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EELE 3332 – Electromagnetic II
Chapter 11
Transmission Lines
Islamic University of Gaza
Electrical Engineering Department
Dr. Talal Skaik
2012 1
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11.1 Introduction
Wave propagation in unbounded media is used in radio or TV
broadcasting, where the information being transmitted is meant for
everyone who may be interested.
Another means of transmitting power or information is by guided
structures. Guided structures serve to guide (or direct) the
propagation of energy from the source to the load.
Typical examples of such structures are transmission lines and
waveguides.
Waveguides are discussed in the next chapter; transmission lines
are considered in this chapter.
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Transmission lines are commonly used in power distribution (at
low frequencies) and in communications (at high frequencies).
A transmission line basically consists of two or more parallel
conductors used to connect a source to a load.
Typical transmission lines include coaxial cable, a two-wire line, a
parallel-plate line, and a microstrip line.
Coaxial cables are used in connecting TV sets to TV antennas.
Microstrip lines are particularly important in integrated circuits.
Transmission line problems are usually solved using EM field
theory and electric circuit theory, the two major theories on which
electrical engineering is based.
Introduction
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Figure 11.1 Typical transmission lines in cross-sectional view: (a) coaxial line, (b) two-wire line, (c) planar line, (d) wire above conducting plane, (e) microstrip line.
Introduction
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Transmission line parameters are:
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11.2 Transmission Line Parameters
R: Resistance per unit length. (Ω/m)
L: Inductance per unit length. (H/m)
G: Conductance per unit length. (S/m)
C: Capacitance per unit length. (F/m)
Distributed parameters of a two-conductor
transmission line
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Transmission Line Parameters
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The line parameters R, L, G, and C are uniformly distributed
along the entire length of the line.
For each line, the conductors are characterized by σc,µc,εc,=ε0,
and the homogeneous dielectric separating the conductors is
characterized by σ,µ,ε.
G ≠1/R; R is the ac resistance per unit length of the conductors
comprising the line and G is the conductance per unit length due to
the dielectric medium separating the conductors.
For each line:
Transmission Line Parameters
and G
LCC
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Fields inside transmission line
•Transmission lines transmit TEM waves.
•V proportional to E,
•I proportional to H
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11.3 Transmission Line Equations
Two-conductor transmission lines support a TEM wave; E and H
are perpendicular to each other and transverse to the direction of
propagation.
E and H are related to V and I:
Using V and I in solving the transmission line problem is simpler
than solving E and H (requires Maxwell’s equations).
Examine an incremental portion of length Δz of a two-conductor
transmission line.
E. , I= H.V dl dl
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Transmission Line Representation
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Transmission Line Equations ( , )
Using KVL:- ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
Taking the limit as z 0 leads to:
( , ) ( , ) ( , )
I z tV z t R zI z t L z V z z t
t
V z z t V z t I z tor RI z t L
z t
V z t I z tRI z t L
z t
equivalent circuit model
of a two-conductor T.L.
of differential length z.
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Using KCL:- ( , ) ( , )
( , ) ( , )= ( , ) ( , )
( , ) ( , ) ( , ) ( , )
Taking the limit as z 0 leads to:
(
I z t I z z t I
V z z tI z t I z z t G zV z z t C z
t
I z z t I z t V z z tor GV z z t C
z t
I z
, ) ( , )( , )
t V z tGV z t C
z t
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The time domain form of the transmission line equations:
( , ) ( , )( , )
( , ) ( , )
( , )
If we assume harmonic time dependence so that:
V( , )
V z t I z tRI z t L
z t
I z t V z tGV z t C
z t
z t
j t
s
j t
s
s s
=Re[ (z) ]
I( , )=Re[ (z) ]
where and are the phasor forms of ( , ) and ( , ),
( )
( )
ss
ss
V e
z t I e
V I V z t I z t
dVR j L I
dz
dIG j C V
dz
13
Transmission Line Equations
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s
2
2
s
2
2
( ) , ( )
To solve the previous equations, take second derivative of gives
( )( )
Now take second derivative of gives
( )(
s ss s
ss
s
dV dIR j L I G j C V
dz dz
V
d VR j L G j C V
dz
I
d IG j C R j
dz
22
2
22
2
)
, the for voltage and current become
0
, where ( )( )
0
s
ss
ss
L I
Hence
d VV
dzj R j L G j C
d II
dz
wave equations
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Transmission Line Equations
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( )( )
: is the propagation constant
: attenuation constant (Np/m or dB/m)
: phase constant ( rad/m)
2wavelength is: = , wave velocity is:
The soluti
j R j L G j C
u f
0 0 0 0
0 0 0 0
ons to the wave equations are:
,
where , , , are wave amplitudes.
wave traveling along +z direction.
- sign wave traveling along -z direction
z z z z
s sV V e V e I I e I e
V V I I
sign
. 15
+z -z +z -z
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0 0
0 0
0 0
In time domain:
( , ) Re[ ( ) ]
( , ) cos( ) cos(
z z
s
z z
s
j t
s
z z
V V e V e
I I e I e
V z t V z e
V z t V e t z V e
0 0
)
Similarly for current:
( , ) cos( ) cos( )z z
t z
I z t I e t z I e t z
Transmission Line Equations
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0 0
0 0
0 0
0 0
0
( ) , ( )
( )since ( ) ( ),
( ) ( )
Characterestic 1 ,
Admittance
z z
z z
o o o
o o o
V z V e I z I e
dV zR j L I z
dz
V e R j L I e
V R j L R j L VZ R jX
I G j C I
R j LZ R jX Y
G j C Z
017
Characteristic Impedance, Z0
The Characteristic Impedance Z0 of the line is the ratio of the
positively travelling voltage wave to the current wave at any point on
the line.
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For lossless line, R=G=0
Since = ( )( )
0, ,
1 2 ,
0o o o
j R j L G j C
j LC
u fLC
LX Z R
C
18
Lossless Line (R=G=0) A transmission line is said to be lossless if the conductors of the line
are perfect (σc ≈ ∞) and the dielectric medium separating them is
lossless (σ ≈ 0)
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Any signal that carries significant information must has some
non-zero bandwidth. In other words, the signal energy (as well as
the information it carries) is spread across many frequencies.
If the different frequencies that comprise a signal travel at
different velocities, that signal will arrive at the end of a
transmission line distorted. We call this phenomenon signal
dispersion.
Recall for lossless lines, however, the phase velocity is
independent of frequency—no dispersion will occur!
Of course, a perfectly lossless line is impossible, but we find
phase velocity is approximately constant if the line is low-loss.
Distortionless Line
1/u LC
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A distorionless line results if the line parameters are such that
, = ( )( )= 1 1
= 1
R G
L C
j L j CThus R j L G j C RG
R G
j CRG j
G
or RG
,
/ 1/ (frequency independent)
LC
u LC
20
Distortionless Line (R/L=G/C)
A distortionless line is one in which the attenuation constant α is
frequency independent while the phase constant β is linearly
dependent on frequency.
α does not depend on
frequency, whereas β is a linear
function of frequency.
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Notes:
Shape distortion of signals happen if α and u are frequency
dependent.
u and Z0 for distortionless line are the same as lossless line.
A lossless line is also a distortionless line, but a distortionless line
is not necessarily lossless.
Lossless lines are desirable in power transmission, and telephone
lines are required to be distortionless. 21
Distortionless Line (R/L=G/C)
0
1 / (Real)
1 /
1u=
R j L RR j L R LZ
G j C G j C G G C
LC
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To achieve the required condition of R/L=G/C for a transmission line,
L may be increased by loading the cable with a metal with high
magnetic permeability (μ).
A common practice is to replace repeaters in long lines to maintain the
desired shape and duration of pulses for long distance transmission. 22
Distortionless Line – Practical use
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Zo
General
Lossless
Distortionless
CjG
LjRZo
o o
LZ R
C
))(( CjGLjR
C
LRZ oo
Summary
0 j LC
RG j LC
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Example 11.1 An air line has characteristic impedance of 70 Ω and a phase
constant of 3 rad/m at 100 MHz. Calculate the inductance per meter
and the capacitance per meter of the line.
0 0
0
6
0
An air line can be regarded as lossless line because 0
. 0 and =0
= =
1Deviding the two equations yields:
3 6
2 100 10 (70)
cand Hence R G
LZ R
C
LC
R
C
or CR
2 2 12
0
8.2 pF
= (70) (68.2 10 ) 334.2 nH/mL R C
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0
0
3
0
2 6
0
1A distortionless line has or ,
, =
(20 10 )(60) 1.2 /m
400 10 = 333 S/m
1.2
1 by gi
RCRC GL G u
L LC
L C RZ RG R
C L Z
R Z
Since RG GR
LDividing Z u
C LC
0
8
ves
60= 333 nH/m
0.6(3 10 )
ZL
u
25
Example 11.2
A distortionless line has Z0=60 Ω, α=20 mNp/m, u=0.6c, where c is
the speed of light. Find R,L,G , C and λ at 100 MHz.
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0
0 8
0
8
8
1 by gives
1 1 1= 92.59 pF/m
0.6(3 10 )60
0.6(3 10 )= 1.8 m
10
LMultiplying Z u
C LC
uZ CC uZ
u
f
Example 11.2 – solution continued
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11.4 Input impedance, standing wave ratio, power
Consider a transmission line of length l, characterised by and Z0,
connected to a load ZL. Generator sees the line with the load as an
input impedance Zin.
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0 0
0 0 0 00
0 0 0 0
( )
At generator terminals (sen
( ) , =
ding e
n
Let
d):
z z
s
z z
s
V z V e V e
V V V VI z e e Z
Z Z I I
0 0
0 0 0 0 0 0 0
0 00
0 0 0 00 0
in
0 0
( 0), ( 0), Substitute in prev. equs.:
1 2
... (1) 1
2
If the input impedance at the terminals is Z ,
then , gin
g
in g
V V z I I z
V V V V V Z I
V VI
V V Z IZ Z
VZV V I
Z Z Z
in gZ 28
Input impedance
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0 00 0
0 0
0 0 0 0
0 0
00 0
( ) , ( )
Let ( ), ( ), Substitute in prev.
At the load:
equs.
1 2
1
2
z z z z
s s
L L
l l lL L L
l l
L
V VV z V e V e I z e e
Z Z
V V z l I I z l
V V e V e V V Z I e
V VI e e
V VZ Z
0
in
... (2)
Now determine the input impedance Z = ( ) / ( ) at any
point on the line.
l
L L
s s
Z I e
V z I z
29
Input impedance
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0 00 0 0 0
0 0
0 0 00in
0 0 0
At the generator, recall , ,
( )Z =
( )
Substituting eq. 2 and utilizing
the fact that:
cosh , sinh , 2 2
or t
s
s
l l l l
V VV V V I then
Z Z
Z V VV z V
I z I V V
e e e el l
00
0
sinhanh
cosh
tanhwe get (General - Lossy Lin
nh)
te
a
l l
l l
Lin
L
l e el
l e e
Z Z lZ Z
Z Z l
30
Input impedance
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00
0
00
0
tanh (General - Lossy Line)
tanh
For a lossless line, = , tanh tan , then
tan
tan
Lin
L
Lin
L
Z Z lZ Z
Z Z l
j j l j l
Z jZ lZ Z
Z jZ l
( )
Note: To find at a distance
Loss
less L
' from load,
e
rep
in
inZ
l
00
0
lace by ' :-
tan '
tan '
Lin
L
l l
Z jZ lZ Z
Z jZ l
31
Input impedance (Lossless Line)
βl is known as electrical length, in degrees or radians
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0
0
0 0
0 0
Define as the voltage reflection coefficient (at the load), as the
ratio of the voltage reflection wave to the incident wave at the load,
1
2 , an
1
2
L
l
L l
l
L L
l
L L
V e
V e
V V Z I e
Since
V V Z I e
0
0
Volt
d
age Reflection coefficient at l
oa d ( )
L L L
LL
L
V Z I
Z Z
Z Z
32
Reflection Coefficient, (at load)
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0
0
0 00 0
0 0
0 0 0 0
0 0
Define (at 0) as the voltage reflection coefficient at the source,
as the ratio of the voltage reflection wave to the incident wave at source,
1
2
1
2
z
V e V
V e V
V V Z I
Since
V V Z
0 0
0 0
00
0
V
, and
oltage Reflection coefficient at
( )source
in
in
in
V Z I
I
Z Z
Z Z
33
Reflection Coefficient, (at generator)
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20 0
0 0
0 0
That is: ( )
Thus the current reflection coefficient at the load is
/
zz
z
l l
L
V e Vz e
V e V
I e I e
34
Reflection Coefficient
The current reflection coefficient at any point on the line is the
negative of the voltage reflection coefficient at that point.
The voltage reflection coefficient at any point on the line is the ratio
of the reflected voltage wave to that of the incident wave.
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max max 0
min min 0
0
The standing wave ratio s is defined as: (as we did for plane waves)
1=
1
When load is ( ) Total Transmission
L LL
L L
L
V I Z Zs
V I Z Z
perfectly matched Z Z
0 1
When load is a :
Total Reflection
1 1
When load is an :
Total Reflection
1 1
L
L L
L L
s
short circuit
s
open circuit
s
35
Whenever there is a reflected wave, a standing wave will form
out of the combination of incident and reflected waves.
Standing Wave Ratio
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36
*
2
20
0
The time-average power flow along the line at the point is:
1Re[ ( ) ( )]. For a line, this can be reduced to:
2
1 , 2
The average powe
ave s s
ave
z
P V z I z lossless
VP
Z
2 2 2
0 0 0 0
r flow is constant at any point on the line.
The total power delivered to the load ( ) is equal to the incident
power ( / 2 ) the reflected power ( / 2 )
If 0, maximum p
minus
av
l
P
V Z V
le s
Z
oss s
ower is delivered to the load, while no power
is delivered for 1.
The above discussion assumes that the generator is matched.
Power
Incident
Power (Pi)
Reflected
Power (Pr)
2 22
0 0 0 0/ 2 / 2aveP V Z V Z
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Short Circuited Line
(ZL=0)
Open Circuited Line
(ZL=∞)
Matched Line
(ZL=Z0)
37
Special Cases , ZL=0, ZL=∞, ZL=Z0
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38
Shorted Line (ZL=0)
00 0
0
0
0
tantan
tan
1, (Total Reflection)
Lin
L
LL
L
Z jZ lZ Z jZ l
Z jZ l
Z Zs
Z Z
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39
Open-Circuited Line (ZL=∞)
00
0
0
00 0
0
0
0
tan, ( )
tan
1 tan
cot tan
tan
1, (Total Reflection)
Lin L
L
L
in
L
LL
L
Z jZ lZ Z Z
Z jZ l
Zj l
Z ZZ Z jZ l
j lZj l
Z
Z Zs
Z Z
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40
Matched Line (ZL=Z0)
0 0
0
0
Most desired case from practical point of view.
0, 1
The whole wave is transmitted, and there is no reflection.
The incident power is fully absorbed by the load.
(
in in
LL
L
Since Z Z Z Z
Z Zs
Z Z
Maximum power transfer)
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( ) Since 1 Np=8.686 dB
8 = 0.921 /
8.686
= + 0.921 1
=2(0.921 1)
Solution
a
Np m
j j
l j
1.84 2j
41
Example 11.3
A certain transmission line 2 m long operating at ω=106 rad/s has
α=8 dB/m, β=1 rad/m, and Z0= 60+j40 Ω. If the line is connected
to a source of 10∟00 V, Zg=40 Ω and transmitted by a load of
20+j50 Ω, determine
(a) The input impedance
(b) The sending end current
(c) The current at the middle of the line.
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00
0
tanh tanh 1.84 2 1.033 0.03929
tanh
tanh
20 50 (60 40)(1.033 0.03929)(60 40)
60 40 (20 50)(1.033 0.03929)
60.25 38.79
( ) The sending end cu
Lin
L
in
in
l j j
Z Z lZ Z
Z Z l
j j jZ j
j j j
Z j
b
0
0
rrent is ( 0)
1093.03 21.15
60.25 38.79 40
g
in g
I z I
VI mA
Z Z j
42
Example 11.3 – Solution continued
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0 0
0
0 0 0
0 0
0 0 0 0
( ) To find the current at any point, we need V and V .
93.03 21.15
(71.66 32.77 )(0.09303 21.15 ) 6.667 11.62
1 1V 6.667 11.62 (60 40)(0.09303 21.15)
2 2
6.687 12.
in
c But
I mA
V Z I
V Z I j
0
0 0 0 0
0
08
1 1V 6.667 11.62 (60 40)(0.09303 21.15)
2 2
0.0518 260
V Z I j
43
Example 11.3 – Solution continued
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0 0
/2 /20 0
0 0
12.08 0.921 1 260 0.921 1
At the middle of the line, / 2, z= / 2 0.921+j1, Hence the
current at this point is:
V V( / 2)
(6.687 ) (0.0518 ) =
60 40 60 40
No
l l
s
j j j j
z l l
I z l e eZ Z
e e e e
j j
0 0 0 0
0 0
0
0
12.08 0.921 57.3 260 0.921 57.3
33.69 33.69
78.91
te that 1 is in radians and is equivalent to j57.3 ,( 1 180/ ):-
(6.687 ) (0.0518 )( / 2)=
72.1 72.1
0.0369 0.001805
j j j j
s j j
j
j j
e e e e e eI z l
e e
e
0283.61
0
6.673 34.456 mA
=35.10 281 mA
je
j
44
Example 11.3 – Solution continued