The Barkhausen Stability Criterion is simple, intuitive, and wrong. During the study of the phase margin of linear systems, this criterion is often suggested by students grasping for an intuitive understanding of stability. Unfortunately, although counterexamples are easy to provide, I do not know of a satisfying disproof to the Barkhausen Stability Criterion that combats this intuition.

Some textbooks even state the Barkhausen Stability Criterion (although none refer to it by name). In their introduction of the Nyquist Stability Criterion, Chestnut and Meyer state

If in a closed-loop control system with sinusoidal excitation the feedback signal from the controlled variable is in phase and is equal or greater in magnitude to the reference input at any one frequency, the system is unstable.[3, page 142]

The history of the Barkhausen Stability Criterion is an unfortunate one. In 1921, during his study of feedback oscillators, Barkhausen developed a ``formula for self-excitation''



The concept, as stated by Chestnut and Mayer, seems intellectually satisfying. In fact, I've often had students (and professors) ask, ``But if the gain around the loop is five, and the total phase shift around the loop is exactly zero, then doesn't that imply that any signal around the loop will grow with time?''

This reasoning, although deeply flawed, seems to make sense. There is no shortage of counterexamples, such as







Using Black's Formula provides one refutation. For a system with unity negative feedback and loop transfer function L(s), the closed-loop transfer function is







Kent H Lundberg

2002-11-14