Barkhausen's stability criterion

The Barkhausen stability criterion yielded a necessary mathematical condition, when an electric circuit consisting of an amplifier and a suitable feedback is, can swing independently. This criterion does not provide any information as to whether the oscillator circuit formed in this way works stably and generates sinusoidal oscillations of constant amplitude.

criteria

Amplifier (above) with feedback

Amplifiers with a gain factor can be excited to stable oscillation by feedback with the (linear) transfer function if the following two conditions are met: ${\ displaystyle A}$ ${\ displaystyle \ beta (\ mathrm {j} \ omega)}$

Amplitude condition: The amount of loop gain is equal to 1, that is: ${\ displaystyle | \ beta (\ mathrm {j} \ omega) \ cdot A | = 1; \,}$
Phase condition: The phase shift must have positive feedback at the oscillator frequency. This is fulfilled when the phase shift has integer multiples of : ${\ displaystyle 2 \ pi}$ ${\ displaystyle \ angle \ beta (\ mathrm {j} \ omega) \ cdot A = 2 \ pi n; \ qquad n \ in 0,1,2, \ dots \,}$

This condition is necessary for stable vibration generation, but not sufficient. As a rule, neither the amplifier nor the transfer function are linear, but the circuit still oscillates. The Nyquist stability criterion provides a necessary and sufficient statement about the instability of the system, but no statement about the stability of the oscillation. A general sufficient stability criterion for generating a stable oscillation is not known.

Limits of applicability

The stability criterion was developed at a time when the cut-off frequency of the amplifier tubes (over 100 MHz) far exceeded the operating frequency of the oscillator circuits of the time (less than 1 MHz) and could not be measured with the means at that time. The above formulation therefore assumes that the output signal of the amplifier follows the changes in the input signal without delay and that no phase shift occurs ( transit time = 0). This assumption is no longer fulfilled with increasing frequency, which leads to the (false) statement that the ring oscillator cannot function, although the loop gain is considerably greater than one. In fact, this circuit oscillates in a stable and very reliable manner , whereby the frequency generated can be calculated from the processing speed in the amplifier stages .

In oscillator topologies such as the relaxation oscillator , the stability criterion cannot be used because these are based on the negative characteristic of a component. There are circuits with transfer functions that meet Barkhausen's stability criterion, but do not oscillate stably. In the super regenerative receiver , for example, an amplifier generates vibrations at two very different frequencies that influence each other. In the case of acoustic feedback , the transfer function is mostly unknown and hardly linear, which is why the frequency of the whistling can only be predicted within rough limits. Nevertheless, the effect can be reproduced well.

Incorrect formulation by Barkhausen

The first formulations and the naming go back to Heinrich Barkhausen , who first formulated this condition in the 1920s and published it in the third volume of his four-volume work Elektron-Röhren . At that time, however, Barkhausen published an incorrect version, which was partially preserved in the following decades, especially in German-language specialist literature.

For the generation of the oscillation, which he called self-excited oscillation , Barkhausen started from the not generally applicable idea that stability is generally present and instability is present. In fact, there is only a need for stable oscillation . The mathematical modeling of the time was not yet so advanced, and Nyquist's stability criterion, which clarifies this point more comprehensively, was formulated only a few years later by Harry Nyquist and Felix Strecker . ${\ displaystyle | \ beta (\ mathrm {j} \ omega) \ cdot A | <1 \,}$ ${\ displaystyle | \ beta (\ mathrm {j} \ omega) \ cdot A | \ geq 1 \,}$ ${\ displaystyle | \ beta (\ mathrm {j} \ omega) \ cdot A | = 1 \,}$

literature

Heinrich Barkhausen: Electron tubes, 3rd volume feedback . 4th edition. S. Hirzel, Leipzig 1931.

Individual evidence

^ ^A ^b Barkhausen Stability Criterion , Kent H Lundberg, November 14, 2002, engl.

[lund1-1] A ^b Barkhausen Stability Criterion , Kent H Lundberg, November 14, 2002, engl.