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hw4_solutions-2

# hw4_solutions-2 - ECE 486 Issued February 17 Reading...

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ECE 486 Assignment # 4 Issued: February 17 Due: February 24, 2011 Reading Assignment: FPE , Sections 5.1 - 5.3 Problems: 9. Sketch the root loci below by hand, applying rules A—F. KG ( s ) = (a) K s 2 + 2 s + 12 s 2 + 2 s + 10 (b) K ( s 2 + s + 1)( s + 1) s ( s + 3)( s 2 + 5) (c) K s + 2 s 5 + 1 (d) K ( s + 2) 4 s 5 + 1 Solution : (a) OL poles: - 1 ± j 3 . 3166 ( n = 2) OL zeros: - 1 ± j 3 ( m = 2) RULES A) Number of branches = 2 B) 2 branch(es) start at OL poles C) 2 branch(es) end at OL zeros D) Real locus: None. No OL poles or zeros on the real axis. E) Asymptotes: n-m = 0 F) jw-crossings 0 = s 2 + 2 s + 10 + K ( s 2 + 2 s + 12) s = = - ω 2 + 2 + 10 - 2 + 2 Kjω + 12 K (1+ K ) ω 2 = 10+12 K and 2 ω +2 = 0 No solution (No crossings!) -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 -4 -3 -2 -1 0 1 2 3 4 Root Locus Real Axis Imaginary Axis

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(b) OL poles: 0 , - 3 , ± 5 j ( n = 4) OL zeros: - 1 , - 0 . 5 ± j 0 . 8660 ( m = 3) RULES A) Number of branches = 4 B) 4 branch(es) start at OL poles C) 3 branch(es) end at OL zeros D) Real locus: [- , -3], [-1, 0] E) Asymptotes: n-m = 4-3 = 1 ϕ = 180 + 360 ( - 1) n - m = 180 F) jw- crossings s ( s + 3)( s 2 + 5) + K ( s 2 + s + 1)( s + 1) s = = 0 Obtain two quadratic equations in the variables (
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