Hi Steve,
Yes that sounds like a good idea. It seems to me that the DC gain less than 1 criterion you had been mentioning kicks in once we go to the practical, because any op amp is bound to have some output DC offset either positive or negative. The theoretical circuit i had been looking at will not have this problem because there is just plain and simple no DC on the output, it's a theoretical zero DC just like the pole pair is a theoretical exactly on the jw axis. In the presents of the tiniest DC though, i think we have to consider your specification that the DC gain has to be lower than 1 because if not it would just ramp the output up (or down) indefinitely until it hit one of the rails.
So for the theoretical model it probably wouldnt matter if the DC gain was 10, as long as there was no DC it cant ever feed back and become a problem.
That's the way i see it so far.
But you know i cant help but think, we've had various opinions come up in this thread now, and they usually turn out to be just different ways of looking at it, and that varies with the person and what they are expecting based partly on what they know already. So i cant help but wonder just who was Barkhausen's audience? If they were beginners in the field or intermediates they might have had little knowledge about these issues and so 1000 questions were bound to come up. But what if they were highly educated and experienced in the field. Maybe he took it for granted that they would surely know that there has to be a little more to it than that (his two conditions for oscillation) such as start up conditions and DC feedback and such. Maybe that's the way he was thinking when he wrote his formulas. He assumed the target audience knew enough about this so he did not have to spell every little thing out in black and white. On the other hand, i can also see that questions come up for almost everyone at one time or another regarding his criterion, and it's often talked about as if he was "wrong". Was he really wrong or was it just that he assumed a minimum level of proficiency, which would then mean he considered everything else obvious.