A Field Procedure to Include Magnetic Saturation Effect on a Three-Phase Induction Machine

Transcription

1 A Feld Procedure to Include Magnetc Saturaton Effect on a Tree-Pase Inducton Macne L.M. eto (Dr. +*, M.S. Mskuln (PD * D. Bspo (MSc +, J.T. de Resende (MSc +, J.R. Camaco (PD + (+ Electrcal Engneerng Department UFU - Unversdade Federal de Uberlânda P.O. Box 59, Uberlânda - MG Brazl e.mal: jrcamaco@ufu.br (* FEEC - Engenara Elétrca e Computação UICAMP - Unversdade Estadual de Campnas P.O. Box: Campnas - SP - Brazl Abstract - A tree-pase nducton macne can be studed more accurately, regardng operatng and steady-state runnng condtons, wt te development of a matematcal model ncludng magnetc saturaton. Ts paper presents a magnetc saturaton model n te A, B, C reference frame. Te model takes two functons as a reference. Saturaton s dentfed n te model from an armonc magnetc functon, obtaned from te model development and laboratory tests. Te model also allows te macne smulaton at no-load syncronous speed, were as an example, currents wll be obtaned for a gven voltage exctaton. Expermental tests are carred out to compare smulaton and laboratory results. Index terms - Inducton macne, magnetc saturaton, turns accumulaton functon. I. ITRODUCTIO Modern applcatons wt electrcal macnes requre te assumpton of non lnear ar-gap flux and feedng currents, and also non-lnear relatonsp between de argap flux and te feedng current. Te matematcal complexty nvolved n te non-lnear study s a major concern wen developng a model snce te electrcal macne model beavor canges wt te complexty, gettng closer to te laboratory observatons as te complexty ncreases. In te case of developng a dynamc study of a capactor self-excted nducton generator te non-lnearty provded by magnetc saturaton s very mportant.[] Current or voltage dstortons due to magnetc saturaton, on voltage or frequency converter drven nducton motors, can ave a substantal sgnfcance n te global beavor of te converter-motor system.[][] Wen consderng te system n balanced steady-state condtons tose dstortons ave low sgnfcance. Furtermore, te best way to nclude te saturaton effect depends on te problem to be solved. A very mportant ssue n ts case, wen saturaton s present, s te cause-effect analyss. Wen developng ts analyss, one of te ways to be followed s te macne smulaton stressng magnetc saturaton and oter components as magnetc feld and conductors spatal dstrbuton. Ts sort of smulaton allow us to dentfy quanttatvely cause and effect. Ts paper ntends to analyze nducton macnes magnetc caracterstcs ncludng magnetc saturaton, represented by ts armonc functons, and also by te conductors accumulaton functon concept, used n obtanng te macne total flux. Te parameters n te electrcal termnals are taken as reference. Ts analyss s made troug a matematcal model wc allows te smulaton of a balanced tree-pase nducton macne at no-load syncronous speed, obtanng as an output ts currents and a value of voltage exctaton. Te magnetc saturaton effect s obtaned troug te soluton of te macne dfferental equatons, usng te saturaton curve. Te armonc functon s obtaned from te macne ysteress loops and from a lnear functon, representng te load, obtaned from te model. Te crossng pont of te two functons wll gve us nformaton for te soluton of ts matematcal model. Teoretcal results for te descrbed matematcal model were tested aganst expermental results. II. MODEL DEVELOPMET Snce our man goal s te modelng of te macne underlnng ts magnetc caracterstcs, t can be consdered operatng at no-load syncronous condton,.e., at syncronous speed. Under suc condton te macne electrcal crcut sows only te equatons wo present relatonsps between stator voltage and currents. For a generc stator "" pase, can be wrtten at te electrcal termnals tat: v = r + (0 were, v,, r, and λ are respectvely voltage, current, resstance, and coupled total flux wt "" pase as a reference, respectvely. Fmm(, Pase axs α π/ α B(, α π/ Fgure - Fmm(α, and B(α, representaton.

2 Te mutual flux λ s te sum of te leakage flux and te magnetc flux for "" pase. Consderng te leakage flux not nfluenced by saturaton, te followng equaton can be wrtten: λ = Ld + λm (0 were, L d e λm - leakage nductance and magnetzng coupled flux, for pase "", respectvely. To solve equaton (0, knowng v, t s necessary to relate wt λ. Ts s possble from equaton (0, beng mandatory te knowledge of magnetzaton flux λm. Fgure s consdered wen obtanng te magnetzaton flux, ts represents te magnetomotve force fmm(α, as a functon of flux densty B(α, for a generc poston related to a reference α. At Fgure t s represented a generc secton of te rotor core materal. Were l, ds, r,, d are respectvely, te rotor lengt, nfntesmal area of te rotor secton, rotor radus, generc poston angle and nfntesmal angle for te rotor angle. Wt te use of te concept of turns accumulaton functon sown at Appendx I, from Fgure t s stragt forward to obtan Equaton (0. and ds = l r d (06 From equatons (0, (04, (05 and (06, λm wll be obtaned gve by: m lr = { cos[ ( } B( α, d (07 It s necessary to develop te flux densty B(α,, wc wll be related wt te magnetomotve force fmm(α,, troug te magnetzaton curve as can be seen Fgure. Samplng te curve at Fgure between ponts A and B can be obtaned Fgure 4 as follows. B j B j- a j B(, A B β j = tgb j = b j Ref. α ds r Fgure - Macne elements representaton. m = FA dφ (0 were, FA, dφ, are respectvely, te turns accumulaton functon, defned as n Appendx I, and nfntesmal flux varaton n a well defned poston. Te turns accumulaton functon s defned as: FA = { cos[ ( ]} (04 And t s known tat: B(, A Fgure - Magnetzaton curve representaton. dφ = B(α, ds (05 B ds l Fmm(, F j- Fgure 4 - Samplng for te magnetzaton curve. Terefore: B(α, = a j + b j fmm(α, (08 were, t s defned: fmm(α, = FM cos( - α (09 Tem, substtutng equaton (09 n (08, we ave: B(α, = a j + b j FM cos( - α (0 Substtutng equaton (0 n (07, can be obtaned: m = lr { cos[ ( α ]}. ( { a j + b j FM cos( α } d By ntegratng dm troug de π radans nterval n order to cover all te macne's argap lengt λm wll be obtaned. π α + ( λm = m π α Evaluatng te ntegral at equaton ( for =, as te fundamental component, and =, as te trd armonc, λm can be obtaned troug equaton (. λm = 4rlK cos( α a j[ sn( j α sn( j α] rlkfm cos( α b j ( j j rlkfm cos( α sn( j j cos( j + j α F j Fmm(,

3 K rl cos(α a j[ sn( j α sn( j α ] 9 j = rlfmk cos( α bj sn( j j cos( j + j α rlfm K ( cos( α b jsn( j j cos( j + j 4α were: - refers to pases a, b, c. FM - s te maxmum value of fundamental component from te macne fmm(α,. - angle tat defnes any poston along te macne argap, from a stator reference, as n Fgure. α - angle wo defnes te nstantaneous poston of FM. K, K - constant wo relates te turns accumulaton functon, fundamental and trd armonc factors. Te tree-pase nducton macne load curve can be obtaned troug matematcal manpulaton of equatons (0, ( and (6, represented by equaton (8. Tese equatons added to te saturaton curves and macne dfferental equatons wll be responsble for solvng te proposed matematcal model. f R = λ cos (4 Ld = a,b,c f I = λ sen (5 Ld = a,b, c ( = f R f I f λ + (6 fi tg α = (7 f R f ( λ FM FFM( FM = (8 A A were, k = k ( = a,b,c, a = 0, b = -0 o, c = 0 o and A = L d. Results expressed n equatons (4 and (7 are only vald wen fundamental and trd armoncs are consdered n equaton (, n anoter words = and. Te process adopted for dgtal smulaton of te nducton macne under syncronous no-load t s presented as follows. From an ntegraton numercal metod equaton (0 s solved appled to te stator tree pases (a, b, c. To every ntegraton step values of λ and v are known. From equatons (4 and (6 can be obtaned f(λ. Te knowledge of te expermental caracterstc FFM (FM, FFM (FM and L d, togeter wt te stragt caracterstc gven by equaton (8, for every ntegraton step wll be obtaned values of FFM, FFM and FM, FM. From FM te value of F (FM can be obtaned, troug ts expermental functon. Terefore, te value of λm can be easly obtaned from equatons ( and (7, and "a posteror", from equaton (, can be obtaned wc wll be part of equaton (0 and consequently closes te numercal soluton cycle. III. CURVE FOR FFM (FM, AD FFM (FM For te nducton macne under study te parameters can be obtaned for te no-load and blocked rotor tests. Wt te obtaned parameters te ysteress loops can be drawn, and wll be easy to obtan te magnetzaton curve from tose loops. Anoter nformaton needed at ts stage are te ron losses wen lookng from te stator, obtanng dfferentally te magnetzaton current as can be seen at Appendx II. In order to estmate losses t s necessary to obtan voltages v and currents at te macne termnals, wle takng te macne under test to ts syncronous speed. Ts can be done by avng a mecancal couplng wt a syncronous motor wt te same syncronous speed and rotatng n te same drecton of te syncronous rotatng feld of te nducton macne. Under tese condtons te syncronzaton can be easly verfed troug a stroboscope. Under tese crcumstances wll be made te data acquston of waveforms for voltage v and current. Wt te acquston of te above waveforms t s computed e, takng as a reference Fgure 5. v RS LS Fgure 5 - Defnton of voltage e. e m Rm d e v R L = S S (9 Te d dervatve can be obtaned wt precson troug te Fourer Seres decomposton, by dervng and decomposng te armoncs. It s well known tat: d ' = [ W sen( Wt + + W cos( Wt + β] = d [ ' (0 = W cos( Wt + β sen( Wt + ] = were a a e ' a β a. Assumng a symmetrcal waveform, te sne terms wll be zero, terefore we ave: d = W [ ' cos( Wt + β] ( = Terefore: d = W [ a cos( Wt + β a + a cos(wt + βa] + 5a 5 cos(wt + βa5] ( were ' a s frst armonc current magntude and βa s te frst armonc pase and ten successvely. Takng Fgure as a reference actve power nput at te magnetzng branc computed gven by:

4 T P = e ( T 0 Wt te power computed troug equaton ( te voltage rms value e or E, and te losses resstance Rp can be obtaned, were: E Rp = (4 P Te loss current tem, wll be gven by: e p = (5 Rp Were: ' = p (6 or e abc abc = abc (7 Rpabc Te ysteress loops for some prevously establsed voltages, can be obtaned, usng e from equaton (9 and ' from equaton (7. Te curve λm aganst FM for te fundamental and trd armonc wll be obtaned connectng te loops far ends. From te samplng of curves λm aganst (FM as n Fgure 4, and usng te sums obtaned n equaton (, for te followng condtons wt F j j α = arccos, FM we ave: [ ( ( ] + ( = a j sn A sn B FMb j sn A B. (8 FFM = ( ( j cos A + B + FMb j A [ ( ( ] + ( = (9 a j sn A sn B FMb jsn A B. FFM cos( A + B + FM sn A + B ( A + B. [( ].cos[ ] were: F, j A = arccos FM, F, j. (0 B = arccos FM, From equatons (8 and (9 a curve s obtaned to buld Fgures 6 and 7. Fgure 6 - Frst armonc summaton curve. IV. THEORETICAL AD EXPERIMETAL RESULTS Some of te macne data are te values of pase resstance and pase leakage reactance presented n te Appendx II, obtaned troug te ordnary process. From functons FFM (FM and FFM (FM, for te frst and trd armonc respectvely, tey are obtaned troug laboratory tests. Terefore, wll be possble wt te presented model to smulate n a dgtal computer te nducton macne n no-load syncronous operaton. Fgure 7 - Trd armonc summaton curve. For an Y connecton te lne voltages are known, and make a balanced tree-pase system and lne currents can be computed by te followng equatons: v R abc abc = s abc + ( ( ab + bc a = ( ( bc + ab b = ( c = ( a b (4 ( λ λ a mab ab = (5 LS ( λb λmbc bc = (6 LS were L s, λ mab, λ mbc are stator leakage nductance and mutual fluxes between lnes ab, and bc respectvely. For a connecton te lne voltage can be obtaned from te dfferences n voltages between pases, current and fluxes represented n equatons (7, (8, (9 and (40. v R ab ab = s ab + (7 v R bc bc = s bc + (8 v R ca ca = s ca + (9 were Wab = Wa Wb Wbc = Wb Wc (40 Wca = Wc Wa were "W" can be v, or λ. Terefore, wt v, and λ for Y and connectons, te computatonal model wll be te same prevously descrbed.

5 waveforms, for te connecton, obtaned expermentally and troug computer smulaton. Fgures from ( to (5 represents te pase voltage and current waveforms, for te star connected macne, obtaned expermentally and troug computer smulaton. V - COCLUSIOS Fgure 8 - Pase voltages obtaned expermentally. Fgure 9 - Pase voltages obtaned toug smulaton. Fgure 0 - Pase currents obtaned expermentally. From te matematcal development of te macne magnetzng flux, ncludng te drawng of armonc functons FFM(FM, wll be possble to solve te dfferental equatons wt te ncorporaton of effects for te magnetc saturaton. Anoter nterestng aspect s tat for te model soluton are used te macne parameters, current and voltages avalable at te macne electrcal termnals. Terefore te model allows te analyss of magnetc saturaton. It s mportant to add also tat separaton of magnetc saturaton, for varous armonc frequences, can be very useful for te knowledge of te nducton motor armonc model. From results obtaned n Fgures 8 to 4, can be seen tat n te smulaton te voltage n Y connected and te current n te connected macne sows dstortons due saturaton effect. Besdes, te peak values are very smlar to te expermental ones, ts clearly valdates te computer model proposed ere. It s mportant to add te mportance of suc conclusons over macne project and n te feld of macne dagnostcs. Ts wll be even more mportant wen te macne s drven by unusual voltage or current wave-sapes. Fgure - Pase currents obtaned toug smulaton. Fgure 4 - Pase currents obtaned expermentally. Fgure - Pase voltages obtaned expermentally. Fgure 5 - Pase currents obtaned toug smulaton. VI - REFERECES Fgure - Pase voltages obtaned toug smulaton. To valdate te model dgtal smulatons were made wt te macne n star and delta connectons respectvely. Fgures from (08 to ( represent te voltage and current [] JESUS, J. M. F; "A Model for Saturaton n Inducton Macnes", IEEE Transactons on Energy Converson, Vol., o -, p , September 988. [] MOREIRA, J. C.; LIPO, T. A.; "Modelng of Saturated AC Macnes Includng Ar Gap Flux Harmonc Components", IEEE Transactons on Industry Applcaton, Vol. 8, o -, p. 4-49, Marc/Aprl 99.

6 [] - LIAO, Y.; Lpo, T. A.; "Effect of Saturaton Trd Harmonc on te Performance of Squrrel Inducton Macnes", Electrc Macnes and Power Systems, Vol., o -, p. 55-7, Marc/Aprl 994. VII. APPEDIX I OBTAIIG THE TURS ACCUMULATIO FUCTIO FROM MAGETOMOTIVE FORCE FMM(α, SPATIAL DISTRIBUTIO. Te turns functon s gven by te followng ntegral: FA = η ( d π ψ (I. were η ( s te dstrbuton functon for a magnetomotve force spatal dstrbuton. η ( = Σsn[ ( ] (I. ψ - π/ s te poston for te wndng frst conductor. To obtan FA t s necessary to defne te ntegraton lmts, as n Fgure I.. FA = η ( = sen[ ( ] d FA = { cos[ ( ]} (I. ψ - π/ - π/ ψ η ( + π/ Fgure I. - Obtanng te ntegraton lmts. VIII - APEDIX II d η( OBTAIIG MACHIE DATA Frst of all, equvalent crcut data was obtaned from a tree-pase nducton macne wt te followng plaque data: HP, 70 rpm, 60 Hz, /Υ - 0/80 V - 6,90/,99 A. Obtaned data were from te ordnary no-load and blocked rotor tests and values n Oms(Ω are: R S =,, R R =,80, X S =,8, X R =,8, Xm = 7,6 were: R S, X S, R R, X R e Xm are, respectvely, te stator and rotor resstance and leakage reactance, and magnetzaton reactance wt te rotor parameters referred to te stator. IX - BIOGRAPHY Lucano Martns eto - Dr. Martns eto was born n Botucatu, SP, Brazl on May nd, 948. 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 980. 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 (UESP - Unversdade Estadual Paulsta at Ila Soltera, SP, Brazl. He s currently workng as a Senor Lecturer at Unversdade Federal de Uberlânda, MG, Brazl. Hs areas of nterest are Electrcal Macnes and Groundng. Mauro Sérgo Mskuln - Dr. Mskuln was born n Santa Rta do Passa Quatro, SP, Brazl on ovember st, 947. He receved s BSc n 97 and MSc degree n 974 from UICAMP - Unversdade Estadual de Campnas, Brazl. He receved s PD degree at Cranfeld Insttute of Tecnology, England, n 980. He s a Senor Lecturer at Unversdade Estadual de Campnas - UICAMP. Dr. Mskuln as been recently on study leave at te Unversty of ew Mexco, USA. Déco Bspo - Mr. Bspo was born n São Vcente, SP, Brazl on ovember 8 t, 95. He got s Electrcal Engneerng BSc degree from UFU n 979, and s MSc degree n Automaton from UICAMP n 985. He s a Lecturer at Unversdade Federal de Uberlânda were e works snce 979, currently e s pursung s doctoral degree n Automaton at UICAMP. Hs areas of nterest are Electrcal Macnes and Industral Applcatons José Tarcso de Resende - Mr. Resende was born n Itumrm, MG, Brazl on June st, 96. He got s Electrcal Engneerng BSc degree from F.E.S. São João del Rey n 987, and s MSc degree n Electrcal Macnes from EFEI n 99. He s currently pursung s doctoral degree n Electrcal Macnes at UFU. Hs areas of nterest are Electrcal Macnes and related felds. José Roberto Camaco - Dr. Camaco was born n Taquartnga, SP, Brazl on ovember rd, 954. Completed s PD degree n te Electrcal and Electronc Engneerng Department at Canterbury Unversty, Crstcurc, ew Zealand, n August 99. He s a Senor Lecturer at Unversdade Federal de Uberlânda were e works snce February 979. Dr. Camaco s a Researcer-Consultant of CPq (Brazlan atonal Councl for Scentfc and Tecnologcal Development and worked as a collaboratormember of Brazlan Commttee of CIGRÉ-JWG /4-09 (Unt Connecton. Hs areas of nterest are Dynamc Smulaton, Electrcal Macnes and HVAC-DC converson.