Transmission Lines - FEUP
67
Faculdade de Engenharia Transmission Lines ELECTROMAGNETIC ENGINEERING MAP – TELE 2008/2009
Transcript of Transmission Lines - FEUP
Faculdade de Engenharia
Transmission Lines
ELECTROMAGNETIC ENGINEERINGMAP – TELE 2008/2009
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
EE 0809Lines 3
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
EE 0809Lines 4
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:
EE 0809Lines 5
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
EE 0809Lines 6
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
EE 0809Lines 7
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
EE 0809Lines 8
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
EE 0809Lines 9
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
EE 0809Lines 10
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
EE 0809Lines 11
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
EE 0809Lines 12
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
EE 0809Lines 13
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
EE 0809Lines 14
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
EE 0809Lines 15
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
=−= −
−
+
+
EE 0809Lines 16
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
EE 0809Lines 17
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
EE 0809Lines 18
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
EE 0809Lines 19
Faculdade de EngenhariaTransmission-line parameters
a
b
a
D
a
hW
2h
coaxial two-wire conductor over ground parallelplate
EE 0809Lines 20
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
EE 0809Lines 21
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(
EE 0809Lines 22
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
++
=
EE 0809Lines 23
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
EE 0809Lines 24
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
EE 0809Lines 25
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
EE 0809Lines 26
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
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
EE 0809Lines 28
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 )
EE 0809Lines 29
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
EE 0809Lines 30
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
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
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
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
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
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
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
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
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
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−=
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 ,
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
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
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
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
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
EE 0809Lines 46
Faculdade de EngenhariaProblem
formulae
EE 0809Lines 47
Faculdade de EngenhariaProblem
formulae
EE 0809Lines 48
Faculdade de EngenhariaProblem
formulae
EE 0809Lines 49
Faculdade de EngenhariaProblem
formulae
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
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
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
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
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
EE 0809Lines 55
Faculdade de EngenhariaSmith chart
reΓ
imΓ
1
xL constant
rL constant
EE 0809Lines 56
Faculdade de EngenhariaSmith chart
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:
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
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
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
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
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
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
EE 0809Lines 64
Faculdade de EngenhariaProblem
EE 0809Lines 65
Faculdade de EngenhariaProblem
EE 0809Lines 66
Faculdade de EngenhariaProblem
EE 0809Lines 67
Faculdade de EngenhariaProblem
