---
title: "Short note: Nyquist stability criterion"
subject: "Control Systems"
chapter: 6
key: "semester_3/control_systems:short-nyquist-criterion"
source: https://www.bctengineer.com/q/semester_3/control_systems/short-nyquist-criterion
site: BCT Engineering
---

# Short note: Nyquist stability criterion

Control Systems · Chapter 6: Frequency Response Techniques

- Asked in: 2076 Chaitra, 3 marks
- Video: [Intro to Control - 16.4 Nyquist Stability Criterion](https://www.youtube.com/watch?v=32RbwQddfQQ) — Engineering with Prof. Kim
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/short-nyquist-criterion
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-6

## Question

Write a short note on Nyquist stability criterion.

## Short answer

Nyquist criterion maps the Nyquist contour (encircling the RHP) through G(s)H(s). Z=P+N, where P=open-loop RHP poles, N=clockwise encirclements of –1. For stability Z=0, so N=–P. It handles delays, experimental data, and open-loop unstable systems.

## Answer

The **Nyquist stability criterion** is a graphical frequency-domain method that finds the number of closed-loop poles in the right half $s$-plane from the plot of the open-loop transfer function $G(j\omega)H(j\omega)$. It is based on Cauchy's **principle of the argument**.

#### Statement

If the Nyquist contour (the whole $j\omega$ axis plus an infinite semicircle enclosing the RHP, traversed clockwise) is mapped through $G(s)H(s)$, then

$$
N = Z - P \quad\text{or}\quad Z = P + N
$$

- $P$ = number of open-loop poles in the RHP
- $Z$ = number of closed-loop poles (zeros of $1+GH$) in the RHP
- $N$ = number of **clockwise** encirclements of the critical point $-1+j0$

The closed-loop system is stable only if $Z=0$, i.e. $N=-P$ (the plot encircles $-1$ anticlockwise $P$ times). For an open-loop stable system ($P=0$): **stable if the Nyquist plot does not encircle $-1+j0$**.

#### Steps

1. Find $P$ from the open-loop poles.
2. Plot $G(j\omega)H(j\omega)$ for $\omega=0\to\infty$, add its mirror image, and close it (poles at origin give infinite clockwise arcs).
3. Count $N$ around $-1+j0$, compute $Z$.

#### Merits

- Works with experimental frequency-response data.
- Handles time delay and open-loop unstable systems.
- Gives relative stability (gain and phase margins) as well.

**Example:** $GH=\dfrac{60}{(s+1)(s+2)(s+5)}$ cuts the real axis at $-0.476$; $P=0$, $N=0$, so $Z=0$: stable.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. On what principle is the Nyquist criterion based?
   A. Cauchy's principle of the argument
   B. Routh's array
   C. The root locus angle condition
   D. Final value theorem
   Answer: A. Mapping the Nyquist contour through GH and counting encirclements follows from the principle of the argument. (Key point: Basis of the criterion)

2. In Z = P + N, what is N?
   A. The number of open-loop RHP poles
   B. The number of clockwise encirclements of -1 + j0
   C. The number of closed-loop RHP poles
   D. The number of anticlockwise encirclements of the origin
   Answer: B. N counts clockwise encirclements of the critical point -1 + j0; Z is the closed-loop RHP poles and P the open-loop ones. (Key point: Encirclement relation)

3. For an open-loop stable system (P = 0), when is the closed loop stable?
   A. When the plot encircles -1 once clockwise
   B. When the plot does not encircle -1 + j0
   C. When the plot passes through the origin
   D. When the plot encircles -1 anticlockwise
   Answer: B. With P = 0, stability needs Z = N = 0, i.e. no encirclement of -1 + j0. (Key point: Stability condition)

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