Why Relaxation Oscillators DON'T satisfy BARKHAUSEN CRITERIA?
Answer:
Relaxation oscillators do satistfy Barkhausen's criteria if they start up. To see this you have to remember what Barkhausen's criteria is: a simple startup condition for a LINEAR system with feedback. Because any physical system's response is continuous, one can model its input/output characteristics in the form of a taylor expansion and for inputs small enough, the system is more or less linear. In this case, Barkhausen's criteria will determine if the oscillation starts to oscillate from the white thermal noise components present in any physical system.
Now when you're talking about a relaxation oscillator (ie: a ring oscillator), you need to be careful about using Barkhausen's criteria. The ring oscillator (as all oscillators are) are highly nonlinear systems once the oscillation amplitude is big enough and are then governed by non-linear dynamics, in which Barkhausen's criteria no longer applies..
Barkhausen's criteria is probably the simplest and least rigorous of all the oscillator startup formulations. It has logical holes in it when you really think about it (see: http://web.mit.edu/klund/www/weblatex/no... ). What Barkhausen's criteria amounts to is a special case of the more comprehensive root-locus and nyquist plot techniques (both of which you should wikipedia), which are much better ways to characterize if and how linear systems start up. Remember that root locus and nyquist plots are still methods for systems that are linear. When systems become non-linear then the Laplace and fourier techniques become less applicable, and then you need to look into phase space and limit cycle techniques (wikipedia these).
Hope this helped! :)
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