If the open-loop transfer function is a ratio of a numerator polynomial of degree 'm' and a denominator polynomial of degree 'n' then the integer (n-m) represents the number ofa)breakaway pointb)unstable polesc)separate root locid)asymptotesCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Open-loop transfer function:
The open-loop transfer function of a control system is the ratio of the output of the system to the input, without any feedback. It is typically represented as G(s), where s is the Laplace variable.

Numerator and denominator polynomials:
The open-loop transfer function can be written as the ratio of two polynomials. The numerator polynomial has a degree of m, while the denominator polynomial has a degree of n. These polynomials can be expressed as:

G(s) = N(s) / D(s)

where N(s) is the numerator polynomial of degree m and D(s) is the denominator polynomial of degree n.

Number of asymptotes:
The number of asymptotes of the open-loop transfer function is given by the difference between the degrees of the numerator and denominator polynomials, i.e., (n - m).

Explanation:
The number of asymptotes represents the number of poles at infinity in the root locus plot of the open-loop transfer function. Poles at infinity are the points where the root locus approaches as s tends to infinity. These points determine the behavior of the system at high frequencies.

In the root locus plot, the asymptotes are straight lines that intersect the real axis at points that are equidistant from each other. The angle of the asymptotes can be calculated using the formula:

θ = (2k + 1) * π / (n - m)

where k is an integer ranging from 0 to (n - m - 1).

The asymptotes provide valuable information about the stability and performance of the control system. They give an indication of the ultimate behavior of the system as the frequency increases. The slope and position of the asymptotes determine the gain and phase margins of the system.

Therefore, the integer (n - m) represents the number of asymptotes in the root locus plot, which is the correct answer option D.

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