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Design >> Analog Design >> Loop gain Stability analysis
https://designers-guide.org/forum/YaBB.pl?num=1297390339

Message started by Rakesh on Feb 10^th, 2011, 6:12pm

Title: Loop gain Stability analysis
Post by Rakesh on Feb 10^th, 2011, 6:12pm

Hi,
The system is a second order system. The bode plot is attached. The question is to estimate the poles and zeros and to comment on the stability of the system...

If i plot the nyquist plot i see that the system is stable.
How to explain with the help of bode plot. as it is seen from the figure that phase only changes by 120 deg approx till UGB.
Thanks

Title: Re: Loop gain Stability analysis
Post by rfidea on Feb 11^th, 2011, 9:49am

Hi Rakesh!

Is the plot for open loop gain or closed loop gain?

Title: Re: Loop gain Stability analysis
Post by Rakesh on Feb 11^th, 2011, 11:41am

Hi rfidea,
The plot is for the loop gain. (GH)
Rakesh

Title: Re: Loop gain Stability analysis
Post by rfidea on Feb 11^th, 2011, 12:23pm

Ok, I suppose by GH you mean the loop gain without the inversion where you get the negative feedback. Then I also suppose you are worried by the -180deg at low freq, which points to a stability problem. I think the reason for the somewhat strange curves is that you have two feedback loops, the main one you know about and one other that you have not think about. Then those curves are typical. See also

http://www.designers-guide.org/Forum/YaBB.pl?num=1294178255

To convice yourself about the stability you can do the complete Nyquist criteria. The simplified one just checking if you not circle the -1 point is not enough. The reason is that it looks like you have two poles in the right half-plate for the open loop. This can be allowed, Nyquist will tell you. It also looks like you have one zero in the right half-plane at lower freq.

Title: Re: Loop gain Stability analysis
Post by Rakesh on Feb 11^th, 2011, 12:46pm

Hi rfidea,
yes you have two poles on right half plane. and its the gain without inversion. It is stable as i have seen from nyquist criteria. There are no internal feedback loops present. But i want to know how to extimate stability from bode plot for these systems .
I am not worried for -180 deg as watever the phase in the loop gain change only by 120 deg.

Title: Re: Loop gain Stability analysis
Post by rfidea on Feb 11^th, 2011, 1:34pm

I'm not sure you can determine stability from the bode plot in such a case. Maybe some other knows a trick. When looking at phase margin and gain margin it is assumed that the loop behaves nice, which is not the case here. What is defining the stability is the lack of poles in the right half plane for the closed loop. Nyquist criteria is a way of finding that out.

Title: Re: Loop gain Stability analysis
Post by Rakesh on Feb 11^th, 2011, 3:40pm

Thats certainly correct.. So in our design is it worth while to use nyquist criteria for stability analysis.
Is it possible to know the limits of bode plot analysis like when we cannot determine the stability of system using bode plot and when we can.
Thanks
Rakesh

Title: Re: Loop gain Stability analysis
Post by vp1953 on Feb 11^th, 2011, 6:06pm

Hi Rajesh,

My quick comments - i am assuming that this is the open loop gain phase plots. It seems to have a bandpass sort of a relationship and looks like there is a single zero and two poles (zero lower than the poles). I also dont think this is an unconditionally stable system - there are two unit gain crossover points - for the first one, there is no issue with phase margin of 120 as you say. For the second one, the phase reaches 180 while the loop gain is greater than one - this could be cause for instability

Title: Re: Loop gain Stability analysis
Post by Rakesh on Feb 11^th, 2011, 6:15pm

Hi,
We cannot say when phase is -180 and loop gain is greater than 1 the system is unstable. This is because consider the loop gain of a PLL or any feed forward compensated system. Near DC we have two poles and the loop gain is very high. However we compensate it by adding a zero so that it is similia to first order system near UGB and to get sufficient phase margin making system stable.
Thanks
Rakesh

Title: Re: Loop gain Stability analysis
Post by vp1953 on Feb 12^th, 2011, 8:10am

HI Rakesh,

Quote:

We cannot say when phase is -180 and loop gain is greater than 1 the system is unstable.

i completely agree with you for small signal analysis. the question is if this ensures stability over all input voltage magnitudes - even when the small signal gain is greater than 1, the large signal gain can be smaller and can be one (eg when the input peak to peak voltage is equal to the output voltage swing). this situation does not arise when the small signal UGC is less than when the phase is 180

Title: Re: Loop gain Stability analysis
Post by Rakesh on Feb 12^th, 2011, 8:31am

Hi vp1953.,
Can u expand ur reason. I didnt follow much...
Rakesh

Title: Re: Loop gain Stability analysis
Post by vp1953 on Feb 13^th, 2011, 11:32am

Hi Rajesh,

Case 1. Small signal gain is greater than 1 at phase =180. As the input signal increases in magnitude, the gain can come down and even unity gain is possible (when input signal peak to peak value is equal to output voltage swing). Even thought the small signal gain is greater than one, here have a situation for large signals the gain becomes unity. So i think that gain crossover (gain going from greater than 1 to less than 1) happening before phase crossover may not ensure stability.

Case 2. Small signal gain is less than 1 at phase 180. in this case the large signal gain will also be less than 1, no issue here.

Title: Re: Loop gain Stability analysis
Post by Rakesh on Feb 13^th, 2011, 12:21pm

Hi,
Assume that i dont use transistors to make these feedback elements. so for the entire region of operation gm of the transistor can be asssumed to be constant. Then how to explain.
Actually i want to know how to explain the stability of second order systems using bode plot, where to apply and where not.
Thanks
Rakesh

Title: Re: Loop gain Stability analysis
Post by vp1953 on Feb 13^th, 2011, 12:54pm

Hi Rakesh,

it does not really matter if your circuit has constant gm. In fact for most oscillators and VCO's the loop gain is chosen to be greater than 1 to ensure adequate startup.

Now lets say that your circuit has ideal components - you still have a finite supply voltage and your output swing cannot be greater than the supply. Your input signal cannot be greater than the supply - when the output swing becomes equal to input swing, you have the unity gain situation.

So i think gain crossover coming after phase crossever can cause instability. Feel free to correct me.

Title: Re: Loop gain Stability analysis
Post by Rakesh on Feb 13^th, 2011, 4:57pm

Assume that your gain cross over frequency comes before phase cross over frequency. Then is it possible to estimate the stability of the system using bode plot.
Thanks
Rakesh

Title: Re: Loop gain Stability analysis
Post by raja.cedt on Feb 13^th, 2011, 6:05pm

hello all,
some told me if open loop is stable then bode can be used to examine closed loop behavior and both phase and magnitude should be monotonic ..)

correct me if i am wrong

Thanks.

Title: Re: Loop gain Stability analysis
Post by vp1953 on Feb 14^th, 2011, 10:57am

Hi Rakesh,

Gain crossover happening before phase crossover with some margin for the phase should ensure stability.

Title: Re: Loop gain Stability analysis
Post by Rakesh on Feb 14^th, 2011, 1:32pm

Hi all,
Can some one give some reference material where i can understand these stability properly.
Thanks
Rakesh

Title: Re: Loop gain Stability analysis
Post by Alexandar on Feb 15^th, 2011, 12:47am

In most analog circuit design/control system books you can find something on stability. Please use google instead of cluttering this forum.

Title: Re: Loop gain Stability analysis
Post by Berti on Feb 15^th, 2011, 1:07am

Alexandar,

The simple bode stability does not apply in all cases. Most real system however are simple enough that they can be analyzed using the bode stability criterion. On the other hand, conditionally stable systems for instance (like the amplifiers design by Xignal for their CT Sigma-Delta or maybe the example given in this thread) need to be analyzed using the Nyquist diagram.

I therefore think that Rakesh's question whether bode stability criterion is enough or the Nyquist plot has to be drawn is not that trivial ... and often ignored in classical text books ... and in my experience often poorly understood by many EE!

Regards

Title: Re: Loop gain Stability analysis
Post by Alexandar on Feb 15^th, 2011, 1:30am

Berti wrote on Feb 15^th, 2011, 1:07am:

Alexandar,

The simple bode stability does not apply in all cases. Most real system however are simple enough that they can be analyzed using the bode stability criterion. On the other hand, conditionally stable systems for instance (like the amplifiers design by Xignal for their CT Sigma-Delta) need to be analyzed using the Nyquist diagram.

I therefore think that Rakesh's question whether bode stability criterion is enough or the Nyquist plot has to be drawn is not that trivial ... and often ignored in classical text books!

Regards

FYI.. Bode plots and Nyquist plots show exactly the same information. It does not mean that if <insert some layman rule> does not apply, we need to use a Nyquist plot. Imho best to use is the complex plane w/ rootlocus method.

BTW strong nonlinear dynamics can be solved using the phase plane, that is if u are really interested into going the theoretical way. These are NOT Nyquist plots...