Engineering College

For Figure P8.3, K (s + 1)(8 + 10) G(s) = (s + 4)(s – 6) Sketch the root locus and find the value of K for which the system is closed- loop stable. Also find the break-in and breakaway points. [Section: 8.5]

Answers

Answer 1

To find the value of K for stability, sketch the root locus by determining the asymptotes, break-in points, and breakaway points, and identify the value of K where the root locus crosses the imaginary axis on the left-hand side of the complex plane.

To sketch the root locus and find the value of K for stability, we need to follow these steps:

Step 1: Determine the open-loop transfer function G(s) based on the given equation:

G(s) = (s + 4)(s - 6) / ((s + 1)(8 + 10))

Step 2: Identify the poles and zeros of the transfer function G(s).

Poles: s = -1, -4, 6

Zeros: None

Step 3: Determine the number of branches of the root locus.

The number of branches is equal to the number of poles minus the number of zeros, which is 3 - 0 = 3.

Step 4: Determine the asymptotes of the root locus.

The asymptotes can be calculated using the formula:

Angle of asymptotes (θa) = (2k + 1) * π / n

where k = 0, 1, 2, ..., n-1 and n is the number of branches. In this case, n = 3.

Step 5: Determine the break-in and breakaway points.

The break-in and breakaway points occur when the root locus intersects the real axis. To find these points, we solve the equation G(s)H(s) = -1, where H(s) is the characteristic equation.

Step 6: Sketch the root locus by plotting the branches, asymptotes, break-in points, and breakaway points.

Step 7: Find the value of K for closed-loop stability.

The value of K for closed-loop stability is the value of K where the root locus crosses the imaginary axis (jω axis) on the left-hand side of the complex plane.

To know more about break-in points,

https://brainly.com/question/17118645

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