Calibration of Bridge-, Charge- and Voltage Amplifiers for Dynamic Measurement Applications

52 PTB-Mi eilungen 125 (2015), No. 2Dynamic Measu emen o Mechanical Quan i ies measu emen esul and an associa ed measu e- men unce ain y has o be assigned o each componen . While such p ocedu es a e well es ablished o s a ic measu emen s, he e a e no s anda dized p ocedu es ye in he ield o dynamic measu e- men s. The equi emen s o he dynamic calib a- ion o b idge ampli ie s a e al eady included in in e na ional s anda ds [4], al hough he e a e no comme cially a ailable calib a ion de ices ye . Cu en ly, s anda ds desc ibing he p ocedu es o a dynamic calib a ion o condi ioning ampli ie s a e unde de elopmen in na ional and in e na- ional s anda diza ion wo king g oups, namely he Ge man DKD Fachausschuss “K a und Beschleuni gung” ( echnical commi ee on “Fo ce and Accele a ion”) and he Wo king G oup 6 o ISO TC 108/SC 3 “Use and calib a ion o ib a- ion and shock measu ing ins umen s”. This will emedy he cu en lack o alida ed p ocedu es wi hin he o eseeable u u e. The subsequen sec ion will p o ide an o e iew o he cu en s a e o knowledge and will u he gi e an ou line o wha is o be expec ed om he u u e documen a y s anda ds. 2 Requi emen s o Condi ioning Ampli ie s In heo y, condi ioning ampli ie s should ha e no in luence on he con en o he measu ed signal in he equency ange o in e es and should only con e he signal in he desi ed manne . Howe e , in eali y in luences o condi ioning ampli ie s exis and need o be de e mined h ough p ope calib a ion. In he ield o accele a ion measu e- 1 In oduc ion Fo he dynamic measu emen o mechanical quan i ies, he Eu opean esea ch p ojec T aceable Dynamic Measu emen o Mechanical Quan i ies wi hin he amewo k o he Eu opean Me o logy Resea ch P og amme (EMRP) ocused on he de elopmen o p ocedu es o aceable measu e- men s o he quan i ies o ce, p essu e and o que [1–3]. This Join Resea ch P ojec (JRP) conjoined expe imen alis s and ma hema icians om nine Eu opean Na ional Me ology Ins i u es (NMIs) o esea ch on he measu ands o ce, p essu e and o que. Fo dynamic measu emen s o any o hese quan i ies, he espec i e senso ypically needs o be complemen ed by a condi ioning ampli ie . The e o e, hese ampli ie s we e included in he esea ch wo k as a majo componen o he ace- abili y chain. The majo i y o measu emen s a e ca ied ou wi h ansduce s ha ing a b idge-, cha ge- o ol age ou pu . These di e en ypes o ou pu signals need o be condi ioned in o de o be digi ized by an analogue- o-digi al con e e o u he p ocessing. Fo he condi ioning o he signals, which may include he ampli ica ion, he decoupling and he con e sion o quan i ies, signal condi ioning ampli ie s – also called measu ing ampli ie s – a e used. Bo h e ms will be used synonymously in he subsequen ex . A ypical measu ing chain ha consis s o a ansduce , an ampli ie and a da a acquisi ion sys em (DAQ) is depic ed in igu e1. To ob ain aceable measu emen s, all compo- nen s o he measu ing chain ha e o be calib a ed and analysed ega ding hei in luence on he Calib a ion o B idge-, Cha ge- and Vol age Ampli ie s o Dynamic Measu emen Applica ions Leona d Klaus*, Thomas B uns and Hen ik Volke s * Leona d Klaus, Wo king G oup “Impac Dynamics”, PTB, e-mail: [email p o ec ed] Rep in o he o iginal A icle in Me ologia 52 (2015) 72–8, DOI:10.1088/0026- 1394/52/1/72 Con en may be used unde C ea i e Commons (CC) 3.0 licence. Any u he dis ibu ion o his wo k mus main ain a ibu ion o he au- ho s and he i le o wo k, jou nal ci a ion, and DOI. Figu e 1: Measu ing chain consis ing o a ansduce , a measu ing ampli ie and a da a acquisi ion sys em. 53 PTB-Mi eilungen 125 (2015), No. 2 Condi ioning Ampli ie s men some expe ience in he dynamic calib a ion o cha ge ampli ie s exis s. This knowledge has now been ex ended o he de elopmen o calib a- ion p ocedu es o a ious ypes o condi ioning ampli ie s. Abo e all, he ollowing ypes o ampli- ie s we e conside ed in he JRP and hence, in his pape : • Vol age ampli ie s • Cha ge ampli ie s • IEPE ampli ie s, and • B idge ampli ie s. In he ollowing pa ag aphs, he peculia i ies o he di e en de ices a e b ie ly in oduced. 2.1 P ope ies o condi ioning ampli ie s Real ampli ie s exhibi a equency-dependen beha iou desc ibed by hei equency esponse unc ion. This esponse may in u n depend o some ex en on he p ope ies o he connec ed sou ce ( ansduce ). These dependencies need o be cha ac e ized by means o calib a ion. To be able o calib a e a measu ing ampli ie , he de ice should be linea and ime-in a ian (LTI). Linea i y: he ou pu scaling ac o should no depend on he le el o he inpu signal o he ampli ie . Time in a iance: he equency esponse unc ion should no change o e ime. The assump ion o an LTI beha iou o condi ion- ing ampli ie s is a p e equisi e o he applicabil- i y o he calib a ion p ocedu es desc ibed in his pape . 2.2 Di e en ypes o ampli ie s used o dynamic measu emen s 2.2.1 Vol age ampli ie Vol age ampli ie s a e used o condi ion inpu ol ages o p opo ional ou pu ol ages. Two main applica ions exis : • Fo small inpu ol ages, ol age ampli ie s a e used o signal ampli ica ion. • They a e also used as uni y-gain ollowe s o a decoupling o inpu and ou pu in case o load-sensi i e ansduce ou pu s o o adap he sou ce impedance o he connec ed da a acquisi ion channel. 2.2.2 Cha ge ampli ie These de ices a e used in conjunc ion wi h piezo elec ic ansduce s o con e he cha ge gene a ed by he ansduce o a p opo ional low impedance ol age ou pu . The in e nal ci cui y o he inpu s age o hese de ices ypically exhibi s ei he high-pass cha ac e is ics, o d i beha iou , i he high pass is compensa ed o [5]. 2.2.3 IEPE ampli ie Piezoelec ic ansduce s may be supplied wi h embedded in eg a ed elec onics (in eg a ed elec- onic piezoelec ic, IEPE), which may be named ICP®, Del a on®, Piezo on®, o simila , depending on he manu ac u e . The speci ica ions o IEPE ansduce s and hei powe supply a e no de ined in a s anda d speci ica ion and may di e in de ail om manu ac u e o manu ac u e . Howe e , he known ypes ollow a common p inciple. The powe supply o such a ansduce is ealized using a wo-wi e connec ion wi h a cons an cu en eed. The ol age be ween he ails used o he powe supply changes, depending on he measu ed quan i y. I has a bias ol age le el o ypically 8V o 12V, which co esponds o he ze o poin o he measu ed quan i y. The ol age will change p opo ionally o he measu ed quan i y in a ange o ypically 0.5 V (minimum ol age, minimum alue o he measu and) o 24 V (supply ol age, maximum alue o he measu and). Besides he cu en supply and he bias ol age le el, he wo king p inciple o an IEPE condi ioning ampli- ie is ela ed o ha o a ol age ampli ie used o decoupling o inpu and ou pu . 2.2.4 B idge ampli ie B idge ampli ie s a e used o he signal condi ion- ing o he Whea s one b idge ou pu s o s ain gauge o o piezo esis i e ansduce s. The ampli- ie eeds he ansduce ’s b idge ci cui wi h a supply ol age. The ol age ou pu o he b idge is dependen on he de uning o he b idge’s esis- o s and addi ionally p opo ional o he supply ol age. The e o e, he ou pu o a ansduce is a a iome ic quan i y and is usually gi en as he a io o he b idge ou pu ol age and supply ol age in mV/V. Fo co ec signal condi ioning, b idge ampli ie s should implemen his a iome - ic p inciple by no only aking he ou pu signal o he ansduce in o accoun , bu by including he supply ol age le el as well. I should be no ed ha o he scope o he wo k de sc ibed he e, only ampli ie s p o iding a DC supply ol age a e con- side ed. So-called ca ie equency b idge ampli- ie s a e ypically dedica ed o s a ic measu emen exe cises and he e o e a e no discussed he e. 54 PTB-Mi eilungen 125 (2015), No. 2Dynamic Measu emen o Mechanical Quan i ies Ca ie equency b idge ampli ie s can be used in a equency ange o only abou 20 % o he ca ie equency, which is usually below 5 kHz. E en hen, de ia ions o mo e han 10 % in he magni- ude esponse ha e o be aken in o accoun [6]. Fu he mo e, i should be no ed ha acco ding o he common uni s o he inpu in mV/V and he ou pu in V, he uni o he equency esponse unc ion o hese de ices is, in ac , V (mV/V) . 3 P ocedu es o Dynamic Calib a ion The goal o a calib a ion is he de e mina ion o he p ope ies o in e es o he de ice unde es (DUT) wi h a known measu emen unce - ain y. Fo he condi ioning ampli ie , which can p esum ably be desc ibed as an LTI sys em, he p ope y o in e es is he equency esponse unc ion. 3.1 F equency esponse unc ion The complex- alued equency esponse unc ion H (iω) desc ibes he ime-dependen inpu –ou pu beha iou o an LTI sys em in he equency domain o an angula equency ω = 2π , whe e is he equency in Hz. Fo con inuous sys ems, he equency esponse unc ion is de ined as he a io o he ou pu Y (iω) o e he inpu X (iω) as H(i )= Y(i ) X(i ) . (1) I s magni ude A(ω) is gi en by AH HH ωω ωω () = () = () () + () () |i|i iRe Im 22 (2) and desc ibes he con e sion and scale- o gain ac o o he de ice. The phase esponse unc ion φ(ω) is gi en by ϕ ( ω )= an−1Im(H(i ω )) Re(H(i ω )) ⎛ ⎝ ⎜⎞ ⎠ ⎟ (3) and cha ac e izes he signal delay be ween inpu and ou pu . To de i e he phase angle om he complex equency esponse unc ion, a ou -quad an in e se angen calcula ion should be applied. I he wo-quad an in e se angen calcula ion is used, a co ec ion o φ o ± π may be necessa y as his unc ion is only de ined in he ange −π/2<φ<π/2. 3.2 Exci a ion signal The dynamic exci a ion o he measu ing ampli- ie ’s inpu quan i y used o calib a ion can be ca ied ou in di e en ways, e.g. by ansien , andom noise o pe iodic exci a ion signals. The mos commonly used exci a ion signal is he mono- equen sinusoidal exci a ion. I s ad an ages a e he selec able exci a ion equency, du a ion and magni ude, bu he compa ably long measu emen ime o ob ain he comple e equency esponse unc ion is disad an ageous. Since non-sinusoidal exci a ions can usually be ela ed o sinusoidal exci- a ions by Fou ie me hods, all subsequen consid- e a ions will be ocused on he la e . Fo an exci a ion signal x( ) o a sinusoidal exci a ion wi h he magni ude Ax, he angula e- quency ω and he phase φx is desc ibed by x( )=Ax · sin(ω +φx)=ax sin (ω )+bx cos(ω ) , (4) wi h Ax= a x 2+b x 2and φx = an−1(ax /bx) . The ou pu signal can be desc ibed acco dingly as y( )=Ay· sin(ω +φy) =ay sin (ω )+by cos(ω ) . (5) Wi h his de ini ion he equency esponse unc- ion can be w i en in he o m H A A y x ie i yx ωω ω ϕω ϕω () = () () ⋅ () − () () . (6) The exci a ion equency should be chosen app op ia ely o he la e applica ion. The ecom- mended equency alues o en used in acous ics and ib a ion calib a ions a e gi en in ISO 266 [7]. These ecommended equencies a e equally spaced in he equency domain on he loga i hmic scale. The wid h be ween he equency s eps can be chosen based on he desi ed numbe o equency s eps o a ixed in e al. 3.3 Ampli ie se ings A calib a ion esul can only be alid o one ce ain se o se ings o he ampli ie unde es , which include gain, co ne equencies o high-pass and low-pass il e s, ansduce sensi i i y and possibly o he pa ame e s. I is essen ial o documen he se ings a which he ampli ie was calib a ed. These se ings should be chosen acco ding o he la e applica ion. 3.4 Linea i y Fo he abo e-men ioned equency esponse unc- ion used o desc ibe he dynamic beha iou , he 55 PTB-Mi eilungen 125 (2015), No. 2 Condi ioning Ampli ie s linea i y o he DUT is manda o y and should be ensu ed du ing calib a ion. Nonlinea i ies cause dis o ions o he sinusoidal shape o he ou pu signal and can be es ima ed by analysing he ha - monic signal componen s. 3.5 S abili y The equi emen o he ime-in a ian beha iou can be p o ed by epea ed calib a ions o e a p o- longed pe iod o ime. 4 Calib a ion Se -Ups 4.1 Vol age ampli ie calib a ion se -up Fo he calib a ion o ol age ampli ie s, a calib a ion se -up as depic ed in igu e2 can be used. The ime- dependen ol age ou pu Ugen o he signal gene a o is coupled di ec ly o he inpu o he ol age ampli- ie unde es (in he case o small gain ac o s) o can be down-con e ed by means o a calib a ed ol age di ide (in he case o la ge gain ac o s). The measu ands o his se -up a e he inpu ol age o he ampli ie Uin( ), which may be cal- cula ed om a calib a ed gene a o ol age and he ou pu ol age Uou ( ) o he ampli ie . In e ms o equa ions (4) and (5), his means x( )=Uin · sin(ω +φin) and (7) y( )=Uou · sin(ω +φou ) . (8) 4.2 Cha ge ampli ie calib a ion se -up In o de o calib a e a cha ge ampli ie i is neces- sa y o con e he gene a o ol age om he p e iously desc ibed se -up o an inpu cha ge q e [8]. Fo his pu pose, a high p ecision capaci- ance C e is employed as shown in igu e3. Assuming ha he inpu impedance o he cha ge ampli ie is negligible, he e e ence cha ge q e o he cha ge ampli ie unde es can be de i ed as q e ( ) = Ugen( ) · C e . (9) In e ms o equa ion (4) he inpu measu and is gi en as x( ) = Ugen · C e · sin(ω + φin) . (10) Unde his assump ion, pa allel capaci ies Cpa (e.g. om he ansduce cable) will no in lu- ence he amoun o cha ge a he ampli ie inpu . Howe e , i was ound ha he o al capaci ance a he inpu (C e + Cpa ) may in luence he cha ge ampli ie ’s equency esponse unc ion [9]. This e ec is ypically small, bu may no be negligible in applica ions wi h a demand o low measu e- men unce ain ies. 4.3 IEPE ampli ie calib a ion se -up IEPE condi ioning ampli ie s can be calib a ed using a measu ing se -up simila o a se -up o low gain ol age ampli ie s. Ins ead o a di ec connec ion o he signal gene a o and he ampli- ie unde es , an IEPE simula o is connec ed be ween he signal gene a o and he IEPE condi- ione [10], con e ing he gene a ed ol age in o an IEPE ansduce -like ou pu signal. This se -up is depic ed in igu e4. Fo he de e mina ion o he equency esponse unc ion o an IEPE signal condi ione , wo connec ion schemes a e possible wi h such a se -up: • Calib a ion wi h a calib a ed IEPE simula o . I he IEPE simula o ’s equency esponse Figu e 2: Schema ic ep esen- a ion o a calib a ion se -up o ol age ampli ie s using a signal gene a o and a ol age di ide (le ), o a signal ge- ne a o only ( igh ). Figu e 3: Schema ic diag am o a calib a ion se -up o cha ge ampli ie s. 56 PTB-Mi eilungen 125 (2015), No. 2Dynamic Measu emen o Mechanical Quan i ies unc ion was de e mined p io o he cali- b a ion o he IEPE condi ioning ampli ie , a known inpu ol age can be used o he exci a ion ollowing x( )=Ugen · sin(ω +φin) (11) and applying a co ec ion acco ding o he p ocedu es desc ibed in sec ion 9. • Calib a ion by measu ing Uin ins ead o Ugen. I he ou pu signal o he IEPE simula o is acqui ed ins ead o he inpu signal, he in lu- ences o he IEPE simula o a e excluded om he measu emen . x( )=Uin · sin(ω +φin) + Uo se (12) Howe e , i has o be kep in mind ha he e will be a ol age o se Uo se due o he IEPE eeding, which can lead o educed ol age measu emen p ecision due o he disad an a- geously la ge inpu ol age anges o he da a acquisi ion de ice. Addi ional in luences due o he ol age measu emen in he measu ing chain mus be analysed as well. 4.4 Dynamic b idge s anda d calib a ion se -up Fo he dynamic calib a ion o b idge ampli ie s, di e en calib a ion se -ups ha e been de eloped a he NMI le el. They all ea u e a a iome ic measu emen p inciple. An olde app oach gene - a es he small exci a ion ol ages by means o an induc i e coupling, which is limi ed a low equen- cies [11]. The e o e, no DC measu emen ( = 0 Hz) is possible. A new app oach gene a es he b idge signals by using wo mul iplying digi al-analogue con e e s (MDACs) and a esis i e ol age di ide [12–13]. The p inciple is depic ed in igu e5. Wi h his p in- ciple, b idge exci a ion equencies down o DC a e possible; in ac , a bi a y signals could be gene a ed as well. As pa o wo k package 4 o he a o emen- ioned EMRP p ojec , he dynamic b idge s anda ds we e u he de eloped o enable phase measu e- men s and a aceable calib a ion was ca ied ou . An in e na ional bila e al compa ison o he di - e en dynamic b idge s anda ds is cu en ly unde way o ensu e compa abili y in he ange o he es ima ed measu emen unce ain ies. The dynamic b idge s anda d simula es he ou pu signal o a s ain gauge ansduce . The de ice is connec ed o he b idge ampli ie unde es and p o ides a simila inpu esis ance o a s ain gauge ansduce . The ou pu ol age o he simula ed b idge depends di ec ly on he b idge supply ol age, because he supply ol age is he e e ence ol age o he mul iplying DACs. In he case o b idge ampli ie s equipped wi h a con ol o he b idge exci a ion ol age by means o a 6-wi e connec ion using sense wi es, hese wi es a e con- nec ed o he MDACs in he b idge s anda d so as o beha e simila ly o a eal ansduce . As he dynamic b idge s anda d gene a es he p ede ined aceable ol age a io, he measu and o a magni ude calib a ion o a b idge ampli ie is solely he ou pu ol age o he ampli ie . In o de o de e mine he phase esponse, he dynamic b idge s anda d supplies a complemen a y no malized signal ou pu , which p o ides a signal synch onous o he ol age a io. Wi h e e ence o his ‘synch o- niza ion signal’, he ime delay and he e o e he phase esponse o he ampli ie can be calib a ed. Figu e 4: Calib a ion se -up o IEPE condi ioning ampli ie s including an IEPE simula o . Figu e 5: Schema ic ep e- sen a ion o a b idge ampli ie calib a ion se -up wi h an MDAC dynamic b idge s anda d. 57 PTB-Mi eilungen 125 (2015), No. 2 Condi ioning Ampli ie s 5 Da a Acquisi ion and Analysis The me hodology p esen ed he e is implemen ed in he espec i e labo a o ies o PTB and oou knowl- edge in many o he NMI labo a o ies, namely in he ield o ib a ion me ology. Ne e heless, o he alid se -ups and me hods a e concei able. To calcula e he equency esponse unc ion (see equa ion (1)), he inpu and ou pu quan i ies ha e o be acqui ed wi h wo measu emen channels. Fo p ope signal acquisi ion, se e al c i e ia should be complied wi h: • Common sample clock o he wo channels (o sample clocks o an in ege a io). • Synch onous s a o sampling. • Common sampling in e al (window wid h) o an in ege mul iple o he mains pe iod. • Sample equency ha co e s he bandwid h o he nominal equency and exis ing ha monics (Nyquis c i e ion). Conside ing his in all o he p e iously desc ibed se -ups, wo sampled ime se ies o sinusoidal ol age signals {xi} and {yi} a e acqui ed1, which is – i espec i e o a cons an ac o (e.g. C e ) – adisc e- ized ealiza ion o he con inuous signals x( ) and y( ) (see equa ions (4) and (5)). Th ough he applica ion o a linea leas squa es i o he named equa ions o he sampled ime se ies, he pa ame e s ax, bx and ay, by can be de e mined easily. Wi h he ans o ma ions gi en in sec ion 3.2, he equency esponse unc ion is hus de i ed acco ding o equa ion (6) by subsequen measu e- men s a all desi ed angula equencies ω. 6 In luence o he Impedance E e y signal inpu , signal ou pu and connec ing cable has i s inhe en complex impedance which, i no aken in o accoun , can add sys ema ic de ia- ions o he measu emen s. Figu e 6 shows he wo se -ups o he ol age and o he cha ge measu e- men s. I he ypically used coaxial cables a e signi i- can ly sho e han he wa eleng h o he signal, hey can be modelled by a pa allel capaci ance Z c. The ou pu ol age Uou o a measu ing ampli ie is dependen on he ou pu scaling ac o de sc ibed by he equency esponse unc ion H as U ou =H · U in (13) o ol age ampli ica ion, and U ou = H · q in (14) o cha ge condi ioning, espec i ely. The ol age a he inpu o he ampli ie U in can de ia e om he senso ou pu ol age U s because o he in luences o he impedances Uin =Us1+Zs1 Zin +1 Zc 1 . (15) Acco dingly, he inpu q in o a cha ge measu emen can de ia e om he ou pu cha ge q s acco ding o θin = θs1+Zs1 Zin +1 Zc 1 . (16) 7 Measu emen Unce ain y Con ibu ions o Realized Calib a ion Facili ies The measu emen unce ain ies o dynamic calib a- ions a e based on he quali y o he de ice unde es as well as on he calib a ion se -up. The majo componen s o he measu emen unce ain y o he se -up can be (see [8–12]): • Non-linea i ies o he A/D con e sion o single equency a io measu emen . • Calib a ion unce ain ies o he used e e ence capaci ance ( o cha ge ampli ie calib a ion). • Calib a ion unce ain ies o he IEPE simula o ( o IEPE ampli ie calib a ion). • Calib a ion unce ain ies o he dynamic b idge s anda d ( o b idge ampli ie calib a ion). • Noise o he inpu ol age (gene a o ) and o he ou pu ol age (condi ioning ampli ie ). 1 In he case o he dynamic b idge s an- da d, he inpu is no acqui ed bu de ined by he p og ammed signal. Howe e , he analysis p ocedu e is easily adap able. Figu e 6. Schema ic diag am including impedan- ces o cu en o cha ge measu e- men s (le ) and o ol age measu e- men s ( igh ). 58 PTB-Mi eilungen 125 (2015), No. 2Dynamic Measu emen o Mechanical Quan i ies Hence, a ough es ima e shows ha he claims o some 10−3 o 10−4 o ela i e combined expanded unce ain y o magni ude a e easible o cha ge ampli ie calib a ion [8–9] wi h he p ope equipmen and unde good condi ions, depending, o cou se, on he de ails o he implemen a ion. The ela i e expanded measu emen unce ain ies o IEPE and he b idge ampli ie s, again o magni ude calib a ion, will be in he anges o en hs o pe cen [10–12]. No only do NMIs ca y ou aceable calib a ions o cha ge ampli ie s, bu acc edi ed labo a o ies also compa e hei calib a ion esul s in a na ional compa ison p og amme [14]. The measu emen unce ain ies o acc edi ed cali b a ion labo a o ies o he a ailable calib a ions (cha ge ampli ie s, ol age ampli ie s, bu no ye b idge ampli ie s) a e ypically in he ange o 0.3 %<U(k=2)<1 % and ypically a ailable o magni ude calib a ion only. 8 Measu emen Resul s o he Calib a ion o Ampli ie s To show he impo ance o a sui able calib a ion o ampli ie s used o dynamic measu emen s, di e en calib a ion esul s o b idge-, cha ge-, ol age- and IEPE condi ioning ampli ie s a e p esen ed. All analysed ampli ie s a e comme - cially a ailable. They we e p oduced by di e en manu ac u e s. The cha ge-, ol age- and IEPE condi ion- ing ampli ie calib a ions we e ca ied ou in a equency ange om 0.1 Hz o 100 kHz. All high- and low-pass il e s we e disabled o se o hei lowes (high-pass il e ) o highes (low-pass il e ) alue, espec i ely. The de ia ions in he magni ude esponses a e gi en wi h espec o he nominal alue, se a he ampli ie . Th ee o he ou cha ge ampli ie s show a ypical high-pass beha iou in hei ampli ude esponse ( igu e7). Cha ge ampli ie 2 has he op ion o disable i s high-pass il e , which esul ed in a much be e low- equency beha iou bu also in a d i ing DC bias ol age (which does no become ob ious om he equency esponse da a). I was ound ha e en wi h high-pass il e s se a he lowes alue, he il e se ings can s ill in luence he measu ing esul s in he equency ange up o 10Hz. The low-pass il e s o h ee ampli ie s (ampli ie s 2, 3, 4) show a signi ican o e shoo ing beha iou , which should be con- side ed wi h cau ion. The co esponding phase esponse o he same ou cha ge ampli ie s is shown in igu e8. The phase and he magni ude esponses can di e subs an ially om ampli ie o ampli ie . The calib a ion esul s o he magni ude ( igu e9) and phase esponse ( igu e10) o ou di e en ol age ampli ie s show he smalles e- Figu e 7: Magni ude espon- ses o ou di e en comme cially a ail- able cha ge ampli- ie s in he equency ange o 0.1 Hz o 100 kHz. Figu e 8: Phase esponses o ou cha ge ampli- ie s in he equency ange o 0.1 Hz o 100 kHz. 59 PTB-Mi eilungen 125 (2015), No. 2 Condi ioning Ampli ie s quency-dependen de ia ions a low equencies. Again, he low-pass il e can p oduce signi ican o e shoo ing e en in a equency ange o a ew hund ed he z ( ol age ampli ie 4), which a ec s he phase esponse acco dingly. All in es iga ed IEPE condi ioning ampli ie s ha e a signi ican high-pass beha iou as depic ed in igu es 11 and 12, which can a ec low e- quency measu emen s. In igu es 13 and 14, he magni ude and phase esponse o ou b idge ampli ie s a e shown. The calib a ion measu emen s we e ca ied ou using he dynamic b idge s anda d o PTB in a equency ange o 10 Hz o 10 kHz. The exci a ion le el was 2 mV/V; all il e s we e swi ched o o se o hei highes alue a ailable. I becomes ob ious om he calib a ion esul s ha b idge ampli ie s can ha e a signi ican a ying magni ude esponse e en in low equency egions, which shows he impo ance o dynamic calib a ions. The phase esponses o he ou analysed ampli ie s show signi ican ly di e ing beha iou . The ou b idge ampli ie s unde es had a phase delay o a leas 15° o 20° a 10 kHz. Bu b idge ampli ie 2 exhibi ed a much s onge phase delay. Figu e 9: Magni ude espon- ses o ou di e en comme cially a aila- ble ol age ampli ie s in he equency ange o 0.1 Hz o 100 kHz. Figu e 10: Phase esponses o ou ol age ampli- ie s in he equency ange o 0.1 Hz o 100 kHz. Figu e 11: Magni ude espon- ses o ou di e en comme cially a aila- ble IEPE condi io- ning ampli ie s in he equency ange o 0.1 Hz o 100 kHz. 60 PTB-Mi eilungen 125 (2015), No. 2Dynamic Measu emen o Mechanical Quan i ies 9 Co ec ion o he F equency-dependen Beha iou The calib a ion esul s can be used o co ec he dynamic in luences o he ampli ie in he meas- u emen chain. In case he magni ude esponse unc ion Acal(ω) and he phase esponse unc ion φcal(ω) o an ampli ie a e known om calib a ion, he magni ude Ameas(ω) and phase φmeas(ω) o he measu ed da a can be co ec ed in he equency domain as Aco (ω) = Ameas(ω) · Acal(ω)−1 (17) and φco (ω) = φmeas(ω) − φcal(ω) . (18) I should be kep in mind ha he equency esponse unc ions o he measu ing ampli ie s ha e o be assigned wi h hei measu emen unce - ain y and will he e o e in luence he co ec ed da a’s unce ain y acco dingly. Figu e 12: Phase esponses o ou IEPE condi io- ning ampli ie s in he equency ange o 0.1 Hz o 100 kHz. Figu e 13: Magni ude espon- ses o ou di e en comme cially a aila- ble b idge ampli ie s in he equency ange o 10 Hz o 10 kHz. Figu e 14: Phase esponses o ou b idge ampli ie s in he equency ange o 10 Hz o 10 kHz.