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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 Winter '09
 IDAN
 Steady State, system S, following systems

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