Nyquist

The Nyquist criterion determines closed-loop stability from the open-loop frequency response — even for open-loop unstable plants and systems with time delay. Learn the polar plot, encirclements of −1, and margins.

The Nyquist plot draws the open-loop frequency response L(jω) as a curve in the complex plane: at each frequency, a point with that magnitude and phase. Whether and how that curve circles the critical point −1 tells you, exactly, how many closed-loop poles are unstable.

Unlike Routh or root locus, it works directly with measured frequency-response data and handles time delay exactly.

An analogy

Imagine walking around the point −1 on a map, following the loop's frequency response. If your path winds around −1 (and the plant was stable), the closed loop will be unstable — the path 'traps' the dangerous point.

The polar plot

For each ω from 0 to ∞, plot the complex number L(jω). A simple lag starts on the positive real axis and curls clockwise toward the origin. The mirror image (ω < 0) completes the closed contour.

Encirclements and the criterion

Count how many times the full contour encircles −1 clockwise (N). With P open-loop RHP poles, Z = N + P closed-loop RHP poles. For a stable open loop (P = 0), stability simply requires no encirclement of −1.

Margins on the Nyquist plot

The gain margin is how far you could scale the plot before it reaches −1 (1/|L| where it crosses the negative real axis). The phase margin is the angle between the unit-circle crossing and −1. The closest distance from the curve to −1 equals 1/M_s — a direct robustness measure.

Open-loop unstable plants

If the plant itself has RHP poles (P > 0), the closed loop is stable only if the plot encircles −1 counter-clockwise P times (N = −P). Magnetic levitation, inverted pendulums and some aircraft rely on this, and the Nyquist criterion is the tool that proves stability.

Time delay

A delay rotates each point by −ωT without changing magnitude, so the plot spirals into the origin, crossing the negative real axis infinitely many times. The Nyquist criterion handles this exactly, while pole-based methods need approximations.

Formulas

• Nyquist criterion: Z = N + P (stable ⇔ Z = 0)
• Gain margin: GM = 1 / |L(jω_180)|
• Phase margin: angle between L(jω_c) and −1 on the unit circle
• Delay: e^(−jωT): magnitude 1, phase −ωT (spirals the plot inward)

Worked example: Critical gain of K/(s + 1)³

Given: L(s) = K/(s + 1)³

Find: Phase-crossover frequency and the K for marginal stability.

• Phase −180° when 3·arctan ω = 180° → arctan ω = 60° → ω = √3 = 1.73 rad/s
• |L(j√3)| = K/(1 + 3)^1.5 = K/8
• Marginal when K/8 = 1 → K = 8

Answer: Stable for K < 8; oscillation at 1.73 rad/s

Worked example: Open-loop unstable plant

Given: L(s) = K/(s − 1) (P = 1)

Find: K range for stability.

• Closed loop: s − 1 + K = 0 → pole at s = 1 − K
• Stable when K > 1
• Nyquist: for K > 1 the circle encloses −1 once counter-clockwise → N = −1 → Z = 0 ✓

Answer: K > 1 (feedback stabilises the plant)

Common mistakes

• Wrong: Ignoring open-loop RHP poles when counting encirclements. — Right: Use Z = N + P; an unstable plant needs counter-clockwise encirclements.
• Wrong: Plotting only ω from 0 to ∞ and counting encirclements. — Right: The full contour includes negative frequencies (mirror image) and any detour around jω-axis poles.

Applications

• Loops with delay: Process-control loops with transport delay (conveyors, pipes) are analysed with Nyquist because the delay is exact.
• Magnetic levitation & balancing: Unstable plants are verified stable in closed loop via counter-clockwise encirclements.
• Robustness: The minimum distance to −1 (1/M_s) is used as a single robustness specification.

Key takeaways

• Plot L(jω) in the complex plane; count encirclements of −1.
• Z = N + P — stable when Z = 0.
• GM = 1/|L| at the negative real axis crossing; PM from the unit-circle crossing.
• Handles delays and unstable plants exactly.

Going further

• Argument principle: Nyquist's criterion follows from Cauchy's argument principle: the number of times a mapped contour encircles the origin equals zeros minus poles of 1 + L inside the contour.
• Nichols charts: Plotting gain (dB) against phase (°) with closed-loop contours overlaid combines Bode and Nyquist insight and is popular in flight control.
• Generalised Nyquist: For MIMO systems, stability is assessed with the eigenvalue loci of L(jω) or det(I + L(jω)), extending the encirclement idea to matrices.