Nyquist Diagram Online Tool Documentation

Introduction

The Nyquist plot is a graphical representation of the frequency response of a dynamic system. It shows the relationship between the real and imaginary parts of the system's transfer function $G(j\omega)$ over a range of frequencies $\omega$.
It is one of the fundamental tools for control system stability analysis and is used in Nyquist's stability criterion to determine the stability of a closed-loop system.

Mathematical Definition

For a linear time-invariant system with a transfer function:

$$G(s) = \frac{N(s)}{D(s)}$$

where the numerator $N(s)$ and the denominator $D(s)$ are functions of a complex variable s. The Nyquist plot represents the values:

$$G(j\omega) = \Re(G(j\omega)) + j \Im(G(j\omega)), \quad \omega \in (-\infty, \infty)$$

on the complex plane.

Nyquist Stability Criterion

Nyquist's criterion determines the stability of a closed-loop system based on the Nyquist plot. It states that the stability depends on the number of encirclements around the point (-1,0) in the complex plane.

A closed-loop system is stable if it satisfies the condition:

$$N = P - Z$$

where:

• $N$ – the number of encirclements of the point $(-1,0)$ by the Nyquist plot,
• $P$ – the number of poles of $G(s)$ in the right half-plane,
• $Z$ – the number of closed-loop poles in the right half-plane.

Interpretation of the Nyquist Plot

• No encirclements of (-1,0) → The system is stable if there are no poles in the right half-plane.
• One or more encirclements of (-1,0) → The system may be unstable.
• Encirclements in the opposite direction → The difference in the number of encirclements affects stability analysis.

Applications of the Nyquist Plot

• Analysis of closed-loop system stability.
• Evaluating stability margins and phase margins.
• Designing PID controllers and compensators in control systems.

How to Use the Nyquist Plot Online Tool?

1. Enter the transfer function $G(s)$. A transfer function is the ratio of the Laplace transform of the output to the Laplace transform of the input. It is typically expressed as:

$$G(s) = \frac{N(s)}{D(s)}$$

The numerator $N(s)$ and denominator $D(s)$ are typically represented as polynomials of a complex variable s. In this application $N(s)$ and $D(s)$ are defined as lists of coefficients, as in the examples below.

Example 1

Given the numerator and denominator lists:

numerator = [1, 2, 1] denominator = [1, 3, 3, 1]

This corresponds to the transfer function:

$$G(s) = \frac{s^2 + 2s + 1}{s^3 + 3s^2 + 3s + 1}$$

where:

• ( s^2 + 2s + 1 ) comes from [1, 2, 1]
• ( s^3 + 3s^2 + 3s + 1 ) comes from [1, 3, 3, 1]

Example 2: First-Order System

For a first-order system:

This represents:

$$G(s) = \frac{1}{s + 5}$$

where:

• The numerator is just 1 (constant gain).
• The denominator represents a first-order lag system. The polynomial (s + 5) comes from [1, 5]

2. Click run. The plot will be generated automatically.
3. Analyze the Nyquist plot. The result section shows if it encircles (-1,0).