Uncertainty and Confidence in Measurements

CHAPTER 10

Uncertainty and Confidence in Measurements

The ideal objective of any measurement is the determination of the true value of a measurand. The real objective is to make the most accurate estimate of this true ~/alue because no measuring operation can exist without an error. Consequently, a measurement has a meaning if, having defined a measuring method and a measuring procedure, it provides the best estimate of the value of the measurand and a related uncertainty, the latter representing the degree of dispersion of the results around such an estimate. At the core of the concept of measurement lies the principle of reproducibility, which implies the possibility to compare results obtained at different times and in different laboratories. It is not only a vital requirement of any scientific investigation, but it also responds to practical needs in various fields, such as industrial production and quality control, commerce, law, health, and environment. In order to make meaningful comparisons, it is necessary that measuring uncertainties be treated through a consistent approach. Although the subject is very old, general consensus on the procedures to be followed for expressing the uncertainty has only been reached in recent times, under the initiative of the Comitf International des Poids et Mesures (CIPM), the highest authority in the field of metrology. Through the active cooperation of the National Metrological Institutes and various international organizations, the International Standard Organization (ISO) undertook the task of preparing a Guide to the Expression of Uncertainty of Measurement, which was eventually published in 1993 [10.1]. We will refer to this Guide in the following.

10.1 E S T I M A T E O F A M E A S U R A N D MEASURING UNCERTAINTY

VALUE AND

Measuring a physical quantity is a very common activity in everyday life and the concepts of measurement accuracy and repeatability do not 581

582

CHAPTER 10 Uncertainty and Confidence in Measurements

require special competence to be appreciated. It is intuitive to recognize that some kind of stochastic behavior inevitably affects any measuring operation, be it some gross evaluation performed through our senses or some sophisticated measure made by specialists in the laboratory. If we go somewhat deeper into the problem, we can easily verify that, by repeating the very same measurement m a n y times under identical conditions, scattered values of the measurand are found. Once ordered according to the customary histogram or frequency representations, these do provide the idea of an underlying probability distribution function [10.2]. Such an idea was made quantitative a long time ago. By denoting with x the generic value of the measurand subjected to direct determination, it can be shown that, for example, if the condition of stationarity is satisfied, the probability of finding it within a prescribed interval (x, x + dx) is given by the normal distribution function:

(X- ~)2) exp 2~2 dr(x) dx -~r2 ~

"

(10.1)

f(x) is a symmetric function, peaked at the mean value x = ~, and satisfies the normalization condition y~-oof(x) dx --- 1./~ is also the most probable value of x and is identified with the true value of the measurand, with the meaning that this term has in a statistical sense. It would be the result of a perfect measurement and cannot be known. If the outcome of a measurement is x, the difference 8 = x - ~ is defined as the measurement error, again an unknowable quantity, cr2 is the second-order m o m e n t about the mean:

~

oo

o .'2 - -

(x -

~)2f(x)

dx

(10.2)

and is called variance. Its square root cr provides a measure of the dispersion of the measured values around the true value and is called

standard deviation. Historically, the normal function was proposed by Gauss in order to represent the error distribution in the astronomical observations. It is the idealized distribution function associated with a truly stochastic variable. Any reading or measurement of this variable can be thought of as affected by m a n y small contributions of r a n d o m sign and amplitude, which are generated by a large n u m b e r of sources of influence. The central limit theorem [10.2] ensures that, in a case like this, the values taken by the variable closely follow a normal law, whatever the distribution function of the contributing variables. It is therefore understood that fix) can be

10.1 ESTIMATE OF A MEASURAND VALUE

583

assumed of normal type [10.3]. Any practical measuring operation, carefully performed and corrected for any possible bias, can only approximate the generation of a truly Gaussian process and what one achieves, in general, is an estimate of the true value of the measurand. If n independent observations of the measurable quantity x are performed, providing the values X(1),X(2)~...~X(n)~ the best estimate is given by the arithmetic mean:

YC--

~.k

=1 F/

x(k) ,

(10.3)

where the individual outcomes X (i) differ because of random effects. In the limit n--* oo~ it is expected that ~ =/~. In reality, it is difficult in most instances to fulfill the condition of stationarity for a sufficiently long time and a convenient number of repetitions is chosen according to specific conditions imposed by the problem under testing. From a sample of measurements, one can make an estimate of the variance cr 2 of the whole population of the possible values of the measurand by defining an experimental variance s2(x(k)). This characterizes the dispersion of the measured values around ~:

11 82(x(k)) = Yk=l

(x(k) --

~)2

n- 1

'

(10.4)

together with its square root, the experimental standard deviation s(x(k)). Notice that the number of degrees of freedom v = n - 1 is used in the definition of the experimental variance in Eq. (10.4). In fact, of the n terms (x (k) -Yc)~ only n - 1 are independent. Since the experiment provides the value ~ as a best estimate of the true value of the measurand, we wish to know how good such an estimation is. ~ is itself a random quantity and, according to Eq. (10.3), its variance and standard deviation are cr2(Yc) = cr2/n and cr(~)= cr/x/~ , respectively (the law of large numbers). The best estimates of 02(~) and cr(~) are

$2(~) __

S2(x(k))

__

n

ylkZ=l (x(k) __ ~)2 n(n - 1) (10.5)

S(x(k)) s(~)

-

-- .. I ~k=l (x(k) --

~

n ( n - 1)

~)2

584

CHAPTER 10 Uncertainty and Confidence in Measurements

the experimental variance and the standard deviation of the mean, respectively, s(~) is also called the standard uncertainty u(yc) of the best estimate of the measurable quantity x u(~) = s(~)

(10.6)

and the corresponding variance is u2(x)~-s2(x). According to this definition, the standard uncertainty u(~) is a parameter providing a quantitative evaluation of the dispersion of values that can be reasonably attributed to the measurand. By making repeated measurements of the same quantity, stochastic effects are thus revealed and can be quantified through the standard uncertainty. There are, however, further sources of uncertainty, whose contribution remains constant while the measurements are repeated. They can derive from the environmental conditions (e.g. temperature, humidity, and electromagnetic interference), calibration and resolution of the equipment, peculiarities of the electrical circuit (e.g. thermoelectromotive forces), drifts and distortions, inaccurate assumptions about constants and parameters to be used in the data treatment procedure, and personal errors. The related uncertainty is traditionally classified as systematic, in contrast with the random uncertainty, associated with repeated measurements. It is recognized, however, that such a classification applies to a specific measurement only, because what is random in a measurement can become systematic in a further measurement at a different level. For example, an instrument calibration made in a standard laboratory will report the combination of random and systematic components as a single value of the total uncertainty. A laboratory at a lower hierarchical level, making use of this calibrated instrument, will introduce this value as a systematic effect in the derivation of its uncertainty budget. Systematic effects are expected to produce a bias on the random distribution of the x (k) values obtained upon repeated measurand determinations. This bias can be quantified and corrected for a good proportion, as schematically illustrated in Fig. 10.1, but a residual contribution to the uncertainty of the measurement, having a systematic origin, is nevertheless expected to remain. This can be evaluated by judicious appraisal of all available information on the physical quantity being measured, the measuring procedure and the measuring setups, previous knowledge on the subject, etc. Notice that, in some instances, the correction for the bias can be estimated to be zero, without implying that the associated uncertainty contribution is also zero. The method by which an uncertainty contribution deriving from systematic effects is obtained is defined as a Type B evaluation method. For repeated measurements, we speak of a Type A evaluation method of the uncertainty. These definitions

10.1 ESTIMATE OF A MEASURAND VALUE

0.201

585

-",,,,

0.10

o 0.00.

i\ ~

95

1O0

(a)

105

x m,L

,

bias i--

12

i _ e A

x

v

Z

i

....

(b)

85

90

i ....

95 x

i ......

100

105

FIGURE 10.1 (a) Normal distribution function f(x) for the probability of finding the value of a measurand in the interval (x,x + dx) (Eq. (10.1)). x is assumed to be a truly stochastic quantity. The mean value /~ is defined as the true value of the measurand, an ideal and unknowable quantity, cr is the standard deviation. (b) Independent repeated measurements generate numbers that, arranged in a histogram, emulate the normal distribution. The raw data are shown on the left, with their mean ~ and the related standard deviation s(2) (not in scale). The standard deviation of the mean is equivalently called standard uncertainty u(~) (see Eqs. (10.5) and (10.6)). After correction for the systematic effects (somewhat exaggerated here for clarity), the best estimate of the true measurand value is characterized by a combined uncertainty uc(x), including the uncertainty on the correction.

586

CHAPTER 10 Uncertainty and Confidence in Measurements

are recommended by the ISO Guide [10.1]. In any case, all components of the uncertainty, be they evaluated with A or B methods, are described by the same statistical methods, characterized by probability densities with variances and standard deviations, and are treated and combined in the same way. For the Type A uncertainties, the probability densities are obtained from observed frequency distributions, while for the Type B uncertainties one makes use of "a priori" probabilities. When applying the B method, an assumption is made regarding the distribution function of the measurand values. If derived from a calibration certificate, this distribution is conveniently assumed as being of the normal type. In other cases, it is only possible to estimate upper and lower bounds x0- and x0+ for the values that x can take in a specific measurement. With no further knowledge on how these values are distributed, one can reasonably assume that they are equally likely to belong to any point of the interval (Xo-,Xo+). The variance and the uncertainty associated with the expectation value ~ = ( x 0 - + x0+)/2 of this rectangular distribution are expressed, posing x 0 - + x0+ = 2a, by the equations: U2(~ ) __ (X0+ -- X 0 _ ) 2

12

10.2 C O M B I N E D

a2

= -~-,

a

u(~) = - ~ .

(10.7)

UNCERTAINTY

In the usual case one does not make a direct determination of the measurand, but a certain number of input quantities are sampled or considered, to which the measurand is related by a functional relationship. Let the functional relationship between the output quantity y and the input quantities Xl,X2, ...,XN be y = g(xl~x2, ..., XN).

(10.8)

If the best estimates of the input quantities a r e Xl~ x2~-..~ XN~ we write for the best estimate of the output quantity:

9 = g(x~,X2,

"", IN).

(10.9)

The identificationof the input quantities is a crucial step of the whole process of uncertainty determination. They can include, besides the quantities subjected to direct measurement, the bias corrections to suggested by the specifically considered measuring procedure. The problem then becomes one of determining variance and uncertainty of from knowledge of the same quantifies for ~i,x2,...,xN, taking into

10.2 COMBINED UNCERTAINTY

587

account possible correlation effects. To this end, we assume that the function g and its derivatives are continuous around the expectation value Y- A Taylor series development, truncated to the first order, provides y-- y = ~ ~ i=1

(10.10)

(Xi - YCi)

for small intervals (X i -- YCi). The square of Eq. (10.10) is

(y_~)2=~.

N (0g)2 N-1 N OR OR ~ ( XOXj ~ (Xi_YCi)2q_2 y . y . ~OXi i=1 i=1 j=i+l

i -- YCi)(Xj -- YCj). (10.11)

By interpreting the differences appearing in Eq. (10.11) as experimental samples and taking the averages, we can express the variance of the output estimate as a combination of the variances u2(xi) and covariances U(YCi,YCj) of the input estimates, according to the law of propagation of

uncertainty: N (Og)2

uc(y)= ~22

i=1

~

N-1N OgOg /,/2(~i)+ 2 y. y. OXi ~u(Yci,Yq). OXi i=1 j=i+l

(10.12)

-

uc(y) is called combined variance and its square root Uc(y) is the combined standard uncertainty. The partial derivatives in Eq. (10.12) are called sensitivity coefficients and by posing ci -~ Og/Oxi we can rewrite Eq. (10.12) as N N-1 N 2Uc(y) = y c2ua(xi)-}-2 y~ y . CiCjU(Xi~YCj). i=1 i=1 j-i+1

(10.13)

Equation (10.13) is of general character and combines variances and covariances of the input quantities irrespective of the type of evaluation method (either A or B) employed in their derivation. An input estimate Xi can be associated with both Type A and Type B uncertainties and the related variance in Eq. (10.13) is written as U2(Xi) = U2A(YCi)-~- U~(YCi).

(10.14)

Notice that the output quantity y is associated in many cases with an approximately normal distribution function, although the distribution functions of the input quantities can be far from normal. One can, in fact, linearize the functional relationship (10.8) around the best estimates of the input quantities by means of a Taylor development truncated to

~8

CHAPTER 10 Uncertainty and Confidence in Measurements

the first order (Eq. (10.10)). The output distribution is then provided by the convolution of the input distributions and, according to the central limit theorem, it can be approximated by a normal distribution, the higher the number of repetitions and the input quantities, the better the approximation. The case where a dominant Type B component of uncertainty exists, with distribution different from normal, is an exception to this rule. 2- -2 In many cases, it is useful to resort to the relative variance Uc(y)/y and the relative standard uncertainty uc(Y)/9. Remarkably enough, if it occurs that the functional relationship relating the output quantity and the input independent quantities has the general form y = m.xPl.x p2, ...,XPNN, with m a constant coefficient and Pl,P2, ...,Pn either positive or negative exponents, we can express the relative variance as

2Uc(y)

N U2(Xi) ~2 __~p2 L-~ 9 i=l Xi

(10.15)

The sensitivity coefficients are sometimes evaluated by experiments by determining the variation of the output quantity y upon a small variation of an input quantity xi, the other input quantities being kept fixed (see Eq. (10.10)). If the input quantities are uncorrelated, the covariance u(Yci,ycj) is zero and the combined uncertainty reduces to

Uc(Y) = i ~'i=1r

(10.16)

A measure of the degree of correlation is provided by the value of the coefficient: u(~i, ~j) r(Yci,Ycj) = u(Yci)u(Ycj),

(10.17)

which varies from 0 to 1 on going from uncorrelated to completely correlated input variables. In the latter case, the combined standard uncertainty becomes

N UcQ~) --- ~ CiU(YCi). i=1

(10.18)

For a Type A evaluation of the uncertainty, the covariance of two correlated input estimates (2i, x]) can be experimentally evaluated by forming the cross-products (xl k)- Yci).(x~k)- 2]) at each repetition and

10.3 EXPANDED UNCERTAINTY AND CONFIDENCE LEVEL

589

averaging them according to the equation: i! u(2i,Yq) = s(2i, 2j)= n(n - 1) y (xlk) -- Xi)'(X~k) -- ~j)"

(10.19)

k=l

For a Type B evaluation, critical analysis of the available information on the reciprocal influence of the input estimates should be carried out. If, for example, it is known that a variation Ai of xi produces a variation &j of ~j, the correlation coefficient can be roughly estimated as r(Yci, Ycj) ~ u(Ycj)a i "

(10.20)

Two input quantities (Xi,Xj) can have their correlation originating from a common set of independent and uncorrelated quantities z,,,. Let gi(Zl, ...,ZM) and gj(zl, ...,zM) be the functional relationships associated with xi and xj, respectively. We can write for the covariance of the best estimates (2i, ~j): M Ogi Ogj U2(~,n). U(YCi'YCJ) = Y" OZ,,, OZ,n llZ--1

(10.21)

10.3 E X P A N D E D U N C E R T A I N T Y A N D C O N F I D E N C E LEVEL. WEIGHTED UNCERTAINTY The discussion of the derivation of the uncertainty budget carried out in the previous sections illustrates the great merit of the procedure recommended by the ISO Guide, which permits one to combine in a consistent way all the contributions to the uncertainty, derived either from repeated measurements or through "a priori" probabilities. The information conveyed by the measurement can then be confidently collected in two parameters: the best estimate (experimental mean) and the combined uncertainty. Except for special cases, the probability distribution of the output quantity, being the convolution of the input distributions, is approximately of normal type (central limit theorem). If z is a quantity described by a normal distribution function, characterized by an expectation value/~ and a standard deviation or, and we define a confidence interval + kcr around/~, by integrating the distribution function over it we will achieve a corresponding confidence level p (the included portion of the area of the distribution). With coverage factors k = 1, 2, and 3, the confidence levels are p = 68, 95.5, and 99.7%, respectively. Let y be a quantity, defined as in Eq. (10.8), subjected to measurement and characterized by an experimental best estimate 9 and

590

CHAPTER 10 Uncertainty and Confidence in Measurements

a combined standard uncertainty Uc(y). We wish to determine the coverage factor k identifying an expanded uncertainty U = kuc(~) , that is, an interval 9 - U --< y - 9 + U, to which the true value of the measurand is expected to belong with a given high confidence level p (e.g. 95%). With knowledge of U, the result of the measurement can be declared in the form: Y = 9 + U.

(10.22)

We do not actually know the expectation value /~ and the standard deviation or of the output distribution, but only the best estimate 9 and the standard uncertainty uc(~). We then consider the quantity t-

y- y

(10.23)

Uc(9)

and its probability distribution function ~ t ) . The integration of q~(t) over a certain interval ( - t p , +tp): f+t~ q~(t) dt

(10.24)

P = d-tp

provides the confidence level for the expanded interval U = t~Uc(~)= kuc(9). In fact, the condition ( - t p ~ t ~ tp) is equivalent, according to Eq. (10.23), to the condition (9 - tpUc(9) ~ y ~ 9 + tpUc(9)). When y is a single quantity subjected to direct measurement and its best estimate 9 = ~ is obtained by means of a series of n independent repeated measurements, q~(t) is described by the Student distribution function (t-distribution): ~b(t) -

~

1

F((v + 1)/2) ~+1)/2, F(u/2) (1 + ta/v) -(

(10.25)

where the properties of the F function are known and v = n - 1 is the n u m b e r of degrees of freedom, q~(t) reduces to the normal distribution in the limit v---+ oo, a condition already well approximated for v --- 50. For the general case where y is a function of two or more input quantities, Eq. (10.25) can be used only as an approximation by introducing an effective number of degrees of freedom /jeff in place of v. /jeff can be calculated in terms of the degrees of freedom/ji of the input uncertainty contributions u2(9) c2ua(xi)~ under the assumption of independent input estimates. It is provided, in particular, by the Welch-Satterthwaite formula: =

4-

/Jeff =

uc(y) 4-

ui (y) /ji

,

(10.26)

where u~(9) = (y../N_au 2i (y)) - ~_. While for the Type A evaluation, z,i = n - 1 , the degrees of freedom in the Type B evaluation can be estimated on

10.3 EXPANDED UNCERTAINTY AND CONFIDENCE LEVEL

591

the principle that the more reliable the standard uncertainty ui(y) the higher ~'i. For the usual case where an "a priori" probability is taken, the uncertainty is completely defined and z,i--* oo. Tabulations are available where, for given ~'eff values, the coverage factors k = tp associated with a confidence level p are provided [10.4]. In most cases, a confidence level p---95% is deemed adequate. The corresponding coverage factors are provided, for different values of ~'eff, in Table 10.1. An application of the concepts discussed in this section is given in Appendix C (Example 2). The procedure for providing the result of a measurement can therefore be summarized as follows: (1) The measurement process is modeled and the mathematical relationship y = g(xl,x2, ...,x N) between the measurand y and the input quantities Xl,X2, ...,XN is expressed. These quantities also include possible bias corrections. (2) The best estimates xl, x2,..., XN of the input quantities are made. (3) The standard uncertainties u(xi) of the input estimates are found. Type A evaluation is applied for input estimates obtained by means of repeated measurements and Type B evaluation for all other kinds of estimates. If there are correlations between input quantities, covariances are considered. (4) Using the functional relationship y = g ( x l , x 2 , . . . , X N ) , the best estimate 9 of the measurand is made. (5) The combined standard uncertainty Uc(y) is calculated by combination, with the appropriate sensitivity coefficients, of the variances and covariances of the input estimates. (6) The expanded uncertainty U = kuc(9) is determined, with the coverage factor k typically ranging from 2 to 3 for a confidence level of 95%. The actual value of k depends on the effective number of degrees of freedom Veff, calculated by means of the Welch-Satterthwaite formula, and are found in generally available tabulations. The result of the measurement is eventually declared a s y = ~/+ U.

TABLE 10.1 Coverage factor k as a function of the effective degrees of freedom Veff

for a confidence level p = 95.45%. Ueff k

1 13.97

2 4.53

3 3.31

4 2.87

5 2.65

7 2.43

10 2.28

20 2.13

50 2.05

oo 2

k is provided by the Student distribution function and coincides, in the limit veff~ o% with the value provided by the normal distribution.

592

CHAPTER 10 Uncertainty and Confidence in Measurements

It may happen that the same quantity is measured by means of different methods or in different laboratories and the problem arises of combining the results in order to obtain the most reliable estimate of the measurand value. It is usually held that, being the different estimates normally associated with different uncertainties, a comprehensive estimate based on the data generated by the whole ensemble of experiments is best obtained by means of weighted averaging. Let us assume that M independent experiments, made of a convenient number of repeated measurements, have produced the best estimates Yl~ Y2~'" "~ YM and the related uncertainties uc(yl),Uc(92),...,Uc(gM). We look for a weighted estimate of the type M

= ZgilJi

(10.27)

i=1

having minimum variance. The weight factors gi must satisfy the condition: M

Z gi = 1.

(10.28)

i=1

We then write the variance of ~ in terms of the variances of the estimates Yl,Y2, ...,YM

M U2(~) = Z gi2 Uc(Yi) 2 i=1

(10.29)

and find the set of factors gl,g2, ...,gM minimizing u2(~) [10.5]. With the use of the Lagrange multiplier k, the variance can be written as u 2 ( Y ) = Z gi2 Uc(Yi) 2 - + ,~ 1 - Z g i i=1 i=1

(lO.3O)

and the minimization conditions

0u2(#) Ogi

= 2giu2(gi)- X = 0

(10.31)

provide the weight factors k

gi = ," cl,y "

(10.32)

as a function of the multiplier JL This is eliminated through the normalization constraint (10.28) and the factors gi are thus obtained as

10.3 EXPANDED UNCERTAINTY AND CONFIDENCE LEVEL

593

a function of the input variances:

U2(~ti)

g~ =

~.iM1

.

(10.33)

1 U2(~/i)

The weighted mean and the associated weighted variance follow from Eqs. (10.27) and (10.29): y.iM1

Yi

u2(yi)

~t ~-

1

,

(10.34)

2 Uc(Yi)

1

u2(~)

M =

1 u2(yi) "

(10.35)

According to these equations, the smaller the uncertainty associated with a result, the higher its role in determining the reference value ~ and the uncertainty u(,0). A confidence interval can be identified with ~ + U r e f = ~t + ku(~t), where the coverage factor k is taken from the t-distribution for v = M - 1 (see Table 10.1). An example of intercomparison of magnetic measurements is shown in Fig. 10.2. Laboratories find the most stringent test of their measuring capabilities in the comparison exercise. At the highest level, the national metrological laboratories organize key comparisons as a technical basis for establishing the equivalence of measurement standards and the mutual recognition of calibration and measurement certificates. The degree of equivalence of each national measurement standard is expressed quantitatively by two terms: its deviation from the key comparison reference value and the uncertainty of this deviation at 95% level of confidence [10.6]. The assumption of the weighted mean (10.34) as the reference value is considered appropriate if the collective measurements are consistent and they can be treated as part of a homogeneous population. Discrepant results often arise in intercomparisons and special approaches have therefore been proposed to deal with the problems, including politically sensitive issues, raised by the presence of inconsistent data [10.7, 10.8]. It is clear that meaningful comparisons can be pursued only where all laboratories follow a common approach to the evaluation of the measuring uncertainty, such as the one provided by the ISO Guide [10.9].

594

CHAPTER 10 Uncertainty and Confidence in Measurements

3.2

t_ - Uref

3.1

cL 3.0

2.9

Laboratories

FIGURE 10.2 Six different laboratories perform the measurement of magnetic power losses on the same set of non-oriented Fe-Si laminations with the SST method [10.10]. They report their best estimates and the related extended uncertainties as shown in this figure. Analysis shows that the result provided by laboratory 6 is to be excluded, because it largely fails the consistency test provided by the calculation of the normalized error (Eq. (10.36)). The reference value Pref solid line) and the expanded uncertainty Uref (delimited by the dashed lines) are then obtained by re-calculating Eqs. (10.34) and (10.35) with the results of laboratories 1-5.

Let us analyze, as an example regarding magnetic measurements, some results derived from an intercomparison of magnetic power losses in non-oriented Fe-Si laminations [10.10]. Six laboratories (i = 1, ..., 6) provide, as shown in Fig. 10.2, their best estimates Pi of 50 Hz losses at 1.5 T (full dots) and the associated expanded uncertainties (at 95% confidence level) Ui = ku~(Pi). These data are all included in a preliminary determination of the weighted mean (reference value Pref) and the expanded weighted uncertainty Uref~ according to Eqs. (10.34) and (10.35). The consistency of the reported uncertainties with the observed deviations of the best estimates Pi from Pref is then verified. To this end, the normalized error

Eni

=

IPi - Prefl ~/U2..}_Ur2;

(10.36)

10.4 TRACEABILITYAND UNCERTAINTY

595

is considered [10.11]. When the dispersion of the individual estimates is in a correct relationship with the correspondingly provided uncertainties, it is expected that Eni < 1 [10.12]. In the present case, the reported (P6, U6) values largely fail to satisfy this condition (En6--3.1), due to both unrecognized bias and unrealistic uncertainty estimate. They are consequently excluded from the analysis. Pref and Ure f are then re-calculated by means of Eqs. (10.34) and (10.35), providing the results reported in Table 10.2. It should be stressed that the estimated expanded relative uncertainty of the reference value Uref~re1 is of the order of 1%, typical of this kind of measurement. It is also noticed that the result of laboratory 5 is not completely satisfactory because IP i - Prefl is higher than the related expanded uncertainty: (10.37)

U(Pi - Pref) -- k~/u2(Pi) if- u2(Pref) 9

10.4 T R A C E A B I L I T Y MAGNETIC

AND UNCERTAINTY MEASUREMENTS

IN

Measurements are indispensable for the manufacturing and trade of products and for any conceivably related research activity. They need to be traceable to the relevant base and derived SI units, that is, related to the corresponding standards through an "unbroken chain of comparisons, all having stated uncertainties" [10.13]. Industrial and research laboratories can achieve traceability for a specific kind of measurement through accredited laboratories or directly to National Metrological Laboratories (NMIs). The mission of NMIs is to ensure that the standards are the most accurate realization of the units, so that these can be disseminated to the national measurement network. To ensure this calibration flow, national calibration and accreditation systems have been developed. The NMIs engage in extensive intercomparisons of standards, organized either by the regional metrological organizations (e.g. EUROMET and NORAMET) or the Consultative Committees of the International Committee for Weights and Measures (CIPM). Supervision of the intercomparison activity is carried out by the Bureau International des Poids et Mesures (BIPM), which has the task of ensuring worldwide uniformity of measurements and their traceability to the SI units (Fig. 10.3). Physical standards for magnetic units, traceable with stated uncertainties to the base SI units, are maintained in several NMIs and used for dissemination to measurement and testing laboratories [10.14, 10.15]. Illustrative examples of magnetic standards and calibration capabilities

596

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~o c~

~b

~

0

0

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c5c5c5c5c5

I

I

0 0 0 0 c : )

I

c~c~c~c~c~

I

qa

x

0

b, lii 0

,,,...

0 0

,'0

l>

0

.,ii

0

.,i,,,z

.~i~

Ii,I

".n

.~

0 .4.a

~,

II

~:

~

II

9~

-~

~~

,4.,

N

v

.,.~

O~

0

r~

Uncertainty and Confidence in Measurements

I

I

o oo

o

xo

i c5

0 LO 0 LCb 0 C~b t'~ ,~ XO t~b 0 0 0 0 0

c5c5c5c~c5

0 I..~ L~ I..~ 0 O0 C~ L~ ~'~

L~b 04 0

i c5

Lr) L~ ~0 t~ O'b O~ t~. ~11 ~0 brb 0 0 0 0 0

c:5 c:5 ~-~ c:5 ~-~

CHAPTER 10

b

bO

I

b~

bO

bO

v

bO

0 0

10.4 TRACEABILITYAND UNCERTAINTY

597

W......................................... I BIPM

......f

NMI

--~

!

NMI

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Calibration& TestingLab. FIGURE 10.3 Traceability chain in measurements. The calibration and testing laboratories can relate their measurements to the SI units through a flow of calibrations starting from the National Metrological Laboratories (NMIs). The NMIs perform mutual comparisons of their standards, under supervision by the Bureau International Poids et Mesures (BIPM). developed by NMIs are given in Table 10.3 [10.6, 10.16]. Examples of recent NMI intercomparisons of magnetic measurements, regarding DC and AC flux density and apparent p o w e r / p o w e r loss in electrical steel sheets, are reported in Refs. [10.17, 10.18]. The importance for industrial customers of measurement traceability and calibrations, as ensured today by the NMIs, is easily appreciated for magnetic measurements. For example, a magnetic steel producer can ensure the quality of the grainoriented laminations delivered by one of its plants only through periodic calibrations of its magnetic equipment, traceable to an NMI laboratory. Since a large plant can produce 105 ton/year of this high-quality material, worth around s 10s, the economic impact of traceable magnetic measurements is apparent. It should be stressed that when ferromagnetic (or ferrimagnetic) materials are characterized, several factors can detrimentally affect the reproducibility of the measurements. For one thing, the magnetization process is stochastic in character and strongly affected by the geometrical properties of the sample and magnetic circuit. In addition, time effects are ubiquitous, either due to aging or various types of relaxation effects, and many alloys (e.g. rapidly quenched magnetic ribbons) display an intrinsically metastable behavior. Measurements must then be carried out under tightly prescribed conditions, such as those defined by written standards, and by means of accurately calibrated setups in order to achieve

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CHAPTER 10 Uncertainty and Confidence in Measurements

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CHAPTER 10 Uncertainty and Confidence in Measurements

good reproducibility. To illustrate this point, we can take some of the results obtained with the previously mentioned intercomparison of grain-oriented steel sheets [10.18], which involved three NMIs and 15 different types of laminations. The statistical analysis of these results (see the discussion in Section 10.3) provides the relative deviations (Pi- Pref)/Pref, where Pi is, for each lamination, the power loss of the ith NMI (J = 1.7 T, f = 50 Hz) and Pref is the associated reference value (Eq. (10.34)) (see the histogram shown in Fig. 10.4a). The relative deviations of the apparent power (Si- Sref)/Sref are represented in Fig. 10.4b. It is observed that, under the excellent measuring conditions attained by these metrological laboratories, (Pi- Pref)/Pref is in the 0.5% range, always in agreement (within the 95% confidence level) with the consistency condition U(Pi- Pref) > [Pi- PrefI. The same condition is satisfied by ( S i - S r e f ) / S r e f , which is found to be in the 1% range. It has been previously emphasized that intercomparisons have meaning if the different laboratories follow a uniform approach to the measuring uncertainty. The ISO Guide provides such an approach. It therefore appears appropriate to discuss a few illustrative cases of correspondingly obtained uncertainty determination in magnetic measurements.

10.4.1 Calibration of a magnetic flux density standard A Helmholtz pair is prepared to serve as a magnetic flux density standard in the range 2 x 10-4T ~ B-----4 x 10-2T. Each winding is made of 2210 turns, with average radius r - 0.125 m, and is provided with a supplementary 200-turn coil, improving the uniformity of the generated axial field. The Helmholtz coil constant kH - - B / i is obtained by the simultaneous measurement of the magnetic flux density in the center of the pair and the circulating current. B is detected by means of a lowfield NMR probe operating in the range 0.034-0.121 T. By placing the Helmholtz pair within a triaxial Helmholtz coil system, the earth's magnetic field is compensated to the level of 0.02 ~T. The calculations show that inside the active volume of the probe (11 m m x 6.5 mm), the produced field is homogeneous enough to permit safe establishment of the resonance conditions. It is observed, in particular, that the maximum relative variation of the field amplitude in this volume is --- 1.5 x 10 -5. The current i (of the order of 1.6 A for B --- 0.036 T) is determined by detecting the voltage drop Vacross a standard resistor R = I f~ (four point contacts). Twenty repeated acquisitions are made over a time span of 10 s, short enough to keep the temperature increase of the windings due to Joule heating within z~T---0.4 ~ The coil temperature is controlled immediately before and after the determination of B by measuring the resistance

10.4 TRACEABILITY AND UNCERTAINTY

601

E ~ NMI1 15.

Jp= 1.7T f = 50 Hz

rrrrrl NMI 2 NMI 3

10.

52 !

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9

o~o

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9

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FIGURE 10.4 An intercomparison of power loss P and apparent power S carried out by three NMIs in 15 different types of grain-oriented Fe-Si laminations tested by the Epstein method identifies, for each lamination, the reference values Pref and Sref. The relative deviations (Pi - P r e f ) / P r e f and (Si - S r e f ) / S r e f of the best estimates from the reference values are in the 0.5 and 1.0% range, respectively (adapted from Ref. [10.18]).

CHAPTER 10 Uncertainty and Confidence in Measurements

602

of the winding. B and V are s i m u l t a n e o u s l y detected and the ratio k H -is calculated. The best estimate is written as

(B/V)R

w

kH = (B/V)R + (~(kH)B -4- (~(kH)V q- ~(kH)R,

(10.38)

w h e r e the first term on the right-hand side is the experimental mean, as p r o v i d e d b y Eq. (10.3) w i t h n = 20 a n d (~(kH)B, (~(kH)v, and (~(kH)R are the bias c o m p e n s a t i o n terms associated w i t h field reading, voltage reading, a n d s t a n d a r d resistance value. The bias terms are d e e m e d negligible a n d w e find, from averaging, kH --0.01952636 T/A. The relative c o m b i n e d u n c e r t a i n t y is provided, according to Eq. (10.16), as a function of the Type A a n d Type B i n p u t uncertainties b y the expression: w

uc(kH) kH

i

u2(kH) +

U2B(BB)A U2B(V)} u2B(R)

~2

e 2

R2

(10.39)

,

as s u m m a r i z e d in Table 10.4. The Type A uncertainty, evaluated for n - - 2 0 , is uA(kH)/kH = 1.5X10 -6. The Type B uncertainty

UB(B)/B

10.4 Uncertainty budget in the calibration of a standard magnetic flux density source

TABLE

Source of uncertainty

Distribution Divisor Relative Sensitivity Degrees of function uncertainty coefficient freedom

Magnetic field reading Voltage reading Standard resistor (calibration) Repeatability Combined relative standard uncertainty Expanded uncertainty (95% confidence level)

Rectangular

x/3

2.1 x 10 -5

1

oo

Rectangular Normal

~/3 2

1.2 x 10 -5 2.5 x 10 -5

1 1

oo oo

Normal Normal

1

1.5 X 10 -6

1

-

3.4 x 10 -5

-

19 137

6.8 x 10 -5

Coverage factor k=2

The source is realized by means of a Helmholtz pair (average winding radius r = 0.125 m) energized by a maximum current of 1.6 A. The calibration is performed via a low field NMR probe at B = 0.036 T. The Helmholtz pair is placed in a field-immune region (residual B--- 20 nT), at the center of a triaxial Helmholtz coil setup providing active compensation of the earth's magnetic field. The maximum field gradient in the active NMR probed central volume 11 mm x 6.5 mm of the standard coil is 3 x 10 -7 T / m m at B = 0.036 T. The result is kH = kH + U = 0.01952636 + 1.33 x 10 -6 T/A.

10.4 TRACEABILITY AND UNCERTAINTY

603

associated with the B reading is the sum of two main independent contributions: (1) spatial inhomogeneity of the generated field and uncertainty in the position of the NMR probe; (2) temperature variation and uncertainty on its value during the measuring time. Regarding contribution (1), we assume a rectangular distribution with upper and lower bounds differing by 2a(1)/B--4x10 -5 and we estimate from Eq. (10.7) u~)(B)/B -- 1.2 x 10 -5. Contribution (2) is estimated on the basis of a series of measurements around room temperature (e.g. between 19 and 30 ~ With measuring temperature fluctuation limits +AT = 0.2 ~ the rectangular distribution of half-width a (2)/B = 3 x 10 -5 is evaluated, leading to U(B2)(B)/B-- 1.7 • 10 -5. Consequently, UB(B)/B= 2.1 x 10 -5. The voltmeter specifications provided by the manufacturer give, in the employed 10 V scale, a 1 year accuracy of 12 p p m of reading + 2 p p m of range. For a 2.3 V read-out on this scale, this corresponds to a semi-amplitude of the distribution a = 4.8 x 10 -5 V. The corresponding relative uncertainty is UB(V)/V = (a/x/3)(1/V)--1.2• 10 -5. The calibration certificate of the standard I f~ resistor provides a 2or uncertainty of + 5 0 p p m . It is then UB(R)/R=2.5xIO -5. The relative combined standard uncertainty is obtained by Eq. (10.39) as uc(kH)/kH = 3.4 X 10 -5 and the expanded uncertainty at 95% confidence level is, with coverage factor k---2, U/fcH = 6.8• 10 -5. We eventually write the Helmholtz coil constant at the temperature T -- 23 ~ as kH = kH + U -- 0.01952636 + 1.33 • 10 -6 T/A.

10.4.2 D e t e r m i n a t i o n of the D C polarization in a ferromagnetic alloy We wish to determine the normal magnetization curve of a non-oriented Fe-(3 wt%)Si lamination with the ballistic method. We want to know, in particular, the uncertainty associated with the determination of the polarization value J at a given applied field. Testing is made, according to standards, on half longitudinal and half transverse Epstein strips on a rig provided with compensation of the air flux. Each point of the curve is obtained, after demagnetization, by switching the field several times between symmetric positive and negative values and eventually recording the flux swing 2 h ~ = 2NSJ, where N is the number of turns of the secondary winding (N = 700) and S is the cross-sectional area of the sample, by means of a calibrated fluxmeter. The measurement is repeated five times, always using the same procedure, and the arithmetic mean h ~ is obtained. The best estimate of the polarization value for

604

CHAPTER 10 Uncertainty and Confidence in Measurements

a given applied field H is A~ J= ~ 4- 3(J)d q- ~)(J)a if- 3(1)s q- 3(J)T,

(10.40)

where 3(J)d, 8(/)a, 3(/)T, and 3(J) s are the bias corrections for fluxmeter reading, residual air flux, temperature, and sample cross-sectional area, respectively. The relative combined standard uncertainty of the polarization value is then expressed through the Type A and Type B contributions (see Eq. (10.15)) as Uc(j[) / U2(A(I )) U2(~-~)d u2(A(I))a U2(~-~)T u2(S) . (10.41) i -- V ~-~2 if- A(I)2 -}- A ~ 2 if- A ~ 2 -} ~ $2

Eight strips (two on each leg of the frame), nominally 305 m m long and 30 m m wide, are tested. The sample cross-sectional area S is determined by the precise measurement of the total mass 8m and the strip length l as S = m/431, assuming the nominal material density 8 = 7650 k g / m 3. It is found S = 2.9529 x 10 -5 m 2. The variance u2(S) is obtained by combination of the variances associated with m, 1 and 3. The uncertainty of the value of 8 is by far the largest and we write

S2

~

(~2

(10.42)

Based on the data provided by the steel producer, it is assumed for 8 a rectangular distribution of semi-amplitude a - 25 k g / m 3. Since UB(3)a/x/-3, we obtain u2(S)/S 2= 3.6• -6. Again, we consider the bias compensation terms equal to zero and from the five repeated measurements, it is obtained, for H - 80 A / m , A ~ - 0.02013 V s, i.e. J = 0 . 9 7 3 8 T . The associated Type A uncertainty is found to be UA(A~)/A(I) = 2 x 10 -3. The fluxmeter, calibrated by means of a standard mutual inductor, is assigned a relative uncertainty (1or) on the employed scale (r/A(I) - 4 x 10 -3, from which UB(A(I))d/A(I) = 4 x 10 -3. The uncertainty for the uncompensated air flux is estimated UB(A(I))a/A(I ) = 5 X 10 -4, while any contribution to the uncertainty of the measured polarization value related to temperature is deemed negligible in these alloys and UB(A(I))T/A(I ) ~" 0. It should be stressed, however, that this last term could become very important in some hard magnets (e.g. N d - F e - B alloys) to which the present discussion clearly applies. The combined and expanded uncertainties are thus obtained by means of Eq. (10.41), as summarized in Table 10.5. This specific result is expressed as J = j + u = 0.9738 + 1.00 x 10 -2 T.

10.4 TRACEABILITY AND UNCERTAINTY

605

10.5 Uncertainty budget in the measurement with the ballistic method of the magnetic polarization in non-oriented Fe-(3 wt%)Si laminations TABLE

Source of uncertainty

Distribution Divisor Relative Sensitivity Degrees of function uncertainty coefficient freedom

Normal 1 Fluxmeter reading and drift Rectangular x/3 Air-flux compensation Rectangular x/3 Cross-sectional area of the sample Rectangular x/3 Sample temperature 1 Repeatability Normal Combined relative Normal standard uncertainty Expanded uncertainty (95% confidence level)

4 x 10 -3

1

co

5X

10 - 4

1

co

1.9 X

10 - 3

1

co

1

co

1

4 20

---0 2X

10 - 3

4.9 X 10 -3

1.03 x

10 - 2

-

Coverage factor k=2.1

The uncertainty components are specified in Eq. (10.41). The measurement is performed on a point of the normal magnetization curve, under an applied field H -- 80 A/m, with eight strips inserted in an Epstein rig. The result is J = j + U = 0.9738 ___1.00 x 10 - 2 T.

10.4.3 Measurement of power losses in soft magnetic laminations Soft magnetic materials are p r o d u c e d and sold for use p r e d o m i n a n t l y in energy applications and have their quality classified according to their p o w e r loss figure. The precise and reproducible m e a s u r e m e n t of the p o w e r losses in these materials is industrially significant and is required in m a n y application-oriented research investigations. Specific measurem e n t standards have therefore been developed and u p g r a d e d over the years [10.19-10.21]. Inter laboratory comparisons have been carried out to validate these standards, settling to a broad extent the m e a s u r e m e n t capabilities of metrological and industrial laboratories. Critical to the appraisal of the reproducibility and degree of equivalence of the measurements p e r f o r m e d by different laboratories is the correct determination of

606

CHAPTER 10 Uncertainty and Confidence in Measurements

the measurement uncertainty. This is quite a complex task because many possible contributions have to be taken into account, as thoroughly discussed in Ref. [10.22]. Unduly optimistic or pessimistic evaluations are not infrequent, as revealed by the analysis of intercomparisons (see Fig. 10.2). We shall discuss here a largely simplified approach, focused on the testing of soft magnetic laminations at power frequencies, by considering only the most relevant contributions to the uncertainty and assuming that the signal treatment is performed by digital methods (see also Section 7.3). Let us therefore express the average magnetic power loss per unit mass at the frequency f as

Ps = ~ H dB = ~

/fH(t)---d--~dt ,

(10.43)

where H and B are applied field and induction in the sample, respectively, and 3 is the material density. Equation (10.43) can equivalently be written in terms of the current iH in the primary circuit and the secondary voltage VB as

f NH fl/d VB(t)iH(t)dt (10.44) 3 NB~,*S Jo having posed H(t) = (NH/~,*)iH(t)and VB(t) -- NBS(dB(t)/dt), NH and NB Ps

_

being the number of turns of the primary and the secondary windings, respectively, s the magnetic path length and S the sample cross-sectional area. We assume that the measurement is performed under sinusoidal induction waveform (i.e. sinusoidal secondary voltage VB(t)) and we consequently write 1

Ps-- 3r

N H

NB

1

~

~

RH VBVH1cos qG

(10.45)

where RH is the resistance value of a calibrated shunt in the primary circuit. 17B and 17m are the rms values of the secondary voltage and of the fundamental harmonic of the voltage drop on the shunt, respectively, which are phase shifted by the angle ~. Let us thus consider a possible approach to the evaluation of the uncertainty in the specific practical case of grain-oriented Fe-(3 wt%)Si laminations, tested by means of an Epstein frame and a digital wattmeter. It is assumed that testing is made at peak polarization Jp--1.7 T and frequency f = 50 Hz, with automatic air-flux compensation. We can therefore assume VB(t)-NBS(dJ(t)/dt) = 2~fNBSJp sin tot. A number of repeated determinations of P are made, each time disassembling and assembling the strips in the same order. Sixteen strips, 305 mm long and 30 mm wide, are used

10.4 TRACEABILITYAND UNCERTAINTY

607

(four in each leg of the frame) and the cross-sectional area of the resulting sample is determined measuring the total mass m (8 = 7650 k g / m 3) and the length of the strips. It is obtained that S--3.3569 x 10 -5 m 2. The voltages VB(t) and Vm (t) are amplified and fed by synchronous sampling (e.g. 2000 points per period, interchannel delay lower than 1 x 10 -9 s, trigger jitter ~- 10-1~ -11 s) into a two-channel acquisition setup and A / D converter and Eq. (10.44) is computed by means of suitable software. By denoting with Pmeas the result of such a calculation, we express the best estimate of the power loss as P -- PmeasF(AJp)F(AFF)F(AT)"4- 8(P)VBa q- 8(P)VBg q- 3(P)vH1

(10.46)

+ 8(P)a + 8(P)s + 8(P)~,

where the first term on the right-hand side is the mean, made over the repeated measures, of the values of Pmeas times the correction factors F(AJp), F(AFF), and F(AT). These factors account for the differences AJp, AFF, and AT between the actual and the prescribed values of peak polarization, form factor of VB(t) (FF = 1.1107), and temperature T (23 ~ respectively. They are recorded and automatically multiplied by Pmeas with each measurement repetition. Experiments [10.23-10.25] suggest the following approximate relationships:

( ,p )18 jp+Ajp'

F(T) =

(1-5 x 10-4AT),

F(AFF)=

( ) 1.8 1 + fl 1.1107 + AFF 1.1107 (10.47)

where fl is the ratio between the dynamic and hysteresis loss components at the measuring frequency. The bias correction terms ~(P)VBa, (~(P)VBg, 8(P)vH1 , 8(P)a, 8(P)s, 8(P), are assumed to have zero value and non-zero uncertainty and are associated with air-flux compensation, gain and offset of the secondary voltage channel, gain of the field channel, material density, cross-sectional area, phase shift between VB(t) and VHl(t), respectively. With the employed measuring setup, the contributions to the uncertainty deriving from frequency setting, synchronization error during signal acquisition, standard shunt resistor in the primary circuit, and quantization of the signal by the A / D converters are deemed negligible. The latter might become important at very high inductions, where the peak amplitude of the fundamental harmonic of the field (i.e. VH1) reduces to a small fraction of the peak field amplitude, with ensuing reduction of the effective dynamic range of the field channel.

608

CHAPTER 10 Uncertainty and Confidence in Measurements

Based on the foregoing discussion a n d Eq. (10.16), w e express the relative c o m b i n e d s t a n d a r d u n c e r t a i n t y of the p o w e r loss m e a s u r e m e n t as

u~(P) P = ~ u2(p)~2 + u2(VB)a ~ +

u2(VB)g V-----~ +

U~_II(VH1) V21

U~(~)T +2

32

U~(q0) +qo2tan2cp qo2 (10.48)

TABLE 10.6 Uncertainty budget in the measurement of the magnetic power losses at 50 Hz and 1.7 T peak polarization in a grain-oriented Fe-(3 wt%)Si lamination (Eq. (10.48)) Source of uncertainty

Distribution function

Divisor Relative uncertainty

Sensitivity coefficient

Degrees of freedom

Air-flux compensation Gain and offset of B channel Gain of H channel Sample cross-sectional area Material density Phase shift between VB and VH1 Repeatability Combined relative standard uncertainty Expanded uncertainty (95% confidence level)

Rectangular

x/3

2 x 10-3

1

oo

Rectangular

~/3

2 x 10-3

1

OO

Rectangular

x/3

1 x 10 -3

1

oo

Rectangular

x/3

1.5 x 10 -3

1

oo

Rectangular Rectangular

x/3 ~/3

1.5 x 10 -3 4 x 10 -3

1 co qotan~ - 0.7 oo

Normal Normal

1 -

I x 10 -3 5.5 x 10 -3

1 -

-

-

11 x 10 -3

6 1536

Coverage factor k=2

The measurement is performed by a digital wattmeter, using an Epstein test frame. For this specific case, the phase shift ~ between secondary voltage and fundamental component of the primary current is 43~. The result is expressed as P = P + U = 1.176 + 13 x 10 -3 W/kg.

REFERENCES

609

2.5 2.0

Ni/

1.5

:2)

!o/o

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.

.

.

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2.0

FIGURE 10.5 Expanded relative uncertainty and its dependence on peak polarization in the measurement of the magnetic power losses at 50 Hz in grainoriented and non-oriented Fe-Si laminations (PTB laboratory [10.18, 10.22]). where it is assumed, according to Eq. (10.42), that the uncertainty of S is chiefly due to the uncertainty of 8. Notice also that we have treated as independent all the input quantities appearing in Eq. (10.48). This is a crude approximation. In particular, for a given Jp value, VB and ~ are expected to correlate and the associated covariance should be considered, via Eqs. (10.17) and (10.20), in Eq. (10.48). Table 10.6 provides, u n d e r s u c h an approximation, the details of the calculation of uc(P)/P and the related expanded uncertainty U. For the specific measurement reported in this example, we find P = P + U - 1 . 1 7 6 + 11x10-BW/kg. Notice that uc(P)/P is observed to increase rapidly on increasing Jp towards saturation, as illustrated for non-oriented and grain-oriented laminations in Fig. 10.5 [10.18, 10.22]. This is chiefly ascribed to the previously stressed detrimental effect of reduced dynamic range in the field channel and the phase error. In fact, for high J~ values, 17B and VH1 are nearly in quadrature and the term ~2 tan2q0(u2(qo)/~2) in Eq. (10.48) tends to diverge.

References 10.1. BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, Guide to the Expression of Uncertainty in Measurement (Geneva, Switzerland: International Organization for Standardization, 1993).

610

CHAPTER 10 Uncertainty and Confidence in Measurements

10.2. In repeated measurements one may find from time to time seemingly inconsistent outcomes (outliers). A good appraisal of the kind of measurement being performed and expertise should guide the person making the experiment in accepting or rejecting these outcomes. 10.3. A. Papoulis, Probability, Random Variables and Stochastic Processes (Tokyo: McGraw-Hill Kogakusha, 1965), p. 266. 10.4. C.F. Dietrich, Uncertainty, Calibration and Probability (Bristol: Adam Hilger, 1991), p. 10. 10.5. S. Rabinovich, Measurement Errors (New York: AIP, 1995), p. 195. 10.6. BIPM, Mutual Recognition of National Measurement Standards and of

10.7. 10.8. 10.9.

10.10.

10.11. 10.12. 10.13. 10.14. 10.15.

10.16.

10.17.

Calibration and Measurement Certificates Issued by the National Metrology Institutes (S6vres: Bureau International des Poids et Mesures, 1999), http:// www.bipm.fr / BIPM-KCDB. M.G. Cox, "A discussion of approaches for determining a reference value in the analysis of key-comparison data," NPL Report CISE 42 (1999). J.W. M(iller, "Possible advantages of a robust evaluation of comparisons," BIPM Report 95/2 (1995). S. D'Emilio and F. Galliana, "Application of the GUM to measurement situations in metrology," in Proc. 16th IMEKO World Congress (A. AfjehiSadat, M.N. Durakbasa, and P.H. Osanna, eds., Wien: ASMA, 2000), 253-258. J. Sievert, M. Binder, and L. Rahf, "On the reproducibility of single sheet testers: comparison of different measuring procedures and SST designs," Anal. Fis. B, 86 (1990), 76-78. EUROMET, "Guidelines for the organisation of comparisons," (EUROMET Guidance Document No. 3). R. Thalmann, "EUROMET key comparison: cylindrical diameter standards," Metrologia, 37 (2000), 253-260. ISO, International Vocabulary of Basic and General Terms in Metrology (Geneva, Switzerland: International Organization for Standardization, 2000). A.E. Drake, "Traceable magnetic measurements," J. Magn. Magn. Mater., 133 (1994), 371-376. M.J. Hall, A.E. Drake, and L.C.A. Henderson, "Traceable measurement of soft magnetic materials at high frequencies," J. Magn. Magn. Mater., 215216 (2000), 717-719. R.D. ShuU, R.D. McMichael, L.J. Swartzendruber, and S. Leigh, "Absolute magnetic moment measurements of nickel spheres," J. Appl. Phys., 87 (2000), 5992-5994. "Intercomparison of magneticflux density by means ofj~'eld coil transfer standard," (EUROMET Project No. 446, Final Report, 2001).

REFERENCES

611

10.18. J. Sievert, H. Ahlers, F. Fiorillo, L. Rocchino, M. Hall, and L. Henderson, "Magnetic measurements on electrical steels using Epstein and SST methods. Summary report of the EUROMET comparison project no. 489," PTB-Bericht, E-74 (2001), 1-28. 10.19. IEC Standard Publication 60404-2, Methods of Measurement of the Magnetic

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