Question 12.20: Objective: Determine the dominant pole required to stabilize...
Objective: Determine the dominant pole required to stabilize a feedback system. Consider a three-pole feedback amplifier with a loop gain given by T (f) = \frac{1000}{\left(1 + j \frac{f}{10^{4}}\right) \left(1 + j \frac{f}{10^{6}}\right) \left(1 + j \frac{f}{10^{8}}\right)}
Insert a dominant pole, assuming the original poles do not change, such that the phase margin is at least 45 degrees.
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.