a question about pll stability

[hhq414]

nyquist condition oscillation phase gain

As we know,when the phase shift at the open loop pll bandwith is bigger than
-180+PM,where PM stands for phase margin and is ususlly >45 degree,we say the pll is stable.I am confused that in a second order pll the phase shift in the frequency near dc is almost -180 degree, but we usually don't care about that. How can it be stable when the frequency is near dc?

 

[saro_k_82]

nyquist criterion of oscillations

The criterion for stability from bode plot is that 'at' the unity gain crossover the phase shift should not be 180 or more. For systems without an observable zero within the UGB(monotonic phase response), it so happens that the phase shift worsens as frequency increases so you were probably made to believe that it should be that way with all systems. The problem is that the bode plot plots the mag and phase of A*beta while the characteristic equation is 1+A*beta. Try capturing the same case with root locus.., you'll be satisfied.

 

[hhq414]

type ii pll stability

saro_k_82.
I still can't understand your explanation clearly.As the figure below shows,when the frequency is near dc,the pll open loop gain is bigger than 1. I think it is not stable if we look it as an amplifer.
Can anyone tell me the difference between an amplifer and a pll system when we analyze their stability?

 

[ravirajdv]

question nyquist stability criterion

Hi All,
Even I can't justify myself about the stability of PLL.
Any explanation for this?

Regards,
RDV

 

[LvW]

2nd order pll stability zero and pole frequency

Quote hhq414:
I still can't understand your explanation clearly.As the figure below shows,when the frequency is near dc,the pll open loop gain is bigger than 1. I think it is not stable if we look it as an amplifer.
Can anyone tell me the difference between an amplifer and a pll system when we analyze their stability?


1.) There is absolutely no difference between an amplifier and a PLL as far as stability properties and stability criteria are concerned.
2.) If you extend the simulation to lower frequencies, you will see that the phase function starts at -90 deg., that means: NOT at dc.
3.) Even if the phase would cross the -180 deg line the system would be stable - if and only if the phase would increase again above that line before the magnitude falls below 0 dB. This is a consequence of the general Nyquist criterion.
4.) Consequence of case 3.): There would be 2 frequencies with phase=180 deg and a gain larger than 1.
Question: Does this condition fulfill the oscillation criterion ?
Answer: NO !
5.) Explanation: It is a common misunderstanding (which can be found even in several textbooks and papers !!!) that the BARKHAUSEN criterion for oscillation is a sufficient one. And that´s not the case. Barkhausen´s sentence is a necessary criterion for a system to be able to oscillate. That means, when the condition as mentioned above under 4.) is fulfilled, a circuit may oscillate or not. It depends on some other conditions - and the NYquist criterion has to be applied.
6.) In general, it is not easy to intuitively UNDERSTAND why the circuit under discussion is stable, as it is not easy to really understand Nyquist. This is because his sentence is based on the CAUCHY residuen theorem on the pole and zero location for complex functions. To really understand the background you should go deep into the system theory.
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Perhaps this helps a bit.
LvW

 

[safwatonline]

pll stability criteria

Quote hhq414:

2.) If you extend the simulation to lower frequencies, you will see that the phase function starts at -90 deg., that means: NOT at dc.
LvW

why is that!

 

[LvW]

how pll stable at dc

why is that!

Is this a question ?
It is because the loop gain includes an integrating function (VCO).

 

[safwatonline]

rdv pll

yes, this is a question
actually i think the original question (the bode plot) provided by hhq414 is about a type II pll with two integrators in the loop at DC (one from loop filter and the other is from the VCO)

so in short at low frquency you will get the -180 phase shift shown in the bode plot

 

[LvW]

what is pll stability

yes, this is a question
actually i think the original question (the bode plot) provided by hhq414 is about a type II pll with two integrators in the loop at DC (one from loop filter and the other is from the VCO)
so in short at low frquency you will get the -180 phase shift shown in the bode plot

OK, agreed ! But it does not matter. Because there is in reality no ideal integrator and no ideal PI controller, the phase response will not start at -180 deg. for very low frequencies.
But I think, that was not the main problem of hhq414.

 

[hhq414]

pll not stable

Quote hhq414:
I still can't understand your explanation clearly.As the figure below shows,when the frequency is near dc,the pll open loop gain is bigger than 1. I think it is not stable if we look it as an amplifer.
Can anyone tell me the difference between an amplifer and a pll system when we analyze their stability?


1.) There is absolutely no difference between an amplifier and a PLL as far as stability properties and stability criteria are concerned.
2.) If you extend the simulation to lower frequencies, you will see that the phase function starts at -90 deg., that means: NOT at dc.
3.) Even if the phase would cross the -180 deg line the system would be stable - if and only if the phase would increase again above that line before the magnitude falls below 0 dB. This is a consequence of the general Nyquist criterion.
4.) Consequence of case 3.): There would be 2 frequencies with phase=180 deg and a gain larger than 1.
Question: Does this condition fulfill the oscillation criterion ?
Answer: NO !
5.) Explanation: It is a common misunderstanding (which can be found even in several textbooks and papers !!!) that the BARKHAUSEN criterion for oscillation is a sufficient one. And that´s not the case. Barkhausen´s sentence is a necessary criterion for a system to be able to oscillate. That means, when the condition as mentioned above under 4.) is fulfilled, a circuit may oscillate or not. It depends on some other conditions - and the NYquist criterion has to be applied.
6.) In general, it is not easy to intuitively UNDERSTAND why the circuit under discussion is stable, as it is not easy to really understand Nyquist. This is because his sentence is based on the CAUCHY residuen theorem on the pole and zero location for complex functions. To really understand the background you should go deep into the system theory.
---------------
Perhaps this helps a bit.
LvW

Thank you for all your replies!
My intuitive undstanding is when there is a low frequency phase perturbation at input, because the phase shift is near -180,and the loop gain is larger than 1, the phase perturbation will become bigger and bigger.So I think it will be unstable.
Do you think so?

 

[LvW]

phase lag pll

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.....................
My intuitive undstanding is when there is a low frequency phase perturbation at input, because the phase shift is near -180,and the loop gain is larger than 1, the phase perturbation will become bigger and bigger.So I think it will be unstable.
Do you think so?

No, it may be unstable or not. It really depends on the results of the Nyquist stability check. In some cases it´s not easy to intuitively decide about stability properties.

 

[saro_k_82]

cauchy index, routh-hurwitz

OK, agreed ! But it does not matter. Because there is in reality no ideal integrator and no ideal PI controller, the phase response will not start at -180 deg. for very low frequencies.
But I think, that was not the main problem of hhq414.

Even if the integrators are ideal, one can prove that the system is still stable. The bode phase will indeed start at 180deg(for an ideal integrator) but it is not right to conclude from the bode plot that the system is unstable. There are several limitations / caveats on using bode plots (Non-monotonic phase systems, systems with more than one gain/phase crossovers., stability cant be predicted by bode plots).
Ultimately only the "closed-loop" poles determine the stability. It is easy to see where the closed loop poles are in any other method (rlocus, nyquist, routh-hurwitz, etc) even for ideal integrators.
Ok., coming to the pll example, let us examine it's stability. Apply a step change at the input and see how the output responds. The PFD sends a command to increase current. This current is perfectly integrated(90deg lag) for very low frequencies, but not so for frequencies near the BW which goes back to correct for the step error at the input withe less than 180 deg phase lag.
The crux here is that the loop is not waiting long enough to get the error perfectly integrated because the zero is well within the bandwidth of the loop. On the other hand, if the zero is not introduced, it is easy to see that for any frequency the phase lag will be 180 deg and the loop will be unstable for any frequency.

 

[cadafa]

To hhq414
Hello. Basically, there is no difference between amplifier stability and pll stability.
In the case of the phase shift in the low frequency or DC, we don't need to care it because the feedback is negatively added.
Therefore, when the loop gain becomes 0 or minus value and the phase shift is lower than -180, the feedback factor becomes harmful to be oscillated.
That's why we concern the point of unit gain frequency. We should think two things(Loop gain and Phase shift) at the same time.