Preface

The goal of acceptable quality, cost, and time is a decisive challenge in every engineering development process. To be familiar with metrology requires choosing the best combination of techniques, standards, and tools to control the project from advanced simulations to final performance measurements and periodic inspections. This book contains a cluster of chapters from international academic authors who provide a meticulous way to discover the impacts of metrology in both theoretical and application fields. The approach is to discuss the key aspects of a selection of untraditional metrological topics, covering the analysis procedures and set of solutions obtained from experimental studies. Each chapter is designed for both the engineering and academic communities and is partitioned as a scientific paper; a congruous list of references is available at the end of each section.

The volume has been divided into five chapters:

### **1. Metrological Traceability at Different Measurement Levels**

*Chapter 1* presents the basis of global metrological traceability and the standards of comparison of different regional metrology organizations. A procedure to evaluate specific interlaboratory results on national and international levels is described. This method can contribute the mutual recognition of measurement and testing results by different countries.

### **2. Self-Calibration of Precision XYθz Metrology Stages**

*Chapter 2* studies the on-axis calibration for precision XYθz metrology stages, and presents a holistic XYθz self-calibration approach. The proposed approach uses an artifact plate specially designed with XY grid mark lines and angular mark lines as a tool, to be measured by the XYθz metrology stages. Computer simulation is conducted and the designed artifact plate is illustrated to meet practical industrial requirements.

### **3. Third-Order Nonlinear Optical Properties of Quantum Dots**

*Chapter 3* introduces quantum dots (QDs), which are semiconducting nanocrystalline particles and attractive photonic media. Third-order nonlinear optical properties and a brief idea of the physics of QDs are discussed; the Z-scan technique and theoretical analysis adopted to obtain nonlinear parameters are detailed. Despite their size, QDs represent a good example of optical limiters with low threshold.

### **4. Analysis of Pulsating White Dwarf Star Light Curves**

*Chapter 4* is focused on analysis techniques for extracting the frequencies contained in the light curves of pulsating white dwarf stars. In several surface temperature regimes, these astronomical objects are unstable to gravity mode pulsations, which result in brightness variations corresponding to the periods of the excited modes.

Mode periods present in the light curve are detected by undertaking a Fourier analysis of the time series light curve.

### **5. Biotoxicological Monitoring of Organic Solvents in the Tunisian Footwear Industry**

*Chapter 5* is oriented to evaluate objectively the health effects of organic solvents that are widely used in the Tunisian footwear industry. The goal of the study is to identify analytical methods for exposure assessment of predominant solvents such as acetone, cyclohexane, hexane, methyl ethyl ketone, and toluene to arrange a process of occupational risk assessment via biotoxicological and airborne monitoring for solvents.

My appreciation goes to all authors for their valuable contributions. I would also like to thank my nephew Alessandro for his contagious energy.

> **Luigi Cocco, PhD** Maserati S.p.A., Italy

Chapter 1

Abstract

scores.

laboratory

1

1. Introduction

Metrological Traceability at

Oleh Velychko and Tetyana Gordiyenko

Different Measurement Levels

The international agreements are the basis for establishing the global metrological traceability at different measurement levels. The concepts and concept relations around metrological traceability are presented. An important element of providing the metrological traceability is the evaluation of measurement uncertainty. The procedure of linking of key and supplementary comparison results is described. Linking of key and supplementary comparison results of the Regional Metrology Organization for some quantities according to the described procedure was

presented. Results for all participants of presented key and supplementary comparisons are satisfactory for chi-square test and En number. The procedure of linking of key or supplementary comparison and national inter-laboratory comparison results is described. This procedure can be used for practical evaluation of specific interlaboratory comparison results on a national level in different countries by means of laboratory results of the National Metrology Institute and Designated Institute. This procedure can contribute the mutual recognition of measurement and testing results by different countries. Linking of key comparison and inter-laboratory comparison results for some quantities according to the described procedure was presented. Results for all participants of presented key comparison and interlaboratory comparison are satisfactory for chi-square test, En number, z scores and ζ

Keywords: metrological traceability, measurement uncertainty, measurement standard, comparison, inter-laboratory comparison, National Metrology Institute,

The Mutual Recognition Agreement (MRA) of the International Committee on Weights and Measures (CIPM) [1] and the MRA of the International Laboratory Accreditation Cooperation (ILAC) play an important role in overcoming technical barriers to international trade. CIPM MRA plays a key role in ensuring the international equivalence of national measurement standards of different countries. ILAC MRA plays a key role in ensuring international recognition of calibration results or test results in accredited calibration and testing laboratories. The main base of these agreements is special documents, guidelines, standards and recommendations [2]. National Metrology Institutes (NMIs) and Designated Institutes (DIs) play an important role in implementation of the CIPM MRA. They take an active part in organizing and conducting international comparisons of national standards.

### Chapter 1

## Metrological Traceability at Different Measurement Levels

Oleh Velychko and Tetyana Gordiyenko

### Abstract

The international agreements are the basis for establishing the global metrological traceability at different measurement levels. The concepts and concept relations around metrological traceability are presented. An important element of providing the metrological traceability is the evaluation of measurement uncertainty. The procedure of linking of key and supplementary comparison results is described. Linking of key and supplementary comparison results of the Regional Metrology Organization for some quantities according to the described procedure was presented. Results for all participants of presented key and supplementary comparisons are satisfactory for chi-square test and En number. The procedure of linking of key or supplementary comparison and national inter-laboratory comparison results is described. This procedure can be used for practical evaluation of specific interlaboratory comparison results on a national level in different countries by means of laboratory results of the National Metrology Institute and Designated Institute. This procedure can contribute the mutual recognition of measurement and testing results by different countries. Linking of key comparison and inter-laboratory comparison results for some quantities according to the described procedure was presented. Results for all participants of presented key comparison and interlaboratory comparison are satisfactory for chi-square test, En number, z scores and ζ scores.

Keywords: metrological traceability, measurement uncertainty, measurement standard, comparison, inter-laboratory comparison, National Metrology Institute, laboratory

### 1. Introduction

The Mutual Recognition Agreement (MRA) of the International Committee on Weights and Measures (CIPM) [1] and the MRA of the International Laboratory Accreditation Cooperation (ILAC) play an important role in overcoming technical barriers to international trade. CIPM MRA plays a key role in ensuring the international equivalence of national measurement standards of different countries. ILAC MRA plays a key role in ensuring international recognition of calibration results or test results in accredited calibration and testing laboratories. The main base of these agreements is special documents, guidelines, standards and recommendations [2].

National Metrology Institutes (NMIs) and Designated Institutes (DIs) play an important role in implementation of the CIPM MRA. They take an active part in organizing and conducting international comparisons of national standards.

Consultative Committees (CCs) of CIPM and the International Bureau of Weights and Measures (BIPM) carry out key comparisons (KCs) of national standards in different fields of measurements. KCs are also being carried out by Regional Metrology Organizations (RMOs), which are equivalent to CC KCs. Only RMO makes supplementary comparisons (SCs) for those measurements that are not covered by KC CC or RMO. Results of all comparisons of standards are published in a special database KC (KCDB) of BIPM [3].

For CC KC and RMO KC, the reference value (RV) of KC and degree of equivalence (DoE) of national standards with corresponding uncertainty are established [4, 5]. DoE derived from an RMO KC has the same status as that derived from a CC KC. RMO SC has the same status as RMO KC. RMOs have a procedure to carry out comparisons, but only the Euro-Asian Cooperation of National Metrological Institutions (CООМЕТ) has guidelines on comparison data evaluation [6, 7].

According to results obtained by the NMI or DI (NMI/DI) in conducted comparisons, Calibration and Measurement Capabilities (CMCs) of NMI/DI are being prepared [8, 9]. The internationally recognized NMI CMCs are those that are published to the KCDB of BIPM. Metrological traceability [10] is important for industrial metrology, because it allows you to compare measurement accuracy in accordance with a standardized procedure for assessing measurement uncertainty [11].

uncertainty, standard, and calibration. Hierarchical generic relations of metrological traceability with a measurement unit and of standard with international standard and national standard are established. Hierarchical partitive relation of calibration

At the modern stage of development of the industrial metrology, the role of NMIs/DIs and CLs increases significantly. This is due to the need to ensure mutual recognition of measurement results in different countries. Global metrological traceability at different measurement levels [15] is provided by the CIPM MRA and ILAC MRA. These agreements set out the basic requirements for ensuring mutual

The general scheme of global metrological traceability at different measurement

International comparisons of national standards of NMIs/DIs are carried out as part of activities of the CIPM consultative committees (CCs) and technical committees of six RMOs. Results of these comparisons are technical basis for the preparation of NMI/DI CMC for publication in KCDB of BIPM. Accredited CLs and testing laboratories participate at the national level in the ILCs as part of activities of

The general scheme of global metrological traceability at different measurement levels.

hierarchy with calibration is also established.

Partial concept diagram around metrological traceability.

Metrological Traceability at Different Measurement Levels

DOI: http://dx.doi.org/10.5772/intechopen.84853

recognition of both measurements and testing.

levels is presented in Figure 2.

Figure 1.

Figure 2.

3

ILAC publication [12] established the need to ensure a continuous calibration chain to international or national standards as the main element for establishing metrological traceability. Important roles for the implementation of this requirement are calibration laboratories (CLs).

Inter-laboratory comparisons (ILCs) are a form of experimental verification of accredited calibration and test laboratories. They must meet the requirements of international standards ISO/IEC 17025 [13] and ISO/IEC 17043 [14]. Their main goal is to determine the technical competence of accredited laboratories for specific activities. The purpose of the ILC is to establish the inter-laboratory differences of their participants. Successful laboratory results in ILC confirm technical competence for certain types of measurements or testing.

Establishment of measurement traceability at the highest metrological level is carried out in accordance with procedures through international comparisons of the national standards of NMI/DI. Establishment of metrological traceability at lower measurement level is carried out in accordance with the calibration procedures of working standards by both NMIs/DIs and accredited CLs.

For the highest level of the metrological traceability, it is advisable to develop a methodology for linking of results of RMO SC to RMO KC, and RMO SC to other RMO SC. For lower level of the metrological traceability, it is advisable to develop a methodology for linking of results of the national ILC to RMO KC or RMO SC. These methodologies can be used for practical assessment of results of specific RMO KC/SC as an extension of the technical basis of confirmation of NMI/DI CMC or specific ILC and at the national level in different countries using the comparison results and CMC NMIs/DIs.

### 2. Bases of metrological traceability

The concept of metrological traceability is important for industrial metrology and is associated with such basic metrological concepts as measurement result, calibration chain, and measurement uncertainty [10]. A partial concept diagram around metrological traceability is shown in Figure 1.

The concept diagram demonstrates associative relations of metrological traceability with metrological traceability chain, measurement result, measurement

Metrological Traceability at Different Measurement Levels DOI: http://dx.doi.org/10.5772/intechopen.84853

#### Figure 1.

Consultative Committees (CCs) of CIPM and the International Bureau of Weights and Measures (BIPM) carry out key comparisons (KCs) of national standards in different fields of measurements. KCs are also being carried out by Regional Metrology Organizations (RMOs), which are equivalent to CC KCs. Only RMO makes supplementary comparisons (SCs) for those measurements that are not covered by KC CC or RMO. Results of all comparisons of standards are published in

For CC KC and RMO KC, the reference value (RV) of KC and degree of equivalence (DoE) of national standards with corresponding uncertainty are established [4, 5]. DoE derived from an RMO KC has the same status as that derived from a CC KC. RMO SC has the same status as RMO KC. RMOs have a procedure to carry out comparisons, but only the Euro-Asian Cooperation of National Metrological Institutions (CООМЕТ) has guidelines on comparison data evaluation [6, 7]. According to results obtained by the NMI or DI (NMI/DI) in conducted comparisons, Calibration and Measurement Capabilities (CMCs) of NMI/DI are being

prepared [8, 9]. The internationally recognized NMI CMCs are those that are

published to the KCDB of BIPM. Metrological traceability [10] is important for industrial metrology, because it allows you to compare measurement accuracy in accordance with a standardized procedure for assessing measurement uncertainty [11]. ILAC publication [12] established the need to ensure a continuous calibration chain to international or national standards as the main element for establishing metrological traceability. Important roles for the implementation of this require-

Inter-laboratory comparisons (ILCs) are a form of experimental verification of accredited calibration and test laboratories. They must meet the requirements of international standards ISO/IEC 17025 [13] and ISO/IEC 17043 [14]. Their main goal is to determine the technical competence of accredited laboratories for specific activities. The purpose of the ILC is to establish the inter-laboratory differences of their participants. Successful laboratory results in ILC confirm technical compe-

Establishment of measurement traceability at the highest metrological level is carried out in accordance with procedures through international comparisons of the national standards of NMI/DI. Establishment of metrological traceability at lower measurement level is carried out in accordance with the calibration procedures of

For the highest level of the metrological traceability, it is advisable to develop a methodology for linking of results of RMO SC to RMO KC, and RMO SC to other RMO SC. For lower level of the metrological traceability, it is advisable to develop a methodology for linking of results of the national ILC to RMO KC or RMO SC. These methodologies can be used for practical assessment of results of specific RMO KC/SC as an extension of the technical basis of confirmation of NMI/DI CMC or specific ILC and at the national level in different countries using the comparison

The concept of metrological traceability is important for industrial metrology and is associated with such basic metrological concepts as measurement result, calibration chain, and measurement uncertainty [10]. A partial concept diagram

The concept diagram demonstrates associative relations of metrological traceability with metrological traceability chain, measurement result, measurement

a special database KC (KCDB) of BIPM [3].

Standards, Methods and Solutions of Metrology

ment are calibration laboratories (CLs).

results and CMC NMIs/DIs.

2

2. Bases of metrological traceability

around metrological traceability is shown in Figure 1.

tence for certain types of measurements or testing.

working standards by both NMIs/DIs and accredited CLs.

Partial concept diagram around metrological traceability.

uncertainty, standard, and calibration. Hierarchical generic relations of metrological traceability with a measurement unit and of standard with international standard and national standard are established. Hierarchical partitive relation of calibration hierarchy with calibration is also established.

At the modern stage of development of the industrial metrology, the role of NMIs/DIs and CLs increases significantly. This is due to the need to ensure mutual recognition of measurement results in different countries. Global metrological traceability at different measurement levels [15] is provided by the CIPM MRA and ILAC MRA. These agreements set out the basic requirements for ensuring mutual recognition of both measurements and testing.

The general scheme of global metrological traceability at different measurement levels is presented in Figure 2.

International comparisons of national standards of NMIs/DIs are carried out as part of activities of the CIPM consultative committees (CCs) and technical committees of six RMOs. Results of these comparisons are technical basis for the preparation of NMI/DI CMC for publication in KCDB of BIPM. Accredited CLs and testing laboratories participate at the national level in the ILCs as part of activities of

Figure 2. The general scheme of global metrological traceability at different measurement levels.

national accreditation bodies. The calibration hierarchy is provided by calibration of the working standards and MIs: CLs—for testing laboratories; NMIs/DIs—for CLs.

RMO organizes KC with a number of joint NMI/DI participants with CC KC. This is necessary in order to link the results of the RMO KC with the results of the CC KC. For this purpose, equivalent technical protocols of both comparisons are used. The procedures for evaluating the data obtained at RMO KC are necessary to establish the DoE of national standards of NMI participants. PL calculates the KC RV and DoE for all NMI participants when preparing draft of comparison reports. The procedures used for evaluating RMO SC data are the same as for RMO KC. SC RMO complements KC CC or RM KC and is not second level comparison. RMO SC

RMO KC and RMO SC data evaluation usually includes determining the following characteristics: determining the RV comparison with the corresponding uncertainty, the DoE with corresponding uncertainties for each NMI/DI participant, and

corresponding uncertainties [6, 7]. RMO KC data evaluation includes the definition of such additional characteristics: converted KC data with corresponding uncertainties and DoE with corresponding uncertainties for each NMI/DI participant,

The RMO KC/SC RV XRV is calculated as the mean of NMI/DI participant results

n i¼1

where xNMIi is the result for i-th NMI/DI participant in RMO KC/SC; u xð Þ NMIi is corresponding standard uncertainty for i-th NMI/DI participant in RMO KC/SC; i ¼ 1, 2, …, n, n is the total number of NMI/DI participants of RMO KC/SC. The DoE of i-th NMI/DI participant DNMIi and corresponding combined

ð Þþ xNMIi <sup>u</sup><sup>2</sup>

Pairs of DoE of i-th NMI/DI participant and j-th NMI/DI participant DNMIij of RMO KC/SC and corresponding combined standard uncertainty u DNMIij are

On the basis of the measurement results of RMO KC/SC and corresponding combined standard uncertainties claimed by NMI/DI participants of RMO KC/SC,

> D2 NMIi u<sup>2</sup>ð Þ xNMIi

If the calculated chi-criterion value does not exceed the chi-square test critical

<sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>∑</sup> n i¼1

value with the coverage level of 0.95 and freedom degrees of n – 1

= ∑ n i¼1

1 u<sup>2</sup>ð Þ xNMIi

1 u<sup>2</sup>ð Þ xNMIi

DNMIi ¼ xNMIi � XRV, (3)

DNMIij ¼ xNMIi � xNMIj, (5)

ð Þþ xNMIi <sup>u</sup><sup>2</sup> xNMIj : (6)

(1)

, (2)

ð Þ XRV : (4)

: (7)

xNMIi u<sup>2</sup>ð Þ xNMIi

ð Þ¼ XRV 1= ∑

a pair DoE of i-th NMI/DI participant and j-th NMI/DI participant with

XRV ¼ ∑ n i¼1

u2

with the combined standard uncertainty

standard uncertainty u Dð Þ NMIi are estimated as

the chi-square test value is calculated [7].

u2

ð Þ¼ DNMIi <sup>u</sup><sup>2</sup>

<sup>u</sup><sup>2</sup> DNMIij <sup>¼</sup> <sup>u</sup><sup>2</sup>

results are also published in KCDB of BIPM [16, 17].

Metrological Traceability at Different Measurement Levels

DOI: http://dx.doi.org/10.5772/intechopen.84853

except for linking NMI/DI.

estimated as

5

from RMO KC/SC data are given by

### 3. The data evaluation of standard comparisons

The diagram of concept relations for standard comparisons is shown in Figure 3. Besides to KCs and SCs, pilot comparisons are also carried out, which all these comparisons can be bilateral. The organization of CC KCs and RMO KCs/SCs is the responsibility of pilot laboratory (PL) whose functions are performed by one of the selected NMI/DI [4, 6, 7]. The main responsibilities of PL include development of technical protocol of comparison, selection, and research of traveling standard, and the development of draft comparison reports. Coordination of the entire work of the PL as part of comparison is carried out by the contact person of PL.

The organizational scheme of standard comparisons is shown in Figure 4. NMI 1 is PL and is responsible for organizing the delivery of traveling standard to NMI participants. This scheme can be circular or radial. In the second case, it is better to provide research of drift of the traveling standard. The most commonly used is a mixed comparison scheme: after several NMI/DI participants of comparison, a traveling standard returns to PL for research of their drift.

#### Figure 3.

The diagram of concept relations for standard comparisons.

Figure 4. The organizational scheme for standard comparisons.

Metrological Traceability at Different Measurement Levels DOI: http://dx.doi.org/10.5772/intechopen.84853

national accreditation bodies. The calibration hierarchy is provided by calibration of the working standards and MIs: CLs—for testing laboratories; NMIs/DIs—for CLs.

The diagram of concept relations for standard comparisons is shown in Figure 3.

The organizational scheme of standard comparisons is shown in Figure 4. NMI 1 is PL and is responsible for organizing the delivery of traveling standard to NMI participants. This scheme can be circular or radial. In the second case, it is better to provide research of drift of the traveling standard. The most commonly used is a mixed comparison scheme: after several NMI/DI participants of comparison, a

Besides to KCs and SCs, pilot comparisons are also carried out, which all these comparisons can be bilateral. The organization of CC KCs and RMO KCs/SCs is the responsibility of pilot laboratory (PL) whose functions are performed by one of the selected NMI/DI [4, 6, 7]. The main responsibilities of PL include development of technical protocol of comparison, selection, and research of traveling standard, and the development of draft comparison reports. Coordination of the entire work of

the PL as part of comparison is carried out by the contact person of PL.

traveling standard returns to PL for research of their drift.

The diagram of concept relations for standard comparisons.

The organizational scheme for standard comparisons.

Figure 3.

Figure 4.

4

3. The data evaluation of standard comparisons

Standards, Methods and Solutions of Metrology

RMO organizes KC with a number of joint NMI/DI participants with CC KC. This is necessary in order to link the results of the RMO KC with the results of the CC KC. For this purpose, equivalent technical protocols of both comparisons are used. The procedures for evaluating the data obtained at RMO KC are necessary to establish the DoE of national standards of NMI participants. PL calculates the KC RV and DoE for all NMI participants when preparing draft of comparison reports. The procedures used for evaluating RMO SC data are the same as for RMO KC. SC RMO complements KC CC or RM KC and is not second level comparison. RMO SC results are also published in KCDB of BIPM [16, 17].

RMO KC and RMO SC data evaluation usually includes determining the following characteristics: determining the RV comparison with the corresponding uncertainty, the DoE with corresponding uncertainties for each NMI/DI participant, and a pair DoE of i-th NMI/DI participant and j-th NMI/DI participant with corresponding uncertainties [6, 7]. RMO KC data evaluation includes the definition of such additional characteristics: converted KC data with corresponding uncertainties and DoE with corresponding uncertainties for each NMI/DI participant, except for linking NMI/DI.

The RMO KC/SC RV XRV is calculated as the mean of NMI/DI participant results from RMO KC/SC data are given by

$$X\_{RV} = \sum\_{i=1}^{n} \frac{\varkappa\_{\text{NMIi}}}{\mathfrak{u}^2(\varkappa\_{\text{NMIi}})} / \sum\_{i=1}^{n} \frac{1}{\mathfrak{u}^2(\varkappa\_{\text{NMIi}})} \tag{1}$$

with the combined standard uncertainty

$$\mathfrak{u}^2(\mathbf{X}\_{RV}) = \mathbf{1} / \sum\_{i=1}^n \frac{\mathbf{1}}{\mathfrak{u}^2(\mathfrak{x}\_{NMIi})},\tag{2}$$

where xNMIi is the result for i-th NMI/DI participant in RMO KC/SC; u xð Þ NMIi is corresponding standard uncertainty for i-th NMI/DI participant in RMO KC/SC; i ¼ 1, 2, …, n, n is the total number of NMI/DI participants of RMO KC/SC.

The DoE of i-th NMI/DI participant DNMIi and corresponding combined standard uncertainty u Dð Þ NMIi are estimated as

$$D\_{\rm NMHi} = \mathcal{X}\_{\rm NMHi} - \mathcal{X}\_{\rm RV},\tag{3}$$

$$
\mu^2(D\_{\rm NMIi}) = \mu^2(\mathfrak{x}\_{\rm NMIi}) + \mathfrak{u}^2(X\_{\rm RV}).\tag{4}
$$

Pairs of DoE of i-th NMI/DI participant and j-th NMI/DI participant DNMIij of RMO KC/SC and corresponding combined standard uncertainty u DNMIij are estimated as

$$D\_{\rm NMMij} = \mathfrak{x}\_{\rm NMMi} - \mathfrak{x}\_{\rm NMMi},\tag{5}$$

$$
\mu^2 \left( D\_{\text{NMij}} \right) = \mu^2 \left( \mathfrak{x}\_{\text{NMij}} \right) + \mu^2 \left( \mathfrak{x}\_{\text{NMij}} \right). \tag{6}
$$

On the basis of the measurement results of RMO KC/SC and corresponding combined standard uncertainties claimed by NMI/DI participants of RMO KC/SC, the chi-square test value is calculated [7].

$$\chi^2 = \sum\_{i=1}^n \frac{D\_{\text{NMli}}^2}{\mathfrak{u}^2(\mathfrak{x}\_{\text{NMli}})}.\tag{7}$$

If the calculated chi-criterion value does not exceed the chi-square test critical value with the coverage level of 0.95 and freedom degrees of n – 1

Standards, Methods and Solutions of Metrology

$$
\chi^2 \prec \chi^2\_{0.95}(n-1),
\tag{8}
$$

comparisons for all NMI/DI participants were checked for En number using Eq. (9). The resulting En number values for all NMI/DI participants do not exceed the

Results for the NMI/DI participants of COOMET.EM-K4 comparison are shown

COOMET.EM-K6.a comparison of AC voltage of 3 V at frequency of 20 kHz at frequencies of 1, 20, 100, and 1 MHz was organized UMTS and carried out in 2013– 2014. KV of COOMET.EM-K6.a of AC/DC voltage transfer of AC voltage of 3 V at frequency of 20 kHz is XKV = �2.0 μV/V, and corresponding combined standard uncertainty is u(XKV) = 1.9 μV/V (k = 2 for coverage level of 0.95). DoE for NMI/DI participants of COOMET.EM-K6.a comparison for AC voltage of 3 V at frequency

Results of COOMET.EM-K6.a comparison of AC/DC voltage transfer of AC voltage of 3 V at a frequency of 20 kHz were checked for the fulfillment of the chicriterion. The obtained value of the chi-square test for all NMI/DI participants can

<sup>0</sup>:95ð Þ¼ n � 1 0:71 without INM data). The same results of COOMET.EM-K6.a comparisons for all NMI/DI participants were checked for En number using Eq. (9). The resulting En number values for all NMI/DI participants

Results for the NMI/DI participants of COOMET.EM-K6.a comparison are shown in Table 2 for AC/DC voltage transfer of AC voltage of 3 V at a frequency of

CMC [8] has three unambiguous characteristics: measurand, measurement range, and measurement uncertainty (generally given at a confidence level of 0.95).

NMI BIM PTB VNIIM KazInMetr UMTS BelGIM DNMI, μF/F 0.43 0.16 �0.06 �0.41 0.05 �0.10 u(DNMI), μF/F 1.16 0.18 0.15 0.33 0.19 1.09 En 0.19 0.46 0.20 0.61 0.13 0.05

Results for NMI/DI participants of COOMET.EM-K4 comparison.

DoE for NMI/DI participants of COOMET.EM-K6.a comparison.

be considered consistent, since the condition of expression (8) is satisfied

in Table 1 for a nominal capacitance of 10 pF at a frequency of 1000 Hz.

value 1.0.

(χ<sup>2</sup> <sup>¼</sup> <sup>0</sup>:<sup>64</sup> <sup>≺</sup>χ<sup>2</sup>

20 kHz.

Table 1.

Figure 6.

7

do not exceed the value 1.0.

of 20 kHz [19] is shown in Figure 6.

Metrological Traceability at Different Measurement Levels

DOI: http://dx.doi.org/10.5772/intechopen.84853

then data can be acknowledged as consistent. This is the objective confirmation of declared uncertainties.

The NMI/DI participants of RMO KC/SC that provides maximum En number are determined [7].

$$\max\_{i} E\_{n} = \frac{|D\_{\text{NMli}}|}{2\sqrt{\mathfrak{u}^{2}(\mathfrak{x}\_{\text{NMli}}) - \mathfrak{u}^{2}(X\_{RV})}}.\tag{9}$$

Then the data of NMI/DI participants with the largest value of En number are temporarily excluded from consideration, and the procedure for checking of consistency of the comparison data is repeated. Sequential data exclusion is repeated until the condition (8) is fulfilled.

The State Enterprise "Ukrmetrteststandard" (UMTS) was PL of several COOMET KCs and SCs in the field of electricity and magnetism (EM) in 2005– 2018. UMTS as PL prepared and agreed with all NMI/DI participants draft reports on comparison COOMET.EM-K4, COOMET.EM-K5, COOMET.EM-K6.a, COOMET.EM-S2, COOMET.EM-S4, COOMET.EM-S13, COOMET.EM-S14, which comparison results are published in the KCDB of BIPM.

COOMET.EM-K4 comparison of national standards of a nominal capacitance of 10 pF at frequencies of 1000 and 1593 Hz was organized UMTS and carried out in 2005–2009. KV of COOMET.EM-K4 is XKV = �0.13 μF/F at a frequency of 1000 Hz, and corresponding combined standard uncertainty is u(XKV) = 0.22 μF/F (k = 2 for coverage level of 0.95). DoE for NMI/DI participants of COOMET.EM-K4 comparison for a nominal capacitance of 10 pF at a frequency of 1000 Hz [18] is shown in Figure 5.

Results of COOMET.EM-K4 comparison for a nominal capacitance of 10 pF at a frequency of 1000 Hz were checked for the fulfillment of the chi-square test. The obtained value of the chi-square test for all NMI/DI participants can be considered consistent, since the condition of expression (8) is satisfactory (χ<sup>2</sup> <sup>¼</sup> <sup>0</sup>:<sup>68</sup> <sup>≺</sup>χ<sup>2</sup> <sup>0</sup>:95ð Þ¼ n � 1 1:15). The same results of COOMET.EM-K4

Figure 5. DoE for NMI/DI participants of COOMET.EM-K4 comparison.

Metrological Traceability at Different Measurement Levels DOI: http://dx.doi.org/10.5772/intechopen.84853

comparisons for all NMI/DI participants were checked for En number using Eq. (9). The resulting En number values for all NMI/DI participants do not exceed the value 1.0.

Results for the NMI/DI participants of COOMET.EM-K4 comparison are shown in Table 1 for a nominal capacitance of 10 pF at a frequency of 1000 Hz.

COOMET.EM-K6.a comparison of AC voltage of 3 V at frequency of 20 kHz at frequencies of 1, 20, 100, and 1 MHz was organized UMTS and carried out in 2013– 2014. KV of COOMET.EM-K6.a of AC/DC voltage transfer of AC voltage of 3 V at frequency of 20 kHz is XKV = �2.0 μV/V, and corresponding combined standard uncertainty is u(XKV) = 1.9 μV/V (k = 2 for coverage level of 0.95). DoE for NMI/DI participants of COOMET.EM-K6.a comparison for AC voltage of 3 V at frequency of 20 kHz [19] is shown in Figure 6.

Results of COOMET.EM-K6.a comparison of AC/DC voltage transfer of AC voltage of 3 V at a frequency of 20 kHz were checked for the fulfillment of the chicriterion. The obtained value of the chi-square test for all NMI/DI participants can be considered consistent, since the condition of expression (8) is satisfied (χ<sup>2</sup> <sup>¼</sup> <sup>0</sup>:<sup>64</sup> <sup>≺</sup>χ<sup>2</sup> <sup>0</sup>:95ð Þ¼ n � 1 0:71 without INM data). The same results of COOMET.EM-K6.a comparisons for all NMI/DI participants were checked for En number using Eq. (9). The resulting En number values for all NMI/DI participants do not exceed the value 1.0.

Results for the NMI/DI participants of COOMET.EM-K6.a comparison are shown in Table 2 for AC/DC voltage transfer of AC voltage of 3 V at a frequency of 20 kHz.

CMC [8] has three unambiguous characteristics: measurand, measurement range, and measurement uncertainty (generally given at a confidence level of 0.95).


Table 1.

χ<sup>2</sup> ≺χ<sup>2</sup>

max En <sup>i</sup>

comparison results are published in the KCDB of BIPM.

of declared uncertainties.

Standards, Methods and Solutions of Metrology

until the condition (8) is fulfilled.

determined [7].

shown in Figure 5.

(χ<sup>2</sup> <sup>¼</sup> <sup>0</sup>:<sup>68</sup> <sup>≺</sup>χ<sup>2</sup>

Figure 5.

6

DoE for NMI/DI participants of COOMET.EM-K4 comparison.

then data can be acknowledged as consistent. This is the objective confirmation

The NMI/DI participants of RMO KC/SC that provides maximum En number are

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2ð Þ� xNMIi u2ð Þ XRV

<sup>¼</sup> DNMIi j j

Then the data of NMI/DI participants with the largest value of En number are temporarily excluded from consideration, and the procedure for checking of consistency of the comparison data is repeated. Sequential data exclusion is repeated

The State Enterprise "Ukrmetrteststandard" (UMTS) was PL of several COOMET KCs and SCs in the field of electricity and magnetism (EM) in 2005– 2018. UMTS as PL prepared and agreed with all NMI/DI participants draft reports

COOMET.EM-S2, COOMET.EM-S4, COOMET.EM-S13, COOMET.EM-S14, which

1000 Hz, and corresponding combined standard uncertainty is u(XKV) = 0.22 μF/F (k = 2 for coverage level of 0.95). DoE for NMI/DI participants of COOMET.EM-K4 comparison for a nominal capacitance of 10 pF at a frequency of 1000 Hz [18] is

Results of COOMET.EM-K4 comparison for a nominal capacitance of 10 pF at a frequency of 1000 Hz were checked for the fulfillment of the chi-square test. The obtained value of the chi-square test for all NMI/DI participants can be considered consistent, since the condition of expression (8) is satisfactory

<sup>0</sup>:95ð Þ¼ n � 1 1:15). The same results of COOMET.EM-K4

COOMET.EM-K4 comparison of national standards of a nominal capacitance of 10 pF at frequencies of 1000 and 1593 Hz was organized UMTS and carried out in

on comparison COOMET.EM-K4, COOMET.EM-K5, COOMET.EM-K6.a,

2005–2009. KV of COOMET.EM-K4 is XKV = �0.13 μF/F at a frequency of

<sup>0</sup>:95ð Þ n � 1 , (8)

<sup>p</sup> : (9)

Results for NMI/DI participants of COOMET.EM-K4 comparison.

Figure 6.

DoE for NMI/DI participants of COOMET.EM-K6.a comparison.


The standard uncertainty sð Þ ΔiLink associated with ΔiLink is calculated by the rootsum-square of the transfer standard uncertainty in CC KC: uT is transfer standard uncertainty in RMO KC; u(pi) is standard uncertainty associated with the imperfect reproducibility of results of NMIiLink in time period spanning two measurements; riLink is uncertainty associated with the imperfect reproducibility of measurement results of NMIiLink in time period spanning its two measurements in CC KC and

Table 3 lists the quantity values used in calculation linking total correction factor Δ and corresponding standard deviation sð Þ Δ for CCEM-K4 and COOMET.- EM-K4 comparisons for nominal capacitance 10 pF at a frequency of 1592 Hz [18].

ð Þ¼ <sup>Δ</sup> <sup>u</sup><sup>2</sup>

EM-K4 results to the CCEM-K4 results for nominal capacitance of 10 pF at frequency 1592 Hz [18, 23, 24] is shown in Figure 7. When linking results of those

S20 comparison (two RMO SCs for an inductance of 100 mH at frequency

PTB �0.00 �0.17 0.17 0.02 0.08 0.07 0.15 0.51

The expanded uncertainty is U dð Þ¼ NMIi ku dð Þ NMIi which is chosen k = 2 for a

An example of linking of EUROMET.EM-K4, APMP.EM-K4.1, and COOMET.-

Results of EUROMET.EM-S26 comparison have been linked to EUROMET.EM-

Linking NMI diLink DiLink Δ<sup>i</sup>Link uT u(pi) riLink sðΔ<sup>i</sup>LinkÞ wiLink Δ sðΔÞ VNIIM �0.12 �0.10 �0.02 0.02 0.08 0.07 0.16 0.49 0.11 0.11

Corrected DoE for participants of CCEM-K4, EUROMET.EM-K4, APMP.EM-K4.1, and COOMET.EM-K4

where u Xð Þ RV is combined standard uncertainty in CC KC RV.

ð Þþ DNMIi s

2

ð Þþ <sup>Δ</sup> <sup>u</sup><sup>2</sup>

ð Þ XRV , (15)

RMO KC; i ¼ 1, 2, …, k, k is total number of linking NMIs/DIs.

The combined standard uncertainty is calculated as:

ð Þþ DNMIi <sup>u</sup><sup>2</sup>

Metrological Traceability at Different Measurement Levels

DOI: http://dx.doi.org/10.5772/intechopen.84853

comparisons, the presented linking procedure was used.

CCEM-K4 and COOMET.EM-K4 data for linking NMIs, μF/F.

u2

Table 3.

Figure 7.

9

comparisons.

coverage level of 0.95.

ð Þ¼ dNMIi <sup>u</sup><sup>2</sup>

Table 2.

Results for NMI/DI participants of COOMET.EM-K6.a comparison.

They also contain a description of the used method or used measuring system, values of influence parameters, and any other relevant information. Normally for CMC, there are four ways in which a complete statement of uncertainty may be expressed: measurement range, equation, fixed measurand, and a matrix of measurement uncertainties.

CMC must be consistent with information from some or all of the following sources: results of KC and SC, knowledge of technical activities by other NMIs/DIs, including publications, other available knowledge and experience, etc. Results of RMO KCs/SCs are the ideal supporting evidence, but they can be used for fixed measurand only.

Methodologies for estimating the measurement uncertainty in a wide range of capacitance from 10 pF to 10 nF at frequencies of 1000 Hz and 1592 Hz and of inductance from 10 μH to 10 Hz at 1000 Hz are described in [20, 21], respectively. In these methodologies, requirements of both GUM [11] and regional recommendation [22] are used.

### 4. Linking procedures for international comparisons

Only CC KC results have a KC RV. Through joint NMI/DI participants, RMO KC must be linked to corresponding CC KC. The complete results of the linked RMO KC are presented in exactly the same form as the corresponding CC KC in KCDB of BIPM [4].

DoE of i-th NMI/DI participant of RMO KC is estimated as

$$d\_{\rm NMIi} = D\_{\rm NMIi} + \Delta,\tag{10}$$

where DNMIi is result for NMI/DI participant from RMO KC only; dNMIi is result for i-th NMI/DI participant which is linked to CC KC.

The correction factor for i-th linking NMI/DI is estimated as

$$
\Delta\_{iLink} = d\_{iLink} - D\_{iLink} \tag{11}
$$

where diLink is result for i-th linking NMI/DI from CC KC; DiLink is result for i-th linking NMI/DI from RMO KC.

The total correction factor Δ is then calculated as the weighted mean of the correction factor for linking NMI/DI participants, that is:

$$
\Delta = \sum\_{iLink}^{k} w\_{iLink} \Delta\_{iLink} \tag{12}
$$

$$w\_{iLink} = \frac{s^2(\Delta)}{s^2(\Delta\_{iLink})},\tag{13}$$

$$s^2(\Delta) = \mathbf{1} / \sum\_{iLink}^k \frac{\mathbf{1}}{s^2(\Delta\_{iLink})}.\tag{14}$$

Metrological Traceability at Different Measurement Levels DOI: http://dx.doi.org/10.5772/intechopen.84853

The standard uncertainty sð Þ ΔiLink associated with ΔiLink is calculated by the rootsum-square of the transfer standard uncertainty in CC KC: uT is transfer standard uncertainty in RMO KC; u(pi) is standard uncertainty associated with the imperfect reproducibility of results of NMIiLink in time period spanning two measurements; riLink is uncertainty associated with the imperfect reproducibility of measurement results of NMIiLink in time period spanning its two measurements in CC KC and RMO KC; i ¼ 1, 2, …, k, k is total number of linking NMIs/DIs.

Table 3 lists the quantity values used in calculation linking total correction factor Δ and corresponding standard deviation sð Þ Δ for CCEM-K4 and COOMET.- EM-K4 comparisons for nominal capacitance 10 pF at a frequency of 1592 Hz [18].

The combined standard uncertainty is calculated as:

$$
\mu^2(d\_{\rm NMIi}) = \mu^2(D\_{\rm NMIi}) + \mu^2(\Delta) = \mu^2(D\_{\rm NMIi}) + \sigma^2(\Delta) + \mu^2(X\_{RV}), \tag{15}
$$

where u Xð Þ RV is combined standard uncertainty in CC KC RV.

The expanded uncertainty is U dð Þ¼ NMIi ku dð Þ NMIi which is chosen k = 2 for a coverage level of 0.95.

An example of linking of EUROMET.EM-K4, APMP.EM-K4.1, and COOMET.- EM-K4 results to the CCEM-K4 results for nominal capacitance of 10 pF at frequency 1592 Hz [18, 23, 24] is shown in Figure 7. When linking results of those comparisons, the presented linking procedure was used.

Results of EUROMET.EM-S26 comparison have been linked to EUROMET.EM-S20 comparison (two RMO SCs for an inductance of 100 mH at frequency


Table 3.

They also contain a description of the used method or used measuring system, values of influence parameters, and any other relevant information. Normally for CMC, there are four ways in which a complete statement of uncertainty may be expressed: measurement range, equation, fixed measurand, and a matrix of mea-

Results for NMI/DI participants of COOMET.EM-K6.a comparison.

Standards, Methods and Solutions of Metrology

NMI VNIIM SMS BelGIM INM UMTS DNMI, μV/V 0.48 13.98 11.98 �1.12 0.38 u(DNMI), μV/V 1.05 10.96 14.47 1.19 2.00 En 0.23 0.64 0.41 0.47 0.10

CMC must be consistent with information from some or all of the following sources: results of KC and SC, knowledge of technical activities by other NMIs/DIs, including publications, other available knowledge and experience, etc. Results of RMO KCs/SCs are the ideal supporting evidence, but they can be used for fixed

Methodologies for estimating the measurement uncertainty in a wide range of capacitance from 10 pF to 10 nF at frequencies of 1000 Hz and 1592 Hz and of inductance from 10 μH to 10 Hz at 1000 Hz are described in [20, 21], respectively. In these methodologies, requirements of both GUM [11] and regional recommen-

Only CC KC results have a KC RV. Through joint NMI/DI participants, RMO KC must be linked to corresponding CC KC. The complete results of the linked RMO KC are presented in exactly the same form as the corresponding CC KC in KCDB of BIPM [4].

where DNMIi is result for NMI/DI participant from RMO KC only; dNMIi is result

where diLink is result for i-th linking NMI/DI from CC KC; DiLink is result for i-th

<sup>2</sup>ð Þ <sup>Δ</sup> s<sup>2</sup>ð Þ ΔiLink

> 1 s<sup>2</sup>ð Þ ΔiLink

k iLink

The total correction factor Δ is then calculated as the weighted mean of the

dNMIi ¼ DNMIi þ Δ, (10)

ΔiLink ¼ diLink � DiLink (11)

wiLinkΔiLink, (12)

, (13)

: (14)

4. Linking procedures for international comparisons

DoE of i-th NMI/DI participant of RMO KC is estimated as

The correction factor for i-th linking NMI/DI is estimated as

Δ ¼ ∑ k iLink

wiLink <sup>¼</sup> <sup>s</sup>

ð Þ¼ Δ 1= ∑

for i-th NMI/DI participant which is linked to CC KC.

correction factor for linking NMI/DI participants, that is:

s 2

linking NMI/DI from RMO KC.

8

surement uncertainties.

measurand only.

Table 2.

dation [22] are used.

CCEM-K4 and COOMET.EM-K4 data for linking NMIs, μF/F.

#### Figure 7.

Corrected DoE for participants of CCEM-K4, EUROMET.EM-K4, APMP.EM-K4.1, and COOMET.EM-K4 comparisons.

1000 Hz) with used special linking procedure [25] which is similar to the described linking procedure. Results of COOMET.EM-S2 comparison have been linked to EURAMET.EM-K5.1 comparison for electrical power [26]; results of COOMET.EM-S1 comparison have been linked to COOMET.EM-K6.a comparison of AC/DC voltage transfer difference [27] (RMO SC to RMO KC for similar values of physical quantities). When linking results of those comparisons, the described linking procedure was used.

Table 4 lists data for calculated total correction factors Δ and corresponding combined standard uncertainties u(Δ) for linking of COOMET.EM-S1 comparison results to COOMET.EM-K6.a comparison results for AC voltage of 3 V at frequencies of 1 kHz, 20 kHz, and 100 kHz [19], where XK6aKV is COOMET.EM-K6.a RV; u (XK6aKV) is combined standard uncertainty of COOMET.EM-K6.a RV.

Linked results of COOMET.EM-S1 (mark \*) and COOMET.EM-K6.a comparison of AC/DC voltage transfer difference of AC voltage of 3 V at frequencies of 1, 20, and 100 kHz [27] are shown in Figure 8. When linking results of those comparisons, the presented linking procedure was used.

For consistency verification of results of COOMET.EM-K6.a and COOMET.EM-S1 comparisons, the value of chi-square test was calculated. The obtained value of chisquare test for all participants can be considered consistent: <sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:58≺χ<sup>2</sup> <sup>0</sup>:95ð Þ n � 1 <sup>¼</sup> <sup>0</sup>:71 (without VNIIM result) at frequency 1 kHz; <sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:46<sup>≺</sup> <sup>χ</sup><sup>2</sup> <sup>0</sup>:95ð Þ¼ n � 1 0:71 at frequency 20 kHz; and <sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:49≺χ<sup>2</sup> <sup>0</sup>:95ð Þ¼ n � 1 0:71 (without VNIIM result) at frequency 100 kHz.

confirmed by Eqs. (8) and (9) accordingly. Results for NMI/DI participants of COOMET.EM-K6.a and COOMET.EM-S1 comparisons are satisfactory (Table 5).

Results for NMI/DI participants of COOMET.EM-K6.a and COOMET.EM-S1.

Metrological Traceability at Different Measurement Levels

DOI: http://dx.doi.org/10.5772/intechopen.84853

NMI VNIIM UMTS BelGIM INM UMTS\*

DNMI, μV/V 1.10 0.00 4.10 1.20 2.60 u(DNMI), μV/V 0.90 2.02 11.97 1.22 2.35 En 0.28 0.00 0.17 0.28 0.44

DNMI, μV/V 0.48 0.38 11.98 1.12 0.20 u(DNMI), μV/V 1.06 2.00 14.47 1.19 2.45 En 0.11 0.07 0.41 0.26 0.03

DNMI, μV/V 1.81 3.19 25.80 5.99 0.60 u(DNMI), μV/V 1.03 3.84 69.50 5.75 3.45 En 0.28 0.32 0.19 0.46 0.06

5. The data evaluation of national inter-laboratory comparisons

considered in [31–33], etc.

The organizational scheme for ILCs.

Figure 9.

11

1 kHz

20 kHz

100 kHz

Table 5.

A number of studies are devoted to urgent questions of the data evaluation of ILC: the use of different methods for inconsistent data evaluation of ILC discussed in [28], suggested approaches to verifying the reliability of measurement results for CL participations of ILC [29], the application of z score test for performance evaluation of CLs recommended instead of En number since this number is not applicable due to the difficulty in determining the assigned value (AV) [30], algorithms for conducting ILC and obtaining precision data for CMC evaluation of laboratories are

The general scheme of ILC is shown in Figure 9. Lab 1 is reference laboratories (RLs) of ILC. This scheme can be either circular or radial. Most often, a mixed

The maximum En number and declared uncertainties for DoE of NMI/DI participants of COOMET.EM-K6.a and COOMET.EM-S1 comparisons are judged as


#### Table 4.

Data for linking of COOMET.EM-S1 comparison results to COOMET.EM-K6.a comparison results, μV/V.

Figure 8. Corrected DoE for participants of COOMET.EM-K6.a and COOMET.EM-S1 comparisons.



Table 5.

1000 Hz) with used special linking procedure [25] which is similar to the described linking procedure. Results of COOMET.EM-S2 comparison have been linked to EURAMET.EM-K5.1 comparison for electrical power [26]; results of COOMET.EM-S1 comparison have been linked to COOMET.EM-K6.a comparison of AC/DC voltage transfer difference [27] (RMO SC to RMO KC for similar values of physical quantities). When linking results of those comparisons, the described linking pro-

Table 4 lists data for calculated total correction factors Δ and corresponding combined standard uncertainties u(Δ) for linking of COOMET.EM-S1 comparison results to COOMET.EM-K6.a comparison results for AC voltage of 3 V at frequencies of 1 kHz, 20 kHz, and 100 kHz [19], where XK6aKV is COOMET.EM-K6.a RV; u

Linked results of COOMET.EM-S1 (mark \*) and COOMET.EM-K6.a comparison of AC/DC voltage transfer difference of AC voltage of 3 V at frequencies of 1, 20, and 100 kHz [27] are shown in Figure 8. When linking results of those compari-

For consistency verification of results of COOMET.EM-K6.a and COOMET.EM-S1 comparisons, the value of chi-square test was calculated. The obtained value of chi-

The maximum En number and declared uncertainties for DoE of NMI/DI participants of COOMET.EM-K6.a and COOMET.EM-S1 comparisons are judged as

Frequency XK6aKV u(XK6aKV) Δ u(Δ) 1 kHz 0.30 0.85 �0.60 1.15 20 kHz �2.00 0.95 1.70 1.30 100 kHz �6.80 1.70 5.60 1.85

Data for linking of COOMET.EM-S1 comparison results to COOMET.EM-K6.a comparison results, μV/V.

Corrected DoE for participants of COOMET.EM-K6.a and COOMET.EM-S1 comparisons.

<sup>0</sup>:95ð Þ n � 1

<sup>0</sup>:95ð Þ¼ n � 1 0:71 at

<sup>0</sup>:95ð Þ¼ n � 1 0:71 (without VNIIM result) at

(XK6aKV) is combined standard uncertainty of COOMET.EM-K6.a RV.

square test for all participants can be considered consistent: <sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:58≺χ<sup>2</sup>

<sup>¼</sup> <sup>0</sup>:71 (without VNIIM result) at frequency 1 kHz; <sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:46<sup>≺</sup> <sup>χ</sup><sup>2</sup>

sons, the presented linking procedure was used.

Standards, Methods and Solutions of Metrology

frequency 20 kHz; and <sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:49≺χ<sup>2</sup>

frequency 100 kHz.

Table 4.

Figure 8.

10

cedure was used.

Results for NMI/DI participants of COOMET.EM-K6.a and COOMET.EM-S1.

confirmed by Eqs. (8) and (9) accordingly. Results for NMI/DI participants of COOMET.EM-K6.a and COOMET.EM-S1 comparisons are satisfactory (Table 5).

### 5. The data evaluation of national inter-laboratory comparisons

A number of studies are devoted to urgent questions of the data evaluation of ILC: the use of different methods for inconsistent data evaluation of ILC discussed in [28], suggested approaches to verifying the reliability of measurement results for CL participations of ILC [29], the application of z score test for performance evaluation of CLs recommended instead of En number since this number is not applicable due to the difficulty in determining the assigned value (AV) [30], algorithms for conducting ILC and obtaining precision data for CMC evaluation of laboratories are considered in [31–33], etc.

The general scheme of ILC is shown in Figure 9. Lab 1 is reference laboratories (RLs) of ILC. This scheme can be either circular or radial. Most often, a mixed

Figure 9. The organizational scheme for ILCs.

scheme of ILCs is used: after several Lab participants, the traveling standard is returned to RL for research of its drift.

ILCs based on fundamental requirements: the repeatability and instability of traveling standard. Main steps common to nearly all ILCs are: the determination of AV, the calculation of performance statistics, the evaluation of performance, and the preliminary determination of ILC traveling standard stability [14].

The RL processes the data received from CL participants according to results of ILC for CL. Verification of ILC data is required for consistency. In the case of uncoordinated data, an analysis is conducted for the purpose of rejecting these data or for further harmonization by correction of the applied indicators. To verify the consistency of data, comparative analyses of the relevant criteria for performance statistics are carried out and the most effective for use in processing of the data is selected [14, 34].

There are various procedures available for the establishment of AV. These procedures involve the use of, in particular AVs—as determined by analysis, the measurement or standard comparison, traceable to a national or an international standard. The general algorithm for data evaluation of ILC is described in [35]. This algorithm allows RL to take into account all the reporting features of ILC.

The laboratory difference Dlabj for j-th CL participant of ILC is calculated using Equation [14, 35, 36].

Dlabj ¼ xlabj � XAV, (16)

that is, normal or robust standard deviation, based on the results of ILC participat-

ζ ¼ Dlabj=

and be given the appropriate performance evaluation [35]. The value of expanded uncertainty U Xð Þ AV is estimated as

U Xð Þ¼ AV 2 �

The value of standard uncertainty u xð Þ stab is estimated as

For checking consistency of the ILC data, a ζ scores is used, which is calculated

where u xlabj � � is the combined standard uncertainty associated with result of the laboratory participating in the ILC; u Xð Þ AV is the combined standard uncertainty of


2.0 |z| < 3.0 and 2.0 < |ζ| < 3.0 indicate a dubious performance characteristic


Obvious blunders, such as those with incorrect units, decimal point errors, and

where u xref � � is the standard uncertainty obtained by calibrating traveling standard with a RL; u xð Þ stab is the standard uncertainty from the instability of traveling

u xð Þ¼ stab <sup>Δ</sup>Xmax<sup>=</sup> ffiffiffi

where ΔXmax is the maximum change in nominal value of traveling standard

that is, the most accurate ILCs are those that are performed by NMIs.

Linking the correspondingly expanded uncertainties of AV UAV when RL of ILC are NMIs, accredited by CLs or accredited RLs that are not NMIs or accredited by

The value of the expanded uncertainty UAV NMI for a case where the NMI is RL can be derived from results of corresponding international comparisons of national standards in which the NMI participated. The value of the expanded uncertainty UAV CL for a case where CL is RL can be derived from corresponding calibration certificates for working standards issued by the NMI using CL in ILC. The value of the expanded uncertainty UAV RL for a case where an RL is an accredited provider can be obtained from corresponding calibration certificates for working standards

An example of the laboratory difference Dlab of lab participants for national ILC of AC/DC voltage transfer difference of AC voltage of 3 V at a frequency of 20 kHz

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>u</sup><sup>2</sup> xref � � <sup>þ</sup> <sup>u</sup><sup>2</sup>ð Þ xstab <sup>q</sup>

3

UAV NMI < UAV CL < UAV RL, (23)

results for a different ILC item will be removed from the data set and treated separately. These results will not be subject to outlier tests or robust statistical methods. If results are removed as outliers, they will be removed only for calculation of summary statistics. These results should still be evaluated within ILC scheme

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>u</sup><sup>2</sup> xlabj � � � <sup>u</sup>2ð Þ XAV <sup>q</sup>

(20)

, (21)

<sup>p</sup> , (22)

ing laboratories, etc.

For a z scores and a ζ scores:

and require precautionary measures;

standard during ILC period.

during ILC period.

13

CLs, is as follows [36]:

issued by accredited CLs that use RLs in ILC.

require adjustment or response measures.

not require adjustment or response measures;

Metrological Traceability at Different Measurement Levels

DOI: http://dx.doi.org/10.5772/intechopen.84853

by the equation

ILC AV.

where xlabj is the measured value fori-th CL; XAV is AV which is determined by RL. The percent laboratory difference D%labj for ILC is calculated using equation

$$D\_{\text{@alobj}} = \left[D\_{\text{labj}} / \mathbf{X}\_{AV}\right] \cdot \mathbf{100}.\tag{17}$$

The criteria for performance evaluation will be established after taking into account whether methods for evaluating the performance characteristics consider the main features, namely: the statistical determination of indicators, i.e. when the criteria must be suitable for each indicator; the compliance with the purpose, given criteria that take into account, for example, technical specifications for characteristics of method and recognized level of participant studies, etc. [14].

The most often to check consistency of ILC data that uses En number which is calculated using equation

$$E\_n = D\_{labj} / \sqrt{U^2(\mathbf{x}\_{labj}) - U^2(\mathbf{X}\_{AV})},\tag{18}$$

where U xlabj � � is the expanded uncertainty of a participant's result; U Xð Þ AV is the expanded uncertainty of RL's AV.

For an En number:



For checking consistency of ILC data, a z scores is also used, which is calculated by the equation

$$z = D\_{\text{labj}} / \sigma\_{\text{\textquotedblleft}} \tag{19}$$

where σ is the standard deviation for qualification assessment.

The value of σ can be calculated based on [14]: estimates from a statistical model (main model) or results of a precision experiment, estimates from previous ILC rounds or assumptions based on experience, results of participating laboratories,

that is, normal or robust standard deviation, based on the results of ILC participating laboratories, etc.

For checking consistency of the ILC data, a ζ scores is used, which is calculated by the equation

$$\mathcal{L} = D\_{\text{labj}} / \sqrt{\mathfrak{u}^2(\mathfrak{x}\_{\text{labj}}) - \mathfrak{u}^2(X\_{AV})} \tag{20}$$

where u xlabj � � is the combined standard uncertainty associated with result of the laboratory participating in the ILC; u Xð Þ AV is the combined standard uncertainty of ILC AV.

For a z scores and a ζ scores:

scheme of ILCs is used: after several Lab participants, the traveling standard is

ILC for CL. Verification of ILC data is required for consistency. In the case of uncoordinated data, an analysis is conducted for the purpose of rejecting these data or for further harmonization by correction of the applied indicators. To verify the consistency of data, comparative analyses of the relevant criteria for performance statistics are carried out and the most effective for use in processing of the data is

algorithm allows RL to take into account all the reporting features of ILC.

D%labj ¼ Dlabj=XAV

tics of method and recognized level of participant studies, etc. [14].

q

where σ is the standard deviation for qualification assessment.

En ¼ Dlabj=


The criteria for performance evaluation will be established after taking into account whether methods for evaluating the performance characteristics consider the main features, namely: the statistical determination of indicators, i.e. when the criteria must be suitable for each indicator; the compliance with the purpose, given criteria that take into account, for example, technical specifications for characteris-

The most often to check consistency of ILC data that uses En number which is

U<sup>2</sup> xlabj

For checking consistency of ILC data, a z scores is also used, which is calculated

The value of σ can be calculated based on [14]: estimates from a statistical model (main model) or results of a precision experiment, estimates from previous ILC rounds or assumptions based on experience, results of participating laboratories,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ XAV

z ¼ Dlabj=σ, (19)

, (18)

� � � <sup>U</sup><sup>2</sup>

� � is the expanded uncertainty of a participant's result; U Xð Þ AV is

the preliminary determination of ILC traveling standard stability [14].

ILCs based on fundamental requirements: the repeatability and instability of traveling standard. Main steps common to nearly all ILCs are: the determination of AV, the calculation of performance statistics, the evaluation of performance, and

The RL processes the data received from CL participants according to results of

There are various procedures available for the establishment of AV. These procedures involve the use of, in particular AVs—as determined by analysis, the measurement or standard comparison, traceable to a national or an international standard. The general algorithm for data evaluation of ILC is described in [35]. This

The laboratory difference Dlabj for j-th CL participant of ILC is calculated using

where xlabj is the measured value fori-th CL; XAV is AV which is determined by RL. The percent laboratory difference D%labj for ILC is calculated using equation

Dlabj ¼ xlabj � XAV, (16)

� � � <sup>100</sup>: (17)

returned to RL for research of its drift.

Standards, Methods and Solutions of Metrology

selected [14, 34].

Equation [14, 35, 36].

calculated using equation

the expanded uncertainty of RL's AV.

where U xlabj

by the equation

12

For an En number:


2.0 |z| < 3.0 and 2.0 < |ζ| < 3.0 indicate a dubious performance characteristic and require precautionary measures;


Obvious blunders, such as those with incorrect units, decimal point errors, and results for a different ILC item will be removed from the data set and treated separately. These results will not be subject to outlier tests or robust statistical methods. If results are removed as outliers, they will be removed only for calculation of summary statistics. These results should still be evaluated within ILC scheme and be given the appropriate performance evaluation [35].

The value of expanded uncertainty U Xð Þ AV is estimated as

$$U(X\_{AV}) = 2 \cdot \sqrt{u^2(\varkappa\_{ref}) + u^2(\varkappa\_{stab})},\tag{21}$$

where u xref � � is the standard uncertainty obtained by calibrating traveling standard with a RL; u xð Þ stab is the standard uncertainty from the instability of traveling standard during ILC period.

The value of standard uncertainty u xð Þ stab is estimated as

$$
\mu(\mathbf{x}\_{\text{stab}}) = \Delta \mathbf{X}\_{\text{max}} / \sqrt{\mathbf{3}},\tag{22}
$$

where ΔXmax is the maximum change in nominal value of traveling standard during ILC period.

Linking the correspondingly expanded uncertainties of AV UAV when RL of ILC are NMIs, accredited by CLs or accredited RLs that are not NMIs or accredited by CLs, is as follows [36]:

$$U\_{AV\ NMI} \le U\_{AV\ CL} \le U\_{AV\ RL} \tag{23}$$

that is, the most accurate ILCs are those that are performed by NMIs.

The value of the expanded uncertainty UAV NMI for a case where the NMI is RL can be derived from results of corresponding international comparisons of national standards in which the NMI participated. The value of the expanded uncertainty UAV CL for a case where CL is RL can be derived from corresponding calibration certificates for working standards issued by the NMI using CL in ILC. The value of the expanded uncertainty UAV RL for a case where an RL is an accredited provider can be obtained from corresponding calibration certificates for working standards issued by accredited CLs that use RLs in ILC.

An example of the laboratory difference Dlab of lab participants for national ILC of AC/DC voltage transfer difference of AC voltage of 3 V at a frequency of 20 kHz

#### Figure 10.

Results of national ILC for AC/DC voltage transfer difference.


In [38], the proposed procedure links RMO KC/SC and ILC results for CL. This procedure can be used for practical estimation of specific ILC results on a national level in different countries by means of NMIs/DIs results from RMO KC/SC. The result of i-th NMI in some specific RMO KC/SC can be determined for linking in a specific ILC. Results of ILC will be expressed in relation to specific RMO KC/SC RV through linking laboratory—RL. For this purpose, the laboratory difference of ILC Dlabj will be corrected by a correction factor dlab, which is determined from the results of participant Lab 1 (RL) in RMO KC/SC and ILC (Lab 1 – NMI i):

ð Þþ DNMIi <sup>u</sup><sup>2</sup>

The corrected DoE for j-th lab participant in ILC with respect to linking to RMO

labj <sup>¼</sup> <sup>u</sup><sup>2</sup> Dlabj <sup>þ</sup> <sup>u</sup><sup>2</sup>

labj 

 =U D<sup>0</sup>

labj 

 

where σlab is the standard deviation, based on the results of ILC participating

with the combined standard uncertainty:

The organizational scheme for linking of RMO KC/SC and national ILC.

Metrological Traceability at Different Measurement Levels

DOI: http://dx.doi.org/10.5772/intechopen.84853

u2

with the combined standard uncertainty:

u<sup>2</sup> D<sup>0</sup>

The values of En number is determined by the equation

Enlabj ¼ D<sup>0</sup>

zlabj ¼ D<sup>0</sup>

The values of z scores is determined by the equation

KC/SC RV is estimated as

Figure 11.

laboratories.

15

ð Þ¼ dlab <sup>u</sup><sup>2</sup>

D0

dlab ¼ DNMIi � Dlab<sup>1</sup> (24)

=2: (25)

labj ¼ Dlabj þ dlab (26)

ð Þ dlab (27)

labj <sup>≤</sup>1:0: (28)

=σlab <sup>&</sup>lt; <sup>2</sup>:0, (29)

ð Þ Dlab<sup>1</sup>

#### Table 6.

Results for all lab participants of ILC.

with respect to the AV with expanded uncertainty U(Dlab) [37] is shown in Figure 10.

For verification of consistency of the ILC results, the value of chi-square test was calculated. The obtained value of chi-square test for lab participants can be considered consistent: <sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>:52≺χ<sup>2</sup> <sup>0</sup>:95ð Þ¼ n � 1 2:73 (without Lab3 and Lab 4 results). Results for lab participants of ILC are satisfactory (Table 6).

### 6. Linking procedures for international comparisons and national interlaboratory comparisons

ILCs for CLs are carried out in different countries. To ensure the mutual recognition of calibration results, it is advisable to establish the relationship between these ILCs. To do this, NMI/DI results of international standard comparisons can be used. In this case, the DoE of NMI/DI standards and their uncertainty may be taken into account. Thus, it is possible to establish the metrological traceability of CL standards to corresponding national standards.

The organizational scheme of linking of international standard comparison and national ILC is shown in Figure 11. The Lab 1 is RL for ILC which is also i-th NMI for RMO KC/SC.

Metrological Traceability at Different Measurement Levels DOI: http://dx.doi.org/10.5772/intechopen.84853

#### Figure 11.

The organizational scheme for linking of RMO KC/SC and national ILC.

In [38], the proposed procedure links RMO KC/SC and ILC results for CL. This procedure can be used for practical estimation of specific ILC results on a national level in different countries by means of NMIs/DIs results from RMO KC/SC.

The result of i-th NMI in some specific RMO KC/SC can be determined for linking in a specific ILC. Results of ILC will be expressed in relation to specific RMO KC/SC RV through linking laboratory—RL. For this purpose, the laboratory difference of ILC Dlabj will be corrected by a correction factor dlab, which is determined from the results of participant Lab 1 (RL) in RMO KC/SC and ILC (Lab 1 – NMI i):

$$d\_{lab} = D\_{NMIi} - D\_{lab1} \tag{24}$$

with the combined standard uncertainty:

$$
\mu^2(d\_{lab}) = \left[\mu^2(D\_{NMIi}) + \mu^2(D\_{lab1})\right]/2. \tag{25}
$$

The corrected DoE for j-th lab participant in ILC with respect to linking to RMO KC/SC RV is estimated as

$$D'\_{labj} = D\_{labj} + d\_{lab} \tag{26}$$

with the combined standard uncertainty:

$$
u^2 \left( D\_{labj}' \right) = \mu^2 \left( D\_{labj} \right) + \mu^2 (d\_{lab}) \tag{27}$$

The values of En number is determined by the equation

$$E\_{nlabj} = \left| D'\_{labj} \right| / U \left( D'\_{labj} \right) \le 1.0. \tag{28}$$

The values of z scores is determined by the equation

$$\mathbf{z}\_{labj} = \left| \mathbf{D}'\_{labj} \right| / \sigma\_{lab} \le \mathbf{2}.\,\mathbf{0},\tag{29}$$

where σlab is the standard deviation, based on the results of ILC participating laboratories.

with respect to the AV with expanded uncertainty U(Dlab) [37] is shown in

Results for lab participants of ILC are satisfactory (Table 6).

For verification of consistency of the ILC results, the value of chi-square test was calculated. The obtained value of chi-square test for lab participants can be consid-

Lab Ref Lab 2 Lab 3 Lab 4 Lab 5 Dlab, μV/V 0.00 �42.00 17.40 28.10 68.20 u(Dlab), μV/V 2.25 32.50 9.60 14.10 1570.00 En 0.00 0.65 0.91 0.99 0.02 z 0.00 1.04 0.43 0.70 1.69 ζ 0.00 0.32 0.45 0.50 0.01

6. Linking procedures for international comparisons and national inter-

ILCs for CLs are carried out in different countries. To ensure the mutual recognition of calibration results, it is advisable to establish the relationship between these ILCs. To do this, NMI/DI results of international standard comparisons can be used. In this case, the DoE of NMI/DI standards and their uncertainty may be taken into account. Thus, it is possible to establish the metrological traceability of CL

The organizational scheme of linking of international standard comparison and national ILC is shown in Figure 11. The Lab 1 is RL for ILC which is also i-th NMI

<sup>0</sup>:95ð Þ¼ n � 1 2:73 (without Lab3 and Lab 4 results).

Figure 10.

Table 6.

Figure 10.

for RMO KC/SC.

14

ered consistent: <sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>:52≺χ<sup>2</sup>

Results for all lab participants of ILC.

laboratory comparisons

standards to corresponding national standards.

Results of national ILC for AC/DC voltage transfer difference.

Standards, Methods and Solutions of Metrology

important elements of providing metrological traceability. The general scheme of the global metrological traceability at different measurement levels is presented. NMIs/ DIs and accredited CLs play an important role in establishing those traceability.

Metrological Traceability at Different Measurement Levels

DOI: http://dx.doi.org/10.5772/intechopen.84853

The organizational scheme for standard comparisons and RMO KC and RMO SC data evaluation procedure is presented. Results of data evaluation for COOMET.- EM-K4 and COOMET.EM-K6.a comparisons are indicated. Results of those comparisons were checked for the fulfillment of the chi-square test. The obtained values of the chi-square test for NMI/DI participants are satisfactory. Results for all NMI/DI participants of those comparisons for En number are also satisfactory.

The procedure of linking of RMO KC and RMO SC results is presented. Linking of COOMET.EM-S1 and COOMET.EM-K6.a comparison results of AC/DC voltage transfer difference at different frequencies is presented. The value of chi-criterion for linked comparison results was calculated. The obtained value of chi-square test for NMI/DI participants of those comparisons is satisfactory. Results for all NMI/DI participants of those comparisons for En number (from 0.03 to 0.46) are also

Results of linking of COOMET.EM-S1 and COOMET.EM-K6.a comparison results can also be used as the technical basis of confirming CMC NMIs/DIs. Such work can be done by PL of RMO KC or RMO SC, as well as by NMI/DI experts. The NMIs/DIs also must implement a full assessment of the uncertainty budget and the metrological traceability for validation of their CMCs in a wide range of used

The organizational scheme for ILs and ILC data evaluation procedure is presented. Results of data evaluation for ILC of AC/DC voltage transfer difference are indicated. Results of this comparison were checked for the fulfillment of the chisquare test. The obtained value of the chi-square test for laboratory participants is satisfactory. Results for all laboratory participants of this comparison for En number

The organizational scheme of linking of international standard comparison and national ILC is indicated. The procedure of linking of RMO KC or RMO SC and national ILC results is presented. This procedure can be used for practical estimation of results specific ILC on a national level by means of the results from NMI/DI laboratories. Linking of COOMET.EM-K6.a comparison and national ILC of AC/DC voltage transfer difference results was presented. The value of chi-square test was calculated and the obtained value of chi-square test for all participants can be considered consistent. Results for all participants of comparisons are satisfactory for En number (from 0.10 to 0.83), z scores (from 0.01 to 2.22), and ζ scores (from 0.05

Results of this linking can be used also for different metrological areas as technical basis of confirming CMC accredited laboratories. Such work can be done by RL of the ILC, as well as by metrological experts. The RL of the ILC can also implement a full assessment of the uncertainty budget and the metrological trace-

ability for validation of their CMCs in a wide range of used quantities.

satisfactory.

quantities.

to 0.41).

17

are also satisfactory.

Figure 12.

The corrected laboratory difference for lab participants of national ILC for AC/DC voltage transfer standards with respect to linking to COOMET.EM-K6.a.


Table 7.

Results for all NMI/DI and lab participants.

The values of ζ scores is determined by the equation

$$\mathcal{L}\_{\text{labj}} = \left| D'\_{\text{labj}} \right| / \mu \left( D'\_{\text{labj}} \right) < 2.0. \tag{30}$$

An example of the corrected laboratory difference D<sup>0</sup> lab of lab participants for national ILC of AC/DC voltage transfer difference of AC voltage of 3 V at a frequency of 20 kHz with respect to linking to COOMET.EM-K6.a with expanded uncertainty [38] is shown in Figure 12. When linking results of those comparisons, the presented linking procedure was used.

For verification of consistency of COOMET.EM-K6.a and ILC results, the value of chi-square test was calculated. The obtained value of chi-square test for lab participants can be considered consistent: <sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:71≺χ<sup>2</sup> <sup>0</sup>:95ð Þ¼ n � 1 0:42. Results for all NMI/DI and lab participants are satisfactory (Table 7).

### 7. Conclusions

CIPM MRA and ILAC MRA are the basis for establishing the global metrological traceability and play an important role in overcoming technical barriers to international trade. The calibration hierarchy and measurement uncertainty evaluation are

### Metrological Traceability at Different Measurement Levels DOI: http://dx.doi.org/10.5772/intechopen.84853

important elements of providing metrological traceability. The general scheme of the global metrological traceability at different measurement levels is presented. NMIs/ DIs and accredited CLs play an important role in establishing those traceability.

The organizational scheme for standard comparisons and RMO KC and RMO SC data evaluation procedure is presented. Results of data evaluation for COOMET.- EM-K4 and COOMET.EM-K6.a comparisons are indicated. Results of those comparisons were checked for the fulfillment of the chi-square test. The obtained values of the chi-square test for NMI/DI participants are satisfactory. Results for all NMI/DI participants of those comparisons for En number are also satisfactory.

The procedure of linking of RMO KC and RMO SC results is presented. Linking of COOMET.EM-S1 and COOMET.EM-K6.a comparison results of AC/DC voltage transfer difference at different frequencies is presented. The value of chi-criterion for linked comparison results was calculated. The obtained value of chi-square test for NMI/DI participants of those comparisons is satisfactory. Results for all NMI/DI participants of those comparisons for En number (from 0.03 to 0.46) are also satisfactory.

Results of linking of COOMET.EM-S1 and COOMET.EM-K6.a comparison results can also be used as the technical basis of confirming CMC NMIs/DIs. Such work can be done by PL of RMO KC or RMO SC, as well as by NMI/DI experts. The NMIs/DIs also must implement a full assessment of the uncertainty budget and the metrological traceability for validation of their CMCs in a wide range of used quantities.

The organizational scheme for ILs and ILC data evaluation procedure is presented. Results of data evaluation for ILC of AC/DC voltage transfer difference are indicated. Results of this comparison were checked for the fulfillment of the chisquare test. The obtained value of the chi-square test for laboratory participants is satisfactory. Results for all laboratory participants of this comparison for En number are also satisfactory.

The organizational scheme of linking of international standard comparison and national ILC is indicated. The procedure of linking of RMO KC or RMO SC and national ILC results is presented. This procedure can be used for practical estimation of results specific ILC on a national level by means of the results from NMI/DI laboratories. Linking of COOMET.EM-K6.a comparison and national ILC of AC/DC voltage transfer difference results was presented. The value of chi-square test was calculated and the obtained value of chi-square test for all participants can be considered consistent. Results for all participants of comparisons are satisfactory for En number (from 0.10 to 0.83), z scores (from 0.01 to 2.22), and ζ scores (from 0.05 to 0.41).

Results of this linking can be used also for different metrological areas as technical basis of confirming CMC accredited laboratories. Such work can be done by RL of the ILC, as well as by metrological experts. The RL of the ILC can also implement a full assessment of the uncertainty budget and the metrological traceability for validation of their CMCs in a wide range of used quantities.

The values of ζ scores is determined by the equation

An example of the corrected laboratory difference D<sup>0</sup>

participants can be considered consistent: <sup>χ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:71≺χ<sup>2</sup>

for all NMI/DI and lab participants are satisfactory (Table 7).

the presented linking procedure was used.

7. Conclusions

16

Figure 12.

D0

Table 7.

u D<sup>0</sup> lab

with respect to linking to COOMET.EM-K6.a.

Standards, Methods and Solutions of Metrology

Results for all NMI/DI and lab participants.

ζlabj ¼ D<sup>0</sup>

labj 

 <sup>=</sup>u D<sup>0</sup> labj 

The corrected laboratory difference for lab participants of national ILC for AC/DC voltage transfer standards

NMI-Lab VNIIM SMS BelGIM INM UMTS Lab 2 Lab 3 Lab 4 Lab 5

lab, μV/V 0.50 14.00 12.00 �1.10 0.40 �46.60 12.80 23.50 63.60

 , μV/V 1.05 10.95 14.45 1.20 2.00 32.50 9.85 14.20 157.00 En 0.23 0.64 0.41 0.47 0.10 0.72 0.65 0.83 0.20 z 0.02 0.49 0.42 0.04 0.01 1.62 0.45 0.82 2.22 ζ 0.11 0.32 0.21 0.24 0.05 0.36 0.33 0.41 0.10

national ILC of AC/DC voltage transfer difference of AC voltage of 3 V at a frequency of 20 kHz with respect to linking to COOMET.EM-K6.a with expanded uncertainty [38] is shown in Figure 12. When linking results of those comparisons,

of chi-square test was calculated. The obtained value of chi-square test for lab

For verification of consistency of COOMET.EM-K6.a and ILC results, the value

CIPM MRA and ILAC MRA are the basis for establishing the global metrological traceability and play an important role in overcoming technical barriers to international trade. The calibration hierarchy and measurement uncertainty evaluation are

< 2:0: (30)

lab of lab participants for

<sup>0</sup>:95ð Þ¼ n � 1 0:42. Results

Standards, Methods and Solutions of Metrology

References

10-01-2019]

1/012017

[1] Text of the CIPM MRA [Internet]. 1999. Available from: https://www.b ipm.org/utils/en/pdf/CIPM-MRA-2003.

DOI: http://dx.doi.org/10.5772/intechopen.84853

Metrological Traceability at Different Measurement Levels

documents/CIPM-MRA/CIPM-MRA-D-

measurement capabilities of metrological institutes: Features of preparation, examination, and publication.

Measurement Techniques. 2010;53(6): 721-726. DOI: 10.1007/s11018-010-9567-x

metrology—Basic and general concepts and associated terms (VIM). 3rd edition. JCGM 200 [Internet]. 2012. Available from: https://www.bipm.org/utils/c ommon/documents/jcgm/JCGM\_200\_ 2012.pdf [Accessed: 10-01-2019]

[11] Uncertainty of measurement—Part 3: Guide to the expression of uncertainty in measurement (GUM). JCGM 100 [Internet]. 2008. Available from: https://www.bipm.org/utils/common/ documents/jcgm/JCGM\_100\_2008\_ E.pdf [Accessed: 10-01-2019]

[12] ILAC Policy on Traceability of Measurement Results. ILAC Р10:01/2013 [Internet]. 2013. Available from: https:// ilac.org/publications-and-resources/ilacpolicy-series/ [Accessed: 10-01-2019]

[13] ISO/IEC 17025:2017. General requirements for the competence of testing and calibration laboratories. Switzerland: ISO/IEC; 2017. p. 30

[14] ISO/IEC 17043:2010. Conformity Assessment. General requirements for proficiency testing. Switzerland: ISO/

[15] Velichko ON. Traceability of measurement results at different levels of metrological work. Measurement Techniques. 2009;52(11):1242-1248. DOI: 10.1007/s11018-010-9428-7

[16] Cox MG. The evaluation of key comparison data. Metrologia. 2002;39: 589-595. DOI: 10.1088/0026-1394/39/6/10

IEC; 2010. p. 39

04.pdf [Accessed: 10-01-2019]

[9] Velichko ON. Calibration and

[10] International vocabulary of

[2] Velychko O, Gordiyenko T. The implementation of general guides and standards on regional level in the field of metrology. Journal of Physics: Conference Series. 2010;238:012044:6. DOI: 10.1088/1742-6596/238/1/012044

[3] The BIPM key comparison database (KCDB) [Internet]. Available from: http://kcdb.bipm.org/ [Accessed:

[4] Measurement comparisons in the context of the CIPM MRA. CIPM MRA-D-05 [Internet]. 2016. Available from: https://www.bipm.org/utils/common/ documents/CIPM-MRA/CIPM-MRA-D-05.pdf [Accessed: 10-01-2019]

[5] Velychko O, Gordiyenko T. The estimation of the measurement results with using statistical methods. Journal of Physics: Conference Series. 2015;588: 012017:6. DOI: 10.1088/1742-6596/588/

[6] CООМЕТ R/GM/14:2016. Guidelines for data evaluation of COOMET key comparison [Internet]. 2016. Available from: http://www.coomet.org/DB/isapi/

[7] CООМЕТ R/GM/19:2016. Guideline

cmt\_docs/2016/5/2BMD1O.pdf

on COOMET supplementary comparison evaluation [Internet]. 2016. Available from: http://www.c oomet.org/DB/isapi/cmt\_docs/ 2016/5/21XQGO.pdf [Accessed:

[8] Calibration and Measurement Capabilities in the context of the CIPM MRA. CIPM MRA-D-04 [Internet]. 2013. Available from: https://www.

bipm.org/utils/common/

[Accessed: 10-01-2019]

10-01-2019]

19

pdf [Accessed: 10-01-2019]

### Author details

Oleh Velychko<sup>1</sup> \* and Tetyana Gordiyenko<sup>2</sup>


\*Address all correspondence to: velychko@ukrcsm.kiev.ua

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Metrological Traceability at Different Measurement Levels DOI: http://dx.doi.org/10.5772/intechopen.84853

### References

[1] Text of the CIPM MRA [Internet]. 1999. Available from: https://www.b ipm.org/utils/en/pdf/CIPM-MRA-2003. pdf [Accessed: 10-01-2019]

[2] Velychko O, Gordiyenko T. The implementation of general guides and standards on regional level in the field of metrology. Journal of Physics: Conference Series. 2010;238:012044:6. DOI: 10.1088/1742-6596/238/1/012044

[3] The BIPM key comparison database (KCDB) [Internet]. Available from: http://kcdb.bipm.org/ [Accessed: 10-01-2019]

[4] Measurement comparisons in the context of the CIPM MRA. CIPM MRA-D-05 [Internet]. 2016. Available from: https://www.bipm.org/utils/common/ documents/CIPM-MRA/CIPM-MRA-D-05.pdf [Accessed: 10-01-2019]

[5] Velychko O, Gordiyenko T. The estimation of the measurement results with using statistical methods. Journal of Physics: Conference Series. 2015;588: 012017:6. DOI: 10.1088/1742-6596/588/ 1/012017

[6] CООМЕТ R/GM/14:2016. Guidelines for data evaluation of COOMET key comparison [Internet]. 2016. Available from: http://www.coomet.org/DB/isapi/ cmt\_docs/2016/5/2BMD1O.pdf [Accessed: 10-01-2019]

[7] CООМЕТ R/GM/19:2016. Guideline on COOMET supplementary comparison evaluation [Internet]. 2016. Available from: http://www.c oomet.org/DB/isapi/cmt\_docs/ 2016/5/21XQGO.pdf [Accessed: 10-01-2019]

[8] Calibration and Measurement Capabilities in the context of the CIPM MRA. CIPM MRA-D-04 [Internet]. 2013. Available from: https://www. bipm.org/utils/common/

documents/CIPM-MRA/CIPM-MRA-D-04.pdf [Accessed: 10-01-2019]

[9] Velichko ON. Calibration and measurement capabilities of metrological institutes: Features of preparation, examination, and publication. Measurement Techniques. 2010;53(6): 721-726. DOI: 10.1007/s11018-010-9567-x

[10] International vocabulary of metrology—Basic and general concepts and associated terms (VIM). 3rd edition. JCGM 200 [Internet]. 2012. Available from: https://www.bipm.org/utils/c ommon/documents/jcgm/JCGM\_200\_ 2012.pdf [Accessed: 10-01-2019]

[11] Uncertainty of measurement—Part 3: Guide to the expression of uncertainty in measurement (GUM). JCGM 100 [Internet]. 2008. Available from: https://www.bipm.org/utils/common/ documents/jcgm/JCGM\_100\_2008\_ E.pdf [Accessed: 10-01-2019]

[12] ILAC Policy on Traceability of Measurement Results. ILAC Р10:01/2013 [Internet]. 2013. Available from: https:// ilac.org/publications-and-resources/ilacpolicy-series/ [Accessed: 10-01-2019]

[13] ISO/IEC 17025:2017. General requirements for the competence of testing and calibration laboratories. Switzerland: ISO/IEC; 2017. p. 30

[14] ISO/IEC 17043:2010. Conformity Assessment. General requirements for proficiency testing. Switzerland: ISO/ IEC; 2010. p. 39

[15] Velichko ON. Traceability of measurement results at different levels of metrological work. Measurement Techniques. 2009;52(11):1242-1248. DOI: 10.1007/s11018-010-9428-7

[16] Cox MG. The evaluation of key comparison data. Metrologia. 2002;39: 589-595. DOI: 10.1088/0026-1394/39/6/10

Author details

Oleh Velychko<sup>1</sup>

18

\* and Tetyana Gordiyenko<sup>2</sup>

1 State Enterprise "Ukrmetrteststandard", Kyiv, Ukraine

\*Address all correspondence to: velychko@ukrcsm.kiev.ua

provided the original work is properly cited.

Standards, Methods and Solutions of Metrology

2 Odesa State Academy of Technical Regulation and Quality, Odesa, Ukraine

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

[17] Mana G, Massa E, Predescu M. Model selection in the average of inconsistent data: An analysis of the measured Planck-constant values. Metrologia. 2012;49:492-500. DOI: 10.1088/0026-1394/49/4/492

[18] Velychko O, Akhmadov O. Final report on COOMET key comparison of capacitance at 10 pF (COOMET.EM-K4). Metrologia. 2017;54(1A):01005. DOI: 10.1088/0026-1394/54/1A/01005

[19] Velychko O, Darmenko Y. Final report on COOMET key comparison of AC/DC voltage transfer reference (COOMET.EM-K6.a). Metrologia. 2016; 53(1A):01011. DOI: 10.1088/0026-1394/ 53/1A/01011

[20] Velychko O, Shevkun S. Support of metrological traceability of capacitance measurements in Ukraine. Eastern-European Journal of Enterprise Technologies. 2017;3(9 (87)):4-10. DOI: 10.15587/1729-4061.2017.101897

[21] Velychko O, Shevkun S. A support of metrological traceability of inductance measurements in Ukraine. Eastern-European Journal of Enterprise Technologies. 2017;5(9 (89)):12-18. DOI: 10.15587/1729-4061.2017.109750

[22] Evaluation of the Uncertainty of Measurement in Calibration. EA-04/02 М [Internet]. 2013. Available from: https://european-accreditation.org/wpcontent/uploads/2018/10/ea-4-02-mrev01-september-2013.pdf [Accessed: 10-01-2019]

[23] Delahaye F, Witt TJ. Linking the results of key comparisons CCEM-K4 with the 10 pF results of EUROMET.EM-K4. Metrologia. 2002; 39:01005

[24] Velychko O. Proposals for linking the results of key comparisons CCEM-K4 and COOMET.EM-K4. In: Proceedings of the Conference on Precision Electromagnetic

Measurements (CPEM 2010); 2010; Daejeon; South Korea: CPEM; 2010; 5545263. pp. 414-415. DOI: 10.1109/ CPEM.2010.5545263

Series. 2018;1065:082007:4. DOI: 10.1088/1742-6596/1065/8/082007

Korea. IMEKO; 2012. p. 6

[32] Sandu I, Dragomir L. Interlaboratory comparison. In: Proceedings of the 15th IMEKO TC 4 Symposium on Novelties in Electrical Measurements and Instrumentations; 2007; Iasi, Romania. IMEKO; 2007. p. 4

[31] Claudio J, Costa M. Brazilian energy interlaboratory program applicative. In: Proceedings of the XX IMEKO World Congress "Metrology for Green Growth"; 2012; Busan, Republic of

DOI: http://dx.doi.org/10.5772/intechopen.84853

Metrological Traceability at Different Measurement Levels

Electrical Engineering and Information

[38] Velychko O, Gordiyenko T. Linking Results of International Comparisons of the National Standard and the National Inter-Laboratory Comparisons. XXII World Congress of the International Measurement Confederation (IMEKO 2018). Journal of Physics: Conference Series. 2018;1065(4):072004. DOI: 10.1088/1742-6596/1065/7/072004

Technologies. 2018;3(1–2):5-12

[33] Sousa JJL, Leitão LTS, Costa MM, Faria MC. Considerations on the influence of travelling standards instability in an interlaboratory

comparison program. In: Proceedings of

[34] ISO 13528:2015. Statistical methods for use in proficiency testing by interlaboratory comparisons. Switzerland: ISO; 2015. p. 89

[35] Velychko O, Shevkun S, Gordiyenko T, Mescheriak O. Interlaboratory comparisons of the calibration results of time meters. Eastern-European Journal of Enterprise Technologies. 2018;1/9

(91):4-11. DOI: 10.15587/ 1729-4061.2018.121089

[36] Velychko O, Gordiyenko T.

DOI: 10.30748/soi.2018.155.10

[37] Velychko O, Isaiev V.

21

Features of the processing of results and estimation of measurement uncertainty of inter-laboratory comparison for calibration laboratories. Information Processing Systems. 2018;4(155):77-83.

Interlaboratory comparison in context of inappropriate results of voltage thermal converter calibration. Journal of

the XX IMEKO World Congress "Metrology for Green Growth"; 2012; Busan, Republic of Korea. IMEKO; 2012.

p. 4

[25] Dierikx E, Nestor A, Melcher J, Kölling A, Callegaro L. Final report on the supplementary comparison EURAMET.EM-S26: inductance measurements of 100 mH at 1 kHz (EURAMET project 816). Metrologia. 2012;49:01002

[26] Velychko O, Karpenko S. Linking results of key and supplementary comparisons of regional metrology organization for electrical power. International Journal of Metrology and Quality Engineering. 2016;7(3). DOI: 10.1051/ijmqe/2016014

[27] Velychko O. Linking results of key and supplementary comparisons of AC/DC voltage transfer standard. International Journal of Metrology and Quality Engineering. 2018;9:4. DOI: 10.1051/ijmqe/2018002

[28] Chunovkina A, Zviagin N, Burmistrova N. Interlaboratory comparisons. Practical approach for data evaluation. In: Proceedings of the XX IMEKO World Congress "Metrology for Green Growth"; 2012; Busan, Republic of Korea. IMEKO; 2012. p. 5

[29] Briggs P. Proficiency testing for calibration laboratories. In: Proceedings of the XX IMEKO World Congress "Metrology for Green Growth"; 2012; Busan, Republic of Korea. IMEKO; 2012. p. 5

[30] Beckert SF, Fischer GE. Interlaboratory comparison of roughness measurement: Application of Algorithm A of ISO 13528:2015 in determining the designated value and the standard deviation. XXII World Congress of the International Measurement Confederation (IMEKO 2018). Journal of Physics: Conference

Metrological Traceability at Different Measurement Levels DOI: http://dx.doi.org/10.5772/intechopen.84853

Series. 2018;1065:082007:4. DOI: 10.1088/1742-6596/1065/8/082007

[17] Mana G, Massa E, Predescu M. Model selection in the average of inconsistent data: An analysis of the measured Planck-constant values. Metrologia. 2012;49:492-500. DOI: 10.1088/0026-1394/49/4/492

Standards, Methods and Solutions of Metrology

Measurements (CPEM 2010); 2010; Daejeon; South Korea: CPEM; 2010; 5545263. pp. 414-415. DOI: 10.1109/

[25] Dierikx E, Nestor A, Melcher J, Kölling A, Callegaro L. Final report on the supplementary comparison EURAMET.EM-S26: inductance measurements of 100 mH at 1 kHz (EURAMET project 816). Metrologia.

[26] Velychko O, Karpenko S. Linking results of key and supplementary comparisons of regional metrology organization for electrical power. International Journal of Metrology and Quality Engineering. 2016;7(3). DOI:

[27] Velychko O. Linking results of key and supplementary comparisons of AC/DC voltage transfer standard. International Journal of Metrology and Quality Engineering. 2018;9:4. DOI:

CPEM.2010.5545263

2012;49:01002

10.1051/ijmqe/2016014

10.1051/ijmqe/2018002

p. 5

p. 5

[28] Chunovkina A, Zviagin N, Burmistrova N. Interlaboratory comparisons. Practical approach for data evaluation. In: Proceedings of the XX IMEKO World Congress "Metrology for Green Growth"; 2012; Busan, Republic of Korea. IMEKO; 2012.

[29] Briggs P. Proficiency testing for calibration laboratories. In: Proceedings of the XX IMEKO World Congress "Metrology for Green Growth"; 2012; Busan, Republic of Korea. IMEKO; 2012.

roughness measurement: Application of Algorithm A of ISO 13528:2015 in determining the designated value and the standard deviation. XXII World Congress of the International

Measurement Confederation (IMEKO 2018). Journal of Physics: Conference

[30] Beckert SF, Fischer GE. Interlaboratory comparison of

[18] Velychko O, Akhmadov O. Final report on COOMET key comparison of capacitance at 10 pF (COOMET.EM-K4). Metrologia. 2017;54(1A):01005. DOI: 10.1088/0026-1394/54/1A/01005

[19] Velychko O, Darmenko Y. Final report on COOMET key comparison of AC/DC voltage transfer reference (COOMET.EM-K6.a). Metrologia. 2016; 53(1A):01011. DOI: 10.1088/0026-1394/

[20] Velychko O, Shevkun S. Support of metrological traceability of capacitance measurements in Ukraine. Eastern-European Journal of Enterprise

Technologies. 2017;3(9 (87)):4-10. DOI:

[21] Velychko O, Shevkun S. A support

inductance measurements in Ukraine. Eastern-European Journal of Enterprise Technologies. 2017;5(9 (89)):12-18. DOI: 10.15587/1729-4061.2017.109750

[22] Evaluation of the Uncertainty of Measurement in Calibration. EA-04/02 М [Internet]. 2013. Available from: https://european-accreditation.org/wpcontent/uploads/2018/10/ea-4-02-mrev01-september-2013.pdf [Accessed:

[23] Delahaye F, Witt TJ. Linking the results of key comparisons CCEM-K4 with the 10 pF results of EUROMET.EM-K4. Metrologia. 2002;

[24] Velychko O. Proposals for linking the results of key comparisons CCEM-

K4 and COOMET.EM-K4. In: Proceedings of the Conference on

Precision Electromagnetic

10.15587/1729-4061.2017.101897

of metrological traceability of

53/1A/01011

10-01-2019]

39:01005

20

[31] Claudio J, Costa M. Brazilian energy interlaboratory program applicative. In: Proceedings of the XX IMEKO World Congress "Metrology for Green Growth"; 2012; Busan, Republic of Korea. IMEKO; 2012. p. 6

[32] Sandu I, Dragomir L. Interlaboratory comparison. In: Proceedings of the 15th IMEKO TC 4 Symposium on Novelties in Electrical Measurements and Instrumentations; 2007; Iasi, Romania. IMEKO; 2007. p. 4

[33] Sousa JJL, Leitão LTS, Costa MM, Faria MC. Considerations on the influence of travelling standards instability in an interlaboratory comparison program. In: Proceedings of the XX IMEKO World Congress "Metrology for Green Growth"; 2012; Busan, Republic of Korea. IMEKO; 2012. p. 4

[34] ISO 13528:2015. Statistical methods for use in proficiency testing by interlaboratory comparisons. Switzerland: ISO; 2015. p. 89

[35] Velychko O, Shevkun S, Gordiyenko T, Mescheriak O. Interlaboratory comparisons of the calibration results of time meters. Eastern-European Journal of Enterprise Technologies. 2018;1/9 (91):4-11. DOI: 10.15587/ 1729-4061.2018.121089

[36] Velychko O, Gordiyenko T. Features of the processing of results and estimation of measurement uncertainty of inter-laboratory comparison for calibration laboratories. Information Processing Systems. 2018;4(155):77-83. DOI: 10.30748/soi.2018.155.10

[37] Velychko O, Isaiev V. Interlaboratory comparison in context of inappropriate results of voltage thermal converter calibration. Journal of Electrical Engineering and Information Technologies. 2018;3(1–2):5-12

[38] Velychko O, Gordiyenko T. Linking Results of International Comparisons of the National Standard and the National Inter-Laboratory Comparisons. XXII World Congress of the International Measurement Confederation (IMEKO 2018). Journal of Physics: Conference Series. 2018;1065(4):072004. DOI: 10.1088/1742-6596/1065/7/072004

Chapter 2

Abstract

stage error

23

1. Introduction

Metrology Stages

meet practical industrial requirements.

Chuxiong Hu, Yu Zhu and Luzheng Liu

Self-Calibration of Precision XYθ<sup>z</sup>

This chapter studies the on-axis calibration for precision XYθ<sup>z</sup> metrology stages and presents a holistic XYθ<sup>z</sup> self-calibration approach. The proposed approach uses an artifact plate, specially designed with XY grid mark lines and angular mark lines, as a tool to be measured by the XYθ<sup>z</sup> metrology stages. In detail, the artifact plate is placed on the uncalibrated XYθ<sup>z</sup> metrology stages in four measurement postures or views. Then, the measurement error can be modeled as the construction of XYθ<sup>z</sup> systematic measurement error (i.e. stage error), artifact error, misalignment error, and random measurement noise. With a new property proposed, redundance of the XYθ<sup>z</sup> stage error is obtained, while the misalignment errors of all measurement views are determined by rigid mathematical processing. Resultantly, a least squarebased XYθ<sup>z</sup> self-calibration law is synthesized for final determination of the stage error. Computer simulation is conducted, and the calculation results validate that the proposed scheme can accurately realize the stage error even under the existence of various random measurement noise. Finally, the designed artifact plate is developed and illustrated for explanation of a standard XYθ<sup>z</sup> self-calibration procedure to

Keywords: XYθ<sup>z</sup> stage, self-calibration, measurement system, least square,

approach especially for micro-/nano-level mechanical systems [11–14].

Precision XYθ<sup>z</sup> motion stages are ubiquitously utilized in industrial mechanical systems to meet the requirement of high-performance manufacture [1]. As automatical servo systems, these stages have both precision linear encoders and angle encoders for measurement and motion feedback control [2–7]. In practice, the measurement accuracy inevitably suffers from surface non-flatness and un-roundness, axis nonorthogonality, scale graduation nonuniformity, encoder installation eccentricity, read-head misalignment, and so on, which resultantly generate systematic measurement error, i.e. stage error. The stage error can in principle be eliminated through calibration technology [8–10]. Due to the difficulty on finding a more accurate standard tool in traditional calibration technologies, self-calibration technology has been developed with utilization of an artifact with mark positions not precisely known. As an alternative of intelligent calibration processes, self-calibration is an effective and economical

Existing self-calibration technologies were developed for X, XY, XYZ, and angular metrology stages, respectively. For example, Takac studied one-dimensional

### Chapter 2
