1 Chapter 10: Frequency Response Techniques. Chapter 10. Frequency Response Techniques

Transcription

1 1 Chapter 10 Frequency Response Techniques

2 2 Figure 10.1 The HP 35670A Dynamic Signal Analyzer obtains frequency response data from a physical system. The displayed data can be used to analyze, design, or determine a mathematical model for the system. Courtesy of Hewlett-Packard.

3 3 Figure 10.2 Sinusoidal frequency response: a. system; b. transfer function; c. input and output waveforms

4 4 Figure 10.3 System with sinusoidal input

5 5 Figure 10.4 Frequency response plots for G(s) = 1/(s + 2): separate magnitude and phase

6 6 Figure 10.5 Frequency response plots for G(s) = 1/(s + 2): polar plot

7 7 Figure 10.6 Bode plots of (s + a): a. magnitude plot; b. phase plot.

8 8 Table 10.1 Asymptotic and actual normalized and scaled frequency response data for (s + a)

9 9 Figure 10.7 Asymptotic and actual normalized and scaled magnitude response of (s + a)

10 10 Figure 10.8 Asymptotic and actual normalized and scaled phase response of (s + a)

11 11 Figure 10.9 Normalized and scaled Bode plots for a. G(s) = s; b. G(s) = 1/s; c. G(s) = (s + a); d. G(s) = 1/(s + a)

12 12 Figure Closed-loop unity feedback system

13 13 Figure Bode log-magnitude plot for Example 10.2: a. components; b. composite

14 14 Table 10.2 Bode magnitude plot: slope contribution from each pole and zero in Example 10.2

15 15 Table 10.3 Bode phase plot: slope contribution from each pole and zero in Example 10.2

16 16 Figure Bode phase plot for Example 10.2: a. components; b. composite

17 17 Figure Bode asymptotes for normalized and scaled G(s) = a. magnitude; b. phase s 2 + 2ζω

18 18 Table 10.4 Data for normalized and scaled logmagnitude and phase plots for (s 2 + 2ζω n s + ω n2 ). Mag = 20 log(m/ω n2 ) (table continues) Freq. ω/ω n Mag (db) ζ = 0.1 Phase (deg) ζ = 0.1 Mag (db) ζ = 0.2 Phase (deg) ζ = 0.2 Mag (db) ζ = 0.3 Phase (deg) ζ = 0.3

19 19 Freq. ω/ω n Mag (db) ζ = 0.5 Phase (deg) ζ = 0.5 Mag (db) ζ = 0.7 Phase (deg) ζ = 0.7 Mag (db) ζ = 1 Phase (deg) ζ = 1 Table 10.4 (continued)

20 20 Figure Normalized and scaled log-magnitude response for

21 21 Figure Scaled phase response for

22 22 Table 10.5 Data for normalized and scaled logmagnitude and phase plots for 1/(s 2 + 2ζω n s + ω n2 ). Mag = 20 log(m/ω n2 ) (table continues) Freq. ω/ω n Mag (db) ζ = 0.1 Phase (deg) ζ = 0.1 Mag (db) ζ = 0.2 Phase (deg) ζ = 0.2 Mag (db) ζ = 0.3 Phase (deg) ζ = 0.3

23 23 Freq. ω/ω n Mag (db) ζ = 0.5 Phase (deg) ζ = 0.5 Mag (db) ζ = 0.7 Phase (deg) ζ = 0.7 Mag (db) ζ = 1 Phase (deg) ζ = 1 Table 10.5 (continued)

24 24 Figure Normalized and scaled log magnitude response for

25 25 Figure Scaled phase response for

26 26 Figure Bode magnitude plot for G(s) = (s + 3)/[(s + 2) (s 2 + 2s + 25)]: a. components; b. composite

27 27 Table 10.6 Magnitude diagram slopes for Example 10.3

28 28 Table 10.7 Phase diagram slopes for Example 10.3

29 29 Figure Bode phase plot for G(s) = (s + 3)/[(s +2) (s 2 + 2s + 25)]: a. components; b. composite

30 30 Figure Closed-loop control system

31 31 Figure Mapping contour A through function F(s) to contour B

32 32 Figure Examples of contour mapping

33 33 Figure Vector representation of mapping

34 34 Figure Contour enclosing right half-plane to determine stability

35 35 Figure Mapping examples: a. contour does not enclose closed-loop poles; b. contour does enclose closed-loop poles

36 36 Figure a. Turbine and generator; b. block diagram of speed control system for Example 10.4

37 37 Figure Vector evaluation of the Nyquist diagram for Example 10.4: a. vectors on contour at low frequency; b. vectors on contour around infinity; c. Nyquist diagram

38 38 Figure Detouring around open-loop poles: a. poles on contour; b. detour right; c. detour left

39 39 Figure a. Contour for Example 10.5; b. Nyquist diagram for Example 10.5

40 40 Figure Demonstrating Nyquist stability: a. system; b. contour; c. Nyquist diagram

41 41 Figure a. Contour for Example 10.6; b. Nyquist diagram

42 42 Figure a. Contour and root locus of system that is stable for small gain and unstable for large gain; b. Nyquist diagram

43 43 Figure a. Contour and root locus of system that is unstable for small gain and stable for large gain; b. Nyquist diagram

44 44 Figure a. Portion of contour to be mapped for Example 10.7; b. Nyquist diagram of mapping of positive imaginary axis

45 45 Figure Nyquist diagram showing gain and phase margins

46 46 Figure Bode log-magnitude and phase diagrams for the system of Example 10.9

47 47 Figure Gain and phase margins on the Bode diagrams

48 48 Figure Second-order closed-loop system

49 49 Figure Representative log-magnitude plot of Eq. (10.51)

50 50 Figure Closed-loop frequency percent overshoot for a two-pole system

51 51 Figure Normalized bandwidth vs. damping ratio for: a. settling time; b. peak time; c. rise time

52 52 Figure Constant M circles

53 53 Figure Constant N circles

54 54 Figure Nyquist diagram for Example and constant M and N circles

55 55 Figure Closed-loop frequency response for Example 10.11

56 56 Figure Nichols chart

57 57 Figure Nichols chart with frequency response for G(s) = K/[s(s + 1)(s + 2)] superimposed. Values for K = 1 and K = 3.16 are shown.

58 58 Figure Phase margin vs. damping ratio

59 59 Figure Open-loop gain vs. open-loop phase angle for 3 db closed-loop gain

60 60 Figure a. Block diagram (figure continues)