[time-nuts] Checking accuracy of Rubidium standards

Bruce Griffiths bruce.griffiths at xtra.co.nz
Mon Nov 10 03:59:32 UTC 2008


Neville Michie wrote:
> Bruce, how does the three cornered hat scheme work?
>
> If I had two LPRO Rubidium oscillators and a TBOLT GPSDO, and I  
> divided each of them down to
> 100KHz, then I could compare pairs of them with D latches and record  
> 3 different analogue
> signals of phase difference.
> One of the LPRO oscillators is in an oven to remove ambient  
> temperature influence.
> If I ran them for several weeks and logged the signals every 10 minutes,
> what could I expect to recover from the data and how would I apply  
> the 3 cornered hat scheme?
> I ask this question because this is about where my building program  
> is taking me.
> cheers, Neville Michie
>
>   
Neville

When the fluctuations of 3 quantities are independent then comparing 2
of them the individual variances of the add:

VAR(1,2) = VAR(1) + VAR(2)
VAR(1,3) = VAR(1) + VAR(3)
VAR(2,3) = VAR(2)+ VAR(3)

Where
VAR(1,2) denotes the variance of the fluctuations in the difference
between quantities 1 and 2.
VAR(1,3) denotes the variance of the fluctuations in the difference
between quantities 1 and 3.
VAR(2,3) denotes the variance of the fluctuations in the difference
between quantities 2 and 3.
VAR(1) denotes the variance of the quantity 1.
VAR(2) denotes the variance of the quantity 2.
VAR(3) denotes the variance of the quantity 3.

Thus
VAR(1) = (VAR(1,2) + VAR(1,3) - VAR(2,3))/2
VAR(2) = (VAR(1,2) + VAR(2,3) - VAR(1,3))/2
VAR(3) = (VAR(1,3) + VAR(1,3) - VAR(1,2))/2

The same results hold for ADEV (used for characterising the stability of
oscillators as  variance of the phase is divergent for oscillators).

ADEV(1,2) = ADEV(1) + ADEV(2)
ADEV(1,3) = ADEV(1) + ADEV(3)
ADEV(2,3) = ADEV(2) + ADEV(3)

Where
ADEV(1,2) denotes the Allen variance of the fluctuations in the phase
difference between oscillators 1 and 2.
ADEV(1,3) denotes the Allen variance of the fluctuations in the phase
difference between oscillators 1 and 3.
ADEV(2,3) denotes the Allen variance of the fluctuations in the phase
difference between oscillators 2 and 3.
ADEV(1) denotes the Allen variance of the phase fluctuations of
oscillator 1.
ADEV(2) denotes the Allen variance of the phase fluctuations of
oscillator 2.
ADEV(3) denotes the Allen variance of the phase fluctuations of
oscillator 3.

Thus
ADEV(1) = (ADEV(1,2) + ADEV(1,3) - ADEV(2,3))/2
ADEV(2) = (ADEV(1,2) + ADEV(2,3) - ADEV(1,3))/2
ADEV(3) = (ADEV(1,3) + ADEV(1,3) - ADEV(1,2))/2


Note it is essential to measure the relative phase fluctuations between
all 3 oscillator pairs simultaneously.

The limitation is that the oscillators should all have similar ADEV.
If the calculations assign negative values to one or more of the
individual variances then the phase fluctuations for the individual
oscillators may be correlated or at least one of the oscillators may be
much quieter than the others.

Eventually common environmental variations such as temperature pressure
and humidity fluctuations introduce correlations invalidating the above
simplified analysis.
However in this case the theory has been extended to include the effect
of correlations which are adjusted to ensure that the calculated
individual variances are positive definite.

Bruce




More information about the Time-nuts_lists.febo.com mailing list