2.transmission line theory
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TRANSMISION LINE THEOR
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Outline2
1. The Lumped-Element Model for a Transmission L
2. Field Analysis of Transmission Lines
3. The Terminated Lossless Transmission Line
4. The Smith Chart
5. The QuarterWave Transformer
6. Generator and Load Mismatches
7. Lossy Transmission Line
8. Transients on Transmission Lines
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Transmission Line3
Transmission Line: two-port network with sending end an
receiving end
Sending end: Generator circuit
Receiving end: Load circuit
Vg
Rg
RL
A
A
B
B
Sending-end
portReceiving-end
port
Generator Circuit
Load Circuit
Transmission Line
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General consideration4
Transmission Line are commonly wires in Low-frequencyelectrical circuits
We often analyze circuits without considering dispersive effect
of wire.
Question: When do we need to consider transmission line effe
Ans. It depends on length of line land frequencyf
of generator circuit
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General consideration5
Vg Rg
RL
A
A
B
B
Generator Circuit
Load Circ
Transmission Line
xl0
Phas
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Outline7
1. The Lumped-Element Model for a Transmission L
Wave Propagation on a Transmission Line
The Lossless Line
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Lumped-Element Model8
R: series resistance per unitlength, for both conductors
[/m]
L: series inductance per unit
length, for both conductors
[H/m] G: shunt conductance per unit
length [S/m]
C: shunt capacitance per unit
length [F/m]
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10
Transmission Line Equation (Cont
For the sinusoidal steady-state condition, with cosine-based phasors, the t
equations simplify to
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11
Wave Propagation on a Transmissio
The two equations can be solve simultaneously to give wave equations for
where is the complex propagation constant which is the function of freque
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12
Wave Propagation on a Tx. Line (C
Traveling wave solution can be found as:
where the e z
term represents wave propagation in the +z direction, and therepresents wave propagation in the z direction. The current on the line:
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Characteristic Impedance13
The characteristic impedance relate to the voltage and current on the li
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Voltage Waveform14
f is the phase angle of the complex voltage V
The wavelength and the phase velocity are
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In many practical cases the loss of the line is very small so can be n
resulting in a simplification of the results. SettingR = G = 0 the propagation constant is given
The characteristic impedance reduce to
The Lossless Line16
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Voltage and Current on a Lossless T17
The wave length and the phase velocity are
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Outline18
1. The Lumped-Element Model for a Transmission L
2. Field Analysis of Transmission Lines
3. The Terminated Lossless Transmission Line
4. The Smith Chart
5. The QuarterWave Transformer
6. Generator and Load Mismatches
7. Lossy Transmission Line
8. Transients on Transmission Lines
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Field Analysis of Transmission Lines19
The self-inductance per unit length:
The time-average stored electric energy per unit length:
The capacitance per unit length:
The time-average stored magnetic energy per unit length:
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Field Analysis of Transmission Lines (C20
The time-average power dissipated per unit length in a lossy dielectri
The power loss per unit length due to the finite conductivity of the co
The series resistance R per unit length:
The shunt conductance per unit length:
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Voltage Reflection Coefficient23
The time-average power flow along the line at the positionz
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Return Loss24
When the load is mismatched, not all of the available power from the
delivered to the load. The loss cause by the reflection at load is call the return loss and is de
Matched load (=0): RL=
Total reflection (=1): RL= 0
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Standing Wave Ratio25
The magnitude of the voltage on the line:
Standing wave ratio is the ratio of Vmax to Vmin as follow
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Reflection Coefficient and Input Impeda26
The reflection coefficient atz = - l
The imput impedance atz = - l
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Transmission Line Impedance E27
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Outline28
1. The Lumped-Element Model for a Transmission Line
2. Field Analysis of Transmission Lines3. The Terminated Lossless Transmission Line:
Terminated lossless transmission line
4. The Smith Chart
5. The QuarterWave Transformer6. Generator and Load Mismatches
7. Lossy Transmission Line
8. Transients on Transmission Lines
Special Cases of Lossless Terminated
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Special Cases of Lossless Terminated Short-circuited Transmission Line
29
Short-circuited Tx. Line:
The voltage and current on the line:
The input impedance:
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Short-circuited Transmission Line30
Open-circuited Transmission Line
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Open-circuited Tx. Line:
The voltage and current on the line:
Open-circuited Transmission Line31
The input impedance:
(
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Open-circuited Transmission Line (Co32
H lf l h T Li
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Half-wavelength Tx. Line33
If the length of the line is half of the wave length
Q f
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Quarter-wavetransformer34
If the length of the line is a quarter wavelength, then the in
impedance is given by
a quarter wavelength transformer
O i H k
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Onsite Homework35
P 2.3, 2.8, 2.10,
O tli
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Outline36
1. The Lumped-Element Model for a Transmission L
2. Field Analysis of Transmission Lines3. The Terminated Lossless Transmission Line
4. The Smith Chart
5. The QuarterWave Transformer
6. Generator and Load Mismatches7. Lossy Transmission Line
8. Transients on Transmission Lines
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N li d L d I d
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Normalized Load Impedance38
0ZZz LL
The nomalized load impedance:
G
G 1
1Lz
LLL jxrz
The nomalized load impedance is a complex quantity compose
normalized load resistance and normalized load reactance
ir
irL
j
jz
GG
GG
1
1
GG
G
GG
GG
22
22
22
1
2
1
1
ir
i
L
ir
ir
L
x
r
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Normali ed Load Impedance (cont
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Normalized Load Impedance (cont41
0LrWhich circle yield
The Smith chart
simply a depiction onr- i plane, of the
families of circles
family rL and xL fa
plotted for sele
values ofrL andxL
Normalized Load Impedance
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42
Consider norma
impedance (1+j2)
Point A: the interse
of the rL = 1 con
resistance circle
the xL = 2 con
reactance circle segm
The norma
impedance of point
Normalized Load Impedance
E Fi d th fl ti ffi i t
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Find point P represents normalized load impedance
zL= (2-j).
Find the correcsponding reflection coefficient.
Ex: Find the reflection coefficient43
Point P: zL= (2-j)OR
OP G
r .26
Input Impedance
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Input Impedance44
G
G
zj
zj
in
e
eZzZ
2
2
0
1
1)(
G
G
zj
zj
inin
e
e
Z
Zzz
2
2
0 1
1)(
L
L
in zz G
G
1
1
)( G
G
1
1
Lz
The same form!!!
On Smith chart, transforming to l means maintainingconstant and decreasing the phase r
Input Impedance on Smith Chart
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Input Impedance on Smith Chart
Consider a 50 Ohm lossless transmission line terminated in a load
impedance (100j50) Ohm. Find Zin at a distance l = 0.1 from l
45
Z
Zz LL 2
0
Point A repre
zL = 2
Point B repre
zin =0.6j0.
SWR Voltage Maxima and Minim
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SWR, Voltage Maxima, and Minim46
Point A represents
zL = 2+j
S = 2.6 (at Pmax
lmax =(0.25-0.213)
=0.037
lmin =(0.037+0.25)
=0.287
A
0.287
SWR Circle
0 Pmax
0.213
0.037
Pmin
Impedance to Admittance Transform
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Impedance to Admittance Transform47
The admittance Y is the reciprocal ofZ
2222211
RXj
XRR
XRjXR
jXRZY
Conductance GSusceptan
The normalized admittancey:
jbgY
Bj
Y
G
Y
Yy
000
Impedance to Admittance Transform
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Impedance to Admittance Transform48
The admittanceyL:
Rotation /4 on the Smith chart transformszL intoyL
GG
111
L
Lz
y
Lj
j
in ye
elz
G
G
G
G
1
1
1
1)4/(
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Single-stub Matching (Cont.)
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g g ( )50
The normalized impedance is
jj
Z
Zz LL
5.0
50
5025
0
(point A in Smith chart)
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Single-stub Matching (Cont.)
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g g ( )52
In the admittance domain , rL circles becomeyL circles,
xL circles become bL circles
The normalized admittance is
8.04.05025
500 jjZ
Zy
L
L
(point B in Smith chart,
0.25 rotate from A)
It is easier to work with admittances
than with impedance
Single-stub Matching
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g g53
Single-stub Matching
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Single stub Matching54
Solution for Point C:
At C,yd = 1+j1.58
Distance between B and C
d1 = (0.178-0.115)=0.063
Need:ys = - j1.58 (point F)
l1 = (0.34-0.25)=0.09
( distance from E to F)
Step 1: Select dso that
yLbecomes :
yd = 1+b
There are 2 solutions:
Point C and Point D
Single-stub Matching
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Single stub Matching55
Solution for Point D:
At D,yd = 1-j1.58Distance between B and D
d2 = (0.178-0.115)
=0.207
Need:
ys = j1.58 (point G)l2 = (0.25+0.16)=0.41
(distance from E to G)
Single-stub Matching (Cont.)
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Two Solutions
Single-stub Matching (Cont.)
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d1 = 0.063
l1 = 0.09
57
d2 = 0.207
l2 = 0.41
Outline
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1. The Lumped-Element Model for a Transmission L
2. Field Analysis of Transmission Lines3. The Terminated Lossless Transmission Line
4. The Smith Chart
5. The QuarterWave Transformer
6. Generator and Load Mismatches7. Lossy Transmission Line
8. Transients on Transmission Lines
Quarter-wave transformer
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Q59
Ex: Quarter-wave transformer
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Q60
A 50 Ohm lossless Tx line is to be matched to a resistive loa
impedance 100 Ohm via a quarter-wave section thereb
eliminating reflections along the feedline. Find the characterist
impedance of the quarter-wave transformer
Ex: Quarter-wave transformer (cont
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Q (61
Solution: To eliminate reflections at terminal AA, the in
impedance Zin looking into the quarter-wave line should be equa
Z01, from following Eq.,
L
inZZZ
2
02
7.701005002 LinZZZ
Outline
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1. The Lumped-Element Model for a Transmission L
2. Field Analysis of Transmission Lines3. The Terminated Lossless Transmission Line
4. The Smith Chart
5. The QuarterWave Transformer
6. Generator and Load Mismatches7. Lossy Transmission Line
8. Transients on Transmission Lines
Input impedance
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Voltage reflection coefficient
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is determined by the ratio:
64
Is real or complex?! complex
For passive load | | 1
Passive load Re[ZL] 0
Also, lossless Z0= Real 0
Re[ZL /Z0] 0
11/
1/
0
0
0
0
G
ZZ
ZZ
ZZ
ZZ
L
L
L
L
Standing point of transmission line
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VSWR
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VSWR: Voltage standing wave ratio
66
);1[ S
S=1 ||=0 No reflected power
S= ||=1 All power reflect
1
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67
Let
Then we will have
Matched Case
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Load match to the line
Generator matched to loaded line
Conjugate matching
Outline
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1. The Lumped-Element Model for a Transmission L
2. Field Analysis of Transmission Lines3. The Terminated Lossless Transmission Line
4. The Smith Chart
5. The QuarterWave Transformer
6. Generator and Load Mismatches7. Lossy Transmission Line (self study)
8. Transients on Transmission Lines
Transients on transmission line
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Treatment of wave propagation on Tx line: Focus on the analysis of s
frequencies, time-harmonic signals under steady-state conditions
Tool: impedance matching & the use of Smith chart
Useful for a wide range of applications
Inappropriate for dealing with digital or wideband signals
Need to exam the transient behavior as a function of time
70
Basic approach
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0,0
0,1)(
t
ttU
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elsewhere,0
0,)(
0 tVtV
)()(
)()()(
00
21
tUVtUV
tVtVtV
V(t) is the sum of 2 unit step functions :
Analyze the response of Tx line
to a unit step function ( DC turn on voltage) V0 U(t)
From the linearity of the line, the pulse transient response is then
obtained by superposition
Consider the simple case of a single
rectangular pulse of amplitude V0 duration :
Transient response
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Consider a step voltage V0 U(t) applied to a Tx line ofZ0terminated in a real loadZL
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g
g
VZR
ZV
0
0
1
0
1ZR
VI
g
g
Switch close at t = 0+
Initial voltage V1+ & currentI1
+
Transient response (cont.)
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What happens as this voltage step echoes off the ends of the
line?
Consider the line withRg = 4Z0 &ZL = 2Z0
73
3
1
0
0
G
ZZ
ZZ
L
LL
5
3
0
0
G
ZR
ZR
g
g
g
The length of the line is l
One way transit time : T=l/up
Transient response (cont.)
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Steady state voltage V
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G
GGGGG
GGGGGGGGG
2
1
22
1
23222
1
332211
11
11
1
xxV
V
V
VVVVVVV
L
gLgLL
gLgLgLgLL
Lx GG
The series inside the square bracket is the binominal series of
funtion
1for11
1 2
xxxxThen we will get
gL
LVVGG
G
1
1
1
Lg
Lg
ZR
ZVV
Bounce diagram
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A simple graphical way to keep track of the bounce history
Axes of bounce diagram represent
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z
t
Bounce diagram (Cont.)
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Voltage vs time atz=l/4 for a circuit
withg =3/5 &L=1/3
Bounce diagram
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Onsite Homework & Homework79
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Onsite P 2.8, 2.12
Home work: P 2.3, 2.10, 2.16, 2.20, 2.21,2.31
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