The characteristic equation of the feedback control system is given as: s3 + 4s2 + (K + 5)s + K = 0Here, K is a scalable variable parameter. In the root loci diagram of the system, the asymptotes of the root locus for large values of K meet at a point in the s-plane whose coordinate is:a)(-1.5, 0)b)(-2, 0)c)(-1, 0)d)(2, 0)Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Given that the characteristic equation of the feedback control system is:

s^3 + 4s^2 + (K + 5)s + K = 0

To find the asymptotes of the root locus for large values of K, we can use the rules of root locus analysis. The asymptotes of the root locus are lines that indicate the approximate direction of the root locus as K approaches infinity.

To find the number of asymptotes, we count the number of poles (or zeros) that move from the left half of the s-plane to the right half as K increases. In this case, we have three poles at the origin (s = 0) and two poles at s = -5. Since the number of poles is greater than the number of zeros, there will be two asymptotes.

The angle of the asymptotes can be calculated using the formula:

θ = (2k + 1)π / N

where θ is the angle of each asymptote, k is an integer ranging from 0 to N-1, and N is the number of asymptotes.

In this case, N = 2, so we have two asymptotes. Plugging in the values of k = 0 and k = 1, we can calculate the angles of the asymptotes:

θ1 = (2(0) + 1)π / 2 = π / 2
θ2 = (2(1) + 1)π / 2 = 3π / 2

So the angles of the asymptotes are π/2 and 3π/2.

To find the intersection point of the asymptotes, we need to find the centroid of the poles and zeros. The centroid is given by the formula:

σc = (sum of poles - sum of zeros) / (number of poles - number of zeros)

In this case, we have three poles at the origin and two poles at s = -5. Since there are no zeros, the formula simplifies to:

σc = (0 + 0 + 0 - 5) / (3 - 0) = -5/3

Therefore, the intersection point of the asymptotes is located at σ = -5/3 on the real axis.

The correct answer is option 'B' (-2, 0) which corresponds to σ = -5/3.

Answer

The given equation for the feedback control system is s³ + 4s² + (K + 5)s + K = 0

The above equation can also be written as:

s³ + 5s² + Ks + 6s + K = 0

s³ + 5s² + 6s + K(s + 1) = 0

s (s² + 5s + 6) + K(s + 1) = 0

1 + K(s + 1) / [s (s² + 5s + 6)] = 0

1 + K(s + 1) / s (s² + 2s + 3s + 6) = 0

1 + K(s + 1) / s (s + 2)(s + 3) = 0

Now, we will calculate the value of centroid, which is equal to:

σ= / (P - Z)

Here, the number of poles and zeroes are 3 and 1.

σ= [(0 - 2 - 3) + 1] / (3 - 1)

σ=(-4)⁄2

σ=-2

Hence, the correct answer is option (b).

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