On Fractional
Parts and the Z3801A GPS Frequency Standard

The Z3801A is a 10 MHz standard. We discuss with great interest
and excitement, incredibly small errors in frequency regarding
these standards. Before, only a few among us had the luxury of
a surplus Cesium or Rubidium standard. Never before has the amateur
community had such relatively easy access to such an instrument
at a cost much less than most of today’s commercial amateur
radios.

It appears that absolute frequency accuracy to some parts in 10^{10}
is quite easily achieved with these instruments. It is possible
that some short term stability might be achievable to parts in
10^{11}. It is quite difficult to measure frequency below
about a part in 10^{10}. One complication is that long
gate times, and / or averaging of many readings is required. And,
if measurements require “long” measuring windows, then
can anything be said about the frequency to this resolution during
short times within that window?

Even
the most sophisticated counters that actually measure time and
then calculate frequency, such as the HP 53132A and Stanford SR620,
still require long gate times to “see” below parts in
10^{10}.This is because the counters can only resolve
to around 100 to 200 picoseconds. They work by measuring the time
from a zero crossing to the end of the “gate” very precisely.
That time gets divided by the number of zero crossings. But, the
catch is, to resolve the very last increment of time at the gate
closing (a fraction of the period of their internal clock) they
use an analog time to voltage circuit!!!. We usually never notice
the analog errors, because we are generally well within the counter
manufacturer’s specs. But, trying to measure down to 10’s
of micro Hz pushes the envelope! Here’s why:

First
consider an error of 10^{-12} in 10 MHz over one cycle
of 10 MHz. The period of one cycle is 100 nanoseconds, or 100
x 10^{-9} seconds. A part in 10^{12} of that is
100 x 10^{-21} seconds!!! But, look what happens if we
consider a long period of cycles (the counter gate time).

In
100 seconds, the time is 100 x 10 million x 10^{-21} (the
error in seconds for one cycle) or about 100 picoseconds longer
or shorter than it should be at absolute 10 MHz.

So,
even 100 seconds is inadequate for most counters to measure to
10 micro Hz at 10 MHz. And, there are other measurement errors
too!

Also, remember that the principle of operation of disciplined
oscillators is that highly accurate control is obtained by considering
very long periods of the one pulse per second (1 PPS) signal from
the GPS receiver. Therefore, the exact output frequency at any
given moment (say on the order of a cycle to 10 cycles of the
10 MHz signal) depends crucially on the ‘jitter’ of
the 10811 oscillator and noise on the Vefc control line.

**What is
a part in 10**^{12}?

With the above cautions in mind, what is a part in 10^{12},
and how does the Z3801A set the oscillator output frequency with
that resolution? A part in 10^{12} means literally to
divide 10 MHz (10^{7}) by 10^{12}. But, first
consider a part in 10^{7} of 10 MHz or 1 Hz. Now it is
more apparent that a part in 10^{12} is 5 orders of magnitude
smaller than 1 Hz or 10 micro Hz! That is 1 x 10^{-5}
Hz.

Note
that when discussing “a part in” the exponent is positive
because we are referring to a fractional element of a total thing
(here 10 MHz). But, when discussing a fractional part, the exponent
(depending on the exact context) is usually negative. For example,
at 10 MHz a change of 10 micro Hz is a frequency change of 10^{-12},
or could be said to be a change of a part in 10^{12.}

What
does a part in 10^{12} mean in regard to the HP 10811 double walled
oscillator?

The designed output frequency of the HP 10811 oscillator is 10
MHz. It is adjusted by a very fine control voltage. By considering
the range of control voltage and the designed frequency change
that results, the frequency change for an incremental change in
control voltage can be determined.

A control voltage change from –5 volts to +5 volts causes
a designed frequency change of just greater than, or equal to
10^{-6} (10 Hz) in output frequency. This means that at
a Vefc of –5 volts, the output frequency is the nominal frequency
of the oscillator is plus 5 Hz, and at +5 volts, the output frequency
is the nominal frequency of the oscillator minus 5 Hz. This means
that the output changes in frequency by 1 Hz per volt. 1 Hz is
a part in 10^{7} of 10 MHz, so you could also say that
the sensitivity is about 1 x 10^{-7} per volt.

But, now we want to talk about parts in 10^{11}! Okay,
so divide the last numbers by 10^{4} to get from a part
in 10^{7} to a part in 10^{11}.

**We get
an HP 10811 oscillator sensitivity of:**

1 x 10^{-11}
Hz per 1 x 10^{-4} volts, or: a part in 10^{11}
per 100 microvolts

Now, getting to the discussion of the Z3801A in particular, what
if we wanted to control the range with a 16 bit DAC? If the Z3801A
directly controlled Vefc with the 16 bit DAC (it doesn’t)
then one could get 2^{16} combinations of the range from –5
volts to +5 volts. The full voltage range is 10 volts. The number
of fractional settings is 1/2^{16}, or 1/65,536. 10 volts / 65,536
= 152 microvolts per bit change.

This
means that a 1 LSB (least significant bit) change in DAC voltage
would make for a 152 microvolt change in Vefc if the DAC output
was directly connected to the hp Vefc input. Now, we already said
that a Vefc change of 100 microvolts causes a frequency change
of 1 part in 10^{11} of the oscillator output frequency.
So, by algebraic ratio, a change of 152 microvolts causes an output
frequency of 1.52 parts in 10^{11}.

The Z3801A analog section following the DAC output of –5
volts to +5 volts attenuates the output by a gain of .662. This
means that out of the attenuator, a 1 LSB change is no longer
152 microvolts, but rather 101 microvolts. This value, for all
practical purposes is a part in 10^{11}! Nothing comes
for free. Here, the trade off is that the Z3801A can no longer
make use of the full control range of –5 volts to +5 volts.

After an offset is introduced, the Z3801A control range is from
–2.06 volts to +4.56 volts. Clearly the available range is
sufficient for the production units.

Looking
at a graph of voltage control plotted against frequency change
for the double walled oscillator, you can see that it is non-linear,
as expected because of the non-linear varactor control element.
The black fine line is the entire range. The red line is the part
of the curve the Z3801A uses. Apparently this range was chosen
for its symmetry about the 10 MHz output frequency (0 on the y
axis). The blue part of the curve is the general area where most
of our Z3801A receivers seem to be operating. Individual oscillators
will vary, but not surprisingly the blue area is near absolute
10 MHz.

One
could consider adding a x10 attenuator after the INA105 stage
to make each DAC LSB weigh about a part in 10^{12} (the resulting
Vefc control range would be from -.21V to +.46V) But, the loop
may no longer be stable, and the resultant dynamic characteristics
may or may not be desirable. An experiment for another day!

©
2002 Joe Geller,
KO2Y