By Dr.M.PadmaLalitha HOD & Professor AITS-RAJAMPET

1. Frequency Response By Dr.M.PadmaLalitha HOD & Professor AITS-RAJAMPET

2. - Bode and Nyquist plots for control analysis - Determination of transfer function - Gain and Phase margins - Stability in frequency response Frequency Response OBJECTIVE ~.

3. Magnitude and Phase Angle Transfer function is R(s) Y(s) By replacing we can find its magnitude and its phase angle Higher order transfer function Can be presented in magnitude-phase form as

4. Example A system with transfer function of G(s) is subjected to a sinusoidal input. Determine the time response of the system. Solution The input in phasor form (manitude-phase form) can be presented as As the transfer function is Hence the output is by replacing In time domain which give the time response of the system, the output is

5. First order Transfer function by replacing Frequency response Its magnitude and phase angle

6. Second order Transfer function Frequency response by replacing Its magnitude and phase angle

7. Higher Order Cascade form Frequency response by replacing Or in phasor form Example: Find the frequency response of the following transfer function

8. Where and Respective magnitude and phase angle and and

9. Bode Plot Consider higher order system Logarithmic form In dB. Phase angle

11. Bode Plot for constant gain Log magnitude in dB Its phase angle Let K=200 >> bode([200],[1]);grid

13. Bode Plot of pole on the origin Log magnitude in dB

14. 1 0

15. Slope of –20dB/decade or –6dB/octave. Its phase angle >>bode([1],[1 0]);grid

17. Bode Plot for Real Pole Frequency response Logarithmic magnitude in dB Phase angle

18. The corner frequency is For low frequency, and Can also be defined as a tenth of the corner frequency i.e. For high frequency, Can also be defined as a tenth of the corner frequency i.e. and

19. A straight line approximation for a first order system of transfer function

20. The actual Bode plot can be obtained by using the exact equation for the log magnitude and phase angle. There are small differences between the actual and approximate as shown by the following table Using MATLAB we can display the Bode plot of an open-loop transfer function of And the MATLAB command used is given by » nr=[1]; » dr=[1 1]; » sys=tf(nr,dr) » bode(sys)

21. Actual and approximate value of log magnitude for an open-loop transfer function of for a frequency range between 0.1-20 rad/s

22. Actual and approximate value of phase angle for an open-loop transfer function of for a frequency range between 0.1-20 rad/s

24. Bode Plot for Complex poles

25. Is the corner frequency For low frequency , and We define the low frequency as a tenth of the corner frequency i.e. For high frequency and While the high frequency as ten time of the corner frequency i.e.

26. Based on straight line approximation the Bode plots for LM and phase angle are shown

27. We can observe the Bode plots in MATLAB by considering In MATLAB for and >>zeta=0.25;wn=1;num=[1];den1=[1 2*zeta*wn wn*wn]; sys1=tf(num,den1); >>zeta=0.5;wn=1;num=[1];den2=[1 2*zeta*wn wn*wn]; sys2=tf(num,den2); >>zeta=0.75;wn=1;num=[1];den3=[1 2*zeta*wn wn*wn];sys3=tf(num,den3); >> bode(sys1,sys2,sys3);grid

29. Example Obtain the bode plot of the following block diagram if + _

30. 1/s 500 s+4 By replacing with s=j and making the open-loop transform function into corner freqeuncy form

31. The intial LM and phse at strat frequency is given by

32. For the magnitude polot of the Bode plot, we build a table to tshow the contribution of gradient/slope by the zero and poles at respective corner frequencies. The pole at origin will provide a slope of -20db/decade for all range of frequency

33. - For the phase plot, we need to know the change of gradient of the phase at low and high frequencies of respective corner frequency- The pole at the origin does not contribute to gradient of the phase angle as its phase angle is a constant 90o

34. The LM vs freq. plot can be determined the LM value at the corner frequency, this can be obtained by simple trigonometry (dB/dec) Using the trigonometry’s formula, we tabulate the values at the initial, corner frequencies and final value.

35. As for the LM slope, we apply a trignometry’s formula to obtain the phase angle slope at the low and high frequency for each corner frequency as in table below (deg/decade)

36. Bode plot for the magnitude and phase using straight line approximations

37. We can obtain the actual Bode plot using MATLAB as >>bode([5000 20000],[1 210 2000 0],{10^-1,10^4});

38. Determination of transfer function Constant gain Example: If dB, determine the ransfer function. dB -  (rad.s-1) dB

39. Pole/zero at origin Example: If rad.s-1, dB and slope of dB/decade. dB dB/dekad -  (rad.s-1) We know rad.s-1, dB

40. Real pole/zero Example: dB dB/dekad 45 dB/dekad  (rad.s-1) 0.01 40 250 dB/dekad

42. Pair of complex poles Example: Assume damping ratio of 0.5. dB 60 dB/dekad  (rad.s-1) 0.1 50 400 dB/dekad

44. Nyquist Plot Nyquist plot is a plot of magnitude, and phase angle for frequency on s-plane.  (rad.s-1)  0 However we can obtain the sketch of the plot by obtaining the following vectors: • at (ii) at (iii) at , crossing on the real axis (iv) at , crossing on the imaginary axis

45. First order Frequency response Magnitude dan Phase angle

46. , , and (i) At , , dan (ii) At . i.e. or (iii) No crossing in the real axis as , (iv) No crossing in the imaginary axis as , is a circle.

47. Example: Nyquist plot of Frequency response , , and . At and At and >> nyquist([5],[.25 1])