Correct Answer - Option 3 : 360 degree/(number of poles - number of zeroes)

__Construction rules of Inverse Root locus:__

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

2. Inverse root locus diagram is symmetrical with respect to the real axis. If the pole-zero plot is symmetrical with respect to any point on the real axis, then the IRLD also symmetrical with respect to that point, provided that the point does not have any poles or zeros.

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

N = P if P ≥ Z

= Z, if P ≤ Z

⇒ Number of branches of the root locus diagram is equal to the order of the system.

Where P & Z are the number of finite poles and zeros of G(s)H(s)

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

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

Centroid may or may not be the part of Inverse Root locus diagram.

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

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: \({\theta _l} = \frac{{\left( {2l } \right)\pi }}{{P - Z}}\)

l = 0, 1, 2, … |P – Z| – 1

7. A segment of the real axis exists on the Inverse root locus diagram if the sum of a total number of poles and zeros of G(s)H(s) is even.

8. Break-in/away points: These exist when there are multiple roots on the inverse root locus diagram.

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

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