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The Barkhausen stability criterion is a mathematical condition to determine when a linear electronic circuit will oscillate. It was put forth in 1921 by German physicist Heinrich Georg Barkhausen (1881-1956). It is widely used in the design of electronic oscillators, and also in the design of general negative feedback circuits such as op amps, to prevent them from oscillating.

Limitations

Barkhausen's criterion applies to linear circuits with a feedback loop. Therefore it cannot be applied to one port negative resistance active elements like tunnel diode oscillators.
Criterion

It states that if A is the gain of the amplifying element in the circuit and β(jω) is the transfer function of the feedback path, so βA is the loop gain around the feedback loop of the circuit, the circuit will sustain steady-state oscillations only at frequencies for which:

The loop gain is equal to unity in absolute magnitude, that is, $$|\beta A| = 1\,;$$
The phase shift around the loop is zero or an integer multiple of 2π: $$\angle \beta A = 2 \pi n, n \in 0, 1, 2,\dots\,.$$

Barkhausen's criterion is a necessary condition for oscillation but not a sufficient condition: some circuits satisfy the criterion but do not oscillate. Similarly, the Nyquist stability criterion also indicates instability but is silent about oscillation. Apparently there is not a compact formulation of an oscillation criterion that is both necessary and sufficient.[1]
Erroneous version

Barkhausen's original "formula for self-excitation", intended for determining the oscillation frequencies of the feedback loop, involved an equality sign: |βA| = 1. At the time conditionally-stable nonlinear systems were poorly understood; it was widely believed that this gave the boundary between stability (|βA| < 1) and instability (|βA| ≥ 1), and this erroneous version found its way into the literature.[2] However, stable oscillations only occur at frequencies for which equality holds.

Nyquist stability criterion

References

^ von Wangenheim, Lutz (2010), "On the Barkhausen and Nyquist stability criteria", Analog Integrated Circuits and Signal Processing (Springer Science+Business Media, LLC) 66 (1): 139–141, doi:10.1007/s10470-010-9506-4, ISSN 1573-1979. Received: 17 June 2010 / Revised: 2 July 2010 / Accepted: 5 July 2010.
^ Lundberg, Kent (2002-11-14). "Barkhausen Stability Criterion". Kent Lundberg faculty website. MIT. Retrieved 2008-11-16.

Lindberg, Erik (26–28 May 2010), "The Barkhausen Criterion (Observation ?)", 18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010) (Dresden, Germany): 15–18. URL leads to 56MB conference proceedings download. Reproduces illustration from Barkhausen's original paper.