Chapter 12, Problem 12.84P

A loop gain function is given by
$T(f)=\frac{500}{\left(1+j\frac{f}{{10}^4}\right)\left(1+j\frac{f}{5\times {10}^4}\right)\left(1+j\frac{f}{{10}^5}\right)}$

(a) Determine the frequency $f_{180}$ (to a good approximation) at which the phase of $T(f)$ is $-180$ degrees. (b) What is the magnitude of $T(f)$ at the frequency $f=f_{180}$ found in part (a)? (c) Insert a dominant pole such that the phase margin is approximately 60 degrees. Assume the original poles are fixed. What is the dominant pole frequency?