the Nyquist stability criterion
/ NY-kwist /
When you wrap a feedback loop around an amplifier or a control system, you face a worry: will the loop settle down, or will it howl and oscillate out of control? You would like to answer this just by looking at the open loop — before you actually close it — and ideally just from measurements you can take experimentally. The Nyquist criterion does exactly that. It is a graphical stability test that reads the answer off a single curve in the complex plane.
It is a direct application of the argument principle. For a feedback system the closed-loop poles are the zeros of 1 + L(s), where L(s) is the open-loop transfer function. The argument principle says that as s traverses a closed contour, the number of times 1 + L(s) winds around the origin equals (number of zeros) minus (number of poles) of 1 + L(s) inside that contour. Nyquist's recipe: let s travel up the entire imaginary axis and close around the right half-plane (the Nyquist contour, which encloses every unstable location), and plot the resulting curve L(i omega) — the Nyquist plot. Count N, the net clockwise encirclements of the critical point -1, and P, the number of open-loop poles already in the right half-plane. Then the number of unstable closed-loop poles is Z = N + P. The system is stable precisely when Z = 0.
The beauty is twofold. First, it works directly from the open-loop frequency response L(i omega), which you can measure on the bench without ever building the closed loop or knowing a model — you just sweep frequency and watch how the response curves around -1. Second, it tells you not merely yes-or-no but how much margin you have: how far the curve passes from the point -1 becomes the gain margin and phase margin that engineers design to. The honest caveat: counting encirclements correctly requires care with poles sitting exactly on the imaginary axis (you must indent the contour around them with small semicircles), and the criterion presumes a linear, time-invariant model — it says nothing about nonlinear limit cycles.
For an open loop with no right-half-plane poles (P = 0) the rule simplifies: the closed loop is stable if and only if the Nyquist plot of L(i omega) does not encircle the point -1 at all (N = 0, so Z = 0). If raising the gain pushes the curve out until it loops around -1, you have crossed into instability — and the gain value at which the curve first touches -1 is the gain margin.
Stability is decided by how the curve L(i omega) encircles the critical point -1.
The magic number is -1, not 0: closed-loop poles are zeros of 1 + L(s), so you watch L for encirclements of -1 (equivalently 1 + L encircling 0). Mixing up which point and which direction (clockwise versus counterclockwise) is the classic mistake.