Counter/timer accurately calibrates frequency
A high-resolution counter/timer with an external frequency reference lets you calibrate frequency to an uncertainty of 10–12 or better.
Staffan Johansson, Pendulum Instruments, Stockholm, Sweden -- Test & Measurement World, 5/1/2002
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An accurate way to calibrate a frequency source is to use a high-resolution counter/timer to measure the beat frequency, or phase difference, between the source to be calibrated and a reference such as a cesium or a GPS-controlled rubidium standard. The method measures how fast the phase difference between the unknown source and the reference increases or decreases. One requirement of this method is that the source and the reference have the same nominal frequency.
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| Figure 1. A counter/timer can measure TIE based on the zero-crossing times of both the unknown frequency and the reference frequency. |
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| Figure 2. Measuring TIE at two points lets you calculate the frequency difference between two signals with the same nominal frequency (Df/f = DTIE/Dt). |
The easiest way to determine this phase difference is to measure the time interval, or time interval error (TIE), between zero crossings of the two signals with a counter/timer (Figure 1). By measuring the TIE at two points several seconds apart, you can calculate a relative frequency difference (Äf/f), as shown in Figure 2:
Äf/f =
TIE2 – TIE1/t 2–t 1
To illustrate, I measured Äf for an unknown 10-MHz signal with moderate short-term frequency fluctuations using an extremely stable 10-MHz signal from a Fluke 910R, a GPS-controlled frequency reference with a built-in rubidium oscillator. I also used a Fluke PM6681 counter/timer with 50-ps resolution to measure the TIE every 20 s. The results are shown in Table 1.
| Time t (s) | TIE (ns) | ÄTIE (ps) | Ät (s) | ÄTIE/Ät = Äf/f |
| t0 = 0 | +4.55 | |||
| t1 = 20 | +4.75 | 200 | 20 | 1.0 x 10–11 |
| t2 = 40 | +4.99 | 440 | 40 | 1.1 x 10–11 |
| t3 = 60 | +5.23 | 680 | 60 | 1.1 x 10–11 |
| t4 = 80 | +5.49 | 940 | 80 | 1.2 x 10–11 |
| t5 = 100 | +5.72 | 1170 | 100 | 1.2 x 10–11 |
You can see that a measurement taken over 100 s shows that the two frequencies differ by 1.2 x 10–11. You should note, though, that even the 20-s reading gives a good indication of the difference. Knowing the difference, you can calibrate the unknown source either by documenting its value or by adjusting it to reduce the difference if that facility exists.
No calibration is complete without knowing the measurement uncertainty. So, what's the uncertainty of this measurement? There are four steps to attaining measurement uncertainty:
- Compensate the result for any known systematic errors.
- Express the remaining uncertainty factors as standard deviations (such as rms or 1-ó values).
- Take the square root of the sum of the squares of all uncertainty contributions to get a standard uncertainty.
- Multiply the standard uncertainty by 2 to reduce the risk of a measurement value being outside the area of uncertainty. For a normal distribution, 2-ó means 96% of all measurements will be within the given area of uncertainty.
In my example, the counter/timer measures TIE with 50-ps resolution and measures t with 100-ns resolution. The relative frequency difference is ÄTIE/Ät. The TIE uncertainty at t 0 and t 5 is resolution (50 ps) plus the same systematic error (t syst) in both TIE measurements. Because you subtract TIE(t 0) from TIE(t 5), the calculation eliminates the systematic error (t syst), and only the uncertainty of the resolution remains in ÄTIE.
The uncertainty of the time difference ÄT = t 5–t 0 (100 ns in 100 s) is negligible. Therefore, the total measurement uncertainty becomes:
The relative frequency difference (Äf/f) is thus (1.2 ±0.14) x 10-11 in the example.
By measuring ÄTIE over longer periods (Ät), you can improve resolution and reduce the measurement uncertainty even more. Other sources of error in the measurement, such as the counter/timer's timebase oscillator stability and trigger errors due to noise, are negligible compared to the resolution of the TIE measurement.
Note that the calculation in the example gives the measurement uncertainty of the difference between the two frequencies. In order to determine the accuracy of the frequency being calibrated, you have to consider the accuracy of the reference itself. For a Fluke 910R, deviations appear in the 12th digit. For cesium standards, deviations appear in the 13th digit.
Direct calibration is simplerIf you don't need such a high-accuracy method for calibrating frequency, you can simply connect the signal you want to calibrate to an already-calibrated frequency counter and read the result. The big advantage of this method is that you can calibrate any frequency within the range of the counter. The disadvantage of using a direct reading is a limit of approximately 10 digits of measurement accuracy even if you use the highest-resolution counters with a built-in rubidium reference. The frequency measurement time—averaging period—is normally limited to 10 s.
To find the measurement uncertainty of direct frequency calibration, you must add internal-timebase error, trigger error due to signal noise and internal noise, and systematic timing error in the counting circuits. Noise errors normally show up in the 9th or 10th digit of a reading, systematic errors usually occur in the 10th or 11th digit.
As an example, if both resolution and timebase uncertainty are 1 x 10–7 rms, and the other sources of errors are negligible, then the total uncertainty will be:
Today's counters have a typical resolution of 9 or 10 digits (in some cases up to 11 digits) for a measuring time of 1 s. It's the built-in timebase oscillator that limits the accuracy (see "Accuracy versus resolution in counter/timers," p. 24).
Theoretically, you could use an external cesium standard or a GPS-controlled rubidium reference to obtain 11- to 12-digit measurement uncertainty, but in practice, trigger noise and systematic errors in the counter usually limit any improvement to 9 to 10 digits.
Even with the best frequency counters, you won't beat 10-digit measurement uncertainty using direct-frequency calibration. To achieve anything better, you must use the phase comparison method.
Note: A version of this article previously appeared in Test & Measurement Europe.
| Author Information |
| Staffan Johansson is marketing manager with Pendulum Instruments, Stockholm, Sweden, and has 20 years experience in the test and measurement business. He has presented many seminars and lectures on time and frequency analysis, and he has written books on measurement techniques. |
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