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

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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

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

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Dispersion (cont.) 4 Matched load Recall:

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

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

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

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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. )

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Signal Propagation (cont.) 9

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

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Summary Signal Propagation (cont.) 11

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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

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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

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Narrow-Band Signal (cont.) 14

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

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

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

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

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

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vgvg vpvp Narrow-Band Signal (cont.) 20

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

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

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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

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

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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

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

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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

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