1. Lecture 7/8 Analysis in the time domain (II) North China Electric Power University Sun Hairong

2. Topics of this class Second-Order Systems  Physical examples  Transfer function and block diagram  Step response  Relationship between system poles and transient response  Time-domain performance specifications Reading: Module 4, Module 5

3. 1.Examples of second-order systems (assuming zero initial conditions) Examples : Electrical circuit & Spring-Damper System

4. 2. Transfer function of seconded-order systems The generalized block diagram The generalized transfer function The parameters: the damping ratio, the undamped natural frequency. The characteristic equation The close-loop poles of the system

5. The generalized block diagram may be show as Where, ξis the damping ratio and ω n is the undamped natural frequency. The symbols ξ and ω n are very important parameter of the second-order systems. They will take on physical meaning. The input and the output may be related by the equation (The generalized transfer function )

6. The characteristic equation is given by Characteristic equation solution is The solution also can be called close-loop poles of the system. It is obviously that the value of ξ and ω n can determine the close-loop poles’s position in the complex plane.

7. Case 1: For ξ <1, the poles of the equation are (a complex conjugate pair) In such a case the system is classified as underdamped system. Case 2: For ξ=1, the poles of the equation are s 1,2 = -ω n In such a case the system is classified as critically damped system. Case 3: Forξ>1, the poles of the equation are In such a case the system is classified as overdamped system.

8. 3. Step response 3.1 Assuming the system is underdamped, that is, ξ<1  The step response oscillates  The frequency is ω d.  The response decays due to the exponent - ξω n

9. 3.2 Assuming the system is overdamped, that is, ξ>1 (t ≥ 0)  The response consists of two damped items, which decay due to the exponent - s 1 and -s 2 respectively.

10. 3.3 Assuming the system is critically damped, that is, ξ=1 (t ≥ 0)  The response decays due to the exponent - ξω n

11. The pole on the negative real axisExponent decay item The poles on the imaginary axis The step response oscillates in frequency ω d with equal amplitude The poles on the left complex plane The step response oscillates with frequency ω d The response decays due to the exponent - ξω n  The response’s form is related to the close-loop poles’s form. That is

12.  The relationship between the location of the system poles on the complex plane and the corresponding transient time response may be summarized as follows: The frequency of oscillation ω d is given by the imaginary part of the pole. The damping ratio is the sine of the angle between the pole- origin line and the imaginary axis. (Page 79, Fig.5.4) : relationship between poles and the time-domain parameters (Page 80, Fig.5.5) : Step response from various poles on the complex plane

13. (Page 65, Fig.4.16)  The step response for various values of damping ratio  The the relationship between overshot and damping ratio (Page 66, Fig.4.17)

14. 4. Time-domain performance specifications The specifications are usually given subjected to a step input. A typical underdamped response is adopted to define the specifications. (Page 82, Fig. 5.6. ) 1). Rise Time (Tr): The time for the system to first achieve the final value of the output. It is a measure of the speed with which the system responds to the input. 2). Peak Time (Tp): The time to the maximum of the first overshoot. It is also a measure of the speed of the system’s response. Since the output oscillates at frequency ωd,the peak time is one half cycle.

15. 3). Equivalent Time Constant: Step response of first-order system is known as The response decays due to the exponent – 1/T. And step response of second order system is known as The response decays due to the exponent – ξωn.

16. 4). Settling Time: This time is defined to be the elapsed time before the output becomes bounded by two equispaced limits on either side of the stead-state output. Typically these limits may be specified as or.

17. 5). Steady-state Error : It is a function of the system transfer function and the type of input.  First-order system:  Second-order system: Stead-state error is: R(s)-C(s)=1/(1+K) Stead-state error is: R(s)-C(s)=0

18. 6). Percentage Overshoot:

19. 5. Other Time-domain response ， Ramp response Harmonic response 6. Sample problems

20. The end