21_SpeedUpEffectOfZero

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

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.

Ask a homework question - tutors are online