# What is the reasoning behind Barkhausen's criteria?

Especially why should loop gain be equal to 1, won't that give us infinite circuit gain?

• I'm only aware of a single criterion. And, as someone who's especially been taught that there is a wealth of literature that reproduces a misinterpretation of the Barkhausen Criterion: Could you please write down exactly the criterion you want us to reason about, as in exactly what it requires from a system and exactly what it then says about that system? – Marcus Müller Nov 10 '19 at 10:12
• This looks like a homework question and nothing else, to me. – TonyM Nov 10 '19 at 12:44
• @TonyM..I am not as sure as you. Answering this question is not a simple thing - when we really try to go deep into the problem. A good indication for the complexity is the following sentence: For a high quality sinusoidal output the oscillator must be as linear as possible - and needs, for this purpose, a certain kind of non-linearity. – LvW Nov 10 '19 at 13:53
• If the loop includes a small amount of random noise, then gain > 1 for small perturbations becomes your friend. – analogsystemsrf Nov 10 '19 at 17:25

Yes - in principle, you are right...the CLOSED-LOOP gain (effective for an external signal) would be, theoretically, infinite. However, it is not the purpose of such a circuit to amplify any external signal.

The oscillator principle is that it produces the required input signal for the amplifier itself by compensating the gain factor for the attenuation in the feedback circuit. This is equivalent to a unity loop gain.

However, due to tolerances and some other uncertainties within each electronic circuit, it is not possible to designa circuit having a loop gain of exactly "1". Hence, we design it for a slightly larger loop gain and use an extra gain control (diodes, thermistor, FET as a resitor,..) which automatically brings the loop gain back to unity as soon as a certain output is available. (In reality, due to the time constant of the regulation mechanism, the loop gain swings around the nominal value of unity).

Comment: In many publications it is not mentioned that the Barkhausen criterion is only a necessary one. That means: It is not a sufficient criterion ....there are circuits which have unity loop gain for one single frequency only - but the circuit does not oscillate.

• Is that actually the case? I remember closed-loop gain actually not being infinite, following multiple arguments. Best I can find on short notice: web.mit.edu/klund/www/weblatex/node4.html – Marcus Müller Nov 10 '19 at 10:35
• @Marcus..The first sentence of the linked document ("The Barkhausen Stability criterion is simple, intuitive, and wrong") is a really stupid statement - and it has been several times proven as wrong. The author did not fully understand that Barkausens rule was and is a necessary condition only. By the way - recently this criterion was supplemened by another condition and now forms a sufficient oscillation criterion. – LvW Nov 10 '19 at 13:43
• @Marcus...In the mentioned contribution the expression "L(s)" is the loop gain and - as I have mentioned - it is the closed-loop gain which could be considered as infinite (for unity loop gain). However, such a statement is a pure formal/theoretical consideration and has no practical meaning. More than that (coming back to the oiriginal question) it has nothing to do with Barkhausens original oscillation condition. In the mentioned paper, the closed-loop gain would be L(s)/[1-L(s)]....and you can see what appens for L(s)=1 – LvW Nov 10 '19 at 14:24
• Correction: Sorry, I have mixed the forward function L(s) with the loop gain LG=-L(s), Therefore, the correct expression for the closed-loop gain is H(s)=L(s)/[1-LG]=L(s)/[1+L(s)]...and H(s) will theoretically approach infinity for LG=1. – LvW Nov 10 '19 at 14:39
• @Marcus M....even the headline of the linked document ("Barkhausen Stability Criterion") shows that the author does not know what he is writing about. H. Barkhausen never has formulated a "stability criterion" - hence, his findings must not be confused with Nyquists criterion. Barkhausens statement was just the following: When we want a circuit with feedback to oscillate it must satisfy the condition of unity loop gain (but he did not use th term "loop gain". He has used the product "gainxfeedback factor".). That means, he has formulated a necessary oscillation condition - nothing else. – LvW Nov 13 '19 at 15:35