1. INC 341PT & BP INC341 Frequency Response Method Lecture 11

2. INC 341PT & BP Design controller to decrease peak time to 2/3 and steady-state error to 0 System has 20% overshoot

3. INC 341PT & BP 3 expressions of sinusoidal signal Starts from a sinusoidal signal,, which can be rewritten as Polar form (showing magnitude and phase shift):

4. INC 341PT & BP 2 expressions of sinusoidal signal (cont.) Rectangular form (complex number): Euler’s formula (exponential):

5. INC 341PT & BP Frequency response of system Magnitude response: –ratio of output mag. To input mag. Phase response: –difference in output phase angle and input phase angle Frequency response:

6. INC 341PT & BP Question What is the output from a known system fed by a sinusoidal command?

7. INC 341PT & BP Basic property of frequency Response ‘mechanical system’ input = force output = distance sinusoidal input gives sinusoidal output with same damped frequency shifted by, mag. expanded by Answer:

8. INC 341PT & BP The HP 35670A Dynamic Signal Analyzer obtains frequency response data from a physical system.

9. INC 341PT & BP Finding frequency response from differential equation Get transfer function Set Write Then the output is composed of

10. INC 341PT & BP Finding frequency response from transfer function ω = 0, G = 0.50.5 ∟0 ω = 2, G = 0.25 – j0.250.35 ∟-45 ω = 5, G = 0.07 - 0.17i0.19 ∟-68.2 ω = 10, G =0.019 - j0.096 0.01∟-78.7 ω = ∞, G = 00 ∟-90 Substitute with

11. INC 341PT & BP What’s next? After getting magnitude and phase of the system, we need to plot them but how???

12. INC 341PT & BP Types of frequency response plots Polar plot (Nyquist plot): real and imaginary part of open-loop system. Bode plot: magnitude and phase of open- loop system (begin with this one!!). Nichols chart: magnitude and phase of open-loop system in a different manner (not covered in the class).

13. INC 341PT & BP Polar plot of so called ‘Nyquist plot’

14. INC 341PT & BP Bode plot Note: log frequency and log magnitude Magnitude Phase

15. INC 341PT & BP Bode plot 1 st order or higher terms that can be written as a product of 1 st order terms –4 cases: 2 nd order terms –2 cases:

16. INC 341PT & BP ω = 0 ω >> a phase = 0 phase = 90 ω = a phase = 45 First order terms Case I: one zero at -a

17. INC 341PT & BP Asymptotes (approximation) Break frequency = freq. at which mag. has changed by 3 db

18. INC 341PT & BP 3 dB at break frequency

19. INC 341PT & BP ω = 0 ω >> a phase = 0 phase = -90 ω = a phase = -45 First order terms Case II: one pole at -a

20. INC 341PT & BP G(s) = 1/s Magnitude depends directly on jω (straight line down passing through zero dB at ω=1) Phase = - 90 (constant) Case III: one zero at 0 G(s) = s Magnitude depends directly on jω (straight line up passing through zero dB at ω=1) Phase = 90 (constant) Case IV: one pole at 0 First order terms

21. INC 341PT & BP G(s) = s G(s) = 1/s G(s) = s+aG(s) = 1/(s+a)

22. INC 341PT & BP It’s convenient for calculation to plot magnitude in log scale!!! What about ??? plot each term separately and sum them up log magnitude (s+2) added with log magnitude (s+3) phase (s+2) added with phase (s+3)

23. INC 341PT & BP Bode Plots Find magnitude and phase of each term and sum them up!!! mag(num)-mag(den) phase(num)-phase(den)

24. INC 341PT & BP Example sketch bode plot of break frequency at 1,2,3

25. INC 341PT & BP Frequencysmall123 s-20 1/(s+1)0-20 1/(s+2)00-20 (s+3)00020 Total Slope-20-40-60-40 Slope at each break frequency for magnitude plot

26. INC 341PT & BP Magnitude Plot

27. INC 341PT & BP Frequency small 0.10.20.3102030 s0000000 1/(s+1)0-45 000 1/(s+2)00-45 00 (s+3)00045 0 Total Slope0-45-90-450450 Slope at each point for phase plot

28. INC 341PT & BP Phase Plot

29. INC 341PT & BP Case I: 2 zeros Small ω = 0 large ω = ∞ log magnitude: set s = jω 2 nd order terms

30. INC 341PT & BP Second-order bode plot

31. INC 341PT & BP Magnitude plot of

32. INC 341PT & BP Phase plot of

33. INC 341PT & BP Magnitude plot of Case II: 2 poles

34. INC 341PT & BP Phase plot of

35. INC 341PT & BP Example sketch bode plot of Set then At DC, set s=0, Break frequency at 2, 3, (or 5)

36. INC 341PT & BP Magnitude Plot

37. INC 341PT & BP Phase plot

38. INC 341PT & BP Phase plot

39. INC 341PT & BP Conclusions Drawing Bode plot Get transfer function Set Evaluate the break frequency Approximate mag. and phase at low and high frequencies, and also at the break frequency –Mag. plot: slope changes for 1 st order, for 2 nd order (at break frequency) –Phase plot: slope changes for 1 st order, for 2 nd order