1、Lecture 56 Analysis in the time domain (I) First-order system,North China Electric Power University Sun Hairong,Topics of this class,First-Order Systems: examples Transfer function of first-order systems Common inputs First-order systems response to some common inputs First-order feedback system Pol
2、es and zeros of the first-order system Examples,Reading: Module 3,1. Examples of first-order systems,Assuming zero initial conditions, Example1：RC Circuit(1) Example2：Spring-Damper system Example3：RC Circuit(2),2. Transfer function of first-order systems,It may be seen from the previous examples tha
3、t 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,3. Common inputs,Unit impulse signal Unit step signal Unit ramp signal Harmonic signal,Expre
4、ssion:,Unit impulse signal,Laplace transforms: R(s)=1,Unit step signal,Expression:,Laplace transforms:,Unit ramp signal,Harmonic signal,r(t)=Asint1(t),Expression:,Laplace transforms:,Expression:,Laplace transforms:,4. First-order systems response to some common inputs,Impulse response Step response
5、Ramp response,Impulse response,Taking the inverse Laplace transform gives,The following figure shows the output of the system,(t0),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
6、0),Ramp response,Taking the inverse Laplace transform gives,The following figure shows the output of the system,In the stead-state the outputlags the input by a timeequal to the time constant.,(t0),Observation of the above responses,The relationship of these responses,Assuming zero initial condition
7、s,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,Lets consider the following quest
8、ion,Whats 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 systems transient and stead-state performance?,6. Poles and zeros of first-order system,Concept ( See page 4546)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,The end,
