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]They continue to insist that ``[t]he Nyquist stability criterion presents this fact in a rigorous mathematical form.'' This introduction is completely wrong, and Chestnut and Mayer even provide the Nyquist plot of a perfect counterexample a mere fourteen pages later (while defining

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''

where

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

Yet quoting a counterexample rarely satisfies the student. Dislodging this intuitive misconception is an uphill battle. Thus, it is worth refuting this argument in every possible way.

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

Clearly, if we know that at some frequency, then we know that the system is unstable. However, knowing that the loop transfer function is negative and larger than one at one particular frequency gives us no information about stability. For example, if all we know is that , then perhaps

Knowing the magnitude and phase at one frequency does not give us enough information to assess stability.