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Produce the Root Locus of the following transfer function using whatever means you wish: G(s) = K(s+3) (s+5)/ (s+1)(s7)

a) With a gain (K) of O is the system stable? Explain?

b) Can the system become stable by adjusting K? Explain?

User AskNilesh
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Final answer:

The system is stable with a gain (K) of 0 as all poles are in the left-half of the s-plane. The system remains stable for all positive values of K since adjusting K will not cause the poles to cross into the right-half of the s-plane.

Step-by-step explanation:

To produce the Root Locus of the given transfer function G(s) = K(s+3)(s+5)/(s+1)(s+7), we need to look at the open-loop poles and zeros and how they move in the complex plane as K varies from 0 to infinity.

(a) When K=0, the transfer function G(s) reduces to zero. In this case, the stability of the system only depends on the poles of the transfer function, which are at s = -1 and s = -7. As all the poles are in the left-half of the s-plane, the system is stable at K=0.

(b) To determine if different values of K affect stability, we can examine how the root locus enters into the right-half of the s-plane. For the system to become unstable, the root locus must cross over the imaginary axis into the right-half plane. By adjusting K, we can vary the position of the closed-loop poles. However, since all the poles and zeros are already on the left side, increasing K will just move the poles along the real axis towards the zeros, but it will not cause the system to become unstable.

User VitalyP
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