# Bode phase plot phase margin

From what I have studied so far, gain margin is defined as how much additional gain the system can be given before it becomes unstable, and this can be measured from the bode magnitude plot corresponding to the frequency at which the phase crosses -180 degrees.

• What if the phase plot is such that it becomes flat-looking at say -75 degrees, and never goes below that? Then how is the gain margin calculated? (i.e then will the phase crossover frequency be infinite?)

• If an open loop system's bode plot says that it has negative gain margin but positive phase margin, is the system unstable (and cannot be stabilized)?

EDIT: How would the -180 degree phase crossover frequency point be determined here? For example, just by a Google search I got a few instances,

• It is difficult to imagine reactances that do not go towards zero or infinity as the frequency goes to DC or infinity. Do you have a particular circuit in mind that stays at -75 degrees from some frequency and then out to infinity?
– jonk
Commented Nov 16, 2022 at 20:52
• As jonk says difficult to imagine. In fact it's impossible given that the speed of light is finite; at some point in the spectrum, the phase will invert. Commented Nov 16, 2022 at 21:06
• @jonk I made some edits to my post, perhaps now it'd be easier to help me understand how to calculate this?
– SM32
Commented Nov 16, 2022 at 21:13
• Notice the first example marks phase and gain margin points and lists the values at the top. Commented Nov 16, 2022 at 21:24
• Think of a circle, 180 = -180. Have another look at the plot, you should see the markings. Commented Nov 16, 2022 at 21:37

From a theoretical point of view...

If the loop phase never reaches -180 degrees then the gain margin is undefined. The gain margin is measured at -180 degrees loop phase. The phase margin is measured at the unity loop gain frequency and if this occurs at a frequency after the phase flattens off at -75 degrees then the phase margin will be +105 degrees. A pretty stable system in the closed loop I would say.

Negative gain margin means that a Nyquist diagram plot passes to the left of the -1+j0 point and so the system is unstable. The system should be stabilised by reducing the forward gain so that the gain margin becomes positive and the Nyquist plot passes to the right of the -1+j0 point when travelling in a clockwise direction.

Whether a system is stable or unstable when the loop is closed depends on what the value of the loop gain is when the loop phase is -180 degrees. This statement doesn't encompass the full Nyquist criteria but is a general guide.

• Thanks for your reply, could you explain then how the GM is found for the pictures in the post? I'm confused because MATLAB gives out a number, but I can't see the -180 deg mark in the phase plot
– SM32
Commented Nov 16, 2022 at 21:34
• Look at the two extra vertical dotted lines, one at the phase crossover (-180 degrees) and one at the gain crossover (0 dB). The gain margin is then the number of dBs that have to be added to the gain at -180 degrees to make the gain 0 dB (in that case -7.93 dB) and the phase margin is how far short of -180 degrees is the phase is when the gain is 0 dB (in that case -10.4 degrees). The gain and phase margins are both negative implying that the closed loop system will be unstable.
– user173271
Commented Nov 16, 2022 at 21:44
• @James I just skimmed that first paper cited by the OP. I think the authors may have in inserted the wrong chart. The text related to the chart says that it "shows only a phase margin of $8.79^\circ$." Other numbers they cite, $\omega_{_0}$ and $\omega_{_\text{RHP}}$ for example, do match their formula results. But I'm having difficulty seeing how they got a positive and different phase margin. Certainly not from what I see in that chart, anyway.
– jonk
Commented Nov 17, 2022 at 0:09
• @James Never mind about being the wrong chart. It's the right one. I just plotted my own and it just the same as their included chart. So perhaps the text is mystifying (at least to me.) That's the only thing that doesn't seem to match up. Here's the larger context in case you can see where they get that number.
– jonk
Commented Nov 17, 2022 at 1:07
• Some clarification is necessary. The gain margin is determined at a frequency where the phase is 180deg off with respect to the phase at very low frequencies. This is important because there are two different loop gain definitions: With or without considering the phase inversion at the summing junction (feedback model). That means: The critical point is at -180deg (-360deg=0deg) taking the phase inversion NOT into account (resp. including the phase inversion). I strongly recommend to use the 360deg-criterion because this is in accordance with loop simulations.
– LvW
Commented Nov 17, 2022 at 9:44