Crystal Oscillators -- accuracy versus temperature

I have done quite a lot of desk research and spoken to a few manufacturers, but I am still not clear about the specifications for crystal oscillators and I hope someone can help me out. It is a multi-part question.

1) The accuracy of an XO is usually specified as plus or minus at a given temperature. I assume that a specific oscillator, if measured, would have a specific accuracy at a specific temperature on a specific occasion. This plus or minus might refer to manufacturing tolerance for different oscillators. Or it might refer to changes of accuracy (retrace) of a specific oscillator when tested on different occasions. Or both. Any idea which it is?

2) The accuracy of an oscillator varies with temperature. The variation is a parabolic coefficient. I have usually seen this shown as negative. The oscillator loses in frequency as the temperature decreases or increases. It never gains, against its nominal frequency. Is that correct? Does a crystal oscillator only lose, and never gain, with changes in temperature away from the turnover temperature?

3) If we put sample variation and temperature variation together, then we would have, for example, +-20ppm at 25 degrees C, and -100 (+-20) ppm at 0-40 degrees C i.e. between -80 and -120 ppm. The parabolic coefficient also has a plus or minus range, so we might have -70 to -130 ppm as the range of variation.

4) Or the manufacturing sample tolerance might change at different temperatures. Samples might be +-20ppm at 25 degrees C, but they might be +-100ppm at 0-40 degrees. Does the sample accuracy stay the same at different temperatures, or does it vary more widely with temperature? Does anyone know?

5) I am aware that TCXO's change this behaviour. I am not asking how to achieve better accuracy. I am trying to understand the behaviour of a standard XO with changes in temperature.

Thanks
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 Have you seen this: http://www.leapsecond.com/pdf/an200-2.pdf There are many different cuts for crystals and you can't generalize so simply about freq vs temp.
 Thank you very much for the reply, much appreciated. Yes, I had seen that paper, but I had missed the part about the temperature effect of different cuts. Here is the specification of a crystal oscillator: Seiko VT-200-F. - "frequency tolerance" +- 20 ppm - turnover temperature 25 degrees C +- 5 - parabolic coefficient -3 (+- 0.8) x 10 -8 / degree C2 Does the frequency tolerance refer to a manufacturing tolerance between different samples? Or is it a variation of accuracy within a given sample? I understand that different cuts will have different effects, and so this is just an example of one type. With this one I can see that the temperature has a significant effect, but it does not seem to be that significant. For example the parabolic curve indicates that even at 0 degrees, the loss of accuracy is less than double the normal inaccuracy.

Crystal Oscillators -- accuracy versus temperature

 Quote by Anthony_ Does the frequency tolerance refer to a manufacturing tolerance between different samples? Or is it a variation of accuracy within a given sample?
I believe it is the manufacturing tolerance. I don't understand what you mean by "variation of accuracy with a given sample."
 Recognitions: Science Advisor You are probably aware that crystals are small chunks of quartz mounted in a small box while crystal oscillators are a complete circuit which oscillates if power is applied and are controlled by the crystal. The crystal itself has a frequency which is specified according to a standard temperature (maybe 20°C) and a standard capacitance. If the crystal is placed in an oscillator, then the frequency should be correct at that temperature and capacitance. Exactly how accurately that frequency actually matches the frequency that was ordered depends on how carefully the crystal dimensions were trimmed at the factory. If you pay more, they make them more carefully. Having got your crystal, you can modify the frequency to some extent by the oscillator circuit you put it in. If you bought a cheap crystal, you may be able to pull it onto the right frequency by adding more or less than the standard capacitance across it, for example. Although 20 parts per million doesn't sound like much, the error at 20 MHz could be as much as 400 Hz which would be almost useless in some applications. Some crystals are designed to be operated in an oven which is a small heating chamber operated at about 50°C. They are placed in an oscillator circuit, brought up to temperature and then the capacitance across them is adjusted to give the right frequency. After that, the device is relatively free of frequency variations due to ambient temperature.
 Thank you both for your replies. Yes, I am aware that there is a manufacturing tolerance for the crystal, and then an optimisation of the circuitry for the oscillator. I had assumed that the "frequency tolerance" refers to manufacturing variation across samples, but I was looking for some confirmation of that. If that is the case, then it suggests that you could test each one in a batch and a) reduce the tolerance by throwing some away or b) tune the circuitry to compensate. If tuning is easy, then it could be cheaper than using a TXCO instead. There's a practical aspect (which might be rather obscure!). If you use one clock to time the start and stop of an event, then the elapsed time will be subject to temperature change during the experiment. But if you use two clocks of the same type, then it will be subject to sample variation (which you can test) but not temperature variation, which you may not know but which should be the same for both. The parabolic coefficient is interesting, because most documents I have read imply that temperature variation is considerable, whereas the coefficient published in the spec for the Seiko shows that, for this XO, over a normal range or 0-40 degrees, it adds less than the sample variance. I know this is material, but so is the sample variance. It suggests to me that the most important thing for an XO is to calibrate it occasionally; and then secondly to keep it at roughly room temperature. If you do that, it suggests that the accuracy will be in the low single figure ppm (based only on small temperature, ageing, voltage variations). Thanks again for pointing me to the graphs for the different crystal cuts. From that its clear that you can't assume a temperature behaviour. You need to see a curve, or perhaps a published accuracy over a given temperature range.
 Recognitions: Science Advisor If you have two clocks that are equally bad at keeping time, then surely you would get equally bad results for each one, although they will both agree with the result. The problem with using inaccurate crystals is that you may not be able to keep them oscillating with the extra capacitance you would have to add to get them on frequency.
 Thanks, yes. I am referring to the variation you should expect within the specs for a given crystal oscillator. In other words, if you buy two Seiko VT crystal oscillators, what behaviour will you actually see? You are right about the two clocks example. It is a bad example.
 Recognitions: Science Advisor When a producer makes a large number of crystals he will say that these crystals are not the same as the nominal frequency but they vary from it by no more than some figure he quotes. So, if you have no option to measure a lot of them and pick out the very good ones, you have to decide if you want to accept a random sample and if it would still be good enough if you we're unlucky enough to get one at the extreme limit of the quoted error range.
 Agreed, although you would never know unless you tested it.
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