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Transmission Lines and Waveguides

Transmission Lines and Waveguides. Mode of guided waves: Modes of wave Propagation along the Lines T ransverse E lectro- M agnetic (TEM) Wave T ransverse E lectric (TE) Wave , h-Wave T ransverse M agnetic (TM) Wave, e-Wave. Transmission Line or Waveguide region is source free:

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Transmission Lines and Waveguides

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  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),

  3. TEM Waves

  4. Wave Impedance of a TEM mode

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

  6. TE Wave (h-wave)

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

  8. TM Wave (e-wave)

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

  10. Complex Propagation Constant (dielectric loss)

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

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

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

  14. TEM mode: 4. Characteristic Impedance and Propagation

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

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

  17. 3. Find transverse Field Components TM_n mode

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

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

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

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

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

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

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

  25. Boundary Condition

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

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

  28. Wave Impedance of TE mode

  29. 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

  30. Dominant Mode TE10 Field Components

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

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

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

  34. TEM mode: 2. Find Fields from Potential

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

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

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

  38. ( ) ( ) ( ) ¢ ¢ r = f µ F f + = e a , C J ( k a ) D Y ( k a ) 0 , f n c n c ( ) ( ) ( ) ¢ ¢ r = f µ F f + = e b , C J ( k b ) D Y ( k b ) 0 f n c n c ¢ ¢ é ù é ù J ( k a ) Y ( k a ) C n c n c Þ = 0 ê ú ê ú ¢ ¢ J ( k b ) Y ( k b ) D ë û ë û n c n c for nontrivial solution , ¢ ¢ J ( k a ) Y ( k a ) n c n c = 0 ¢ ¢ J ( k b ) Y ( k b ) n c n c ¢ ¢ ¢ ¢ Þ - = J ( k a ) Y ( k b ) J ( k b ) Y ( k a ) 0 n c n c n c n c Can be determined Approximate Solution for n=1

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

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

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

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