Weighing Accuracy. Estimating Measurement Bias & Uncertainty Of A Weighing

Transcription

1 Weighing Accuracy Estimating Measurement Bias & Uncertainty Of A Weighing METTLER TOLEDO Arthur Reichmuth V1 Sept 001

2 Abstract Contents In many applications, the weighing result needs to be qualified, ie, the measurement bias and uncertainty accompanying the weighing process are required These values are usually not readily available, not least because they are dependent on the application at hand At other times, the operator needs to know the minimal amount of mass (aka minimal weight ) that can be weighed conforming to a required ative uncertainty and confidence level This paper discusses influences on the balance and the weighing object, and explains how the weighing uncertainty and minimal sample mass can be estimated The assumptions and restrictions, under which this deduction is valid, and under which conditions influences may be neglected, are explained Two examples with actual data from analyti balances are given as illustrations The theory and examples provided here enables the user to asses the uncertainty and estimate figures of uncertainty of, or minimal sample mass for, a weighing Introduction 1 Influences Originating From The Balance 1 Influences Affecting The Weighing Object Influences by the Environment 3 Modelling The Weighing 4 Characterization Of The Individual Influences 6 Combined Weighing Bias 17 Combined Weighing Standard Uncertainty 18 Expanded Weighing Uncertainty 19 Example 1 1 Uncertainty Charts 5 Minimal Sample Mass 31 Example 33 Chart of Minimal Sample Mass 33 Conclusions 35 Appendices 37 References 4

3 1 Introduction The primary aim of a weighing is to determine the mass of the weighing object Measurement processes are generally subject to influences of all kinds of origins The weighing is no exception, as it may get distorted by many influences which introduce bias and uncertainty in its result Such influences may affect the weighing object, the balance, or both The distortions may be caused by properties of the weighing object, the balance, the environment and the procedure, by which the weighing is performed Influences originating from balance are manifold They include effects introduced by the balance s quantized digital display, the balance s limited capability to repeat, its nonlinear characteristic, its sensitivity to eccentric loading, its deviation of sensitivity from the correct value, as well as its temperature dependence Among the influences that afflict the weighing object, the most prominent is buoyancy, which is caused by the fact that usually weighings are carried out in air (instead of an empty space, the vacuum) Other influences are magnetic and electrostatic forces Ambient conditions like air temperature, humidity, pressure and air velocity will influence the weighing object and the balance Influences caused by vibration, inclination, or other conditions may be present Influences Originating From The Balance Sources of weighing deviations and uncertainties with laboratory balances are: Readability Rounding of the weighing value to the last digit inherently introduces quantization noise Repeatability is introduced by noise of the electronic circuits (especially by the A/D converter s reference)

4 thermal agitation caused by power dissipated by electronic circuits wind draft at the site of the balance (especially with resolutions of 1mg and below) vibrations of the support on which the balance stands temperature differences pressure fluctuations of the ambient air Non-Linearity is caused by load dependent deformations of the weighing cell the electrodynamic transducer s inherent non-linearity between current and force a non-linear characteristic of the A/D conversion Eccentric Load is caused by finite tolerances in manufacturing, assembly and adjustment deformations of the weighing cell due to the position of the weighing object ative to the weighing pan load dependent deformations of the weighing cell Sensitivity Accuracy & Drift is caused by adjustment tolerance, or determination accuracy, of the ibration weight buoyancy of the ibration weight spontaneous and temperature induced drift, mainly of the lever's mechani advantage, the electrodynamic transducer s magnetic flux, and the A/D converter's reference (if the balance is not re-ibrated and adjusted) mechani shocks Zero Point Drift is caused by spontaneous or temperature induced drift of the weighing cell, mainly of the weighing cell s mechani and electronic components mechani shocks air buoyancy Influences Affecting The Weighing Object The most prominent influence that acts on the weighing object is air buoyancy If a weighing is carried out in a

5 3 natural environment, ie, in the atmosphere that surrounds the earth, a buoyancy force acts on the weighing object This force is equal to the weight of the air displaced by the weighing object, and is opposed to its weight force; that way always diminishing it The factor by which the weight force is reduced is dependent on the densities of the weighing object and the air Sometimes, the weighing object s mass is not well defined in the first place 1 ) Mass gain or loss may take place, for example caused by hygroscopic exchange of water, or by evaporation Thus, we have to be aware that the measurand of a weighing, the mass of an object, may not be the unalterable physi quantity as it is often believed to be Varying or contradicting weighing values need not necessarily be caused by the measuring instrument, the balance, but are often resulting from the fact that the mass (or its weight force) is not well defined and varying We do assume here that such spurious influences are recognized before the weighing is carried out, or that the measurement result is corrected for if the amount of influence is known The handling of the weighing object, ie, its preparation, the way how it is loaded onto the balance, may have an influence on the weighing result There may also be influences brought about by the operator ) Last not least, it has to be defined whether the true mass, the conventional (ie, apparent) mass, or just the weighing value shall be reported [R33] All three values are generally different for one and the same weighing object Influences by the Environment Environmental conditions are criti for every weighing The air surrounding the weighing object and the balance has a major influence on the weighing result Air temperature, humidity, pressure and draft (caused by ventilation of fans) may deteriorate the weighing process Heat radiation is also criti Direct sunlight, the rays of incandescent lights, or other radiation sources should be prevented from reaching the balance or the weighing objects With high-resolution 1) see [Niels] ) see [Clark]

6 4 balances, even the radiation of the operator s body may, after 1/ hour to one hour, disturb the weighing Modelling The Weighing The measurand of a weighing process is the mass of the weighing object, the result is the indication of the balance Ideally, the reading R should be identi to the mass m of the weighing object For this case, we can write R = m This ationship largely holds, indeed At a closer look, however, the influences discussed do affect, although to a small extent, the weighing Some of the influences such as readability, repeatability, non-linearity, zero drift are additive influences, ie, their contribution simply adds to the reading We can model them mathematily as follows R = m+y RD +y RP +y NL +y Z, Where y RD, y RP, y NL and y Z are the contributions of readability, repeatability, non-linearity and zero drift Others such as eccentric load, sensitivity offset, sensitivity drift with time, sensitivity temperature drift and air buoyancy are multiplicative in their effect as they are proportional to the sample mass and therefore can be considered to influence the sensitivity of the transfer characteristic Sensitivity is the proportionality factor between the mass of the weighing object and the displayed value A sensitivity drift of the balance can be caused by long-term drift, or by temperature change Internally, the balance is equipped with a ibration factor c which ates the weighing force F W acting upon the transducer to the displayed value, according to R = cf W On the surface of the earth, the weight force of an object originates from the attraction force caused by gravity g, according to F W = gm, where g 981 N/kg It is well known that the earth is covered by a layer of air, at the bottom of which an object is usually weighed Its weight

7 5 force is therefore reduced by buoyancy, and hence the object appears to have less mass 3 ) Modelling this effect, we have R = cg 1 a ρ m, where a and ρ are the densities of air, and the object weighed, respectively, where a 1 1 kg/m 3 Combining these influences, we are able to model the weighing process with the following mathemati expression R = cg 1 a ρ m+y RD +y RP +y NL +y Z We did not consider eccentric load in this model, for two reasons: first, the mathemati expression to estimate the influence of eccentric load is a rather complicated one, and second, the effect is easily suppressed by placing the weighing object (ie, its center of gravity) in the center of the weighing pan Having thus determined the transfer function of a balance from load to reading, we now direct our interest to the process of weighing A regular weighing consists of two loadings (load instances) and readings, namely 1) one with the empty pan (or with a tare), which constitutes the weighing of the tare mass, and ) one with the object (or with the tare and the object), which constitutes the weighing of the gross mass The mass of the object is obtained by subtracting the tare reading from the gross reading, resulting in the (net) mass of the object 4 ), namely R = R R 1 3) see Buoyancy in the appendix 4) Even though one might argue that it it possible to weigh by simply loading the object and reading once, this does but suppress the fact that the state of no load (or tare mass) represents a load instant of the balance, which serves as a reference value for the subsequent reading(s) with the object placed on the pan The subtraction of the two readings can also be achieved, at the operators discretion, by re-zeroing the balance as soon as the tare mass is on the pan

8 6 We obtain the tare reading R 1 from the balance while either no load, or the tare load, is on the on the weighing pan, constituting the tare load m t 5 ) R 1 = cg 1 1 a 1 ρ m t t +y +y RD 1 RP1 +y NL1 +y Z1, with the parameters of the influences acting on the weighing at the time of this tare loading and reading Analogously, placing the sample m in addition on the weighing pan, we get for the gross reading R = c g 1 a ρ m t t +g 1 a ρ m +y +y RD RP +y NL +y Z, again with the parameter values at the time of this gross loading and reading The sample mass is obtained by subtracting the tare reading from the gross reading Before we execute the subtraction of these readings, we will take a closer look at these expressions We do assume that the loadings are executed on the same balance in sequence, within a short time interval between the two readings From this assumption we conclude that the same air density and gravity apply to both loadings, namely a 1 = a = a and g 1 = g = g Unfortunately, this equality does not apply for the readability, repeatability and non-linearity contributions of the respective readings; the contributions of non-linearity will be dealt with separately later We thus obtain for the net value of the sample mass R = cg 1 a ρ m+y RD y RD1 +y RP y RP1 +y NL y NL1 +y Z y Z1 This expression allows to determine the effects on the weighing by the influences considered The next step is to characterize the behavior of the individual contributing influences Characterization Of The Individual Influences What we are interested in is the question whether, and how the individual influences affect accuracy and precision of the weighing result We thus have to ascertain whether they 5) Even with no load on the weighing pan, there is always weight acting on the balance, constitued by the mass of the weighing pan, and other dead loads, such as mechani parts of the weighing cell This defines the no load condition

9 7 provide a bias, ie, have an average value different from zero, and what their contribution to uncertainty is Readability concerns the term y y RD RD1 To indicate a value on a display with a finite number of digits it must be rounded If the step with of the display is d, and if the value is rounded halfway between two steps 6 ), this operation adds quantization noise y RD to the value The average value of this noise is zero δ RD = y RD =0, while its variance can be approximated by s RD = 1 1 d 7) The average contribution of the readability term y RD y RD1, comprising two readings, is therefore zero As the two roundings may be considered uncorated, their uncertainty variance evaluates to u RD =s RD = 1 6 d The quantization noise is a random contribution Repeatability concerns the term y y RP RP1 Repeatability is the capability of a balance to produce the same reading, provided the same load is placed several times in sequence on the platform The bias of this measurement series is the average of all net readings R = R R 1, namely n Σ n Σ R = n 1 R i =1 i = n 1 R R i =1 1 i and represents the best estimate for the sample mass that has been repeatedly weighed As a consequence, the contribution of the repeatability term y y RP is considered free of bias RP1 δ RP = y RP =0 Or stated in different words: its average deviation is the measurement value The variance of this series, however, is 6) aka 4/5-rounding 7) This is valid for a 4/5-rounding, with a busy measurement value, ie, a weighing value that bounces at least some steps around its average value

10 8 generally nonzero, and is obtained from the weighing series as s RP = n 1 1 n Σ R i R i =1 Hence, repeatability does not introduce a bias, but it contributes to uncertainty, namely u RP = s RP Repeatability is a random contribution Sensitivity 8 ) concerns the proportionality term between sample mass and displayed value Sensitivity is the ratio of change in reading versus change of load The slope of a straight line through the readings at the no-load and full load points of a balance may therefore be interpreted as its (global 9 ) sensitivity S = m R To evaluate sensitivity deviations, the following decomposition will prove useful S = S 0 1+ S The sensitivity of a balance is accurate, if its reading coincides with the value of the mass loaded Therefore, the following ationship must hold R = m Provided the sensitivity deviation S is zero, it follows that a balance s accurate value of sensitivity S 0 is S 0 =1 We thus have S =1+ S The transfer function of a balance whose sensitivity has been adjusted is R = g 1 a ρ g 1 a 1 m 1+ µ 10 ) ρ The parameters with the index denote values at the time of ibration 8) Definition of term: see [VIM], item 510 9) Sensitivities measured over small(er) intervals are an issue of linearity 10) see chapter Sensitivity Adjustment in the appendix

11 9 Relating this expression with the definition of sensitivity, we obtain for the sensitivity S = g 1 a ρ g 1 a 1 =1+ S, 1+ µ ρ from which the sensitivity deviation can be expressed as S = g 1 a ρ g 1 a µ ρ g 1 a g ρ 1+ a ρ 1 µ 1 11) From this expression we conclude that gravity and buoyancy both at the time of adjustment, as well as at the time of weighing and ibration weight accuracy, influence sensitivity accuracy Gravity The gravity g at the location of weighing does influence the sensitivity of the balance We do assume here, however, that the sensitivity of the balance has been adjusted at the site of use 1 ) Thus, we have g = g, thereby eliminating the influence of lo gravity Calibration Weight The sensitivity proper, namely the sensitivity of the balance, depends on the ibration weight deviation, as S µ 1 µ 1= µ 13 ) Its average deviation vanishes δ µ = µ =0, 11) This approximation is based on the assumptions that g g, a << ρ and µ << 1 1) see chapter Sensitivity Adjustment in the appendix 13) We do assume for this and the following derivations that g g, a << ρ, a << ρ, and µ << 1 apply, meaning that the nominal values of the terms not investigated in the formula are close to 1

12 10 while its variance can be estimated from the ibration weight tolerance µ lim max µ as s µ = 1 3 µ lim with µ lim = m lim m 0 This influence is systematic in nature, although it will vary from balance to balance Air Buoyancy Buoyancy does not influence the sensitivity proper of a balance 14 ), but it does influence the overall sensitivity of a weighing Both the ibration standard, as well as the sample is affected by buoyancy A first order approximation of the influence of buoyancy on the overall sensitivity is S buoy 1 a ρ 1+ a ρ 1 a ρ a ρ Sensitivity deviation thus depends on the densities of the ibration standard ρ and the sample ρ, as well as the air densities prevailing at the time of sensitivity adjustment a and at the time of weighing a The density of a ibration weight (internal or external) is kept around the conventional density of 8000kg/m 3 15 ) If a sample being weighed happens to be of the same density, and the air density at the time of weighing is the same as it was at the time of sensitivity adjustment, the buoyancy influence cancels as can be concluded from the expression of sensitivity deviation 16 ) However, this assumption does hold only in few special applications, the dissemination of mass standards being the most important of them For other applications, where the weighing object has a density different from the conventional one, buoyancy does influence the weighing result 14) With the exception, when weighing the ibration standard for the purpose of adjusting the balance s sensitivity 15) see [R33] 16) This is not to say that buoyancy has ceased to occur It means, that the effect on the weighing result is greatly reduced and can be neglectec but in a few cases

13 11 If the conventional value of mass of the sample is to be expressed, then the influence of buoyancy is to be forgone 17 ) However, if the true, physi mass of the sample is to be determined, buoyancy must be corrected for According to this expression, the average value of sensitivity deviation due to buoyancy is δ buoy = S buoy = a ρ a ρ The variance resulting from buoyancy at the time of weighing the reference standard can be derived by taking the partial derivatives of the sensitivity expression This yields s buoy = a ρ s ρ + 1 a ρ s a + ρ s ρ + 1 ρ s 18 a ) While the density variance of the ibration standard may be obtained from the balance manufacturer, the user must provide the density information of the object weighed A completely different issue is the determination of the density variance of air Air density depends largely on air pressure and temperature a = nk m p T 19 ) If no specific air density data is available, one could estimate it from pressure and temperature As a first order approximation, the air density depends on air pressure a p = a p and on air temperature a T = a T From this, the variance of air density can be determined to be 17) The conventional value of mass is understood to be the value obtained from a weighing in air of density 1kg/m3 on a balance, whose sensitivity was ibrated with a mass standard of 8000kg/m3 Hence, the exact conventional value is only obtained if the air density at the time of weighing the sample is 1kg/m3 Details are given in [R33] 18) see cahpter Variance Of Relative Buoyancy in the appendix 19) To a lesser extent, air density is also dependent on air humidity, its compressibility and its composition See Dependence Of Air Density On Temparature & Pressure in the appendix

14 1 s a = a p s a p + T s s T = a p + s T T The standard uncertainty of air density is then s a = a s pp + s T T 0 ) Buoyancy introduces both a bias and an uncertainty, mainly stemming from the limited knowledge of the density of the sample and the air density at the time of weighing Its influence is systematic as far as the average buoyancy goes, and is random to the extent, as the true sample and air densities are unknown 1 ) Sensitivity Temperature Drift Temperature may also influence sensitivity, as the weighing transducer of the balance may exhibit a temperature dependent characteristic This influence is expressed by the (sensitivity) temperature coefficient (TC) of the balance and is defined as the ratio of ative change of sensitivity S/S 0 and the temperature difference T causing it, namely S S0 TC S = T The TC of sensitivity only states how sensitive the slope of the balance s characteristic curve reacts to temperature To provoke a slope change, a change of ambient temperature, with respect to the temperature at the last sensitivity adjustment, must occur The sensitivity change can be expressed as S TC = TC S T S 0 The influence of ambient temperature on sensitivity is of systematic nature However, because the actual TC of a balance are usually unknown to the operator, this influence is treated as a random contribution ) 0) see chapter Variance Of Relative Buoyancy in the appendix 1) Unless special means are provided, such as a controlled atmoshperes in an enclosed volume ) uncertainty of systematic influence, type B uncertainty, according to [GUM], chapter 43

15 13 If at least some information about the TC and the behavior of ambient temperature is available, this can be used to estimate their influence Typily, the operator may get some tolerance band of the TC from the data sheet, TC max TC 3 S lim S ), and from experience or observation, a maximum excursion of the ambient temperature may be known T lim max T As a tolerance specification, the average value of the sensitivity TC of a set of balances (not of a single one) is zero, ie, TC S =0 The average value of sensitivity drift due to temperature of this set of balances is therefore zero as well δ STC S = TC =0 Its variance, however, is not zero As an approximation it can be obtained as the product of the variances of TC and temperature excursion s STC = s TCS s T To obtain the variance of the TC, we assume for lack of other information that the TCs of a set of balances be uniformly distributed within the interval of the specified limits We thus get for the variance s TCS = 1 3 TC 4) S lim The variance of the ambient temperature can be obtained likewise from its maximum excursion, with the same reasoning This yields s T = 1 3 T lim Sensitivity Long-Term Drift Except for the influences modeled mathematily, there is the balance s long-term drift of sensitivity that may occur This means that the sensitivity over time may change spontaneously, if only slightly This property is sometimes specified in the data sheet as a tolerance value S max S LTD lim LTD 3) the subscript lim refers to the specified limit value 4) see [GUM], chapter 437

16 14 As with other influences before, the average value of longterm drift can be assumed to vanish δ SLTD S = LTD =0, while its variance may be estimated by s SLTD = 1 3 S LTD, lim assuming a uniform probability distribution Sensitivity Drift: Summary When the sensitivity of a balance is adjusted with an internal or external ibration standard, the long-term drift and temperature drift of sensitivity are eliminated The influence of long-term drift starts anew, and any temperature change with respect to the temperature at the time of adjustment produces a temperature drift again We assume here that the balance is adjusted occasionally Some balance models provide sensitivity adjustment means, by which the balance automatily adjusts sensitivity at proper time intervals, or when the ambient conditions change We therefore drop the influence of long-term drift here; only if a balance does not get adjusted anymore, it should be considered On the other hand, we will still consider temperature drift, for two reasons: First, temperature changes may occur within adjustment intervals, for example daily fluctuations, and second, even with automatic adjustment features, there has to occur a certain temperature change to trigger an automatic adjustment The total sensitivity bias thus amounts to δ = δ S = a buoy ρ a ρ, while the total sensitivity variance is equal to u S = s µ + s buoy + s STC Sensitivity is the proportionality factor between load and reading Therefore, sensitivity deviation must be multiplied by the sample mass to obtain absolute bias δ S = m δ S

17 Because the variance is a quadratic measure, it must be multiplied with the square of the sample mass to obtain absolute variance s S = m s S Non-Linearity concerns the term y y NL NL1 15 The ationship between the input (the load) and the output (the reading) of a balance is led its characteristic curve In an ideal case, this curve is a straight line, as defined in the chapter about sensitivity Non-linearity is the deviation of the characteristic curve from this straight line going through the no-load and full-load (maximum capacity) point This deviation is systematic in nature for a given balance unit, although generally different from balance to balance However, linearity deviation as a function of load is generally unknown to the user; in the best case a maximum deviation may be known from the data sheet NL lim max NL The actual linearity deviations of the tare and the gross readings for large sample loads may be considered random values that are weakly corated, if at all Without additional information, non-linearity deviations are considered to be evenly distributed on both sides of the ideal straight line Thus, the bias introduced by linearity deviation y NL vanishes δ NL = y NL 0 Without further information about the actual shape of the linearity deviation, we assume a uniform distribution (generally an overly pessimistic assumption 5 ) within the tolerance interval specified in the data sheet From this we can estimate the non-linearity variance to be s = 1 NL 3 NL lim valid for one reading For the net weighing value, obtained by taking the difference of two readings, we therefore have to take twice this variance u NL =s = NL 3 NL lim 5) A triangularly shaped distibution might be appropriate in some cases See [Rth1] und [Rth]

18 16 For small sample loads, the coration between the two linearity deviations increases It can be shown that the difference of the two non-linearity deviations, picked up with the two readings required for a sample weighing, decreases with diminishing sample mass, at least in a statisti sense We will make use of this behavior when discussing the issue of minimal sample mass Zero Drift & Zero Temperature Drift concerns the term y y Z Z1 The majority of weighings is probably carried out according to one of the following procedures: i) the weighing takes place within a short interval of time, without re-zeroing the balance between the two readings If this is the case, virtually no zero drift will occur, signifying that the two contributions of zero are about equal, namely y y NL 1 Z ; ii) the two readings take place at different times, yet the balance is re-zeroed before the net reading, as well as before the gross reading In this case, there might have occurred a zero drift; however, because the balance was rezeroed, the two contributions of zero vanish, namely y y NL 1 Z 0 In either case, the difference y y Z Z1 = 0 of the two contributions will be zero Die Varianz der Differenz der Nullpunktdrift verschwindet ebenfalls, da die Differenzen indentisch Null sind s Z =0 It is a different issue, if for example a container is weighed, and after a long time hours or days say, this container with the sample added is weighed again, and from the two readings the net value is determined and assigned to the sample mass Here, there might have occurred a zero drift, and the two zero contributions may not longer be equal y NL 1 y Z, thereby introducing a difference into the net value that stems from zero drift δ Z = y NL 1 y Z 0

19 17 The same is true when the measurement value needs to be monitored continuously, as in sedimentation, evaporation or other applications, where the sample mass versus time is of interest These considerations apply independently of the source of the zero drift Combined Weighing Bias The combined bias picked up in a sample weighing is obtained by arithmetily adding up all individual biases discussed in the previous chapters We thus have for the combined bias δ = δ RD +δ RP +δ NL +δ Z +δ S, where we have used the indices already introduced Contributions from readability, repeatability and non-linearity and zero point to bias are all zero Whereas bias, resulting from the sensitivity proper of the balance, is also zero (both initial and temperature induced), apparent sensitivity bias is caused by buoyancy, since δ S = δ buoy = m a ρ a ρ We thus obtain for the combined bias of a sample weighing δ = δ S = m a ρ a ρ, stemming exclusively from air buoyancy The reason the balance does not provide bias is owed to the fact that systematic deviations of the balance s transfer characteristic provided, they are of systematic origin and invariable are eliminated either through adjustment after assembly, or measured and stored in the balance for on-line compensation These include: non-linearity correction correction of temperature influence (with on-line measurement of the temperature) correction of the ibration weight s adjustment deviation on-site adjustments for sensitivity and sometimes non-linearity The remaining deviations after adjustment or compensation are too small by definition to be compensated (had they been large enough, they would have been compensated);

20 18 are time dependent in an unknown manner (unknown systematic deviations); are caused by unknown ambient conditions (such as temperature or humidity); or are of entiy unknown origin neither their source or amount, nor their course over time are known and therefore are by definition not identifiable as systematic deviations Hence, these influences, although of systematic origin, must be regarded as uncertainties and are included in the uncertainty estimate Only air buoyancy introduces a significant bias into the weighing result Combined Weighing Standard Uncertainty To obtain the combined variance of a sample weighing, all individual variances earlier discussed are added This provides a reasonable estimate of the combined variance if the individual contributions are mutually independent, or at least statistily uncorated 6 ) This seems to be a reasonable assumption, since the individual causes for the balance s deviation from its ideal performance may be considered independent from each other, indeed We thus have for the combined variance u = u RD +u RP +u NL +u Z +u S The variance caused by rounding (readability) is u RD =s RD = 1 6 d, the variance caused by repeatability u RP = s RP, and the variance caused by non-linearity u NL = s NL = 3 NL lim The variance due to the sensitivity proper of the balance is s µ = 1 3 µ, lim and the variance due to sensitivity drift caused by temperature is s S,TC = s TCS s t m, with 6) see [GUM], chapter 51

21 s TCS = 1 3 TC and s S lim T = 1 3 T, lim while the variance due to apparent sensitivity, caused by buoyancy, is s buoy = a s ρ + 1 ρ s a + ρ a ρ s ρ + 1 ρ s a 19 Eventually the combined variance equals u = u RD +u RP +u NL +m s µ +s S,TC + s buoy Taking the square root of the uncertainty variance yields the standard uncertainty u = u RD +u RP +u NL +m s µ Expanded Weighing Uncertainty +s S,TC + s buoy It is common practice to express the result of a measurement as Y = y±u, where y is the best estimate of the measurand, and U the expanded uncertainty The expanded uncertainty is obtained by multiplying the combined standard uncertainty by a coverage (or expansion) factor k, such that U = k u The expanded uncertainty ±U is an interval that may be considered to encompass a large fraction of the distribution of values that could reasonably be attributed to Y 7 ) The coverage probability or level of confidence P is the measure of how large that fraction is The coverage factor and the coverage probability stay in ation to each other This ation is mediated by the probability distribution of the measured value Unless there is a clear indication for a different distribution, a normal distribution is adequate This may be assumed for the weighing result After all, some of the uncertainty contributions are themselves normally distributed, eg the one of repeatability Moreover, as there are multiple sources of uncertainty contributions, their combined distribution tends to a normal distribution The normal probability distribution is therefore used to define the ationship 7) [GUM], chapter 61

22 0 k = kp between coverage factor k and coverage probability) P 8 ) Sometimes, the missing probability is used, according to Q =1 P This number states the fraction of those values which are expected to lie outside the uncertainty interval Frequently used coverage factors are and 3, which corresponds to a coverage probability of about 95% and 997%, respectively Thus, between a sample of mass m and the result R (single observation) of its weighing, the following ationship holds R U m R +U, where the expanded uncertainty U = k u 9 ), is obtained by multiplying the standard uncertainty u with the coverage factor k, the latter being an appropriate choice ( or 3, for example), from which the coverage probability k > P = f k follows, or vice versa, the coverage probability is appropriately chosen, and from it the coverage factor is determined P > k = f P, according to the table given Eventually, the weighing result can be expressed as 8) Coverage Factor Level of Confidence Missing Probability (Single Sided) (Coverage Probability) k P Q =1 P 1 687% 3173% % 10% % 5% 9545% 455% % 1% % 07% % 0006% % % (A normal probability distribution is assumed for this table) 9) To keep things simple, we have consequently refrained from determining, or correcting for, the degree of freedom Of course, nothing stands against the notion of correcting for the degree of freedom, if it is known of all individual contributions An instruction for how to determine the correction factor can be found in [GUM]

23 m = R δ ±U We now have all pieces in place to discuss the weighing result of an example 1 Example 1: Determination Of Weighing Bias And Uncertainty The mass of approximately 1g of aluminum is weighed in a 190g container on a 00g semi-micro balance 30 ) What are the resulting bias and uncertainty of the weighing result (considering a coverage factor of )? The temperature in the laboratory is controlled to ± C Additional information: Density of aluminum (standard dev) 700kg/m 3 ( 1%) Density of surrounding air 1kg/m 3 Laboratory Temperature (±) C Bias First, we determine the bias of the weighing result We have δ = m a ρ a ρ The density of the sample and the ibration standard are given with ρ = 700 kg/m 3 and 30) AT01 semi-micro balance: Excerpt from data sheet specifications: Property Specification Readability 001mg Repeatability up to 50g 0015mg 50-00g 004mg Non-Linearity within 10g 003mg within 00g 01mg Temperature Coefficient 15ppm/K Sensitivity Long Term Drift 15ppm Additional information about this balance Property Specification Calibration Weight Tolerance *) 15ppm Calibration Weight Density 8006kg/m3 Std Dev of Cal Weight Density 10kg/m3 *) adjusted to conventional value

24 ρ = 8006 kg/m 3, respectively, while we apply a density of air at normal conditions and assume it to be the same at the time of ibration, as well as at the time of weighing a = a 1 kg/m 3 The sample weighs m =1g Hence, we get for the bias of the weighing result 1 kg/m δ =1g 3 1 kg/m kg/m3 700 kg/m 3 = =1g 95ppm 030mg While the result R obtained from the balance represents approximately the conventional mass m con of the sample 31 ), its bias corrected result R δ renders the physi mass m In this example, this means that sample actually holds 30mg more mass than what is reported from the balance This loss is resulting from buoyancy Uncertainty Second, we determine the uncertainty of the weighing result We have for the standard uncertainty u = u RD +u RP +u NL +m s µ The individual contributions are: +s S,TC + s buoy Readability d = 001mg u RD = mg = g Repeatability As the repeatability specification at 190g is unavailable, we use the 00g specification instead: s RP = 004mg 00g u RP = s RP = 004mg = g Non-Linearity The sample mass is smaller than the interval of the more favorable non-linearity specification (10g) We therefore may use it 31) exactly, if the air density is 1kg/m3

25 3 NL lim 10g = 003mg u NL =s = NL 3 NL lim = 3 003mg = g Calibration weight tolerance From the specification, we have µ = 15ppm = 1 3 µ = 1 lim 3 15ppm = 075ppm s µ Temperature drift From the balance specifications we know the TC TC = 15ppm/K S lim s TCS = 1 3 TC = 1 S lim 3 15ppm/K = 075 ppm/k and from the environmental conditions the temperature excursion we have T lim =K s T = 1 3 T = 1 lim 3 K = 133K Hence, the variance of sensitivity temperature drift amounts to s S,TC = s TCS s T = 075 ppm/k 133K = 1ppm Buoyancy From the environmental conditions, we have the laboratory temperature C T = 95K, and we know already the laboratory temperature variance s T = 133K Considering air pressure, no information is available We consider a standard pressure of p = 1bar Assuming that the laboratory is essentially in free pressure exchange with the outside atmosphere, we could try and find some data about the latter From weather data we can estimate a pressure standard deviation of atmospheric pressure fluctuations to be in the order of s p = 10mbar We thus get for the air density standard deviation s a = a s pp + s T T = 1 kg/m 3 10mbar 1bar 1 kg/m g/m K 95K

26 4 (In this example the variance introduced by pressure fluctuation dominates the variance introduced by temperature change) The density of the sample is given above, and its standard deviation is s ρ =1% 700 kg/m 3 =7kg/m 3 The density of the ibration standard is also given above, and its standard deviation is specified as s ρ =10kg/m 3 = From these givens we culate the ative variance of buoyancy = a s ρ + 1 a ρ s a + ρ s ρ + 1 ρ s a = 1 kg/m kg/m 3 s buoy 10 kg/m3 ρ + 19g/m kg/m kg/m kg/m 3 7 kg/m = 46ppm The corresponding standard deviation amounts to s buoy 68ppm + 19g/m3 700 kg/m 3 The sample amounts to m 1g Adding all variances yields the combined variance u = u RD +u RP +u NL +m s +s µ + s S,TC buoy = = g g g +1g 075ppm +1ppm g +1g 46ppm 10 9 g g 10 9 g, from which we obtain the standard uncertainty u 47µg With the required coverage factor k =, corresponding to a coverage probability of P 95%, we get for the expanded uncertainty U 0094mg Conclusion Of Example 1 The major contributors to weighing uncertainty in this example are repeatability and non-linearity, in that order

27 5 Negligible effects, in their order of increasing irevance, are readability, and all the influences which are proportional to the sample mass, namely air buoyancy, temperature drift and ibration weight tolerance Air buoyancy per se is dominated by the uncertainties of sample density and air density The expanded weighing uncertainty of the 1g sample (weighed in a 190g container) thus amounts to 0094mg Besides, the mass obtained from the weighing is too small by 03mg because of buoyancy; the actual mass of the sample is larger by that amount We thus we obtain for the true mass of the sample m = R δ ±U = R+030mg ±0094mg If we were to forego buoyancy correction, the weighing result reflects the apparent mass of the sample m app = R±U = R±0094mg, which approximately corresponds to the conventional mass In this example, measurement bias exceeds measurement uncertainty by about a factor of three 3 ) Uncertainty Charts The example discussed just pinpoints one weighing situation, given by circumstances of the particular weighing object and container, as delineated in example 1 Instead, we are interested in the uncertainties for all possible combinations of sample masses and gross weights applicable to a given balance Uncertainty Chart Of A Semi-Micro Balance To chart the weighing uncertainty obtained with a balance, we use the formula of standard uncertainty, derived in the previous chapter, and multiply it with the coverage factor This yields for the expanded uncertainty +s S,TC U = k u RD +u RP +u NL +m s µ To prevent the cluttering of the chart with too many parameters, in the above expression the contribution of buoyancy was dropped However, the influence of buoyancy is individually shown in the chart for several sample densities U buoy = ku buoy = km u buoy = 3) siehe [Kehl]

28 6 = km a ρ s ρ + 1 ρ s a + a ρ s ρ + 1 ρ s a As instrument we will use here the AT01 semi-micro balance introduced in the previous example, hence we will continue to employ the same specifications We have to be aware that sometimes the data sheet may state several values for a single property, as these may depend on the circumstances of the weighing This is the case for the AT01, since its repeatability and linearity depend on the gross weight (tare plus sample) and the sample weight, respectively: Repeatability up to 50g 0015mg 50-00g 004mg Non-Linearity within 10g 003mg within 00g 01mg This constitutes four operational cases: I) gross below 50g & sample less than 10g II) gross 50 00g & sample less than 10g III) gross below 50 & sample 10 50g IV) gross 50 00g & sample 10 00g In cases I) and II), the more favorable non-linearity specification applies, while in cases I) and III) the more favorable repeatability specification applies As it turns out, the numeri values for case III) are almost identi with those of case IV); case III) is therefore omitted in the chart All other parameters and assumptions in example 1, such as specifications of the balance and environmental conditions and assumptions, are used unaltered for this chart (except sample mass, which is the independent variable, and container mass, which is a parameter, of the chart) Relative Uncertainty Chart Of A Semi-Micro Balance Often it is more convenient, or it may even be required, to use the ative uncertainty, ie, the uncertainty normalized to the sample mass u = m u This yields +s S,TC u = m 1 u RD +u RP +u NL +m s µ + s buoy = 1 u m RD +u RP +u NL + s µ +s S,TC + s buoy =

29 7 Diagram 1: Absolute weighing uncertainty on an AT01 semi-micro balance as a function of sample mass

30 8 and U = ku = k 1 u m RD +u RP +u NL + s µ +s S,TC + s buoy Again, in this chart we dropped the influence of buoyancy U = k 1 u m RD +u RP +u NL + s µ +s S,TC to represent buoyancy separately in the chart U buoy = ku buoy = ks buoy These uncertainty charts of the AT01 illustrate that there are three characteristic behaviors of the uncertainty functions that are ated to three ranges with respect to the sample mass First, for small sample masses, below 10g, uncertainty stays constant This is attributable to the contributions which dominate uncertainty and are independent of sample mass These are mainly repeatability, non-linearity and readability, usually in that order of importance This means that the weighing uncertainty stays essentially constant and is not a function of the sample weight Therefore, as the sample mass decreases, the ative uncertainty increases in this range Second, for large sample masses, larger than half the maximal weighing capacity, the contributions that are proportional to sample mass dominate uncertainty These are temperature drift 33 ), initial accuracy (ibration weight tolerance), long-term drift and buoyancy Therefore, the ative uncertainty stays essentially constant in this range If the sample density is low, buoyancy may become by far the largest contributor to uncertainty in this region Third, between the two ranges is a transition range where the constant contributions as well as the ones proportional to sample mass are of the same order of magnitude Although there may be specific differences between balance models, this behavior is typi for most laboratory balances For small samples ie, sample masses below 1/10 to 1/0 of the weighing capacity the (absolute) uncertainty stays constant, while the ative decreases with increasing sample mass) For large samples ie, sample masses of 1/4 to 1/ of the weighing capacity and larger the 33) Unless the balance provides an means to automatily re-adjust its sensitivity, or there is no temperature change

31 9 Diagram : Relative weighing uncertainty on an AT01 semi-micro balance as a function of sample mass

32 30 (absolute) uncertainty increases proportionally with sample mass, while the ative uncertainty remains constant Having said this, it becomes clear that for small samples it is usually unnecessary to evaluate the contributions that are proportional to the sample mass They may be neglected in favor of the contributions that are constant In the original expression of the standard uncertainty +s S,TC u = u RD +u RP +u NL +m s µ + s buoy we may therefore drop the term that gets multiplied with the sample mass, and we obtain a much simpler expression for the standard uncertainty, namely u = u RD +u RP +u NL (valid for m << m max ), which is valid for small samples From this we get for the ative standard uncertainty u = m u = m 1 u RD +u RP +u NL (valid for m << m max ) On the other hand, with large samples, the constant terms may be dropped in favor of the one that gets multiplied with the sample mass We then have +s S,TC u = m s µ + s buoy = = m s µ +s S,TC + s buoy (valid for m m max ) In this case, the ative standard uncertainty becomes s +s µ S,TC u = m u = + s buoy (valid for m m max ) Moreover, the diagrams illustrate that the uncertainty which is caused by air buoyancy especially with large sample masses can easily exceed all other uncertainty contributions A perfectly adjusted balance with a high resolution is no guarantee for a small weighing uncertainty, especially if the sample has a low mass density Measuring the air density (usually inferred from air temperature, humidity and pressure) and using it for the compensation of air buoyancy, or more sophisticated, a pressure and temperature controlled atmosphere, reduces this uncertainty

33 Minimal Sample Mass 31 Often, a user is interested in the amount of mass that can be weighed on a balance under the condition that the ative uncertainty meets a required level For example, the US Pharmacopeia (USP) requires that weighings have to be performed with a ative uncertainty of smaller than 01%, observing an coverage factor of three 34 ) The smallest possible amount of mass that meets such a requirement is known as minimal sample mass, minimal sample weight, or simply minimal weight The minimal sample mass can be determined from the formula of ative uncertainty Instead of determining the ative uncertainty, we start with the required expanded ative uncertainty U and coverage factor k 35 ) From these givens we determine the standard uncertainty to be met u < U k We insert this expression into the formula of the ative standard uncertainty and we use the approximation formula, as the minimal sample mass clearly is a small mass u = m 1 u RD +u RP +u NL This yields U k > m 1 u RD +u RP +u NL Finally, we solve this equation for the sample mass We get for the minimal sample mass (minimal weight) m min > U k u RD +u RP +u NL 34) Excerpt from the US Pharmacopeia, USP4-NF19, <41> Weights And Balances: «Unless otherwise specified, when substances are to be accurately weighed for Assay the weighing is to be performed with a weighing device whose measurement uncertainty (random plus systematic error) does not exceed 01% of the reading Measurement uncertainty is satisfactory if three times the standard deviation of not less than ten replicate weighings divided by the amount weighed, does not exceed 0001» 35) If instead of the coverage factor the coverage probability is given, the coverage factor can be obtained by aid of the table given

34 3 Considering non-linearity, we have here the situation, that the sample mass m is, by definition, rather small In a previous chapter we have pointed out that for small sample masses the linearity deviations picked up by the tare and the gross reading are highly corated It is fair to expect that, with decreasing sample mass, the difference of these two deviations decreases, too, at least in a statisti sense This means that for small sample weight, the difference tends towards zero y y NL NL1 0, or that it may at least be neglected with respect to other contributions In fact, linearity measurements with laboratory balances support this assumption 36 ) It can be concluded, then, that the uncertainty contribution of non-linearity is smaller than the remaining contributions, especially the one of repeatability s NL << s RP We thus get as the following simplified expression for the minimal mass m min > k U u RD +u RP (valid for m << m max ) If we have the situation, as is mostly the case with highresolving laboratory balances, that the repeatability is at least equal to, but usually larger than, the readability s RP d, then the variance of repeatability is at least 6 times larger than the variance of readability s RP d =6s RD One might thus even neglect the contribution of readability 37 ) This leads to an even simpler expression, 36) see [Rth1] and [Rth] 37) It might be argued why the contribution of readability was considered in the first place If repeatability is evaluated from the readings of a measurement series, then clearly, because the indicated values were displayed by the balance, they were rounded, and hence, repeatability already includes the effect of readability From this point of view, the variance of readability should not be included On the other hand, it sometimes happens with balances that have not that high a resolution, such a precision balance, for example, when evaluation repeatability, all values of a measurement series have the