**Nichols Chart Definition**: A Nichols Chart is a graphical representation used to analyze and design feedback control systems by depicting stability and frequency responses.**Functionality**: The chart works by transforming complex frequency responses into a simpler gain phase plane, making it easier to analyze system behavior.**Applications**: Nichols charts are particularly useful in designing compensators for devices like DC motors, emphasizing their role in practical engineering.**Advantages**: One of the major benefits of using a Nichols chart is its ability to graphically determine gain and phase margins, facilitating adjustments to the system’s gain.**Challenges**: Despite its usefulness, the Nichols chart can be less effective for minor gain adjustments due to the deformation of constant magnitude and phase circles.

## What is a Nichols Chart?

A Nichols Chart (also known as a Nichols Plot) is a plot used in signal processing and control system design to determine the stability and closed-loop frequency response of a feedback system. The Nichols chart is named after its founder, Nathaniel B. Nichols.

## How Does A Nichols Chart Work?

The fundamental components of the Nichols chart are the M-circles (constant magnitude loci) and N-circles (constant phase angle loci).

In the G (jω) plane, the constant M and N circles are crucial for analyzing and designing control systems.

However, the constant M and constant N circles in the gain phase plane are prepared for system design and analysis as these plots supply information with fewer manipulations.

The gain phase plane is the graph having gain in decibels along the ordinate (vertical axis) and phase angle along the abscissa (horizontal axis).

The M and N circles of G (jω) in the gain phase plane are transformed into M and N contours in rectangular coordinates.

A point on the constant M loci in G (jω) plane is transferred to the gain phase plane by drawing the vector directed from the origin of the G (jω) plane to a particular point on the M circle and then measuring the length in dB and angle in degree.

A critical point in the G (jω) plane corresponds to zero decibels and -180 degrees in the gain phase plane. The plot of M and N circles in the gain phase plane is known as the Nichols chart (or Nichols plot).

Compensators can be designed using a Nichols plot.

The Nichols plot technique finds applications in DC motor design and is widely used in signal processing and control system design.

The related Nyquist plot in the complex plane shows how the phase of the transfer function and frequency variation of magnitude are related. We can find out the gain and phase for a given frequency.

The angle of the positive real axis determines the phase and the distance from the origin of the complex plane determines the gain. There are some advantages of Nichols’ plot in control system engineering.

They are:

- Gain and phase margins can be determined easily and also graphically.
- Closed loop frequency response is obtained from open loop frequency response.
- The gain of the system can be adjusted to suitable values.
- Nichols chart provides frequency domain specifications.

However, the Nichols plot has drawbacks, including difficulty in assessing small changes in gain.

Constant M and N circles in the Nichols chart are deformed into squashed circles.

The complete Nichols chart extends for the phase angle of G (jω) from 0 to -360^{o}. The region of ∠G(jω) is used for the analysis of systems between -90^{o} to -270^{o}. These curves repeat after every 180^{o} interval.

If the open loop T.F of unity feedback system G(s) is expressed as

Closed loop T.F is

Substituting s = jω in the above eq. frequency functions are,

and

Eliminating G(jω) from the above two eq.

and