This document outlines the syllabus for a course on transmission lines and waveguides. The course objectives are to introduce various transmission line types and associated losses, impart an understanding of impedance transformation and matching using tools like the Smith chart, and cover topics like filter theories and waveguide principles. The five units cover transmission line theory, high frequency transmission lines, impedance matching, passive filters, and waveguides and cavity resonators. Key concepts taught include propagation of signals on transmission lines, signal analysis at radio frequencies, guided radio propagation, and the use of cavity resonators.
2. Overview of syllabus
OBJECTIVES
To introduce the various types of transmission lines and to
discuss the losses associated.
To give thorough understanding about impedance
transformation and matching.
To use the Smith chart in problem solving.
To impart knowledge on filter theories and waveguide
theories
3. Overview of syllabus
OUTCOMES
Upon completion of the course, students will be able
to Discuss the propagation of signals through transmission
lines.
Analyze signal propagation at Radio frequencies.
Explain radio propagation in guided systems.
Utilize cavity resonators
4. Overview of syllabus
TEXT BOOK
1. John D Ryder, “Networks lines and fields”, Prentice Hall of India,
New Delhi, 2005
REFERENCES
1.William H Hayt and Jr John A Buck, “Engineering Electromagnetics”
Tata Mc Graw-Hill Publishing Company Ltd, New Delhi, 2008
2.David K Cheng, “Field and Wave Electromagnetics”, Pearson
Education Inc, Delhi, 2004
3.John D Kraus and Daniel A Fleisch, “Electromagnetics with
Applications”, Mc Graw Hill Book Co,2005
4.GSN Raju, “Electromagnetic Field Theory and Transmission Lines”,
Pearson Education, 2005
5.Bhag Singh Guru and HR Hiziroglu, “Electromagnetic Field
Theory Fundamentals”, Vikas Publishing House, New Delhi, 2001.
6. N. Narayana Rao, “ Elements of Engineering Electromagnetics” 6
5. Overview of syllabus
UNIT I -TRANSMISSION LINE THEORY
General theory of Transmission lines - the transmission line
- general solution - The infinite line - Wavelength, velocity
of propagation - Waveform distortion - the distortion-less
line - Loading and different methods of loading - Line not
terminated in Z0 - Reflection coefficient - calculation of
current, voltage, power delivered and efficiency of
transmission - Input and transfer impedance - Open and
short circuited lines - reflection factor and reflection loss.
6. Overview of syllabus
UNIT II -HIGH FREQUENCY TRANSMISSION LINES
Transmission line equations at radio frequencies - Line of
Zero dissipation - Voltage and current on the dissipation-
less line, Standing Waves, Nodes, Standing Wave Ratio -
Input impedance of the dissipation-less line - Open and
short circuited lines - Power and impedance measurement
on lines - Reflection losses - Measurement of VSWR and
wavelength.
7. Overview of syllabus
UNIT III-IMPEDANCE MATCHING IN HIGH
FREQUENCY LINES
Impedance matching: Quarter wave transformer -
Impedance matching by stubs - Single stub and
double stub matching - Smith chart - Solutions of
problems using Smith chart - Single and double
stub matching using Smith chart.
8. Overview of syllabus
UNIT IV-PASSIVE FILTERS
Characteristic impedance of symmetrical networks
- filter fundamentals, Design of filters: Constant K -
Low Pass, High Pass, Band Pass, Band
Elimination, m- derived sections - low pass, high
pass composite filters.
9. Overview of syllabus
UNIT V -WAVE GUIDES AND CAVITY RESONATORS
General Wave behaviours along uniform Guiding
structures, Transverse Electromagnetic waves, Transverse
Magnetic waves, Transverse Electric waves, TM and TE
waves between parallel plates, TM and TE waves in
Rectangular wave guides, Bessel‟s differential equation
and Bessel function, TM and TE waves in Circular wave
guides, Rectangular and circular cavity Resonators.
11. Transmission Line
Has two conductors running parallel
Can propagate a signal at any frequency (in theory)
Becomes lossy at high frequency
Can handle low or moderate amounts of power
Does not have signal distortion, unless there is loss
May or may not be immune to interference
Does not have Ez or Hz components of the fields (TEMz)
Properties
Coaxial cable (coax)
Twin lead
(shown connected to a 4:1
impedance-transforming balun)
11
12. Transmission Line (cont.)
CAT 5 cable
(twisted pair)
The two wires of the transmission line are twisted to reduce interference and
radiation from discontinuities.
12
15. Fiber-Optic Guide
Properties
Uses a dielectric rod
Can propagate a signal at any frequency (in theory)
Can be made very low loss
Has minimal signal distortion
Very immune to interference
Not suitable for high power
Has both Ez and Hz components of the fields
15
16. Fiber-Optic Guide (cont.)
Two types of fiber-optic guides:
1) Single-mode fiber
2) Multi-mode fiber
Carries a single mode, as with the mode on a
transmission line or waveguide. Requires the fiber
diameter to be small relative to a wavelength.
Has a fiber diameter that is large relative to a
wavelength. It operates on the principle of total internal
reflection (critical angle effect).
16
18. Waveguides
Has a single hollow metal pipe
Can propagate a signal only at high frequency: > c
The width must be at least one-half of a wavelength
Has signal distortion, even in the lossless case
Immune to interference
Can handle large amounts of power
Has low loss (compared with a transmission line)
Has either Ez or Hz component of the fields (TMz or TEz)
Properties
http://en.wikipedia.org/wiki/Waveguide_(electromagnetism) 18
19. Lumped circuits: resistors, capacitors, inductors
neglect time delays (phase)
account for propagation and
time delays (phase change)
Transmission-Line Theory
Distributed circuit elements: transmission lines
We need transmission-line theory whenever the length of
a line is significant compared with a wavelength.
19
20. Transmission Line
2 conductors
4 per-unit-length parameters:
C = capacitance/length [F/m]
L = inductance/length [H/m]
R = resistance/length [/m]
G = conductance/length [ /m or S/m]
Dz
20
21. 21
Transmission Line (cont.)
z
D
,
i z t
+ + + + + + +
- - - - - - - - - -
,
v z t
x x x
B
21
RDz LDz
GDz CDz
z
v(z+ z,t)
+
-
v(z,t)
+
-
i(z,t) i(z+ z,t)
22. 22
( , )
( , ) ( , ) ( , )
( , )
( , ) ( , ) ( , )
i z t
v z t v z z t i z t R z L z
t
v z z t
i z t i z z t v z z t G z C z
t
D D D
D
D D D D
Transmission Line (cont.)
22
RDz LDz
GDz CDz
z
v(z+ z,t)
+
-
v(z,t)
+
-
i(z,t) i(z+ z,t)
23. 23
Hence
( , ) ( , ) ( , )
( , )
( , ) ( , ) ( , )
( , )
v z z t v z t i z t
Ri z t L
z t
i z z t i z t v z z t
Gv z z t C
z t
D
D
D D
D
D
Now let Dz 0:
v i
Ri L
z t
i v
Gv C
z t
“Telegrapher’s
Equations”
TEM Transmission Line (cont.)
23
24. 24
To combine these, take the derivative of the first one with
respect to z:
2
2
2
2
v i i
R L
z z z t
i i
R L
z t z
v
R Gv C
t
v v
L G C
t t
Switch the
order of the
derivatives.
TEM Transmission Line (cont.)
24
25. 25
2 2
2 2
( ) 0
v v v
RG v RC LG LC
z t t
The same equation also holds for i.
Hence, we have:
2 2
2 2
v v v v
R Gv C L G C
z t t t
TEM Transmission Line (cont.)
25
26. 26
2
2
2
( ) ( ) 0
d V
RG V RC LG j V LC V
dz
2 2
2 2
( ) 0
v v v
RG v RC LG LC
z t t
TEM Transmission Line (cont.)
Time-Harmonic Waves:
26
27. 27
Note that
= series impedance/length
2
2
2
( )
d V
RG V j RC LG V LC V
dz
2
( ) ( )( )
RG j RC LG LC R j L G j C
Z R j L
Y G j C
= parallel admittance/length
Then we can write:
2
2
( )
d V
ZY V
dz
TEM Transmission Line (cont.)
27
28. 28
Let
Convention:
Solution:
2
ZY
( ) z z
V z Ae Be
1/2
( )( )
R j L G j C
principal square root
2
2
2
( )
d V
V
dz
Then
TEM Transmission Line (cont.)
is called the "propagation constant."
/2
j
z z e
j
0, 0
attenuationcontant
phaseconstant
28
29. 29
TEM Transmission Line (cont.)
0 0
( ) z z j z
V z V e V e e
Forward travelling wave (a wave traveling in the positive z direction):
0
0
0
( , ) Re
Re
cos
z j z j t
j z j z j t
z
v z t V e e e
V e e e e
V e t z
g
0
t
z
0
z
V e
2
g
2
g
The wave “repeats” when:
Hence:
29
30. 30
Phase Velocity
Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the
crest.
0
( , ) cos( )
z
v z t V e t z
z
vp (phase velocity)
30
31. 31
Phase Velocity (cont.)
0
constant
t z
dz
dt
dz
dt
Set
Hence p
v
1/2
Im ( )( )
p
v
R j L G j C
In expanded form:
31
32. 32
Characteristic Impedance Z0
0
( )
( )
V z
Z
I z
0
0
( )
( )
z
z
V z V e
I z I e
so 0
0
0
V
Z
I
+
V+(z)
-
I+ (z)
z
A wave is traveling in the positive z direction.
(Z0 is a number, not a function of z.)
32
33. 33
Use Telegrapher’s Equation:
v i
Ri L
z t
so
dV
RI j LI
dz
ZI
Hence
0 0
z z
V e ZI e
Characteristic Impedance Z0 (cont.)
33
34. 34
From this we have:
Using
We have
1/2
0
0
0
V Z Z
Z
I Y
Y G j C
1/2
0
R j L
Z
G j C
Characteristic Impedance Z0 (cont.)
Z R j L
Note: The principal branch of the square root is chosen, so that Re (Z0) > 0.
34
35. 35
0
0
0 0
j z j j z
z z
z j z
V e e
V z V e V
V e e e
e
e
0
0 cos
c
, R
os
e j t
z
z
V e t
v z t V z
z
V z
e
e t
Note:
wave in +z
direction
wave in -z
direction
General Case (Waves in Both Directions)
35
36. 36
Backward-Traveling Wave
0
( )
( )
V z
Z
I z
0
( )
( )
V z
Z
I z
so
+
V -(z)
-
I - (z)
z
A wave is traveling in the negative z direction.
Note: The reference directions for voltage and current are the same as
for the forward wave.
36
37. 37
General Case
0 0
0 0
0
( )
1
( )
z z
z z
V z V e V e
I z V e V e
Z
A general superposition of forward and
backward traveling waves:
Most general case:
Note: The reference
directions for voltage
and current are the
same for forward and
backward waves.
37
+
V(z)
-
I(z)
z
38. 38
1
2
1
2
0
0 0
0 0
0 0
z z
z z
V z V e V e
V V
I z e e
Z
j R j L G j C
R j L
Z
G j
Z
C
I(z)
V(z)
+
-
z
2
m
g
[m/s]
p
v
guided wavelength g
phase velocity vp
Summary of Basic TL formulas
38
39. Lossless Case
0, 0
R G
1/ 2
( )( )
j R j L G j C
j LC
so 0
LC
1/2
0
R j L
Z
G j C
0
L
Z
C
1
p
v
LC
p
v
(indep. of freq.)
(real and indep. of freq.)
39
40. 40
Lossless Case (cont.)
1
p
v
LC
In the medium between the two conductors is homogeneous (uniform)
and is characterized by ( , ), then we have that
LC e
The speed of light in a dielectric medium is
1
d
c
e
Hence, we have that p d
v c
The phase velocity does not depend on the frequency, and it is always the
speed of light (in the material).
(proof given later)
40
41. 41
0 0
z z
V z V e V e
Where do we assign z = 0?
The usual choice is at the load.
I(z)
V(z)
+
-
z
ZL
z = 0
Terminating impedance (load)
Ampl. of voltage wave
propagating in negative z
direction at z = 0.
Ampl. of voltage wave
propagating in positive z
direction at z = 0.
Terminated Transmission Line
Note: The length l measures distance from the load: z
41
42. What if we know
@
V V z
and
0 0
V V V e
z z
V z V e V e
0
V V e
0 0
V V V e
Terminated Transmission Line (cont.)
0 0
z z
V z V e V e
Hence
Can we use z = - l as
a reference plane?
I(z)
V(z)
+
-
z
ZL
z = 0
Terminating impedance (load)
42
43.
( ) ( )
z z
V z V e V e
Terminated Transmission Line (cont.)
0 0
z z
V z V e V e
Compare:
Note: This is simply a change of reference plane, from z = 0 to z = -l.
I(z)
V(z)
+
-
z
ZL
z = 0
Terminating impedance (load)
43
44. 0 0
z z
V z V e V e
What is V(-l )?
0 0
V V e V e
0 0
0 0
V V
I e e
Z Z
propagating
forwards
propagating
backwards
Terminated Transmission Line (cont.)
l distance away from load
The current at z = - l is then
I(z)
V(z)
+
-
z
ZL
z = 0
Terminating impedance (load)
44
45.
2
0
0
1 L
V
I e e
Z
2
0
0
0
0 0 1
V
V e
V V e e
e
V
V
Total volt. at distance l
from the load
Ampl. of volt. wave prop.
towards load, at the load
position (z = 0).
Similarly,
Ampl. of volt. wave prop.
away from load, at the
load position (z = 0).
0
2
1 L
V e e
L Load reflection
coefficient
Terminated Transmission Line (cont.)
I(-l )
V(-l )
+
l
ZL
-
0,
Z
l Reflection coefficient at z = - l
45
46.
2
0
2
2
0
0
2
0
1
1
1
1
L
L
L
L
V V e e
V
I e e
Z
V e
Z Z
I e
Input impedance seen “looking” towards load
at z = -l .
Terminated Transmission Line (cont.)
I(-l )
V(-l )
+
l
ZL
-
0,
Z
Z
46
47. At the load (l = 0):
0
1
0
1
L
L
L
Z Z Z
Thus,
2
0
0
0
2
0
0
1
1
L
L
L
L
Z Z
e
Z Z
Z Z
Z Z
e
Z Z
Terminated Transmission Line (cont.)
0
0
L
L
L
Z Z
Z Z
2
0 2
1
1
L
L
e
Z Z
e
Recall
47
48. Simplifying, we have
0
0
0
tanh
tanh
L
L
Z Z
Z Z
Z Z
Terminated Transmission Line (cont.)
2
0
2
0 0 0
0 0 2
2 0 0
0
0
0 0
0
0 0
0
0
0
1
1
cosh sinh
cosh sinh
L
L L L
L L
L
L
L L
L L
L
L
Z Z
e
Z Z Z Z Z Z e
Z Z Z
Z Z Z Z e
Z Z
e
Z Z
Z Z e Z Z e
Z
Z Z e Z Z e
Z Z
Z
Z Z
Hence, we have
48
49.
2
0
2
0
0
2
0 2
1
1
1
1
j j
L
j j
L
j
L
j
L
V V e e
V
I e e
Z
e
Z Z
e
Impedance is periodic
with period g/2
2
/ 2
g
g
Terminated Lossless Transmission Line
j j
Note:
tanh tanh tan
j j
tan repeats when
0
0
0
tan
tan
L
L
Z jZ
Z Z
Z jZ
49
50. For the remainder of our transmission line discussion we will assume that the
transmission line is lossless.
2
0
2
0
0
2
0 2
0
0
0
1
1
1
1
tan
tan
j j
L
j j
L
j
L
j
L
L
L
V V e e
V
I e e
Z
V e
Z Z
I e
Z jZ
Z
Z jZ
0
0
2
L
L
L
g
p
Z Z
Z Z
v
Terminated Lossless Transmission Line
I(-l )
V(-l )
+
l
ZL
-
0 ,
Z
Z
50
51. Matched load: (ZL=Z0)
0
0
0
L
L
L
Z Z
Z Z
For any l
No reflection from the load
A
Matched Load
I(-l )
V(-l )
+
l
ZL
-
0 ,
Z
Z
0
Z Z
0
0
0
j
j
V V e
V
I e
Z
51
52. Short circuit load: (ZL = 0)
0
0
0
0
1
0
tan
L
Z
Z
Z jZ
Always imaginary!
Note:
B
2
g
sc
Z jX
S.C. can become an O.C.
with a g/4 trans. line
0 1/4 1/2 3/4
g
/
XSC
inductive
capacitive
Short-Circuit Load
l
0 ,
Z
0 tan
sc
X Z
52
53. Using Transmission Lines to Synthesize Loads
A microwave filter constructed from microstrip.
This is very useful is microwave engineering.
53
54.
0
0
0
tan
tan
L
in
L
Z jZ d
Z Z d Z
Z jZ d
in
TH
in TH
Z
V d V
Z Z
I(-l)
V(-l)
+
l
ZL
-
0
Z
ZTH
VTH
d
Zin
+
-
ZTH
VTH
+
Zin
V(-d)
+
-
Example
Find the voltage at any point on the line.
54
55. Note:
0
2
1 j
L
j
V V e e
0
0
L
L
L
Z Z
Z Z
2
0 1
j d j d in
TH
in TH
L
V d
Z
Z
e V
Z
V e
2
2
1
1
j
j d
in L
TH j d
m TH L
Z e
V V e
Z Z e
At l = d:
Hence
Example (cont.)
0 2
1
1
j d
in
TH j d
in TH L
Z
V V e
Z Z e
55
56. Some algebra:
2
0 2
1
1
j d
L
in j d
L
e
Z Z d Z
e
2
2
0 2
0
2 2
2
0
0 2
2
0
2
0 0
2
0
2
0 0
0
1
1
1
1 1
1
1
1
1
1
j d
L
j d
j d
L
L
j d j d
j d
L TH L
L
TH
j d
L
j d
L
j d
TH L TH
j d
L
j d
TH TH
L
TH
in
in TH
e
Z
Z e
e
Z e Z e
e
Z Z
e
Z e
Z Z e Z Z
e
Z
Z
Z
Z Z
Z Z Z
e
Z Z
Z
2
0
2
0 0
0
1
1
j d
L
j d
TH TH
L
TH
e
Z Z Z Z
e
Z Z
Example (cont.)
56
57.
2
0
2
0
1
1
j
j d L
TH j d
TH S L
Z e
V V e
Z Z e
2
0
2
0
1
1
j d
in L
j d
in TH TH S L
Z Z e
Z Z Z Z e
where 0
0
TH
S
TH
Z Z
Z Z
Example (cont.)
Therefore, we have the following alternative form for the result:
Hence, we have
57
58.
2
0
2
0
1
1
j
j d L
TH j d
TH S L
Z e
V V e
Z Z e
Example (cont.)
I(-l)
V(-l)
+
l
ZL
-
0
Z
ZTH
VTH
d
Zin
+
-
Voltage wave that would exist if there were no reflections from
the load (a semi-infinite transmission line or a matched load).
58
59.
2 2
2 2 2 2
0
0
1 j d j d
L L S
j d j d j d j d
TH L S L L S L S
TH
e e
Z
V d V e e e e
Z Z
Example (cont.)
ZL
0
Z
ZTH
VTH
d
+
-
Wave-bounce method (illustrated for l = d):
59
60. Example (cont.)
2
2 2
2
2 2 2
0
0
1
1
j d j d
L S L S
j d j d j d
TH L L S L S
TH
e e
Z
V d V e e e
Z Z
Geometric series:
2
0
1
1 , 1
1
n
n
z z z z
z
2 2
2 2 2 2
0
0
1 j d j d
L L S
j d j d j d j d
TH L S L L S L S
TH
e e
Z
V d V e e e e
Z Z
2
j d
L S
z e
60
61. Example (cont.)
or
2
0
2
0
2
1
1
1
1
j d
L s
TH
j d
TH
L j d
L s
e
Z
V d V
Z Z
e
e
2
0
2
0
1
1
j d
L
TH j d
TH L s
Z e
V d V
Z Z e
This agrees with the previous result (setting l = d).
Note: This is a very tedious method – not recommended.
Hence
61
62. I(-l )
V(-l )
+
l
ZL
-
0 ,
Z
At a distance l from the load:
*
*
2
0 2 2 * 2
*
0
1
Re 1 1
1
R
2
e
2
L L
V
e e
Z
V I
e
P
2
2
0 2 4
0
1
1
2
L
V
P e e
Z
If Z0 real (low-loss transmission line)
Time- Average Power Flow
2
0
2
0
0
1
1
L
L
V V e e
V
I e e
Z
j
*
2 * 2
*
2 2
L L
L L
e e
e e
pure imaginary
Note:
62
63. Low-loss line
2
2
0 2 4
0
2 2
2
0 0
2 2
* *
0 0
1
1
2
1 1
2 2
L
L
V
P d e e
Z
V V
e e
Z Z
power in forward wave power in backward wave
2
2
0
0
1
1
2
L
V
P d
Z
Lossless line ( = 0)
Time- Average Power Flow
I(-l )
V(-l )
+
l
ZL
-
0 ,
Z
63
64. 0
0
0
tan
tan
L T
in T
T L
Z jZ
Z Z
Z jZ
2
4 4 2
g g
g
0
0
T
in T
L
jZ
Z Z
jZ
0
2
0
0
0
in in
T
L
Z Z
Z
Z
Z
Quarter-Wave Transformer
2
0T
in
L
Z
Z
Z
so
1/2
0 0
T L
Z Z Z
Hence
This requires ZL to be real.
ZL
Z0 Z0T
Zin
64
65. 2
0 1 L
j j
L
V V e e
2
0
2
0
1
1 L
j j
L
j
j j
L
V V e e
V e e e
max 0
min 0
1
1
L
L
V V
V V
max
min
V
V
VoltageStanding WaveRatio VSWR
Voltage Standing Wave Ratio
I(-l )
V(-l )
+
l
ZL
-
0 ,
Z
1
1
L
L
VSWR
z
1+ L
1
1- L
0
( )
V z
V
/ 2
z
D
0
z
65
66. Coaxial Cable
Here we present a “case study” of one particular transmission line, the coaxial cable.
a
b ,
r
e
Find C, L, G, R
We will assume no variation in the z direction, and take a length of one meter in the z
direction in order top calculate the per-unit-length parameters.
66
For a TEMz mode, the shape of the fields is independent of frequency, and hence we
can perform the calculation using electrostatics and magnetostatics.
67. Coaxial Cable (cont.)
- l0
l0
a
b
r
e
0 0
0
ˆ ˆ
2 2 r
E
e e e
Find C (capacitance / length)
Coaxial cable
h = 1 [m]
r
e
From Gauss’s law:
0
0
ln
2
B
AB
A
b
r
a
V V E dr
b
E d
a
e e
67
68. - l0
l0
a
b
r
e
Coaxial cable
h = 1 [m]
r
e
0
0
0
1
ln
2 r
Q
C
V b
a
e e
Hence
We then have
0
F/m
2
[ ]
ln
r
C
b
a
e e
Coaxial Cable (cont.)
68
69. ˆ
2
I
H
Find L (inductance / length)
From Ampere’s law:
Coaxial cable
h = 1 [m]
r
I
0
ˆ
2
r
I
B
(1)
b
a
B d
S
h
I
I
z
center conductor
Magnetic flux:
Coaxial Cable (cont.)
69
Note: We ignore “internal inductance”
here, and only look at the magnetic field
between the two conductors (accurate
for high frequency.
70. Coaxial cable
h = 1 [m]
r
I
0
0
0
1
2
ln
2
b
r
a
b
r
a
r
H d
I
d
I b
a
0
1
ln
2
r
b
L
I a
0
H/m
ln [ ]
2
r b
L
a
Hence
Coaxial Cable (cont.)
70
71. 0
H/m
ln [ ]
2
r b
L
a
Observation:
0
F/m
2
[ ]
ln
r
C
b
a
e e
0 0 r r
LC e e e
This result actually holds for any transmission line.
Coaxial Cable (cont.)
71
72. 0
H/m
ln [ ]
2
r b
L
a
For a lossless cable:
0
F/m
2
[ ]
ln
r
C
b
a
e e
0
L
Z
C
0 0
1
ln [ ]
2
r
r
b
Z
a
e
0
0
0
376.7303 [ ]
e
Coaxial Cable (cont.)
72
73. - l0
l0
a
b
0 0
0
ˆ ˆ
2 2 r
E
e e e
Find G (conductance / length)
Coaxial cable
h = 1 [m]
From Gauss’s law:
0
0
ln
2
B
AB
A
b
r
a
V V E dr
b
E d
a
e e
Coaxial Cable (cont.)
73
74. - l0
l0
a
b
J E
We then have leak
I
G
V
0
0
(1)2
2
2
2
leak a
a
r
I J a
a E
a
a
e e
0
0
0
0
2
2
ln
2
r
r
a
a
G
b
a
e e
e e
2
[S/m]
ln
G
b
a
or
Coaxial Cable (cont.)
74
75. Observation:
F/m
2
[ ]
ln
C
b
a
e
G C
e
This result actually holds for any transmission line.
2
[S/m]
ln
G
b
a
0 r
e e e
Coaxial Cable (cont.)
75
76. G C
e
To be more general:
tan
G
C
e
tan
G
C
Note: It is the loss tangent that is usually (approximately)
constant for a material, over a wide range of frequencies.
Coaxial Cable (cont.)
As just derived,
The loss tangent actually arises from
both conductivity loss and polarization
loss (molecular friction loss), ingeneral.
76
This is the loss tangent that would
arise from conductivity effects.
77. General expression for loss tangent:
c
c c
j
j j
j
e e
e e
e e
tan c
c
e
e
e
e
Effective permittivity that accounts for conductivity
Loss due to molecular friction Loss due to conductivity
Coaxial Cable (cont.)
77
78. Find R (resistance / length)
Coaxial cable
h = 1 [m]
Coaxial Cable (cont.)
,
b rb
a
b
,
a ra
a b
R R R
1
2
a sa
R R
a
1
2
b sb
R R
b
1
sa
a a
R
1
sb
b b
R
0
2
a
ra a
0
2
b
rb b
Rs = surface resistance of metal
78
79. General Transmission Line Formulas
tan
G
C
0 0 r r
LC e e e
0
lossless
L
Z
C
characteristic impedance of line (neglectingloss)
(1)
(2)
(3)
Equations (1) and (2) can be used to find L and C if we know the material
properties and the characteristic impedance of the lossless line.
Equation (3) can be used to find G if we know the material loss tangent.
a b
R R R
tan
G
C
(4)
Equation (4) can be used to find R (discussed later).
,
i
C i a b
contourof conductor,
2
2
1
( )
i
i s sz
C
R R J l dl
I
79
80. General Transmission Line Formulas (cont.)
tan
G C
0
lossless
L Z e
0
/ lossless
C Z
e
R R
Al four per-unit-length parameters can be found from 0 ,
lossless
Z R
80
81. Common Transmission Lines
0 0
1
ln [ ]
2
lossless r
r
b
Z
a
e
Coax
Twin-lead
1
0
0 cosh [ ]
2
lossless r
r
h
Z
a
e
2
1 2
1
2
s
h
a
R R
a h
a
1 1
2 2
sa sb
R R R
a b
a
b
,
r r
e
h
,
r r
e
a a
81
82. Common Transmission Lines (cont.)
Microstrip
0 0
1 0
0
0 1
eff eff
r r
eff eff
r r
f
Z f Z
f
e e
e e
0
120
0
0 / 1.393 0.667ln / 1.444
eff
r
Z
w h w h
e
( / 1)
w h
2
1 ln
t h
w w
t
h
w
er
t
82
83. Common Transmission Lines (cont.)
Microstrip ( / 1)
w h
h
w
er
t
2
1.5
(0)
(0)
1 4
eff
r r
eff eff
r r
f
F
e e
e e
1 1 1
1 /
0
2 2 4.6 /
1 12 /
eff r r r
r
t h
w h
h w
e e e
e
2
0
4 1 0.5 1 0.868ln 1
r
h w
F
h
e
83
84. At high frequency, discontinuity effects can become important.
Limitations of Transmission-Line Theory
Bend
incident
reflected
transmitted
The simple TL model does not account for the bend.
ZTH
ZL
Z0
+
-
84
85. At high frequency, radiation effects can become important.
When will radiation occur?
We want energy to travel from the generator to the load, without radiating.
Limitations of Transmission-Line Theory (cont.)
ZTH
ZL
Z0
+
-
85
86. r
e a
b
z
The coaxial cable is a perfectly
shielded system – there is never
any radiation at any frequency, or
under any circumstances.
The fields are confined to the region
between the two conductors.
Limitations of Transmission-Line Theory (cont.)
86
87. The twin lead is an open type of transmission
line – the fields extend out to infinity.
The extended fields may cause
interference with nearby objects.
(This may be improved by using
“twisted pair.”)
+ -
Limitations of Transmission-Line Theory (cont.)
Having fields that extend to infinity is not the same thing as having radiation, however.
87
88. The infinite twin lead will not radiate by itself, regardless of how far apart
the lines are.
h
incident
reflected
The incident and reflected waves represent an exact solution to Maxwell’s
equations on the infinite line, at any frequency.
*
1
ˆ
Re E H 0
2
t
S
P dS
S
+ -
Limitations of Transmission-Line Theory (cont.)
No attenuation on an infinite lossless line
88
89. A discontinuity on the twin lead will cause radiation to occur.
Note: Radiation effects
increase as the
frequency increases.
Limitations of Transmission-Line Theory (cont.)
h
Incident wave
pipe
Obstacle
Reflected wave
Bend h
Incident wave
bend
Reflected wave
89
90. To reduce radiation effects of the twin lead at discontinuities:
h
1) Reduce the separation distance h (keep h << ).
2) Twist the lines (twisted pair).
Limitations of Transmission-Line Theory (cont.)
CAT 5 cable
(twisted pair)
90
98. 98
Maxwell’s Equations in Large Scale Form
S
d
D
t
S
d
J
l
d
H
S
d
M
S
d
B
t
l
d
E
S
d
B
dv
S
d
D
S
S
l
S
S
l
S
S
V
0
99. ENEE482 99
•Maxwell’s Equations for the Time - Harmonic Case
D
j
J
H
M
B
j
E
B
D
E
E
t
E
E
e
E
E
e
jE
E
E
jE
E
a
jE
E
a
jE
E
a
z
y
x
E
e
z
y
x
E
t
z
y
x
E
xr
xi
xi
xr
j
t
j
xi
xr
t
j
xi
xr
x
zi
zr
z
yi
yr
y
xi
xr
x
t
j
t
j
,
0
,
)
/
(
tan
,
)
cos(
]
Re[
]
)
Re[(
)
(
)
(
)
(
)
,
,
(
]
)
,
,
(
Re[
)
,
,
,
(
:
then
,
variations
e
Assume
1
2
2
2
2
100. 100
Boundary Conditions at a General Material Interface
s
s
s
s
t
t
n
n
s
s
n
n
s
t
t
J
H
H
n
M
E
E
n
B
B
n
D
D
n
J
H
H
B
B
D
D
M
E
E
)
(
ˆ
)
(
ˆ
0
)
(
ˆ
)
(
ˆ
;
Density
Charge
Surface
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
D1n
D2n
h
Ds
h
E1t
E2t
1,e1
2,e2
108. 108
The field amplitude decays exponentially from its surface
According to e-u/
s where u is the normal distance into th
Conductor, s is the skin depth
H
n̂
,
1
:
Impedance
surface
The
E
J
,
2
m
s
m
t
s
m
s
Z
J
Z
E
j
Z
111. 111
Under steady-state sinusoidal time-varying
Conditions, the time-average energy stored in the
Electric field is
V
e
V
V
e
dV
E
E
W
dV
E
E
dV
D
E
W
*
*
*
4
then
real,
and
constant
is
If
4
1
4
1
Re
e
e
e
115. 115
volume.
in the
stored
energy
reactive
the
times
2
and
)
(
volume
in the
heat
lost to
power
the
,
P
surface
he
through t
smitted
power tran
the
of
sum
the
to
equal
is
)
(P
sources
by the
delivered
power
The
)
(
2
)
(
2
4
4
2
2
1
Im
)
(
2
1
P
0
s
0
*
*
*
*
s
e
P
W
W
j
P
P
P
W
W
dV
E
E
H
H
dS
H
E
dV
M
H
J
E
e
m
s
e
m
V
S
s
s
V
s
loss
power
average
Time
2
1
)
(
2
*
*
*
dV
E
E
dV
E
E
H
H
P
V
V
e
118. 118
Solution For Vector Potential
(x,y,z)
(x’,y’, z’) R
r
r’
J
V
d
R
e
r
J
r
A
r
r
z
z
y
y
x
x
R
V
d
R
e
r
J
r
A
z
y
x
A
jkR
V
jkR
)
(
4
)
(
)
(
)
(
)
(
current
al
infinitism
an
for
)
(
4
)
(
)
,
,
(
2
2
2
124. 124
Transmission Lines & Waveguides
Wave Propagation in the Positive z-Direction is Represented By:e-jz
,
,
)
(
)
(
)
(
)
(
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
z
t
t
z
t
t
z
t
t
z
t
t
z
j
z
t
z
z
z
t
z
t
z
j
z
t
z
j
z
t
z
t
z
j
z
z
j
t
z
t
z
j
z
z
j
t
z
t
e
j
e
h
j
h
e
j
h
h
j
e
e
h
h
j
e
a
j
e
e
a
j
e
e
h
h
j
e
e
e
a
j
E
e
y
x
h
e
y
x
h
z
y
x
H
z
y
x
H
z
y
x
H
e
y
x
e
e
y
x
e
z
y
x
E
z
y
x
E
z
y
x
E
e
125. 125
Modes Classification:
1. Transverse Electromagnetic (TEM) Waves
0
z
z H
E
2. Transverse Electric (TE), or H Modes
0
but
,
0
z
z H
E
3. Transverse Magnetic (TM), or E Modes
0
But
,
0
z
z E
H
4. Hybrid Modes
0
,
0
z
z E
H
126. 126
TEM WAVES
z
j
z
z
j
t
t
z
j
t
z
j
t
t
t
t
t
t
t
t
t
t
e
e
a
Y
e
h
H
e
y
x
e
e
E
y
x
y
x
y
x
e
h
e
e
h
e
ˆ
)
,
(
0
,
ential
Scalar Pot
0
,
,
e
h
â
,
0
h
e
â
,
0
0
,
0
0
2
t
0
t
z
t
t
0
t
z
t
137. Qc OF COAXIAL LINE AS A FUNCTION OF Zo
137
Q-Copper
of
Coaxial
Line
2000
2200
2400
2600
2800
3000
3200
3400 0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
er Zc
GHz
c
f
b
Q
140. 140
0
n
for
2
)
Re(
0
n
for
4
)
Re(
2
1
ˆ
2
1
2
,
Z
:
is
modes
TM
the
of
impedance
wave
The
2
2
0
E
,
e
cos
)
,
,
(
2
2
2
2
*
0 0
0 0
*
0
g
TM
x
z
j
-
n
c
n
c
x
w
x
d
y
y
w
x
d
y
p
x
y
c
c
y
n
c
y
A
k
d
A
k
d
dydx
H
E
dydx
z
H
E
P
v
k
H
E
d
k
f
H
y
d
n
A
k
j
z
y
x
E
e
e
e
e
e
144. 144
COUPLED LINES EVEN & ODD
MODES OF EXCITATIONS
AXIS OF EVEN SYMMETRY AXIS OF ODD SYMMETRY
P.M.C. P.E.C.
EVEN MODE ELECTRIC
FIELD DISTRUBUTION
ODD MODE ELECTRIC
FIELD DISTRIBUTION
e
Z0 o
Z0
=EVEN MODE CHAR.
IMPEDANCE
=ODD MODE CHAR.
IMPEDANCE
Equal currents are flowing
in the two lines
Equal &opposite currents are
flowing in the two lines
