# Understanding Nyquist criterion and plot

• Vyse007
In summary, the Nyquist criterion is a criterion for stability of a closed loop system. Different sources use different terms and nomenclature, but the criterion is based on the Cauchy's Principle of Argument. The direction of encirclement matters, and opposite direction indicate a negative encirclement.
Vyse007
In my course for Control Systems Engineering, I came across the Nyquist criterion for stability of a closed loop system, which confused me to no end. I thought the Internet would provide me some relief, but alas, I ended up getting even more confused. Different sources use different terms and nomenclature.
Here is what I understood so far:

Nyquist criterion is based on the Cauchy's Principle of Argument. It says that after a contour has been transformed to a new plane, it encircles the origin of that plane N times, where N is (no. of zeroes - no. of poles) of the transforming function. The direction of encirclement matters, and opposite direction indicate a negative encirclement. So for a control system, where the gain is G(s) and feedback element is H(s) we plot the poles of the open loop transfer function G(s)H(s), since the poles of this function is same as the poles of the characteristic equation of the closed loop transfer function.

That is all I got so far. Most pages talk about the encirclement of the point -1+j0, and say that the no. of encirclements of this point is the no. of poles on the RHP (or something like that.) Some books plot the open loop function, while others plot the characteristic equation. I am really confused as to why the point (-1,0) matters, and what actually is the Nyquist criterion, and how does it help us in forming a Nyquist plot.

If possible please explain with a simple example ( 1/(s+1), or something of that sort). Help is appreciated.

@Omkar13
OK I understood the criterion by referring to Nise. But I am unable to understand how the actual plot is made. I know how to start the plot: Plot the magnitude as the frequency increases. But can you please tell me how does the peculiar shape arise. For eg, in the link that you gave, in the example 1/[(s+1)^2], the plot resembles a cardiode(kinda). How does that come about?

Sorry for late reply.I understood that you are confused because you are thinking that we Plot magnitude Vs frequency.It's not the case.We are plotting Re(G(jw)H(jw)) on X axis and Im(G(jw)H(jw)) on Y axis as Frequency increases(we are not showing frequency).We are interested in stability etc. of system rather than values.Just try to plot roughly so that you will get it.

## 1. What is the Nyquist criterion?

The Nyquist criterion is a principle used to determine the minimum sampling rate required for a continuous-time signal to be accurately represented in the digital domain. It states that the sampling frequency must be at least twice the highest frequency component of the signal in order to avoid aliasing and accurately reconstruct the original signal.

## 2. How is the Nyquist criterion represented graphically?

The Nyquist criterion is represented graphically by plotting the amplitude spectrum of the signal and its replicas. The plot is typically symmetrical around the origin, with the original signal centered at 0 and its replicas spaced at multiples of the sampling frequency. The criterion is satisfied when the replicas do not overlap and the last replica is located at the Nyquist frequency, which is half the sampling frequency.

## 3. What is the relationship between the Nyquist criterion and the sampling rate?

The Nyquist criterion states that the sampling rate must be at least twice the highest frequency component of the signal. This means that as the sampling rate increases, the criterion is more easily satisfied. However, increasing the sampling rate beyond the minimum required does not necessarily result in a more accurate representation of the signal.

## 4. Can the Nyquist criterion be violated?

Yes, the Nyquist criterion can be violated if the sampling rate is not high enough. This can result in aliasing, where high frequency components of the signal are incorrectly represented as lower frequency components. This can lead to distortion and loss of information in the digital signal.

## 5. How is the Nyquist criterion used in practical applications?

The Nyquist criterion is used in many practical applications, such as digital audio and video processing, telecommunications, and data acquisition systems. It is essential for ensuring accurate and reliable digital representations of analog signals. Engineers and scientists must consider the Nyquist criterion when designing and implementing these systems to avoid aliasing and ensure high-quality signal processing.

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