1. Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 8 ECE 6340 Intermediate EM Waves 1

2. Signal Propagation on Line Introduce Fourier transform: v i (t) +vo (t)-+vo (t)- + - Z 0 ,  Semi-infinite line z = lz = l z = 0z = 0 2

3. Signal Propagation on Line Goal: To get this into a form that looks like 3 The transform variable  can then be interpreted as (radian) frequency.

4. Signal Propagation on Line (cont.) Since it follows that Next, use We start by considering a useful property of the transform. Transform definition: 4 (assuming that  is real)

5. Signal Propagation on Line (cont.) Use  = -  5

6. Signal Propagation on Line (cont.) Hence or 6 The transform variable is now interpreted as radian frequency.

7. Signal Propagation on Line (cont.) 7 t Any signal can be expressed as a collection of infinite sinusoidal signals.

8. Signal Propagation on Line (cont.) Denote Then 8 The integral is approximated as a sum.

9. Signal Propagation on Line (cont.) Taking the limit, Therefore, 9

10. Signal Propagation on Line (cont.) Final result: 10

11. Signal Propagation on Line (cont.) Compare: Output voltage expression Output voltage expressed in inverse transform form 11 Conclusion:

12. Signal Propagation on Line (cont.) The most general scenario: 12 T()T() Vi ()Vi () Vo ()Vo () T(  ) is the frequency-domain transfer function. Then we have

13. Lossless Line Denote Then 13

14. Compare with Hence or Lossless Line (cont.) 14

15. Lossless Line (cont.) The pulse moves at the phase velocity without distortion. 15

16. Lossless Line (cont.) 16 Note that the shape of the pulse as a function of z is the mirror image of the pulse shape as a function of t. z = 0z > 0 t Sawtooth pulse Note the time delay in the trace!

17. Lossless Line (cont.) 17 The sawtooth pulse is shown emerging from the source end of the line. z = 0 t=0 t = t 1 t = t 2 t = t 3 t = t 4 t Sawtooth pulse

18. Signal Propagation with Dispersion (dispersion) Assume S(t) = slowly varying envelope function v i (t) + v o (t) - + -+ - Low-loss line z = l z = 0 18

19. Signal Propagation with Dispersion (cont.) 19 This could be called a “wave packet”: The signal consists of not a single frequency, but a group of closely- spaced frequencies, centered near the carrier frequency  0.

20. The spectrum of S(t) is very localized near zero frequency: The envelope function is narrow in the  (transform) domain. Signal Propagation with Dispersion (cont.) 20

21. Use Signal Propagation with Dispersion (cont.) Hence 21

22. Peak near  0 Peak near -  0 Hence, we can write Signal Propagation with Dispersion (cont.) 22

23. Hence Neglect (second order) Next, use with Signal Propagation with Dispersion (cont.) 23

24. Signal Propagation with Dispersion (cont.) Hence 24 Denote

25. Signal Propagation with Dispersion (cont.) Multiply and divide by exp ( j  0 t ): 25

26. Let Signal Propagation with Dispersion (cont.) 26

27. Extend the lower limit to minus infinity: (since the spectrum of the envelope function is concentrated near zero frequency). Signal Propagation with Dispersion (cont.) 27 Hence

28. Signal Propagation with Dispersion (cont.) 28

29. or Define group velocity: Signal Propagation with Dispersion (cont.) 29

30. Summary vgvg vpvp Signal Propagation with Dispersion (cont.) 30

31. t = 0 t =  t = 2  Example: v g = 0, v p > 0 31 Phase velocity Group velocity

32. Example 32 Example from Wikipedia (view in full-screen mode with pptx) This shows a wave with the group velocity and phase velocity going in different directions. (The group velocity is positive and the phase velocity is negative.) http://en.wikipedia.org/wiki/Group_velocity “Backward wave” (The phase and group velocities are in opposite directions.)

33. Example 33 Example from Wikipedia (view in full-screen mode with pptx) Frequency dispersion in groups of gravity waves on the surface of deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red dot overtakes two green dots when moving from the left to the right of the figure. New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front. For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases. http://en.wikipedia.org/wiki/Group_velocity

34. Notes on Group Velocity  In many cases, the group velocity represents the velocity of information flow (the velocity of the baseband signal).  This is true when the dispersion is sufficiently small over the frequency spectrum of the signal, and the group velocity is less than the speed of light. Example: a narrow-band signal propagating in a rectangular waveguide. In some cases the group velocity exceeds the speed of light. In such cases, the waveform always distorts sufficiently as it propagates so that the signal never arrives fast than light. 34

35. Notes on Group Velocity (cont.) Sometimes v g > c (e.g., low-loss TL filled with air) t = 0 t > 0 lossless low-loss The first non-zero part of the signal does not arrive faster than light. 35

36. Signal Velocity Relativity: V0V0 + v o (t) - + -+ - TL z = l t = 0 l z = 0 36

37. Energy Velocity Definition of energy velocity v E : = time-average energy stored per unit length in the z direction. 37 vEvE zz V <W><W>

38. Energy Velocity Note: In many systems the energy velocity is equal to the group velocity. 38 vEvE zz V <W><W> where

39. Example Rectangular Waveguide b a TE 10 39 Multiply top and bottom by c /  = 1 / k 0 :

40. Example (cont.) To calculate the group velocity, use Hence Note: This property holds for all lossless waveguides. 40

41. Example (cont.) 41

42. Example (cont.) Graphical representation (  -  diagram): 42

43. Dispersion Theorem: If there is no dispersion, then v p = v g. Proof: Hence: 43 An example is a lossless transmission line (no dispersion). Note: For a lossless TL we have

44. Dispersion (cont.) Theorem: If v p = v g for all frequencies, then there is no dispersion. Proof: 44

45. Dispersion (cont.) 45 EXAMPLES: - Plane wave in free space - Lossless TL - Distortionless (lossy)TL

46. Dispersion (cont.) No dispersion Attenuation is frequency independent No distortion and Note: Loss on a TL causes dispersion and it also causes the attenuation to be frequency dependent. 46 (The phase velocity is frequency independent.)

47. Dispersion (cont.) “Normal” Dispersion: “Anomalous” Dispersion: Example: waveguide Example: low-loss transmission line This is equivalent to 47

48. Backward Wave Definition of backward wave: The group velocity has the opposite sign as the phase velocity. This type of wave will never exist on a TEM transmission line filled with usual dielectric materials, but may exist on a periodic artificial transmission line. Note: Do not confuse “backward wave" with "a wave traveling in the backward direction." 48