdI
IW
bj
dzdV
εω
µω
−=
−= ( )H/mW
bL
µ=
( )C/mbW
Cε
= VCjdzdI
ILjdzdV
ω
ω
−=
−=
0
0
22
2
22
2
=+
=+
LCIdz
Id
LCVdz
Vd
ω
ω
eqs. for V e I in a losslesstransmission line
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Faculdade de EngenhariaEquivalent circuit of a lossless transmission line
differential length ∆z of a transmission line:
zL∆
zC∆
z∆
i(z+∆z,t)i(z,t)
+ +
--v(z,t) v(z+∆z,t)
( )t
tzizLvL ∂
∂∆=
,
( )t
tzzvzCiC ∂
∆+∂∆=
,
( ) ( ) ( )
( ) ( ) ( ) 0,,
,
,,
,
=∆++∂
∆+∂∆+−
∆++∂
∂∆=
tzzit
tzzvzCtzi
tzzvt
tzizLtzv
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Faculdade de EngenhariaEquivalent circuit of a lossless transmission line
zL∆
zC∆
z∆
i(z,t)
+
-v(z,t)
( ) ( ) ( )
( ) ( ) ( ) 0,,
,
,,
,
=∆++∂
∆+∂∆+−
∆++∂
∂∆=
tzzit
tzzvzCtzi
tzzvt
tzizLtzv
( ) ( )
( ) ( )t
tzvCz
tzit
tziLz
tzv
∂∂=
∂∂−
∂∂=
∂∂−
,,
,, ( ) ( )
( ) ( )zVCjdz
zdI
zILjdz
zdV
ω
ω
=−
=−
0lim →∆z
phasor notation0
0
22
2
22
2
=+
=+
LCIdz
Id
LCVdz
Vd
ω
ω
same as before
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Faculdade de EngenhariaEquivalent circuit of a lossy transmission line
differential length ∆z of a transmission line:
zR∆ zL∆
zG∆zC∆
z∆
i(z+∆z,t)i(z,t)
+ +
--v(z,t) v(z+∆z,t)
( )( )
ttzi
zLv
tzizRv
L
R
∂∂
∆=
∆=,
,
( )( )
ttzzv
zCi
tzzvzGi
C
G
∂∆+∂
∆=
∆+∆=,
,
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 0,,
,,
,,
,,
=∆++∂
∆+∂∆+∆+∆+−
∆++∂
∂∆+∆=
tzzit
tzzvzCtzzvzGtzi
tzzvt
tzizLtzizRtzv( ) ( ) ( )
( ) ( ) ( )t
tzvCtzvG
ztzi
ttzi
LtziRz
tzv
∂∂
+=∂
∂−
∂∂
+=∂
∂−
,,
,
,,
,0lim →∆z
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Faculdade de EngenhariaGeneral transmission line equations
( ) ( ) ( )
( ) ( ) ( )t
tzvCtzvG
ztzi
ttzi
LtziRz
tzv
∂∂
+=∂
∂−
∂∂
+=∂
∂−
,,
,
,,
,
( ) ( ) ( )
( ) ( ) ( )zVCjGdz
zdI
zILjRdz
zdV
ω
ω
+=−
+=−
general solution
i(z,t)
+
-v(z,t)
( )( )CjGLjR ωωγ ++=
( ) ( )
( ) ( )zIdz
zId
zVdz
zVd
22
2
22
2
γ
γ
=
=
βαγ j+= ( )( ) zz
zz
eIeIzI
eVeVzVγγ
γγ
−−+
−−+
+=
+=
00
00
propagation constant
attenuation constant
phase constant
phasor notation
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Faculdade de Engenharia
( ) zz eVeVzV γγ −−+ += 00
Attenuation and phase constants
±gV
−0V
+0V
gZ
z
+
−( )zV
( )zI
( ) ( ) tjzz eeVeVtzv ωγγ −−+ += 00Re,
( ) ( ) ztjzztjz eeVeeV βωαβωα +−−−+ += 00Re
−+00 and VVif are real
( ) ( )zteVzteV zz βωβω αα ++−= −−+ coscos 00
z
atenuation phase
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Faculdade de Engenharia
( ) zz eVeVzV γγ −−+ += 00
Voltage and current in transmission line
( ) ( ) ( )
( ) ( ) ( )zVCjGdz
zdI
zILjRdz
zdV
ω
ω
+=−
+=−
( ) zz eIeIzI γγ −−+ += 00
γωLjR
IV +
=+
+
0
0 only 2 constantsare required
4 constants required to define voltage and current
−
−
−=0
0
IV
++
+= 00 V
LjRI
ωγ
−−
+−= 00 V
LjRI
ωγ
±gV
−−00 , IV
++00 , IV
gZ
z
+
−( )zV
( )zI
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Faculdade de EngenhariaCharacteristic impedance
Characteristic impedanceà
( )Ω++
=CjGLjR
ωω
infinite lineà no reflections
ratio between voltage and current for an infinite length transmission line
( ) zeIzI γ−+= 0
( ) zeVzV γ−+= 0
γωLjR +
=
characteristic impedance
( )( )CjGLjR ωωγ ++=
±gV
−−00 , IV
++00 , IV
gZ
z
+
−( )zV
( )zI
+
+
=0
0
IV
( )zZ
( ) ( )( )zIzV
zZ = ( ) 0ZzZ =
note: in general 00
0
0
0 ZIV
IV
=−= −
−
+
+
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Faculdade de EngenhariaSummary
Propagation constantà ( )( ) ( )1m−++=+= CjGLjRj ωωβαγ
( )Ω++
=CjGLjR
Zωω
0
Propagation velocityà
Characteristic impedanceà
( )1ms −=βω
v
Wavelengthà ( )m2βπ
λ =
General case
•frequency dependent attenuation
•frequency dependent velocity
SIGNAL DISTORTION
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Faculdade de EngenhariaTransmission lines – special cases
LCjωγ =
CL
Z =0 LCv
1=
Lossless lines
NO DISTORTION
0== GR ( )( )CjGLjRj ωωβαγ ++=+=
CjGLjR
Zωω
++
=0 βω
=v
LCωβ
α
=
= 0
Distortionless linesCG
LR
=
( )LC
LjR ωγ +=
CL
Z =0LC
v1
=LC
LC
R
ωβ
α
=
=
•zero or constant attenuation•constant velocity•constant and real characteristic impedance
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Faculdade de EngenhariaTransmission-line parameters
In turn, these parameters depend on the line geometry and on the materials thatconstitute the line
Letσ à dielectric conductivityσC à conductor conductiviityε à electric permitivitty of the dielectricµ à magnetic permeability of the dielectricµC à magnetic permeability of the conductor
The behaviour of a transmission line depends on the operating frequency andon parameters R, L, G and C
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Faculdade de EngenhariaTransmission-line parameters
a
b
a
D
a
hW
2h
coaxial two-wire conductor over ground parallelplate
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Faculdade de EngenhariaFinite transmission lines
LLL IZV =
( ) ( ) ( )[ ]( ) ( ) ( )[ ]z
Lz
LL
zL
zLL
eZZeZZIZ
zI
eZZeZZIzV
γγ
γγ
000
00
21
21
−−+=
−++=
−
−
( )
( ) zozo
zo
zo
eZV
eZV
zI
eVeVzV
γγ
γγ
00
−−
+
−−+
−=
+=
±gV
gZ
0
+
−( )zV
( )zI
( )zZ
LZ+
−( )zV
( )zI
+
−LV
LI
zl−0
0
0
0
0 ZIV
IV
=−=−
−
+
+
( )( ) zz
zz
eIeIzI
eVeVzVγγ
γγ
−−+
−−+
+=
+=
00
00
00
0
0
ZV
ZV
I
VVV
oL
oL
−+
−+
−=
+= ( )
( )00
00
2121
ZZIV
ZZIV
LL
LL
−=
+=
−
+
0=z
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Faculdade de EngenhariaImpedance along the transmission line
z
LZ±gV
gZ
+
−( )zV
( )zI
( )zZ
+
−( )zV
( )zI
+
−LV
LI
( ) ( )( )
( ) ( )( ) ( ) z
Lz
L
zL
zL
eZZeZZeZZeZZ
ZzIzV
zZ γγ
γγ
00
000
−−+−++
== −
−
( ) ( ) ( )[ ]( ) ( ) ( )[ ]z
Lz
LL
zL
zLL
eZZeZZIZ
zI
eZZeZZIzV
γγ
γγ
000
00
21
21
−−+=
−++=
−
−
( ) ( ) ( )( ) ( ) L
zzzz
zzL
zz
ZeeZeeZeeZee
ZzZ γγγγ
γγγγ
−−+−−+
= −−
−−
0
00
( ) ( )( )zZZ
zZZZzZ
L
L
γγ
tanhtanh
0
00 −
−=
'z
( ) ( )( )'tanh
'tanh'
0
00 zZZ
zZZZzZ
L
L
γγ
++
=
xx
xx
eeee
x −
−
+−
=)tanh(
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Faculdade de EngenhariaInput impedance – lossless transmission line
z
LZ±gV
gZ
+
−( )zV
( )zI
( )zZ
+
−( )zV
( )zI
+
−LV
LI
'z
lossless line βγ j=( ) ( )xjjx tantanh =
( ) ( )( )'tan
'tan'
0
00 zjZZ
zjZZZzZ
L
L
ββ
++
=
length l
( )( )ljZZ
ljZZZZ
L
Lin β
βtantan
0
00 +
+=
( ) ( )( )'tanh
'tanh'
0
00 zZZ
zZZZzZ
L
L
γγ
++
=
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Faculdade de EngenhariaInput impedance of lossless transmission lines – special cases
lossless transmission line of length l ( )( )ljZZ
ljZZZZ
L
Lin β
βtantan
0
00 +
+=
0ZZ L = 0ZZ in =
∞=LZ ( )ljZZ in βcotg0−=
2λ
nl = Lin ZZ =
0=LZ ( )lanjZZ in βt0=
( )4
12λ
−= nlL
in ZZ
Z20=
always imaginary
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Faculdade de EngenhariaReflection coefficient at the load
Reflection coefficient (voltage)à ratio between reflected and incident voltages
( )( ) +
−
==
==Γ
00
0
VV
zV
zV o
inc
refL
at the load:
0
0
ZZZZ
L
LL +
−=Γ
( )
( )00
00
2121
ZZIV
ZZIV
LL
LL
−=
+=
−
+
Special cases:
0ZZ L = 0=ΓL
∞=LZ 1=ΓL
0=LZ 1−=ΓL
no reflections MATCHED LINE
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Faculdade de EngenhariaReflection coefficient at the load
0
0
ZZZZ
L
LL +
−=Γ
Notes:
1. For current
2. Most often, is complex àLΓ ΓΓ=Γ θjLL e||
Linc
refI
VV
II
I
IΓ−=−===Γ +
−
+
−
0
0
0
0
1
1
0
0
+
−=Γ
ZZZZ
L
L
L 11
+−
=ΓL
LL z
z
LL z
ZZ
=0
( )( ) LL
LL
jxrjxr
+++−
=11
LL jxr +=
1||, ≤ΓL
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Faculdade de EngenhariaReflection coefficient along the line
( )( ) +
−
==
==Γ
00
0
VV
zV
zV o
inc
refLat the load: ΓΓ=
+−
=Γ θjL
L
LL e
ZZZZ
0
0
along the line:( )( )
zLz
zo
inc
ref eeVeV
zV
zVz γ
γ
γ2
0
)( Γ===Γ −+
−
zz −='
'2)'( zLez γ−Γ=Γ
lossless line: βγ j= ( )'2)'( zjL ez βθ −ΓΓ=Γ absolute value is constant
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EE 0809Lines 27
Faculdade de EngenhariaVoltage along the line
( ) zjzj eVeVzV ββ −−+ += 00
z
( ) ( ) ( )zjzjzj eeVeVVzV βββ −−−−+ ++−= 000
( ) ( ) ( )zVeVVzV zj ββ cos2 000−−−+ +−=
( )2
cosjxjx ee
x−+
=
propagating wave
stationary wave
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Faculdade de EngenhariaNote – propagating and stationary waves
•let ( ) zjAezV β−=•let ( ) ( ) ( )ztAAeeAetzv ztjtjzj βωβωωβ −=== −− cosReRe,
zpropagating wave
( ) ( )zAzV βcos=•let ( ) ( ) ( ) ( )tzAezAtzv tj ωββ ω coscoscosRe, ==
stationary wavez
nodes( v=0 for every t )
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Faculdade de EngenhariaVoltage along the line
( ) zjzj eVeVzV ββ −−+ += 00
z
propagating + stationary waves
( ) ( )zjL
zj eeVzV ββ 20 1 Γ+= −+
( ) ( )( ) ( )( )( )'2cos21
'2sin'2cos1'
20
220
zV
zzVzV
LL
LL
βθ
βθβθ
−Γ+Γ+=
−Γ+−Γ+=
Γ+
ΓΓ+
periodic termperiod=λ/2
( ) ( )( )( )'2'
0
'2'0
1
1'zj
Lzj
zjL
zj
eeV
eeVzVβθβ
ββ
−+
−+
ΓΓ+=
Γ+=
'z
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Faculdade de EngenhariaVoltage along the line - example
( ) ( )'2cos21'2
0 zVzV LL βθ −Γ+Γ+= Γ+
Let
( )m2m1
5.0
V1
1
4
0
πλβ
π
=⇒=
=Γ
=
−
+
j
L e
V
0123456789100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2λ
minV
MAXV
![Page 31: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/31.jpg)
EE 0809Lines 31
Faculdade de EngenhariaVoltage maxima and minima
•voltage minima: ( ) 1'2cos −=−Γ zβθ
( ) ( )'2cos21'2
0 zVzV LL βθ −Γ+Γ+= Γ+
•location: πβθ nzM 22 / −=−Γ ( )Γ+= θπβ
nzM 221/
n
z 0'≥
integer
•location: ( )πβθ 122 / +−=−Γ nzm ( )[ ]Γ++= θπβ
1221/ nzm
n
z 0'≥
integer
•value: LLVV Γ−Γ+= + 212
0min( )LVV Γ−= + 10min
•value: LLMAXVV Γ+Γ+= + 21
20 ( )LMAX
VV Γ+= + 10
•voltage maxima: ( ) 1'2cos +=−Γ zβθ
![Page 32: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/32.jpg)
EE 0809Lines 32
Faculdade de EngenhariaVoltage along the line - example
( ) ( )'2cos21'2
0 zVzV LL βθ −Γ+Γ+= Γ+
Let
( )m2m1
5.0
V1
1
4
0
πλβ
π
=⇒=
=Γ
=
−
+
j
L e
V
0123456789100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
minV
MAXV
( ) 5.110 =Γ+= +LMAX
VV
( ) 5.010min=Γ−= +
LVV
8π
85π
πλ
=28
/ ππ += nzM
85/ π
π += nzm
![Page 33: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/33.jpg)
EE 0809Lines 33
Faculdade de EngenhariaSWR
SWR (Voltage Standing Wave Ratio)à ratio between voltage maxima and minima
( )( )L
LMAX
V
V
V
VSWR
Γ−
Γ+==
+
+
1
1
0
0
min L
LSWRΓ−
Γ+=
1
1
11
+−
=ΓSWRSWR
L
Note: 1≥SWR
![Page 34: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/34.jpg)
EE 0809Lines 34
Faculdade de EngenhariaSWR – particular cases
0
0
ZZZZ
L
LL +
−=Γ
L
LSWRΓ−
Γ+=
1
111
+−
=ΓSWRSWR
L
Particular cases:
0ZZ L = 0=ΓL minVV MAX =
no reflections
no stationary wave
1=SWR 0=ΓL
matched line 1=SWR
1=SWR
![Page 35: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/35.jpg)
EE 0809Lines 35
Faculdade de EngenhariaSWR – particular cases
0
0
ZZZZ
L
LL +
−=Γ
L
LSWRΓ−
Γ+=
1
111
+−
=ΓSWRSWR
L
Particular cases:
∞=LZ 1=ΓL
0=LZ 1−=ΓL ∞=SWR
( ) ++ =Γ+= 00 21 VVV LMAX
( ) 010min=Γ−= +
LVV
∞=SWR
+= 02VVMAX
0min
=V
![Page 36: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/36.jpg)
EE 0809Lines 36
Faculdade de EngenhariaCurrent along the line
( ) zjzj eIeIzI ββ −−+ += 00
z
propagating + stationary waves
( ) ( )zjL
zj eeZV
zI ββ 2
0
0 1 Γ−= −+
( ) ( )( ) ( )( )
( )'2cos21
'2sin'2cos1'
2
0
0
22
0
0
zZ
V
zzZ
VzI
LL
LL
βθ
βθβθ
−Γ−Γ+=
−Γ−+−Γ−=
Γ
+
ΓΓ
+
periodic termperiod=λ/2
( ) ( )
( )( )'2'
0
0
'2'
0
0
1
1'
zjL
zj
zjL
zj
eeZV
eeZV
zI
βθβ
ββ
−+
−+
ΓΓ−=
Γ−=
'z
![Page 37: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/37.jpg)
EE 0809Lines 37
Faculdade de EngenhariaCurrent maxima and minima
•current minima: ( ) 1'2cos =−Γ zβθ
( ) ( )'2cos21' 2
0
0z
Z
VzI LL βθ −Γ−Γ+= Γ
+
•location: ( )πβθ 12'2 +−=−Γ nz ( )[ ]Γ++= θπβ
1221
' nzn
z 0'≥
integer
•location: πβθ nz 2'2 −=−Γ ( )Γ+= θπβ
nz 221
'n
z 0'≥
integer
•value: LLZ
VI Γ−Γ+=
+
21 2
0
0
min( )LZ
VI Γ−=
+
10
0
min
•value: LLMAX Z
VI Γ+Γ+=
+
21 2
0
0 ( )LMAX Z
VI Γ+=
+
10
0
•current maxima: ( ) 1'2cos −=−Γ zβθ
![Page 38: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/38.jpg)
EE 0809Lines 38
Faculdade de EngenhariaVoltage and current – maxima and minima location
( ) 1'2cos =−Γ zβθ
( ) ( )'2cos21' 2
0
0z
Z
VzI LL βθ −Γ−Γ+= Γ
+
( )[ ]Γ++= θπβ
1221/ nzm
n
z 0'≥
integer
( )Γ+= θπβ
nzM 221/
n
z 0'≥
integer
( ) 1'2cos −=−Γ zβθ
( ) ( )'2cos21' 20 zVzV LL βθ −Γ+Γ+= Γ+
máximos de tensãoe
mínimos de corrente
voltage maximaAND
current minima
voltage minimaAND
current maxima
![Page 39: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/39.jpg)
EE 0809Lines 39
Faculdade de EngenhariaTransmission lines in circuits
±gV
gZ
0
LZ+
−inV
inI
+
−LV
LI
( )( )lzII
lzVV
VZIV
in
in
inging
−==−==
+=
( )
( ) zozo
zo
zo
eZV
eZV
zI
eVeVzV
γγ
γγ
00
−−
+
−−+
−=
+=
zl−+− Γ= 00 VV L
[ ][ ]l
Ll
in
lL
lin
eeZV
I
eeVV
γγ
γγ
2
0
0
20
1
1
−+
−+
Γ−=
Γ+=
( ) ( )[ ]lL
lLg
lg eZeZe
ZV
V γγγ 20
2
0
0 11 −−+
Γ++Γ−=
![Page 40: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/40.jpg)
EE 0809Lines 40
Faculdade de EngenhariaTransmission lines in circuits
±gV
gZ
0
LZ+
−inV
inI
+
−LV
LI
zl−
( ) ( )[ ]lL
lLg
lg eZeZe
ZV
V γγγ 20
2
0
0 11 −−+
Γ++Γ−=
( )[ ]lLgg
gl
eZZZZ
VZeV
γ
γ
200
00 −
+
Γ−++=
0
0
ZZZZ
g
gg +
−=Γ (reflection coefficient at the source)
[ ]lLg
g
g
l
e
V
ZZZ
eVγ
γ
20
00
1 −
+
ΓΓ−+=
( )
( )
ΓΓ−Γ−
+=
ΓΓ−Γ+
+=
−−
−
−−
−
lLg
zLz
g
lg
lLg
zLz
g
lg
ee
eZZ
eVzI
ee
eZZ
eVZzV
γ
γγ
γ
γ
γγ
γ
2
2
0
2
2
0
0
11
11
voltage and current as functions of
LZload:
line:
source:
lZ ,,0 γ
gg ZV ,
![Page 41: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/41.jpg)
EE 0809Lines 41
Faculdade de EngenhariaTransmission lines in circuits
±gV
gZ
0
LZ+
−inV
inI
+
−LV
LI
zl−
( )
( )
ΓΓ−Γ−
+=
ΓΓ−Γ+
+=
−−
−
−−
−
lLg
zLz
g
lg
lLg
zLz
g
lg
ee
eZZ
eVzI
ee
eZZ
eVZzV
γ
γγ
γ
γ
γγ
γ
2
2
0
2
2
0
0
11
11
( ) ( )( ) 122
0
0 11−−−
−
ΓΓ−Γ++
= lLg
zL
z
g
lg eee
ZZ
eVZzV γγγ
γ
( ) ( ) ( ) ( )
+ΓΓΓ+ΓΓ+ΓΓΓ+ΓΓ+Γ+
+= −−−−−−−
−
LzL
lLg
zlLg
zL
lLg
zlLg
zL
z
g
lg eeeeeeeeee
ZZ
eVZ γγγγγγγγγγγ
222222
0
0
( ) ( )
+ΓΓ+ΓΓ+Γ+
+= −−−
−
L222
0
0 1 lLg
lLg
zL
z
g
lg eeee
ZZ
eVZ γγγγγ
L++++=−
3211
1xxx
x
![Page 42: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/42.jpg)
EE 0809Lines 42
Faculdade de EngenhariaTransmission lines in circuits
±gV
gZ
0
LZ+
−inV
inI
+
−LV
LI
zl−
( ) ( ) ( ) ( ) ( )
+ΓΓΓ+ΓΓ+ΓΓΓ+ΓΓ+Γ+
+= −−−−−−−
−
LzL
lLg
zlLg
zL
lLg
zlLg
zL
z
g
lg eeeeeeeeee
ZZ
eVZzV γγγγγγγγγγ
γ222222
0
0
( ) L++++++= −−+−−+−−+ zzzzzz eVeVeVeVeVeVzV γγγγγγ332211
−−+ Γ= 12
2 VeV lg
γ
+− Γ= 33 VV L
−−+ Γ= 22
3 VeV lg
γ
+2V
+− Γ= 22 VV L
−2V
g
lg
ZZ
eVZV
+=
−+
0
01
γ
+1V
+− Γ= 11 VV L
−1V
![Page 43: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/43.jpg)
EE 0809Lines 43
Faculdade de EngenhariaTransmission lines in circuits
±gV
gZ
0
LZ+
−inV
inI
+
−LV
LI
zl−
( ) L++++++= −−+−−+−−+ zzzzzz eVeVeVeVeVeVzV γγγγγγ332211
+2V
−2V
+1V
−1V
( ) ( ) ( ) zzzz eVeVeVVVeVVVzV γγγγ −−+−−−−+++ +=+++++++= 00321321 LL
![Page 44: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/44.jpg)
EE 0809Lines 44
Faculdade de EngenhariaPower in lossless transmission lines
( ) ( )( )( ) ( )( )'2'
0
0
'2'0
1'
1'
zjL
zj
zjL
zj
eeZV
zI
eeVzV
βθβ
βθβ
−+
−+
Γ
Γ
Γ−=
Γ+=
(lossless transmission line)
( ) ( ) ( ) ''Re21
' * zIzVzPav =
( ) ( )( ) ( ) ( )( )
Γ−Γ+= −−−+
−+ ΓΓ '2'
0
*0'2'
0 11Re21
' zjL
zjzjL
zjav ee
ZV
eeVzP βθββθβ
( ) ( )( ) '2'22
0
2
01Re
2zjzj
LL eeZ
Vβθβθ −−−
+
ΓΓ −Γ+Γ−=
( ) '2sin21Re2
2
0
2
0zj
Z
VLL βθ −Γ+Γ−= Γ
+
( ) ( ) constant12
' 2
0
2
0=Γ−=
+
Lav Z
VzP
incident reflected
![Page 45: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/45.jpg)
EE 0809Lines 45
Faculdade de EngenhariaPower in transmission lines – general case
( ) ( )( )( ) ( )( )'2'2''
0
0
'2'2''0
1'
1'
zjzL
zjz
zjzL
zjz
eeeeZV
zI
eeeeVzV
βθαβα
βθαβα
−−+
−−+
Γ
Γ
Γ−=
Γ+=( ) ( ) ( ) ''Re
21
' * zIzVzPav =
( ) ( )( ) ( ) ( )( )
Γ−Γ+= −−−−+
−−+ ΓΓ '2'2''
0
*0'2'2''
0 11Re21
' zjzL
zjzzjzL
zjzav eeee
ZV
eeeeVzP βθαβαβθαβα
( ) '2sin21Re2
'2'42'2
0
2
0zejee
R
Vz
Lz
Lz βθααα −Γ+Γ−= Γ
−−+
( ) ( )'22'2
0
2
0
2' z
Lz
av eeR
VzP αα −
+
Γ−= ( ) ( )2
0
2
0, 1
20' LavLav R
VzPP Γ−===
+
( ) ( )lL
lavinav ee
R
VlzPP αα 222
0
2
0, 2
' −+
Γ−===
00 RZ =if
![Page 46: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/46.jpg)
EE 0809Lines 46
Faculdade de EngenhariaProblem
formulae
![Page 47: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/47.jpg)
EE 0809Lines 47
Faculdade de EngenhariaProblem
formulae
![Page 48: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/48.jpg)
EE 0809Lines 48
Faculdade de EngenhariaProblem
formulae
![Page 49: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/49.jpg)
EE 0809Lines 49
Faculdade de EngenhariaProblem
formulae
![Page 50: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/50.jpg)
EE 0809Lines 50
Faculdade de Engenharia
( )
( ) 22
22
22
1
21
1
imre
imL
imre
imreL
x
r
Γ+Γ−
Γ=
Γ+Γ−
Γ−Γ−=
Load impedance ó reflection coefficient
11
+−
=ΓL
LL z
zwhere
0ZZ
z LL = (normalized load impedance)
00 RZjXRZ LLL
=+=
(lossless line)
LLL jxrz +=
imrej
LL je Γ+Γ=Γ=Γ Γθ
L
LLz
Γ−Γ+
=11
( )( ) imre
imreLL j
jjxr
Γ−Γ−Γ+Γ+
=+11
![Page 51: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/51.jpg)
EE 0809Lines 51
Faculdade de Engenharia
reΓ
imΓ
( )
( ) 22
22
22
1
21
1
imre
imL
imre
imreL
x
r
Γ+Γ−
Γ=
Γ+Γ−
Γ−Γ−=
Load impedance ó refelction coefficient
22
2
11
1
+
=Γ+
+
−ΓL
imL
Lre rr
r
Lr+11
L
L
rr+1
( ) ( ) 220
20 Ryyxx =−+−
( )0
1=Γ
+=Γ
im
LLre rrcentered at
circle of radius ( )Lr+11
the reflection coefficients of all ZLwhose real part is rL are in this circle
![Page 52: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/52.jpg)
EE 0809Lines 52
Faculdade de Engenharia
reΓ
imΓ
Load impedance ó reflection coefficient
22
2
11
1
+
=Γ+
+
−ΓL
imL
Lre rr
r
Note:
curve does not depend on xL
0=Γim 111
,, =Γ∨+−
=Γ rreL
Llre r
r
111
+−
L
L
rr
0=Lr 1, −=Γ lre
for any ZL
∞=Lr 1, =Γ lre
1−
open circuit
![Page 53: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/53.jpg)
EE 0809Lines 53
Faculdade de Engenharia
reΓ
imΓ
( )
( ) 22
22
22
1
21
1
imre
imL
imre
imreL
x
r
Γ+Γ−
Γ=
Γ+Γ−
Γ−Γ−=
Load impedance ó reflection coefficient
( )2
2 111
=
−Γ+−Γ
LLimre xx
( ) ( ) 220
20 Ryyxx =−+−
circle of radius Lx1
Lim
re
x11
=Γ=Γcentered at
Lx1
Lx1
1
1≤ΓL
the reflection coefficients of all ZLwhose imaginary part is xL are here
![Page 54: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/54.jpg)
EE 0809Lines 54
Faculdade de Engenharia
reΓ
imΓ
Load impedance ó reflection coefficient
Lx1
Lx1
1
Note:
curve does not depend on rL
( )2
2 111
=
−Γ+−Γ
LLimre xx
Lx1
−
0=Lx
0=Lx infinite radius
symmetrical curves for xL < 0
![Page 55: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/55.jpg)
EE 0809Lines 55
Faculdade de EngenhariaSmith chart
reΓ
imΓ
1
xL constant
rL constant
![Page 56: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/56.jpg)
EE 0809Lines 56
Faculdade de EngenhariaSmith chart
![Page 57: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/57.jpg)
EE 0809Lines 57
Faculdade de Engenharia
Γθ
reΓ
imΓ
Smith chart
1
LΓ
LZ
•from:
point in chart ( intersection of curves corresponding to rL and xL )
ΓθandLΓ
rL and xL
Lx
Lr
LΓ
•from:
![Page 58: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/58.jpg)
EE 0809Lines 58
Faculdade de EngenhariaReflection coefficient along the line
along the line:
zz −='
'2)'( zLez γ−Γ=Γ
lossless line: βγ j= ( )'2)'( zjL ez βθ −ΓΓ=Γ
reΓ
imΓ
1
constant magnitude
phase decreases with z’
toward generator
toward load
( )( ) 0
0)(ZzZZzZ
z+−
=Γ
Note:
( )( )
zLz
zo
inc
ref eeVeV
zV
zVz γ
γ
γ2
0
)( Γ===Γ −+
−
Smith chart can be used to obtain from ( )zZ )(zΓ
![Page 59: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/59.jpg)
EE 0809Lines 59
Faculdade de EngenhariaDistances in the Smith chart
in Smith chart the distances are measured as fractions of λ
reΓ
imΓ
1
toward generator
toward load
( )'2)'( zjL ez βθ −ΓΓ=Γ when πβ 2'2 =z
222
'λ
βπ
==z
a complete turn (360º)
corresponds to a distance = λ/2
initial position
![Page 60: Transmission Lines - FEUP](https://reader031.vdocuments.net/reader031/viewer/2022012920/61c73c5b874df22b7c58dc8c/html5/thumbnails/60.jpg)
EE 0809Lines 60
Faculdade de EngenhariaInput impedance
1. draw the point corresponding to the normalized load impedance zL à point P1
2. draw the circle centered at the origin with radius OP1
3. draw the straight line from O to P1
4. draw the straight line from O that corresponds to a rotation of l toward the generator
5. intersection of this line with previous circle à point P2
6. obtain , where zin is read from P2
reΓ
imΓ
1
0ZzZ inin ⋅=
P1
P2
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EE 0809Lines 61
Faculdade de EngenhariaAdmittance
reΓ
imΓ
1
( ) ( )( )'tan
'tan'
0
00 zjZZ
zjZZZzZ
L
L
ββ
++
= ( )( ) LL
L
ZZ
jZZjZZ
ZzZ20
0
00 2tan
2tan4
' =++
=
=
ππλ
( )LZ
ZZ
Z 0
0
4=
λ( ) Lyz =4λ
º3602 ⇔λ
º1804 ⇔λ
1. draw zL
2. rotate 180º
Ly
Lz
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EE 0809Lines 62
Faculdade de EngenhariaMaxima and minima location
( ) ( )'2cos21'2
0
0z
Z
VzI LL βθ −Γ−Γ+= Γ
+
( ) ( )'2cos21' 20 zVzV LL βθ −Γ+Γ+= Γ+
( ) 1'2cos =−Γ zβθ à voltage maxima and current minima
à voltage minima and current maxima( ) 1'2cos −=−Γ zβθ
( ) ( )'2' zjL ez βθ −ΓΓ=Γ
voltage maxima where ( ) πnz 2' =Γ∠
voltage minima where ( ) ( )π12' +=Γ∠ nz
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EE 0809Lines 63
Faculdade de EngenhariaMaxima and minima location
reΓ
imΓ
1
voltage maxima where ( ) πnz 2' =Γ∠
voltage minima where ( ) ( )π12' +=Γ∠ nz
voltage maxima
voltage minima
Note:
1. maxima and minima where input
impedance is real
2. maxima (minima) points are separated
by nλ/2
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EE 0809Lines 64
Faculdade de EngenhariaProblem
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EE 0809Lines 65
Faculdade de EngenhariaProblem
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EE 0809Lines 66
Faculdade de EngenhariaProblem
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EE 0809Lines 67
Faculdade de EngenhariaProblem