[time-nuts] Thermal impact on OCXO

Bob Camp kb8tq at n1k.org
Wed Nov 16 20:06:53 EST 2016


Hi

The issue in fitting over short time periods is that the noise is very much 
*not* gaussian. You have effects from things like temperature and warmup 
that *do* have trends to them. They will lead you off into all sorts of dark
holes fit wise. 

Bob

> On Nov 16, 2016, at 6:48 PM, Scott Stobbe <scott.j.stobbe at gmail.com> wrote:
> 
> A few different plots. I didn't have an intuitive feel for what the B
> coefficient in log term looks like on a plot, so that is the first
> plot. The same aging curve is plotted three times, with the exception
> of the B coefficient being scaled by 1/10, 1, 10 respectively. In hand
> waving terms, it does have an enormous impact during the first 30 days
> (or until Bt >>1), but from then on, it is just an additive offset.
> 
> The next 4 plots are just sample fits with noise added.
> 
> Finally the 6th plot is of just the first 30 days, the data would seem
> to be cleaner than what was shown as a sample in the paper, but the
> stability of the B coefficient in 10 monte-carlo runs is not great.
> But when plotted over a year the results are minimal.
> 
>          A1              A2            A3
>     0.022914       6.8459   0.00016743
>     0.022932       6.6702   0.00058768
>     0.023206       5.7969    0.0026103
>     0.023219       4.3127    0.0093793
>      0.02374       2.8309     0.016838
>     0.023119       5.0214    0.0061557
>     0.023054       5.8399    0.0031886
>     0.022782       9.8582   -0.0074089
>     0.023279       3.7392     0.012161
>      0.02345       4.1062    0.0095448
> 
> The only other thing to point out from this, is that the A2 and A3
> coefficients are highly non-orthogonal, as A2 increases, A3 drops to
> make up the difference.
> 
> On Wed, Nov 16, 2016 at 7:38 AM, Bob Camp <kb8tq at n1k.org> wrote:
>> Hi
>> 
>> The original introduction of 55310 written by a couple of *very* good guys:
>> 
>> http://tycho.usno.navy.mil/ptti/1987papers/Vol%2019_16.pdf
>> 
>> A fairly current copy of 55310:
>> 
>> https://nepp.nasa.gov/DocUploads/1F3275A6-9140-4C0C-864542DBF16EB1CC/MIL-PRF-55310.pdf
>> 
>> The “right” equation is on page 47. It’s the “Bt+1” in the log that messes up the fit. If you fit it without
>> the +1, the fit is *much* easier to do. The result isn’t quite right.
>> 
>> Bob
>> 
>> 
>>> On Nov 15, 2016, at 11:58 PM, Scott Stobbe <scott.j.stobbe at gmail.com> wrote:
>>> 
>>> Hi Bob,
>>> 
>>> Do you recall if you fitted with true ordinary least squares, or fit with a
>>> recursive/iterative approach in a least squares sense. If the aging curve
>>> is linearizable, it isn't jumping out at me.
>>> 
>>> If the model was hypothetically:
>>>    F = A ln( B*t )
>>> 
>>>    F = A ln(t) + Aln(B)
>>> 
>>> which could easily be fit as
>>>    F  = A' X + B', where X = ln(t)
>>> 
>>> It would appear stable32 uses an iterative approach for the non-linear
>>> problem
>>> 
>>> "y(t) = a·ln(bt+1), where slope = y'(t) = ab/(bt+1) Determining the
>>> nonlinear log fit coefficients requires an iterative procedure. This
>>> involves setting b to an in initial value, linearizing the equation,
>>> solving for the other coefficients and the sum of the squared error,
>>> comparing that with an error criterion, and iterating until a satisfactory
>>> result is found. The key aspects to this numerical analysis process are
>>> establishing a satisfactory iteration factor and error criterion to assure
>>> both convergence and small residuals."
>>> 
>>> http://www.stable32.com/Curve%20Fitting%20Features%20in%20Stable32.pdf
>>> 
>>> Not sure what others do.
>>> 
>>> 
>>> On Mon, Nov 14, 2016 at 7:15 AM, Bob Camp <kb8tq at n1k.org> wrote:
>>> 
>>>> Hi
>>>> 
>>>> If you already *have* data over a year (or multiple years) the fit is
>>>> fairly easy.
>>>> If you try to do this with data from a few days or less, the whole fit
>>>> process is
>>>> a bit crazy. You also have *multiple* time constants involved on most
>>>> OCXO’s.
>>>> The result is that an earlier fit will have a shorter time constant (and
>>>> will ultimately
>>>> die out). You may not be able to separate the 25 year curve from the 3
>>>> month
>>>> curve with only 3 months of data.
>>>> 
>>>> Bob
>>>> 
>>>>> On Nov 13, 2016, at 10:59 PM, Scott Stobbe <scott.j.stobbe at gmail.com>
>>>> wrote:
>>>>> 
>>>>> On Mon, Nov 7, 2016 at 10:34 AM, Scott Stobbe <scott.j.stobbe at gmail.com>
>>>>> wrote:
>>>>> 
>>>>>> Here is a sample data point taken from http://tycho.usno.navy.mil/ptt
>>>>>> i/1987papers/Vol%2019_16.pdf; the first that showed up on a google
>>>> search.
>>>>>> 
>>>>>>       Year   Aging [PPB]  dF/dt [PPT/Day]
>>>>>>          1       180.51       63.884
>>>>>>          2       196.65        31.93
>>>>>>          5          218       12.769
>>>>>>          9       231.69       7.0934
>>>>>>         10       234.15        6.384
>>>>>>         25        255.5       2.5535
>>>>>> 
>>>>>> If you have a set of coefficients you believe to be representative of
>>>> your
>>>>>> OCXO, we can give those a go.
>>>>>> 
>>>>>> 
>>>>> I thought I would come back to this sample data point and see what the
>>>>> impact of using a 1st order estimate for the log function would entail.
>>>>> 
>>>>> The coefficients supplied in the paper are the following:
>>>>>   A1 = 0.0233;
>>>>>   A2 = 4.4583;
>>>>>   A3 = 0.0082;
>>>>> 
>>>>> F =  A1*ln( A2*x +1 ) + A3;  where x is time in days
>>>>> 
>>>>>   Fdot = (A1*A2)/(A2*x +1)
>>>>> 
>>>>>   Fdotdot = -(A1*A2^2)/(A2*x +1)^2
>>>>> 
>>>>> When x is large, the derivatives are approximately:
>>>>> 
>>>>>   Fdot ~= A1/x
>>>>> 
>>>>>   Fdotdot ~= -A1/x^2
>>>>> 
>>>>> It's worth noting that, just as it is visually apparent from the graph,
>>>> the
>>>>> aging becomes more linear as time progresses, the second, third, ...,
>>>>> derivatives drop off faster than the first.
>>>>> 
>>>>> A first order taylor series of the aging would be,
>>>>> 
>>>>>   T1(x, xo) = A3 + A1*ln(A2*xo + 1) +  (A1*A2)(x - xo)/(A2*xo +1) + O(
>>>>> (x-xo)^2 )
>>>>> 
>>>>> The remainder (error) term for a 1st order taylor series of F would be:
>>>>>    R(x) = Fdotdot(c) * ((x-xo)^2)/(2!);  where c is some value between
>>>> x
>>>>> and xo.
>>>>> 
>>>>> So, take for example, forward projecting the drift one day after the
>>>> 365th
>>>>> day using a first order model,
>>>>>   xo = 365
>>>>> 
>>>>>   Fdot(365) =  63.796 PPT/day, alternatively the approximate derivative
>>>>> is: 63.836 PPT/day
>>>>> 
>>>>>   |R(366)| =  0.087339 PPT (more than likely, this is no where near 1
>>>>> DAC LSB on the EFC line)
>>>>> 
>>>>> More than likely you wouldn't try to project 7 days out, but considering
>>>>> only the generalized effects of aging, the error would be:
>>>>> 
>>>>>   |R(372)| = 4.282 PPT (So on the 7th day, a 1st order model starts to
>>>>> degrade into a few DAC LSB)
>>>>> 
>>>>> In the case of forward projecting aging for one day, using a 1st order
>>>>> model versus the full logarithmic model, would likely be a discrepancy of
>>>>> less than one dac LSB.
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> <AGING_30DAYS_0p5ppb.png><AGING_30DAYS_0p5ppb_simple.png><AGING_30DAYS_0p5ppb_zoomin.png><AGING_30DAYS_5ppb.png><AGING_30DAYS_5ppb_simple.png><AGING_SCALE_A2.png>_______________________________________________
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