Goseeko blog

# What is the Nyquist Stability Criterion?

Nyquist Stability Criterion gives the frequency response as well as comments on the stability and the relative stability of the system.

## Nyquist Stability Criteria:

The semi graphical method which we use to determine the stability of a closed loop system by investigating the properties of the frequency domain plot (Polar plot). The Nyquist plot of the open loop transfer function G(S) H(S) is L(S)

L(S) = G(S)H(S)

The Nyquist plot of L(S) is a plot drawn by substituting S=jw and varying the value of w as per the polar plot. We plot the polar plot ranging 0 to – ∞. But in the Nyquist plot the range of frequency is varied from ( -∞ to 0 ) and (0 to ∞ ). Therefore, the Nyquist plot gives us the plot for the entire frequency range.

Inferences from Nyquist Criteria :-

(1). The Nyquist criterion mainly provides information on relative stability of a stable system and also the instability of any unstable system. It also provides information about the absolute stability of  the system.

(2). It also provides information about the improvement of system stability.

(3). As we know, the Nyquist plot of G(S) H(S) is the polar plot of G(S) H(S) with a wider range of frequency ( -∞ to ∞ ) and along the Nyquist path.

(4). The additional information like bandwidth, gain margin and phase margin is also provided by Nyquist plot.

## Construction of Nyquist Plot

### Encircled :

A point or region in a complex function phase i.e. The S-plane is said to be encircled by a closed path if it is found inside the path.

Assumptions :

Let us consider the above figure where point A is encircled by the closed path Y. The point B being outside the closed path is not encircled by path Y. The encirclement made by the closed path Y are in either clockwise or in an anticlockwise direction.

In figure b where again point A is encircled by Y in an anticlockwise direction. Therefore we can conclude that the region which lies inside the path is encircled and the outside path is not encircled.

### Enclosed :-

A point or region is said to be enclosed by a closed path if it is encircled in the counterclockwise direction. In other cases a point or a region is said to be enclosed when it always lies to the left of the path, when the path has a definite direction.

In the figure considered below the shaded area is said to be enclosed by a closed path Y. From the figure it is very clear that point A is enclosed by Y in fig a. but is not enclosed by Y in fig b. and for point B it is vice versa.

No. of encirclements and enclosures : For A line is cut once as it’s overlapping but two times in same direction

When a point is encircled by a closed path Y, a no. N can be assigned to the no. of times it is encircled. By drawing an arrow around closed path Y we can determine the magnitude of N.

Taking an arbitrary point S, and moving around in clockwise direction and anti-clockwise direction respectively. We are getting a direction.

The direction is determined by the path followed by S1. The total number of revolutions by S1 is N. This is only the net angle ‘ 2 π N ’.

For N = 2 = B   for A = 1 = N

Now, point A is encircled (or 2 π radians) by Y and point B is encircled twice (or 4 π radians). Both are in clockwise direction.

In diagram B again A and B are encircled but in counter clockwise direction. Thus for this diagram A is enclosed and B is enclosed twice.

By definition M is positive for anticlockwise(direction) encirclement and negative for clockwise encirclement.