The formula
N = Z - P
Reading: the number of clockwise encirclements of the point -1 + j0 (call it N) equals the number of unstable closed-loop poles (Z) minus the number of unstable open-loop poles (P). Everything to the right of the imaginary axis counts as "unstable."
L(s) = G(s)·H(s) (open-loop / loop transfer function)
Reading: the loop transfer function is the plant times the sensor/feedback path. The Nyquist plot is L(s) evaluated as s walks the boundary of the entire right half-plane.
1 + L(s) = 0 (closed-loop characteristic equation)
Reading: closed-loop poles are the zeros of 1 + L(s). Encircling -1 in the L-plane is the same as encircling 0 in the 1 + L plane — which is why the magic number is minus one, not zero.
For stability with P = 0: N = 0 (no encirclements of -1)
For stability with P > 0: N = -P (P counter-clockwise encirclements)
Reading: if the open loop is already stable, you just need the curve to stay clear of -1. If the open loop has P unstable poles, the curve must wrap -1 exactly P times the other direction to pull them into the left half-plane.
Gain margin = 1 / |L(jω)| at the frequency where phase(L) = -180°
Phase margin = 180° + phase(L(jω)) at the frequency where |L(jω)| = 1
Reading: how much you can crank the gain, or how much extra phase lag you can eat, before the curve lands on -1 and the system sings.
Where you meet it
- Servo and actuator loops on a test stand. You measured the open-loop frequency response of a hydraulic or electric actuator with a swept sine — you have magnitude and phase versus frequency, not a clean transfer function. Nyquist takes that measured data directly and tells you your stability margins. Bode does the same job for minimum-phase plants; Nyquist is what you reach for when there's a transport delay or a right-half-plane zero that makes Bode ambiguous.
- Loops with pure time delay. Data links, thermal plants, and anything with
e^(-sT)in the loop add phase without touching magnitude. That extra phase rotates the whole Nyquist curve clockwise toward-1. A delay a Bode gain plot barely notices can be the thing that swallows your phase margin. - A design review board. When you present a control loop for sign-off, the reviewer wants two numbers: gain margin and phase margin. Both come straight off the Nyquist (or Bode) plot at the axis crossings. A loop with 6 dB and 45° is a defensible design; one that skates past
-1at 1.5 dB gets sent back. - Power-electronics and grid-tie stability. Source-load interactions in DC buses and inverters are checked with the impedance-ratio Nyquist test, and the multivariable version handles the three-phase case.
How it works
The whole thing rests on Cauchy's argument principle. Walk s once clockwise around a closed contour, and the image L(s) will encircle the origin a number of times equal to (zeros inside) minus (poles inside) of L. Nyquist's move was to make that contour the entire right half-plane — up the imaginary axis from -j∞ to +j∞, then a giant semicircle closing off to the right. Because real hardware has real coefficients, the s = jω piece for negative ω is just the mirror image of the positive-ω piece, so in practice you sweep frequency from 0 to ∞ and reflect.
Apply the argument principle to 1 + L(s) and its origin becomes -1 in the L-plane. That's the entire trick: N = Z - P, and you want Z = 0.
Here is a case worth carrying in your head. Take a unity-feedback loop with L(s) = K/(s+1)³. The open loop is stable, so P = 0; you just need the curve to stay off -1. The phase hits -180° when 3·arctan(ω) = 180°, i.e. ω = √3 ≈ 1.732 rad/s. There the magnitude is 1/(1+3)^1.5 = 1/8. So the curve crosses the negative real axis at -K/8. It touches -1 exactly when K = 8. Below that, no encirclement, stable. Above it, two clockwise encirclements, Z = 2 — two unstable closed-loop poles. Solve the closed-loop cubic and you can confirm it: at K = 8 two poles sit dead on the imaginary axis at ±j1.732 (marginal, the system oscillates at √3 rad/s); at K = 9 those poles have crossed to real part +0.04. The plot and the pole-solve agree to the decimal. At K = 1 the gain margin is 8, which is 18.06 dB.
The mistakes people make, in order of how often they bite:
- Ignoring open-loop instability. When
P > 0, no encirclements is not what you want — you want P counter-clockwise ones. Engineers trained on the "keep it away from -1" shortcut get this exactly backwards on unstable plants. The shortcut only holds whenP = 0. - Poles on the imaginary axis. An integrator (
1/s) puts a pole right on your contour. You detour around it with a small semicircle, which maps to a large arc at infinity on the plot. Miss the detour and you'll miscount encirclements. - Reading margins off one crossing when there are several. A curve that crosses the negative real axis more than once has multiple gain margins; the binding one is the crossing nearest
-1. Conditionally-stable loops can go unstable when you lower the gain, which stuns anyone expecting gain to only ever help. - Trusting Bode on non-minimum-phase plants. Right-half-plane zeros and time delays break the clean gain-phase relationship Bode assumes. Nyquist has no such blind spot — it counts encirclements regardless.
- Forgetting margins are a distance, not a guarantee. Good gain and phase margins individually can still leave the curve passing close to
-1on a diagonal. The honest robustness number is the shortest distance from the curve to-1(the peak of the sensitivity function), not either margin alone.
Limits of validity: the criterion is exactly true for linear time-invariant systems. Real actuators saturate and have backlash; a Nyquist result is a small-signal statement about the loop around an operating point. And the plot is only as good as the frequency response behind it — a swept-sine that skipped the frequency where the curve nears -1 will lie to you.
History
By 1927 Harold Black at Bell Telephone Laboratories had invented the negative-feedback amplifier — the thing that made long-distance telephony possible by trading raw gain for stability and low distortion [2]. It worked on the bench, but there was a catch nobody could pin down: pile on enough feedback and the amplifier would break into oscillation at some frequency, and no one had a reliable way to predict which designs would sing and which would behave [2]. Gain and phase both drifted with frequency, and the interaction was hard to reason about with the tools of the day.
Harry Nyquist, a Swedish-born physicist who'd come to the United States in 1907 and taken a Yale PhD before joining the Bell system, took the problem on. In January 1932 he published "Regeneration Theory" in the Bell System Technical Journal, volume 11, pages 126–147 [3][4]. His insight was geometric: draw the loop's gain and phase as a single curve in the complex plane, and a simple encirclement count of the critical point tells you whether the closed loop is stable [2]. The tangled question of feedback stability collapsed into looking at one picture.
Nyquist wasn't quite alone. Felix Strecker at Siemens in Germany worked out an equivalent criterion around the same period, arriving independently at the same encirclement idea [1]. Nyquist's paper is the one the English-speaking world learned from, and his name is the one that stuck.
The idea kept growing. In 1977 A. G. J. MacFarlane and I. Postlethwaite generalized it to multi-input, multi-output systems in the International Journal of Control, volume 25, replacing the single curve with the characteristic loci of the return-ratio matrix and grounding the whole thing on the argument principle applied to a Riemann surface [5]. That generalized Nyquist criterion is what power-systems and multivariable-control engineers still reach for today. Ninety-plus years on, a test engineer with a swept-sine measurement and no clean model is using the exact tool Nyquist built to keep Black's amplifiers from howling.
Related tools
- /tools/rc-filter
- /tools/opamp-gain
- /tools/lc-resonance
- /tools/frequency-period