Question 12.20: Objective: Determine the dominant pole required to stabilize...

By inserting a dominant pole, we change the loop gain function to
$T_{P D}( f ) = \frac{1000}{\left(1  +  j \frac{f}{f_{PD}}\right)\left(1  +  j \frac{f}{10^{4}}\right) \left(1  +  j \frac{f}{10^{6}}\right) \left(1  +  j \frac{f}{10^{8}}\right)}$

We assume that $f_{P D} \ll 10^{4}  Hz$. A phase of −135 degrees, giving a phase margin of 45 degrees, occurs approximately at $f_{135} = 10^{4}  Hz$.
Since we want the loop gain magnitude to be unity at this frequency, we have
$|T_{P D}(f_{135})| = 1 = \frac{1000}{\sqrt{1  +  \left(\frac{10^{4}}{f_{P D}} \right)^{2}} \sqrt{1  +  \left(\frac{10^{4}}{10^{4}} \right)^{2}} \sqrt{1  +  \left(\frac{10^{4}}{10^{6}} \right)^{2}} \sqrt{1  +  \left(\frac{10^{4}}{10^{8}} \right)^{2}}}$
or
$1 = \frac{1000}{\sqrt{1  +  \left(\frac{10^{4}}{f_{P D}}\right)^{2}}  (1.414)(1)(1)}$
Solving for the dominant pole frequency $f_{P D}$, we find
$f_{P D} = 14.14  Hz$
Comment: With high-gain amplifiers, the dominant pole must be at a very low frequency to ensure stability of the feedback circuit.