The Dynamic Modelling and Electrical Profiles of Three Phase Induction Motor Using MATLAB Program

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1 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No he Dynamc oellng an Electcal ofle of hee hae nucton oto Ung AAB ogam E.S. Oluwaogo 1* O.D. Oune A.K. Babane.K. Okakwu 4 1.Depatment of Electcal an Compute Engneeng Kwaa State Unvety alete Ngea.Depatment of Electcal/Electonc Engneeng Unvety of ago Ngea.Depatment of Electcal/Electonc Engneeng Feeal Unvety Oye-Ekt Ngea 4.hD Schola Depatment of Electcal Engneeng. Unvety of Benn Ngea *E-al of the Coeponng Autho: emmanueloluwaogo@yahoo.com Abtact h pape aee the ynamc behavou of -φ nucton machne bae on a qualty mathematcal moel an compute pogam mulaton n a tatonay efeence fame to avo the complexty nvolve n the coue of olvng tme-vayng ffeental equaton obtane fom the ynamc moel. aplace tanfom of the ynamc equaton wa cae out to eve analytcal oluton the cuent vaable. 4 th e unge-kutta metho aopte to gve numecal oluton of the ynamc moel. he bac electcal chaactetc of the - φ nucton machne ae conee n the two-phae appoxmate equvalent moel. he gaphcal cuve wee obtane fom AAB pogam coe evelope. Keywo: -ϕ nucton achne Dynamc oel efeence Fame -Coonate Analytcal an Numecal Soluton AAB pogam coe 1. ntoucton Wth the avancement n moton contol of an automaton ytem ue n evey ecton of nutal an commecal actvte nucton moto AC ve ae beng pefee n contat to the DC ve countepat becaue of t low mantenance cot whle offeng equal an often upeo ynamc pefomance n tem of pee an toque chaactetc when compae wth the DC ve. Howeve quel cage oto of nucton moto the mot commonly ue AC ve. he actual oto cage contan a elatvely lage numbe of conucto oto ba whch ae hot cculate by the en ng. t clea that the ynamc moel of an nucton machne whch compe ffeental equaton an algebac expeon elatng the machne vaable voltage cuent an flux an paamete ung tanent poce an alo n the teay tate gven by the voltage equaton Kchoff an Faaay law an Newton ffeental equaton of moton that ecbe the electcal an mechancal ubytem of the nucton moto epectvely. But thee et of ffeental equaton an expeon ecbng the ynamc behavo of the nucton moto ae tme-vayng ue to elatve moton n the electc ccut of an nucton moto except fo the locke oto. Owng to complexty nvolve when olvng thee tme-vayng ffeental equaton even when they ae beng tanfome to complex fequency oman.e aplace tanfome whch hown late a change of vaable often aopte to mnmze uch complexty ae fom the ffeental equaton n the coue of electc machne analy. Seveal tanfomaton of vaable appoach have been evelope pak Stanley Clake Beeton an kon et al to elmnate tme vayng nuctance by efeng the tato an oto vaable to a fame of efeence that may otate at an angula velocty ef o eman tatonay.e ef an th known a abtay efeence fom. Becaue of the uncetanty n the choce of efeence fame to gve accuate an pece eult whle nvetgatng the electcal behavo of nucton machne ynamc moelng an teay tate analy of an nucton machne have gane popula eeach nteet. heefoe one of the efeence fame ue n th pape to valate the ynamc epone of thee phae nucton moto a efeence fame fxe n the tato tatonay efeence fame. n th tanfomaton metho the tato AC vecto vaable efne by aal an tangental pace vecto of cylncal coonate ae epeente by ectangula coonate ytem of two othogonal axe enote by an on the ame plane Clake E. By th appoach the ognal vaable f abcx of the thee phae machne moel can be euce to a ample clea an ntutve whch the bac pupoe of any eeach wok two-phae equvalent moel havng the ame flux toque an powe a the ognal thee phae nucton machne. Note that the ubcpt x n the vaable f abcx coul ethe be o enotng vaable an paamete aocate wth ethe tato o oto ccut epectvely.. nucton moto mathematcal moel t geneally known that the thee phae nucton moto ha thee et of wnng patally place by electcal egee. f thee tato wnng ta Y connecte ae beng fe by a balance thee phae 8

2 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No upply voltage efne a; a b co co co ωt co ωt max ωt π co max ωt π ωt π co π max ωt c A ee magnetc fel wthn the thee phae nucton moto etablhe by a et of thee-phae cuent of the ame ampltue an angula fequency ω expee a; a b co co co ωt co ωt max ωt π co max ωt π ωt π co π max ωt c Snce the tato wnng ae conee to be thee entcal nuoally bute wnng place by wth N equvalent tun an etance pe phae. kewe oto wnng ae alo conee a thee entcal muoally bute wnng place by wth N equvalent tun an etance. 1 he voltage equaton n the moto vaable may be expee a; Snce; abcx xabcx Nxφabcx N φ x abcx abcx heefoe; the voltage equaton ae; abc abc abc abc abc abc Whee; vaable the flux lnkage n the wnng. he poceeng ecton ummaze the conveon of thee phae wnng moel to t two phae equvalent moel..1 wo hae Equvalent oel of the -Φ nucton oto n oe to obtan the two-phae equvalent moel of the thee phae nucton moto moel vaable; tatonay efeence fame of plane ue a hown n equaton 5a. f K x f abcx 5a Whee; f f f ] f f f f ] an x x x abcx ax bx cx 4 9

3 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No he ntantaneou value of the thee-phae ytem vaable can be obtane fom the tatonay efeence fame ung: f k f 5b abcx Whee; k x Dynamc Equaton of the wo hae Equvalent oel he - tanfomaton apple to the thee-phae tato voltage of equaton 1 ; max max co n ωt ωt 6 o obtan the ynamc equaton of the moto t convenent to efe all oto vaable to the tato e by calng oto vaable wth appopate tanfomaton ato mla to tanfome ccut. t vey mpotant to note that all voltage tem ue to angula pee of efeence fame ccut ae exempte nce a tatonay efeence fame aopte n th pape. he equvalent ccut of the moto efee to the tato hown n Fg.. A. oltage equaton: Applyng K an Faaay law aoun the tato an oto loop epectvely. 4

4 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Stato oop: 7a oto oop: 8a But oto wnng ae hot ccute by the en ng at of quel cage oto. heefoe. hu oto voltage equaton can be euce to; 8b Hence; the tanfome ynamc voltage equaton can be expee n matx fom a; 9 B. Flux nkage Equaton he flux lnkage wthn the tato an oto of the moto equvalent ccut on -plane ae gven by: co n n co co n n co 1 e-wtng the flux lnkage of equaton 1 n a tate-vaable fom: Whee; ] ] uch that; co n n co co n n co an the nuctance matx efne a; C. oque Equaton he total toque on the oto ; ] co n n co 11 Wth the loa toque the Newton equaton ae:

5 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No J ω 1 An ω Whee the tanfomaton angle ue to poton change of the moto moton wth epect to the tato. N.B: All the ynamc equaton eve above ae fo a pae of pole.e one Noth pole an one South ole.. Dynamc Equaton of a ultple a of ole nucton f the oto ha pa of pole the flux lnkage ae mofe a; ] co n ] n co ] co n ] n co 1 Snce N an N wee efne a the tato an oto wnng pe a pole pae epectvely. heefoe the total numbe of wnng n a tato phae wnng ae; N an N epectvely. hen the ynamc equaton fo -multple pa of pole ae; ]] co n ]] n co ]] co n ]] n co 1 J ω ω an ] co n n co 14. athematcal moel oluton ung aplace tanfomaton Expanng equaton 1 above; co n n co co n n co 15 f n equaton 15:. n an co hen takng the aplace tanfom of the eultng equaton we have;

6 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No ] ] ] ] ] ] ] ] ] ] ] ] 16 Whee; f the ntal conon ae ubttute n equaton 16 an collectng the lke tem; ] ] ] ] 17 Wtng th n a tate matx fom; 17 h coul be epeente by; ] ] Z 18 Whee the mpeance matx ZZ gven by; Z Z Solvng equaton 17 ung Came ule; 1

7 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Such that; analytcal oluton to the cuent ae; n equel to the above aplace tanfom whch coul only pove analytcal oluton to the unknown vaable n the ynamc equaton thee a nee to evelop a quanttatve metho that can cetze the oluton of the equaton 1 & 14 ntea of analytcal oluton. Snce th wok ame at nvetgatng the ynamc behavo of the moto. heefoe the numecal value of the vaable obtane n a cete fom coul be ue to play th n gaphcal fom. he numecal metho aopte n olvng the ffeental equaton n th pape Fouth 4 th -Oe unge-kutta metho becaue t an excellent geneal pupoe ffeental equaton olve Geenbeg 1998]. Hee we peent 4 th oe unge Kutta metho fo geneatng appoxmate numecal oluton value to an ntal value poblem; f x y y x y at electe pont y wth ntal conon x o x n.x N. f the ance between ucceve pont at whch appoxmate value of the oluton ae compute tep zeh. o compute pont x n y n eque gaent of f f x n y whch can be ummaze a follow; n 44

8 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No f hen; Fnally whee the x n k 1 k k k 4 ncement x y y n 1 hf hf hf hf n x n 1 x x x y n 1 n 1 n 1 n y h n 1 n y k k K k ] 1 fo n 1 1 h 1 h y k n 1 ; y n n 1 efne 1... y n 1 y ; n 1 a; 1 k1 ; 1 k ;...N So fom the ynamc equaton 1 an 14 of multple pole pa of an nucton moto; f the elatve poton of the oto to the tato et at.e axmum. magnetc couplng an the numbe of pole pa ; we have: a b c 4 an J f the etang toque pouce by the fcton n the ball beang of the moto moele by fω whee fcoeffcent of cou Fcton. heefoe the equvalent Newton ffeental equaton ; J fω ultply equaton a an c by an epectvely; Subtactng equaton f fom e to mplfy; kewe multply equaton b an by an epectvely we have; e f g 45

9 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Subtactng equaton fom h we have; Alo f equaton a an c ae multple by an epectvely we have; h j k l Subtactng equaton l fom k; Alo f equaton b an ae multple by an epectvely the eultng equaton ae o an p; Subtactng p fom o we have q; Aangng equaton g m j an q; Alo; m o p q g m j q f ] ω J ω Fom equaton g m j q an equaton 1 an 4 ae obtane epectvely; ] ] ] ] 1 46

10 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No An; ] ] ] ] ω fω ω ] J 4 n oe to mulate the two-phae equvalent of the thee phae ynamc moel of nucton moto the ffeental equaton -5 wll be olve ung the 4 th Oe unge-kutta ecbe above wth the a of ABAB coe pogam -fle. 4. Compute mulaton an eult he hp 415volt 5H Z 144pm 6pole thee phae nucton moto choen fo th ynamc moelng an mulaton ha the followng opeatng chaactetc paamete: e phae tato etance.15ω e phae oto etance efee to tato e.96ω e phae tato nuctance.5h e phae oto efee to tato e 1.7H e phae mutual nuctance.117h oment of netal J.1kg-m oa oque.8 Nm he AAB coe pogam evelope to olve the ynamc ffeental equaton of the thee-phae nucton moto wa acheve ung the algothm tate n the flow-chat of fg. below. 5 47

11 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Stat nput nucton moto paamete Compute: -efeence oltage ung eqn-6 ea the ntal value of the unknown vaable n the D. Equaton Fo tpan.5]; %tme tep h N1; %No of tme tep Fo n1:n t n t n-1 h Solve the ffeental equaton 5 ung 4 th Oe unge-kutta Evaluate the thee-phae ntantaneou value v an fo the tato an oto ung eqn-5b Ye n<n? No nt; v v ω a a b b c c e.t.c. Fg. : he Flow-Chat fo the Dynamc oel of the hee-hae nucton oto En 4.1 eult an Dcuon he AAB compute mulaton of the moel wa cae out une the followng bac aumpton:- umpe aamete Aumpton A an Effect of magnetc atuaton neglecte. he gaphcal eult obtane fom the compute mulaton of the nucton moto fo ynamc analy n a tatonay efeence fame ae hown n the fgue below: 48

12 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Fg.4: Stato cuent n tatonay efeence fame an 49

13 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Fg. 5: oto cuent n tatonay efeence fame an 5

14 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Fg. 6: Stato voltage n tatonay efeence fame an 51

15 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Fg. 7: oto voltage n tatonay efeence fame an 5

16 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Fg. 8: Stato phae cuent A B an C epectvely. 5

17 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Fg. 9: oto phae cuent A B an C epectvely. Fg.1: oto Angula Spee ω pm-me t ofle 54

18 nnovatve Sytem Degn an Engneeng SSN -177 ape SSN -871 Onlne ol.6 No Fg.11: oto Angula Dplacement Deg.-me t ofle Fgue 4-9 how the electcal pofle of the thee-phae nucton moto moel bae on Faaay law an Fg. 1&11 how the mechancal behavo of the machne bae of Newton equaton. Concluon he pape gve a compehenve mathematcal ynamc moel of a thee-phae nucton moto ung a efeence fame - tanfomaton fxe n the tato. he tatonay efeence fame o - axe moel peent a mple an clea analy of the machne uch that all ntenal electcal vaable acceble fo the tuy of ynamc an teay tate behavo of the nucton moto. Alo t mut be mentone hee that the 4 th Oe unge-kutta etho ha poven utable fo povng a pece numecal oluton to ffeental equaton wth ntal conon. Fnally the mathematcal moel an AAB pogam coe evelope gve a elable ynamc behavo of nucton moto ung a efeence fame theoy. efeence E.S. Oluwaogo an.k. Okakwu 14 efomance Analy of a Sngle-hae AC oltage Contolle une nucton oto oa ntenatonal Jounal of eeach n Engneeng an echnology JE. ol.: Ael Aktab et al Dynamc Smulaton of a hee-hae nucton oto Ung atlab Smulnk Sfat Shah A. ah K Bhatt 1 Dect Quaate D-Q oelng of -hae nuctonoto Ung atab / Smulnk Canaan Jounal on Electcal an Electonc Engneeng ol. No. 5 Chaon J. 5 oelng an Hgh efomance Contol of Electc achne EEE See on owe Engneeng Kt Jatne Snghvek ahwa 1 Dynamc an Steay State Analy of nucton achne ntenatonal Jounal of Emegng Scence an Engneeng JESE ol-1 ue-1 Sloboan N. ukoavc 1 Electcal achne Spnge New Yok Heelbeg Doecht onon A.K. Babane et al 14 Effect of Electcal aamete aaton on the Dynamc Behavou of wo- hae nucton oto ntenatonal Jounal of Engneeng Scence nventon. ol. ue chael D. Geenbeg 1998 Avance Engneeng athematc n Eon entce Hall Uppe Sale ve New Jeey. 55

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