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l21 - 16.06 Lecture 21 Root Locus Rules Karen Willcox...

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16.06 Lecture 21 Root Locus Rules Karen Willcox October 23, 2003 Today’s Topics 1. Angle and magnitude conditions 2. Root locus rules Reading : 6.1, 6.2, 6.3 1
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1 Angle and Magnitude Conditions Recall what we saw last lecture. The closed-loop C.E. is G c GH + 1 = 0 G c GH = 1 ( s + a 1 )( s + a 2 ) · · · ( s + a m ) K = 1 ( s + b 1 )( s + b 2 ) · · · ( s + b n ) The closed-loop poles are the values of s for which the vector G c GH has a length of one and a phase angle of ± (2 n + 1)180 . The vector G c GH can be computed graphically by considering appropriate products of the factors s + a i and s + b k . 1. Angle condition 2. Magnitude condition Note that the A i may be taken to be 1 if there are no open-loop zeroes. 2
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2 Applying the angle and magnitude condi- tions We see that there is a two-stage process: I (a) Select (guess) a trial point in the s-plane for the locus (b) Check the angle condition (c) Probably not right - move point until it is - now you have a closed-loop root (d) Repeat to develop the complete locus II Select a desired closed-loop root and calculate the necessary K 3
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