On the Uncertainty of Sensors Based on Magnetic Effects. E. Hristoforou, E. Kayafas, A. Ktena, DM Kepaptsoglou
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1 On the Uncertinty of Sensors Bsed on Mgnetic Effects E. ristoforou, E. Kyfs, A. Kten, DM Kepptsoglou Ntionl Technicl University of Athens, Zogrfou Cmpus, Athens 1578, Greece Tel: , Fx: , e-mil: Abstrct- In this pper we illustrte tht mgnetic hysteresis ffects the uncertinty of sensors bsed on mgnetic phenomen nd mterils. Testing vrious mgnetic techniques, we conclude tht sensor uncertinty cn be optimized by both, the optimiztion of mgnetic hysteresis nd the opertion in the mgneticlly reversible re of the mteril. Preliminry modelling results re in greement with our experimentl findings. I. Introduction Mgnetic sensors re importnt in mny fields of engineering [1]. They cn be minly used in the mesurement of physicl sizes, such s position, stress nd field []. Their chrcteristics nd rnge of mesurement mke them ttrctive for pplictions where other kinds of sensors cnnot operte. They cn be mnufctured using hybrid or micromechnicl techniques nd cn be esily pckged in miniturized set-ups. Mny of them hve the intrinsic property of self-clibrtion, llowing ccurte nd repetble mesurements. The most significnt property of sensor is its uncertinty of mesurement, which is determined s the totl devition of the mesured vlue from the ner true vlue of mesurement. The effort of ll R&D groups working in the field of sensor development is minly trgeted towrds the optimiztion of such uncertinty. The optimiztion of the mgnetic sensors response with respect to hysteresis hs been our motivtion for this work. In this pper we briefly present our theory on the sensor uncertinty optimiztion, bsed on the optimiztion of the mgnetic hysteresis of the used mteril. We pply this theory to the clibrtion of vrious kinds of mgnetic sensors, proving tht the control of mgnetic hysteresis is essentil for the sensor uncertinty. The principle of opertion of mgnetic sensors lies in the dependence on the pplied field of the mgnetic flux density B, the mgnetostriction λ, nd the mgneto-impednce Z, yielding the B- loop, the λ- loop nd the Z- loop respectively. The Z- loop response corresponds to the dc nd c resistnce response, i.e. the MR nd MI response. Considering sensor bsed on the B- loop response, e.g. liner vrible differentil trnsformer (LVDT), two prmeters re of importnce: the dc bising point nd the c periodic excittion. Usully, such sensor excittion is fixed in mplitude, while chnge of the dc bising point results in chnge of the sensor response, relized s dynmic chnge of mgnetiztion with respect to time. When operting in the irreversible re of mgnetiztion, such dynmic chnge of mgnetiztion is lrger but hysteretic. In the reversible re of the B- loop, hysteresis is prcticlly bsent nd sensitivity is considerbly decresed. The sme response cn be observed in sensors bsed on the λ- loop, s well s in sensors bsed on the Z- loop, like the mgneto-inductive sensors. From the description of these three min mgnetic effects, it cn be seen tht mgnetic hysteresis determines the hysteresis of sensor. The sensor uncertinty is function of both the sensor sensitivity nd the sensor hysteresis. igher sensitivity reduces the uncertinty. owever, both our theoreticl nd experimentl results gree tht higher sensitivity is chieved when operting in the hysteretic prt of the curves ginst operting in the reversible nhysteretic region but it is lso there tht the uncertinty is higher due to hysteresis. ence, good sensor design should tke into ccount the hysteresis response of the mteril nd either use it or void it. II. Experimentl We hve developed severl kinds of sensors bsed on mgnetic mterils, llowing for ll three kinds of mgnetic responses presented in the previous section. For illustrtion purposes we shll provide the results of three kinds of sensors, using correspondingly the three bove-mentioned different types of
2 mgnetic response. Ech type of sensor ws bsed on three different mgnetic mterils, performing t different mgnetic hysteresis levels. The first type of sensor ws n LVDT [3], performing displcement mesurements. The cores used to relize this sensor were polycrystlline Fe nd Ni wires s well s morphous field nneled FeSiB wires. The sensor operted in both the irreversible nd reversible re of the B- loop, resulting in the response shown in Fig. 1, illustrting results of Fe wires. From these results it cn be seen tht the sensor hysteresis is negligible for the cse of the reversible re of opertion of the mteril. In the irreversible re the signl is significntly enhnced but so is the hysteresis. For exmple, t 6mm displcement the signl is 1mV in the cse of the reversible re nd 1mV or 5mV in the irreversible. So, the signl is doubled but exhibits ±7% devition. It is noted tht the hysteresis of the sensor using FeSiB wires ws negligible in both cses of reversible nd irreversible opertion. The second type of sensor ws mgnetostrictive dely line (MDL) sensor [4], bsed on the λ- loop, performing stress mesurements. The sme cores were used to relize this sensor, lso operting in the reversible nd irreversible res of the λ- loop. The response of the Fe nd FeSiB wire, operting in the irreversible nd hysteretic region of the λ- loop, is shown in Fig.. It cn be seen tht hysteresis is smller, lmost negligible in the cse of the FeSiB wire. ysteresis in the sensor using Fe wire contributes ±1% uncertinty. The third type of sensor ws mgneto-inductive field sensor [5], lso using Fe nd morphous FeSiB wires. The response of the Fe nd FeSiB wire, operting in the irreversible nd hysteretic region of the Z- loop, is shown in Fig. 3. In this cse lso it is cler tht hysteresis in the sensor using FeSiB is negligible. We note here, tht in the cse of the FeSiB mgneto-inductive sensor the signl ws mplified by 5 nd the offset ws deducted. Following the bove experimentl results we hve developed mcroscopic mthemticl model of hysteresis describing the sensor sensitivity dependence on hysteresis. Vo (mv) 4 3 Fe (irreversible) Fe(reversible) Displcement (mm) Vo (x1 mv) Fe FeSiB Tensile Stress (MP) Fig. 1. ysteretic response of LVDT, using Fe sensing cores, operting in the reversible nd irreversible re of mgnetiztion. Fig.. Tensile stress hysteretic dependence of MDLs under tensile stress. 3 1<<3 Voltge Output (Volts),5 1,5 1 Fe wire FeSiB wire B,λ 1 3 λ(η),5 B() Mesurble Field (A/cm) Fig. 3. Response of mgneto-inductive sensors operting in the irreversible re. Fig 4. Clculted M- nd λ- curves for 1<<3.
3 III. The model The phenomenologicl pproch to the modeling of the M-, λ- nd Z- curves is bsed on the x gussin probbility density function, f ( x) e. The M- curve cn be modeled by the integrl of f(x) while the other two curves by f(x) itself. To void numericl difficulties rising from the clcultion of the integrl of f(x) we use the "sigm" probbility density function insted: 1 σ ( x) = < x < + x 1+ e, (1) Its sigmoidl shpe is pproprite for the modeling of the M- loop: ± c M( ) = σ 1 = 1 () ± c 1+ e where is the pplied field, c is the coercive field defined s M( c ) =, nd is field relted to the coercivity squreness of the loop. The plus or minus sign indictes the scending or descending brnch, respectively. Using the ssumption tht the minor loops of given M- loop belong to the sme fmily of curves, eq. () cn be prmeterized nd used to reproduce higher order curves s well. We hve determined tht ± c λ( ) = λ s 1 e which cn be pproximted by the first derivtive of the sigm function: 1 ± c ± c λ( ) = σ σ 1 (3) Accordingly, the Z- curve is given by the following Gussin eqution: ( ) ± c mxe Z = Z which cn be pproximted by: ± c ± c Z( ) = σ 1 σ (4) Fig. 4 shows M- nd λ- loops produced by () nd (3) for different -vlues. Fig. 5 shows set of minor scending curves for given M- curve. Similr minor loops cn be obtined for the other two curves s well. The response of the sensors bsed on these three different loops my be clculted s db V, for the LVDT (5) dt d λ V, for the MDL dt nd (6) V, for the MI sensor (7) where V is the induced voltge nd B the mgnetic induction, B = μ ( + M). (8) Combining () nd (8), for sinusoidl excittion () t = dc + c sin ωt (9) eq. (5) yields the response of the LVDT: db μ c = 1 σ() [ σ() 1] dt ω (1) where: dc + c sin ωt ± c σ() = σ (11) We note tht the induced voltge is proportionl to the mgnitude of the c excittion nd inversely proportionl to the frequency ω=πf. Depending on the region of opertion on the M- loop, the following three cses rise: Cse I: dc >> c nd c << c.
4 This cse corresponds to opertion in the reversible prt of the curve, ner sturtion nd eq. (1) db μ yields, c. dt ω The induced voltge is pprently smll since c-field is smll. Cse II: dc c nd c < c +. This cse corresponds to opertion in the highly hysteretic prt of the M- curve. The condition < ensures tht the opertion is limited to the pproximtely liner prt of the curve nd c c + sturtion is voided. Becuse mx{ ( σ 1) } =. 5 db μ mx = dt ω c.5 1 +, σ,.5 i.e. the sensitivity of the sensor is incresed by fctor of. The dependence on prmeter is twofold: smll suggests steep loops nd therefore dicttes the ppliction of smll c while, on the. 5 other hnd, the fctor is incresed. In ny cse, the sensitivity is higher thn in Cse I. We lso note tht the sinωt dependence of σ( ), becuse of the prominent hysteresis effect, contributes to the uncertinty. Cse III: c >> c. This cse corresponds to opertion between positive nd negtive sturtion nd db μ c. dt ω The result is the sme s in Cse I but c is now much lrger nd hence the response is enhnced. The sme nlysis hs been crried out for the other two types of sensors with similr results. The response of the MDL sensor s given by (6) is illustrted in Fig. 6 where the first nd second time derivtive of λ ( ) is pproximted by the first nd second derivtive of (3), respectively. The bove nlysis grees with the experimentl results presented in the previous section. λ'(η) Μ Fig 6. Clculted first nd second time derivtives of λ(). λ(η) Fig 5. Clculted M- loop with set of minor scending curves. Fig. 6. The first nd second time derivtive of λ ( )
5 13 th Workshop on ADC Modelling nd Testing Sep. -4, 8, Florence, Itly References [1] R. Boll nd L. Borek, Mgnetic Sensors of New Mterils, Siemens Forch.-u. Entwickl.-Ber.Bd. 1, p. 83-9, [] Ε.Ο. Doebelin, Mesurement Systems: pplictons nd design. Fourth Edition. McCrw-ill, 199. [3]. Chiric, E. ristoforou, M. Negu, M. Pertnrin, F.J. Cstno, J. Appl. Phys., 87, p ,. [4] E. ristoforou,. Chiric, M. Negu nd I. Drie, Sensors nd Actutors A, 67, p , [5] K. Mohri, K. Ksi, T. Kondo,. Fujiwr, nd M. Msumoto, IEEE Trns. Mgn., 17, p
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