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Unformatted text preview: tory but is enclosed
by two boundaries described by ±e t . The associated time constant is 1/ = 0.33, which is larger
than the time constant of A. Therefore, A has the fastest convergence.
19. Consider a second order system having a pair of complex poles at 3 ± 4j . What are the damping
ratio ⇣ and undamped natural frequency !n ?
Solution: The undamped natural frequency !n is the distance between the origin and the pole
+
p
2 + ! 2 . With numerical values
j ! , that is, !n =
= 3 and ! = 4, we have !n = 5 . The damping
ratio is given by sin where is the angle between the imaginary axis and the line connecting the
origin and the pole
+ j ! . Drawing the ﬁgure, complex poles at
19. Consider a second order system having a pair of we see that sin 3= 4j .!n = 3/5. Hence, ⇣ = 0.6 .
± / What are the damping
ratio ⇣ and undamped natural frequency !n ? 20. Consider the step responses of the following systems:
Solution: The undamped natural frequency !n is the distance between the origin and t...
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 Winter '09
 IDAN

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