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 10¹⁰ is quite easily achieved with these instruments. It is possible that some short term stability might be achievable to parts in 10¹¹. It is quite difficult to measure frequency below about a part in 10¹⁰. 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 10¹⁰.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 10¹² 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 10¹²?

With the above cautions in mind, what is a part in 10¹², and how does the Z3801A set the oscillator output frequency with that resolution? A part in 10¹² means literally to divide 10 MHz (10⁷) by 10¹². But, first consider a part in 10⁷ of 10 MHz or 1 Hz. Now it is more apparent that a part in 10¹² 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 10¹² 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 10⁷ 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 10¹¹! Okay, so divide the last numbers by 10⁴ to get from a part in 10⁷ to a part in 10¹¹.

We get an HP 10811 oscillator sensitivity of:

1 x 10^-11 Hz per 1 x 10^-4 volts, or: a part in 10¹¹ 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 2¹⁶ combinations of the range from –5 volts to +5 volts. The full voltage range is 10 volts. The number of fractional settings is 1/2¹⁶, 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 10¹¹ of the oscillator output frequency. So, by algebraic ratio, a change of 152 microvolts causes an output frequency of 1.52 parts in 10¹¹.

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 10¹¹! 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 10¹² (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