1. Transmission Lines and Waveguides
Mode of guided waves:
Modes of wave Propagation along the Lines
Transverse Electro-Magnetic (TEM) Wave
Transverse Electric (TE) Wave , h-Wave
Transverse Magnetic (TM) Wave, e-Wave
Transmission Line or Waveguide region is source free:
=> Maxwell Curl Equations are

2. Solution for Wave Propagation Modes
For time harmonic waves propagating along the lines( z-axis),
Electric and magnetic field can be written as,
in cartesian coordinate system (x,y,z),

6. TEM Waves

8. Wave Impedance of a TEM mode

9. Voltage: Potential Difference between two Conductors
Current: from Ampere’s Circuital law,
Characteristic Impedance

10. TE Wave (h-wave)

11. TE Wave (h-wave)
Wave Impedance of a TE mode

12. TM Wave (e-wave)

13. TM Wave (e-wave)
Wave Impedance of a TM mode

14. Complex Propagation Constant (dielectric loss)

15. Parallel Plate Waveguide
Boundary Condition
TEM mode:
TE mode:
TM mode:

16. TEM mode:
1. Solve the Laplace Equation for Electrostatic Potential with Boundary Condition

17. TEM mode:
2. Find Fields from Potential
3. Compute V and I

18. TEM mode:
4. Characteristic Impedance and Propagation

19. Time average Poynting Vector
TEM mode:
5. Transmitted Power
Time average Poynting Vector
Time average Power transmitted to (+z) direction along the line,

20. TM mode:
1. Solve the scalar Helmholtz Eq. for axial electric field
2. Find Constants by Applying B.C.

21. 3. Find transverse Field Components TM_n mode

22. Time average Poynting Vector
Time average Power transmitted to (+z) direction along the line,

23. TE mode:
1. Solve the scalar Helmholtz Eq. for axial magnetic field
2. Find transverse Field Components TEn mode

24. 3. Find Constants by Applying B.C. on

25. Time average Poynting Vector
Time average Power transmitted to (+z) direction along the line,

27. Cut-off frequency for TM and TE mode
Minimum Cut-off Frequency

29. Rectangular Waveguide Rectangular Waveguide can’t propagate TEM waves
Propagate only TE & TM wave
For TEM,
With Boundary Condition,
Equipotential Surface (a Conductor Surface)
Rectangular Waveguide can’t propagate TEM waves

30. Rectangular Waveguide (TM modes)
Scalar Wave Equation for electric field axial component
Separation of variables

31. Boundary Condition

33. The TM mode with lowest cutoff frequency: TM11
lowest cutoff frequency of TM11:
Wave Impedance of TM mode

35. Rectangular Waveguide (TE modes)
Scalar Wave Equation for magnetic field axial component
B.C on tangential electric fields:

36. Wave Impedance of TE mode

38. The TE mode with lowest cutoff frequency: TE10
lowest cutoff frequency of TE10:
TE10
+TE11
+TM11
Only TE10
No propagation
The Dominant Mode of Rectangular Waveguide is TE10
Only TE10 mode can propagate when

39. Dominant Mode TE10
Field Components

40. Dominant Mode TE10
Time average Poynting Vector
Time average Power transmitted to (+z) direction along the line,

41. Coaxial Line Boundary Condition TEM mode can propagate TEM mode:
TE mode:
TM mode:

42. TEM mode:
1. Solve the Laplace Equation for Electrostatic Potential with Boundary Condition

43. TEM mode:
2. Find Fields from Potential

44. Wave Impedance
3. Compute V and I
Characteristic Impedance

45. Scalar Wave Equation for magnetic field axial component
Higher Order Mode (TE mode):
Scalar Wave Equation for magnetic field axial component
Separation of variables

46. Bessel’s Differential Equation
1st kind
2nd kind

47. ( ) , = ¢ - Þ ú û ù ê ë é + F µ a k Y b J solution nontrivial for D C, = - Þ ú û ù ê ë é + F a k Y b J
solution
nontrivial
for D C e c n f r
Can be determined
Approximate Solution for n=1

48. Circular Waveguide can’t propagate TEM waves
Propagate only TE & TM wave
For TEM,
With Boundary Condition,
Equipotential Surface (a Conductor Surface)
Circular Waveguide can’t propagate TEM waves

49. Scalar Wave Equation for axial component
--same with Higher order mode of Coaxial Line

50. B.C on tangential electric fields:
1. TE mode

51. B.C on tangential electric fields:
2. TM mode

52. Dominant Mode of the Circular Waveguide:TE11
Wave Impedance

53. Stripline
Boundary Condition
TEM mode can propagate

54. TEM mode:
1. Solve the Laplace Equation for Electrostatic Potential with Boundary Condition

56. General solution for B.C at y=b/2 can be written as linear combination of general solutions each region.

57. On Strip, Equipotential as V0
Not exact
Erroneous Approx.

58. On Strip,
Total Charge On Strip of unit length,

59. On Strip, given surface charge density
Not exact,
Reasonable Approx.

60. Characteristic Impedance

61. Phase unbalance for TEM mode
Microstripline
Phase unbalance for TEM mode
1. TEM mode can’t propagate
2. Hybrid TE-TM mode -> Quasi TEM mode Approx. when d<<wavelength

62. Fourier Series Solution
Quasi TEM mode Approx.
Fourier Series Solution

64. On Strip,

65. On Strip, given surface charge density

66. Characteristic impedance for microstrip transmission lines
(assumes nonmagnetic dielectric)

67. Wave Velocity and Dispersion
- Distortionless transmission
- Transfer Function for Distortionless transmission
Distortion
Amplitude distortion
Phase distortion
- Dispersion

68. Wave Velocity and Dispersion
f(t): Band limited signal with highest freq. fm

71. Wave-front for TEM
Energy Transmission

72. Power transmitted
along the Line
Standing Wave
Across the Line
For TEm0 mode

73. Definition of Voltage and Current for Network Analysis and Synthesis
1)TEM mode

74. TE10 of Rectangular Waveguide
2)non-TEM mode
TE10 of Rectangular Waveguide
Transverse Electric Field for TE10
can’t define unique Voltage & Current waves

75. For a non-TEM mode with wave impedance Zw, Transverse Fields are
Define Voltage wave proportional to Transverse Electric field,
Current wave proportional to Transverse Magnetic field.
Not Unique Definition

76. Forward power transmission
Arbitrary

77. The Concept of Impedance
1. Intrinsic Impedance of the Medium
the ratio of Electric Field to Magnetic Field
2. Wave Impedance of the particular Mode
the ratio of transverse Electric Field to transverse Magnetic Field
Characteristic Impedance of Transmission Line
the ratio of Voltage Wave to Current Wave
Impedance at some point of Transmission Line
the Ratio of Voltage to Current