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Theory to design a pierce oscillator (tough question) (Read 4262 times)
Pictou
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Theory to design a pierce oscillator (tough question)
May 21st, 2013, 2:19am
 
Hello,
I'm trying to design a crystal oscillator (Pierce schematic). On the simulator (cadence) it works fine. However I would like to validate the simulation with some theory but I can't. My oscillator is oscillating (on simulation) at 40.1 Mhz, and I can't find why.

I'm just interested in knowing if the steps I took to calculate this oscillation frequency is correct or not. In the link below you will find a PDF showing my methodology. I decided to write it in a pdf because otherwise it would not be readable.

http://www.pdfhost.net/index.php?Action=Download&File=ea91ab67d9637ad314ac1238e8...

If someone could check my pdf and tell me if I made a mistake somewhere I would really appreciate.

2 options, either my calculations are wrong in theory, or my simulator is wrong....

Thank you very much for your help.
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ywguo
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Re: Theory to design a pierce oscillator (tough question)
Reply #1 - May 23rd, 2013, 2:09am
 
HI Pictou,

First, you must know something about crystal model. What is the parameter of your crystal model? and its Q value?


Best Regards,
Yawei
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Pictou
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Re: Theory to design a pierce oscillator (tough question)
Reply #2 - May 23rd, 2013, 3:19am
 
Hello ywguo,
Yes I asked the manufacturer he gave me the R1, L1, C1, C0. From this, Q is approximately 100 000.

Thank you.
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ywguo
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Re: Theory to design a pierce oscillator (tough question)
Reply #3 - May 24th, 2013, 1:07am
 
Hi Pictou,

The Q value is good. What's the series resonant frequency with your parameter?

fs=1/(2*PI*sqrt(L1*C1))

Best Regards,
Yawei
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Pictou
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Re: Theory to design a pierce oscillator (tough question)
Reply #4 - May 24th, 2013, 1:25am
 
fs = 40.026 MHz.

Thing is, I can't find this result back from the analysis of the schematic.

I mean, wherever I read something about oscillators, it says that it's really hard to start oscillation; that C1 and C2 (for load capacitance) must be chosen carefully.
BUT I can't chose them wisely if I don't even know how my schematic works (theoretically at least).

What I would like to do is find this expression of fs by analyzing my schematic. Also, I would like to be able to calculate the voltage across the oscillator, the current through it, etc etc...


I think my mistake is that I assumed that X1 + X2 + X3 = 0, it can't be 0 as the crystal is not a perfect inductor... still investigating though.
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ywguo
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Re: Theory to design a pierce oscillator (tough question)
Reply #5 - May 24th, 2013, 5:11pm
 
Hi Pictou,

I think that X1+X2+X3=0 is eventually correct if the oscillator starts up. However, it seems that you have not read some literatures about crystal oscillator design. This is a link that I googled just now.  :)
http://electronicdesign.com/analog/fundamentals-crystal-oscillator-design

Quote:
I mean, wherever I read something about oscillators, it says that it's really hard to start oscillation; that C1 and C2 (for load capacitance) must be chosen carefully.

Yes, the designer must choose C1 and C2 carefully, but usually not very very carefully. The available values of C1 and C2 vary in a very big range. It says that it's really hard to start oscillation just because the authors have to say something to emphasize the importance of their analysis and circuit design skills.  :)

I think the most important paper on crystal oscillator deisgn is High-performance crystal oscillator circuits: theory and application by Eric Vittoz, etc, in JSSC 1998.

Best Regards,
Yawei
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Pictou
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Re: Theory to design a pierce oscillator (tough question)
Reply #6 - May 27th, 2013, 6:50am
 
Okay I'll read this article, I'm sure it will help.

Thank you very much.
EDIT : I've been trying to design an oscillator for 3 months now... I haven't found anything relevant in most of the things I read. And you managed to find a reliable source of information with just one google search... I've been on google for a while... must have typed something wrong.
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raja.cedt
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Re: Theory to design a pierce oscillator (tough question)
Reply #7 - May 28th, 2013, 12:14am
 
Hello,
You could think of think of this problem like a Colpitts oscillator as well. Calculate how much inductance your crystal has at it's series or // resonance frequency.

Normally C1/C2 start-up requirement as well noise(i suppose this is less critical for you). Initial guess would be C1/C2 is around 1, which gives easy start up (around 4 loop gain you need).

Thanks,
Raj.
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Pictou
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Re: Theory to design a pierce oscillator (tough question)
Reply #8 - Jun 10th, 2013, 1:55am
 
Hello, well after a few days trying to fit the theory to my simulation... it doesn't work one bit, I'm sure I'm doing something wrong and I don't know what.

Anyway, I'm getting short on "free" time and need to work on something else.

The article and the colpitts approach allowed me to understand a little better. I'm also sure that the results they get are correct, so it must be my numerical application that doesn't fit.

Thank you all for your help!
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Kevin Aylward
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Re: Theory to design a pierce oscillator (tough question)
Reply #9 - Jul 6th, 2013, 3:11am
 
First, what you posted is the generalised Colpitts oscillator. That is, ANY single device configuration that has a capacitor, effectively across its drain source and a capacitor across its gate source. A Pierce is defined as having a resister from the drain in series with the drain capacitor, to form a pi network. This resister is important. It reduces both sensitivity to DC supply, and lowers 1/f up converted phase noise. Always use it, period, unless you simply don’t have enough gain at frequency.

Ignore all the equations. There are useless. All xtals in the known 3 universes from manufactures are specified to run on only a handful of standard load capacitances. These will be, 30p/2, 20p/2 or 13.5pf/2. i.e.  make the caps the same. To check enough loop gain stick a voltage source in the gate lead and run Cadence stb analysis. Run transients with a “DeQed” xtal of say, 100 times. i.e. multiply the c1, series capacitor by 100. C1 is typically 1ff to 20ff. Use 10ff X 100 as a starter. The bigger the cap, the more frequency stable but harder to get the gain at high frequencies.

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Pictou
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Re: Theory to design a pierce oscillator (tough question)
Reply #10 - Jul 8th, 2013, 12:27am
 
Thank you for your help. I abandoned the equations, they don't seem to work and there is no "real application on a real case" to be found... even on the internet.

About the resistor in series, do you mean like the figure 2 page 3 of this link?

http://www.freescale.com/files/microcontrollers/doc/app_note/AN3208.pdf

It says that I only need it in case I have too much drive level, however nobody knows how to define this resistor as there is no equation to prove anything. So I just tossed it and it works fine as I managed to get oscillations without consuming much.


To calculate power, I used the mean value of V * I going through the C1 capacitor (as it gave me the biggest power). Thing is, I don't know why in every books and on the internet they use P = R * I² as V and I are not even in phase...
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« Last Edit: Jul 8th, 2013, 4:32am by Forum Administrator »  
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Kevin Aylward
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Re: Theory to design a pierce oscillator (tough question)
Reply #11 - Jul 8th, 2013, 10:25am
 
Yes, that’s the resistor. As I noted, the “Pierce resistor” was invented as a design technique to get better frequency stability. It is not necessary, but if cost and space are not an issue, it is better to use it. Yes, it can also be used to control drive current, but that was not why the topology was invented. A starter value is 1k. Depends on frequency. Too large and it will stop oscillating. To get xtal power, it’s the power in the esr resistor of the xtal. That is, the model has a L C1 ESR in series and a parallel Co. The voltage and current are always in phase in a resistor, irrespective of what else it may be attached to. In checking oscillation in simulation, use a value of esr up to 3 times the rated esr. This is because start up esr can be higher than running esr. Do not use the esr as a way of lowering Q for transient runs. Q lowering is always done by way of changing c1. Most spices will allow a equations with parameter to do this easily.
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Kevin Aylward
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Pictou
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Re: Theory to design a pierce oscillator (tough question)
Reply #12 - Jul 9th, 2013, 12:30am
 
Thank you very much for your help, I understand now better why some schematic includes the resistor and some doesn't.

About the resistor power, I still have a problem.  If I can do P = R*I² I can also do : P = U²/R. So I decided to plot the U across the R.

Please find below V1 and V2 (Voltage level across the resistance) and then V1 - V2 AND the current.

http://www.hostingpics.net/viewer.php?id=640763voltageacrossESR.png


http://www.hostingpics.net/viewer.php?id=723730voltagedropandcurrent.png


As you can see on the first image, even the voltage is not in phase after crossing the resistor.

In the second screenshot, I made V1-V2 and plotted the current crossing the R resistance and it's not in phase either. So, I'm kind of lost... My guess is, it's not in phase because L and C are in series with R. Maybe I'm wrong, could you help me again please? Thank you again.
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Kevin Aylward
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Re: Theory to design a pierce oscillator (tough question)
Reply #13 - Jul 9th, 2013, 10:57am
 
The voltagedropandcurrent.png show the current and voltage perfectly in phase!
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Pictou
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Re: Theory to design a pierce oscillator (tough question)
Reply #14 - Jul 10th, 2013, 12:26am
 
Actually they are not, I just made the trace thick so it wouldn't be too ugly with my screenshot. I should have let it as "fine trace".

If you look at the 0 level for instance, you can see that the green curve is too much on the right (or the red one is too much on the left).

All I'm saying is, I would like to use P = R*I², but for now, I can't justify why it would be correct. Even the first trace doesn't make sense to me, the two voltages are not in phase...

In all books I read and on the internet, almost everyone is using P = R*I² but they don't justify why. There is a phase to consider in all these signals but they don't seem to care.
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