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# N at 3 4j what ratio dampingfollowing the natural

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Unformatted text preview: ratio ⇣ dampingfollowing the natural frequency ! undamped a pair of complex poles and damping are the for the ratio ⇣ and undamped natural frequency !n ? transfer Solution: The undamped natural frequency !n is the distance between the origin and the pole function: + p 2 + ! 2 . With numerical values j ! , that is, !n = =Faster= 4, we constant >> fastest 3 and ! time have !n = 5 . The damping s+1 P (ratio is 2 s) = given by sin . where is the angle between theonverging rate the line connecting the c imaginary axis and origin s + 4spole and the + 20 + j ! . Drawing the ﬁgure, we see that sin = /!n = 3/5. Hence, ⇣ = 0.6 . 18. Which system has the step response converging to a steady state value at the fastest rate? 20. Consider the step responses of the following systems: (A) 1 4 1 12 1 (C) 1 2 2 (C) (D) (A) (B) 2 (B) 2 (s + 1)2 + 1 100 (s +s + + 9 + 9 s + 5 (s + 2) s + 11 + ( 1) 3) s 19. Consider a second order system to meet the followingcomplex poles atrise3time4is .less...
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