Nyquist Stability Calculator

Argument principle sampled positively oriented full contour winding around−1, Z=P+counterclockwise winding; no stability assertion from a partial frequency plot. Caller independently establishes units, regime, calibration and applicability; this does not establish design-code compliance, authenticity, operational safety, treatment efficacy or a validated forecast.

Description

A Nyquist polygon's winding about minus one connects contour geometry with a declared open-loop pole count.

A Nyquist polygon's winding about minus one connects contour geometry with a declared open-loop pole count.^[1]

Contour orientation, detours and sampling adequacy must be established before interpreting the stability count. Use this tool to count winding and the declared argument-principle result on a supplied contour.

When to use Nyquist Stability Calculator

Use this tool to count winding and the declared argument-principle result on a supplied contour. Establish the SI units or caller unit basis, load signs, measurement interval, model conditions and independently chosen coefficients or references.

How to use Nyquist Stability Calculator and understand its results

Establish the SI units or caller unit basis, load signs, measurement interval, model conditions and independently chosen coefficients or references.
Enter closedNyquistReal, closedNyquistImaginary, callerKnownOpenLoopRhpPoleCount, callerFullContourAdequatelySampledVerified, using the field meanings and rules below, then run the tool.
Read sampledCounterclockwiseCriticalWinding, argumentPrincipleClosedLoopRhpZeros. Interpret the fields using their units, signs and evidence conventions below.
If input validation fails, correct the reported field and run again. Before applying the result, check reference scope, model assumptions and any missing evidence or placeholder values.

Input field	Meaning	Value type	Requirement
closedNyquistReal	Closed nyquist real	list of number	Required
closedNyquistImaginary	Closed nyquist imaginary	list of number	Required
callerKnownOpenLoopRhpPoleCount	Caller known open loop rhp pole count; binary 1=known, 0=unknown, in the order described below	number	Required
callerFullContourAdequatelySampledVerified	Caller full contour adequately sampled verified	boolean	Required

Enter at least four aligned finite real/imaginary contour samples with exact first=last, integer open-loop right-half-plane pole count 0–100 and verified positive-orientation/full-detour/adequate-sampling boolean true. Sum signed segment angles around −1 and round to a winding integer; samples or segments through the critical point and unresolved winding fail. Return counterclockwise winding and Z=P+winding, requiring 0≤Z≤10000. This declared positive orientation differs from conventions using clockwise encirclements; the tool does not generate or verify the frequency contour.

CCWwinding=round(Σatan2(cross(z_i+1,z_(i+1)+1),dot(z_i+1,z_(i+1)+1))/(2π)); RHPclosedPoles=P+CCWwinding

Use the exact units, boundary rules, ordering and missing-evidence conventions above.

Output field	Meaning	Value type
sampledCounterclockwiseCriticalWinding	Sampled Counterclockwise Critical Winding; follow the units, sign, ordering and evidence convention stated above	number
argumentPrincipleClosedLoopRhpZeros	Argument Principle Closed Loop Rhp Zeros; follow the units, sign, ordering and evidence convention stated above	number

json

{
  "input": {
    "closedNyquistReal": [
      0,
      -1,
      -2,
      -1,
      0
    ],
    "closedNyquistImaginary": [
      0,
      1,
      0,
      -1,
      0
    ],
    "callerKnownOpenLoopRhpPoleCount": 0,
    "callerFullContourAdequatelySampledVerified": true
  },
  "output": {
    "sampledCounterclockwiseCriticalWinding": 1,
    "argumentPrincipleClosedLoopRhpZeros": 1
  }
}

Limitations and assumptions

Contour orientation, detours and sampling adequacy must be established before interpreting the stability count. Interpretation depends on the units, sampling scope and exact model described above.

All listed fields require explicit values. Unexpected fields, incorrect types, nonfinite numbers and sparse evidence arrays fail validation; the specific bounds and alignment requirements are stated above. Extremely large inputs or outputs can also be rejected. An input error is distinct from a successful result reporting false, no matches or missing evidence.

Alternative or Complementary methods

Use Root Locus Breakaway Calculator for this related task: A real root-locus stationary point satisfies a logarithmic gain-derivative condition.

Use Robot Jacobian Determinant Calculator when its separate inputs and assumptions fit the question: A planar two-link Jacobian determinant reflects local geometric sensitivity.

References

Nyquist Stability Calculator — tool.io

Similar or alternative tools

Root Locus Breakaway Calculator
A real root-locus stationary point satisfies a logarithmic gain-derivative condition.
Robot Jacobian Determinant Calculator
A planar two-link Jacobian determinant reflects local geometric sensitivity.