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7/31/2019 13 Transmission Line Theory
1/25
Transmission Line Theory
7/31/2019 13 Transmission Line Theory
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Transmission Line Equations (1)
At low frequencies:
connect two components
At high frequencies:
Cannot simply use a wire to
connect two components
m105Hz60 6
0.5m103
MHz6006
8
f
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Transmission Line Equations (2)
RzLz
Gz
Differential Length
CzSource Load
z z + zz
R: Series resistance per unit length ( /m)
G: Shunt conductance per unit length (S/m)
L: Series inductance per unit length (H/m)
C: Shunt capacitance per unit length (F/m)
Lossless Line: R = G = 0
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Transmission Line Equations (3)
),( tzi ),( tzzi
Gz
z
Lz
Cz
+ +),( tzv ),( tzzv Source Load
z z + zz
Differential Length
Cannot apply Kirchhoff's Laws to the whole line
Can apply Kirchhoff's Laws to the differential length
KVL: tzzi ),(
tzziLtzzRi
tzzvtzv
t
),(
),(),(),(
,,,
t
tzi
LtzRiz
tzv
z
),(
),(
),(
0
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Transmission Line Equations (4)
),( tzi ),( tzzi
Gz
z
Lz
Cz
+ +),( tzv ),( tzzv Source Load
z z + zz
Differential Length
tzzit
tzvzCzGtzvtzi
),(
),(),(),(:KCL
t
tzvCtzGv
z
tzzitzi
),(),(
),(),(
t
tzv
CtzGvz
tzi
z
),(
),(
),(
0
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Transmission Line Equations (5)
t
tziLtzRi
tzv)1(
),(),(
),(
ttzvCtzGv
ztzi )2(),(),(),(
t
tzi
L
tzv
)3(
),(),(
neoss ess
t
tzvC
z
tzi)4(
),(),(
tzvtzv
ttzvLC
ztz
tL
ztzv
1,1,
,),(,(3)From
22
22
LCv
tvzp
p
222
Wave Equation
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Phasor Form Representation for
)cos( tu
Re eAe
2
j
AeU
)cos(
2t
kztu
jt
Re
jkzAeU
tje
jkzAe
U
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Sinusoidal Excitation of
),(),(
),(
tziLtzRi
tzv
),(),(
),(
t
tzvCtzGv
z
tzi
)( IZLIRIzdV
)()()()(
zVYzCVjzGVzdI
dz
Len ther UnitIm edanceSeries LjRZ
Lengthper UnitAdmittnaceShunt CjGY
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Sinusoidal Excitation of
)(
zdV
)()()()()(
zVCjGzCVjzGvzdI
dz
From
2
z
)(
0)())((
2222
2
zVd
zVCjGLjRdz
)(
22
2
zVd
zdz
2
dz
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Sinusoidal Excitation of
kzkz
zz eVeVzV
00 )(
ee
00
jkzjkzeVVeVV
00
Define
WaveBackwardWaveForward
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Sinusoidal Excitation of
:osolution tabovetheSubstitute
)()()( zILjRd
zdV
)(00 zz
e
V
e
V
zI
II
where0
0
0
00
zze
ZVIe
ZVI
CjGLjRZ
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Characteristic Impedance
0 1LjRZ
If R=G=0 lossless case
CZ 0
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Propagation Constant
))((22 CjGLjRk
rwavenumbe:constant,npropagatio: k-
rad/mconstant,phase:nep/mconstant,nattenuatio:
zzzkz .
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Reflection Coefficient (1)
)(z
Source Load
Reflection Coefficient (or: Voltage Reflection Coefficient) at
zz0
any po nt a ong t e transm ss on ne s e ne as:
)()(
zVz
)()()( zVzVzV z jkz
eVzV 0)(
))(1)(( zzV jkz
eVzV
0)(
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Reflection Coefficient (2)
)( 1z )( 2z
Source Loadd
z0 1z 2z
11 201)( jkzjkzeVzV
z
1
01
1)(
jkzeVzV
22 2
2
2
2 )()0()0()(jkzjkz
ezez
jkdzzjkezezz
2
2
)(2
21 )()()(12
djdeez
22
2 )(
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Input Impedance (1)
)(zZin
SourceLoad
)(z
zz0
Defined as: )(zV )(zI
in
)](1[)()()( zVzVzVzV
)](1[)()()(0
zZVzIzIzI
0
0
0 )(
)(
)(or)(1
)(1
)( ZzZ
ZzZ
zz
z
ZzZin
in
in
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Input Impedance (2)
)( 1zZin )( 2zZin
SourceLoad
1 2
d
z0 1z 2z
From 01 )( ZzZ in 01
1)( ZzZ in
02
022
)(
)()(
ZzZ
ZzZz
in
in
dezz
2
21 )()(
)tanh()()( 0201
dZzZZzZ inin
20 in
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Input Impedance (3)
inZ
Sourced LZ
z0
Im edance Transformation Formula:
d 0
)tanh(00
dZZZZ Lin
0 L
)tanh(0 dZZin Note: as d
)tanh(00
dZZ inL
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Input Impedance (4)
inin ZY /1
Sourced LZ
z0
z
d 0
)(10
zY
ZY
in
in
)tanh()tanh(
0
00
dYYdYYYY
L
Lin
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Lossless Terminated Transmission Line (1)
inZ
Sourced LZ
z0
zd 0
real,,,0 0C
Zjk
02 ZZindj
0ZZinL constantL
tan dZZ
)tan(00
djZZ Lin
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Lossless Terminated Transmission Line (2)
inZ
Sourced LZ
z0
)tan( dZZ
d 0
)tan(00
djZZ Lin
Discussions:
2)12(or,
4When(1)
2
Z
ndnd
.0,if
;,
LL
inL
L
in
ZZ
ZZZ
Z
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Lossless Terminated Transmission Line (2)
inZ
Sourced LZ
z0
tan dZZ
d 0
)tan(00
djZZZZ
L
in
Lin ZZndn
d ,or,When(2)
coto.c.When4
)tan((s.c.)0When(3) 0
dZZZ
djZZZ inL
0,When(5)00
ZZZZinL
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Average Power on Lossless
Transmission Line)(zI
)()()( zVzVzV
SourceLoad
)(zV
00
)()()( Z
zV
Z
zVzI
zz0
)])(Re[(2
1
)]()(Re[2
)(
**
*
Z
V
Z
VVV
zIzVzP
]||||
Re[2
1
22
0
**
0
2
0
2
Z
VVVV
Z
V
Z
V
(constant)22
0
0
0
0 PPZZ
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Chain Matrix of Transmission Line (1)
)0(I )(dI
SourceLoad
)0(V )(dV
zd0
00)0( VVV
jkdjkd
eVeVdV
00)(
0
0
0
0)0(Z
V
Z
VI
0
0
0
0)(Z
eV
Z
eVdI
jkdjkd
)0(
)0(
cossin
sincos
)(
)( 0
I
V
kdkd
j
kdjZkd
dI
dV
0
Chain Matrix A
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Chain Matrix of Transmission Line (2)
Z01k1
Z03k3
Z02k2
Z0,N-2kN-2
Z0,N-1kN-1
Z0NkN
d3d
2 dN-2d
N-1
2321tot AAAAA NN