1. EC6503 TRANSMISSION LINES AND WAVEGUIDES
BY H.UMMA HABIBA, Professor SVCE,Chennai.

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
William H Hayt and Jr John A Buck, “Engineering Electromagnetics” Tata Mc Graw-Hill Publishing Company Ltd, New Delhi, 2008
David K Cheng, “Field and Wave Electromagnetics”, Pearson Education Inc, Delhi, 2004
John D Kraus and Daniel A Fleisch, “Electromagnetics with Applications”, Mc Graw Hill Book Co,2005
GSN Raju, “Electromagnetic Field Theory and Transmission Lines”, Pearson Education, 2005
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.

10. TRANSMISSION LINE THEORY UNIFORM PLANE WAVES
UNIT I
TRANSMISSION LINE THEORY UNIFORM PLANE WAVES

11. (shown connected to a 4:1 impedance-transforming balun)
Transmission Line
Properties
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)
Twin lead
(shown connected to a 4:1 impedance-transforming balun)
Coaxial cable (coax) 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

13. Transmission Line (cont.)
Transmission lines commonly met on printed-circuit boards
Microstrip h w er er w
Stripline h
Coplanar strips h er w
Coplanar waveguide (CPW) h er w 13

14. Transmission Line (cont.)
Transmission lines are commonly met on printed-circuit boards.
Microstrip line
A microwave integrated circuit 14

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
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.
2) Multi-mode fiber
Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect). 16

17. Fiber-Optic Guide (cont.)
Higher index core region 17

18. Waveguides Properties
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) 18

19. Transmission-Line Theory
Lumped circuits: resistors, capacitors, inductors
neglect time delays (phase)
Distributed circuit elements: transmission lines
account for propagation and time delays (phase change)
We need transmission-line theory whenever the length of a line is significant compared with a wavelength. 19

20. Transmission Line C = capacitance/length [F/m]
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. Transmission Line (cont.) x B
RDz
LDz
GDz
CDz z
v(z+z,t) + -
v(z,t)
i(z,t)
i(z+z,t) 21

22. Transmission Line (cont.)
RDz
LDz
GDz
CDz z
v(z+z,t) + -
v(z,t)
i(z,t)
i(z+z,t) 22

23. TEM Transmission Line (cont.)
Hence
Now let Dz  0:
“Telegrapher’sEquations” 23

24. TEM Transmission Line (cont.)
To combine these, take the derivative of the first one with respect to z:
Switch the order of the derivatives. 24

25. TEM Transmission Line (cont.)
Hence, we have:
The same equation also holds for i. 25

26. TEM Transmission Line (cont.)
Time-Harmonic Waves: 26

27. TEM Transmission Line (cont.)
Note that
= series impedance/length
= parallel admittance/length
Then we can write: 27

28. TEM Transmission Line (cont.)
Let
Then
Solution:
 is called the "propagation constant."
Convention: 28

29. TEM Transmission Line (cont.)
Forward travelling wave (a wave traveling in the positive z direction):
The wave “repeats” when:
Hence: 29

30. Phase Velocity
Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest. vp
(phase velocity) 30

31. Phase Velocity (cont.)
Set
In expanded form:
Hence 31

32. Characteristic Impedance Z0+
V+(z) -
I+ (z) z
A wave is traveling in the positive z direction. so
(Z0 is a number, not a function of z.) 32

33. Characteristic Impedance Z0 (cont.)
Use Telegrapher’s Equation: so
Hence 33

34. Characteristic Impedance Z0 (cont.)
From this we have:
Using
We have
Note: The principal branch of the square root is chosen, so that Re (Z0) > 0. 34

35. General Case (Waves in Both Directions)
wave in +z direction
wave in -z direction
Note: 35

36. Backward-Traveling Wave+
V -(z) -
I - (z) z
A wave is traveling in the negative z direction. so
Note: The reference directions for voltage and current are the same as for the forward wave. 36

37. General Case Most general case: I (z) + V (z) - z
A general superposition of forward and backward traveling waves:
Note: The reference directions for voltage and current are the same for forward and backward waves. 37

38. Summary of Basic TL formulas
guided wavelength  g
phase velocity  vp 38

39. Lossless Case so
(real and indep. of freq.)
(indep. of freq.) 39

40. Lossless Case (cont.)
In the medium between the two conductors is homogeneous (uniform) and is characterized by (, ), then we have that
(proof given later)
The speed of light in a dielectric medium is
Hence, we have that
The phase velocity does not depend on the frequency, and it is always the speed of light (in the material). 40

41. Terminated Transmission Line
Terminating impedance (load)
Ampl. of voltage wave propagating in positive z direction at z = 0.
Ampl. of voltage wave propagating in negative z direction at z = 0.
Where do we assign z = 0?
The usual choice is at the load.
Note: The length l measures distance from the load: 41

42. Terminated Transmission Line (cont.)
Terminating impedance (load)
What if we know
Can we use z = - l as a reference plane?
Hence 42

43. Terminated Transmission Line (cont.)
Terminating impedance (load)
Compare:
Note: This is simply a change of reference plane, from z = 0 to z = -l. 43

44. Terminated Transmission Line (cont.)
Terminating impedance (load)
What is V(-l )?
propagating forwards
propagating backwards
The current at z = - l is then
l  distance away from load 44

45. Terminated Transmission Line (cont.)
Total volt. at distance l from the load
Ampl. of volt. wave prop. towards load, at the load position (z = 0).
Ampl. of volt. wave prop. away from load, at the load position (z = 0).
L  Load reflection coefficient
l  Reflection coefficient at z = - l
Similarly, 45

46. Terminated Transmission Line (cont.)
Input impedance seen “looking” towards load at z = -l . 46

47. Terminated Transmission Line (cont.)
At the load (l = 0):
Recall
Thus, 47

48. Terminated Transmission Line (cont.)
Simplifying, we have
Hence, we have 48

49. Terminated Lossless Transmission Line
Impedance is periodic with period g/2
tan repeats when
Note: 49

50. Terminated Lossless Transmission Line
For the remainder of our transmission line discussion we will assume that the transmission line is lossless. 50

51. Matched Load A Matched load: (ZL=Z0) No reflection from the load
For any l 51

52. Short-Circuit Load B Short circuit load: (ZL = 0) Note:
Always imaginary!
S.C. can become an O.C. with a g/4 trans. line 52

53. Using Transmission Lines to Synthesize Loads
This is very useful is microwave engineering.
A microwave filter constructed from microstrip. 53

54. Example
Find the voltage at any point on the line. 54

55. Example (cont.)
Note:
At l = d :
Hence 55

56. Example (cont.)
Some algebra: 56

57. Example (cont.) Hence, we have where
Therefore, we have the following alternative form for the result: 57

58. Example (cont.)
Voltage wave that would exist if there were no reflections from the load (a semi-infinite transmission line or a matched load). 58

59. Example (cont.)
Wave-bounce method (illustrated for l = d ): 59

60. Example (cont.)
Geometric series: 60

61. Example (cont.) Hence or
This agrees with the previous result (setting l = d ).
Note: This is a very tedious method – not recommended. 61

62. Time- Average Power Flow
At a distance l from the load:
Note:
If Z0  real (low-loss transmission line) 62

63. Time- Average Power Flow
Low-loss line
Lossless line ( = 0) 63

64. Quarter-Wave TransformerZL Z0
Z0T
Zin
This requires ZL to be real. so
Hence 64

65. Voltage Standing Wave Ratio65

66. Coaxial Cable a b Find C, L, G, R
Here we present a “case study” of one particular transmission line, the coaxial cable. a b
Find C, L, G, R
For a TEMz mode, the shape of the fields is independent of frequency, and hence we can perform the calculation using electrostatics and magnetostatics.
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

67. Coaxial Cable (cont.) h = 1 [m] Find C (capacitance / length)
From Gauss’s law:
-l0
l0 a b 67

68. Coaxial Cable (cont.) h = 1 [m] Coaxial cable Hence We then have -l068

69. Coaxial Cable (cont.) h Find L (inductance / length) h = 1 [m] I
From Ampere’s law:
Note: We ignore “internal inductance” here, and only look at the magnetic field between the two conductors (accurate for high frequency. S h I z
center conductor
Magnetic flux: 69

70. Coaxial Cable (cont.)
Coaxial cable
h = 1 [m] I
Hence 70

71. Coaxial Cable (cont.) Observation:
This result actually holds for any transmission line. 71

72. Coaxial Cable (cont.)
For a lossless cable: 72

73. Coaxial Cable (cont.) h = 1 [m] Find G (conductance / length)
From Gauss’s law:
-l0
l0 a b 73

74. Coaxial Cable (cont.)
-l0
l0 a b
We then have or 74

75. Coaxial Cable (cont.) Observation:
This result actually holds for any transmission line. 75

76. Coaxial Cable (cont.) As just derived, To be more general:
This is the loss tangent that would arise from conductivity effects.
The loss tangent actually arises from both conductivity loss and polarization loss (molecular friction loss), ingeneral.
Note: It is the loss tangent that is usually (approximately) constant for a material, over a wide range of frequencies. 76

77. Coaxial Cable (cont.) General expression for loss tangent:
Effective permittivity that accounts for conductivity
Loss due to molecular friction
Loss due to conductivity 77

78. Coaxial Cable (cont.) h = 1 [m] Find R (resistance / length)
Rs = surface resistance of metal a b 78

79. General Transmission Line Formulas
(1)
(2)
(4)
(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.
Equation (4) can be used to find R (discussed later). 79

80. General Transmission Line Formulas (cont.)
Al four per-unit-length parameters can be found from 80

81. Common Transmission Lines
Coax a b
Twin-lead h a a 81

82. Common Transmission Lines (cont.)
Microstrip h w er t 82

83. Common Transmission Lines (cont.)
Microstrip h w er t 83

84. Limitations of Transmission-Line Theory
At high frequency, discontinuity effects can become important.
transmitted
incident
Bend
reflected
The simple TL model does not account for the bend.
ZTH ZL Z0 + - 84

85. Limitations of Transmission-Line Theory (cont.)
At high frequency, radiation effects can become important.
We want energy to travel from the generator to the load, without radiating.
ZTH ZL Z0 + -
When will radiation occur? 85

86. Limitations of Transmission-Line Theory (cont.)
The coaxial cable is a perfectly shielded system – there is never any radiation at any frequency, or under any circumstances. a b z
The fields are confined to the region between the two conductors. 86

87. Limitations of Transmission-Line Theory (cont.)
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.”)
Having fields that extend to infinity is not the same thing as having radiation, however. 87

88. Limitations of Transmission-Line Theory (cont.)
The infinite twin lead will not radiate by itself, regardless of how far apart the lines are. h
incident
reflected S
No attenuation on an infinite lossless line
The incident and reflected waves represent an exact solution to Maxwell’s equations on the infinite line, at any frequency. 88

89. Limitations of Transmission-Line Theory (cont.)
A discontinuity on the twin lead will cause radiation to occur. h
Incident wave
pipe
Obstacle
Reflected wave
Note: Radiation effects increase as the frequency increases.
Bend h
Incident wave
bend
Reflected wave 89

90. Limitations of Transmission-Line Theory (cont.)
To reduce radiation effects of the twin lead at discontinuities:
Reduce the separation distance h (keep h << ).
Twist the lines (twisted pair). h
CAT 5 cable
(twisted pair) 90

91. Two conductor
wire
Coaxial line
Shielded
Strip line
Dielectric 91

92. Two conductor
wire
Coaxial line
Shielded
Strip line
Dielectric 92

93. Common Hollow-pipe waveguides
Rectangular
guide
Ridge guide
Circular
guide 93

94. STRIP LINE CONFIGURATIONSW
SINGLE STRIP LINE
COUPLED LINES
COUPLED STRIPS
TOP & BOTTOM
COUPLED ROUND BARS 94

95. MICROSTRIP LINE CONFIGURATIONS
SINGLE MICROSTRIP
TWO COUPLED MICROSTRIPS
TWO SUSPENDED
SUBSTRATE LINES
SUSPENDED SUBSTRATE
LINE 95

96. TRANSVERSE ELECTROMAGNETIC (TEM): COAXIAL LINES
TRANSMISSION MEDIA
TRANSVERSE ELECTROMAGNETIC (TEM):
COAXIAL LINES
MICROSTRIP LINES (Quasi TEM)
STRIP LINES AND SUSPENDED SUBSTRATE
METALLIC WAVEGUIDES:
RECTANGULAR WAVEGUIDES
CIRCULAR WAVEGUIDES
DIELECTRIC LOADED WAVEGUIDES
ANALYSIS OF WAVE PROPAGATION ON THESE
TRANSMISSION MEDIA THROUGH MAXWELL’S
EQUATIONS 96

97. Auxiliary Relations: 97

98. Maxwell’s Equations in Large Scale Form98

99. Maxwell’s Equations for the Time - Harmonic Case
ENEE482 99

100. Boundary Conditions at a General Material Interfaceh
E1t
E2t
m1,e1
m2,e2
D1n
D2n h Ds
100

101. Fields at a Dielectric Interface
101

102. + + + n rs Js Ht
102

103. 103

104. Wave Equation
104

105. 105

106. 106

107. 107

108. The field amplitude decays exponentially from its surface
According to e-u/ds where u is the normal distance into the
Conductor, ds is the skin depth
108

109. Parallel Polarizationx Er n2 e0 Et n3 q2 q3 q1 z n1 Ei
109

110. 110

111. Under steady-state sinusoidal time-varying
Conditions, the time-average energy stored in the
Electric field is
111

112. 112

113. 113

114. 114

115. 115

116. L R I V C
116

117. 117

118. Solution For Vector PotentialJ
(x’,y’, z’) R
(x,y,z) r’ r
118

119. Lumped element circuit model for a transmission lineRg z
Lumped element circuit model for a transmission line
Ldz
I(z,t)+dI/dz dz
I(z,t)
V(z,t)
Cdz
V(z,t)+dv/dz dz
119

120. 120

121. 121

122. ZL Zc Z
To generator
122

123. 123

124. Wave Propagation in the Positive z-Direction is Represented By:e-jbz
Transmission Lines & Waveguides
Wave Propagation in the Positive z-Direction is Represented By:e-jbz
124

125. Modes Classification: 1. Transverse Electromagnetic (TEM) Waves
2. Transverse Electric (TE), or H Modes
3. Transverse Magnetic (TM), or E Modes
4. Hybrid Modes
125

126. TEM WAVES
126

127. 127

128. TE WAVES
128

129. 129

130. TM WAVES
130

131. TEM TRANSMISSION LINES
Coaxial
Two-wire
Parallel -plate a b e
131

132. COAXIAL LINES a b e
132

133. THE CHARACTERISTIC IMPEDANCE OF A COAXIAL IS Z0
133

134. Zc OF COAXIAL LINE AS A FUNCTION OF b/a1 10
100 20 40 60 80
120
140
160
180
200
220
240
260 = er Zo
134

135. 135

136. 136

137. Qc OF COAXIAL LINE AS A FUNCTION OF Zoer Zc
137

138. TEM Modes y d x w
138

139. 139

140. 140

141. 141

142. 142

143. 143

144. COUPLED LINES EVEN & ODD
MODES OF EXCITATIONS
AXIS OF EVEN SYMMETRY
AXIS OF ODD SYMMETRY
P.M.C.
P.E.C.
ODD MODE ELECTRIC
FIELD DISTRIBUTION
EVEN MODE ELECTRIC
FIELD DISTRUBUTION
=ODD MODE CHAR.
IMPEDANCE
=EVEN MODE CHAR.
IMPEDANCE
Equal currents are flowing
in the two lines
Equal &opposite currents are
flowing in the two lines
144