hw3 - EML 4312 Fall 2009 Root Locus and Continued Fraction...

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EML 4312 Fall 2009 Root Locus and Continued Fraction Expansion The due date for this assignment is Wednesday 10/14. Show your work. 1. Determine the root locus for the following problems. For full credit, you need to show the hand drawn plot, and all calculations such as centroid, breakaway points, relative degree, asymptote angles (at least draw the asymptotes), etc. a. (10 points) F ( s ) = s + 2 ( s + 1) 3 Solution: -2.5 -2 -1.5 -1 -0.5 0 0.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 Root Locus Real Axis Imaginary Axis Poles = 1 , 1 , 1 Zero = 2 Locus on real line is between 2 and 1 (found by counting number of poles and zeroes which are odd in this interval). No of asymptotes rd = n m = 2 Angle of asymptotes ± 180 2 = ± 90 1
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Point of intersection of asymptotes with real axis s = 3 2 2 s = 0 . 5 Calculate Break away point = ( s + 1) 3 + k ( s + 2) = 0 k = ( s + 1) 3 s + 2 Calculate dk ds dk ds = 3 ( s + 1) 2 s + 2 + ( s + 1) 3 ( s + 2) 2 Equate dk ds = 0 s = 1 , 1 , 5 2 5 2 does not lie on the real root locus, so Break away point is 1 . b. (10 points) F ( s ) = ( s + 1) s 2 Solution: Poles = 0 , 0 Zero = 1 Locus on real line is between −∞ and 1 (found by counting number of poles and zeroes which are odd in this interval). No of asymptotes = rd = n m = 1 Angle of asymptote 180 Calculate Break away and Break in point = s 2 + k ( s + 1) = 0 k = s 2 s + 1 Calculate dk ds dk ds = 2 s s + 1 + s 2 ( s + 1) 2 Equate dk ds = 0 s ( s + 2) = 0 s = 0 , 2 2
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Breakaway point = 0 Breakin Point = 2 .
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