Nyquist Plots
The Nyquist plot traces the loop gain in the complex plane as frequency varies, encoding closed-loop stability in its encirclements.
One Curve in the Complex Plane
A Nyquist plot is the parametric curve traced by the open-loop transfer function L(j*omega) in the complex plane as frequency omega runs from minus infinity to plus infinity. Where a Bode plot separates magnitude and phase into two graphs, the Nyquist plot combines them into a single locus.
What it reveals
The Nyquist plot's value lies in how it relates to the critical point at minus one on the real axis. Because the closed-loop poles are the roots of 1 + L(s) = 0, the behavior of L(s) relative to minus one governs stability. This connection is made rigorous by the Nyquist stability criterion, which counts encirclements of that point.
Reading the shape
- Distance from the minus-one point indicates robustness; a curve passing close to it means thin margins.
- The point where the curve crosses the negative real axis gives the gain margin.
- The point where the curve crosses the unit circle gives the phase margin.
- Loops around the origin at low frequency reflect integrators in the loop.
Advantages over Bode
The Nyquist plot handles cases Bode struggles with: open-loop-unstable plants, systems with multiple gain crossings, and pure time delays. Because it works with the full mapping rather than asymptotes, it gives an exact stability verdict for these harder cases, which is why it is the tool of choice when a plant is itself unstable and must be stabilized by feedback.
Time delay
A time delay multiplies L(j*omega) by e^(-j*omega*T), which rotates each point by an angle growing with frequency without changing its magnitude. The Nyquist curve spirals inward, and the criterion still applies directly, unlike polynomial tests that cannot represent delay.
Learning to read a Nyquist plot builds intuition for how proximity to the critical point trades off against speed, which is the core tension of every feedback design.