###### Authored By: Ankita Prajapati

Root locus is a graphical representation of the closedloop poles of a feedback system as a parameter of the system is varied. It can be used to determine the stability of the system, the location of the poles in the splane, and the system‘s performance characteristics. The root locus is constructed by plotting the positions of the system poles as a parameter changes.

The root locus method is based on studying the effects of feedback on the openloop transfer function of a linear system.

The openloop transfer function is usually expressed in terms of the splane, which is a complex plane in which the real and imaginary parts of the svariable are plotted.

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The splane is divided into regions, which are referred to as the locus of the poles. As the parameter of the system is varied, the poles of the transfer function move around the splane and the locus of the poles is traced out.

Root locus diagrams are used to analyze the stability of a system by determining the locus of the poles.

The stability of a system can be determined by examining the locus of the poles. A system is considered to be stable if all the poles of the system are located in the left half of the splane.

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Example:

Consider the following transfer function:

G(s)= (2s + 3)/(s2 + 5s + 6)

The roots of the denominator are the poles of the system. For example, when s = -2 and s = -3, the denominator is equal to zero.

We can graph these roots over a range of values for s to create a root locus.

## Applications

1. Root locus is used in control systems engineering to understand the behavior of a system when the gain is changed.
2. It can be used to design controllers for servo systems.
3. It can be used to analyze condition monitoring systems.

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1. It can be used to analyze the stability and transient response of a system.
2. It can be used to determine the gain and phase margin of the system.
3. It can be used to determine the system’s frequency response.

1. It is difficult to analyze the stability of a system for non-linear systems.
2. It does not provide an accurate representation of the transient response of a system.
3. It does not provide an accurate representation of the frequency response of a system.

## Conclusion

Root locus diagrams are a powerful tool for analyzing the stability, performance, and sensitivity of linear, timeinvariant systems.

They can be used to determine the stability of the system, the location of the poles in the splane, and the system‘s performance characteristics. They can also be used to determine the sensitivity of a system to changes in parameters.