UDC 006.91 EXTENDED METHOD OF ESTIMATION UNCERTAINTIES OF INDIRECT MULTIVARIABLE MEASUREMENTS ON THE TWOPORT EXAMPLE

The new extended mathematical model for evaluation uncertainties of indirect multivariable measurements, which upgrades the method given in Supplement 2 of guide GUM, is presented. In this model the uncertainties and correlations of parameters of the processing function are taken also into account. This model can be used for multivariable measurements and to describe the accuracy of instruments and systems that perform such measurements. The estimation of uncertainties of voltage and current on the output of a twoport network from indirect measurements on its input with considering influences the uncertainties of twoport elements is included.


Introduction
In the basic and technical research, in monitoring and technical diagnostics, many physical quantities and parameters have to be measured for characterize the object under the test. In many cases there is no possibilities to carry out direct measurements. Then indirect methods must be applied. The international recommendations for application of the method of determination estimators of values, uncertainties and correlations in multivariable measurements are described in Supplement 2 to the GUM.
Directly measured on input the n-element measurand X is l processed to the output m-measurand Y by relation Y= F(X, P) (1) In this paper are used such designations: X, Y -input and output measurands; X 0 ,Y 0 -their initial values; ,vectors of estimators of n values i and of m values j ; u x , u y and u δx ,, u δytheir absolute and relative standard uncertainties; (X), (X,P)ideal and real multivariable functional of processing to Y; U F , U δF -its covariance matrices; = / , -sensitivity matrices for absolute and relative uncertainties; U X , U Y , U Y0 , U δX , U δY , U YPcovariance matrices of X, Y; U P , U(P,X)covariance and correlation matrices of k parameters P of the processing functional F of the measurement circuit For the ideal functional F(·), i.e. when uncertainties of its parameters P are negligible, the propagation of absolute and relative variances of the t measurand X to Y ones are: .Even in the case when the basic multivariable relation (1) is nonlinear, in the most cases for uncertainties of X and Y as small deviations, their scatter regions can be defined by a model of joint multidimensional normal probability distributions. Then, for a given probability density p 0 the distribution region for p ≥p 0 takes the form of a n-and mdimensional hyper-ellipsoids, if covariance matrix is positive definite, with centers at the ends of their averages.
Matrices U X and U Y describe the cover regions of n and m dimension probabilities if their det(U i ) > 0 or det(R i ) > 0 as equal condition for matrix of the correlation coefficients, called the correlator R. All, or some components of the vector of output results can be next used separately or jointly. In the latter case it is necessary to find and take in considerations also the correlations between pairs of variables y i .
Relations (2a,b) for covariance matrices , of absolute and relative uncertainties of estimators j of results, obtained for indirectly observed m-dimensional measurand , are the same whether the measurement functional (X,P) is linear or linearized by the first derivative. All, or some components of measurement results can be next used separately or jointly. In the latter case it is also necessary to take in considerations the correlations between variables y i of output measurand .
In many cases the indirect measurements of mcomponents of the measurand are made now by automatic measurement systems. If not, then a collection of individual quantities of X should be synchronically measured, and from above data both, of and covariance matrices U X , U Y externally calculated.

Basic formulas of the extended method
The actual GUM-S2 method [1] and earler literature, e,g. [2 -5] do not cover situations of the not accurate multivariable functional F(X, P), for example due to approximation, the limited frequency range of transfer function, uncertainties of passive and active elements, AC/DC converters and analogue multipliers, and also when measurements are possible only indirectly via other nonideal internal parameters of the tested object. In precise measurements, the rounding of calculations also becomes essential, including ones resulting from the precision of the digital part of the circuit. In the instrumental measuring systems, the real multivariable processing function is = ( , )eq. (3) given in Table 1 [8,11]. The accuracy of indirect measurements of the multivariable measurand Y depends on the uncertainty and correlations of X and also on uncertainties and correlations of its parameters P -formula (5a). The relative uncertainty propagation is also given. Developed is the extended formula (5) for the covariance matrix U Y , which includes all influences on uncertainties u y and its simpler cases (6) -(9a-c) given also in Table 1.   (8e, f, g) The relationships between small deviations of the values of n-elements of the input measurand X and melements of the indirectly measured measurand Y are described by the formula (4). The deviations of the measuring system P parameters are determined from their nominal values on the basis of the maximum permissible errors (MPE) known from their technical data or as deviations from the estimators of their values determined in measurements. Sensitivity matrices S X , S δ and S P (4a-c) express the influence of deviations , and on the output deviations , of the initial quantities. The deviations of known values or their course during the period of measurement experiment, are removed from the results by corrections. Other, which are not known and not determinate, are randomized. For the single-parameter measurand, the statistical properties of a set of deviations of each quantity are described in GUM [1] by the standard uncertainty u as the geometric sum of its components u A and u B . For multidimensional measurands, the equivalent of the variance of single variable, are their covariance matrices, e.g. symmetrical matrices , and (5ce). They contain on the main diagonal squares of standard uncertainties (variances) of individual quantities, and on other places, products of the corresponding one from n(n-1)/2 correlation coefficients and uncertainties of both type correlated quantities.
Sets of random deviations from estimators of the output measurand Y variables are the result of multiparameter distributions of the deviations of the input measurand X variables and deviations of parameters P of the measurement system performing the multivariable functional F(X, P). When linearizing each of its functions for small deviations, the general formula (5) for the U Y covariance matrix in multi-parameter measurements and its subsequent developed forms (5a), (5b) is obtained from the propagation law of variance. Uncertainties and correlation coefficients of n variables of the measurand X and of k system parameters P are included in the and covariance matrices (5c,d), In general case, variables X can be also correlated with parameters P of measuring system. This is described by the matrix U with the size [n x k], given in the formula (5f). Such relationship may appear under the influence of a common external random effect on X and P, e.g. a variable outside temperature.
The number of independent correlation coefficients in the U matrix is smaller by the number of m equations elements of measurand Y. In the measurement practice, including electrical measurement systems, there is usually a simpler case when the directly measured quantities X and deviations of parameters P of the measuring system are not correlated (e.g. X is differently located then P and they do not affect themselves and their external influences are also not related). Then the covariance matrix U does not occur and = = . The propagation equation of variance (5b) has then a simpler two-component form (6). The first component depends on the uncertainties and correlations of elements of the input measurand X, similar as in the classic approach according to GUM-S2 [1]. The second component, depending on the uncertainty of the processing function, appeared in the extended method and constitutes its essence. It expresses the influence of uncertainties and correlation coefficients ρ p of P parameters of the system processing function F(X,P), analog or digital.
In the papers [6][7][8][9][10][11], the authors stated that only sets of deviations with uncertainties of the same type, i.e. only of u A or only of u B , can be correlated with each other, for variables of the same or of different multi-measurands. Covariance matrices of multi-measurands, similarly as the variance 2 = 2 + 2 of each single measured variable, can be presented also as the sum of two component matrices of type A and B, i.e.U X = U XA +U XB , U Y = U YA +U YB -formula (8). The elements of component matrices type A and B are given in (8a)-(8d) and method of their calculations in (8e,f,g).
For the measurand X, only the correlation coefficients ρ xA in the U XA matrix can be experimentally determined by synchronous measurements of variables of X. On the other hand, the coefficients ρ B of the U XB matrix, similarly as the uncertainties of type B, have to be estimated heuristically. If two quantities are measured with the same or similar instrument and under the same conditions, then the correlation coefficient ρ xB equal to 1 [4], [5] can be assumed. For different instruments and in different operating conditions this coefficient is closer to 0. The correlation coefficient ⎼1 is rather rare. It occurs e.g. when changes of both correlated variables from common interactions, have the opposite signs.
Type A and type B uncertainties for individual quantities of the output measurand Y should be carried out separately from the component U YA , U YB of the covariance matrix U Y , according to formulas (8), (8a-g). If during the measurements the values of P system parameters are constant, the matrix does not occur, and the U P =U PB matrix is estimated heuristically for the long period changes of deviations ΔP. However, if ΔP is changing randomly in the duration of the measurement experiment, then the elements of their component matrix must also be estimated heuristically based on technical data and own knowledge. It is also valuable to perform additional measurements in the specially created influencing conditions to estimate the level of the short time random changes in P parameters.
In several papers, authors described how to use this new method. Few examples of implementation this model to indirect measurement of a two-terminal circuit parameters through a four-terminal T type network, considering the uncertainties and correlation of its impedances in general case and for U= 0 are presented in detail in [8][9][10][11]. One of these examples, the indirect measurements of voltage and current at the output of a loaded four-terminal circuit is considered in detail below.

Uncertainties of indirect measurements of the twoport output variables
Let us consider the indirect measurements of the voltage and current of an inaccessible branch on the output of a linear passive twoport network based on measurements of these variables on its input terminals. It was assumed that this twoport has the T-type structure given in Fig. 1. Then observed is the 2D measurand Y = [ , ] T . The accuracy of Y are determined from measurements of the input voltage and current, i.e.: X = [ , ] T. Accuracy of results depends also from values, uncertainties and correlations of twoport impedances Z 1 , Z 2 , Z 3 .

Fig 1. Diagram of a passive twoport T-type circuit
For the T-type passive twoport from Kirchhoff laws: = 2 + ; = 2 2 + 1 ; 2 2 = + 3 the following relations for the output variables are obtained 3 3 1 13 22 The relation between , and directly measured , of the T-type twoport circuit has the form of matrix function Y = B·X , i.e.: If the twoport is passive and reversible, then the determinant of the matrix B satisfies the equation det( ) = 11 22 − 12 21 = 1 (11) and only three of the matrix B elements are independent, the fourth one follows from (11).
In the opposite situation, when tested are variables on input of the twoport and measured are on its output, the matrix A is used, which for the twoport T is like B with replaced impedances 3 and 1 .

Matrix
when U P = 0 (GUM-S2 case) If uncertainties of impedances Z i (i=1,2,3) are negligible, then the matrix U P = 0 and from (6) results that the output covariance matrix = , as it is in GUM-S2 method. If the measured twoport quantities X are uncorrelated, its input covariance and the sensitivity matrices are and its sign depend on the sign of expression in module: for plus = −1 and for minus = 1.

Component of covariance matrix U Y for uncorrelated impedances of twoport T
Covariance matrix U P for uncorrelated impedances 1 , 2 , 3 of the twoport T and sensitivity matrix have the following forms The equation (6) shows that the component of the covariance matrix = + of the output quantities, depending on the uncertainties of uncorrelated impedances of the T twoport is (23) Despite the non-correlation of impedance Z i , when the matrix is diagonal, U out , I out as variables of the output measurand Y, will be correlated. From (15b),(19a-c) and (20) for uncorrelated quantities X, obtained are the resultant variances for the output voltage and current are:

Uncertainties and correlation in the output
The formula for uncertainty of current I out if correlated are only impedances, obtained from (28) Figure 3 shows examples of the dependence of the uncertainty of the twoport T output voltage as a function of the output current for different values of the correlation coefficients 12 = 13 = 23 = 0; 0,1; 0,5; 0,7; 1, U in =25 V; and relative uncertainties = = = 0,2%.

Summary and conclusions
Formulas summarized in Table 1, extend the method of GUM Supplement 2 [1] for determining the uncertainty of indirect multi-parameter measurements. The model which considers uncertainties and correlations of system parameters that implement the multivariable processing function, proposed by Z. Warsza, is used.
As an example of the application of this method in indirect 2D measurements of the voltage and current of the two-terminal circuit branch, available only through the twoport T, is presented. Such measurement occur in the identification of voltages and currents of inaccessible directly elements forming the electrical systems, and in multi-sensor measurements and technical diagnostics.
Matrix relationships were derived considering the uncertainty of processing functions performed by twoport. The formulas of increased total uncertainty of the estimated voltage and current due to the impedances uncertainties of this system are find. It did not exceed the sum of uncertainties of input variables and twoport impedances.
In the presented variants of the twoport, the uncertainties at the output also depend on the value and sign of the correlation coefficients of the input quantities and circuit parameters, as well as the current at the output of this circuit. The possibilities of minimizing these uncertainties were also discussed.
The proposed method can be usefully used both for the evaluation of indirect multi-parameter measurements made with a set of instruments, as well as for the assessment of the accuracy of measuring instruments and systems with an integrated measurement system for multivariable measurements. This method may be also the basis for development a new extended version of the Guide GUM Supplement 2 or included in its new version GUM2.
This method will allow the assessment of the accuracy of multivariable instrumental measuring systems by means of uncertainties. Therefore, other interesting partial methods, e.g. given in [5], [9, 10], are not discussed here.
It is possible to analyze by this method the determination of uncertainty of several other multiparameter measurement systems, e.g. AC networks as in [2], [3], [5], [7], power component measurements in three-phase networks with different waveforms, and then to examine the statistical properties of multi-variable systems with non-Gaussian probability distributions and various processing functions of measurands.