# Nyquist Plot: What is it? (And How To Draw One)

Contents

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Key learnings:
• Nyquist Plot Definition: A Nyquist plot is a graphical representation used in control systems to analyze stability by plotting the complex frequency response.
• Nyquist Stability Criterion: This criterion determines system stability by assessing the number of encirclements of the point (-1, 0) in the plot.
• Drawing Process: To draw a Nyquist plot, one must determine system poles on the jω axis, select a suitable contour, and map each segment to visualize the system’s response.
• Encirclement Rule: The stability of the system is determined by the direction and number of encirclements around the critical point.
• Mapping Function: The mapping function F(s) transforms points from the s-plane to the G(s) H(s) plane, essential in visualizing system dynamics.

## What is a Nyquist Plot?

A Nyquist plot is a graphical tool in control engineering and signal processing to evaluate feedback system stability. It plots the transfer function real part on the X-axis and the imaginary part on the Y-axis in Cartesian coordinates.

The frequency is swept as a parameter, resulting in a plot based on frequency. The same Nyquist plot can be described using polar coordinates, where the gain of the transfer function is the radial coordinate, and the phase of the transfer function is the corresponding angular coordinate.

The stability analysis of a feedback control system is based on identifying the location of the roots of the characteristic equation on the s-plane.

The system remains stable when its roots are located on the left side of the s-plane. Relative stability is evaluated using frequency response methods like Nyquist, Nichols plot, and Bode plot.

The Nyquist stability criterion is used to identify the presence of roots of a characteristic equation in a specified region of the s-plane.

To understand a Nyquist plot we first need to learn about some of the terminologies. Note that a closed path in a complex plane is called a contour.

## Nyquist Path or Nyquist Contour

The Nyquist contour is a closed path in the s-plane that encircles the entire right-hand half.

In order to enclose the complete RHS of the s-plane, a large semicircle path is drawn with a diameter along jω axis and center at the origin. The radius of the semicircle is treated as Nyquist Encirclement.

## Nyquist Encirclement

A point is said to be encircled by a contour if it is found inside the contour.

## Nyquist Mapping

Mapping involves transforming a point from the s-plane to the F(s) plane using the mapping function F(s).

### How to Draw Nyquist Plot

A Nyquist plot can be drawn using the following steps:

• Step 1 – Check for the poles of G(s) H(s) of the jω axis including that at the origin.
• Step 2 – Select the proper Nyquist contour – a) Include the entire right half of the s-plane by drawing a semicircle of radius R with R tends to infinity.
• Step 3 – Identify the various segments on the contour with reference to the Nyquist path
• Step 4 – Perform the mapping segment by segment by substituting the equation for the respective segment in the mapping function. Basically, we have to sketch the polar plots of the respective segment.
• Step 5 – Mapping of the segments are usually mirror images of mapping of the respective path of +ve imaginary axis.
• Step 6 – The semicircular path which covers the right half of s plane generally maps into a point in G(s) H(s) plane.
• Step 7- Interconnect all the mapping of different segments to yield the required Nyquist diagram.
• Step 8 – Note the number of clockwise encirclements about (-1, 0) and decide stability by N = Z – P

is the Open loop transfer function (O.L.T.F)

is the Closed loop transfer function (C.L.T.F)
N(s) = 0 is the open loop zero and D(s) is the open loop pole
From a stability point of view, no closed loop poles should lie on the RH side of the s-plane. Characteristics equation 1 + G(s) H(s) = 0 denotes closed-loop poles .

Now as 1 + G(s) H(s) = 0 hence q(s) should also be zero.

Therefore, from the stability point of view zeroes of q(s) should not lie in the RHP of the s-plane.
To define the stability entire RHP (Right-Hand Plane) is considered. We assume a semicircle that encloses all points in the RHP by considering the radius of the semicircle R tends to infinity. [R → ∞].

The first step to understanding the application of the Nyquist criterion in relation to the determination of stability of control systems is mapping from the s-plane to the G(s) H(s) – plane.

s is considered an independent complex variable and the corresponding value of G(s) H(s) is the dependent variable plotted in another complex plane called the G(s) H(s) – plane.

Thus for every point in the s-plane, there exists a corresponding point in G(s) H(s) – plane. During the process of mapping, the independent variable s is varied along a specified path in the s-plane, and the corresponding points in G(s)H(s) plane are joined. This completes the process of mapping from the s-plane to the G(s)H(s) – plane.

Nyquist stability criterion says that N = Z – P. Where, N is the total no. of encirclement about the origin, P is the total no. of poles and Z is the total no. of zeroes.
Case 1: N = 0 (no encirclement), so Z = P = 0 and Z = P
If N = 0, P must be zero therefore the system is stable.
Case 2: N > 0 (clockwise encirclement), so P = 0, Z ≠0 and Z > P
In both cases the system is unstable.
Case 3: N < 0 (counter-clockwise encirclement), so Z = 0, P ≠0 and P > Z
The system is stable.

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