Nyquist · controls

Nyquist asks a topological question of the open-loop curve $L (j o m e g a)$ : does it encircle the point $(- 1, 0)$ ? If yes, the closed loop is unstable. If no, it's stable. Margins are just "how close did you come?" Drag the plant's poles and zeros below, slide the loop gain $K$ , and watch the curve shrink, grow, and eventually wrap around the red point.

Encircle the −1 point, lose the loop

K (loop gain) 1.00

gain margin —

phase margin —

encirclements (N) —

closed-loop —

what to try

Turn K up slowly. The Nyquist curve grows radially. At some critical K it kisses the (−1, 0) point — that's the stability boundary.
Drag a pole rightward (toward unstable). Watch the curve start encircling (−1) even at small K — some plants need negative compensation.
At the gain-crossover frequency (where |L| = 1, i.e. the curve crosses the unit dashed circle), the angle to (−1, 0) is the phase margin. 30°–60° is the comfort zone.

show the math

Nyquist's stability criterion: count the clockwise encirclements $N$ that $L (j o m e g a)$ makes of $(- 1, 0)$ . Let $P$ be the number of open-loop right-half-plane poles. Then the number of closed-loop right-half-plane poles is

Z = N + P

Stable closed loop ⟺ $Z = 0$ . For an open-loop-stable plant $(P = 0)$ we just need $N = 0$ — don't encircle the point.