How to use the Nyquist Plot Generator
- 1. Enter G(s)H(s)Numerator and denominator, for example 1 and s(s+1)(s+2).
- 2. Set the gain KK multiplies the numerator, so you can test several gains.
- 3. DrawSee the plot, the crossings, GM, PM, N, P, Z and the verdict.
The Nyquist stability criterion
The Nyquist plot traces G(jω)H(jω) as ω goes from −∞ to ∞. The part for negative frequencies is the mirror image of the part for positive frequencies. If the open loop has P poles in the right half-plane and the plot circles the −1 point N times clockwise, then the closed loop has Z = N + P right half-plane poles. A stable closed loop needs Z = 0. Poles on the jω axis (such as an integrator 1/s) are bypassed with small semicircles, and each one adds an arc of infinite radius to the plot.
Worked example: G(s) = 1/(s(s+1)(s+2)). The imaginary part is zero at ω = √2, where G = −1/6, so the gain margin is 6 (15.56 dB). |G| = 1 at ω = 0.446 rad/s, where ∠G = −126.6°, so the phase margin is 53.4°. The plot does not circle −1 and P = 0, so the closed loop is stable. With K = 10 the plot circles −1 twice, so Z = 2 and the system is unstable.
Exam tips
- Find the real-axis crossing by setting Im G(jω) = 0. The gain margin is 1/|G| at that point.
- For an unstable open loop (P > 0), stability needs anticlockwise encirclements, N = −P.
- The phase margin is 180° + ∠G where the plot crosses the unit circle.
- Type 1 and type 2 systems start at infinity. Remember the closing arcs of 180° and 360°.
Frequently asked questions
How does the tool count encirclements?
It follows the full Nyquist contour numerically, including the small detours around jω-axis poles and the large arc, and adds up the change in angle of 1 + G(s). It then checks the result against the closed-loop poles.
Why do the arcs at infinity not appear on the plot?
They have infinite radius, so they cannot be drawn to scale. The working tells you how many there are and in which direction they go.
Can I use it for an unstable open-loop system?
Yes. That is where the Nyquist criterion is most useful. The tool reports P, the number of open-loop right half-plane poles, and uses Z = N + P.
Is this tool free?
Yes. You fill in a short form once to see your result, and the other Uncodemy tools then open without it. All the maths runs in your browser.