Zeros and Poles of Transfer Functions

1. Time Response*, ME451 Instructor: Jongeun Choi * This presentation is created by Jongeun Choi and Gabrial Gomes

2. Zeros and poles of a transfer function • Let G(s)=N(s)/D(s), then • Zeros of G(s) are the roots of N(s)=0 • Poles of G(s) are the roots of D(s)=0 Im(s) Re(s)

3. Theorems • Initial Value Theorem • Final Value Theorem • If all poles of sX(s) are in the left half plane (LHP), then

4. DC gain or static gain of a stable system 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3

5. DC Gain of a stable transfer function • DC gain (static gain) : the ratio of the steady state output of a system to its constant input, i.e., steady state of the unit step response • Use final value theorem to compute the steady state of the unit step response

6. Pure integrator • ODE : • Impulse response : • Step response : • If the initial condition is not zero, then : Physical meaning of the impulse response

7. First order system • ODE : • Impulse response : • Step response : • DC gain: (Use the final value theorem)

8. First order system • If the initial condition was not zero, then Physical meaning of the impulse response

9. Matlab Simulation • G=tf([0 5],[1 2]); • impulse(G) • step(G) • Time constant

10. First order system response System transfer function :

11. First order system response System transfer function : Impulse response :

12. First order system response System transfer function : Impulse response :

13. First order system response System transfer function : Impulse response : Step response :

14. First order system response Im(s) Re(s)

15. First order system response Im(s) Unstable Re(s)

16. First order system response Im(s) Unstable Re(s) -1

17. First order system response Im(s) Unstable Re(s) -2

18. First order system response Im(s) Unstable faster response slower response Re(s) constant

19. First order system – Time specifications.

20. Time to go from to First order system – Time specifications. Time specs: Steady state value : Time constant : Rise time : Settling time :

21. First order system – Simple behavior. No overshoot No oscillations

22. Second order system (mass-spring-damper system) • ODE : • Transfer function :

23. Polar vs. Cartesian representations. Cartesian representation : … Imaginary part (frequency) … Real part (rate of decay)

24. Polar vs. Cartesian representations. Cartesian representation : … Imaginary part (frequency) … Real part (rate of decay) Polar representation : … natural frequency … damping ratio

25. Polar vs. Cartesian representations. Cartesian representation : … Imaginary part (frequency) … Real part (rate of decay) Polar representation : … natural frequency … damping ratio

26. Polar vs. Cartesian representations. Cartesian representation : … Imaginary part (frequency) … Real part (rate of decay) Polar representation : … natural frequency … damping ratio Unless overdamped

27. Polar vs. Cartesian representations. System transfer function : Significance of the damping ratio : … Overdamped … Critically damped … Underdamped … Undamped

28. Polar vs. Cartesian representations. System transfer function : Significance of the damping ratio : … Overdamped … Critically damped … Underdamped … Undamped

29. Polar vs. Cartesian representations. System transfer function : Significance of the damping ratio : … Overdamped … Critically damped … Underdamped … Undamped

30. Polar vs. Cartesian representations. System transfer function : All 4 cases Unless overdamped Significance of the damping ratio : … Overdamped … Critically damped … Underdamped … Undamped

31. Underdamped second order system • Underdamped • Two complex poles:

32. Underdamped second order system

33. Impulse response of the second order system

34. Matlab Simulation • zeta = 0.3; wn=1; • G=tf([wn],[1 2*zeta*wn wn^2]); • impulse(G)

35. Unit step response of undamped systems • Unit step response : • DC gain :

36. Unit step response of undamped system

37. Matlab Simulation • zeta = 0.3; wn=1; G=tf([wn],[1 2*zeta*wn wn^2]); • step(G)

38. Second order system response. Stable 2nd order system: 2 distinct real poles A pair of repeated real poles A pair of complex poles Im(s) Unstable Re(s)

39. Second order system response. Stable 2nd order system: 2 distinct real poles A pair of repeated real poles A pair of complex poles Im(s) Unstable Re(s)

40. Second order system response. Stable 2nd order system: 2 distinct real poles A pair of repeated real poles A pair of complex poles Im(s) Unstable Re(s)

41. Second order system response. Stable 2nd order system: 2 distinct real poles A pair of repeated real poles negative real part A pair of complex poles zero real part Im(s) Unstable Re(s)

42. Second order system response. Stable 2nd order system: 2 distinct real poles A pair of repeated real poles negative real part A pair of complex poles zero real part Im(s) Unstable Re(s)

43. Second order system response. Stable 2nd order system: 2 distinct real poles A pair of repeated real poles negative real part A pair of complex poles zero real part Im(s) Unstable Re(s)

44. Second order system response. Im(s) 2 distinct real poles = Overdamped Unstable Re(s)

45. Second order system response. Im(s) Repeated real poles = Critically damped Unstable Re(s)

46. Second order system response. Im(s) Complex poles negative real part = Underdamped Unstable Re(s)

47. Second order system response. Im(s) Complex poles zero real part = Undamped Unstable Re(s)

48. Second order system response. Im(s) Underdamped Unstable Undamped Overdamped or Critically damped Re(s) Underdamped

49. Overdamped system response System transfer function : Impulse response : Step response :

50. Overdamped and critically damped system response.