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

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) Dispersion 2

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

Dispersion (cont.) 4 Matched load Recall:

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

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

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

We can then show (See the derivation on the next slide.) Signal Propagation (cont.) 8 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. )

Signal Propagation (cont.) 9

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

Summary Signal Propagation (cont.) 11

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. Dispersionless System 12

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

Narrow-Band Signal (cont.) 14

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

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

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

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

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

vgvg vpvp Narrow-Band Signal (cont.) 20

Recall Phase velocity: Group velocity: Example: TE 10 Mode of Rectangular Waveguide After simple calculation: Observation: 21

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

Filter Response Input 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. Output signal 23 From the previous results, we have

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

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

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

Linear-Phase Filter Response (cont.) A linear-phase filter does not distort the signal. We then have 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. 27