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Phase locked Loop Bandwidth for second order system

Started by chess 3 years ago11 replieslatest reply 3 years ago873 views

Hi everyone 

I would like to know can anyone tell me how to determine the #PLL bandwidth ? I read that it can be set to 1/10 or 1/20 of reference frequency however I do not know the reason for it. However I am thinking if I find the closed loop transfer function for the second order system and according to that I find the 3dB bandwidth as a function of damping ratio and natural frequrency that should give me the loop bandwidth. Can anyone please help me to cralify thses concepts since I would like to define other parameters as a function of loop bandwidth ?

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Reply by Tim WescottJuly 12, 2016

Yes, you can find the (linearized) closed-loop transfer function and from that you can find the loop bandwidth.

Setting the loop bandwidth to 1/10 or 1/20 of the reference frequency isn't universal: it's more of a guide to the maximum loop bandwidth that you can realistically expect to attain without many trials and tribulations.  It's roughly the same rule of thumb, and for mostly the same reasons, that you don't expect to set the bandwidth of a sampled-time system to greater than 1/10 or so of the sampling rate.

As a not-randomly-chosen counter-example, if you have a loop that is locking to a signal in a great deal of noise, and if both the PLL reference and the incoming signal have very steady frequencies, then PLL bandwidths can profitably go as low as a few PPM of the reference frequency.

Given the questions that you're asking, it sounds like you're flying blind.  Giving us more information about what you're really trying to do (building a radio, building a data link, etc.) would go a long way to helping us to help you in a sensible way.

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Reply by chessJuly 12, 2016

Thanks. I am trying to simulate a generic second order PLL and need to express the coefficients of my filters and other parameters based on the loop bandwidth. my loop filter is in the form Kp+ KI/s. I have a VCO with a gain KVCO and multiplier with the gain Kd. I need to have everything expressed in terms of loop bandwidth if it is possible according to close loop transfer function.

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Reply by all4dspJuly 12, 2016

The PLL bandwidth can be computed from the closed loop transfer function, where the roll-off passes through -3 dB. The 1/10 rule of thumb people talk about comes from Gardner's famous paper talking about stability requirements for PLL analysis [1]. 

I believe the story goes [2], if the PLL bandwidth is less than one tenth of the reference frequency, you can assume linearity and use a continuous time analysis, meaning, LaPlace (s-domain) analysis. Otherwise, if the bandwidth is more than one tenth of the reference, then you can't ignore the non-linearity, and must use discrete-time analysis, meaning z-transform (which is much more complicated).

There are many references how to compute the 3 dB frequency from damping factor and natural frequency (e.g. see equation A.34 in [3]).

[1] F.M. Gardner, “Charge-Pump Phase-lock Loops,” IEEE Transactions on Communications, vol. 28, no. 11, pp. 1849–1858, Nov. 1980.

[2] ISSC2006-briand paper

[3] Compute PLL loop bandwidth from damping factor and natural frequency

 

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Reply by Tim WescottJuly 12, 2016

I think you're confused about when to use z-domain analysis.  Like Laplace domain analysis, z-domain analysis only makes sense when you're analyzing a linear system.  This means that if the real system you're analyzing is nonlinear, you first have to approximate it with a linear system, then you have to analyze that approximated linear system.

If you have a system that is periodically time varying then you might be able to use z-domain analysis, if you can pick some point in the period and analyzing it from period to period.  But that doesn't mean that you can analyze a nonlinear, time-varying system without first approximating it as a linear system.

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Reply by all4dspJuly 12, 2016

Thanks for straighting me out Tim.

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Reply by Tim WescottJuly 12, 2016

It's easy to get confused.  I didn't actually get it completely straight in my head until I was doing a lot of sampled-time control, pondering switching power supplies (which could be treated with the z-tranform if only they were linear), and teaching various control system bits to various people.

When you're trying to communicate this to people, it suddenly becomes very important to make the various distinctions.

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Reply by shafie7July 12, 2016

Hello Chess,

There are number of ways to determine the loop bandwidth of PLL, it depends whether you are specifying PLL requirements based on application, having a transfer function, or measuring the PLL.

If you have the PLL, just by measuring the performance on spectrum analyzer you can see the loop bandwidth.

If you have the transfer function of Loop Filter, the Bode plots would help to determine the bandwidth.

If you are specifying the requirements of the PLL for given application, then there are some considerations that you have to evaluate and trade off, such as; phase noise budget vs. lock time.

Anyway, let us know what is/are the constraints, then it shed light how to proceed.

Best regards,

Shahram 

ortenga.com

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Reply by chessJuly 14, 2016

Thanks Shahram. I have a closed loop transfer function and I would like to find the analytical expression for loop bandwidth. I got it by finding the 3-dB bandwidth but I am wondering can other parameters like VCO gain and multiplier gain be represented as loop bandwidth?

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Reply by shafie7July 16, 2016

Hello Chess,

Here is a link to a paper with derivations in terms of PLL parameters;

http://users.ece.gatech.edu/pallen/Academic/ECE_64...

Keep in mind, these type of equations are using linear model for PLL which obviously has its limitations.  Anyway, hope it helps.

Best regards, Shahram    ortenga.com

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Reply by chessJuly 17, 2016

Thanks shahram that was very helpful.

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Reply by shafie7July 20, 2016

You are welcome, Chess.

If you want to learn more about PLL, look up the following webinar link;

https://ortenga.com/ortenga800/

Best regards,

Shahram