Non Integer Identification of Rotor Skin Effect in Induction Machines

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1 eaoal oual of Eleccal ad Compue Egeeg ECE Vol. 3, No. 3, ue 3, pp SSN: No ege defcao of Roo S Effec duco Mache Abdelhamd ALLOUL *, haled ELASS *, ea-claude TRGEASSOU ** * Naoal Egeeg School, Eleccal Syem Laboaoy LSE, Tu, Tua ** Laboaoe égao du Maéau au Syème MS-LAPS, UMR 58, Uveé Bodeau, Face Acle fo Acle hoy: Receved Feb 7, 3 Reved Ap, 3 Acceped May 3, 3 eyod: Facoal mpedace duco mache Ladde eo No ege ode dffeeal Oupu eo defcao S effec ABSTRACT Facoal defcao of oo effec duco mache peeed h pape. Pa afomao ued o oba a yem of dffeeal equao hch allo o clude he effec he oo ba of aychoou mache. A afe fuco h a facoal devave ode ha bee eleced o epee he admace of he ba by he help of a o ege egao hch appomaed by a + dmeoal modal yem. The mache paamee ae emaed by a oupu-eo echque ug a o lea eave opmzao algohm. Epemeal eul ho he pefomace of he modal appoach fo modelg ad defcao. Copygh 3 ue of Advaced Egeeg ad Scece. All gh eeved. Coepodg Auho: Abdelhamd ALLOUL, Naoal Egeeg School, Eleccal Syem Laboaoy LSE,Tu, Tua Emal: elloulabd@yahoo.f. NTRODUCTON Accuae modelg of eleccal mache vey mpoa fo he dege of he mache, facg mpoveme. O he ohe had, he oledge of paamee eceay o ealze ealc mulao of he mache ad mpoa fo he opeao of mode dve ho mpleme cool yem. Moeove, he cae of he aocao of a ac covee o a eleccal mache, he aoal ue of he hole pae by a pefec cool of he global dyamc behavo. Wh PWM poe upple, he eleccal mache have o o o a vey lage fequecy age. Thu, he epeeao of h mache by a mplfed model, oly vald o a lmed fequecy age, he ouce of uafacoy eul. The uffcecy of hee model moe acceuaed he he eleccal mache have a mave ucue le aychoou mache h cage, deep oche o mave oo chaacezed by effec o fequecy effec. duco cue he oo ba ae goveed by a dffuve pheomeo. A lo fequece, cue have a dey hch ufom ad equal eveyhee ove he ee co ecoal aea. f he fequecy hgh eough, cue dey ed o be hghe a he uface of he ba. The hghe he fequecy, he geae he edecy fo h effec o occu. Th pheomeo called «effec» oo ba, o «fequecy effec» moe geeally. Thee ae hee poble eao e mgh cae abou effec []:. The effec caue he effecve co ecoal aea o deceae. Theefoe, he effec caue he effecve eace of he coduco o ceae.. The effec a fuco of fequecy. Theefoe, he effec caue he eace of a coduco o become a fuco of fequecy. 3. f he effec caue he effecve co ecoal aea of a ba o deceae ad eace o ceae, he he ba ll hea fae ad o a hghe empeaue a hghe fequece fo he ame level of cue. oual homepage: hp://aeoual.com/ole/de.php/ece

2 ECE SSN: eleccal egeeg, h pheomeo paculaly mpoa mave oo o quel-cage duco moo. dffuve chaace lead o oce a og modfcao of he mpedace boh eace ad eacace accodg o he fequecy []. hu eeg o ue a afe fuco h a facoal devave ode o epee he admace of he ba o a boad fequecy cale, le ha bee demoaed a ece pape [3]. he coe of paamee emao of he admace, he devave ode hould be emaed he ame ay ha he ohe coeffce. Baed o he oupu eo mehod, he model ued ae o lea he paamee ad opmzao algohm volve o lea pogammg NLP h pape, e popoe o defy he paamee of he aychoou mache model ag o accou he facoal feaue of he oo model ad emag paamee ug a oupu eo defcao echque. Afe a emde of defo elaed o facoal egao opeao pa ad 3, he Pa model of aychoou mache h facoal mpedace peeed pa 4. Pa 5 devoed o pee he oupu eo mehod. We popoe, pa 6 epemeal eul of facoal defcao fo paamee emao.. FRACTONAL DFFERENTATON AND NTEGRATON Facoal egao defed by he Rema-Louvlle egal [6], []-[]. The h ode egal eal pove of he fuco f defed by he elao: f f d Whee Γ e d he gamma fuco. f epeed a he covoluo [] of he fuco f h he mpule epoe: h of he facoal egao opeao hoe Laplace afom : Lh 3 Facoal dffeeao he dual opeao of he facoal egao. Code he facoal egao opeao hoe pu/oupu ae epecvely ad y. The: y 4 o Y X 5 Recpocally, he h ode facoal devave of y defed a: D y 6 o X Y 7 Whee epee he Laplace afom of he facoal dffeeao opeao h zeo al codo. No ege defcao of Roo S Effec duco Mache Abdelhamd ALLOUL

3 346 SSN: FRACTONAL NTEGRATON OPERATORS The facoal egao opeao he ey eleme fo FDE mulao. Hoeve, he ealzao of o a mple poblem a he ege ode cae. poble o code he fequecy ad modal appoache. Ou obecve o compae he mpac of hee appoache fo he mulao ad he defcao of he aychoou mache. 3.. Fequecy Appoach Syhe 3... Pcple Le u code he Bode plo of a facoal egao ucaed lo ad hgh fequece Fgue [3]-[5]. Fgue. Bode dagam of he facoal egao compoed of hee pa. The emeday pa coepod o o-ege aco, chaacezed by he ode. he o ohe pa, he egao ha a coveoal aco, chaacezed by ode equal o. h ay, he opeao defed a a coveoal egao, ecep a lmed bad [ b; h ] hee ac le. The opeao defed ug a facoal phae-lead fle [6] ad a egao. G 8 The coeffce G a omalzed faco, uch a ad ae decal o [ b; h ]. Th opeao compleely defed by he follog elao [6]:, 9, log log ad ae ecuve paamee elaed o he o ege ode. Whe uffcely lage, he bode dagam of ed oad he deal oe of Fgue Sae pace model of covee o aocae a ae-pace epeeao o ode o mulae facoal yem. Thee a fe umbe of poble o epee by a ae pace model. Paccally, e have choe he oe hee he ae vaable coepod o he oupu of he elemeay cell of. ECE Vol. 3, No. 3, ue 3:

4 ECE SSN: No ege defcao of Roo S Effec duco Mache Abdelhamd ALLOUL 347 X X o o fo 3 Wh V G X. Whee v he pu of ad oupu. The coepodg ae pace model : v B A M 4 Wh M A G B ; Becaue oe of ou obecve o emae he paamee e have pvleged pamoou model ode o faclae he defcao pocedue [4], [7]. Ohe appomao ca be ued ad bg mpoveme o mplfy he calculao of he fequecy doma appoach, le he modal appoach. 3.. Modal Appoach 3... Fequecy dbued model The facoal ode egao a lea yem uch a: u h y 5 Wh h L H 6 Th yem ca be epeeed by a fequecy dbued model; alo o a a dffuve model efe o apped ad [8]-[]:,,, d y u 7 Ad d h, 8 Wh, 9 called fequecy eghg fuco.

5 348 SSN: Fequecy dcezed dbued model Th couou fequecy eghed model o decly uable. A paccal model eceay fo mulao applcao obaed by fequecy dcezao of, hee he fuco eplaced by a mulple ep fuco h ep. Fo a elemeay ep, hegh, ad dh. Le c be he egh of he h eleme: c Fgue. Fequecy dcezao of Thu, he couou dbued model become a coveoal ae model h dmeo equal o. d u ;... d y c o equvalely: X A X Bu T y C X Wh X A ; T T B, C c c Wh h appoach, e oba a dcee ae-pace model hch fequecy dbued h he coa:, ad 3.3. Compao h Fequecy Model eay o afom he model 4 of o a modal fom becaue he ae o a po. Th afomao baed o he follog decompoo mple eleme: c c 3 ECE Vol. 3, No. 3, ue 3:

6 ECE SSN: Whee c ad c coeffce ae led o G, ad by he elao: c G 4 c G 5 Th ecod defo of coepod o a modal ae model: X A X B T y C X u 6 Wh: A T C c c c ; X ; B he fequecy doma appoach, he mode ae decly obaed by he ; eval, hey coepod o he mode of he modal appoach. The ee of h la epeeao ha he mode ae decoupled, hch allo fa compuao. Moeove, a mpoa ee of o eec ac eo mulao applcao. b h 4. PAR S MODEL OF NDUCTON MACHNE The mo mpoa aumpo o deve he Pa model ae:. The a gap beee he ao magec ucue ad he oo magec ucue ufom. All magec vaao due o lo ae egleced.. The magec feld aumed o have a uodal paal dbuo. 3. The ao ad oo dg ae cocde h he magec ae of he phae. 4. The pemeably of he o fe. The Pa afomao eablhe a equvalece beee a hee-phae epeeao ad oo efeece fame. The coveoal equvale dagam [] of Pa model epeeed o Fgue 4 : Fgue 3. Coveoal equvale dagam of Pa model Wh: R ad R epeeg he eace of he ao ad he eace of he oo ba epecvely. he oo peed, l ae he ao ad oo leaage ducace, L m he magezg ducace. No ege defcao of Roo S Effec duco Mache Abdelhamd ALLOUL

7 35 SSN: Ladde Model To ae o accou effec oo ba, he aumpo made ha each equvale oo dg compoed of lce paallel. The Pa equvale dagam h ladde model efe o [] ad [3] fo moe deal epeeed o Fgue 5. Fgue 4. Pa ladde model l epee he lage ducace of each elemeay lce. A comple oao ued: d X 7 q The mahemacal model of quel cage duco moo ca be e a: d U R d d R d l Lm Lm l Thee ae eveal epeo ha ca decbe he developed elecomechacal oque of a duco mache [], [3], e pefe o ue he follog becaue efe oly o ao vaable: 8 C em d q q d duco mache equvalece h facoal mpedace Ug equvalece beee a ladde eo ad facoal mpedace [3], oe ca defe he Pa facoal model of he duco mache: Fgue 5. Pa facoal model The equao decbg elecomagec pocee duco mache cludg a quel-cage oo ae a follo: d U R 3 d Lm l L m ECE Vol. 3, No. 3, ue 3:

8 ECE SSN: m We defe he magezg flu L 3 m m We ca e: Z 3 m The, a m 33 b b Whch coepod o he facoal ode dffeeal equao: d a m D d b b 34 Becaue, l 35 m e oba a dffeeal yem allog he mulao of he aychoou mache: d U R d d D a b U R l d L m Lm l The mechacal epeo of he oo peed obaed ha o he elao: 36 d d C C f 37 em C em epeed 9, : mome of ea f : fco coeffce 5. OUTPUT ERROR DENTFCATON Ne, e emd he pcple of a mehod allog he emao of he paamee of he Pa model of duco mache h facoal mpedace 36. Wheea paamec emao ca be pefomed by a lea opmzao echque cae [4] he model lea he paamee, he emao of he devave ode ad of he coeffce eque he ue of a olea pogammg algohm. The mehod uggeed by Tgeaou, L ad Poo [3], [7], baed o he defo of a o ege egao opeao lmed fequecy fequecy appoach. The model of he yem couou me epeeao ad e ue a oupu eo echque OE o emae paamee [5], [6]. Fo he facoal ae-pace model of he duco mache, he paamee veco defed by: T R L a b 38 m The ae-pace model mulaed ug a umecal egao algohm, hu oe ge: No ege defcao of Roo S Effec duco Mache Abdelhamd ALLOUL

9 35 ˆ SSN: f u, ˆ 39 Whee ˆ a emao of a eao. The opmal value of ˆ obaed by mmzao of he quadac ceo: op * d ˆ d * q ˆ q 4 e oba: ˆ ˆ 4 Whee deped o he opmzao algohm. We ca ue a blac bo echque povded by he Malab oolbo fuco ode o mmze. h cae e a o oba he opmal op hou oyg of ho e oba h emae. Bu h echque pee ome dabac uch a he abece of dec fomao o he ceo a he opmum, hu pacula o he peco evy of h egad o he dffee emae. To emedy hee dabac, e ue evy fuco of he mulaed oupu [5], [6]. Becaue ˆ o lea ˆ, a No Lea Pogammg echque ued o emae eavely ˆ : ˆ ˆ " 4 Wh [5], [ 6]: ˆ, : gade ", : hee : Maquad paamee yˆ, : ev y fuco 43 Th algohm, o a Maquad oe [7], ofe ued o lea opmzao, eue obu covegece pe of a bad alzao of ˆ. A good peco of he oupu ebly fuco, hoeve eceay o eue a good covegece ad peco of he algohm. 6. EXPERMENTAL DENTFCATON ode o appecae he ee of he Pa facoal model h he modal epeeao of he facoal egao, e ue h mehod o defy hee poble model, ug pu/oupu daa povded by a a e bech cludg he duco mache, he daa acquo yem ad he PWM geeao gve hee-phae volage ad cue ad he poo of he moo ale a dffee peed. The amplg peod Te=.7m. Ug he Pa efeece fame led o he oo, e oba daa u ad. 6.. defcao of he Coveoal Model * Le d be he meaued cue ad îd be he mulaed cue, ug coveoal o facoal model. ECE Vol. 3, No. 3, ue 3: dq dq

10 ECE SSN: he quadac ceo hch mmzed accodg o he oupu eo echque ee [3], [5] ad [8] fo moe deal. The paamee h he coveoal Pa model ae defed by: R R l l ] T [ m d.5 meaued emaed 4 R emao R emao L m emao l emao eao eao Fgue 6. Meaued ad emaed cue h coveoal model Fgue 7. Paamee emao h coveoal model 6.. defcao of he Facoal Model H The modal fomulao o adaped o he eac calculao of becaue he ad c ae complcaed fuco of. poble o mplfy ad poceed decly he calculao of he evy fuco [5], [6], [9] by umecal dffeeao, he fom: ˆ, lm ˆ ˆ, ˆ, 44 A pelmay udy eeal fo he choce of. he geeal cae, dffcul o chooe becaue ca vay fom - o. Becaue, eay o fd a opmal value of, hch ll be alay he ame. The he calculao become moe mple. The paamee h he facoal model T H ae defed by: [ R l m a b ]. A ehbed by Fgue 6 ad 8, hee a good f beee meaued ad emaed cue h boh coveoal ad facoal model. H d d 9 meaued emaed eao R emao 5 5 L m emao a emao b emao emao 5 5 eao Fgue 8. Meaued ad emaed cue h facoal model H Fgue 9. Paamee emao h facoal model H No ege defcao of Roo S Effec duco Mache Abdelhamd ALLOUL

11 354 SSN: ode o appecae he mpoveme of H model, eceay o compae he epecve quadac ceo ee Table. obvou ha he facoal model povde a bee appomao of meaueme ha he coveoal Pa model defcao of he Facoal Model H, The model 36 gve a good appomao oly a lo ad medum fequece [8]. ode o mpove he facoal model 36 ad paculaly hgh fequecy appomao, a ecod model popoed. H, b b 45 a a ha bee demoaed ee fo eample [8] ha he phae of he facoal model ha o be b b equal o a hgh fequece,.e h a facoal ode equal o.5. f, H, 4. The f. 5, model ll povde a good appomao a hgh fequece. The paamee of he facoal model H, ae defed by: T [ R l m a a b b ]. Becaue e equal o.5, oly eceay o emae. A pevouly, hee a good f beee meaued ad emaed cue demoaed by Fgue. The Fgue 7, 9 ad epee he paamee vaao dug he defcao. The coepodg quadac ceo of Table dcae ha H, pefom a bee appomao ha he ohe model. We pee he follog able all he eul of epemeal paamee emao d meaued emaed eao Fgue. Meaued ad emaed cue h facoal model H, R emao 3 a emao 3 b emao 3 emao.5 3 eao 3 a emao Fgue. Paamee emao h facoal model.5 L m emao 3 b emao eao H, Table. Emaed paamee Clacal model R R l l m Facoal model H R l m a b Facoal model H, R lm a a b b Clacal model : Quadac ceo = Facoal model H : Quadac ceo = Facoal model H, : Quadac ceo = 7.86 ECE Vol. 3, No. 3, ue 3:

12 ECE SSN: CONCLUSON h pape, e have peeed ad compaed ome model fo he defcao of oo effec duco mache. Tha o Pa afomao e have obaed a coveoal model efeece fame dq elaed o oo. To ae o accou he dffuve pheomea of he effec, he Pa equvale dagam h ladde model ha bee popoed. The, e have eplaced he ladde model by a facoal mpedace. The defcao of he Pa model h a facoal mpedace ha bee pefomed by he oupu eo mehod. Fudameally, h mehod baed o he mulao of he model ad of evy fuco. We have ued he modal appoach o compae hee model h epemeal daa. The eul ho clealy ha he facoal model gve bee appomao ha he coveoal Pa model. Moeove, e have ho ha a e facoal model h o devave able o mpove hee epemeal appomao. APPENDX Ug he comple fomula, he vee Laplace afom L gve by: h H e d A. We ue he Bomch coou ho Fgue. Fg.. Bomch coou C Thu, he mpule epoe h of ay yem ca be calculaed fom afe fuco H. Becaue H a mulfom fuco, a cu eceay he comple plae, coepodg o he coou C of Fgue. Thu e ca e: e d A. π π C AB BDE EH H L LNA Refeg o Cauchy heoem: C A.3 Becaue, No ege defcao of Roo S Effec duco Mache Abdelhamd ALLOUL

13 356 SSN: BDE H LNA A.4 A.5 he: h lm - R AB EH L A.6 Fally e evaluae he egal alog he pah EH ad L. Alog EH, e e d d A.7 ad a goe fom o R, goe fom o R. e e d e EH R R d A.8 Alog L, e e d d A.9 R R e e d d A. e L We hu oba: h lm R R R e e d e e R e e e d d h lm R A. ad fally: h e d A. Becaue coepod o a fequecy, le u defe. Noce ha e he mpule epoe z, e of he pu v. Thu, a moe geeal uao, he epoe vefe he dffeeal equao: z, of he elemeay yem o a pu v ECE Vol. 3, No. 3, ue 3:

14 ECE SSN: z, z, v A.3 ad he oupu of he facoal yem he eghed egal h egh of all he cobuo z, agg fom o : y z, d A.4 A.5 h ACNOWLEDGEMENT Th o a uppoed by he Tua My of Hgh Educao, Reeach ad Techology. REFERENCES [] C Cha, Dougla Boo. S effec abdged copy of h acle appeaed Ped Ccu Deg ad Maufacug, UP Meda, 9. [] S Caa. Cobuo à la modélao dyamque d ode o ee de la mache aychoe à cage. Thèe de l NPT, Face, 5. [3] C Tgeaou, T Poo, L, A Oualoup, F Levo. Modelg ad defcao of a ege ode yem. Poceedg of ECC 99, Euopea Cool Cofeece, aluhe, Gemay, 999. [4] L, T Poo, Tgeaou, R Ouva. Paamee emao of facoal yem: applcao o he modelg of a lead-acd baey. SYSD, h FAC Sympoum o Syem defcao, USA.. [5] L. Modélao e defcao de yème d ode o ee. Thèe de Docoa, Uveé de Poe, Face,. [6] A Oualoup. La dévao o eèe: héoe, yhèe e applcao. Edo Hemè, Pa, 995. [7] T Poo, C Tgeaou. Paamee emao of facoal model: applcao o he modelg of dffuve yem. Poc. 5h FAC Wold Coge, Baceloa, Spa.. [8] D Helechez, D Mago. Dffuve ealzao of facoal ego-dffeeal opeao: ucual aaly ude appomao. Cofeece FAC, Syem, Sucue ad Cool. Face. 998; : [9] G Moey. Dffuve epeeao of peudo dffeeal me opeao. Poceedg ESSAM 998; 5: [] Sabae. O a epeeao of facoal ode yem: ee fo he al codo poblem. 3 d FAC ohop, FDA 8, Ahaa, Tuey, 8. [] G Teode. Eleccal Dve ad Dve ad Cool Techque. Leuve/Belgum: Acco, 4. [] H abba. defcao d u modèle ype ccu pea e compe le effe de féquece da ue mache aychoe à cage d écueul. Thèe de l NPT, Face, 997. [3] A alloul, ella, P Melcho, Tgeaou. Facoal modelg of oo effec duco mache. FAC Wohop o Facoal Dffeeao ad Applcao. Uvey of Eemadua, Badaoz, Spa.. [4] L Lug. Syem defcao heoy fo he ue. Pece-Hall, c., Egleood Clff, Ne eey 763, 987. [5] Rchale, A Raul, R Poulque. defcao de poceu pa la méhode du modèle. Godo ad Beach, 97. [6] C Tgeaou. Recheche de modèle epémeau aée pa odaeu. Lavoe- Tec e Doc Pa, 988. [7] DW Maquad. A algohm fo Lea-Squae emao of No-Lea Paamee.. Soc. du. Appl. Mah., 963; : [8] A Bechellal. Modélao de eface de dffuo à l ade d opéaeu d égao facoae. Thèe de Docoa, Uveé de Poe, Face, 8. [9] T Damah, S Deoue, M Beayeb. Facoal ode yem defcao. TEA 8 May Hammame Tua. 8. [] Mago, D Repéeao e vaable d éa de modèle de gude d ode avec dévao facoae. Thèe de Docoa. Uveé de Pa X, ORSAY, 994. [] Mlle S, Ro B. A oduco o he facoal calculu ad facoal dffeeal equao. oh Wley ad So Ne-Yo, 993. [] Oldham B. Spae. The facoal calculu. Academc Pe Ne-Yo, 974. No ege defcao of Roo S Effec duco Mache Abdelhamd ALLOUL

15 358 SSN: BOGRAPHES OF AUTHORS Abdelhamd ALLOUL a bo Moa 979. He eceved he egee dploma ad mae degee fom Naoal Egeeg School of Sfa ad Moa, Tua, epecvely 4 ad 6. He cuely og fo he PhD degee a Tu Uvey, Tua. H eeach ee ae he facoal modelg of oo effec duco mache h applcao o cool ad dago. haled ELASS a bo 96, Tua. He eceved PhD Eleccal Egeeg 99. He cuely Pofeo a Tu Uvey, Tua. H eeach ee ae maly he aea of modelg ad dago of he faul of he elecomechacal yem. ea-claude TRGEASSOU a bo Lboue 33 Face o decembe, 946. He eceved he Ph.D. degeee auomac cool fom ENSM Nae 98 ad he Docoa d Ea E Scece auomac cool fom Poe Uvey 987. Fom 9888 o 6, he ha bee pofeo a ESP, a egeeg chool a Poe Uvey. Sce 6, he eed ad Hooay Pofeo. H mao eeach ee have bee he mehod of mome h applcao o defcao ad cool ad he paamee emao of couou yem h applcao o he dago of eleccal mache. A pee, he h aocaed o he acve of he MS-LAPS a Bodeau Uvey ad h eeach o deal h modellg, ably, defcao ad cool of facoal ode yem. ECEE Vol. 3, No. 3, ue 3: