21_SpeedUpEffectOfZero

# 21_SpeedUpEffectOfZero - Speed-up Effect of Zero...

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Unformatted text preview: Speed-up Effect of Zero (Overdamped 2nd-order) Step Response So … a zero “speeds-up” the response of a process = but Why? and How? Simple example: poles at -0.2 and -1 with a zero at -0.5, thus = 5, τ z 2, τ 2 1 then for a gain of 1: τ1 = = K (τ s + 1) 2s + 1 2s + 1 = = 2 2 z 2 (5s + 1)(1s + 1) 5s + 6s + 1 τ s + 2τζ s + 1 1 2s Gc ( s ) = Ga ( s ) + Gb ( s ) = 2 + 2 5s + 6 s + 1 5s + 6 s + 1 = Gc ( s ) "The presence of a zero changes the shape (coefficients) but not the speed (exponentials)." −t −t τ τ y0/2 (t ) = e τ1 + 2 eτ 2 for simple second-order overdamped response 1− 1 KM τ1 − τ 2 τ1 − τ 2 −t 5 −t 1 1 − 0.4598 + 0.0017 = at t = 0.5419 5 = e + 0.25e = 1 − 1.25 y0/2 d KM and 2 dt −t −t = 2 0.25e 5 − 0.25e 1 = 0.5(0.36788) − 0.5(0.00674) = 0.1805 at t = 5 −t τ −τ τ −τ y1/2 (t ) 1− 1 z = e τ1 + 2 z KM τ1 − τ 2 τ1 − τ 2 −t τ2 e for 1st/2nd overdamped step response −t −t 5 τ2 = e − 0.25e = 1 − 0.75 1 − 0.2759 − 0.0017 = at t = 0.7224 5 which is = the 0.5419 + 0.1805 from above. Speed-Up Effect of Zero on 2nd-order Overdamped Response SpeedUpEffectOfZero.mdl Gc = Ga + Gb 2s+1 2 +6s+1 5s 1/2_Gc(s) 1 5s2 +6s+1 0/2_Ga(s) Mux PC_graphY 2s 5s2 +6s+1 2xDerivOf_0/2_Gb(s) Mux Dash 0/2 order + Dotted deriv. = Solid 1/2 order U_step 1 0.9 0.8 0.7 0.6 2 s2 + 1 5s 2 + 6 s + 1 Gc(s) ( s ) = G = (2s+1)/(5s +6s+1) c 0.5 0.4 Gc = 1st/2nd Ga = 0th/2nd Gb = 2 x time derivative of Ga step response 0.3 0.2 0.1 0 0 5 10 Time 15 20 ...
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## This note was uploaded on 03/01/2012 for the course CHE 361 taught by Professor Staff during the Winter '08 term at Oregon State.

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