The Complex Plane Test
Harry Nyquist's 1932 paper introduced a remarkable stability test: plot the open-loop frequency response G(jω) as a curve in the complex plane, and count how many times it circles the point -1+j0. This single topological property — the encirclement count — determines whether closing the feedback loop creates an unstable system. No root-finding, no characteristic polynomial factoring, just geometry.
Drawing the Contour
The Nyquist contour traces G(jω) for ω from 0 to +∞ (the upper half), then mirrors it for negative frequencies. At ω=0, the plot starts at G(0) on the real axis (the DC gain). As frequency increases, the curve spirals inward due to pole roll-off, and time delay adds additional spiraling. The plot closes through an arc at infinity, completing the contour needed for the encirclement count.
Stability Margins Visualized
On the Nyquist plot, the gain margin is the reciprocal of the distance from the origin to where the contour crosses the negative real axis. The phase margin is the angle between the negative real axis and the point where the contour crosses the unit circle. Both margins are visible geometrically — the farther the contour stays from -1+j0, the more robust the stability.
Time Delay — The Stability Killer
Pure time delay e^(-sT) adds phase lag proportional to frequency without changing magnitude. On the Nyquist plot, this rotates each frequency point clockwise, causing the contour to spiral. Even a small delay can cause encirclement at moderate gains, which is why dead-time compensation (Smith predictor) is critical in process control. This simulator shows how increasing delay progressively wraps the contour around the critical point.