Nyquist stability criterion
Nyquist stability criterion.
The root locus method provides information on the roots of the characteristic equation of a linear feedback system as the loop gain is varied. This information is then used to access the stability of the system and also the transient response of the system.
However, to make this method work, we first need to find the poles and zeros of the system’s loop transfer function, this can be difficult to do in certain cases. In such cases, the Nyquist criterion is used as an alternative method for evaluating the stability of the system.
The Nyquist stability criterion can be defined as a frequency-domain method that is based on the polar coordinates of the loop transfer function L(s) for s = jw. Nyquist criterion is preferred for the following reasons:
- It provides information on the stability of the system, the degree of stability, and how to stabilize the system if it is unstable.
- It helps us know more about the frequency-domain response of the system.
- It is used to study the stability of the linear feedback system in accordance with the time delay that arises due to the presence of distributed components.
Though, there is also a limitation to the Nyquist criterion. In contrast to the root locus technique, the Nyquist method does not provide the exact location of the roots of the system’s characteristic equation.
Enclosures and encirclements
Enclosures and encirclements are recurring and important terminologies that will help us know more about the Nyquist stability criterion. They are related to contour mapping. In the study of signals and systems, we are used to representing the matters relating to s in a complex plane of its own called the s-plane.
Similarly, assuming a complex value F(s), we will be representing it on a complex plane of its own known as the GH-plane. And let C denote a closed contour traversed by the complex variable s in the s-plane.
A contour is said to be closed if it terminates onto itself and does not intersect itself as it is traversed by the complex variable s.
There are two different situations that can arise in the GH-plane:
- The interior of the contour C in the s-plane is mapped onto the interior of the contour in the GH-plane. In this case, the contour is traversed in the anti-clockwise direction.
- The interior of the contour C in the s-plane is mapped onto the exterior of the contour in the GH-plane. In this case, the contour is traversed in a clockwise direction.
Image from Electronics club
So from the above discussion, we can say that a region or point in a plane is said to be “enclosed” by a closed contour if the region or point is mapped inside the contour traversed in the anticlockwise direction.
Another thing to remember is a point is said to be encircled by a closed contour if it lies inside the contour. It is also possible for a point of intersection in the F-plane to be encircled more than once in a positive or negative direction.
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