Question

A closed-loop system has the characteristic function ($s^2-4$) (s+1) + K(s-1) = 0. Its root locus plot against K is

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Characteristic function

$\Rightarrow$ ($s^2-4$)(S+1) + K(s-1) = 0

$\Rightarrow$ 1+$\frac{K(s-1)}{(s^2-4)(s+1)}$ = 1+ G(s) H(s)

Open loop transfer function

= G(s) H(s)

= $\frac{K(s-1)}{(s^2-4)(s+1)}$

Zero of OLTF s = 1; z = 1

Poles of OLTF s = -1, -2, +2, P = 3



The root locus starts from open-loop poles and terminates either on open-loopz zero or infinity.

Root locus exist on a section of real axis it the sum of the open-loop poles and zeros to the right of the section is odd.

Number of branches terminating on infinity.

= P-Z = 3-1 = 2

Angle of asymptotes

= $\frac{(2k+1)x{180}^0}{P-Z}=\frac{(2k+1)x{180}^0}{2}$

= ${90}^0$ and ${270}^0$

Intersection of asymptotes on real axis (centroid)

= $\frac{\sum poles-\sum zeros}{P-Z}=\frac{(-1-2+2)-\left(1\right)}{2}$ = -1

Option (b) is correct on the basic of above analysis.