Question

Direction: It consists of two statements, one labelled as Statement (I) and the other as Statement (II). Examine these two statements carefully and select the answers to these items using the codes given below:

Statement (I): Centroid is the point where the root loci break from the real axis.

Statement (II): Centroid is the point on the real axis where all the asymptotes intersect.

1. Both statement (I) and Statement (II) are individually true and Statement (II) is the correct explanation of Statement (I)
2. Both statement (I) and Statement (II) are individually true and Statement (II) is not the correct explanation of Statement (I)
3. Statement (I) is true but Statement (II) is false
4. Statement (I) is false but Statement (II) is true

Answer 1

Correct Answer - Option 4 : Statement (I) is false but Statement (II) is true

Root Locus Diagram:

1. Every branch of a root locus diagram starts at a pole (K = 0) and terminates at a zero (K = ∞) of the open-loop transfer function.

2. Root locus diagram is symmetrical with respect to the real axis.

3. Number of branches of the root locus diagram are:

N = P if P ≥ Z

= Z, if P ≤ Z

4. Number of asymptotes in a root locus diagram = |P – Z|

5. Centroid: It is the intersection of the asymptotes and always lies on the real axis. It is denoted by σ.

\(\sigma = \frac{{\sum {P_i} - \sum {Z_i}}}{{\left| {P - Z} \right|}}\)

ΣPi is the sum of real parts of finite poles of G(s)H(s)

ΣZi is the sum of real parts of finite zeros of G(s)H(s)

6. Angle of asymptotes: 

l = 0, 1, 2, … |P – Z| – 1

7. On the real axis to the right side of any section, if the sum of total number of poles and zeros are odd, root locus diagram exists in that section.

8. Break in/away points: These exist when there are multiple roots on the root locus diagram. It is the point where the root loci break from the real axis.

At the breakpoints gain K is either maximum and/or minimum.

So, the roots of \(\frac{{dK}}{{ds}}\) are the break points.

Therefore, Statement (I) is false but Statement (II) is true.