Allan Variance Analysis of Measurement D

IMTC 2006-Instrumentation and Measurement Technology Conference

Warsaw,Poland, 1-3 May 2007

Allan Variance Analysis of Measurement Data Series for Instrument Verification

M.

Bertoccol, A.

Monetti2,

E.

Mottin1,

C.

Narduzzi1,

E.

Sienil

'Departmentof Information Engineering, University of Padova, viaGradenigo 6/b-35131Padova, Italy

Phone: +39 0498277500, Fax: +39 049 8277699, e-mail:matteo.bertocco, claudio.narduzzi, elisabetta.sieni@unipd.it

2InfineonTechnologies Italia Srl, Development Center Padova, via N. Tommaseo,65/b-35131Padova, Italy.

Abstract- The paper is concernedwith the analysis of measurand

and instrumentstability during the calibration verification process,

with a view on ensuring that comparison is carried out under suitable

assumptions. It will be shown howanalysis of the Allan variance

allowsstability characterization ofthe measurement circuit and yields

information about measurement noise, so that a correct procedure for the verification of the calibration status can be defined. This is

of particularsignificance in the context of an industrial laboratory,

where oneof the main purposes of the analysis would be to identify

improvements to the measurement system, that could allow toperform

calibrationverification within the limits of available equipment.

Keywords-Allanvariance, measurement noise, calibration labora-tory

I. INTRODUCTION

Quality control of production processes requires the peri-odic calibration of instrumentation used intestsand in process measurements[1], [2]. Using an instrument in a non-valid cal-ibration status invalidates measurements and wouldprobably require that they be repeated at the laboratory own cost.

Generally, instrument calibration is performed at suitably competentlaboratories, however thisoperationis rather expen-sive. Tomake the bestuseofresources whilereducing the risk ofanon-valid calibration status, alaboratory may perform in-ternal calibration verification in the time interval between two

calibrations, using procedures and instruments in accordance with the indications givenin [3]. In an industrial laboratory

environment themetrological support for this kind ofactivity

isgenerally far less comprehensive than in a dedicated calibra-tionlaboratory. This suggests the usefulness ofinvestigating

some aspects of the verification process more closely, in the hope of contributing some useful criteria to assure its correct-ness.

Verification of the calibration status ofan instrument can

beperformed by comparingmeasurementresults obtained for the same measurand using, respectively, a calibrated

instru-mentactingas aworking standard and theoneunder-test. The

comparison criterion can be seen as a mathematical rule trans-forming measurement results into an indexthat,by comparison with athreshold, allows a decision about the compatibility of measurements. Most common criteria are based on statistical indicators [4], [5], [6], that rely on hypotheses made about the statistics of the data sets, such as theircorrelation, probability distribution anduncertainty.

In common practice, data are supposed to be random, un-correlated and, possibly, having a Gaussian probability dis-tribution. Under these assumptions it can be considered that accuracy in the estimation of the mean value and variance of measurements improves as the number of values, N, in the data set getslarger. However, if N is large, deviations from the assumed conditions may occur, as a result of noise and trends superposed on the measured values. These effects limit the accuracy in the estimation ofa quantity, since variance may actually tend to diverge as the data set size N increases. Con-sequently, a moredetailedanalysis of the statistical properties of themeasurementdata series withregardstotimestability is required.

The paper is mainly concerned with the analysis of mea-surand and instrumentstability during the verification process, with aview onensuring thatcomparison is carriedoutunder suitableassumptions. Itshould beemphasizedthat the

investi-gation intomeasurement fluctuations needsto accountfor the wholemeasurementsystem [11],because contributionscanbe duetothe measurandaswellas toinstruments.

Twodifferent buttypical cases will be discussed: first,

com-parison of adigital multimeter with a calibrated one having

better resolutionby anorder ofmagnitude(61 _{versus 7}1

dig-its) will be considered. Then, comparison between multimeters having similar performances (6 digits) will be analyzed: the ra-tionale here isthat,withinapopulationof several similar multi-meters, one is maintained as aworking standard and kept under strictermetrologicalcontrol. Inthe former case, the aim is to

about measurement noise, so thata correct procedure for the verification of the calibration status can be defined. This is ofparticular significance in thecontext ofan industrial

labo-ratory,where oneof the mainpurposes of the analysis would

betoidentify improvements tothemeasurement system, that could allowtoperform calibration verification within the limits of availableequipment.

II. ALLAN VARIANCE ANALYSIS

Measurement variability can be due to the variability of

the measurand itself, to variations in some influence

quan-tity (e.g. environmental temperature) and, possibly, to noise and drift within themeasuring instrument itself. Forinstance,

Fig.1 shows a sequence of 5000 resistance measurements for

astandard commercial resistor with anominal value of 10Q,

obtained by adigital multimeter (Agilent 34401A).

Measure-ments weretaken in a room with no temperature control. It

is difficulttodetermine a meanvalue thatcanbeagood

rep-resentation of themeasurand, as trends and local fluctuations

are rather clearly present inthe time series of data. In other words, thesemeasurementsshould be dealtwith, more

appro-priately, as a time series, whose relevant features need tobe analyzed. Ofcourse,theassumption ofaGaussian distribution wouldcertainlynothold in thiscasefor the whole dataset.

Fluctuationsarecommonly described byapower-law

spec-traldensity model [14]:

S(f) =

Eh,f

ae

(2)

whereaisaninteger and it is assumed that -2 <a <+2. Al-though the model originates from theanalysis of noisesources

in high stability frequency standards, it is in fact fairly

gen-eral and has been extended to other kinds of standards (e.g., [10]). Thetermin(2)forwhicha=0, representswhite noise.

If its contribution dominates the model, a standard statistical approachmay beapplicable. On the otherhand, if thepower

spectral density of the observedquantity isnotwhite and

val-ues arecorrelated,useof standard methodstoestimate TypeA

uncertainty from acquireddata, referredtoin [7], ishardly jus-tifiable [8], [9], [10]. Forthispurposethe Allan variance and

thepowerspectraldensity of the the series should be analyzed

[12],[13], [14].

The empirical plot ofor (T) canbe compared with the Al-lan variance model underlying (2), which is described by the following expression [12]:

72(T) =

J

kltT.

/1

The coefficients kft are constants and u is relatedto aby: (3)

{ -a 1 for: a<I

j =

-2 for:

a>1

1.OE+O

a)

C)

CZ 1.OE-1

1.OE-2

Fig.1. Set of5000measurementresults for a 10 Q resistor(Agilent 34401A).

The Allan variance is commonly used in metrologyto

de-fine the stability ofa reference. Given a time series of data

indicatedby y(kT), where T is theconstantinterval between

consecutivemeasurements,Allan variance is calculatedby the

expression:

koZ0+N-1

kzo+2N-1

-2

2(T)=

I2N

[E

x

S(

)

x(kT)

₍₁₎

k=ko k=ko+N

with T = NT and N apositive integer. Analysis ofan

em-pirical Allan variance plotcan, inmoregeneralterms,provide

useful information about thestabilityofanykind of

measure-mentsystemunderanalysis. Thus, itmay proveauseful tool

[image:2.595.96.252.436.563.2]

intheanalysis of the calibration verificationprocess.

Fig. 2. Time-variance plot for the Allan variance model (3) with:

k-I =0.015,ko=0.04,k+1 =0.002.

Different values of,u indicate different noise components:

,u =-1 refers to white noise, ,u = 0 refers to _{a 1/f} noise

component, ,u =+1 to arandom walk noise component and

=+2_toadeviation of themeanvalue, [10], [13]. Their

rel-ativeweights, that depend on theconstants kft, determine the

shape of the Allan variance plot.

Using plots of Allan variance versus averaging time, the

structureof the noisecontributioncanbe evidenced anda

max-imum time intervalinwhich noisecanbe considered white

in-dicated, asdiscussedin[9]. Ideally, aU-shaped plotwould

re-sult: this is shownintheexampleofFig. 2,whereaminimum

is obtained foran averaging time between 1 and 10 s. Inthis

(4)

10 100

[image:2.595.351.551.441.587.2] [image:2.595.65.298.664.701.2]

case further averaging would in no way improve the results.

Inthe following, someexamples of the proposed useof Allan varianceplots in the analysis of themeasurementprocedure in normal laboratory practiceareshown.

III. PRACTICAL EXAMPLES

A. Verification based on TUR criteria

This subsection considers calibration verification of a

61

₂

digit multimeter (Keithley 2000) by comparison with the indi-cations of a 71 _{digit multimeter (Keithley 2001). Following}

recommended practice [3], the value TUR= 4 is taken as a

reference.

Presented results refertothe DC voltage function: a

volt-age sourceisappliedtothe referencevoltmeter andmeasured,

then thesame sourceis measuredby the multimeter undertest

and readings compared. Results presented here refer to the

1 V range of the instrument under test, whose resolution is

1 ,uV; for the reference instrument the correspondingrangeis

2 V and the resolution 100 nV. It should be noticed that the specified uncertainty of the K2000 multimeter in the 1 V range

is ±37,uV atfull scale; therefore the specification TUR= 4

yields arequirement fora verificationuncertainty notgreater

than±9,uV.

The reference multimeter is assumedtobefully

calibrated,

thereforeonly TypeAuncertainty will be accounted for. How-ever this should make allowance for the measurement sys-temstability, considering also the dead time between

measure-mentstaken with thetwomultimeters. Calibration verification

therefore requires the availability ofa suitably stable voltage

source. This may be a critical point, in view of the

resolu-tionspecificationsof the instruments. Allan variance analysis

canthenhelpassesswhether calibration verification is feasible with themeasurementsystem athand.

Figure 3 shows the Allan variance plots formeasurementsin

the2 V range of the reference instrument. Allplots werebuilt

by acquiringa setof 8192values; averaging involved at most

256 consecutivereadings, sothat each value is determinedby

averaging atleast 32 estimates. The curverepresented bythe dashed line referstothemeasurementof theoutputvoltageofa

commercialadjustable laboratorypowersupply. Itisusefulto

compareit with that obtainedby acquiringmeasurementswith

ashortedinput, represented bythe continuous line. The former

accountsfor instabilities and noise of the whole measurement

system,while the latter referstothe instrument alone.

Comparison between thetwoplots canprovide an assess-mentof the relativeimportance of the instrument contribution

[15]. Inthiscasethemeasurementsystem as awhole is char-acterized by a considerably larger Allanvariance, about four orders ofmagnitude higher. This canbe attributed tovoltage

fluctuations of thepowersupplyoutput, while the continuous linetraceshows that intrinsicstabilityof themeasuring

instru-mentis much better.

Using the laboratory powersupplyas areference does not

allowto reach the specified uncertaintytarget. Therefore, we

analyzed the feasibility of using a low-pass filterto stabilize

the power supply output. The first-order filter employed for

thispurpose hadatime constant of 560s, resulting ina very

long settling time: it takes anestimated21 _{hours for the filter}

output to settletowithin 0.1 ppmof the final value.

Introduc-tion of such unsophisticated, butvery narrow-band low pass

filter achievesasurprising improvement: thecurvemarked by 'x' refersto measurementsof the filteredpowersupplyoutput

andpractically coincides with the continuous line. Thismeans

that,overtime intervalsup to200ms,the Allan deviation is of the order of 100nV,well below thetargetuncertainty.

108

10 10

1 0

10 16

C)-10 2 _{1o-1 100}

[image:3.595.325.550.283.459.2]

integration timeX[s] 101 102

Fig. 3. Allan variance plots for measurement of unfiltered voltage (dashed line), filtered voltage('x')and shorted input (continuous line), 2 V range

(Keithley 2001).

However, even after such a long time, only a relatively

short-termstability canbe ensured for thegenerated voltage. Theupward slope in the rightpartof theplot is duetothe slow voltage variations relatedtothelow-pass filter. These limit the long-term stability of themeasurement,puttinganupperbound ofafew secondsonthe time availabletoperformcomparisons

with the instrument undertest.

B. Dataanalysis forstatistical tests

Results shown here are based on resistance measurement

dataacquired usinganAgilent34401Amultimeter. Inthiscase

Allan varianceanalysis is mainly usedtostudy noise contribu-tions and theirsources, as apreliminarystep totheapplication

of statisticaltestsforcompatibilityofmeasurements. A setof

commoncommercial resistors mountedon asupportprovided

with bananaplugshave beenemployedina roomwithout

tem-perature control. Connections are provided by equal length

cables. The autozerofunction of the instrument has been dis-abled in ordertoobtain the maximumsamplingrate,resulting

inasampling interval of 200ms. Foreach series of

measure-ments5000samples have been taken; acquisitionsareabout 16

Ii:

.... 60 0#0. ::I

minutes

long

andare

performed automatically using

aPCand

aLabVIEW program.

xlO0 Dati

1.0118

1.0116-

1.0114-

1.0112-,1.011 U51.0108 1.0106

1.0104

1.0102

1.01

0

100 200 300 400 500 600 700 800 900 1000

time(s) (a) time series. 10.

10

..1 0.

101

101 10o 101

time[s]

(b) Allan varianceplot.

10'

Fig.4. Measurement results fora1MQresistor(Agilent 34401A).

Measured values for a 1 MQ resistor are shown in

Fig. 4(a),

with the

corresponding

Allan variance

being plotted

in

Fig. 4(b).

The initial

slope

u 1 is followed

by

aflattract

evidencing, respectively,

white and flicker noise contributions.

'The curve

slope

next shows arandom walk noise contribution

(,u=

+

1)

and,

finally,

the effect of atrend in the measured values becomes dominant

(,u=

+2).

An

important

_aspect of the

analysis

is that the

length

of the time window

during

which the

assumption

of a white noise model is

acceptable

canbe determined

by evaluating

the

posi-tion of the first knee in the Allan variancecurve,

starting

from the

slope

of -1. Forinstance, the

plot

in

Fig. 4(b)

showsthat,

foran

acquisition

time shorter than2smeasurement

variability

canbe described as a white noise contribution

superposed

on

the measured values.

In this case the

probability

distribution of measured

val-ues

might reasonably

be assumed to be Gaussian, therefore

instrument

comparison by

statisticaltests can

rely

onwell

es-tablished

techniques.

The window size

corresponds

to

just

20

samples,

but

taking

morevalues cannot

improve

measurement

accuracy [10]. On the other hand, this limitation also

influ-encesthe confidence level that anystatisticaltest onthe

com-patibility

ofmeasurements canhave.

Foran

acquisition

window between2and 5sthe Allan

vari-anceremains constant, howeveraGaussian distributioncan no

longer

be assumed. If the

acquisition

is

longer

than 5 sdrift ef-fects areevidenced. Therefore, with themeasurement _system

considered in this case the

comparison

ofmeasurements ob-tained fromtwodifferent instrumentscanbe

performed only

if

acquisition

time is shorter than 5 s,thatmeans a

comparatively

small dataset. Itshould also be remembered that initial

mea-surementconditionsmustbe restored before the secondset of

measurement

acquisitions

starts. Care has to be taken to

en-surethat conditions

(e.g.,

resistor

heating)

in two

subsequent

acquisitions

canbe assumedtoremain

reasonably

constant.

Fig.

5 shows a

comparison

between the Allan variance

plots

evaluated for two series ofmeasurement values, obtained

re-spectively

with a 10 Q resistor and a short circuit connected

to the same instrument

input.

It can be noticed that the flat

tracts

(,u= 0)

associated with flicker noise coincide in thetwo

curves. This means that

Type

A measurement

uncertainty

is bounded

by

the instrument contribution and

corresponds

toan

Allan variance of2x 10 ' _{Q2. The final partof the}

plot

_for

the 10 Q resistor shows alinear trend with

slope

,u= +2 that

highlights

adrift in measuredvalues, as seenin

Fig.

1. Thecurves in

Fig.

5 also showa

peculiar

behavior for small

values ofT, where a

slope

,u= +1 is evidenced. This is

asso-ciated with band limited white noise and is aconsequence of the instrument bandwidth limitation [14]. Itshould be noticed that the effect is also

dependent

onthe

sampling frequency

em-ployed

to

acquire

measurement data. It

progressively

disap-pears ifmeasurements aretaken witha

longer sampling

inter-val,which has the effect of

decorrelating

consecutive

samples.

10-'

10 lo short

10-6

107_

1081

101 10o 101

time_[s] 102

Fig.5. Comparisonof Allan variances for resistancemeasurementbya

6-digitmultimeter(Agilent 34401A).

[image:4.595.68.283.157.537.2] [image:4.595.334.544.570.740.2]

This is practically demonstrated in Fig. 6, that shows the Allan variance plots of the 10 Q resistor, calculated byvarying the sampling time between 0.2 s and4 s. It can be seenthat, by increasing the sampling time, the initial slope of the curve is reduced, while the level of flicker noise is not modified.

0.

[image:5.595.68.281.193.363.2]

-.-:..-.-:-.-:.

... ... 0.2

Tc =1s

Tc = 2 s

::. Tc=4s

10v

-10-7_

108-10-1 10 10

time[s] 102 103

Fig. 6. Evaluation of Allan variance for a 10 Qresistor,with different

samplingtimes.

[6] EA-2-03,(EAL-P7),EALInterlaboratory Comparisons,1996. [7] ENV 13005, Guide to the expression of uncertainty in measurement,

1999.

[8] T.J. Witt, "Low-frequency spectral analysisof DCnanovoltmeters and voltage referencestandards," IEEETrans. on Instrument.nand Meas., vol.46, pp.318-321,1997.

[9] P.Helist6and H. Seppa, "Measurementuncertainty in the presence of low-frequency noise," IEEE Trans. on Instrument.n and Meas., vol. 50, pp.453-456,2001.

[10] T.J. Wittand D. Reymann, "Using power spectra and Allan variance to characterise the noise of Zener-diodevoltagestandards,"IEEEProc-Sci Meas. Technol., vol. 147, pp. 177-182, 2000.

[11] J. Valdes, M. E. Porfiri, E. E. Lobbe, F. Kornblit, M. N. Passarino de Marques, and J. A.Leiblich, "Longtermfluctuations associated with differentstandards," IEEETrans. onInstrument.n and Meas., vol. 42,

pp.269-272, 1993.

[12] D. W.Allan, "Should the classical variance be used as a basic measure instandardsmetrology?," IEEE Trans. on Instrument.n and Meas., vol. 147, pp.646-654,1987.

[13] T.J. Witt, "Using the Allan variance and powerspectral densityto char-acterize DCnanovoltmeters," IEEETrans. on Instrument.nandMeas.,

vol.50,pp.445-448,2001.

[14] J. Rutman andF.L.Walls, "Characterization offrequency stabilityin precisionfrequency sources," Proc.s of the IEEE, vol.79,pp.952-960,

1991.

[15] T.J.Witt, "Allan variances andspectral densities for DCvoltage mea-surements withpolarity reversals," IEEE Trans. onInstrument.n and Meas.,vol.54,pp.550-553, 2005.

[16] AgilentTechnologies, Agilent 34401A,Service'sGuide, 1996.

IV. CONCLUSIONS

Allan variance results have been used toanalyze the mea-surementsystems usedtoverifythe calibrationstatusofdigital

multimeters. Inparticular, resistance measurementdata series have been analyzed to decide about the maximum number of

samplesthatcanbeacquiredtoobtain uncorrelated realizations of the measurand. Ithas been shown that the effects ofchanges

inthe measurement system set-upcanbe studiedby the

analy-sis of Allan varianceplots,evidencingdifferent contributions. Theproposed approach can be usedbytestlaboratoriesas a

help in the design of the proceduresemployedtoverifythe cal-ibration status of their instrumentation. Inparticularthe

analy-siscanbe usedtodetermine better waystoreduce noise effects

or measurement trends and assess whether noise is primarily

duetothe instrumentorifthe measurand noise contribution is important.

REFERENCES

[1] [2] [3]

[4] [5]

ISO9001,Qualitymanagement systems.Requirements,2000.

ISO/IEC 17025, General requirementsforthe competenceof testingand calibrationlaboratories,1999.

UNI ENISO10012, Measurement management systems.Requirements formeasurementprocesses andmeasuring equipment,2004,

(Interan-tional reference: ISO 10012:2003).

G.Zingales, "Compatibilityinindustrialmeasurements,"Proc.ofIMTC,

pp.377-380,1999.

E.Arri,F.Cabiati,S.D'Emilio,and L.Gonella, "On theapplicationof theexpressionofuncertaintyinmeasurement tomeasuringinstruments,"