Barkhausen stability criterion

In

, to prevent them from oscillating.

Limitations

Barkhausen's criterion applies to

oscillators.

The kernel of the criterion is that a

oscillations should take place. In the real world, it is impossible to balance on the imaginary axis, so in practice a steady-state oscillator is a non-linear circuit:

• It needs to have positive feedback.
• The loop gain is at unity (${\displaystyle |\beta A|=1\,}$).

Criterion

It states that if A is the

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, ${\displaystyle |\beta A|=1\,}$ and
2. The
   phase shift
   around the loop is zero or an integer multiple of 2π: ${\displaystyle \angle \beta A=2\pi n,n\in \{0,1,2,\dots \}\,.}$

Barkhausen's criterion is a necessary condition for oscillation but not a sufficient condition: some circuits satisfy the criterion but do not oscillate.^[5] 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.^[6]

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.^[7] However, sustained oscillations only occur at frequencies for which equality holds.

See also

• Nyquist stability criterion

References