[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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