[time-nuts] Thermal impact on OCXO

Scott Stobbe scott.j.stobbe at gmail.com
Wed Nov 16 18:48:55 EST 2016


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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