Signals and Systems II
Signals and Systems II ECE 450
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This 2 page Class Notes was uploaded by Fredy Okuneva on Thursday October 22, 2015. The Class Notes belongs to ECE 450 at University of Idaho taught by Richard Wells in Fall. Since its upload, it has received 7 views. For similar materials see /class/227721/ece-450-university-of-idaho in ELECTRICAL AND COMPUTER ENGINEERING at University of Idaho.
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Date Created: 10/22/15
CE 450 Root Locus Handout Some Rules of Root Locus Construction We assume the system has n poles and m lt n zeroes We let K be the gain term of the compensator RULE l Loci branches Continuous curves which comprise the branches of the locus start at each of the n poles for which K 0 As K a 00 the branches of the locus approach the m zeroes Zeroes do not move Locus branches for excess poles ie pole m l to 11 approach in nity RULE 2 Real axis segments The locus includes all points along the real axis to the left of an odd number of poles plus zeroes RULE 3 Asymptotic angles As K a 00 the branches of the locus become asymptotic to straight lines with angles 6180 i360 71 i m for i 0 i l i 2 until all n 7 m angles not differing by 360 are obtained RULE 4 Centroid of asymptotes The starting point on the real axis from which the asymptotic lines radiate is given by 20pm 100p poles 7 2 open loop zeroes 039 n 7 m This point is called the centroid of the asymptotes RULE 5 Breakaway points Roots leave the real axis at a gainK which is the maximum possible value of K in that region of the real axis Roots enter the real axis at a gain K which is the minimum possible value of K in that region of the real axis Two roots leave or strike the real axis at the breakaway points at angles of r 90 Actually applying rule 5 by hand is generally somewhat dif cult and requires a trialanderror process However the root locus tools in the MATLAB control system tool box make this very easy to do For initial design purposes eg deciding upon what kind of compensator to use rules 1 through 4 are usually suf cient to rough ou the initial design The MATLAB root locus tool kit can then be used to nish off the design There are a few additional root locus rules not discussed here These are not described because nowadays it is generally easier to use MATLAB than to apply these additional rules by hand and they are not particularly useful for roughing out an initial design CE 450 Root Locus Handout TABLE 5 1 Some root locus plots Im 1m GsH GsHs s z z sPIPz E PIXS Pz K O Q K HQ p2 2 p Re 72 p1 2 Re 1m Im GGWW GsH 5 l 1175 1112 5 plx p2sip3 I i w p l Re 1 3 P2 P1 Re lm Im a 1 GsHs a it GHs S gt 1 l S r Ix naHBXHa i fafj Xa i K zl Re b Re 1 5 ol i5 1 139quot a i m Gs1x Gm m u 15 1 1 5 psai3aim spsai smi H P1 Re P1 Re a il3 a i
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