
In
electronics
Electronics is a scientific and engineering discipline that studies and applies the principles of physics to design, create, and operate devices that manipulate electrons and other Electric charge, electrically charged particles. It is a subfield ...
, the Barkhausen stability criterion is a mathematical condition to determine when a
linear electronic circuit will
oscillate
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulu ...
.
It was put forth in 1921 by
German physicist
Heinrich Barkhausen (1881–1956).
It is widely used in the design of
electronic oscillator
An electronic oscillator is an electronic circuit that produces a periodic, oscillating or alternating current (AC) signal, usually a sine wave, square wave or a triangle wave, powered by a direct current (DC) source. Oscillators are found ...
s, and also in the design of general
negative feedback
Negative feedback (or balancing feedback) occurs when some function (Mathematics), function of the output of a system, process, or mechanism is feedback, fed back in a manner that tends to reduce the fluctuations in the output, whether caused ...
circuits such as
op amps, to prevent them from oscillating.
Limitations
Barkhausen's criterion applies to
linear circuits with a
feedback loop
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause and effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handle ...
. It cannot be applied directly to active elements with
negative resistance like
tunnel diode
A tunnel diode or Esaki diode is a type of semiconductor diode that has effectively " negative resistance" due to the quantum mechanical effect called tunneling. It was invented in August 1957 by Leo Esaki and Yuriko Kurose when working ...
oscillators.
The kernel of the criterion is that a
complex pole pair must be placed on the
imaginary axis of the
complex frequency plane if
steady state
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
oscillations should take place. In the real world, it is impossible to balance on the imaginary axis; small errors will cause the poles to be either slightly to the right or left, resulting in infinite growth or decreasing to zero, respectively. Thus, in practice a steady-state oscillator is a non-linear circuit; the poles are manipulated to be slightly to the right, and a nonlinearity is introduced that reduces the loop gain when the output is high.
Criterion
It states that if ''A'' is the
gain of the amplifying element in the circuit and β(''j''ω) is the
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, models the system's output for each possible ...
of the feedback path, so β''A'' is the
loop gain around the
feedback loop
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause and effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handle ...
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,
and
#The
phase shift
In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a s ...
around the loop is zero or an integer multiple of 2π:
Barkhausen's criterion is a ''necessary'' condition for oscillation but not a ''sufficient'' condition: some circuits satisfy the criterion but do not oscillate.
[ discusses reasons for this. (Warning: large 56MB download)] Similarly, the
Nyquist stability criterion
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry ...
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.
Erroneous version
Barkhausen's original "formula for self-excitation", intended for determining the oscillation frequencies of the feedback loop, involved an equality sign: , β''A'', = 1. At the time conditionally-stable nonlinear systems were poorly understood; it was widely believed that this gave the boundary between stability (, β''A'', < 1) and instability (, β''A'', ≥ 1), and this erroneous version found its way into the literature.
However, ''sustained'' oscillations only occur at frequencies for which equality holds.
See also
*
Nyquist stability criterion
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry ...
References
{{reflist, 30em
Oscillation
Electronic circuits