Nyquist Plot Generator

Generate Nyquist plots from transfer functions. Analyze encirclements of the critical point for stability assessment.

  • Math & Calculators
  • Runs in your browser
  • Free, no signup

The guide

Tested tool guide Tested browser tools Checked August 16, 2026

What Nyquist Plot Generator does and how it behaves

Convert an open-loop transfer function into the complex-plane path traced by L(j omega), then judge how that path winds around the Nyquist critical point, -1 + 0j. Real and imaginary response form the horizontal and vertical coordinates, while frequency determines position along the curve. The stability interpretation applies to the loop transfer function in a negative-feedback characteristic equation of the form 1 + L(s) = 0, not automatically to an arbitrary plant transfer function. A frequent mistake is counting visible loops without accounting for direction and open-loop right-half-plane poles.

How the result is produced

1

Frequency mapping

The generator substitutes s = j omega into the entered transfer function over a frequency range. Each complex value becomes one point: Re[L(j omega)] on the horizontal axis and Im[L(j omega)] on the vertical axis. Sweeping omega joins those points into the locus. For real-coefficient transfer functions, the negative-frequency branch is the complex-conjugate reflection of the positive-frequency branch across the real axis.

2

Encirclement interpretation

Nyquist stability reasoning compares the oriented winding of the complete mapped contour around -1 + 0j with the number of right-half-plane poles of L(s). Together, those quantities determine how many right-half-plane zeros 1 + L(s) has, corresponding to unstable closed-loop poles. A curve image without direction, pole count, and a valid contour is not a complete stability test.

Good uses

  • Checking whether a proposed controller and plant loop transfer function has the encirclement pattern required for closed-loop stability.
  • Comparing controller gains by seeing how positive scaling of L(s) moves the Nyquist locus relative to -1 + 0j and changes its closest approach.
  • Finding frequencies where phase is near -180 degrees and magnitude is near one, which places the frequency response close to the critical point.

Limits and checks

  • No visible encirclement does not by itself prove stability. The conclusion also depends on the number of open-loop right-half-plane poles and on the orientation used when counting encirclements.
  • A displayed curve may conceal a close pass, crossing, or small loop if the chosen frequency range or point spacing does not resolve rapid changes in the response.
  • Poles on the imaginary axis require a modified Nyquist contour with indentations around those poles. Their mapped branches should not be interpreted as ordinary finite-frequency crossings.

Common questions

Why does the Nyquist curve sometimes appear mirrored?

For a transfer function with real coefficients, L(-j omega) is the complex conjugate of L(j omega). The negative-frequency locus therefore reflects the positive-frequency locus across the real axis. This symmetry does not generally follow for transfer functions with complex coefficients, and a positive-frequency-only display is only one branch of the complete contour.

Does avoiding the -1 point mean the closed loop is stable?

No. Merely avoiding -1 rules out that specific boundary condition but does not settle the required winding count. Under negative unity feedback, stability depends on the net oriented encirclements of -1 together with the number of right-half-plane poles of L(s). Imaginary-axis poles and exact passages through -1 require separate treatment.

References and verification

The behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.