PracticeProblemsMidterm_solu

The associated time constant is 1 033 which is larger

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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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