Nyquist stability criterion — encirclements of −1: Foundations

This page builds every symbol the parent note uses, starting from things a 12-year-old already knows. Read it top to bottom: each brick sits on the one below it.

1. What is $s$ ? The complex-frequency plane

Before any transfer function makes sense, you need to know what lives inside the parentheses of $L (s)$ .

The picture. Draw a flat map. Left–right is $σ$ , up–down is $ω$ . Every point on that map is one value of $s$ . This map is called the $s$ -plane.

Figure — Nyquist stability criterion — encirclements of −1

Figure 1 — The $s$ -plane. The vertical (bold) line is the imaginary axis $σ = 0$ . The mint-shaded left half ( $σ < 0$ ) is where signals decay (safe); the coral-shaded right half ( $σ > 0$ ) is the RHP where signals grow (unstable). The lavender dot on the axis marks a pure wiggle $s = j ω$ .

A point on the right half ( $σ > 0$ ) means "grows over time" → this is the Right-Half Plane (RHP), the danger zone.
A point on the left half ( $σ < 0$ ) means "decays over time" → safe.
A point exactly on the vertical axis ( $σ = 0$ , so $s = j ω$ ) means "pure oscillation, neither growing nor shrinking" — a steady wiggle at frequency $ω$ .

Why the topic needs this. The whole Nyquist question is "are any poles in the RHP?" You cannot ask that until you can point at the RHP on a map. The $s$ -plane is that map.

2. Transfer function $L (s)$ , poles, and zeros

The picture. On the $s$ -plane, mark each pole with a small $\times$ and each zero with a small $\circ$ . Poles are "spikes" where the function screams to infinity; zeros are "holes" where it drops to nothing.

Figure 2 — Poles ( $\times$ ) and zeros ( $\circ$ ) plotted on the $s$ -plane. The two lavender $\times$ at $- 1$ and $- 2$ are safe LHP poles; the coral $\times$ at $s = + 1$ is a RHP pole (an explosive natural motion); the teal $\circ$ is a zero where $L = 0$ .

Why the topic needs this. The parent's letters are all pole-counts: $P$ = RHP poles of the open loop $L$ ; $Z$ = RHP poles of the closed loop. If you cannot picture a pole as a spike whose position decides stability, none of $Z = N + P$ means anything.

3. Feedback: open loop → closed loop, and where $1 + L$ comes from

Before the formula, meet its two ingredients:

The one thing you MUST take from this:

The picture. Whatever value of $s$ makes the open-loop $L$ equal exactly $- 1$ becomes a closed-loop pole. So the number $- 1$ is not arbitrary — it is the exact numerical fingerprint of "this frequency turned into a natural motion of the closed loop."

Why the topic needs this. This single line, $L = - 1$ , is why we hunt the point $- 1$ instead of any other point on the map.

4. Size and angle of a complex number (magnitude & argument)

The Nyquist plot draws $L (j ω)$ as a point that moves. Each point is a complex number, and every complex number is fully described by two things — you need both words.

The picture. Draw the arrow from $0$ to $w$ . Its length is $∣ w ∣$ ; the angle it opens from the rightward horizontal is $ar g w$ . This is the same right-triangle idea as $θ = arctan (y / x)$ — the arrow is the hypotenuse, $x$ is adjacent, $y$ is opposite.

Figure 3 — A complex number $w = x + j y$ drawn as a lavender arrow. Its length is the magnitude $∣ w ∣ = x^{2} + y^{2}$ (mint horizontal = adjacent $x$ , coral vertical = opposite $y$ ), and the teal wedge at the origin is its argument $ar g w$ .

Where the counting formula comes from (the Argument Principle, in words)

Set $F = 1 + L$ and let $C$ fence the whole RHP: enclosed zeros $= Z$ (unstable closed-loop poles), enclosed poles $= P$ (open-loop RHP poles), and turns of $1 + L$ about the origin = turns of $L$ about $- 1$ (shift by one). That is exactly $Z = N + P$ .

Why the topic needs this. Bode Plot & Gain/Phase Margins splits the same two ingredients: gain = magnitude $∣ L ∣$ , phase = argument $ar g L$ .

5. Frequency response and the FULL Nyquist contour

Figure 4 — The Nyquist contour in the $s$ -plane. Lavender arrows go up the positive imaginary axis (leg 1); the coral dashed arc is the large $R \to \infty$ semicircle sweeping the RHP (leg 2); mint arrows come down the negative imaginary axis (leg 3, the mirror). The whole loop is traversed clockwise and fences the shaded RHP.

Definition Zeros on the imaginary axis — the mirror problem

A zero of $1 + L$ (a closed-loop pole) sitting exactly on the $j ω$ -axis means the closed loop has a pure, undamped oscillation — marginally stable, right on the knife's edge. Two things follow: (1) The plot passes through $- 1$ . A zero of $1 + L$ on the axis means $1 + L (j ω_{0}) = 0$ , i.e. $L (j ω_{0}) = - 1$ exactly — the Nyquist curve touches the danger point. (2) The Argument Principle breaks there, because $F = 1 + L$ hits $0$ on the contour, and the formula only counts zeros strictly inside. Fix / reading: just as with an axis pole, indent the contour with a tiny semicircle around $s = j ω_{0}$ ; whether you bulge left or right decides whether that borderline root counts as inside (RHP) or outside. Practically, a curve that lands on $- 1$ is the definition of a marginally stable design — the encirclement count is ambiguous precisely because the system is on the boundary, and you must nudge the gain to resolve which side it falls.

6. The Nyquist contour direction and encirclements ( $N$ )

How to read a wrap. Stand at the point $- 1$ . Watch the arrow that points from you to the moving curve as $ω$ runs over the whole contour. If that arrow completes one full clockwise spin (odometer $- 360°$ ), $N$ went up by one.

Figure 5 — Counting an encirclement. The lavender loop is the Nyquist curve; small lavender arrows show it is traversed clockwise. The coral $\times$ is the danger point $- 1$ . Watch the teal arrow from $- 1$ to the curve: over the full loop it spins once clockwise, so $N = + 1$ here.

Why the topic needs this. $N$ is the middle letter of $Z = N + P$ . It is the only quantity you get by looking at the picture; $P$ you get from the plant, and $Z$ you deduce.

7. Putting the letters together

Symbol	Plain words	Picture	Where it comes from
$s = σ + j ω$	complex frequency	point on the map	§1
RHP	$σ > 0$ region	right half of map	§1
$L (s)$	open-loop response	fraction of polynomials	§2
pole $\times$	where $L = \infty$	spike = natural motion	§2
proper	$de g N \leq de g D$	bottom no shorter than top	§2
$G, H$	forward / feedback path	the two loop blocks	§3
$1 + L = 0$	characteristic eq.	closed-loop poles	§3
$- 1$	"flipped, full-strength"	danger point	§3
$	L	,\arg L$	size & angle
$Δ_{C} (ar g F)$	total angle turned	odometer of the arrow	§4
contour $C$	RHP-fencing loop	3 legs + detours	§5
$L (j ω)$	frequency response	the Nyquist curve	§5
$P$	RHP open-loop poles	spikes trapped in fence	§2,§6
$N$	wraps of $- 1$	arrow spins	§6
$Z$	RHP closed-loop poles	the answer	§3,§6

Everything culminates in the parent's one boxed line: $Z = N + P .$ Compare with Routh–Hurwitz Criterion (an algebra-only way to get $Z$ ) and Root Locus (which tracks where the closed-loop poles move as gain changes).

Prerequisite map

The diagram below renders in Obsidian as a top-down flow; read it as "each box feeds the arrow it points to, all streams converging on $Z = N + P$ ."

Equipment checklist

Cover the right side and see if you can answer each before revealing.

What are the two parts of

s = σ + j ω

and what does each measure?

σ

= growth/decay (real part),

ω

= oscillation speed (imaginary part).

Where on the

s

-plane is the unstable region?

The Right-Half Plane,

σ > 0

(right of the vertical axis).

What is a pole, in one phrase?

A value of

s

where

L

blows up to infinity — one natural motion of the system.

Why is a RHP pole "unstable"?

Its natural motion grows like

e^{σ t}

with

σ > 0

, exploding over time.

What does "strictly proper" mean and why does it matter?

de g N < de g D

; then

∣ L ∣ \to 0

at infinity so the big closing semicircle collapses to a point and adds no encirclements.

What do

G (s)

and

H (s)

stand for?

G

= forward path (controller+plant that acts);

H

= feedback path (sensor that measures and returns the output);

L = G H

What equation defines the closed-loop poles?

1 + L (s) = 0

, i.e.

L (s) = - 1

Why is the number

- 1

special?

It has size

1

(full-strength) and angle

180°

(flipped) — the echo that self-reinforces.

What are the magnitude and argument of a complex number

w = x + j y

∣ w ∣ = x^{2} + y^{2}

and

ar g w =

the angle of the arrow from origin to

w

Why must

ar g w

be tracked continuously, not read off a calculator?

The principal argument jumps

360°

at a branch cut; only the continuous (odometer) total counts encirclements correctly.

How does Cauchy's CCW baseline reconcile with Nyquist's CW walk?

Reversing to CW flips the turn-sign to

P - Z

; defining

N

as CW wraps (

N = Z - P

) restores

Z = N + P

What are the three legs of the Nyquist contour?

Positive

j ω

-axis up, the large RHP semicircle, and the negative

j ω

-axis down (mirror of the first).

Why draw the negative-frequency half at all?

The contour must be closed for the Argument Principle;

L (- j ω) = \overline{L (j ω)}

so it is just the mirror of the positive half.

How do you handle a pole sitting on the

j ω

-axis?

Detour with a tiny semicircle bulging into the RHP so the pole stays outside the fence (uncounted); it maps to a large arc on the plot.

What does a Nyquist curve passing exactly through

- 1

signify?

A zero of

1 + L

on the axis — an undamped closed-loop oscillation, i.e. a marginally stable design on the boundary.

State the criterion and its sign convention.

Z = N + P

with the contour clockwise and

N

= CW wraps of

- 1

; stable iff

Z = 0