1. WAVEGUIDES AND RESONATORS
UNIT V
WAVEGUIDES AND RESONATORS

2. Transmission Lines and Waveguides
Waveguide and other transmission lines for the low-loss transmission of microwave power.
Early microwave systems relied on waveguide and coaxial lines for transmission line media.
Waveguide: high power-handling capability, low loss, but bulky and expensive
Coaxial line: high bandwidth, convenient for test applications, difficult medium in which to fabricate complex microwave components.
Planar transmission lines: stripline, microstrip, slotline, coplanar waveguide  compact, low cost, easily integrated with active devices 2

3. 3.1 General Solutions for TEM, TE and TM waves
General solutions to Maxwell’s equations for the specific cases of TEM, TE and TM wave propagation in cylindrical transmission lines or waveguides.
Uniform in z direction and infinitely long
Figure 3.1 (p. 92) (a) General two-conductor transmission line and (b) closed waveguide. 3

4. Assume ejωt dependence
where e(x,y) and h(x,y): transverse (x,y) E & H field components, ez and hz: longitudinal E & H field components.
Assume source free, 4

5. 5

6. where kc: cutoff wavenumber,
: the wavenumber of the material filling with the transmission line or waveguide region.
TEM Waves
TEM waves are characterized by Ez = Hz = 0.
From (3.3a) and (3.4b)
 the cutoff wavenumber kc = 0 for TEM waves.
Helmholtz equation for Ex from (1.42) 6

7. Similar result also applies to Ey ( )
For e-jβz dependence,
Similar result also applies to Ey ( )
Transverse fields e(x,y) of a TEM wave satisfy Laplace’s equation.
Similarly,
In the electrostatic case, E field can be expressed as 7

8. In order for (3.13) to be valid, the curl of e must vanish:
The voltage between 2 conductors and current flow on a conductor:
TEM waves can exist when 2 or more conductors are present. (ex: Plane waves) 8

9. The wave impedance of a TEM mode9

10. The procedure for analyzing a TEM line:
Solve Laplace equation, (3.14) for Φ(x,y)
Find these constants by applying the B.C. for the known voltages on the conductors
Compute e and E form (3.13) & (3.1a). Compute h and H from (3.18) and (3.1b).
Compute V from (3.15), I from (3.16).
The propagation constant is given by (3.8), Z0 is given by Z0 = V/I. 10

11. TE Waves Characterized by Ez = 0, Hz ≠ 0.
In this case, kc ≠ 0, and the propagation constant
is generally a function of frequency
and the geometry of the line or guide. 11

12. The Helmholtz equation
Since Hz(x,y,z) = hz(x,y)e-jβz, and kc2 = k2 – β2
TE wave impedance can be 12

13. TM Waves
Characterized by Hz = 0, Ez ≠ 0. 13

14. The procedure for analyzing TE and TM waveguides
Solve the reduced Helmholtz equation, (3.21) or (3.25) for hz or ez. The solution will contain several unknown constants, and the unknown cutoff wavenumber, kc.
Use (3.19) or (3.23) to find the transverse fields from hz or ez.
Apply the B.C. to the appropriate field components to find the unknown constants and kc.
The propagation constant is given by (3.6) and the wave impedance by (3.22) or (3.26). 14

15. Attenuation due to Dielectric Loss
Using the complex dielectric constant
In practice, most dielectric materials have a very small loss (tan δ <<1). Using the Taylor expansion,
for x << a 15

16. (3.27) reduces to
For TE or TM wave
For TEM line, kc = 0, k = β 16

17. 3.2 Parallel Plate Waveguide
Figure 3.2 (p. 98) Geometry of a parallel plate waveguide. 17

18. TEM mode solution can be obtained by solving Laplace’s equation.
The simplest type of guide that can support TM and TE modes; can also support a TEM mode.
TEM Modes
TEM mode solution can be obtained by solving Laplace’s equation.
Assume
Since there is no variation in x,
The transverse E-field from (3.13), 18

19. The voltage of the top plate with respect to the bottom plate
The total current on the top plate 19

20. TM Modes
Hz = 0, Ez satisfies (3.25) with
B.C.  20

21. The cutoff frequency fc
TM0 mode = TEM mode
The cutoff frequency fc
TM1 mode is the lowest TM mode with a cutoff frequency 21

22. TMn mode propagation is analogous to a high-pass filter response.
At frequencies below the cutoff frequency of a given mode, the propagation constant is purely imaginary, corresponding to a rapid exponential decay of the fields.  cutoff or evanescent modes.
TMn mode propagation is analogous to a high-pass filter response.
The wave impedance
pure real for f > fc, pure imaginary for f < fc.
The guide wavelength is defined
 the distance between equiphase
planes along the z-axis. 22

23. λg > λ = 2π/k, the wavelength of a plane wave in the material.
The phase velocity and guide wavelength are defined only for a propagation mode, for which β is real.
A cutoff wavelength for the TMn mode may be defined as
Poynting vector 23

24. Consider the dominant TM1 mode, which has a propagation constant,
2 plane waves traveling obliquely in the –y, +z and +y, +z directions. 24

25. f  fc: β1  0: 2 plane waves up and down, no real power flow.
Figure 3.3 (p. 102) Bouncing plane wave interpretation of the TM1 parallel plate waveguide mode. 25

26. Conductor loss can be treated using the perturbation method.
where Po: the power flow down the guide in the absence of conductor loss given by (3.54), Pl: the power dissipated per unit length in the 2 lossy conductors 26

27. TE Modes
Ez = 0, Hz satisfies (3.21) with
B.C.  27

28. The cutoff frequency fc
The wave impedance 28

29. If n = 0, Ex = Hy = 0,  P0 = 0  no TE0 mode.29

30. Figure 3.4 (p. 105) Attenuation due to conductor loss for the TEM, TM, and TE1 modes of a parallel plate waveguide. 30

31. Figure 3.5 (p. 106) Field lines for the (a) TEM, (b) TM1, and (c) TE1 modes of a parallel plate waveguide. There is no variation across the width of the waveguide. 31

32. 3.3 Rectangular Waveguide
TE Modes
Ez = 0
Hz must satisfy the reduced wave equation (3.21)
Can be solved by separation of variables 32

33. Figure 3.6 (p. 107) Photograph of Ka-band (WR-28) rectangular waveguide components. Clockwise from top: a variable attenuator, and E-H (magic) tee junction, a directional coupler, an adaptor to ridge waveguide, an E-plane swept bend, an adjustable short, and a sliding matched load. 33

34. Figure 3.7 (p. 107) Geometry of a rectangular waveguide.34

35. We define separation constant kx and ky
Boundary conditions
Using (3.19c) and (3.19d) 35

36. From B.C , D = 0, and ky = nπ/b, B = 0 , and kx = mπ/a
The transverse field components of TEmn mode 36

37. The mode with the lowest cutoff frequency is called the dominant mode;
is real when
The mode with the lowest cutoff frequency is called the dominant mode; 37

38. If more than one mode is propagating, the waveguide is overmoded.
For f < fc, all field components will decay exponentially  cutoff or evanescent modes
If more than one mode is propagating, the waveguide is overmoded.
The wave impedance
The guide wavelength (λ: the wavelength of a plane wave in the filling medium) 38

39. For the TE10 mode 39

40. The power flow down the guide for the TE10 mode:
Attenuation can occur because of dielectric loss or conductor loss. 40

41. There are surface currents on all 4 walls.
The surface current on the x = 0 wall is
The surface current on the y = 0 wall is 41

42. The attenuation due to conductor loss for TE10 mode42

43. TM Modes
Hz = 0
BC: ez(x,y) = x = 0, a and y = 0, b 43

44. 44

45. Figure 3.8 (p. 112) Attenuation of various modes in a rectangular brass waveguide with a = 2.0 cm.45

46. Figure 3.9 (p. 114) Field lines for some of the lower order modes of a rectangular waveguide.46

47. Ex 3.1
a = 1.07 cm, b = 0.43 cm, f = 15 GHz
Solution: for Teflon εr = 2.08, tan δ =
Mode m n
fc (GHz) TE 1
9.72 2
19.44
24.19
TE, TM
26.07
31.03 47

48. At 15 GHz 48

49. Figure 3.10 (p. 115) Geometry of a partially loaded rectangular waveguide.49

50. 50

51. 3.4 Circular Waveguide
Figure (p. 117) Geometry of a circular waveguide. 51

52. TE Modes
Ez = 0 52

53. The general solution is53

54. Since
BC: Since Ez = 0,
But, D = 0 54

55. If the roots of Jn' (x) are defined as p'nm, so that Jn'(p'nm) = 0, where p'nm is the mth root of Jn', then kc must have the value.
See Table 3.3
The Temn modes are defined by the cutoff wavenumber, kcmn = p'nm/a, where n refers to the number of circumferential (φ) variations, and m refers to the number of radial (ρ) variations. 55

56. TE11 mode: dominant mode 56

57. Consider the dominant TE11 mode with an excitation such that B = 0.
The wave impedance
Because of the azimuthal symmetry of the circular waveguide, both sin nφ and cos nφ are terms are valid solutions, and can be present in a specific problem to any degree.
The actual amplitudes of these terms will be dependent on the excitation of the waveguide.
Consider the dominant TE11 mode with an excitation such that B = 0. 57

58. The power flow down the guide58

59. 59

60. 60

61. TM Modes 61

62. 62

63. Figure 3.12 (p. 123) Attenuation of various modes in a circular copper waveguide with a = 2.54 cm.63

64. Ex 3.2
Figure (p. 123) Cutoff frequencies of the first few TE and TM modes of a circular waveguide, relative to the cutoff frequency of the dominant TE11 mode. 64

65. Figure 3.14 (p. 125) Field lines for some of the lower order modes of a circular waveguide.65

66. 3.5 Coaxial Line
TEM Mode
Boundary conditions 66

67. Figure 3.15 (p. 126) Coaxial line geometry.67

68. The general solution to (3.148)
Since the boundary conditions do not vary with φ, the potential Φ should not vary with φ.  n = 0  kρ = 0. 68

69. Higher Order Modes
The coaxial line also support TE & TM waveguide modes in addition to a TEM mode.
In practice, these modes are usually cutoff (evanescent), only a reactive effect near discontinuities or sources, where they are excited.
For TE modes, Ez = 0, and
The general solution (from Sec. 3.4), 69

70. Nontrivial solution for C & D   Characteristic equation for kc
Boundary conditions:
Nontrivial solution for C & D 
 Characteristic equation for kc 70

71. Figure (p. 129) Normalized cutoff frequency of the dominant TE11 waveguide mode for a coaxial line. 71

72. Figure 3.17 (p. 129) Field lines for the (a) TEM and (b) TE11 modes of a coaxial line.72

73. Ex 3.3
a = 0.035”, b = 0.116”, εr = 2.2. What is the highest usable frequency, before the TE11 waveguide mode starts to propagate? 73

74. Photograph on Page 134. 74

75. Figure 3.18 (p. 131) Geometry of a grounded dielectric slab.75

76. Figure (p. 133) Graphical solution of the transcendental equation for the cutoff frequency of a TM surface wave mode of the grounded dielectric slab. 76

77. Figure (p. 135) Graphical solution of the transcendental equation for the cutoff frequency of a TE surface wave mode. Figure depicts a mode below cutoff. 77

78. Figure (p. 136) Surface wave propagation constants for a grounded dielectric slab with εr = 2.55. 78

79. 79

80. 3.7 Stripline
A planar-type of transmission line that lends itself well to microwave integrated circuitry and photolithographic fabrication.
Since stripline has 2 conductors and a homogeneous dielectric, it can support a TEM wave.
The stripline can also support higher order TM and TE modes, but these are usually avoided in practice. 80

81. Figure 3. 22 (p. 137) Stripline transmission line. (a) Geometry
Figure (p. 137) Stripline transmission line. (a) Geometry. (b) Electric and magnetic field lines. 81

82. Figure (p. 138) Photograph of a stripline circuit assembly, showing four quadrature hybrids, open-circuit tuning stubs, and coaxial transitions. 82

83. The phase velocity of a TEM mode:
Formulas for Propagation Constant, Characteristic Impedance and Attenuation
The phase velocity of a TEM mode:
The propagation constant of the stripline is
The characteristic impedance of a transmission line is
Laplace’s equation can be solved by conformal mapping to find the capacitance per unit length of the stripline.  complicated special function 83

84. where We is the effective width of the center conductor
For practical computations simple formulas have been developed by curve fitting to the exact solution.
where We is the effective width of the center conductor
Given the characteristic impedance, the strip width is
where 84

85. The attenuation due to the dielectric loss is the same as (3.30).
The attenuation due to the conductor loss
with
Ex 3.5 85

86. Figure 3.24 (p. 141) Geometry of enclosed stripline.
An Approximate Electrostatic Solution
Modified the geometry truncating the plates beyond some distance and placing metal walls on the sides.
Figure (p. 141) Geometry of enclosed stripline. 86

87. 3.8 Microstrip
Microstrip line is one of the most popular types of transmission lines, primarily because it can be fabricated by photolithographic process and is easily integrated with other passive and active microwave devices.
Microstrip line cannot support a pure TEM wave.
In most practical applications, the dielectric substrate is electrically very thin (d<<λ), and so the fields are quasi-TEM. 87

88. Figure 3. 25 (p. 143) Microstrip transmission line. (a) Geometry
Figure (p. 143) Microstrip transmission line. (a) Geometry. (b) Electric and magnetic field lines. 88

89. Figure (p. 145) Equivalent geometry of quasi-TEM microstrip line, where the dielectric slab of thickness I and relative permittivity εr has been replaced with a homogeneous medium of effective relative permittivity, εe. 89

90. The effective dielectric constant of a microstrip line:
Formulas for Propagation Constant, Characteristic Impedance and Attenuation
The effective dielectric constant of a microstrip line:
The characteristic impedance of a microstrip line is 90

91. Given Z0, and εr, the strip width is
where
The attenuation due to dielectric loss 91

92. The attenuation due to the conductor loss
where is the surface resistivity of the conductor.
Ex 3.7 92

93. An Approximate Electrostatic Solution
Figure (p. 146) Geometry of a microstrip line with conducting sidewalls. 93

94. The potential Φ(x,y) satisfying Laplace’s equation:
with BC
Since there are 2 regions defined by air/dielectric interface, with a charge discontinuity on the strip, we will have separate Φ(x,y). 94

95. Applying BC & even function on x & y,
Φ must be continuous at y = d  95

96. 96

97. The surface current density at y = d,
By a good guess,
Taking the orthogonalization for both sides of ρs, 97

98. C = capacitance per unit length of the microstrip line with εr ≠ 1
HW2. Use MATLAB to calculate the Z0 with εr = 2.2, a = 100d, for W/d = 0 to 10 (step = 0.01) and compare these results with (3.195) & (3.196) 98

99. Figure (p. 150) A rectangular waveguide partially filled with dielectric and the transverse resonance equivalent circuit. 99

100. 3.10 Wave Velocities and Dispersion
The speed of light in a medium is the velocity at which a plane wave would propagate in that medium.
The phase velocity is the speed at which a constant phase point travels.
The phase velocity is different for different frequencies, then the individual frequency components will not maintain their original phase relationships as they propagate down the transmission line or waveguide, and signal distortion will occur.  dispersion
100

101. Group Velocity The velocity at which a narrow band signal propagates.
101

102. For a lossless, matched transmission line or waveguide,
The time-domain output signal,
Now if |Z(ω)| = A is a constant, and the phase of ψ is a linear function of ω (ψ = aω),
 dispersionless (not distortion for a lossless TEM wave)
102

103. For a lossy TEM line, consider a narrow band input signal of the form representing an AM modulated carrier wave of frequency ω0.
103

104.  a time-shifted replica of the original modulation envelope f(t).
In general, β may be a complicated function of ω. But if F(ω) is narrowband (ωm<<ωo),
Then, y = ω- ω0,
 a time-shifted replica of the original modulation envelope f(t).
104

105. The velocity of this envelope is the group velocity, vg:
Ex 3.9
105

106. Figure 3.31 (p. 155) Cross section of a ridge waveguide.
106

107. Figure 3.32 (p. 155) Dielectric waveguide geometry.
107

108. Figure 3.33 (p. 156) Geometry of a printed slotline.
108

109. Figure 3.34 (p. 156) Coplanar waveguide geometry.
109

110. Figure 3.35 (p. 157) Covered microstrip line.
110