3.4.3 Responses of measuring system to typical excitations y 1.0 y(t) 864 00 10.632 0.865 02 580 0 0 eq1 01000 10.368 0.135 0.6 0050 d0018 0 420 Fig 3.22 Nondimensional step-function response of first-order system
3.4.3 Responses of measuring system to typical excitations Fig. 3.22 Nondimensional step-function response of first-order system
3.4.3 Responses of measuring system to typical excitations ig 3.22(b) displays the curve of error versus time constant T Obviously y(t=0.982 when t=4T The difference between the output and the response in steady state is less than 2%0 since the step excitation is easy to carry out it is often employed to measure dynamic performances of measuring systems
Fig. 3.22 (b) displays the curve of error em versus time constant τ. Obviously y(t)=0.982 when t=4τ. The difference between the output and the response in steady state is less than 2%. ❖Since the step excitation is easy to carry out, it is often employed to measure dynamic performances of measuring systems. 3.4.3 Responses of measuring system to typical excitations
3.4.3 Responses of measuring system to typical excitations A second-order system H(s)= S 2 Its unit step response under zero initial conditions in nondimensional form are e -gont SIn Son,t+o)(underdamp ed) (367) y(o=1-1+@, De-o '(criticall y damped (3.68) 0()=1-5+y52-12(5-v2 1-s+√=2-1)on1 (overdamp d)(3.69) where P=arctan
A second-order system: Its unit step response under zero initial conditions, in nondimensional form, are where 3.4.3 Responses of measuring system to typical excitations ( ) 1 2 1 2 2 + + = n n s s H s ( ) sin ( 1 ) (underdamp ed) 1 1 2 2 − + − = − − t e y t n t n ( ) 1 (1 ) (criticall y damped) t n n y t t e − = − + ( ) (overdampe d) 2 1 1 2 1 1 1 1 2 2 1 2 2 2 2 t t n n y t e e − + − − − − − − − + − + − = − (3.67) (3.68) (3.69) 2 1 arctan − =
3.4.3 Responses of measuring system to typical excitations y(t) 2.0 1.8 N■■■■■■■■■■■ N 口■■■■■■■■■■□■ 口■■■■■■■■■L 1.2 ■■■■■■ 1.0 08 06出击 十十 0,4 ■■■ 0.2 TTTTNXLLI 0中个 6 0 Fig 3.23 Nondimentional step-function response of second-orderinstrument
3.4.3 Responses of measuring system to typical excitations Fig. 3.23 Nondimentional step-function response of second-order instrument
3.4.3 Responses of measuring system to typical excitations OAll the terms of error in the equations of step function response contains the factor e-At, thus the dynamic error is zero when t->00 UThe response of system depends largely on its damping ratio s and the natural frequency @, the higher the @n, the faster the systems response
❑All the terms of error in the equations of stepfunction response contains the factor e -At, thus the dynamic error is zero when t→∞. ❑The response of system depends largely on its damping ratio ς and the natural frequency ωn , the higher the ωn , the faster the system’ s response. 3.4.3 Responses of measuring system to typical excitations