What is special about the point −1?

Closed-loop poles relate to 1 + L(s) = 0, so L = −1 is the stability boundary for the unity-feedback loop. The Nyquist plot shows whether L(jω) gets near or encircles that point as frequency runs. Students often confuse −1 with the origin; the origin is L=0 (open-loop gain zero), not the feedback critical point.

How do gain and phase margins appear on the plot?

Gain margin is tied to how far the locus is from −1 when it crosses the negative real axis; phase margin to the angle from −1 when |L|=1. Both measure distance to the critical point along different directions—this lab emphasizes that geometric picture.

Does crossing into the left half-plane always mean instability?

Not by itself. Stability depends on encirclements of −1 relative to open-loop right-half-plane poles (Nyquist criterion), not merely whether Re{L} is negative. A locus can enter the left half-plane safely if it stays clear of −1 with the correct winding.

Nyquist Plot (linear systems)

Open-loop L(jω) in the complex plane vs frequency; critical point −1 and distance teaching aid.

About this model

This lab plots the open-loop frequency response L(jω) as a curve in the complex plane while ω sweeps from low to high. Stability intuition comes from how the locus approaches the critical point −1 + 0j: gain and phase margins relate to the distance and angle of that approach. The display is a teaching aid for classical loop shaping, not a full MIMO or nonlinear analyzer. Assumptions include a linear time-invariant single-loop plant and a frequency-domain open-loop transfer function evaluated along the jω axis. You can inspect how the Nyquist image changes with the underlying L(jω) and relate encirclements or proximity to −1 to closed-loop stability margins.

Who it's for: Advanced undergraduate control systems, mechatronics, and electrical engineering courses on classical feedback.

Key terms

Nyquist plot
open-loop transfer function
critical point −1
gain margin
phase margin
frequency response

How it works

Polar plot of the open-loop frequency response L(jω). Compare with the Bode diagram of the same transfer function: magnitude and phase become radius and angle here. The critical point −1 marks where unit gain meets −180° phase.

Key equations

LP1: L = 1/(1 + jωτ)

HP1: L = jωτ/(1 + jωτ)

Integrator: L = K/(jω)

2nd LP: L = ωₙ² / ((jω)² + 2ζωₙ(jω) + ωₙ²)

Frequently asked questions

What is special about the point −1?: Closed-loop poles relate to 1 + L(s) = 0, so L = −1 is the stability boundary for the unity-feedback loop. The Nyquist plot shows whether L(jω) gets near or encircles that point as frequency runs. Students often confuse −1 with the origin; the origin is L=0 (open-loop gain zero), not the feedback critical point.
How do gain and phase margins appear on the plot?: Gain margin is tied to how far the locus is from −1 when it crosses the negative real axis; phase margin to the angle from −1 when |L|=1. Both measure distance to the critical point along different directions—this lab emphasizes that geometric picture.
Does crossing into the left half-plane always mean instability?: Not by itself. Stability depends on encirclements of −1 relative to open-loop right-half-plane poles (Nyquist criterion), not merely whether Re{L} is negative. A locus can enter the left half-plane safely if it stays clear of −1 with the correct winding.

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About this model

How it works

Key equations

Frequently asked questions

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How it works

Key equations

Frequently asked questions