Conversion from geometrical to electrical model of LVDT

Transcription

1 6 t IMKO TC4 Symum xlrng New Frnter f Intrumentatn and Metd fr lectrcal and lectrnc Meaurement Set. -4, 008, Flrence, Italy Cnvern frm gemetrcal t electrcal mdel f VDT Clenln Prtá de Suza, Mcel Brun Wanderley Federal Center f Tecnlgcal ducatn f Maranã-Brazl (CFT-MA) Av. Getúl Varga, 04 Mnte Catel Sã uí-ma, Brazl, , rta@cefet-ma.br Reearc Grant urted by Reearc Surt Fundatn f Maranã-Brazl (FAPMA), Av.Bera-Mar, 4, Centr Sã uí-ma, Brazl, , mbwander@gmal.cm Abtract- Te near Varable Dfferental Tranfrmer (VDT) an nductve enr wc ued t meaure lnear dlacement and fnd ue n mdern macne-tl, rbtc, avnc, and cmuterzed manufacturng. It bac tructure cnt f a rmary cl and tw ecndary cl lke an electrcal tranfrmer. Hwever, VDT a a mvable magnetc cre tat wen te rmary cl excted wt an AC vltage urce, nduced ecndary vltage vary wt te dlacement f te cre. In general, t accurate and relable dlacement-t-electrcal enr can be mdeled n tw frm: gemetrcal-arameter-baed mdel and electrcal-arameter-baed mdel. Bt are very ued. Hwever, reearc reult baed n gemetrcal-baed mdel may becme uele wen nly arameter frm electrcal ne are knwn. In t aer, t wn a way f cnvern frm gemetrcal- t electrcal-baed mdel n rder t allw te nteractn frm ne t ter. I. Intrductn Te near Varable Tranfrmer Dfferental (VDT) ued t meaure lnear dlacement r tn. VDT can be ued n everal alcatn a: autmble uenn [], emc ck meaurement [], rtdnta [], femral rte n medcne [4], defrmatn n cncrete frame n cvl engneerng [5], tn ytem f rbtc arm [6], bede ter ycal meaurement. Te bac tructure f VDT cnt f a rmary cl and tw ecndary cl lke an electrcal tranfrmer. Hwever, VDT a a mvable magnetc cre tat wen dlcated t vare te mutual nductance amng ecndary cl and te rmary cl. In eratn, t aled an AC vltage, n te rmary cl, a wn n Fgure. Te ecndary cl are cnnected n ere wt ng ae, wen te magnetc cre exactly at te ycal center between tem, ten te utut vltage,, zer, a te net utut te dfference between te tw ecndary vltage,. If te cre mved frm te center, t btaned an utut vltage tat rrtnal t te cre dlacement, becaue tere a mbalance f magnetc flux ntente amng te rmary cl and te ecndary cl and ne ecndary cl receve mre flux tan te ter. T mbalance rduce nn-zer utut vltage rrtnal t te cre dlacement [7]. On te ter and, matematcal mdel are very mrtant tl fr engneerng and fr revu develment. Fr VDT, te determnatn f a mdel and te etmatn f t arameter an mrtant way t devel new alcatn ung t enr. Al, wen t cmmercalzed tere a deendence f te manufacturer wt regard t te calbratn and te recn f te enr. Fgure. VDT: (a) Internal Scematc. (b) Internal Mdel.

2 6 t IMKO TC4 Symum xlrng New Frnter f Intrumentatn and Metd fr lectrcal and lectrnc Meaurement Set. -4, 008, Flrence, Italy In general, VDT mdeled n tw way: electrcal-arameter-baed mdel (P) r gemetrcalarameter-baed mdel (GP). P mdel ue electrcal arameter a retance and nductance, and GP mdel ue gemetrcal arameter baed n ycal dmenn f te VDT. It ntced tat, me centfc wrk ue P mdel and ter ued GP ne. Tu, reearc reult baed n a certan mdel can becme uele wen nly arameter frm anter mdel are knwn. T cnder te reult f a mdel avng arameter frm ter mdel, t neceary t ave tl t ermute t arameter. In t way, t reented a way f cnvern frm GP mdel t P mdel n rder t allw te nteractn between t mdel and ue reult frm ne t ter. II. lectrcal-parameter-baed Mdel f VDT Frm Fgure, an P mdel f te VDT can be btaned a fllw, cnderng te fllwng arameter. : ttal retance f te rmary cl, and R : retance f te ecndary cl, P : elf-nductance f te rmary, and : elf nductance f te ecndary, and M and M : mutual nductance between rmary cl and te ecndary cl [8] [9] [0]. Cnderng R a te ttal retance f te ecndary crcut, t btaned R R + R + R were R a lad retance nt wng. Te nduced vltage n te rmary cl gven by: R I + I + M I M I were M, M are deendent f te cre dlacement and, te exctatn vltage. Te net equatn n te ecndary crcut gven by: I M I + I R + I I 0 M + () () Fgure. VDT lectrcal Scematc. Frm (), t btaned M I + M I I () R + and frm () I R + + I (4) M M Cmarng () and (4) and after me algebrac manulatn, t btaned were I X M M (5) X + Y + R R + M M (6) Y R + R (7) + Te utut vltage,, gven by I R (8)

3 6 t IMKO TC4 Symum xlrng New Frnter f Intrumentatn and Metd fr lectrcal and lectrnc Meaurement Set. -4, 008, Flrence, Italy Relacng (5) n ( 8), t btaned M M R X + Y + R R (9) T a ecnd-rder ytem wc ndcate tat te ae angle vare frm +90º n lw frequence t 90º n g frequence. Wen te cre n te ycal center, M M, and accrdng t (9), O 0, a exected. III. Gemetrcal-Parameter-Baed Mdel f VDT A general tructure tat w te gemetrcal arameter (GP) f VDT wn n Fgure (a) were: te rmary cl lengt; te ecndary cl lengt; D te external dameter f te cylnder frmed by te cl; d te cre dameter; t te cre lengt tat nerted n te ecndary cl ; t te cre lengt tat nerted n te ecndary cl ; δ e δ are te arga amng ecndary cl and te cre; and R te flux effectve radu. (a) (b) Fgure. (a) VDT Gemetrcal Scematc. (b) VDT equvalent magnetc crcut. In [], t wa ued magnetc crcut tery t devel a mat mdel tat nvlve nternal arameter lke: number f cl turn, ycal dmenn f te cl and materal ermeablty f te cre. T reearc a nvlved exermental rcedure ung knwn VDT dmenn and cmarng btaned teretcal calculatn f utut vltage and meaured value. Ung tee dmenn, t ble t calculate elf-nductance and mutual nductance between rmary cl and te ecndary cl. Te equvalent magnetc crcut fr an VDT llutrated n Fgure (b) were MMF te magnetmtve frce and G 0, G, G m, G 0, G, m are magnetc cnductance related t t reectve magnetc fluxe: φ, φ, φ m, φ, φ, φ m were φ te man flux wc ge t ecndary ; φ te ecndary deflux; φ m te man deflux wc cre te ecndary ; φ te man flux wc ge t ecndary ; φ te ecndary deflux; and φ m te man deflux wc cre te ecndary. Wen δ r δ vare lgtly due t a cre dlacement, R vare, and f te dlacement mall, ten G 0 and G 0 are nt cnderedd (becaue tey are deendent f δ and R). Al, te flux φ m de nt cr te ecndary cl and t nt able t rduce mutual nductance. In t way, G m and Gm are nt cndered a well []. S, nly G and G are cndered, wc are te magnetc cnductance cred by te flux φ, and tey can be calculated, accrdng t [], by G δ g t g (0) G δ g t g () were g te ecfc magnetc cnductance between cylndrcal urface and gven by µ 0 π/ln D/d [].

4 6 t IMKO TC4 Symum xlrng New Frnter f Intrumentatn and Metd fr lectrcal and lectrnc Meaurement Set. -4, 008, Flrence, Italy After cmute te magnetc cnductance, te elf nductance f te rmary cl P gven by N G () were N te number f turn f te rmary cl and G te ttal magnetc cnductance tat gven by G G //G () Te mutual nductance M and M, cred by te flux φ m, amng te rmary cl and te ecndary cl, can be calculated by M N N gt / 4 (4) N N gt / 4 (5) M were N te number f turn f te ecndary cl. Accrdng t [0], ( M M ). S, relacng (), (4) and (5) n t, ten gven by N gt 0 Δt (6) N G IV. Gemetrcal- t lectrcal-parameter-baed Mdel Cnvern A relatn between electrcal and gemetrcal arameter n a drect way reented a fllw. Cnderng a deal te retance f rmary and ecndary cl,.e. R P R S 0, n frequency dman (relacng by w,), frm (9) t btaned R w( + ) + R S (7) were +. Te magntude f (7) gven by 4 R ( w ( + + ) + R (8) reultng n R (9) 4 R w w 4 Dentng Z + +, t btaned (0) R w ( Z + ) R w and () wz + R

5 6 t IMKO TC4 Symum xlrng New Frnter f Intrumentatn and Metd fr lectrcal and lectrnc Meaurement Set. -4, 008, Flrence, Italy In rder t btan a relatn ndeendent f te lad retance, t dne R, ten wz / R +, ten () M M Hence, relacng (), (4) and (5) n (), t btaned A P N P G, N N ( t t ) () 4 ) Ng( t t (4) 4 N G Due t te VDT ymmetrcal tructure: t t 0 + Δt e t t 0 Δt, were t 0 te lengt f te magnetc cre nerted n te ecndary cl, wen te cre ut at te VDT center, and Δt t dlacement. In t way, t btaned wc te ame a (6). Ngt 0 Dentng k, ten N G Ng [ t 0 + t 0Δt + Δt t 0 + t 0Δt Δt ] 4N G Ng Ngt 0Δt ( 4t 0Δt) 4 N G N G KΔt KΔt CΔt (5) were C K and a cndered cntant, t w te lnearty f te tranducer. Cnderng at balance, Δt 0, and knwng tat M and M tat cme frm tery f culed magnetc crcut between rmary cl and te ecndary cl, t btaned and M (6) Relacng () and (4) n (6), t btaned In te ame way, relacng () and (5) n (7), t btaned M (7) N g t + t t (8) 6t N g t + t t (9) 6t

6 6 t IMKO TC4 Symum xlrng New Frnter f Intrumentatn and Metd fr lectrcal and lectrnc Meaurement Set. -4, 008, Flrence, Italy V. Reult A a reult, accrdng t (), (4), (5), (8) and (9), t ble t cmute te electrcal arameter frm gemetrcal ne by ung te cnvern frmula gven n Table. Prmary Inductance Mutual Inductance Mutual Inductance N N gt N N gt N G M M 4 4 Secndary Inductance Secndary Inductance N g t + t t N g t + t t 6t 6t Table. Cnvern frmula frm GP t P. xamle: cnderng a real VDT wt gemetrcal arameter gven by d mm, D 9mm, 4mm, t t mm (cre n te ntal tn) e N P N S 000. Calculatng g µ 0 π/ln D/d and G, ee q., t cmuted te fllwng electrcal arameter P M M S S 7,mH,8mH,8mH 0,mH 0,mH VI. Cnclun In t wrk wa decrbed te mdel baed n electrcal arameter (P) and n gemetrcal arameter f VDT. It wa reented a way f cnvern frm GP mdel t P mdel n rder t allw te nteractn between bt mdel and ue reult frm ne t ter. Ung te cnvern frmula, reearc reult baed n a certan mdel can be ued wen nly arameter frm anter mdel are knwn. Reference [] Rajaman, R.; Hedrck, J. Adatve berver fr actve autmtve uenn: tery and exerment. I Tranactn n Cntrl Sytem Tecnlgy, ():869, marc 995. [] Salaaka, S. et al. A elf cmenated mart VDT tranducer. Prceedng f Amercan Cntrl Cnference, (8-0):96697, May 00. [] Büler D, Oxland T. R., Nlte. P. Degn and evlutn f a devce fr meaurng treedmennal mcrmtn f a re-ft femral tem rtee. Medcal ngneerng Pyc, 9():87{99, 997. [4] Delng, R.; Dugla, W. An artfcal ral envrnment fr tetng dental materal. I Tranactn n Bmedcal ngneerng, 8(4):9-45, Arl 99. [5] H, C.; Pam, H. Inelatc degn f lw-axally laded g-trengt renfrced cncrete clumn. Jurnal f ngneerng Structure, 5(8):08096, 00. [6] Gwam, A. Q. A.; Pekn, M. Identfyng rbt arameter ung artal e nfrmatn. Cntrl Sytem Magazne, (5):6(4), Octber 99. [7] Cta Junr, A. G. Medçã de delcament utlzand um Tranfrmadr Dferencal Varável near baeada em técnca de etmaçã. Mater Te frm Federal Unverty f Camna Grande, Brazl, 005. [8] Debeln,. O. Meaurement Sytem: Alcatn and Degn. McGraw-Hll, New Yrk, 4 edtn, 990. [9] Antnell, K. et al. Dlacement Meaurement, near and Angular. Te Meaurement, Intrumentatn and Senr Handbk. CRC Pre, Bca Ratn, 999. [0] Palla-Areny, R.; Webter, J. Senr and Sgnal Cndtnng. Wley InterScence, New Yrk, 00. [] Tan, G. et al. Cmutatnal algrtm fr near Varable Dfferental Tranfrmer (VDT). I Prceedng Scence, Meaurement & Tecnlgy, 44(4):89{9, July 997. [] Neubert, H. K. P. Intrument Tranducer. Oxfrd Unverty, ndn, 975.