1. Prof. David R. Jackson Dept. of ECE Notes 1 ECE 5317-6351 Microwave Engineering Fall 2015 Transmission Line Theory 1

2.  In this set of notes we first overview different types of waveguiding systems, including transmission lines.  We then develop transmission line theory in detail. Overview 2

3. A waveguiding structure is one that carries a signal (or power) from one point to another. There are three common types:  Transmission lines  Fiber-optic guides  Waveguides Waveguiding Structures Note: An alternative to using waveguiding structures is wireless transmission using antennas. (Antennas are discussed in ECE 5318.) 3

4. Transmission Line  Has two conductors running parallel (one can be inside the other)  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 (but there always is)  May or may not be immune to interference (shielding property)  Does not have E z or H z components of the fields (TEM z ) Properties Coaxial cable (coax) Twin lead (shown connected to a 4:1 impedance-transforming balun) 4

5. Transmission Line (cont.) CAT 5 cable (twisted pair) The two wires of the transmission line are twisted to reduce interference and radiation from discontinuities. 5

6. Transmission Line (cont.) Microstrip h w rr rr w Stripline h Transmission lines commonly met on printed-circuit boards Coplanar strips h rr w w Coplanar waveguide (CPW) h rr w 6

7. Transmission Line (cont.) Transmission lines are commonly met on printed-circuit boards. A microwave integrated circuit. Microstrip line 7

8. Waveguides  Consists of a single hollow metal pipe  Can propagate a signal only at high frequency: f > f 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 E z or H z component of the fields (TM z or TE z ) Properties http://en.wikipedia.org/wiki/Waveguide_(electromagnetism) 8

9. Fiber-Optic Guide Properties  Uses a dielectric rod  Can propagate a signal at any frequency (in theory)  Can be made to have very low loss  Has minimal signal distortion  Has a very high bandwidth  Very immune to interference  Not suitable for high power  Has both E z and H z components of the fields (“hybrid mode”) 9

10. 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). 10 Note: The carrier is at optical frequency, but the modulating (baseband) signal is often at microwave frequencies (e.g., TV, internet signal).

11. Fiber-Optic Guide (cont.) http://en.wikipedia.org/wiki/Optical_fiber Higher index core region 11 Multi-mode Fiber

12.  Lumped circuits: resistors, capacitors, inductors Neglects time delays (phase change) 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 to a wavelength. 12

13. 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 ]  zz 13

14. Transmission Line (cont.) + + + + + + + - - - - - xxx B 14 RzRz LzLz GzGz CzCz z v(z+z,t)v(z+z,t) + - v(z,t)v(z,t) + - i(z,t)i(z,t) i(z+z,t)i(z+z,t) Note: There are equal and opposite currents on the two conductors. (We only need to work with the current on the top conductor, since we have chosen to put all of the series elements there.)

15. Transmission Line (cont.) 15 RzRz LzLz GzGz CzCz z v(z+z,t)v(z+z,t) + - v(z,t)v(z,t) + - i(z,t)i(z,t) i(z+z,t)i(z+z,t)

16. Hence Now let  z  0: “Telegrapher’s Equations” TEM Transmission Line (cont.) 16

17. To combine these, take the derivative of the first one with respect to z : Switch the order of the derivatives. TEM Transmission Line (cont.) 17

18. The same differential equation also holds for i. Hence, we have: TEM Transmission Line (cont.) 18

19. TEM Transmission Line (cont.) Time-Harmonic Waves: 19

20. Note that = series impedance/length = parallel admittance/length Then we can write: TEM Transmission Line (cont.) 20

21. Let We have: Solution: Then TEM Transmission Line (cont.)  is called the “propagation constant”. 21 Question: Which sign of the square root is correct?

22. Principal square root: TEM Transmission Line (cont.) 22 We choose the principal square root. Hence (Note the square-root (“radical”) symbol here.)

23. Denote: TEM Transmission Line (cont.) 23

24. TEM Transmission Line (cont.) 24 There are two possible locations for  : it must be in the first quadrant (  > 0 ). Hence: Re Im Re Im Re Im

25. TEM Transmission Line (cont.) Forward travelling wave (a wave traveling in the positive z direction): The wave “repeats” when: Hence: 25

26. Phase Velocity Let’s track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest of the wave. vpvp ( phase velocity ) 26

27. Phase Velocity (cont.) Set Hence In expanded form: 27

28. Characteristic Impedance Z 0 so +V+(z)-+V+(z)- I + (z) z Assumption: A wave is traveling in the positive z direction. (Note: Z 0 is a number, not a function of z.) 28

29. Use first Telegrapher’s Equation: so Hence Characteristic Impedance Z 0 (cont.) 29 Recall:

30. From this we have: Use: Characteristic Impedance Z 0 (cont.) 30 Both are in the first quadrant (principle square root)

31. Hence, we have Characteristic Impedance Z 0 (cont.) 31

32. In the time domain: Wave in +z direction Wave in -z direction General Case (Waves in Both Directions) 32

33. Backward-Traveling Wave so + V - (z) - I - (z) z A wave is traveling in the negative z direction. Note: The reference directions for voltage and current are chosen the same as for the forward wave. 33

34. General Case 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. 34 +V (z)-+V (z)- I (z) z

35. Guided wavelength Phase velocity Summary of Basic TL formulas 35

36. Lossless Case so (independent of freq.) (real and independent of freq.) 36

37. Lossless Case (cont.) If the medium between the two conductors is homogeneous (uniform) and is characterized by ( ,  ), then we have that (proof omitted): 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). (proof given later) 37

38. Where do we assign z = 0 ? The usual choice is at the load. Amplitude of voltage wave propagating in negative z direction at z = 0. Amplitude of voltage wave propagating in positive z direction at z = 0. Terminated Transmission Line 38 V (z) + - z ZLZL z = 0 Terminating impedance (load) I (z)

39. Terminated Transmission Line (cont.) Hence Can we use z = z 0 as a reference plane? 39 V (z) + - z ZLZL z = 0 Terminating impedance (load) I (z) z = z 0

40. Terminated Transmission Line (cont.) Compare: Note: This is simply a change of reference plane, from z = 0 t o z = z 0. 40 V (z) + - z ZLZL z = 0 Terminating impedance (load) I (z) z = z 0

41. What is V(-d) ? Propagating forwards Propagating backwards Terminated Transmission Line (cont.) d  distance away from load The current at z = -d is 41 V (-d) + - z ZLZL z = 0 Terminating impedance (load) I (-d) z = -d (This does not necessarily have to be the length of the entire line.)

42. Similarly,  L  Load reflection coefficient Terminated Transmission Line (cont.) 42 V (-d) + - z ZLZL z = 0 I (-d) z = -d or

43. Z(-d) = impedance seen “looking” towards load at z = -d. Terminated Transmission Line (cont.) 43 V (-d) + - z ZLZL z = 0 I (-d) z = -d Z(-d) Note: If we are at the beginning of the line, we will call this “input impedance”.

44. At the load ( d = 0 ): Thus, Terminated Transmission Line (cont.) Recall 44

45. Simplifying, we have: Terminated Transmission Line (cont.) Hence, we have 45

46. Impedance is periodic with period g /2: Terminated Lossless Transmission Line Note: The tan function repeats when 46 Lossless:

47. Note: For the remainder of our transmission line discussion we will assume that the transmission line is lossless. Summary for Lossless Transmission Line 47 V (-d) + - z ZLZL z = 0 I (-d) z = -d Z(-d)

48. No reflection from the load Matched Load ( Z L =Z 0 ) 48 V (-d) + - z ZLZL z = 0 I (-d) z = -d Z(-d)

49. Always imaginary! Short-Circuit Load ( Z L =0 ) 49 V (-d) + - z z = 0 I (-d) z = -d Z(-d)

50. Note: S.C. can become an O.C. with a g /4 transmission line. Short-Circuit Load ( Z L =0 ) 50 0 1/41/2 3/4 X sc Inductive Capacitive d/ g

51. Always imaginary! Open-Circuit Load ( Z L =  ) 51 V (-d) + - z z = 0 I (-d) z = -d Z(-d) or

52. O.C. can become a S.C. with a g /4 transmission line. Open-Circuit Load ( Z L =  ) 52 Note: 0 1/4 1/2 X oc Inductive Capacitive d/ g

53. Using Transmission Lines to Synthesize Loads A microwave filter constructed from microstrip line. This is very useful in microwave engineering. 53 We can obtain any reactance that we want from a short or open transmission line.

54. 54 Find the voltage at any point on the line. Z Th V Th + Z in + - - V (- d ) At the input: Voltage on a Transmission Line I (z) + Z L - Z Th V d Z in + - V (z) z=0

55. At z = -d : Hence 55 Voltage on a Transmission Line (cont.)

56. Let’s derive an alternative form of the result. 56 Start with: Voltage on a Transmission Line (cont.)

57. where Therefore, we have the following alternative form for the result: Hence, we have 57 (source reflection coefficient) Voltage on a Transmission Line (cont.) Recall: Substitute

58. Voltage wave that would exist if there were no reflections from the load (a semi-infinite transmission line or a matched load). 58 Voltage on a Transmission Line (cont.) This term accounts for the multiple (infinite) bounces.

59. Wave-bounce method (illustrated for z = -d ): 59 ZLZL Z Th V d + - Voltage on a Transmission Line (cont.)

60. Geometric series: 60 Group together alternating terms: Voltage on a Transmission Line (cont.)

61. or This agrees with the previous result (setting z = -d ). Note: The wave-bounce method is a very tedious method – not recommended. Hence 61 Voltage on a Transmission Line (cont.)

62. At a distance d from the load: If Z 0  real (low-loss transmission line) Time-Average Power Flow Note: 62 V (z) + - z ZLZL z = 0 I (z) z

63. Low-loss line Lossless line (  = 0 ) Time-Average Power Flow (cont.) 63 V (-d) + - z ZLZL z = 0 I (-d) z = -d Note: In the general lossy case, we cannot say that the total power is the difference of the powers in the two waves.

64. Quarter-Wave Transformer so Hence (This requires Z L to be real.) 64 Matching condition ZLZL Z0Z0 Z 0T Z in d

65. Voltage Standing Wave 65 V (z) + - z ZLZL z = 0 I (z) z

66. Voltage Standing Wave Ratio 66 V (z) + - z ZLZL z = 0 I (z) z

67. Coaxial Cable Here we present a “case study” of one particular transmission line, the coaxial cable. a b 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 to calculate the per-unit-length parameters. 67 For a TEM z mode, the shape of the fields is independent of frequency, and hence we can perform the calculation using electrostatics and magnetostatics.

68. Coaxial Cable (cont.) -l0-l0 l0l0 a b Find C (capacitance / length) Coaxial cable h = 1 [m] From Gauss’s law: 68

69. -l0-l0 l0l0 a b Coaxial cable h = 1 [m] Hence We then have: Coaxial Cable (cont.) 69

70. Find L (inductance / length) From Ampere’s law: Coaxial cable h = 1 [m] I S h I I z Center conductor Magnetic flux: Coaxial Cable (cont.) 70 Note: We ignore “internal inductance” here, and only look at the magnetic field between the two conductors (accurate for high frequency.

71. Coaxial cable h = 1 [m] I Hence Coaxial Cable (cont.) 71

72. Observation: This result actually holds for any transmission line (proof omitted). Coaxial Cable (cont.) 72

73. For a lossless cable: Coaxial Cable (cont.) 73

74. -l0-l0 l0l0 a b Find G (conductance / length) Coaxial cable h = 1 [m] From Gauss’s law: Coaxial Cable (cont.) 74

75. -l0-l0 l0l0 a b We then have or Coaxial Cable (cont.) 75

76. Observation: This result actually holds for any transmission line (proof omitted). Coaxial Cable (cont.) 76

77. Hence: Coaxial Cable (cont.) As just derived, 77 This is the loss tangent that would arise from conductivity.

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

79. General Transmission Line Formulas (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 then be used to find G if we know the material loss tangent. (4) Equation (4) can be used to find R (discussed later). 79 These formulas hold for any shape of line.

80. General Transmission Line Formulas All four per-unit-length parameters ( R, L, G, C ) can be found from 80 Note:

81. Complex Permittivity Accounting for Dielectric Loss 81 The permittivity becomes complex when there is polarization (molecular friction) loss. Loss term due to polarization (molecular friction) Example: distilled water heats up in a microwave oven, even though there is essentially no conductivity.

82. Effective Complex Permittivity Effective permittivity that accounts for conductivity 82 The effective permittivity accounts for conductive loss. Hence We then have Accounting for Dielectric Loss (cont.)

83. Most general expression for loss tangent: Loss due to molecular friction Loss due to conductivity 83 Accounting for Dielectric Loss (cont.)

84. For most practical insulators (e.g., Teflon), we have 84 Note: The loss tangent is usually (approximately) constant for practical insulating materials, over a wide range of frequencies. Typical microwave insulating material (e.g., Teflon): tan  = 0.001. Accounting for Dielectric Loss (cont.)

85. General Transmission Line Formulas (1) (2) (3) (4) 85 Here we allow for dielectric polarization loss. Note:

86. General Transmission Line Formulas (cont.) 86 Here we allow for dielectric polarization loss. All four per-unit-length parameters ( R, L, G, C ) can be found from Note:

87. Common Transmission Lines Coax Twin-lead a b h a a 87

88. Common Transmission Lines (cont.) Microstrip h w rr t 88 Approximate CAD formula

89. Common Transmission Lines (cont.) Microstrip h w rr t 89 Approximate CAD formula

90. 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. Z Th ZLZL Z0Z0 + - 90

91. 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.) 91 Z Th ZLZL Z0Z0 + - This is explored next.

92. a b z The coaxial cable is a perfectly shielded system – there is never any radiation at any frequency, as long as the metal thickness is large compared with a skin depth. The fields are confined to the region between the two conductors. Limitations of Transmission-Line Theory (cont.) 92 Coaxial Cable

93. 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! 93

94. The infinite twin lead will not radiate by itself, regardless of how far apart the lines are (this is true for any transmission line). h Incident Reflected The incident and reflected waves represent an exact solution to Maxwell’s equations on the infinite line, at any frequency. S Limitations of Transmission-Line Theory (cont.) No attenuation on an infinite lossless line 94

95. A discontinuity on the twin lead will cause radiation to occur. Note: Radiation effects usually 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 95

96. 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) 96

97. Baluns 97 Baluns are used to connect coaxial cables to twin leads. 4:1 impedance-transforming baluns, connecting 75  TV coax to 300  TV twin lead Coaxial cable: an “unbalanced” transmission line Twin lead: a “balanced” transmission line Balun: “Balanced to “unbalanced” https://en.wikipedia.org/wiki/Balun They suppress the common mode currents on the transmission lines.

98. Baluns (cont.) 98 https://en.wikipedia.org/wiki/Balun “Baluns are present in radars, transmitters, satellites, in every telephone network, and probably in most wireless network modem/routers used in homes.” Baluns are also used to connect coax (unbalanced line) to dipole antennas (balanced).

99. 99 Ground (zero volts) + - + 0 The voltages are unbalanced with respect to ground. The voltages are balanced with respect to ground. Coax Twin Lead Baluns (cont.)

100. 100 Baluns are necessary because, in practice, the two transmission lines are always both running over a ground plane. If there were no ground plane, and you only had the two lines connected to each other, then a balun would not be necessary. (But you would still want to have a matching network between the two lines if they have different characteristic impedances.) Coax Twin Lead

101. Baluns (cont.) 101 When a ground plane is present, we really have three conductors, forming a multiconductor transmission line, and this system supports two different modes. + - Differential mode (desired) + + Common mode (undesired) The differential and common modes on a twin lead over ground

102. Baluns (cont.) 102 Differential mode (desired) Common mode (undesired) The differential and common modes on a coax over ground + - + -

103. Baluns (cont.) 103 A balun prevents common modes from being excited at the junction between a coax and a twin lead. Coax Twin Lead Ground Common mode No Balun Differential mode

104. Baluns (cont.) 104 From another point of view, a balun prevents currents from flowing on the outside of the coax. Coax Twin Lead Ground No Balun I1I1 I1I1 I2I2 I1I1 I2- I1I2- I1 (direct current path to ground)

105. Baluns (cont.) 105 A simple model for the mode excitation at the junction The coax is replaced by a solid tube with a voltage source at the end. Solid tube Twin Lead Ground + - V0V0

106. Baluns (cont.) 106 Next, we use superposition. + Solid tube Twin Lead Ground + - + - V 0 / 2 Differential mode Solid tube Twin Lead Ground + - + - V 0 / 2 Common mode

107. Baluns (cont.) 107 A balun prevents common modes from being excited at the junction between a coax and a twin lead. With Balun Coax Twin Lead Ground Differential mode Balun

108. Baluns (cont.) 108 One type of balun uses an isolation transformer. https://en.wikipedia.org/wiki/Balun The input and output are isolated from each other, so one side can be grounded while the other is not. Circuit diagram of a 4:1 autotransformer balun, using three taps on a single winding, on a ferrite rod

109. Baluns (cont.) 109 Here is a more exotic type of 4:1 impedance transforming balun. 0 V0V0 V0V0 0 0 V0V0 -V 0 0 0 I 0 /2 I0I0 I0I0 V0V0 This balun uses two 1:1 transformers to achieve symmetric output voltages. “Guanella balun”

110. Baluns (cont.) 110 Inside of 4:1 impedance transforming VHF/UHF balun for TV Ferrite core

111. Baluns (cont.) 111 Another type of balun uses a choke to “choke off” the common mode. https://en.wikipedia.org/wiki/Balun A coax is wound around a ferrite core. This creates a large inductance for the common mode, while it does not affect the differential mode (whose fields are confined inside the coax).

112. Baluns (cont.) 112 Baluns are very useful for feeding differential circuits with coax on PCBs. No balun Without a balun, there would be a common mode current on the coplanar strips (CPS) line.

113. Baluns (cont.) 113 + - + - + - + - - + = + V 0 / 2 V0V0 Excites desired differential mode Excites undesired common mode Superposition allows us to see what the problem is.

114. Baluns (cont.) 114 Baluns can be implemented in microstrip form. Tapered balun Balun feeding a microstrip yagi antenna

115. Baluns (cont.) 115 Baluns can be implemented in microstrip form. Marchand balun 180 o hybrid rat-race coupler used as a balun Input (unbalanced) Output #2 Output #1

116. Baluns (cont.) 116 If you try to feed a dipole antenna directly with coax, there will be a common mode current on the coax. I 1 = I 2 I3 0I3 0 No balun

117. Baluns (cont.) 117 Baluns are commonly used to feed dipole antennas. A balun first converts the coax to a twin lead, and then the twin lead feeds the dipole. A “sleeve balun” directly connects a coax to a dipole. Coaxial “sleeve” region Coaxial “sleeve balun”

118. Baluns (cont.) 118 Feeding a dipole antenna over a ground plane This acts like voltage source for the dipole, which forces the differential mode. The dipole is loaded by a short-circuited section of twin lead – an open circuit because of the /4 height.