ACTA IMEKO ISSN: 2221-870X July 2017, Volume 6, Number 2, 89-92 ACTA IMEKO | www.imeko.org July 2017 | Volume 6 | Number 2 | 89 Improved evaluation of uncertainty for indirect measurement Valeriy Didenko Department of Information-Measuring Technique, National Research University “MPEI”, Krasnokazarmennaya 14, 105568 Moscow, Russian Federation Section: RESEARCH PAPER Keywords: Indirect measurement; accuracy specifications; measurement uncertainty Citation: Valeriy Didenko, Improved evaluation of uncertainty for indirect measurement, Acta IMEKO, vol. 6, no. 2, article 16, July 2017, identifier: IMEKO- ACTA-06 (2017)-02-16 Section Editor: Eric Benoit, University Savoie Mont Blanc, France Received February 13, 2016; In final form June 30, 2017; Published July 2017 Copyright: © 2017 IMEKO. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited Corresponding author: Valeriy Didenko, e-mail: didenkovi@mail.ru 1. INTRODUCTION It is well-known [1] that the result of an indirect measurement is a function of several variables: ),...,2,1( nXXXfX = . (1) The values of these variables are found by direct measurements. The maximum possible absolute error of the indirect measurement as a function of maximum possible errors for the direct measurements ( ...1 X nX ∆∆ ) can be found from (1) approximately as [1]: nX nX f X X f X ∆ ∂ ∂ ++∆ ∂ ∂ =∆ ...1 1 . (2) It is usually suggested that the accuracy of (2) is found by means of high order derivatives in Taylor’s expansion of f [1]. Another way is to use simulation methods giving changes of , ...,1X X n in (1). According to modern metrology, (2) can be explained as the uncertainty with 100 % confidence level (the worst-case uncertainty) [2]. The worst-case uncertainty means that errors higher than found by (2) are absent. In practice, a real maximum possible absolute error of the indirect measurement can be a little higher than one found by (2). One reason of this fact has already been discussed (errors of Taylor’s expansion). Other reasons are elevated values of the errors of the direct measurements with regard to , ...,1X X n∆∆ . The worst-case method supposes that elevated errors of the indirect measurements with regard to (2) are negligible, for example, lower than five percent from (2). The result found by (2) is usually much higher than the real values. The reason is that all the errors of the direct measurements are calculated independently. The exception to the rule is given in [2]. All the direct measurements are supposed to use an analog to digital converter (ADC) with the same values of the maximum offset error 0U , the maximum gain error GU , and the maximum linearity error inlU . The first two errors are supposed to have the same sign (positive or negative), while the linearity error can change the sign for ABSTRACT The paper gives formulae for uncertainty evaluation of an indirect measurement based on direct measurements made by different types of measuring devices. The first type of the measuring devices has the specifications of a total error (e.g. digital instruments), while the second type has the specifications of offset, gain and linearity errors (e.g. analog to digital converters). The choice of a device range and the configuration of measuring circuits for decreasing uncertainty are considered. The conversion of the specifications for the first type to the specifications for the second type is discussed. ACTA IMEKO | www.imeko.org July 2017 | Volume 6 | Number 2 | 90 different direct measurements. The uncertainty of the indirect measurements is found in accordance with [2], [3] for four cases: the standard uncertainty and the worst- case uncertainty (both absolute and relative). The absolute worst-case uncertainty (the maximum possible absolute error) of the indirect measurement with negligible quantization error is then [2]: ∑ = +∑ =+∑ == n i i kinlU n i iXikGU n i ikUXU 1 .110)( (3) Coefficients ik can have different signs. For example, the indirect measurement 21 XXX −= gives .121 =−= kk Therefore, (3) can show a lower value than the one found by (2) for the devices with the same errors of each direct measurement. Direct measurements of nXX ...1 are often fulfilled by data acquisition devices. Three most popular sampling architectures are: multiplexed structures, simultaneous sample and hold structures and multi-ADC structures [4]. Corresponding structures are shown in Figure 1. For the same speed of the used ADC, the multi-ADC architecture gives a higher scan rate per channel and therefore is preferable according to the recommendations in [4]. The following abbreviations are used in Figure 1: Mux – multiplexer, Amp – instrumentation amplifier, ADC - analog to digital converter, SSH – simultaneous sample and hold. The advantage of the multiplexed architecture in comparison with the multiplexed structure in terms of uncertainty is shown in Section 4. Besides the values of direct measurements, the values reproduced by material measures (standard electric resistors, standard signal generators etc.) are also included in (1). For several direct measurements, the following variants are discussed in terms of uncertainty: the application of the same ADC at the same range, or different ranges; the choice of the same ADC, or different ADCs at the same range. For simplicity, several phenomena (dynamic errors, e.g.) ignored in [2] are not considered here either. Additionally, the quantization error and noise are supposed to be negligible. All these approximations do not usually influence the main conclusions given in the paper. 2. ACCURACY SPECIFICATIONS FOR DIFFERENT TYPES OF MEASURING DEVICES Accuracy specifications of digital instruments (DI) are usually presented by the total (maximum) absolute error. As a first approximation, the maximum possible absolute measurement error for each variable iX found by a digital instrument is ( ),X a b Xi iDI∆ = + (4) where a is a positive number of the same unit as iX and b is a positive non-dimensional number. The maximum absolute error of a digital instrument as a function of an input signal X is shown in Figure 2. Two values of the input signal (X1 and X2) are considered. The corresponding absolute errors ( 1∆ and 2∆ ) are shown in Figure 2. Sometimes a is given in % of the full scale ( iFSX . ) and b is given in % of a reading. Accuracy specifications of the ADC and data acquisition (DA) devices are usually presented by the maximum offset, gain and linearity errors. The quantization error and noise (the random error) are usually included in a and b for digital instruments but can be specified separately for other devices. For simplicity, we will not consider them in this paper. Then the maximum absolute error of the DA is .0X U U X Ui iG inlDA∆ = + + (5) For example, (5) is used for finding the maximum absolute error of a single measurement in [5]. The maximum absolute error of the ADC as a function of input signal X is shown in Figure 3. Positive errors are considered only as an example. Two values of the input signal (X1 and X2) are used. The linearity error is supposed to be zero at the ends of the range (X=0 and X=XFS) but is equal to the maximum value Uinl with any sign at any other points. The linearity error is negative near X1 and is positive near X2 (the worst-case method). Let us Figure 1. Simultaneous sampling architecture – simultaneous sample and hold (ssh). Figure 2. Maximum absolute measurement error vs. input signal for digital instruments. ACTA IMEKO | www.imeko.org July 2017 | Volume 6 | Number 2 | 91 suppose that 1∆ and 2∆ errors are the same for a digital instrument (Figure 2) and an ADC (Figure 3). Then the difference between specifications of the two devices mentioned is not important for the evaluation of the direct measurement uncertainty. The situation changes dramatically if we investigate indirect measurements. Let us consider the simplest indirect measurement 2 1X X X= − . The uncertainty of the measurement for the digital instrument (Figure 2) is .1 2 X DI ∆ + ∆∆ = The uncertainty of the measurement for the ADC (Figure 3) is .2 1 X DA ∆ −∆∆ = It is clear from Figures 2 and 3 that the second value can be much lower. Because of this result it is necessary to find ways to transform the specifications of digital instruments into the form specified for ADCs. If (4) and (5) show the same results, then a and b can be found with given inlUGUU ,,0 as inlUUa += 0 , (6) GUb = . (7) Let a and b be specified. If we consider X=0, then 0U = a .It is possible to find the following inequalities (see Figure 2): 2 / , 2( ),G FS inlU b a X V a b X≤ + ≤ + (8) where FSX is the full scale of the DI. The evaluation of the linearity error by (8) is usually much higher than the true value. Therefore (8) does not give any advantages for calculation of the uncertainty of indirect measurements in comparison with initial equation (4). Fortunately, some digital instruments have the additional specification of the linearity error. For example, the linearity error is specified in [6] as ,XU A X BL LFSinl = + (9) where AL and BL are positive non-dimensional numbers. Now the maximum possible absolute error of the digital instrument can be written practically in the same way as it was given for ADCs or data acquisition devices: 2 ( . )a inlX FS X a X Ui iDI b∆ = + ++ (10) In accordance with Figure 3, inlU is supposed to be equal to zero at the ends of the scale, but can produce both positive and negative errors at any other points. The signs of errors produced by a and b are constant for the given indirect measurement. The linearity error found by (9) is approximately 10 times less than one found by (8) for the instrument 3401A [6]. If the linearity error for the DI is not specified, then it can be found approximately from the experiment [7]. The accuracy of material measures (standard electric resistors, standard signal generators etc.) can be specified by both (4) and (5). For example, the accuracy of voltage calibrators is usually specified as (4), while data acquisition devices with analog output [5] have specifications (5). It means that all the following theory can be used both for results of the direct measurements and quantities reproduced by material measures. 3. GENERAL FORMULAE FOR INDIRECT MEASUREMENT UNCERTAINTY Direct measurements n used for the indirect measurement can be divided into three parts. Then the absolute worst-case uncertainties of the indirect measurement ,) 121 ( 21 11 . 21 11 . 21 11 .0 1 1 1 1 1 10 )( ikiXib n nni i a nn ni i k iinl U nn ni i XikiGU nn ni i kiU n i i kinlUiX n i i kGU n i i kUXU +∑ ++= + +∑ + += ∑ + += +∑ + += ++ ∑ = ++∑ = +∑ = = (11) The simplest indirect measurement is described by the function 2 1X X X= − , where 11 2k k= − = − . The absolute worst-case uncertainty of the indirect measurement when one device is used in the same range ( 21 == nn ) and 012 ≥≥ XX in accordance with (11) is inlUXXGUXIU 2)12()( +−= . (12) The absolute worst-case uncertainty of the indirect measurement when two devices of the same type are applied in the same range ( 2,2n n= = ,02.01.0 UUU == GUGUGU == 2.1. , inlUinlUinlU == 2.1. ) in accordance with (11) is inlUXXGUUXIIU 2)21(02)( +++= . (13) The maximum difference of the results found by (12) and (13) is at :21 FSXXX ≈≈ inlU FSXGUU XIU XIIU ++= 01 )( )( . (14) Let us use (14) to compare the uncertainties of the multiplexed and multi-ADC structures (discussed in Section 1) Figure 3. Maximum absolute error vs. input signal for ADC. ACTA IMEKO | www.imeko.org July 2017 | Volume 6 | Number 2 | 92 for the implementation of the function 2 1X X X= − . Model PCI-6250, used in the multiplexed structure, includes the ADC with the following parameters [5]: =FSX 10 V, =0U 2∙10 -4 V, =GU 6∙10 -5, =inlU 6∙10 -4 V. If we use the same ADC in the multi-ADC structure, then, according to (14), the worst-case uncertainty is 2.3 times more. It means that recommendations [4] can be not true for the uncertainty of the indirect measurement. If 1X << 2X , then two ranges can be used for each direct measurement. In this case, the absolute worst-case uncertainty of the indirect measurement 2 1X X X= − is .2.1.22.11.2.01.0)( inlUinlUXGUXGUUUXIIIU +++++= (15) The application of one range for both direct measurements will be better if (12) gives a lower result in comparison with (15). Let us consider the example PCI-6250 [5] used for the indirect measurement 2 1X X X= − with 1X ≈ 5 V and 2X ≈10 V. The corresponding specifications for =≈ 1.1 FSXX 5 V are =1.0U 1∙10 -4 V, =1.GU 7∙10 -5 V and =1.inlU 3∙10 -4 V. The specifications for =≈ 1.2 FSXX 10 V were given before. Using the PCI-6250 at 10 V range only, we get from (12) that )( XIU = 1.5 mV. If we use two ranges (5 V for 1X and 10 V for 2X ), the result is )( XIIIU =2.15 V. It means that the application of only one range gives 1.4 times better result, and the well-known recommendation to use the lowest range (5 V in this example) is valid only for the uncertainty of the direct measurements but can be incorrect for the uncertainty of the indirect measurements. 4. CONCLUSIONS There are two main types of the specifications for measuring devices: the maximum possible total error (4) and the maximum offset, the gain, and the linearity errors (5). The inequalities (8) are offered to find the maximum gain and linearity errors approximately. The result of the evaluation of the linearity error by (8) can be much higher than the true value. If the linearity error is specified besides (4), then the gain error can be found from (4) by (8). General formulae for the absolute (11) and the relative (12) worst-case uncertainties of the indirect measurement are found as functions of three parts of variables. These parts include application for the indirect measurement of one or several devices in one or several ranges with the specifications of the total error or the maximum offset, gain, and linearity errors separately. Only the first parts of (11) can give a lower value of the uncertainty for some types of the indirect measurements in comparison with the approach (2). The formulae published before [1], [2] are special cases of (11). The second part of (11) was not discussed in [1], [2]. Formula (11) can also be used to take into account application of material measures (standard electric resistors, standard signal generators etc.) for the indirect measurements. Applications of the offered approaches are given in Section 4. The advantage of the multiplexed structure vs. the multi- ADC structure from the uncertainty point of view is shown though the multi-ADC structure is better from the dynamics point of view [4]. Conditions for the choice of one range of a device instead of two ranges for two direct measurements are found from the uncertainty point of view. REFERENCES [1] M. Sedlacek, V. Haasz, “Electrical Measurement and Instrumentation”, Vydavatelstvi CVUT, Prague, 1996. [2] F. Attivissimo, N. Giaquinto, M. Savino, “Worst-case uncertainty measurement in ADC-based instruments”, Computer Standard & Interfaces, 29 (2007), pp. 5-7. [3] ISO, Guide to the Expression of Uncertainty in Measurement, 1 st edition 1993, 1995. [4] Simultaneous Sampl. Data Acquisition Architectures. National Instrument, Publish Date 2010. [5] High-Speed Data Acquisition, National Instruments, 2011. [6] Agilent Technologies, User’s Guide, 3401A. [7] V. Didenko, A. Minin. A. Movchan, “Polynomial and piece-wise linear approximation of smart transducer errors”, Measurement, 31 (2002), pp. 61-69. Improved evaluation of uncertainty for indirect measurement The simplest indirect measurement is described by the function, where . The absolute worst-case uncertainty of the indirect measurement when one device is used in the same range () and in accordance with (11) is The absolute worst-case uncertainty of the indirect measurement when two devices of the same type are applied in the same range (, ) in accordance with (11) is The maximum difference of the results found by (12) and (13) is at The application of one range for both direct measurements will be better if (12) gives a lower result in comparison with (15). There are two main types of the specifications for measuring devices: the maximum possible total error (4) and the maximum offset, the gain, and the linearity errors (5). The inequalities (8) are offered to find the maximum gain and linearity errors a... General formulae for the absolute (11) and the relative (12) worst-case uncertainties of the indirect measurement are found as functions of three parts of variables. These parts include application for the indirect measurement of one or several device... Applications of the offered approaches are given in Section 4. The advantage of the multiplexed structure vs. the multi-ADC structure from the uncertainty point of view is shown though the multi-ADC structure is better from the dynamics point of view ...