ANDRZEJ MUCIEK 1), FRANCO CABIATI 2)

Transcription

1 NDRZEJ MCIEK, FRNCO CITI Wocław niveity of Technology Chai of Electonic and Fotonic Metology Poland, Itituto Elettotecnico Nazionale Galileo Feai Toino, Italy NLYSIS OF THREE-VOLTMETER MESREMENT METHOD DESIGNED FOR LOW-FREQENCY IMPEDNCE COMPRISONS The pinciple of the thee voltmete method deigned fo pecie impedance compaion i peented. Genealization of the claical appoach i popoed and analyed. Fomula fo impedance paamete detemination ae deived and meauement uncetainty analyed fo vaiou ind of compaion, and net optimal meauement condition pecified. Impotant voltmete paamete, uch a andom eo and non-lineaity eo, ae epeimentally invetigated. The peented eult how that by uing the method and commecially available appaatu it i poible to compae two impedance with a elative uncetainty of ( in the low fequency ange. Keywod: impedance meauement, impedance tandad, voltage meauement, impedance compaion. INTRODCTION In ecent yea a geat impovement of digital voltmete paamete i obeved, uch a accuacy, lineaity, tability and fequency ange. Thi poge i paticulaly evident in DC meauement but alo a geat tep in impoving C meauement ha been made. lthough the lineaity of peciion C digital voltmete i till lowe than that of inductive voltage divide (IVD, it i high enough to be taen a a bai fo development of impedance meauement ytem moe accuate than commecially available intument. Thi i the cae fo the thee-voltmete method, aleady developed fo paticula pupoe [,, 3], but uitable to genealiation. In the pape popetie of the thee-voltmete method ae analyed. It i hown that two majo poblem mae it difficult to contuct a pecie ytem baed on the method and limit the cope of it application. The fit one follow fom the demand of pecie voltage meauement that one point mut be at low potential. Thi cannot be accomplihed diectly fo all thee meaued voltage. The econd one i inevitable tay impedance of the lead connecting both compaed tandad: meauement impedance and a efeence impedance tandad. To olve the fit poblem Cabiati popoed a olution [], which applie an inductive voltage divide (IVD et to a value of 0.5. It i poible to meaue all thee voltage with one end at a low potential without looing accuacy, and fo the econd one the pecial cicuit deign to inject a cuent which compenate the tay paamete of the lead. In the pape a new genealied appoach i popoed, which ue the IVD et to any value (0,. Thi inceae the method' fleibility and mae it poible to find optimal meauement condition fo compaing vaiou ind of impedance, a fo eample inductance in efeence to capacitance. The poblem have been cutinized and fo vaiou voltage atio and vaiou ind of the compaed impedance an optimal etting of the IVD ha been found. lo a geneal meauement uncetainty analyi ha been caied out and fomula deived fo eamination of paticula meauement cae. Diffeent hypothee have been conideed about the calibation of the voltmete, uch a: no pecial calibation of

2 voltmete and voltmete with calibation of lineaity. Impotant paamete of digital voltmete: noie, lineaity and tability have been invetigated and value aeed.. THE PRINCIPLE OF THE METHOD The baic, implified cheme of the thee-voltmete method deigned fo impedance compaion i peented in Fig. a. The ame inuoidal cuent, I = I m e jωt, i diven though two compaed impedance: the impedance unde tet Z = R jx (IT and the efeence impedance tandad Z = R jx. a b Fig.. The cheme of thee voltmete method applied fo impedance compaion: a the baic cicuit; b the vecto diagam. To detemine the elation between the impedance Z (R, X and the impedance Z (R,X, thee voltage =, =, and = ae meaued. vecto diagam of thee voltage i peented in Fig. b, fo the cae of linea impedance. nalying the elation between the meaued voltage,,, the cuent I = I and the paamete R, X, R, X of the compaed impedance we get the ytem of equation ( R R R ( X X X = ( R = ( R X X, whee = / and = / ae elative voltage, efeed to the voltage. y ubtituting the econd equation to the fit one we obtain the linea equation, with epect to R and X : whee the coefficient i defined by the fomula R R X X = (R X, (, = ( - - /. ( To obtain a econd linea equation we compae the aea of the tiangle P OP calculated, by the Heon fomula, diectly fom it leg, to it aea calculated by ubtacting fom the

3 aea of the ight tiangle P QP the aea of the ight tiangle P Q O, OQ P, P OP, and the ectangula OQ QQ. Fom thi, afte tanfomation, we get the econd linea equation -X R R X = (R X, (3 whee = / i a coefficient popotional to the aea, the value of which can be calculated fom the fomula [ ( ][( ]. = (4 The Equation ( and (3 fom a ytem of two linea equation and it olution povide fomula R X = R = R X X (5 fo the paamete R and X, detemination povided the paamete R and X of the efeence impedance and voltage,,, ae nown. The fomula (5 ae imila to thoe obtained by Cabiati et al. [] and alo eemble to a cetain degee the balance equation of a Wheattone type bidge. In the Wheattone bidge, coefficient and ae equal to the atio of voltage, which ae equal to atio of impedance, wheea in the thee-voltmete method the coefficient and ae function of quaed atio of voltage. the eult of thi quadatic elation the compaion eo depend on many agent and to obtain the highet accuacy the optimal meauement condition hould be aeed. Thi will be done fathe in the pape. 3. THE CTL CONFIGRTION OF MESREMENT SYSTEMS To apply thi method fo pecie impedance compaion, two majo poblem hould be olved: the fit one i pecie meauement of the voltage affected by a lage common mode voltage and the econd one i the influence of the tay impedance of the lead connecting the compaed impedance. To olve the fit poblem Cabiati [] connected the IVD between point P and P, Fig., with voltage output to input atio,, et to 0.5 and meaued the voltage M between the output tap of the IVD and the gound (intead of the voltage. Let M = M / and m = m / = M / = M be the elative voltage efeed to the voltage. To detemine the elation between and M (o between and m which i moe convenient fo futhe analyi we compae the aea of the P OP tiangle to the aea of the tiangle P OO'. oth aea ae calculated by uing Heon' fomula. fte ome tanfomation we obtain the equation = - m (. Net, by ubtituting thi fomula to Eq. ( we get a new fomula fo the coefficient = ( - m /. (7

4 Similaly by uing the Eq. (7 in (4 we get the new fomula fo [ ( ][( ] = m m, (8 which i imila to (4. The only diffeence i that the coefficient ha been ubtituted by m. To achieve high accuacy meauement, the compaed impedance hould wo unde condition pecified in the definition of the fou pot impedance, intoduced by Cutoy [4]. It i conideed to be the mot igoou definition fo hielded impedance o admittance equipped with fou coaial connecto [5]. In the fou-pot impedance, the voltage i defined at the high-voltage pot in open cicuit condition while the cuent i defined at the lowvoltage cuent pot whoe voltage i equal to zeo. One way of olving thi poblem, uggeted by Cabiati [], i baed on uing a cuent injection nea the middle ection of the yoe, which connect the compaed impedance, to compenate the voltage dop on the conducto. Thi olved the poblem but wa inconvenient fo opeation and needed complicated manual adjutment. Poliano and othe [, 3] developed the olution which opeate automatically, peeving high accuacy. Howeve the olution i complicated. 4. GENERLIZTION OF THE CIRCIT WITH IVD In the olution peented in [] the IVD ha the atio of voltage et to the value equal to = 0.5 (the voltage i divided by. We hall genealize thi condition by letting the IVD to be et to any value of between 0 and and meauing the voltage intead of M (ee Fig.. Now the ta i how, nowing the voltage,, and the etting of. Fig.. Geometical peentation of the genealied thee voltmete method. the IVD (o what i equivalent nowing elative voltage = /, = /, and, to detemine value of the coefficient and. The geometical peentation of the poblem, in the Gauian plane, i hown in Fig.. The ta can be tated geometically a follow: given P O =, OP =, OP = and = P O / P O find the elative voltage P P =. The comple numbe z atifie equation z - =, (9 z- =. (0 Solving thi ytem of equation of comple vaiable, with epect to = z we obtain

5 =. ( ( Net, by etting thi value to the Eq. ( we get the geneal fomula fo =. ( ( Similaly, applying the Eq. ( to (8, afte ome tanfomation, we get the new geneal fomula fo =. 4 ( (3 It can be eaily checed that by etting fo the value = 0.5 we get the fomula (7 and (8 a a paticula cae of the genealized appoach. The geneal fomula (5 fo detemination of the paamete R and X emain the ame a fo = 0.5. The only diffeence i that now the coefficient and ae calculated fom genealized fomula ( and (3. 5. MESREMENT OF IMPEDNCES The cicuit i uitable fo impedance compaion. We hall now detemine fomula fo meauement of paticula impedance. We aume that the efeence tandad i the eitance and deived fomula fo detemination of eito, inducto and capacito. The meaued impedance, IT, will be peented in the two element eie equivalent cicuit: eito: Z = R jωl = R ( jωτ, whee τ = L /R i the time contant, inducto: Z = jωl R = jωl (-j/q, whee Q = ωl /R i the toage facto, capacito: Z =/jωc R =( j D /jωc, whee D = ωr C i the diipation facto. The impedance of the tandad eito i equal to Z = R jωl = R ( jωτ, whee τ = L /R i the time contant of the efeence eito. It can be hown that paamete of the IT can be detemined fom the fomula: eito: inducto: R = ( ωτ R, (4 ωτ τ =, (5 ω ωτ L = ( τ R, (6 ω

6 capacito: Q C ωτ =, (7 ωτ = ω( ωτ R, (8 D ωτ =. (9 ωτ The net poblem i to ae the uncetainty of detemination of thee paamete a function of uncetaintie of the voltage meauement. 6. NCERTINTY NLYSIS The paamete R and X of the IT ae detemined fom the fomula (5 and it uncetainty depend mainly on uncetaintie of detemination of the coefficient and. The coefficient and ae detemined fom: a meauement of voltage,, and (o m, M, b meauement of voltage atio and (o m, M. The econd cae occu when the lineaity of voltmete i calibated, e.g. in efeence to the IVD. We hall deive fomula fo uncetaintie fo both mentioned condition and net apply them fo paticula meauement. The following notation i ued: the abolute uncetainty of a quantity i denoted by and the elative, by = /. We deive fomula fo uncetaintie fo the geneal configuation fit and analyze the paticula cae when = ncetainty of detemination of Fit, we conide cae a when voltage ae meaued. To ae the uppe limit of the elative uncetainty, we ubtitute = / and = m / into Eq. (3, calculate patial deivative with epect to, and m and divide both ide of the equation by. Fom thi we get the fomula fo the uppe limit of elative uncetainty. To analyze how thi uncetainty depend on the value of the compaed impedance paamete it i eential to epe fomula in tem of the compaed impedance paamete, R, X and R (we aume that the eactance X of the efeence impedance Z i negligible in compaion to R. It can be poved that, unde thee condition, the following equation hold: =, =, m = ( -, (0 = [( - ] =, whee = X /R i elative eactance efeed to the eitance R and = R /R i elative eitance efeed to R.

7 y ubtituting them to the equation fo the elative uncetainty we get the following fomula ( ( ( ( (, ( fo it uppe limit. In the cae of independent meauement we can ae the value of fom the equation. ( ( ( = ( The epective fomula fo cae b, when, and m ae meaued, can be obtained fom cae a imply by ubtituting = to the above fomula. Then we get: = / = and m = m, and in conequence = 0, = and m = m, whee and m ae uncetaintie of detemination of the voltage atio and m, epectively. 6.. ncetainty of detemination of ing the ame technique we can deive fomula fo the uncetainty fo the cae a when voltage ae meaued and fom thi we get the fomula. ( ( = (3 Now having thee eult we hall analye paticula cae of compaion: eitance veu eitance (R v. R. Inductance veu eitance (L v. R and capacitance veu eitance (C v. R. We now dicu the poblem in tun ncetainty of compaion R v. R The value of R i calculated fom fomula (5. Fo the eo analyi we conide only the main pat of the ight hand ide of thi equation, which mean that R R. Thu the elative uncetainty of detemination of R of R depend mainly on uncetainty which i given by Eq. (3. Fo typical condition, when the eito unde tet ha a mall time contant τ, and we can aume that = X /R <<, we get the aement. m R R = (4 gaph of the elative uncetainty of compaion R of the R to R i fo the cae when voltage, and M ae meaued ( = 0.5 i peented in Fig. 3.

8 een, the uncetainty ha a minimum fo equal appoimately to 0.5. If, fo intance, we aume that the compaion eo hould be malle than.4 then the atio = R /R hould be within the inteval (0.35, ncetainty of compaion of L v. R The value of inductance i calculated fom Eq. (4. We aume that the efeence tandad R ha a vey mall (negligible time contant, then the econd tem in the paenthee of the ight hand ide of (4 i vey mall and we can neglect it fo the eo analyi. Thu the value of inductance i detemined fom the implified fomula L = R /ω (5 and the elative uncetainty L i appoimately equal to (povided that ω, R <<. 6 R [ppm] = R /R Fig. 3. Relative uncetainty R of R v. R compaion, a a function of the of compaed eitance atio = R /R. We hall analyze the dependence of on the atio of the compaed element. If the toage facto Q i ubtantially lage than one (Q >> and the eactance of the IT, X i of the ame ode of magnitude a R, then the atio = R /R i much malle than and the fomula ( fo the uncetainty implifie to the fom =. (6 m The uncetainty depend mainly on uncetaintie of meauement of voltage and (o. If the toage facto Q i ubtantially malle than one (Q << and << then the fit tem

9 6 L [ppm] 5 4 Q = Q = 3 Q = 3 Q = 3 Q = 0 Q = =X /R Fig. 4. ncetainty L of inductance detemination a a function of = ωl /R fo vaiou value of the toage facto Q and fo = = M = ppm. in the quae oot of the above fomula vanihe. The uncetainty of detemination of inductance L i appoimately equal to the uncetainty of detemination of the atio = / and doe not depend on the value of R. Howeve fo inductance tandad meaued at low fequencie the toage facto Q i mall. Fo intance, fo the often ued GenRad inductance tandad, type 48, the value of the toage facto Q i about 0 fo a fequency of Hz and can be even malle than fo 00Hz. The uncetainty L depend, in thi cae, on the value of the efeence tandad R, Fig 4, and we can obeve that fo the lage Q value, the eo i appoimately contant - it doe not depend on = X /R but depend ignificantly on fo malle Q value. The minimal meauement eo i obtained fo. Thu fo the low value of toage facto it i impotant to efe the inductance L to uch eito R whoe value atifie the condition R ωl ncetainty of compaion of C v. R The fomula fo the detemination of capacitance C i given by (8. Fo the eo analyi we hall conide only the main pat of the denominato, then C /(ω R. The elative uncetainty C i appoimately equal to (uncetainty ω i negligible and R i the efeence tandad value. Diipation facto, D, of eal capacito atifie the condition D ( and fom thi we have << and <<. Then the value of uncetainty, C, fo the low value of the diipation facto, i imply given by the fomula C (ee Eq. (. Fom the above analyi we can daw two majo concluion: that in the cae of capacitance meauement in efeence to eitance the voltage m (and can be meaued with much lowe accuacy than the othe voltage, and that the uncetainty of detemination of depend diectly on uncetaintie of detemination of voltage and (o the atio.

10 7. OPTIMIZTION OF MESREMENT CONDITIONS The uncetainty of compaion depend on the value of the IVD etting (ee ( and (3. We hall analyze now how to choe a value of in paticula meauement. 7.. Choice of optimal in inductance and capacitance meauement The uncetainty of inductance and capacitance meauement i appoimately equal to uncetainty of detemination of the coefficient. In meauement of capacito with a low diipation facto, D, value and inducto with a high value of the toage facto Q, the coefficient i vey mall, <<, and uncetainty implifie to the fomula, (7 which doe not depend on. Howeve, fo inducto meaued at low fequencie, the fit pat in fomula (6 i quite ignificant and it impotant to detemine the optimal value fo fo each paticula cae. In Figue 5 the uncetainty L of L i peented fo a 0 mh GenRad inductance tandad, efeed to 0Ω, 60Ω and 00Ω eitance tandad, R, fo meauement at a fequency of f = Hz. The uncetainty change ubtantially with the value of and fo the efeence tandad R = 60Ω, the minimal meauement eo i obtained fo Fo a lowe fequency (f < Hz the influence of the value of on the meauement eo i getting lage and it i eential to calculate an optimal value of fo each R. L [ppm] R =00 Ω.3. R = 0 Ω. R = 60 Ω Fig. 5. ncetainty L a a function of the IVD etting fo diffeent value of efeence R tandad. 7.. Choice of optimal in eitance meauement The uncetainty of eitance meauement i appoimately equal to the uncetainty of detemination of the coefficient. Fo compaion of R v. R, with mall value of time contant of the compaed tandad we can neglect the value of the atio = X /R. The plot

11 of meauement uncetainty a a function of fo vaiou value of = R /R i peented in Fig. 6. The eult 0 R [ppm] = 0 = 3 = = 0.3 = Fig. 6. ncetainty of R v. R compaion a a function of the IVD etting fo vaiou atio = R /R. how a vey tong dependence of uncetainty on the value of and thi mean that the optimal value of hould be calculated fo each meauement and the IVD etting. imila analyi hould be pefomed when lage capacitance ae meaued. 8. NCERTINTY OF VOLTGE MESREMENT The accuacy of meauement i detemined by uncetainty of voltage meauement. Thi depend on the paamete of voltmete and the geneato. The mot impotant facto ae: tability of voltmete and geneato and lineaity of voltmete. In the cae of geneato we need high tability, paticulaly fo the configuation in which. only one voltmete i ued to meaue all thee voltage. Two voltmete of type C Standad 5790 and one geneato Calibato 5700, manufactued by FLKE wee eamined. Mot meauement wee made in the. V ange and at a fequency of 000 Hz. 8.. Stability Two ind of tability have been eamined: mutual tability of one voltmete v. anothe one and mutual tability of a voltmete v. the calibato. Fom the obtained eult the following concluion can be dawn: a elative diffeence between ucceive eading doe not eceed ±5 ppm, b elative diffeence between "moving aveage" of 0 eading uually doe not eceed ±. ppm, c elative diffeence between moving aveage of 00 eading uually doe not eceed ± 0.3 ppm d dift of moving aveage of 00 eading uually doe not eceed ± 0.3 ppm/minute. Sometime it happen that thee i a bigge change in ucceive eading. In thi cae we hould emove thee eult a outlie. Mutual tability of the calibato and voltmete wa eamined fo the. V ange. Fom thee meauement the following concluion can be dawn: the ytem (paticulaly the

12 calibato hould opeate at leat 4 h befoe the meauement. It i not enough to eep it in the tand by mode. fte witching fom the tand by mode to opeate mode the output of the calibato difted fo many hou and duing thi time quic change of eading eulting in ± ppm "jump" of 00 eading aveage, have been obeved. It wa pobably due to the intenal ytem which contol the level of the calibato output. Such jump wee aely obeved afte 4 hou wo and the aveage of 500 eading wa vey table - unde 0. ppm/4 hou. To get the highet accuacy of meauement the following pocedue i ecommended: a et the ytem in opeating mode fo at leat 4 h and obeve if ucceive eading ae tabilied, b tat the meauement fom the highet voltage of, 0 and, ay and tae 500 eading of each voltage meauement, c duing meauement, contol the jump of eading and in cae they hould occu emove them fom the data, d etun to the meauement of the fit voltage,, and if the aveage of 500 eading did not change moe fom the fit eie than an aumed value, ay ppm, the meauement eult can be accepted. 8.. Lineaity The lineaity eo of the voltmete wa checed in efeence to the IVD, Model 73 manufactued by ESI, with a elative uncetainty of 0.5 ppm. Two voltmete wee eamined, C STNDRD manufactued by FLKE, at Hz fequency and ma =.V ange and (ome meauement wee made alo at lowe ange the following eult have been obtained: a The non-lineaity eo, in the voltage ange (0.6 ma to ma efeed to the meaued value, i malle than ppm. Fo the lowe voltage value the eo i geate and fo /ma = 0.3 can be a big a 6.5 ppm. b Non-lineaity eo of both checed voltmete have a imila hape and the diffeence between them, fo the ange. V, i malle than 0. ppm. c Fo the ange of 0 mv the non-lineaity eo i geate and can be a big a 30 ppm. d The non-lineaity eo duing 5 day of eamination did not change ubtantially. In the voltage ange (0,6 -ma the obeved change wa malle than 0.3 ppm. It i woth to notice that the non-lineaity eo of the eamined voltmete, fo the ange of. V, i malle than ppm. Thu the eo of voltmete i about 4-5 time geate than the non-lineaity eo of typical IVD. 9. CONCLSIONS The peented analyi of the thee-voltmete method and invetigation of commecially available intument poved that the method i uitable fo impedance compaion with elative uncetainty up to a few ppm. To obtain uch high accuacy the cicuit hould be optimized by: uing a pope value of the efeence tandad, Z, etting an optimal value,, of the IVD; waming up the appaatu duing at leat 4 hou befoe meauement and uing an aveage of leat 500 eading with outlie emoved. REFERENCES

13 . Cabiati F., oco G. Soo., Impedance compaion in the low-medium ange though peciion voltage meauement, in Poceeding of the the XIII IMEKO Wold Conge, Toino, Septembe 5-9, 994, vol., pp Callegao L., D Elia V., utomated ytem fo inductance ealization taceable to C eitance with a thee-voltmete method, IEEE Tan. on Intum. and Mea., vol. 50, n. 6, Dec. 00, pp Callegao L., Galzeano G., Svelto C., Peciion impedance meauement by the thee-voltage method with a novel high-tability multiphae DDS geneato, IEEE Tan. on Intum. and Mea., Vol 5, n. 4, ug. 003, pp Cutoy R.D., Fou-teminal-pai netwo a peciion admittance and impedance tandad, IEEE Tan. Commun. Electon, 9, 964, Januay 964, pp Kibble.P. and Rayne G.H., Coaial C bidge, dam Hilge, itol, 984. NLIZ METODY TRZECH WOLTOMIERZY PRZEZNCZONEJ DO KOMPRCJI IMPEDNCJI W ZKRESIE MŁYCH CZĘSTOTLIWOŚCI S t e z c z e n i e Pzedtawiono podtawy metody tzech woltomiezy pzeznaczonej do pecyzyjnych ompaacji impedancji. Opacowano uogólnienie laycznego uładu i pzeanalizowano jego właności. Wypowadzono wyażenia pozwalające na wyznaczanie paametów impedancji i analizowano niepewność pomiaów dla óżnych odzajów ompaacji. Oeślono tąd optymalne wauni pomiaów. Ważne paamety woltomiezy, taie ja błąd loowy i błąd nieliniowości były epeymentalnie badane. Pzedtawione wynii wazują, ze pzy użyciu metody tzech woltomiezy z wyozytaniem dotępnej w handlu apaatuy pomiaowej, jet możliwe doonywanie ompaacji impedancji, w zaeie niich czętotliwości, ze względną niepewnością na poziomie (