# [time-nuts] Thermal impact on OCXO

Scott Stobbe scott.j.stobbe at gmail.com
Sun Nov 13 22:59:08 EST 2016

```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.

```