Research on the national standard of the unit of mass in the range from 2 kg to 20 kg

It is difficult to imagine modern science, industry, and economics without mass measurements in a wide range: from a few tenths of a milligram up to hundreds or even thousands of tons. So, Ukraine, in this case, is not an exception. National sci - ence, medicine, industry, and other fields of activity increasingly feel the need for reliable and traceable results of mass measurements in the range of more than 1 kg. Considering this, a national standard of the unit of mass in the range from 2 kg to 20 kg was established in SE «Ukrmetrteststandart». This article provides information on the general structure of the national standard, its metrological characteristics, and the methods of its application. It is also described the dissemi - nation procedure of the unit of mass from the 1kg reference weight to the weights in the range from 2 kg to 20 kg, as well as the processing of the obtained measurement results by means of the least squares method. In addition, the article presents the results of the national standard’s research, the results of the calculation of the masses of the standard weights, and their uncertainty assessment. The obtained results confirm compliance of the metrological characteristics of the national standard to the declared ones. It has been demonstrated that the national standard provides calibration of weights of accuracy class E 1 in the range from 2 kg to 20 kg.


Introduction
Mass is an important and fundamental physical quantity. Its unit of measurement, the kilogram, along with six other units (meter, second, ampere, kelvin, candela, mole), form the basis of the International System of Units (SI). In addition, there are many other physical quantities whose measurements are closely related to mass measurements, such as force, torque, pressure, density, etc.
Mass measurements have accompanied modern society since ancient times. Some scientific sources [1] cite examples of the existence of weighing instruments in 3000 BC. Other autors [2] suggest that people used weighing instruments even earlier. For the most part, in those days, mass mea-surements were closely related to trade transac-

tions.
Modern science, medicine, industry, trade, etc., cannot be imagined without mass measurements: from a few tenths of a milligram (in fundamental scientific research and medicine) to hundreds and even thousands of tons (for example, in heavy industry and trade). An important task for scientific metrological institutions is to ensure the reliability, validity, and traceability of mass measurement results in a wide measurement range for the needs of various fields of activity.
If we talk about the needs of modern Ukraine in accurate measurements, today it is absolutely necessary to disseminate unit of mass to the E 1 accuracy class weights in the range from 1 mg to mass U, which for the weights of E 1 accuracy class should not exceed from 1 0 10 9 , ⋅ − kg to 3 3 10 6 , ⋅ − kg in the range from 1 mg to 20 kg.
Established in 1996 on the basis of the NSC «Institute of Metrology», the national standard of the unit of mass, which was granted national status in 2020, ensures the dissemination of a unit of mass in the range from 1 mg to 1 kg, with an expanded uncertainty U of 2 0 10 9 , ⋅ − kg to 57 10 9 ⋅ − kg (with coverage coefficient k = 2 and confidence interval P = 0 95 , ) 1

The national standard structure
The standard includes a complex of measuring equipment, technical devices, and auxiliary tools.   [4] .
After the completion in 2020 of the first inter-  Table 1, was increased after February 2021 to 50 10 6 ⋅ − g.
The mass comparators are used to accurately disseminate the unit of mass from the reference standard weights (refer to Table 2).
The comparators shown in Table 2  In order to realize the weighing scheme described in Section 4, additional standard weights are involved during the measurement process (Table 3).

Metrological characteristics of the national standard
The nominal mass of the reference standard weights is 1 kg.
The national standard provides the dissemination of the unit of mass in the range from 2 kg to 20 kg.
The expanded uncertainty of the dissemination of the unit of mass U with the coverage factor k = 2 with a confidence interval of P = 0 95 , as follows: -for 2 kg weights is less than 0 2 10 6 , − · kg; -for 5 kg weights is less than 0 5 10 6 , − · kg; -for 10 kg weights is less than 1 0 10 6 , − · kg; -for 20 kg weights is less than 3 3 10 6 , − · kg.

The dissemination procedure
The multiplication method is used to disseminate the unit of mass from the 1kg reference standard weights. This method is used when there is a need to determine the mass of each standard weight in a set (for example, a set of standard weights from 1 kg to 20 kg) with the help of one reference standard weight (for example, a standard weight of 1 kg) [3] , [5] , [6] , [7] . This is realized with the help of an appropriate weighing scheme (see Table 4), which from the mathematical perspective is represented by an overdetermined system of equations. According to this weighing scheme, mass comparisons of certain combinations of weights are carried out using the method of differential weighing according to the ABBA cycle (where A represents a standard weight and B represents a test weight or combination of weights), as described in [3] . To assess the magnetic properties of weights, the Sartorius Susceptometer YSZ02C is used. If the magnetic properties of the weights (the permanent magnetic polarization and the magnetic susceptibility) are lower than the values specified in [3] , it is assumed that the uncertainty due to the magnetic properties of the weights is small and can be neglected. Otherwise, additional actions are required, for example, as it is described in [3] .
The standard is kept in a specially designed, ther- -PC and appropriate software shall be suitable to solve the system of equations by the least square method.
The system of equations representing the weighing scheme may be written by the following equation in matrix form [6] , [7] : For the weighing scheme shown in Table 4, the left side of equation 1 is given by: Since the measurements according to the weighing scheme shown in Table 4 are performed on different mass comparators (with unequal accuracy), the system of equations (1) is solved by the least squares method using weighting coefficients to each equation. Further calculations are carried out according to the procedure described in [6] , [7] .

The research on the metrological characteristics of the national standard
In order to verify and confirm its stated metrological characteristics, research on the national standard was carried out. The research, in particular, was subject to: -standard weights; -mass comparators; -the weighing scheme, as well as the procedure for processing measurement results and uncertainty assessment.
Two reference standard weights were periodically calibrated in the Physikalisch-Technische Each mass comparator is annually investigated in order to evaluate and check its pooled standard deviation s p , as described in [8] , [9] . The pooled standard deviation of each mass comparator does not exceed the value shown in Table 2.
In 2019 a series of measurements were carried out to verify the declared metrological characteristics of the national standard. The weighing scheme provided in Table 4 β 10 β 11 β 12 1 ref 1 1* 1** 2 2* 2** 5 5* 10 10* 10** At the beginning of 2023, the periodic study of the national standard was carried out in the range from 1 kg to 10 kg according to the weighing scheme shown in Table 5. Some of the standard weights from the initial research in 2019 were involved in this periodic study. In particular, these were the 2 kg, 5 kg, and 10 kg standard weights marked in Table 4 as β 3 , β 4 , β 6 , β 7 , β 8 , and in Table 5 as β 6 , β 7 , β 9 , β 11 , β 12 respectively. These standard weights were used in order to compare the results of the initial research and the periodic study.
The summary of the periodic study (the conventional mass and associated expanded uncertainty) is given below: It is shown in Table 6 that the expanded uncertainties obtained from the results of the research are smaller than the maximum permissible expanded uncertainties for the E 1 class weights according to [3] .
Thus, it confirms that the national standard ensures the calibration of E 1 class weights in the range from 2 kg to 20 kg. This fully meets the needs of national science, medicine, trade, and industry in accurate mass measurements.
The results of the initial research and the periodic study carried out in 2021 and 2023 are shown in Table 7.
The expanded uncertainties of measurement results during the latest study are also less than the maximum permissible uncertainty for E 1 class weights and are very close to the uncertainty values obtained from the results of initial research. It demonstrates not only the high stability of the national standard and the repeatability of measurement results but also confirms its stated metrological characteristics.

Nominal weight mass, kg
The maximum permissible expanded uncertainty for