Polar Plots of Transfer Functions in Control Systems (How to Draw Nyquist Plot Examples)

In this post we will learn what are polar plots. Here we will also trace polar plots of some transfer functions also.
Here we will see the general shapes of the polar plots of some important transfer functions. You will learn in this post, how the shape of the polar plot changes on adding poles (non zero poles and poles at origin) or zeros to the transfer function.
But before this it is important to understand what we mean by polar plots-

Watch the Complete Video Here-

What is Polar Plot in Control System

A polar plot is a plot of a function that is expressed in polar coordinates, with radius as a function of angle.
Polar plots are drawn between magnitude and phase.
If we have a sinusoidal transfer function G(jw), which is a complex function.
Therefore we can write-
G(jw)= Re[G(jw)] + jIm[G(jw)]
G(jw)= lG(jw)l angle(G(jw))

Therefore we can represent the transfer function G(jw), as a phasor of magnitude M and phase angle phi. This phase angle is measured positively in the counter clockwise direction.
The magnitude and the phase angle changes as the input frequency (w), is varied from zero to infinity.
So the locus obtained in the complex plane by the tip of the phasor G(jw) is called the polar plot.
Polar plots are also known as a Nyquist Plots.

Now we will understand the effect on shape of the polar plot on adding poles or zeros to the transfer function.

Polar Plots of Transfer Functions (Adding Poles and Zeros)

There are following three rules that are followed to trace Polar Plots in control systems, on adding poles and zeros to the transfer function-

1- Addition of a non-zero pole to a transfer function, results in further rotation of the polar plot through an angle of -90 degrees as w tends to infinity.
2- Adding a pole at the origin to a transfer function, rotates the polar plot at 0 and infinite frequencies by a further angle of -90 degrees.
3- When we add a Zero to a transfer function, then the high frequency portion of the polar plot rotates by 90 degrees in counterclockwise direction.

We will use these rules to trace polar plots of different transfer functions with the help of a standard polar plot.
Here we will add non zero poles or poles at origin to the transfer function and will see how it affects the shape of the polar plots.
Now look at the images shown below-

Polar Plots of Transfer Functions (Nyquist Plots) Image 1

Polar Plots in Control Systems (Polar Plots of Transfer Functions) Image 2

Here you will see total 7 polar plots. Observe that, all the polar plots shown in the images are traced with the help of polar plot number 1. 
Here we have added either non zero pole or pole at origin to the transfer function of the first polar plot. The rules described above are used to draw other polar plots.

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