The
concept of calibration implies the comparison of a DUT (Device Under
Test) parameter
(the unknown quantity) with a standard (the known quantity) and documentation
of the difference. Moreover, the concept of adjustment implies that we
affect the DUT in such a way that the difference between the controlled value
and the desired value will decrease. Consequently we need a reference and
a comparator to perform a calibration. See Figure 1.
Especially
for frequency calibration we mainly use a frequency counter,
characterized by
the fact that it contains the frequency reference (a 10 MHz timebase oscillator)
as well as the comparator. See Figure 2. However, most counters canaccommodate
an external reference, if necessary.
There
are two basic methods for frequency calibration, direct frequency
measurement
and phase comparison.
Direct
Frequency Measurement
The
normal method of frequency calibration is simply to connect a frequency counter
to the DUT and read the result. See Figure 2 again. The advantage of this
method is that we can calibrate an arbitrary frequency within the frequency
range of the counter (typically from mHz to GHz). The disadvantage is that the
accuracy of the measurement is limited to approximately 10 digits, even with
the counters having the highest resolution on the market and built-in atomic clock
(Rubidium reference).
Measurement
Uncertainty:
What
measurement uncertainty can we expect? The most important factors to consider
are the counter’s timebase oscillator and resolution, see Basics
in Brief. Then we must add trigger error due to signal noise and/or
internal noise, and systematic timing error in the counting circuits.
The latter sources of errors are usually negligible compared to the former and
normally appear in the 9th – 10th
digit
of the result. According to the theories of uncertainty calculation, we should
calculate the total uncertainty using the following steps:
1.
Compensate the result for known systematic errors.
2.
Express all the remaining uncertainty factors as standard deviation (”rms
value”
or ”1 sigma value”).
3.
Add up the squares of all uncertainties to be considered and extract the square
root of the sum. This will give us a
standard uncertainty ”rms” (rms = root mean square).
4.
Reduce the risk of a measurement value being outside the area of uncertainty
by
multiplying the standard uncertainty (from step 3) by 2. For a normaldistribution,
96 % of all measurements will be within the given area of
uncertainty.
5.
If, for instance, both resolution and timebase uncertainty are 1x10-7 (rms),
and
the other sources of errors are negligible, then the total uncertainty will be:
Modern
counters have a resolution of typically 9-10 digits, and in certain cases up to
11 digits, for a measuring time of 1 s. It is the built-in timebase oscillator
that limits the accuracy to 5-10 digits, depending on the type of timebase
oscillator used (Standard/TCXO/OCXO/Rubidium).
Theoretically
we can use an external Cesium standard or a GPS-controlled Rubidium reference
to obtain measurement results with 11-12 digits, but it is not sufficient in practice
for the counter to attain a total uncertainty on this level of accuracy. The systematic
error of the counter usually puts an end to improvement at 9-10 digits. Even
with the best instruments on the market (e. g. CNT-81 from Pendulum) we cannot
get beyond a measurement uncertainty in the 10th digit when using direct frequency
measurement. If we want to make frequency measurements with an uncertainty only
in the 11th, 12th or 13th digit, we have to turn to phase comparison.
Phase
Comparison (TIE Method)
By
means of phase comparison we can compare the phase difference of two signals having
the same nominal frequency. One signal is the reference with ”known” frequency
and the other signal comes from the DUT, the ”unknown” frequency. By measuring
how fast the phase difference increases or decreases, we can estimate the
frequency difference between the signals.
We
can make an analogous comparison to tuning a musical instrument. A guitar string,
for instance, is best ”calibrated” by simultaneously plucking two strings of
the same pitch (the "reference" and the one to be tuned). One listens
to the beat frequency created by the two strings and adjusts one string (the
DUT) until the beat frequency ceases (the change of phase approaches zero). The
beat method (phase comparison) gives the tuning quite a different precision
compared to listening to just one string and trying to put its absolute
frequency in tune (direct frequency measurement).
The
easiest way to estimate the phase difference between the reference and the DUT is
to measure the time interval between the zero crossings of the signals by means
of a timer/counter. This time interval is usually called TIE (Time Interval
Error), see Figure 3.
The frequency
difference between the signals in Figure 4 will then be:
The
uncertainty of a single TIE measurement is partly the resolution, partly a systematic
error that can be up to one nanosecond (or more in certain models) if we have
an uncalibrated input on the timer/counter. As we take the difference between two
TIE measurements at different points of time, all systematic sources of error
will be eliminated, since these errors are subtracted from themselves.
With
an accurate frequency reference and a high-resolution time interval meter
(timer/counter)
we can attain uncertainties of 10-12 or better.
Let
us give an example. We are measuring the frequency difference between an unknown
10 MHz signal and an extremely stable 10 MHz reference from the GPScontrolled frequency
reference GPS-89 with a built-in Rubidium oscillator. A CNT-81 with 50 ps
resolution is being used for the time interval measurement. We are measuring
the time interval between the zero crossings of the signals every 20 seconds and
get the following measurement results.
Time
(T) TIE Delta
(TIE) Delta (f)
T0=0s +4.55 ns
T1=20s
+4.75
ns 200
ps 200ps/20s =
1,0x10-11
T2=40s
+4.99
ns 440 ps
440ps/40s = 1,1x10-11
T3=60s
+5,23
ns 680
ps 680ps/60s
=
1,1x10-11
T4=80s
+5,49
ns 940 ps 940ps/80s = 1,2x10-11
T5=100s
+5,72 ns 1170
ps 1170ps/100s = 1,2 x10-11
In
this example we can draw attention to the fact that after 100 s the two
compared frequencies differ by 1.2 x 10-11. After only 20 seconds we get a very
good measure of the difference.
Measurement
Uncertainty
What
is the uncertainty of this measurement? The CNT-81 measures time interval (TIE)
with 50 ps resolution and point of time (T) with 100 ns resolution. In this example
we have a TIE uncertainty at time T0 and T5 resp. of 50 ps, plus the same systematic
error in both TIE measurements. The uncertainty of the time difference T5-T0
(100 ns in 100 s) is negligible. Since we subtract TIE(T0) from TIE(T5) the systematic
error will be completely eliminated and only the uncertainty of the resolution
will remain. The total uncertainty is calculated as before as ”the double rms
value” of the relevant uncertainty factors, and we get in this case:
Note
that the uncertainty in the example refers to how accurately we can measure the
difference between the two frequencies. In order to draw conclusions as
to the accuracy of the DUT frequency, we have then to consider the accuracy of
the reference itself. For the GPS-89, deviations appear in the 12th digit.
By
measuring for quite a long time we can improve the resolution and reduce the uncertainty
of the measurement even more. The other sources of errors in the measurement,
like timebase oscillator stability and trigger errors due to noise, are negligible
compared to the resolution of the TIE measurement.
Conclusion
By
utilizing a high-resolution timer/counter to measure the beat frequency
generated by two nominally equal frequencies, we can calibrate frequency with a
very high degree of accuracy. The condition is, of course, that we have an
extremely accurate reference, e. g. a Cesium reference or a GPS-controlled
Rubidium standard.
The
combination of the frequency reference GPS-89, Figure 5, and the timer/counter CNT-81,
Figure 6, from Pendulum Instruments, is a very powerful package for calibrating
the most usual reference frequencies, e. g. 10, 5 and 2.048 MHz. Furthermore,
the reference GPS-89 has a programmable reference output where you can set an
arbitrary reference frequency (the period time is set as N x 100 ns).
The
counter CNT-81 is the fastest on the market and has also the highest
resolution. When measuring frequency by means of the phase comparison method,
the CNT-81 reaches a given level of accuracy, for instance 12 digits, 15 times faster
than the most usual timer/counter on the market.
Basics
in Brief – Timebase Oscillator
Internal
References
The
timebase oscillator of a counter is a built-in frequency reference, most often
designed as a crystal oscillator, with varying degrees of accuracy:
-
UCXO = Un-Compensated X-tal Oscillator (standard)
-
TCXO = Temperature Compensated X-tal Osc.
-
OCXO = Oven Controlled X-tal Oscillator
Where
the utmost performance characteristics are required we can also see built-in Rubidium oscillators,
which then utilize the resonance properties of the Rubidium atom to give extremely
good frequency stability.
A
built-in timebase oscillator in a counter must be calibrated and, if need be,
adjusted regularly, typically once a year, to compensate for the so-called
aging of the oscillator, i. e. long-term drift. Furthermore, variations of the
ambient temperature affect the oscillator frequency.
A
counter can have a display with 10-12 digits, but depending on the type of
timebase oscillator, a major or a minor part of these may be reliable. Typical
values for use in normal room temperature and with 1-year calibration interval
are:
UCXO : 5-6
reliable digits
TCXO : 6-7
reliable digits
OCXO : 7-9
reliable digits
Rubidium
: 10-11 reliable digits
In
principle a newly calibrated timebase oscillator belongs to the upper part of
the interval.
External
References
For
even higher stability we have to use ultra-stable external frequency
references, e. g. Cesium standards or GPS-controlled references. The latter
lack aging, in principle, since the built-in local oscillator is locked to the
Cesium standards in the GPS satellites. However, the locking process causes
increased short-term instability. The frequency stability that can be 1 x 10-12
over 24 h can be 100 – 1000 times worse over short times, e. g. 1 s – 10 s,
normal measuring times in frequency counters. Note that this instability
concerns locking a local oven oscillator. If we lock a local Rubidium
oscillator (like the GPS-89 from Pendulum, for instance), the locking process
is not measurable, but we have a frequency reference comparable to the Cesium
reference at a fraction of the cost. See Figure 7.
Basics
in Brief – Resolution
Depending
on the design of the counter with conventional, reciprocal or interpolating measurement
technique, the measurements will have low or high resolution. Conventional
frequency counting does not exist any more in modern counters. With this method the
number of cycles at the input was counted during a fixed time, usually one second.
The resolution of the measurement is then one cycle per second (c/s), i. e. 1
Hz. The relative resolution then depends on what frequency we are measuring.
For 100 Hz, for instance, the relative resolution is 1 Hz / 100 Hz or ”2 digits”
(10-2). For 10 MHz the relative resolution is 1 Hz / 10 MHz or ”7 digits” (10-7).
Reciprocal
counters measure the period time of the input signal by counting clock
pulses (NCP) from a 10 MHz reference clock for an integer number of periods (NP)
of the input signal. Usually we measure during a measuring time of 1 s. After
that the period time T is
calculated:
T = NCP/NP x 100 ns
and
the frequency f is the reciprocal of T:
f = 1/ T = NP/NCP x 10 MHz
The
resolution with this method is always 7 digits for 1 s measuring time (100 ns
resolution
in 1
s), independent of the frequency.
Interpolating
counters are basically reciprocal counters with improved time measurement resolution.
Not only the integer number of clock pulses given by the time measurement is counted,
but also the incomplete parts of the clock pulses at the beginning and at the
end of the time measurement are calculated by means of interpolation. The
result for modern counters is an improvement of the resolution by 100-400
times. Time measurement can then be
made with a resolution of hundreds of ps.
In
this way modern interpolating counters will attain a resolution of 10 digits in
1 s measuring time.
For commercially available counters, the absolutely highest resolution on the
market today is found in the CNT-81 from Pendulum Instruments, having a
resolution of 50 ps or 11 digits
for frequency measurements with 1 s measuring time.
Independent
of the design of the counter (conventional, reciprocal or interpolating), it is
true that resolution is proportional to measuring time. If we increase the
measuring time from 1 s to 10 s, we will get another digit in the actual
resolution. And vice versa, if we decrease the measuring time from 1 s to 100
ms we will get one digit less. Most counters on the market cannot measure with
longer measuring times than 10 s. An exception is the CNT-80 family from
Pendulum Instruments, capable of measuring up to 400 s. See Figure 8.
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