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Chapter 6 Frequency Response

Chapter 6 Frequency Response. Hieronymus Bosch Garden of Delights . http://kunst.gymszbad.de/kunstgeschichte/motivgeschichte/altaere/frame-menue.htm. Master of Flemalle ( Robert Campin) Mérode-Altar . The central panel shows the Annunciation.

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Chapter 6 Frequency Response

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  1. Chapter 6 Frequency Response

  2. Hieronymus BoschGarden of Delights • http://kunst.gymszbad.de/kunstgeschichte/motivgeschichte/altaere/frame-menue.htm

  3. Master of Flemalle ( Robert Campin)Mérode-Altar The central panel shows the Annunciation. The child is already on his way on golden rays… • http://kunst.gymszbad.de/kunstgeschichte/motivgeschichte/altaere/frame-menue.htm

  4. Master of Flemalle ( Robert Campin)Mérode-Altar The Cloisters New York, NY • http://kunst.gymszbad.de/kunstgeschichte/motivgeschichte/altaere/frame-menue.htm Detail: The child is already on his way on golden rays, carrying the cross of the passion with him.

  5. Master of Flemalle ( Robert Campin)Mérode-Altar The Metropolitan Museum of Art, The Cloisters New York, NY Another Detail: St. Joseph the carpenter (right panel) has just completed a mousetrap (on the table), possibly to trap the devil.

  6. An Example: Analysis of Sound Waves Time domain signals: Square Wave and triangular wave.

  7. Analysis of Sound Waves Fundam. freq HARMONICS The second harmonic is twice the fundam-ental frequency, the third harmonic is three times the fundam. frequency, and so forth. Time domain signal analysis: Spectrum ofSquare Wave

  8. Fourier Transform: Let period T  infinity The interval between Discrete frequencies  0 The Fourier series becomes the Fourier Transform

  9. The A(w) and B(w) terms of the Fourier Transform can be combined into the complex term C(jw) becomes

  10. where Compare with the definition of the Laplace Transform

  11. A Sound Wave and its Spectrum

  12. Question: How do we recognize voices or musical instruments?

  13. Answer: Our brains perform a real time spectral analysis of the incoming sound signal. The spectrum, not the signal itself, informs us about the source.

  14. Question: How do we recognize color?

  15. Bode Plots: • Same content as polar plot, just a different mode of presentation. • Bode Plots: • Logarithmic w-axis. Logarithmic |F| (magnitude axis) Why? Phase values are entered directly Why?

  16. G(s)= K/(ts+1) Basic Bode Plot (First Order) Break Frequency wb f = -45deg. at wb 2. wb at -45 deg. And |F| = 0.707

  17. Bode Magnitude Plot K = 2 wb =5 1. Note K and wb 2. Draw |F| from low freq to wb 3. Draw |F| from wb , slope -1/decade

  18. Bode Phase Plot 1. Phase = -450 at wb 2. Draw f from 0 to wb/10, slope =0 3. Draw f from wb/10 freq to 10*wb 4. Min Phase is -900 from 10*wb

  19. Decibels • An alternate unit of Magnitude or Gain • Definition: xdB = 20* lg(x) • dB Notation is widely used in Filter theory and Acoustics

  20. Decibels • An alternate unit of Magnitude or Gain • Definition: xdB = 20* lg(x)

  21. Bode Plot of Integrator G(s) = 1/(s) |F|= 1/w f = -tan-1(w/0) = -900 Memorize!

  22. Underdamped second order systems and Resonance

  23. Underdamped second order systems and Resonance Asymptote Slope = -2 Phase is -90 deg. at wn

  24. Bode Plot Construction G(s) = 2/(s)(s+1) 1. Construct each Element plot 2. Graphical Summation Gain = 2. Integrator Slope = -1 Slope = -2 Integrator Phase = -90 deg.

  25. Bode Plot of 1/(s(s+1)): Matlab Plot

  26. Bode Plot Construction G(s) = 5*(s+1)/(10s+1)(100s+1) 1. Construct each Element plot Slope = -1 K = 5 Slope = -2 2. Graphical Summation: Complete plot. Note beginning and final values Slope = -1

  27. Phase Plot Construction 2. Graphical Summation of phase angles. Note beginning and final phase values. Here: f = 0 at w = 0, and f = -90 final angle G(s) = 5*(s+1)/(10s+1)(100s+1) K = 5 Initial Phase is zero to 0.001, follows the first Phase up to 0.01 Final phase: Constant - 90 deg - 90 deg./decade +45 deg./decade 0 deg./decade

  28. Bode Plot Construction: Matlab Plot

  29. Nyquist Criterion:Closed Loop Stability: Evaluate Frequency response at Phase of -180 degrees

  30. Nyquist Stability Criterion

  31. Nyquist Criterion:Stability in the Frequency Domain

  32. Nyquist Criterion in the Bode Plot:Gain Margin and Phase Margin Gain Margin Phase Margin

  33. Nyquist Criterion in the Bode Plot:Gain Margin and Phase Margin

  34. Bode Lead Design 1. Select Lead zero such that the phase margin increases while keeping the gain crossover frequency as low as reasonable. 2. Adjust Gain to the desired phase margin.

  35. Lead compensator |p| = 10*z G(s) = 1. Construct each Element plot 2. Graphical Summation Slope = 0 Slope = +1 Note Break Frequencies Slope = 0 Gain = 1 Slope = 0 Slope = 0 Slope = 0 Phase = 0

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