Please help me clear up this confusion regarding gain margin and phase margin and unity feedback

Question

I understand the physical significance of gain and phase margin. However, I would like some clarification regarding the mathematical aspects. Videos such as the following:

https://www.youtube.com/watch?v=ThoA4amCAX4 (time: 6:28)

https://www.youtube.com/watch?v=ThoA4amCAX4 (time: 7:12)

define phase margin and gain margin for unity feedback. Even in Matlab when we plot bode of any transfer function, it assumes unity feedback. What's so special about unity feedback?

My next question is, if I want to find gain and phase margin of system with gain G(s) and feedback H(s), whose gain and phase should I plot among the following:

1. G(s)?

2. G(s)/(1+G(s)) ? (unity feedback transfer function)

3. G(s)/(1+G(s)H(s)) ? (closed loop transfer function)

4. G(s)H(s) ? (loop gain)

I feel it should be option 4 but I would like to confirm since nothing seems to mention which bode plot they are checking to obtain PM and GM, and unity gain feedback seems to be extremely popular for some reason.

Answer 1

Unity feedback is the most critical case regarding the stability of a system with feedback. All other applications (higher closed-loop gain) are less critical. For this reason (and to get a "feeling" for the margins) some companies specify the gain/phase margin for unity feedback only. This enables the user to compare different opamps regarding their stability properties.

Yes - your feeling is correct. All the stability margins (gain, phase or vector) are defined for the loop gain function only. (The vector margin can be found in the Nyquist plot for the loop gain only).

In this context, you should know that in some cases the inversion at the summing junction (negative sign) is included in the loop gain definition - and in some cases it is not. Therefore, there are two options for finding the phase margin: Find the phase difference (at unity loop gain) with respect to -360deg (0 deg) or -180deg.

However, in any case the phase difference between the phase at very low frequencies (0 deg resp. -180deg) and the phase at unity gain must not be larger than 180deg (when the closed loop has to be stable)