Barkhausen stability criterion
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Barkhausen stability criterion 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.
Block diagram of a feedback oscillator circuit to which the Barkhausen criterion applies. It consists of an amplifying element A whose output vo is fed back into its input vf through a feedback network ß(jω).
To find the loop gain, the feedback loop is considered broken at some point and the output vo for a given input vi is calculated:G = \frac {v_o}{v_i} = \frac{v_f}{v_i}\frac {v_o}{v_f} = \beta A(j \omega)\,
Barkhausen stability criterion
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Criterion It states that if feedback path, so
is the gain of the amplifying element in the circuit and
is the transfer function of the
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, 2. There must be a positive feedback i.e., the phase shift around the loop is zero or an integer multiple of 2π: 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. Unfortunately these can not be distinguished with the Nyquist stability criterion. The Nyquist stability criterion in its general form only indicates instability but cannot provide any information if this instability will cause oscillations or not. Thus, there seems to be still no compact formulation of an oscillation criterion that is both necessary as well as sufficient, ref. [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.[1] However, stable oscillations only occur at frequencies for which equality holds.
See also • Nyquist stability criterion
Notes [1] Lundberg, Kent (2002-11-14). "Barkhausen Stability Criterion" (http:/ / web. mit. edu/ klund/ www/ weblatex/ node4. html). Kent Lundberg faculty website (http:/ / web. mit. edu/ klund/ www/ ). MIT. . Retrieved 2008-11-16.
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/
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