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adesign

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PLL stability
Feb 18^th, 2008, 8:34pm

Dear all,

I've a very basic question about PLL stability. In the attached figure, the PLL loop gain characteristics are shown. In figure(a) and (b), we can see that phase is -180deg near low frequencies. With an additional -180deg phase-shift due to PFD, the total phase shift in close loop is 360deg and also loop-gain is very high at these frequencies. According to Barkheusen criteria, this signifies an unstable system. Then how come PLL is a stable system.

Could anyone explain this?

Best Regards,

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PLL_loop_gain.GIF

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buddypoor

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Re: PLL stability
Reply #1 - Feb 18^th, 2008, 11:57pm

Hi,

Fig. (a) clearly is unstable whereas the BODE response in Fig (b) indicates a stable system, because the phase lag at the cross-over frequency (0 dB-point) is below 180 deg. The reason is the additional zero in the low pass loop filter.
By the way, you should not misuse the Barkhausen criterion: It is only a necessary condition for oscillation - opposite to a sufficient condition which can be used as a stabilty criterion (Nyquist, Routh,..)

Lutz

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LvW (buddypoor: In memory of the great late Buddy Rich)

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adesign

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Re: PLL stability
Reply #2 - Feb 19^th, 2008, 12:43am

Try to imagine a system with loop-gain > 1 and having total phase shift of 360deg or its multiples. This will come out to be unstable. I think the systems in the figure also denote this.

Could you please highlight more of your explanation?

Also, Barkheusen criteria is the very basic criteria and if the system fails this then it'll oscillate. Please correct me if I'm wrong.

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sheldon Community Fellow Offline Posts: 751	Re: PLL stability Reply #3 - Feb 19^th, 2008, 2:42am Adesign, If loop gain is greater than 1, wouldn't you have a latch and not an oscillator. Thought an oscillator needed a gain of 1 for stable oscillation? Best Regards, Sheldon
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buddypoor

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Re: PLL stability
Reply #4 - Feb 19^th, 2008, 2:54am

Hi, Adesign

Quote: Try to imagine a system with loop-gain > 1 and having total phase shift of 360deg or its multiples. This will come out to be unstable.

No, it is not as simple as it looks. For example there are systems which are “conditionally stable”. The answer depends on the whole system behaviour. If (a) the loop gain is equal to or greater than 1 and (b) having a total phase shift of 360deg and if (c) the phase lag monotonically further increases with rising frequencies, the system is unstable. Otherwise (e.g. if the system inclues zeros) you have to apply one of the universal stability criterions (Nyquist, Routh, Hurwitz…)

Quote: Also, Barkhausen criteria is the very basic criteria and if the system fails this then it'll oscillate.

No, that is not correct. The Barkhausen rule is not a stability criterion. It is nothing more than a necessary condition for a system being able to oscillate. That means, you are allowed to apply it only in this direction. It is not a sufficient condition for a system to be unstable.

One further remark to the asymptotic phase response as shown in Fig (b): The real phase of the integrating parts of your system will never exactly reach the value of - 180 deg. This may be considered as another reason for stability.

I hope this helps a bit.
Regards
Lutz

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LvW (buddypoor: In memory of the great late Buddy Rich)

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safwatonline

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Re: PLL stability
Reply #5 - Feb 19^th, 2008, 10:40am

buddypoor wrote on Feb 19^th, 2008, 2:54am:

Hi, Adesign

Quote: Try to imagine a system with loop-gain > 1 and having total phase shift of 360deg or its multiples. This will come out to be unstable.

No, it is not as simple as it looks. For example there are systems which are �conditionally stable�. The answer depends on the whole system behaviour. If (a) the loop gain is equal to or greater than 1 and (b) having a total phase shift of 360deg and if (c) the phase lag monotonically further increases with rising frequencies, the system is unstable. Otherwise (e.g. if the system inclues zeros) you have to apply one of the universal stability criterions (Nyquist, Routh, Hurwitz�)

Quote: Also, Barkhausen criteria is the very basic criteria and if the system fails this then it'll oscillate.

No, that is not correct. The Barkhausen rule is not a stability criterion. It is nothing more than a necessary condition for a system being able to oscillate. That means, you are allowed to apply it only in this direction. It is not a sufficient condition for a system to be unstable.

One further remark to the asymptotic phase response as shown in Fig (b): The real phase of the integrating parts of your system will never exactly reach the value of - 180 deg. This may be considered as another reason for stability.

I hope this helps a bit.
Regards
Lutz

Hi Lutz,
i was wondering if you could provide any reference about the conditional stability as i searched a lot for a good explanation but i didn't find any good reference.
regards,
Safwat

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buddypoor

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Re: PLL stability
Reply #6 - Feb 19^th, 2008, 12:31pm

Hi, Safwat !

In response to your question concerning systems which are conditionally stable I can recommend two
textbooks:
1.) B. J. Lurie, P. J. Enright: Classical feedback control
2.) G. F.Franklin, J. D. Powell, A. Emami-Naeini: Feedback control of dynamic systems.

Among other methos, such systems can be investigated with the "Lyapunov stability criterion".

I am not sure if you can find some relevant contributions via "google", however, some good books about control theory should cover this item.
Regards
Lutz

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LvW (buddypoor: In memory of the great late Buddy Rich)

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buddypoor Community Fellow Offline Posts: 529 Bremen, Germany	Re: PLL stability Reply #7 - Feb 19^th, 2008, 12:56pm Safwat, here are two additional references: 3.) V. Y. Belozyorov: On an invariant Design of feedbacks........ (Int. J. Appl. Math. Comput. Sci., 2001, Vol.11, No.2, 377-389 4.) V. Y. Belozyrov: Design of linear feedback for bilinear control systems (same magazin, 2002, Vol. 12, No. 4, 493-511) Lutz
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safwatonline Junior Member Offline Posts: 20	Re: PLL stability Reply #8 - Feb 19^th, 2008, 2:18pm Thanks a lot Lutz
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Frank Wiedmann Community Fellow Offline Posts: 678 Munich, Germany	Re: PLL stability Reply #9 - Feb 20^th, 2008, 12:54am An example with conditional stability has also been discussed in this forum at http://www.designers-guide.org/Forum/YaBB.pl?num=1182388268;start=all.
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