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UNIT-II TIME RESPONSE ANALYSIS

UNIT-II TIME RESPONSE ANALYSIS. Contents. Introduction Influence of Poles on Time Response Transient Response of First-Order System Transient Response of Second-Order System. Introduction.

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UNIT-II TIME RESPONSE ANALYSIS

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  1. UNIT-II TIME RESPONSE ANALYSIS

  2. Contents • Introduction • Influence of Poles on Time Response • Transient Response of First-Order System • Transient Response of Second-Order System

  3. Introduction • The concept of poles and zeros, fundamental to the analysis of and design of control system, simplifies the evaluation of system response. • The poles of a transfer function are: • Values of the Laplace Transform variables s, that cause the transfer function to become infinite. • Any roots of the denominator of the transfer function that are common to roots of the numerator. • The zeros of a transfer function are: • The values of the Laplace Transform variable s, that cause the transfer function to become zero. • Any roots of the numerator of the transfer function that are common to roots of the denominator.

  4. Influence of Poles on Time Response • The output response of a system is a sum of • Forced response • Natural response System showing an input and an output Pole-zero plot of the system

  5. Influence of Poles on Time Response Evolution of a system response. Follow the blue arrows to see the evolution of system component generated by the pole or zero

  6. Influence of Poles on Time Response Effect of a real-axis pole upon transient response First-order system Pole plot of the system

  7. First-Order System • General form: • Problem: Derive the transfer function for the following circuit

  8. First-Order System • Transient Response: Gradual change of output from initial to the desired condition. • Block diagram representation: • By definition itself, the input to the system should be a step function which is given by the following: • Where, • K : Gain •  : Time constant R(s) C(s)

  9. First-Order System • General form: • Output response:

  10. First-Order System • Problem: Find the forced and natural responses for the following systems

  11. First Order System First-order system response to a unit step

  12. Transient Response Specifications • Time constant,  • The time for e-at to decay 37% of its initial value. • Rise time, tr • The time for the waveform to go from 0.1 to 0.9 of its final value. • Settling time, ts • The time for the response to reach, and stay within 2% of its final value.

  13. Transient Response Specifications • Problem: For a system with the transfer function shown below, find the relevant response specifications • Time constant,  • Settling time, ts • Rise time, tr

  14. Second-Order System • General form: • Roots of denominator: • Where, • K : Gain • ς : Damping ratio • n : Undamped natural frequency

  15. Second-Order System • Natural frequency, n • Frequency of oscillation of the system without damping. • Damping ratio, ς • Quantity that compares the exponential decay frequency of the envelope to the natural frequency.

  16. Second-Order System • Problem: Find the step response for the following transfer function • Answer:

  17. Second-Order System • Problem: For each of the transfer function, find the values of ς and n, as well as characterize the nature of the response.

  18. Second-Order System

  19. Second-Order System

  20. Second-Order System • Step responses for second-order system damping cases

  21. Second-Order System • Pole plot for the underdamped second-order system

  22. Second-Order System • Second-order response as a function of damping ratio

  23. Second-Order System • Second-order response as a function of damping ratio

  24. Second-Order System • When 0 < ς < 1, the transfer function is given by the following. • Pole position: Where,

  25. Second-Order System • Second-order response components generated by complex poles

  26. Second-Order System • Second-order underdamped responses for damping ratio value

  27. Transient Response Specifications • Second-order underdamped response specifications

  28. Transient Response Specifications • Rise time, Tr • The time for the waveform to go from 0.1 to 0.9 of its final value. • Peak time, Tp • The time required to reach the first or maximum peak. • Settling time, Ts • The time required for the transient’s damped oscillation to reach and stay within ±2% of the steady-state value.

  29. Transient Response Specifications • Percent overshoot, %OS • The amount that the waveform overshoots the steady-state, or final value at peak time, expressed as a percentage of the steady-state value.

  30. System Performance • Percent overshoot versus damping ratio

  31. System Performance • Lines of constant peak time Tp, settling time Ts and percent overshoot %OS Ts2 < Ts1 Tp2 < Tp1 %OS1 < %OS2

  32. System Performance • Step responses of second-order underdamped systems as poles move With constant real part With constant imaginary part

  33. System Performance • Step responses of second-order underdamped systems as poles move With constant damping ratio

  34. STEADY STATE RESPONSE • The steady state response is that part of the output response where the output sognal remails constant. • The parameter that is important in this is the steady state error(Ess) • Error in general is the difference between the input and the output. Steady state error is error at t→∞

  35. STEADY STATE ERROR • Static error coefficient • The response that remain after the transient response has died out is called steady state response • The steady state response is important to find the accuracy of the output. • The difference between the steady state response and desired response gives us the steady state error. • Kp = positional error constant • Kv = velocity error constant • Ka = acceleration error constant • These error constant called as static error co efficient. they have ability to minimize steady state error.

  36. STEADY STATE ERROR

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