# Loop Gain and Phase Margin Correlation

I have found a definition of phase margin of amplifier system from Texas Instruments application report. This definition looks like this: $$\phi = tan^{-1} (A \beta)$$ where $A$ is amplifiers open-loop gain (aka direct gain) and $\beta$ is feedback return signal ratio - or $A \beta$ known as loop gain. Now, $A\beta$ would typically be a value ranging $1000000$ to $10000$ (in opamp amplifier systems, where open-loop gain is usually around $120 dB$).

Such values of $A\beta$ inserted into upper definition of phase margin always equals (approximately) $\phi = 90°$. So, using that equation for definition of phase margin must be definitely wrong, because it is not possible, for amplifier's phase margin to be $90°$ in all scenarios possible. Unless we would be discussing an example with $A\beta < 100$, which is very unlikely to happen.

Also, it would seem more logical if phase margin definition equation would be described as a function, dependent on poles of amplifier or $s$, damping factor or $\zeta$, frequency or $\omega$, etc.

I know how to find phase margin (and gain margin) from already drawn Bode plot, but I cannot solve it, using mathematical ways, not graphical.

Can anyone tell me, if this is the actual formula for calculation of phase margin? Or are there more data needed to solve such case? Would "fully defined" transfer function provide enough data for proper calculation of phase margin?

• I do not know which Texas application report you are referring to. However, the "definition" as given by you cannot be true. You say that you are able to find the phase margin of a system with feedback. But in this case, it should be clear to you that the mentioned definition is not correct. Are you aware that the lopp gain will be unity at a certain frequency? – LvW Aug 27 '18 at 14:32
• For a second-order system, there is a fixed relation between phase margin and damping factor. – LvW Aug 27 '18 at 14:34
• @LvW Here: ti.com/lit/an/sloa021a/sloa021a.pdf Pages 6-7 are explaining about stability of current feedback amplifier, including definition of phase margin. – Keno Aug 27 '18 at 15:02
• Yes - of course. But did you realize that the definition of the phase margin is not in accordance with the first two lines of your question? I am afraid you misunderstood something. – LvW Aug 27 '18 at 15:12
• Try to watch this youtube.com/watch?v=kC8FYL8gr3E and Lec44 and 45 – G36 Aug 29 '18 at 17:44