A loop gain function is given by [T(f)=frac{500}{left(1+j frac{f}{10^{4}} ight)left(1+j frac{f}{5 times 10^{4}} ight)left(1+j frac{f}{10^{5}} ight)}] (a) Determine the frequency (f_{180}) (to a good approximation)

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?