STRATEGIES FOR BUILDING AN AC-DC TRANSFER SCALE

Transcription

1 Smposo de Metrología al 7 de Octbre de 00 STRATEGIES FOR BUILDING AN AC-DC TRANSFER SCALE Héctor Laz, Fernando Kornblt, Lcas D Lllo Insttto Naconal de Tecnología Indstral (INTI) Avda. Gral Paz, 0 San Martín, Argentna laz@nt.gov.ar Abstract: Ths paper presents the comparson between two strateges to bld an AC-DC transfer scale. It s shown that an overdetermned scheme has some advantages over a smple one, thogh more tme consmng. Three statstcal tools are ntrodced to dscard bad standards or measrements and to calclate ncertanty contrbtons. We compared the contrbted ncertantes for both methods and reslts of ther to an AC-DC crrent transfer scale.. INTRODUCTION AC-DC crrent (or voltage) scales are blt sng the well known step-p-and-down procedres. At a medm level (typcally 0 ma or V), a set of well characterzed thermal converters are taken as the bass of the system. For them, the AC-DC transfer dfference s evalated theoretcally or determned n another lab. At other crrent (or voltage) levels, standards are calbrated aganst the standards of the neghborng range. The only assmpton made s that the ac-dc transfer dfference of each standard remans constant along ts crrent (or voltage) range, from the redced crrent (or voltage) at whch t s calbrated aganst the neghborng standard to ts hgher rated crrent (or voltage). A decson to be taken s the selecton of the step-p strategy. Two basc approaches are possble: a drect one, where the largest possble jmp s made wth only one standard or an overdetermned scheme n whch the jmps are made wth more than one standard and the AC-DC transfer dfference are overdetermned at each level. Fg shows the two schemes for a crrent step-p that wll be sed frther as examples.. THE DIRECT SCHEME The AC-DC transfer dfference of the standard beng calbrated at level,, s calclated as = + () where - s the AC-DC transfer dfference of the standard comng from the prevos step, and d s the measred dfference between both standards. The standard ncertanty of I s calclated as ( ) = ( ) + ( ) + ( ) + ( ) + ( )() A where ( I- ) s the ncertanty of the standard comng from the prevos step, ( A ) s the standard d C S L devaton of the measrements, ( C ) s the ncertanty contrbted by the comparson system (.e, lnearty of nanovoltmeters, exponents of TCs), ( S ) s the ncertanty contrbted by the measrement set-p (.e, gardng, connectors) and ( L ) s the standard ncertanty contrbted by the level dependence of the standard. In ths method ( S ) and ( L ) are estmated from prevos experence or ndrectly evalated, bt, to some extent, sbjectvely.. THE OVERDETERMINED SCHEME The overdetermned scheme s crrently sed at INTI. At 0 ma, fve PTB thn-flm mltjncton thermal converters (PMJTCs), two of them together wth shnts, are the bass of the system. The determnaton of the AC-DC transfer dfference of the PMJTCs depends on the freqency range. At ado freqences ( 00 Hz < f 0 khz ), the fve PMJTC are compared among them and the mean vale or the AC-DC transfer dfference of three (, and ) of the fve PMJTCs are taken as zero. Hence, the followng system of eqatons reslts a b c () d = e f g h where a,b,c,d,e,f, and h are the measred AC-DC transfer dfferences. The best solton [ ] s obtaned sng a modfed least sqare method []. In ths method the comparson statstc ncertanty and the lack of agreement of the fttng process are

2 Smposo de Metrología al 7 de Octbre de 00 nclded n the ncertanty calclaton. Ths lack of agreement s attrbted to the changes of connectons and postons of the standards n the measrng setp. At hgher freqences ( 0 khz < f MHz ) the assgned vale to TC- n a calbraton aganst the PTB standards s sed as a reference. To obtan the AC-DC transfer dfference vale of each standard, the eqaton system s solved sng the least sqare method. Below 00 Hz, a dfferent strategy s sed ot of the scope of ths paper []. To step-p, we sed two standards to jmp from one range to the other. At the hghest range of the leap, one of these standards s at ts rated power and the other one s at a qarter of t. The redndancy s necessary for the statstcal tools that wll be ntrodced. At each crrent level a system of eqatons s obtaned. For nstance, at 0 ma we get, 0 a 0 b 0 = c e [ A ] [ ] = [ B] () () where and are the vales obtaned for these standards n the prevos steps. Usng the least sqare method T T ( A A) A B = C B ' = () The resdal vector B A represents the lack of ft of the model. Its assocated ncertanty can be qantfed from the resdal sm of sqares and the resdal varance SS = B A ; ˆ σ step = SS / df (7) where df, the nmber of degrees of freedom sed to estmate the resdal varance, s calclated as the nmber of rows mns the nmber of colmns of A. The ncertantes of are calclated as the sqare root of the dagonal terms of the covarance matrx T cov( ') = C cov( B) C (8) and cov(b) s the covarance matrx of B. The dagonal terms of cov(b) are: var( b ) = ( ) + ( ) + ( ) =a,b,c (9) A C var () = ( p ) =d,e (0) M 0 m A ma 0 ma 7 00 ma ma ma 0 8 A 0 A A 0 A () Fg - Overdetermned AC-DC crrent step-p. In lght ble s also shown one possble drect scheme wth only one TC jmpng between levels. where ( A ) s the Type A standard ncertanty assocated to the repeatablty of each blateral comparson, ( C ) s the Type B standard ncertanty assocated to the comparson system, ( M ) s the standard ncertanty of the scheme of the comparson, whch, n or approach, can be estmated as the resdal standard devaton of the least sqare ft, that s, M ( ) = ˆ σ () step

3 Smposo de Metrología al 7 de Octbre de 00 and ( p ) s the standard ncertanty of the AC-DC transfer dfference of the standards comng from the prevos step []. The standard ncertanty of s ' ' ( ) cov( ) = () Both schemes reqre thermal converters wth level ndependent ac-dc dfferences and good stablty of ther ac-dc dfference. The overdetermned scheme allows for the se of statstcal tests to check these reqrements qanttatvely and objectvely... Statstcal Tests... Testng the Level Dependence. We propose a way to test whether the AC-DC transfer dfference of a standard does not depend on the crrent level. Ths test shold be appled to all the pars of standards sed to go p ( or down) from one range to another. Let s sppose that two of these transfers, A and B, are sed to go from a crrent level to another level, and let s call A and B ther vales n level, and A and B the correspondng vales n level (Fg.). Both standards have been compared n tmes at both levels, and the averages and standard devatons of the measred dfferences have been calclated. level level A A y, s y, s B B Fg.. Step wth two transfers as reference If both standards are eqally affected by the level change, the averages wll be smlar at both levels. If not so, we cold conclde that one of them s more affected than the other one. Both PMJTCs are of smlar desgn and technology bt are sed at qte dfferent power, that s, dfferent nternal temperatre. Ths, f t exsts a change between y and y t can be assgned to the most powered PMJTC. To test f the dfference between y and y s statstcally sgnfcant, a smple two-sample t-test for mean dfferences [] s appled, based on the statstc s + ( y y ) n s T = () A vale of T greater than a crtcal vale t n-;α/ (whch depends on a prevosly stated type of rsk α ) leads to conclde that the dfference between the averages s statstcally sgnfcant and, therefore, the level dependence mst be consdered an ncertanty component, assgned to the most powered standard. Otherwse, t can be assmed that the dfference s neglgble or attrbted to random errors, whch are contemplated n the least sqare calclaton. For example, wth the data from step 0 ma to ma at 00 khz, wth n=, y =0,9, y =,, s =0,79, s =0,, we obtan T =,99. If we se an α = 0,0, t n-α/ =,07. Ths, T > t n-α/, and we conclde that the dfference between the averages s statstcally sgnfcant. Therefore, the ncertanty cased by ths factor mst be consdered. Its standard ncertanty s estmated as y y ld = () and ncorporated to the ncertanty of the most powered standard.... Testng the Consstency between Pars of Standards. To verfy that the vales assgned to both reference standards A and B at the same step are consstent, we propose to compare the reslts obtaned by solvng the step twce, accordng to the followng procedre: - Frst, the step s solved, consderng both standards provdng a lnk condton to the prevos step. Let s call AB the otpt vector of the step - Then, one of the lnk condtons s elmnated from the model (deletng one of the two last lnes n the desgn matrx A). Therefore, other vales wll be obtaned for all the transfers, B Fnally, both estmatons are compared by means of the parameter E n [], AB ( ) B ( ) ( () () ) En () = () AB B The standard ncertanty n the denomnator mst be calclated sppressng all the correlatons between AB and AB. Vales of E n () greater than for any express lack of consstence.. For example, at 0 ma 00 khz, we obtan, ΑΒ ={.07; 9.;.8} for the transfers PMJTC-+SH-, PMJTC-+SH-, PMJTC-+SH-, respectvely. Β ={.;.7;.} ΑΒ Β = {.7;.8;.08} ( ΑΒ Β ) = {0.; 0.; 0.} Therefore, E n ={0.; 0.; 0.}

4 Smposo de Metrología al 7 de Octbre de 00 As E n s always smaller that, we conclde that there s consstency between the two reference standards at ths step.... Testng the Stablty of a Standard A statstcal method to test the nner consstency of each step s proposed. If a transfer s not stable enogh along the tme when the measrements are performed, the least sqare ft wll be poor and the resdal standard devaton (7) wll be too hgh. So, to test the hypothess of consstency, t can be compared wth the ft acheved from a redced model. If one of the transfers s sspected of beng nstable, t s dscarded from the scheme. A redced matrx A r s obtaned from A, by elmnatng the colmn correspondng to the dscarded standard, and all the rows related to the measrements n whch ths standard was nvolved. Also, a redced vector of observatons B r s obtaned from B, and new estmatons for the non-dscarded transfers can be calclated. Followng the same procedre than for the fll model, the redced sm of sqares SS r, the redced degrees of freedom df r and the redced resdal varance ˆσ r are obtaned. Then, an F-statstc can be calclated as ( SS SS ) ( df df ) r r F = () SS r df r It can be shown that SS - SS r and SS r are dstrbted accordng to χ dstrbtons wth df - dfr and dfr degrees of freedom, respectvely, and that both qanttes are statstcally ndependent []. Ths, F s dstrbted accordng to a Fsher-Snedecor dstrbton wth df-df r degrees of freedom n the nmerator and df r degrees of freedom n the denomnator []. A type one rsk α (the rsk of detectng a non-exstng nstablty) s prevosly stated. Ths, f the calclated vale of F s greater than the tablated crtcal vale f df-dfr,dfr,α we can conclde that the model consstency s sgnfcantly weaker for the fll model than for the redced one. Then, the lack of stablty of the separated transfer can be consdered sgnfcant. The power of the F-test that s, the probablty of detectng an actal lack of consstence- was evalated by Monte Carlo smlatons. As an example, Fg. shows the 0 ma step wth transfers, where transfer was evalated as possbly nstable. Smlated reslts of measrements are obtaned assgnng random nmbers to each par-comparson n the step. Sch random nmbers are generated from gassan dstrbtons wth a common mean vale 0 and a common standard devaton σ step, whch s the combnaton of the Type A sorces of ncertanty assocated to the step (lack of ft and repeatablty). Drng the smlaton, one of the comparsons n whch the sspected transfer partcpates was contamnated, addng ncreasng constant bases between 0 and σ step. Then, for each vale of contamnaton, the smlaton was repeated M=000 tmes, recordng when the contamnaton was detected by the F-test. Fg. depcts that the F-test power s not good. For nstance, the test for α=0.0 detects a σ contamnaton wth a probablty close to %. To ncrease the power of the method, we ntrodce a modfcaton of the test based on the Monte Carlo smlaton of the measrements. Each one of the comparsons presented n the step s repeated by the generaton of N random nmbers wth gassan dstrbtons centered n the average of an actal measrement. Those generatons are performed wth a common standard devaton. Smlated versons of the F statstcs F F N are compted by means of the same procedre that for the F-test. These copes of F cold be sed for statstcal calclatons. However, as the mathematcal propertes of F are hard to work wth, we compte log(f ) whch has a probablty dstrbton not so far from the gassan one. So, the followng statstcs can be obtaned log ( F ) µ log ( F ) T = (7) s( log ( F )) N where s ( log ( F )) s the sample standard devaton of log(f ) log(f N ), and µ log ( F ) s the theoretcal expected vale of log(f ), whch can be calclated followng the pdf of the F dstrbton [], as follows n+ m n m / Γ ( ) n m n / log( x) x (8) µ log ( F ) = ( ) ( ) ( ) ( ) dx = 0 Γ n Γ m m+ n / m + n x 0 beng n = df - dfr and m = dfr. For nstance, for the step as n Fg., the test for transfer nmber gves df =, dfr =, n = m =, and log ( x) µ log ( F ) = dx = 0 (9) ( + x) 0 The dstrbton of T can be approxmated by a t one, wth N- degrees of freedom. So, the condton to conclde nstablty or lack of consstence n the step s T > t N,α (0)

5 Smposo de Metrología al 7 de Octbre de 00 The power of the T-test was evalated for the same case and n a smlar way than for the F-test. The reslts for smlatons are shown n Fg. P 00% 80% 0% 0% 0% 0% Test T Test F 0 c (σ step nts) Fg. Power of the F anf T-tests. P s the percentage of detecton and c s the contamnaton n σ nts. Note that, for each contamnaton, M N smlatons were needed: M vales smlatng the measrement reslts mst be generated, and, for each one of these, N smlated F mst be obtaned., (0,), (0,),7 (,),0 (0,), (0,),9 (0,) -0,(0,), (0,) Fg. Comparson of standards at 0 ma, 0 khz. The vales near the arrows are the measred vales wth ther standard devaton n brackets Fg. shows the reslts of the measrements at 0 A at khz. If we apply eq. () to each standard sspected of beng nstable we get T =0,, T =0,, T =0,9, T =,, T =0,88. If we choose a type one of rsk of 0%, we get crtcal t,0. =, []. As T >., we conclde that the transfer s nstable and shold be replaced.. RESULTS The ncertanty was evalated for the two schemes shown n Fg.. To show the contrbton of the step-p clearly, the ncertanty of the basc standards at 0 ma was taken as zero. As an example, Table I depcts the contrbton for the drect scheme at A. The contrbtons for the overdetermned scheme are calclated from eq. (). Table II shows the calclated ncertantes. Table I Examples of ncertantes components for the drect method at A n µa A - Component f = khz f = 0 khz f =00 khz ( A ) 0, 0, 0, ( C ) 0, 0, 0, ( S ) ( L ) Table II Standard ncertantes calclated for the drect (D) and the overdetermned (O) schemes n µa A - Standards f = khz f = 0 khz f =00 khz D O D O D O 0,0 A 0,9 0,,,0,, 0, A, 0,7,8,,9,7 A,8 0,9,7,0 7,,0. CONCLUSIONS The se of an overdetermned scheme to step p allows for the se of statstcal tests to assess the qalty of a step-p scheme. Unstable or level dependant standards can be dscarded wth a base on objectve nmbers. Therefore, the contrbton of the step-p to the ncertanty can be redced. Besdes, the ncertanty components can be calclated from the measrements. The drect method needs less measrements, bt some ncertanty components mst be estmated from the prevos experence. REFERENCES [] H. Laz, M. Klonz, New AC-DC Transfer Step-p and Calbraton n PTB and INTI, Conference of Precson Electromagnetc Measrements, CPEM 000 Conf. Dg., pp 90-9, 000. [] H. Laz, Low Freqency Behavor of Thn-Flm Mltjncton Thermal Converters, Doctoral Thess, TU Branschweg, PTB-E-, 999. [] NIST/SEMATECH e-handbook of Statstcal Methods. NIST, USA. [Onlne]. Avalable: 00 [] Hornkova, A.,Zhang, N.,F The relaton between the E n vales ncldng covarances and the exclsve statstc, METROLOGIA, vol., L-L, Jan. 00. [] Scheffé H., The Analyss of Varance, John Wley & Sons, 99.