How to Draw Root Locus of a System?

To draw the root locus of a system, you can follow these steps:

Identify the system’s transfer function. The transfer function is a mathematical representation of the system’s behaviour in the frequency domain. It is typically written as the ratio of the Laplace transform of the system’s output to the Laplace transform of the system’s input.

Determine the system’s open-loop transfer function. The open-loop transfer function is when the system’s feedback loop is open, meaning that the output is not fed back to the input.

Determine the system’s closed-loop transfer function. We use the closed-loop transfer function if we feed the output back to the input. It is the transfer function when the system’s feedback loop is closed.

Now find the poles and the zeros of the open-loop transfer function. We get the poles by determining the roots of the denominator of the transfer function. Similarly, we get the zeros by selecting the roots of the numerator.

Determine the location of the poles and zeros of the closed-loop transfer function. The poles and zeros of the closed-loop transfer function will be the same as those of the open-loop transfer function, except that the poles will be shifted by the negative of the feedback gain.

Plot the poles and zeros of the closed-loop transfer function on the complex plane. The root locus is the path traced out by the poles of the closed-loop transfer function as the feedback gain is varied.

Sketch the root locus. To sketch the root locus, you can start by plotting the poles and zeros of the closed-loop transfer function and then drawing lines connecting the poles and zeros. The root locus will typically have branches corresponding to the transfer function’s different poles and zeros. You can then use the rules of root locus analysis to determine the shape of the root locus.

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