29. Sketch the root locus for the system of Figure P8.10 and find the following: [Section: 8.7] a. The range of gain to yield stability b. The value of gain that will yield a damping ratio of 0.707 for the system's dominant poles c. The value of gain that will yield closed-loop poles that are critically damped K / [s(s + 3)(s + 7)(s + 9)] (s + 30) / (s^2 + 20s + 225)
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The open-loop transfer function \( G(s)H(s) \) is given by: \[ G(s)H(s) = \frac{K}{s(s + 3)(s + 7)(s + 9)} \cdot \frac{s + 30}{s^2 + 20s + 225} \] Show more…
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GH(s) = 1 / ((s^2 + 4s + 3)(s^2 + 10s + 24)) sketch the root locus and find the following: a. The breakaway and break-in points b. The jω-axis crossing c. The range of gain to keep the system stable d. The value of K to yield a stable system with second-order complex poles, with a damping ratio of 0.5
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23. For the system of Figure P8.8(a), sketch the root locus and find the following: [Section: 8.7] a. Asymptotes b. Breakaway points c. The range of K for stability d. The value of K to yield a 0.7 damping ratio for the dominant second-order pair To improve stability, we desire the root locus to cross the jω-axis at j5.5. To accomplish this, the open-loop function is cascaded with a zero, as shown in Figure P8.8(b). e. Find the value of α and sketch the new root locus. f. Repeat Part c for the new locus. g. Compare the results of Part c and Part f. What improvement in transient response do you notice?
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