Barkhausen stability criterion

The stability criterion of Barkhausen provided a necessary mathematical condition, when an electric circuit consisting of an amplifier and an appropriate feedback can swing independently. This criterion does not provide information as to whether the oscillator circuit thus formed operates stably and produces sinusoidal oscillations of constant amplitude.

Criterion

Amplifier having a gain factor can be excited by a feedback to the (linear) transfer function for stable vibration, if the following two conditions are met:

This condition is necessary for the stable oscillation generation, but not sufficient. As a rule, neither amplifiers nor transfer function are linear, still swinging the circuit. The Nyquist stability criterion of supplying a necessary and sufficient indication of the instability of the system, but no information on the stability of the oscillation. A general criterion for sufficient stability to produce a stable oscillation is not known.

Limits of applicability

The stability criterion was developed at a time when the cutoff frequency of the amplifier tubes ( about 100 MHz) (less than 1 MHz ) exceeded the operating frequency of that oscillator circuits by far and with the former agents was not measurable. Therefore, the above formulation assumes that without delay the output signal of the amplifier follows the changes of the input signal and no phase shift occurs (duration = 0). This assumption is no longer met with increasing frequency, leading to statement ( false ), the ring oscillator can not work, even though the loop gain is much greater than one. In fact, this circuit oscillates stable and very reliable, with the frequency generated from the processing speed can be calculated in the amplifier stages.

In oscillator topologies such as the relaxation oscillator, the stability criterion is not applicable, because these are based on the negative characteristics of a component. There are circuits with transfer functions which satisfy the stability criterion of Barkhausen, but do not swing stable. In the super-regenerative receiver, for example, produces an amplifier oscillations at two very different frequencies that affect each other. In the case of acoustic feedback, the transfer function is usually unknown and unlikely to be linear, which is why the frequency of the whistle can be predicted only in rough boundaries. Nevertheless, the effect is highly reproducible.

Incorrect formulation of Barkhausen

First formulations and the naming goes back to Heinrich Barkhausen, who in the 1920s first formulated this condition and published in the third volume of his work four Restraining electron tubes. Barkhausen published at that time, however, a faulty version, which has been partially in the following decades, especially in German-language literature was obtained.

Barkhausen went for the creation of what he called self-excited oscillation of the oscillation is not generally valid idea of that stability and instability in general at present ( | ≥ 1 | βA ). In fact, the need for a stable oscillation is only when ( | βA | = 1) before. The former mathematical modeling was not yet so far advanced and the stability criterion of Nyquist, which clarifies this point comprehensive, was only some years later formulated by Harry Nyquist and Felix Strecker.