Electronic – What does it mean for amplifiers to be stable only down to N gain, where N > unity

I think the keyword is "loop gain" - and the most important point (as far as stability is concerned) is the frequency where the loop gain Alo is unity (0 dB).

Because the loop gain is the product of the amplifier´s open-loop gain Alo and the feedback factor Hr (Alo=Hr*Aol) you can find the BODE diagram for the loop gain response very easily:

Draw Aol=f(w) and 1/Hr(w) as a BODE plot and the difference between both curves gives you the loop gain Alo (in dB). The stability criterion requires that at the frequency where both curves meet (Alo=0 dB) the phase shift of the loop gain must not equal to (or even more than) 180 deg.

That means: The "rate of closure" (slope) of the loop gain at this crossing frequency must not be -40dB/dec. (Rule of thumb: If the second pole is identical to the frequency with Alo=0 dB we have app. 45 deg phase margin).

Based on this requirement you can derive the necessary/desired location of the second pole.

Remark ("It's often said that the more feedback the better"): You strictly must distinguish between DC and dynamic stability. Heavy feedback improves the stability of the operating point but - at the same time - degrades the dynamic stability (against oscillations).

EDIT: Sorry for the large picture. I don´t know if/how the size could be reduced.

Some comments from my side:

1.) The stability check in the BODE diagram concerns the LOOP GAIN response only (because once you did mention "closed-loop system" in your text.)

2.) The shown system is "conditionally stable". That means: It is stable - regardless the properties at the frequency A. However, if you REDUCE the gain within the loop until the gain crosses the point A (the phase remains unchanged) the closed-loop system will be unstable.

Such conditional stable system should be avoided because a gain reduction can happen due to aging or other damping effects. Remember: Classical feedback systems with a continuos decreasing loop phase will become unstable (under closed-loop conditions) for rising gain values (beyond a certain limit) only.

As to your next question - the input signal Vi does not influence stability properties at all. Stability is determined by the loop components only. That is the reason, we investigate the loop gain only.

EDIT: Here is an explanation why the closed loop (your example) will be stable: If a closed-loop system is unstable, this point of instability also must be "stable". That means - either we will have "stable" and continuous oscillations or the output is latched at one of the supply voltage rails. In both cases, this point of instability is fixed.

Now - what happens at the point A in your example? Here we have a rising phase which is identical to a NEGATIVE group delay at this point (group delay is defined as the negative phase slope). This is an indication for the unability of the closed-loop system to let the amplitudes rise (oscillations or latching at the supply rail). Rather, the system returns to a stable operating point.

A final information: The stability check investigates either (a) the -180deg line or (b) the -360 deg line. This depends on what you are investigating: (a) Either the simple product GH or (b) the loop gain LG which is LG=-GH.