An op amp having a lowfrequency gain of 103 and a singlepole

An op amp having a low-frequency gain of 10^3 and a single-pole rolloff at 10,000 rad/s is connected in a negative-feedback loop via a feedback network having a transmission k and a two-pole rolloff at 10,000 rad/s. Find the value of k above which the closed-loop amplifier becomes unstable.

Solution

A(s)=1000/(1+0.0001s)

B(s)=k/(1+0.0001s)²

Ang(AB)=-tan^-1 (w/10000)-2 tan^-1 (w/10000)=-3 tan^-1 (w/10000)

For Ang(AB)=180^o , W₁₈₀=1.732*10000 rad/s

We require |AB|<1

1000/(1+3)^1/2*k/(1+3)^1/2<1

k<0.008.