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# 9 solution 2 1 2 1 1 5 2 1 1 5 2 1 5 2 2 2 2 j s j s

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9. Solution 0 ) 2 1 )( 2 1 ( 1 0 ) 5 2 ( 1 0 ) ( 1 ) 5 2 ( ) ( 1 ) ( ) 5 2 ( ) ( 2 2 2 = + + + + = + + + = + + + = = + + = j s j s s k s s s k s kp s s s k s GH s H s s s k s G (a) Asymptote center 0 3 ) 2 1 ( ) 2 1 ( 0 + + + = = j j n n z p zero pole A σ = 67 . 0 3 2 3 2 1 2 1 = + j j Asymptote angle ,... 2 , 1 , 0 9 , 0 18 * 1 2 = + = & z p A n n q φ 1 A σ ω j 0 = k 1 37 . 0 = A σ -1+2j -2j +2j -j +j j 5 j 5 +

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= 0 6 0 18 * 3 1 0 18 * 0 3 1 0 * 2 & & & = = + (b) Departure angle 5 4 5 25 0 18 0 18 ) 0 9 5 13 ( 0 36 0 18 ) ( 0 36 0 18 2 1 & & & & & & & & & & = = = + + ± = + + ± = d d d d pole zero θ θ θ θ θ θ (c) Routh-Hurwrtz 0 5 2 0 5 2 1 2 3 2 3 = + + = + + + s s s s s s k 0 1 2 3 s s s s k k 2 10 2 1 0 0 5 k k s s U k k k + = = = = 2 2 ) ( 10 0 10 0 2 10 1 ω j -1+2j -2j +2j d θ 0 9 2 & = θ 5 13 1 & = θ -1-2j A σ
= 18 . 32 23 . 2 ) 5 2 ( 0 5 2 1 0 10 5 2 10 ) 5 )( 5 ( 2 ) 5 ( 2 10 2 2 3 2 3 2 3 2 2 = ± = + + = = + + + = + + + = + + = + k s s s s s k s s s k s s s k s s s s s s 10. Consider a unity feedback system with G(s) = 5 4 ) 1 ( 2 + + + s s s k (16 Marks) (a) Find the angle of departure of the root focus from the complex poles. (b) Find the entry point for the root locus as it enters the real axis. Solution 5 4 ) 1 ( ) ( 2 + + + = s s s k s G ,H(s) =1 5 4 ) 1 ( ) ( 2 + + + = s s s k s G 1+kp(s) =0 0 ) 2 )( 2 ( ) 1 ( 1 0 5 4 ) 1 ( 1 2 = + + + + + = + + + + j s j s s k s s s k 0 = k A σ -1+2j -1-2j -1 -2 -3 -4 α k 0 = k 0 ω j

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