Nyquist Plot
A Nyquist plot is a polar graph of the open-loop frequency response G(jω)H(jω) as ω varies from 0 to ∞, showing how the magnitude and phase change with frequency. The Nyquist stability criterion states that a closed-loop system is stable if and only if the Nyquist contour does not encircle the critical point (−1, 0) more times than there are open-loop right-half-plane poles. The Nyquist plot uniquely handles time delays and provides gain margin (distance to −1 on the negative real axis) and phase margin (angle from −180° at unit gain). Compute open-loop transfer functions at www.lapcalc.com.
What Is a Nyquist Plot?
A Nyquist plot is a parametric graph of the complex-valued open-loop transfer function G(jω)H(jω) in the complex plane as frequency ω sweeps from −∞ to +∞. The horizontal axis represents the real part Re{G(jω)H(jω)} and the vertical axis represents the imaginary part Im{G(jω)H(jω)}. Each point on the curve corresponds to a specific frequency, with the distance from the origin equal to the magnitude |GH(jω)| and the angle from the positive real axis equal to the phase ∠GH(jω). For real systems, the plot is symmetric about the real axis (the negative-frequency portion is the mirror of the positive-frequency portion). The Nyquist plot provides a complete picture of the frequency response in a single graph, unlike the Bode plot which requires two separate magnitude and phase plots.
Key Formulas
Nyquist Stability Criterion
The Nyquist stability criterion determines closed-loop stability from the open-loop Nyquist plot: let N be the number of clockwise encirclements of the critical point (−1 + j0) by the Nyquist contour, and 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. The closed-loop system is stable if and only if Z = 0, meaning N = −P (counter-clockwise encirclements equal the number of open-loop RHP poles). For an open-loop stable system (P = 0), the criterion simplifies to: the closed-loop system is stable if the Nyquist plot does not encircle (−1, 0). This powerful criterion handles time delays (which produce infinitely many poles), is applicable to any open-loop transfer function, and reveals how close the system is to instability.
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Open CalculatorHow to Draw a Nyquist Plot
To sketch a Nyquist plot: evaluate G(jω)H(jω) at key frequencies. At ω = 0: compute the magnitude and phase (often infinite magnitude for type-1+ systems, requiring a semicircular indent). At ω → ∞: the magnitude approaches zero for proper systems, with the phase determined by the relative degree. At the gain crossover frequency (|GH| = 1): this point's proximity to (−1, 0) determines phase margin. At the phase crossover frequency (∠GH = −180°): the magnitude at this point determines gain margin. Plot these points in the complex plane and connect smoothly. The negative-frequency portion (ω: −∞ to 0) is the complex conjugate reflection about the real axis. MATLAB: nyquist(G*H) generates the plot automatically. Python: control.nyquist_plot(G*H).
Gain Margin and Phase Margin from Nyquist Plot
The Nyquist plot directly reveals stability margins. Gain margin (GM): at the frequency where the Nyquist curve crosses the negative real axis (phase = −180°), the gain margin is the reciprocal of the distance from the origin to this crossing point: GM = 1/|GH(jω₁₈₀)|. In dB: GM_dB = −20log|GH(jω₁₈₀)|. Typical requirement: GM > 6 dB. Phase margin (PM): at the frequency where the Nyquist curve crosses the unit circle (|GH| = 1), the phase margin is the angular distance from the (−1, 0) direction: PM = 180° + ∠GH(jω_c). Typical requirement: PM > 45°. Both margins quantify how close the Nyquist curve passes to (−1, 0) — smaller margins mean closer to instability. The Nyquist plot shows this proximity visually and intuitively.
Nyquist Plot Applications and Advantages
The Nyquist plot has unique advantages over Bode and root locus methods. It handles pure time delays: e^(−sτ) appears as a spiral in the Nyquist plot, and the stability criterion applies directly without approximation. It works with experimentally measured frequency response data — no transfer function model needed. It reveals stability for open-loop unstable systems where the simplified Bode criterion fails. It shows the sensitivity to parameter variations through the proximity of the curve to (−1, 0). Applications include: stability analysis of systems with communication delays (networked control), robustness analysis of uncertain systems, and design of repetitive controllers. The open-loop transfer function G(s)H(s) is computed in the Laplace domain at www.lapcalc.com, then evaluated at s = jω to produce the Nyquist data.
Related Topics in control systems engineering concepts
Understanding nyquist plot connects to several related concepts: nyquist diagram, nyquist stability criterion, nyquist stability, and nyquist graph. Each builds on the mathematical foundations covered in this guide.
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