1. Microwave Engineering
Adapted from notes by Prof. Jeffery T. Williams
ECE
Microwave Engineering
Fall 2018
Prof. David R. Jackson
Dept. of ECE
Notes 13
Waveguiding Structures Part 8:
Dispersion and Wave Velocities

2. Dispersion
Frequency dispersion: The phase velocity is not constant with frequency.
=> Phase relationships in the original signal spectrum are changed as the signal propagates down the structure.
In waveguiding structures, signal distortion is due to:
Frequency dispersion
Frequency-dependent attenuation
Propagation of multiple modes that have different phase velocities (modal dispersion)

3. Dispersion (cont.)
Consider a signal applied to the input of a transmission line:
Transmission line
Matched load
The transmission line as a transfer function:

4. Signal Propagation (cont.)
Input signal
Output signal
Fourier transform pair
Note:
The signal will normally be assumed to represent the voltage.
Property of real-valued signal:
(Please see the proof on the next slide.)

5. Signal Propagation (cont.)
Proof of Property:

6. Signal Propagation (cont.)
We can then show:
(Please 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 .)

7. Signal Propagation (cont.)
Derivation:

8. Signal Propagation (cont.)
Hence, we have
Interpreted as a phasor.

9. Signal Propagation (cont.)
Using the transfer function, we have:

10. Signal Propagation (cont.)
Summary
Waveguiding system:

11. Signal Propagation (cont.)
Propagation on a microstrip line
CAD formulas are used to get (), ().

12. Signal Propagation (cont.)
Propagation on a microstrip line

13. Dispersionless System with Constant Attenuation
No dispersion:
The phase velocity is constant
(not a function of frequency).
The output is a delayed and scaled version of the input.
The output signal has no distortion.

14. 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.

15. Narrow-Band Signal (cont.)

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

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

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

19. Narrow-Band Signal (cont.)
Define:
phase 0
Define:
group 0

20. Narrow-Band Signal (cont.)
Summary
Envelope travels with group velocity
Carrier travels with phase velocity

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

22. Narrow-Band Signal (cont.)
Example from Wikipedia (view in full-screen mode with pptx)
Red dot: phase velocity
Green dot: group velocity
Phase velocity > group velocity

23. Narrow-Band Signal (cont.)
Note on dispersion
Assume:
No dispersion
In this case the envelope and carrier are delayed the same.
Example: lossless transmission line

24. Example: TE10 Mode of Rectangular Waveguide
Recall:
After simple calculation:
Phase velocity:
Group velocity:
Observation:
(This final result is valid for any mode of a lossless waveguide or transmission line.)

25. (This is the “light line” that describes a TEM wave.)
Example (cont.)
Lossless Case
(This is the “light line” that describes a TEM wave.)
Operating point

26. 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

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

28. Filter Response (cont.)
Input signal
Output signal
This motivates the following definitions:
Phase delay:
Group delay:

29. Filter Response (cont.)
Summary
Input signal
Output signal
Phase delay:
Group delay:

30. Linear-Phase Filter Response
Input signal
Output signal
The attenuation of the ideal filter is constant, at least over the bandwidth of the filter.
Ideal linear phase filter:
The envelope and carrier are delayed the same.
Recall:
Hence

31. Linear-Phase Filter Response (cont.)
We then have: so
An ideal 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.