Gain margin and Phase Margin Physical Meaning

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?

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.

• I think this is the most stellar explanation of gain and phase margin I've seen, and that's after graduate classes in control theory. – Chuck Aug 28 '15 at 17:57
• Thanks a lot.However I still have doubt regarding the second part of my question, how do we relate phase of the system to instability. ie how is a system with Magnitude=1 and phase =-190degrees unstable? – Fawaz Aug 28 '15 at 18:09
• @Fawaz: Note that we're talking about negative feedback and 180 deg phase shift, which makes it positive feedback. A system with gain above 1 and its output fed back into its input will be unstable. If this occurs at DC, then it will simply latch up. The output goes up a little, so the input goes up a little via feedback, so the output goes up a little more, etc. When these conditions don't occur at DC but at some other frequency, the system will oscillate at the frequency. This is really the basics of what a oscillator is. – Olin Lathrop Aug 28 '15 at 20:26
• @Fawaz, usually, gain and phase reduce as frequency increases, so if the phase is -190 when the gain is unity, the gain must have been >1 when the phase was -180. This is the condition for instability. – Chu Aug 28 '15 at 23:24
• Oscillations are technically marginally unstable or stable. Instability in a linear system means the system is running off towards infinite bounds. – docscience Sep 2 '15 at 21:32

1.) A circuit with feedback is unstable in case the loop gain has a phase shift of 360deg at a frequency where the loop gain magnitude is still larger the 0 dB. Note that this phase shift includes the inverting properties of the inverting terminal. Taking this phase inversion NOT into account (as this is done, normally, in the Nyquist plot) the criterion for instability regarding the phase reduces to -180deg phase shift of the loop gain function. This explains the case of positive feedback (360deg) because we have input phase=output phase (which is critical if the loop gain is larger than unity under this condition).

Note that in case the stability check is performed using a simulation program, the additional 180deg. phase is normally included - provided that the loop gain is determined correctly (which sometimes is a bit involved). In this case, the loop phase must start at -180deg (at low frequencies) - and both margins are related to the frequency where the loop phase is -360deg.

2.) Interpretation (for a good understanding): Phase margin PM is the additional loop phase which would be necessary to bring the closed-loop system to the stability limit. Gain margin is the additional loop gain which would be necessary to make the closed-loop unstable.

3.) UPDATE/EDIT: "May please correct if I have made conceptual mistake anywhere during the course of the Question"

Yes - you have made a severe "conceptual mistake" in speaking always of the "systems phase and gain". Normally, we use the term "system" for a working system - that means: Closed-loop. However, the stability margins (PM and GM) are defined for the LOOP GAIN. Hence, for determining the margins you must open the loop at a suitable point and inject a test signal to find the gain and the phase response of the open-loop circuit.

People tend to make this way too complicated and difficult to understand. Stability margins are only defined for an ideal, linear transfer function model - a model expressed in terms of rational function of polynomials in the complex variable, s. In a feedback loop with a forward transfer function G(s) and feedback transfer function H(s), the input/output closed loop transfer function is $$\frac{y(s)}{x(s)}=\frac{G(s)}{1+G(s)H(s)}$$ The closed loop system is unstable if the characteristic equation (the denominator) is such that $$G(s)H(s)=-1$$ and that happens when $$|G(s)H(s)|=1$$ and at the same time $$\angle G(s)H(s)=-180^{\circ} = 180^{\circ}$$ since G(s)H(s) is complex.

These comprise the stability margins of gain and phase which ask how much additional gain can be added to the closed loop to reach this condition or how much phase shift must be imposed in the closed loop to reach this condition.

This can be determined directly by solving these equations but more often by using graphical tools such as the Bode, Nyquist or Nichol's plots.

Here is simplest answer At -180 degrees, gain must be below 0dB to avoid positive feedback and oscillation. The amount of dB below 0dB at -180 degree is the gain margin. If the amp is -15dB at -180. The gain margin would be 15dB

Phase margin is simple the phase difference between the phase angle at the 0dB crossover point and -180. Eg If the amp measures -140 degrees at 0dB then the phase margin would simply 180-140 = 40 degrees of phase margin.

• Jeff - you are speaking of "gain" and "phase". It would be helpful (better: necessary) to state WHICH gain you are speaking of. There are tree alternatives: (1) Closed-loop gain, (2) Loop gain and (3) Gain of all loop components (without the sign inversion for neg. feedback). Because your critical phase shift is 180deg. it is clear that you are referring to case (3) only! Nevertheless, I recommend to use the 360deg criterion only because there are several examples where the sign inversion takes place WITHIN the feedback loop (and NOT at the summing node). This requires the 360deg criterion. – LvW Jan 18 '17 at 9:41

The feedback is always negative, thus subtracted to the setpoint: epsilon=(setpoint-feedback).
Once you have feedback -1 (-180 deg, A=1) you get a positive feedback. This makes the whole system as stable harmonic oscillator, an undesirable feature.
Therefore with adjusting gain you can modify the curve looking in Nyquist plot, if you add gain the curve is inflating, to that point that has still some margin, not to be attracted to a point-of-no-return (-1,0)

The confusion here is created by the following equation =A/(1+AB). This tells us that the system will be unstable when AB = -1 or a magnitude of 1 and a phase of 180 degrees . However if we also have this explained as a loop phase of 360 ( 180 degrees from inverting terminal plus 180 degrees from feedback network to produce positive feedback when the loop gain magnitude is 1 . This is confusing ! In one case we have 180 degrees loop phase shift presented as the loop phase shift that will cause instability and in the other 360 degrees loop phase shift required to meet the condition for positive feedback . The explanation for the apparent anomaly is that 180 degree phase shift is already implied by negative feedback making and additional 180 degrees loop gain enough to give an overall phase shift around the loop of 360 degrees -enough to cause instability .

To understand its concept, let assume the system as an amplifier, For -ve feedback t/f= AB/(1+AB). Now Gain margin, as we know = 1/gain of system, at -180 degree of phase i.e. at phase cross over frequency. Now if this happen then this leads to AB=1, as phase is -180 degree, then this leads to AB/(1+AB) to 1/(1-1), which is infinite, so system become unstable after this point. And, we know Phase margin is difference in phase at gain crossover, i.e. when gain of system is 1. Now what happen in this case is when phase reaches to -180 degree, same t/f become AB/(1-AB), and as gain is unit here, then this will also leads to infinity, So in both case we are calculating the one of two variables i.e. Gain and Phase, assuming one of them is at the edge i.e either gain=1, or phase =-180 degree, that will lead our system response to infinite i.e unstable.

• prem, sorry to say but your answer causes more confusion than it can help clarifying things. This starts with your first sentence: AB/(1+AB) is wrong! You are mixing closed-loop gain with loop gain (see other answers). – LvW Dec 7 '16 at 13:01
• Also, the formatting and lack of paragraphs makes it difficult to follow. – dim Dec 7 '16 at 13:25
• @ LvW: actually i had taken it for convinience, as it is simple to understand from amplifier point of view, and as for your doubt, we generally solve for unit feedback,Which leads to t/f=G(s)/(1+G(s)H(s) ). The point is, in both cases when phase is -180 degree, and G(s)H(s) leads to a magnitude of 1, then due to phase denominator of t/f become zero, leads to infinte response or undefined response. – prem Dec 7 '16 at 15:46
• Actually in frequency analysis we do take open loop t/f but our main aim is to find the stability of system, which totally depends on the response of the system. – prem Dec 7 '16 at 15:52
• And response of system is dependent of t/f, which depend on one variable G(s)H(s). Thats why we consider open loop gain, just conclude the results that system will be stable or not. – prem Dec 7 '16 at 15:55