Electronic – Gain margin and Phase Margin Physical Meaning

bode plotcontrol systemphase marginstabilitysystem

I have been trying to understand the physical concept of Gain and Phase Margin.

What I understand about this is that a relative comparison around the critical point \$(-1,0)\$, which when converted to magnitude and phase form turns out Magnitude = 1 and phase = -180°.

Also for a negative feedback system the Gain and Phase Margin should be positive, i.e., a system is unstable under the following 2 cases:

  1. When the System/OLTF phase is -180° but System Magnitude \$>1\$. Thereby making Gain Margin negative.
    I was able to correlate a physical meaning to this condition as the same would lead to a positive feedback condition with Gain \$>1\$ thereby leading to Unbounded output and hence instability.

  2. When the System Magnitude = \$1\$ but System Phase \$>-\$ 180°. I'm not able to get a physical understanding of this unstablility case.

My questions:

  • How is after all phase used to comment about unstability of a closed loop system?

  • In this case after accounting for the negative feedback inherently present due to negative feedback the net phase might turn out to be positive, so how does that make the system unstable?

Best Answer

Gain and phase margin are usually applied to systems that are amplifiers of some sort with negative feedback around them. The more negative feedback, the tighter the system is controlled. However, you don't want to provide feedback in such a way that the system will oscillate. The gain and phase margin are two metrics to tell you how close the system is to oscillation (instability).

A system with over-unity gain will oscillate with positive feedback. Usually the intent is to stabilize a system by using negative feedback. However, if this is phase shifted by 180°, then it becomes positive feedback, and the system will oscillate. This can happen due to various characteristics of the system itself or what happens to the feedback signal.

Note the two criteria for oscillation: a gain greater than 1, and positive feedback. Since we are usually trying to provide negative feedback, we think of positive feedback as what happens when there is a 180° phase shift in the loop. This therefore gives us two metrics to decide how close to oscillation the system is. These are the phase shift at unity gain, and the gain at 180° phase shift. The first had better be below 180°, and the second had better be below 1. The extent they are less than 180° and less than 1 is how much room, or margin, there is. 180° minus the actual phase shift at unity gain is the phase margin, and 1 divided by the gain at 180° phase shift is the gain margin.

Since the main problem is usually that the overall phase and gain change as a function of frequency, loop gain and phase shift are often plotted as a function of Log(frequency). The gain curve is then basically a Bode plot. You have to examine the two curves carefully to see that the system stays away from the combination of characteristics that will make it oscillate. When this is the main point, something called a stability diagram shows you more directly how close the system is to instability and at what operating point. That closest approach to instability is called the stability margin.