This Criterian is necessary to make a successfull
oscillator and the conditions are
1. loop gain must be unity
2. total phase shift should be zero or 360 degrees

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''

KF(jw)=1

where K is an amplifier gain factor and F(jw)is the
frequency dependence of the feedback loop. This equation
was originally intended for the determination of the
oscillation frequency for use in radio transmitters.
However, before conditionally-stable nonlinear systems were
understood, it was widely believed that only a single value
of K separated stable and unstable regions of behavior.
Thus, Barkhausen's Criterion was incorrectly used as a
stability criterion, especially in the German literature

magnitude of AB is greater than or equal to 1.a is the open
loop gain,b is the feedback factor and ab is the closed loop
gain.also,the totl phase shift around the closed loop is 0
or 360 degrees.

A criterion used to determine the stability of an oscillator
circuit which states that, if the circuit is seen as a loop
consisting of an amplifier with gain A and a linear circuit
whose gain β(jω) depends on frequency ω, then the loop will
oscillate with a perfect sine wave at some frequency ω0 if
at that frequency Aβ(jω0) = 1 exactly, that is, if the
magnitude of Aβ(jω0) is exactly 1 and its phase is 0° or 360°.

Barkhausen's Criterion may be found in his book "Lehrbuch
der Elektronen-Röhren", 3. Band:, “Rückkopplung”, Verlag S.
Hirzel, 1935, Achte Auflage, pp. 4–5, 1960. He pointed out
that an oscillator may be described as an inverting
amplifier (a vacuum tube) with a linear frequency
determining feedback circuit. The non-linear amplifier is a
two-port with a static gain-factor equal to the ratio
between the signals at the ports. The linear feedback
circuit is a two-port with a feed-back-factor equal to the
ratio between the port signals. It is obvious that the
product of the two factors becomes equal to one. The product
is called the Barkhausen Criterion or the "Allgemeine
Selbsterregungsformel" in German language.