# Bode criterion for stability - applicability conditions

While checking Bode criterion for stability, I found several sources disagreeing on what is needed to be able to use it.

In general, every source I checked agrees that the open-loop system must be stable and minimum-phase; however, this is were I found different additional conditions:

• the gain diagram must have one and only one crossover frequency

• the phase diagram must have one and only one crossing at -180 ° (while another source not only didn't use this condition, it calculated the margin for the first crossing without saying why)

• the gain diagram must be decreasing (i.e., diagrams like this and this are out, since the diagram has a increasing part); this one I think makes sense given what I understood about the criterion, since a phase margin less than zero means - if the diagram is decreasing - that going towards smaller frequencies there must be a frequency with φ = -180 ° and gain > 1

• the crossover slope must be decreasing (i.e., from positive to negative gain in the direction of increasing frequency: this is fine, this is not)

• the open loop function can't have imaginary poles

• the function's gain must be positive

Some are linked (like having a decreasing diagram means there can't be more than one crossover frequency and it has to be with negative slope; but a diagram can have an increasing part and a negative slope crossover, so they aren't the same condition), but in general I wasn't able to understand under which conditions the criterion can be used. I even found some books that didn't list any condition at all - they just explain briefly the criterion and went on on the next topic.

Which of those additional conditions - beside being open-loop stable and minimum-phase - are needed to be able to use Bode criterion?

A couple of examples I'm not sure can be analyzed with Bode (first, is increasing and has a positive slope crossover; second, it has two crossovers at -180 °):

I found this paper that I think helps: it redefines Bode criterion by adding conditions and some analysis of the results (in the last page there is a summary).

• "there must be a frequency with φ = -180 ° and gain > 1" That's not necessarily unstable. The phase margin is measured ONLY at the point where the gain crosses 0 dB. The system can be stable if that condition is met but there is φ = -180 ° and gain > 1 at lower frequencies for example. That's called conditional stability. I have this argument often, but you can read about it here: ocw.mit.edu/courses/… Commented Apr 17 at 15:59
• @JohnD neat resource! Thanks for sharing. Commented Apr 17 at 17:45
• That's interesting: as far as I know, every source I checked state that if the gain margin is negative (i.e., the gain @ -180 ° is positive) the system is unstable; could this be because in the source you provide there are two crossing of -180 °? Is Bode applicable in this case? If so, when is the gain margin relevant, if the system can be stable with a negative gain margin? Commented Apr 17 at 19:25
• @Mauro That's not correct in the case of conditional stability. Here are more references: ti.com/lit/an/slva947/… electronics.stackexchange.com/questions/72900/… venableinstruments.com/venable-vault/… The possible issues are instability during startup or during system saturation when conditional stability might become instability (or might not) There are lots of C. S. voltage mode DC-DC converters out there. Commented Apr 17 at 19:40
• Is there a way to know in advance if the gain margin is reliable or not? Like excluding cases with more than one crossover at -180 °, since all those cases crossed that phase more than once. Also, I added a couple of diagrams I'm not sure can be analyzed with Bode. Commented Apr 17 at 22:42

I think, there is nothing like a "Bode criterion for stability". Most probably, what you are referring to is the "Nyquist stability criterion" applied to a Bode plot.

In this context, it should be pointed out that there is no fundamental difference between the Nyquist plot for the loop gain and the corresponding Bode plot (magnitide and phase). Both plots contain exactly the same information.

More than that, it should be pointed out that there is

(a) a general Nyquist criterion which applies to all kinds of functions for the loop gain, and

(b) a simplified version of this criterion which applies to loop gain functions only which have no zeroes in the right half of the s-plane.

In both cases, we also can use the Bode diagram for stability analyses.

• However, in some cases (where the general Nyquist diagram is required), we must not use the simplified version of the Bode diagram analysis (phase margin at the 0 dB crossing).

In such a case, the number and the direction (slope) of the 180deg - phase crossings is relevant and must be analyzed - dependent on the number "p" of poles in the right half of the s-plane (p=0, 1 or 2)

• In the Bode diagram, the simplified version of the criterion can be used only when (a) the loop gain has no pole in the right half of the s-plane, (b) there is only one 0dB crossing for the magnitude function and (c) there is only one 180deg crossing of the phase response.

• Thanks; I always saw it called in that way, I didn't know it was a misnaming. So it'd be possible to use the simplified version with the first diagram in my question? Commented Apr 25 at 21:46
• @Mauro - Which gain is shown in the diagrams? Loop gain or closed-loop gain?
– LvW
Commented Apr 26 at 6:30
• They are loop gain. Commented Apr 26 at 7:47
• Where is the phase diagram for the 1st magnitude diagram?
– LvW
Commented Apr 26 at 10:15
• Due to negative feeedback, the real phase response must always start at -180deg . Therefore, the critical point for stability is the 0 deg-crossing. However, a second-order system will reach the 0 deg at infinite frequencies only. Hence - no stability problem.
– LvW
Commented Apr 26 at 12:10

You can gain some insights if you look at it from a different view and ask yourself: what are the conditions under which the system might be oscillating? The Barkhausen criteria for oscillation provides two simultaneous conditions for sustained oscillation: unity loop gain (0 dB gain) and positive feedback which could also mean 180° phase shift in a negative feedback system.

Using the bode plot and the stability criteria you can gage how large the margin of the system to this oscillation condition is. Phase margin meaning the margin in phase when the oscillation criterion for gain is met. Gain margin meaning how large the margin in gain is when the phase condition for oscillation is met.

To gage a systems stability you need to know the frequency range that is susceptable to oscillation. Bode plots only show a limited frequency range. So the question you have to ask is: Is the Bode plot showing the relevant frequency range to gage stability? Or in other words: Are there frequencies beyond the displayed frequency range that are susceptable to oscillation?

The conditions that you mention aim towards a reasoning why there cannot be any frequency susceptable to oscillation beyond the displayed frequency range. E.g. decreasing gain makes sure that there is no "hidden" frequency point of unity gain.

So in my book, these conditions can be considered soft conditions that actually mean: you might have to increase the frequency range you are looking at to say anything about stability.

This is one of your examples which has 20dB gain at the upper frequency limit of the Bode diagram. But any amplifier has some bandwidth limit where gain will start to decrease and will eventually drop to unity gain and below. This frequency is not shown so you cannot tell if the system might be oscillating or not.

• If it were to drop it'd then have two crossover frequencies; could then be analyzed with Bode, or due to the double crossover I'd have to use another method? "Bode can't be used to analyze the stability given this diagram" could be the reason behind (some of?) those conditions, but I'm not sure if all of them make sense (for example, can I analyze a system with imaginary poles?). Commented Apr 21 at 17:27
• In other words, if you had to analyze the stability using Bode, which conditions would you expect to be needed to be able to consider the analysis reliable? Could the diagram you commented be analyzed with Bode, assuming an ideal situation with no bandwidth limit and no decrease at higher frequencies? Commented Apr 24 at 9:16
• I found a paper that could help, it redefines Bode criterion by adding some conditions for it to be used, and referring to Nyquist in other cases; I linked it in the question. Commented Apr 25 at 14:24