Closed-loop Stable for Negative Phase Margin?

celebrevida

Junior Member level 1
Closed-Loop Stable for Negative Phase Margin?

Consider the following T(s)=1/(1+G(s))
Let G=K(s+1)^2/s^3
Let's define wc=unity gain frequency & w180= -180 crossing freq

The loopgain G is very interesting.

If K=10, we see that w180=1 but Gain(w180)=26dB. The GM is negative but if we look at poles of T(s) or Nyquist, it is stable!
This means that you can be stable even with negative GM!

But if K=0.25, we see that wc=0.725 and the phase(wc)=-198deg. The PM is -18deg as we are -18deg pass -180deg. And indeed we see that the closed loop is unstable (poles of T(s) are RHP & Nyquist unstable)

This leads me to wonder, is it possible to be stable if we have negative PM with a different example?
(Since it is clearly possible to be stable with negative GM, I am wondering if it is also possible to be stable with negative PM instead.)

If not, then are we always unstable if phase is more negative than -180deg at unity gain and why?

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LvW

Yes...for K=10 the closed loop is stable. It is stable down to K=0.5.
In these cases, the gain margin GM is NOT negative (instead, we have GM=+25dB for K=10)
For K<0.5 the closed loop is unstable.

You have made the following error: The function G is NOT identical to the loop gain which is to be evaluated for stability proprties.
The loop gain is LG=-G.
That means: You have forgotten to consider the negative sign at the summing junction where the feedback signal must be subtracted from the input signal (for negative feedback).
The denominator of the closed-loop function is D(s)=(1-LG)

celebrevida

Junior Member level 1
Yes...for K=10 the closed loop is stable. It is stable down to K=0.5.
In these cases, the gain margin GM is NOT negative (instead, we have GM=+25dB for K=10)
For K<0.5 the closed loop is unstable.

You have made the following error: The function G is NOT identical to the loop gain which is to be evaluated for stability proprties.
The loop gain is LG=-G.
That means: You have forgotten to consider the negative sign at the summing junction where the feedback signal must be subtracted from the input signal (for negative feedback).
The denominator of the closed-loop function is D(s)=(1-LG)
My closed loop function is just this one but with H(s)=1:
https://en.wikipedia.org/wiki/Closed-loop_transfer_function

If you do the math, it is clear that Closed_Loop = G/(1+GH) and denominator is 1+GH (or 1+G if H=1).
Not sure where you are getting 1-G. In fact you would get 1-G only in positive feedback.

And with that when K>0.5, the gain of G is indeed higher than 1 (or 0db) at w180. GM is indeed negative. Therefore it IS possible to be stable when loopgain is higher than 1 at w180.

I was just wondering though if there was another example (not this one obviously) where it is closed loop stable but you have negative PM. That's actually my real question.

Thanks!

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LvW

OK - I have to apologize...my answer was not quite correct. I have made an error - I did not consider the fact that the loop gain has three poles in the origin.
(By the way- this is an ideal case, which never can happen in reality).
In this case, the simple stability check in the BODE dfiagram (using PM and GM) must not be applied.
Instead, you have to use the complete Nyquist theorem (Nyquist plot in the complex s-plane).
The closed-loop system

T(s)=1/[1+G(s)] with G(s)=K(1+2s+s²)/s³

will show rising oscillations for K< 0.5 and will be stable for K>0.5

Quote: If you do the math, it is clear that Closed_Loop = G/(1+GH) and denominator is 1+GH (or 1+G if H=1).

Just for clarification (to avoid misunderstandings) : The above equation is correct if G and H are the transfer functions of the blocks within the loop - if we have negative feedback (a minus sign at the summing junction).
However, the producr GH is NOT identical to the loop gain because - for negative feedback - we must consider the minus sign.
That means: Loop Gain LG=-GH. Therefore, the denominator can also be written as (1-LG).

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SunnySkyguy

I agree if loop gain is greater than 2nd order near f limit , Bode Plots are unreliable for margin estimates.

celebrevida

Junior Member level 1
I agree if loop gain is greater than 2nd order near f limit , Bode Plots are unreliable for margin estimates.
Can you elaborate this? It is too cryptic.
If possible, what is the general "rule" for when Bode is unreliable (and why?)

Thanks

FvM

Super Moderator
Staff member
Some previous discussion about Bode and Nyquist

A typical case where simple positive PM criterion doesn't work are loop gain functions with multiple crossings of the 0 dB line. But that's not the case here.

In the present circuit, PM criterion still "works", although GM is unusual. You can still derive stability from the Bode plot, but looking at the Nyquist curve makes things clearer.

FvM

Super Moderator
Staff member
Additional comment. The reason why the usual Bode plot gain margin criterion can't be applied is that phase is rising instead of falling when crossing phi = -180° line.