1. Prof. David R. Jackson Dept. of ECE Fall 2015 Notes 11 ECE 6340 Intermediate EM Waves 1

2. Waveguides Waveguide geometry: Assumptions: 1)The waveguide boundary is PEC 2)The material inside is homogeneous (it may be lossy) 3)A traveling wave exists in the z direction: exp ( -jk z z ). C cc z A hollow pipe waveguide is assumed here. 2

3. Comparison of Systems Transmission lines  Can propagate a signal at any frequency  Have minimal dispersion (only due to loss for TEM mode) – good for signal propagation  May or may not be shielded (e.g., coax vs. twin lead)  Become very lossy at high frequencies  Cannot handle high power (unless the cable are very big) Waveguides  Can handle high power  Have low loss (lower than transmission lines)  Are completely shielded  Can only be used above the cutoff frequency (not suitable for low frequencies)  Have large amounts of dispersion (not the best for signal propagation) Fiber Optic Cables  Have very low loss (lower than transmission lines or waveguides)  Have very low dispersion (lower than transmission lines)  Cannot handle high power  Require electro-optic interfaces with electronic and RF/microwave equipment 3

4. Free Space Propagation (Wireless System with Antennas)  No need for transmitter and receiver to be physically connected  Free space can propagate a signal at any frequency  Free space has no dispersion  Requires antennas, which have a limited bandwidth and may introduce dispersion  Has 1/r 2 propagation loss (worse then guiding systems for short distances, better than guiding systems for longer distances (kr >>1)  Usually not used for high power transfer due to propagation loss and interference Comparison of Systems (cont.) Guiding system: Free-space system: 4

5. No TEM Mode Theorem: There is no TEM z mode Proof: Assume TEM mode ( k z = k ): 5 C cc z

6. Uniqueness theorem of electrostatics: Hence trivial field C cc S No TEM Mode (cont.) 6

7. Helmholtz Equation for WG Assume TM z : so 7

8. Helmholtz Equation for WG (cont.) Then Factoring out the z dependence, Define (similarly for H z0 ) “cutoff wavenumber” Note: For a homogeneously-filled WG, k c will be number, not a function of position. 8 (2-D Helmholtz equation)

9. Helmholtz Equation for WG (cont.) The transverse field components are given by: 9

10. Property of k c Theorem k c = real number (even if  c is complex) Proof Let 2-D divergence theorem: C S 10

11. On C : Hence Therefore or Property of k c (cont.) Use 11

12. or Hence k c 2 = real positive number Note: This is also a variational equation for k c 2 (proof omitted). Hence Property of k c (cont.) 12

13. WG Eigenvalue Problem Assume TM z Denote Then Note: is independent of frequency and permittivity (from the form of the eigenvalue problem). 13 C cc z

14. WG Eigenvalue Problem (cont.) Assume TE z Let Then Note: B.C. follows from Ampere’s law (proof on next slide). Once again: is independent of frequency and permittivity (from the form of the eigenvalue problem). 14

15. Note: We also have H n = 0 on PEC. Hence WG Eigenvalue Problem (cont.) PEC x (local coordinates) 15

16. Summary of Properties for k c  The value k c is always real, even if the waveguide is filled with a lossy material.  The value of k c does not depend on the frequency or the material inside the waveguide.  The value of k c only depends on the geometrical shape of the waveguide cross section. 16

17. “Real” Theorem for Field where R 1 (x,y) is a real function, and c 1 is a complex constant. (This is valid even if the medium is lossy.) We assume that the waveguide shape is such that it does not support degenerate modes (more than one mode with the same value of k c, and hence the same value of wavenumber k z ). Assume TM z ( A similar result holds for H z0 of a TE z mode.) The shape of the waveguide field corresponding to a particular k c is unique. 17

18. “Real” Theorem for Field Proof Hence Note that Take the real part of both sides. and 18

19. Similarly, The solution of the eigenvalue problem for a given eigenvalue is unique to within a multiplicative constant (we assumed nondegenerate modes). Hence “Real” Theorem for Field (cont.) A = real constant 19 (same eigenvalue problem)

20. Therefore (proof complete) “Real” Theorem for Field (cont.) 20

21. Corollary Proof where R 2 (x,y) is a real vector function, and c 2 is a complex constant. Assume a TM z mode (similar proof for TE z mode). From the guided-wave equations: Note: The transverse field may be out of phase from the longitudinal field (they cannot both be assumed to be real at the same time, in general). 21

22. Corollary (cont.) Hence Therefore with 22

23. Wavenumber so The wavenumber is written as Phase constant Attenuation constant (The value of k c is independent of frequency and material.) 23

24. Wavenumber: Lossless Case Conclusion: k z = real or imaginary real 24 No conductivity No polarization loss

25. Lossless Case: Cutoff Frequency Cut-off frequency  c : k z = 0 Hence Note: (This is a physical interpretation of k c ) 25

26. Lossless Case: Wavenumber f > f c Above cutoff 26

27. f < f c Below cutoff Lossless Case: Wavenumber (cont.) 27

28. Lossless ( k = real) Lossless Case: Cutoff Frequency Cutoff frequency f c : 28

29. Lossy ( k = k- jk) Define “quasi-cutoff” frequency f c lossy : Mainly reactive attenuation Mainly dissipative attenuation 29 Lossy Case: Cutoff Frequency (This is a homework problem.)

30. Lossless Case: Phase Velocity f > f c where so Above cutoff: 30

31. Lossless Case: Group Velocity so f > f c Above cutoff: 31

32. Example: Rectangular Waveguide Assume TE z mode Eigenvalue Problem: where 32 The waveguide may be filled with a lossy material. b a x y

33. Separation of variables: so Hence Assume: or Rectangular Waveguide (cont.) or 33

34. Rectangular Waveguide (cont.) Hence B.C.’s: 34 Note: We could also have k x = 0, but this is included in the m = 0 result. (See note below.) (setting B = 1 )

35. Similarly, Hence so Rectangular Waveguide (cont.) 35

36. Summary Rectangular Waveguide (cont.) 36

37. TE 10 Mode Rectangular Waveguide (cont.) 37

38. TE 10 Mode Rectangular Waveguide (cont.) 38 Note that the transverse fields are both 90 o out of phase from the longitudinal field, If the material is lossless and we are above cutoff ( Z TE is real). (after simplifying)

39. TE 10 Mode Rectangular Waveguide (cont.) 39 We see that the results are consistent with the “Real” theorem and its corollary.