Positive feedback evaluation

[Rodrigo Franchi]

Hello,

I´m trying to figure out the following issue:

If we have a feedback system:

Af = A /(1 + βA)

In the case that phase of βA = 180°, we have 3 cases:

1_ | βA| < 1 then, if we choose arbitrary βA = 0,5 (with phase 180°)  Af = A/(1-0,5) = 2*A ….. AMPLIFIER WITH POSITIVE FEEDBACK. STABLE AND REGENERATIVE.

2_| βA| = 1 (with phase 180°)  Af = infinite = A/(1-1) …. OSCILLATOR

3_ | βA| > 1 we choose arbitrary βA = 1,5 (with phase 180°)  Af = A/(1-1,5)= -2*A
According to nyquist we encircle point -1 what is the condition for instability. Now mathematically in equation Af=A/(1+ βA) I got a -2 value for Af module. Not an unstable value for Af?

How can I see the instability mathematically in equation: Af = A /(1 + βA)????

Thanks
Rodrigo

 

[hacksaw]

you've just defined the stability boundary,

 

[Rodrigo Franchi]

Ok. Then it is not a valid equation for the instability region?. In fact the equation has a discontinuity when BA cross the -1 point, it changes from +infinite to -infinite.