I have a very basic question about stability that has been bothering me lately.

Considering a negative feedback system, we know that \[ T_{CL}(s)=\frac{A(s)}{1+A(s)B(s)} \]

Clearly, if \(\ A(s)B(s)=-1\), then \(T_{CL}(s)\) goes to infinity and the system is unstable. However, with this reasonning, we conclude there is no other value of the loop gain that makes the system instable.

However, Nyquist criterion, or Phase/Gain margins, tell us that if \[|A(s)B(s)|\geq 1\] when \[\angle A(s)B(s)=0\], the system is also instable. Therefore, the system is instable for value like \[A(s)B(s)=-2, -3, \dots\].

Can someone explain this apparent paradox?

Thank you so much!