Loop Gain and Phase Margin Correlation

Question

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?

Answer 1

Now, Aβ would typically be a value ranging 1000000 to 10000 (in opamp amplifier systems, where open-loop gain is usually around 120dB).

That's at DC where nobody really worries about phase margin because it's never going to be an issue. Look at a typical open-loop response of an op-amp: -

Picture source and other relevant information that could be useful.

At 1 kHz the open loop response might have dropped to 60 dB (G = 1000). At 1 MHz, the o/l gain is only 10 and this is an area where quite a few op-amp circuits have problems. Arctan of 10 (assuming a unity gain situation) is 84 degrees and consistent with the graph above. At 10 MHz the gain is unity and arctan of 1 is 45 degrees i.e. not a million miles off.

If you know the T.F. of the forward gain device and you know the feedback T.F. then certainly you can calculate phase margin by considering the loop broken with a signal being injected at the input and the output being taken from where the break is. But you have to respect impedances and loading when you open the loop and sometimes it can be difficult to realize.