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  "slug": "nyquist",
  "title": "Nyquist Plot & Stability Analyzer",
  "category": "control-systems",
  "category_label": "Control systems",
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  "description": "The Nyquist Plot tool draws the complex locus H(jω) for ω from −∞ to +∞, and counts encirclements of the critical point (−1, 0) to determine closed-loop stability via the Nyquist stability criterion: Z = N + P, where N is the clockwise encirclements, P is the open-loop unstable poles, Z is the closed-loop unstable poles. Use it to analyze systems where the Bode plot is ambiguous (e.g., conditionally stable systems with multiple crossover frequencies).",
  "keywords": [
    "Nyquist plot",
    "Nyquist criterion",
    "stability criterion",
    "gain margin",
    "phase margin",
    "control stability"
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    "formulas": [
      "Nyquist criterion: Z = N + P (closed-loop unstable = clockwise encirclements + open-loop unstable)",
      "Critical point: (−1, 0) on the real axis",
      "Gain margin (from Nyquist): GM = 1 / |H(jω_pc)| where Nyquist crosses negative real axis",
      "Phase margin (from Nyquist): PM = 180° + ∠H(jω_gc) where |H| = 1",
      "Conditional stability: Nyquist encircles −1 in one direction at low gain, opposite at high gain — system is stable only in a gain range"
    ],
    "citations": [
      "Nyquist, H., \"Regeneration Theory,\" Bell System Technical Journal, 1932.",
      "Ogata, K., \"Modern Control Engineering,\" 5th ed., 2010, Ch. 8."
    ],
    "faq": [
      {
        "q": "My Bode plot says the system is stable but Nyquist says otherwise — why?",
        "a": "Conditional stability. Bode gain/phase margin looks at the first crossover; Nyquist looks at the entire locus. If the system is stable in a narrow gain range but unstable outside, the margins are positive but the locus still encircles −1 at some gain."
      },
      {
        "q": "How do I count encirclements?",
        "a": "Draw a ray from (−1, 0) to infinity. Count how many times the Nyquist contour crosses the ray clockwise minus counterclockwise. The workbench shows this as N (clockwise encirclements of −1)."
      }
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  "last_updated": "2026-07-05"
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