1. Lecture 5\6 Analysis in the time domain (I) —First-order system North China Electric Power University Sun Hairong

2. Topics of this class  First-Order Systems: examples  Transfer function of first-order systems  Common inputs  First-order system’s response to some common inputs  First-order feedback system  Poles and zeros of the first-order system  Examples Reading: Module 3

3. 1. Examples of first-order systems Assuming zero initial conditions, Example1 ： RC Circuit(1) Example2 ： Spring-Damper system Example3 ： RC Circuit(2)

4. 2. Transfer function of first-order systems It may be seen from the previous examples that many different systems may be represented in first-order form. The generalized block diagram may be show as The generalized transfer function between the input and the output may be related by the equation

5. 3. Common inputs Unit impulse signal Unit step signal Unit ramp signal Harmonic signal

6. Expression: Unit impulse signal Laplace transforms: R(s)=1 Unit step signal Expression: Laplace transforms:

7. Unit ramp signal Harmonic signal r(t)=Asinωt  1(t) Expression: Laplace transforms: Expression: Laplace transforms:

8. 4. First-order system’s response to some common inputs Impulse response Step response Ramp response

9. Impulse response Taking the inverse Laplace transform gives The following figure shows the output of the system (t≥0)

10. Step response Leading to Taking the inverse Laplace transform gives The following figure shows the output of the system c(T)=0.632 ； c(2T)=0.865 ； c(3T)=0.95 ； c(4T)=0.98 。 (t≥0)

11. Ramp response Taking the inverse Laplace transform gives The following figure shows the output of the system In the stead-state the output lags the input by a time equal to the time constant. (t≥0)

12. Observation of the above responses ImpulseStepRamp response transient part stead- state part 01

13. The relationship of these responses Assuming zero initial conditions

14. 5. First-order feedback system Suppose a first order system is considered to be the plant in a feedback control system with a variable amplifier gain as controller, as shown in the following figure The close-loop transfer function If the input is a unit step, then

15. Let’s consider the following question, What’s the relationship of the two systems? What would be the response figure of the feedback system like? How about the stead-state error? And what is the influence of the variable K on the system’s transient and stead-state performance?

16. 6. Poles and zeros of first-order system  Concept ( See page 45~46)  Dominant poles( See page 47) Consider the case; the transfer function is given by There are two real-axis poles, far from each other. Assuming that the system is subjected to a unit impulse input ， Leading to Taking the inverse Laplace transform gives

17. The end