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I am interested in finding the gain margin, phase margin and bandwidth of a system via the bode frequency response.

I do know how to find each. For each one of them, first you have to find a certain frequency based on where the bode plot passes through a value. Specifically:

  • Gain margin: that frequency is the one at which the phase of the open loop system is at -180 degrees

  • Phase margin: that frequency is the one at which the gain of the open loop system is at 0 db

  • Bandwidth: the frequency is the one from which the gain of the closed loop system passes through -3db

My question:

In all the above cases, if the bode passes multiple times through a specific db/degree, which do we choose?

For example, in the case of the bandwidth. At the bode plot of the closed loop system, in the gain plot, if the system passes through the -3db at ω=10, 50 and 105 rad/second (so it goes up and down a few times) which frequency do we chose as the bandwidth? (and this question extended to the gain and phase margin as well.)

Kind regards,

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First of, some people define the bandwidth differently. For example at my university the bandwidth is also defined as the frequency where the magnitude of the openloop transfer function passes through the 0 dB. But in practice these different definitions do not make a big difference of the exact value of the bandwidth frequency.

But lets get back to your question of when there are multiple instances where the magnitude equals 0 dB or the phase equals -180° (or a multiple of 360° added to -180°). The gain margin, phase margin and bandwidth would be the one with the smallest absolute value. This is also why I prefer the use the Nyquist plot instead of a Bode plot, because from that it is much easier to spot the smallest values. Only reading the bandwidth is a little inconvenient because the plot itself does not contain any frequency information in the curve itself.

One other thing that is nice about a Nyquist plot is that it allows you to identify the modulus margin (a term I have not seen being used a lot outside my university). The modulus margin is the smallest distance between the curve and the minus one point and is also equal to the inverse of the magnitude of the largest peak of the sensitivity transfer function. So which frequency of the disturbance applied to the system gets amplified the most. The gain margin is basically the shortest distance to the minus one point along the real axis and the phase margin is basically the smallest angle to the minus one point from the curve that crosses the unit circle around the origin. So the modulus margin by definition gives a lower bound of both the gain and phase margin. Below there is an example of how to read these margins, but also shows that the gain and phase margins can be misleading for the stability of the system and disturbance attenuation.

Nyquist plot

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  • $\begingroup$ Adding a link to a Nyquist plot containing what you said would be good (higheredbcs.wiley.com/legacy/college/nise/0471794759/justask/…). $\endgroup$ – MrYouMath Oct 20 '17 at 15:48
  • $\begingroup$ @MrYouMath thanks for your suggestion. Because of your comment I also noticed I wrote the definition wrong for the phase margin from Nyquist plot, so fixed that as well. $\endgroup$ – Kwin van der Veen Oct 20 '17 at 16:04
  • $\begingroup$ @KwinvanderVeen Thanks a lot for the tips. I am familiar with Nyquist plot and tend to use that one more often. But would like to focus on Bode for now. You say that chosen frequency (for each one) is the smallest absolute value. The smallest absolute value of which value? So the most left value on the frequency axis? (10 rad/sec in my example) $\endgroup$ – Dimitris Pantelis Oct 21 '17 at 0:09
  • $\begingroup$ @DimitrisPantelis No, the smallest value of each potential margin. But for the bandwidth that does indeed comes down to the smallest frequency, so 10 rad/s. But for the gain and phase margins it can be other values. The margins are just the largest amount of gain or phase shift you can apply to the openloop with before it potentially becomes unstable. $\endgroup$ – Kwin van der Veen Oct 21 '17 at 0:32
  • $\begingroup$ @KwinvanderVeen ohh yeah now I see. It should have been more obvious to me :P. Thanks ^^ $\endgroup$ – Dimitris Pantelis Oct 21 '17 at 12:07

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