What is a root locus plot?

In a feedback control system, the characteristic equation is obtained by setting the denominator of the system's closed-loop transfer function equal to zero. This equation is generally a polynomial in terms of the complex variable s (where s = σ + jω, with σ being the real part and ω the imaginary part).

The root locus plot is constructed by varying the value of the parameter (often the gain, K) along the real axis and observing how the roots of the characteristic equation move in the complex plane. The plot shows the trajectories of the roots as the parameter changes, illustrating how the system's stability and performance characteristics are affected by different parameter values.

Key features of the root locus plot include:

Asymptotes: Root loci often exhibit straight-line asymptotes that extend to infinity. The number of asymptotes is equal to the number of open-loop poles minus the number of open-loop zeros.

Breakaway and break-in points: These points represent locations where the roots enter or exit the real axis as the gain varies.

Departure angles and arrival angles: These angles show the angle at which the root loci intersect the imaginary axis as the gain increases or decreases.

Crossing points: These are points where the root loci cross the imaginary axis, representing complex conjugate poles.

Root locus plots provide valuable insights into the system's stability and performance characteristics. Engineers use root locus techniques to design controllers by placing the closed-loop poles in desirable locations to achieve desired system behavior, such as damping ratio, settling time, and overshoot. Additionally, root locus plots help engineers to understand the trade-offs between different control system parameters and to ensure system stability.