COMPUTER MODELING IN THE STUDY OF THE EFFECT OF NORMALIZED QUANTITIES ON THE MEASUREMENT ACCURACY OF THE QUADRATIC TRANSFORMATION FUNCTION

The research of the systems of equations of quantities describing, respectively, 5 and 6 measurement cycles revealed the peculiari- ties of redundancy formation. It is proved that the normalized temperature T 1 has the greatest effect on the measurement result for both systems. In addition, it was found that in both sys- tems, an increase in the reproduction accuracy of the normalized temperature T 1 (with a con- stant reproduction error of T 2 ) does not lead to a significant improvement in the results. Due to this, it can be argued on the use of non-pre-cision normalized sources to reproduce the temperature T 1 . However, an order of magnitude increase in the reproduction accuracy of both normalized quantities T 1 and T 2 also increases the measurement accuracy by an order of magnitude. Computer modeling confirmed that for the redundant measurement equation (11) at the ratio Т 1 =Т і (0.0005 • Т і +1) in the range (10 ÷ 200) °С, measurement with a relative error (0.01 ÷ 0.00003) % is provided. When applying the redundant measure- ment equation (13), the accuracy increases to 0.0059 % only at the end of the range. Based on the results obtained, it was found that the accuracy of redundant measurements is influ- enced by the type of equation itself, not their number. Processing of the results based on the redundant measurement equation, by the way, ensures the independence of the measurement result from the influence of absolute values of the transformation function parameters, as well as their deviations from nominal values under the influence of external destabilizing factors. Thus, there is reason to believe that it is possible to increase the accuracy of measure- ment in a wide range by observing the ratio between normalized and controlled quantities measurement equations, accuracy improvement, nor- malized quantities, reproduction errors of quantities


Introduction
The vast majority of production and technical processes involve measuring one or more desired physical quantities, the accuracy and reliability of which determine the quality of finished products or the final result of research. Therefore, for laboratory research or industrial application, sensors are used that have high requirements for accuracy, linearity and sensitivity. In this regard, relevant research and techniques are aimed at improving and further developing existing measurement methods and means to ensure the quality of measurement [1][2][3]. Thus, digital or analog methods are used to reduce the error from the nonlinearity of the transformation function. Linearization algorithms are applied when using digital methods, additional linearization schemes are applied when using analog methods. However, the use of these methods leads to a narrowing of the measurement range. The accuracy of the measurement result is also affected by the instability of the sensor transformation function caused by external destabilizing factors and aging of the materials of the sensitive elements. This, in turn, leads to the need for frequent sensor checks or calibration, which requires additional material and time costs.
Thus, research aimed at a comprehensive solution of the problem of improving measurement accuracy with nonlinear and unstable sensor transformation function is considered relevant.

Literature review and problem statement
The process of improving sensors and related research methods determines the overall quality of measurement.

COMPUTER MODELING IN THE STUDY OF THE EFFECT OF NORMALIZED QUANTITIES ON THE MEASUREMENT ACCURACY OF THE QUADRATIC TRANSFORMATION FUNCTION V o l o d y m y r S h c h e r b a n '
Corresponding author Doctor of Technical Sciences, Professor, Head of Department* Е-mail: shherban.vy@knutd.com.ua

G a n n a K o r o g o d
PhD, Associate Professor*

O k s a n a K o l y s k o
PhD, Associate Professor*

A n t o n i n a V o l i v a c h
PhD, Senior Lecturer*

Y u r y S h c h e r b a n '
Doctor of Technical Sciences, Professor, Head of Department**

G a n n a S h c h u t s k a
Doctor of Technical Sciences, Associate Professor, Director** *Department of Computer Science Kyiv National University of Technologies and Design Nemirovicha-Danchenko str., 2, Kyiv, Ukraine, 01011 **Department of Light Industry Technologies State Higher Educational Establishment "Kyiv College of Light Industry" Dzhona Makkeina str., 29, Kyiv, Ukraine, 01601 The research of the systems of equations of quantities describing, respectively, 5 and 6 measurement cycles revealed the peculiarities of redundancy formation. It is proved that the normalized temperature T 1 has the greatest effect on the measurement result for both systems. In addition, it was found that in both systems, an increase in the reproduction accuracy of the normalized temperature T 1 (with a constant reproduction error of T 2 ) does not lead to a significant improvement in the results. Due to this, it can be argued on the use of non-precision normalized sources to reproduce the temperature T 1

. However, an order of magnitude increase in the reproduction accuracy of both normalized quantities T 1 and T 2 also increases the measurement accuracy by an order of magnitude. Computer modeling confirmed that for the redundant measurement equation (11) at the ratio Т 1 =Т і (0.0005•Т і +1) in the range (10÷200) °С, measurement with a relative error (0.01÷0.00003) % is provided. When applying the redundant measurement equation (13), the accuracy increases to 0.0059 % only at the end of the range. Based on the results obtained, it was found that the accuracy of redundant measurements is influenced by the type of equation itself, not their number. Processing of the results based on the redundant measurement equation, by the way, ensures the independence of the measurement result from the influence of absolute values of the transformation function parameters, as well as their deviations from nominal values under the influence of external destabilizing factors. Thus, there is reason to believe that it is possible to increase the accuracy of measurement in a wide range by observing the ratio between normalized and controlled quantities Keywords: redundant methods, measurement equations, accuracy improvement, normalized quantities, reproduction errors of quantities
Therefore, it is important to create high-precision sensors and improve methods for reliable measurements. Various measures and methods are applied to improve the accuracy of measurements in a wide range: stable materials and calibration are used, systematic errors are corrected, linearization measures are carried out, sensitivity is increased, and so on. Thus, in [4], an increase in sensitivity was achieved using low-current base-collector voltage, and in [5] elements of solid microelectronics and multifunctional signal converters based on calorimetric methods were used. Ways to increase the speed of transistors based on materials science structures and schematic aspects of selectively doped heterotransistors are given in [6]. However, the issues related to reducing the systematic measurement error remained unresolved. An option to overcome these difficulties may be to use recursive methods. This approach is used in [7], where obtaining individual values of the resistances of resistors in an array of sensors made it possible to reduce cross-noise. The issue of reducing the systematic error of the measuring channel was also considered in [8], where the accuracy was improved by introducing corrections. However, the issue of obtaining reliable and useful information from sensors was not considered. So, in [9], to effectively obtain reliable information from an array of gas sensors, hardware data processing (analog interface) was used. Improving the efficiency of signal processing was also considered in [10], where the received signals were processed according to the proposed HD (hyperdimensional) computing. In [11], the informativeness of sensor signals was increased by combining thermal and capacitive measurement methods. However, there are still unresolved issues related to the nonlinearity of sensor characteristics, which leads to a narrowing of the range or an increase in the measurement error. A way to overcome these difficulties may be to use the linearization method. This approach is used in [12], which proposed to form a compensatory measuring current and change the transformation factor of the output scaling amplifier in certain temperature measurement ranges. In [13], by introducing the correction of nonlinear curvature at several reference temperatures, an increase in the sensor accuracy was achieved. Ways to expand the linear range by improving the signal processing algorithm were presented in [14]. However, the issues of improving accuracy by minimizing the impact of errors caused by the influence of destabilizing factors on measurement elements remain unresolved. An option to overcome these difficulties may be to use calibration methods. In [15], the calibration process is based on continuous temperature deployment while recording calibration data. The issue of accuracy improvement by calibration was also considered in [16]. The work found that the quality of calibration is affected by the number of coefficients in the calibration equation. Therefore, based on nine additional calibration equations, a new calibration equation was proposed, which provides high accuracy and reliability of the information obtained. The issue of improving accuracy through linearization and calibration was presented in [17], where a CMOS temperature sensor with integrated calibration for curve linearization was proposed. But such accuracy improvement approaches require the use of exemplary sources (materials) or high-precision components, as in [18], which require certain material costs for their implementation. Despite the practical significance of the results obtained, the issue of the scattering of characteristics from measurement to measurement on the one hand, and the nonlinearity of characteristics on the other, was not sufficiently considered. As an option to overcome difficulties in improving accuracy for unstable and nonlinear transformation functions in a wide range of values, redundant measurement methods can be recommended. Thus, the main theoretical aspects of redundant methods were considered in [19]. In [20], a structural analysis of the equations of redundant and super-redundant measurements describing redundant methods was performed in order to study the statistical characteristics before and after modification of their structure. However, of practical interest is also the mathematical form of redundant measurement equations. In [21], a universal measurement equation is mathematically presented. Despite the fact that various forms of redundant measurement equations were given, their practical implementation for various sensor transformation functions in order to obtain a highly accurate result was not demonstrated. Further development of measurement redundancy was obtained in [22], where the possibility of high-precision measurement of the resistance of resistive sensors by obtaining not one but several basic equations of redundant measurements was considered. However, such results were obtained with a linear transformation function. Therefore, to solve the problem of applying the redundant measurement method for a nonlinear transformation function, a sensor with a logarithmic transformation function was considered in [23]. The efficiency of the presented methods for improving the measurement accuracy and the possibility of providing metrological self-control are shown. However, the effect of the random component of the measurement error on the measurement result was not considered. An option to overcome the relevant difficulties may be the approach proposed in [24]. Processing of measurement results according to the proposed approaches allows us to assert an increase in measurement accuracy by eliminating the systematic component of the error caused by changing the parameters of the transformation function, as well as reducing the random component of the error. However, the issue of the influence of the values of normalized quantities and their relationship with the desired quantity on the final measurement result for a nonlinear transformation function remains unresolved. Therefore, further research on redundant measurement methods presented in [25] established a relationship between the normalized and desired quantities, which significantly expands the range of high-precision measurements. The results obtained demonstrated the high efficiency of redundant measurement methods in improving measurement accuracy by adjusting the values of normalized quantities for the logarithmic transformation function. However, sensors such as a transistor and a thermistor have quadratic transformation functions. Thus, it is of scientific interest to further identify ways to improve measurement accuracy for quadratic and unstable transformation functions.
This allows us to argue about the feasibility of research to determine the effect of normalized quantities and their relationships with the controlled quantity on the measurement accuracy with a quadratic sensor transformation function.

The aim and objectives of the study
The aim of the study is to determine the relationship between controlled and normalized quantities, which provides an increase in accuracy for the quadratic transformation function of the sensor. This will make it possible to control the measurement process with high accuracy over a wide measurement range by adjusting the values of normalized quantities.
To achieve the aim, the following objectives were accomplished: -to carry out computer modeling of a system of nonlinear equations of quantities consisting of 5 equations and the corresponding equation of redundant measurements to study the effect of normalized quantities on the measurement result; -to conduct a similar computer modeling of a system of nonlinear equations of quantities consisting of 6 equations and the corresponding equation of redundant measurements to study the effect of normalized quantities on the measurement result; -to make a comparative analysis of the results obtained applying each of the proposed systems with the corresponding equation of redundant measurements.

1. Research materials and modeling tools
The primary temperature converter, KT3132 A-2 silicon bipolar transistor (Ukraine), was chosen as the test material.
The Mathcad15.0 (USA) and MS Excel (USA) applications were used for mathematical modeling.

2. Method for studying redundant measurements
Redundant measurement methods (RMM) have proven themselves well for a comprehensive solution of the problem of improving accuracy over the entire measurement range. The essence of RMM is a time-separated measurement transformation, in addition to the desired physical quantity, of several normalized physical quantities, which are associated with the desired specific pattern. The number of such measurements depends on the number of parameters of the transformation function (TF). So, if the TF consists of (n) parameters, the number of measurement cycles must be at least (n+1). A feature of using RMM for nonlinear TF is the presence of measurement cycles in which the desired and normalized quantity (quantities) are simultaneously measured. In general, the mathematical model of RMM is a system of equations of quantities describing the measurement cycles, the solution of which gives the equations of redundant measurements of the desired physical quantity. Generally, a nonlinear TF: has 4 parameters, so it is necessary to make a system of at least 5 equations of quantities: , , , , .
The solution of this system relative to the desired quantity x i allows obtaining its value: ,..., , , , , where Hj y′ -sensor output signals ( j=5); x i -desired physical quantity; x 1 , …, x 3 -normalized physical quantities, the values of which are related by a certain law, from a source with normalized characteristics; , H S ′ L S ′ -sensitivity (steepness) of the transformation of the nonlinear and linear components of the transformation function; ∆y' -parameter (offset) of the transformation function taking into account the additive component of the error.
As can be seen from the redundant measurement equation (3) The application of RMM for quadratic TF and the effect of normalized quantities on the measurement result in cases where the system consists of (n+1) and (n+2) redundant measurement equations describing measurement cycles are considered below.
A quadratic TF has the form: Since the quadratic TF (4) has 4 parameters, two variants of systems (5), (6) were studied, consisting of 5 and 6 measurement cycles, described by the corresponding equations of quantities. Thus, if it is possible to form normalized physical quantities х 1 , 2х 1 and х 2 , a system of nonlinear equations of quantities takes the form: If it is possible to form normalized physical quantities х 1 , 2х 1 and 2х 2 , a system of nonlinear equations of quantities takes the form: As a result of solving systems (5) and (6), the corresponding equations of redundant measurements of the desired physical quantity x i were obtained: In equations of redundant measurements (7), (8) due to differences in the sensor output signals, the additive component of the measurement error is excluded, and due to their division, the multiplicative component of the systematic error is excluded. Thus, the influence of the parameters , H S ′ , L S ′ ∆y' on the measurement result is excluded, which confirms the effectiveness of using RMM in quadratic TF.

1. Computer modeling of a system of equations consisting of 5 equations of quantities
As is known [11], the dependence of the base-emitter voltage of a transistor on temperature is described by the equation: where U bet0 -value of the base-emitter voltage at t=0 °С; If the temperature values T 1 , 2T 1 and T 2 are formed using standard sources with normalized characteristics based on system (5), we obtain the following system of equations: .
As a result of solving system (10), the equation of redundant measurements of the desired temperature T i was obtained: As can be seen from the redundant measurement equation (11), it does not include the parameters , i. e. their influence on the measurement result is excluded.
For computer modeling, the limits of change for the parameters , ∆ ′ and 0 bet U ′ lie within ±10.0 %. The values of the reproduction error of the normalized temperatures T 1 and T 2 were chosen as ∆t=0.001 °C. Thus, the system (10) takes the form: During the modeling, a KT3132 A-2 transistor with a measurement range from 10 °С to 200 °С, at U bet0 =0.6 V and at ∆U A =1.882 mV/°С, ∆U B =0.41 μV/°С 2 was used [11].
To study the effect of the value of each of the normalized temperatures Т 1 and Т 2 on the measurement result of T i by equation (12), 3 cases were considered: As a result of computer modeling in the Mathcad15.0 environment, the following values of relative measurement errors were obtained in three sections of the operating range (at the beginning, middle and end), presented in Tables 1-3.
As can be seen from the tables above (Tables 1-3), there is a certain relationship between T i and normalized temperatures Т 1 and Т 2 . It was found that a more significant relationship was found between the values of T 1 and T i than between T 2 and T i . Thus, the lowest value of the relative error is obtained when T 1 is close to T i . The best accuracy results were obtained for the second case (Table 2) at T 1 =(10÷100) °С, T 2 =(1÷10) °С. For this case, the influence of reproduction errors of normalized temperatures T 1 and T 2 at different values was considered. It was found that increasing the reproduction accuracy of the normalized temperature T 1 (with a constant reproduction error of T 2 ) does not significantly improve the results, which gives grounds to use a non-precision normalized source for T 1 reproduction. On the other hand, an order of magnitude increase in the reproduction accuracy of both normalized temperatures Т 1 and Т 2 also increases the measurement accuracy by an order of magnitude.
Since the best accuracy results were obtained for the second case (Table 2) at Т 1 =(10÷100) °С, Т 2 =(1÷10) °С, it was considered in more detail. Studies were performed at Т 1 =(10÷100) °С with a step of 10 °С and at Т 2 =(1÷10) °С with a step of 1 °С, as well as at T i =50 °С. The obtained calculations of the relative measurement error for the second case are given in Table 4. As can be seen from Table 4, the results obtained confirm the fact that the value of T 1 should be equal to T i or higher, while the value of T 2 has little effect under these conditions. Thus, in subsequent calculations, the value Т 2 =10 °С was chosen. To find the desired relationship between the values of T 1 and T i , which provide high-precision measurement results, the MS Excel data analysis package for the nonlinear function was used. Using the solution search option and constructing a trend line, the optimal ratio between T 1 and T i was found, at which the measurement error decreases.
As can be seen from Fig. 1, there is a certain undesirable relationship between T 1 and T 2 , at which measurements become unacceptable. Based on Fig. 1 and calculations performed in the Mathcad15.0 environment, it was found that T 2 ≠2·T 1 . Moreover, as the difference between T 1 and T i increases, this ratio will not be point, but have a certain range with a maximum error at the point T 2 ≠2·T 1 . This indicates the expediency of observing the previously found relationship between the values of the normalized temperature T 1 and controlled temperature T i , which is T 1 =T i (0.0005·T i +1). In addition, when measuring at the beginning of the range at T 1 =T 2 =(10÷100) °С and T i =100 °С, it is necessary to adhere more precisely to the desired ratio between the normalized temperatures T 1 and Т 2 .
The worst accuracy results were obtained for the third case (Table 3) at Т 1 =(1÷10) °С, Т 2 =(10÷100) °С, where even the proximity of T 2 to T i does not lead to the desired result. Thus, it was found that when applying the redundant measurement equation (11), the value of T 2 should not exceed the value of Т 1 .

2. Computer modeling of a system consisting of 6 equations of quantities
Based on the dependence of the base-emitter voltage of a transistor on temperature described by equation (9), the system of nonlinear equations of quantities (6) has the following form: As a result of solving system (13), the equation of redundant measurements of the desired temperature T i was obtained: For computer modeling, the limits of changes similar to those for system (10) were set for the parameters ∆ ′ , lying within ±10.0 %. The value of the reproduction error of the normalized temperatures T 1 and T 2 was selected as ∆ t =0.001 C. Thus, the system (13) takes the form:    Table 4 Relative measurement errors (%) at Т 1 =(10÷100) °С, Т 2 =(1÷10) °С

Fig. 1. Graph of the relative error against the values of normalized temperatures
To logically compare the results obtained by equations (11) and (14), studies were conducted with the same initial data for three cases: 1) Т 1 =Т 2 =(10÷100) °С with a step of 30 °С; 2) Т 1 =(10÷100) °С, Т 2 =(1÷10) °С with a step of 30 °С; 3) Т 1 =(1÷10) °С, Т 2 =(10÷100) °С with a step of 30 °С. As a result of computer modeling in the Mathcad15.0 environment, the following values of relative measurement errors were obtained, presented in Table 5-7. Table 5 Relative measurement errors (%) at Т 1 =Т 2 =(10÷100) °С  Table 6 Relative measurement errors (%) at Т 1 =(10÷100) °С,  Table 7 Relative measurement errors (%) at Т 1 =(1÷10) °С, Т 2 =(10÷100) °С As can be seen from Tables 5-7, the best accuracy results were obtained for the second case (Table 6) at Т 1 =(10÷100) °С, Т 2 =(1÷10) °С in the middle and end of the range. The worst accuracy results, as for equation (11), were obtained for the third case (Table 7) at Т 1 =(1÷10) °С, Т 2 =(10÷100) °С even at the end of the range. Thus, we can summarize that the relationship between T i and T 1 is more significant than between T i and T 2 , and the best accuracy results are obtained at a high (more than 100 °С) value of T 1 . The lowest value of the relative error is obtained when the total value of T 1 and T i increases. It was found that for the second case (at Т 1 =(10÷100) °С, Т 2 =(1÷10) °С) when the value of T i increases, the relative measurement error decreases. In addition, when the reproduction accuracy of the normalized temperature Т 1 increases (with a constant reproduction error of T 2 ), there was no significant improvement in the results, which gives grounds to use a non-precision normalized source for T 1 reproduction. On the other hand, an order of magnitude increase in the reproduction accuracy of both normalized temperatures T 1 and T 2 also increases the measurement accuracy by an order of magnitude.
As a result of computer modeling in the Mathcad15.0 environment, calculations of the relative measurement error at Т 1 =(100÷190) °С and Т і =(100÷190) °С and with step of 10 °С were carried out, presented in Table 8. Since the parameter T 2 does not have a significant effect on the measurement result, T 2 =10 °С was taken for calculations (as for the redundant measurement equation (11)). Table 8 Relative measurement errors, % Thus, processing of measurement results according to the redundant measurement equation (14) is recommended for measuring high temperatures T i ≥100 °С and Т 1 =190 °С. During the analysis of Table 8, it was found that the best accuracy results were obtained at Т 1 =Т i =190 °С.
It should be noted that for the case shown in Table 8, an order of magnitude increase in the reproduction accuracy of the normalized temperature T 1 (with a constant reproduction error of T 2 ) does not increase the measurement accuracy. So, this gives grounds to use a non-precision normalized source for T 1 reproduction.
The calculations confirmed that an order of magnitude increase in the reproduction accuracy of normalized temperatures T 1 and T 2 leads to an order of magnitude increase in the measurement accuracy.

3. Comparative analysis of the two systems
Comparisons of the results obtained in the study of redundant measurement equations (11) and (14) were made according to the following criteria: 1) measurement capability over the entire operating range; 2) measurement accuracy; 3) number of measurement cycles; 4) ease of use. The redundant measurement system (10) is considered below, which describes and consists of 5 measurement cycles in relation to the specified criteria: 1) applies to the entire range provided that Т 1 =Т і (0.0005·Т і +1) and Т 1 >Т 2 (recommended range of normalized temperatures Т 1 =(10÷100) °С and Т 2 =(1÷10) °С); 2) in the measurement range from 10 °С to 200 °С, provides high-precision measurement with a relative error of 0.0100 % to 0.00003 % at Т 1 =Т і (0.0005·Т і +1); 3) to implement it, you need to perform 5 measurement cycles; 4) high-precision measurement in a wide range requires the relationship between the desired temperature T i and the normalized temperature T 1 =T i (0.0005·T i +1). The redundant measurement system (13) is considered below, which describes and consists of 6 measurement cycles in relation to the specified criteria: 1) it is recommended to use for measuring high temperatures Т i ≥100 °С; 2) in the measurement range from 100 °С to 190 °С at Т 1 =190 °С, provides high-precision measurement with a relative error from 0.0117 % to 0.0059 %; 3) to implement it, you need to perform 6 measurement cycles; 4) high-precision measurement of high temperatures requires compliance with the condition for setting the normalized temperature value T 1 , which corresponds to the end of the operating range (Т 1 =190 °С).
Thus, when using a sensor with an unstable quadratic TF to increase the accuracy in the whole range, a system of nonlinear equations of quantities (10) and the equation of redundant measurements (11) should be used to process the obtained measurement results.

Discussion of the results of computer modeling of the quadratic transformation function
When applying RMM, it is necessary to generate redundancy or additional data, as in [16], which, when processed according to a certain algorithm, as in [9,14], will improve accuracy. Similarly, for quadratic TF, two variants of redundancy formation systems were studied, consisting of 5 and 6 equations of quantities, each of which describes the measurement cycle. It should be noted that in each cycle, there is a measurement of the desired temperature, normalized temperatures T 1 and T 2 , and their combination. To create normalized quantities (temperatures), it is necessary to have standardized sources with normalized characteristics, because accurate knowledge of normalized quantities during their further processing helps to improve accuracy, as noted in [12,13]. In turn, the use of high-precision elements in measurements, as in [18], reduces the methodological error of RMM.
As a result of solving the proposed systems (10) and (13), the corresponding equations of redundant measurements (11) and (14) were obtained. Processing of measurement results in accordance with the proposed equations provides automatic exclusion of the additive and multiplicative components of the error, i.e. error correction, which does not contradict the data presented in [19][20][21][22][23][24][25]. In addition, due to the application of the proposed redundant measurement equations (11) and (14), it is possible to directly use a nonlinear TF without special measures for its linearization, which are typical for studies published in [12][13][14].
When studying the effect of normalized quantities on the measurement result for systems (10) and (13), as can be seen from Tables 1-3 and Tables 5-7, it was found that T 1 has a greater impact than T 2 . This is especially true for the system (13) with the corresponding redundant measurement equation (14). This effect of normalized quantities on the measurement result, according to the authors, is due to the type of equations of quantities: so in equation (11) there are two quantities T 1 and T 2 in the denominator, and for equation (14) -only T 1 . This means that such a mechanism of influence of the normalized quantity T 1 is the process regulation factor that can affect the relative measurement error. The influence of the values of certain parameters on the measurement result does not differ from the practical data presented in [7,12]. It was also found that for both systems (10) and (13), the best accuracy results (Tables 2, 6) were obtained for the case of Т 1 =(10÷100) °С, T 2 =(1÷10) °С. For the best accuracy, studies were conducted on the effect of reproduction errors of normalized temperatures, which are part of the methodological error of RMM on the accuracy of the result. The studies were carried out first with an order of magnitude increase in the reproduction accuracy of the normalized temperature T 1 , and then with an order of magnitude increase in the reproduction accuracy of both normalized temperatures T 1 and T 2 . As a result, it was found that increasing the reproduction accuracy of the normalized temperature T 1 (with a constant reproduction error of T 2 ) does not lead to a significant improvement in the results. This gives grounds to use a non-precision normalized source for T 1 reproduction. On the other hand, an order of magnitude increase in the reproduction accuracy of both normalized temperatures T 1 and T 2 also increases the measurement accuracy by an order of magnitude.
Since the best accuracy results were obtained at Т 1 =(10÷100) °С, Т 2 =(1÷10) °С, more detailed studies of the system (10) and the corresponding equation (11) are of particular interest. In further research (Table 4), a certain relationship was found for the controlled T i and normalized T 1 quantities, at which the smallest error was observed.
Thus, it becomes possible to measure with a relative error of 0.002 %, with the proximity of T 1 to T i , namely Т 1 =Т i =50 °С, and T 2 =(1÷10) °С. Therefore, to identify the relationship between the controlled T i and normalized T 1 quantities, at which an increase in accuracy is observed, computer analysis was performed. Thus, it was found that the ratio between the values of T 1 and T i is Т 1 =Т i (0.0005·Т i +1), which provides high-precision measurement with a relative error of 0.01 % to 0.00003 % in the range of measured temperatures from 10 to 200 °С. Thus, compliance with the ratio between the controlled T i and normalized T 1 quantities provides high-accuracy results in a wide range of values. This also agrees with the conclusions of [8], where an increase in measurement accuracy is also associated with compliance with the established parameter corrections.
For system (10) with equation (11), detailed studies were also conducted for another relationship between T 1 and T 2 : Т 1 =Т 2 =(10÷100) °С and at the value of the controlled temperature T i =100 °С, i.e. without observing the found ratio T 1 =T i (0.0005·T i +1). Thus, the functional dependence of the relative measurement error on the normalized values of T 1 and T 2 and at the value of the controlled temperature T i =100 °С was constructed (Fig. 1). The analysis of the resulting surface revealed a certain undesirable ratio between T 1 and T 2 , at which measurements become unacceptable. Therefore, to identify the undesirable ratio, computer analysis was performed, which found that T 2 ≠2·T 1 . It should be noted that when the difference between T 1 and T i increases, this ratio has a certain range with a maximum error at the point T 2 ≠2·T 1 . This indicates the expediency of observing the found relationship between the values of the normalized T 1 and controlled T i temperature, which is When determining the nature of the effect of the normalized quantities T 1 and T 2 for the system (13) with the corresponding redundant measurement equation (14), the study was conducted with similar initial data as for equation (11). The best accuracy results (Tables 5-7), in contrast to the results obtained for equation (11), are obtained only at large (more than 100 °С) values of T 1 . But, at the same time, the range of normalized temperatures T 1 =(10÷100) °С and T 2 =(1÷10) °С, which provides the smallest measurement error, was the same as for equation (11). For this range, studies were also conducted on the effect of reproduction errors of normalized temperatures T 1 and T 2 on the measurement accuracy. The results of this study were similar to those for equation (11). This indicates the possibility of using a non-precision normalized source for T 1 reproduction. In the further analysis of the obtained data (Tables 5-7), it was found that when the total value of T 1 and T i increases, the relative error value decreases. Therefore, to identify the relationship between the normalized T 1 and controlled T i temperature, computer modeling was performed at T 1 =(100÷190) °С and T i =(100÷190) °С. Since, as can be seen from Tables 1-3, 5-7, the parameter T 2 does not have a significant effect on the measurement result, T 2 =10 °С was taken for calculations. As a result of computer modeling, it was found that the smallest error (0.0059 %) is obtained at Т 1 =Т і =190 °С (Table 8). This means that when applying the redundant measurement equation (14), it is necessary to set the value of the normalized temperature T 1 , which corresponds to the end of the operating range (Т 1 =190 °С).
The analysis of the results obtained, presented in p. 5. 3, showed that the system of equations of quantities (10) with 5 equations proved to be the best compared to the sys-tem (13) with 6 equations. It should be noted that in terms of statistical data processing, the opposite results were expected. This indicates that the increase in the accuracy of RMM measurements is influenced not by the number of measurements, but by the type of redundant measurement equation. In addition, the ability to use systems with a different number of equations of quantities indicates the flexibility of using RMM depending on the possibility of their implementation, as well as the possibility of reproducing normalized quantities.
Thus, the obtained data on the application of the system (10) with the redundant measurement equation (11) for an unstable quadratic TF allow us to state the following: 1) by observing the relationship between the values of the normalized T 1 and controlled T i temperature, which is Т 1 =Т і (0.0005·Т і +1), high-precision measurement in a wide range becomes possible, 2) there is no reason to use a precision normalized source for T 1 reproduction.
Such conclusions can be considered expedient from the practical point of view as they allow high-precision measurement in a wide range while observing the found ratio between the values of normalized T 1 and controlled T i temperature. In addition, the use of a non-precision normalized source to reproduce the normalized temperature T 1 allows reducing the requirements for its power and accuracy. It should also be noted that the application of the redundant measurement equation (11) ensures the independence of measurement results from the absolute values of the parameters of the transformation function and their deviations from the nominal values. Due to this, inexpensive sensors can be used for RMM (with low stability requirements). From a theoretical point of view, the relationship between T 1 and Т і of the form Т 1 =Т і (0.0005·Т і +1) allows us to argue about the possibility of improving the accuracy of measurement in a wide range of values, which is a certain advantage of this study. Another advantage is determining the impact of T 1 and T 2 on the measurement result. It was found that T 1 has a greater influence on the measurement result than T 2 . In addition, the best accuracy results for equations (11) and (14) were obtained for the range of normalized temperatures T 1 =(10÷100) °С and T 2 =(1÷10) °С. However, it should be noted that such accuracy results are obtained when changes in the parameters of the transformation function remain constant during the measurement cycles. Given the found ratio between T 1 and T i , knowledge of the current value of the controlled temperature is required, which is not always known. In addition, when applying RMM, there is a methodological error due to the error of reproduction of the normalized temperature T 2 . The inability to remove these limitations in this study creates a potentially interesting direction for further research.

Conclusions
1. Computer modeling of the presented system of nonlinear equations of quantities consisting of 5 equations was carried out. The modeling was performed at different values of normalized temperatures T 1 and T 2 and with changes in the parameters of the transformation function within ±10.0 %. The studies have shown that T 1 has the greatest influence on the measurement error among the normalized quantities. Based on the research, it was found that Т 1 >Т 2 , and the recommended range of normalized temperatures is T 1 =(10÷100) °С and T 2 =(1÷10) °С. In addition, it was found that for the redundant measurement equation (11), the relationship between T 1 and T 2 should be observed, where Т 2 ≠2·Т 1 . The studies of the effect of reproduction errors of normalized temperatures found that increasing the reproduction accuracy of the normalized temperature T 1 (with a constant reproduction error of T 2 ) does not significantly improve the results. So, it can be argued on the use of non-precision normalized sources for T 1 reproduction. An order of magnitude increase in the reproduction accuracy of both normalized temperatures T 1 and T 2 also increases the measurement accuracy by an order of magnitude, which indicates the relationship of these normalized quantities with controlled T i . The relationships between the controlled and normalized quantities were determined, which reduce the measurement error in a wide range. It is found that when applying the redundant measurement equation (11), it is recommended to adhere to such a relationship between T 1 and T i at which Т 1 =Т і (0.0005·Т і +1). This provides high-precision measurement with a relative error of 0.01 % to 0.00003 % in the range of measured temperatures from 10 °С to 200 °С. This indicates the possibility of increasing the measurement accuracy with an unstable quadratic transformation function by correctly setting the normalized quantity T 1 .
2. Computer modeling of the presented system of nonlinear equations of quantities consisting of 6 equations, with the same initial data as for the system of 5 equations was carried out. The studies have shown that T 1 has the greatest influence on the measurement error among the normalized quantities, as for the system of 5 equations. The studies found that Т 1 >Т 2 , and the recommended range of normalized temperatures is Т 1 =(10÷100) °С and Т 2 =(1÷10) °С. It was found that an order of magnitude increase in the reproduction accuracy of the normalized temperature T 1 (with a constant reproduction error of T 2 ) does not increase the measurement accuracy. Therefore, we can conclude on the use of a non-precision normalized source for T 1 reproduction, as for the system of 5 equations. An order of magnitude increase in the reproduction accuracy of both normalized temperatures T 1 and T 2 also increases the measurement accuracy by an order of magnitude, which indicates the relationship of these normalized quantities with controlled T i . The studies have shown that when applying the redundant measurement equation (14) and at Т 1 =Т і =190 °С, the error is reduced to 0.0059 %. This indicates the need to set the value of the normalized temperature T 1 , which corresponds to the end of the operating range. Thus, it is recommended to use a system of 6 equations of quantities and the corresponding redundant measurement equation (14) when measuring large values of the controlled temperature T i .
3. A comparative analysis of the proposed systems with the corresponding redundant measurement equations was carried out. When comparing the results obtained according to equations (11) and (14), the prospects of conducting 5 measurement cycles in accordance with the system (10) were evaluated. Compared to the system of 6 equations, processing of the results using equation (11) makes it possible to obtain high-precision results over the entire measurement range of temperatures, and also requires fewer measurement cycles.