Time and frequency measurements
A direct-counting counter with a 1 s gate measures 10 Hz with a 10% error. A reciprocal one, with one part in ten billion: the difference lies in what is counted.
01The best-measured quantity
Time is the quantity that is measured with the greatest accuracy. Since 1967, the second of the International System has been defined by the radiation of cesium 133: 9,192,631,770 periods of a transition of its atom. The cesium atomic clocks of the national laboratories realize it with uncertainties of 10⁻¹⁵ or less, and that reference reaches measurement laboratories through radio signals, networks and satellites.
| Reference | Typical accuracy | Use |
|---|---|---|
| Ordinary crystal (XO) | 10 to 50 ppm | Microcontrollers, clocks |
| Temperature-compensated crystal (TCXO) | 0.5 to 2 ppm | Phones, GPS receivers |
| Oven-controlled crystal (OCXO) | 0.001 to 0.1 ppm | Laboratory instruments |
| Rubidium | 10⁻¹⁰ to 10⁻¹¹ | Base stations, laboratories |
| GPS-disciplined oscillator | 10⁻¹² over long averages | Traceable laboratory reference |
Besides accuracy, stability matters: how much the frequency changes over a given interval. It is expressed with the Allan deviation for each averaging time; a crystal is very stable over one second but drifts over hours, and GPS the other way around: that is why they are combined in disciplined oscillators.
02The direct-counting counter
The simplest counter opens a gate for a time \( T_g \), derived from its timebase, and counts the edges of the signal. Since the gate is not synchronized with the signal, one edge more or one less can get in: the ±1 count error. The relative resolution is:
03The reciprocal counter
The reciprocal counter synchronizes the gate with the edges of the signal: it counts an integer number of periods of the input and, at the same time, measures with the timebase, and a high-resolution interpolator, the exact time they lasted. The frequency is the quotient. Its resolution no longer depends on the input frequency, but on the time resolution \( t_{res} \) of the interpolator:
No method outdoes its timebase: the relative error of the reference adds directly to the result. That is why laboratory counters have an external 10 MHz reference input, to share a better standard.
The relative resolution of the two methods as a function of frequency, for the chosen gate time (the reciprocal one with a 100 ps interpolator). The horizontal line is the timebase error; the uncertainty of each method is the sum.
04In the laboratory
Measure a 50 Hz signal with a direct-counting counter and a reciprocal one, with several gates, and compare the scatter of the readings.
Measure the frequency of a generator with the counter free-running and with both instruments locked to the same 10 MHz reference.
Record for an hour the frequency of an ordinary crystal oscillator while warming it with your hand, against a GPS-disciplined reference.
05Common mistakes
- Measuring low frequencies by direct counting. The ±1 count error dominates; it is better to measure the period or use a reciprocal counter.
- Trusting every digit on the display. The timebase limits the accuracy even if the resolution is high.
- Confusing accuracy with stability. An oscillator can be very stable and be off frequency.
- Forgetting the noise at the trigger. Signal noise shifts the edges and adds uncertainty.
06Self-assessment
What resolution does a direct counter give with a 0.1 s gate?
10 Hz.
A counter with a 1 ppm TCXO measures 100 MHz. What is the timebase error?
100 Hz.
Why does the reciprocal counter have the same resolution at any frequency?
Because it measures the time of an integer number of periods, and its resolution depends on the interpolator, not on the frequency.
How many cesium periods define the second?
9,192,631,770.
07Further reading
- Michael A. Lombardi. Fundamentals of Time and Frequency. In The Mechatronics Handbook, CRC Press, 2002; also published by NIST. A clear introduction to standards, stability and traceability.
- Keysight Technologies. Fundamentals of the Electronic Counters. Application Note 200, Keysight. Direct counting, reciprocal counting and time interval measurement.
- David W. Allan. Statistics of Atomic Frequency Standards. Proceedings of the IEEE, 54(2), 1966. The paper that introduced the Allan variance.