The number of individual loci in a root locus plot is equal to:

The number of branches/loci of the root locus diagram is:

N = P if P ≥ Z

= Z, if P ≤ Z

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

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

3. 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)

4. Angle of asymptotes: 

l = 0, 1, 2, … |P – Z| – 1

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

6. Break-in/away points:

These exist when there are multiple roots on the root locus diagram.

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

So, the roots of are the breakpoints.