Ph.D. in Electrical Engineering XXV Cycle

Transcription

1 Ala Mate Studiou Univeity of Bologna DEPARTMENT OF ELECTRICAL, ELECTRONIC AND INFORMATION ENGINEERING GUGLIELMO MARCONI Ph.D. in Electical Engineeing XXV Cycle Electical Enegy Engineeing (9/E) Powe electonic convete, electical achine and dive (ING-IND/3) Non-Linea Analyi and Deign of Synchonou Beaingle Multihae Peanent Magnet Machine and Dive Ph.D. Thei of STEFANO SERRI Tuto: Pof. Eng. Giovanni Sea Coodinato: Pof. Eng. Doenico Caadei July 3, Bologna, Italy

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5 Table of content Table of content Table of content I Intoduction.. CHAPTER TWO-DIMENSIONAL ANALYSIS OF MAGNETIC FIELD DISTRIBUTIONS IN THE AIRGAP OF ELECTRICAL MACHINES. Intoduction.. 9. Analytical ethod in liteatue....3 Main aution and cae tudy....4 Analytical olution of the oble Analyi in the Region Analyi in the Region Coon bounday condition Cuent heet ditibution of the agnet Concluion. 7.7 Refeence.. 8 I

6 Table of content CHAPTER AN ALGORITHM FOR NON-LINEAR ANALYSIS OF MULTIPHASE BEARINGLESS SURFACE-MOUNTED PM SYNCHRONOUS MACHINES. Intoduction. 9. The agnetic cicuit odel Analytical odel of the eluctance The nueical olving oce Siulating the oveent Co-enegy, toque and adial foce Reult and coaion with FEA oftwae Machine A Machine B Concluion 57.8 Refeence. 58 Aendix A. The Pogaing Code - Pat A.. The ain oga. 6 Aendix A. The Pogaing Code - Pat A.. The MMF aay (ilified veion)... 9 A.. The MMF aay (oiginal veion) CHAPTER 3 PRINCIPLES OF BEARINGLESS MACHINES 3. Intoduction.. 99 II

7 Table of content 3. Geneal incile of agnetic foce geneation Beaingle achine with a dual et of winding Beaingle achine with a ingle et of winding Roto eccenticity. 3.6 Concluion Refeence. 4 CHAPTER 4 AN ANALYTICAL METHOD FOR CALCULATING THE DISTRIBUTION OF FORCES IN A BEARINGLESS MULTIPHASE SURFACE-MOUNTED PM SYNCHRONOUS MACHINE 4. Intoduction Definition of vaiable Analyi of flux denity ditibution in the aiga Calculation of the foce acting on the oto Noal coonent of the foce Tangential coonent of the foce Pojection of the tangential foce Siulation and eult Concluion Refeence.. 8 Aendix A4. Magnetic field ditibution in the aiga of ultihae electical achine A4.. Intoduction. 83 A4.. The ultihae otating agnetic field. 84 III

8 Table of content CHAPTER 5 DESIGN AND DEVELOPMENT OF A CONTROL SYSTEM FOR MULTIPHASE SYNCHRONOUS PERMANENT MAGNET BEARINGLESS MACHINES 5. Intoduction Mechanical equation Geneal tuctue of the contol yte Detailed analyi of the contol yte Levitation foce bloc.... A. Poition eo B. PID contolle.. C. Foce Contolle bloc.. D. Electoagnetic odel bloc 6 E. Foce to oent atix bloc Eule equation bloc... 8 A. Alied oent bloc. 8 B. Eule equation bloc The etting of PID contolle A Equilibiu along the y-axi... 3 B Equilibiu along the z-axi Siulation and eult Concluion... 5 Concluion Lit of ae 59 IV

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11 Intoduction Intoduction The ue of ultihae oto ove conventional thee-hae oto give a eie of benefit that can be uaized a follow: oibility of dividing the owe between ultile hae, highe eliability in cae of failue of a hae, ue of vaiou haonic ode of aiga agnetic field to obtain bette efoance in te of electoagnetic toque and oibility to ceate ultioto dive by connecting eveal achine in eie contolled by a ingle owe convete []-[9]. Thee featue ae aeciated when high owe, high eliability and low dc bu voltage ae equeted a it haen in hi oulion, electical vehicle and aeoace alication. In ecent yea, uitable technique have been alied in ode to educe the owe loe in ultihae IGBT invete []. Beaingle oto ae eading becaue of thei caability of oducing oto uenion foce and toque avoiding the ue of echanical beaing and achieving in thi way uch highe axiu eed. Thee ae two tyologie of winding configuation: dual et and ingle et of winding. The fit categoy coie two eaated gou of thee-hae winding, with a diffeence in thei ole ai nube equal to one: the ain one caie the oto cuent fo diving the oto, while the othe caie the levitation cuent, to uend the oto []. The winding belonging to the latte categoy oduce toque and adial foce by ean of injecting diffeent cuent equence to give odd and even haonic ode of agnetic field, uing the oetie of ultihae cuent yte, which have ultile othogonal d-q lane. One of the can be ued to

12 Intoduction contol the toque. The additional degee of feedo can be ued to oduce levitation foce []. The ain advantage of beaingle oto with a ingle et of winding (i.e., the aet of beaingle and ultihae oto togethe) lead to a ile contuction oce, bette efoance in contol tategy and toque oduction with elatively low owe loe [3]. Thi ind of technology i exected to have vey lage develoent in the futue, aticulaly in the deign of high owe denity geneato, actuato and oto of Moe Electic Aicaft (MEA), ainly fo the ability of achieving highe eed in coaion to conventional electical achine [4]. In addition, it can be uoed that the cooeation of beaingle contol technique and the adotion of agnetic beaing could be of lage inteet in the MEA field. An iotant taget in the deign of electical achine i the analyi and coaion of a lage nube of olution, ending le tie than i oible but alo oviding an accuate decition of electoagnetic henoena. The ain oble ae elated to the calculation of global and local quantitie lie linage fluxe, outut toque, flux denitie in vaiou aea of the device. The difficultie inceae eecially in eence of agnetic atuation, in fact in the cae of non-linea agnetic oble it would be neceay to ovide in-deth analye by uing colex oftwae baed on accuate analytical ethod, lie Finite Eleent Analyi (FEA). Siultaneouly, it would be ueful to ave tie, not only in te of educing couting tie, but ainly fo the need of edeigning the odel of the achine in a CAD inteface when changing oe electical o geoetical aaete. In ode to olve thi oble, oe autho eent analye baed on equivalent agnetic o lued aaete cicuit odel [5], [6], [7]. In thi thei, a ethod fo non-linea analyi and deign of Suface- Mounted Peanent Magnet Synchonou Moto (SPMSM) i eented. The

13 Intoduction elevant edge conit in the oibility of defining the achine chaacteitic in a ile ue inteface. Then, by dulicating an eleentay cell, it i oible to contuct and analyze whateve tyology of winding and aee-tun ditibution in a ole-ai. Futheoe, it i oible to odify the agnet width-to-ole itch atio analyzing vaiou configuation o iulating the oto oveent in inuoidal ultihae dive o in a ue-defined cuent ditibution. Peviou ae ooed the analyi of oen-lot configuation with efixed tuctue of the oto, with given nube of ole and lot, o fo only a aticula oition of the oto with eect to the tato. The efoance of the ooed non-linea odel of SPMSM have been coaed with thoe obtained by FEA oftwae in te of linage fluxe, co-enegy, toque and adial foce. The obtained eult fo a taditional thee-hae achine and fo a 5-hae achine with unconventional winding ditibution howed that the value of local and global quantitie ae actically coinciding, fo value of the tato cuent u to ated value. In addition, they ae vey iila alo in the non-linea behavio even if vey lage cuent value ae injected. When develoing a new achine deign the ooed ethod i ueful not only fo the eduction of couting tie, but ainly fo the ilicity of changing the value of the deign vaiable, being the nueical inut of the oble obtained by changing oe citical aaete, without the need fo edeigning the odel. Fo a given oto oition and fo given tato cuent, the outut toque a well a the adial foce acting on the oving at of a ultihae achine can be calculated. The latte featue ae the algoith aticulaly uitable in ode to deign and analyze beaingle achine. Fo thee eaon, it contitute a ueful tool fo the deign of a beaingle ultihae ynchonou PM achine contol yte. Anothe iotant ection of thi thei concen an analytical odel fo adial foce calculation in ultihae beaingle SPMSM. It allow to edict 3

14 Intoduction alitude and diection of the foce, deending on the value of the toque cuent, of the levitation cuent and of the oto oition. It i baed on the ace vecto ethod, letting the analyi of the achine not only in teady-tate condition but alo duing tanient. When deigning a contol yte fo beaingle achine, it i uual to conide only the inteaction between the ain haonic ode of the tato and oto agnetic field. In ultihae achine, thi can oduce itae in deteining both the odule and the atial hae of the adial foce, due to the inteaction between the highe haonic ode. The eented algoith allow to calculate thee eo, taing into account all the oible inteaction; by eeenting the locu of adial foce vecto, it allow the aoiate coection. In addition, the algoith eit to tudy whateve configuation of SPMSM achine, being aaeteized a a function of the electical and geoetical quantitie, a the coil itch, the width and length of the agnet, the oto oition, the alitude and hae of cuent ace vecto, etc. The deign of a contol yte fo beaingle achine contitute anothe contibution of thi thei. It ileent the above eented analytical odel, taing into account all the oible inteaction between haonic ode of the agnetic field to oduce adial foce and ovide in thi way an accuate electoagnetic odel of the achine. Thi latte i at of a thee-dienional echanical odel whee one end of the oto haft i contained, to iulate the eence of a echanical beaing, while the othe i fee, only uoted by the adial foce develoed in the inteaction between agnetic field, to iulate a beaingle yte with thee degee of feedo. The colete odel eeent the deign of the exeiental yte to be ealized in the laboatoy. 4

15 Intoduction Refeence [] D. Caadei, D. Dujic, E. Levi, G. Sea, A. Tani, and L. Zai, Geneal Modulation Stategy fo Seven-Phae Invete with Indeendent Contol of Multile Voltage Sace Vecto, IEEE Tan. on Indutial Electonic, Vol. 55, NO. 5, May 8, [] Fei Yu, Xiaofeng Zhang, Huaihu Li, Zhihao Ye, The Sace Vecto PWM Contol Reeach of a Multi-Phae Peanent Magnet Synchonou Moto fo Electical Poulion, Electical Machine and Syte (ICEMS), Vol., , Nov. 3. [3] Ruhe Shi, H.A.Toliyat, Vecto Contol of Five-hae Synchonou Reluctance Moto with Sace Pule Width Modulation fo Miniu Switching Loe, Induty Alication Confeence, 36th IAS Annual Meeting. Vol. 3,. 97-3, 3 Set.-4 Oct.. [4] M. A. Abba, R.Chiten, T.M.Jahn, Six-hae Voltage Souce Invete Diven Induction Moto, IEEE Tan. on IA, Vol.IA-, No. 5,. 5-59, 984. [5] E. E. Wad, H. Hae, Peliinay Invetigation of an Invete fed 5-hae Induction Moto, IEE Poc, June 969, Vol. 6(B), No. 6, , 969. [6] Y. Zhao, T. A. Lio, Sace Vecto PWM Contol of Dual Thee-hae Induction Machine Uing Vecto Sace Decooiton, IEEE Tan. on IA, Vol. 3, , 995. [7] Xue S, Wen X.H, Siulation Analyi of A Novel Multihae SVPWM Stategy, 5 IEEE Intenational Confeence on Powe Electonic and Dive Syte (PEDS), , 5. [8] Paa L, H. A. Toliyat, Multihae Peanent Magnet Moto Dive, Induty Alication Confeence, 38th IAS Annual Meeting. Vol.,. 4-48,.-6 Oct. 3. [9] H. Xu, H.A. Toliyat, L.J. Peteen, Five-Phae Induction Moto Dive with DSP-baed Contol Syte, IEEE Tan. on IA, Vol. 7, No. 4, ,. [] L. Zai, M. Mengoni, A. Tani, G. Sea, D. Caadei: "Miniization of the Powe Loe in IGBT Multihae Invete with Caie-Baed Pulewidth Modulation," IEEE Tan. on Indutial Electonic, Vol. 57, No., Novebe, [] A. Chiba, T. Deido, T. Fuao and et al., "An Analyi of Beaingle AC 5

16 Intoduction Moto," IEEE Tan. Enegy Conveion, vol. 9, no., Ma. 994, [] M. Kang, J. Huang, H.-b. Jiang, J.-q. Yang, Pincile and Siulation of a 5-Phae Beaingle Peanent Magnet-Tye Synchonou Moto, Intenational Confeence on Electical Machine and Syte,. 48 5, 7- Oct. 8. [3] S. W.-K. Khoo, "Bidge Configued Winding fo Polyhae Self-Beaing Machine" IEEE Tan. Magnetic, vol. 4, no. 4, Ail. 5, [4] B. B. Choi, Ulta-High-Powe-Denity Moto Being Develoed fo Futue Aicaft, in NASA TM 3-96, Stuctual Mechanic and Dynaic Banch Annual Reot,., Aug. 3. [5] Y. Kano, T. Koaa, N. Matui, Sile Nonlinea Magnetic Analyi fo Peanent-Magnet Moto, IEEE Tan. Ind. Al., vol. 4, no. 5,. 5 4, Set./Oct. 5. [6] B. Sheih-Ghalavand, S. Vaez-Zadeh and A. Haanou Ifahani, An Ioved Magnetic Equivalent Cicuit Model fo Ion-Coe Linea Peanent-Magnet Synchonou Moto, IEEE Tan. on Magnetic, vol. 46, no.,., Jan.. [7] S. Vaez-Zadeh and A. Haanou Ifahani, Enhanced Modeling of Linea Peanent-Magnet Synchonou Moto, IEEE Tan. on Magnetic, vol. 43, no., , Jan. 7. 6

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19 Two-dienional analyi of agnetic field ditibution in the aiga of electical achine Chate TWO-DIMENSIONAL ANALYSIS OF MAGNETIC FIELD DISTRIBUTIONS IN THE AIRGAP OF ELECTRICAL MACHINES. Intoduction The ai of thi chate i the develoent of a ethod ooed in liteatue [] to tudy the ditibution of the agnetic vecto otential, agnetic field and flux denity in the aiga of axial flux eanent agnet electical achine by alying a two-dienional odel. With eect to [], the contibution of thi chate conit in the execution of the colete calculation, not eoted in the oiginal wo, to get the olution of the oble. They wee conducted by uing the technique of atheatical analyi alied to hyical 9

20 Chate and engineeing oble, with aticula efeence to []. In the oigin, the ethod ha been alied to the deign of axial flux PM achine, but it can be genealized to the analyi of any tyology of electical achine in the cae of neglecting lotting effect and with the aution of develoing the achine linealy in coeondence of the ean aiga adiu.. Analytical ethod in liteatue The wo [3]-[6] eeent a eie of ae fo a colete -d analyi of the agnetic field ditibution in buhle PM adial-field achine. In [3] i eented an analytical ethod fo deteining the oen-cicuit aiga field ditibution in the intenal and extenal oto tyologie. The olution i given by the govening field equation in ola coodinate alied to the annula agnet and aiga egion of a ulti-ole lotle oto, with an unifo adial agnetization in the agnet. In [4] the analyi i conducted to deteine the aatue eaction field oduced by a 3-hae tato cuent and to tae into account the effect of winding cuent haonic ode on the aiga field ditibution. In [5], the ethod develoed in [3], [4] i integated with a odel to edict the effect of tato lotting on the agnetic field ditibution, uing a -d eeance function which ealize a uch highe accuacy than the conventional -dienional odel. Finally, [6] eent a odel to analyze the load oeating condition of the oto, by cobining the aatue eaction field coonent with the oen-cicuit field coonent oduced by the agnet, tudied in [3]. All the cae [3]-[6] wee coaed with the eult of FE analyi, howing an excellent ageeent. The ae [7] eent an analytical ethod to tudy agnetic field in eanent-agnet buhle oto, taing into conideation the effect of tato

21 Two-dienional analyi of agnetic field ditibution in the aiga of electical achine lotting, by tudying the agnetic field ditibution in the ituation whee the agnet ae ove the lot oening. In uch ituation it i difficult to inteet the coect ethod fo deteining, with the oely accuacy, the flux denity ditibution and, conequently, the agnetic foce and cogging toque. In [8] the effect of lotting in a buhle dc oto (BLDCM) ae deteined by calculating the aiga eeance ditibution uing the Schwaz- Chitoffel tanfoation. The analytical calculation of no-load ai-ga agnetic field ditibution, aatue field ditibution, and hae electootive foce, ae ileented. Then, a thee-hae cicuital odel i ealized fo deteining the hae cuent wavefo and the intantaneou agnetic field ditibution in load condition, duing the actual oeation of the dive. The coutation of electoagnetic toque and the analyi of toque ile colete the featue of the algoith. The ae [9] eent a ethod fo the accuate calculation of agnetic field ditibution in the oto with big aiga, by ean of the agnetic otential ueioed calculation, ince in the exained cae the couting eo eulting by conventional foula can t be neglected a haen in the all aiga achine. In [] a geneal analytical ethod to edict the agnetic field ditibution in uface-ounted buhle eanent agnet achine i eented, conideing a two-dienional odel in ola coodinate which olve the Lalacian equation in the aiga and agnet aea, with no containt about the ecoil eeability of the agnet. The analyi i alicable to intenal/extenal oto tyologie, to adial/aallel agnetization of the agnet, to lotle/lotted oto.

22 Chate.3 Main aution and cae tudy In the following, the ain aution of the cae tudy ae eented: I) The eeability of ion i infinite; II) The conideed odel i a lotle achine, o a lotted one with lotoening uoed of infiniteial width, o that the lotting effect ae negligible; III) In coeondence of the oto and tato bounday uface, the agnetic field line have only noal coonent; IV) The ean aiga adiu i aued infinite, o that the aiga ath can be conideed a having a linea develoent, ignoing cuvatue; V) Exteity effect ae neglected. VI) The effect of the leaage fluxe ae neglected. The Aee-tun ditibution ae analyzed by ean of the cuent heet technique; the innovative aect of the analyi, eented in [], i the oce of olving the electoagnetic oble deending on a genealized cuent ditibution, whateve be the geneating ouce, and then alying to the geneal olution the cuent-heet elated to the aticula cae tudy (aee-tun ditibution of the tato, equivalent ditibution of the agnet, etc.). Conide the -D odel eented in Fig..: Fig..

23 Two-dienional analyi of agnetic field ditibution in the aiga of electical achine The lowe uface, laced at y, eeent the oto ion; the highe one, laced at y Y, eeent the tato ion. A genealized cuent heet ditibution, given by K ( x) Kˆ in( ux) n n, i laced at y Y coodinate: thi aaete can be aued a a vaiable height, dividing the aiga in two aea and deteining diffeent olution of the agnetic vecto otential in eveyone of the. In thi way, the cuent heet K n ( x) can be conideed in one cae the tato cuent ditibution (in the eented exale, by ubtituting Y Y ) in the othe cae the equivalent aee-tun ditibution oduced by oto agnet (in the eented exale, by ubtituting Y Y, being thi latte the agnet height). So, it i oible fitly to olve the oble fo a genealized ditibution and then to aly it to the aticula cae to eeent. 3

24 Chate 4.4 Analytical olution of the oble Conide by aution that the agnetic vecto otential A ha the only nonzeo coonent z A, not deendent on z-coodinate (i.e., the analyi i caied out by oeating on xy -lane whee all the agnetic and electical quantitie ae uoed invaiant with eect to the z -axi). With thee aution, the Lalace oeato A can be witten a: ( )ˆ y x, A A z (.) ( ) y A x A ĵ y A î x A ˆ z ĵ y î x A A A z z z z z z (.) The x - and y -coonent of flux denity and agnetic field ditibution can be deteined a: ĵ x A î y A A z y x ˆ ĵ î A B z z z (.3), x A B H, y A B H, x A B, y A B z y y z x x z y z x μ μ μ μ (.4) The equation to be olved in the doain of tudy (.5), with it elated bounday condition (.6), i the chaacteitic Lalace equation conideed in a twodienional doain: y A x A A z z (.5)

25 Two-dienional analyi of agnetic field ditibution in the aiga of electical achine ) H x( y ) ) H x( y Y ) ) H x( y Y ) H x( y Y ) Kn( x) ) H ( y Y ) H ( y Y ) 3 4 y y (.6) Since the bounday condition ae hoogeneou, it i oible to aly the ethod of eaation of vaiable. Let u aue, theefoe, that fo: ( x) ( y) A z i of the X Y Az X ( x) Y ( y) Az Y ( y) X ( x) (.7) x y hence, ultilying both ide by [ X ( x) Y ( y) ] deivative in a diffeent way fo bevity, we obtain: and ewiting the econd X ( x) X x ( x) Y ( y) Y y ( y) X ( x) X xx ( x) Y ( y) Y yy ( y) (.8) By iolating in diffeent ebe the te eectively deendent on x and y we obtain: Y Y ( y) ( y) yy ( x) ( x) xx X (.9) X Note that the ebe of the equation ae abolutely indeendent fo each othe, ince the fit one i a function of the vaiable x only, the econd one of the y only: having to be equivalent fo any value aued by the two vaiable, it i deduced that they have to be both equal to a contant te, which we define a u, aued oitive. By futhe develoing the calculation, two eaate diffeential equation ae obtained, each one a a function of a ingle vaiable: ( x) ( x) xx X X u X xx ( x) u X ( x) (.) 5

26 Chate Y Y ( y) ( y) yy ( y) u Y ( ) u Y y yy (.) Ae obtained fo (.), (.) the eective chaacteitic equation and thei olution: u, ± j u ± ju (.) q u q, ± u u (.3) Recalling the geneal exeion of the olution aociated with the chaacteitic equation (.), (.3): X Y jux jux ( x) A e B e (.4) x uy uy ( y) A e B e y y x (.5) whee A x, B x, A y, B y, ae contant te to be evaluated uing the bounday condition. Recalling (.7) i oible to wite: A z jux jux uy uy ( x, y) X ( x) Y ( y) ( A e B e )( A e B e ) (.6) x By intoducing the Eule foula, eented in the follow: x y y e e jux uy co coh jux ( ux) j in( ux), e co( ux) j in( ux) uy ( uy) inh( uy), e coh( uy) inh( uy) (.7) (.8) and uing (.7) and (.8) in (.6), the geneal olution can be exeed in a tigonoetic fo: ( x,y) X ( x) Y ( y) [ Ain( ux) Bco( ux) ][ C inh( uy) Dcoh( uy) ] A z (.9).4. Analyi in the Region Aue fo egion the following geneal exeion fo the agnetic vecto 6

27 Two-dienional analyi of agnetic field ditibution in the aiga of electical achine otential: ( x,y) X ( x) Y ( y) [ A in( ux) B co( ux) ][ A inh( uy) B coh( uy) ] A z (.) The value of the x -coonent of the agnetic field in the lowe bounday of the egion, lead to the fit bounday condition: ( y ) H x (.) H A y u μ z ( x,y) [ A in( ux) B co( ux) ][ A coh( uy) B inh( uy) ] x μ (.) By alying the condition (.) in (.) and conideing that the equation ha to be veified fo any value of x and y, we obtain (.3): u H x (.3) ( y ) [ A in( ux) B co( ux) ] A A μ By ubtituting the eult of (.3) into (.): ( x,y) B coh( uy) [ A in( ux) B co( ux) ] A z (.4) whee, defining the contant te B A and BB : ( x, y) coh( uy) [ in( ux) co( ux) ] A z (.5) By ubtituting the eult of (.3) into (.): u (.6) μ ( x, y) inh( uy) [ in( ux) co( ux) ] H x By executing iila calculation i oible to wite the H y coonent of the agnetic field a: H A u (.7) μ x μ z ( x, y) coh( uy) [ in( ux) co( ux) ] y 7

28 Chate.4. Analyi in the Region Aue fo egion the following geneal exeion fo the agnetic vecto otential: ( x, y) X ( x) Y( y) [ C in( ux) D co( ux) ][ C inh( uy) D coh( uy) ] A z (.8) The value of the x -coonent of the agnetic field in the highe bounday of the egion, lead to the econd bounday condition: ( y Y ) H x (.9) H A y u μ z ( x, y) [ C in( ux) D co( ux) ][ C coh( uy) D inh( uy) ] x μ (.3) By alying the condition (.9) in (.3) and conideing that the equation ha to be valid fo any value of x and u, we obtain (.3), (.3): u ( y Y ) [ C in( ux) D co( ux) ][ C coh( uy ) D inh( uy )] H x μ C ( uy ) D inh( uy ) D C coth( ) coh uy (.3) (.3) By ubtituting (.3) in (.8): ( x, y) C [ C in( ux) D co( ux) ][ inh( uy) coh( uy) coth( )] A z uy (.33) whee, in a iila way to what wa done fo the egion, by intoducing the contant te 3 CC and 4 DC, it give: ( x,y) [ in( ux) co( ux) ][ inh( uy) coh( uy) coth( )] A z 3 4 uy (.34) 8

29 Two-dienional analyi of agnetic field ditibution in the aiga of electical achine which can be witten, by exlicating coth ( uy ), a: A z ( x, y) [ in( ux) co( ux) ] By conideing that: 3 4 inh ( uy) inh( uy ) coh( uy) coh( uy ) inh( uy ) ( Y y) coh( uy uy) coh( uy ) coh( uy) inh( uy ) inh( uy) coh u The elationhi (.35) can be ilified a in (.37): A z ( x, y) [ in( ux) co( ux) ] ( Y y) ( uy ) (.35) (.36) coh u 3 4 (.37) inh and, by ean of (.38), the elated coonent of the agnetic field in egion can be calculated a in (.39), (.4): H x A y z ( x, y), H ( x, y) Az x y μ μ (.38) H x H y ( x, y) [ in( ux) co( ux) ] ( Y y) ( uy ) u inh u 3 4 (.39) μ inh ( x, y) [ co( ux) in( ux) ] ( Y y) ( uy ) u coh u 3 4 (.4) μ inh.4.3 Coon bounday condition Conideing a cuent heet decibed by ean of an haonic ditibution given in the geneic fo: ( x) Kˆ in( ux) Kn n (.4) whee Kˆ n deend on the actual cuent ditibution and ha to be evaluated in 9

30 Chate any aticula conideed cae, while u i defined a follow: π u n (.4) τ The dicontinuity between the x -coonent value of the agnetic field in the cuent heet egion, lead to the thid bounday condition: H ( y Y ) H ( y Y ) K ( x) x x n (.43) which can be exeed by calculating (.6) and (.39) in coeondence of the aticula value y Y. By ubtituting the in (.43) it give: u μ [ in( ux) co( ux) ] 3 4 inh u inh ( Y Y ) ( uy ) By collecting the coon te in (.44): in ( ux) 3 inh u inh ( Y Y ) ( uy ) inh ( uy ) co ( ux) u μ μkˆ u 4 [ in( ux) co( ux) ] inh( uy ) n inh u inh ( Y Y ) ( uy ) Kˆ inh n ( ux) in (.44) (.45) ( uy ) Note that (.45) ha to be veified fo any value of u and x, o the only oibility i that both the coefficient of in ( ux) and ( ux) ( Y Y ) ( uy ) ( uy ) 3 inh inh u co ae equal to zeo: inh u μ Kˆ n (.46) ( Y Y ) ( uy ) inh u (.47) ( uy ) 4 inh inh Afte a few te (.46) and (.47) give, eectively, the elation 3 f ( ) and f ( ): 4

31 Two-dienional analyi of agnetic field ditibution in the aiga of electical achine 3 [ ] ( uy ) μkˆ n u inh( uy ) u inh u( Y Y ) inh (.48) ( uy ) inh( uy ) ( Y Y ) 4 inh u inh (.49) Putting (.48) and (.49) in (.4): H y ( x,y) u coh u μ ( Y y) μkˆ n u inh( uy ) co( ux) u inh u( Y Y ) inh( uy ) inh u( Y Y ) in ( ux) (.5) The continuity between the y -coonent of the agnetic field in the cuent heet egion, lead to the fouth bounday condition: H ( y Y ) H ( y ) (.5) y y Y By calculating (.5) and (.7) in y Y it eectively give (.5) and (.53); by ubtituting the in (.5), it give (.54): H y ( y Y ) u coh u μ ( Y Y ) μkˆ n u inh( uy ) co( ux) u inh u( Y Y ) inh( uy ) inh u( Y Y ) in ( ux) (.5) u (.53) μ ( y Y ) coh( uy )[ in( ux) co( ux) ] H y u coh u μ ( Y Y ) μkˆ n u inh( uy ) u inh u( Y Y ) By collecting coon te in (.54): co u μ ( ux) coh inh inh u ( uy ) ( Y Y ) in ( ux) ( uy )[ in( ux) co( ux) ] (.54)

32 Chate μ Kˆ n u μ inh ( uy ) u coth u inh ( Y Y ) coh( uy ) co( ux) ( uy ) coth u μ ( Y Y ) u μ u μ coh ( uy ) in( ux) (.55) A een befoe, (.55) ha to be veified fo any value of u and x, o the only oibility i that both the coefficient of in ( ux) and ( ux) Fo (.55) the equation (.56): ( Y Y ) ( uy ) co ae equal to zeo. u inh u Kˆ n coth u (.56) ( Y Y ) coh( uy ) coth u μ μ which eult afte a few te in (.57): coth u( Y Y ) ( uy ) coth u( Y Y ) coh( uy ) μkˆ n (.57) u inh and alo the equation (.58): ( Y Y ) coth u u u inh( uy ) coh( uy ) (.58) μ μ which eult in (.59): (.59) Note that (.57) can be ilified, by ilifying the te u( ) nueato and denoinato. Afte a few te, it give: ( Y Y ) ( uy ) n coh u u inh coth Y Y in the μ Kˆ (.6) By ubtituting (.6) in (.48) and efoing oe iila calculation, a ilified fo fo 3 can be obtained: μkˆ n 3 coh ( uy ) (.6) u

33 Two-dienional analyi of agnetic field ditibution in the aiga of electical achine Finally, uing the eult of (.59) in (.49), it give: 4 (.6) All the coefficient ae now nown; thu, i oible to deteine the exeion of agnetic vecto otential and of agnetic field in the egion of the achine. By ubtituting (.59) and (.6) in (.5), it iediately give: A z ( x, y) ( Y Y ) ( uy ) μkˆ n coh u in( ux) coh( uy) (.63) u inh Siilaly, by ubtituting (.6) and (.6) in (.37): ( uy ) ( uy ) μkˆ n coh Az ( x, y) in( ux) coh u( Y y) (.64) u inh Fo (.63) and (.64) ae deived the following elationhi (.65)-(.68): H H x y ( x, y) ( x, y) ( Y Y ) ( uy ) Az coh u Kˆ n in( ux) inh( uy) (.65) μ y inh ( Y Y ) ( uy ) Az coh u Kˆ n co( ux) coh( uy) (.66) μ x inh H H x y ( x, y) ( x, y) ( uy ) ( uy ) Az coh Kˆ n in μ y inh ( uy ) ( uy ) ( ux) inh u( Y y) Az coh Kˆ n co μ x inh ( ux) coh u( Y y) (.67) (.68).5 Cuent heet ditibution of the agnet A a aticula exale of a cuent heet ditibution K n ( x), will be exained the equivalent cuent denity ditibution of the agnet. Each agnet i eeented by two cuent ule at it edge, auing to flow in a tending to 3

34 Chate zeo thicne, having an angula width equal to δ. J ( x) Ĵ in( nθ) Ĵ in n x Ĵ in( ux) n n, 3, 5.. n, 3, 5.. n π τ n n, 3, 5.. (.69) The function i eeented by ean of the Fouie haonic eie ditibution, the coefficient of which ae calculated in the following: Ĵ n π π π π j θ δ θ δ J π n J nπ π ( θ) in( nθ) dθ j( θ) in( nθ) J in π ( nθ) dθ J in( nθ) θ δ πθ δ {[ co( nθ) ] [ co( nθ) ] } θ δ π { in( nθ ) in( nδ ) in( nπ nθ ) in( nδ )} { in( nδ )[ in( nθ ) in( nπ nθ )]} 4J nπ 4J in nπ πθ δ πθ δ ( nδ ) dθ dθ πθ δ nπ nθ in co nπ Conideing that n i an odd nube, the value of ( nπ ) zeo: nθ co nπ co (.7) co in (.7) i alway π π π ( nθ ) co n in( nθ ) in n in( nθ ) in n By ubtituting the eult of (.7) in (.7), it give: (.7) Ĵ n 8J in nπ π 8J ( nδ ) in n in( nθ ) in( nδ ) in( nθ ) nπ (.7) The equivalent uface cuent denity elated to the agnet i exeed by (.73): 4

35 Two-dienional analyi of agnetic field ditibution in the aiga of electical achine [ A / ] Hcπ Be π J (.73) δ τ μ δ τ By ubtituting (.73) in (.7), i alo neceay to calculate the liit a δ tend to zeo, conideing evey edge of the agnet a a cuent ule: Ĵ n 8 B e li δ π μ π τ in nδ ( nδ ) in( nθ ) (.74) Being: li δ in nδ It eult: ( nδ ) n (.75) Ĵ n 4B μ τ e in 4Be ( ) π τ nθ μ τ in n τ (.76) By ubtituting the elationhi (.76) in (.69), it give: J n, 3, 5.. 4B μ τ e ( ) x in n in( nθ) π τ τ (.77) Conideing that: π u n n π θ n x ux (.78) τ τ By ubtituting (.78) in (.77), it eult: J n, 3, 5.. 4B μ τ e ( ) x in n in( ux) π τ τ (.79) To define the function of ditibution K n ( x), i iotant to note that the agnet ae contituted by a ucceion of cuent heet, each one of 5

36 Chate infiniteial width dy, thu chaacteized by a linea cuent denity given a: ( x) Ĵ dy in( ux) [ A / ] K n n (.8) The exeion of agnetic vecto otential in the egion, given by the agnet ditibution (.8) can be obtained by integating (.64) ove the agnet thicne Y : A z ( x, y) Y μĵ u n coh inh ( uy ) ( uy ) μ Ĵ u in n ( ux) coh u( Y y) inh inh ( uy ) ( uy ) in dy ( ux) coh u( Y y) (.8) Note that the aticula fo of equation (.8), which eeent in thi cae the agnet ditibution, elace the geneal function in( ux) (.64). Kˆ n in the equation 6

37 Two-dienional analyi of agnetic field ditibution in the aiga of electical achine.6 Concluion In thi chate a ethod ooed in liteatue wa develoed to tudy the ditibution of the agnetic vecto otential, agnetic field and flux denity in the aiga of axial flux eanent agnet electical achine by alying a twodienional odel. The contibution of thi chate with eect to the exained wo, conit in the execution of the colete calculation, which ae not eented in the oiginal ae, to get the olution of the oble. They wee conducted by uing the technique of atheatical analyi alied to hyical and engineeing oble. Thi ethod can be genealized to the analyi of any tyology of electical achine in the cae of neglecting lotting effect and with the aution of develoing the achine linealy in coeondence of the ean aiga adiu. 7

38 Chate.7 Refeence [] J.R. Buby, R. Matin, M.A Muelle, E. Soone, N.L. Bown and B.J. Chale, Electoagnetic deign of axial-flux eanent agnet achine, IEEE Poc.-Elect. Powe Al., Vol. 5, No., Mach 4 [] P. Zecca, Poblei al bodo e le Equazioni Diffeenziali, Diene dei coi univeitai, htt:// [3] Z. Zhu, D. Howe, E. Bolte, and B. Aceann, Intantaneou agnetic field ditibution in buhle eanent agnet dc oto, Pat I: Oencicuit field IEEE Tanaction on Magnetic, vol. 9, no.,. 4-35, Jan [4] Z. Q. Zhu and D. Howe, Intantaneou agnetic field ditibution in buhle eanent agnet dc oto, Pat II: Aatue eaction field, IEEE Tan. Magn., vol. 9, no.,. 36-4, Jan [5] Z. Q. Zhu and D. Howe, Intantaneou agnetic-field ditibution in buhle eanent agnet dc oto, Pat III: Effect of tato lotting, IEEE Tan. Magn., vol. 9, no.,. 43-5, Jan [6] Z. Q. Zhu and D. Howe, Intantaneou agnetic field ditibution in buhle eanent agnet dc oto, Pat IV: Magnetic field on load, IEEE Tan. Magn., vol. 9, no.,. 5-58, Jan [7] Z. J. Liu and J. T. Li, Analytical olution of ai-ga field in eanentagnet oto taing into account the effect of ole tanition ove lot, IEEE Tan. Magn., vol. 43, no., , Oct. 7. [8] X. Wang, Q. Li, S. Wang, and Q. Li, Analytical calculation of ai-ga agnetic field ditibution and intantaneou chaacteitic of buhle dc oto, IEEE Tan. Enegy Conve., vol. 8, no. 3, , Se. 3. [9] G. Meng, H. Li and H. Xiong, Calculation of big ai-ga agnetic field in oly-hae ulti-ole BLDC oto, Intenational Confeence on Electical Machine and Syte, ICEMS 8, [] Z.Q. Zhu, D. Howe and C.C. Chan, Ioved Analytical Model fo Pedicting the Magnetic Field Ditibution in Buhle Peanent-Magnet Machine, IEEE Tan. Magnetic, vol. 38, no.,,

39 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine Chate AN ALGORITHM FOR NON- LINEAR ANALYSIS OF MULTIPHASE BEARINGLESS SURFACE-MOUNTED PM SYNCHRONOUS MACHINES. Intoduction In ecent yea, oe and oe advanced technologie and an ieive ie in the ue of electonic, both in civil a in the indutial ecto, given a contibution to educe the cot of the coonent, allowing the ue of colex technologie which in the at had high cot and theefoe of little indutial 9

40 Chate inteet. In the field of electical achine thi evolution led not only to the ealization of owe dive contolled by an invete, caable of enuing efoance ignificantly bette than thoe obtained with the eviou contol yte, but alo the advent of a new tye of achine with a diffeent nube of hae fo the taditional thee-hae, uually eloyed in geneation and ditibution of electic enegy. Thi ha eawaened the inteet in the tudy of ulti-hae electical achine. In [], a geneal odulation tategy i eented to be ued in ultioto dive and in ultihae oto dive fo ioving the toque denity. In [] a chee, functional to ileent a ace vecto PWM contol of a twelve-hae eanent agnet ynchonou oto i analyzed, to educe the witching loe without affecting efoance. A oto field oiented baed on the ace vecto PWM (SVPWM) technique fo a 5-hae ynchonou eluctance oto i develoed in [3] and veified uing a dedicated invete. In [4] the tato of an induction achine i ewound with two thee-hae winding et dilaced fo each othe by 3 electical degee, howing that thi winding configuation eliinate oto coe loe and toque haonic of aticula ode and the ixth haonic doinant toque ile. In [5] an inveto-fed 5-hae induction oto i coaed with a coeonding 3-hae oto, howing that the alitude of the toque fluctuation i educed to aoxiately one thid. The ace vecto decooition technique i eented in [6], whee the analytical odeling and contol of the achine ae develoed in thee - dienional othogonal ubace which eit to decoule the vaiable elated to the contol of haonic contibution. In [7] a novel ultihae SVPWM tategy i eented, able to yntheize the d-q ubace voltage vecto to accolih the contol equieent and 3

41 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine ae null the eultant voltage vecto on othe ubace, iniizing the witching loe. The advantage of ultihae achine ae exlained and dicued in [8]: caability of iove the toque oduction by injecting haonic of cuent in the oto, a bette toque and flux adjutent in DTC contol, the fault eilient cuent contol of ulti hae dive unde lo of hae and the oibility of contolling ulti oto though a ingle invete. The ace vecto contol and diect toque contol (DTC) chee ae eented in [9], alied to the oeation of a 5-hae induction oto and uing a fully digital ileentation. Exeiental eult how that an otial contol caability i obtained fo both ethod, futhe validating the theoetical concet. In the lat yea, vaiou technique have been alied in ode to educe the owe loe in ultihae IGBT invete []. The ultihae featue eult aticulaly uitable in beaingle achine, caable of oducing oto uenion foce and toque avoiding the ue of echanical beaing and achieving in thi way uch highe axiu eed []. Thee ae two tyologie of winding configuation: dual et and ingle et of winding. The fit categoy coie two eaated gou of thee-hae winding, with a diffeence in thei ole ai nube equal to one: the ain one caie the oto cuent fo diving the oto, while the othe caie the levitation cuent, to uend the oto []. The winding belonging to the latte categoy oduce toque and adial foce by ean of injecting diffeent cuent equence to give odd and even haonic ode of agnetic field, uing the oetie of ultihae cuent yte, which have ultile othogonal d-q lane. One of the can be ued to contol the toque, the additional degee of feedo can be ued to oduce levitation foce [], a will be exlained in chate 3. The ain advantage of beaingle oto with a ingle et of winding (i.e. of ultihae tye) conit 3

42 Chate of a ile contuction oce, bette efoance in contol tategy and toque oduction with elatively low owe loe [3]. Thi ind of technology i exected to have vey lage develoent in the futue, aticulaly in the deign of high owe denity geneato, actuato and oto of Moe Electic Aicaft (MEA), ainly fo the ability of achieving highe eed in coaion to conventional electical achine [4]. In addition, it can be uoed that the chaacteitic of beaingle contol technique and the ue of agnetic beaing could be of lage inteet in the MEA field. The oibility of aing quic analye, with the coaion of a lage nube of olution, nevethele oviding an accuate calculation of electoagnetic quantitie, eeent a elevant goal in the deign of electical achine, by analyzing global and local quantitie a outut toque, agnetic enegy and co-enegy, linage fluxe, agnetic field and flux denitie in any at of the achine. The difficultie inceae eecially in eence of agnetic atuation; in ode to olve thee oble, the equivalent agnetic cicuit ethod let fat odification of the geoetical and electical aaete ily by vaying nueical inut and, at the ae tie, obtaining an high accuacy in calculation with eect to othe oftwae baed on oe in-deth analytical ethod, a Finite Eleent Analyi (FEA). Peviou ae ooed the analyi of oen-lot configuation with a efixed tuctue of the oto, with a given nube of ole and lot [5], o by tudying only aticula oition of the oto with eect to the tato without elative oveent [6], [7]. Thi chate eent an algoith fo nonlinea agnetic analyi of ultihae uface-ounted eanent-agnet achine with ei-cloed lot. The elevant edge of the ethod conit in the oibility of defining the achine chaacteitic in a ile ue inteface. Then, by dulicating an eleentay cell, it i oible to contuct and analyze 3

43 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine whateve tyology of winding and aee-tun ditibution in a ole-ai. Futheoe, it i oible to odify the agnet width-to-ole itch atio analyzing vaiou configuation in ode to iniize the cogging toque, o iulating the oto oveent in inuoidal ultihae dive o in a uedefined cuent ditibution. Finally, the caability of adial foce calculation allow to deteine the otial aee-tun ditibution in the deign of a beaingle contol of the oto. The choice of uing a oftwae baed on the equivalent agnetic cicuit ethod allow elevant tie aving fo thi ind of analyi with eect to a FEA oftwae, not only due to the eduction of couting tie, but ainly fo the ile change of electical and geoetical aaete (i.e. the nueical inut of the oble), without the need of e-deigning the odel in a CAD inteface. 33

44 Chate. The Magnetic Cicuit Model The baic eleent of the agnetic netwo i hown in Fig.., whoe the elated eluctance ae highlighted. Fig... The baic eleent of the agnetic netwo It conit of one tooth and the adjacent two ei-lot, being cooed of 8 eluctance eeenting ub-doain of the achine, i.e. volue of teeth, ection of the aiga, of the agnet, banche of yoe, ei-lot, etc. Beide of conideing longitudinal coonent, in the odel wee ovided tanvee coonent of the agnetic fluxe a, fo exale, in the lot aea and in the lot-oening to tae into account the leaage ath [8], in the ti of the tooth and in the banche of tato and oto yoe. To contuct the whole odel of a oto, the i-th baic cell i connected to the eviou one though fou tanvee eluctance: N cv i -, N cv i - 34

45 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine (tato and oto yoe), 3N cv i - (lot aea), 8N cv i - (lot-oening aea). Futheoe, the i-th baic cell i connected to the following one though the eleent N cv i, N cv i, 3N cv i, and 8N cv i. Conideing one ole ai of the odel, that coie two ga between the agnet (each one of the ovide 4 additional te), the whole netwo eult in a nube of eluctance, i.e. unnown te, equal to 8N cv 8, being N cv the nube of lot e ole ai... Analytical Model of the Reluctance In thi ubection oe foula and citeia ued to deteine the ot elevant aaete of the agnetic cicuit ae decibed. Fig... Configuation of the netwo in the cae of unifo agnet Eleent i to N cv, i 5N cv to 6N cv To ovide a oe ealitic eeentation of the flux line coing the aiga, 35

46 Chate the total agnetic flux in a lot itch, aing though the agnet, wa divided into thee tube (Fig..): the one in the iddle eent the tooth uface a co-ectional aea, deending on the ean adiu R fo the agnet zone (.) and on the ean adiu R g fo the aiga zone (.). The elated eluctance ae eectively calculated a: R i μ R L α Ldg L (.) R i μ R g g α Ldg L (.) whee the eaning of the ybol i hown in Fig. 4 and in Tab. I. Eleent i 7N cv to 8N cv, i 9N cv to N cv, eleent i N cv to N cv, i N cv to N cv The two tube in left and ight ideway oition with eect to the tooth, develo thei ath aco the aiga in a ucceion of a taight line and a cicufeential ac, cloing in the tooth ti [4], [8]. The elated eluctance ae calculated a: R i Lμ hcl g π d μ π g πh Lln g cl (.3) In the agnet, flux ath ae develoed in adial diection, uing the ile conventional foula: R i μ R L α a L (.4) 36

47 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine Eleent i N cv to 4N cv Futheoe, a decition of the leaage flux in the ga between the agnet i ovided by uing tanvee eluctance. Thee ceate a cloed loo including the agnet, the aiga and the tooth ti. Thi ituation ha a not negligible effect on PM achine [5] and i decibed in the liteatue [9]. The eluctance ued to decibe a tooth ti i given by the eie of two eleent, a ectangulahaed one and a taezoidal-haed one (Fig..3 a): Fig..3. Refeence yte fo calculating the tanvee eluctance: a) tooth ti, b) lot aea. R i μ i L h dt L μ h L h h ln L h ( ) ( ) ( ) B,H bd i B,H cl bd bd L dg dt cl (.5) Note that the eluctance laced in the ion have a value of the eeability which deend on the agnetic chaacteitic of the ateial, thu chaacteized a μ i( B,H ). Eleent i 3N cv to 4N cv, i 8N cv to 9N cv Fo evaluating the leaage flux oduced by the tato cuent wee ued two tanvee eluctance: one though the ai of the lot (.6), calculated a the 37

48 Chate aallel connection of two eluctance (Fig..3 b), the othe aco the lot oening (.7): R i μ h L ln L μ avv tc L ( Ltc L fc ) fc L ( hbd hcl ) Lcl ( ) ln Lcl Ltc Ltc (.6) Lcl R i (.7) μ h L cl Othe eleent The eluctance elated to othe eleent ae not eoted becaue of thei ile fo. Note that, in geneal, the non-linea ub-doain have a value of eeability that deend on the B-H cuve, a in (.5)..3 The Nueical Solving Poce The oble i decibed though a non-linea yte of n equation (n 8N cv 8) whee the unnown ae the value of the agnetic fluxe φ, φ, φ n in evey ub-doain of the achine. Two incile of electoagneti ae ued to wite the equation: Hoinon law, alied along cloed ath identified in the achine, and Gau law, i.e. conevation of the agnetic fluxe incoing in and outgoing fo the node of the netwo (continuity equation). Oveall, the yte include 9N cv 5 equation of the fit tyology and 9N cv 3 of the econd tyology, with a atix fo defined a follow: [ A ] ϕ F M (.8) 38

49 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine Matix [A] in (.8) can be een a cooed by oe bloc deending on the following chaacteization: it coefficient a ij eeent agnetic eluctance in the ow elated to Hoinon law equation, wheea they have the value ± in the ow elated to Gau law equation, being eentially algebaic u of fluxe: a a ij ij ( μ, θ ) R x i.. 5N cv, N cv.. 4 N cv, 6 N cv....8 N cv, 8 N cv, 8 N cv, 8 N cv 5, 8 N cv 6 ± i 5N cv.. N cv, 4 N cv.. 6 N cv, 8 N 3, 8 N 4, 8 N 7, 8 N 8 cv cv cv cv (.9) The yte, divided in gou of equation accoding to the diffeent aea of the achine, i ecified in detail a follow (..8). Note that the index i vaie in evey gou fo to N cv deending on the baic cell elated to the exained equation, excet fo gou (.6), whee the fit equation of the gou i ubtituted by (.5): Tooth to tooth aco the aiga ( i to ) N cv R( i) ϕ( i) R( i ) ϕ( i ) R( i) ϕ( i) R( i) ϕ( i) R( 4 i) ϕ( 4 i) R( 4 i ) ϕ( 4 i ) R( 5 i) ϕ( 5 i) Ncv R( 5 i ) ϕ( 5 i ) R( 6 i) ϕ( 6 i) N cv R( 6 i ) ϕ( 6 i ) FM ( i) Ncv Ncv N cv Ncv Ncv Ncv N cv Ncv Ncv N cv Ncv Ncv Ncv Ncv (.) Highe lot aea between two teeth ( i to ) N cv N cv R( i) ϕ( i) R( 3 i) ϕ( 3 i) R( 6 i) ϕ( 6 i) N cv R( 6 i ) ϕ( 6 i ) F M ( i) N cv N cv N cv N cv N cv N cv N cv N cv (.) 39

50 Chate Tooth to tooth aound the lot aea ( i to 3 ) N cv N cv R( i) ϕ( i) R( 4 i) ϕ( 4 i) R( 4 i ) ϕ( 4 i ) N cv R( 6 i) ϕ( 6 i) R( 6 i ) ϕ( 6 i ) R( 8 i) ϕ( 8 i) Ncv R( i ) ϕ( i ) R( 3 i) ϕ( 3 i) F M ( i) Ncv N cv Ncv Ncv N cv Ncv N cv Ncv Ncv Ncv N cv Ncv N cv Ncv Ncv (.) Right tooth ti aco the aiga ( i 3 to 4 ) N cv N cv R( i) ϕ( i) R( 5 i) ϕ( 5 i) R( 9 i) ϕ( 9 i) R( i) ϕ( i) R( 3 i) ϕ( 3 i) N cv Ncv N cv Ncv N cv Ncv N cv Ncv (.3) Left tooth ti aco the aiga ( i 4 to 5 ) N cv N cv R() i ϕ() i R( 5 i) ϕ( 5 i) R( 7 i) ϕ( 7 i) R( i) ϕ( i) R( i) ϕ( i) N cv Ncv N cv Ncv N cv Ncv N cv Ncv (.4) Cloing equation ( i 5 N ) N cv cv R ( N i) ϕ( i). (.5) cv Ncv i Node ( ) ( i 5 to 6 ) N cv N cv ϕ cv Ncv Ncv Ncv Ncv ( N i ) ϕ( i) ϕ( 6 i) ϕ( 6 i) ϕ( 7 i) (.6) 4

51 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine ( i 6 to 7 ) N cv N cv ϕ( 3 i) ϕ( 3 i) ϕ( 4 i) ϕ( 6 i) ϕ( 4 i) Ncv ϕ( 5 i) ϕ( 6 i) ϕ( 7 i) Ncv Ncv Ncv Ncv Ncv Ncv Ncv (.7) ( i 7 to 8 ) N cv N cv ϕ cv Ncv Ncv Ncv ( 4 N i) ϕ( 5 i) ϕ( i) ϕ( 3 i) (.8) ( i 8 to 9 ) N cv N cv ϕ Ncv i () i ϕ( 5 ) (.9) ( i 9 to ) N cv N cv ϕ Ncv Ncv Ncv Ncv () i ϕ( i) ϕ( i) ϕ( i) ϕ( i) (.) ( i to ) N cv N cv ϕ cv Ncv Ncv Ncv ( 7 N i) ϕ( 8 i) ϕ( i) ϕ( 4 i) (.) ( i to ) N cv N cv ϕ cv Ncv Ncv Ncv ( 8 N i) ϕ( 9 i) ϕ( 3 i) ϕ( 5 i) (.) Right ei-lot aea ( i to 3 ) N cv N cv R( 4 i) ϕ( 4 i) R( 6 i) ϕ( 6 i) R( 3 i) ϕ( 3 i) N cv R( 5 i) ϕ( 5 i) R( 7 i) ϕ( 7 i) F M ( i) Ncv N cv Ncv N cv Ncv N cv Ncv N cv Ncv N cv (.3) 4

52 Chate Lowe tooth aea between two lot ( i 3 to 4 ) N cv N cv R( i) ϕ( i) R( 3 i) ϕ( 3 i) Ncv F Ncv R( 4 i) ϕ( 4 i) R( 5 i) ϕ( 5 i) M ( 3 i) Ncv Ncv Ncv Ncv Ncv Ncv Ncv (.4) Node (6-8) ( i 4 to 5 ) N cv N cv ϕ cv Ncv ( 7 N i) ϕ( i) (.5) ( i 5 to 6 ) N cv N cv ϕ cv Ncv ( 9 N i) ϕ( i) (.6) Highe left ei-lot aea ( i 6 to 7 ) N cv N cv cv Ncv Ncv Ncv 6Ncv R( 6 N i) ϕ( 6 i) R( 6 i) ϕ( 6 i) F M ( i) (.7) Highe ight ei-lot aea ( i 7 to 8 ) N cv N cv R cv Ncv Ncv Ncv 7Ncv ( 6 N i) ϕ( 6 i) R( 7 i) ϕ( 7 i) F M ( i) (.8) The fo of the eaining eight equation i decibed in the next ection: they ae ued fo decibing additional banche which ae foed duing the oveent of the oto. The nown te F M(i), i to n, eeent the aeetun lined by the ath elated to the Hoinon law equation. Note that the ow 5N cv (4) eeent the equation which ae the yte 4

53 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine olvable: it can be een a a bounday condition equation, cloing the ath along the banche of the tato yoe. The olving oce i baed on the ethod of Gauian eliination, fo educing the atix of coefficient to a tiangula one. It i alied iteatively - tie, being oe coefficient deendent on the oto oition (a ij() f(θ x )), oe othe alo deendent on the value of agnetic eeability of the i-th ubdoain (a ij() f(μ i(), θ x )). Fo a given oto oition θ x the length and thicne fo the flux tube that change dienion o oition ae ecalculated. Stating with initial ando value of R i(), i to n, the olution of the yte in the -th ode of iteation i obtained in te of φ i(), i to n, by olving the yte (.8): it i then oible to deteine the value of flux denitie B i() being nown the flux tube co-ectional aea S i. An inteolation of the agnetic chaacteitic H i f(b i ) i ileented fo doain occuied by nonlinea agnetic ateial, while a contant value of μ i i ued fo linea doain. It i then oible to calculate the -th value μ i() by cobining the actual value ˆμ i( ) and the eviou value μ i(-) and, fo evey ode of iteation, e-calculating the agnetic eeabilitie μ i() elated to non-linea doain, following the geneal citeion, to facilitate the convegence oce [5]: Bi ( ) μ ˆ i( ) (.9) Hi( ) d d μi( ) μˆ i ( ) μi( ) (.3) whee the value of d, the daing contant, i choen equal to.. Conequently, the eluctance R i() f(μ i(), θ x ) ae udated, leading to a futhe te fo the Gau ethod, until the following condition i atified [5]: μi( ) μi( ) μi( ) δ (.3) 43

54 Chate being δ the equeted accuacy. At thi te, all the agnetic quantitie elated to each ub-doain of the achine ae nown: eeabilitie μ i, agnetic field H i, fluxe φ i, flux denitie B i..4 Siulating the Moveent The analyi in the eence of oveent oceed with an extenal loo that et the oto angula oition θ x, eeenting the N-agnet oition in a geneic tie intant, with eect to a fixed efeence yte. The oigin of the efeence yte lie on the axi of yety of a choen lot. Fig..4. Configuation of the netwo in eence of ga between N- and S- agnet. By vaying the value of θ x, the geoetical aaete of the flux tube ae odified. Conequently, the elated coefficient a ij () f(μ i (), θ x ) and the continuity equation in the node involved in change ae odified alo. The olution oce i then eeated by olving evey te in the ae way decibed in Section.3.When the ga between the agnet i coied unde a tooth, a 44

55 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine hown in Fig..4, the oftwae odifie the configuation of the agnetic netwo by adding two new banche and fou new eluctance of vaiable co ection, deending on the value of θ x in a geneic tie intant. Conequently, thee ae fou new unnown e ga. Intead of only one flux tube, a in the cae of unifo agnet (Fig..), in thi ituation the aea unde the tooth can be divided into thee flux tube elated, eectively, to N-agnet otion, agnet ga and S-agnet otion, a hown in Fig..4.The index i dt aue an intege value to identify the tooth that coie the ga, o that the eluctance R (idt), R (5Ncv idt), R (8Ncv), R (8Ncv3) involved in the oveent, change thei efeence angle θ A f(θ x ) and θ B f(θ x ), that ubtend the elated co-ectional aea, accoding to: [ θ α ( ) ] θa i α α x a dt cv (.3) R [ α α θ α ( ) ] θb i α (.33) cv a x dt cv (.34) The thee flux tube unde the tooth ae chaacteized by ix eluctance: two of the alo exiting in the cae of unifo agnet, R (idt) and R (5Ncv idt), but in the eent cae odified in the co-ection and fou additional eluctance, fo the R (8Ncv) to the R (8Ncv4). R( i dt ) μ R θ A L ( θ,i )L x dt (.35) ( 5Ncv idt ) μ R ( 8N cv ) μ R R R g θ θ A B g ( θ,i )L x dt L ( θ,i )L x dt (.36) (.37) 45

56 Chate ( 8Ncv 3) μ R ( 8Ncv ) μ R ( 8Ncv 4) μ R R R R g g θ B L α g α g ( θ,i )L x L L dt (.38) (.39) (.4) The co-ectional aea in the foula ae calculated at the efeence adiu of evey ub-doain. Fou additional equation (.4)-(.44) ae added to the yte fo taing into account the new unnown fluxe with eect to the cae of unifo agnet: the Hoinon law alied to N- and S-agnet, the continuity equation alied to node 7 () and 7 (3) : R( ) ϕ( ) R( 5 ) ϕ( 5 ) R( 8 ) ϕ( 8 ) idt idt R( 8 4) ϕ( 8 4) FM ( 8 ) N cv Ncv N cv idt Ncv idt N cv R( 8 ) ϕ( 8 ) R( 8 ) ϕ( 8 ) R( 8 3) ϕ( 8 3) Ncv R( 8 4) ϕ( 8 4) FM ( 8 ) N cv Ncv N cv Ncv N cv Ncv Ncv Ncv Ncv Ncv (.4) (.4) cv cv ϕ( 8 N ) ϕ( 8N 4) (.43) cv cv ϕ( 8 N ) ϕ( 8N 3). (.44) Obviouly, othe equation aleady coied in the yte have to be odified accoding to the vaiation in the agnetic cicuit, a fo exale in the node 4, whee the continuity equation witten in the eence of ga (.45) i coaed with the ituation of unifo agnet (.46): ϕ( 4 ) ϕ( 5 ) ϕ( ) ϕ( 3 ) N cv i dt ϕ( 8 3) ϕ( 8 4) N cv N cv N cv i dt N cv i dt N cv i dt (.45) ϕ cv Ncv Ncv Ncv ( 4 N i) ϕ( 5 i) ϕ( i) ϕ( 3 i) (.46) 46

57 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine The ae logic can be alied to the othe additional equation (with indexe fo 8N cv 5 to 8N cv 8) conideing the econd ga between the agnet, deending on the elated efeence angle and on a new index i dt which identifie the tooth coiing the ga, with eluctance R (idt), R (5Ncv idt), R (8Ncv 5), R (8Ncv 6), R (8Ncv 7), R (8Ncv 8)..5 Co-enegy, Toque and Radial Foce One of the featue of the ooed algoith i the caability of deteining the total agnetic co-enegy of the achine in a eie of diffeent oto oition: thi allow calculating the electoagnetic toque acting between tato and oto. In ode to do thi, the volue τ i and flux ath of evey ubdoain ae develoed in cicula hae, baed on the aveage adiu of the elated at of the achine [5], [8]. The equation ued to deteine the agnetic co-enegy W i of the i-th ub-doain i []: H Wi B i( H )dhidτ τi (.47) To evaluate (.47), in cae of non-linea ub-doain, the algoith ue the nueical integation ethod of taezoid by inteolating the agnetic chaacteitic of the ateial in ode to find the coule of conequent value B j and B j in - te, whee deend on the deied accuacy: W i( NL) j [ B ( H ) B ( H )] j j j j ΔH In the cae of linea ub-doain, the following elationhi i ued: j τ i (.48) 47

58 Chate Bi W i( L) τi μ i (.49) The electoagnetic toque T i calculated though the finite diffeence aoxiation of the fit deivative of the total co-enegy W obtained a uation of all the W i []: T li Δθ ΔW' Δθ i f cont. W' θ i f cont. (.5) whee Δθ i the angula oto dilaceent between two diffeent te and the calculation i done by aintaining contant value of the hae cuent. The adial coonent F of the foce wa deteined by alying the Maxwell Ste Teno ethod (4) on a cloed uface enveloing the fontal co ection of the teeth in the tato, taing into account the noal and tangential coonent of flux denity and of agnetic field acting in evey tooth. T H ( B nˆ ) ( B H ) nˆ H B tˆ ( H B H B )nˆ t n n n t t (.5) Integating (.5) on the above decibed uface and conideing the dicetization of the odel, the contibution elated to the i-th tooth i given, fo adial coonent acting along the axi of the tooth (noal to the uface), a F n μ () i ( Sdg B5 i SaB7 i SaB9 i ) μ ( S B S B ) a Ncv 8Ncv i a Ncv 8Ncv i Ncv (.5) and, fo adial coonent acting eendiculaly to the axi of the tooth (tangential to the uface), i given a () i ( SaB7 ib8 i SaB9 ib8 i ) F t N cv N cv N cv N cv μ (.53) 48

59 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine Note that, in (.5) and (.53), S dg eeent the fontal co-ectional aea of the tooth, S a the aea elated to a ei-lot oening, while the conideed flux denitie value ae decibed in Fig The ojection of the noal and tangential coonent on a efeence yte centeed on the oto axi ae finally given a F F x y ( ) () ( ()) () i Fn() i co αdt() i Ft i in αdt i (.54) ( ) () ( ()) () i Fn() i in αdt() i Ft i co αdt i (.55) whee α dt(i) eeent the angula oition of the axi of the i-th tooth with eect to the x axi; by uing the te in (.54), (.55) fo all the teeth, it give the coonent F x and F y of the eultant adial foce alied on the oto..6 Reult and Coaion with FEA Softwae The D FEA oftwae FEMM 4. [] wa ued in ode to veify the accuacy of the ooed analyi. Two diffeent tyologie of ynchonou PM achine wee conideed, a hown in Fig..5, ulied by inuoidal dive. Fig..5. The PMSM achine conideed in the analyi 49

60 Chate Machine A: -ole ai, 3-hae, 4 lot, taditional winding (only odd f haonic coonent), - 6 double laye hotened itch. Machine B: -ole ai, 5-hae, 3 lot, ecial winding (odd and even f haonic coonent), 6 lot e hae e ole in each laye, one hae occuying 7 in angula ace of the tato. In ode to obtain an accuate electoagnetic analyi, about 8 node wee ued fo ehing the odel in FEMM. In Tab. I the ain geoetical dienion of the achine ae given. Fig..6. The B-H chaacteitic of the ateial The non-linea B-H cuve of the agnetic ateial i hown in Fig..6, the ae fo Machine A and fo Machine B. The initial elative eeability i 438. In Tab. I the ain data of the two achine ae eented. In the following the eult of the coaion ae eented in gahical fo. 5

61 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine TABLE I. MAIN DATA OF MACHINES A E B Paa. Decition A B N l nube of lot 4 3 ole ai of the achine nube of hae 3 5 I n ated hae cuent (A ) T n ated toque (N) g aiga width () D e tato oute diaete () 3 D tato inne diaete () D ean diaete of the agnet () 6 6 D cv_ext diaete at the botto of the lot () 7 7 D cv_int diaete at the to of the lot () D oto oute diaete () 4 4 D alb oto inne diaete () 74 6 α Ldg angle undelying the tooth uface 3.. α a ei-angle undelying the lot oening α angle undelying the agnet ga 4 8 α cv lot itch angle 5 a dt tato lot height () 5 5 h cl lot oening height () L axial length of the achine () 8 8 L agnet width () L dt tooth-body width () L cl lot oening width () L tc lot width at the to lot adiu () L fc lot width at the botto lot adiu () τ cv lot itch at the inne tato adiu () Machine A With efeence to Machine A, Fig..7,.8 how the eult of the coaion fo the ae oto oition. In aticula, Fig..7 how the linage fluxe. A can be een, the linage fluxe of the thee hae calculated uing the ooed ethod ae in vey good ageeent with thoe obtained by FEA analyi. 5

62 Chate Fig..7. Phae linage fluxe v tato RMS cuent (Machine A) Sall diceancie aea only fo vey high value of the cuent (highly atuated achine). In Fig..8 the toque value (.u.) and the agnetic co-enegy value, calculated uing the ooed ethod, ae coaed with thoe obtained by FEA. Fig..8. Toque and agnetic co-enegy v tato RMS cuent (Machine A) Fo low value of the cuent the two aoache give the ae eult. Fo 5

63 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine highe cuent value (non-linea behavio) oe all diffeence aea. Howeve, the toque eduction with eect to the uoed linea behavio i vey well eeented..6. Machine B With efeence to the 5-hae Machine B, Fig..9 to. how the eult of the coaion fo the ae oto oition. Fig..9. Phae linage fluxe v tato RMS cuent (Machine B) In aticula, Fig..9 how the linage fluxe of the five hae. Alo in thi cae, the eult obtained uing the ooed ethod ae in vey good ageeent with thoe obtained by FE analyi. Fig.. how the toque (.u.) and the agnetic co-enegy value of the yte calculated uing the ooed and the FEA aoach. Refeing to the agnetic co-enegy, a vey good ageeent aea, with only all diffeence when the achine i highly atuated; about the toque, i oible to obeve a vey good ageeent fo low and ediu (ated) cuent value. Soe all diffeence aea in the higly atuated behavio. 53

64 Chate Fig... Toque and agnetic co-enegy v tato RMS cuent (Machine B) A futhe coaion ha been ade by ulying the five hae of Machine B with a balanced yte of inuoidal cuent with a tie-hae dilaceent of 4π/5 (equence ). The conideed winding aangeent oduce an aiga ditibution of the f having odd and even haonic coonent. Fig... Magnitude of the adial foce v tato RMS cuent (Machine B) 54

65 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine Taing the winding aangeent into account and the aticula yte of cuent, it i oible to align the 4 ole haonic coonent of the tato f with the agnet axi. In thee condition a eultant adial foce exit acting on the oto along the diection of the agnet axi []. Fig.. how the calculated agnitude of the adial foce coaed with that obtained by FEA. A can be een, in thi cae alo thee i a good ageeent between the ooed ethod and the eult of FEA. In Fig.. the x- and y-coonent of the adial foce ae hown. The foce ha been calculated fo 9 value, equally aced of π/6, of the hae angle of the econd haonic coonent of the tato f with eect to the agnet axi (y-axi in Fig..), and fo 3 value of the tato cuent alitude: the ated value, two-tie and fou-tie the ated value. Fig... y-coonent v x-coonent of the adial foce (Machine B) Alo in thi ind of analyi, the obtained eult ae vey iila to thoe obtained by FEA. It i inteeting to note that, fo the ae cuent alitude, 55

66 Chate the agnitude of the adial foce i not contant but change with the hae angle of the f. Thi behavio i due to the eence of highe haonic ode in the agnet and tato f ditibution. 56

67 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine.7 Concluion In thi chate an algoith fo the non-linea agnetic analyi of ultihae uface-ounted PM achine with ei-cloed lot ha been eented. The baic eleent of the geoety i dulicated allowing to build and analyze whateve tyology of winding and aee-tun ditibution in a ai of ole. The efoance of the ooed ethod have been coaed with thoe of a well nown FEA oftwae in te of linage fluxe, co-enegy, toque and adial foce. The obtained eult fo a taditional thee-hae achine and fo a 5-hae achine with unconventional winding ditibution howed that the value of local and global quantitie ae actically coinciding fo value of the tato cuent u to ated value. Futheoe, they ae vey iila alo in the non-linea behavio even if vey lage cuent value ae injected. When develoing a new achine deign the ooed ethod i ueful not only fo the eduction of couting tie, but ainly fo the ilicity of changing the value of the deign vaiable, being the nueical inut of the oble obtained by changing oe citical aaete, without the need fo edeigning the odel in a CAD inteface. It can be concluded that the ooed ethod ovide an accuate decition of electoagnetic henoena taing agnetic atuation into account. Fo a given oto oition and fo given tato cuent, the outut toque a well a the adial foce acting on the oving at of a ultihae achine can be eaily and quicly calculated. The latte featue ae the algoith aticulaly uitable in ode to deign and analyze beaingle achine. 57

68 Chate.8 Refeence [] D. Caadei, D. Dujic, E. Levi, G. Sea, A. Tani, and L. Zai, Geneal Modulation Stategy fo Seven-Phae Invete with Indeendent Contol of Multile Voltage Sace Vecto, IEEE Tan. on Indutial Electonic, Vol. 55, NO. 5, May 8, [] Fei Yu, Xiaofeng Zhang, Huaihu Li, Zhihao Ye, The Sace Vecto PWM Contol Reeach of a Multi-Phae Peanent Magnet Synchonou Moto fo Electical Poulion, Electical Machine and Syte (ICEMS), Vol., , Nov. 3. [3] Ruhe Shi, H.A.Toliyat, Vecto Contol of Five-hae Synchonou Reluctance Moto with Sace Pule Width Modulation fo Miniu Switching Loe, Induty Alication Confeence, 36th IAS Annual Meeting. Vol. 3,. 97-3, 3 Set.-4 Oct.. [4] M. A. Abba, R.Chiten, T.M.Jahn, Six-hae Voltage Souce Invete Diven Induction Moto, IEEE Tan. on IA, Vol.IA-, No. 5,. 5-59, 984. [5] E. E. Wad, H. Hae, Peliinay Invetigation of an Invete fed 5-hae Induction Moto, IEE Poc, June 969, Vol. 6(B), No. 6, , 969. [6] Y. Zhao, T. A. Lio, Sace Vecto PWM Contol of Dual Thee-hae Induction Machine Uing Vecto Sace Decooiton, IEEE Tan. on IA, Vol. 3, , 995. [7] Xue S, Wen X.H, Siulation Analyi of A Novel Multihae SVPWM Stategy, 5 IEEE Intenational Confeence on Powe Electonic and Dive Syte (PEDS), , 5. [8] Paa L, H. A. Toliyat, Multihae Peanent Magnet Moto Dive, Induty Alication Confeence, 38th IAS Annual Meeting. Vol.,. 4-48,.-6 Oct. 3. [9] H. Xu, H.A. Toliyat, L.J. Peteen, Five-Phae Induction Moto Dive with DSP-baed Contol Syte, IEEE Tan. on IA, Vol. 7, No. 4, ,. [] L. Zai, M. Mengoni, A. Tani, G. Sea, D. Caadei: "Miniization of the Powe Loe in IGBT Multihae Invete with Caie-Baed Pulewidth Modulation," IEEE Tan. on Indutial Electonic, Vol. 57, No., Novebe, [] A. Chiba, T. Deido, T. Fuao and et al., "An Analyi of Beaingle AC 58

69 An algoith fo non-linea analyi of ultihae beaingle uface-ounted ynchonou achine Moto," IEEE Tan. Enegy Conveion, vol. 9, no., Ma. 994, [] M. Kang, J. Huang, H.-b. Jiang, J.-q. Yang, Pincile and Siulation of a 5-Phae Beaingle Peanent Magnet-Tye Synchonou Moto, Intenational Confeence on Electical Machine and Syte,. 48 5, 7- Oct. 8. [3] S. W.-K. Khoo, "Bidge Configued Winding fo Polyhae Self-Beaing Machine" IEEE Tan. Magnetic, vol. 4, no. 4, Ail. 5, [4] B. B. Choi, Ulta-High-Powe-Denity Moto Being Develoed fo Futue Aicaft, in NASA TM 3-96, Stuctual Mechanic and Dynaic Banch Annual Reot,., Aug. 3. [5] Y. Kano, T. Koaa, N. Matui, Sile Nonlinea Magnetic Analyi fo Peanent-Magnet Moto, IEEE Tan. Ind. Al., vol. 4, no. 5,. 5 4, Set./Oct. 5. [6] B. Sheih-Ghalavand, S. Vaez-Zadeh and A. Haanou Ifahani, An Ioved Magnetic Equivalent Cicuit Model fo Ion-Coe Linea Peanent-Magnet Synchonou Moto, IEEE Tan. on Magnetic, vol. 46, no.,., Jan.. [7] S. Vaez-Zadeh and A. Haanou Ifahani, Enhanced Modeling of Linea Peanent-Magnet Synchonou Moto, IEEE Tan. on Magnetic, vol. 43, no., , Jan. 7. [8] J. M. Konena and D. A. Toey, Magnetic cicuit odel fo utually couled witched eluctance achine, in Conf. Rec. IEEE-IAS Annual. Meeting, vol., 997, [9] R. Qu and T. A. Lio, Analyi and Modeling of Aiga & Zigzag Leaage Fluxe in a Suface-Mounted-PM Machine, in Conf. Rec. IEEE- IAS Annu. Meeting, vol. 4, ,. [] D. C. White, H. H. Woodon, Electoechanical Enegy Conveion, MIT Pe, Dec [] N. Matui, M. Naaua, and T. Koaa, Intantaneou Toque Analyi of Hybid Steing Moto, IEEE Tan. Ind. Al., vol. 3, no. 5,. 76 8, Se./Oct [] D. C. Meee, Finite Eleent Method Magnetic, Veion 4. (OctBuild), htt:// 59

70 Chate 6

71 The ogaing code - Pat Aendix A. THE PROGRAMMING CODE Pat In the following, the ogaing code of the algoith i eented. Thi i the VBA veion, ince thee i an advanced vaiant, ileented in Viual Baic 6. and not eoted hee. Soe inut file ae neceay to un the oga: od.txt, odc.txt, which decibe eectively the dioition in the tato lot of the fit and econd winding laye; am().txt, which decibe the oition of the equivalent aee-tun ditibution of the agnet. The agnetic chaacteitic of the ateial i ovided though the.xl ain file and the tato aee-tun ditibution i ceated by the oga with the uboutine eented in Aendix A.. A.. The ain oga Cont i Cont uzeo A Double.5664 * ^ -6 Public a, ai, ai_, ai, ai, am, ASP, ainiz, noti, x, x_, y, B, Hdt, Flux_dt, deltabh, nox A Vaiant Public Fluxc, u, u_, u_dt, Rildt, Rilc, Bc, Hc, change_r, change_c A Vaiant Public Fluxcv, Hg_dt, Bg_dt, Acv, Bdt_, Hdt_, u_dt_ A Vaiant Public Bla, Hla, Flux_tdt, Flux_d, dflux_f, Flux_f, If A Vaiant Public iv_r, iv_c, e, cv A Vaiant Public n,, z, fault, undteeth_, zeo_ctl A Vaiant Public ivot, a_logic, b_logic A Vaiant Public xec, Bec, diffx, Flux_tcv, Flux_tcv_ A Vaiant 6

72 Aendix A. Public denm, EnM, EnM_, deltaenm, Vol, Ril, Sez A Vaiant Public H A Vaiant Public Ncv, nc, Ia, Ib, Ic, Hc, u_ag, L, Ldt, Lcv, Lcl, hcl, adt, ac, ac, L, g, havv, tollx, ta, x,, dx, dx_cot, ix_ax, w A Vaiant Public Taucv, Taucv_R, Taucv_int, Ldg, Ldg_R, hbd, alfa_tcv, Ltd, u_ag, Tau, B, x A Double Public ind_dt, ind_dt A Vaiant Public od, odc, Ife, Ife_ A Vaiant Public n, nc, nfai, w, fi_in, I A Vaiant Public fileath A Sting Public, w, t A Intege Public LdA_, LdA_, LdB_, LdB_, Ldx_, Ldx_, Lddx_, Lddx_, LdA R, LdA R, LdB R, LdB R A Double Public Fx, Fy, F_n, F_tx, F_tdx, F_od, alfa_dt, alfa_dt_deg A Vaiant Public Fx_, Fy_, F A Double Public,, 3 A Double Public Rcv_ext, Lfc, Ltc, Rcv_int, thcv, Vol_etcv, VolH_etcv, VolL_etcv, Lcvx_ed, Re, R, R, Ralb, alfa_cv, Lc_ed, Lc_ed, Vol_c, Vol_c A Double Public Rcv_efz, Rcv_ifz, Rcv_fz, LcvxH, LcvxL A Double Public VolHcv, VolLcv A Vaiant 'NUOVI PARAMETRI *********************************************************************** Public R, Re, Rg, alfa_a, alfa_ldg, Vol_Ldg, Vol_a, Vol_gLdg, Vol_ga, alfa_, th_x, th_x, tha_, tha_, thb_, thb_ A Double Public dth_, C_EM A Double Public x_r, x_r, _R A Double ' ************************************************************************************** Pivate Sub OtionButton_Clic() If OtionButton.Value Tue Then MgBox "Peaae il file ASP_TOT.txt eendo" & Ch(3) & "il ulante nel foglio ucceivo", vbokonly, "SATSOLVER" End If End Sub Public Sub Tiang_Clic() Range("AE5:AG6549").CleaContent Range("AH5").CleaContent Range("AI37:IV6549").CleaContent Range("E5:AB6549").CleaContent ReDi Bla( To 45), Hla( To 45), Flux_tdt(3 To 5), Flux_d(3 To 5), dflux_f( To 3), If( To 3) A Double '*************************************************************************************** 'INPUT 'nueo di cave e coia olae Ncv Int(Cell(4, 4)) 'conduttoi in cava 'I tato n Cell(5, 4) 'II tato nc Cell(6, 4) 'ao olae Tau Cell(7, 4) Tau Tau *. 'coente efficace I Cell(8, 4) 'cao coecitivo intineco del agnete (A/) Hc Cell(9, 4) 'eeabilità elativa agnete u_ag Cell(, 4) 'eoe agnete L Cell(, 4) L L *. 'eoe del coo del dente 6

73 The ogaing code - Pat Ldt Cell(, 4) Ldt Ldt *. 'aetua di cava (eale) Lcl Cell(4, 4) Lcl Lcl *. 'altezza collaino hcl Cell(5, 4) hcl hcl *. 'altezza dente tatoe adt Cell(6, 4) adt adt *. 'eoe coona tatoe ac Cell(7, 4) ac ac *. 'eoe coona otoe ac Cell(8, 4) ac ac *. 'ofondità di acchina L Cell(9, 4) L L *. 'eoe tafeo g Cell(, 4) g g *. 'altezza dell'avvolgiento havv Cell(, 4) havv havv *. 'cato aio aeo fa due oluzioni ucceive del itea tollx Cell(, 4) tollx tollx *. 'aaeto adienionale che definice il aoto fa 'la iua della ate baa e quella dell'inteo dente ta Cell(3, 4) 'oizione iniziale del agnete N x Cell(4, 4) x x *. 'te di oenione del oviento (< dx) dx_cot Cell(5, 4) dx_cot dx_cot *. 'te di otaento otoe dx Cell(6, 4) dx dx *. 'nueo di te nel oviento ix_ax Cell(7, 4) 'fae iniziale fi_in Cell(8, 4) 'aiezza del agnete w Cell(9, 4) w w *. 'nueo di fai nfai Cell(3, 4) '*************************************************************************************** 'INPUT CARATTERISTICA DI MAGNETIZZAZIONE Fo i To 45 Bla(i) Cell(4 i, 9) Hla(i) Cell(4 i, 3) '*************************************************************************************** 'fequenza elettica f 5 w t 'aggio al fondo cava Rcv_ext.85 'coie olai 'laghezza cava al fondo 63

74 Aendix A. Lfc.97 'laghezza cava in teta Ltc.53 fileath CSt(TextBox.Text) If ta < - havv / adt Then MgBox ("Valoe di ta too bao!") Exit Sub End If 'angolo di cava (ad. ecc.) alfa_cv * i / / Ncv 'aggio eteno tatoe Re Rcv_ext ac 'aggio otoe alla bae del agnete R Rcv_ext - adt - g - L 'aggio otoe inteno (aggio albeo) Ralb R - ac 'aggio inteno tatoe R Rcv_ext - adt 'aggio di cava in teta Rcv_int Rcv_ext - havv 'angolo coiondente alla cava (ad.) thcv * Atn((Lfc - Ltc) / / havv) 'ao di cava tatoe Taucv * Tau / Ncv 'ao di cava al aggio inteno tatoe Taucv_R * i * R / / Ncv 'ao di cava al aggio di cava in teta Taucv_int * i * Rcv_int / / Ncv 'laghezza dente al diaeto di ifeiento Ldg Taucv - Lcl 'laghezza dente al aggio inteno tatoe Ldg_R * i * R / / Ncv - Lcl 'laghezza cava tatoe in teta Lcv Taucv_int - Ldt 'azio inteagnetico Tau - w 'NUOVI PARAMETRI ********************************************************************** 'aggio edio agnete R R - g - L / 'aggio eteno agnete Re R - g 'aggio edio tafeo Rg R - g / 'angolo coi. alla eiaetua di cava alfa_a Lcl / / R 'angolo coi. alla teta del dente alfa_ldg alfa_cv - * alfa_a ' ************************************************************************************* 'conduttoi in cava nc n nc 'ulazione elettica w * i * f 'altezza della bae del dente hbd adt - havv - hcl 'angolo teta di cava alfa_tcv Atn((Lcv - Lcl) / / hbd) 'linea di ezione teta del dente Ltd hbd / Co(alfa_tcv) hcl 'eeabilità aoluta del agnete u_ag u_ag * uzeo 'eeabilità elativa del agnete B u_ag * Hc 'oizione iniziale del agnete S 64

75 The ogaing code - Pat x x w '*************************************************************************************** 'CONTROLLO DELLA POSIZIONE INIZIALE DEL MAGNETE 'Alla vaiabile "undteeth_" viene aegnato valoe Tue 'e la dicontinuità N-S i tova otto un qualiai dente undteeth_ Fo i To Ncv a_logic (x > (Lcl / (i - ) * Taucv) And x < (Lcl / Ldg (i - ) * Taucv)) b_logic ((x ) > (Lcl / (i - ) * Taucv) And (x ) < (Lcl / Ldg (i - ) * Taucv)) If a_logic And b_logic Tue Then undteeth_ i GoTo End If MgBox "ATTENZIONE: uno dei due o entabe i agneti" & Ch(3) & "i tovano otto l'aetua di cava", vbcitical, "SATSOLVER" Exit Sub 'Alla vaiabile "undteeth_" viene aegnato valoe Tue 'e la dicontinuità S-N i tova otto un qualiai dente undteeth_ Fo i To Ncv c_logic (x > (Lcl / (i - ) * Taucv) And x < (Lcl / Ldg (i - ) * Taucv)) d_logic ((x ) > (Lcl / (i - ) * Taucv) And (x ) < (Lcl / Ldg (i - ) * Taucv)) If c_logic And d_logic Tue Then undteeth_ i GoTo 3 End If MgBox "ATTENZIONE: uno dei due o entabe i agneti" & Ch(3) & "i tovano otto l'aetua di cava", vbcitical, "SATSOLVER" Exit Sub 3 '*************************************************************************************** If undteeth_ <> And undteeth_ <> Then n 8 * Ncv 8 Ele n 8 * Ncv End If Int(n ) ReDi a(n, ), ai(n), ai_(n), ai(n), ai(n), am(n), ASP(ix_ax * n), ainiz(n, ), noti(n), x( To n), x_( To n), y(n, ), B(n), Hdt(n), Flux_dt( To Ncv), deltabh(n), nox(99) A Double ReDi Fluxc(n), u(n), u_(n), u_dt(n), Rildt(n), Rilc(n), Bc(n), Hc(n), change_r(), change_c(n) A Double ReDi Fluxcv(n), Hg_dt(n), Bg_dt(n), Acv(n), Bdt_(n), Hdt_(n), u_dt_(n) A Double ReDi iv_r(n), iv_c(n), e(n), cv( To Ncv) A Intege ReDi xec( To n), Bec( To n), diffx( To n), Flux_tcv( To Ncv), Flux_tcv_( To Ncv) A Double ReDi denm( To n), EnM( To n), EnM_( To n), deltaenm( To n), Vol( To n), VolHcv( To * Ncv), VolLcv( To * Ncv), Ril( To n), Sez( To n) A Double ReDi H( To n) A Double ReDi od( To Ncv), odc( To Ncv) A Intege ReDi Ife( To nfai), Ife_( To nfai), Flux_f( To nfai) A Double 65

76 Aendix A. ReDi Fx( To Ncv), Fy( To Ncv), F_n( To Ncv), F_tx( To Ncv), F_tdx( To Ncv), F_od( To Ncv), alfa_dt( To Ncv), alfa_dt_deg( To Ncv) A Double 'PARAMETRI CIRCOLARI DI MACCHINA *************************************** 'CAVA 'NB: i aggi "fittizi" eguono i fianchi della cava, ' NON convegendo al cento del otoe (,) 'aggio eteno fittizio Rcv_efz Lfc / / Tan(thcv / ) 'aggio inteno fittizio Rcv_ifz Ltc / / Tan(thcv / ) 'aggio edio fittizio Rcv_fz Rcv_efz - ( - ta) * adt 'volui nella zona alta di cava (H) e baa (L) VolH_etcv thcv * L / * (Rcv_efz ^ - Rcv_fz ^ ) VolL_etcv thcv * L / * (Rcv_fz ^ - Rcv_ifz ^ ) 'volue dell'aea di cava ta il aggio TCV e FCV Vol_etcv thcv * L / * (Rcv_efz ^ - Rcv_ifz ^ ) 'volue cava (ecluo il collaino) Vol_cv (Lcv Lcl) / * hbd * L Vol_etcv 'laghezza di cava al aggio edio Lcvx_ed thcv * Rcv_fz If Lcvx_ed < Then MgBox ("Auentae ta!") Sto End If LcvxH thcv / * (Rcv_efz Rcv_fz) LcvxL thcv / * (Rcv_fz Rcv_ifz) Fo i To * Ncv VolHcv(i) VolH_etcv / VolLcv(i) ((Lcv Lcl) * (hbd - hcl) / * L VolL_etcv) / 'CORONE Vol_c alfa_cv * L / * (Re ^ - Rcv_ext ^ ) Vol_c alfa_cv * L / * (R ^ - Ralb ^ ) Lc_ed alfa_cv / * (Re Rcv_ext) Lc_ed alfa_cv / * (R Ralb) 'NUOVI PARAMETRI ********************************************************************** 'MAGNETI Vol_Ldg alfa_ldg * L / * (Re ^ - R ^ ) Vol_a alfa_a * L / * (Re ^ - R ^ ) 'TRAFERRO Vol_gLdg alfa_ldg * L / * (R ^ - Re ^ ) Vol_ga alfa_a * L / * (R ^ - Re ^ ) ' ************************************************************************************** 'PARAMETRI AUSILIARI (e calcolo iluttanze) ****************************************** (adt - havv) * L * hbd * L / (Ldg - Ldt) 3 * hbd * L / (Lcvx_ed - Lcl) '*************************************************************************************** 'RILUTTANZE COSTANTI 'Riluttanze MAGNETE SOTTO IL DENTE d(i) Fo i To Ncv Sez(i) R * alfa_ldg * L Vol(i) Vol_Ldg Ril(i) / u_ag * L / Sez(i) 'Riluttanze CORONA STATORICA Fo i Ncv To * Ncv 66

77 The ogaing code - Pat Sez(i) ac * L Vol(i) Vol_c Ril(i) Rnd() * 3 'Riluttanze CORONA ROTORICA Fo i * Ncv To 3 * Ncv Sez(i) ac * L Vol(i) Vol_c Ril(i) Rnd() * 3 'Riluttanze taveali CAVE cvx(i) Fo i 3 * Ncv To 4 * Ncv Sez(i) (Rcv_ext - Rcv_int) * L Vol(i) Vol_cv Ril(i) / uzeo * / 3 * Log((adt - hcl) / havv) / uzeo * Lcl / (adt - hcl) / L 'Riluttanze DENTE LOW Fo i 4 * Ncv To 5 * Ncv Sez(i) Ldt * L Vol(i) (adt * (ta - ) havv) * Ldt * L Ril(i) Rnd() * 3 'Riluttanze TRAFERRO SOTTO IL DENTE gd(i) Fo i 5 * Ncv To 6 * Ncv Sez(i) Rg * alfa_ldg * L Vol(i) Vol_gLdg Ril(i) / uzeo * g / Sez(i) 'Riluttanze DENTE HIGH Fo i 6 * Ncv To 7 * Ncv Sez(i) Ldt * L Vol(i) adt * ( - ta) * Ldt * L Ril(i) Rnd() * 3 'Riluttanze TRAFERRO SOTTO SEMICAVA SX gc(i) Fo i 7 * Ncv To 8 * Ncv Sez(i) Rg * alfa_a * L Vol(i) Vol_ga i * hcl ^ / 4 * L Ril(i) i / ( * uzeo * L * Log(( * g i * hcl) / / g)) 'Riluttanze taveali COLLARINO clx(i) Fo i 8 * Ncv To 9 * Ncv Sez(i) hcl * L Vol(i) hcl * Lcl * L Ril(i) / uzeo * Lcl / (hcl * L) 'Riluttanze TRAFERRO SOTTO SEMICAVA DX gcd(i) Fo i 9 * Ncv To * Ncv Sez(i) Rg * alfa_a * L Vol(i) Vol_ga i * hcl ^ / 4 * L Ril(i) i / ( * uzeo * L * Log(( * g i * hcl) / / g)) 'Riluttanze MAGNETE SOTTO SEMICAVA SX c(i) Fo i * Ncv To * Ncv Sez(i) R * alfa_a * L Vol(i) Vol_a Ril(i) / u_ag * L / Sez(i) 'Riluttanze MAGNETE SOTTO SEMICAVA DX cd(i) 67

78 Aendix A. Fo i * Ncv To * Ncv Sez(i) R * alfa_a * L Vol(i) Vol_a Ril(i) / u_ag * L / Sez(i) 'Riluttanze nel SEMIDENTE SX e DX Fo i * Ncv To 4 * Ncv Sez(i) Ldg / / (Ldt / / (adt - havv) / L / * Log( / ( - * (Ldg - Ldt) / ))) Vol(i) (Ldg / Ldt / ) * hbd / * L Ldg / * hcl * L Ril(i) Rnd() * 3 'Riluttanze longitudinali BASSE SEMICAVE Lcvy(i), Lcvyd(i) Fo i 4 * Ncv To 6 * Ncv Sez(i) LcvxL / * L Vol(i) VolLcv(i - 4 * Ncv) Ril(i) / uzeo * hcl / (Lcl * L) / uzeo * (hbd - hcl) / (Ltc - Lcl) / L * Log(Ltc / Lcl) / uzeo * (ta * adt - hbd) / (Lcvx_ed - Ltc) / L * Log(Lcvx_ed / Ltc) 'Riluttanze longitudinali ALTE SEMICAVE Hcvy(i), Hcvyd(i) Fo i 6 * Ncv To 8 * Ncv Sez(i) LcvxH / * L Vol(i) VolHcv(i - 6 * Ncv) Ril(i) / uzeo * adt * ( - ta) / (Lfc - Lcvx_ed) / L * Log(Lfc / Lcvx_ed) '*************************************************************************************** 'ARRAY DEI TERMINI NOTI '*************************************************************************************** Fo i To n ai(i) am(i) 'Lettua del file elativo al I tato dell'avvolgiento tatoico Oen fileath & "\od.txt" Fo Inut A #5 Fo i To Ncv Inut #5, od(i) Cloe #5 'Lettua del file elativo al II tato dell'avvolgiento tatoico Oen fileath & "\odc.txt" Fo Inut A #6 Fo i To Ncv Inut #6, odc(i) Cloe #6 'Lettua del file contenente le aeie equivalenti del agnete Oen fileath & "\am().txt" Fo Inut A # Fo i To n Inut #, am(i) Cloe # If OtionButton.Value Tue Then 'Lettua del file contenente le aeie con andaento SINUSOIDALE Oen fileath & "\ASP_TOT.txt" Fo Inut A # Fo i To (n * ix_ax) Inut #, ASP(i) Cloe # Ele 'Lettua del file contenente le aeie COSTANTI Oen fileath & "\ASP_COST.txt" Fo Inut A #8 Fo i To n Inut #8, ai(i) 68

79 The ogaing code - Pat Cloe #8 End If 'Ceazione file di outut Oen fileath & "\PP_8Ncv_FX(.7t).txt" Fo Outut A #3 Call SUBR_ End Sub Public Sub SUBR_() 'Azzeaento coenegia agnetica EnM_t Fo i To n EnM_(i) 'Inizializzazione vaiabile di contollo delle oluzioni nulle zeo_ctl 'CICLO DI POSIZIONAMENTO DEL MAGNETE ' NB: e le aeie cabiano di oto nel vettoe dei teini noti al vaiae ' della oizione del agnete, è neceaio ineie l'inut nel ciclo "ix" Fo ix To ix_ax Range("AE5:AG6549").CleaContent Fo i To Ncv a_logic (x > (Lcl / (i - ) * Taucv) And x < (Lcl / Ldg (i - ) * Taucv)) b_logic ((x ) > (Lcl / (i - ) * Taucv) And (x ) < (Lcl / Ldg (i - ) * Taucv)) If a_logic And b_logic Fale Then MgBox "ATTENZIONE: uno dei due o entabe i agneti" & Ch(3) & "i tovano otto l'aetua di cava", vbcitical, "SATSOLVER" GoTo 5 End If c_logic (x > (Lcl / (i - ) * Taucv) And x < (Lcl / Ldg (i - ) * Taucv)) d_logic ((x ) > (Lcl / (i - ) * Taucv) And (x ) < (Lcl / Ldg (i - ) * Taucv)) If c_logic And d_logic Fale Then MgBox "ATTENZIONE: uno dei due o entabe i agneti" & Ch(3) & "i tovano otto l'aetua di cava", vbcitical, "SATSOLVER" GoTo 5 End If 'INDIVIDUAZIONE POSIZIONE MAGNETE 'Alle vaiabili ind_dt, ind_dt viene aegnato l'indice 'nueico dei denti otto i quali i tovano i "buchi" ind_dt ind_dt Fo i To Ncv If x > (Lcl / (i - ) * Taucv) And x < (Lcl / Ldg (i - ) * Taucv) Then ind_dt i GoTo End If Fo i To Ncv If x > (Lcl / (i - ) * Taucv) And x < (Lcl / Ldg (i - ) * Taucv) Then ind_dt i GoTo 3 End If 69

80 Aendix A. 3 If ind_dt ind_dt Then MgBox "ATTENZIONE: ind_dt ind_dt!" & Ch(3) & "CONDIZIONE IMPOSSIBILE!", vbcitical, "SATSOLVER" GoTo 5 End If '*************************************************************************************** 'PARAMETRI DIPENDENTI DALLA POSIZIONE 'PARAMETRI GEOMETRICI f(x,ind_dt) ************************************* LdA_ x - Lcl / - (ind_dt - ) * Taucv LdA_ x - Lcl / - (ind_dt - ) * Taucv LdB_ Taucv - Lcl / - x - (ind_dt - ) * Taucv LdB_ Taucv - Lcl / - x - (ind_dt - ) * Taucv Ldx_ x - Lcl / / - (ind_dt - ) * Taucv Ldx_ x - Lcl / / - (ind_dt - ) * Taucv Lddx_ Taucv - Lcl / - x - / (ind_dt - ) * Taucv Lddx_ Taucv - Lcl / - x - / (ind_dt - ) * Taucv 'aaeti e il calcolo della foza adiale ifeiti 'al diaeto inteno di tatoe x_r R / R * x x_r R / R * x _R R / R * LdA R x_r - Lcl / - (ind_dt - ) * Taucv_R LdA R x_r - Lcl / - (ind_dt - ) * Taucv_R LdB R Taucv_R - Lcl / - x_r - _R (ind_dt - ) * Taucv_R LdB R Taucv_R - Lcl / - x_r - _R (ind_dt - ) * Taucv_R 'NUOVI PARAMETRI ******************************************************************** 'angolo coiondente allo azio inteagnetico alfa_ / R 'angoli coi. alle oizioni dei agneti th_x x / R th_x x / R 'angoli coi. ai aaeti vaiabili tha_ th_x - alfa_a - (ind_dt - ) * alfa_cv tha_ th_x - alfa_a - (ind_dt - ) * alfa_cv thb_ alfa_cv - alfa_a - th_x - alfa_ (ind_dt - ) * alfa_cv thb_ alfa_cv - alfa_a - th_x - alfa_ (ind_dt - ) * alfa_cv ' *********************************************************************************** '*********************************************************************** 'RICALCOLO DI RILUTTANZE, SEZIONI E VOLUMI NELLE ZONE SOGGETTE A MODIFICHE DELLA GEOMETRIA If ind_dt <> And ind_dt <> Then 'DISCONTINUITA' N-S Vol(ind_dt) tha_ / * L * (Re ^ - R ^ ) Sez(ind_dt) tha_ * (Re R) / * L Ril(ind_dt) / u_ag * L / Sez(ind_dt) Vol(5 * Ncv ind_dt) tha_ / * L * (R ^ - Re ^ ) Sez(5 * Ncv ind_dt) tha_ * (R Re) / * L Ril(5 * Ncv ind_dt) / uzeo * g / Sez(5 * Ncv ind_dt) Vol(8 * Ncv ) thb_ / * L * (Re ^ - R ^ ) Sez(8 * Ncv ) thb_ * (Re R) / * L Ril(8 * Ncv ) / u_ag * L / Sez(8 * Ncv ) 7

81 The ogaing code - Pat Vol(8 * Ncv ) alfa_ / * L * (Re ^ - R ^ ) Sez(8 * Ncv ) alfa_ * (Re R) / * L Ril(8 * Ncv ) / uzeo * L / Sez(8 * Ncv ) Vol(8 * Ncv 3) thb_ / * L * (R ^ - Re ^ ) Sez(8 * Ncv 3) thb_ * (R Re) / * L Ril(8 * Ncv 3) / uzeo * g / Sez(8 * Ncv 3) Vol(8 * Ncv 4) alfa_ / * L * (R ^ - Re ^ ) Sez(8 * Ncv 4) alfa_ * (R Re) / * L Ril(8 * Ncv 4) / uzeo * g / Sez(8 * Ncv 4) 'DISCONTINUITA' S-N Vol(ind_dt) tha_ / * L * (Re ^ - R ^ ) Sez(ind_dt) tha_ * (Re R) / * L Ril(ind_dt) / u_ag * L / Sez(ind_dt) Vol(5 * Ncv ind_dt) tha_ / * L * (R ^ - Re ^ ) Sez(5 * Ncv ind_dt) tha_ * (R Re) / * L Ril(5 * Ncv ind_dt) / uzeo * g / Sez(5 * Ncv ind_dt) Vol(8 * Ncv 5) thb_ / * L * (Re ^ - R ^ ) Sez(8 * Ncv 5) thb_ * (Re R) / * L Ril(8 * Ncv 5) / u_ag * L / Sez(8 * Ncv 5) Vol(8 * Ncv 6) alfa_ / * L * (Re ^ - R ^ ) Sez(8 * Ncv 6) alfa_ * (Re R) / * L Ril(8 * Ncv 6) / uzeo * L / Sez(8 * Ncv 6) Vol(8 * Ncv 7) thb_ / * L * (R ^ - Re ^ ) Sez(8 * Ncv 7) thb_ * (R Re) / * L Ril(8 * Ncv 7) / uzeo * g / Sez(8 * Ncv 7) Vol(8 * Ncv 8) alfa_ / * L * (R ^ - Re ^ ) Sez(8 * Ncv 8) alfa_ * (R Re) / * L Ril(8 * Ncv 8) / uzeo * g / Sez(8 * Ncv 8) End If If OtionButton.Value Tue Then If Int(ix / ) <> ix / Then 'te di otaento DISPARI: le coenti vengono calcolate 'econdo l'eatto valoe itantaneo coelato alla oizione t i * x / w / Tau 'Meoizzazione del valoe delle coenti allo te ecedente Fo i To nfai Ife_(i) Ife(i) Fo j To nfai 'Andaento coinuoidale delle coenti Ife(j) -Sq() * I * Co(w * w * t - (j - ) * w * t * * i / nfai fi_in) Next j Ele Fo i To nfai Ife(i) Ife_(i) End If End If '*************************************************************************************** 'COSTRUZIONE DELLA MATRICE AB '*************************************************************************************** Fo i To n Fo j To a(i, j) Next j 7

82 Aendix A. If OtionButton.Value Tue Then Fo i To n a(i, ) ASP((ix - ) * n i) Ele Fo i To n a(i, ) ai(i) am(i) End If 'PARTE INVARIANTE DELLA MATRICE (A) '5Ncv to 6Ncv *********************************************************************** 'NODO : CONTINUITA' DENTE H - COR.STAT. Fo i To Ncv a(5 * Ncv i, Ncv i - ) a(5 * Ncv i, Ncv i) - a(5 * Ncv i, 6 * Ncv i) - a(5 * Ncv i, 6 * Ncv i) - a(5 * Ncv i, 7 * Ncv i) - '6Ncv to 7Ncv *********************************************************************** 'NODO : CONTINUITA' DENTE L - DENTE H a(6 * Ncv, 4 * Ncv) a(6 * Ncv, 3 * Ncv ) - a(6 * Ncv, 4 * Ncv ) a(6 * Ncv, 6 * Ncv ) - a(6 * Ncv, 4 * Ncv ) a(6 * Ncv, 5 * Ncv ) a(6 * Ncv, 6 * Ncv ) - a(6 * Ncv, 7 * Ncv ) - Fo i To Ncv a(6 * Ncv i, 3 * Ncv i - ) a(6 * Ncv i, 3 * Ncv i) - a(6 * Ncv i, 4 * Ncv i) a(6 * Ncv i, 6 * Ncv i) - a(6 * Ncv i, 4 * Ncv i) a(6 * Ncv i, 5 * Ncv i) a(6 * Ncv i, 6 * Ncv i) - a(6 * Ncv i, 7 * Ncv i) - '7Ncv to 8Ncv *********************************************************************** 'NODO 4 'Equazioni tandad Fo i To Ncv a(7 * Ncv i, 4 * Ncv i) a(7 * Ncv i, 5 * Ncv i) - a(7 * Ncv i, * Ncv i) - a(7 * Ncv i, 3 * Ncv i) 'Vaiazioni If ind_dt <> Then a(7 * Ncv ind_dt, 8 * Ncv 3) - a(7 * Ncv ind_dt, 8 * Ncv 4) - End If If ind_dt <> Then a(7 * Ncv ind_dt, 8 * Ncv 7) - a(7 * Ncv ind_dt, 8 * Ncv 8) - End If 7

83 The ogaing code - Pat '8Ncv to 9Ncv *********************************************************************** 'NODO 7() Fo i To Ncv a(8 * Ncv i, i) a(8 * Ncv i, 5 * Ncv i) - '9Ncv to Ncv ********************************************************************** 'NODO 9: CONTINUITA' MAGNETE - COR.ROT. a(9 * Ncv, ) a(9 * Ncv, * Ncv ) a(9 * Ncv, 3 * Ncv) - a(9 * Ncv, * Ncv ) a(9 * Ncv, * Ncv ) Fo i To Ncv a(9 * Ncv i, i) a(9 * Ncv i, * Ncv i) a(9 * Ncv i, * Ncv i - ) - a(9 * Ncv i, * Ncv i) a(9 * Ncv i, * Ncv i) If ind_dt <> Then a(9 * Ncv ind_dt, 8 * Ncv ) a(9 * Ncv ind_dt, 8 * Ncv ) End If If ind_dt <> Then a(9 * Ncv ind_dt, 8 * Ncv 5) a(9 * Ncv ind_dt, 8 * Ncv 6) End If 'Ncv to Ncv ********************************************************************* 'NODO 3 a( * Ncv, 9 * Ncv) a( * Ncv, 7 * Ncv ) a( * Ncv, * Ncv ) - a( * Ncv, 4 * Ncv ) - Fo i To Ncv a( * Ncv i, 8 * Ncv i - ) a( * Ncv i, 7 * Ncv i) a( * Ncv i, * Ncv i) - a( * Ncv i, 4 * Ncv i) - 'Ncv to Ncv ********************************************************************* 'NODO 5 Fo i To Ncv a( * Ncv i, 8 * Ncv i) a( * Ncv i, 9 * Ncv i) - a( * Ncv i, 3 * Ncv i) - a( * Ncv i, 5 * Ncv i) '4Ncv to 5Ncv ********************************************************************* 'NODO 6 Fo i To Ncv a(4 * Ncv i, 7 * Ncv i) a(4 * Ncv i, * Ncv i) - '5Ncv to 6Ncv ********************************************************************* 'NODO 8 73

84 Aendix A. Fo i To Ncv a(5 * Ncv i, 9 * Ncv i) a(5 * Ncv i, * Ncv i) - '*************************************************************************************** If OtionButton.Value Fale Then Fo i To n a(i, ) ai(i) am(i) Ele Fo i To n a(i, ) ASP((ix - ) * n i) End If '*************************************************************************************** 'Ai coefficienti del itea che diendono dalle iluttanze agnetiche vengono aegnati 'valoi di io tentativo cauali non nulli e evitae che nello te iniziale della 'oluzione vengano ilevate equazioni lineaente diendenti Fo i To (5 * Ncv ) Fo j To n a(i, j) Rnd() * 3 Next j Fo i ( * Ncv ) To (4 * Ncv) Fo j To n a(i, j) Rnd() * 3 Next j Fo i (6 * Ncv ) To n Fo j To n a(i, j) Rnd() * 3 Next j '*************************************************************************************** 'chiaata della uboutine e la oluzione del itea di equazioni Call SOLVESYS '*************************************************************************************** 'CALCOLO DI FLUSSI NELLE VARIE ZONE DI MACCHINA 'fluo nel ao di cava Fo i To Ncv Flux_tcv_(i) x(i 5 * Ncv) x(i 7 * Ncv) x(i 9 * Ncv) 'fluo e olo al tafeo Flux_g Fo i To Ncv Flux_g Flux_g Flux_tcv(i) 'fluo nel dente bao - alto Flux_dL Flux_dH Fo i To Ncv Flux_dL Flux_dL x(4 * Ncv i) Flux_dH Flux_dH x(6 * Ncv i) 74

85 The ogaing code - Pat 'fluo all'altezza dei denti Flux_d ta * Flux_dL ( - ta) * Flux_dH 'flui nei denti Fo i To Ncv Flux_dt(i) ta * x(4 * Ncv i) ( - ta) * x(6 * Ncv i) 'flui nel ao di cava (inteo coe eicava x - dente - eicava dx) Fo i To Ncv Flux_tcv_(i) Flux_dt(i) x(4 * Ncv i) x(5 * Ncv i) 'FLUSSI CONCATENATI CON LE FASI Fo i To nfai Flux_f(i) 'Flui concatenati I tato Fo j To nfai Fo i To Ncv If od(i) j Then Flux_f(j) Flux_f(j) n * x(ncv i) EleIf od(i) -j Then Flux_f(j) Flux_f(j) - n * x(ncv i) End If Next j 'Flui concatenati II tato Fo j To nfai Fo i To Ncv If odc(i) j Then Flux_f(j) Flux_f(j) nc * x(ncv i) EleIf odc(i) -j Then Flux_f(j) Flux_f(j) - nc * x(ncv i) End If Next j 'FORZA RADIALE Fo i To Ncv Fy(i) Fx(i) F_n(i) F_tx(i) F_tdx(i) Fy_ Fx_ 'NB: il itea di ifeiento conideato nel calcolo delle foze ha gli ai coincidenti con il itea di if. ' x-y tandad di FEMM. L'ae della fae, econdo l'inteetazione del egno delle coenti in FEMM (- e ' ENTRANTI nello cheo, e USCENTI), è uotato invece di 8 g.. in eno oaio ietto al itea di ' ifeiento in quetione. Gli angoli ono valutati ietto all'ae (x-tandad). 'angolo dell'ae del dente (g..) Fo i To Ncv alfa_dt(i) i / - (i - ) * alfa_cv alfa_dt_deg(i) 8 / i * alfa_dt(i) 'Suefici fontali dei DENTI Fo i To Ncv Select Cae i Cae ind_dt 'dente ind_dt 75

86 Aendix A. If ind_dt Then 'Coonente NORMALE della foza adiale F_n(i) / / uzeo * L * (LdA R * B(5 * Ncv i) ^ _R * B(8 * Ncv 4) ^ LdB R * B(8 * Ncv 3) ^ Lcl / * B(7 * Ncv i) ^ Lcl / * B(9 * Ncv i) ^ ) F_n(i) F_n(i) - / / uzeo * Lcl / * L * B(9 * Ncv) ^ - / / uzeo * Lcl / * L * B(8 * Ncv i) ^ 'Coonenti TANGENZIALI della foza adiale F_tx(i) / uzeo * Lcl / * L * B(7 * Ncv i) * B(9 * Ncv) F_tdx(i) / uzeo * Lcl / * L * B(9 * Ncv i) * B(8 * Ncv i) Ele 'Coonente NORMALE della foza adiale F_n(i) / / uzeo * L * (LdA R * B(5 * Ncv i) ^ _R * B(8 * Ncv 4) ^ LdB R * B(8 * Ncv 3) ^ Lcl / * B(7 * Ncv i) ^ Lcl / * B(9 * Ncv i) ^ ) F_n(i) F_n(i) - / / uzeo * Lcl / * L * B(8 * Ncv i - ) ^ - / / uzeo * Lcl / * L * B(8 * Ncv i) ^ 'Coonenti TANGENZIALI della foza adiale F_tx(i) / uzeo * Lcl / * L * B(7 * Ncv i) * B(8 * Ncv i - ) F_tdx(i) / uzeo * Lcl / * L * B(9 * Ncv i) * B(8 * Ncv i) End If Cae ind_dt 'dente ind_dt If ind_dt Then 'Coonente NORMALE della foza adiale F_n(i) / / uzeo * L * (LdA R * B(5 * Ncv i) ^ _R * B(8 * Ncv 8) ^ LdB R * B(8 * Ncv 7) ^ Lcl / * B(7 * Ncv i) ^ Lcl / * B(9 * Ncv i) ^ ) F_n(i) F_n(i) - / / uzeo * Lcl / * L * B(9 * Ncv) ^ - / / uzeo * Lcl / * L * B(8 * Ncv i) ^ 'Coonenti TANGENZIALI della foza adiale F_tx(i) / uzeo * Lcl / * L * B(7 * Ncv i) * B(9 * Ncv) F_tdx(i) / uzeo * Lcl / * L * B(9 * Ncv i) * B(8 * Ncv i) Ele 'Coonente NORMALE della foza adiale F_n(i) / / uzeo * L * (LdA R * B(5 * Ncv i) ^ _R * B(8 * Ncv 8) ^ LdB R * B(8 * Ncv 7) ^ Lcl / * B(7 * Ncv i) ^ Lcl / * B(9 * Ncv i) ^ ) F_n(i) F_n(i) - / / uzeo * Lcl / * L * B(8 * Ncv i - ) ^ - / / uzeo * Lcl / * L * B(8 * Ncv i) ^ 'Coonenti TANGENZIALI della foza adiale F_tx(i) / uzeo * Lcl / * L * B(7 * Ncv i) * B(8 * Ncv i - ) F_tdx(i) / uzeo * Lcl / * L * B(9 * Ncv i) * B(8 * Ncv i) End If Cae Ele If i Then 'Coonente NORMALE della foza adiale F_n(i) / / uzeo * L * (Ldg_R * B(5 * Ncv i) ^ Lcl / * B(7 * Ncv i) ^ Lcl / * B(9 * Ncv i) ^ ) F_n(i) F_n(i) - / / uzeo * Lcl / * L * B(9 * Ncv) ^ - / / uzeo * Lcl / * L * B(8 * Ncv i) ^ 'Coonenti TANGENZIALI della foza adiale F_tx(i) / uzeo * Lcl / * L * B(7 * Ncv i) * B(9 * Ncv) F_tdx(i) / uzeo * Lcl / * L * B(9 * Ncv i) * B(8 * Ncv i) Ele 'Coonente NORMALE della foza adiale F_n(i) / / uzeo * L * (Ldg_R * B(5 * Ncv i) ^ Lcl / * B(7 * Ncv i) ^ Lcl / * B(9 * Ncv i) ^ ) F_n(i) F_n(i) - / / uzeo * Lcl / * L * B(8 * Ncv i - ) ^ - / / uzeo * Lcl / * L * B(8 * Ncv i) ^ 'Coonenti TANGENZIALI della foza adiale F_tx(i) / uzeo * Lcl / * L * B(7 * Ncv i) * B(8 * Ncv i - ) F_tdx(i) / uzeo * Lcl / * L * B(9 * Ncv i) * B(8 * Ncv i) End If End Select 76

87 The ogaing code - Pat 'Coonenti econdo (y) della foza adiale in ogni dente Fy(i) F_n(i) * Sin(alfa_dt(i)) - F_tx(i) * Co(alfa_dt(i)) - F_tdx(i) * Co(alfa_dt(i)) 'Coonenti econdo (x) della foza adiale Fx(i) F_n(i) * Co(alfa_dt(i)) F_tx(i) * Sin(alfa_dt(i)) F_tdx(i) * Sin(alfa_dt(i)) 'Modulo della foza adiale in ogni dente F_od(i) Sq(Fx(i) ^ Fy(i) ^ ) 'Calcolo della iultante delle foze RADIALI Fo i To Ncv Fy_ Fy_ Fy(i) Fx_ Fx_ Fx(i) 'RISULTANTE F Sq(Fx_ ^ Fy_ ^ ) '*************************************************************************************** 'CICLO DI CALCOLO COENERGIA MAGNETICA DEL SISTEMA If OtionButton.Value Tue Then 'Meoizzazione dei valoi te ecedente 'Coenegia agnetica coleiva EnM_t EnM_t 'Meoizzazione dei valoi te ecedente 'Coenegia agnetica nel ingolo volue Fo i To n EnM_(i) EnM(i) EnM_t nh 64 'MAGNETE (SOTTO IL DENTE) Fo i To Ncv denm(i) / * B(i) ^ / u(i) EnM(i) denm(i) * Vol(i) EnM_t EnM_t EnM(i) 'CORONA STAT., CORONA ROT. Fo i Ncv To 3 * Ncv denm(i) dh Ab(H(i)) / nh Fo j To nh H (j - ) * dh H j * dh B Induction(n, H, Hla, Bla) B Induction(n, H, Hla, Bla) da (B B) * dh / denm(i) denm(i) da Next j EnM(i) denm(i) * Vol(i) EnM_t EnM_t EnM(i) 'CAVE (Coonenti oizzontali CVX) Fo i 3 * Ncv To 4 * Ncv denm(i) / * B(i) ^ / u(i) EnM(i) denm(i) * Vol(i) EnM_t EnM_t EnM(i) 'DENTE BASSO Fo i 4 * Ncv To 5 * Ncv 77

88 Aendix A. denm(i) dh Ab(H(i)) / nh Fo j To nh H (j - ) * dh H j * dh B Induction(n, H, Hla, Bla) B Induction(n, H, Hla, Bla) da (B B) * dh / denm(i) denm(i) da Next j EnM(i) denm(i) * Vol(i) EnM_t EnM_t EnM(i) 'TRAFERRO otto il DENTE Fo i 5 * Ncv To 6 * Ncv denm(i) / * B(i) ^ / u(i) EnM(i) denm(i) * Vol(i) EnM_t EnM_t EnM(i) 'DENTE ALTO Fo i 6 * Ncv To 7 * Ncv denm(i) dh Ab(H(i)) / nh Fo j To nh H (j - ) * dh H j * dh B Induction(n, H, Hla, Bla) B Induction(n, H, Hla, Bla) da (B B) * dh / denm(i) denm(i) da Next j EnM(i) denm(i) * Vol(i) EnM_t EnM_t EnM(i) 'TRAFERRO otto la SEMICAVA SX 'COLLARINO (TRASV.) 'TRAFERRO otto la SEMICAVA DX 'MAGNETE SOTTO APERTURA CAVA SX e DX Fo i 7 * Ncv To * Ncv denm(i) / * B(i) ^ / u(i) EnM(i) denm(i) * Vol(i) EnM_t EnM_t EnM(i) 'TESTA DENTE SX e DX Fo i * Ncv To 4 * Ncv denm(i) dh Ab(H(i)) / nh Fo j To nh H (j - ) * dh H j * dh B Induction(n, H, Hla, Bla) B Induction(n, H, Hla, Bla) da (B B) * dh / denm(i) denm(i) da Next j EnM(i) denm(i) * Vol(i) EnM_t EnM_t EnM(i) 'SEMICAVE SX e DX Fo i 4 * Ncv To 6 * Ncv denm(i) / * B(i) ^ / u(i) EnM(i) denm(i) * Vol(i) EnM_t EnM_t EnM(i) 78

89 The ogaing code - Pat 'CONTRIBUTI AGGIUNTIVI If ind_dt <> And ind_dt <> Then Fo i To 8 denm(8 * Ncv i) / * B(8 * Ncv i) ^ / u(8 * Ncv i) EnM(8 * Ncv i) denm(8 * Ncv i) * Vol(8 * Ncv i) EnM_t EnM_t EnM(8 * Ncv i) End If 'VARIAZIONE COENERGIA MAGNETICA TOTALE denm_t EnM_t - EnM_t 'VARIAZIONE COENERGIA MAGNETICA SINGOLI VOLUMI If ix Then Fo i To n deltaenm(i) Ele Fo i To n deltaenm(i) EnM(i) - EnM_(i) End If Cell(5 ix, 3 * nfai ) EnM_t * End If '*************************************************************************************** If OtionButton.Value Tue And Int(ix / ) ix / Then Fx_EM denm_t / dx_cot 'Coia elettoagnetica dth_ i / / Tau * dx_cot C_EM denm_t / dth_ Cell(5 ix, 3 * nfai ) C_EM * End If '*************************************************************************************** 'OUTPUT ARRAY Fo i To Ncv Cell(4 i, 5) CSt("d") Cell(4 Ncv i, 5) CSt("c") Cell(4 * Ncv i, 5) CSt("c") Cell(4 3 * Ncv i, 5) CSt("cvx") Cell(4 4 * Ncv i, 5) CSt("dL") Cell(4 5 * Ncv i, 5) CSt("gd") Cell(4 6 * Ncv i, 5) CSt("dH") Cell(4 7 * Ncv i, 5) CSt("gc") Cell(4 8 * Ncv i, 5) CSt("clx") Cell(4 9 * Ncv i, 5) CSt("gcd") Cell(4 * Ncv i, 5) CSt("c") Cell(4 * Ncv i, 5) CSt("cd") Cell(4 * Ncv i, 5) CSt("dx") Cell(4 3 * Ncv i, 5) CSt("ddx") Cell(4 4 * Ncv i, 5) CSt("Lcvy") Cell(4 5 * Ncv i, 5) CSt("Lcvyd") Cell(4 6 * Ncv i, 5) CSt("Hcvy") Cell(4 7 * Ncv i, 5) CSt("Hcvyd") Cell(5 8 * Ncv, 5) CSt("dB") Cell(6 8 * Ncv, 5) CSt("") Cell(7 8 * Ncv, 5) CSt("gdB") Cell(8 8 * Ncv, 5) CSt("gd") Cell(9 8 * Ncv, 5) CSt("dB") 79

90 Aendix A. Cell( 8 * Ncv, 5) CSt("") Cell( 8 * Ncv, 5) CSt("gdB") Cell( 8 * Ncv, 5) CSt("gd") Fo i To n Cell(4 i, 6) H(i) Cell(4 i, 7) B(i) Cell(4 i, 8) u(i) / uzeo Cell(4 i, 9) x(i) Cell(5 ix, ) x Cell(5 ix, 3) Hc Fo i To nfai Cell(5 ix, 3 i) Ife(i) Fo i To nfai Cell(5 ix, 3 nfai i) Flux_f(i) * Cell(5 ix, 3 * nfai 3) Fx_ Cell(5 ix, 3 * nfai 4) Fy_ Cell(5 ix, 3 * nfai 5) F If OtionButton.Value Tue Then Pint #3, x Pint #3, Pint #3, "i VOL(^3) denm(j/^3) EnM(J) H(A/) B(T) deltaenm(j) Ril(H-) x(wb)" Pint #3, Fo i To 8 * Ncv Val_ Foat(i, "" & " ") Val_ Foat(Vol(i), ".E" & " ") Val_ Foat(dEnM(i), ".E" & " ") Val_3 Foat(EnM(i), ".E" & " ") Val_4 Foat(H(i), "." & " ") Val_5 Foat(B(i), "." & " ") Val_6 Foat(deltaEnM(i), ".E" & " ") Val_7 Foat(Ril(i), ".E" & " ") Val_8 Foat(x(i), ".E" & " ") Pint #3, Val_ & Val_ & Val_ & Val_3 & Val_4 & Val_5 & Val_6 & Val_7 & Val_8 'ELEMENTI AGGIUNTIVI Fo i 8 * Ncv To n Val_ Foat(i, "" & " ") Val_ Foat(Vol(i), ".E" & " ") Val_ Foat(dEnM(i), ".E" & " ") Val_3 Foat(EnM(i), ".E" & " ") Val_4 Foat(H(i), "." & " ") Val_5 Foat(B(i), "." & " ") Val_6 Foat(deltaEnM(i), ".E" & " ") Val_7 Foat(Ril(i), ".E" & " ") Val_8 Foat(x(i), ".E" & " ") Pint #3, Val_ & Val_ & Val_ & Val_3 & Val_4 & Val_5 & Val_6 & Val_7 & Val_8 Pint #3, Pint #3, Pint #3, "Fy(N) Fx(N) alfa_dt(g..) F_od(N)" Pint #3, Fo i To Ncv Val_9 Foat(Fy(i), "." & " ") Val_ Foat(Fx(i), "." & " ") Val_ Foat(alfa_dt_deg(i), "." & " ") Val_ Foat(F_od(i), "." & " ") Pint #3, Val_9 & Val_ & Val_ & Val_ 8

91 The ogaing code - Pat Pint #3, Pint #3, Pint #3, End If 'aggionaento della oizione dei agneti If Int(ix / ) <> ix / Then dx dx_cot Ele dx dx End If x x dx x x w Cell(5, 34) zeo_ctl x 5 Cloe #3 OtionButton.Value Fale OtionButton.Value Fale End Sub Public Sub SOLVESYS() Fo i To n Fo j To 'atice iniziale del itea ainiz(i, j) a(i, j) Next j '*************************************************************************************** axdiffx z Fo i To n u(i) uzeo u_(i) Do While axdiffx > tollx '*************************************************************************************** 'TRASFORMAZIONE DELLA MATRICE a(i,j) IN TRIANGOLARE SUPERIORE ivot Fale fault ie Fo To n - 'lo cabio avviene con le ighe ucceive a quella conideata (-eia) Do Until a(, ) <> 'ciclo che cabia la iga -eia con la -eia 'nel cao in cui un eleento diagonale ia nullo ivot Tue Fo icol To change_r(icol) a(, icol) a(, icol) a(, icol) a(, icol) change_r(icol) 8

92 Aendix A. col Fo i To n iv_r(i) iv_r() iv_r() If > n Then MgBox ("Un eleento A(,) iane nullo doo ave coo tutte le ighe ( > n)") Sto End If ie ie Loo If ivot Tue Then e() Ele e() End If Fo i To n y(i, ) -a(i, ) / a(, ) a(i, ) y(i, ) Fo j To a(i, j) a(i, j) a(i, ) * a(, j) Next j ivot Fale Next '*************************************************************************************** 'SOLUZIONE DEL SISTEMA DI EQUAZIONI CON IL METODO DIRETTO Fo i To n noti(i) a(i, ) x(n) noti(n) / a(n, n) Fo i n - To Ste - Fo n To i Ste - a(i, ) * x() Next x(i) (noti(i) - ) / a(i, i) '*************************************************************************************** 'eoizzazione eeabilità allo te ecedente Fo j To n u_(j) u(j) Next j 'deteinazione delle induzioni dai flui Fo i To n B(i) x(i) / Sez(i) 'ichiaa la function e il calcolo delle eeabilità agnetiche Fo j To n Select Cae j Cae To Ncv 8

93 The ogaing code - Pat u(j) u_ag H(j) / u_ag * B(j) Cae Ncv To 3 * Ncv H(j) Magfield(n, B(j), Hla, Bla) If B(j) < Then H(j) -H(j) If B(j) Then u(j) (Bla() - Bla()) / (Hla() - Hla()) zeo_ctl zeo_ctl Ele u(j) B(j) / H(j) End If Cae 3 * Ncv To 4 * Ncv u(j) uzeo H(j) / uzeo * B(j) Cae 4 * Ncv To 5 * Ncv H(j) Magfield(n, B(j), Hla, Bla) If B(j) < Then H(j) -H(j) If B(j) Then u(j) (Bla() - Bla()) / (Hla() - Hla()) zeo_ctl zeo_ctl Ele u(j) B(j) / H(j) End If Cae 5 * Ncv To 6 * Ncv u(j) uzeo H(j) / uzeo * B(j) Cae 6 * Ncv To 7 * Ncv H(j) Magfield(n, B(j), Hla, Bla) If B(j) < Then H(j) -H(j) If B(j) Then u(j) (Bla() - Bla()) / (Hla() - Hla()) zeo_ctl zeo_ctl Ele u(j) B(j) / H(j) End If Cae 7 * Ncv To * Ncv u(j) uzeo H(j) / uzeo * B(j) Cae * Ncv To * Ncv u(j) u_ag H(j) / u_ag * B(j) Cae * Ncv To 4 * Ncv H(j) Magfield(n, B(j), Hla, Bla) If B(j) < Then H(j) -H(j) If B(j) Then u(j) (Bla() - Bla()) / (Hla() - Hla()) zeo_ctl zeo_ctl Ele u(j) B(j) / H(j) End If Cae 4 * Ncv To 8 * Ncv u(j) uzeo H(j) / uzeo * B(j) End Select Next j If ind_dt <> Then u(8 * Ncv ) u_ag H(8 * Ncv ) / u_ag * B(8 * Ncv ) u(8 * Ncv ) uzeo H(8 * Ncv ) / uzeo * B(8 * Ncv ) u(8 * Ncv 3) uzeo H(8 * Ncv 3) / uzeo * B(8 * Ncv 3) u(8 * Ncv 4) uzeo H(8 * Ncv 4) / uzeo * B(8 * Ncv 4) End If If ind_dt <> Then u(8 * Ncv 5) u_ag H(8 * Ncv 5) / u_ag * B(8 * Ncv 5) 83

94 Aendix A. u(8 * Ncv 6) uzeo H(8 * Ncv 6) / uzeo * B(8 * Ncv 6) u(8 * Ncv 7) uzeo H(8 * Ncv 7) / uzeo * B(8 * Ncv 7) u(8 * Ncv 8) uzeo H(8 * Ncv 8) / uzeo * B(8 * Ncv 8) End If 'Ridefinizione delle eeabilità agnetiche Fo j To n u(j) u(j) ^. * u_(j) ^.9 Next j '************************************************************************************ ' **** CRITERIO DI CONVERGENZA **** axdiffx 'Solo eleenti NON LINEARI Fo j Ncv To 3 * Ncv diffx(j) Ab((u(j) / uzeo - u_(j) / uzeo) / (u_(j) / uzeo)) If diffx(j) > axdiffx Then axdiffx diffx(j) End If Next j Fo j 4 * Ncv To 5 * Ncv diffx(j) Ab((u(j) / uzeo - u_(j) / uzeo) / (u_(j) / uzeo)) If diffx(j) > axdiffx Then axdiffx diffx(j) End If Next j Fo j 6 * Ncv To 7 * Ncv diffx(j) Ab((u(j) / uzeo - u_(j) / uzeo) / (u_(j) / uzeo)) If diffx(j) > axdiffx Then axdiffx diffx(j) End If Next j Fo j * Ncv To 4 * Ncv diffx(j) Ab((u(j) / uzeo - u_(j) / uzeo) / (u_(j) / uzeo)) If diffx(j) > axdiffx Then axdiffx diffx(j) End If Next j '*************************************************************************************** 'RIDEFINIZIONE MATRICE "A" DEL SISTEMA 'vengono aggionati i coefficienti vaiabili in cui coaiono i valoi di eeabilità 'agnetica delle vaie ati di acchina '*************************************************************************************** 'azzeaento di tutti i coefficienti Fo i To n Fo j To a(i, j) Next j 'RIDEFINIZIONE DELLE RILUTTANZE VARIABILI f(u) 'coona tatoe Fo i Ncv To * Ncv Ril(i) / u(i) * Lc_ed / Sez(i) 'coona otoe Fo i * Ncv To 3 * Ncv Ril(i) / u(i) * Lc_ed / Sez(i) 84

95 The ogaing code - Pat 'dente LOW Fo i 4 * Ncv To 5 * Ncv Ril(i) / u(i) * (adt * (ta - ) havv) / Sez(i) 'dente HIGH Fo i 6 * Ncv To 7 * Ncv Ril(i) / u(i) * (adt * ( - ta)) / Sez(i) 'eidente SX e DX Fo i * Ncv To 4 * Ncv Ril(i) / u(i) / * Log( * / ( * - * (Ldg - Ldt))) / u(i) * Ldt / / (adt - havv) / L 'le ighe con coefficienti cotanti della atice del itea vengono icoote coe 'quelle iniziali e iatie con la oluzione del itea, aggionato aggiungendo 'le eeabilità calcolate Fo i 5 * Ncv To * Ncv Fo j To n a(i, j) ainiz(i, j) Next j Fo i 4 * Ncv To 6 * Ncv Fo j To n a(i, j) ainiz(i, j) Next j 'l'ultia colonna della atice (teini noti) 'iaue i valoi iniziali Fo i To n a(i, ) ainiz(i, ) 'PARTE VARIABILE DELLA MATRICE (A) ' to Ncv ***************************************************************************** 'CIRCUITAZIONI DENTE - DENTE Fo i To Ncv - a(i, i) Ril(i) a(i, i ) -Ril(i ) a(i, i Ncv) -Ril(i Ncv) a(i, i * Ncv) -Ril(i * Ncv) a(i, i 4 * Ncv) Ril(i 4 * Ncv) a(i, i 4 * Ncv ) -Ril(i 4 * Ncv ) a(i, i 5 * Ncv) Ril(i 5 * Ncv) a(i, i 5 * Ncv ) -Ril(i 5 * Ncv ) a(i, i 6 * Ncv) Ril(i 6 * Ncv) a(i, i 6 * Ncv ) -Ril(i 6 * Ncv ) a(ncv, Ncv) Ril(Ncv) a(ncv, ) -Ril() a(ncv, * Ncv) -Ril( * Ncv) a(ncv, 3 * Ncv) -Ril(3 * Ncv) a(ncv, 5 * Ncv) Ril(5 * Ncv) a(ncv, 4 * Ncv ) -Ril(4 * Ncv ) a(ncv, 6 * Ncv) Ril(6 * Ncv) a(ncv, 5 * Ncv ) -Ril(5 * Ncv ) a(ncv, 7 * Ncv) Ril(7 * Ncv) a(ncv, 6 * Ncv ) -Ril(6 * Ncv ) Select Cae ind_dt 85

96 Aendix A. Cae 'Cicuitaz. ecedente a(ncv, ) a(ncv, 5 * Ncv ) a(ncv, 8 * Ncv ) -Ril(8 * Ncv ) a(ncv, 8 * Ncv 4) -Ril(8 * Ncv 4) 'Cicuitaz. attuale a(, ) a(, 5 * Ncv ) a(, 8 * Ncv ) Ril(8 * Ncv ) a(, 8 * Ncv 4) Ril(8 * Ncv 4) Cae To (Ncv - ) 'Cicuitaz. ecedente a(ind_dt -, ind_dt) a(ind_dt -, ind_dt 5 * Ncv) a(ind_dt -, 8 * Ncv ) -Ril(8 * Ncv ) a(ind_dt -, 8 * Ncv 4) -Ril(8 * Ncv 4) 'Cicuitaz. attuale a(ind_dt, ind_dt) a(ind_dt, ind_dt 5 * Ncv) a(ind_dt, 8 * Ncv ) Ril(8 * Ncv ) a(ind_dt, 8 * Ncv 4) Ril(8 * Ncv 4) Cae Ncv 'Cicuitaz. ecedente a(ncv -, Ncv) a(ncv -, 6 * Ncv) a(ncv -, 8 * Ncv ) -Ril(8 * Ncv ) a(ncv -, 8 * Ncv 4) -Ril(8 * Ncv 4) 'Cicuitaz. attuale a(ncv, Ncv) a(ncv, 6 * Ncv) a(ncv, 8 * Ncv ) Ril(8 * Ncv ) a(ncv, 8 * Ncv 4) Ril(8 * Ncv 4) Cae Ele Sto End Select Select Cae ind_dt Cae 'Cicuitaz. ecedente a(ncv, ) a(ncv, 5 * Ncv ) a(ncv, 8 * Ncv 6) -Ril(8 * Ncv 6) a(ncv, 8 * Ncv 8) -Ril(8 * Ncv 8) 'Cicuitaz. attuale a(, ) a(, 5 * Ncv ) a(, 8 * Ncv 6) Ril(8 * Ncv 6) a(, 8 * Ncv 8) Ril(8 * Ncv 8) Cae To (Ncv - ) 'Cicuitaz. ecedente a(ind_dt -, ind_dt) a(ind_dt -, ind_dt 5 * Ncv) a(ind_dt -, 8 * Ncv 6) -Ril(8 * Ncv 6) a(ind_dt -, 8 * Ncv 8) -Ril(8 * Ncv 8) 'Cicuitaz. attuale a(ind_dt, ind_dt) a(ind_dt, ind_dt 5 * Ncv) a(ind_dt, 8 * Ncv 6) Ril(8 * Ncv 6) a(ind_dt, 8 * Ncv 8) Ril(8 * Ncv 8) Cae Ncv 'Cicuitaz. ecedente a(ncv -, Ncv) a(ncv -, 6 * Ncv) a(ncv -, 8 * Ncv 6) -Ril(8 * Ncv 6) a(ncv -, 8 * Ncv 8) -Ril(8 * Ncv 8) 'Cicuitaz. attuale a(ncv, Ncv) a(ncv, 6 * Ncv) 86

97 The ogaing code - Pat a(ncv, 8 * Ncv 6) Ril(8 * Ncv 6) a(ncv, 8 * Ncv 8) Ril(8 * Ncv 8) Cae Ele Sto End Select 'Ncv to Ncv ************************************************************************* 'CIRCUITAZIONI DENTE H Fo i To Ncv - a(ncv i, Ncv i) -Ril(Ncv i) a(ncv i, 3 * Ncv i) -Ril(3 * Ncv i) a(ncv i, 6 * Ncv i) Ril(6 * Ncv i) a(ncv i, 6 * Ncv i) -Ril(6 * Ncv i) a( * Ncv, * Ncv) -Ril( * Ncv) a( * Ncv, 4 * Ncv) -Ril(4 * Ncv) a( * Ncv, 7 * Ncv) Ril(7 * Ncv) a( * Ncv, 6 * Ncv ) -Ril(6 * Ncv ) 'Ncv to 3Ncv ************************************************************************ 'CIRCUITAZIONI DENTE - DENTE (without AIRGAP) Fo i To Ncv - a( * Ncv i, Ncv i) -Ril(Ncv i) a( * Ncv i, 4 * Ncv i) Ril(4 * Ncv i) a( * Ncv i, 4 * Ncv i) -Ril(4 * Ncv i) a( * Ncv i, 6 * Ncv i) Ril(6 * Ncv i) a( * Ncv i, 6 * Ncv i) -Ril(6 * Ncv i) a( * Ncv i, 8 * Ncv i) -Ril(8 * Ncv i) a( * Ncv i, * Ncv i) -Ril( * Ncv i) a( * Ncv i, 3 * Ncv i) -Ril(3 * Ncv i) a(3 * Ncv, * Ncv) -Ril( * Ncv) a(3 * Ncv, 5 * Ncv) Ril(5 * Ncv) a(3 * Ncv, 4 * Ncv ) -Ril(4 * Ncv ) a(3 * Ncv, 7 * Ncv) Ril(7 * Ncv) a(3 * Ncv, 6 * Ncv ) -Ril(6 * Ncv ) a(3 * Ncv, 9 * Ncv) -Ril(9 * Ncv) a(3 * Ncv, * Ncv ) -Ril( * Ncv ) a(3 * Ncv, 4 * Ncv) -Ril(4 * Ncv) '3Ncv to 4Ncv ************************************************************************ 'CIRCUITAZIONI BASE DENTE - TRAFERRO - MAGNETE DX Fo i To Ncv a(3 * Ncv i, i) Ril(i) a(3 * Ncv i, * Ncv i) -Ril( * Ncv i) a(3 * Ncv i, 5 * Ncv i) Ril(5 * Ncv i) a(3 * Ncv i, 9 * Ncv i) -Ril(9 * Ncv i) a(3 * Ncv i, 3 * Ncv i) Ril(3 * Ncv i) If ind_dt <> Then a(3 * Ncv ind_dt, ind_dt) a(3 * Ncv ind_dt, 5 * Ncv ind_dt) a(3 * Ncv ind_dt, 8 * Ncv ) Ril(8 * Ncv ) a(3 * Ncv ind_dt, 8 * Ncv 3) Ril(8 * Ncv 3) End If If ind_dt <> Then a(3 * Ncv ind_dt, ind_dt) a(3 * Ncv ind_dt, 5 * Ncv ind_dt) a(3 * Ncv ind_dt, 8 * Ncv 5) Ril(8 * Ncv 5) a(3 * Ncv ind_dt, 8 * Ncv 7) Ril(8 * Ncv 7) End If '4Ncv to 5Ncv ************************************************************************ 87

98 Aendix A. 'CIRCUITAZIONI BASE DENTE - TRAFERRO - MAGNETE SX Fo i To Ncv a(4 * Ncv i, i) -Ril(i) a(4 * Ncv i, * Ncv i) Ril( * Ncv i) a(4 * Ncv i, 5 * Ncv i) -Ril(5 * Ncv i) a(4 * Ncv i, 7 * Ncv i) Ril(7 * Ncv i) a(4 * Ncv i, * Ncv i) Ril( * Ncv i) '5Ncv ******************************************************************************** 'CIRCUITAZIONE HcS Fo i To Ncv a(5 * Ncv, Ncv i) Ril(Ncv i) 'Ncv to 3Ncv ********************************************************************** 'CIRCUITAZIONI DENTE - SEMIDENTE - SEMICAVA DX Fo i To Ncv a( * Ncv i, 4 * Ncv i) Ril(4 * Ncv i) a( * Ncv i, 6 * Ncv i) Ril(6 * Ncv i) a( * Ncv i, 3 * Ncv i) -Ril(3 * Ncv i) a( * Ncv i, 5 * Ncv i) -Ril(5 * Ncv i) a( * Ncv i, 7 * Ncv i) -Ril(7 * Ncv i) '3Ncv to 4Ncv ********************************************************************** 'CIRCUITAZIONI SEMICAVA SX LOW Fo i To Ncv 'a(3 * Ncv i, 4 * Ncv i) -Ril(4 * Ncv i) a(3 * Ncv i, * Ncv i) -Ril( * Ncv i) a(3 * Ncv i, 3 * Ncv i) -Ril(3 * Ncv i) a(3 * Ncv i, 4 * Ncv i) Ril(4 * Ncv i) a(3 * Ncv i, 5 * Ncv i) -Ril(5 * Ncv i) '6Ncv to 7Ncv ********************************************************************* 'CIRCUITAZIONI ALTA SEMICAVA - ALTO SEMIDENTE SX Fo i To Ncv a(6 * Ncv i, 6 * Ncv i) Ril(6 * Ncv i) a(6 * Ncv i, 6 * Ncv i) -Ril(6 * Ncv i) '7Ncv to 8Ncv ********************************************************************* 'CIRCUITAZIONI ALTA SEMICAVA - ALTO SEMIDENTE DX Fo i To Ncv a(7 * Ncv i, 6 * Ncv i) Ril(6 * Ncv i) a(7 * Ncv i, 7 * Ncv i) -Ril(7 * Ncv i) '8Ncv to 8Ncv8 ******************************************************************* 'EQUAZIONI AGGIUNTIVE If ind_dt <> Then 'Cicuitazione PRIMA dicontinuità del agnete a(8 * Ncv, ind_dt) Ril(ind_dt) a(8 * Ncv, 5 * Ncv ind_dt) Ril(5 * Ncv ind_dt) a(8 * Ncv, 8 * Ncv 4) -Ril(8 * Ncv 4) a(8 * Ncv, 8 * Ncv ) -Ril(8 * Ncv ) 'Cicuitazione SECONDA dicontinuità del agnete a(8 * Ncv, 8 * Ncv ) Ril(8 * Ncv ) a(8 * Ncv, 8 * Ncv ) -Ril(8 * Ncv ) a(8 * Ncv, 8 * Ncv 4) Ril(8 * Ncv 4) a(8 * Ncv, 8 * Ncv 3) -Ril(8 * Ncv 3) 88

99 The ogaing code - Pat 'NODI 7() a(8 * Ncv 3, 8 * Ncv ) - a(8 * Ncv 3, 8 * Ncv 4) 'NODI 7(3) a(8 * Ncv 4, 8 * Ncv ) - a(8 * Ncv 4, 8 * Ncv 3) End If If ind_dt <> Then 'Cicuitazione PRIMA dicontinuità del agnete a(8 * Ncv 5, ind_dt) Ril(ind_dt) a(8 * Ncv 5, 5 * Ncv ind_dt) Ril(5 * Ncv ind_dt) a(8 * Ncv 5, 8 * Ncv 8) -Ril(8 * Ncv 8) a(8 * Ncv 5, 8 * Ncv 6) -Ril(8 * Ncv 6) 'Cicuitazione SECONDA dicontinuità del agnete a(8 * Ncv 6, 8 * Ncv 6) Ril(8 * Ncv 6) a(8 * Ncv 6, 8 * Ncv 5) -Ril(8 * Ncv 5) a(8 * Ncv 6, 8 * Ncv 8) Ril(8 * Ncv 8) a(8 * Ncv 6, 8 * Ncv 7) -Ril(8 * Ncv 7) 'NODI 7() a(8 * Ncv 7, 8 * Ncv 6) - a(8 * Ncv 7, 8 * Ncv 8) 'NODI 7(3) a(8 * Ncv 8, 8 * Ncv 5) - a(8 * Ncv 8, 8 * Ncv 7) End If Cell(4 z, 3) z Cell(4 z, 3) axdiffx * z z Loo End Sub Public Function Induction(n, x_, Hla, Bla) 'NON LINEARE Fo i To 45 If Ab(x_) > Hla(i) And Ab(x_) < Hla(i ) Then deltabh (Bla(i ) - Bla(i)) / (Hla(i ) - Hla(i)) Induction deltabh * (Ab(x_) - Hla(i)) Bla(i) End If If Ab(x_) > Hla(45) Then deltabh (Bla(45) - Bla(45)) / (Hla(45) - Hla(45)) Induction deltabh * (Ab(x_) - Hla(45)) Bla(45) End If End Function Public Function Magfield(n, x_, Hla, Bla) 'NON LINEARE Fo i To 45 If Ab(x_) > Bla(i) And Ab(x_) < Bla(i ) Then deltahb (Hla(i ) - Hla(i)) / (Bla(i ) - Bla(i)) Magfield deltahb * (Ab(x_) - Bla(i)) Hla(i) End If 89

100 Aendix A. If Ab(x_) > Bla(45) Then deltahb (Hla(45) - Hla(45)) / (Bla(45) - Bla(45)) Magfield deltahb * (Ab(x_) - Bla(45)) Hla(45) End If End Function 9

101 The ogaing code - Pat Aendix A. THE PROGRAMMING CODE Pat In the following, the ogaing code of the uboutine to deteine the aay of the tato aee-tun ditibution, to ave in the file ASP_TOT.txt cobined with the agnet ditibution. Two veion ae eented, a ile one and a oe elaboate one. A.. The MMF aay (ilified veion) Cont i Pivate Sub CoandButton_Clic() 'INPUT Ncv Foglio.Cell(4, 4) n Foglio.Cell(5, 4) nc Foglio.Cell(6, 4) Tau Foglio.Cell(7, 4) Tau Tau *. I Foglio.Cell(8, 4) adt Foglio.Cell(6, 4) adt adt *. havv Foglio.Cell(, 4) havv havv *. ta Foglio.Cell(3, 4) x Foglio.Cell(4, 4) x x *. dx_cot Foglio.Cell(5, 4) dx_cot dx_cot *. dx Foglio.Cell(6, 4) dx dx *. ix_ax Foglio.Cell(7, 4) fi_in Foglio.Cell(8, 4) nfai Foglio.Cell(3, 4) 9

102 Aendix A. fileath CSt(Foglio.TextBox.Text) If ta < - havv / adt Then MgBox ("Valoe di ta too bao!") Sto End If f 5 w t 'Gandezze deivate w * i * f n 8 * Ncv 8 ci adt * ( - ta) / havv ReDi od( To Ncv), odc( To Ncv) A Intege ReDi ai( To n), ai( To n), ai( To n), ai_( To n), am( To n), ASP( To n) A Double ReDi Ife( To nfai), Ife_( To nfai) A Double 'Lettua del file elativo al I tato dell'avvolgiento tatoico Oen fileath & "\od.txt" Fo Inut A # Fo i To Ncv Inut #, od(i) Cloe # 'Lettua del file elativo al II tato dell'avvolgiento tatoico Oen fileath & "\odc.txt" Fo Inut A # Fo i To Ncv Inut #, odc(i) Cloe # 'Lettua del file contenente le aeie equivalenti del agnete Oen fileath & "\am().txt" Fo Inut A #3 Fo i To n Inut #3, am(i) Cloe #3 Oen fileath & "\ASP_TOT.txt" Fo Outut A #4 Fo ix To ix_ax If Int(ix / ) <> ix / Then 'te di otaento DISPARI: le coenti vengono calcolate 'econdo l'eatto valoe itantaneo coelato alla oizione 'inizializzazione dell'aay Fo i To n ai(i) 'Andaento inuoidale delle coenti t i * x / w / Tau 'Meoizzazione del valoe delle coenti allo te ecedente Fo i To nfai Ife_(i) Ife(i) Fo j To nfai Ife(j) -Sq() * I * Co(w * w * t - (j - ) * w * t * * i / nfai fi_in) Next j 'ARRAY AMPERSPIRE STATORICHE Fo i To Ncv ai(i) n * Sgn(od(i)) * Ife(Ab(od(i))) 9

103 The ogaing code - Pat ai(i) nc * Sgn(odc(i)) * Ife(Ab(odc(i))) Fo i To Ncv ai(ncv i) ai(ncv i) nc * Sgn(odc(i)) * Ife(Ab(odc(i))) Fo i To Ncv ai( * Ncv i) n * Sgn(od(i)) * Ife(Ab(od(i))) ai( * Ncv i) nc * Sgn(odc(i)) * Ife(Ab(odc(i))) Fo i To Ncv ai( * Ncv i) n / * Sgn(od(i)) * Ife(Ab(od(i))) ai( * Ncv i) nc / * Sgn(odc(i)) * Ife(Ab(odc(i))) ai(3 * Ncv ) n / * Sgn(od(Ncv)) * Ife(Ab(od(Ncv))) n / * Sgn(od()) * Ife(Ab(od())) ai(3 * Ncv ) Fo i To Ncv ai(3 * Ncv i) n / * Sgn(od(i - )) * Ife(Ab(od(i - ))) n / * Sgn(od(i)) * Ife(Ab(od(i))) ai(3 * Ncv i) ai(6 * Ncv ) ai(6 * Ncv ) nc / * Sgn(odc(Ncv)) * Ife(Ab(odc(Ncv))) Fo i To Ncv ai(6 * Ncv i) ai(6 * Ncv i) nc / * Sgn(odc(i - )) * Ife(Ab(odc(i - ))) Fo i To Ncv ai(7 * Ncv i) ai(7 * Ncv i) nc / * Sgn(odc(i)) * Ife(Ab(odc(i))) 'cooizione dei vettoi dei due tati in un unico (ASP TOTALI STATORICHE) Fo i To 3 * Ncv ai(i) ai(i) ai(i) Fo i * Ncv To 4 * Ncv ai(i) ai(i) ai(i) Fo i 6 * Ncv To 8 * Ncv ai(i) ai(i) ai(i) 'eoizzazione e lo te ucceivo '(nel quale le coenti etano invaiate) Fo i To n ai_(i) ai(i) dx dx_cot Ele 'te di otaento PARI: le coenti vengono 'antenute cotanti e il calcolo della Fx Fo i To n ai(i) ai_(i) End If dx dx 93

104 Aendix A. 'cooizione vettoe aeie totali Fo i To n ASP(i) ai(i) am(i) 'cittua u file Fo i To n Pint #4, ASP(i) x x dx x Cloe #4 End Sub A.. The MMF aay (oiginal veion) Cont i Pivate Sub CoandButton_Clic() 'INPUT Ncv Foglio.Cell(4, 4) n Foglio.Cell(5, 4) nc Foglio.Cell(6, 4) Tau Foglio.Cell(7, 4) Tau Tau *. I Foglio.Cell(8, 4) adt Foglio.Cell(6, 4) adt adt *. havv Foglio.Cell(, 4) havv havv *. ta Foglio.Cell(3, 4) x Foglio.Cell(4, 4) x x *. dx_cot Foglio.Cell(5, 4) dx_cot dx_cot *. dx Foglio.Cell(6, 4) dx dx *. ix_ax Foglio.Cell(7, 4) fi_in Foglio.Cell(8, 4) nfai Foglio.Cell(3, 4) fileath CSt(Foglio.TextBox.Text) If ta < - havv / adt Then MgBox ("Valoe di ta too bao!") Sto End If f 5 w t 94

105 The ogaing code - Pat 'Gandezze deivate w * i * f n 8 * Ncv 8 ci adt * ( - ta) / havv ReDi od( To Ncv), odc( To Ncv) A Intege ReDi ai( To n), ai( To n), ai( To n), ai_( To n), am( To n), ASP( To n) A Double ReDi Ife( To nfai), Ife_( To nfai) A Double 'Lettua del file elativo al I tato dell'avvolgiento tatoico Oen fileath & "\od.txt" Fo Inut A # Fo i To Ncv Inut #, od(i) Cloe # 'Lettua del file elativo al II tato dell'avvolgiento tatoico Oen fileath & "\odc.txt" Fo Inut A # Fo i To Ncv Inut #, odc(i) Cloe # 'Lettua del file contenente le aeie equivalenti del agnete Oen fileath & "\am().txt" Fo Inut A #3 Fo i To n Inut #3, am(i) Cloe #3 Oen fileath & "\ASP_TOT.txt" Fo Outut A #4 Fo ix To ix_ax If Int(ix / ) <> ix / Then 'te di otaento DISPARI: le coenti vengono calcolate 'econdo l'eatto valoe itantaneo coelato alla oizione 'inizializzazione dell'aay Fo i To n ai(i) 'Andaento inuoidale delle coenti t i * x / w / Tau 'Meoizzazione del valoe delle coenti allo te ecedente Fo i To nfai Ife_(i) Ife(i) Fo j To nfai Ife(j) -Sq() * I * Co(w * w * t - (j - ) * w * t * * i / nfai fi_in) Next j 'ARRAY AMPERSPIRE STATORICHE Fo i To Ncv ai(i) n * Sgn(od(i)) * Ife(Ab(od(i))) ai(i) nc * Sgn(odc(i)) * Ife(Ab(odc(i))) Fo i To Ncv If ( < ci) And (ci <.5) Then ai(ncv i) ai(ncv i) * nc * ci * Sgn(odc(i)) * Ife(Ab(odc(i))) EleIf (.5 < ci) And (ci < ) Then ai(ncv i) ( * n * ci - n) * Sgn(od(i)) * Ife(Ab(od(i))) ai(ncv i) nc * Sgn(odc(i)) * Ife(Ab(odc(i))) Ele ai(ncv i) n * Sgn(od(i)) * Ife(Ab(od(i))) 95

106 Aendix A. ai(ncv i) nc * Sgn(odc(i)) * Ife(Ab(odc(i))) End If Fo i To Ncv ai( * Ncv i) n * Sgn(od(i)) * Ife(Ab(od(i))) ai( * Ncv i) nc * Sgn(odc(i)) * Ife(Ab(odc(i))) Fo i To Ncv ai( * Ncv i) n / * Sgn(od(i)) * Ife(Ab(od(i))) ai( * Ncv i) nc / * Sgn(odc(i)) * Ife(Ab(odc(i))) If ( < ci) And (ci <.5) Then 'concatena in ogni cao ee tutto il (I) ai(3 * Ncv ) n / * Sgn(od(Ncv)) * Ife(Ab(od(Ncv))) n / * Sgn(od()) * Ife(Ab(od())) 'concatena una ate del (II), a econda del valoe di ta ai(3 * Ncv ) (nc - * nc * ci) / * Sgn(odc(Ncv)) * Ife(Ab(odc(Ncv))) (nc - * nc * ci) / * Sgn(odc()) * Ife(Ab(odc())) EleIf (.5 < ci) And (ci < ) Then ai(3 * Ncv ) * n * ( - ci) / * Sgn(od(Ncv)) * Ife(Ab(od(Ncv))) * n * ( - ci) / * Sgn(od()) * Ife(Ab(od())) ai(3 * Ncv ) Ele ai(3 * Ncv ) ai(3 * Ncv ) End If Fo i To Ncv If ( < ci) And (ci <.5) Then 'concatena in ogni cao ee tutto il (I) ai(3 * Ncv i) n / * Sgn(od(i - )) * Ife(Ab(od(i - ))) n / * Sgn(od(i)) * Ife(Ab(od(i))) 'concatena una ate del (II), a econda del valoe di ta ai(3 * Ncv i) (nc - * nc * ci) / * Sgn(odc(i - )) * Ife(Ab(odc(i - ))) (nc - * nc * ci) / * Sgn(odc(i)) * Ife(Ab(odc(i))) EleIf (.5 < ci) And (ci < ) Then ai(3 * Ncv i) * n * ( - ci) / * Sgn(od(i - )) * Ife(Ab(od(i - ))) * n * ( - ci) / * Sgn(od(i)) * Ife(Ab(od(i))) ai(3 * Ncv i) Ele ai(3 * Ncv i) ai(3 * Ncv i) End If If ( < ci) And (ci <.5) Then ai(6 * Ncv ) ai(6 * Ncv ) * nc * ci / * Sgn(odc(Ncv)) * Ife(Ab(odc(Ncv))) EleIf (.5 < ci) And (ci < ) Then ai(6 * Ncv ) ( * n * ci - n) / * Sgn(od(Ncv)) * Ife(Ab(od(Ncv))) ai(6 * Ncv ) nc / * Sgn(odc(Ncv)) * Ife(Ab(odc(Ncv))) Ele ai(6 * Ncv ) n / * Sgn(od(Ncv)) * Ife(Ab(od(Ncv))) ai(6 * Ncv ) nc / * Sgn(odc(Ncv)) * Ife(Ab(odc(Ncv))) End If Fo i To Ncv If ( < ci) And (ci <.5) Then ai(6 * Ncv i) ai(6 * Ncv i) * nc * ci / * Sgn(odc(i)) * Ife(Ab(odc(i))) EleIf (.5 < ci) And (ci < ) Then ai(6 * Ncv i) ( * n * ci - n) / * Sgn(od(i)) * Ife(Ab(od(i))) ai(6 * Ncv i) nc / * Sgn(odc(i)) * Ife(Ab(odc(i))) 96

107 The ogaing code - Pat Ele ai(6 * Ncv i) n / * Sgn(od(i)) * Ife(Ab(od(i))) ai(6 * Ncv i) nc / * Sgn(odc(i)) * Ife(Ab(odc(i))) End If Fo i To Ncv If ( < ci) And (ci <.5) Then ai(7 * Ncv i) ai(7 * Ncv i) * nc * ci / * Sgn(odc(i)) * Ife(Ab(odc(i))) EleIf (.5 < ci) And (ci < ) Then ai(7 * Ncv i) ( * n * ci - n) / * Sgn(od(i)) * Ife(Ab(od(i))) ai(7 * Ncv i) nc / * Sgn(odc(i)) * Ife(Ab(odc(i))) Ele ai(7 * Ncv i) n / * Sgn(od(i)) * Ife(Ab(od(i))) ai(7 * Ncv i) nc / * Sgn(odc(i)) * Ife(Ab(odc(i))) End If 'cooizione dei vettoi dei due tati in un unico (ASP TOTALI STATORICHE) Fo i To 3 * Ncv ai(i) ai(i) ai(i) Fo i * Ncv To 4 * Ncv ai(i) ai(i) ai(i) Fo i 6 * Ncv To 8 * Ncv ai(i) ai(i) ai(i) 'eoizzazione e lo te ucceivo '(nel quale le coenti etano invaiate) Fo i To n ai_(i) ai(i) dx dx_cot Ele 'te di otaento PARI: le coenti vengono 'antenute cotanti e il calcolo della Fx Fo i To n ai(i) ai_(i) dx dx End If 'cooizione vettoe aeie totali Fo i To n ASP(i) ai(i) am(i) 'cittua u file Fo i To n Pint #4, ASP(i) x x dx x Cloe #4 End Sub 97

108 Aendix A. Fig. A.-. The flow-chat of the algoith 98

109 Pincile of beaingle achine Chate 3 PRINCIPLES OF BEARINGLESS MACHINES 3. Intoduction A beaingle oto i an electical achine whee the uenion and the centeing of the oto i ovided by adial foce geneated by the inteaction between the agnetic field acting in the aiga, avoiding the ue of echanical beaing and achieving in thi way uch highe axiu eed [], a in [] whee i ooed a 6 oto fo coeo and ecial u. In thi way, the oto i uended and centeed by a adial foce ditibution, uitably ceated by the inteaction between diffeent haonic ode of the agnetic field oduced by the tato and oto ouce, whateve they ae: in fact, the 99

110 Chate 3 incile can be alied to the eanent agnet uface-ounted ynchonou achine [3], [4], [5], intenal PM oto, induction oto. An inteeting geneal ethod fo a coaion of beaingle achine i eented in liteatue [6]: PM ynchonou oto have the advantage that the contol of otation and levitation ae indeendent, while the levitation foce i wea. Conveely, induction oto oduce a tong levitation foce, but thei efficiency i oo and the contol of otation and levitation ae couled. The intenal eanent agnet (IPM) tye beaingle oto eeent a cooie, being chaacteized by tong levitation foce and elatively eay contol oetie. The adial foce ae geneated by ceating an unbalanced flux denity ditibution in the aiga, which eult in a agnetic foce acting on the oto. In fact, in thi ituation by uing the foce vecto elated to evey ole, they give a not null eultant. On the contay, in the electical achine of conventional tyology, the agnetic ole have equal flux denity and hence equal agnitude of the attactive foce, with a null vecto u of the adial foce. An unbalanced agnetic field ditibution in a beaingle achine can be obtained by two diffeent winding ditibution: ) Dual et of winding, chaacteized by two yte of thee-hae winding hyically eaated, one dedicated to the geneation of tangential foce which oduce toque, the othe to the geneation of the adial foce of levitation; a beaingle oto eviouly ooed in liteatue, eent a -ole adial foce winding wounded in the tato of a 4-ole oto [], could be alied to ue high eed oto a well a induction and ynchonou eluctance achine, a theoized in [7], [8]. An analyi and claification of 3-hae eaated and concentated winding beaingle achine i ooed by [9], which contitute a elief in the deign of thi tyology of oto.

111 Pincile of beaingle achine ) Single et of ultihae winding, in which the two tyologie of foce ae obtained fo a ingle winding, exloiting the otential of ultihae oto to oduce atial haonic of odd and even ode, by injecting diffeent cuent ace vecto: in thi way two tato agnetic field of diffeent haonic ode, by acting with the oto agnetic field, enue one the uenion of the oto, the othe the geneation of toque. In [] i ooed a 6-hae induction achine with one et of winding intead of two et of 3-hae winding, taing advantage of the ultile contol degee of feedo given by ultihae achine; [] eent a 5-hae beaingle oto, exlicating the incile of geneating the toque and uenion foce by feeding two gou of cuent ojected into othogonal d-q lane. In the ae ae i alo eented a contol yte which etiate the aaete of the cuent levitation yte by ean of PID contolle, acquiing the oition eo of the oto and etiate the aaete of the cuent oto yte by ean of a PI contolle, acquiing the angula eed eo. 3. Geneal Pincile of Magnetic Foce Geneation Fig. 3. how the co ection of a geneal beaingle oto unde diffeent oeative condition [], []. In Fig. 3.(a), the flux ditibution i a yetical 4-ole, the flux ath aound the conducto 4a being hown: the fou ole flux wave ψ 4a oduce altenating agnetic ole in the aiga. Since the flux ditibution i yetical, the flux denity ditibution i identical, aat fo the ign, in the aiga ection,, 3 and 4. Thee ae attactive agnetic foce between the oto ole and tato ion, which have identical alitude, but with equally ditibuted diection, o that the u of adial foce

112 Chate 3 acting on the oto i zeo. Fig. 3.(b) how the incile of adial foce geneation: a -ole winding, eeented by conducto a, oduce a agnetic flux ψ a having the ae diection than the one geneated fo the 4-ole winding in the ection, but having the ooite diection in ection 3. In thi way, the flux denity will inceae in ection, while will be educed in ection 3, geneating a adial foce F accoding to the x -axi diection. It follow that the alitude of the adial foce inceae a the cuent value in conducto a inceae. Fig. 3.(c) how how a negative adial foce in the x -axi diection i geneated. The cuent in conducto a i eveed o that the flux denity in aiga ection now deceae while that in aiga ection 3 inceae. Hence the agnetic foce in aiga ection 3 i lage than that in aiga ection, oducing a adial foce in the negative x -axi diection. Fig. 3. Fig. 3. how the adial foce geneation in the y -axi diection. The conducto b, which have an MMF diection along the y -axi, oduce a flux though the aiga ection and 4, thu eulting in a foce along the y -axi. The olaity of the cuent will dictate the diection of the foce. Thee ae the incile of the adial foce geneation, being it value alot ootional to the cuent in the

113 Pincile of beaingle achine winding a and b (auing a contant 4-ole cuent). The vecto u of thee two eendicula adial foce can oduce a adial foce in whateve diection, with any alitude. A an eential condition to geneate adial foce, the diffeence of ole ai nube of the otoing and levitation field ha to be ±. Fig Beaingle Machine with a Dual Set of Winding In [3] i eented an inteeting analyi of a beaingle eanent agnet oto with a dual et of winding, eoted in the following. The 4-ole agnetic field (oduced by the toque cuent yte) and the -ole agnetic field (oduced by the levitation cuent yte) ae geneated by eaated, hyically ditinct, winding. Thi ethod allow to deign the two winding indeendently, but ha the diadvantage of eeving at of the coe uface in the lot to the levitation winding (Fig. 3.3), ued to oduce adial foce, cauing in thi way highe Joule loe to give the ae toque outut with eect to the 3

114 Chate 3 conventional electical achine. Fig. 3.3 With efeence to Fig. 3.4, the additional -ole winding N α and N β ae wound in the tato lot with the conventional 4-ole winding. The adial foce i caued by the unbalanced ditibution of the flux denity in the aiga, caued by the exiting inteaction between the excitation flux ψ P, which flow in the 4 ole and in the eanent agnet, and the flux geneated by the -ole winding cuent i α and i β. flux The cuent i α, with a diection and oientation a in Fig. 3.4, geneate the ψ α. The flux denity inceae in the aiga ection and deceae in the aiga ection, the adial foce F i geneated in the negative diection of the α axi. Fo ilicity, it will analyzed a odel of two-hae achine. The cuent able to geneate a agneto-otive foce equal to that of the eanent agnet i eeented a an equivalent cuent in the oto winding. The cuent i a and i b, in the oto winding N a and N b, ae the u of the actual coonent of the oto cuent with alitude I q and of the equivalent cuent 4

115 Pincile of beaingle achine of eanent agnet with alitude I. Fig PM beaingle oto with a dual et of winding The cuent can be witten a: i i a b I in ωt I co ωt (3.) q I co ωt I in ωt (3.) q Whee ω eeent the angula fequency If the beaingle oto i in oencicuit oeating condition I q i about zeo and can be neglected: i i a b I co ωt (3.3) I in ωt (3.4) By defining ψ a, ψ b, ψ α, ψ β eectively a the 4-ole fluxe elated to 5

116 Chate 3 winding N a, N b and the -ole fluxe elated to winding N α, N β, the elationhi between the and the cuent that flow in the oto and levitation winding can be witten a: ψ ψ ψ ψ a b α β L4 M' α M' β L 4 M' β M' α M' α M' β L M' β i M' α i i L i a b α β (3.5) Being L 4 and L eectively the elf-induction coefficient of oto and levitation winding, α e β the oto dilaceent along the x and y axe, the deivative of the utual-induction coefficient, elated to the couling between oto and levitation winding with eect to oto dilaceent. L 4, L and M ae function of the aiga lenght, nube of tun and oto dienion. By auing a agnetic linea yte, ( l l g ) ( l l ) M ' can be witten a: μ πn n4l M' (3.6) 8 g whee n and n 4 ae the eal nube of tun of the winding, l i the axial length of the achine, i the intenal tato adiu, l the agnet thicne, l g the aiga length. Conequently, ( l l g ) eeent the ditance between the intenal tato uface and extenal oto uface. The toed agnetic enegy can be exeed a: M ' W W [ i i i i ] a b α β L4 M' α M' β L 4 M' β M' α M' α M' β L M' β i M' α i i L i a b α β (3.7) The adial foce oduced by the inteaction between the two winding and the 6

117 Pincile of beaingle achine oto agnetic field can be calculated in te of it coonent F α and F β along the axe diection α e β : W F α (3.8) α W F β (3.9) β By executing the calculation i obtained: F F α β M' I co ω in ωt t in ωt i co ωt i α β (3.) It can be een that the adial foce i ootional to cuent of the eanent agnet I ; thu, if the facto M ' and to the equivalent M ' I ha an high value, the levitation winding cuent can be educed. In ode to do thi, I and the agnet thicne ae inceaed but, conequently, the total ga between oto and tato gow cauing a deceaing of M '. So, it eult vey iotant to chooe the ight cooie between the otiu thicne of the eanent agnet and the oto efoance. Fig. 3.5(a). 7

118 Chate 3 Fig. 3.5(b)-(c). In ode to do thi, wa defined a efoance index which coelate the adial foce-veu-cuent atio to the flux denity in the aiga. In Fig. 3.5 i hown the co-ection of the exained eanent agnet beaingle oto, being R the oto adiu, φ and φ the angula oition along the tato and oto eihey with eect to the α axi, ω t the oto angula dilaceent. The oto ion coe i ade fo the lainated ilicon teel with the all ojection. Peanent agnet ae ounted on the uface of the oto coe a hown in Fig. 3.5(b). w i the width of eanent agnet. w ' i the width between the all ojection. Then, an aea S ' between all ojection and an aea S of a eanent agnet can be eeented a w ' l and wl, eectively. Fig. 3.5(c) how a agnetic equivalent cicuit of the eanent agnet having width w. The agneto-otive foce F of eanent agnet i given a F l B (3.) μ whee B i the eanent flux denity of the eanent agnet, μ i the ai 8

119 Pincile of beaingle achine eeability, R g and R, in Fig. 3.5(c) ae the eluctance of echanical aiga cleaance and eanent agnet, eectively: R R g l g (3.) S μ l (3.3) S μ Neglecting agnetic atuation, lot ile and eeance in the all ojection, the flux linage ψ, caued by eanent agnet can be witten a ψ R g F R l B μ l g l μ S μ S Sl l l g B (3.4) Theefoe, ea value of the flux denity B,, in the aiga can be witten a B g ψ l B (3.5) S' l l S S' g Let u define the te l n, and l gn, a l and l g,eectively. Thee noalized length with eect to the tato inne adiu can be ued to deive the geneal exeion of the ea ai-ga flux denity. Thu, ewitten a B g can be B g l S ln B B (3.6) l l g S' ln lgn S S' Fig. 3.6 how the elationhi between l n, and B g, with aaete of l gn. The te SB S' in the vetical axi i a contant deteined by the eanent 9

120 Chate 3 flux denity of eanent agnet and the aea denity of ounted eanent agnet. In the cae of S-Co agnet, B T. The atio S S' i equal to if a cylindical agnet i ued, but it i lightly alle than if all eanent agnet ae ounted on the uface of oto ion coe a hown in Fig. 3.5(b). The deceae in l gn, i.e. a eduction of the atio of echanical ai-ga width l g to adiu of tato inne uface, and the inceae in l n, i.e. the atio of eanent agnet thicne l to eult in an inceae in aiga flux denity. The deied value of the ea aiga flux denity i geneally deteined at a ated otational eed and heat diiation. Fig An iotant eult achieved in [3], i given by analyzing the quantity adial foce fo unit cuent F I a a function of the achine aaete: α α n( ln lgn ) ( l l ) F l α nlbs (3.7) I S' α n gn The fit te of (3.7) deend on the nube of tun of levitation winding,

121 Pincile of beaingle achine the oto tac length, the eanent flux denity of eanent agnet, and the aea denity of ounted eanent agnet. Thu, i not oible to inceae it without an inceae in dienion o an ioveent in ateial. The econd te i deteined by atio of eanent agnet thicne and echanical aiga width to the adiu of the tato inne uface. Radial foce can be oduced ot efficiently when the econd te ha it axiu value. The quetion i if the aiga flux denity value i to deteine at ated oto eed and ated heat diiation o fo the otial condition to oduce adial foce. Howeve, by diffeentiating (3.7) with eect to l n, it can be found that F α I α i axiu when n lgn l, i.e. adial foce can be oduced ot efficiently when eanent agnet thicne i equal to echanical aiga width (Fig. 3.7). Fig. 3.7.

122 Chate Beaingle Machine with a Single Set of Winding It i nown that a ultihae yte of cuent can be eeented by uing teoal equence (if in inuoidal altenating egie) o, oe geneally, by uing ace vecto of diffeent ode (if in non-eiodic egie). By contolling eaately and in an aoiate anne vaiou ode of the cuent ace vecto it i oible, fo exale, to geneate and contol the toque oduced by the oto with a cetain ode and to oduce and contol the uenion adial foce with othe ode. The advantage of the ingle et of winding conit in the oibility of ileenting both the above function uing a ingle winding, with a ile contuction oce, without deigning anothe one which ubtact a ueful ection of coe, with a eduction of owe loe. In thi tye of achine i howeve neceay to ovide ayetic hotened itch winding, in ode to geneate even haonic ode in the agnetic field that eit to ceate a adial foce ditibution. Thi tyology of winding ha the diadvantage of educing the available toque, o the deigne ut then find a cooie between the intenity of the adial foce and the oto efoance. In [] i eented an inteeting analyi of a beaingle eanent agnet oto with a ingle et of 5-hae winding, eoted in the following. The two needed agnetic field ae oduced by feeding two gou of cuent which ae ojected into othogonal d-q lane eectively. The adial foce acting on the oto can be obtained fo Maxwell te teno: σ ( b b ) μ o n t (3.8)

123 Pincile of beaingle achine whee the contibution of b t can be neglected with eect to that thee ae two evolving agnetic field in the aiga B and B : b n. By uoing b n B B (3.9) B co( ω t θ ) B (3.) ϕ B co( ω t θ ) B (3.) ϕ whee and ae thei coeonding nube of ole ai, ω and ω ae thei coeonding angula fequencie, and θ i an abitay angle in tato uface. The hoizontal and vetical foce coonent ae given by: F σ co( θ ) ldθ (3.) α F σ in( θ ) ldθ (3.3) β whee l i the length of the tac, and i the adiu of aiga. By cobining the equation (3.9)-(3.), then ubtituting in (3.8) and integating by ean of (3.), (3.3), the ojection of the foce on the hoizontal α axi and vetical β axi ae obtained. F α lπ μ, B B co ( ω t ω t ϕ ϕ ), ± ± (3.4) F β lπ μ, lπ μ B B B B in in ( ω t ω t ϕ ϕ ) ( ω t ω t ϕ ϕ ),, ± (3.5) So, it eult that a adial foce can be obtained when, and the foce i ± 3

124 Chate 3 table when ω ω. The Modified Winding Function Method i alied in ode to analyze the agneto-otive foce. In an n-hae yetic yte, the winding function of each hae can be witten a: N N N M N a b c n N N N N N co θ N co( θ ξ ) N co( θ ξ ) N co θ N co( θ ( n ) ξ ) N 3 co( ( θ ξ )) N co( 3( θ ( n ) ξ )).. 3 co 3θ.. co( ( θ ξ )) N co( 3( θ ξ )).. co( ( θ ( n ) ξ )) 3 3 co( 3( θ ξ )).. (3.6) whee ξ π n and N, N, N 3 ae coefficient of the winding function. The gou of cuent i defined a I : ia co( ωt φ ) ib co( ωt φ ξ ) I ic I co( ωt φ ξ ) (3.7) M M i co( ωt φ ( n )ξ ) n whee I i the alitude of I andφ i the initial hae of i a. The MMF wave of an n-hae yetic yte fed by I can be exeed a F I n i I n i v co N v [ ωt φ ( i ) ξ] N co v[ θ ( i ) ξ] { co [ ωt φ vθ ( v )( i ) ξ] co [ ωt φ vθ ( v )( i ) ξ]} v v (3.8) Thu, the v -th haonic f wave eult: 4

125 Pincile of beaingle achine F v nn v I nn v I co( ωt φ co( ωt φ vθ ) vθ ), v μn, ± v μn, v μn, μ, ±,..., μ, ±,..., μ, ±,... (3.9) Fo (3.9), the eultant MMF of 5-hae yetical winding can be hown in Tab. I. F and B indicate the haonic ode of MMF otating fowad and bacwad eectively, ± denote a ulating MMF. TABLE I. SPACE HARMONICS OF A 5-PHASE MOTOR In a 5-hae winding oto, I geneate ole-ai evolving agnetic field in the aiga, I geneate ole-ai evolving agnetic field when N.So the uenion foce can be oduced by the inteaction of I and I. In the analyzed achine, the 5-hae winding ae identical and each of the i 7 dilaced in angula ace aound the tato. Fig. 3.8 how the winding configuation and the lot whee a hae goe in, in caital lette, whee goe out, in lowecae lette. The hae ae ta-connected, a can be een in the ae Fig

126 Chate 3 Fig Voltage equation in tationay efeence fae ae given by ( L I ) U (3.3) R I Ψ whee U, I, R, L and Ψ ae atice of voltage, cuent, eitance, inductance and flux linage eectively, ubcit and eeent tato and oto eectively, and i the diffeential oeato. A tanfoation ha to be alied to equation (3.3) in ode to exe the voltage and cuent of tato to ynchonou efeence fae, given by a atix C not eoted hee. The fo of the voltage equation in the new efeence fae eult: ψ d ψ q U ω t Rt I t Lt I t W Lt I t (3.3) whee ψ q, ψ d ae the q- and d-axi flux linage of oto eectively. The 6

127 Pincile of beaingle achine 7 adial foce acting on oto can be calculated with Vitual Dilaceent Method: α α d q d d q q t T d q d d q q i i i i i i d d i i i i i i F L (3.3) β β d q d d q q t T d q d d q q i i i i i i d d i i i i i i F L (3.33) being d d L i ψ, q q L i ψ, equivalent oto cuent and L, L leaage winding inductance elated to ole ai and ole ai field. By develoing (3.3), (3.33), the adial foce can be een a cooed of two at. One at i elated to eccenticity: [ ] 4 ) i ( i g L ) i ( i ) i ( i g L F d q d d q q α α α (3.34) [ ] 4 ) i ( i g L ) i ( i ) i ( i g L F d q d d q q β β β (3.35) while the othe eult indeendent of eccenticity, ued to contol the oto uenion: ) ( L L g F q q d d ψ ψ ψ ψ α (3.36) ) ( L L g F q q d d ψ ψ ψ ψ β (3.37) whee g i the equivalent adial length of a unifo ai ga, α and β ae the dilaceent of oto cente. The fluxe which aea in (3.36), (3.37) ae

128 Chate 3 defined a: ψ ψ ψ ψ d q d q L L L L i d i q i d i q ψ ψ d q (3.38) (3.39) The oto dilaceent in tationay efeence fae ae given by: α&& β && F β F α G (3.4) whee and G ae the a and weight of oto eectively. The oto cuent coand i * q i geneated fo eed contolle. Suenion foce coand * F α, * F β ae geneated by oto dilaceent contolle, afte that, levitation cuent coand * q i * d i ae given by (3.4), (3.4), which ae deived by (3.36), (3.37). Phae cuent ae given by the invee tanfoation of * q i * d i and * q i * d i. i i g L L F ( ψ F ψ ) q q α d ψ q ψ d g L L F ( ψ F ψ ) d d α q ψ q ψ d β β (3.4) (3.4) In the following, the ain data of the ooed oto: H, L.3 H, L H,. Ω, L 36 H, L 6.3 ψ f.5 Wb, J. g, aiga thicne:, agnet thicne 3, g 5. The aiga length between the oto haft and the touchdown beaing i.6. In Fig. 3.9 i hown the contol diaga of the 5-hae PM beaingle oto. 8

129 Pincile of beaingle achine Fig Fig. 3. how angula eed, toque and oto dilaceent duing the tat u oeation. The haft eult uccefully uended and the dynaic of oto uenion i table with adial dilaceent vaiation le than 5 μ. It i alo obviou that the yte ha good eed-egulation efoance.. Fig

130 Chate Roto eccenticity In beaingle achine, due to the fact that the oto i not utained by echanical beaing, a oto eccenticity i deteined while the oto i oeating and ha to be coenated by the contol yte. Thi henoenon lead to a vaiation in the value of the aiga flux denity, a thee eulting deceae in the aea whee the aiga widen, and inceae in the aea whee the aiga i educed, cauing a iila behavio in the adial foce ditibution. Fig. 3. In Fig. 3. i hown the efeence yte in which a ilified calculation [] eit to tae into account the eccenticity by witing the aiga width g a a function of the angle φ, that the line joining the effective oto cente with the oigin, decibe with the x -axi: g ( ) g x co φ y in φ φ (3.43) being x and y the actual coodinate of the oto cente. With the aution of all dilaceent coaed to the noinal aiga length g, i oible to

131 Pincile of beaingle achine wite: x g co φ in φ ( φ ) g y g g (3.44) and, conequently, to calculate the eeance P in the geneic angula oition φ : μ Rl x y P (3.45) ( φ ) co φ in φ g g g being R the oto adiu and l the axial length. By uing oely (3.45) in the equation of the achine, i oible to tae into account the foce vaiation with eect to oto oition. Two tyologie of eccenticity can be defined in beaingle achine: Static eccenticity: it occu when the oto i not cented in the tato boe; Dynaic eccenticity: it occu when the oto i not otating on the oto axi but i otating on the cente axi. Vaiou autho develoed odel able to inteet and calculate the effect of eccenticity on the adial foce and on the oto oition. In [4] i eented, by uing the nonlinea FEM and a theoetical analyi, an analytical odel fo calculating the levitation foce unde aiga eccenticity by ean of the inteaction between haonic field coonent and a ilified odeling fo levitation foce contol in beaingle induction achine; [5] decibe a ethod fo odeling a beaingle IPM oto, fo calculating the foce on the oto by uing colex winding analyi and otating field theoy and coaing the eult to FEA analyi; the odel allow the intoduction of levitation and ain winding and oto eccenticity. In [6] an analytical exeion of the levitation foce fo an induction-tye beaingle oto i ooed, taing into

132 Chate 3 account the oto eccenticity, being it coutation accuacy veified by ANSOFT. A eal-tie obevation of agnetic levitation foce can be ealized, i then ileented a cloed-loo contol of levitation foce on the bai of aigaflux-oiented decouling contol yte of the beaingle oto. A vey inteeting wo about the analyi of tatic and dynaic eccenticity i eented in [7], which efe to an analytical odeling technique fo calculating the adial foce on the oto of a beaingle conequent-ole eanent agnet oto. The flexibility of the ethod eit to identify vibation coonent and calculate the foce in vaiou ituation when the oto i not cented in the boe, in cae of eithe dynaic eccenticity, tatic eccenticity o when the oto i vibating. It alo allow a calculation of the foce in eence of a vaying load o with load ibalance. The ae alo tudie the effect of winding deign and give a validation of the eult uing D finite eleent analyi. Anothe aticula wo which could be alied alo to the foce calculation in beaingle achine i [8], whee a geneal analytical odel, foulated in -D ola coodinate, i develoed to edict the unbalanced agnetic foce, which eult in eanent-agnet buhle ac and dc achine having a diaetically ayetic dioition of lot and hae winding. The unbalanced agnetic foce can be ignificant in achine having a factional atio of lot nube to ole nube, aticulaly when the electic loading i high. The develoed odel i validated by FE calculation on 9-lot/8-ole and 3-lot/-ole achine. Finally, [9] ooe a novel aoach to contol the oto adial dilaceent in beaingle eanent-agnet-tye ynchonou oto, baed on the elationhi between adial dilaceent and adial uenion foce. The oto flux oientation i adoted to decoule the electoagnetic toque and the adial uenion foce. Thi aoach, which diectly contol the oto adial dilaceent, wa alied by deigning a

133 Pincile of beaingle achine uitable contol yte. 3.6 Concluion The ain featue and iue of beaingle achine wee eented in thi chate, highlighting the otential of the ultihae ingle et of winding tye, which uely eeent the ot effective deign ethodology to adot in the futue. In fact, it eit valid contol tategie fo uending the oto, fo contolling the oto and, at the ae tie, fo geneating toque by uing the oetie of ultihae cuent yte, without the need of alteing the hyical tuctue of the achine by deigning othe gou of winding in addition to the ain one. Thu, the focu of the activity in the next chate will be on the analyi of thi tyology of beaingle achine. 3

134 Chate Refeence [] A. Chiba, T. Deido, T. Fuao, M.A. Rahan, "An Analyi of Beaingle AC Moto," IEEE Tan. Enegy Conveion, vol. 9, no., Ma. 994, [] T. Schneide and A. Binde, Deign and evaluation of a 6 RPM eanent agnet beaingle high eed oto, Poc. Conf. on Powe Electonic and Dive Syte, Bango, Thailand, Nov. 7 3, 7,. -8. [3] M. Ohia. S. Mivazawa. T. Deido, A. Chiba. F. Naaua, and T. Fuao, Chaacteitic of a eanent agnet tye beaingle oto, in IEEE IAS Annu. Meet., 994, [4] K. Dejia, T. Ohihi, and Y. Oada, Analyi and contol of a eanent agnet tye levitated otating oto, in ZEEJ Poc. Sy. Dynaic of Electo Magn. Foce, June 99, [5] M. A. Rahan, T. Fuao, and A. Chiba, Pincile and develoent of beaingle AC oto, in Poc. IPEC-95, Yoohaa, 995, [6] Y. Oada, S. Miyaoto and T. Ohihi, Levitation and-toque contol of intenal eanent agnet tye heaingle oto, IEEE Tanaction on Contol Syte Technology, Vol. 4, NOS, 996, [7] A. Chiba, K. Chida, and T. Fuao, Pincile and chaacteitic of a eluctance oto with winding of agnetic beaing, in Poc. IPEC- Toyo,99, [8] A. Chiba, D. T. Powe, and M. A. Rahan, No load chaacteitic of a beaingle induction oto, in IEEE IAS Annu. Meet., 99, [9] W.-R. Cande, D. Hulann, Analyi and claification of beaingle achine with yetic 3-hae concentated winding XIX Intenational Confeence on Electical Machine (ICEM),, 6. [] M. Kang, J. Huang, J.-q. Yang and H.-b. Jiang Analyi and Exeient of a 6-hae Beaingle Induction Moto, Int. Conf. on Electical Machine and Syte ICEMS 8, Oct. 8, [] M. Kang, J. Huang, H.-b. Jiang, J.-q. Yang, Pincile and iulation of a 5-hae beaingle eanent agnet-tye ynchonou oto Intenational Confeence on Electical Machine and Syte (ICEM) 8., 8, Page():

135 Pincile of beaingle achine [] A. Chiba, T. Fuao, O. Ichiawa, M. Ohia, M. Taeoto and D. G. Doell "Magnetic Beaing and Beaingle Dive", Elevie, Ma. 5. [3] M. Oohia, A. Chiba, T. Fuao, M.A. Rahan, "Deign and Analyi of Peanent Magnet-Tye Beaingle Moto," IEEE Tan. On Indutial Electonic, vol. 43, no., A. 996, [4] W. Baoguo, W. Fengxiang, Modeling and Analyi of Levitation Foce Conideing Ai-ga Eccenticity in a Beaingle Induction Moto, Poc. of the fifth Intenational Confeence on Electical Machine and Syte,, [5] D. G. Doell, M. Ohia, A. Chiba, Foce Analyi of a Buied Peanent Magnet Beaingle Moto IEEE IEMDC, Poc., June -4 3, Madion Wiconin USA, [6] H. Yiang and N. Heng, Analytical odel and feedbac contol of the levitation foce fo an induction tye beaingle oto, in IEEE Poc. PEDS, Nov. 3, vol., [7] D. G. Doell, J. Aeiya, A. Chiba, and T. Taenaga, Analytical Modeling of a Conequent-Pole Beaingle Peanent Magnet Moto, in Poc. IEEE Powe Electonic and Electic Dive Conf., Singaoe, Nov. 5, 3, [8] Z. Q. Zhu, D. Iha, D. Howe, and J. Chen, Unbalanced agnetic foce in eanent-agnet buhle achine with diaetically ayetic hae winding, IEEE Tan. Magn., vol. 43, no. 6, Nov./Dec. 7, [9] Shaou Zhang and Fang Lin Luo, Diect Contol of Radial Dilaceent fo Beaingle Peanent-Magnet-Tye Synchonou Moto, IEEE Tanaction on Indutial Electonic, vol. 56, no., Feb. 9,

136 Chate 3 6

137 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine Chate 4 AN ANALYTICAL METHOD FOR CALCULATING THE DISTRIBUTION OF FORCES IN A BEARINGLESS MULTIPHASE SURFACE-MOUNTED PM SYNCHRONOUS MACHINE 4. Intoduction Beaingle oto have the caability to achieve uch highe axiu eed in coaion to conventional electical achine [],[]. The tyology dual et winding, ha a ain one which caie the toque cuent fo diving the oto and oducing toque, while the othe caie the levitation 7

138 Chate 4 cuent, to uend the oto [3],[4]. It advantage conit of a ile contuction oce, highe flexibility in contol tategy and elatively low owe loe [5]. The tyology ingle et winding oduce toque and adial foce by ean of injecting diffeent cuent ace vecto within the ae winding to give odd and even haonic ode of agnetic field, uing the oetie of ultihae cuent yte with ultile othogonal d-q lane. One of the can be ued to contol the toque, a in [6]. The additional degee of feedo can be ued to oduce levitation foce [7],[8]. An iotant develoent of thi technology i exected to be in the deign of electoechanical device fo Moe Electic Aicaft (MEA), ainly fo the oibility of achieving highe eed in coaion to conventional electical achine [9]. Alo, it would be alicable in the aeoace field, whee the lubicant of echanical beaing evaoate in the eence of vacuu []. The ultihae oto with eect to conventional thee-hae oto give a eie of advantage [], in aticula in the cae of high owe, high eliability, low dc bu voltage and eduction of owe loe in IGBT invete [] a it haen in hi oulion, electical vehicle and MEA alication. In the contol yte of ultihae beaingle oto i neceay to calculate the levitation cuent able to coenate the actual eo in the oto haft oition, thu the analytical function which coelate the alied cuent to the eulting uenion foce on the oto. In ode to do thi and to ilify the oble, oe autho conide only the inteaction between the ain haonic ode of the tato and oto agnetic field, in the aticula cae of teady-tate AC condition, with inuoidal yte of cuent, a done in [7]; nevethele, by oceeding in thi way, a elevant eo in the ediction of the foce vecto can be coitted, becaue the inteaction between highe haonic ode of the agnetic field acting on the oto and the effect of the toque 8

139 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine cuent on the adial foce ae neglected. The ai of thi chate i to eent a odel allowing to edict alitude and diection of the foce fo given value of the toque cuent, of the levitation cuent and of the oto oition [3], baed on the ace vecto ethod, being able to include in thi way alo the analyi of tanient. Fo thi uoe, a genealized analytical odel fo the calculation of adial foce in ultihae beaingle Suface-Mounted Peanent Magnet Synchonou Moto (SPMSM) i eented. The tato agnetic field i eeented a the u of eaated contibution given by the diffeent cuent ace vecto. In thi way it i oible to analyze the inteaction between the toque cuent yte, the levitation cuent yte and the oto agnetic field. In fact, in ultihae achine the cobined effect of two tato cuent ace vecto lead to a eulting levitation foce which i enibly diffeent fo that, foeeeable, oduced by the cuent in the eaated winding of taditional beaingle achine. In Tab. I ae hown all the oible inteaction between the haonic ode of the agnetic field The eult ae coaed with thoe of FE analyi to deontate thei accuacy. 9

140 Chate 4 4. Definition of vaiable In the following, i eented a lit of the vaiable ued in the equation and elationhi of thi chate: θ () Angula abcia in the tato efeence yte Nube of tato ole ai Nube of oto ole ai Satial haonic ode of the ace vecto () tato field ditibution h Satial haonic ode of the oto field ditibution φ () Phae of the cuent ace vecto () n Half of the nube of conducto in a lot δ Total aiga height g Aiga height L Magnet height q Nube of lot e ole e hae d() -th haonic coonent of the winding ditibution facto elated to the ace vecto () γ Half of the coil itch angle v() Ode of the ace vecto () Δθ Angula dilaceent between hae- axi and agnet axi B e Reanent flux denity of the agnet K c Kate facto μ Magnetic eeability of the ai μ Relative agnetic eeability of the agnet α Angle undelying the agnet itch i v() Alitude of the cuent ace vecto () Mean aiga adiu L Axial length of the achine B S() ea value of the -th haonic ode of flux denity ditibution oduced by cuent ace vecto () B Rh ea value of the h-th haonic ode of flux denity ditibution oduced by oto agnet n ac Nube of lot foing the coil itch N Set of natual nube including zeo N odd Set of odd natual nube 3

141 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine 4.3 Analyi of flux denity ditibution in the aiga Conideing 5 eal vaiable x,.., x l,.., x 5, an aociated et of colex vaiable x,x, x can be obtained by ean of the following yetical linea tanfoation: 5 ( l), ( ρ,, ) xρ xl α 5 (4.) l whee α ex( j π / 5) x ρ. The invee tanfoation of (4.) ae a follow: ρ( l x ) x α, ( l,,.., 5) ρ (4.) l ρ, whee the ybol eeent the cala oduct, defined a the eal at of the oduct between the fit oeand and the colex conjugate of the econd. Fig. 4. 3

142 Chate 4 The conideed efeence yte i a cateian one, whoe axe x- (hoizontal) and y- (vetical) have the oigin in the oto haft cente, the y-axi coinciding with hae axi (Fig. 4.). The aution taen into account ae: infinite ion eeability, cuent ditibution concentated at the lot oening, alot unitay elative eeability of the agnet. The ethod will be alied to SPMSM achine with 5 ta-connected tato hae winding, yetically ditibuted within the tato lot: the winding ae uoed to be hifted by π/5 electical adian, with a ingle neutal oint. Accoding to (4.) and (4.), it i oible in thi cae to eeent the cuent yte uing ace vecto in diffeent α - β lane, being the zeoequence coonent null becaue of the ta-connection of the hae: jϕ () t j ( t ) () t e ϕ, i i () t e iv i v v v (4.3) By alying (4.3) to (4.) and by exlicating oe vaiable, the l-th hae cuent can be witten a: π 4π l ρ, i j ( ) ( l ) j ( l ) l i 5 5 v ρ ρ α iv e iv e, ( l,,.., 5) (4.4) By develoing (4.4), it give: i π 5 4π 5 () t i co ϕ () t ( l ) i co ϕ () t ( l ), ( l,,.., 5) l v v (4.5) When the v-th ace vecto cuent i flowing in the coil, the l-th hae of the oto oduce a agnetic field ditibution whoe the adial coonent i given by (4.6):,, 3.. π π H l ( θ,t) H M co θ ( l ) co ϕ() t v ( l ) 5 5 (4.6) 3

143 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine with H M defined a follow: H M 4 niv qd in π δ ( γ) (4.7) Note that the exanded Fouie haonic eie (4.6) contain odd and even ode, diffeently fo the eeentation of the agnetic field ditibution in the uual electical achine, becaue of the aticula location of the coil in the tato lot; (4.6) i alo a function of the angula abcia θ whoe oigin coincide with the y-axi, a hown in Fig. 4.. By cobining the elationhi (4.6) conideing all the hae cuent, the -th haonic ode H of the 5-hae tato agnetic field, elated to the cuent ace vecto v, can be exeed a: H ( θ,t) 5 H M co [ θ ϕ() t ] if if 5 5 v v N N (4.8) The total tato agnetic field i the uation of all the exiting te given by (4.8), deending on the value of. The oto agnetic field geneated by the agnet can be witten a: H Rh h, 3, 5.. ( θ,t) H co[ h θ h Δθ] (4.9) whee H Rh, B RM ae defined a follow: H Rh 4 hπ B μ RM α in h, B RM L L μ K c g B e (4.) Two diffeent cuent ace vecto, one fo toque oduction and the othe fo levitation (4.5), ae injected: eectively v and v, o that the eulting adial coonent of flux denity in the aiga can be witten by ean of the incile 33

144 Chate 4 of ueoition: B n ( θ,t) [ B ( θ,t) B ( θ,t) B ( θ,t)] B co[ θ ϕ () t ],,h B S co [ θ ϕ () t ] B co[ h θ h Δθ] h (4.) whee the value of and follow the condition exlained in (4.8), and the alitude of the thee ditibution ae defined a follow: niv BS μ H M μ qd in( γ) (4.) π δ niv BS μ H M μ qd in( γ) (4.3) π δ α B 4 Rh BRM in h (4.4) hπ h Rh S 34

145 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine 4.4 Calculation of the foce acting on the oto The eulting foce acting on the oto can be deteined by alying the Maxwell Ste Teno ethod on a cloed uface enveloing the fontal co ection of the teeth in the tato: T Bn Bt Bn ( H nbn Ht Bt ) nˆ Ht Bntˆ nˆ tˆ (4.5) μ μ Taing the noal and tangential coonent of flux denity and of agnetic field into account and neglecting the te elated to the oduct of tangential coonent, the exeion of the eleentay foce abcia θ on the eleentay uface ds Ldθ, eult: df (,t) T ( θ,t) ( θ,t) B ( θ,t) B ( θ,t) d F, acting at the angula Bn t n θ ds dsnˆ dstˆ (4.6) μ μ By ubtituting the exeion of ds in (4.6), it give: df LB μ 443 LB B μ 443 n t n ( θ,t) dθ nˆ dθ tˆ dfn dft whee can be ecognized the eleentay noal coonent tangential coonent df t. (4.7) df n and the 4.4. Noal Coonent of the Foce With the eviou aution the eulting foce acting on the oto can be exeed a: F xn L π π dfn in θ μ B n ( θ,t) in θdθ (4.8) 35

146 Chate 4 F yn L π π dfn co θ μ B n ( θ,t) co θdθ (4.9) Reebe that the efeence axi fo the angle θ i the y-axi. Conideing the intege value of, and h that ae non-zeo the elated haonic coonent of the flux denitie in (4.), we obtain ix te eeenting the quae of B n : (,t) B B Bh BB BBh B Bh Bn θ (4.) When integating all the te in (4.), taing (4.8)-( 4.9) into account, it i iotant to conide that the coonent of foce have to be added in uation only fo the exiting haonic ode of, ( '), which ae elated to the tato agnetic field eectively given by the cuent ace vecto v and v, and the exiting haonic ode of h, which ae elated to the oto agnetic field. Thi fact i exeed by ean of the e-condition (4.)- (4.3): v 5 5 v N N, ' 5 v N (4.) (4.) h N odd (4.3) In (4.) it i highlighted the inteaction between diffeent haonic ode of the ae cuent ace vecto v : in fact, a will be clea late, oe coonent of the foce deend on thi henoenon. In Tab. I ae eented all the oible inteaction, in thi cae until the 8-th ode fo eaon of bevity, deteined by uing the equation (4.), (4.), (4.3). By efoing the calculation (4.9) and conideing te by te of (4.): 36

147 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine TABLE I. INTERACTIONS BETWEEN HARMONIC ORDERS F L L μ μ π π ( a) yn B ( θ,t) co θdθ BS co [ θ ϕ() t ] co θdθ By conideing that: ( ) (4.4) co α co ( α) (4.5) Alying (4.5) to (4.4), it give: ( a) LB π S LB π S Fyn co θdθ co[ θ ϕ() t ] co θdθ μ μ (4.6) 4 4 By conideing that: ( α β) co( α β) co co ( α) co( β) (4.7) Alying (4.7) to (4.6), it give: 37

148 Chate 4 F ( a) S S co[ ( ) θ ϕ( t) ] dθ co[ ( ) θ ϕ( t) ] dθ yn π π LB LB 8μ 8μ (4.8) F F LB 8μ ( a) S [ in[ ( ) θ ϕ ] yn LB π [ in[ ( ) θ ϕ ] ( a) S [ in[ π( ) ϕ] in[ ϕ ] yn 8μ π [ in[ π( ) ϕ] in[ ϕ ] (4.9) (4.3) Note that, being both and intege nube, the angula aguent in (4.3) diffe by intege ultile of π, thu the inuoidal function aue the ae value and thei diffeence i equal to zeo. Hence, the contibution of the te in the fo (4.4) to the y-coonent of the foce i zeo: F L π L μ μ ( a) B ( θ,t) co θ dθ B co [ θ ϕ ] co θ dθ yn π S (4.3) Fo the ae eaon, the iila quadatic te elated to the ace vecto v and the one elated to the oto agnetic field give a null eult (4.3), (4.33): F F L π L μ μ ( b) B ( θ,t) co θ dθ B co [ θ ϕ ] co θ dθ yn L π L μ h μ π S (4.3) ( c) B ( θ,t) co θ dθ B co [ h ( θ Δθ) ] co θ dθ yn π Rh (4.33) 38

149 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine 39 Now, exaine the fouth te of (4.): ( ) ( ) ( ) [ ] [ ] S S d yn d co co co B B L d co,t B,t B L F θ θ ϕ θ ϕ θ μ θ θ θ θ μ π π (4.34) By alying (4.7): ( ) ( ) [ ] ( ) [ ] θ θ ± ϕ ϕ θ θ θ ϕ ϕ θ μ π π S S d yn d co co d co co B B L F (4.35) Reeating the ae oce (4.7) in both the integal of (4.35): ( ) ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] θ ± ϕ ϕ θ θ ± ϕ ϕ θ θ ϕ ϕ θ θ ϕ ϕ θ μ π π π π S S d yn d co d co d co d co B B L F (4.36) We ae the aution (4.37), that accoding to (4.36) give the elationhi (4.38) and conequently (4.39): (4.37) ( ) ( ) [ ] [ ] [ ] [ ] θ ± ϕ ϕ θ θ ± ϕ ϕ θ ϕ ϕ θ θ ϕ ϕ θ μ π π π π S S d yn d co d co d co d co B B L F (4.38)

150 Chate 4 ( d ) L Fyn 4μ B S [ in[ ( ) θ ϕ ϕ ] ( ) π (4.39) π [ in[ θ ϕ ϕ ] πco( ϕ ± ϕ ) in[ θ ϕ ± ϕ ] B S A obeved above, the fit, econd and fouth te in (4.39) have a null eult, being the diffeence of ine function with ultile aguent of π. The only te diffeent fo zeo i the thid, not deending on the integation vaiable θ. Finally, the eult of the integal (4.34), unde the hyothei (4.37) i: ( d ) πl Fyn BSBS co( ϕ ± ϕ ) if (4.4) μ We ae the aution (4.4), that accoding to (4.36) give the elationhi (4.4): (4.4) π F F L μ ( d ) B B co[ θ ϕ ϕ ] yn L 4μ S S π π π co co dθ [ ( ) θ ϕ ϕ ] dθ [ θ ϕ ± ϕ ] dθ co[ ϕ ± ϕ ] π ( d ) B B [ in( θ ϕ ϕ )] yn S [ in[ ( ) θ ϕ ϕ ] ( ) S in π ( θ ϕ ± ϕ ) πco( ϕ ± ϕ ) π π dθ (4.4) (4.43) With vey iila concluion of the othe aution, the eult of the integal (4.34), unde the hyothei (4.4) i: 4

151 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine ( d ) πl Fyn BSBS co( ϕ ± ϕ ) if (4.44) μ Note that the aution egading the u of the nube of ole ai, hown in (4.45), (4.46) ae not exained becaue they don t coeond to eal cae, being ioible to occu. (4.45) (4.46) So, the final fo of the y-ojection of the noal coonent of the foce, elated to the d facto of (4.), i given by: F πl μ ( d ) B B co( ϕ ± ϕ ) yn S with : S v N if if v N (4.47) It i vey iotant to obeve that the ign of the condition of exitence have to be accoded to ign of the angula hae in the ae ode that aea in (4.47). Pefoing iila calculation, by integating the e and f facto in (4.) which efe to the inteaction between evey ingle tato cuent ace vecto and the oto field oduced by the agnet, the following elationhi can be found: F ( e) B B co( h Δθ ϕ ) yn πl μ S with : Rh v N if if h N odd h h (4.48) 4

152 Chate 4 4 ( ) ( ) odd v Rh S f yn N h N h h h co B B L F ϕ Δθ μ π with : if if (4.49) By adding u all the non-zeo te of the y-ojection of the noal coonent of the foce, i obtained: ( ) ( ) with : N N N N co B B L F v v S S, yn ± ϕ ϕ μ π (4.5) ( ) [ ] with : N N h N h h co B B L F v odd odd Rh S,h yn ϕ Δθ μ π (4.5) ( ) [ ] with : N N h N h h co B B L F v odd odd Rh S,h yn ϕ Δθ μ π (4.5) ( ) [ ] with : N N N N co B B L F v ' v ' ' ' ' S S ', yn ± ϕ ϕ μ π (4.53)

153 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine The equation (4.53) eeent a tyology of inteaction not exlicated in (4.): it i the influence between diffeent haonic ode belonging to the ae cuent ace vecto v, denoted with and '. In fact, it haen in the analyzed beaingle achine that oe haonic ode of v exit, whoe nube of ole ai diffe of ± : in thi cae, they alo give a contibution to the eultant foce. I iotant to note that the hae angle ϕ ' in (4.53) eeent exactly the ae hae angle ϕ, being only a foal ditinction which indicate that the ign of ϕ ' deend on the exiting haonic ode ', while the ign of ϕ i obviouly elated to : in thi way the aguent of the function coine in (4.53) ha to be undetood a the algebaic u of the ae vaiable ϕ, eulting in a, ϕ o ϕ. Finally, the eultant y-ojection of the noal coonent of the foce tanitted to the oto i the u of the equation (4.5), (4.5), (4.5), (4.53): (, ) (,h ) (,h ) (, ' ) F F F F (4.54) F yn yn yn yn yn A iila calculation can be efoed to deteine the eultant x-ojection, by conideing te by te of (4.) integating with (4.8): F ( a) B ( θ,t) inθdθ BS co [ θ ϕ() t ] inθdθ xn L L μ μ π π By alying (4.5) to (4.55), it give: (4.55) ( a) LB π S LB π S Fxn inθdθ co[ θ ϕ() t ] inθdθ μ μ (4.56) 4 4 By conideing that: ( α β) in( α β) in co ( α) in( β) (4.57) 43

154 Chate 4 By alying (4.57) to (4.56), it give: F ( a) S S in[ ( ) θ ϕ( t) ] dθ in[ ( ) θ ϕ( t) ] dθ xn π π LB LB 8μ 8μ (4.58) F F LB 8μ ( a) S [ co[ ( ) θ ϕ ] xn LB μ π [ co[ ( ) θ ϕ ] ( a) S [ co[ π( ) ϕ] co[ ϕ ] xn 8 π [ co[ π( ) ϕ] co[ ϕ ] (4.59) (4.6) In a vey iila way to the equation (4.3), being both and intege nube, the aguent of the coine in (4.6) diffe by intege ultile of π, thu they aue the ae value and thei diffeence i equal to zeo. Hence, the contibution of the te in the fo (4.55) to the x-coonent of the foce i zeo: F L π L μ μ ( a) B ( θ,t) inθ dθ B co [ θ ϕ ] inθ dθ xn π π ( c) L L Fxn ( θ ) θ θ [ ( θ Δθ) ] θ θ μ Bh,t in d μ BRh co h in d (4.63) π S (4.6) Fo the ae eaon, the iila quadatic te elated to the ace vecto v and the one elated to the oto agnetic field give a null eult (4.6), (4.63): F L π L μ μ ( b) B ( θ,t) inθ dθ B co [ θ ϕ ] inθ dθ xn π S (4.6) 44

155 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine 45 By integating the fouth te of (4.): ( ) ( ) ( ) [ ] [ ] S S d xn d in co co B B L d in,t B,t B L F θ θ ϕ θ ϕ θ μ θ θ θ θ μ π π (4.64) By alying (4.7) to (4.64): ( ) ( ) [ ] ( ) [ ] θ θ ± ϕ ϕ θ θ θ ϕ ϕ θ μ π π S S d xn d in co d in co B B L F (4.65) By alying (4.57) to both the integal of (4.65): ( ) ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] θ ± ϕ ϕ θ θ ± ϕ ϕ θ θ ϕ ϕ θ θ ϕ ϕ θ μ π π π π S S d xn d in d in d in d in B B L F (4.66) We ae the aution (4.67), that accoding to (4.66) give the elationhi (4.68) and, conequently, (4.69): (4.67) ( ) ( ) [ ] [ ] [ ] [ ] θ ± ϕ ϕ θ θ ± ϕ ϕ θ ϕ ϕ θ θ ϕ ϕ θ μ π π π π S S d xn d in d in d in d in B B L F (4.68)

156 Chate 4 ( d F ) xn L 4μ B S B S [ co[ ( ) θ ϕ ϕ ] ( ) π in [ co[ θ ϕ ϕ ] π ( ϕ ± ϕ ) co[ θ ϕ ± ϕ ] π π (4.69) The fit, econd and fouth te in (4.69) have a null eult, being diffeence between coine with aguent ultile of π. The only te diffeent fo zeo i the thid, not deending on the integation vaiable θ. Finally, the eult of the integal (4.64), unde the hyothei (4.67) i: ( d ) πl Fxn BSBS in( ϕ ± ϕ ) if (4.7) μ We ae the aution (4.7), that accoding to (4.66) give the elationhi (4.7) and, conequently, (4.73): (4.7) F F L μ ( d ) B B in[ θ ϕ ϕ ] xn L 4μ S π S π in π in dθ [ ( ) θ ϕ ϕ ] dθ [ θ ϕ ± ϕ ] dθ in[ ϕ ± ϕ ] ( d ) B B [ co( θ ϕ ϕ )] xn S S π [ co[ ( ) θ ϕ ϕ ] ( ) co dθ π ( θ ϕ ± ϕ ) π in( ϕ ± ϕ ) π π (4.7) (4.73) With vey iila concluion of the othe aution, the eult of the integal 46

157 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine (4.64), unde the hyothei (4.7) i: ( d ) πl Fxn BSBS in( ϕ ± ϕ ) if (4.74) μ A done befoe, the aution egading the u of the nube of ole ai, elicated in (4.75), (4.76) ae not exained becaue they don t coeond to eal cae. (4.75) (4.76) So, the final fo of the x-ojection of the noal coonent of the foce, elated to the d facto of (4.), i given by: πl BSB ( d ) μ Fxn πl BSB μ with : S S in ( ϕ ± ϕ ) in ( ϕ ± ϕ ) v N if if v N (4.77) A aid befoe, the ign of the condition of exitence have to be accoded to ign of the angula hae in the ae ode that aea in (4.77). Note that the ign of the x-coonent change deending on the condition to be veified (4.67) o (4.7), diffeently fo the y-coonent (4.47). By integating the e and f facto of (4.) in a vey iila way to what ha been done fo d, the inteaction between evey ingle tato cuent ace vecto and the oto field oduced by the agnet can be exeed a: 47

158 Chate 4 πl BSBRh in ( e) μ Fxn πl BSBRh in μ with : [ h Δθ ϕ ] v πl BS BRh in ( f ) μ Fxn πl BS BRh in μ with : [ h Δθ ϕ ] N [ h Δθ ϕ ] [ h Δθ ϕ ] v N h N if if odd if if h N odd h h h h (4.78) (4.79) By adding u all the non-zeo te of the x-ojection of the noal coonent of the foce, i obtained: (, ) F xn (,h ) F xn πl B μ with : S πl B μ πl μ B πl μ S B B S S B B S with : Rh B S v Rh in N in [ ϕ ± ϕ ] in [ ϕ ± ϕ ] [ h Δθ ϕ ] in [ h Δθ ϕ ] v N v N h N h odd h N N N (4.8) odd N odd (4.8) 48

159 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine (,h) F xn πl μ B πl μ S B B S Rh B in Rh [ h Δθ ϕ ] in [ h Δθ ϕ ] h h N odd N odd with : v N h N odd (4.8) (, ) F ' xn πl B μ S πl B μ B S S' B S' in [ ϕ ± ϕ ] in [ ϕ ± ϕ ] ' ' ' ' N N with : v N ' v N (4.83) The ae conideation done fo the equation (4.53), elated to the eaning of and ', ae valid fo (4.83). So, the eultant x-ojection of the noal coonent of the foce tanitted to the oto i the u of the equation (4.8), (4.8), (4.8), (4.83), taing into account all the eented aution: (, ) (,h) (,h) (, ' ) F F F F (4.84) F xn xn xn xn xn 49

160 Chate Tangential coonent of the foce An analyi to deteine the contibution of the tangential coonent of the agnetic field to the adial foce i eented, with efeence to the equivalent uface cuent denity ( x,t) ditibution of the agnetic field: G ( x,t) G, eonible fo the noal coonent H n (4.85) x Note that x i the linea abcia coeonding to the angula one θ. By conideing the ilified aution that give the tangential coonent of agnetic field coelation between H t equal to the linea cuent denity J, it i oible to find a H t and H n : H n H n H t ( x,t) J ( x,t) Gδ δ Ht δ (4.86) x x In ode to deteine the exeion of (4.86), baed on (4.), it i oible to wite: H n ( θ,t) H ( θ,t) H ( θ,t) H ( h θ, t) whee h (4.87) h π θ θ x (4.88) τ Note that a ecified in (4.88), and have the ae value, being the ole ai nube of tato and oto field fo the otoing toque. The deivative of H n by uing (4.87), (4.88) can be exeed a: H x n π H τ ( θ,t) ( θ ) π H τ ( θ,t) ( θ ) h hπ H τ ( hθ,t) ( h θ ) h (4.89) 5

161 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine By ubtituting (4.89) in (4.86), calculating the deivative and exlicating the value of the flux denitie, H t can be finally exeed a: H t ( θ,t) B in[ θ ϕ () t ] δ μ π B τ δ μ S π τ in S δ μ hπ τ [ θ ϕ () t ] B in[ h θ h Δθ] h Rh (4.9) Accoding to the Maxwell te teno exeion (4.5), the tangential coonent of T i given by H tbn, calculated by ultilying (4.) and (4.9), eulting in the u of the following nine te:,,h,,h h, δπ BS in[ θ ϕ() t ] co[ θ ϕ() t ] (4.9) μ τ δπ BSBS in[ θ ϕ() t ] co[ θ ϕ() t ] (4.9) μ τ δπ BSBRh in[ θ ϕ() t ] co[ hθ hδθ] (4.93) μ τ δπ BSBS co[ θ ϕ() t ] in[ θ ϕ() t ] (4.94) μ τ δπ BS in[ θ ϕ() t ] co[ θ ϕ() t ] (4.95) μ τ δπ BS BRh in[ θ ϕ() t ] co[ hθ hδθ] (4.96) μ τ hδπ B μ τ S B Rh co [ θ ϕ () t ] in[ h θ h Δθ] (4.97) 5

162 Chate 4 h, hδπ B μ τ S B Rh co [ θ ϕ () t ] in[ h θ h Δθ] (4.98) h δπ B μ τ h Rh in [ h θ h Δθ] co[ h θ h Δθ] (4.99) A done in ection 4.4., evey uation decibed in (4.9)-(4.99) ha to be ojected along the axe diection and integated between and π in ode to deteine the x- and y-coonent of the eulting tangential foce in the cateian efeence yte (4.), (4.): F F xt yt π π dft co θ L Ht π π dft ( θ,t) Bn ( θ,t) co θdθ ( inθ ) L Ht ( θ,t) Bn ( θ,t) inθdθ (4.) (4.) A een in the eviou, the eult of the integal i non-zeo when the diffeence between the ole ai nube of two field in exa i equal to ±. Futheoe, the te in uation ae conideed fo the exiting haonic ode of,, h. Let u efo the calculation (4.), conideing the te (4.9), (4.95): ( a,e ) δπ Fyt L π BS in[ θ ϕ() t ] co[ θ ϕ() t ] inθdθ (4.) μ τ By conideing that: in ( α) co ( α) in ( α) (4.3) Alying (4.3) to (4.), it give: ( a,e ) δπ Fyt BS L in[ θ ϕ() t ] inθdθ μ τ π (4.4) By auing that: 5

163 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine ( α β) co( α β) co in ( α) in( β) (4.5) Alying (4.5) to (4.4), it give: F F δπ π S μ τ 4 ( a,e ) B L co[ ( ) θ ϕ] dθ co[ ( ) θ ϕ] yt δπ 4 μ τ ( a,e ) B L [ in[ ( ) θ ϕ ] yt S Poceeding by analogy, it can be aued that: π π π [ in[ ( ) θ ϕ ] dθ (4.6) (4.7) F δπ 4 μ τ h () i hb L [ in[ ( h ) θ h Δθ ] yt Rh h π π [ in[ ( h ) θ h Δθ ] (4.8) Thu, the integal elated to the te of the tyologie a, e, i, ae zeo. Let u calculate the te of the tyology c, f, coeonding to equation (4.93), (4.96): ( c, f ) δπ Fyt L π BS BRh in[ θ ϕ() t ] co[ h ( θ Δθ) ] in θdθ (4.9) μ τ By conideing (4.) and alying it to (4.9), it give (4.): ( α β) in( α β) in in ( α) co( β) (4.) F δπ μ τ ( c, f ) B B L in[ ( h ) θ ϕ h Δθ] yt S Rh π π in [( h ) θ ϕ h Δθ] inθ dθ inθdθ (4.) 53

164 Chate 4 By uing (4.5) in (4.): F ( c, f ) B B L co[ ( h ) θ ϕ h Δθ] yt δπ 4 μ τ π co [( h ) θ ϕ h Δθ] S π Rh co dθ [( h ) θ ϕ h Δθ] π co π dθ [( h ) θ ϕ h Δθ] dθ dθ (4.) Conide, a done above, that the u of the ole ai of the tato and oto field cannot be equal to ±, condition ioible to veify, thu the hyothei (4.3), (4.4) aen t aditted: h h (4.3) h h (4.4) Let u uoe that (4.5): h h (4.5) By uing (4.5) in (4.), it give (4.6) and conequently (4.7): F δπ 4 μ τ ( c, f ) B B L co[ ( ) θ ϕ h Δθ] yt ( c, f F ) yt 4 π co δπ B μ τ S π ( θ ϕ h Δθ) dθ co( ϕ h Δθ) S B Rh Rh L ( θ ϕ h Δθ) dθ π π co dθ dθ [ in[ ( ) θ ϕ hδθ ] ( ) π [ in( θ ϕ h Δθ) ] πco( ϕ h Δθ) [ in( θ ϕ h Δθ) ] π π (4.6) (4.7) 54

165 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine Fo the eaon aleady entioned eveal tie, all the integal deendent on θ calculated on the inteval [ π] contibution to the foce:, ae zeo; o, the thid te i the only ( c, f ) π δl Fyt BS BRh co[ hδθ ϕ() t ] if h (4.8) μ τ Let u uoe that (4.9) i veified: h h (4.9) By uing (4.9) in (4.), it give: F δπ 4 μ τ ( c, f ) B B L co( θ ϕ h Δθ) yt π co π [( ) θ ϕ h Δθ] dθ co( θ ϕ h Δθ) S Rh π π co dθ dθ ( ϕ h Δθ) dθ (4.) By executing the integal in (4.): F δπ 4 μ τ ( c, f ) B B L [ in( θ ϕ h Δθ) ] yt S Rh π [ in[ ( ) θ ϕ hδθ ] ( ) π [ in( θ ϕ h Δθ) ] πco( ϕ h Δθ) π (4.) The only not null contibution to the foce i the fouth te of (4.): ( c, f ) π δl Fyt BS BRh co[ hδθ ϕ() t ] if h (4.) μ τ So, it can be concluded that the contibution of the te of the tyologie c and f to the y -coonent of the tangential foce acting on the oto i given a: 55

166 Chate 4 π δl BS BRh co[ hδθ ϕ() t ] ( c, f ) μτ Fyt π δl BS BRh co hδθ ϕ t μτ [ ()] if if h h (4.3) Obviouly, thi contibution ha to be conideed in the inteaction between eveyone of the cuent ace vecto v, v, and the oto agnetic field oduced by the agnet: π δl BSBRh co[ hδθ ϕ() t ] ( c) μτ Fyt π δl BSBRh co hδθ ϕ t μτ [ ()] if if h h (4.4) π δl BS BRh co ( f ) μτ Fyt π δl BS BRh co μτ [ h Δθ ϕ () t ] [ h Δθ ϕ () t ] if if h h (4.5) Let u calculate the te of the tyology g, h, coeonding to equation (4.97), (4.98): ( g,h ) hδπ Fyt L π BS BRh co[ θ ϕ() t ] in[ h ( θ Δθ) ] inθdθ (4.6) μ τ By conideing (4.7) and alying it to (4.6), it give (4.8): ( α β) in( α β) in co ( α) in( β) (4.7) F δπ μ τ π ( g,h ) hb B L in[ ( h ) θ ϕ h Δθ] yt S Rh π in [( h ) θ ϕ h Δθ] inθ dθ inθdθ (4.8) 56

167 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine By uing (4.5) in (4.8): F δπ 4 μ τ ( g,h ) hb B L co[ ( h ) θ ϕ h Δθ] yt π co [( h ) θ ϕ h Δθ] π S co dθ [( h ) θ ϕ h Δθ] π Rh co π dθ [( h ) θ ϕ h Δθ] dθ dθ (4.9) A done in the eviou, the hyothei (4.3), (4.4) ae to be conideed not valid, o uoe that: h h (4.3) By uing (4.3) in (4.9), it give: F δπ 4 μ τ ( g,h ) hb B L co[ ( ) θ ϕ h Δθ] yt π co S π ( θ ϕ h Δθ) dθ co( ϕ h Δθ) Rh π π co dθ dθ ( θ ϕ h Δθ) dθ (4.3) By executing the integal in (4.3): ( g,h F ) yt 4 δπ hb μ τ S B Rh L [ in[ ( ) θ ϕ hδθ ] ( ) π [ in( θ ϕ h Δθ) ] πco( ϕ h Δθ) [ in( θ ϕ h Δθ) ] π π (4.3) The thid te of (4.3) i the only contibution to the foce: 57

168 Chate 4 ( g,h ) π δl Fyt h BS BRh co[ hδθ ϕ() t ] if h (4.33) μ τ Let u uoe that (4.34) i veified: h h (4.34) By uing (4.34) in (4.9), it give: F δπ 4 μ τ ( g,h ) hb B L co( θ ϕ h Δθ) yt π co π [ ( ) θ ϕ h Δθ] dθ co( θ ϕ h Δθ) S Rh π π co dθ dθ ( ϕ h Δθ) dθ (4.35) By executing the integal in (4.35): F δπ 4 μ τ ( g,h ) hb B L [ in( θ ϕ h Δθ) ] yt S Rh π [ in[ ( ) θ ϕ hδθ ] ( ) π [ in( θ ϕ h Δθ) ] πco( ϕ h Δθ) π (4.36) The only not null contibution to the foce i the fouth te of (4.36): ( g,h ) π δl Fyt h BS BRh co[ hδθ ϕ() t ] if h (4.37) μ τ So, it can be concluded that the contibution of the te of the tyologie g and h to the y -coonent of the tangential foce acting on the oto i given a: π δl h BS BRh co[ hδθ ϕ() t ] if h ( g,h ) μτ Fyt (4.38) π δl h BS BRh co[ hδθ ϕ() t ] if h μτ 58

169 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine A in the eviou, thi contibution ha to be conideed in the inteaction between eveyone of the cuent ace vecto v, v, and the oto agnetic field oduced by the agnet (4.39), (4.4): π δl h BSBRh co hδθ ϕ t ( g ) μτ Fyt π δl h BSBRh co[ hδθ ϕ() t ] μτ [ ()] if if h h (4.39) π δl h BS BRh co ( h) μτ Fyt π δl h BS BRh co μτ [ h Δθ ϕ () t ] [ h Δθ ϕ () t ] if if h h (4.4) The lat tyologie to analyze ae b and d, elated to the equation (4.9), (4.94): F π δπ μ τ ( b) yt L BS BS in[ θ ϕ ] co[ θ ϕ ] inθdθ By alying (4.) to (4.4), it give (4.4): (4.4) F δπ μ τ π ( b) B B L in[ ( ) θ ϕ ϕ ] yt S S π in [( ) θ ϕ ± ϕ ] inθ dθ inθdθ (4.4) By uing (4.5) in (4.4): F δπ 4 μ τ ( b) B B L co[ ( ) θ ϕ ϕ ] yt π co π [( ) θ ϕ ϕ ] dθ co[ ( ) θ ϕ ± ϕ ] S S π co[ ( ) θ ϕ ± ϕ] dθ (4.43) π dθ dθ 59

170 Chate 4 A done in the eviou, the aution (4.44), (4.45) ae to be conideed not valid: (4.44) (4.45) So, let u to exaine the hyothei (4.46): (4.46) By uing (4.46) in (4.43), it give: F δπ 4 μ τ ( b) B B L co[ ( ) θ ϕ ϕ ] yt π co π π dθ [ θ ϕ ϕ ] dθ co[ ϕ ± ϕ ] dθ co[ θ ϕ ± ϕ ] S S π dθ (4.47) By executing the integal in (4.47): ( b F ) yt 4 δπ B μ τ S B S L [ in[ ( ) θ ϕ ϕ ] ( ) π [ in( θ ϕ ϕ )] πco( ϕ ± ϕ ) [ in( θ ϕ ± ϕ )] π π (4.48) The thid te of (4.48) i the only not null contibution to the foce: ( b) π δl Fyt BSBS co[ ϕ() t ± ϕ() t ] if (4.49) μ τ Let u uoe that (4.5) i veified: (4.5) By uing (4.5) in (4.43), it give: 6

171 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine F δπ 4 μ τ ( b) B B L co[ θ ϕ ϕ ] yt π co π co [ ( ) θ ϕ ϕ ] dθ co[ θ ϕ ± ϕ ] S [ ϕ ± ϕ ] S dθ π π dθ dθ (4.5) By executing the integal in (4.5): F ( b) B B L [ in[ θ ϕ ϕ ] yt δπ 4 μ τ S S π π [ in[ ( ) θ ϕ ϕ ] [ in( θ ϕ ± ϕ )] ( ) πco( ϕ ± ϕ )} The only not null contibution to the foce i the fouth te of (4.5): π (4.5) ( b) π δl Fyt BSBS co[ ϕ() t ± ϕ() t ] if (4.53) μ τ So, it can be concluded that the contibution of the te of the tyology b to the y -coonent of the tangential foce acting on the oto i given a: π δl BSBS ( b) μτ Fyt π δl BSB μτ co S [ ϕ () t ± ϕ () t ] co [ ϕ () t ± ϕ () t ] if if (4.54) Without the need to execute the whole calculation, the contibution of the te of the tyology d can be deteined by analogy with eect to (4.54): π δl BSB ( d ) μτ Fyt π δl BSBS μτ S co co [ ± ϕ () t ϕ () t ] [ ± ϕ () t ϕ () t ] if if (4.55) 6

172 Chate Pojection of the tangential foce The coonent elated to the x -ojection wee calculated in a vey iila way to that hown in the cae of y -ojection; only the final eult being eoted hee. In the following ae lited all the contibution to the y-ojection (4.56)-(4.6) and x-ojection (4.6)-(4.67) of the tangential coonent of the foce. Note that the equation (4.6), (4.6) and (4.66), (4.67) elated to the inteaction between diffeent haonic ode of the ae cuent ace vecto v, wee deteined by analogy. The eultant tangential foce i given by adding all the te, eectively along the x and y axi. ( ) [ ] [ ] with : N N N co B B L N co B B L F v v S S S S, yt ± ϕ ϕ τ μ δ π ± ϕ ϕ τ μ δ π (4.56) ( ) [ ] [ ] with : N N N co B B L N co B B L F v v S S S S, yt ϕ ± ϕ τ μ δ π ϕ ± ϕ τ μ δ π (4.57)

173 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine 63 ( ) ( ) [ ] ( ) [ ] odd v odd Rh S odd Rh S,h yt N h N N h h co B B L h N h h co B B L h F ϕ Δθ τ μ δ π ϕ Δθ τ μ δ π with : (4.58) ( ) ( ) [ ] ( ) [ ] odd v odd Rh S odd Rh S,h yt N h N N h h co B B L h N h h co B B L h F ϕ Δθ τ μ δ π ϕ Δθ τ μ δ π with : (4.59) ( ) [ ] [ ] with : N N N co B B L N co B B L F v ' v ' ' ' S S ' ' ' S S ', yt ± ϕ ϕ τ μ δ π ± ϕ ϕ τ μ δ π (4.6)

174 Chate 4 64 ( ) [ ] [ ] with : N N N co B B L N co B B L F v ' v ' ' ' S S ' ' ' ' S S ', ' yt ϕ ± ϕ τ μ δ π ϕ ± ϕ τ μ δ π (4.6) ( ) [ ] with : N N N N in B B L F v v S S, xt ± ϕ ϕ τ μ δ π (4.6) ( ) [ ] with : N N N N in B B L F v v S S, xt ϕ ± ϕ τ μ δ π (4.63)

175 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine 65 ( ) ( ) [ ] odd v odd odd Rh S,h xt N h N N h N h h in B B L h F ϕ Δθ τ μ δ π with : (4.64) ( ) ( ) [ ] odd v odd odd Rh S,h xt N h N N h N h h in B B L h F ϕ Δθ τ μ δ π with : (4.65) ( ) [ ] with : N N N N in B B L F v ' v ' ' ' ' S S ', xt ± ϕ ϕ τ μ δ π (4.66) ( ) [ ] with : N N N N in B B L F v ' v ' ' ' ' S S ', ' xt ϕ ± ϕ τ μ δ π (4.67)

176 Chate 4 The ae conideation done fo the equation (4.53), elated to the eaning of and ', ae valid fo equation (4.6), (4.6), (4.66), (4.67). 66

177 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine 4.5 Siulation and eult An electical achine uitable fo beaingle alication (Tab. II) wa analyzed in ode to validate the elationhi eented in the eviou chate, aing a coaion with the eult of the D FEA oftwae FEMM 4. [4]. TABLE II. DATA OF THE MACHINE Paa. Decition Value N l nube of lot 3 ole ai of the achine nube of hae 5 I n ated hae cuent (A ) T n ated toque (N) 3.9 g aiga width () D e tato oute diaete () 3 D tato inne diaete () D ean diaete of the agnet () 6 D cv_ext diaete at the botto of the lot () 7 D cv_int diaete at the to of the lot () 6.3 D oto oute diaete () 4 D alb oto inne diaete () 6 α Ldg angle undelying the tooth uface. α a ei-angle undelying the lot oening.95 α angle undelying the agnet 7 α cv lot itch angle a dt tato lot height () 5 h cl lot oening height () L axial length of the achine () 8 L agnet width () L dt tooth-body width () 8 L cl lot oening width () L tc lot width at the to lot adiu () 5.3 L fc lot width at the botto lot adiu () 9.7 τ cv lot itch at the inne tato adiu ().57 67

178 Chate 4 The analyi wa caied out conideing only the noal coonent of the foce, afte having veified that the contibution of tangential coonent ae negligible, by alying hae cuent given by the u of the two ace vecto v and v. A a geneal citeion, the toque cuent ace vecto i aintained in leading by 9 electical degee with eect to the oto oition in ode to each the axiu toque e aee behavio of the oto. In the following, the foce i calculated by vaying the hae angle ϕ elated to the levitation cuent ace vecto. Fo (4.5), (4.8) it i oible to deteine the elationhi between the diection of the foce ϕ F, eaued with eect to y - axi and oitive when clocwie-oiented, the hae ϕ and the angula oition of the oto Δ θ, obtaining: ( ) ( ),h,h F xn F atn (,h ) hδθ ϕ, h Nodd (4.68) F yn ϕ ( ) ( ),h,h F xn F atn h,h F yn ϕ ( ) ( Δθ ϕ ), h Nodd (4.69) Conideing the effect of the econd atial haonic coonent of the tato agneto-otive foce ( ) which inteact with the fit atial haonic coonent of the PM ( h ), (4.69) allow to calculate the diection of the foce ϕ F by ean of the following elationhi: (, ) F ϕ Δθ (4.7) ϕ In the equation i conideed only the inteaction between the ain haonic ode of the levitation cuent ace vecto and the oto agnetic field: thi i the uual aoach when deigning a contol yte fo beaingle achine, following the elationhi (4.7). In ultihae achine thi can oduce itae in deteining both the odule and the atial hae of the adial foce, 68

179 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine due to the inteaction between the highe haonic ode. The elationhi (4.5)-(4.53) and (4.8)-(4.83), taing into account all the oible inteaction, allow to calculate thee eo in te of diffeence, in odule and hae of the adial foce, between the ilified ediction (4.7) and the actual function, eeenting the locu of adial foce vecto and allowing the aoiate coection. Fig how a coaion between the ilified equation ain haonic ode (4.7) and the actual adial foce vecto deteined by uing both the ooed ethod, both the FEA oftwae. The analyi wa conducted by vaying the hae angle of levitation cuent ϕ by ultile value of.5 electical degee. Paticulaly, in the Fig ae hown the x - and y - coonent of the calculated adial foce and the good ageeent with the FE oftwae eult, eeented by the ed dot. Fig. 4.. y-coonent v x-coonent of the adial foce: i v A, i v A, Δθ ech. degee, n ac (/5) 69

180 Chate 4 Fig y-coonent v x-coonent of the adial foce: i v A, i v A, Δθ ech. degee, n ac (/5) In thi cae, the agnet axi i aligned to the hae axi, conequently Δ θ and the hae ϕ of the toque cuent ace vecto i 9 electical degee. By coaing Fig. 4. to Fig. 4.3, the value of i v wa changed fo to the ated value of A. A it i oible to ee, the effect of the toque cuent ace vecto v deteine a counteclocwie otation of the locu that decibe the oition of the eultant adial foce vecto. In fact, fo a null value of the toque cuent ace vecto, the actual diection of the adial foce i actically coinciding with that of the hae angle ϕ elated to the ace vecto v given by (4.7), a hown in Fig. 4.; the influence of i v deteine a ignificant change with eect to the actual value of ϕ and the edicted one. The itch winding i anothe iotant facto which influence the locu of adial foce vecto, that i eeented by a nealy ellioidal hae only in the aticula cae of a hotened itch winding, able to eliinate the thid haonic ode 7

181 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine (coeonding to a n ac value of ). Fig y-coonent v x-coonent of the adial foce: i v A, i v A, Δθ ech. degee, n ac 9 (/5) In geneal, by changing thi value, the locu ee to be waed with eect to thi ideal hae, a can be een in Fig. 4.4 fo a n ac value of 9. In the Fig , odulu and hae of the adial foce ae eented in te of diffeence with eect to the value of the ain haonic ode locu: in thi way, they give the coection to be ade in ode to obtain the actual adial foce vecto. 7

182 Chate 4 Fig Modulu of the diffeence of the adial foce: i v A, i v A, Δθ ech. Degee Fig Phae diffeence of the adial foce: i v A, i v A, Δθ ech. Degee 7

183 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine The influence of the oto oition, fo a fixed ace vecto i v, eeent anothe analyi tool, hown in Fig In Fig. 4.7 the toque cuent i null, while in Fig. 4.8 i equal to the ated value of A. In evey ictue the agnet axi i otated in thee diffeent oition with eect to hae axi:, 45 and 9 echanical degee. The analyi how a clocwie otation of the locu by the ae echanical angle than the agnet axi, but i iotant to note that the oition of the coeonding oint, chaacteized by a ae value of the hae ϕ, change by the ae aount but in a counteclocwie otation. Fig y-coonent v x-coonent of the adial foce: i v A, i v A, Δθ, 45, 9 ech. degee, n ac (/5) 73

184 Chate 4 Fig y-coonent v x-coonent of the adial foce: i v A, i v A, Δθ, 45, 9 ech. degee, n ac (/5) In the next will be hown an in-deth analyi, conducted with the eented algoith by vaying the ain aaete in a lage nube of oible cobination, to highlight the influence of the agnet itch, the oto oition and the coil itch ( α, Δ θ, n ac ); the value of the odulu of cuent ace vecto i v, i v ae fixed to thei axiu value. The eult ae eented in te of odulu and hae diffeence, howing in thi way the coection which can be alied to the ilified function ain haonic ode to obtain the abolute value of the odulu and diffeence of the foce. All the quantitie ae deteined a function of the levitation cuent ace vecto hae angle ϕ. 74

185 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine α ech. ech., nac 7, iv A,iv A, Δθ vaie Fig Fig

186 Chate 4 α ech. ech., nac 7, iv A,iv A, Δθ 45 vaie Fig. 4.. Fig

187 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine α ech. ech., nac, iv A,iv A, Δθ vaie Fig Fig

188 Chate 4 α ech. ech., nac 9, iv A,iv A, Δθ vaie Fig Fig

189 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine α ech. ech., nac 9, iv A,iv A, Δθ 9 vaie Fig Fig

190 Chate Concluion In thi chate an analytical odel fo adial foce calculation in ultihae beaingle Suface-Mounted Peanent Magnet Synchonou Moto (SPMSM) i eented. It allow to edict alitude and diection of the foce, deending on the value of the toque cuent, of the levitation cuent and of the oto oition. It i baed on the ace vecto ethod, letting the analyi of the achine not only in teady-tate condition but alo duing tanient. When deigning a contol yte fo beaingle achine, it i uual to conide only the inteaction between the ain haonic ode of the tato and oto agnetic field: in ultihae achine thi can oduce itae in deteining both the odule and the atial hae of the adial foce, due to the inteaction between the highe haonic ode. The eented algoith allow to calculate thee eo, taing into account all the oible inteaction; by eeenting the locu of adial foce vecto, it allow the aoiate coection. In addition, the algoith eit to tudy whateve configuation of SPMSM achine, being aaeteized a a function of the electical and geoetical quantitie, a the coil itch, the width and length of the agnet, the oto oition, the alitude and hae of cuent ace vecto, etc. Finally, the eult of the ooed ethod have been coaed with thoe of a ot ued FEA oftwae, obtaining vey iila value of the analyzed quantitie. 8

191 An analytical ethod fo calculating the ditibution of foce in a beaingle ultihae uface-ounted ynchonou achine 4.7 Refeence [] T. Schneide and A. Binde, Deign and evaluation of a 6 RPM eanent agnet beaingle high eed oto, Poc. Conf. on Powe Electonic and Dive Syte, Bango, Thailand, Nov. 7 3, 7,. -8. [] A. Salaza, A. Chiba, T. Fuao, "A Review of Develoent in Beaingle Moto", 7th Int. Sy. on Magn. Beaing, Zuich, Switzeland, Aug. 3-5,, [3] A. Chiba, T. Deido, T. Fuao, M.A. Rahan, "An Analyi of Beaingle AC Moto," IEEE Tan. Enegy Conveion, vol. 9, no., Ma. 994, [4] M. Oohia, A. Chiba, T. Fuao, M. A. Rahan, Deign and Analyi of Peanent Magnet-Tye Beaingle Moto IEEE Tan. on Indutial Electonic, vol. 43, no., A. 996, [5] S. W.-K. Khoo, "Bidge Configued Winding fo Polyhae Self-Beaing Machine" IEEE Tan. Magnetic, vol. 4, no. 4, Ail. 5, [6] H. Xu, H. A. Toliyat, L. J. Peteen, "Roto Field Oiented Contol of a Five-Phae Induction Moto with the Cobined Fundaental and Thid Haonic Cuent", Poc. APEC ' Conf., Vol.,, [7] M. Kang, J. Huang, H.-b. Jiang, J.-q. Yang, Pincile and Siulation of a 5-Phae Beaingle Peanent Magnet-Tye Synchonou Moto, Intenational Confeence on Electical Machine and Syte,. 48-5, 7- Oct. 8. [8] Y. Oada, S. Miyaoto, T. Ohihi, Levitation and Toque Contol of Intenal Peanent Magnet Tye Beaingle Moto, IEEE Tanaction on Contol Syte Technology, Vol. 4, No. 5, Setebe 996, [9] B. B. Choi, Ulta-High-Powe-Denity Moto Being Develoed fo Futue Aicaft, in NASA TM 3-96, Stuctual Mechanic and Dynaic Banch Annual Reot,., Aug. 3. [] Y. Chiti and M. Moo-Young, Clean-in-lace Syte fo Indutial Bioeacto: Deign, Validation and Oeation, Jounal of Indutial Micobiology and Biotechnology, Vol. 3,. -7, July 994. [] D. Caadei, D. Dujic, E. Levi, G. Sea, A. Tani, and L. Zai, Geneal Modulation Stategy fo Seven-Phae Invete with Indeendent Contol 8

192 Chate 4 of Multile Voltage Sace Vecto, IEEE Tan. on Indutial Electonic, Vol. 55, NO. 5, May 8, [] L. Zai, M. Mengoni, A. Tani, G. Sea, D. Caadei: "Miniization of the Powe Loe in IGBT Multihae Invete with Caie-Baed Pulewidth Modulation," IEEE Tan. on Indutial Electonic, Vol. 57, No., Novebe, [3] S. Sei, A. Tani, G. Sea, A Method fo Non-linea Analyi and Calculation of Toque and Radial Foce in Peanent Magnet Multihae Beaingle Moto, Int. Sy. on Powe Electonic, Electical Dive, Autoation and Motion SPEEDAM, Soento, Italy, June -,, [4] D. C. Meee, Finite Eleent Method Magnetic, Veion 4. (OctBuild), htt:// 8

193 Magnetic field ditibution in the aiga of ultihae electical achine Aendix A4. MAGNETIC FIELD DISTRIBUTION IN THE AIRGAP OF MULTIPHASE ELECTRICAL MACHINES A4.. Intoduction In ecent yea, oe and oe advanced technologie and an ieive ie in the ue of electonic, both in civil a in the indutial ecto, given a contibution to educe the cot of the coonent, allowing the ue of colex technologie which in the at had high cot and theefoe of little indutial inteet. In the field of electical achine thi evolution led not only to the ealization of owe dive contolled by an invete, caable of enuing efoance ignificantly bette than thoe obtained with the eviou contol yte, but alo the advent of a new tye of achine with a diffeent nube of hae fo the taditional thee-hae, uually eloyed in geneation and ditibution of electic enegy. Thi ha eawaened the inteet in the tudy of ulti-hae electical achine. 83

194 Aendix A4. The field of tudy of olyhae achine i elatively new and in aid evolution, but it i aleady oible to ay that thee achine ae able to ovide bette efoance of the claical one, and eciely fo thi eaon, ae cuently a atte of geat inteet. Indeed, the ultihae achine have eveal advantage coaed to the taditional thee-hae achine, uch a the alitude eduction and the inceae of the fequency of ulating toque, the eduction of tato hae cuent, the inceae of the fault toleance. In addition, the ultihae achine offe a lage nube of degee of feedo with eect to the thee-hae achine, which can be ued to iove the efoance. Futheoe, the ceation of oftwae fo analyzing the behavio of agnetic field in electoechanical device, baed on nueical ethod a the FEA, ha geatly contibuted to the ioveent in the deign of electical achine by intoducing, in addition to an excellent accuacy of the eult, alo a conideable aving of tie and oney. A4.. The ultihae otating agnetic field Fo an in-deth undetanding on the toic of ulti-hae achine, aing a bief efeence to the theoy of the otating agnetic field, in teady tate condition, to bette undetand the equation that will be ooed in the following ection. By uoing to oeate in linea egie, with an ion agnetic eeability of infinite value, it can be concluded that thee ae no aeciable do of agneto-otive foce (f) in the ion, conideing in thi egion a null value fo the agnetic field. Thu the tudy i conducted only in the aiga, conideing the hyothei of the eeentation along a taight line, a in Fig. A4.-, the ditibution of 84

195 Magnetic field ditibution in the aiga of ultihae electical achine the cuent denity J ( x,y,z) being only by ean of z -coonent (A4.-6); the cuent ouce ae located in the lot and conideed a concentated in a oint. With efeence to the Maxwell equation, develo the fit of (A4.-): oth J divb (A4.-) Fig. A4.-. Refeence yte in the aiga oth î ĵ ˆ (A4.-) x y z H H H x y z By executing the calculation, the fit equation of (A4.-) give thee cala coonent: H y z H z y J x (A4.-3) H z x H x z J y (A4.-4) H x y H y x J z (A4.-5) J î ĵ J ( x)ˆ (A4.-6) z Anothe vey iotant hyothei about the agnetic field ditibution, give 85

196 Aendix A4. the only coonent in the y diection, being (A4.-3)-(A4.-5), it i obtained: H y z H y J z x H H ĵ : by alying thi to y (A4.-7) Fo the fit equation of (A4.-7) it eult that H y doe not deend on z, vaying only a a function of the x coodinate. Fig. A4.-. Reeentation of agnetic field ditibution in the aiga H x y J z ( x) (A4.-8) In the Fig. A4.-3 and A4.-4 ae hown the ot nown tyologie of cuent denity and the elated agnetic field ditibution, by following the elationhi A4.-8. The eented analyi of agnetic field ditibution i baed on the following aution: I) The eeability of ion i infinite; II) The lot of the achine ae ei-cloed, having an infiniteial lot oening width and height; 86

197 Magnetic field ditibution in the aiga of ultihae electical achine Fig. A4.-3. Sinuoidal (left) and quae wave (ight) ditibution Fig. A4.-4. Rectangula (left) and iulive (ight) ditibution III) The agnetic field line ae adial and eendicula to the oto and tato bounday uface; IV) The ean aiga adiu of cuvatue i infinite, o that the aiga ath can be conideed a a taight line; V) Exteity effect ae neglected. 87

198 Aendix A4. VI) The effect of the leaage fluxe ae neglected. The analyi will be conducted tating fo the agnetic field ditibution oduced by a one lot-e-ole winding (Fig. A4.-5). Conideing, with no lac of geneality, a -ole achine and by alying the econd equation of (A4.-): divb B nˆ ds Bτ Sc μ H τ L μ H L B τ τ L L H H ( H ) (A4.-9) Fig. A4.-5. Magnetic field ditibution ( lot-e-ole winding) By alying Aee law on a ath coing the aiga which include a gou of n conducto, neglecting the x -coonent of H and taing into account (A4.-9) it give: H dl ni H δ Hδ ni Hδ ni (A4.-) whee it i undetood that H tay fo H y. Afte a ile te: ni H (A4.-) δ A hown in Fig. A4.-5, the conideed efeence yte ha the oigin in the iddle of the coil. So, the atial ditibution of the agnetic field i a eiodic function of θ and can be develoed in Fouie haonic eie: H, 3,... ( θ) H co( θ) (A4.-) 88

199 Magnetic field ditibution in the aiga of ultihae electical achine whee: π π H H ( θ) co( θ) dθ π (A4.-3) Having all the coil the ae itch, equal to π electical adian, it can be eaily ove that the Fouie eie ha only odd haonic ode. Futheoe, it eent only coine te due to the choice of the efeence yte. The calculation of (A4.-3) give: H π π π H ( θ) co( θ) dθ co( θ) dθ co( θ) ni π in δ π ni π δ 4 ni π δ ( θ) in( θ) in π π ni δ π π π ni δ π dθ π (A4.-4) To obtain: 4 ni π H in (A4.-5) π δ By ubtituting (A4.-5) in (A4.-): H, 3,... 4 ni π δ π ( θ) in co( θ) (A4.-6) By aution the lot oening width and height ae conideed to be infiniteial, without ceating leaage fluxe; the coil of one hae ae eieconnected with the ae cuent flowing in. The equation (A4.-6) contain the following te: ( ) 4 ni π δ eeent the alitude of fundaental haonic ode, in π i the haonic facto whoe value i ±, the deendence on the function co ( θ) eeent the atial haonic ditibution. 89

200 Aendix A4. The atial ditibution of the agnetic field i contituted by the ueoition of inuoidal ditibution of deceaing alitude with the haonic ode. Fo each haonic the axiu value of the field occu at the cente of the coil. The alitude of each haonic i ootional to the value of the cuent flowing in the coil. Conide a agnetic field ditibution geneated by a q lot-e-ole winding, a hown in Fig. A4.-6: Fig. A4.-6. Magnetic field ditibution geneated by a q lot-e-ole winding. The field ditibution i foed by the ueoition of q contibution, elatively dilaced by an electical angle α (the angle between two adjacent lot): the elated eeentation, by ean of the Fouie haonic eie, becoe: H q q ( θ) in co( [ θ ( j ) α] ) j, 3,... 4 ni π δ π (A4.-7) By vaying the index j fo to q it i obtained a uation of coine function which can be calculated a: co ( θ) co( [ θ α] ) co( [ θ α] )... co( [ θ ( q ) α] ) qα in q co θ α qk α in q (A4.-8) co θ α 9 d

201 Magnetic field ditibution in the aiga of ultihae electical achine whee the te K d i defined ditibution facto of Blondel (A4.-9): qα in K d (A4.-9) α q in By adoting a uitable winding ditibution in the lot, K d allow to educe, even eaably, oe haonic ode. By ubtituting the elation (A4.-8) in the exeion (A4.-7), it give: H q, 3,... 4 ni π δ π ( θ) qk in co θ α d q (A4.-) q Note that the diaga of the agnetic field ditibution ( θ) H oved in the oiginal efeence yte with eect to (A4.-6) due to the odification of the ditibution, that now i elated to q lot-e-ole-e-hae: thu, i neceay to hift the efeence yte by an angle ( q ) α, coeonding with the new q ea value of the ( θ) H ditibution a the cente of the hae. H q, 3,... 4 ni π δ π ( θ) qk in co( θ) d (A4.-) In the coaion between (A4.-) and (A4.-6) the contibution of the q lot-e-ole-e-hae can be ecognized in the te qk d. Fig. A4.-7. A lot with double laye winding 9

202 Aendix A4. Conide now the ditibution of agnetic field oduced by a winding of q lot-e-ole-e-hae in double-laye. It i hown in Fig. A4.-7 the eeentation of a lot in double laye, while in Fig. A4.-8 the eeentation of a double laye winding. Fig. A4.-8. Double laye winding eeentation A can be een, in a double laye winding it could haen to find, in the ae lot, gou of conducto belonging to diffeent coil. The agnetic field ditibution oduced by diffeent laye of the ae hae, can be een a identical to (A4.-) but hifted by a β electical angle. Alo, i iotant to note that in (A4.-) the nube of conducto-e-lot n i divided by due to the eence of the laye. Thu, the eultant ditibution can be witten a: H q ( θ) qk in co( θ), 3,... 4 n i π δ, 3,... d 4 n i qk π δ d π π in co ( [ θ β] ) (A4.-) The even haonic ode cancel each othe fo the two laye. Both the botto laye than the to one, oduce a field of q lot-e-ole: being a hae hift between the, it i neceay to add the two inuoidal function taing into account of it. The eult, by alying again an angula hift to the efeence yte on the new cente of the hae, i equivalent to ultilying by and by the hotened itch facto (A4.-3): 9

203 Magnetic field ditibution in the aiga of ultihae electical achine β K co (A4.-3) The elationhi (A4.-4) eeent the agnetic field ditibution oduced by a double laye winding of one hae, having q lot-e-ole-e-hae: H, 3,... 4 ni π δ π ( θ) qk K in co( θ) d (A4.-4) The oduct of the Blondel facto K d and the hotened itch facto K give the winding facto K a elated to -th haonic ode: qα in β Ka Kd K co (A4.-5) α q in Conide an intantaneou hae cuent i ( t) which vaie following a inuoidal law (A4.-6), being I the value and ω the angula fequency: i () t I co( ωt) (A4.-6) By ubtituting it in (A4.-4) it give: H, 3,... 4 π ni δ ( θ,t) qk in co( θ) co( ωt) a π (A4.-7) The equation (A4.-7) can be witten a: H M, 3,... ( θ,t) H co( θ) co( ωt) whee: (A4.-8) 4 ni π H M qka in (A4.-9) π δ The ditibution given by (A4.-8) eeent, fo each value of, a tationay 93

204 Aendix A4. wave with ole ai, hown in Fig. A4.-9 in a ucceion of tie intant. Fig. A4.-9. The tationay wave hown in diffeent tie intant The equation (A4.-8) can be een alo a the u of two counte-otating field deending on a ace-tie vaiable f ( θ ωt), each one oving in o ooite to the θ axi diection without waing, a hown in Fig. A4.- and in the elationhi (A4.-3). Fig. A4.-. A talating wave H H M, 3,... ( θ,t) H [ co( θ ωt) co( θ ωt) ] ( θ,t) H co( θ ωt) H co( θ ωt), 3,... M, 3,... M (A4.-3) (A4.-3) Each ditibution in (A4.-3) eeent, fo each value of, a otating wave with ole ai: in aticula, the fit uation i a wave which otate in 94

205 Magnetic field ditibution in the aiga of ultihae electical achine the ae diection of θ, while the econd one i a wave which otate in the ooite diection of θ. In Fig. A4.- i hown the dual inteetation of thi henoenon: the concet of the tationay wave and the counte-otating field. Fig. A4.-. The tationay and the counte-otating ditibution The ace-tie ditibution oduced by one of the two otating field i eeated identical in the following ituation: in a oition θ at the intant of tie t and in the oition θ at the intant of tie t, uch that: ω ω θ ± t θ ± t (A4.-3) Afte oe ile calculation: θ ω ω ω ω ( t t ) ( t ) (A4.-33) θ t ± t ± t The electical angula velocity ec., i given by: ω ce of the ditibution, eaued in adian e ω ce θ θ ω (A4.-34) t t The echanical angula velocity ω c of the otating field, eaued in adian 95

206 Aendix A4. e ec., i given by: ω ω c ± (A4.-35) Fo which follow the eed of the -th ode of the agnetic field: 6 f n ± (A4.-36) Conide a yte of cuent chaacteized by a hae diffeence ( π ) S t : S t i defined a equence of tie. Soe exale ae eented in the Fig. A4.- - A4.-4 (in the following, will be aued equal to ): i T t (A4.-37) () t I co ω t S ( j ) j Fig. A4.-. Sequence of tie with 3 Fig. A4.-3. Sequence of tie with 5 96

207 Magnetic field ditibution in the aiga of ultihae electical achine Fig. A4.-4. Sequence of tie with 5 The equence of tie ut coly with the following containt: S t (A4.-38) If the hae ae dioed by following diffeent atial configuation, in thi cae one ea of equence of ace S. Conide a achine with ole ai, hae hifted in ace by an angle equal to ( π ) balanced yte of cuent. S, in which flow a In the efeence yte centeed in the hae, the agnetic field ditibution oduced by the j -th hae i given by: H j M, 3,... π ( θ,t) H co θ S ( j ) co ωt S ( j ) t π (A4.-39) The eulting agnetic field oduced by all the hae of the achine can be exeed a: H ( θ,t) H co θ S ( j ) co ωt S ( j ) M j, 3,... j, 3,... H M π co θ ωt ( j )( S S ) t t π π (A4.-4) 97

208 Aendix A4. Finally i obtained: H π M t (A4.-4) ( θ,t) H co θ ωt ( j )( S S ) j, 3,... The eulting agnetic field i given by adding inuoidal contibution with a hae diffeence equal to ( S S ) π. t Thee ae two oible cae: if all the contibution have the ae hae, the eult i tie the contibution; if the contibution ae hifted by the ae angle, thei eultant i null. Thi eult i exlained in (A4.-4): H ( θ,t) H M co ( θ ωt) S St if S S if t intege intege (A4.-4) Fo a fixed value of, the agnetic field ditibution ha a diection of otation: diect, if evee, if S S t S S intege t intege A yetical olyhae winding in which flow a balanced yte of cuent, oduce a ditibution of agnetic field in the aiga decibed by the following elation: H,... 4 n I π δ π ( θ,t) qk in co( θ ωt) a (A4.-43) The te in uation (A4.-43) have to be conideed not null only fo the value of uch that S S t i intege. Thi ditibution can theefoe be conideed foed by the ovela of diect and invee haonic otating agnetic field. 98

209 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Chate 5 DESIGN AND DEVELOPMENT OF A CONTROL SYSTEM FOR MULTIPHASE SYNCHRONOUS PERMANENT MAGNET BEARINGLESS MACHINES 5. Intoduction In thi chate a contol yte fo beaingle ultihae ynchonou PM achine i eented, integating the electoagnetic odel een in Chate 4 with a thee-dienional echanical odel develoed baed on the Eule equation. One end of the oto haft i contained, to iulate the eence of a echanical beaing, while the othe i fee, only uoted by the adial foce develoed in the inteaction between agnetic field, to iulate a beaingle 99

210 Chate 5 yte with thee degee of feedo. The body in Fig. 5. eeent the oto and the haft of the achine, otating aound an axi with a fixed oint. The inteaction between the levitation cuent ace vecto i v and the othe ouce of agnetic field, i.e. oto agnet and toque cuent ace vecto i v, ovide to geneate the levitation foce, a een in the eviou chate by ean of the analytical foulation. The yte wa ileented on a SIMULINK odel, eeenting the concetual deign of the exeiental device and elated contol yte that could be ealized in a tet bench alication. Fig. 5. Fig. 5.

211 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Conide thee efeence yte in the ace: Abolute (ab), chaacteized by the unit vecto (, ĵ,ˆ ) Relative (el), chaacteized by the unit vecto ( Î,Ĵ,Kˆ ) Fixed-to-oto (ft), chaacteized by the unit vecto ( Î,ˆε, ηˆ ) î, Fig. 5.;, Fig. 5.(a);, Fig. 5.(b). The lat two yte have a coon unit vecto becaue thei fit axe coincide. 5. Mechanical equation The oto angula eed in the abolute efeence yte i calculated by alying the incile of cooition of angula eed : ω Ω ω el (5.) whee Ω i the angula eed of the elative efeence yte with eect to the abolute and ω el i the angula eed of the oto with eect to the elative efeence yte. They can be exeed a: Ω ψ& ˆ ϕĵ & (5.) ω el θî & (5.3) By ubtituting (5.), (5.3) in (5.), it give the oto angula eed vecto: ω ψ& ˆ ϕ& Ĵ θ& Î (5.4) The equation of the otion ae given by: dγ dt o M o (5.5)

212 Chate 5 Being Γ o, M o eectively the oent of oentu and the oent of the extenal foce evaluated with eect to the oint O. Γo i given by: Γ o I o ω (5.6) The fixed-to-oto efeence yte i obtained by otation of an angle θ aound the X-axi, and the elationhi between it unit vecto and the one of the elative yte ae (note that the unit vecto Î i the ae becaue the X-axi i in coon): Ĵ co θε ˆ in θηˆ (5.7) Kˆ in θε ˆ co θηˆ (5.8) In ode to wite the angula eed vecto with eect to the fixed-to-oto efeence yte, i neceay fitly to wite it in exlicit way, with eect to the elative yte: ˆ in ϕî co ϕkˆ (5.9) By ubtituting (5.9) in (5.4): ω ( ψ in ϕ θ& ) Î ϕ& Ĵ ψ& co ϕkˆ & (5.) By ubtituting (5.7) and (5.8) in (5.): ω ( ψ in ϕ θ& ) Î ( ψ& co ϕ in θ ϕ& co θ) ˆε ( ϕ& in θ ψ& co ϕco θ)ηˆ & (5.) which can be exeed in atix fo in the following: ω ω ω I ε η in ϕ co ϕ in θ co ϕco θ co θ in θ ψ& ϕ& θ& (5.) A the yte choen i incial of inetia, the I o i a diagonal atix thu,

213 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine with efeence to (5.6), i oible to wite: Γ o I o I ω oi I oε ωi ωε I oη ωη I oi ω I Î I oε ω ˆε I ε oη ω η ηˆ (5.3) Reinding that ω i defined with eect to a tationay obeve, the deivative dγ o ha to be deteined with eect to the ae obeve. It i oible to dt wite: dγ dt o dγo ( ab) dt ( ft) ω Γ o (5.4) By conideing the equation (5.6) and that the atix of inetia change, it can be calculated: I o doen t dγo dt ( ft ) I o ω& (5.5) By ubtituting (5.6) and (5.5) in (5.4): dγo dt ( ab) I o ω & ω I ω (5.6) o dγ dt o ( ab) I oi Î ˆε ηˆ ω& Î I ω& ˆε I ω& η ˆ ω ω ω (5.7) I oε ε oη η I oi I ω I I oε ε ω ε I oη η ω η By executing the calculation: dγ dt o ( ab) [ I ω& ( I I ) ω ω ] Î [ I ω& ( I I ) ω ω ] oi I oη oε ε η oε ε ˆε [ I ω& ( I I ) ω ω ]η ˆ oη η oi oε oη oi I I η ε (5.8) 3

214 Chate 5 Witing the oent of the extenal foce vecto O in the ft efeence yte: M o with eect to the oint M o M oi Î M oεˆε M oηηˆ (5.9) By ubtituting (5.8) and (5.9) in (5.5) i oible to deteine the equation of the otion, alo called Eule equation : I I I oi oε oη ω& I ω& ε ω& η ( Ioη Ioε ) ( I I ) oi ω ( I I ) oε oη oi ω ω ω ε I I ω ω η η ε M M M oi oε oη (5.) It i now ueful to define the elationhi between the oent of the extenal foce vecto evaluated in the ft efeence yte and the one evaluated in the ab efeence yte. Thu, i neceay to wite the unit vecto (, ĵ,ˆ ) eect the unit vecto ( Î,ˆε, ηˆ ) atix fo: î with. Fitly, the equation (5.7), (5.8) ae eented in Î Ĵ Kˆ co θ in θ Î in θ ˆ ε co θ ηˆ (5.) The unit vecto nˆ i defined a the ojection of the unit vecto Î on the xy lane: nˆ co ϕî in ϕkˆ (5.) By ubtituting (5.) in (5.3), (5.4), the elationhi between the unit vecto ( î, ĵ,ˆ ) and the unit vecto (, Ĵ,Kˆ ) Î ae found: î co ψnˆ in ψĵ co ψ co ϕî in ψĵ co ψ in ϕkˆ (5.3) ĵ in ψnˆ co ψĵ in ψ co ϕî co ψĵ in ψ in ϕkˆ (5.4) 4

215 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine ˆ in ϕî co ϕkˆ (5.5) By exeing (5.3)-(5.5) in the atix fo: î co ψ co ϕ ĵ in ψ co ϕ ˆ in ϕ in ψ co ψ co ψ in ϕ Î in ψ in ϕ Ĵ co ϕ Kˆ (5.6) By cobining (5.) and (5.6), ae eented the elationhi between the unit vecto ( î, ĵ,ˆ ) elated to abolute efeence yte and the unit vecto ( Î,ˆε, ηˆ ) elated to fixed-to-oto efeence yte: î co ψ co ϕ ĵ in ψ co ϕ ˆ in ϕ in ψ co ψ co ψ in ϕ in ψ in ϕ co ϕ co θ in θ Î in θ ˆ ε co θ ηˆ (5.7) By ultilying the atice in (5.7) it give: co ψ co ϕ in ψ co θ co ψ in ϕ in θ in ψ in θ co ψ in ϕco θ in ψ co ϕ co ψ co θ in ψ in ϕ in θ co ψ in θ in ψ in ϕco θ in ϕ co ϕ in θ co ϕco θ B ( θ, ϕ, ψ) B The atix obtained in (5.8), called ( θ, ϕ, ψ) (5.8), eit to diectly convet the ab unit vecto in the ft unit vecto. It i entioned a an invee atix becaue the ain atix ( θ, ϕ, ψ) extenal foce vecto B i that one which define the oent of M o in the ft efeence yte, being the ot ued. In fact, the latte can be iediately ubtituted in the Eule equation (5.). By cobining (5.9) and (5.3), i obtained (5.3): 5

216 Chate 5 M M M ox oy oz co ψ co ϕ in ψ co ψ in ϕ in ψ co ϕ co ψ in ψ in ϕ in ϕ M ox M oy (5.9) co ϕ M oz M M M oi oε oη M co θ in θ M in θ co θ M Θ( θ) ox oy oz (5.3) In (5.3), (5.3) i hown the ( θ, ϕ, ψ) oent of the extenal foce vecto B atix, which eit to calculate the M o in the fixed-to-oto efeence yte: co ψ co ϕ in ψ co ϕ in ϕ in ψ co θ co ψ in ϕ in θ co ψ co θ in ϕ in ψ in θ co ϕ in θ 4 in ψ in θ co ψ in ϕco θ co ψ in θ in ϕ in ψ co θ co ϕco θ B( θ, ϕ, ψ) (5.3) M M M oi oε oη B ( θ, ϕ, ψ) M M M ox oy oz (5.3) The elationhi (5.3) i eally iotant becaue the oent of extenal foce ae given in the abolute efeence yte, but i neceay to exe the in the fixed-to-oto efeence yte in ode to aly equation (5.). 5.3 Geneal tuctue of the contol yte The extenal tuctue of the contol yte, ealized with SIMULINK, i cooed by fou ain bloc, the othe being coe bloc and integato : Levitation foce bloc, Eule equation bloc, Lagangian vaiable bloc, 6

217 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Axi coodinate bloc. Fig Geneal tuctue of the contol yte ) Levitation foce bloc: the action of PID contolle i baed on the oition eo along the y and z axe, by calculating the foce, of odulu F and atial hae fi_f, equeted to aintain the oto in the cente of the tato boe. Togethe with the angula oition of the agnet theta, the Foce Contolle bloc deteine the odulu and the hae of the i v cuent ace vecto, called a vaiable I_ and fi_in_, to oduce the equeted foce. Thi coule of value i neceaily oviional, being the analytic elationhi that give the i v aaete univocal only conideing the ain haonic ode of tato and oto agnetic field. The next bloc, Electoagnetic Model, deteine the eal value of the foce to uend the oto, by taing into account all the oible inteaction between haonic ode of the agnetic field. Alo, thi odel conide the effect of both cuent ace vecto i v, i v, eectively toque and 7

218 Chate 5 levitation. It give in thi way an exact ediction of the adial foce and electoagnetic toque oduced by the oto. The dive aintain contantly the cuent ace vecto i v in leading by 9 electical degee with eect to agnet axi, in ode to give the equeted toque in whateve oeating condition. The outut give in aticula the coonent of the foce (vaiable F y, F z ) which ae ued to calculate the oent with eect to the abolute efeence yte, by ean of the Foce to Moent atix bloc. FORCE CONTROLLER Fig Diaga of the Levitation Foce bloc ) Eule equation bloc: the ub-bloc Alied Moent calculate the oent of extenal foce with eect to the fixed-to-oto efeence yte and alie the to the Eule equation (5.) in the fo (5.33), achievable afte oe ile calculation. The outut i contituted by the coonent of angula eed vecto, evaluated in the fixed-to-oto efeence yte and deteined by integating (5.33), a can be een in the SIMULINK diaga of Fig

219 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine 9 ω ω ω ω ω ω ω ω ω ε η ε η η η η ε η ε ε ε η ε ε η I o oi o o o I o o oi o o oi o o oi oi I I I I I M I I I I M I I I I M & & & (5.33) Fig Diaga of the Eule equation bloc 3) Lagangian vaiable bloc: alie the invee atix of (5.), decibed in the equation (5.34), to the angula eed vecto ω in ode to deteine the deivative with eect to tie [ ] θ ϕ ψ & & &,, of the lagangian vaiable [ ] θ ϕ ψ,,, which ae obtained by ean of integation: they contitute the coonent of the angula eed vecto ω evaluated in the abolute efeence yte. ω ω ω θ ϕ θ ϕ θ θ ϕ θ ϕ θ θ ϕ ψ η ε I co tan in tan in co co co co in & & & (5.34)

220 Chate 5 Fig Diaga of the Lagangian vaiable bloc 4) Axi coodinate bloc: it convet the lagangian vaiable [ ψ ϕ, θ] contituted by angula coodinate, in the linea coodinate [, y, z],, x which decibe the oition of the oto haft end oint. Thee ae then ued in a cloed loo feedbac to etun a an inut in the Levitation foce bloc, cloing in thi way the loo of the beaingle achine contol yte. Fig Diaga of the Axi coodinate bloc

221 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine 5.4 Detailed analyi of the contol yte 5.4. Levitation Foce Bloc A) Poition Eo Thee bloc deteine the eo along the y and z axe, calculating the diffeence between the actual oition, on the y-z lane, of the end haft oint and the efeence value y and z, which obviouly have to be et to zeo equiing a centeed oto (Fig. 5.8). Fig Poition eo B) Pid Contolle In Fig. 5.9 i eeented the geneal tuctue of the PID contolle, whee i oible to ditinguih the thee action: ootional, integato, deivative. Fig Diaga of the PID contolle

222 Chate 5 The aaete atd in the deivative banch i a tie contant which eit to et the duation of the tanitional egie. The outut of the PID give the y- and z-coonent of the foce, neceay to tabilize the oto. C) Foce Contolle Bloc The inut tadiu tanfo the coonent of the foce fo cateian eeentation to ola, oviding the odulu F and the atial hae fi_f. In addition, the angula oition of the oto theta i a equied inut vaiable. A aid above, the Foce Contolle deteine the odulu and the hae of the levitation cuent ace vecto i v (eectively identified by the outut vaiable I_ and fi_in_), equied to geneate the inut foce. Thee value eult aoxiate becaue the analytic elationhi that allow the calculation of the cuent ace vecto aaete fo the value of the foce, i invetible only conideing the inteaction between the ain haonic of the tato levitation field and the ain one of the oto field. In thi way, all the highe haonic ode ae neglected. Howeve, thi i not a oble becaue the conequent bloc tae into account all the oible inteaction between the agnetic field, and the PID contolle ovide to tabilize the yte with thei feedbac action; but, the function of the Foce Contolle i neceay to give oe initial value of the i v aaete, in abence of which would not be oible to ileent the egulation oce. FORCE CONTROLLER Fig. 5.. The Foce Contolle of the odel

223 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine In the following, the ogaing code of the Foce Contolle i hown, being a Matlab function. function ABF_inv fcn(f,fi_f,theta) %definizione cotanti Pi ; uzeo.56; %*************************************************************************************** %INPUT DATI (begin) %*************************************************************************************** %%odulo della foza adiale % F 6.6; %%fae aziale della foza adiale, iuata ietto all'ae y (FEMM) % fi_f -9 * Pi / 8; %%aio odine aonico indagato _ax ; B_zeo(,); B_zeo(,); I_zeo(,); fi_in_zeo(,); _zeo(,); hn_zeo(,); h_zeo(,); %dichiaazione aay ctzeo(_ax,); otzeo(_ax,); Kdzeo(_ax,); B_hzeo(_ax,); B_hnzeo(_ax,); B_hzeo(_ax,); B_hnzeo(_ax,); fi_in_hzeo(_ax,); fi_in_hnzeo(_ax,); Fqzeo(_ax,); Fdzeo(_ax,); %Fzeo(_ax,); %fi_fzeo(_ax,); I_hzeo(_ax,); I_hnzeo(_ax,); hzeo(_ax,); hnzeo(_ax,); %angolo eccanico fa ae M ed ae fae (g.ecc.) % deltath 9; % deltath deltath * Pi / 8; %nueo di cave ottee dalla bobina nac ; %nueo di fai 5; %nueo di cave/olo/fae q 6; %nueo di conduttoi in cava n ; %nueo cave tatoiche %Ncv 3; %coie olai STATORE (ifeite al cao PRINCIPALE) N_ ; %coie olai ROTORE M_ ; 3

224 Chate 5 %angolo decitto dal agnete (ad.ecc.) alfa_ag 7; alfa_ag alfa_ag * Pi / 8; %eoe del agnete L.; %eoe del tafeo g.; %eeabilità elativa agnete u_.45; %fattoe di Cate c ; %induzione eidua agnete Be.5; %ofondità di acchina L.8; %aggio edio al tafeo Rg.595; %angolo di cava (ad.el.) alfa_c ; alfa_c alfa_c * Pi / 8; %alfa_c Pi / / q %alfa_c * Pi / Ncv * N_ %alfa_c * Pi / Ncv * M_ %equenza teoale di coente t ; %angolo elettico coenti (ulazione * teo - g.el.) wt ; wt wt * Pi / 8; %*************************************************************************************** %INPUT DATI (end) %*************************************************************************************** %eoe tafeo coleivo deltag L g; %eiaiezza di una bobina (ad.el.) gaa nac * alfa_c / ; %induzione al tafeo geneata dal agnete BgM L / (L u_ * c * g) * Be; %ao olae (al tafeo) %Tau Pi * Rg / N_; %odine aio equenza di coente t_ax - ; if t > t_ax etun end fo : _ax ct() ; ot() ' '; Kd() ; %coonenti della foza adiale definite ietto all'ae y (d) (FEMM) Fq() F * in(fi_f); Fd() F * co(fi_f); end fo : _ax if ( t) / int3(( t) / ) ct() ; ot() 'I'; eleif ( - t) / int3(( - t) / ) ct() ; ot() 'D'; ele ct() ; ot() ' '; end 4

225 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine if ct() ct() %fattoe di ditibuzione Kd() in(q * * alfa_c / ) / q / in( * alfa_c / ); %valoe h oitivo h() ( * N_) / M_; %valoe h negativo hn() (- * N_) / M_; %CASO h INTERO e DISPARI % (ovaonibile con hn INTERO) if h() int3(h()) && h() / ~ int3(h() / ) B_h() 4 / h() / Pi * BgM * in(h() * M_ * alfa_ag / ); B_h() * uzeo / Pi / L / Rg / B_h() * F; I_h() * * Pi * * deltag / uzeo / / 4 / n / qt() / q / Kd() / in( * gaa) * B_h(); if ct() fi_in_h() h() * M_ * theta - wt - atan(fq(),fd()); eleif ct() fi_in_h() -h() * M_ * theta - wt atan(fq(),fd()); end end %CASO hn INTERO e DISPARI % (ovaonibile con h INTERO) if hn() int3(hn()) && hn() / ~ int3(hn() / ) B_hn() 4 / hn() / Pi * BgM * in(hn() * M_ * alfa_ag / ); B_hn() * uzeo / Pi / L / Rg / B_hn() * F; I_hn() * * Pi * * deltag / uzeo / / 4 / n / qt() / q / Kd() / in( * gaa) * B_hn(); if ct() fi_in_hn() hn() * M_ * theta - wt atan(fq(),fd()); eleif ct() fi_in_hn() -hn() * M_ * theta - wt - atan(fq(),fd()); end end %il ciclo i inteoe quando l'odine aonico di otoe è INTERO, DISPARI ed %aue il valoe INFERIORE ta i due: hn, h (in ealtà queto è ee 'hn' %e definizione) if (hn() int3(hn()) && hn() / ~ int3(hn() / )) && (hn() < h()) B_B_hn(); B_B_hn(); I_I_hn(); fi_in_fi_in_hn(); _; hn_hn(); h_( _ * N_) / M_; bea eleif (h() int3(h()) && h() / ~ int3(h() / )) && (h() < hn()) B_B_h(); B_B_h(); I_I_h(); fi_in_fi_in_h(); _; hn_(- _ * N_) / M_; h_h(); bea ele B_; B_; I_; fi_in_; _; hn_; h_; end end end %(fo : _ax) 5

226 Chate 5 ABF_inv [B_,B_,I_,fi_In_,_,hn_,h_]; Note that the code goe to a bea when the fit value of an exiting haonic ode i found, both fo the agnetic field oduced by levitation cuent ace vecto i v, both fo the agnetic field oduced by oto agnet, taing in thi way into account only the ain ode. D) Electoagnetic Model Bloc The Electoagnetic Model bloc eeent a colete odel of the oto, by the electical oint of view. It deteine the effective adial foce neceay to uot the oto, in te of the y- and z-coonent Fy and Fz, taing into account all the oible inteaction between the haonic ode of tato and oto agnetic field, accoding to the condition of exitence aleady een in chate 4 and evived heeunde (5.35)-(5.39): ' v v v N, N, N, h N odd (5.35) h ± (5.36) h ± (5.37) (5.38) ± ' ± (5.39) Thi odel alo conide the effect of both ace vecto i v, i v (eectively, toque and levitation), thu ovide an accuate ediction of the adial foce and toque geneated by the oto. The dive aintain the cuent ace vecto i v in leading by 9 electical degee with eect to the agnet axi, o that the oto oduce the equeted toque in each oeating condition of the i v ace vecto: aticulaly, thi ta i efoed by the bloc 6

227 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine eeented in the following Fig. 5., which add an angle of 9 electical degee to the electical angle coeonding to theta, angula oition of the agnet axi. The obtained eult, togethe with the value of I bloc, coletely define the cuent ace vecto i v. Fig. 5. The inut of Electoagnetic odel bloc i given by the i v odulu and hae I and fi_in, the i v odulu and hae I and fi_in, the oto oition deltath (it i the ae vaiable theta ued in the Foce Contolle, it nae changed only becaue of foal eaon), Fig. 5.. Fig. 5. In the following, the ogaing code of the Electoagnetic Model i hown, being a Matlab function. 7

228 Chate 5 %ALGORITMO COMPLETO % inteazioni t-otoe, t-otoe, t-t, t-t decitte taite % viluo in eie di Fouie function ABF fcn(i,i,fi_in,fi_in,deltath) %Izeo(,); %fi_inzeo(,); %tzeo(,); %equenze di coente t; t; %definizione cotanti Pi ; uzeo.56; %nueo di cave ottee dalla bobina nac ; %nueo di fai 5; %nueo di cave/olo/fae q 6; %nueo di conduttoi in cava n ; %coie olai STATORE (ifeite al cao PRINCIPALE) N_ ; %coie olai ROTORE M_ ; %angolo decitto dal agnete (ad.ecc.) alfa_ag 7; alfa_ag alfa_ag * Pi / 8; %eoe del agnete L.; %eoe del tafeo g.; %eeabilità elativa agnete u_.45; %fattoe di Cate c ; %induzione eidua agnete Be.5; %ofondità di acchina L.8; %aggio edio al tafeo Rg.595; %angolo di cava (ad.el.) alfa_c ; alfa_c alfa_c * Pi / 8; %aio odine aonico indagato _ax 3; _ax _ax; _ax _ax; %angolo elettico coenti (ulazione - g.el.) wt ; wt wt * Pi / 8; %*************************************************************************************** %INPUT DATI (end) %*************************************************************************************** %eoe tafeo coleivo deltag L g; %eiaiezza di una bobina (ad.el.) gaa nac * alfa_c / ; %induzione al tafeo geneata dal agnete BgM L / (L u_ * c * g) * Be; 8

229 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine %ao olae (al tafeo) Tau Pi * Rg / N_; ctzeo(_ax,); otzeo(_ax,); Kdzeo(_ax,); Bzeo(_ax,); Bzeo(_ax,); Fq_hzeo(_ax,); Fd_hzeo(_ax,); Fq_hnzeo(_ax,); Fd_hnzeo(_ax,); Fnq_hzeo(_ax,); Fnd_hzeo(_ax,); Fnq_hnzeo(_ax,); Fnd_hnzeo(_ax,); fi_fzeo(_ax,); fi_fnzeo(_ax,); Fq_hzeo(_ax,); Fd_hzeo(_ax,); Fq_hnzeo(_ax,); Fd_hnzeo(_ax,); Fnq_hzeo(_ax,); Fnd_hzeo(_ax,); Fnq_hnzeo(_ax,); Fnd_hnzeo(_ax,); fi_fzeo(_ax,); fi_fnzeo(_ax,); Fq_zeo(_ax,); Fd_zeo(_ax,); Fq_zeo(_ax,); Fd_zeo(_ax,); Toquezeo(_ax,); Toquezeo(_ax,); Fqzeo(,); Fdzeo(,); %*************************************************************************************** % SEQUENZA DI CORRENTE ST fo i : _ax ct(i) ; ot(i) ' '; Kd(i) ; B(i) ; Fq_h(i) ; Fd_h(i) ; Fq_hn(i) ; Fd_hn(i) ; fi_f(i) ; fi_fn(i) ; end fo : _ax if ( t) / int3(( t) / ) ct() ; ot() 'I'; eleif ( - t) / int3(( - t) / ) ct() ; ot() 'D'; ele ct() ; ot() ' '; end 9

230 Chate 5 if ct() ct() %fattoe di ditibuzione Kd() in(q * * alfa_c / ) / q / in( * alfa_c / ); %aiezza della -eia aonica di cao B() uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd() * in( * gaa); %valoe h oitivo h ( * N_) / M_; %valoe h negativo hn (- * N_) / M_; %CASO h INTERO % (ovaonibile con hn INTERO) if h int3(h) && h / ~ int3(h / ) B_h 4 / h / Pi * BgM * in(h * M_ * alfa_ag / ); if ct() Fnq_h() Pi * L * Rg / / uzeo * B() * B_h * in(h * M_ * deltath - wt - fi_in); Fq_h() Fnq_h(); % Ftq_h() - Fttq_h(); Fnd_h() Pi * L * Rg / / uzeo * B() * B_h * co(h * M_ * deltath - wt - fi_in); Fd_h() Fnd_h(); % Ftd_h(); eleif ct() Fnq_h() Pi * L * Rg / / uzeo * B() * B_h * in(h * M_ * deltath wt fi_in); Fq_h() Fnq_h(); % Ftq_h() - Fttq_h(); Fnd_h() Pi * L * Rg / / uzeo * B() * B_h * co(h * M_ * deltath wt fi_in); Fd_h() Fnd_h(); % Ftd_h(); end end %CASO hn INTERO % (ovaonibile con h INTERO) if hn int3(hn) && hn / ~ int3(hn / ) B_hn 4 / hn / Pi * BgM * in(hn * M_ * alfa_ag / ); if ct() Fnq_hn() -Pi * L * Rg / / uzeo * B() * B_hn * in(hn * M_ * deltath - wt - fi_in); Fq_hn() Fnq_hn(); % Ftq_hn() - Fttq_hn(); Fnd_hn() Pi * L * Rg / / uzeo * B() * B_hn * co(hn * M_ * deltath - wt - fi_in); Fd_hn() Fnd_hn(); % Ftd_hn(); eleif ct() Fnq_hn() -Pi * L * Rg / / uzeo * B() * B_hn * in(hn * M_ * deltath wt fi_in); Fq_hn() Fnq_hn(); % Ftq_hn() - Fttq_hn(); Fnd_hn() Pi * L * Rg / / uzeo * B() * B_hn * co(hn * M_ * deltath wt fi_in); Fd_hn() Fnd_hn(); % Ftd_hn(); end end ele Fq_h() ; Fd_h() ; Fq_hn() ; Fd_hn() ; end end %(next ) %RISULTANTE fo i : _ax Fq Fq Fq_h(i); Fq Fq Fq_hn(i); Fd Fd Fd_h(i); Fd Fd Fd_hn(i); end %*************************************************************************************** % SEQUENZA DI CORRENTE ST

231 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine fo i : _ax ct(i) ; ot(i) ' '; Kd(i) ; B(i) ; Fq_h(i) ; Fd_h(i) ; Fq_hn(i) ; Fd_hn(i) ; fi_f(i) ; fi_fn(i) ; end fo : _ax if ( t) / int3(( t) / ) ct() ; ot() 'I'; eleif ( - t) / int3(( - t) / ) ct() ; ot() 'D'; ele ct() ; ot() ' '; end if ct() ct() %fattoe di ditibuzione Kd() in(q * * alfa_c / ) / q / in( * alfa_c / ); %aiezza della -eia aonica di cao B() uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd() * in( * gaa); %valoe h oitivo h ( * N_) / M_; %valoe h negativo hn (- * N_) / M_; %CASO h INTERO % (ovaonibile con hn INTERO) if h int3(h) && h / ~ int3(h / ) B_h 4 / h / Pi * BgM * in(h * M_ * alfa_ag / ); if ct() Fnq_h() Pi * L * Rg / / uzeo * B() * B_h * in(h * M_ * deltath - wt - fi_in); Fq_h() Fnq_h(); % Ftq_h() - Fttq_h(); Fnd_h() Pi * L * Rg / / uzeo * B() * B_h * co(h * M_ * deltath - wt - fi_in); Fd_h() Fnd_h(); % Ftd_h(); eleif ct() Fnq_h() Pi * L * Rg / / uzeo * B() * B_h * in(h * M_ * deltath wt fi_in); Fq_h() Fnq_h(); % Ftq_h() - Fttq_h(); Fnd_h() Pi * L * Rg / / uzeo * B() * B_h * co(h * M_ * deltath wt fi_in); Fd_h() Fnd_h(); % Ftd_h(); end end %CASO hn INTERO % (ovaonibile con h INTERO) if hn int3(hn) && hn / ~ int3(hn / ) B_hn 4 / hn / Pi * BgM * in(hn * M_ * alfa_ag / ); if ct() Fnq_hn() -Pi * L * Rg / / uzeo * B() * B_hn * in(hn * M_ * deltath - wt - fi_in); Fq_hn() Fnq_hn(); % Ftq_hn() - Fttq_hn(); Fnd_hn() Pi * L * Rg / / uzeo * B() * B_hn * co(hn * M_ * deltath - wt - fi_in); Fd_hn() Fnd_hn(); % Ftd_hn();

232 Chate 5 eleif ct() Fnq_hn() -Pi * L * Rg / / uzeo * B() * B_hn * in(hn * M_ * deltath wt fi_in); Fq_hn() Fnq_hn(); % Ftq_hn() - Fttq_hn(); Fnd_hn() Pi * L * Rg / / uzeo * B() * B_hn * co(hn * M_ * deltath wt fi_in); Fd_hn() Fnd_hn(); % Ftd_hn(); end end ele Fq_h() ; Fd_h() ; Fq_hn() ; Fd_hn() ; end end % Next %RISULTANTE fo i : _ax Fq Fq Fq_h(i); Fq Fq Fq_hn(i); Fd Fd Fd_h(i); Fd Fd Fd_hn(i); end %*************************************************************************************** %INTERAZIONE fa i due cai STATORICI di equenza ST, ST fo : _ax %cao INVERSO if ( t) / int3(( t) / ) %aiezza della -eia aonica di cao Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); ( * N_) / N_; n (- * N_) / N_; if ( t) / int3(( t) / ) Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); Fq_() Pi * L * Rg / / uzeo * B_ * B_ * in((wt fi_in) - (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_ * co((wt fi_in) - (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if ( - t) / int3(( - t) / ) Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); Fq_() Pi * L * Rg / / uzeo * B_ * B_ * in((wt fi_in) (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_ * co((wt fi_in) (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if (n t) / int3((n t) / ) Kd_n in(q * n * alfa_c / ) / q / in(n * alfa_c / ); B_n uzeo * / * 4 / n / Pi * n * qt() * I / / deltag * q * Kd_n * in(n * gaa); Fq_() -Pi * L * Rg / / uzeo * B_ * B_n * in((wt fi_in) - (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_n * co((wt fi_in) - (wt fi_in)); Fq Fq Fq_();

233 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Fd Fd Fd_(); end if (n - t) / int3((n - t) / ) Kd_n in(q * n * alfa_c / ) / q / in(n * alfa_c / ); B_n uzeo * / * 4 / n / Pi * n * qt() * I / / deltag * q * Kd_n * in(n * gaa); Fq_() -Pi * L * Rg / / uzeo * B_ * B_n * in((wt fi_in) (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_n * co((wt fi_in) (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end end %cao DIRETTO if ( - t) / int3(( - t) / ) %aiezza della -eia aonica di cao Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); ( * N_) / N_; n (- * N_) / N_; if ( t) / int3(( t) / ) Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); Fq_() Pi * L * Rg / / uzeo * B_ * B_ * in(-(wt fi_in) - (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_ * co(-(wt fi_in) - (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if ( - t) / int3(( - t) / ) Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); Fq_() Pi * L * Rg / / uzeo * B_ * B_ * in(-(wt fi_in) (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_ * co(-(wt fi_in) (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if (n t) / int3((n t) / ) Kd_n in(q * n * alfa_c / ) / q / in(n * alfa_c / ); B_n uzeo * / * 4 / n / Pi * n * qt() * I / / deltag * q * Kd_n * in(n * gaa); Fq_() -Pi * L * Rg / / uzeo * B_ * B_n * in(-(wt fi_in) - (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_n * co(-(wt fi_in) - (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if (n - t) / int3((n - t) / ) Kd_n in(q * n * alfa_c / ) / q / in(n * alfa_c / ); B_n uzeo * / * 4 / n / Pi * n * qt() * I / / deltag * q * Kd_n * in(n * gaa); Fq_() -Pi * L * Rg / / uzeo * B_ * B_n * in(-(wt fi_in) (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_n * co(-(wt fi_in) (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end end end %Next 3

234 Chate 5 %*************************************************************************************** %INTERAZIONE fa divei odini aonici della equenza ST fo : _ax %cao INVERSO if ( t) / int3(( t) / ) %aiezza della -eia aonica di cao Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); ( * N_) / N_; n (- * N_) / N_; if ( t) / int3(( t) / ) Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); Fq_() Pi * L * Rg / / uzeo * B_ * B_ * in((wt fi_in) - (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_ * co((wt fi_in) - (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if ( - t) / int3(( - t) / ) Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); Fq_() Pi * L * Rg / / uzeo * B_ * B_ * in((wt fi_in) (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_ * co((wt fi_in) (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if (n t) / int3((n t) / ) Kd_n in(q * n * alfa_c / ) / q / in(n * alfa_c / ); B_n uzeo * / * 4 / n / Pi * n * qt() * I / / deltag * q * Kd_n * in(n * gaa); Fq_() -Pi * L * Rg / / uzeo * B_ * B_n * in((wt fi_in) - (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_n * co((wt fi_in) - (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if (n - t) / int3((n - t) / ) Kd_n in(q * n * alfa_c / ) / q / in(n * alfa_c / ); B_n uzeo * / * 4 / n / Pi * n * qt() * I / / deltag * q * Kd_n * in(n * gaa); Fq_() -Pi * L * Rg / / uzeo * B_ * B_n * in((wt fi_in) (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_n * co((wt fi_in) (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end end %cao DIRETTO if ( - t) / int3(( - t) / ) %aiezza della -eia aonica di cao Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); ( * N_) / N_; n (- * N_) / N_; if ( t) / int3(( t) / ) Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); 4

235 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); Fq_() Pi * L * Rg / / uzeo * B_ * B_ * in(-(wt fi_in) - (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_ * co(-(wt fi_in) - (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if ( - t) / int3(( - t) / ) Kd_ in(q * * alfa_c / ) / q / in( * alfa_c / ); B_ uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd_ * in( * gaa); Fq_() Pi * L * Rg / / uzeo * B_ * B_ * in(-(wt fi_in) (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_ * co(-(wt fi_in) (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if (n t) / int3((n t) / ) Kd_n in(q * n * alfa_c / ) / q / in(n * alfa_c / ); B_n uzeo * / * 4 / n / Pi * n * qt() * I / / deltag * q * Kd_n * in(n * gaa); Fq_() -Pi * L * Rg / / uzeo * B_ * B_n * in(-(wt fi_in) - (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_n * co(-(wt fi_in) - (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end if (n - t) / int3((n - t) / ) Kd_n in(q * n * alfa_c / ) / q / in(n * alfa_c / ); B_n uzeo * / * 4 / n / Pi * n * qt() * I / / deltag * q * Kd_n * in(n * gaa); Fq_() -Pi * L * Rg / / uzeo * B_ * B_n * in(-(wt fi_in) (wt fi_in)); Fd_() Pi * L * Rg / / uzeo * B_ * B_n * co(-(wt fi_in) (wt fi_in)); Fq Fq Fq_(); Fd Fd Fd_(); end end end %Next %CALCOLO RISULTANTI %odulo della foza adiale F qt(fd ^ Fq ^ ); %angolo iuato ietto all'ae y (FEMM) fi_f atan(fq, Fd); %*************************************************************************************** % equenza ST: CALCOLO COPPIA Toque() fo : _ax if ( t) / int3(( t) / ) ct() ; ot() 'I'; eleif ( - t) / int3(( - t) / ) ct() ; ot() 'D'; ele ct() ; ot() ' '; end 5

236 Chate 5 if ct() ct() %fattoe di ditibuzione Kd() in(q * * alfa_c / ) / q / in( * alfa_c / ); %aiezza della -eia aonica di cao B() uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd() * in( * gaa); if / ~ int3( / ) B() 4 / / Pi * BgM * in( * M_ * alfa_ag / ); if ct() Toque() / uzeo * deltag * L * N_ ^ * Tau * B() * B() * in( * M_ * deltath wt fi_in); eleif ct() Toque() / uzeo * deltag * L * N_ ^ * Tau * B() * B() * in( * M_ * deltath - wt - fi_in); ele Toque() ; end ele B() ; Toque() ; end ele Kd() ; B() ; end end %CALCOLO RISULTANTE Toqueu ; fo : _ax Toqueu Toqueu Toque(); end % equenza ST: CALCOLO COPPIA Toque() fo : _ax if ( t) / int3(( t) / ) ct() ; ot() 'I'; eleif ( - t) / int3(( - t) / ) ct() ; ot() 'D'; ele ct() ; ot() ' '; end if ct() ct() %fattoe di ditibuzione Kd() in(q * * alfa_c / ) / q / in( * alfa_c / ); %aiezza della -eia aonica di cao B() uzeo * / * 4 / / Pi * n * qt() * I / / deltag * q * Kd() * in( * gaa); if / ~ int3( / ) B() 4 / / Pi * BgM * in( * M_ * alfa_ag / ); if ct() Toque() / uzeo * deltag * L * N_ ^ * Tau * B() * B() * in( * M_ * deltath wt fi_in); eleif ct() Toque() / uzeo * deltag * L * N_ ^ * Tau * B() * B() * in( * M_ * deltath - wt - fi_in); ele Toque() ; end ele B() ; Toque() ; 6

237 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine end ele Kd() ; B() ; end end %(fo : _ax) %CALCOLO RISULTANTE Toqueu ; fo : _ax Toqueu Toqueu Toque(); end ABF [Fq,Fd,F,fi_F,Toqueu,Toqueu,I,fi_In,I,fi_In,deltath]; Note that the vaiable Fq, Fd, etituted a outut aguent of the Matlab function, coeond to the eached Fy, Fz coonent of the adial foce. E) Foce To Moent Matix Bloc Thi bloc ily ovide to calculate the oent given by the eultant adial foce in the abolute efeence yte, by ean of the contibution of it coonent (5.4)-(5.4): M Fy F y L h in ϕî F y L h co ϕco ψˆ (5.4) M Fz F z L h co ϕ in ψî F z L h co ϕco ψĵ (5.4) Fig

238 Chate 5 The elationhi wee ileented in SIMULINK by decibing evey ingle te, a hown in Fig Eule Equation Bloc A) Alied Moent Bloc In the Alied Moent bloc, the oent of the eultant adial foce ae ecalculated fo the ab to the ft efeence yte. The iotant featue in thi bloc, i the ABSOLUTE to FTR function which ovide to ealize the above action; othe bloc ae ued fo auxiliay function, a the ignal coe. Fig. 5.4 The aay MFyz eeent the oent aleady calculated in the Levitation Foce bloc, which ae added in uation to the oent oduced by the weight foce, given in (5.4) with eect to the ab efeence yte. M g L h î co ϕ co ψ L h ĵ ˆ Lh co ϕ in ψ in ϕ g (5.4) Lh g ( co ϕ in ψî co ϕ co ψĵ) 8

239 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Conide the a of the oto and the haft, the cente of gavity located on the axi with a fixed oint at the oigin O of the yte at a ditance L h / fo the latte. The total oent i then exeed, by ean of the ( θ, ϕ, ψ) B atix (5.3), with eect to the ft efeence yte and aed to the next bloc. B) Eule Equation Bloc Fig. 5.5 Thi at of the yte ileent the Eule equation, een in (5.) a eaanged in (5.33): the outut of the bloc i contituted by the angula eed vecto with eect to the ft efeence yte (Fig. 5.5). The vaiable aed with a caital lette which coae in Fig. 5.5 coeond to the oent of inetia een in (5.33). 9

240 Chate The etting of PID contolle To et u the PID contolle, fitly the yte wa analyzed by diabling the Foce Contolle and Electoagnetic odel bloc, becaue of the high colexity of thee Matlab function which intoduce the Fouie haonic eie ditibution of the vaiou agnetic field: in fact, conideing the whole haonic contibution, it would be vey difficult to define the coeonding tanfe function. On the contay, in thi way the PID contolle outut give diectly the F y, F z coonent of the eultant foce. Obviouly, thi eeent only an inteediate te to oduce oe oviional value of the PID coefficient K i, K and K d, being not the oiginal yte, but it eitted to obtain the actual value in an eaie way. In fact, afte having calibated the value of the PID coefficient by ean of thi ilified analyi, the two function wee eintoduced to et u and ecalculate the coefficient in the actual, oiginal configuation. The analytical aoach to the oble wa foulated by exeing the foce, eultant of the inteaction between agnetic field by ean of it y- and z- coonent, a the diect eult of the PID contolle egulation, thu a a cobination of ootional, integal and deivative action (5.43), (5.5). Alo, the oto i conideed ubject to the weight foce, obviouly acting along the z- axi: A) Equilibiu along the y-axi: f y () t K Δy() t K Δy( τ) [ Δy( t) ] t d i dτ K d (5.43) dt Whee Δy eeent the eo with eect to y coodinate, a defined in (5.45). Alying the Lalace tanfoation, it give: 3

241 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine F Y K d (5.44) i () K ΔY () ΔY () K [ ΔY () Δy() ] y Δ y() t y() t y( ) ΔY () Y ( ) (5.45) By ubtituting (5.45) in (5.44) it give: F Y () K Y () ( ) ( ) y K i K i K Y () y( ) K Y ( ) K y( ) d d (5.46) By alying the econd law of otion along the y axi: [ ] d y & y () t f y () t L L f y () t (5.47) dt By ubtituting the eective L-tanfo in the equation (5.47) it give: [ Y () y() y& () ] y () ( ) K Y K Conideing that ( ) y( ) K i Y K i () y( ) K Y ( ) K y( ) y &, by ubtituting into (5.48) it give: d d (5.48) Y K i () K Y () Y () K Y () d (5.49) By collecting the coon te in (5.49), it give the tivial olution (5.5): () Y (5.5) Siilaly, oceed to witing the equation along the z- axi: B) Equilibiu along the z-axi: f z () t K Δz() t K Δz( τ) [ Δz( t) ] t d i dτ K d (5.5) dt 3

242 Chate 5 3 () () () () () [ ] z Z K Z K Z K F d i z Δ Δ Δ Δ (5.5) () () ( ) () ( ) ( ) z Z Z z t z t z Δ Δ (5.53) () () ( ) () ( ) ( ) ( ) z K Z K z K Z K z K Z K F d d i i z (5.54) Thi tie the equation include the weight foce, a aid above: () () () [ ] g t f L dt z d L g t f t z z z & & (5.55) ( ) ( ) ( ) [ ] () ( ) () ( ) ( ) ( ) g z K Z K z K Z K z K Z K z z Z d d i i & (5.56) () () () () g Z K Z K Z K Z d i (5.57) By collecting the coon te in (5.57), it give (5.58), (5.59): ()[ ] g K K K Z i d 3 (5.58) () K K K g K K K g Z i d i d 3 3 (5.59) The L-tanfo of the z coodinate (5.59) decibe the height of the haft ending oint with eect to the cente of the oto. By tudying the tability of the equation (5.59) i oible to obtain the ode of agnitude of the PID coefficient. It i not the exact olution becaue, a entioned befoe, a ilified yte configuation i exained. To find the coelation between the thee ole and the coefficient of the olynoial equation in, given by utting the denoinato of (5.59) equal to zeo, a geneic exeion of a thid degee

243 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine 33 olynoial i witten: ( )( )( ) ( ) ( ) (5.6) By equating the coefficient of the olynoial (5.6) to thoe of the denoinato in the equation (5.59), it give: K K K i d (5.6) To olve the oble, the eaiet way i to chooe only one ole of ultilicity equal to thee and of negative value if eal o, if colex, having a eal at of negative value (5.6), to aue the tability of the yte: { } < R R < C e if if 3 (5.6) Finally, by ubtituting (5.6) in (5.6) and develoing the equation, it oible to obtain the elationhi between the PID coefficient and the ole; chooing uitably it value, ae deteined K i, K and K d : K K K K K K i i d d (5.63) In the next, i choen a a eal nube: the geneal citeion i to vay the value of until the axiu excuion of the oto haft ending oint fall within the

244 Chate 5 deied toleance, which can be eaonably fixed in one-tenth of the alitude of the aiga, o le if neceay. 34

245 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine 5.6 Siulation and eult The oftwae SIMULINK wa ued in ode to colete the iulation; the analyzed achine i chaacteized by the following aaete (Tab. I): TABLE I. DATA OF THE MACHINE Paa. Decition Value N l nube of lot 3 ole ai of the achine nube of hae 5 I n ated hae cuent (A ) 59.8 T n ated toque (N) 3.9 g aiga width () D e tato oute diaete () 3 D tato inne diaete () D ean diaete of the agnet () 6 D cv_ext diaete at the botto of the lot () 7 D cv_int diaete at the to of the lot () 6.3 D oto oute diaete () 4 D i oto inne diaete () 6 α cv lot itch angle a dt tato lot height () 5 h cl lot oening height () L axial length of the achine () 8 L h total length of the haft () 3 D h haft diaete () 4 L agnet width () L dt tooth-body width () 8 L cl lot oening width () L tc lot width at the to lot adiu () 5.3 L fc lot width at the botto lot adiu () 9.7 τ cv lot itch at the inne tato adiu ().57 oto and haft a (g) 6.75 I I oent of inetia, I axi (g ).7 I ε oent of inetia, ε axi (g ) 5.6 I η oent of inetia, η axi (g )

246 Chate 5 Uing the ethod decibed in the eviou ection, a lit of value fo K i, K and K d i obtained, hown in Tab. II. By alying thee value in the iulation, it can be een that by inceaing the abolute value of, the axiu excuion of the oto haft ending oint i ogeively educed. TABLE II. VALUES OF PID COEFFICIENTS K d K K i.5e3.e4.34e5 4.E3 8.4E4.7E6 8 4.E3 3.6E E6 In the following, will be analyzed the iulation eult fo -8 by eeenting in the Fig. 5.7, 5.8, the oition of the haft ending oint and of the axi oint coeonding to the oto tac length in the y, z coodinate of the abolute efeence yte (Fig. 5.). It i iotant to note that, with efeence to the contained exteity of the haft, the oto tac length extend u to 5 and it coeonding axi oint oition eeent the aaete to be veified. The haft ending oint extend u to 3, thu it excuion will be obviouly geate than the latte (Fig. 5.6). Shaft ending oint Roto tac length axi oint Fig

247 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine The iulation wee efoed at a oto angula eed ω I 4.8 ad/, coeonding to and the toque cuent ha the ated value of 59.8 A. Fig. 5.7 Fig

248 Chate 5 A exlained above, all the oible inteaction between agnetic field acting in the aiga ae conideed, by ean of the Electoagnetic odel. Thu, the analyzed ituation can be conideed a a colete and ealitic oeating condition of the beaingle achine. In addition, the locu occuied by the ae oint on the y-z lane, eeented in Fig. 5.9, ovide a cleae eeentation of the oto axi oition. It i alo inteeting to obeve the behavio of the toque and levitation cuent ace vecto, i v and i v and hae, Fig :, in te of value Fig

249 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Fig. 5. Fig

250 Chate 5 The cuent ace vecto ha an ioed contant value (59.8) to geneate continuouly the equeted toque, it hae which vaie eaining in leading by 9 electical degee with eect to agnet axi. The odulu of cuent ace vecto, afte having eached a axiu value of alot 4.5 A in the ealy intant of tie, ocillate between 3 and 3.3 A, while the hae continuouly change it value in the whole ange ( to 36 electical degee), having to follow the atial hae of the equied foce neceay to countebalance the weight and the othe foce geneated by the inteaction between haonic ode of the agnetic field. To give a oe ealitic idea, the eultant foce vecto acting on the oto, in the tie inteval fo to, change it oition on the yz lane a hown in Fig. 5.. Fig. 5. A can be een by coaing the Fig , the axiu value of the excuion of the oto tac length axi oint i about tenth of illiete in the 4

251 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine negative z-axi diection and.5 tenth of illiete in the negative y-axi diection, ove the efixed value of tenth, which eeent tenth of the aiga width. So, it i neceay to et a diffeent calibation of the PID contolle. The fit attet, baed on the value of Tab. II coeonding to -8, i ade by vaying the coefficient K d, K, K i and veifying the eult. Poceeding in thi way thee i no oe coelation with geneal citeion (5.63), but lowe abolute value fo aaete could be found with, conequently, an eaie way to actically ealize the contolle. A good cooie i found by acting only on K, ultilying by thee it value in Tab. II. Thu, the value of Tab. III wee ued and the iulation eult ae hown in the following: TABLE III. VALUES OF PID COEFFICIENTS K d K K i 4.E E E6 Fig

252 Chate 5 Fig. 5.4 Fig

253 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Fig. 5.6 Fig

254 Chate 5 The odulu of cuent ace vecto ha an ocillating behavio with a ogeive eduction tending to the egie, tanding in the ange of 3 to 3.4 A, a can be noted in Fig. 5.7 and, clealy, in Fig. 5.8 whee the ae function i eeented with an extended tie axi. Fig. 5.8 A can be een by coaing the Fig , now the axiu excuion of the oto tac length axi oint i about tenth of illiete in the negative z-axi diection and about.75 tenth of illiete in the negative y-axi diection. Thu, it i oible to ay that the taget ha been achieved. A done befoe, in Fig. 5.9 i hown the ucceion of the diffeent oition occuied by the eultant foce vecto on the yz lane in the tie inteval fo to. In the Fig. 5.3, 5.3 the tie axi ha been caled u to the value of econd with eect to Fig , to highlight the table tate achieved by the yte. 44

255 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Fig. 5.9 Fig

256 Chate 5 Fig. 5.3 The econd way to oceed i by continuing to ue the geneal citeion (5.63), which would obably eit a finet egulation of the PID coefficient. Finally, the value in Tab. IV wee found with -: TABLE IV. VALUES OF PID COEFFICIENTS K d K K i 6.3E3 7.36E5.8944E7 The eult of the iulation ae hown in the following, fo Fig. 5.3 to Fig

257 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Fig. 5.3 Fig

258 Chate 5 Fig Fig

259 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Fig Fig

260 Chate 5 A can be een by coaing the obtained eult with the eviou, a enible eduction of the ovehoot and a fate attainent of egie condition ae achieved, even if the abolute value of the PID coefficient ae in geneal geate than in the eviou attet. It can be noted that the yte behave a if in the intant intant t the oto would be efectly centeed and the oto i off; in the t the weight foce and the othe foce, oduced by the inteaction between tato and oto agnetic field, begin thei action on the oto, with the contol yte tying to bing it in the equeted oition. In the Fig. 5.38, 5.39 i hown the analyzed 5-hae beaingle oto. Fig

261 Deign and develoent of a contol yte fo ultihae ynchonou PM beaingle achine Fig

262 Chate Concluion In thi chate a contol yte fo beaingle ultihae ynchonou PM achine i eented, integated by a thee-dienional echanical odel baed on the Eule equation. The electoagnetic odel of the achine, een in Chate 4, tae into account all the oible inteaction between haonic ode of the agnetic field oduced by the cuent ace vecto i v, which give the toque, oduced by the cuent ace vecto i v, which give the levitation foce, and oduced by oto agnet. Diffeently fo othe autho, which ooe odel that tae into account only the ain haonic ode inteaction between agnetic field, the develoed yte i a colete one, giving in thi way a oe accuate odeling of the echanical and electoagnetic henoena. Fo thee eaon, it contitute an iotant tool fo the deign of a beaingle ultihae ynchonou PM achine contol yte and eeent the deign of the exeiental device with elated contol yte to ealize in a tet bench alication. 5

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265 Concluion Concluion The ain coe of thi Ph.D thei i contituted by the non linea analyi and deign of beaingle ultihae achine and dive. The thei wo began with the develoent of a ethod to analyze the ditibution of the agnetic vecto otential, agnetic field and flux denity in the aiga of a eanent agnet electical achine by alying a twodienional odel. The oiginal contibution of the aoach, inied by a liteatue ae, conited in the colete calculation to get the olution of the oble, conducted by uing the technique of atheatical analyi alied to hyical and engineeing oble. Thi odel i chaacteized by a linea analyi. The eviou containt of agnetic lineaity i ovecoe by the econd chate of thi Thei, whee an algoith fo the non-linea agnetic analyi of ultihae uface-ounted PM achine with ei-cloed lot ha been eented. Peviou ae ooed the analyi of oen-lot configuation with a efixed tuctue of the oto, with a given nube of ole and lot, o by tudying only a aticula oition of the oto with eect to the tato. In thi wo, the PM achine i eeented by uing a odula tuctue geoety. The baic eleent of the geoety i dulicated allowing to build u and analyze whateve tyology of winding and aee-tun ditibution in a ai of ole. In the thid chate the thee of the beaingle achine ha been intoduced, analyzing and decibing the ain concet and idea develoed in the liteatue. The fouth chate eent an analytical odel fo adial foce 55

266 Concluion calculation in ultihae beaingle Suface-Mounted Peanent Magnet Synchonou Moto (SPMSM). The odel allow to edict alitude and diection of the foce, deending on the value of the toque cuent, of the levitation cuent and of the oto oition. It i baed on the ace vecto ethod, letting the analyi of the achine not only in teady-tate condition but alo duing tanient. The calculation ae conducted by develoing the analytical function in Fouie eie, taing all the oible inteaction between tato and oto f haonic coonent into account. The ooed ethod allowed to ehaize the effect of electical and geoetical quantitie lie the coil itch, the width and length of the agnet, the oto oition, the alitude and hae of cuent ace vecto, etc. In the lat chate a thee-dienional echanical odel odel of a beaingle ultihae ynchonou PM achine ha been analized. The echanical odel i baed on the Eule equation, while the electoagnetic odel of the achine, develoed in the eviou chate, tae into account all the oible inteaction between haonic ode of the agnetic field oduced by the cuent ace vecto ainly eonible fo the toque, and by the cuent ace vecto injected fo oducing levitation foce. In the contol odel, ileentd in MATLAB SIMULINK, the eo in the oto oition ae ued in ode to calculate the coonent of the adial foce neceay to contol the oto axi oition of the achine. The efoance of the ooed non linea odel of SPMSM have been coaed with thoe obtained by FEA oftwae in te of linage fluxe, coenegy, toque and adial foce. The obtained eult fo a taditional thee-hae achine and fo a 5-hae achine with unconventional winding ditibution howed that the value of local and global quantitie ae actically coinciding, fo value of the tato cuent u to ated value. In addition, they ae vey iila alo in the non-linea behavio even if vey lage cuent value ae 56

267 Concluion injected. The elevant edge of the ethod conit in the oibility of defining the achine chaacteitic in a ile ue inteface. Then, by dulicating an eleentay cell, it i oible to contuct and analyze whateve tyology of winding and aee-tun ditibution in a ole-ai. Futheoe, it i oible to odify the agnet width-to-ole itch atio analyzing vaiou configuation in ode to iniize the cogging toque, o iulating the oto oveent in inuoidal ultihae dive o in a ue-defined cuent ditibution. When develoing a new achine deign the ooed ethod i ueful not only fo the eduction of couting tie, but ainly fo the ilicity of changing the value of the deign vaiable, being the nueical inut of the oble obtained by changing oe citical aaete, without the need fo edeigning the odel in a CAD inteface. Fo a given oto oition and fo given tato cuent, the outut toque a well a the adial foce acting on the oving at of a ultihae achine can be calculated. The latte featue ae the algoith aticulaly uitable in ode to deign and analyze beaingle achine. Fo thee eaon, it contitute a ueful tool fo the deign of a beaingle ultihae ynchonou PM achine contol yte. With efeence to the contol yte fo beaingle achine the eented odel allow to calculate the adial foce avoiding the eo intoduced by the ue of only the baic f haonic coonent. In fact, when deigning a contol yte fo beaingle achine, any autho conideed only the inteaction between the ain haonic ode of the tato and oto f. In ultihae achine thi can oduce itae in deteining both the odule and the atial hae of the adial foce, due to the inteaction between the highe haonic ode. In addition, the ooed algoith eit to tudy whateve configuation of SPMSM achine, being aaeteized a a function of the electical and geoetical quantitie, lie the coil itch, the width and length of the agnet, the oto oition, the alitude and hae of cuent 57

268 Concluion ace vecto, etc. Finally, the eult of the ooed ethod have been coaed with thoe of a ot ued FEA oftwae, obtaining vey iila value of the analyzed quantitie. In concluion, thi thei ai to be a colete efeence fo the deign ethodologie of ultihae beaingle achine and dive, in the linea and non-linea field of alication. 58

269 Lit of ae Lit of ae Lit of ae by Stefano Sei [] S. Sei, A. Tani, G. Sea, A Method fo Non-linea Analyi and Calculation of Toque and Radial Foce in Peanent Magnet Multihae Beaingle Moto, Poc. of SPEEDAM, Intenational Syoiu on Powe Electonic, Electical Dive, Autoation and Motion, Soento, Italy - June,, IEEE Cat. No. CFP48A-CDR, ISBN , [] S. Sei, A. Tani, G. Sea, Analytical Model of Radial Foce Conideing Mutual Effect Between Toque and Levitation Cuent Sace Vecto in 5-hae PM Beaingle Moto, acceted fo eentation in IECON 3, 39th Annual Confeence of the IEEE Indutial Electonic Society, Wien, Autia, -3 Nov., 3. 59

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