[time-nuts] Excel logarithmic function (was Thermal impact on OCXO)

[Bob Camp]

Hi


> On Nov 23, 2016, at 11:21 PM, Scott Stobbe <scott.j.stobbe at gmail.com> wrote:
> 
> Hi Lars,
> 
> There are a few other pieces I have yet to fully appreciate. One of which
> is that Aln(Bt+1) isn't a time-invariant model. In the most common case
> (for the mfg) the time scale aligns with infancy of the OCXO, when it's hot
> off the line. However after pre-aging, perhaps some service life, what time
> reference is best? Sometime I will try adding an additional parameter for
> infancy time and see how that goes.


The biggest challenge is to take out the “early stuff”. One approach is to fit 
the same equation twice with the time constant restricted to a range on each.
For most OCXO’s (90%) the equation when fit early represents an upper limit
to the drift. You might get a another element that comes in and is apparent 
after a year or two. It might be replaced by another element after five or ten years.
They generally (~80%) represent a change in sign (negative drift vs positive). 

If you look at the “other 10%” some have really poor aging and are not shipped. 
Some are very erratic and simply can not be fit. Some of the 90% are fit with a
“upper limit” because they exhibit no measurable aging over the 30 days (or whatever)
of testing. 

If you take the bad aging (out of spec) parts out of the pile, those are the ones
with the best fit. They have very pretty curves and they stick to those curves
for a *long* time. They have a single dominant cause for their aging ( = the defect). 
The rest of the parts have all of the causes bashed down by the process so that
over a 20 or 30 year span, there probably is no single dominant cause. 

Bob

> 
> A fit of the full ten year data-set, attached in the two plots
> "Lars_10Year.png", "Lars_10Year_45Day.png".
> 
> I would agree to your description of 1/sqrt(t) aging for the first 1000
> days, but sometime after, it follows 1/t. Attached is plot of age rate
> "Lars_AgeRate.png". You can see during the first 1000 days the age rate
> declines at 1 decade for 2 decades time indicating t^(-1/2), but eventually
> it follows 1/t.
> 
> On Wed, Nov 23, 2016 at 3:57 PM, Lars Walenius <lars.walenius at hotmail.com>
> wrote:
> 
>> Hi Scott.
>> 
>> 
>> 
>> Here is a textfile with data for the 10 years (As in the graph 2001-2011).
>> 
>> 
>> 
>> Also the ln(bt+1) fit, as Magnus said, has the derivate b/(b*t+1) that
>> with b*t >>1 is 1/t. But my data has the aging between 1 and 10 years more
>> like 1/sqrt(t) If I just have a brief look on the aging graph.
>> 
>> 
>> 
>> Lars
>> 
>> 
>> 
>> *Från: *Scott Stobbe <scott.j.stobbe at gmail.com>
>> *Skickat: *den 19 november 2016 04:11
>> 
>> Hi Lars,
>> 
>> 
>> 
>> I agree with you, that if there is data out there, it isn't easy to find,
>> 
>> many thanks for sharing!
>> 
>> 
>> 
>> Fitting to the full model had limited improvements, the b coefficient was
>> 
>> quite large making it essentially equal to the ln(x) function you fitted in
>> 
>> excel. It is attached as "Lars_FitToMil55310.png".
>> 
>> 
>> 
>> So on further thought, the B term can't model a device aging even faster
>> 
>> than it should shortly after infancy. In the two extreme cases either B is
>> 
>> large and (Bt)>>1 so the be B term ends up just being an additive bias, or
>> 
>> B is small, and ln(x) is linearized (or slowed down) during the first bit
>> 
>> of time.
>> 
>> 
>> 
>> You can approximated the MIL 55310 between two points in time as
>> 
>> 
>> 
>> f(t2) - f(t1) = Aln(t2/t1)
>> 
>> 
>> 
>> A = ( f(t2) - f(t1) )/ln(t2/t1)
>> 
>> 
>> 
>> Looking at some of your plots it looks like between the end of year 1 and
>> 
>> year 10 you age from 20 ppb to 65 ppb,
>> 
>> 
>> 
>> A ~ 20
>> 
>> 
>> 
>> The next plot "Lars_ForceAcoef", is a fit with the A coefficient forced to
>> 
>> be 2 and 20. The 20 doesn't end-up fitting well on this time scale.
>> 
>> 
>> 
>> Looking at the data a little more, I wondered if the first 10 day are going
>> 
>> through some behavior that isn't representative of long-term aging, like
>> 
>> warm-up, retrace (I'm sure bob could name half a dozen more examples). So
>> 
>> the next two plots are fits of the 4 data points after day10, and seem to
>> 
>> fit well, "Lars_FitAfterDay10.png", "Lars_1Year.png".
>> 
>> 
>> 
>> If you are willing to share the next month, we can add that to the fit.
>> 
>> 
>> 
>> Cheers,
>> 
>> 
>> 
>> On Fri, Nov 18, 2016 at 1:26 PM, Lars Walenius <lars.walenius at hotmail.com>
>> 
>> wrote:
>> 
>>> 
>> 
>>> Hopefully someone can find the correct a and b for a*ln(bt+1) with
>> 
>> stable32 or matlab for this data set:
>> 
>>> Days ppb
>> 
>>> 2       2
>> 
>>> 4       3.5
>> 
>>> 7       4.65
>> 
>>> 8       5.05
>> 
>>> 9       5.22
>> 
>>> 12     6.11
>> 
>>> 13     6.19
>> 
>>> 25     7.26
>> 
>>> 32     7.92
>> 
>> 
>> 
> <Lars_10Year.png><Lars_10Year_45Day.png><Lars_AgeRate.png>_______________________________________________
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