TECHNICAL NOTE
Establishing Traceability for
Quantities Derived from Multiple
Traceable Quantities
Jian Liu and Alberto Campillo
Abstract: Some quantities are traceable to a single quantity with the same unit but a lower uncertainty. For instance, a weight
of 50 g ± 1 g might be calibrated with a reference weight of 50 g ± 0.1 g. Meanwhile, other quantities are calibrated by using
multiple quantities with different units as references. A simple example would be measuring DC current with a volt meter and a
standard resistor. In this case, the current measurement is derived from DC voltage and resistance. A more complicated example
of a derived quantity is phase noise. The question is: how do we establish traceability for such derived quantities? This paper
discusses the International System of Units (SI), base and derived quantities, and traceability; and presents a general approach for
establishing traceability for a quantity that is derived from multiple traceable quantities. The traceability of phase noise measurements is then considered as an example, based upon two common measurement techniques that utilize a spectrum analyzer and
a phase detector.
1. Introduction
There are many quantities measured in the
world of metrology, such as length, voltage
and phase noise, and numerous units are
used for their quantitative descriptions. The
International System of Units (SI) [1] defines
those units that are “universally agreed for
the multitude of measurements that support
today’s complex society”. Seven of these
units are defined as “base units”, and the seven
quantities they describe are called “base quantities”, as listed in Table 1. By contrast, other
SI units are defined as “derived units” which
can be expressed as products of powers of
base units, and their corresponding quantities
are similarly called “derived quantities”. In
addition, there are also non-SI units. Some of
them are accepted for use with SI (e.g., hour
and decibel), and others are not recommended
for use in modern scientific and technical
work (e.g., the “foot” which is 0.3048 m and
the Chinese “jin” which is 0.5 kg).
The choice of each of the base units has
been made with special care so that they are
independent, “are readily available to all,
are constant throughout time and space, and
are easy to realize with high accuracy” [1],
because they provide the foundation for the
entire system.
The base quantities and units are the
foundation of the SI system, but not all
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quantities can necessarily be traced to them;
in some cases being traceable to derived
quantities is enough. A typical example is
described in Appendix 2 of the SI brochure
[2] and Section 2.1.5 of the NCSLI Catalog
of Intrinsic and Derived Standards [3]. The
high accuracy of electric current is obtained
through a combination of the realizations of
voltage and resistance. In this case the base
quantity is derived from (and traced to) the
derived quantities. In general, it doesn’t
matter if a traceability chain to the SI leads to
a base quantity or to a derived quantity.
It is accuracy that matters. The general
model of a traceability chain implies that a
quantity can be measured more accurately
at higher levels and less accurately at lower
levels. For example, an iron weight of 50 g
± 1 g can be calibrated with another alloy
steel weight of 50 g ± 0.1 g. The accuracy
relationship in this case is simple and
obvious. However, traceability cannot always
be established by comparison to single
standards of higher accuracy. In some cases,
traceability must be derived from multiple
quantities, making the traceability chain
much more complicated.
In the following sections, the general
approach of establishing traceability for
a quantity that is derived from multiple
traceable quantities is introduced. The
complex derived quantity of phase noise is
then reviewed as an example.
2. General Approach of Establishing
Traceability for Derived Quantities
Metrological traceability is defined as
“property of a measurement result whereby
the result can be related to a reference through
a documented unbroken chain of calibrations,
each contributing to the measurement
uncertainty” [4]. This definition discloses
that the establishment of traceability is
through a quantitative description of the
measurement result and the measurement
uncertainty (MU). Thus, the approach
of establishing traceability is actually a
sequential translation from the physical
Authors
Jian Liu
Agilent Technologies
3 Wangjing North Road
Chaoyang District,
Beijing 100102, China
[email protected]
Alberto Campillo
Agilent Technologies
Carretera Nacional VI, km 18, 200
Las Rozas, Madrid 28230, Spain
[email protected]
www.ncsli.org
TECHNICAL NOTE
Base quantity
Base unit
Name
Symbol
Name
Symbol
Length
l, x, r, etc.
Meter
m
Mass
m
Kilogram
kg
time, duration
t
Second
s
electric current
I, i
Ampere
A
thermodynamic
temperature
T
Kelvin
K
amount of
substance
n
Mole
mol
luminous
intensity
Iv
Candela
cd
Table 1. SI base quantities and base units.
principle of the measurement (calibration) into a calculated number.
Several items are necessary in this translation sequence.
The first thing that should be in place is the measurand - the
derived quantity of interest, including the ranges to be determined. If
the quantity is determined by a particular instrument, the instrument
should be specified. Sometimes the concept of the quantity and other
aspects that may confuse the quantity also need be clarified, depending
upon the complexity of the quantity.
The next item to consider is the measurement method, including the
measurement principle, measuring and test equipment (M&TE) and
ancillary apparatus used, and a step-by-step procedure of connections,
settings, and data processing.
Thirdly are the measurement equation(s) and uncertainty
equation(s) that relate the derived quantity of interest to other
traceable quantities. The uncertainty equations are generally based
on measurement equations, but are much more complicated due to
the inclusion of MU contributors from all of the traceable quantities.
Usually the final measurement result is calculated from several sets
of intermediate data and the final MU can only be calculated from a
set of equations. The physical principle of the measurement method is
translated here into an algebraic format.
Finally, the values and uncertainties of the traceable parameters and
the specifications of the M&TEs are substituted into the uncertainty
equation(s) so that the MU of the derived parameter can be obtained.
However, to ensure the validity of the substituted numbers, other types
of supporting evidence are also needed, such as the environmental
conditions for storing the M&TEs and performing the measurement,
the stabilization of the measurement setup, and the validation method
of the measurement results.
The measurement principle is fundamental to the whole approach
and should receive the most attention, because misunderstanding
the physical principle could result in significant MU contributors
either being omitted or under or over estimated. The next section
discusses the traceability of phase noise measurements. It focuses
on the physical principle of the measurements without detailing the
mathematical calculations.
Vol. 7 No. 2 • June 2012
Figure 1. Phase noise unit of measure.
3. Traceability of Phase Noise Measurements, an Example
3.1 Phase Noise Overview
Phase noise is one of the most significant characteristics of a signal
generating device and can often be a limiting factor in its use. It
has thus drawn intensive attention from researchers and numerous
measurement techniques have been developed [5-14]. In this section,
the definition and metrics of phase noise are briefly discussed. Then,
two methods are presented that illustrate how the derived quantity of
phase noise can be traced to multiple quantities.
One key specification of an oscillating source is its ability to
produce the same frequency for a specified period, a characteristic
known as the frequency stability. Phase noise describes, in the
frequency domain, the frequency stability (or “frequency instability”
[5]) for periods of less than a few seconds.
Historically, the most common unit of measure for phase noise has
been the single sideband (SSB) power within a 1 Hz bandwidth at a
frequency f offset from the carrier, referenced to the carrier frequency
power (Fig. 1) [15]. The unit of measure is represented as script
in units of dBc/Hz,
.
(1)
When measuring phase noise directly with an RF spectrum
ratio is the noise power in a 1 Hz bandwidth at
analyzer, the
a desired offset frequency f away from the carrier, Pn, relative to the
carrier signal power, Ps (Fig. 2) [15]. This is calculated as
. (2)
, the two-sided
Another measure of phase noise often used is
spectral density of mean-squared phase fluctuations, expressed in
per 1 Hz bandwidth [16], as given by:
units of phase variance
.
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(3)
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TECHNICAL NOTE
Figure 2.
signal power.
ratio, noise power density relative to
can also be expressed in terms of dB when referenced to
.
When the total phase fluctuation in the modulation sideband is
as
much less than 1 radian,
can be directly related to
1
.
(4)
When the phase variation exceeds the small angle criterion, the
becomes confusing, because it is possible to have
historical
phase noise density values that are larger than 0 dBc/Hz, even though
the power in the modulation sideband is less than the carrier power.
To eliminate this confusion, the historical definition of
has been
changed to the one shown in Eq. (4) [17]. The new definition allows
in dB is relative to 1 radian,
to be greater than 0 dB. Since
results greater than 0 dB simply means that the phase variations being
measured are larger than 1 radian.
Two common techniques for phase noise measurements are chosen
to show different traceability paths. One technique simply uses a
spectrum analyzer. This technique is often seen, but is not generally
recommended and can only be used under limited conditions. The
other technique employs a phase detector together with a reference
source and a phase-locked loop (PLL). The second technique is more
complicated but more appropriate, and is broadly applied for phase
noise metrology and research. The principles of the two methods as
well as their traceability paths are briefly described in the following
sections.
3.2 Traceability of Phase Noise Measured with a Spectrum
Analyzer
The most direct and probably the oldest method for the phase noise
measurement of oscillators is the direct spectrum technique. As shown
in Fig. 3, the signal from the unit-under-test (UUT) is input to a
spectrum analyzer that is tuned to the UUT frequency. The sideband
noise power can be directly measured and compared to the carrier
signal power to obtain
[18]. The measurement equation was
previously shown in Eq. (2).
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This approach has disadvantages and is not recommended for high
accuracy phase noise measurements for several reasons. Firstly, it
is unable to distinguish phase noise from AM noise. Secondly, the
measurement sensitivity is limited by the analyzer’s internal local
oscillator (LO) noise. And thirdly, the inability to track any signal drift
limits the close-to-carrier measurement capability.
Therefore, the technique is only suitable under limited
circumstances, where the UUT is known to have negligible AM noise
and moderate phase noise – neither too high to break the small angle
restriction, nor too low to be covered by the analyzer’s noise floor,
and when the close-to-carrier phase noise is of no concern [19]. For
such measurements, however, it is still meaningful to understand the
traceability.
A spectrum analyzer used for such phase noise measurements
needs to be calibrated for all frequencies and all power levels and its
resolution bandwidth (RBW) behavior must be known. As can be seen
from Fig. 3, the level of performance is determined by the amplitude
measurement accuracy of the detector, the bandwidth normalization of
the intermediate frequency (IF) filter (i.e., the deviation of the IF filter
from an ideal rectangular filter), the attenuation accuracy of the step
attenuator at the input port (not shown in Fig. 3), the mismatch and
flatness of the whole signal path (especially when the offset frequency
is greater than 1 MHz), the RBW switching uncertainty, and so on.
The calibration is normally done periodically against the analyzer’s
specifications. The calibration standards include frequency counters,
microwave signal generators, RF power sensors and power meters,
step attenuators and power splitters. Therefore, phase noise measured
by spectrum analyzers can be traced to the quantities of frequency, RF
power (calibration factor), attenuation, and reflection and transmission
coefficients.
3.3 Traceability of Phase Noise Measured with a Phase Detector
and a Reference Source
A more appropriate phase noise measurement technique utilizes a
phase detector. The phase detector, normally a mixer, receives two
input signals of the same frequency and converts their phase difference
into a voltage at its output [18, 20]. When the two signals are in phase
quadrature, i.e., 90 degrees out of phase, the output voltage is zero.
Any phase fluctuation will result in an output voltage fluctuation as
,
(5)
where
is the phase detector constant.
One of the most widely used techniques based on the phase detector
concept utilizes a reference source and a PLL as shown in Fig. 4 [20].
Here, the PLL is used to control the reference source and establish
phase quadrature at the phase detector. The phase noise that is
measured at the phase detector is the sum of the mean square phase
fluctuations. The reference source has much less phase noise than the
UUT so the measurement results reflect the higher phase noise of the
UUT. The baseband signal is normally filtered, amplified and input to
a baseband spectrum analyzer. A low-noise microwave downconverter
can be used to translate the carrier frequency of the UUT to a lower IF
frequency for measurement purposes.
This technique overcomes the defects of the spectrum analyzer
method. The phase detector can reject the AM noise, and a reference
source with low phase noise can offer excellent dynamic range. The
PLL can track and limit the frequency drift and thus measure much
www.ncsli.org
TECHNICAL NOTE
Figure 3. Block diagram of phase noise measurement with a spectrum analyzer.
Figure 4. Block diagram of phase noise measurement with phase detectors.
better close-to-carrier phase noise. Another benefit is that the level of
the phase noise under test can exceed the small angle criterion.
The measurement equation [21] is
,
(6)
where
GLin is the baseband analyzer linearity,
K PSD is the baseband analyzer power spectral density conversion
coefficient, and
is the baseband analyzer gain flatness.
The five variables on the right side of Eq. (6) contribute to the
uncertainty of this technique. The analyzer’s accuracy of measuring Vol. 7 No. 2 • June 2012
VRMS is calibrated with an AC calibrator and an RF source (which could
be monitored by AC voltmeter and power sensor and meter). GLin is
tested with a standard step attenuator and a stable signal source. K PSD
is obtained from
is characterized with a known noise source.
linearity measurement and is affected by signal path mismatch and
flatness, especially when far-from-carrier (offset > 1 MHz) phase noise
is concerned. Finally,
of the phase detector is evaluated by letting
the two sources free run to produce a beat frequency at the detector’s
output, and then measuring the amplitude of the output voltage. More
details of the five uncertainty contributors can be found in [21].
Phase noise measured with the phase detector technique can
be traced to the quantities of frequency, AC voltage, RF power
(calibration factor), noise, attenuation, and reflection and transmission
coefficients.
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TECHNICAL NOTE
4. Summary
Many metrology laboratories only provide calibration for quantities
where direct traceability can be established. However, if a quantity
can be derived from other traceable quantities, the traceability of the
new quantity may also be established.
The term “derived quantity” on a traceability chain is different than
that defined in the SI since it is not necessarily derived from or traced
to the SI “base quantities”. Instead, it is derived from and linked to
other traceable quantities that are readily available. Even an SI base
quantity (e.g., electric current) can be derived from (and thus traced
to) other quantities (e.g., voltage and resistance).
Generally speaking, the approach of establishing traceability for
a derived quantity is similar to that for a directly traced quantity. The
details of the measurement method, the measurement equations, and
the uncertainty analysis should be documented. Special attention
should be given to the translation from the physical principle of the
measurement technique to the final uncertainty. It is essential to
ensure that all potential contributors to the measurement uncertainty
are being considered.
To illustrate the concepts presented here, it was shown how phase
noise measurements can be traced to different quantities corresponding
to different measurement techniques. With the direct spectrum method,
measurements can be traced to frequency, RF power, attenuation, and
reflection and transmission coefficients. With the phase detector and
reference source technique the traced quantities include frequency,
AC voltage, RF power (calibration factor), noise, attenuation, and
reflection and transmission coefficients.
5. Acknowledgements
The authors thank Scott Arrants, Ken Harrison, Mike Hutchins, BeeChoo Lim, Richard Lippincott, Michitoshi Noguchi, Richard Ogg,
Wolfgang Trester, Randolph Yamamoto and Tse-Tse Ypma from
Agilent Technologies for their helpful discussion and comments.
6. References
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