Barkhausen stability criterion

Block diagram of a feedback oscillator circuit to which the Barkhausen criterion applies. It consists of an amplifying element A whose output v_o is fed back into its input v_f through a feedback network ß(jω).

To find the loop gain, the feedback loop is considered broken at some point and the output v_o for a given input v_i is calculated:
 :$G={\frac {v_{o}}{v_{i}}}={\frac {v_{f}}{v_{i}}}{\frac {v_{o}}{v_{f}}}=\beta A(j\omega )\,$

For the noise in the output of a ferromagnet upon a change in the magnetizing force, see Barkhausen effect.

The Barkhausen stability criterion is a mathematical condition to determine when an 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.

Applications

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

Theory

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

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

Single-way condition

Barkhausen's criterion is a necessary condition for oscillation, not sufficient. This means there are some circuits which satisfy the criterion but do not oscillate. These can be distinguished with the Nyquist stability criterion, which is both necessary and sufficient.

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. However, stable oscillations only occur at frequencies for which equality holds.

See also

• Nyquist stability criterion

References

[1] Lutz von Wangenheim, "On the Barkhausen and Nyquist stability criteria", Analog Integrated Circuits and Signal Processing, DOI 10.1007/s10470-010-9506-4 Received: 17 June 2010 / Revised: 2 July 2010 / Accepted: 5 July 2010 Springer Science+Business Media, LLC 2010 ISSN: 1573-1979 (electronic version)

[2] Lindberg, Erik, "The Barkhausen Criterion (Observation ?)", part of: Proceedings of NDES 2010, pages: 15-18, IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010), 26 – 28 May 2010, Dresden, Germany, http://www.ndes2010.org/