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What follows are several examples of Nyquist plots. In general each example has five sections: 1 A definition of the loop gain, 2 A Nyquist plot made by the NyquistGui program, 3 a Nyquist plot made by Matlab, 4 A discussion of the plots and system stability, and 5 a video of the output of the NyquistGui program.

In some cases, one or more of these sections will be omitted because they are unnecessary to the understanding of the example. This means that the characteristic equation of the closed loop transfer function has no zeros in the right half plane the closed loop transfer function has no poles there.

The system is stable. This Nyquist Diagram is a little hard to decipher because the branches go off towards infinity.

## Nyquist plot

This system has a pole at the origin i. This is clear on the NyquistGui plot, but is not shown on the Matlab plot.

Whenever a detour around a pole is required, this is not shown on the Matlab plot, and the user must understand what happens going around the detour. In this case, since there is a single pole at the origin and the detour in "s" has radius approaching zero and moves in the counterclockwise direction, you know that the part of the Nyquist plot that is not shown must be a semicircle at infinity in the clockwise direction.

Since the gain margin is 3. Let's test this.

If we multiply L s by 5. But it becomes clear if we zoom in and display the stability margins, which are both negative, indicating instability. This means that the characteristic equation of the closed loop transfer function has two zeros in the right half plane the closed loop transfer function has two poles there.

The system is unstable. This plot is hard to understand because the two branches go off to infinity.

Note: this is the same transfer function that was used in example 1 , with the addition of a time delay. The shape of this plot is significantly different from those that preceded it, as is evident from the spiral as the path in L s approaches the origin.

The cause of this spiral is the delay term.

Note: magnitude and phase of a time delay are discussed here. This term has a magnitude of 1, but a phase of So as we move up the imaginary axis, the magnitude of the Nyquist plot doesn't change but the phase is constantly decreasing; this causes the spiral.

## Nyquist Plot Examples

Note: It may be helpful to look at the Nyquist plots above alongside those of example 1. Because of the time delay, it is impossible to find the roots of the characteristic equation, but simulation using Simulink or some other program will show that the system is stable.

The NyquistGui plot above clearly shows the plot in "L s " spiraling towards the origin because of the negative phase added by the time delay.

This is even more clear in the Matlab plot. Because of the time delay, it is impossible to find the roots of the characteristic equation, but simulation using Simulink or some other program will show that the system is unstable.

NyquistGui animation. NyquistGui animation Discussion This system has a pole at the origin i.

The Matlab plot is initially quite hard to decipher, But it becomes clear if we zoom in and display the stability margins, which are both negative, indicating instability.

Example 5: L s has double pole at the origin Loop gain: NyquistGui plot Matlab plot This plot is hard to understand because the two branches go off to infinity.

Example 6: L s has time delay Loop gain: Note: this is the same transfer function that was used in example 1 , with the addition of a time delay.