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Establishing Traceability forQuantities Derived from Multiple Traceable QuantitiesJian 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 measure-ments is then considered as an example, based upon two common measurement techniques that utilize a spectrum analyzer and a phase detector.

1. IntroductionThere 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 quan-tities”, 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

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

AuthorsJian LiuAgilent Technologies3 Wangjing North RoadChaoyang District,Beijing 100102, Chinajian_liu@agilent.com

Alberto CampilloAgilent TechnologiesCarretera Nacional VI, km 18, 200Las Rozas, Madrid 28230, Spainalberto_campillo@agilent.com

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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.

3. Traceability of Phase Noise Measurements, an Example3.1 Phase Noise OverviewPhase 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 analyzer, the ratio is the noise power in a 1 Hz bandwidth at 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)

Another measure of phase noise often used is  , the two-sided spectral density of mean-squared phase fluctuations, expressed in units of phase variance  per 1 Hz bandwidth [16], as given by:

. (3)

Base quantity Base unit

Name Symbol Name SymbolLength l, x, r, etc. Meter mMass m Kilogram kgtime, duration t Second selectric current I, i Ampere Athermodynamic temperature

T Kelvin K

amount of substance

n Mole mol

luminous intensity

Iv Candela cd

Table 1. SI base quantities and base units. Figure 1. Phase noise unit of measure.

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 can also be expressed in terms of dB when referenced to 1 .

When the total phase fluctuation in the modulation sideband is much less than 1 radian, can be directly related to  as

. (4)

When the phase variation exceeds the small angle criterion, the historical becomes confusing, because it is possible to have 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

to be greater than 0 dB. Since  in dB is relative to 1 radian, 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).

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 SourceA 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

Figure 2. ratio, noise power density relative to signal power.

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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 LinG is the baseband analyzer linearity,

PSDK 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

is calibrated with an AC calibrator and an RF source (which could be monitored by AC voltmeter and power sensor and meter). LinG is tested with a standard step attenuator and a stable signal source. PSDK is characterized with a known noise source.   is obtained from 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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Figure 3. Block diagram of phase noise measurement with a spectrum analyzer.

Figure 4. Block diagram of phase noise measurement with phase detectors.

RMSV

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4. SummaryMany 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. AcknowledgementsThe authors thank Scott Arrants, Ken Harrison, Mike Hutchins, Bee-Choo Lim, Richard Lippincott, Michitoshi Noguchi, Richard Ogg, Wolfgang Trester, Randolph Yamamoto and Tse-Tse Ypma from Agilent Technologies for their helpful discussion and comments.

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