Effects of Rotor Air-gap Eccentricity on the Power Factor of Squirrel Cage Induction Machines

Transcription

1 Effect of Roto A-gap Eccentcty on the Powe Facto of Squel Cage Inducton Machne H. Mehgn-Kelk, J. Mlmonfaed Electc Machne and Dve Laboatoy Depament of Electcal Engneeng Amkab Unvety of echnology ehan 594, Ian Phone: +98 (2) Fax: +98 (2) E-mal: Abtact- A ubtantal poon of nducton machne fault eccentcty elated. In th pape, we have tuded the powe facto vaaton of nducton machne due to the tatc and dynamc eccentcte. Accodng to th tudy, a-gap aymmety lead to nductance coeffcent vaaton and, conequently, nducton machne powe facto vaaton. Expemental eult ae alo povded to vefy the mulaton. Keywod: nducton moto, powe facto, a gap eccentcty, and magnetc equvalent ccut. Intoducton Eccentcty a common poblem n nducton machne. It well known that an nheent level of tatc eccentcty ext n nducton machne due to the manufactung and aembly method []. In geneal, thee ae two type of agap eccentcty,.e., tatc and dynamc. In both, the ax of oto not on the ax of tato. In fact, ax of otaton the ax of oto, n tatc a-gap eccentcty; yet, t not n dynamc a-gap eccentcty. hee a-gap aymmete poduce unque gnatue patten n the tato cuent. Fuhemoe, they ceate Unbalanced Magnetc Pull (UMP) that caue mechancal vbaton of the moto fame. Pevou wok done by many eeache have motly focued on the detecton of thee type of aymmete [-6]. In deed, le attenton ha been pad to the pefomance of moto and powe facto degadaton due to thee aymmete. In th pape, we tudy the vaaton of nductance coeffcent and powe facto of nducton moto due to the a-gap eccentcte. Analy baed on ou mplfed magnetc equvalent ccut appoach. Expemental eult ae alo povded. 2. Devaton of Inductance Equaton he well-known elaton between cuent and flux lnkage n an nducton machne ae wtten a the followng equaton: L + L = λ () L + L = λ (2) whee the cuent vecto, λ the flux lnkage vecto, L the matx of nductance, and ubcpt and ae agned to tato and oto, epectvely. o tudy the ubject of th pape f we need to deve appopate equaton fo machne nductance. In Fg., a pa of Magnetc Equvalent Ccut (MEC) model of an nducton machne hown. o fomulate the ytem of algebac machne equaton, nodal analy technque that atfe magnetc flux contnuty appled. he node potental equaton fo the netwok of Fg. ae gven a A u = Φt (3) A u2 + Au3 = Φt (4) A32 u2 + A33u 3 = Φ (5) A44 u4 = Φ (6) u 2 = u R t Φt + Ft (7) u 3 = u 4 R Φ + F (8) whee u, u 2,u 3, and u 4 ae the vecto of magnetc node potental n tato back on, tato teeth, oto teeth, and oto back on, epectvely. Alo Φ t and Φ ae the vecto of tato and oto tooth fluxe, R t and R ae the tato and oto tooth eluctance matce, F t and F ae the vecto of mmf ouce n tato de and oto de, epectvely. A 32 33, and A 44 ae node pemeance matce. he element of A and A 44 matce depend only on tato and oto back on egment pemeance of Fg., epectvely. In Fg., thee pemeance ae hown by G y, and G y,j. he element of A 32, and A33 depend on tato lot, oto lot and a-gap pemeance. G σ tato lot openng leakage pemeance and a contant. G σ,j oto lot leakage pemeance of lot j, and fo cloed lot, due to atuaton effect, t a nonlnea pemeance. G j the agap pemeance between tato tooth and oto tooth j. G j the mot mpoant paamete n magnetc equvalent ccut modelng. Devatve of a-gap pemeance between tato tooth and oto tooth j wth epect to oto angle when multpled by the quae of mmf dop ove the ame pemeance gve the value of electomagnetc foce between them. If thee ae n tato teeth and n oto teeth the numbe of a-gap pemeance ae (n n ). But mot of thee pemeance wll be zeo nce n any ntant of tme each oto tooth can face only a few tato teeth and vce vea.

2 he magntude of G j between tato tooth and oto tooth j dectly popoonal to the common aea and nveely popoonal to the a-gap length of tato tooth and oto tooth j when thee two face each othe. Gy - Gy Gy + u u (+) t poble to change the value of tooth eluctance by ome pecent to compenate the emoval of yoke eluctance. Flux path n oto teeth Rt Rt + Gσ u 2 Ft Ft + Gσ u 2(+) Gσ Flux path n oto back on G j G (+)(j+) Fg. 2 A typcal oto lamnaton. Gσ j- Gy j- u 3j u 4j Gσ j F j F j+ R j R j+ u 4(j+) u 3(j+) Gσ j+ Gy j Gy j+ Fg. A pa of MEC model of an nducton machne. F t and F vecto n equaton (7) and (8) ae elated to tato phae cuent and oto meh cuent though the followng equaton, F t = W (9) F = W (0) whee W geneated ung the ame appoache peented n [2] o [7]. W called magneto-motve foce tanfom matx and depend on tato wndng confguaton and numbe of tun pe col of tato wndng. W an dentty matx and t ze depend on the numbe of ndependent oto meh cuent. In an nducton machne the geometc hape of yoke pa ae uch that they have lage co ecton aea and nealy ho length wth epect to tooth pa of tato and oto. Fg. 2 how a typcal oto lamnaton of an nducton machne. heefoe, mmf dop on thee yoke pa ae geneally eveal tme malle than mmf dop on tooth pa. It hould be noted that n Magnetc Equvalent Ccut model, decton of fluxe n a tooth and n a yoke egment ae pependcula to each othe. Although yoke pa eluctance have ome effect on machne nductance, mulaton eult how that neglectng mmf dop on yoke pa ha mall effect on moto nductance coeffcent. Alo Neglectng back on eluctance n the tato lead to equalty of u element and neglectng back on eluctance n the oto lead to equalty of u4 element. On the othe hand, due to the fact that: n n Φ t = 0 and Φ = 0 () j = j= heefoe, u and u 4 ae zeo vecto and equaton (7) and (8) educe to: u 2 = RtΦt + W (2) u 3 = R Φ + (3) he eult of uch aumpton lead to emoval of equaton (3) and (6) fom the ytem of algebac equaton of magnetc equvalent ccut model of nducton machne (equaton (3) though (0)). By ubttutng (9), (0), (2), and (3) n (4) and (5), and then eaangng of paamete, we have: A W + A = ( Ιn n + ARt ) Φt + ARΦ (4) A32 W + A33 = ( Ιn n + A33R ) Φ + A32RtΦt (5) By ntoducng matce C and D a follow: C = ( Ιn n + AR t ) D = ( Ιn n + A33R ) and by fuhe mplfcaton the followng equaton wll be obtaned, C A W + C A = Φt + C ARΦ (6) D A32 W + D A33 = Φ + D A32RtΦt (7) Multplyng the two de of (6) by W and defnng the ght hand de of the eult a the ente tato flux lnkage and alo defnng the ght hand de of (7) a λ, the followng wll be obtaned: W CA W + W CA = λ (8) D A32 W + D A33 = λ (9) Compang thee equaton wth () and (2) eult n

3 L = W CAW (20) L = W CA (2) L = DA32W () L = DA 33 () Mot of element n Matce A 32 and A33 ae dependent to a-gap pemeance between each tato tooth and each oto tooth. Fo an eccentc oto, a gap pemeance depend on elatve poton of thee teeth and type of eccentcty. Ung equaton (20-), one can calculate vaou nductance coeffcent wth epect to the vaaton of a-gap length. 3. Calculaton of Inductance Coeffcent (Lnea Magnetc Chaactetc) Accodng to deved equaton, nductance coeffcent of a 3-hp thee-phae nducton moto wth the paamete gven n Appendx I have been calculated wth epect to the vaaton of a-gap length. able and Fg. 3 how the vaaton of aveage of L aa (elf-nductance of phae a) wth epect to the level of tatc and dynamc eccentcty. A lnea magnetc cuve ha been condeed. hee eult how that a the level of both type of eccentcty nceae, aveage value of thee nductance nceae that lead to powe facto mpovement of moto. able Aveage value of tato elf and mutual nductance extacted fom deved equaton SAIC ECCENRICIY DYNAMIC ECCENIRICIY Level of L aa (nh) L ab (nh) L aa (n H) L ab (n H) eccentcty(%) n H) Laa ( de e Level of eccentcty (%) Fg.3 Vaaton of (Laa) v level of a gap eccentcty (de: dynamc eccentcty, e : tatc eccentcty). 4. Powe Facto Condeaton he IEEE Standad 2 teady tate model of an nducton machne appopate fo powe facto condeaton. Fg 4 how th model. R X X 2 Z n X m R 2 / Fg 4 Steady tate model of an nducton machne Fo ou pupoe, appoxmate model of nducton machne model uffcent. By neglectng the tato mpedance and the oto leakage eactance (that moe vald epecally n no-load condton),.e. R + jx 0 (24) X 2 << ( / ) (25) nput mpedance of nducton moto obtaned a follow: j X m Z n = + jx m whee the angle of nput mpedance (26) X m φ = Zn = 90 tan (27) In teady tate condton, powe facto of nducton machne the cone of φ, pf = coφ (28) It clealy een that by nceang the value of X m, the angle of Z n deceae and th n tun lead to hghe powe facto. A hown befoe, a-gap eccentcty nceae the tato elfnductance and theefoe nceae the magntude of magnetzng nductance. heefoe hghe a-gap eccentcty lead to hghe powe facto.

4 5. Complete ytem of dffeental and algebac equaton of an nducton machne Pa. Dffeental Equaton he well-known dffeental ytem of equaton of an nducton machne dλ v = R + (29) dλ v R = + (30) whee v the voltage vecto, the cuent vecto, λ the flux lnkage vecto, R the matx of etance and ubcpt and ae agned to tato and oto, epectvely. Fo tato, v and ae a the followng: v = [ v v v ] a b [ ] c = a b c whee a, b, and c denote fo phae quantte. Stato dffeental equaton can be conveed fom phae quantte nto lne-lne quantte. h can be done by ubtacton of ow 2 fom ow and ow 3 fom ow 2 n equaton (29). hee ubtacton coepond to multplyng both de of equaton (29) by a tanfom matx M yn that ha the followng fom: M yn = 0 0 Fo oto, v and ae a the followng: [ v ]... vj vn [... ] v =... = j... n whee fo a cage nducton moto v a zeo voltage vecto, and the vecto of oto meh cuent. n the numbe of ndependent oto meh cuent. he mechancal dffeental equaton of an nducton machne ae dω e m = J (3) dθ ω = (32) n n 2 dgj e = ( u2 u ). 3j = j= dθ (33) whee e electomechancal toque of machne, m load toque, θ the mechancal angle, J the nea, ω mechancal peed, and ( dθ ) the devatve of a dg j gap pemeance wth epect to mechancal angle. Pa 2. Complete Sytem of Algebac Equaton Algebac equaton of an nducton machne n matx notaton ae a the followng M 0 M ynw AW AW A32W 0 M ynw A A A M ynw ARt M Φ ynλ M ynw AR = + ( I n A Rt RΦ ) Φ t A32Rt ( I + Φ n A R 33 ) (34) u 2 = RtΦt + W (35) u 3 = R Φ + (36) M 0 a matx that eflect the type of wndng connecton (delta, ta, wth o wthout a neutal connecton). Fo example, fo a thee-phae ta connected wthout neutal, M 0 a the followng: [ ] M 0 = 6. Smulaton (Nonlnea BH Cuve) and Powe Facto Meauement Ung the popoed method, a 3-hp moto ha been modeled and mulated. o obtan moe accuate eult, a nonlnea BH cuve ha been condeed. A healthy opeaton a well a an tatc eccentc opeaton ha been tuded. Zeo cong of the phae voltage and cuent have been ued fo the powe facto calculaton, n mulaton and expement. Aveage value of powe facto fom mulaton a well a expemental eult fo a healthy and a tatc eccentc opeaton (nealy 70%) unde no-load and full load opeaton ae hown n table 2. In pte of ome dffeence between mulaton and meaued eult, they confmed each othe. It can be een that aveage of powe facto n healthy moto n no load opeaton lowe wth epect to eccentc oto. able 2 calculated and meaued powe facto Smulaton Expement No-load Full-load No-load Full-load Healthy moto % SEC Inductance coeffcent depend on the tato and oto fluxe epecally when thee deep atuaton (efe to equaton (20)-(), nce they depend to tato and oto eluctance though matce C and D). Fg. 5 depct th vaaton obtaned fom an on-lne tang of moto n a complete mulaton. h fgue how the vaaton of L aa wth epect to the oto poton n one oto evoluton that ha been calculated n each tep of mulaton. A hown, aveage of L aa n teady tate condton (0.380 H fo 40% tatc eccentcty) vey cloe to the mla cae ( H)

5 calculated wth a lnea magnetc cuve (able ). Of coue fo deep atuaton n teady tate, educton of the aveage value of L aa hghe wth epect to ou mulaton condton. Laa (H) tme (ec) aveage=.380 (H) machne IEEE an. on Induty Applcaton, vol. 32, no. 4, pp , July/Aug [3] J. R. Cameon, W.. homon, and A. B. Dow, Vbaton and cuent montong fo detectng a-gap eccentcty n lage nducton moto, IEE Poceedng, vol. 33, pt. B, no. 3, pp , May 986. [4] S. Nand, R. Bhaadwaj, H. A. olyat, and A. G. Palo, Pefomance analy of a thee nducton moto unde mxed eccentcty condton, IEEE 998, Page -28. [5] J. M. Cadoo and E. S. Saava, Pedctng the level of a-gap eccentcty n opeatng thee phae nducton moto, by Pak vecto appoach, Poc. of the Induty Applcaton Socety Annual Meetng, 992, pp [6] J. M. Cadoo, E. S. Saava, M. L. S. Mateu, and A. L. Ramalho, On-lne detecton of a-gap eccentcty n 3-phae nducton moto, by Pak vecto appoach, Poc. of the 5 th Intenatonal Confeence on Electcal Machne and Dve, 99, pp [7] V. Otovc, Dynamc of atuated electc machne, Spnge Velag, New Yok, 989. Fg. 5 Vaaton of (Laa) fo 40% tatc eccentcty afte on-lne tang 7. Concluon In th pape the equaton decbng moto nductance coeffcent ncludng mot mpoant featue of nducton moto have been deved. Accodng to thee equaton, vaaton of moto nductance and theefoe moto powe facto wth epect to the level a-gap eccentcty ha been tuded. Ung the popoed method an nducton moto ha been mulated. Vaaton of tato elfnductance n an on-lne ta up opeaton ncludng atuaton effect ha been epeented. o vefy the theoetcal apect expemental eult have been povded. Appendx I Machne Paamete hee hp, 460/0V, 4 pole Roto length = 2 n Stato lot openng: 0.2 n Inne tato damete = n; A-gap length = 0.03 n Numbe of tato lot, n = 36; Numbe of oto lot, n = 44 Stato wndng confguaton: ngle laye concentated wndng and numbe of col pe lot N=54 Stato etance: R = 3.7 Ω/phae Roto ba etance: Rb =50.06 µω Roto end ng etance: Re = µω Inea: 0.03 fo no-load condton Refeence [] D. G. Doell, W.. homon, and S. Roach, Analy of a-gap flux, cuent, and vbaton gnal a a functon of combnaton of tatc and dynamc a-gap eccentcty n 3-phae nducton moto IEEE an. on Induty Applcaton, vol. 33, no., Jan./Feb [2] H. A. olyat, M. S. Aefeen, and A. G. Palo, A method fo dynamc mulaton of a-gap eccentcty n nducton