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Lecture 5\6 Analysis in the time domain (I) —First-order system North China Electric Power University Sun Hairong
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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
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1. Examples of first-order systems Assuming zero initial conditions, Example1 : RC Circuit(1) Example2 : Spring-Damper system Example3 : RC Circuit(2)
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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
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3. Common inputs Unit impulse signal Unit step signal Unit ramp signal Harmonic signal
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Expression: Unit impulse signal Laplace transforms: R(s)=1 Unit step signal Expression: Laplace transforms:
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Unit ramp signal Harmonic signal r(t)=Asinωt 1(t) Expression: Laplace transforms: Expression: Laplace transforms:
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4. First-order system’s response to some common inputs Impulse response Step response Ramp response
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Impulse response Taking the inverse Laplace transform gives The following figure shows the output of the system (t≥0)
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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)
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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)
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Observation of the above responses ImpulseStepRamp response transient part stead- state part 01
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The relationship of these responses Assuming zero initial conditions
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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
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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?
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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
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