Weighing Accuracy 2 - Mettler Toledo · Weighing Accuracy Estimating ... A. Reichmuth Weighing...

Weighing Accuracy

Estimating Measurement Bias & UncertaintyOf A Weighing

© METTLER TOLEDOArthur Reichmuth V2.1 Sept. 2001

© Mettler Toledo, A. Reichmuth Weighing Accuracy 2.1 Prtd.: September 25, 2001

AbstractIn many applications, the weighing result needs to bequalified, i.e., the measurement bias and uncertaintyaccompanying the weighing process are required. Thesevalues are usually not readily available, not least becausethey are dependent on the application at hand. At othertimes, the operator needs to know the minimal amount ofmass (a.k.a. ”minimal weight”) that can be weighedconforming to a required relative uncertainty and confidencelevel.This paper discusses influences on the balance and theweighing object, and explains how the weighing uncertaintyand minimal sample mass can be estimated. The assumptionsand restrictions, under which this deduction is valid, andunder which conditions influences may be neglected, areexplained. Two examples with actual data from analyticalbalances are given as illustrations.The theory and examples provided here enables the user toasses the uncertainty and estimate figures of uncertainty of,or minimal sample mass for, a weighing.

Contents

Introduction ........................................................... 1Influences Originating From The Balance ................. 1Influences Affecting The Weighing Object ................ 2Influences by the Environment ................................. 3Modelling The Weighing ........................................ 4Characterization Of The Individual Influences .......... 6Combined Weighing Bias ....................................... 17Combined Weighing Standard Uncertainty.............. 18Expanded Weighing Uncertainty ............................ 19Example 1............................................................. 21Uncertainty Charts ................................................. 25Minimal Sample Mass ............................................ 31Example 2............................................................. 33Chart of Minimal Sample Mass ............................... 33Conclusions ........................................................... 35Appendices ........................................................... 37References............................................................. 42

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IntroductionThe primary aim of a weighing is to determine the mass ofthe weighing object.Measurement processes are generally subject to influences ofall kinds of origins. The weighing is no exception, as it mayget distorted by many influences which introduce bias anduncertainty in its result. Such influences may affect theweighing object, the balance, or both. The distortions may becaused by properties of the weighing object, the balance, theenvironment and the procedure, by which the weighing isperformed.Influences originating from balance are manifold. Theyinclude effects introduced by the balance’s quantized digitaldisplay, the balance’s limited capability to repeat, its non-linear characteristic, its sensitivity to eccentric loading, itsdeviation of sensitivity from the correct value, as well as itstemperature dependence.Among the influences that afflict the weighing object, themost prominent is buoyancy, which is caused by the fact thatusually weighings are carried out in air (instead of an emptyspace, the vacuum). Other influences are magnetic andelectrostatic forces.Ambient conditions like air temperature, humidity, pressureand air velocity will influence the weighing object and thebalance. Influences caused by vibration, inclination, or otherconditions may be present.

Influences Originating From The BalanceSources of weighing deviations and uncertainties withlaboratory balances are:

ReadabilityRounding of the weighing value to the last digit inherentlyintroduces quantization noise.

Repeatabilityis introduced by…•noise of the electronic circuits (especially by the A/Dconverter’s reference)

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•thermal agitation caused by power dissipated by electroniccircuits•wind draft at the site of the balance (especially with resolu-tions of 1mg and below)•vibrations of the support on which the balance stands•temperature differences•pressure fluctuations of the ambient air

Non-Linearityis caused by…•load dependent deformations of the weighing cell•the electrodynamic transducer’s inherent non-linearitybetween current and force•a non-linear characteristic of the A/D conversion

Eccentric Loadis caused by…•finite tolerances in manufacturing, assembly and adjustment•deformations of the weighing cell due to the position of theweighing object relative to the weighing pan•load dependent deformations of the weighing cell.

Sensitivity Accuracy & Driftis caused by…•adjustment tolerance, or determination accuracy, of thecalibration weight•buoyancy of the calibration weight•spontaneous and temperature induced drift, mainly of thelever's mechanical advantage, the electrodynamictransducer’s magnetic flux, and the A/D converter's reference(if the balance is not re-calibrated and adjusted).•mechanical shocks

Zero Point Driftis caused by…•spontaneous or temperature induced drift of the weighingcell, mainly of the weighing cell’s mechanical and electroniccomponents.•mechanical shocks•air buoyancy

Influences Affecting The Weighing ObjectThe most prominent influence that acts on the weighing objectis air buoyancy. If a weighing is carried out in a

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”natural”environment, i.e., in the atmosphere that surroundsthe earth, a buoyancy force acts on the weighing object. Thisforce is equal to the weight of the air displaced by theweighing object, and is opposed to its weight force; that wayalways diminishing it. The factor by which the weight force isreduced is dependent on the densities of the weighing objectand the air.Sometimes, the weighing object’s mass is not well defined inthe first place 1). Mass gain or loss may take place, forexample caused by hygroscopic exchange of water, or byevaporation. Thus, we have to be aware that the measurandof a weighing, the mass of an object, may not be theunalterable physical quantity as it is often believed to be.Varying or contradicting weighing values need notnecessarily be caused by the measuring instrument, thebalance, but are often resulting from the fact that the mass (orits weight force) is not well defined and varying. We doassume here that such spurious influences are recognizedbefore the weighing is carried out, or that the measurementresult is corrected for if the amount of influence is known.The handling of the weighing object, i.e., its preparation, theway how it is loaded onto the balance, may have aninfluence on the weighing result. There may also beinfluences brought about by the operator 2).Last not least, it has to be defined whether the true mass, theconventional (i.e., apparent) mass, or just the weighing valueshall be reported [R33]. All three values are generallydifferent for one and the same weighing object.

Influences by the EnvironmentEnvironmental conditions are critical for every weighing. Theair surrounding the weighing object and the balance has amajor influence on the weighing result. Air temperature,humidity, pressure and draft (caused by ventilation of fans)may deteriorate the weighing process. Heat radiation is alsocritical. Direct sunlight, the rays of incandescent lights, orother radiation sources should be prevented from reachingthe balance or the weighing objects. With high-resolution

1) see [Niels]2) see [Clark]

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balances, even the radiation of the operator’s body may,after 1/2 hour to one hour, disturb the weighing.

Modelling The WeighingThe measurand of a weighing process is the mass of theweighing object, the result is the indication of the balance.Ideally, the reading R should be identical to the mass m ofthe weighing object. For this case, we can write

R = m .This relationship largely holds, indeed. At a closer look,however, the influences discussed do affect, although to asmall extent, the weighing.Some of the influences—such as readability, repeatability,non-linearity, zero drift—are additive influences, i.e., theircontribution simply adds to the reading. We can model themmathematically as follows

R = m+yRD+yRP+yNL+yZ ,Where yRD , yRP , yNL and yZ are the contributions ofreadability, repeatability, non-linearity and zero drift.Others—such as eccentric load, sensitivity offset, sensitivitydrift with time, sensitivity temperature drift and airbuoyancy—are multiplicative in their effect as they areproportional to the sample mass and therefore can beconsidered to influence the sensitivity of the transfercharacteristic. Sensitivity is the proportionality factor betweenthe mass of the weighing object and the displayed value.A sensitivity drift of the balance can be caused by long-termdrift, or by temperature change. Internally, the balance isequipped with a calibration factor c which relates theweighing force FW acting upon the transducer to thedisplayed value, according to

R = c FW .On the surface of the earth, the weight force of an objectoriginates from the attraction force caused by gravity g ,according to

FW = gm , where g ≈ 9.81 N/kg .It is well known that the earth is covered by a layer of air, atthe bottom of which an object is usually weighed. Its weight

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force is therefore reduced by buoyancy, and hence the objectappears to have less mass 3). Modelling this effect, we have

R = cg 1– aρ m ,

where a and ρ are the densities of air, and the objectweighed, respectively, where

a ≈ 1 .. 1.2 kg/m3 .Combining these influences, we are able to model theweighing process with the following mathematical expression

R = cg 1– aρ m+yRD+yRP+yNL+yZ .

We did not consider eccentric load in this model, for tworeasons: first, the mathematical expression to estimate theinfluence of eccentric load is a rather complicated one, andsecond, the effect is easily suppressed by placing theweighing object (i.e., its center of gravity) in the center of theweighing pan.Having thus determined the transfer function of a balancefrom load to reading, we now direct our interest to theprocess of weighing. A regular weighing consists of twoloadings (load instances) and readings, namely1) one with the empty pan (or with a tare), which constitutesthe weighing of the tare mass, and2) one with the object (or with the tare and the object), whichconstitutes the weighing of the gross mass.The mass of the object is obtained by subtracting the tarereading from the gross reading, resulting in the (net) mass ofthe object 4), namely

R = R2–R1 .

3) see ”Buoyancy” in the appendix4) Even though one might argue that it it possible to weigh by simply

loading the object and reading once, this does but suppress the factthat the state of no load (or tare mass) represents a load instant of thebalance, which serves as a reference value for the subsequentreading(s) with the object placed on the pan.The subtraction of the two readings can also be achieved, at theoperators discretion, by re-zeroing the balance as soon as the taremass is on the pan.

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We obtain the tare reading R1 from the balance while eitherno load, or the tare load, is on the on the weighing pan,constituting the tare load mt 5)

R1 = cg1 1–a1ρt

mt+yRD1+yRP1

+yNL1+yZ1

,with the parameters of the influences acting on the weighingat the time of this tare loading and reading. Analogously,placing the sample m in addition on the weighing pan, weget for the gross reading

R2 = c g2 1–a2ρt

mt+g2 1–a2ρ m +yRD2

+yRP2+yNL2

+yZ2 ,

again with the parameter values at the time of this grossloading and reading. The sample mass is obtained bysubtracting the tare reading from the gross reading.Before we execute the subtraction of these readings, we willtake a closer look at these expressions. We do assume thatthe loadings are executed on the same balance in sequence,within a short time interval between the two readings. Fromthis assumption we conclude that the same air density andgravity apply to both loadings, namely

a1 = a2 = a and g1 = g2 = g .Unfortunately, this equality does not apply for the readability,repeatability and non-linearity contributions of the respectivereadings; the contributions of non-linearity will be dealt withseparately later. We thus obtain for the net value of thesample mass

R = cg 1– aρ m+yRD2

–yRD1+yRP2

–yRP1+yNL2

–yNL1+yZ2

–yZ1.

This expression allows to determine the effects on theweighing by the influences considered. The next step is tocharacterize the behavior of the individual contributinginfluences.

Characterization Of The Individual InfluencesWhat we are interested in is the question whether, and howthe individual influences affect accuracy and precision of theweighing result. We thus have to ascertain whether they

5) Even with no load on the weighing pan, there is always weight actingon the balance, constitued by the mass of the weighing pan, and otherdead loads, such as mechanical parts of the weighing cell. Thisdefines the ”no load” condition.

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provide a bias, i.e., have an average value different fromzero, and what their contribution to uncertainty is.

Readabilityconcerns the term yRD2

–yRD1.

To indicate a value on a display with a finite number of digitsit must be rounded. If the step with of the display is d , and ifthe value is rounded halfway between two steps 6), thisoperation adds quantization noise yRD to the value. Theaverage value of this noise is zero

δRD = yRD = 0 ,while its variance can be approximated by

sRD2 = 1

12d2 7).

The average contribution of the readability term yRD2–yRD1,comprising two readings, is therefore zero. As the tworoundings may be considered uncorrelated, their uncertaintyvariance evaluates to

uRD2 = 2 sRD

2 = 16d2 .

The quantization noise is a random contribution.

Repeatabilityconcerns the term yRP2

–yRP1.

Repeatability is the capability of a balance to produce thesame reading, provided the same load is placed severaltimes in sequence on the platform.The bias of this measurement series is the average of all netreadings R = R2–R1 , namely

R = 1

n RiΣi = 1

n= 1

n R2–R1 iΣi = 1

n

and represents the best estimate for the sample mass that hasbeen repeatedly weighed. As a consequence, the contributionof the repeatability term yRP2

–yRP1 is considered free of bias

δRP = yRP = 0 .Or stated in different words: its average ”deviation” is themeasurement value. The variance of this series, however, is

6) a.k.a. ”4/5-rounding”7) This is valid for a ”4/5-rounding”, with a busy measurement value,

i.e., a weighing value that ”bounces” at least some steps around itsaverage value.

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generally nonzero, and is obtained from the weighing seriesas

sRP

2 = 1n–1 Ri–R 2Σ

i = 1

n .

Hence, repeatability does not introduce a bias, but itcontributes to uncertainty, namely

uRP2 = sRP

2 .Repeatability is a random contribution.

Sensitivity 8)concerns the proportionality term between sample mass anddisplayed value.Sensitivity is the ratio of change in reading versus change ofload. The slope of a straight line through the readings at theno-load and full load points of a balance may therefore beinterpreted as its (global 9) sensitivity.

S = Rm .

To evaluate sensitivity deviations, the following decompositionwill prove useful

S = S0 1+∆S .The sensitivity of a balance is accurate, if its readingcoincides with the value of the mass loaded. Therefore, thefollowing relationship must hold

R = m .Provided the sensitivity deviation ∆S is zero, it follows that abalance’s accurate value of sensitivity S0 is

S0 = 1 .We thus have

S = 1+∆S .The transfer function of a balance whose sensitivity has beenadjusted is

R = g

gcal

1– aρ

1–acalρcal

11+∆µcal

m 10 ).

The parameters with the index ”cal”denote values at the timeof calibration.

8) Definition of term: see [VIM], item 5.109) Sensitivities measured over small(er) intervals are an issue of linearity.10) see chapter ”Sensitivity Adjustment” in the appendix

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Relating this expression with the definition of sensitivity, weobtain for the sensitivity

S = g

gcal

1– aρ

1–acalρcal

11+∆µcal

= 1+∆S ,

from which the sensitivity deviation can be expressed as

∆S = ggcal

1– aρ

1–acalρcal

11+∆µcal

–1 ≈

≈ ggcal

1– aρ 1+

acalρcal

1–∆µcal –1 11)

From this expression we conclude that gravity andbuoyancy—both at the time of adjustment, as well as at thetime of weighing—and calibration weight accuracy, influencesensitivity accuracy.GravityThe gravity g at the location of weighing does influence thesensitivity of the balance. We do assume here, however, thatthe sensitivity of the balance has been adjusted at the site ofuse 12). Thus, we have

g = gcal ,thereby eliminating the influence of local gravity.Calibration WeightThe sensitivity proper, namely the sensitivity of the balance,depends on the calibration weight deviation, as

∆S∆µ ≈ 1–∆µcal –1= –∆µcal 13).Its average deviation vanishes

δ∆µ = ∆µcal = 0 ,

11) This approximation is based on the assumptions that g ≈ gcal , acal << ρcal and ∆µcal << 1 .

12) see chapter ”Sensitivity Adjustment” in the appendix13) We do assume for this and the following derivations that g ≈ gcal ,

a << ρ , acal << ρcal , and ∆µcal << 1 apply, meaning that thenominal values of the terms not investigated in the formula are close to1.

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while its variance can be estimated from the calibrationweight tolerance

∆µlim ≤ max ∆µcalas

s∆µ rel

2 = 13∆µlim

2 with

∆µlim=∆mlim

m0.

This influence is systematic in nature, although it will varyfrom balance to balance.Air BuoyancyBuoyancy does not influence the sensitivity proper of abalance 14), but it does influence the overall sensitivity of aweighing. Both the calibration standard, as well as thesample is affected by buoyancy.A first order approximation of the influence of buoyancy onthe overall sensitivity is

∆Sbuoy ≈ 1– aρ 1+

acalρcal

–1 ≈acalρcal

– aρ .

Sensitivity deviation thus depends on the densities of thecalibration standard ρcal and the sample ρ , as well as theair densities prevailing at the time of sensitivity adjustment

acal and at the time of weighing a .The density of a calibration weight (internal or external) iskept around the conventional density of 8000kg/m3 15). If asample being weighed happens to be of the same density,and the air density at the time of weighing is the same as itwas at the time of sensitivity adjustment, the buoyancyinfluence cancels as can be concluded from the expression ofsensitivity deviation 16). However, this assumption does holdonly in few special applications, the dissemination of massstandards being the most important of them. For otherapplications, where the weighing object has a densitydifferent from the conventional one, buoyancy does influencethe weighing result.

14) With the exception, when weighing the calibration standard for thepurpose 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 neglectecbut in a few cases.

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If the conventional value of mass of the sample is to beexpressed, then the influence of buoyancy is to beforgone 17). However, if the true, physical mass of the sampleis to be determined, buoyancy must be corrected for.According to this expression, the average value of sensitivitydeviation due to buoyancy is

δbuoy rel= ∆Sbuoy =

acalρcal

– aρ .

The variance resulting from buoyancy at the time of weighingthe reference standard can be derived by taking the partialderivatives of the sensitivity expression. This yields

sbuoy rel

2 =acalρcal

2 sρcal

2+ 1ρcal

sacal

2+ a

ρ2sρ2+ 1ρsa

2 18 ).

While the density variance of the calibration standard maybe obtained from the balance manufacturer, the user mustprovide the density information of the object weighed.A completely different issue is the determination of the densityvariance of air. Air density depends largely on air pressureand temperature

a = mnk

pT 19 ).

If no specific air density data is available, one could estimateit from pressure and temperature. As a first orderapproximation, the air density depends on air pressure

∂a∂p = a

pand 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 obtainedfrom a weighing in air of density 1.2kg/m3 on a balance, whosesensitivity was calibrated with a mass standard of 8000kg/m3. Hence,the exact conventional value is only obtained if the air density at thetime of weighing the sample is 1.2kg/m3.Details are given in [R33].

18) see cahpter ”Variance Of Relative Buoyancy” in the appendix19) To a lesser extent, air density is also dependent on air humidity, its

compressibility and its composition. See ”Dependence Of Air DensityOn Temparature & Pressure” in the appendix.

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sa

2 = ap sp

2+ – a

T sT2= a2

spp

2

+sTT

220 ).

The standard uncertainty of air density is then

sa = aspp

2

+sTT

2.

Buoyancy introduces both a bias and an uncertainty, mainlystemming from the limited knowledge of the density of thesample and the air density at the time of weighing. Itsinfluence is systematic as far as the average buoyancy goes,and is random to the extent, as the true sample and airdensities are unknown 21).Sensitivity Temperature DriftTemperature may also influence sensitivity, as the weighingtransducer of the balance may exhibit a temperaturedependent characteristic. This influence is expressed by the(sensitivity) temperature coefficient (TC) of the balance and isdefined as the ratio of relative change of sensitivity ∆S/S0and the temperature difference ∆T causing it, namely

TCS =

∆SS0∆T .

The TC of sensitivity only states how sensitive the slope of thebalance’s characteristic curve reacts to temperature. Toprovoke a slope change, a change of ambient temperature,with respect to the temperature at the last sensitivityadjustment, must occur. The sensitivity change can beexpressed as

∆STC = TCS∆T S0 .The influence of ambient temperature on sensitivity is ofsystematic nature. However, because the actual TC of abalance are usually unknown to the operator, this influence istreated as a random contribution 22).

20) see chapter ”Variance Of Relative Buoyancy” in the appendix21) Unless special means are provided, such as a controlled atmoshperes

in an enclosed volume22) uncertainty of systematic influence, type B uncertainty, according to

[GUM], chapter 4.3

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If at least some information about the TC and the behavior ofambient temperature is available, this can be used to estimatetheir influence. Typically, the operator may get sometolerance band of the TC from the data sheet,

TCSlim≤ max TCS

23 ) ,and from experience or observation, a maximum excursionof the ambient temperature may be known

∆T lim ≤ max ∆T .

As a tolerance specification, the average value of thesensitivity TC of a set of balances (not of a single one) is zero,i.e.,

TCS = 0 .The average value of sensitivity drift due to temperature ofthis set of balances is therefore zero as well

δSTC rel= ∆STC = 0 .

Its variance, however, is not zero. As an approximation itcan be obtained as the product of the variances of TC andtemperature excursion

sSTC rel

2 = sTCS2 sT

2 .

To obtain the variance of the TC, we assume—for lack ofother information—that the TCs of a set of balances beuniformly distributed within the interval of the specified limits.We thus get for the variance

sTCS2 = 1

3 TCS lim2 24).

The variance of the ambient temperature can be obtainedlikewise from its maximum excursion, with the samereasoning. This yields

sT2 = 1

3 ∆T lim2 .

Sensitivity Long-Term DriftExcept for the influences modeled mathematically, there is thebalance’s long-term drift of sensitivity that may occur. Thismeans that the sensitivity over time may changespontaneously, if only slightly. This property is sometimesspecified in the data sheet as a tolerance value

∆SLTD lim≤ max ∆SLTD .

23) the subscript ”lim” refers to the specified limit value24) see [GUM], chapter 4.3.7

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As with other influences before, the average value of long-term drift can be assumed to vanish

δSLTD rel= ∆SLTD = 0 ,

while its variance may be estimated by sSLTD rel

2 = 13 ∆SLTD lim

2 ,assuming a uniform probability distribution.Sensitivity Drift: SummaryWhen the sensitivity of a balance is adjusted with an internalor external calibration standard, the long-term drift andtemperature drift of sensitivity are eliminated. The influence oflong-term drift starts anew, and any temperature change withrespect to the temperature at the time of adjustment producesa temperature drift again.We assume here that the balance is adjusted occasionally.Some balance models provide sensitivity adjustment means,by which the balance automatically adjusts sensitivity atproper time intervals, or when the ambient conditionschange.We therefore drop the influence of long-term drift here; onlyif a balance does not get adjusted anymore, it should beconsidered.On the other hand, we will still consider temperature drift, fortwo reasons: First, temperature changes may occur withinadjustment intervals, for example daily fluctuations, andsecond, even with automatic adjustment features, there has tooccur a certain temperature change to trigger an automaticadjustment.The total sensitivity bias thus amounts to

δS rel= δbuoy rel

=acalρcal

– aρ ,

while the total sensitivity variance is equal to uS rel

2 = s∆µ rel

2 + sSTC rel

2 + sbuoy rel

2 .

Sensitivity is the proportionality factor between load andreading. Therefore, sensitivity deviation must be multiplied bythe sample mass to obtain absolute bias

δS = m δS rel.

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Because the variance is a quadratic measure, it must bemultiplied with the square of the sample mass to obtainabsolute variance

sS2 = m2 sS rel

2 .

Non-Linearityconcerns the term yNL2

–yNL1.

The relationship between the input (the load) and the output(the reading) of a balance is called its characteristic curve. Inan ideal case, this curve is a straight line, as defined in thechapter about sensitivity. Non-linearity is the deviation of thecharacteristic curve from this straight line going through theno-load and full-load (maximum capacity) point. Thisdeviation 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 generallyunknown to the user; in the best case a maximum deviationmay be known from the data sheet.

NLlim ≤ max NL .The actual linearity deviations of the tare and the grossreadings for large sample loads may be considered randomvalues that are weakly correlated, if at all. Without additionalinformation, non-linearity deviations are considered to beevenly distributed on both sides of the ideal straight line.Thus, the bias introduced by linearity deviation yNL vanishes

δNL = yNL ≈ 0 .Without further information about the actual shape of thelinearity deviation, we assume a uniform distribution(generally an overly pessimistic assumption 25) within thetolerance interval specified in the data sheet. From this wecan estimate the non-linearity variance to be

sNL2 = 1

3 NLlim2 .

valid for one reading. For the net weighing value, obtainedby taking the difference of two readings, we therefore have totake twice this variance

uNL2 = 2sNL

2 = 23 NLlim

2 .

25) A triangularly shaped distibution might be appropriate in some cases.See [Rth1] und [Rth2].

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For small sample loads, the correlation between the twolinearity deviations increases. It can be shown that thedifference of the two non-linearity deviations, picked up withthe two readings required for a sample weighing, decreaseswith diminishing sample mass, at least in a statistical sense.We will make use of this behavior when discussing the issueof minimal sample mass.

Zero Drift & Zero Temperature Driftconcerns the term yZ2

–yZ1.

The majority of weighings is probably carried out accordingto 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. Ifthis is the case, virtually no zero drift will occur, signifyingthat the two contributions of zero are about equal, namely

yNL1≈ yZ2

;

ii) the two readings take place at different times, yet thebalance is re-zeroed before the net reading, as well asbefore the gross reading. In this case, there might haveoccurred a zero drift; however, because the balance was re-zeroed, the two contributions of zero vanish, namely

yNL1≈ yZ2

≈ 0 .In either case, the difference

yZ2–yZ1

= 0of the two contributions will be zero.Die Varianz der Differenz der Nullpunktdrift verschwindetebenfalls, da die Differenzen indentisch Null sind

sZ2= 0.

It is a different issue, if for example a container is weighed,and after a long time—hours or days—say, this containerwith the sample added is weighed again, and from the tworeadings the net value is determined and assigned to thesample mass. Here, there might have occurred a zero drift,and the two zero contributions may not longer be equal

yNL1≠ yZ2

,thereby introducing a difference into the net value that stemsfrom zero drift

δZ = yNL1–yZ2

≠ 0 .

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The same is true when the measurement value needs to bemonitored continuously, as in sedimentation, evaporation orother applications, where the sample mass versus time is ofinterest.These considerations apply independently of the source of thezero drift.

Combined Weighing BiasThe combined bias picked up in a sample weighing isobtained by arithmetically adding up all individual biasesdiscussed in the previous chapters. We thus have for thecombined bias

δ = δRD+δRP+δNL+δZ+δS ,where we have used the indices already introduced.Contributions from readability, repeatability and non-linearityand zero point to bias are all zero. Whereas bias, resultingfrom the sensitivity proper of the balance, is also zero (bothinitial and temperature induced), apparent sensitivity bias iscaused by buoyancy, since

δS = δbuoy = macalρcal

– aρ .

We thus obtain for the combined bias of a sample weighing δ = δS = m

acalρcal

– aρ ,

stemming exclusively from air buoyancy.The reason the balance does not provide bias is owed to thefact that systematic deviations of the balance’s transfercharacteristic—provided, they are of systematic origin andinvariable—are eliminated either through adjustment afterassembly, or measured and stored in the balance for on-linecompensation. These include:• non-linearity correction• correction of temperature influence (with on-line measure-

ment of the temperature)• correction of the calibration weight’s adjustment deviation• on-site adjustments for sensitivity and sometimes non-line-

arityThe remaining deviations after adjustment or compensation…• are too small by definition to be compensated (had they

been large enough, they would have been compensated);

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• are time dependent in an unknown manner (unknownsystematic deviations);

• are caused by unknown ambient conditions (such as tem-perature or humidity);

• or are of entirely unknown origin—neither their source oramount, nor their course over time are known—andtherefore are by definition not identifiable as systematicdeviations.

Hence, these influences, although of systematic origin, mustbe regarded as uncertainties and are included in theuncertainty estimate.Only air buoyancy introduces a significant bias into theweighing result.

Combined Weighing Standard UncertaintyTo obtain the combined variance of a sample weighing, allindividual variances earlier discussed are added. Thisprovides a reasonable estimate of the combined variance ifthe individual contributions are mutually independent, or atleast statistically uncorrelated 26). This seems to be areasonable assumption, since the individual causes for thebalance’s deviation from its ideal performance may beconsidered independent from each other, indeed. We thushave for the combined variance

u2 = uRD2 +uRP

2 +uNL2 +uZ

2+uS2 .

The variance caused by rounding (readability) is uRD2 = 2 sRD

2 = 16d2 ,

the variance caused by repeatability uRP2 = sRP

2 ,and the variance caused by non-linearity

uNL2 = sNL

2 = 23 NLlim

2 .The variance due to the sensitivity proper of the balance is

s∆µ rel

2 = 13∆µlim

2 ,and the variance due to sensitivity drift caused bytemperature is

sS,TC2 = sTCS

2 s∆t2 m2 ,

with

26) see [GUM], chapter 5.1

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sTCS2 = 1

3 TCS lim2 and sT

2 = 13∆T lim

2 ,while the variance due to apparent sensitivity, caused bybuoyancy, is

sbuoy rel

2 =acalρcal

2 sρcal

2+ 1ρcal

sacal

2+ a

ρ2sρ2+ 1ρsa

2.

Eventually the combined variance equals u2 = uRD

2 +uRP2 +uNL

2 +m2 s∆µ rel

2 +sS,TC2 + sbuoy rel

2 .

Taking the square root of the uncertainty variance yields thestandard uncertainty

u = uRD2 +uRP

2 +uNL2 +m2 s∆µ rel


2 .

Expanded Weighing UncertaintyIt is common practice to express the result of a measurementas

Y = y±U ,where y is the best estimate of the measurand, and U theexpanded uncertainty. The expanded uncertainty is obtainedby multiplying the combined standard uncertainty by acoverage (or expansion) factor k , such that

U = k⋅u .

The expanded uncertainty ” ±U is an interval that may beconsidered to encompass a large fraction of the distributionof values that could reasonably be attributed to Y ” 27).The coverage probability or level of confidence P is themeasure of how large that fraction is. The coverage factorand the coverage probability stay in relation to each other.This relation is mediated by the probability distribution of themeasured value. Unless there is a clear indication for adifferent distribution, a normal distribution is adequate.This may be assumed for the weighing result. After all, someof the uncertainty contributions are themselves normally distr-ibuted, e.g. the one of repeatability. Moreover, as there aremultiple sources of uncertainty contributions, their combineddistribution tends to a normal distribution.The normal probability distribution is therefore used to definethe relationship

27) [GUM], chapter 6.2.1

— 20 —


k = k Pbetween coverage factor k and coverage probability) P 28).Sometimes, the missing probability is used, according to

Q = 1–P .This number states the fraction of those values which areexpected to lie outside the uncertainty interval. Frequentlyused coverage factors are 2 and 3, which corresponds to acoverage probability of about 95% and 99.7%, respectively.Thus, between a sample of mass m and the result R (singleobservation) of its weighing, the following relationship holds

R –U ≤ m ≤ R +U ,where the expanded uncertainty

U = k⋅u 29),is obtained by multiplying the standard uncertainty u withthe coverage factor k , the latter being an appropriate choice(2 or 3, for example), from which the coverage probability

k —> P = f kfollows, or vice versa, the coverage probability isappropriately chosen, and from it the coverage factor isdetermined

P —> k = f P ,according to the table given.Eventually, the weighing result can be expressed as

28) Coverage Factor Level of Confidence Missing Probability(Single Sided) (Coverage Probability)

k P Q = 1–P——————— ———————–– ———————

1 68.27% 31.73%1.645 90% 10%1.960 95% 5%2 95.45% 4.55%2.576 99% 1%3 99.73% 0.27%4 99.994% 0.006%5 99.99994% 0.00006%

(A normal probability distribution is assumed for this table.)

29) To keep things simple, we have consequently refrained from deter-mining, or correcting for, the degree of freedom. Of course, nothingstands against the notion of correcting for the degree of freedom, if itis known of all individual contributions. An instruction for how to de-termine the correction factor can be found in [GUM].

— 21 —


m = R –δ ±U .We now have all pieces in place to discuss the weighingresult of an example.

Example 1: Determination Of Weighing Bias AndUncertainty

The mass of approximately 1g of aluminum is weighed in a190g container on a 200g semi-micro balance 30). What arethe resulting bias and uncertainty of the weighing result(considering a coverage factor of 2)? The temperature in thelaboratory is controlled to ±2°C.Additional information:Density of aluminum (standard dev.) ≈ 2700kg/m3 (≈1%)Density of surrounding air ≈ 1.2kg/m3

Laboratory Temperature (22±2)°CBiasFirst, we determine the bias of the weighing result. We have

δ = macalρcal

– aρ .

The density of the sample and the calibration standard aregiven with

ρ = 2700 kg/m3 and

30) AT201 semi-micro balance:

Excerpt from data sheet specifications:Property SpecificationReadability 0.01mgRepeatability up to 50g 0.015mg

50-200g 0.04mgNon-Linearity within 10g 0.03mg

within 200g 0.12mgTemperature Coefficient 1.5ppm/KSensitivity Long Term Drift 1.5ppm

Additional information about this balanceProperty SpecificationCalibration Weight Tolerance *) 1.5ppmCalibration Weight Density 8006kg/m3Std. Dev. of Cal. Weight Density 10kg/m3_____________________*) adjusted to conventional value

— 22 —


ρcal = 8006 kg/m3 ,respectively, while we apply a density of air at normalconditions and assume it to be the same at the time ofcalibration, as well as at the time of weighing

acal = a ≈ 1.2 kg/m3 .The sample weighs

m = 1g .Hence, we get for the bias of the weighing result

δ = 1g

1.2 kg/m3

8006 kg/m3 –1.2 kg/m3

2700 kg/m3 =

= 1g – 295ppm ≈ – 0.30mgWhile the result R obtained from the balance representsapproximately the conventional mass mcon of the sample 31),its bias corrected result R–δ renders the physical mass m .In this example, this means that sample actually holds 30mgmore mass than what is reported from the balance. This lossis resulting from buoyancy.UncertaintySecond, we determine the uncertainty of the weighing result.We have for the standard uncertainty

u = uRD2 +uRP



2 .

The individual contributions are:Readability

d = 0.01mg uRD2 = 1

6 0.01mg 2 = 17×10–12g2

RepeatabilityAs the repeatability specification at 190g is unavailable, weuse the 200g specification instead:

sRP 200g= 0.04mg

uRP2 = sRP

2 = 0.04mg 2 = 1.6×10–9g2

Non-LinearityThe sample mass is smaller than the interval of the morefavorable non-linearity specification (10g). We therefore mayuse it.

31) exactly, if the air density is 1.2kg/m3

— 23 —


NLlim 10g= 0.03mg

uNL2 = 2sNL

2 = 23 NLlim

2 = 23 0.03mg 2 = 600×10–12g2

Calibration weight toleranceFrom the specification, we have

∆µcal = 1.5ppm s∆µ rel

2 = 13∆µlim

2 = 13 1.5ppm 2= 0.75ppm2

Temperature driftFrom the balance specifications we know the TC

TCS lim= 1.5ppm/K

sTCS2 = 1

3 TCS lim2 = 1

3 1.5ppm/K 2 = 0.75 ppm/K 2

and from the environmental conditions the temperatureexcursion we have

∆T lim = 2K sT2 = 1

3∆T lim2 = 1

3 2K 2 = 1.33K2 .Hence, the variance of sensitivity temperature drift amounts to

sS,TC2 = sTCS

2 sT2 = 0.75 ppm/K 2⋅1.33K2= 1ppm2 .

BuoyancyFrom the environmental conditions, we have the laboratorytemperature

22°C → T = 295K ,and we know already the laboratory temperature variance

sT2 = 1.33K2 .

Considering air pressure, no information is available. Weconsider a standard pressure of

p = 1bar .Assuming that the laboratory is essentially in free pressureexchange with the outside atmosphere, we could try and findsome data about the latter. From weather data we canestimate a pressure standard deviation of atmosphericpressure fluctuations to be in the order of

sp = 10mbar .We thus get for the air density standard deviation

sa = a

spp

2

+sTT

2= 1.2 kg/m3 10mbar

1bar

2+

1.33K2

295K 2 ≈

≈ 1.2 kg/m3 10–4+1.5×10–5≈ 12.9g/m3 .

— 24 —


(In this example the variance introduced by pressurefluctuation dominates the variance introduced by temperaturechange.)The density of the sample is given above, and its standarddeviation is

sρ = 1%⋅2700 kg/m3 = 27 kg/m3 .The density of the calibration standard is also given above,and its standard deviation is specified as

sρcal= 10 kg/m3 .

From these givens we calculate the relative variance ofbuoyancy

sbuoy rel

2 =acalρcal

2 sρcal

2+ 1ρcal

sacal

2+ a

ρ2sρ2+ 1ρsa

2=

=

1.2 kg/m3

8006 kg/m3cal2 10 kg/m3

2

+12.9g/m3

8006 kg/m3

2

+1.2 kg/m3

2700 kg/m3 227 kg/m32

+12.9g/m3

2700 kg/m3

2

≈

≈ 35×10–15+2.6×10–12+20×10–12+23×10–12 ≈ 46×10–12= 46ppm2 .The corresponding standard deviation amounts to

sbuoy rel≈ 6.8ppm .

The sample amounts to m ≈ 1g .

Adding all variances yields the combined variance u2 = uRD

2 +uRP2 +uNL

2 +m2 s∆µ2

rel+sS,TC

2 + sbuoy rel

2 = = 17×10–12g2+1.6×10–9g2+600×10–12g2+ 1g 2 0.75ppm2+1ppm2+46×10–12 ≈ ≈ 2.2×10–9g2+1g2 46ppm2 ≈ 2.2×10–9g2+24×10–12g2 ≈ .

≈ 2.2×10–9g2 ,from which we obtain the standard uncertainty

u ≈ 47µg .With the required coverage factor

k = 2 ,corresponding to a coverage probability of

P ≈ 95% ,we get for the expanded uncertainty

U ≈ 0.094mg .Conclusion Of Example 1The major contributors to weighing uncertainty in thisexample are repeatability and non-linearity, in that order.

— 25 —


Negligible effects, in their order of increasing irrelevance, arereadability, and all the influences which are proportional tothe sample mass, namely air buoyancy, temperature drift andcalibration weight tolerance. Air buoyancy per se isdominated by the uncertainties of sample density and airdensity. The expanded weighing uncertainty of the 1g sample(weighed in a 190g container) thus amounts to 0.094mg.Besides, the mass obtained from the weighing is too small by0.3mg because of buoyancy; the actual mass of the sample islarger by that amount.We thus we obtain for the true mass of the sample

m = R–δ ±U = R+0.30mg ±0.094mg .If we were to forego buoyancy correction, the weighing resultreflects the apparent mass of the sample

mapp = R±U = R±0.094mg ,which approximately corresponds to the conventional mass.In this example, measurement bias exceeds measurementuncertainty by about a factor of three 32).

Uncertainty ChartsThe example discussed just pinpoints one weighing situation,given by circumstances of the particular weighing object andcontainer, as delineated in example 1. Instead, we areinterested in the uncertainties for all possible combinations ofsample masses and gross weights applicable to a givenbalance.

Uncertainty Chart Of A Semi-Micro BalanceTo chart the weighing uncertainty obtained with a balance,we use the formula of standard uncertainty, derived in theprevious chapter, and multiply it with the coverage factor.This yields for the expanded uncertainty

U = k uRD2 +uRP


2 +sS,TC2 .

To prevent the cluttering of the chart with too manyparameters, in the above expression the contribution ofbuoyancy was dropped. However, the influence of buoyancyis individually shown in the chart for several sample densities.

Ubuoy = kubuoy = km ubuoy rel=

32) siehe [Kehl]

— 26 —


= km

acalρcal

2 sρcal

2+ 1ρcal

sacal

2+ a

ρ2sρ2+ 1ρsa

2

As instrument we will use here the AT201 semi-micro balanceintroduced in the previous example, hence we will continue toemploy the same specifications.We have to be aware that sometimes the data sheet may state severalvalues for a single property, as these may depend on thecircumstances of the weighing. This is the case for the AT201, since itsrepeatability and linearity depend on the gross weight (tare plussample) and the sample weight, respectively:

Repeatability up to 50g 0.015mg 50-200g 0.04mg

Non-Linearity within 10g 0.03mg within 200g 0.12mg

This constitutes four operational cases:I) gross below 50g & sample less than 10gII) gross 50…200g & sample less than 10gIII) gross below 50 & sample 10…50gIV) gross 50…200g & sample 10…200gIn cases I) and II), the more favorable non-linearity specificationapplies, while in cases I) and III) the more favorable repeatabilityspecification applies. As it turns out, the numerical values for case III)are almost identical with those of case IV); case III) is therefore omittedin the chart.All other parameters and assumptions in example 1, such asspecifications of the balance and environmental conditionsand assumptions, are used unaltered for this chart (exceptsample mass, which is the independent variable, andcontainer mass, which is a parameter, of the chart).

Relative Uncertainty Chart Of A Semi-Micro BalanceOften it is more convenient, or it may even be required, touse the relative uncertainty, i.e., the uncertainty normalized tothe sample mass

urel = um .

This yields urel = 1

m uRD2 +uRP



2 = = 1

m2 uRD2 +uRP

2 +uNL2 + s∆µ rel


2

— 27 —


Diagram 1: Absolute weighing uncertainty on an AT201 semi-micro balanceas a function of sample mass.

— 28 —


and Urel = kurel = k 1

m2 uRD2 +uRP



2

Again, in this chart we dropped the influence of buoyancy. Urel = k 1

m2 uRD2 +uRP


2 +sS,TC2

to represent buoyancy separately in the chart Ubuoy rel

= k ubuoy rel= k sbuoy rel

.These uncertainty charts of the AT201 illustrate that there arethree characteristic behaviors of the uncertainty functions thatare related to three ranges with respect to the sample mass.First, for small sample masses, below 10g, uncertainty staysconstant. This is attributable to the contributions whichdominate uncertainty and are independent of sample mass.These are mainly repeatability, non-linearity and readability,usually in that order of importance. This means that theweighing uncertainty stays essentially constant and is not afunction of the sample weight. Therefore, as the sample massdecreases, the relative uncertainty increases in this range.Second, for large sample masses, larger than half themaximal weighing capacity, the contributions that areproportional to sample mass dominate uncertainty. These aretemperature drift 33), initial accuracy (calibration weighttolerance), long-term drift and buoyancy. Therefore, therelative uncertainty stays essentially constant in this range. Ifthe sample density is low, buoyancy may become by far thelargest contributor to uncertainty in this region.Third, between the two ranges is a transition range where theconstant contributions as well as the ones proportional tosample mass are of the same order of magnitude.Although there may be specific differences between balancemodels, this behavior is typical for most laboratory balances.For ”small”samples—i.e., sample masses below 1/10 to1/20 of the weighing capacity—the (absolute) uncertaintystays constant, while the relative decreases with increasingsample mass). For ”large”samples—i.e., sample masses of1/4 to 1/2 of the weighing capacity and larger—the

33) Unless the balance provides an means to automatically re-adjust itssensitivity, or there is no temperature change.

— 29 —


Diagram 2: Relative weighing uncertainty on an AT201 semi-micro balanceas a function of sample mass.

— 30 —


(absolute) uncertainty increases proportionally with samplemass, while the relative uncertainty remains constant.Having said this, it becomes clear that for small samples it isusually unnecessary to evaluate the contributions that areproportional to the sample mass. They may be neglected infavor of the contributions that are constant. In the originalexpression of the standard uncertainty

u = uRD2 +uRP



2

we may therefore drop the term that gets multiplied with thesample mass, and we obtain a much simpler expression forthe standard uncertainty, namely

u = uRD2 +uRP

2 +uNL2 (valid for m << mmax ),

which is valid for small samples. From this we get for therelative standard uncertainty

urel = um = 1

m uRD2 +uRP

2 +uNL2 (valid for m << mmax ).

On the other hand, with large samples, the constant termsmay be dropped in favor of the one that gets multiplied withthe sample mass. We then have

u = m2 s∆µ rel


2 =

= m s∆µ rel


2 (valid for m ≈ mmax ).

In this case, the relative standard uncertainty becomes urel = u

m = s∆µ rel


2 (valid for m ≈ mmax ).

Moreover, the diagrams illustrate that the uncertainty which iscaused by air buoyancy—especially with large samplemasses—can easily exceed all other uncertaintycontributions. A perfectly adjusted balance with a highresolution is no guarantee for a small weighing uncertainty,especially if the sample has a low mass density. Measuringthe air density (usually inferred from air temperature,humidity and pressure) and using it for the compensation ofair buoyancy, or more sophisticated, a pressure andtemperature controlled atmosphere, reduces this uncertainty.

— 31 —


Minimal Sample MassOften, a user is interested in the amount of mass that can beweighed on a balance under the condition that the relativeuncertainty meets a required level. For example, the USPharmacopeia (USP) requires that weighings have to beperformed with a relative uncertainty of smaller than 0.1%,observing an coverage factor of three 34). The smallestpossible amount of mass that meets such a requirement isknown as ”minimal sample mass”, ”minimal sample weight”,or simply ”minimal weight”.The minimal sample mass can be determined from theformula of relative uncertainty. Instead of determining therelative uncertainty, we start with the required expandedrelative uncertainty Urel and coverage factor k 35). Fromthese givens we determine the standard uncertainty to be met

urel <

Urelk .

We insert this expression into the formula of the relativestandard uncertainty—and we use the approximationformula, as the minimal sample mass clearly is a small mass.

urel = 1m uRD

2 +uRP2 +uNL

2 .This yields

Urelk > 1

m uRD2 +uRP

2 +uNL2 .

Finally, we solve this equation for the sample mass. We getfor the minimal sample mass (minimal weight)

mmin > kUrel

uRD2 +uRP

2 +uNL2 .

34) Excerpt from the US Pharmacopeia,USP24-NF19, <41> Weights And Balances:«Unless otherwise specified, when substances are to be ”accuratelyweighed” for Assay the weighing is to be performed with a weighingdevice whose measurement uncertainty (random plus systematic error)does not exceed 0.1% of the reading. Measurement uncertainty issatisfactory if three times the standard deviation of not less than tenreplicate weighings divided by the amount weighed, does not exceed0.001.»

35) If instead of the coverage factor the coverage probability is given, thecoverage factor can be obtained by aid of the table given.

— 32 —


Considering non-linearity, we have here the situation, that thesample mass m is, by definition, rather small. In a previouschapter we have pointed out that for small sample masses thelinearity deviations picked up by the tare and the grossreading are highly correlated. It is fair to expect that, withdecreasing sample mass, the difference of these twodeviations decreases, too, at least in a statistical sense. Thismeans that for small sample weight, the difference tendstowards zero

yNL2–yNL1

→ 0 ,or that it may at least be neglected with respect to othercontributions. In fact, linearity measurements with laboratorybalances support this assumption 36). It can be concluded,then, that the uncertainty contribution of non-linearity issmaller than the remaining contributions, especially the oneof repeatability

sNL2 << sRP

2 .We thus get as the following simplified expression for theminimal mass

mmin > kUrel

uRD2 +uRP

2 (valid for m << mmax ).

If we have the situation, as is mostly the case with high-resolving laboratory balances, that the repeatability is at leastequal to, but usually larger than, the readability

sRP ≥ d ,then the variance of repeatability is at least 6 times largerthan the variance of readability

sRP2 ≥ d2 = 6sRD

2 .One might thus even neglect the contribution ofreadability 37). This leads to an even simpler expression,

36) see [Rth1] and [Rth2]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 measurementseries, then clearly, because the indicated values were displayed bythe balance, they were rounded, and hence, repeatability alreadyincludes the effect of readability. From this point of view, the varianceof readability should not be included.On the other hand, it sometimes happens with balances that have notthat high a resolution, such a precision balance, for example, whenevaluation repeatability, all values of a measurement series have the

— 33 —


whereby the minimal sample mass depends only on oneproperty of the balance, namely its repeatability

mmin > kUrel

sRP2 = k

UrelsRP (valid for sRP ≥ d ).

Example 2: Determination Of The Minimal Sample MassA substance is weighed in a container. What is the requiredminimal amount of this substance, if a relative weighinguncertainty of 0.1% with an expansion factor of 2 must beobserved? The weighing is performed on a 200g semi-microbalance 38). The container weighs 190g.To solve this task, we use the simplified expression for theminimal sample mass. The coverage factor k = 2 and therequired relative uncertainty Urel are given. Beyond that weneed the readability of the balance, which is d = 0.01mg .As the container weighs 190g, we have to use therepeatability specification valid for 200g, namely

sRP 200g= 0.04mg .

Repeatability is clearly larger than readability; we thereforeresort to the simpler expression of minimal sample mass.From the data we calculate the minimal amount of substancein this example to be

mmin > k

UrelsRP = 2

0.1%0.04mg ≈ 80mg .

Chart of Minimal Sample MassAs example 1 depicts just one solution in the uncertaintydomain, so does example 2 give just one solution in the

same value. This would lead to a repeatability equal to zero, a valuewhich is certainly not realistic. In fact, it might even have beenpartially provoked by the rounding of the indicated result. In any case,including the readability variance in this case makes sens, as it reflectsat least the contribution of rounding.It might therefore be sensible to include the variance of readability inany case: if repeatability is larger than readability, then the varianceof the former negligible compared to the latter; if repeatabilityhappens to be very small or to vanish, then the variance of readabilityis a valid representation of the situation.

38) The specification of the AT201 semi-micro balance is given in aprevious footnote.

— 34 —


domain of minimal sample mass. Here as there, it is desirableto have a chart from which the actual requirement may bederived. Therefore, we might want to create a chart for theminimal sample mass, as well as get an overview.There is no need for a new chart of sample mass, as the chartof relative uncertainty already contains the requiredinformation, indeed. We only have to interpret is the otherway round. Choosing the required relative uncertainty (giventhe chart reflects the correct coverage factor) on the y-axis,the corresponding minimal sample weight can be read fromthe x-axis. We thus may choose the 0.1% level of uncertaintyin the chart of relative uncertainty shown above, and find thecrossing of this gridline with the bold curve (which representsthe 200g repeatability). We read a value of approximately100mg at this crossing, a value slightly higher than what wehave just calculated. The reason for this is that the uncertaintycharts as given above include the full influence of non-linearity even for small weights, something we have neglectedin example 2. These uncertainty curves are thereforesomewhat conservative for small sample masses; however,they still will deliver fair estimates of uncertainty for smallsample mass.

— 35 —


ConclusionsWeighings, i.e., the determination of the mass of a weighingobject, are subject to influences from various sources.Generally, they cause a measurement bias as well as ameasurement uncertainty.The influences can originate from the balance and from theweighing object. Thereby, the environment plays a majorrole. The weighing result is influenced by non-idealcharacteristics of the balance, such as readability,repeatability, corner load error, non-linearity and long-termdrift, or by disturbances caused by environmental conditionssuch as temperature drift. The weighing sample gets affectedmainly by ambient conditions, such as air temperature, airhumidity, air draft, and substantially so, by air buoyancy.Often, the mass of the object weighed can be the culprit, as itmay vary. One should be aware that the mass of an objectmay not be the unchanging physical quantity, for which it isoften taken.Whether these influences are of relevance for the weighingresult depends on several conditions, which should bechecked for every application. As a rule of thumb one cansay that the higher the required measurement resolution ofthe weighing is, the more likely the influences will manifestthemselves.Contrary to a maintained balance which as a rule does notintroduce a measurement bias that is to the detriment of themeasurement accuracy, air buoyancy can cause a substantialmeasurement bias. There is an international regulation aboutconventional mass as how to deal with this influence.However, it was primarily issued to establish rules for thedissemination of mass standards. For weighing objects withdistinctly different densities than the conventional density(8000kg/m3 ) defined is this regulation it is not applicable.Dependent on the application, the measurement bias causedby air buoyancy can exceed measurement uncertainty by far.The measurement uncertainty is determined by theperformance of the balance, the properties of the weighingobject and the state of the environment. The properties of thebalance are partly of random nature (such as readability,repeatability) and partly of systematic nature (such as non-

— 36 —


linearity, sensitivity offset, temperature drift or long-termdrift). The individual values of the systematic deviations areunknown. However, within a set of balances large enough,their expectation is zero, as these deviations get adjusted fora zero target value. Therefore the systematic deviations canbe treated like random deviations and may be included intothe uncertainty determination. It turns out that for smallsample weights (compared to the weighing capacity of thebalance) the (absolute) uncertainty is virtually constant , whilefor large sample masses uncertainty increases with sampleload. Therefore, the relative uncertainty, i.e., the uncertaintynormalized to sample mass, remains constant for largesample masses, while it increases for decreasing samplemasses. With small sample masses, uncertainty is dominatedmainly by repeatability, and to some extent by non-linearity,while with large sample masses the influences of sensitivityoffset, sensitivity drift with temperature and air buoyancyprevail. Especially with small sample densities, variations ofthe ambient air density, caused by atmospheric conditions,can influence buoyancy heavily, such that this influencedominates uncertainty with large sample masses.Relative uncertainty increases with decreasing sample mass.On the other hand, with large sample masses, uncertainty isdominated by sensitivity accuracy and, depending on thesample density and the conditions of the ambient air, airbuoyancy. Relative uncertainty rolls off and stays essentiallyconstant with increasing sample mass.Sometimes, when a weighing is performed, a given weighinguncertainty should be reached, while at the same time only aslittle substance as required should be weighed. The minimalmass of a sample (also called ”minimal sample mass” orsimply ”minimal weight”) is the amount of mass that justmeets this condition. This amount can be derived from theestimation of uncertainty and is—being a small mass—essentially a function of repeatability, determined by theperformance of the balance, the properties of the weighingobject and the environmental conditions.

— 37 —


AppendicesBuoyancy

If a weighing takes place in a vacuum on the surface ofearth, a body with mass m produces a weight force of

G = mg .If the same body is weighed in the atmosphere, it will besubject to an air buoyancy force which is opposed to theweight force and diminishes the latter. According toArchimedes’ principle 39), a body receives a buoyant forceequal to the weight force of the fluid displaced. This buoyantforce can be determined from the density of the air a and thevolume V displaced, according to

GF = aVg .Since the volume of displacement is equal to the volume ofthe body weighed, we can determine it from the density ρand the mass m of the body

V = mρ

which yields for the buoyant force GF = am

ρ g .

For the sum of all forces acting on the buoyant body(apparent weight) we have

GA = G–GF = mg–amρ g = g 1– aρ m .

Sometimes, within the above expression the term mA = 1– a

ρ mis referred to as ”apparent mass”.

Sensitivity Adjustment (Determination of the calibration factor)The sensitivity of a balance is established with the aid of acalibration mass (a.k.a. calibration standard).Upon loading a mass m onto a balance, it will display avalue

R = cg 1– aρ m .

39) Archimedes, 287?-212 B.C. Greek mathematician, engineer, andphysicist.

— 38 —


Loading the same balance with a known calibration mass mcal will produce the following result

Rcal = cgcal 1–acalρcal

mcal ,where the parameters with index ”cal” denote values at thetime of this calibration weighing. From this measurement thecalibration factor c can now be determined

c =

Rcal

gcal 1–acalρcal

mcal

.

The calibration mass may deviate from its nominal value. Theactual mass of the calibration standard may be expressed as

mcal = mcal 01+∆µcal ,

where mcal 0is the nominal value of the standard, with

∆µcal being its relative deviation from the nominal value,according to

∆µcal=

∆mcalmcal 0 ,

where ∆mcal is the (absolute) mass deviation.Inserting this calibration mass into the expression for thecalibration factor provides

c =

Rcal

gcal 1–acalρcal

mcal 01+∆µcal

.

It seems so obvious that we almost forget to mention that weexpect the displayed value to be equal to the nominal mass ofthe calibration standard loaded on the balance, namely

Rcal = mcal 0 .

We thus obtain for the calibration factor

c =1

gcal 1–acalρcal

1+∆µcal

.

Putting this calibration factor to work will adjust the sensitivityof the balance to its correct value, and we get for the transfercharacteristic the adjusted balance

— 39 —


R = g

gcal

1– aρ

1–acalρcal

11+∆µcal

m .

Dependence Of Air Density On Temperature & PressureThe density of air depends mainly on air temperature andpressure; to a lesser extent also on its relative humidity and itscomposition. For the sake of uncertainty estimation, we givehere a simplified theory for the dependence of air density ontemperature and pressure.If we consider air to be an ideal gas—not a severesimplification for our purposes here—the parameters of anamount m of air, such as its volume V , absolutetemperature T and pressure p , obey the law of an idealgas, according to

pV = nkT ,where n is the number of molecules in that amount, and k isBoltzman’s constant. By definition, the density a of this airequals

a = mV ,

and substituting the volume of air from the gas law, we get a = m

nkpT .

From this expression we take the partial derivative withrespect to pressure

∂a∂p = m

nk1T

and with respect to temperature ∂a

∂T = mnk

–pT2 .

Substituting the constant term m/ nk in this expressions withthe aid of the original formula for air density, we get

∂a∂p = aT

p1T = a

pand

∂a∂T = aT

p–pT2 = – a

T 40 ).

40) As a rule of thumb, at normal conditions the relative density changeamounts to about 1/1000 per mbar and about 1/300 per °C.

— 40 —


Thus, at approximately normal atmospheric conditions with p0 ≈ 1 bar and T0 ≈ 290K = 17°C ,

resulting in an air density of a0 ≈ 1.20 kg/m3 (at 50%rH) 41),

we get for its sensitivity with respect to pressure ∂a

∂p 0=

a0p0

≈ 1.20 kg/m3 /bar ,

and with respect to temperature ∂a

∂T 0= –

a0T0

≈ –4.1×10–3 kg/m3 /K .

For a comprehensive theory on air density see [Giac] and[Davis].

Variance Of Relative BuoyancyThe relative change of the weight force under the influence ofair buoyancy on a sample with density ρ , being weighed ona balance, including the balance’s sensitivity adjustment bymeans of a calibrations standard with density ρcal is an(apparent) sensitivity bias given by

b ≈ 1– aρ 1+

acalρcal–1 ≈

acalρcal– a

ρ ,where a and acal are the air densities at the time ofweighing and at the time of sensitivity adjustment,respectively, and b is the relative change in sensitivity (orapparent mass).If the corresponding standard deviations sρcal

, sacal, sρ and

sa of the densities are known, the variance of the buoyancyinfluence can be determined. From the partial derivatives

∂b∂ρcal

= –acalρcal

2

∂b∂acal

= 1ρcal

∂b∂ρ = a

ρ2

∂b∂a = – 1

ρ

41) Air density calculated from [Davis].

— 41 —


we can now determine the combined variance by adding upall individual variances, multiplied by the squares of theirrespective derivatives, assuming that the contributinginfluences are statistically uncorrelated. This yields

sb

2 =acalρcal

2 sρcal

2+ 1ρcal

sacal

2+ a

ρ2sρ2+ 1ρsa

2.

— 42 —


References[Niels] Henrik S. Nielsen: How long is a piece of string?

NCSL Workshop and Conference, Toronto, Canada, July 17-20, 2000.

[Clark] John P. Clark: Evaluation Of Methods For Estimating TheUncertainty Of Electronic Balance Measurements.Weighing 2000, South Yorkshire Trading Standards Unit,Sheffield (UK), June 13-14, 2000.

[R33] Conventional value of the result of weighing in air.International Organization of Legal Metrology (OIML), Paris,France. International Rrecommendation No 33, 1979.

[VIM] International Vocabulary Of Basic And General Terms InMetrologyInternational Organization for Standardization, Geneva,1993ISBN 92-67-01075-1

[GUM] Guide To The Expression Of Uncertainty In MeasurementFirst edition 1995International Organization for Standardization, GenevaISBN 92-67-10188-9

[Rth1] Arthur Reichmuth: Non-Linearity Of High ResolutionLaboratory Balances.Weighing 2000, South Yorkshire Trading Standards Unit,Sheffield (UK), June 13-14, 2000.

[Rth2] Arthur Reichmuth: Non-Linearity Of Laboratory Balances AndIts Impact On Uncertainty.NCSL Workshop & Symposium, Toronto (CAN), July 17-20,2000.

[Kehl] Karl Kehl, Klaus Weirauch, Samuel Wunderli, VeronikaMeyer: The influence of variations in atmospheric pressure onthe uncertainty budget of weighing results.The Analyst, 2000, Vol. 125, P. 959-962.

[Giac] P. Giacomo: Equation for the Determination of the Density ofMoist Air (1981). Metrologia [1982], 18, pp. 33-40.

[Davis] R.S. Davis: Equation for the Determination of the Density ofMoist Air (1981/91). Metrologia [1992], 29, pp. 67-70.

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