ANALYSIS OF A THREE-PHASE INDUCTION MACHINE INCLUDING TIME AND SPACE HARMONIC EFFECTS: THE A, B, C REFERENCE FRAME.

ANALYSIS OF A THREE-PHASE INDUCTION MACHINE INCLUDING TIME AND SPACE HARMONIC EFFECTS: THE A, B, C REFERENCE FRAME. L.M. Neto, J.R. Camaco, C.H. Salerno (Non-Memer) (Memer) (Non-Memer) Unversdade Federal de Uerlânda Electrcal Engneerng Department P.O.Box; 593 - e.mal: rcamaco@ufu.r 38400.90 - UBERLÂNDIA - MG - Brazl Keywords: armoncs, nducton motor, tree-pase model Astract - Ts paper presents a matematcal modelng tat uses te concept of armonc nductances to derve voltage and torque equatons of a tree-pase nducton macne usng a,, c and A, B, C formulaton. Pase currents and electromagnetc torque are otaned y numercal ntegraton of equatons consderng any supply voltage waveform. Two cases were studed - operaton of nducton motor suppled wt snusodal waveform, and operaton of an nducton motor suppled y a tree-pase nverter. I. INTRODUCTION Some ndustral applcatons requre speed control of nducton macnes, ts can e done troug electronc converters. A converter-fed electrcal macne operates wt a g armonc content of voltage and current. Te macne eavor wen appled a non-snusodal AC source s qute dfferent wen applyng a snusodal one. Ts queston sould e andled carefully wen defnng desgn plosopes of nducton macnes and n te development of electronc converters. Inducton macnes wen fed y snusodal currents sow dstortons n te electromagnetc torque, tese dstortons ave t s orgns from te armonc content of magnetc flux densty spatal dstruton produced y ts cols along te argap. Harmonc dstortons produce some effects n te macne operaton, as for example, torque oscllatons wt tme and space armonc components. As B.P. Alvarenga (Non-Memer) Unversdade Federal de Goás Electrcal Engneerng Department GOIÂNIA - GO - Brazl a consequence t s common practce te manpulaton of col ptc and dstruton factors n te macne desgn. Te oectve of ts practce s to mnmze suc dstortons. In te case of nducton macnes fed y nonsnusodal currents, te armonc content n tme sow an nteracton wt te magnetc feld space armoncs content estalsed n te macne produced y col dstruton. From ts nteracton, t s possle to otan comned dstortons very dfferent from tat resultng from a snusodal AC source. An mportant dfference s te fact tat an amount of suc dstortons produce desrale effects n te macne, suc as constant torque components. However, t s not possle yet to pont precsely wc armoncs nteract wt eac oter n order to produce te desrale effects. But ts fact canges te desgner s practce wen workng wt dstruton and col ptc factors. Te queston moves from dstorton mnmzaton to torque optmzaton. In ts way, a model tat allows a more precse evaluaton of nducton macnes electromagnetc torque ecome an mportant matematcal tool for te tree-pase nducton macne desgner. Ts paper presents a matematcal a,, c (stator) and A, B, C (rotor) modelng tat allows te study of tme and space armoncs n nducton and oter macnes. Ts model s made ascally y ordnary voltage and current electrcal equatons tat sows te relatonsps of torque, current and rotatonal speed. Cases of feedng a tree-pase nducton macne from snusodal and non-snusodal AC sources were smulated to empasze dfferences n te macne eavor. Teoretcal and practcal results are compared to valdate te model. II. MODEL DEVELOPMENT In order to otan voltage aganst current electrcal equatons for te nducton macne, frstly sould e otaned te spatal dstruton of magnetomotve force for only one col. In sequence s otaned te spatal dstruton for a generc pase of a wndng. Usng ts dstruton formulaton t s possle to get te armonc components of magnetc flux lnkage etween generc

pases and. Tese armoncs of flux lnkage create te armonc nductances. Self nductances are otaned for and mutual nductances for. Matematcal equatons need te nstantaneous expresson of electromagnetc torque as functon of currents. Ts equaton can e otaned from te varaton of magnetc co-energy. A. Spatal Dstruton of Magnetomotve Force (MMF) Consderng ntally only one col (N turns) of a wndng for a generc pase. Te dstruton of MMF troug te macne ar-gap (δ) can e descred y equaton (1) wen sumtted to a current. 1 MMF( ). N... sn(.. ).cos[ ( )] (3) For any wndng of pase, te armonc component of te spatal dstruton of MMF n te ar-gap (MMF ) can e otaned y superposton of space armoncs wt te same order from te pase cols. Tus, can e otaned equaton (3) from equaton (), consderng te superposton of armonc components from q cols can e otaned te MMF dstruton for pase to te armonc order as eng: MMF q J J 1 1 N.. sn (.. ) cos[ ( )] (4) N.. > < N... MMF ( ) > < < + N... > + < + (1) were q s te numer of dstruted cols n pase were s flowng current. Fgure - Col dstruton n a pase wndng. Consderng te centre for pase located at an angle as sown n Fgure, te summaton n equaton () can e solved and fnally equaton (5) s otaned. MMF N J. q J. k p. k d. 1...cos[ ( )] (5) Fgure 1 - (a) Generc col n a wndng and () ts MMF dstruton. A decomposton of MMF () n ts Fourer seres gave us te followng expresson: 1 MMF N sn ( ) J J.. (.. ).cos[ ( )] () 1( odd) were: - represents genercally an nducton motor pase, - odd armonc ndex, k p - col ptc factor for te t armonc, - angular poston wc locates any pont along te crcumference of te ar-gap from a fxed reference; - value for n troug te centre of col. Eac of te q cols s te orgn of an order armonc component of MMF wc can e extracted from equaton () as sown n equaton (3). were k p and k d are te pase step and dstruton factors respectvely to te t armonc. B. Harmonc Inductance Magnetc feld densty dstruton B produced y MMF () s otaned troug te applcaton of Ampére law n te ar gap only, n our case te magnetc crcut reluctance n te ron parts s neglected. So we ave: B µ 0. MMF ( ) (6) δ were µ 0 s te ar magnetc permealty and δ s te ar gap lengt. Susttutng (5) n (6), we ave:

µ 0 1 B ( ). N. q. k p. k d...cos[ ( )] (7) δ To otan mutual flux etween pases and ntally s necessary to ave te magnetc flux of pase wc emraces pase. Wen we ave te pase flux and wen we ave te mutual flux for eac value of. Space dstruton B () sown n equaton (7) gves orgn to a component of armonc order of te mutual flux n an order col for te pase wndng gven y: +. p. N. B. L. r. d (8) were: p - numer of par of poles for te stator wndngs, B - armonc component of te spatal dstruton of magnetc feld densty produced y pase, N - numer of turns for col, L - magnetc lengt for te rotor, r - average radus for te argap δ. Consderng one pase producng a spatal dstruton of magnetc feld densty wc s lnked wt one col of pase. A component of armonc order for ts flux lnkage can e computed usng equaton (8). Also takng n consderaton pase made y a dstruton q of cols, te flux lnkage etween pases and can e otaned y equaton (9). a q q p. N. B L R d... 1 1 a Integraton lmts e are otaned from te magnetc nducton waveform at te argap, were for every order col we ave: R ( q + 1). (10) and R ( q + 1). + (11) Takng nto consderaton tat MMF s te parameter responsle for te magnetc feld dstruton B on te macne s ar-gap, equaton (9) can e wrtten n a dfferent way. From equatons (5), (8) and (9): k. k w. k w..cos[ ( )] (1) were: (9) and:. p. L. R.. Q. Q. N. N k 8 µ 0. δ k w k p. k d - pase col ptc factor, k w k p. k d - pase col ptc factor, - angular poston of pase ax. Te t order armonc nductance etween two pases and s gven y equaton (9). ζ (13) Applcaton of equaton (1) n equaton (13) gve: ζ L..cos[ ( )] (14) were: k k wk w L or n anoter form: ζ k L 1.cos[ ( )] (15) were k can e te normalzed wndng factor wt respect to te fundamental frequency. D. Electrcal Equatons Instantaneous values for voltage n one pase for te nducton macne can e gven y equaton (16). d v R + (16) were R and are respectvely te resstance and total magnetc flux lnkage for pase. Te total magnetc flux lnkage s otaned accordng to equaton (17). L + L (17) Consderng te leakage nductance n pase (L ) constant, equatons (16) and (17) are te orgn for equaton (18) elow. v R L d d d ζ + + [ ζ + ] (18) E. Electromagnetc torque equaton Te nducton macne electromagnetc torque can e otaned y te magnetc co-energy varaton of te system (W c ) relatve to te electrcal dsplacement of te rotor (θ R ). T p W c θ (19) R I const Te magnetc co-energy s related wt nductance and currents troug equaton (0):

W 1 c ζ (0) and a comnaton of equatons (19) and (0) gve te electromagnetc torque equaton as: T ( p ) 4 e dζ dθ R (1) and θ R s te rotor electrcal angular poston relatve to a stator fxed reference. [L] s te nductance matrx were te elements for te man dagonal are gven y L + Σ ζ for a,, c, A, B and C. Oter elements n te nductance matrx are gven y Σ ζ for a,, c, A, B and C wt. Fgure 3 - Teoretcal pase voltage at te varale frequency nverter. Fgure 5 - Teoretcal result, pase current at te varale frequency nverter(6n±1 armoncs, n9). Fgure 4 - Pase voltage at te varale frequency nverter. F. Dynamc Equatons System for a Tree-Pase and Symmetrcal Inducton Macne Takng a, and c as stator pases and A, B and C as rotor pases (true or equvalent on te squrrel cage rotor case) for a tree-pase and symmetrcal nducton macne. Wt and assumng a,, c, A, B, C from equaton (18) te matrx equaton (19) can e otaned. Applyng te Newton Law n te macne axs te speed varaton n tme s gven y equaton (3). d[ I ] 1 [ L] ([ V ] [ D][ I]) () dwr p ( J T T ) (3) L dθ R wr (4) were [I] and [V] are respectvely column vectors of currents and voltages of pases a,, c, and A, B and C; J s te moment of nerta for te rotatve parts; T s te electromagnetc torque otaned from equaton (1) and T L s te load torque; w R s te rotor electrcal angular speed Fgure 6 - Practcal result, pase current at te varale frequency nverter(6n±1 armoncs, n9). Matrx [D] n equaton () s formed y te followng matematcal manpulaton: d[ L] [ D] + [ R] (5) were [R] s te resstance dagonal matrx n eac stator and rotor pases. Fgure 7 - Teoretcal result, pase current at te varale frequency nverter(6n±1 armoncs, n0). Formaton of matrx [L] requres te calculaton of armonc nductances from equaton (15). Te reference used n ts case s te axs for pase a, values of - are calculated from te resultng angles: a 0; /3; c -/3; A θ R ; B θ R +/3 and C θ R -/3.

Wt te knowledge of macne parameters, load caracterstcs, source voltages (V a, V, V c ) and te fact tat VA VB VC 0 t wll e easer to solve numercally te dfferental equatons system gven y equatons (), (3) and (4). A dgtal computer program accomplses ts task smulatng a tree-pase nducton macne. wen tme and space armoncs are not taken nto consderaton. III. INDUCTION MOTOR DATA AND LOAD CHARACTERISTIC Te tree-pase nducton motor used for ts smulaton n te expermental tests presented te followng caracterstcs: squrrel cage type, two pars of poles, 60 Hz, 3 HP, 0/380 Volts, -Y connecton and rotatng magnetc feld at nomnal speed of 1800 rpm. Fgure 9 - Torque wt snusodal source consderng up to te 55 t armonc order (6n±1 armoncs, n9). Troug tests te followng parameters were otaned: magnetzng nductance of 66.0 mh, stator and rotor leakage nductances represented n te stator were ot equal 3.94 mh, stator pase resstance equal 6.8 Ω and te rotor pase resstance represented n te stator s 4.845 Ω. Fgure 10 - Torque wt nverter source consderng only up to te frst armonc order (6n±1 armoncs, n0). Fgure 8 - Torque wt snusodal source consderng only up to te frst armonc order (6n±1 armoncs, n0). Load n ts case as a formulaton dependng on te square of te rotatng speed as can e seen n equaton (6). T A + B. w (6) Wt te purpose of sowng etter torque oscllatons wt nteracton of tme and space armoncs of magnetc feld Fgures 8 to 11 sow te nducton macne torque n t s steady-state and startng perods, takng nto consderaton dfferent cases. Fgure 8 assumes a alanced tree-pase snusodal source at 60 Hz, only wt te 1 st spatal armonc consdered. Fgure 9 as te same source as n Fgure 8, ut wt armonc content up to 55 t order. In Fgures 10 and 11 our macne as a varale frequency nverter as a source, at 60 Hz, consderng respectvely up to te 1 st and up to te 55 t spatal and tme armoncs. Te tree-pase nducton motor was studed under nomnal speed wt ts data for two dfferent source conons: A - Snusodal and alanced tree-pase source. B - Varale frequency nverter at 60 Hz. Our motor was oserved n a computer smulaton and expermentally n te laoratory for te two conons. IV. RESULTS Teoretcal and expermental graps for pase voltages and currents are presented respectvely n Fgures 3 to 7. In te case of computer smulated currents te results n te smulaton are otaned consderng up to 55 t armonc order. It can also e oserved tat comparng Fgures 5, 6 and 7 te resultng pase current wavesape s dfferent Fgure 11 - Torque wt nverter source consderng up to te 55 t armonc order (6n±1 armoncs, n9). Results from Fgures 8 to 11 were all otaned up to ts moment only troug dgtal computer smulaton. Ts s a study tat s stll under laoratory development and furter conclusons for ts work wll e te suect of anoter pulcaton.

V. CONCLUSIONS Te man advantages of ts formulaton are: - te posslty of ts use not only for nducton macnes ut also for syncronous, drect current macnes and all knds of varale frequency nverters drvng tree pase and sngle pase nducton motors; - te posslty of usng ts model as an ad to te desgn of macnes tat wll e workng under te presence of eavy armonc sources; - ts model s an step furter n te modelng not only te nducton macnes due to te easer mplementaton of te macne geometry n te a,, c (stator) and A, B, C (rotor) formulaton. Tme and spatal armonc content suggests tat our nducton macne can e optmzed n terms of ts geometry to reduce armonc oscllatons n ts mecancal torque. Lookng at results presented n Fgures 5, and 6 anyone can easly conclude tat n te developed model, for te same armonc order, computed pase current waveform sows great smlartes wt te laoratory waveform. However, n Fgure 7, te current waveform s dfferent snce te armonc order used to compute t was restrcted to te fundamental waveform, n ts case n0 n 6n±1. Electromagnetc torque sows a dfferent pattern wen comparng feedng te tree-pase nducton motor troug a snusodal source and troug an nverter source. Ts clearly can e seen n Fgures 8 to 11. In Fgures 10 and 11 can also e oserved tat torque oscllatons n steady state are ger wt snusodal source tan wt te use of an nverter as a source, consderng te calculatons n ot cases up to 6n±1 (n9) tme and spatal armoncs. VI. BIOGRAPHY Lucano Martns Neto - Dr. Martns Neto was orn n Botucatu, SP, Brazl n /05/48. He as a Doctoral degree n Mecancal Engneerng from Escola de Engenara de São Carlos at Unversdade de São Paulo (USP), São Carlos, Brazl snce 1980. Worked as a lecturer at Faculdade de Engenara de Lns, Lns, SP, Brazl, at Escola de Engenara de São Carlos ( USP), São Carlos, Brazl and at te Electrcal Engneerng Department (UNESP - Unversdade Estadual Paulsta) at Ila Soltera, SP, Brazl. He s workng as a Senor Lecturer at Unversdade Federal de Uerlânda, MG, Brazl. Hs areas of nterest are Electrcal Macnes and Groundng. José Roerto Camaco - Dr. Camaco was orn n Taquartnga, SP, Brazl n 03/11/54. Completed s PD degree n te Electrcal and Electronc Engneerng Department at Canterury Unversty, Crstcurc, New Zealand, n August 1993. He s a Senor Lecturer at Unversdade Federal de Uerlânda were e works snce Feruary 1979. Dr. Camaco s a Researcer-Consultant of CNPq (Brazlan Natonal Councl for Scentfc and Tecnologcal Development) and collaorator-memer of Brazlan Commtee of CIGRÉ-JWG 11/14-09 (Unt Connecton). Hs areas of nterest are Dynamc Smulaton, Electrcal Macnes and HVAC-DC converson. Carlos Henrque Salerno - Dr. Salerno was orn n Uerlânda, MG, Brazl n 31/05/61. Completed s Doctoral degree at UNICAMP - Faculdade de Engenara Elétrca n Decemer 199. He s a Senor Lecturer n te Electrcal Engneerng Department at Unversdade Federal de Uerlânda, MG, Brazl were e works snce January 199. Hs areas of nterest are Electrcal Macnes and Dynamc Smulaton. Bernardo Pnero de Alvarenga - Mr. Alvarenga was orn n Ueraa, MG, Brazl n 1/0/66. Completed s Msc degree at UFU - Electrcal Engneerng Department n 1993. He s an Assstant Lecturer n te Electrcal Engneerng Department at Unversdade Federal de Goás, GO, Brazl were e works snce Marc 1994. Hs areas of nterest are Electrcal Macnes and Drves. VII. REFERENCES [1] Langsdorf, A.S., Teory of Alternatng Current Macnery, New York, McGraw Hll, 1955. [] Slemon, G.R.; Magnetoelectrc Devces - Transducers, Transformers, and Macnes, New York, Jon Wley & Sons, 1966. [3] J. Stepna, Matrx Analyss of Space Harmoncs of Asymmetrcal Stator Wndngs, IEE Proceedngs, Vol. 134, Pt. B, No. 4, pp. 07-10, July 1987. [4] L.A.M. Neto, C.H. Salerno and B.P. Alvarenga, Harmonc Inductances n te Lnear Analyss of te Inducton Motor, ICEMA - Internatonal Conference on Electrcal Macnes n Australa, Unversty of Sout Australa, Septemer 1993. [5] L.M. Neto, C.H. Salerno, D. Bspo and B.P. de Alvarenga, Inducton Motor Torque: An Approac Includng Wndngs and Saturaton Effects, ICEMA - Internatonal Conference on Electrcal Macnes n Australa, Unversty of Sout Australa, Septemer 1993. [6] L.M. Neto, B.P. de Alvarenga, C.H. Salerno and J.R. Camaco, Analyss of te Inducton Macne Includng te Tme and Spatal Harmonc Effects, PEMC 94 - Power Electroncs and Moton Control Conference, Warsaw, Poland, Septemer 1994. [7] T.A. Lpo, Introducton to AC Macne Desgn, Volume 1, Wsconsn Power Electroncs Researc Center, Unversty of Wsconsn, Madson, 1996.