Prof. David R. Jackson Dept. of ECE

1. ECE 5317-6351 Microwave Engineering Fall 2011 Prof. David R. Jackson Dept. of ECE Notes 11 Waveguides Part 8: Dispersion and Wave Velocities

2. Dispersion Dispersion => Signal distortion due to “non-constant” z phase velocity => Phase relationships in original signal spectrum are changed as the signal propagates down the guide. • In waveguides, distortion is due to: • Frequency-dependent phase velocity (frequency dispersion) • Frequency-dependent attenuation => distorted amplitude relationships • Propagation of multiple modes that have different phase velocities (modal dispersion)

3. Dispersion (cont.) Consider two different frequencies applied at the input: Matched load

4. Dispersion (cont.) Matched load Recall:

5. Dispersion (cont.) No dispersion (dispersionless) Dispersion Phase relationship at end of the line is different than that at the beginning.

6. Signal Propagation Consider the following system: The system will represent, for us, a waveguiding system. Waveguiding system: Amplitude Phase

7. Signal Propagation (cont.) Input signal Output signal Fourier transform pair Proof: Property of real-valued signal:

8. Signal Propagation (cont.) We can then show (See the derivation on the next slide.) The form on the right is convenient, since it only involves positive values of . (In this case,  has the nice interpretation of being radian frequency:  = 2 f . )

9. Signal Propagation (cont.)

10. Signal Propagation (cont.) Hence, we have Interpreted as a phasor Using the transfer function, we have (for a waveguiding structure)

11. Signal Propagation (cont.) Summary

12. Dispersionless System A) Dispersionless System with Constant Attenuation Constant phase velocity (not a function of frequency) The output is a delayed and scaled version of input. The output has no distortion.

13. Narrow-Band Signal B) Low-Loss System with Dispersion and Narrow-Band Signal Now consider a narrow-band input signal of the form Narrow band (Physically, the envelope is slowing varying compared with the carrier.)

14. Narrow-Band Signal (cont.)

15. Narrow-Band Signal (cont.) Hence, we have

16. Narrow-Band Signal (cont.) Since the signal is narrow band, using a Taylor series expansion about 0 results in: Low loss assumption

17. Narrow-Band Signal (cont.) Thus, The spectrum of E is concentrated near = 0.

18. Narrow-Band Signal (cont.) Define phase velocity @ 0 Define group velocity @ 0

19. Narrow-Band Signal (cont.) Carrier phase travels with phase velocity Envelope travels with group velocity No dispersion

20. Narrow-Band Signal (cont.) vg vp

21. Example: TE10 Mode of Rectangular Waveguide Recall After simple calculation: Phase velocity: Group velocity: Observation:

22. Example (cont.) Lossless Case (“Light line”)

23. Filter Response Input signal Output signal What we have done also applies to a filter, but here we use the transfer function phase directly, and do not introduce a phase constant. From the previous results, we have

24. Filter Response (cont.) Input signal Output signal Let z - Assume we have our modulated input signal: where The output is:

25. Filter Response (cont.) Input signal Output signal This motivates the following definitions: If the phase is a linear function of frequency, then Phase delay: Group delay: In this case we have no signal distortion.

26. Linear-Phase Filter Response Input signal Output signal Linear phase filter: Hence

27. Linear-Phase Filter Response (cont.) We then have A linear-phase filter does not distort the signal. It may be desirable to have a filter maintain a linear phase, at least over the bandwidth of the filter. This will tend to minimize signal distortion.