Nyquist's Stability Criterion

Black's feedback amplifiers worked, and some of them sang. Adding more feedback made every measurable property better right up to the point where the amplifier turned into an oscillator, and nobody could say in advance which side of that line a new design was on. Maxwell's method cannot help. It needs the coefficients of the differential equation, and a Bell Labs repeater full of hand-wound transformers, valves and stray capacitance does not come with coefficients - you would have to guess them, and the guess is exactly what is in question. Harry Nyquist's paper of January 1932 asks a question you can answer with a signal generator instead. Send a sine wave once around the OPEN loop and look at what comes back. If at some frequency it returns the same size and exactly upside down, then joining the loop lets it sustain that sine wave with no input at all, which is what singing is. Plot what comes back over the whole frequency range and the test becomes a count: how many times does the curve go around the point minus one? Two numbers fall straight out and are still how amplifiers are specified. Gain margin is how much more gain the loop would take before it sang; phase margin is how much more lag. Both are read off a measurement of a machine you never modelled. The criterion also handles the case Maxwell's cannot touch at all: a pure time delay. A microphone a few metres from its own loudspeaker has no polynomial to write down, and Nyquist predicts the pitch of the howl and how it changes when you move the microphone.