Theory :
Root locus theory is a technique used in control systems engineering to analyze the behavior of the poles of a system as the gain K varies. It provides insights into the stability and transient response characteristics of a control system.
Consider a control system with an open-loop transfer function of the form:

$${ G(s) = \frac{N(s)}{D(s)} }$$

where N(s) is the numerator polynomial and D(s) is the denominator polynomial. The characteristic equation of the system is obtained by setting the denominator D(s) equal to zero.

To perform root locus analysis, we vary gain (K) and examine how the roots of the characteristic equation change as this parameter varies.
Step-by-step procedure to obtain the root locus is:

1. Determine the poles and zeros: Identify the poles (α, β, γ, ...) of the open-loop transfer function G(s) = N(s) / D(s). These are the roots of the characteristic equation when K = 0.
2. Define the locus equation: The locus equation is derived from the characteristic equation and the parameter K. It is given by:

$${D(s) + K * N(s) = 0}$$

This equation represents the locus of all possible roots of the characteristic equation for different values of K.
3. Determine the angles and magnitudes: For each point on the locus, calculate the angles and magnitudes of the vectors connecting the poles and zeros to that point.

The angle of a vector from a pole or zero to a point on the locus is given by:

$${θ = Σ (P - Z) + (2n + 1)π}$$

where P represents the angles from poles, Z represents the angles from zeros, and n is an integer that ensures the total angle is a multiple of 2π.

The magnitude of a vector from a pole or zero to a point on the locus is given by:

$${|G(s)| = |K * \frac{N(s)}{D(s)}|}$$

4. Construct the root locus plot: Plot the locus in the complex plane using the calculated angles and magnitudes. The locus represents the paths of the roots of the characteristic equation as K varies.

5. Determine the branches and breakaway/intersection points: Analyze the root locus plot to identify the branches of the locus. Each branch starts at a pole and ends at a zero (or at infinity). Breakaway points are the points on the locus where the branches depart from the real axis or intersect with each other.