# Distribution of the frequency drift of oscillators

I'm investigated the effects of oscillator inaccuracy on timing related applications and I'm therefore interested in the distribution of the frequency drift for common oscillators such as LC or Crystal-based ones. Surely they are different for each oscillator and each oscillator type but I'm just interested in the rough characteristics of the distribution.

Assuming e.g. that the frequency drift can be expressed as $$\tilde{f} = f * ( 1 + n_f )$$ where f is the nominal frequency, I'm looking for the distribution of n_f. Does it e.g. make sense to assume it to be Gaussian?

Or alternatively for the estimate t_hat of the real time t and assuming it to be given by $$\hat{t} = t*(1 + n_t) + d$$ where d is the clock offset, I'm looking for the characteristics of n_t.

Sketch of scenario:

I have an oscillator/clock (very small, i.e. < 1.5 mm^3, and low energy consumption) that I want to operate for, say, 48 hours.

Before operation and after operation I synchronize with an atomic clock. I'm interested in the inaccuracy of the time measurements in the operation time if I measure time intervals, which are in order of a second, and the absolute time when I started to measure that interval.

What are the most relevant error components that lead to the inaccuracy in the two timing measurements (the interval time and the absolute time)?

• The main frequency drift is usually temperature related and not gaussian I would say. Commented Oct 18, 2016 at 18:20
• L if made from copper with a PTC unless core is NTC , cap can be ceramic with opposite, but poor choice compare to 1k x better stability of low cost ceramic MEMs and quartz resonators. Commented Oct 18, 2016 at 19:38

Start with your requirement for ppm error frequency and desired operating temperature range then define or choose the resonator that meets this criteria. The most common AT cut quartz crystal comes with different angle cuts that affect the 3rd order temp vs ppm error curve and can be chosen over a very wide temp range or a narrow range.

I once computed the 3rd order coefficients to make a TCXO for under \$1 using testers to bin XTAL curves in 10 seconds and varicaps in 1 second to match with a thermistor to computed DAC Vf to compensate for 3rd order tempco over -40 to +70'C within 1ppm . Now you can buy the 2ppm TCXO oscillators for a few bucks or less.

Room temp watch crystals use a parabolic tempco crystal centred at 28'C

Oscillator frequency drift is NOT a random event, so ditch all of the statistical analysis tools.

There are a number of causes of oscillator frequency drift depending on the specific type of oscillator and its circuit topology. E.g. Colpitts, Hartley, crystal, RC, resonant cavity, ceramic resonator, etc.

The biggest cause is always operating temperature change regardless of the type of oscillator. The next biggest cause is usually supply voltage related. Oscillators usually have an amplifier at their core. Amplifier maladies such as gain drift, gain instability, noise, saturation effects, etc. are also a big cause of "drift".

Aging of components is another factor, as are environmental effects such as humidity and vibration.

It's all about Cause & Effect. The best analyses of oscillator frequency drift usually start with building a detailed and realistic mathematical or P-Spice model of the oscillator circuit. This involves a bit of detective work to identify the actual circuit-specific factors which are capable of changing the output frequency of the oscillator. Usually there are many such factors and sorting them out can be quite confusing - to the point that they appear to be random.

• Thanks a lot for your comment. I have edited the main post with a sketch of the scenario and would appreciate if you could help me also with the more detailed question. Also I have another question: Wouldn't it make sense to model it as a random event if I say that I don't know all the variations that lead to the inaccuracy, i.e. I have some statistical assumptions on the temperature change during the operation and assume some distribution how the supply voltage fluctuates etc. Commented Oct 19, 2016 at 8:00
• The inclusion of the "Sketch of Scenario" provides a much better perspective on your problem. It seems you aren't trying to improve your clock device, so much as you are trying to characterize or model it. Correct? So perhaps a mathematical approach may be in order here. As far as collecting data from the clock-under-test, you could probably use both clocks to drive divider circuits, then read the dividers with a micro. The micro could output the delta time data to an Excel file or similar. Is this an academic exercise, or a real world problem you are solving? My atomic clock knowledge is 0 Commented Oct 19, 2016 at 11:31