Fundamentals of Measurement Technology Prof Wang Boxiong
Fundamentals of Measurement Technology (7) Prof. Wang Boxiong
3. 4.3 Responses of measuring system to typical excitations u Both the transfer function and the frequency response function describe the response of a measuring instrument or system to sinusoidal excitation u But the frequency response describes only the transfer characteristics of a system with steady-state input and output Ua transient output will reduce gradually to zero, and the system will then reach the steady-state stage. For describing the whole process of the two stages, the transfer function must be employed. and the frequency response is only a special case of the transfer function
❑Both the transfer function and the frequency response function describe the response of a measuring instrument or system to sinusoidal excitation. ❑But the frequency response describes only the transfer characteristics of a system with steady-state input and output. ❑A transient output will reduce gradually to zero, and the system will then reach the steady-state stage. For describing the whole process of the two stages, the transfer function must be employed, and the frequency response is only a special case of the transfer function. 3.4.3 Responses of measuring system to typical excitations
3.4.3 Responses of measuring system to typical excitations UThe dynamic response of a measuring system can be also obtained through applying other excitations to the system UThe most commonly used excitation signals are: unit impulse, unit step, and ramp signals
❑The dynamic response of a measuring system can be also obtained through applying other excitations to the system. ❑The most commonly used excitation signals are: unit impulse, unit step, and ramp signals. 3.4.3 Responses of measuring system to typical excitations
3.4.3 Responses of measuring system to typical excitations 1. Unit impulse response For a unit impulse function d(t,its Fourier transform AGj@)=I and the laplace transform of S(: 4(s=LS(/1. The output of a measuring instrument with d(t) as its excitation: Y(S=H(SX(S=H(S)4(S)=H(S) Making inverse Laplace transform of Y(s) en y()=L[Y(s)]=h(t (3.44) h(t is referred to as the impulse response function or weighting function of a measuring system
1. Unit impulse response For a unit impulse function δ(t), its Fourier transform Δ(jω)=1 and the Laplace transform of δ(t): Δ(s)=L[δ(t)]=1. The output of a measuring instrument with δ(t) as its excitation: Y(s)=H(s)X(s)=H(s)Δ(s)=H(s). Making inverse Laplace transform of Y(s), then h(t) is referred to as the impulse response function or weighting function of a measuring system. 3.4.3 Responses of measuring system to typical excitations y(t) = L Y (s) = h(t) −1 (3.44)
3.4.3 Responses of measuring system to typical excitations The first-order system H( +1 its impulse response h(t) (345) where t△ time constant h (t) Fig 3. 18 Impulse response of first-order system
The first-order system its impulse response h(t) where time constant. 3.4.3 Responses of measuring system to typical excitations ( ) 1 1 + = s H s ( ) t h t e − = 1 (3.45) Fig. 3.18 Impulse response of first-order system
3.4.3 Responses of measuring system to typical excitations A second-order system H(S) 2 +1 (assuming its static sensitivity K-1) h sIn sant (underdamp ed, s1) (3.46)
A second-order system (assuming its static sensitivity K=1) 3.4.3 Responses of measuring system to typical excitations ( ) 1 2 1 2 2 + + = n n s s H s ( ) ( ) ( ) ( ) (overdampe d, 1) 1 (criticall y damped, 1) sin 1 (underdamp ed, 1) 1 1 1 2 2 2 2 2 2 − − = = = − − = − + − − − − − − t t n t n n n t n n n n h t e e h t t e h t e t (3.46)
3.4.3 Responses of measuring system to typical excitations 0.8 15=0.65 1.0 0.2 0.4 ig. 3. 19 Impulse responses for second-order system with different dampimgs
3.4.3 Responses of measuring system to typical excitations Fig. 3.19 Impulse responses for second-order system with different dampimgs
3.4.3 Responses of measuring system to typical excitations UThe unit impulse does not exist in reality. Often in engineering, an approximation is made by use of a pulse signal with very short time duration for the impulse signal Example: a shock to a system, if the shock duration is shorter than t/10, where t is the systems time constant then the shock can be considered as a unit mpule
❑The unit impulse does not exist in reality. Often in engineering, an approximation is made by use of a pulse signal with very short time duration for the impulse signal. ▪ Example: a shock to a system, if the shock duration is shorter than τ/10, where τ is the system’s time constant, then the shock can be considered as a unit impulse. 3.4.3 Responses of measuring system to typical excitations
3.4.3 Responses of measuring system to typical excitations a/(x/) qo/u/rI 000 1.0 0000.9990.100 Approximate 0010.9900.995 0.9 01|09050913 208190826 0.7 Exac↑- 500.00674000681 1000.0000500000505 0.6 0 0.4 0.3 0.2 0. 0.20.3040.50.60.7080.9 Fig. 3.2 1 Exact and approximate impulse response
3.4.3 Responses of measuring system to typical excitations Fig. 3.21 Exact and approximate impulse response
3.4.3 Responses of measuring system to typical excitations 2. Step response The unit impulse function d()= (363) The step function is 5()=( (3.64 一 The response of first-order system to a unit step input y()=1-e (365) The related laplace transform is (3.66 s(as
2. Step response The unit impulse function The step function is The response of first-order system to a unit step input The related Laplace transform is 3.4.3 Responses of measuring system to typical excitations ( ) ( ) dt d t t = (3.63) ( ) ( ) − = t ' t t dt (3.64) ( ) t y t e − = 1− (3.65) ( ) ( 1) 1 + = s s Y s (3.66)
