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I have a basic question regarding negative feedback and stability. As I've understood it, negative feedback is supposed to promote equilibrium.

But suppose we have negative feedback loop block diagram, with open loop gain A, and feedback \$\beta\$. Both positive greater than 0.

\$Output = \dfrac{A}{1+\beta\ A} * Input\$

Now suppose there's an instability at the output... say the output is higher than the above value by \$ \Delta \$.

Won't this instability grow at the output until some rail voltage is hit. ie: the instability becomes: \$ \Delta (-\beta A)^n\$ , where n is the number of iterations through the loop.

If \$ \beta A\$ is greater than 1, won't this instability keep growing till some rail voltage is hit.

Is there something wrong with my above analysis? how is equilibrium reached? Thanks.

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    \$\begingroup\$ What do you mean by 'an instability at the output'? If the output increases by \$\small\Delta\$, the error becomes \$\small -\Delta\$, and hence the output reduces until the error is again zero. That's negative feedback. \$\endgroup\$ – Chu Mar 1 '16 at 13:48
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Yes that's true, it would hit a rail voltage but this does not mean that it is working in a stable condition. Even a small shift then there is chance that it might move to the negative rail and back and forth. This swing could happen due to anything such as noise signals in surrounding and so on. The only way to ensure stable operating point is by having a phase margin of less than 180 degrees. If this is accomplished then the output would not saturate at the rails but rather exponentially decay to a stable actual operating point.

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I see my mistake was in thinking that the output changes instantaneously in response to the input. But the output will change at some finite rate (eg: slew rate for opamp). This does indeed lead to equilibrium.

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Your problem is in the signs. You asked about negative feedback, but then stated a positive value for beta. If, however, you use a negative beta value, then the negative feedback (so long as beta is <=1) will always tend to push back towards 0 from whatever A or Delta value you start with.

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