Dynamics and Control of Electrical Drives

Transcription

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2 Dnac and Contol of Electcal Dve

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4 Pot Wach Dnac and Contol of Electcal Dve ABC

5 Autho Pof. Pot Wach Poltechna Opola Inttute of Electoechancal Ste and Appled Infoatc Mołajcza 5 t Opole Poland E-al: p.wach@po.opole.pl ISBN e-isbn DOI.7/ ba of Conge Contol Nube: 9564 c Spnge-Velag Beln Hedelbeg Th wo ubject to copght. All ght ae eeved, whethe the whole o pat of the ateal concened, pecfcall the ght of tanlaton, epntng, eue of llutaton, ectaton, boadcatng, epoducton on cofl o n an othe wa, and toage n data ban. Duplcaton of th publcaton o pat theeof petted onl unde the povon of the Gean Copght aw of Septebe 9, 965, n t cuent veon, and peon fo ue ut alwa be obtaned fo Spnge. Volaton ae lable to poecuton unde the Gean Copght aw. The ue of geneal decptve nae, egteed nae, tadea, etc. n th publcaton doe not pl, even n the abence of a pecfc tateent, that uch nae ae eept fo the elevant potectve law and egulaton and theefoe fee fo geneal ue. Tpeet & Cove Degn: Scentfc Publhng Sevce Pvt. td., Chenna, Inda. Pnted on acd-fee pape pnge.co

6 To dea wfe Iena

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8 Acnowledgent The autho would le to epe wa gattude to all who contbuted n vaou wa o that th boo fnall ha been copleted n t fo. Ftl, I would le to epe eebance and pa tbute to Pof. Aaduz Puchała Acade of Mnng and Metallug n Cacow, who wa pncpal tuto and upevo of Ph.D. the. Howeve, ou contact tenated eal, when he ded qute oung n 97. Then I would le to than wal colleague fo the Inttute of Electoechancal Ste and Appled Infoatc, who encouaged e and ceated favoable condton and nce, fendl atophee to wo and eeach: Pof. Ktna Mace-Kańa Decto of the Inttute, Pof. Maan Łuanzn ou peent Facult Dean a well a Pof. Jez Hcewcz and Pof. Sławo Szanec patne n eveal eeach undetang. Then I would le to than D. Kzztof Toczew, D. Rzad Bena, D. Andzej Wtow, D. Kzztof Wóbel foe Ph.D tudent, wo wth who gave e a lot of epeence, echange of dea and ecellent oppotunt to dcu. Fnall I would le to than dea on Szon Wach fo h ve good I a ue, tanlaton of the boo nto Englh and M Eugenuz Głowenow fo pepaaton of thoe techncal dawng that wee not poduced autoatcall b MAPE, a an outcoe of copute ulaton. Pot Wach

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10 Notaton Inde a π / - phae hft between -phae etcal ne cuve a v& & - acceleaton vecto δ A, δa, δa e - vtual wo, t echancal and electcal coponent A - vecto potental of a agnetc feld A,A - ew etc atce: - and - dentonal epectvel B - agnetc nducton vecto C - electcal capact D - vcou dapng facto e - electootve foce EMF nduced n -th wndng E - total eneg of a te f K - analtcal notaton of holonoc contant functon F, F - vecto of etenal foce, -th coponent of th vecto f, f, f - fequenc of voltage cuent: feedng lne, tato, oto g - acceleaton vecto of eath gavtaton foce g - nube of banche of electc netwo h - nube of holonoc contant Q& - electc cuent a a devatve of electcal chage, f a - ectaton cuent, aatue cuent - bolc value of nuodal cuent I I - at of neta of a gd bod T [ ] - vecto of a -phae tato cuent T [ ] - vecto of a -phae tato cuent n a ta connected te T [ ] - vecto of a -phae oto cuent n a ta connected te T [ K ] - vecto of a -phae oto cuent, - vecto of tanfoed tato, oto cuent n,u, v uv uv ae uv, - vecto of tanfoed tato, oto cuent to u, v uv ae

11 X Notaton Inde, - vecto of tanfoed tato and oto cuent to αβ αβ α, β ae, - vecto of tanfoed tato and oto cuent to d, q dq dq ae, - vecto of tanfoed tato and oto cuent to, ae [ ] - vecto of tato cuent tanfoed to the feld ρ ρ ρ oented, ae ρ J, J - oent of neta, oent of neta of a oto oto j - cuent dent vecto J - Jacob at of a tanfoaton - tffne coeffcent of an elatc eleent, - agnetc couplng coeffcent fo tato and oto wndng - pule wdth facto n PWM contol ethod u,, - agange functon, t echancal and electcal e coponent σ, σ - leaage nductance of tato and oto wndng epectvel,, - elf-nductance of tato and oto wndng, agnetzng nductance - angula oentu of a bod - nube of ba phae n a quel-cage oto wndng, a f - odulaton coeffcent: of an apltude and fequenc epectvel, M - a of -th patcle and total a of a bod M - oent of foce M, M - an feld nductance coeffcent of tato, oto phae wndng M, M - nductance atce fo a tato and oto phae ph ph wndng te M, M - atce of utual nductance between tato and oto phae wndng ph ph M, M, M - atce of utual nductance between tato and uv uv uv oto phae wndng tanfoed to,u, v ae N, N - nube of lot n a tato and oto of electcal achne nh - nube of nonholonoc contant n a te p - nube of pole pa n electcal achne p - vecto of oentu of echancal te P - electc powe P - genealzed foce actng along the -th coodnate ρ

12 Notaton Inde XI Q - electc chage of the -th eleent chaactezed b electcal capact Q & - electc cuent n -th wndng a a devatve of epectve chage Q, Q & - vecto of electcal chage of a te, vecto of electc cuent q - nube of pule of powe electonc convete q, q&, q& - -th genealzed coodnate, - veloct, - acceleaton epectvel q, q&,& q& - vecto of genealzed coodnate, velocte, acceleaton of a te δ q - vecto of vtual dplaceent fo genealzed coodnate δ q - vtual dplaceent fo -th genealzed coodnate R, R - phae wndng etance fo tato and oto epectvel, - adu-vecto pontng -th patcle, adu-vecto fo whole te n Catean coodnate δ - vecto of vtual dplaceent of a te n Catean coodnate R - vecto of eacton foce of contant n a te - lp of an nducton oto oto oton n epect to agnetc feld - total nube of degee of feedo, - nube of echancal, electcal degee of feedo e of a te S - acton functon of a te T, T el, T e - netc eneg, t electcal and echancal coponent T, T el, T e - netc co-eneg, t electcal and echancal coponent T e, T - electoagnetc toque of a oto, load toque l T, T, b t T - bea toque, tatng toque, ated toque of an nducton oto n T - peod of a ngle pulaton equence n PWM contol p ethod - fcton foce T f T, T - othogonal atce of tanfoaton fo tato and oto vaable u, - electc voltage, voltage uppled to -th wndng u u, u - tato and oto voltage epectvel U, - ated voltage, tato ated voltage U n n, U, U, u, u U - phae to phae voltage n -phae electcal te u - tato phae to phae voltage n -phae te

13 XII Notaton Inde U - phae voltage of tato wndng ph - feedng lne voltage U U d U - aveage value of ectfed voltage n a 6 pule -phae π te U - bolc value of nuodal voltage U, U, U - potental eneg of a te, t echancal and electcal coponent e el T U U [ u, u, u ] - tato voltage vecto, tato phae voltage vecto, ph T U, Uph [ u, u, K, u ] - vecto of oto voltage, vecto of phae voltage of oto wndng u - vecto of tato voltage tanfoed to,u, v te uv of ae u uv - vecto of tato voltage tanfoed to u, v te of ae u - vecto of tato voltage tanfoed to, te of ae u - vecto of tato voltage tanfoed to, feld ρ ρ ρ oented ae v & - vecto of velocte of a te w - nube of node of an electc netwo X - eactance of a wndng X, X, X - eactance of a tato, oto and agnetzng one epectvel Χ χ, Kχ n - vecto of coodnate n a pa coodnate te Z, - nube of tato, oto teeth of SRM achne Z α, - angle detenng a poton of wndng α l α on, α off - wtch on and wtch off contol angle of SRM achne γ - phae hft angle δξ - vtual dplaceent of -th Catean coodnate n unfed coodnate te Ξ ξ, Kξ n - vecto of Catean coodnate n unfed te of coodnate ξ η - eneg effcenc facto of a te - otaton angle θ & θ Ω - otatonal peed of a oto ρ - nube of a agnetc feld haonc ρ - feld oentaton angle of ρ, ρ ae vecto contol σ - leaage coeffcent of wndng φ - cala potental of electoagnetc feld

14 Notaton Inde XIII K - analtcal notaton of nonholonoc contant j - feld angle DTC T ψ - flu lnage of -th wngng ψ - ectaton flu f T Ψ [ ψ, ψ, ψ ] - flu lnage vecto of tato wndng ph Ψ - flu lnage vecto of oto wndng ph Ψ uv, Ψ - u, v tanfoed flu lnage vecto of tato and oto wndng uv Ψ uv, Ψ -,u, v tanfoed flu lnage vecto of tato and uv oto wndng Ψ, Ψ - d, q tanfoed flu lnage vecto of tato and dq dq oto wndng Ψ, Ψ -, tanfoed flu lnage vecto of tato and oto wndng Ψ αβ, Ψ - α, β tanfoed flu lnage vecto of tato and αβ oto wndng ω πf - AC uppl lne pulaton ω, ω - pulaton of tato, oto nuodal voltage, cuent ω - efeence pulaton angula peed n tanfoed c,u,v te ω e pω - electcal angula peed of a oto ω f Ω ω / p - nchonou peed of otatng agnetc feld, dle un peed ω - vecto of angula veloct of gd bod Ω ω / p - deal dle un peed of an nducton oto oto Ω - angula peed of a oto

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16 Content Content Intoducton... Refeence...6 Dnac of Electoechancal Ste...9. Mechancal Ste Bac Concept Contant, Clafcaton of Contant and Effect of The Ipoton..... Eaple of Contant Etenal Foce and Reacton Foce; d Alebet Pncple Etenal Foce and Reacton Foce Vtual Dplaceent Pefect Contant d Alebet Pncple Nube of Degee of Feedo and Genealzed Coodnate agange Equaton Potental Mechancal Eneg Genealzed Foce, Echange of Eneg wth Envonent Eaple of Applcaton of agange Equaton Method fo Cople Ste wth Patcle..... Moton of a Rgd Bod Fundaental Noton Moton of a Ma Pont on a Rgd Bod Knetc Eneg of a Rgd Bod Moton of a Rgd Bod aound an A Rgd Bod Ibalance n Moton aound a Contant A Eaple of Applng agange Equaton fo Moton of Rgd Bode Geneal Popete of agange Equaton aw of Conevaton Chaactetc of agange Functon and Equaton Electoechancal Ste Pncple of eat Acton: Nonlnea Ste...59

17 XVI Content... Electcall Unchaged Patcle n Relatvtc Mechanc Electcall Chaged Patcle n Electoagnetc Feld agange Equaton fo Electoechancal Ste n the Noton of Vaance Electc Vaable Vtual Wo of an Electcal Netwo Co-eneg and Knetc Eneg n Magnetc Feld Convete Cae of Sngle Nonlnea Inducto The Cae of a Ste of Inducto wth Magnetc Feld nage Potental Eneg n Electc Feld Convete Magnetc and Electc Te of agange Functon: Electoechancal Couplng Eaple of Applcaton of agange Equaton wth Regad to Electoechancal Ste...8 Refeence...6 Inducton Machne n Electc Dve...9. Matheatcal Model of Inducton Machne Intoducton Contucton and Tpe of Inducton Moto..... Fundaental of Matheatcal Modelng Tpe of Model of Inducton Machne Nube of Degee of Feedo n an Inducton Moto Matheatcal Model of an Inducton Moto wth nea Chaactetc of Coe Magnetzaton Coeffcent of Wndng Inductance Model wth nea Chaactetc of Magnetzaton n Natual phae Coodnate Tanfoaton of Co-odnate Ste Tanfoed Model of Inducton Moto wth nea Chaactetc of Coe Magnetzaton Model n Cuent Coodnate Model n Med Coodnate Model n Flu Coodnate Specal Cae of Selectng Aal Ste u,v Matheatcal Model of Inducton Moto wth Untanfoed Vaable of the Stato/Roto Wndng Model wth Untanfoed Vaable n the Electc Ccut of the Stato Model wth Untanfoed Vaable of Electc Ccut of the Roto...4

18 Content XVII. Dnac and Statc Chaactetc of Inducton Machne Dve Standadzed Equaton of Moton fo Inducton Moto Dve Tpcal Dnac State of an Inducton Machne Dve Eaple of Tajectoe of Moton Stat-Up dung Dect Connecton to Netwo Reconnecton of an Inducton Moto Dve Reveal Cclc oad of an Inducton Moto Soft-Stat of an Inducton Moto fo Non-Sultaneou Connecton of Stato Wndng to the Netwo DC Bang of an Inducton Moto Reducton of a Matheatcal Model to an Equvalent Ccut Daga Statc Chaactetc of an Inducton Moto Method and Devce fo Fong Chaactetc of an Inducton Moto Contol of Suppl Voltage Slp Contol Addtonal Retance n the Roto Ccut Schebu Dve Suppl Fequenc f Contol Dect Fequenc Convete Ccloconvete Two-evel Voltage Souce Invete Inducton Moto Suppled fo -evel Voltage Invete Thee-evel Dode Neutal Pont Claped VSI Invete Cuent Souce Invete wth Pule Wdth Modulaton PWM....4 Contol of Inducton Machne Dve Vecto Contol Matheatcal Model of Vecto Contol Realzaton of the Model of Vecto Contol Foalzed Model of Vecto Contol Dect Toque Contol DTC Decpton of the Method Eaple of Dect Toque Contol DTC on the Ba of a Matheatcal Model Obeve n an Inducton Machne Roto Flu Obeve n Coodnate α, β Roto Feld Obeve n Ψ, ρ Coodnate Roto Feld Obeve n, Coodnate wth Speed Meaueent...68

19 XVIII Content.4..4 Obeve of Inducton Moto Speed Baed on the Meaueent of Roto Poton Angle Flu, Toque and oad Toque Obeve n, Coodnate Stato Flu Obeve Ψ wth Gven Rate of Eo Dapng...7 Refeence Buhle DC Moto Dve BDC Intoducton Peanent Magnet Bac Decpton n the Matheatcal Model Matheatcal Model of BDC Machne wth Peanent Magnet Tanfoed Model Tpe d-q Untanfoed Model of BDC Machne wth Electonc Coutaton Electonc Coutaton of BDC Moto Suppl Voltage of BDC Moto n Tanfoed Model u, u....4 q d 4... Modelng of Coutaton n an Untanfoed Model of BDC Chaactetc of BDC Machne Dve Stat-Up and Reveal of a Dve Dve Stat-Up Reveng DC Moto Chaactetc of BDC Machne Dve Contol of Rotatonal Speed n BDC Moto Contol of BDC Moto Dve Contol Ung PID Regulato Contol wth a Gven Speed Pofle Contol fo a Gven Poton Pofle Foal neazaton of BDC Moto Dve Regulaton of BDC Moto wth Invee Dnac...69 Refeence Swtched Reluctance Moto Dve Intoducton Opeatng Pncple and Suppl Ste of SRM Moto Magnetzaton Chaactetc and Toque Poducng n SRM Moto Matheatcal Model of SRM Moto Foundaton and Aupton of the Matheatcal Model Equaton of Moton fo the Moto Functon of Wndng Inductance Dnac Chaactetc of SRM Dve Eepla Moto fo Sulaton and Tet...4

20 Content XIX 5.5. Statng of SRM Dve Stat-Up Contol fo Swtched Reluctance Moto b Pule Sequence Cuent Deltaton dung Dect Moto Statng Bang and Geneatng b SRM Chaactetc of SRM Machne Contol Sgnal and Tpcal Stead-State Chaactetc Effcenc and Toque Rpple evel of SRM Shape of Cuent Wave of SRS Contol of SRM Dve Vaable Stuctue Sldng Mode Contol of SRM Cuent Contol of SRM Dve Dect Toque Contol DTC fo SRM Dve Seno- and Senole Contol of SRM Dve State Obeve Applcaton fo Senole Contol of SRM Refeence Inde...449

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22 Chapte Intoducton Intoducton Abtact. Ft Chapte an ntoducto one and t geneall peent the cope of th boo, ethodolog ued and dentfe potental eade. It alo develop nteelaton between oden electc dve, powe electonc, echatonc and applcaton of contol ethod a the boo to oe degee cove all thee feld. The content concentated aound electoechancal eneg conveon baed on agange ethod and t clea and ubequent applcaton to contol of electc dve wth nducton achne, buhle DC oto and SRM achne. It doe not cove teppe oto and nchonou PM dve. All copute ulaton eult ae outcoe of ognal atheatcal odel and baed on the coputaton caed out wth ue of MAPE atheatcal pacage. Electcal dve fo a contnuoul developng banch of cence and technolog, whch date bac fo d-9th centu and pla an nceangl potant ole n ndut and coon eveda applcaton. Th o becaue eve da we have to do wth dozen of houehold applance, offce and tanpotaton equpent, all of whch contan electcal dve alo nown a actuato. In the ae anne, ndut and tanpot to a lage etent el on the applcaton of electcal dve fo the pupoe of effectve and pece opeaton. The electcal dve have taen ove and tll tae on a lage hae of the phcal effot that wee pevoul undetaen b huan a well a pefo the tpe of wo that wa ve needed but could not be pefoed due to phcal o othe ltaton. Th potant ole taen on b the electcal dve contnuoul epandng and the ta pefoed b the dve ae becong oe and oe ophtcated and veatle [,,,,]. Electcal dve tend to eplace othe devce and ean of dong phcal wo a a eult of the nueou advantage. Thee nclude a coon acceblt of electcal uppl, eneg effcenc and poveent n te of the contol devce, whch ecue the eental qualt of wo and ft n the ecologcal equeent that appl to all new technolog n a oden ocet []. The up-to-date electcal dve ha becoe oe and oe ntellgent, whch ean fulfllent of the nceangl cople equeent egadng the hapng of the tajectoe of oton, elable opeaton n cae of ntefeence and n the ntance of defcenc o lac of eaueent nfoaton aocated wth the eecuted contol ta [4,8,6,]. Th alo ean that a lage nube of coponent ae nvolved n nfoaton gatheng and poceng, whoe ole to

23 Intoducton enue the pope opeaton, dagnotc and potecton of the dve. The eaon fo uch ntenve developent of electcal dve ae nueou and the bac eaon fo that aocated wth the need of ntellgent, effectve, elable and undtubed eecuton of echancal wo. Such dve ae degned n a ve wde ange of electcal capact a electoechancal devce wth the powe atng fo [W] to [MW]. Addtonall, the pocee ae accopaned b ultpaaete oton contol and the pa ole can be attbuted to peed egulaton along wth the equed level of foce o toque poduced b the dve. The peent capablte of fulfllng uch cople equeent otl eult fo the developent of two technologcal banche, whch ade huge poge at the end of the th centu and the contnuou followng developent. One of uch aea nvolve the banch of technologcal ateal ued fo the poducton of electcal achne and evooto. What eant hee the poge n the technolog of anufactung and acceblt of nepenve peanent agnet, n patcula the one contanng ae eath eleent uch a aau S and neodu Nd. In addton, poge n te of nulaton ateal, the evce lve and all loe fo hgh fequence of electc feld tength, whch eult fo contol nvolvng the wtchng of the uppl voltage. Moeove, condeable poge ha occued n te of the popete of feoagnetc ateal, whch ae contantl ndpenable fo electoechancal conveon of eneg. The othe banch of technolog whch ha enabled uch condeable and quc developent of electcal dve the poge ade n coelectonc and powe electonc. A a eult of the developent of new ntegated ccut coelectonc ha ade t poble to gathe huge aount of nfoaton n a cofotable and nepenve wa, accopaned b t fat poceng, whch n tun offe the applcaton of cople ethod of dve contol. Moeove, up-to-date powe electonc aet new cuent flow wtche that allow the contol ove lage electc powe wth hgh fequenc thu enablng the te to eecute cople contol ta. Th occu wth ve all loe of eneg aocated wth wtchng, hence plang a decve ole n the applcablt of uch devce fo hgh wtchng fequence. The veatlt and wde ange of voltage and cuent opeatng n the up-to-date econducto wtche ae t poble to develop electc powe convete able to adapt the output of the ouce to fulfll the paaete eultng fo ntantaneou equeent of the dve [5,7,]. Aong othe th capablt ha led to the etenve applcaton of ldng ode contol n electcal dve whch ve often nvolve apd wtchng of the contol gnal n ode to follow the gven tajecto of the dve oton []. Such etenve and effectve poblt of the developent of electcal dve, whch eult fo the advanceent n electonc and a apd nceae n the applcaton ange of the actuatng devce, ha gven e to the aea of echatonc. Mechatonc can ethe be thought of a a epaate centfc dcplne o a elevant and oden dvon of the electcal dve patculal elatng to electonc, contol and lage equeent wth egad to the dnac paaete of the dve [9]. It alo poble to dcu th dtncton n te of the nube of degee of feedo of the devce appled fo the poceng of nfoaton followed b

24 Intoducton electc powe conveon nto echancal wo. In a tadtonal electcal dve we have to do wth a nube of degee of feedo fo the electc tate vaable and a ngle one fo the echancal oton. In tandad electcal achne the vaable nclude: angle of the otaton of the oto o tanlatonal oton n a lnea oto. In echatonc t aued that the nube of the degee of echancal feedo can be lage,.e. fo the echancal vewpont the devce eecute a oe cople capablte than pl a otaton o tanlatonal oton. In addton, n echatonc t not necea that the edu evng fo the conveon of eneg the agnetc feld. Mechatonc devce a opeate unde electc feld, whch the cae n electotatc convete [4,5]. In concluon, t can be tated that echatonc ha etended the et of tadtonal devce n the goup of electcal dve to cove a wde choce of the a well a ha uppleented eta ethod and cope of eeach. Howeve, t can be tated beond doubt that echatonc conttute nowada a epaate centfc dcplne whch ealze t a n an ntedcplna anne whle applng equall the fndng of copute engneeng, electonc and electoechanc n ode to ceate a ult-denonal tajecto of the echancal oton. Such undetandng of echatonc bng u cloe to anothe oe geneal centfc dcplne a obotc. What potant to note that f a anpulato o a obot ha electcal jont dve, n t electoechancal natue t conttute a echatonc devce. Concuentl, obotc ha even oe to t [8]. Not to ente the defnton and tadton n th dcplne, what geneall eant the autonoou natue of the obot n te of t capablt of ecognton of t envonent and cope of decon ang,.e. the applcaton of atfcal ntellgence. B loong at a anpulato o a obot poduced n accodance wth up-to-date technolog we tat to ealze t capablte wth egad to t oentaton n pace and oganzaton of the poed contol ta. Howeve, one hould alo gve et to t peed, pecon, epeatablt and elablt of opeaton, all of whch elate to echatonc. The efeence to obotc n a boo devoted to electcal dve eult fo the fact that n t pat devoted to theo and n the peented eaple a efeence ade to the ethod and oluton ognatng fo obotc, the focu n whch ha often been on the oton n a ult denonal echancal te wth contant []. The followng paagaph wll be devoted to the peentaton of the ovevew of the cuent boo, whch contan 4 chapte bede the ntoducton devoted to the ue of the up-to-date electcal dve and the contol. Chapte cove the ue aocated wth the dnac of echancal and electoechancal te. The ubject of the ubequent ecton n th chapte focu on echancal te wth a nube of degee of feedo a well a holonoc and non-holonoc contant. The peented concept cove a phcal te whch educed to a et of ateal pont and a te defned a a et of gd bode. A detaled ethod of the developent of a atheatcal odel ntoduced nvolvng agange functon and equaton depatng fo the pncple of leat acton fo a chaged ateal patcle n the agnetc feld. Subequentl, th concept ha been etended to cove acocopc te capable

25 4 Intoducton of accuulaton of eneg of the agnetc and electc feld n the fo of netc and potental eneg, epectvel. The dpaton of eneg and tanfoaton of the dpaton coeffcent nto te of the tate vaable ae taen down to the negatve te of the vtual wo of the te. Th a wa that foall coect fo the pont of ethodolog of eeach. Addtonall, t pove effectve n the pactce of the foulaton of the equaton of oton. The ecton devoted to electoechancal te ha been llutated b nueou eaple whoe dffcult level nteedate. In geneal, the eaple of the applcaton of theo n the boo ae qute nueou and have been elected n a anne that hould not poe eceve dffcult whle antanng the at a level that can eve fo the pupoe of llutatng pecfc chaactetc of the appled ethod but ae neve elected to be tval. Chapte focue on nducton achne dve. The peented atheatcal odel have been developed wth the ad of agange ethod fo electoechancal te. The odel tanfoed nto othogonal ae ae peented n a clacal anne along wth the odel of an nducton achne fo whch the vaable on one de,.e. tato o oto ae untanfoed and ean n the natual phae coodnate. Th pla an potant ole n the dve te uppled fo powe electonc convete. The adequate and oe detaled odelng of the convete te eque natual vaable of the tate,.e. untanfoed one n ode to oe pecel ealze the contol of the dve. Thee odel,.e. odel wthout the tanfoaton of the vaable on one de of the nducton oto ae peented n the applcaton n the futhe ecton n th boo. The peentaton focue on vaou apect of the uppl, egulaton and contol wth the applcaton of convete. The clacal ubject atte nclude peentaton of DC bang, Schebu dve, a well a the opeaton of a oft-tate. Concuentl, the up-to-date ue aocated wth nducton achne dve cove two - level and thee-level Voltage Souce Invete VSI, Snuodal Pule Wdth Modulaton SPWM, Space Vecto Modulaton SVM, Dcontnuou Space Vecto Modulaton DSVM and PWM Cuent Souce Invete CSI contol. Futhe on, bede VC ethod the Dect Toque Contol DTC peented n theo and n eaple. The fnal ecton of Chapte devoted to the peentaton of tuctual lneazaton of a odel of nducton oto dve along wth eveal tate obeve applcable fo the nducton oto. Chapte 4 devoted to peanent agnet buhle DC oto dve and contol of uch dve. Ftl, the chaactetc and popete of the up-to-date peanent agnet PM ae peented togethe wth plfed ethod appled fo the odelng. The eaple of a pendulu col wngng ove a tatona PM eve fo the pupoe of peentng the effect of plfcaton n the odel of the agnet on the tajectoe of the oton of uch an electoagnetc te. Futhe on, the tanfoed d-q odel of a BDC achne deved along wth an untanfoed odel n whch the coutaton occu n accodance wth the coue of the natual vaable of the achne. The peentaton of the atheatcal odel of BDC doe not cove the ubject of nonholonoc contant n th tpe of achne. In a clacal DC achne wth a echancal coutato

26 Intoducton 5 the occuence of uch contant nvolvng the dependence of the confguaton of electcal ccut on the angle of the otaton of the oto qute evdent. Concuentl, n an electoncall couted achne the uppl of the patcula wndng of the aatue tll elatve to the angle of the otaton of the oto; howeve, the ntoducton of nonholonoc contant n the decpton no longe necea. Coutaton occu wth the peevaton of the fed tuctue of electcal ccut of the electonc coutato coupled wth the phae wndng of the achne aatue and the wtchng of the cuent eult fo the change of the paaete of the pedance of the econducto wtch n the functon of the angle of otaton. Th chapte focue on the chaactetc and dnac coue llutatng the opeaton of BDC oto wth a copaon between the eult of odelng dve wth the ad of two-aal d-q tanfoaton a well a untanfoed odel. On th ba t poble to elect a odel fo the ulaton of the ue aocated wth the dve dependng on the denon of the whole te and the level of the detal of the output fo odelng. The peented tatc chaactetc and dnac coue of BDC dve focu on adequate chaactezng the capablte and opeatng paaete of uch dve wthout contol. Subequentl, eeach focue on the contol of BDC dve and the peentaton of the contol ung PID egulato, contol wth the gven peed pofle and the gven pofle of the poton a well a nvee dnac contol. The llutaton n the fo of dnac coue ae etenve and conducted fo two dffeent tandad BDC oto. The fnal chapte,.e. Chapte 5 devoted to the peentaton of wtched eluctance oto SRM dve. Befoe the developent of the atheatcal odel, agnetzaton chaactetc of SRM oto ae peented and the potant ole of non-lneat of chaactetc n the conveon of eneg b the eluctance oto eaed. Subequentl, the peentaton follow wth the atheatcal odel accountng fo the agnetc atuaton eflected b agnetzaton chaactetc wth egad to the utual poton of the tato and oto teeth. Th pefoed n a wa that ognal nce the nductance chaactetc that ae elatve to two vaable ae peented hee n the fo of a poduct of the functon of the agnetc atuaton and the functon of the oto poton angle. Such an appoach ha a nube of advantage nce t enable one to analze the effect of patcula paaete on the opeaton of the oto. The deved odel of the SRM oto doe not account fo agnetc couplng between phae wndng; howeve, fo the eaple of two tandad SRM oto t wa poble to ndcate a lttle effect of uch couplng on the chaactetc and opeaton of the oto. In th anne uch plfcaton ncluded n the atheatcal odel ae jutfed. The futhe ecton of th chapte focu on a nube of ue egadng the dnac and contol of SRM dve. The peentaton nclude a oluton to the poble of the pule baed detenaton of the tatng equence dung tatng SRM dve fo the elected decton of the otaton of the oto, dect tat-up wth the ltaton of the cuent a well a bang and dcuon of the ue of ve pecfc geneato ege of opeaton. The peentaton alo cove the electon of the egulaton paaete fo SRM wth the a of ganng hgh eneg effcenc and educng toque pple level. The ecton devoted to the

27 Refeence Refeence 6 Intoducton contol nvolve the peentaton of ldng contol appled fo th dve tpe, cuent contol a well a DTC contol and the poblt of ltng the pulaton of the toque a a eult of applng pecfc contol ode. In addton, the peentaton befl cove eno and enole contol of SRM dve and the applcaton of tate obeve whle povdng fo the ecluon of the poton eno. Followng th bef ovevew of the content t valuable that the eade note that the boo contan a lage nube of eaple n the aea of dnac and contol of pecfc dve, whch eflected b wavefo llutatng the pecfc ue that ae peented n the fgue. All eaple a well a llutaton coe fo copute ulaton pefoed on the ba of atheatcal odel developed thoughout the boo. Th ha been pefoed fo tandad eaple of oto the detaled data and paaete of whch ae ncluded n the patcula ecton of the boo. Copute ulaton and gaphcal llutaton ganed on th ba wee pefoed n MAPE atheatcal pogang te, whch ha poved t patcula applcablt and fleblt n th tpe of odelng. All calculaton wee pefoed on the ba of poga ognall developed b the autho. A one can ee fo th hot ovevew, the cope of th boo lted and doe not nvolve oe tpe of dve ued n pactce,.e. teppe oto dve and nchonc achne dve wth peanent agnet. The ng tpe of dve have la chaactetc n te of the pncple of eneg conveon and atheatcal odel to SRM oto dve and BDC dve, epectvel. Howeve, the detal of contucton and opeaton ae dla and onl a lttle effot can enable one to appl the coepondng odel n th boo n ode to develop dedcated poga fo copute ulaton and eeach of the two ng dve tpe. A fnal ea concen the taget goup of th boo, whch n the autho opnon nclude tudent of potgaduate coue and Ph.D. tudent along wth engnee eponble fo the degn of electcal dve n oe cople ndutal te. Refeence [] Aleda, A.T., Feea, F.J., Both, D.: Techncal and Econocal Condeaton on the Applcaton of Vaable-Speed Dve wth Electc Moto Ste. IEEE Tan. Ind. Appl. 4, [] Boldea, I., Naa, A.: Electc Dve. CRC Pe, Boca Raton 999 [] Canuda de Wt, C., Sclano, B., Batn, G.: Theo of Robot Contol. Spnge, Beln 997 [4] Dawon, D.M., Hu, J., Bug, T.C.: Nonlnea Contol of Electc Machne. Macel Dee, New Yo 998 [5] El-Hawa, M.E.: Pncple of Electc Machne wth Powe Electonc Applcaton, nd edn. John Wle & Son Inc., New Yo [6] Ftzgeald, A.E., Kngle, C., Kuo, A.: Electc Machne. McGaw-Hll, New Yo 998

28 Refeence 7 [7] Hee, S.: Wnd Eneg Conveon Ste, nd edn. John Wle & Son, Chchete 6 [8] Id, A.: Nonlnea Contol Ste. Spnge, New Yo, Pat I-995, Pat II- 999 [9] Kootovc, P., Aca, M.: Contuctve nonlnea contol: a htocal pepectve. Autoatca 7, [] Kaue, P.C.: Anal of Electcal Machne. Mc Gaw Hll, New Yo 986 [] Kaue, P.C., Wanczu, O., Sudhoff, S.D.: Anal of Electc Machne and Dve Ste, nd edn. John Wle & Son Inc., New Yo [] Khnan, R.: Electc Moto Dve: Modelng, Anal and Contol. Pentce-Hall, Uppe Saddle Rve [] eonhad, W.: Contol of Electcal Dve, d edn. Spnge, Beln [4], G., Aluu, N.R.: agangan appoach fo electotatc anal of defoable con-ducto. IEEE J. Mcoelectoech. S., 45 5 [5], Y., Hoowtz, R.: Mechatonc of electotatc actuato fo copute dc dve dual-tage evo te. IEEE Tan. Mechat. 6, [6] Macenoth, U.: Robut Contol Ste. Spnge, Hedelbeg 4 [7] Mohan, N.: Advanced Electc Dve. NMPERE, Mnneapol [8] Scavcco,., Sclano, B.: Modelng and Contol of Robot Manpulato. Spnge, ondon [9] de Slva, C.W.: Mechatonc: An Integatve Appoach. CRC Pe, Boca Raton 4 [] Sleon, G.R.: Electc Machne and Dve. Addon-Wele, Readng 99 [] Slotne, J.J.,, W.: Appled Nonlnea Contol. Pentce Hall, New Jee 99 [] Utn, V.I.: Sldng Mode n Contol and Optzaton. Spnge, Beln 99 [] Zhu, Z., Howe, D.: Electcal Machne and Dve fo Electc, Hbd and Fuel Cell Vehcle. Poc IEEE 95,

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30 Chapte Dnac of Electoechancal Ste Abtact. Chapte devoted to dnac of echancal and electoechancal te. Secton dealng wth echancal te concen holonoc and nonholonoc object wth ultple degee of feedo. The concept of an object epeented b a te of connected ateal pont and the concept of a gd bod and connected bode ae peented. The agange ethod of dnac foulaton thooughl coveed, tatng fo d Alebet vtual wo pncple. Caefull elected eaple ae ued to llutate the ethod a well a the applcaton of the theo. Electoechancal te theo alo ntoduced on the ba of the agange equaton ethod, but tatng fo the pncple of leat acton fo a electcall chaged patcle n a tatona electoagnetc feld. Subequentl, the ethod genealzed fo acocopc te whoe opeaton baed on electc eneg and agnetc co-eneg conveon. Nonlnea te ae dcued and the concept of netc co-eneg eplaned. Eneg dpaton ntoduced a a negatve te of the vtual wo of the te, and tanfoaton of dpaton coeffcent to the te of genealzed coodnate ae peented n accodance wth agange ethod. Fnall a nube of eaple peented concenng electoechancal te wth agnetc and electc feld and alo elected obotc tuctue.. Mechancal Ste.. Bac Concept Dcete te - a te whoe poton defned b a countable nube of vaable. In oppoton to dcete te a contnuou te o dtbuted paaete te defned a a te wth contnuoul changng vaable along coodnate n pace. Both thee concept ae a nd of dealzaton of eal ateal te. Patcle an dealzed object that chaactezed onl b one paaete a. To defne t poton n a thee-denonal pace D thee vaable ae necea. Th dealzaton acceptable fo an object whoe a focue cloel aound the cente of the a. In that cae t netc eneg elatve the to lnea tanlatonal oton tongl donant ove the netc eneg of the otatonal oton. Bede, t poble to conde lage bode a patcle

31 Dnac of Electoechancal Ste ateal pont n pecfc ccutance, fo eaple when the do not otate o the otaton doe not pla gnfcant ole n a gven condeaton. Th the cae n the eanaton of nueou atonocal poble of oveent of ta and planet. Rgd bod a ateal object, fo whch one hould tae nto account not onl the total a M, but alo the dtbuton of that a n pace. It potant fo the otatng object whle the netc eneg of uch oveent pla an potant ole n a gven dnacal poble. In an dealzed wa a gd bod can be condeed a a et of patcle, of whch each ha an pecfc a, uch that Σ M. Foall, the bod gd f the dtance d j between patcle,j ae contant. The phcal paaete that chaacteze a gd bod fo the echancal pont of vew ae the total a M, and a etcal at of the denon, called the at of neta. Th at account fo oent of neta on t dagonal and devaton oent, whch chaacteze the dtbuton of ae wthn the gd bod n a Catean coodnate te. Contant ae phcal ltaton on the oton of a te, whch etct the feedo of the oton of that te. The te te ued hee denote a patcle, et of connected patcle, a gd bod o connected bode a well a othe echancal tuctue. Thee ltaton defned a contant ae dvee: the can etct the poton of a te, the veloct of a te a well a the nd of oton. The can be contant, te dependent o pecfc onl wthn a lted ub-pace. Foall, the contant hould be defned n an analtcal fo to enable the ue n atheatcal odel and copute ulaton of oton. Hence, the ae denoted n the algebac fo a equaton o nequalte. Catean coodnate te - the bac, coonl ued coodnate te, whch n a thee denonal pace D ntoduce thee pependcula taght ae. On thee ae t poble to eaue the actual poton of a gven patcle n an unabguou wa ung thee eal nube. Poton of the patcle P n that te gven b a thee denonal vecto, o called adu-vecto :,, z. It coodnate ae, epectvel:,, z ; ee Fg... In cople echancal te, contng of a nube of patcle:,,,n the genealzed poton vecto fo the whole te defned a follow:,,, N,, z,,, z,, N, N, z N whch placed n an abtact N pace. Fo a oe convenent opeaton of th nd of notaton of the te poton, epecall n applcaton n vaou uaton foulae, a unfo Gee lette ξ j ntoduced: ξ, ξ, ξ A a eult of above, the poton vecto fo the whole te of patcle tae the followng fo:

32 . Mechancal Ste,,, Ξ ξ, ξ, ξ, ξ,, ξ N 4 N, ξ N, ξ N. Fg.. Catean coodnate of a patcle Veloct and acceleaton of a te The foal defnton of a veloct of a te gven below: t Δt t v l Δt. Δt Becaue te t a paaete of an oton, dffeentaton and calculaton of devatve n epect to te a fequent opeaton n dnac. Hence tadtonall, the te devatve befl noted b a dot above a vaable that dffeentated n epect to te, n the followng fo: d v.4 dt Acceleaton the te devatve of the veloct, whch ean t the econd devatve of the poton n epect to te: a d v dt.5 Accodng to the Newton Second aw of Dnac, whch decbe the elaton between oton and t caue,.e. the appled foce o toque, n the decpton of dnac thee no need o place fo hghe ode devatve of the poton of a bod than one of the econd ode.e. acceleaton. Th alo ean that n dnac one ha to do onl wth the poton, veloct and the acceleaton, te t a a paaete of oton, and foce toque a caue of oton.

33 Dnac of Electoechancal Ste.. Contant, Clafcaton of Contant and Effect of The Ipoton Contant ae phcal ltaton of oton that educe the feedo of oton of a gven te. The te denote hee a echancal unt uch a a patcle, a et of connected patcle, a gd bod o a et of connected gd bode. The ltaton poed b the contant ae vaou n natue o the a etct the feedo of poton, tpe of oton, a well a veloct; the act n a lted pace and even ae vaable n te. Fo foal pupoe, n ode to pefo analtcal decpton of oton the contant ae denoted a equaton o nequalte and a clafcaton of the contant ntoduced. The geneal fo of an analtcal notaton ued to peent contant actng n a te followng: f,, t R o f Ξ, Ξ, t R.6 whee: f - the analtcal fo of contant functon,,ξ - poton vecto of a te,,ξ - veloct vecto of a te, R - the elaton belongng to the et R {, <, >, } Stff o blateal contant v. eleang o unlateal contant. Th a clafcaton n epect to a elaton R. Stff contant ae epeed b the equalt elaton f,, f,, { <, >,, }.7 whle eleang contant ae one that contan the elaton of nequalt. Geoetc v. neatcal contant. Th clafcaton account fo the abence o peence of veloct n a elaton of contant. In cae that the veloct thee the contant ae called neatcal f,, R f, R.8 and wthout eplct peence of veloct the ae naed geoetc contant. Te dependng cleonoc v. te ndependent eonoc contant. Th a dvon that tae nto account the eplct peence of te n the elaton of contant: f, R f,, t R.9 In that epect the ft elaton of.9 peent cleonoc contant and the econd one eonoc contant. Holonoc v nonholonoc contant. It the bac clafcaton of contant tpe fo the theo of dnacal te pont of vew. The dvon of echancal te nto holonoc and nonholonoc te follow. Holonoc contant ae all geoetc contant and thoe neatcal contant that can be conveted nto geoetc contant b ntegaton.

34 . Mechancal Ste f, R - geoetc contant f,, R - thoe of neatcal contant, fo whch et:. F, - uch that: df f Nonholonoc contant ae all neatcal contant that could not be ntegated and hence cannot be conveted nto geoetc one. The foal dvon clea, but t oe dffcult to offe the phcal eplanaton fo th dtncton. Spl peang one can a that holonoc contant etct the poton of a te and the veloct of that te n a unfo anne, whle nonholonoc one poe etcton on the veloct wthout etctng the poton. Conequentl, one can a that nonholonoc contant etct the anne of oton wthout ltng the poton n whch uch oton can eult. Fo futhe applcaton, nonholonoc contant wll be denoted n the followng fo: ξ, ξ, ξ, ξ,, ξ, t j, b. j n n and n a pecfc cae of the lnea nonholonoc contant: n j h jξ D j j, b. - whee h j n the geneal cae ae functon of poton coodnate and te. To vef whethe t poble to ntegate lnea neatcal contant t uffcent to chec f the ae n the fo of the Pfaff dffeental equaton wth total dffeental... Eaple of Contant Eaple.. In a plana te Fg.. two teel ball ae connected b a tff od. One ha to defne the analtcal fo fo contant notaton and to defne the n accodance wth the peented clafcaton. Fg.. Ste of two ave ball contaned b a tff od

35 4 Dnac of Electoechancal Ste a fo ball connected wth a tff od wth the length l l hence: l. A a conequence, the eaned cae peent geoetc, tff, te ndependent contant. b fo ball connected wth a cod of length l whoe thcne neglgble the equaton. eplaced wth an nequalt n the fo: l.4 Hence the cae epeent eleang unlateal contant. It poble to dffeentate the equaton fo contant. wth epect to te, hence the followng fo obtaned:.5 Th epeent neatc contant eultng fo geoetc contant., whch can tae alo oe geneal fo: f ξ f ξ f ξ f 4ξ4 Fo the above equaton the followng condton fulflled: f ξ f ξ,,,,4.6 whch ean that the condton fo the total dffeental ae et. The equaton.5 tae the fo of the Pfaff dffeental equaton, whch qute elfevdent due to t ogn. A a eult, contant gven b.5 ae holonoc. Eaple.. A clacal eaple of nonholonoc contant can be llutated b the lple oton of a flat plate on a plane. The elaton between the coodnate n th cae ae peented n Fg... The decpton of the lple oton of a plate on a uface Π apple 6 coodnate:,,z, whch detene pont of tangenc of the plate and the plane and angle α,υ,φ whch defne: otatonal angle of the plate, nclnaton of the plate uface and angle of nteecton between the plate uface and the Catean coodnate te, epectvel. Th te lted b the followng contant: f f f : : : z Rdα n d Rdα co d.7 The equaton f fo the contant obvoul peent holonoc contant, whle the contant f, f can alo tae the followng fo: R α n R α co.8 whch epeent nonholonoc contant, nce angle φ conttute a coodnate of the te n oton and doe not fo a value that nput o a functon.

36 . Mechancal Ste 5 Fg.. Flat plate n lple oton on a plane Theefoe, the two equaton.8 cannot be ntegated epaatel wthout po etablhent of a oluton to the te of the equaton of oton. One can note that the contant equaton f enable one to elnate vaable z fo the te of equaton and; thu, t can be degaded n the vecto of te poton. At the ae te, contant equaton f and f do not poe a ltaton on the poton of the te, whle the contan the oton, o that t lple. One can futhe obeve that that the equaton fo nonholonoc contant.8 can be tanfoed to tae the fo: R α v.9 whch elnate angle φ fo contant equaton and denote veloct of the oton of the tangenc pont P ove a plane n whch a plate oll. Th equaton enable one to ntepet nonholonoc contant but doe not offe gound fo the elnaton. Equaton fo nonholonoc contant ae alo encounteed n electcal and electoechancal te n uch a fo that the electcal node n whch the banche of electcal ccut convege ovable and t poton elatve to a echancal vaable. Th tpe of nonholonoc contant encounteed e.g. n electcal pantogaph of al vehcle and echancal coutato n electcal achne nvolvng ldng contact...4 Etenal Foce and Reacton Foce; d Alebet Pncple..4. Etenal Foce and Reacton Foce Etenal foce ae foce toque actng upon the coponent of a te. In th fo the conttute the caue of oton n accodance wth the Newton econd law. Reacton foce of contant Fg..4 fo the ntenal foce actng along the appled contan and opeate o that the te peeve the tate whch eult fo the poed contant. Hence, eacton foce of contant do not conttute the caue of the oton but eult n the peevaton of the te n confot wth the contant. In deal ccutance the foce of contant eacton do not eet an wo aocated wth the oton of a te, whch appled n d Alebet pncple dcued late.

37 6 Dnac of Electoechancal Ste Fg..4 Equlbu between eacton foce of contant R,R eultng n contant dtance l between ball n oton..4. Vtual Dplaceent The ntoducton of the noton of vtual dplaceent,.e. one that ae copatble wth contant ndpenable n analtcal dnac due to the ole n elnaton of contant eacton foce occung n contan baed te [,,6]. The vecto of vtual dplaceent denoted analogcall to the contucton of the poton vecto. whee: δ δ, δ, δ,, δ. N δ δ, δ, δz δξ, δξ, δξ The vecto of vtual dplaceent contucted b the nceent of vaable whch fulfll the followng condton: - poe nfnteal value - ae copatble wth contant - the dplaceent occu wthn fed a oent of te Thee condton alo ean that vtual dplaceent ae alo efeed to a nfnteal dplaceent,.e. all tetng dplaceent whch occu contentl wth appled contant wthout accountng fo the duaton. A a eult, t poble to copae wo eeted b a te fo vaou vecto of vtual dplaceent. Vtual dplaceent do not neceal have to ovelap wth ecton of actual path of oton but need to be content wth potental path fo the neatc pepectve. Fo the tateent of contenc between vtual dplaceent and contant the followng elaton can be etablhed: f δ f whch upon eolvng nto Talo ee elatve to δ and oon of hghe powe δ, δ, lead to the tateent of the elaton between vtual dplaceent fo a gven j-th equaton of contant

38 . Mechancal Ste 7 N f j δ o f j δξ ξ N. The elaton. alo ean that an equaton fo holonoc contant, f j, j,,h enable one to fnd an epeon fo a patcula vtual dplaceent b ue of the eanng one δξ a j,δξ a j,δξ a j, δξ a j, δξ a j, nδξn. a j, whee:..4. Pefect Contant a j, f j ξ It onl poble to defne pefect contant n a te n whch fcton foce ae ethe ng o n the cae whee the nheent fcton foce can be condeed a etenal foce. Afte th peequte fulflled, t poble to defne pefect contant. Such contant atf the condton that total wo eeted on the vtual dplaceent equal to zeo: N R δ. An eaple of pefect contant nclude a gd connecton of ateal pont whch not ubjected to tenon o bendng. Htocall, the concept of pefect contant ognate fo d Alebet pncple and fo a potulate confed b nueou eaple d Alebet Pncple It conttute the ft analtcal tateent of the oton of a te n whch patcle ae contaned. In ode to elnate foce of contant eacton the pncple apple the noton of pefect contant. Fo a ateal pont patcle wth a the equaton of oton dectl eult fo Newton econd law of oton: F Fo a te wth N ateal pont lted b contant, the above equaton can be etated fo eve ateal pont to account fo the eultng foce of contant eacton R bede the etenal foce F F R N.4

39 8 Dnac of Electoechancal Ste The unnown contant eacton foce do not eld t poble to dectl appl equaton.4. Afte uaton of the equaton t poble to elnate contant eacton foce on the ba of the noton of the pefect contant. N R F δ whch gve: N F δ.5 Vtual dplaceent ae not epaate entte but ae elated to one anothe b equaton eultng fo the contant. Hence, d Alebet pncple epeed b the te of equaton: h j f N j N F δ δ.6 Th a et of dffeental - algebac equaton on the ba of whch t poble to obtan equaton of oton e.g. ung agange ndefnte ultple ethod. Th can be pefoed a follow: each of h algebac equaton n.6 ultpled b ndefnte facto λ j and ued up: h j N j j f δ λ.7 The epeon n.7 ubequentl ubtacted fo the equaton of oton, thu obtanng: h j j j N f F δ λ.8 Fo the eultng u of N paenthetcal epeon ultpled b ubequent vtual dplaceent δ, the followng pocedue followed: fo the ft h epeon n paenthee,,h the electon of ultple λ j hould be uch that the value of the epeon n paenthe equal to zeo. In conequence, fo the eanng paenthetcal epeon the vtual dplaceent δ, h,,n ae alead ndependent, hence, the paenthetcal epeon ut be equal to zeo. The fnal equaton of oton tae the fo: N h f h j j j F λ.9

40 . Mechancal Ste 9 h f The te theeof: j λ j conttute the eacton foce of nonholonoc j contant fo equaton.4. In a la anne t poble to etend d Alebet pncple to cove te lted b nonholonoc contant.-.. A a eult the followng epeon obtaned: h nh f j l F λ j l h N μ. j l whee: l - nonholonoc functon of contan tpe. μ l - ndefnte ultple fo nonholonoc contant d Alebet pncple lead to the tateent of a te of equaton wth contant; howeve, th pocedue te-conung and qute budenoe nce the obtaned fo of equaton ae etenve and cople due to the electon of coodnate of oton that fa fo optu. Th can be deontated n a ple peentaton. Eaple.. In a plana te peented n Fg..5 a et of two ball of a and ae connected b a tff od. The ae put n oton unde the effect of etenal foce F and F, n whch gavt pull and fcton foce ae alead accounted fo. The equaton of oton ae ubequentl tated n accodance wth d Alebet pncple. Soluton: the ngle equaton of contant tated n accodance wth. tae the fo: f : l The vecto of Catean coodnate fo th te a follow:, Ξ ξ, ξ, ξ, ξ4 Fg..5 Set of two ball connected b a tff od

41 Dnac of Electoechancal Ste The te of dffeental - algebac equaton wtten n accodance wth.9 tae the fo: l F F F F λ λ λ λ. The equaton of holonoc contant elnate one of the Catean coodnate nce t an dependent vaable n the decpton of dnac. At th pont, let u aue that t, hence: ; l F λ B ntoducng thee vaable nto. we obtan l F F F F F F α α α α. whee: α α The eultng te of equaton of oton tll eque the elnaton of,, whch onl an algebac poble. Th et of equaton ve cople n t analtcal notaton depte the fact that t peent a ve ple echancal te. Th aocated wth the necet of applcaton of Catean coodnate, whch not the ot adequate choce fo the cae of equaton contanng contant, n patcula fo the pont of vew of ple notaton of dnac equaton. A favoable opton n th epect offeed b the ntoducton of genealzed coodnate and epeon of the equaton of oton n the fo of agange equaton...5 Nube of Degee of Feedo and Genealzed Coodnate The ot geneal defnton tate that the nube of degee of feedo n a te ade up of the nube of ndependent vtual dplaceent δξ.. Fo a holonoc te t alo epeent the nube of coodnate vaable

42 . Mechancal Ste necea and uffcent n ode to defne the poton of a te. In accodance wth th decpton eve equaton fo holonoc contant educe the nube of degee of feedo b one ee.,.. Th can be defned b the elaton n h. whee: - the nube of degee of feedo of a holonoc te n - the nube of coodnate necea fo decpton of the poton of an uncontaned te h - the nube of holonoc contant Unde uch aupton egadng the nube of degee of feedo, the equaton of nonholonoc contant.,. alo lead to the educton of the degee of feedo depte the fact that the poton of the te not lted. Th alo ean that the nube of degee of feedo of nonholonoc te lowe than the nube of coodnate necea fo the decpton of the poton of uch a te. Fo the te beng we hall, howeve, focu on holonoc te. Genealzed coodnate fo the vecto of q q,q,,q, and the coponent of th vecto nclude an vaable that fulfll thee pe-equte: the nube of genealzed coodnate equal to the nube of degee of feedo genealzed coodnate ae elected n uch a anne that the ae copatble wth contant peent n the te,.e. the fulfll the condton of dentt wth the equaton of contant f q.4 j genealzed coodnate need to be lneal ndependent, whch ean that the electon of the ha to enable one to unfol epe Catean coodnate q, altenatvel Ξ Ξq, o coodnate of the pa decpton Χ Χq, whch gve ξ ξ q,, q o χ χ q,, q.5 j j Foall t ean that the functonal Jacoban at ξ o ele q j χ q j.6 - of ode n the ente aea of the vaaton of coodnate. The econd of the equaton.5 defne the o called pa decpton coodnate, whch fo an altenatve to the Catean coodnate te, a the nvolve an abta et of vaable fo the decpton of the poton of a te, wthout an poed ltaton on the nube of coodnate ued n uch a decpton. The pactcal electon of genealzed coodnate can be pefoed n a nube of wa and tend to be uch eae than t pled fo the tud of foal equeent Aong Catean, pola, phecal o othe

43 Dnac of Electoechancal Ste vaable ued n the decpton of a phcal odel of a te whch ean all coodnate of the pa decpton t necea to elect uch of ndependent vaable whch ae copatble wth contant and offe a cofotable ouce fo the decpton of the poton of a te. Fo the cae of holonoc contant the geoet of the contant often ugget the electon of uch vaable. Afte an appopate electon of the vaable the eultng equaton ae uccnct and hot, whle fo othe electon the eultng equaton of oton ght be cople and nvolve a lot of othe coponent. Howeve, the total nube of equaton of oton ean contant o the total ode of a te of equaton, whch aount to equaton of the econd ode fo a holonoc te. The appopate electon of the genealzed coodnate n uch a anne that ple and hot fo of equaton enue can be found late n the tet. Tanfoatonal foulae ae functonal elaton whch epe the elaton between Catean coodnate of oton, Ξ o coodnate of pa decpton Χ and the vecto of genealzed coodnate. Sla tanfoatonal foulae account fo the elaton between velocte, whch can be ganed fo holonoc contant b dffeentaton of elaton egadng poton wth epect to te. The tanfoatonal foulae whch ae epeed b equaton.5 fo poton could be copleted b eplct elaton to te fo the pupoe of the geneal condeaton. Such ntance ae non-olated te, e.g. Ξ Ξ q q,, q, o Χ Χ q, q, q,.7 t t Fo thee elaton tanfoatonal foula fo veloct enue n the fo ξ ξ ξ q and q t χ χ χ q.8 q t Slal, dung the calculaton of the vaaton of vaable.5, the eult tae the fo of vtual dplaceent of Catean coodnate of the pa decpton epeed n te of vtual dplaceent vaaton of genealzed coodnate ξ q δξ δq o χ δχ δq.9 q One can note that the tanfoatonal foulae fo vtual dplaceent.9 ae the ae a the one eultng fo the calculaton of total dffeental of vaable fo tanfoatonal foulae.5 not accountng fo te. One alo hould note at th pont that ndependence of vtual dplaceent fo genealzed coodnate coe a a conequence of the fulfllent of contant equaton b the genealzed coodnate δ q δq, δq,, δq.4 and hence the can aue abta value wth the ole of ndefnte ultple.

44 . Mechancal Ste..6 agange Equaton agange equaton ae baed on genealzed coodnate q and appl the ndependence a well a ndependence of vtual dplaceent δq. agange equaton fo a echancal te can be deved fo d Alebet pncple fo holonoc te b calculatng vaaton fo tanfoatonal foulae fo Catean coodnate.5. Afte ntoducton of the tanfoaton of vtual dplaceent nto the ft of equaton.6 q q δ δ.4 we obtan: N q q δ F.4 Concuentl, the econd equaton n d Alebet pncple.6 dappea due to the ndependence of vaaton δq, whch foall ean that genealzed coodnate fulfll thee contant. A a eult of the tanfoaton of.4 we obtan: N N q q q δ F.4 The econd of the epeon n paenthe denote genealzed foce not accountng fo fcton foce actng along the genealzed coodnate N q P F ~ o l N l l q F P ξ ~.44 Th conttute the total of pojecton of all etenal foce epeed n a Catean te towad the genealzed coodnate. Snce the tanfoatonal foulae ae n the geneal cae non-lnea, patal devatve n.44 gve the foulae of foce pojecton. Concuentl, the ft coponent n the bacet n.4 can be tanfoed a follow N N N N N q T q T dt d q q dt d q q dt d q dt d q dt d q.45

45 4 Dnac of Electoechancal Ste The epeon n bacet denote netc eneg of the te n the Catean coodnate T.46 Th eneg aocated wth velocte of te ae ha to be epeed n elaton to genealzed velocte and genealzed coodnate, b eplong tanfoatonal foulae Afte thee tanfoaton the followng obtaned T T q, q, t.47 A a eult fo the ntal equaton.4 we get d dt T q, q, t T q, q, t ~ P q q q δ,,,.48 whch n the condeaton of ndependence of vtual dplaceent lead to epaate equaton n the fo d T q, q, t T q, q, t ~ P dt q.49 q Potental foce.5 actng n the te, whch ae devatve of a potental Uq,t wth epect to poton, can be eal ntegated nto equaton of oton. In th cae thee foce ae otted n the condeaton of etenal foce F. U q, t Pu q The eult tae the followng fo: d T U T U P dt q q.5 d o fnall P, dt q.5 q whch called agange equaton fo an eaned dnac te. Genealzed foce P doe not nclude the coponent eultng fo potental etenal foce, whch wa alead ncopoated n the fo of.5. The coponent Uq,t denotng the potental eneg of the te ha been ncopoated alo nto the ft epeon of agange equaton.5 fo et, depte the fact that t dffeentaton wth epect to q etun the eult zeo. Functon defned a q, q, t T q, q, t U q, t.5

46 . Mechancal Ste 5 called agange functon fo a echancal te. In the clacal echanc t detene the dffeence between the netc and potental eneg of a te. agange equaton ae wdel appled n the tude of dnac popete and contol of echancal te, a well a electoechancal one, ncludng evoechan, anpulato and obot. Bede, the fo one of the two fundaental ethod ued fo the tateent of dnac odel. agange equaton n the fo.5 could be etended [,6] to cove te lted b nonholonoc contant n the fo.,.. f ξ, ξ,, ξ, ξ, ξ,, ξ, t.5 n n If.5 epeent holonoc contant, t poble to ntegate t and the functon tae the fo F ξ, ξ,, ξ, t uch that F f. It notable that n the condton that thee contant poe on vtual dplaceent ae dentcal nce: N F δξ ξ N f o δξ.54 ξ N F F f F ξ hence ξ t f F ξ ξ Afte the ntoducton of genealzed coodnate thee foulae ae otted a a eult of the fulfllent of the equaton of contant. Howeve, f neatc contant of the tpe.5 wee not ntegated, the ntoducton of genealzed coodnate would not eult n the oon of the equaton tpe.5; n fact, t then tanfoed nto the equaton of nonholonoc contant n genealzed coodnate: ψ q, q,, q, q, q,, q, t j nh.55 and n the cae of lnea one: ψ j c c j c h whee: nh - the nube of nonholonoc contant c n-h - the nube of genealzed coodnate n-h-nh - the nube of degee of feedo A a eult of the epanon of the equaton.55 nto ultple vaable Talo ee o b analog to.54 one can deontate that thee equaton can be ued to elate the vtual dplaceent of genealzed coodnate δq j q d c ψ j q δq o c h δ q j nh.56 j

47 6 Dnac of Electoechancal Ste The peent vtual dplaceent ae not ndependent a t wa the cae fo holonoc contant and, hence, the value of the patcula paenthee.48 need not be equal to zeo, whch ha gven the equaton of oton.49 n cae of holonoc contant. In thee ccutance, a well a n the devaton of equaton of oton on the ba of d Alebet pncple.8 we wll appl the ethod of ndefnte ultple. All nh coponent n elaton.56 wll be ultpled b ucceve ndefnte ultple μ j and ubequentl added to equaton.5. A a eult, we obtan c epeon n paenthe n.57. c d dt q, q, t q, q, t Pl q q l l l j nh ψ j μ j δq q l l.57 Fo eaple, n the ft nh coponent of the u n.57 the electon of μ j ade o that the value of epeon n paenthe equal to zeo. The coepondng dependent vaaton ae δq j, j,,nh. In that cae the eanng vaaton of genealzed vaable ae alead ndependent and the coepondng paenthe of the u.57 have to be equal to zeo. A a eult, the equaton of oton tae the fo d dt q, q, t q, q, t P q q, j nh nh j ψ j μ j q.58 The eultng te of equaton cont of agange equaton n the fo.58 and nh equaton of nonholonoc contant n genealzed coodnate.55, n whch the unnown nclude n q vaable and nh n μ j ultple. In equaton.58 the potental foce n the fo.5 have alead been epaated fo genealzed foce P ~ and ntegated nto agange functon, whle the non potental coponent of genealzed foce P ae peeved. Fo the cae of nonholonoc lnea contant the equaton of oton fo a te wth nonholonoc contant, accountng fo.56, tae the followng fo: d dt q, q, t q, q, t P q q, j nh nh j μ h j j Potental Mechancal Eneg Potental echancal eneg,.e. accuulated eneg egadle of veloct, can be toed n two wa: n gavtatonal feld and n the fo of elatc tenon. In the gavtatonal feld the potental eneg accuulated dung the dplaceent of a towad nceang potental. Th eneg equal to the wo eeted dung th dplaceent :

48 . Mechancal Ste 7 b b U F dl gad dl b a.6 a a Fo the cae of contant gavtatonal feld,.e. one wth contant vecto of gavtatonal acceleaton g, potental eneg epeed a b a g h b h a gh U gh.6 whee h the vecto called the heght of a pont above the efeence level eaued n the paallel decton to the vecto of gavtatonal acceleaton g. Fo a old of a M the heght of the cente of a S aocated wth the potental eneg of the old a a total. Potental eneg n elatc eleent aocated wth the wo accuulated dung t elatc tan. The bol of th coponent a png Fg..6. U b F d.6 a Fo a png wth lnea chaactetc the foce that a png eet popotonal to defoaton etenon, copeon, toon n elaton to the tatona tate of the png denoted a. The elatc odulu of an png [N/] defned a the lope of t te-tan cuve n the elatc defoaton egon. A a conequence F Δ hence U ~ d ~.6 Fg..6 Accuulaton of potental eneg n an elatc eleent Fo the cae of paallel connecton between png the total etenon of all png dentcal and the foce affectng the png the u of the foce needed fo the elatc tan, thu F F F F Δ Δ Δ n n n and the eultant elatc tan Δ

49 8 Dnac of Electoechancal Ste F Δ Fo the cae of eal connecton of png, we have: F F F n Fn Δ Δ Δ Δ n.64 hence: F Δ F Δ F n n Δ n Δ n F Genealzed Foce, Echange of Eneg wth Envonent Genealzed foce eultng fo the tanfoaton of Catean foce actng on a te.44 ae peented n detal eale n the boo.e. dung the devaton of agange equaton. In th ecton the wll be etended to cove fcton foce, whch equed a a eult of condeaton of pefect contant ung d Alebet pncple. The echange of eneg wth the envonent occu a a eult of wo eeted on the te b etenal foce and fcton foce, the effect of the latte alwa negatve,.e. eult n dpaton of eneg of a te. Potental foce a be ntegated nto agange functon b condeaton of potental eneg.6-.6, whle the ght hand de of agange equaton educed onl to condeaton of actve foce. Howeve, potental eneg alo capable of plang the ole of echange of eneg wth the envonent, f t eplctl elatve to te Uq,t - th the cae n non-olated te. We hall, howeve, tae nto condeaton fcton foce, whch occu n vtuall an te, have often nonlnea chaactetc and tend to eque notaton n a cople wa. The ot bac te ued fo the notaton of fcton foce aocated wth oton baed on an aupton that the ae popotonal to the veloct. Th cae denoted wth the te vcou fcton, whch tae the fo F ξ t D.66 The echange of eneg wth the envonent due to etenal foce calculated on the ba of vtual wo δa eeted b thee foce on the patcula vtual dplaceent: N δa F D ξ δξ.67 The tanfoaton of vtual wo nto genealzed coodnate apple tanfoatonal foulae of veloct.8 and vtual dplaceent.9 unde the aupton that the efeence te netal and, a a eult, ξ / t.

50 . Mechancal Ste 9 Hence: N N ξ ξ ξ δa F D q δq.68 q q q The ft coponent of vtual wo.68 alead fala to u and nown a genealzed foce.44 actng n the decton of -th genealzed coodnate, whle the econd coponent denote tanfoed fcton foce. Futheoe, t poble to dentf the tanfoed fcton coeffcent D N ξ ξ D q q.69 Genealzed foce accountng fo fcton fo -th genealzed coodnate could be noted a: ~ P P D q.7 The vtual wo of a dnac te.68 can alo be wtten n the fo of a at ung Jacoban at of the tanfoaton of a coodnate te whee: T T T T T δq J F DJq δq P F DJq J δq P δq δa.7 ξ ξ q q ξ J - the Jacoban at of tanfoaton q ξn ξ n q q,,n,, [ D D ] D dag - dagonal at of coeffcent of vcou fcton [ F F ] T F n D n F - vecto of etenal foce n Catean coodnate. A a eult of the tateent of vtual wo n the fo.7 we obtan the vecto of genealzed foce accountng fo vcou fcton wth coponent fo.68. It patculal cople to tanfo fcton foce n whch thee ae coeffcent of utual fcton.69, o n the vecto fo of epeon J DJq n T elaton.7. In nueou pactcal cae fcton coeffcent can be deved dectl fo genealzed vaable q wthout efeence to tanfoaton of loe eultng fo fcton epeed n Catean coodnate.

51 Dnac of Electoechancal Ste..9 Eaple of Applcaton of agange Equaton Method fo Cople Ste wth Patcle Eaple.4. We hall one oe te conde a te of two ball connected wth a tff od of length l Eaple., Fg..5. The equaton of oton fo th te obtaned fo d Alebet pncple n the Catean coodnate. ae ve cople and of lttle ue fo coputng the tajecto of th te. At peent fo the developent of a atheatcal odel we hall ue the ethod of agange equaton wth a dffeent electon of coodnate Fg..7. Fg..7 Plana te contanng two ae connected wth a tff od Fo th te the vecto of poton tae the fo,,,, Ξ ξ, ξ, ξ, ξ4, whch ean that the nube of coodnate n the decpton of uncontaned te aount to n 4. In th te we have one equaton of holonoc contant: f : l o l.7 Hence, the nube of the degee of feedo : n h 4. Catean coodnate of patcle of a and angle φ ae aued a the genealzed coodnate: q q, q, q,,.7 Tanfoatonal foulae of the tpe.5 ae followng: ξ q 4 ξ ξ ξ q l co q l n q l n q l coq

52 . Mechancal Ste and fo velocte, epectvel: 4 n n co co q lq q l q lq q l q q ξ ξ ξ ξ.74 Genealzed coodnate ae copatble wth contant, whch can be vefed b dect ubttuton of.74 nto contant equaton.7; th alo deve fo the ndependence of vtual dplaceent of genealzed coodnate condeed a nfnteal dplaceent. The netc eneg of th te tae the followng fo T, whch afte ntoducton of genealzed velocte fo.74 lead to the epeon n co l l T.75 U hence T U T The vecto of genealzed foce n the ght hand de of agange equaton wll be detened fo elaton.7. The vecto of etenal foce and at of dapng coeffcent ae [ ] T F F F F F [ ] T D D D D dag D wheea Jacoban at of tanfoaton ξ n co l l q J.76 et u addtonall aue that: D D D D D D. In th cae on the ba of.7 we hall obtan the vecto of genealzed foce: n co n co n co l l D F F l l D D D F F l D D D F F P P P P.77 Hence, ung.5, we ae able to tate agange equaton:

53 Dnac of Electoechancal Ste fo q P dt d co P l dt d co n co l D D D F F l fo q P dt d n P l dt d n co n l D D D F F l fo q P dt d co n n co P l l l dt d n co n co / n co l l D F F l l l The above equaton of oton can be etated oe pl b ntoducton of / μ : fo q : μ μ co / n co l D D D F F l fo q μ μ n / co n l D D D F F l.78 fo q / n co n co / n co l l D F F l

54 . Mechancal Ste The eultng te of equaton of oton.78 can be vefed fo pecfc cae, fo ntance b oblzng a patcle of a, then: cont. In th cae the ft two equaton of oton.78 enable one to calculate eacton foce at the faton pont: fo q : F l co n F Dl co fo q : F n co l F Dl n The equaton º afte ntoducton of gavtatonal foce F - g and unde the aupton that F conttute the equaton of daped oton fo pendulu: g D fo q : n.79 l The conducted ntepetaton of equaton lead to the dentfcaton of the patcula te and ndcate how ueful the odel could be depte the cople fo of the equaton. Howeve, the equaton ae uch oe ple than the one n Catean coodnate, a hown n the dcuon of d Alebet pncple.. Eaple.5. A plana echancal te gven whoe phcal odel peented n Fg..8. The te cont of a pendulu of a attached to a ldng along a hozontal ba. The oton beng lted b png of tffne and fee length of d, and coeffcent of vcou fcton D. The coeffcent of vcou fcton of the pendulu equal to D. The two foce opeatng n the te nclude gavt foce G and the foce of lateal peue Q. The ta nvolve the developent of a atheatcal odel of the te oton ung agange equaton. Fg..8 Plana te of pendulu wth oble pvot Th te cont of two oble ae, whch n uncontaned tate eque n 4 coodnate n ode to decbe the poton:,,,, ξ, ξ, ξ, ξ4

55 4 Dnac of Electoechancal Ste In th cae, thee ae two equaton of holonoc contant: : : l f f.8 Hence, the nube of degee of feedo 4 h n The followng vaable ae adopted a genealzed coodnate:,,q q q.8 whch ae ndependent and copatble wth contant. The tanfoatonal foulae.5 tae the followng fo ξ ξ ξ ξ ξ ξ co n n co 4 4 l l q l l.8 The netc eneg of the te T whle the potental eneg of the te aocated wth the png d U Afte the ntoducton of genealzed coodnate.8 agange functon tae the followng fo: co d l l U T.8 The vtual wo eeted b the te account fo etenal foce and foce of vcou fcton [ ] [ ] T T Q g F F F F F δ δ δ δ δ co n P P D Ql gl D Q D Q g D Q g A.84 The equaton of oton tated b ad of agange ethod ae a follow: fo q P dt d co P d l dt d

56 . Mechancal Ste 5 fo q d dt d dt l l co l n P Afte calculatng te devatve and ntoducton of genealzed foce P, P, we obtan P l co n d co l g n Ql co D / l Q D.85 The above equaton have been ulated n the atheatcal pacage. The elected oton cuve fo nput data and ntal condton gven below ae peented n Fg..9 Fg [ g]. [ N / ]. [ g] l. [ ] g 9.8[ / d.5 [ ] ] Q. [ N] 5. [ N / ].5 [ ]. [ ].7 [ ad]. [/ D D ] Fg..9 Poton of the pendulu attachent pont Fg.. Swa angle φ fo the pendulu

57 6 Dnac of Electoechancal Ste Fg.. Angula veloct of the pendulu wng Fg.. Tajecto of the pendulu.. Moton of a Rgd Bod... Fundaental Noton Rgd bod n a concept of dcete te t a et of ateal pont patcle whoe dtance ean contant. Th dealzaton tend to be coect wthn a cetan ange of etenal foce affectng uch a te both fo fleble bode wth hgh gdt and bttle one. In ode to detene the poton of a bod t uffcent to detene thee pont on uch a bod that ae not tuated along a ngle taght lne. Each ucceve pont tuated n uch a bod eque the detenaton of thee coodnate fo the dentfcaton of t poton; howeve, t ept at thee contant dtance fo the pevoul detened pont on th bod. Hence, the nube of degee of feedo n accodance wth. equal to n h jut a t the cae fo thee patcle whoe elaton ae detened b contant dtance Fg... A a conequence, the nube of degee of feedo of uch a bod aount to 6.86; howeve, the decpton of the poton of the bod doe not uuall appl Catean coodnate elated to thee elected pont on the bod. The tpcal pocedue followed n ode to obtan the equed 6 coodnate nvolve the detenaton of thee Catean coodnate fo a elected pont

58 . Mechancal Ste 7 on the bod O b, wth a local te of coodnate O b b b z b aocated wth th bod a well a thee angle fo the decpton of the oentaton of the te on the bod n elaton to the bac Oz te. In bef, one can a that the ft thee coodnate detene the poton of the bod whle the ucceve thee defne the oentaton of the bod n pace. In ode to etablh the angle of the oentaton of the bod o the local te O b b b z b on the gd bod a tandad technque to be appled, whch nvolve the detenaton of Eule angle o navgaton angle te RPY oll, ptch, aw [,6,]. The angle detene the eleenta evolute oton: n the ft cae wth epect to cuent ae n the ucceve ode of z, whle n the latte cae wth epect to contant ae z. Fg.. Rgd bod wth local coodnate te and thee pont detenng the poton of the bod... Moton of a Ma Pont on a Rgd Bod The dffeental dplaceent of a pont on a gd d could be ade up of the dplaceent of the ntal poton of a local te on gd dr O and the dffeental vecto of evoluton dτ whch accopane the evolute oton of a gd bod. The dffeental vecto of evolute oton Fg..4 epeed a Hence: dτ n θ d dτ d.87 b b b dr O dτ o d dr O d b d A a eult of dvdng the dffeental dplaceent b the dffeental of te dt we obtan the veloct of the oton of an -th pont on a gd bod v v ω.88 O whee: v O a lnea veloct of O b pont wth epect to the bac Oz te and d ω.89 dt b

59 8 Dnac of Electoechancal Ste the angula peed of the gd bod. The a detened b the decton of the dffeental angle dφ of the evolute oton of a gd, o angula peed ω, called the ntantaneou a of gd bod evolute oton. Fg..4 Moton of a pont on a gd bod eultng fo t evolute oton The cente of a S on a gd bod a pecfc pont whch, can ethe belong to the gd o not; howeve, t ha to fulfll the followng condton MR S o R S.9 M The cente of a on a gd bod ha oe patcula chaactetc: The potental eneg U of a gd could be epeed a the potental eneg of the total a M of gd focued at the cente of a at pont S. Th egad a gd bod tuated n gavtatonal feld wth contant vecto g of gavtatonal acceleaton Fg..5. U gh g h S g b gh S gh S M MghS Fnall: U MghS.9 Cente of a S alo the cente of the bod weght unde contant gavtatonal feld n the tate of balance between toque due to gavt foce. Hence t epeent the equlbu tate of a tatona gd n gavtatonal feld. M b g b g.9 Afte dffeentaton.9 wth epect to te, we obtan v M S v.9

60 . Mechancal Ste 9 Fg..5 Rgd bod n gavtatonal feld potental eneg Th ean that the total oentu of a gd bod can be epeed a the poduct of old a and veloct of the cente of a v S p Mv S.94 If the local te aocated wth a gd bod tuated n the cente of a, and v, whch, conequentl, gve p. Th ean that the cente of a the pont aound whch the ntenal oentu of a gd equal to zeo.... Knetc Eneg of a Rgd Bod It poble to epe netc eneg of a gd bod n a oe nthetc anne than though the u of netc enege of conventonal a pont wth the total a aountng to M, whch ae dtbuted n pace but ean at contant Fg..6 Dependent and ndependent eleent n elaton to coodnate te tuated on a gd bod

61 4 Dnac of Electoechancal Ste dtance. Howeve, one can note that angula peed ω efe to the gd a a whole, whch ean that t not elatve to the locaton of the coodnate te on a gd Fg..6. Fo an pont of a fo.88 t follow that Ung anothe te on a gd wth ogn n v v ω O O b b O b, th veloct v can be noted a v v ω B ubttuton n a we obtan: b b The copaon of the eult n wth gve: v v ω a ω O b v O vo ω a ω ω.95 Hence, t eult that angula peed ω of gd evolute oton the ae along the ente aea of the bod and not elatve to the poton of the local coodnate te aocated wth t. The netc eneg of the gd equal to T Ung.88 we obtan T v O v O ω v b O ω b ω b ω b.96 The ft te n.96 equal to T Mv O and denote the netc eneg of a a pont equal to the a M of the total gd bod ovng at the veloct of the pont O b, n whch the local coodnate te located. The econd te of the epeon.96 equal to zeo fo the cae when pont O b ovelap wth the cente of a S b S. A a eult: T v Oω b S v S ω v ω S S Concuentl, the thd te.97

62 . Mechancal Ste 4 ω I ω T z z z z z z z z z z T ω ω ω ω ω ω.98 denote the netc eneg of the evolute oton aound cente of a S wth angula peed ω [ ω ω ω z ] T. The eleent peent along the an dagonal of the at of neta I,.e. I, I, I z ae called the oent of neta of a gd wth epect to,,z ae, epectvel. A a eult z I.99 the oent of neta wth epect to a and calculated a the u of the patal ae ultpled b the quae of the dtance of the patcula ae fo a. Apat fo the an ae thee ae the o called oent of devaton, e.g. D. The at of neta I etcal and potvel detened z z z z z I D D D I D D D I I. The netc eneg of a gd bod n fee oton, unde the aupton of a local coodnate te n the cente of a S, tae the fo: ω I ω v S T S M T. whee I S denote the at of neta of a gd bod wth epect to ae of Catean coodnate te nteectng at the gd bod a cente S. Fo an egulal haped gd bod t poble to elect uch decton of the ae of the Catean coodnate te O b b b z b fo whch all oent of devaton ae equal to zeo; a a eult, the dappea. Fo uch ae the te pncpal ae of neta ued

63 4 Dnac of Electoechancal Ste I I I. I z In th cae the netc eneg of a gd bod aocated wth t evolute oton ha onl thee te: T z z I ω I ω I ω.4 Fo an egulal haped bod the dentfcaton of the decton coepondng to the pncpal ae of neta nvolve fndng a oluton to the chaactetc equaton fo a quae at I.,.e. the etablhent of egenvecto and egenvalue ung the ethod of lnea algeba. The te aetc top ha been coned fo uch egulal haped gd bode fo whch the thee pncpal oent of neta I, I, I z ae dffeent. The te etcal top ued wth egad to a gd bod whoe two pncpal oent of neta ae dentcal and the thd denoted wth anothe value, whle the te phecal top ued fo a gd whoe all oent of neta ae equal. The dentfcaton of the cente of a and pncpal ae of gd bod neta uch plfed when the gd dpla the chaactetc of et unde the aupton of contant dent and egula dtbuton of eleenta ae. Fo the cae of a bod poeng an a of et the cente of a tuated along th a and, hence, t epeent one of the pncpal ae of neta. Fo the cae of a gd bod wth a et plane th plane contan two pncpal ae of neta, whle the thd one pependcula to the nteecton of the two ae. The fnal pat of the peent condeaton egadng the oton of a gd bod wll focu on the elaton between the at of neta I S fo the cae when the ntal pont of the local te ovelap wth the cente of a and the at of neta I calculated wth epect to the coodnate te wth paallel ae, whle the ntal pont of the te dplaced fo the cente of a b vecto a, uch that b S a, whch gve: a a a aaz I I S M aa a a az.5 a a a a a z z In th cae the epeon ued fo the decpton of the total eneg of a gd bod, accodng to.97 nclude v ω T S a v O ω Ma O.6 and the total netc eneg account fo thee coponent: T Mv Ma v ω ω I.7 ω T O O

64 . Mechancal Ste Moton of a Rgd Bod aound an A Fee oton of a gd bod o a te of gd bode epeented n a nube of engneeng ue n avaton, balltc, pace eeach, etc.; howeve, the ot nueou goup of engneeng poble concened wth the oton of a gd bod aound a contant a. Th the cae n otatng achne, wheel beang n a ca, ndutal anpulato and obot. In uch cae the decton of the vecto of angula peed ω n confot wth the a of evolute oton detened b the lne of the beang and eacton foce n beang ecue the antenance of the contant decton of angula peed vecto. The epeon accountng fo netc eneg n oton aound a contant a condeabl plfed a t onl nvolve one te of T.4 epeentng a ngle coponent of angula peed ω I O T.8 whee I O the oent of neta of a gd bod wth epect to the cuent a of evolute oton. Fo the cae when the a of evolute oton doe not nteect wth the cente of a S the oent of neta I O can be detened fo the elaton.5, whle angula peed ha jut a ngle coponent I I Ma.9 O S whee: I S - oent of neta fo an a nteectng the cente of a S, paallel to the a of the gd bod evolute oton a - dtance between the two ae. The ae eult can be deved fo elaton.7 unde the aupton of the coodnate te n the cente of a S, o that S S T Mω a ω I ω I Ma. Th eult nown n lteatue unde the te Köng theoe....5 Rgd Bod Ibalance n Moton aound a Contant A In the condeaton of the oton of a gd bod aound a contant a we have to do wth poble aocated wth tatc and dnac balance. The cae of tatc balance Fg..7 dealt wth n the cae when the a of the gd bod evoluton doe not co t cente of a S, but paallel to one of the pncpal ae of neta. In th cae centfugal foce otatng along the cente of a eeted on the beang Mv Mω F Mω. and t poe two paallel foce on the beang.

65 44 Dnac of Electoechancal Ste Fg..7 State of tatc balance of a gd bod n oton aound a contant a In a oe cople cae of dnac balance the a of gd bod evoluton ovelap t cente of a, but not paallel to the pncpal a of neta. To plf thng, one can a that the beang ae of evoluton ae aew n elaton to the a of the gd bod et. Howeve, t can be tctl etablhed that dnac balance the cae when angula oentu aew n elaton to the vecto of angula peed ω. In ode to calculate the effect of dnac balance t necea to appl the noton of angula oentu. Fo a te of patcle the angula oentu defned a v. ung the epeon whch detene the veloct fo a patcle n a gd bod.88 and ltng the cope to evolute oton we obtan: I ω. whee I the at of gd bod neta., and ω the vecto of angula peed epeed n the ae ae a the at of neta. It notable that an change n the angula oentu fo a te, not accountng fo eneg loe, aocated wth the necet of applng oent of foce n the fo d Μ.4 dt In a te accountng fo loe the appled oent of foce ovecoe the oent of fctonal etance and potentall the oent aocated wth the eeted wo; howeve, t onl t uplu o defct that affect a change n the oentu n the te. The cae of dnac balance fo a gd wth ccula ecton, whch ha been lghtl eaggeated, peented n Fg..8. Fo th cae the angula oentu doe not ovelap wth the vecto of angula peed ω due to vaou value of the oent of neta n th d fo the pncpal ae of neta.

66 . Mechancal Ste 45 Fg..8 Dnac balance of a otatng d The angula peed ω of a d dtbuted along the pncpal ae of neta ω ω coϑ ω ω nϑ A a conequence, the coponent of the angula oentu can be detened a > Iω I ω I I In th cae oent M aocated wth antenance of a tead decton of angula peed ω and eult n a pa of eacton foce n beang. It epeed a d Μ ω.5 dt The value of th oent Μ ωn ωn ϑ δ B applcaton of elaton we fnall obtan: co δ nδ Μ ω I I n ϑ.6 Fo foula n.6 t te that the au value of th oent, whch tend to othogonalze oentaton of a d, occu fo the ewne angle ϑ 45, whch the geate the bgge the dffeence between the oent of neta fo the pncpal ae. Cetanl the oent eultng fo centfugal foce elatve to the quae of angula peed. Th oent act on the beang wth foce whoe decton ae oppote a pa of foce. The foce otate a a eult of the evoluton of the d Fg..8, accodng to vecto poduct.5.

67 46 Dnac of Electoechancal Ste.. Eaple of Applng agange Equaton fo Moton of Rgd Bode Eaple.6. Fo the thd te the te of two ball connected b a tff od, efeed to n Eaple. and Eaple.4 wll be eaned. Th te the te depte beng plana, wll be condeed a a gd bod. Due to twodenonal natue of th eanaton and dtbuton of the ae along a taght lne the nube of degee of feedo aount to n contat to 6 n the thee denonal cae. The netc eneg of the te epeed n a wa pecfc fo a gd bod. S S T I ϑ.7 Unde the aupton of genealzed coodnate q,, ϑ, ncludng the poton of the cente of a and angle of gd bod evoluton, the netc eneg epeed n.7 not aocated wth the necet of tanfong vaable. In the ccutance of the lac of eleent whch accuulate potental eneg gavtatonal foce ae accounted fo n the etenal foce, agange functon fo the te equal to the netc eneg.7: T U T S Fg..9 Ste of two ball connected b tff od condeed a a gd bod Concuentl, tanfoatonal foulae ae needed n ode to detene genealzed foce on the ba of data fo etenal foce epeed n Catean coodnate F T T [ F F F F4 ] [ F F F F] Fo th pupoe we have to detene the poton of the cente of a S along the od whch connect the two ball l l l l l l l l l

68 . Mechancal Ste 47 and the oent of neta fo the te l l l I S.8 o that the tanfoaton of coodnate.5 n th cae tae the fo ϑ ξ ϑ ξ ϑ ξ ϑ ξ n n co co 4 l l l l S S S S.9 Subequentl, followng fo.44 ϑ ϑ ϑ ξ ξ ξ ϑ co n l F l F l F l F F Q F F F Q F F F Q S S S S we obtan equaton of oton n the followng fo Fo S q : S S S S F F Q dt d fo S q : S S S S F F Q dt d. fo ϑ q : ϑ ϑ ϑ ϑ ϑ ϑ co n l F l F l F l F I Q dt d S Th the plet fo of the equaton of oton aong the thee eaned odel.,.78,. nce genealzed coodnate ae elected n uch a wa that the fo of notaton of netc eneg that ot uccnct. Eaple.7. It nvolve a odel of dnac fo a phecal anpulato. The neatc odel of the anpulato, contanng vaable and paaete, peented n Fg.. and.. Th anpulato hown n t nonal poton, whch ean that the value of coodnate n the jont ae equal to zeo.

69 48 Dnac of Electoechancal Ste Fg.. Kneatc odel of a phecal anpulato Fg.. Moent of neta n patcula ebe of anpulato n Fg.. The ft two jont n th anpulato ae evolute jont, whle the thd one a tanlatonal jont. The ucceve jont of the local oton of the anpulato and the end effecto n th odel ae epeented b a M at the end of the fnal ebe. Fg.. peent the patcula jont and contan a epeentaton of the oent of neta n epect to the ae along whch the anpulato oton occu. The oent of neta J S, J Sz, J S, J Sz ae detened fo the ae cong the cente of a S and S. Hence, the netc eneg of the anpulato can be detened a the u of fou te whch coepond to the fou ebe: T T T T T M, whch can be epanded to co co M M M S Sz S Sz z z M z J J J z J J T.

70 . Mechancal Ste 49 whee:,, ;,, ;,, M M M M S S z z z ae the epectve Catean coodnate of the cente of a S, S and the a M. Th te ha degee of feedo and the genealzed coodnate ae,,,q, q q q q. whch denote angula vaable n the ft two jont and a lnea vaable n a tanlatonal jont. Po to the peentaton of tanfoatonal foulae t would be ueful to ntoduce the conventon of abbevaton ued fo the notaton of tgonoetc functon whch ae coonl ued n obotc, nael:, n ; co c etc.. whch tend to condeabl hoten the notaton aocated wth tanfoaton of coodnate. The tanfoatonal foulae fo the coodnate of cente of a S, S and M, whch appl the notaton a n. ae the followng: q H z a q H z l H z c q c a q c l c c q c c a q c c l M M M.4 and fo peed, epectvel: c q q z c a q q z c l z f q c q f a q c q f l f q c c q f a q c c q f l M M M.5 whee:,,,,,, c c f f c c f f The potental eneg n th te, aocated wth gavt foce actng on ebe and of the anpulato and on a M, q H Mg a q H g l H g U whch can be tanfoed to tae the fo Mq a q l g cont U.6 agange functon fo the anpulato accountng fo.,.5 and.6 tae the fo: { } { } Mq a q l g M q Mq a q J J Mq a q J J c J U T S O z S z O z.7 whee: S O z S z O J l J J l J

71 5 Dnac of Electoechancal Ste The echange of eneg wth the envonent uoundng the anpulato occu a a eult of vtual wo elatve to etenal foce F-F, the dvng toque M, M and foce F ognatng fo dve on the jont and fcton foce. Ft, we wll calculate the genealzed foce fo the patcula genealzed coodnate eultng fo the etenal foce. The tae the fo c c F q F Q c q F F Q c q F F Q M M M.8 whle the oveall vtual wo of the te : F q q D c c F D c q F D c q F q Q Q Q A δ δ δ δ δ δ δ Μ Μ.9 Fala wth the epeon.7,.9 we ae able to tate the equaton of oton: fo q : Q dt d and followng the opeaton we obtan: { } D c q F Mq a q c q Mq a q J J Mq a q J J c J z S z O z S z O z Μ A a eult: { } D c q F q M c q q J q J c J z z z Μ fo q : Q dt d { } D c q F q M l gc q J q M q q J z Μ.

72 . Mechancal Ste 5 fo lnea vaable q : d dt q q Q q whee: q M M q c g M F F cc D q J q J z q M q J J Oz O q J J Sz S q a Mq q a a Mq Mq The eultng equaton ae qute cople depte the fact that the anpulato ha onl thee degee of feedo. The odel coplet becong geate followng an nceae n the nube of degee of feedo and wth the geate nube of evolute jont n the neatc chan of the anpulato. Hence, n ode to avod laboou tanfoaton and calculaton n the foalzed coue of odel tateent ung agange equaton t poble to poga the odel n a pogang language handlng bolc opeaton [,] and nput the data egadng the neatc and paaete of the eaned object. The eultant equaton of anpulato oton can be vefed fo a nube of patcula ntance b detenng the value of elected vaable and ntepetng the te n equaton whoe u not equal to zeo. et u aue fo ntance that the angula peed of the evoluton of the vetcal a of the anpulato cont, whle the thd jont the tanlatonal one oble: q cont and fo plfcaton pupoe zeo value of foce F. In th cae the equaton of oton fo the econd ebe tae the fo { J q } J z q gc l M q Μ D. It can be concluded that the a of the anpulato loweed unde the effect of gavt foce, the oent of centfugal foce elatve to te to peeve the a n the hozontal poton φ and that th oent the hghet fo angle φ 45, whch qute logcal. An actve ole plaed b the dvng oent M and the pave ole attbuted to the oent of fcton. The ebe of the equaton have phcal ntepetaton and the coect gn denotng the ene of a oent. Fo the ft of the equaton. t poble to calculate oent M needed n ode to antan contant angula peed cont of anpulato oton Μ z D. J q

73 5 Dnac of Electoechancal Ste Th equaton eve n ode to detene the toque equed to ovecoe the fctonal etance and changng the angula oentu aocated wth the oton of the econd ebe wth the veloct equal to. The fnal equaton fo vaable Q ae t poble to detene the value of equed foce F, necea at the pont whee the tanlatonal jont fed, n ode to ae t oble, whch F q c g M. M The cae dealt wth hee nvolve eacton to the centfugal foce elatve to both angula peed n evolute jont and to the gavt foce. The deved equaton of anpulato oton. could well be applcable fo the electon of dve n anpulato jont and n dcuon of ue aocated wth t contol... Geneal Popete of agange Equaton... aw of Conevaton An olated te a te whch nethe affect no affected b othe etenal te. Fo uch a te agange functon not eplctl elatve to te q, q.4 In an olated te we don t have to do wth etenal foce, nethe n an actve fo o a fctonal foce, hence, agange equaton tae the followng fo: d dt q, q q, q q q.5 et u aue that fo a coodnate q j agange functon not elatve to th vaable, but onl depend on peed q, then j d dt q, q q j o q, q cont q j.6 and th nd of coodnate denoted wth the te cclc coodnate. Th contant value, whch a devatve of agange functon accodng to genealzed veloct called genealzed oentu q, q p j.7 q Fo equaton.6 t te that the genealzed oentu fo a cclc coodnate contant. The occuence of cclc coodnate n agange functon depend on the electon of genealzed coodnate coodnate te; conequentl, the contanc of cetan coponent of genealzed oentu elatve j

74 . Mechancal Ste 5 to the adopted efeence te. It ha been deontated [8,,] that the total oentu fo an olated te contant. Th alo ean that fo cclc coodnate the coponent of oentu ae contant, whle fo the eanng coponent of oentu the echange of oentu occu n uch a anne that the total oentu of a te ean contant. et u efe agan to agange functon fo an olated te.4 and calculate the total dffeental fo th functon wth epect to te d dt q q q q The ubttuton of q accodng to agange equaton.5 gve d dt d dt q q q q d dt q q whch eult n the elaton: d dt q q Th can be etated ung the defnton of oentu.7: p q cont.8 Hence, n an olated te the followng value contant E p q.9 and called the eneg of a te. We can tanfo the epeon ued n.9 T p q q q T q.4 q Th o due to the fact that the functon of netc eneg T q, q a hoogenou functon of the econd ode and, conequentl, quadatc fo of the veloct fo the cae when t not eplctl elatve to te. In accodance wth the Eule foula fo hoogenou functon of n ode the followng elaton atfed f ξ ξ nf ξ.4 ξ

75 54 Dnac of Electoechancal Ste and fo a functon of the ode n the elaton n.4 atfed. B applcaton of.4 n the foula tatng eneg.9 we obtan E T T U.4 Th eult could have been antcpated a t ean that the total eneg of an olated te q, q, q cont of the u of netc and potental eneg. In an olated te the law of conevaton of genealzed oentu fulflled [8,], wheea th popet can eal be deontated n a Catean coodnate te both fo patcle and gd bode n evolute oton. Howeve, f a cetan genealzed coodnate a te epeed n te of evolute angle: q j φ, then equaton.7 denote genealzed angula oentu. Fo a cclc angula coodnate the appopate coponent of angula oentu n an olated te contant. A a conequence, agange equaton.5,.5 denote ethe the equaton of foce fo the cae when the genealzed coodnate q ha a lnea eaue.e. t tanlatonal o equaton fo oent of foce toque fo genealzed coodnate n the fo of angula vaable chaactetc fo evolute oton.... Chaactetc of agange Functon and Equaton The equaton fo both echancal and electoechancal te tated n the fo of agange equaton poe a nube of pecfc popete ueful fo the pupoe of tatng and contollng the a well a n vaou ue aocated wth the dnac applng agange equaton, and n patcula fo a contol of dnac object. agange functon doe not have a unabguou epeon. Fo functon and fo functon d F q, t.4 dt the equaton of oton epeed ung agange ethod ae n the ae fo. It poble to vef b dect ubttuton of and n.44. agange equaton peeve an nvaable geneal fo. Fo a holonoc te t followng d dt q q P.44 egadle of the electon of the vecto of genealzed coodnate q q,q,,q. If unabguou tanfoaton wee to be ued u j u q,, q j j havng a non-zeo detenant of Jacobean at fo th tanfoaton n a pecfc aea of vaance, thee a evee tanfoaton uch that q q u,u,,u.

76 . Mechancal Ste 55 Then agange functon u, u tae anothe pecfc fo, howeve, the geneal fo of agange equaton d dt u j u j V j, j ean the ae. The pecfc fo of the patcula equaton fo the ucceve vaable of the two vecto of genealzed coodnate q q,q,,q and u u,u,,u evdentl dffeent. The eultng fo of agange equaton fo a holonoc te wth degee of feedo, whle agange functon not eplctl elatve to te,.e. q, q, can be tated n the fo of a at equaton, whee the epectve atce have the denon of : The atce dpla the followng popete []: B q q C q, q q G q τ.45 a the at of neta Bq a etcal and potvel detened one and elatve onl to the vecto of genealzed coodnate q. Fo that t eult that at of neta alwa eveble. Moeove, thee ae uch cala quantte η q and η q that the followng ltaton atfed η q I < B q < q I.46 η b the at W B q C q, q.47 - ew-etc. Th pla a ole n the contol of vaou te and can eve n ode to contol the coectne of the developed equaton of oton. c the equaton of oton.45 eultng fo agange equaton ae lnea due to the tuctual paaete. Th alo ean that that thee a contant vecto Θ of denon p and a at functon Y q, q, q of denon n p uch that B q q C q, q q G q Y q, q, q Θ τ.48 The functon Y q, q, q denoted wth the te egeo, whle the vecto Θ cont of appopate cobnaton of tuctual paaete uch a denon, ae and oent of neta. The denon p of the vecto Θ not unfol defned and, hence, the dentfcaton of the adequate paaete et fo th te n a wa that denon p nzed pla a pactcal ole. d The equaton of oton.45 poe the popet of pavt. Th ean that the followng appng atfe the elaton: T q u τ u du β,.49

77 56 Dnac of Electoechancal Ste fo the cae of contant β. The evdence of that [,7] gven b dffeentaton wth epect to te of the total eneg of a te.4 T E q B q q U q and deontaton that due to the popet.47 E q T τ.5 whch afte ntegaton of.49 gve HT - H -H -β nce the total eneg HT alwa geate than zeo. The popet of pavt.49 elevant n the evdence of the tablt of dnac te dung the contol [7]. Eaple.8. et u undetae the at notaton of the odel contng Eaple.5 of a pendulu of a attached to a ldng along a hozontal ba. The dnac equaton fo th object.85 n the at fo ae the followng: l co l n l co l B q C q, q d Q D gl n Ql co D G q τ n Snce l B, l n.5 B C l n and hence the lat at ew-etc. The at of neta Bq potvel detened, whch eult fo the value of t detenant det B q l l co > A a eult, t alwa eveble egadle of the value of angle φ. The equaton of oton fo the pepectve of the lnea cobnaton of the paaete.48 can be epeented fo the followng vecto of paaete [ Θ Θ Θ Θ ] [ l ] T 4 l Θ.5 In th cae the egeo tae the fo: co n d Y q, q, q.5 co g n If the paaete Θ 4 l, whch la to Θ l, wee to be otted fo.5, whch hould be aocated wth the equeent of good falat wth denon l, the vecto of paaete would tae a hote fo

78 . Mechancal Ste 57 [ ] [ ] l T Θ Θ Θ Θ.54 Conequentl, the at of the egeo followng n co n co,, g l d q q q Y.55 Howeve, dung the pactcal poce of etatng paaete of the atheatcal odel a condeable poble a be aocated wth the detenaton of the dapng of tanent. Hence dapng coeffcent a be ncopoated nto at, q q C. A a eult n, D l D q q C. In th cae the popet.47 doe not hold. Fo the new vecto of paaete [ ] [ ] 5 4 D D l T Θ Θ Θ Θ Θ Θ the egeo aount to n co n co,, g l d q q q Y.56 In concluon, fo the atheatcal odel.5 of a pendulu ldng along a hozontal ba Fg..8 thee opton fo lneat of agange equaton.48 wee deontated on the ba of vaou tuctual paaete. Eaple.9. In a la anne we hall analze the equaton of anpulato oton fo Eaple.7. The coepondng atce.45 foed on the ba of equaton. ae peented below. The at of neta M q J q J J z co q B.57 dagonal, potvel detened and, hence, t poble to tate that t atfe the equeent.46 M q J q J J M J J J z S O a a Bq

79 58 Dnac of Electoechancal Ste The at, q M q M c q M q M q q J q M c q J q M c q q J z z z q C q.58 whee: Mq a q q M Mq a q J J q J Mq a q J J q J S O z S z O z In accodance wth.47 the at C B W a ew-etc one: q M q M c q M q J q M c q J z z W.59 Moeove, M q M l c g q G F M M q D c c F D c q F D c q F τ.6 The equaton of oton epeed n te of the lnea egeon, n epect to paaete.48, a be peented fo vaou copoton of the vecto of paaete Θ n the contet of the ntended poga fo paaete etaton o fo the pupoe of degn of contol. Fo a -denonal vecto wth paaete [ ] M l J J J J J S O z S z O z T Θ.6 the functon of the egeo tae the fo:,, g c q q gc q q q c q q c q g c a q q gc q a q a q gc q a q c c a q c q q q Y.6

80 . Electoechancal Ste 59 Othe opton fo electng the content of the Θ vecto ae alo avalable. The electon of the elatve to whethe oe of the paaete n a odel ae nown wth hgh pecon o whch hould be the ubject of a foal etaton and contaned n Θ.. Electoechancal Ste.. Pncple of eat Acton: Nonlnea Ste The pupoe of th chapte to povde gound fo the tateent about the elevance of agange equaton wth egad to electoechancal te. Anothe objectve to ndcate the vaablt of paaete n the agange functon fo te wth non-lnea paaete,.e. the one n whch the paaete ae elatve to the vecto of the te vaable. The pncple of leat acton, alo nown a pncple of tatona acton o Halton pncple found n the ajot of handboo n the feld of theoetcal echanc [8,,6] to be cental to the nthe of the law of oton. In clacal echanc t deved fo geneal popete of pace []. Eve echancal te chaactezed b functon q, q, t elatve to the vecto of veloct and genealzed coodnate and pehap te peent n the eplct fo, nown a agange functon. Th the ae functon that wa obtaned fo the devaton of agange equaton but t can alo be deved fo Galleo theo of elatvt on the ba of the geneal popete of hoogenou and otopc pace. Fo an te n oton a functonal defned, whch epeent a et of functon apped onto a et of eal nube Fg.. called an acton. Fg.. Gaphcal epeentaton of a functonal: appng S and a et of tajectoe q ~, q ~ { }

81 6 Dnac of Electoechancal Ste t S q, q, t dt.6 Th functonal a appng of a et of tajectoe { q, q ~ } t ~ nto a et of eal nube { R } whch eult fo the ntegaton of agange functon along the patcula tajectoe. All tajectoe belongng to the et have the ae cla of contnut C, whch ean that the ae contnuou and poe a contnuou devatve; n addton to whch, at the ntant detened a t,t the convege at the ae pont n pace q,q ; ee Fg... The pncple of leat tatona acton tate that fo an actual path of oton qt an acton.6 aue the q ~, q ~. Th alo allet value of all poble tajectoe belongng to the et { } ean foall, that n the tateent of peequte of an oton ealzed n natue that the ft vaaton of the functonal of acton dappea. t δ S δ q, q, t dt.64 Fo the vaaton of ntegal.64 one can futhe tate that: t t t q q dt q δ q δ.65 d Unde the aupton that δ q δq, the econd te of th ntegal can be dt ntegated b pat t t d δq dt δq q dt q t t t t d dt δqdt q.66 Snce at ntant t,t all path of oton convege at the ubequent ntal and fnal pont, then: δq t and δq t,,,. Afte condeaton of th fact the ft te.66 equal to zeo, and ntegal.65 tae the fo t t q d dt δq q dt The ndependence of vaaton of vaable vtual dplaceent, δq lead to the elaton d dt q q.67

82 . Electoechancal Ste 6 whch a fala fo of agange equaton fo a tatona te wthout an echange of eneg wth the envonent. Th eult ean that the actual ealzaton of oton occu along a tajecto fo whch the ependtue of acton.6 of the te the leat poble one. Thu, n the ene of dffeental decpton of oton fo th cae the dffeental equaton of the econd ode n the fo of agange equaton.67 ha to be fulflled. Th a fala eult; howeve, t oe lted n copaon to the equaton n.5 whch account fo etenal foce opeatng n the te. Nevethele, t wa deved on the ba of a geneal pncple, whch vald n clacal a well a elatvtc echanc [,,6] and n electoagnet a well. Th gve a old foundaton to etend the ethod of agange equaton onto calculaton of electoechancal te, whch the fundaental object of the tud of electcal dve.... Electcall Unchaged Patcle n Relatvtc Mechanc Devng fo the geneal potulate aocated wth elatvtc echanc and n patcula fo the potulate that acton S of a echancal te a not be elatve to the electon of an netal efeence te and, hence, t ha to be an nvaant of oentz tanfoaton [,6], agange functon fo a patcle can be tated n the followng fo v c c.68 Concuentl, the acton on the patcle S t t dt c t t v / c dt.69 whee: - the et a of a patcle c - the veloct of lght n vacuu Snce alwa the veloct of a patcle v condeabl lowe than the peed of lght, t poble to epand agange functon.68 o that t tae the fo of Talo powe ee of a all quantt ν/c. A a eult of uch epanon we obtan: v v c v v v c c Fo all velocte ν << c onl the ft two te of the powe epanon of agange functon ae elevant n copaon to the followng one; howeve, the ft te n the, a a contant value, doe not contbute anthng due to the ndefnte fo of agange functon.4. Hence the eult tae the fo of fala clacal eult well nown n echanc ½ ν, whch denote 4

83 6 Dnac of Electoechancal Ste netc eneg of a patcle. The calculaton of patcle oentu accodng to the geneal defnton.7 eult n: p q v v v / c whch when epanded nto powe ee wth epect to ν /c, eld that.7 v v p v v v.7 c 8 c Fo all velocte ν << c th eult n the clacal hape 4 p v.7 The calculaton of the total eneg of a patcle n accodance wth elaton.9 gve the eult, whch elevant to ou condeaton E q p vp c.74 v / c B epanon nto powe ee.74, th gve v E c v v.75 8 c Th eneg cont of et a eneg E c and the eneg aocated wth the veloct of patcle oton. The elevance of th eult aocated wth the fact that the copaon between epeon n.7 and.75 offe a concluon that agange functon fo a patcle n oton equal to t netc eneg onl fo the cae of clacal echanc. In the elatvtc echanc the epeon n.68 and.74 ae copletel dffeent. The foal eaon fo th elated to nonlneat of paaete, o oe pecel, nonlneat of a, whch nceae along wth peed a llutated b the foula fo the patcle oentu. An llutaton of th found n Fg... Fo Fg.. one can conclude that uface aea that llutate netc eneg and the uppleenta te denoted a co-eneg ovelap onl fo all velocte of a patcle n oton. Th the cae when the functon of netc eneg a hoogenou functon of veloct.4. Accodngl, fo foula.75 one can conclude that th o onl when the epanon nto powe ee a ot the te contanng ν /c. In pactce th ean that equalt between netc eneg and netc te n agange functon tae place onl n cae when the a of the patcle contant, whch epeented b a lnea te n the ene of nvolveent of contant paaete. In te wth non-lnea paaete elatve to veloct q, agange functon doe not account fo netc

84 . Electoechancal Ste 6 Fg.. Knetc eneg and co-eneg of a patcle n oton eneg; howeve, n accodance wth.7 the ntegal of te oentu tae the fo T pdq.76 pdv Th ntegal denote the netc te of agange functon wtten a T' and called netc copleenta eneg: co-eneg. It gaphcal epeentaton Fg.. tae the fo of the uface aea unde the cuve denotng the elaton between the oentu of a te and veloct. Fo both the foal pont of vew and t gaphcal llutaton t clea that netc eneg T equal to netc co-eneg T' onl fo a te wth contant paaete. Concuentl, the total eneg of patcle equal to T T cont.77 vdp T q dp.78 whch can be ntegated and the eult tae the fo.74, and the one n.75 afte powe epanon. Both functon of co-eneg.76 and netc eneg.78 add up to fo a ectangle, whch eult fo foula.74 and ha alo a gaphcal epeentaton n Fg... Th alo jutfe the defnton attbuted to co-eneg, whch tate that t coplete the functon of eneg to the bound of a ectangle. T T pq.79 The eult of thee condeaton conducted fo oton of a patcle n elatvtc echanc how that netc co-eneg T'.76 nvolved n the agange functon fo a te wheea netc eneg T not. The dtncton between netc co-eneg and netc eneg onl necea n non-lnea te whoe paaete ae elatve to genealzed velocte. In the eaned cae of a patcle

85 64 Dnac of Electoechancal Ste n oton a elatve to the veloct fo the onl vaable paaete. In engneeng pactce aocated wth ue of electc dve we have to do wth uch low velocte that the elatvtc vaaton n a ngnfcant and, a a eult, we aue that T' T fo echancal vaable. The tude conducted hee have an even oe copehenve applcaton bede theoetcal condeaton, a n electoagnet we have to do wth te wth non-lnea paaete, n patcula wth nductance of the wndng contanng feoagnetc coe, whch elatve to the cuent appled to the wndng.... Electcall Chaged Patcle n Electoagnetc Feld In contat to the pevoul condeed eaple we hall aue hee that a patcle wth et a ha an electc chage Q. The etenal electoagnetc feld o tong and t ouce o eote that the chage Q, whch caed along wth the patcle, doe not affect a change of the feld. Scala potental, t decbe the nteacton between the feld and oble chage, whle vecto potental AA, A, A z decbe the nteacton wth electc chage n oton electc cuent. It poble to defne the fou quantte decbng electoagnetc feld n the fo of a fou-vecto that tanfoed n accodance wth to the ule of oentz tanfoaton []. Fo an electoagnetc feld called a fou-potental, whee the cala potental condeed a the zeo te of th fou-potental μ A φ, A A A, t.8 Such foal notaton efeed to n theoetcal phc pove ve ueful n tanfoaton of feld and tude devoted to geneal popete of electoagnetc nteacton. Fo the pupoe of ou tud t elevant to note that agange functon fo a chaged patcle n electoagnetc feld epeed wth the ad of fou-potental of feld n the followng fo: v c QvA Qφ.8 c The ft two te of agange functon nclude netc co-eneg of a patcle wth an electc chage of Q. Th the fala echancal netc co-eneg.68 whle the econd te the agnetc netc co-eneg of the chage n oton. The te netc co-eneg ued fo the eaon of t elaton to the veloct v. Fo the two netc te we ubtact the potental electc eneg of the chaged patcle, a n the epeon of agange functon. On the ba of agange functon we can calculate patcle oentu n the followng fo v p ~ QA.8 v v / c The oentu of a chaged patcle, hence, nvolve two te,.e. the fala echancal oentu.7 and the agnetc oentu of a patcle. The te aocated wth electc feld abent fo the foula, whch coe at no

86 . Electoechancal Ste 65 upe, nce the nteacton wth the cala potental tatc. Concuentl, the total eneg.74 of a chaged patcle d c E v Qφ.8 dv v / c In concluon, the total eneg nclude the potental eneg aocated wth the cala potental of the feld bede the eneg of the patcle a. We wll poceed to ee what wll eult fo agange equaton fo a chaged patcle n the feld, whoe agange functon epeed n the fo n.8. The eaned te tatona doe not account fo echange of eneg wth envonent and the equaton of oton can tae the fo d dt.84 The ght-hand de of the equaton equal to v p ~ Q gadva Q gadφ The ft te n th equaton, n accodance wth vecto dentt [8], a be epeed b the elaton AB A B B A A B B A The epeon whch condeed hee v cont. A a conequence and hence gad va v A v ot A At the ae te, the left hand de of the equaton.84 tae the fo d dp A p QA Q Q v A dt dt t Afte the cobnaton and plfcaton of the two de of the equaton we obtan dp A Q gad φ v ot A dt t A B defnng: E gad φ H ot A.85 t to be the ubequent vecto of electcal and agnetc feld, we fnall obtan dp Q dt E v H.86

87 66 Dnac of Electoechancal Ste whch gve an equaton of oton fo a chaged patcle n the electoagnetc feld. The ght hand de of the equaton epeent oentz foce wth whch a feld nteact wth an electc chage. The content n th chapte ndcate how noton of dnac n the fo of agange equaton can be appled to poble n electodnac. The ue dealt wth hee wll be tanfeed to acocopc cale fo the applcaton n electoagnetc luped paaete te... agange Equaton fo Electoechancal Ste n the Noton of Vaance Chapte.. peent the pncple of vaance and the applcaton n epect to a ngle patcle wth an a of ndcatng that pncple of leat acton nvolve both echanc and electoechancal te. Th tateent can be futhe etended fo a ngula patcle to cove electoechancal te, ncludng acocopc techncal devce evng often fo powe geneaton and conveon. Bede achne uch a electc powe geneato and oto ued n electc dve, electoechancal devce nclude electoechancal eaueent te, evo-dve, ndutal anpulato and obot a well. In addton, oden engneeng tool nclude Mco/Nano-Electo-Mechancal Ste MEM, NEM [], whch attact a teadl gowng nteet. Such devce ae dffeent fo the tandad electoechancal dve n eveal wa depte the fact that the pncple of opeaton the ae. Ft of all, the ajo ta attbuted to the not aocated wth tanfoaton of electc powe. Intead, the tend to be ued to pefo pece acton n a coodnated and contolled anne, ncludng anpulaton and eaueent on a aco- o co- cale. In addton, the have a geate nube of degee of echancal feedo than the cae fo tandad electc dve, whee the nube of degee of feedo noall equal to one. In contat, n tandad anpulato the nube of degee of feedo aount to 5-6 [,,]. Indutal anpulato, and n patcula oble obot, have a ve cople contol and nfoaton poceng technolog ebedded n the, whch baed on a nube of ntenal and etenal eno. In addton, the have te, whch appl atfcal ntellgence algoth to poote autonoc decon egadng contol paaete. Moeove, the dvng te ae degned n a wa that poote eneg avng n ode to enable peanent opeaton ung an ntenal ouce of eneg n oe cae. Such dve ae oete efeed to a actuato, whch ephaze the ole of the dve baed on atculated jont of the devce. Moe and oe attenton and eeach dedcated to the applcaton of actuato on a co- and nano-cale n copute technolog, edcne and even botechnolog. Such electoechancal te appl a geat nube of electonc and powe electonc devce a a eult of the need to eet the equeent aocated wth fat changng oton and necet of avng powe. In fact, th banch of engneeng whch concuentl nvolve echanc, electc dve, electonc and copute engneeng fo the pupoe of contol nowada efeed to a echatonc [9,4,].

88 . Electoechancal Ste 67 The tanfe fo the pncple of the oton of patcle n electoagnetc feld to electoechancal devce ued n technolog aocated wth the need of tatng aupton egadng qua-tatona chaacte of electoechancal pocee and devng volue ntegal fo feld quantte. Thu, a tanfe can be ade to the dceet odel of the te. Th poble dealt wth n the lteatue n th feld [6,4]. The pncple of tatona leat acton ependtue fo non-tatona electoechancal te tate that t δ δa dt.87 t whee: δ - the vaance of agange functon fo an electoechancal te, δa- not a vaance of an functon but epeent wo ependtue pefoed on vtual dplaceent, whch ealze an echange of eneg wth the envonent, called vtual wo of an electoechancal te. Unde the aupton that pocee ae qua-tatona n natue, agange functon fo an electoechancal te nclude onl two tuctual coponent: a echancal one aocated wth the a and elatc tan of the eleent of the te and an electoechancal one elated to the electc chage electc feld and electc cuent agnetc feld. Howeve, t doe not nvolve a te that account fo the electoagnetc feld. In ua, t pl a total of agange functon fo the echancal and the electcal pat of the te e.88 The echancal te n agange functon epeented b the dffeence between netc echancal co-eneg T and the potental eneg U. In a la anne, electcal te of agange functon epeed a the dffeence between netc co-eneg T e aocated wth agnetc feld electc cuent and potental electcal eneg U e aocated wth electc chage. Hence, the pecfc fo of equaton.88 tae the fo T T U U.89 e e whee: T, T e - ae netc co-enege fo the echancal and electc vaable U, U e - ae potental enege: echancal and electcal. Mechancal netc co-eneg and potental eneg have alead been coveed n detal. In the cuent ecton we wll dcu agnetc netc co-eneg and electcal potental eneg whch coonl epeent the electcal pat of agange functon of the ente te. The nube of degee of feedo the algebac u, whch can be obtaned b addng the nube of degee of feedo fo the echancal and electcal pat: e.9

89 68 Dnac of Electoechancal Ste Th o nce thee aen t an pecfc vaable of electoechancal natue; the vaable ae dceet and epaate. The utual nteacton between the echancal and electcal pat of the te a well a pocee of eneg conveon ae ealzed b the o called electoechancal couplng, whch eult fo the fact that electoagnetc eneg elatve to the echancal vaable and, a a conequence, the equaton ae lned. Th ue coveed n oe detal late n th chapte.... Electc Vaable The electon of electc vaable fo chaactezng an electoechancal te pled b the fo of agange functon fo a ngle patcle n the feld.8. The decpton of potental eneg nvolve chage Q, whle the epeon of netc eneg nvolve a chage vq n oton. Hence, the popoed vaable n acocopc decpton of electoagnetc phenoena ae Q [ C].9 Q [ A] whch denote electc chage and electc cuent, epectvel. The functon of the chage n the electcal te of agange functon la to the ole of the poton n a echancal te, and Q - electc cuent an equvalent to echancal veloct, whch n fact the veloct of the chage. Such a electon of vaable natual and haonc both n te of foal lat and phcal ole n a te accountng fo coodnate n the decpton of echancal oton. The electcal vaable, ncludng electc chage Q and the te devatve,.e. electc cuent Q ae peent n electc ccut whch ae alo nown a electc netwo. The ole of the vaable of the pa decpton n electc netwo plaed b chage on the banche Q b, whch occu on the patcula banche of the netwo Q b Q Q, Q b, b, bg whee: g - a nube of banche. At the ae te, the genealzed coodnate nclude the elected chage along the banche, whch fo the vecto of genealzed coodnate wth the length e q Q Q, Q.9 e,, e The ole of contant n an electcal netwo plaed b equaton foed n accodance wth Kchhoff' ft law, whch tate that the algebac u of cuent flowng though a node of an electc netwo equal to zeo Fg..4 j Q.9 bj

90 . Electoechancal Ste 69 Fg..4 Repeentaton of contant n electcal netwo foed n accodance wth Kchhoff' ft law The neatc equaton of contant.9 ognatng fo Kchhoff' ft law can be ntegated. Snce the node n the netwo do not accuulate electc chage the eult Qbj δ Qbj.94 j The nube of the degee of feedo e n an electcal netwo elated to the nube of banche g and the nube of node w and equal to j e g w.95 The electon of genealzed coodnate ubjected to the tandad condton detened b the elaton n The tanfoatonal foulae tpe.7 ae lnea, whch eult fo lneat of the equaton of contant A a eult, the tae the ae fo fo electc chage Q, electc cuent Q and vtual dplaceent δq: Qb f qe Q f q.96 b δ Q f δq... Vtual Wo of an Electcal Netwo b Vtual wo epeent the echange of eneg wth the envonent. The ole of etenal foce plaed b electc voltage u o electootve foce e. A dpaton of eneg tae place unde the effect of cuent flow though etance R, whch fo the flowng cuent Q pla the la ole a the fcton coeffcent n etng echancal oton. e e

91 7 Dnac of Electoechancal Ste g u RQb δa δq.97 e The uaton pefoed along all banche of electcal netwo. Afte the applcaton of tanfoatonal foulae.96 and plfcaton of algebac epeon, we obtan vtual wo n the fo of the u of the te n paenthe, each of whch epeent genealzed foce fo a gven genealzed coodnate e e δa e u RlQ l δq.98 l whee: u - the total voltage fo a -th eh n a netwo R l - the utual etance of,l ehe of the netwo. Eaple.. et u appl the peented ethod fo the detenaton of the nube of degee of feedo, electon of genealzed coodnate and vtual wo fo an electcal netwo, whch coepond to the topolog of a -phae bdge ectfe peented n Fg..5. Fo the eaned netwo the nube of node w 6 and the nube of banche equal to g. Hence, the nube of degee of feedo aount to e g w 5. The banche ae aed wth aow to how the decton of cuent flow ndcated wth a plu gn. The vecto of cuent n the banche, whch pla the ole of veloct n the pa coodnate te, the followng: Q b Q, Q,, Q Fve lneal ndependent chage ae ndcated a the genealzed coodnate. Fo eaple uch coodnate nclude q q q, q, q, q Q Q Q, Q, e, 4 5,, 4 6 Q and the coepondng vecto of cuent and vtual dplaceent ae: q Q Q Q, Q, Q δ q δq δq δq, δq, δ.99 e,, 4 6 e,, 4 6 Q b Fg..5 Daga of electc netwo epeentng a -phae bdge ectfe

92 . Electoechancal Ste 7 The vecto of genealzed cuent.99 pla the ole of a et of eh cuent n the ethod of anal of electcal netwo fala fo electcal engneeng. The eh cuent coepondng to vecto e q of genealzed cuent ae ndcated n Fg..5. Th te contan 6 node eldng the elaton.9 eultng fo Kchhoff' ft law: Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q. One of the above equaton lneal dependent on the eanng one and can eve to pla the ole of a contol node. The eultng tanfoatonal foulae.96 fo the cuent tae the fo , Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q. In accodance wth.96 the ae functonal elaton ae fulflled fo vtual dplaceent , Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ. et u aue that eve banch ha a cetan etance R l, whee l denote the nube of th banch and uppl voltage n banche,, ae peented along wth the ene n Fg..5. The vtual wo n the coodnate of pa decpton,.e. wth the ue of banch cuent and voltage equal to 9 4 Q Q R e Q R Q Q R Q e A l l l l l l l l l δ δ δ δ Snce tanfoatonal foulae.-. ae lnea, the vtual wo of th te can be eal obtaned b dect ubttuton. Afte ubttuton we have: Q Q Q Q Q A δ δ δ δ δ δ Snce vtual dplaceent.99 ae ndependent, each of the te n paenthee coepond to the ght-hand de of agange equaton fo a -th genealzed coodnate. Aung at the cuent tage of the tud that the te doe not account fo pave eleent but ouce and etance, we obtan A δ. Fo the ndependence of δq,,, e t eult that all te n have to be equal to zeo. The eult fo the tateent of the equaton of oton fo the te of an electcal netwo gven n the daga n Fg..5. The equaton ae found below:

93 7 Dnac of Electoechancal Ste : : : : : R R R R R R e R R R R R R R R R R R R R R R R R R R R R R R R R R R e e R R R R R R R R R e e. Thee equaton appl the tadtonal notaton Q fo the electc cuent. The vefcaton of the equaton n. qute pobleatc. Howeve, one can eal note the et of the netwo and the coepondng electon of genealzed vaable. Th et can be peceved n equaton., whch conf the coectne of the equaton ganed. It uffcent to ubttute the ubcpt, 4 6, 7 9 n equaton, and a change n the poton wll be followed b an unchanged eult. In a la anne we can undetae the pocedue n 4, 6. Equaton can alo be tanfoed n a la anne whle eanng n the ae fo... Co-eneg and Knetc Eneg n Magnetc Feld Convete In electoagnetc convete we have to do wth low fequence of electc cuent and, a a eult, wth lowl vaable electoagnetc feld. Th enable one to conde feld a tatona and epaatel conde each of the feld coponent,.e. agnetc feld and electc one. The ouce of the ft one the vecto potental A elatve to electc cuent, whle the ouce of the latte one the cala potental dependng on electc chage.... Cae of Sngle Nonlnea Inducto et u ft conde an ndvdual nducto wth the cuent Q, whch fo the ouce of a agnetc lnage Ψ wth t col Fg..6. Depatng fo the ntantaneou powe uppled to the col eq P whee dt d e Ψ - electootve foce nduced b agnetc feld on the col, we obtan the delveed powe a the ntegal of the ntantaneou powe

94 . Electoechancal Ste 7 T e t t ~ dψ P dt eq ~ ~ dt ~ Q dt dt The above epeon ndcate that eneg of feld T e not elatve to te; howeve, t elatve to the cuent value of vaable Ψ and Q. Hence, t a functon of the tate. In the eaned ntegal t poble to undetae a change of the vaable, whch gve t Ψ ~ T Q e dψ.4 whee Ψ ~ ~ an ndependent vaable and Q QΨ. Subequentl, we have to detene agange functon fo the eaned nducto. Th functon epeented b the ntegal of agnetc feld oentu wth epect to te. The oentu of the feld fo a ngle chaged patcle n accodance wth.8 equal to QA. Fo the chage avel ovng n a conducto t poble to deve the noton of the dent of electc cuent j[a/ ] and the agange functon epeed a j e d dv dv ~ ~ α j j A j j.5 V V The foula n.5 alo elevant wth egad to non-lnea envonent wth electoagnetc feld nce the epeon fo vecto potental Aj doe not aue the upepoton fo the cuent n pace V. Th potental nvolve the agnetc popete of a ateal n pace, o that otaj B, n contat to vacuu.85, whee vecto B, H ae lneal dependent. It wa ndcated n oe detal n [6] that th vecto baed appoach can lead to the dceet odel of agange functon a a eult of the followng coue of eaonng V A j jdv A j dl jd Ψ j jd S l S whee S denote the ntenal co-ecton of the conducto whch cae the electc cuent. A a eult, we obtan Q e e Ψ ~ T Q ~ dq.6 Th agange functon equal to the co-eneg of the agnetc feld e T. Fo the elaton.6 t eult that n the dceet te the agnetc lnage Ψ pla the ole of agnetc feld oentu and, hence, fo an equvalent of the te fo agnetc oentu of a chaged patcle.8. The eneg and

95 74 Dnac of Electoechancal Ste co-eneg of the agnetc feld u up to fo a ectangle jut a the cae of the oton of a patcle.79. An llutaton of th found n Fg..6. T T ΨQ.7 e e The oentu of the agnetc feld agnetc lnage the functon of the cuent Q and can be detened wth the ue of a paaete called nductance coeffcent Ψ M Q Q.8 whee: M Q Ψ - an nceental nductance coeffcent. Q The cae la fo that of the echancal oentu of a patcle, whch can be detened ung a paaete patcle a. Fo the cae of the lnea agnetzaton chaactetc the nductance coeffcent contant. A a eult, Ψ M Q and the eneg of the agnetc feld equal to: T e Q d MQ MQ Thu, co-eneg : ΨdQ MQ.9 T e and the elaton T T fulflled. e e Fg..6 Illutaton of the elaton between the eneg T e and co-eneg T e of the agnetc feld fo an nducto wth non-lnea agnetzaton cuve One can note at th pont that thee a coplete analog to oentu, eneg and netc co-eneg n echanc fo the cae of a patcle n oton. The non-lnea cae n echanc tae place fo uffcentl hgh velocte.7, , whee the a condeabl elatve to the veloct v.

96 . Electoechancal Ste The Cae of a Ste of Inducto wth Magnetc Feld nage A ve lage nube of electoechancal devce uch a electcal achne and tanfoe have a nube of agnetcall lned wndng. Th aocated wth the uch oe effectve tanfoaton of eneg fo the cae of applcaton of ult-phae te n electc netwo and achne. Such lnage do not pla a all, o even a agnal ole; howeve, t the functon to decde on the opeatng pncple of uch devce. Hence, t eental to tae the uffcentl nto condeaton n the atheatcal odel and dung the degn of uch te. In the genealzed cae agnetc lnage aocated wth all wndng, whch can be peented n an abtact fo n Fg..7. The calculaton of eneg o co-eneg fo uch a lned te uuall poe a dffcult ta and often conducted ung poga fo electoagnetc calculaton n non-lnea envonent, whch ae baed on fnte eleent ethod o edge vaable ethod. Effcent dedcated D and D poga have been developed to handle both ethod bede acceble fee and tal oftwae. Such poga ae t poble to detene the feld n the fo of a patal dtbuton of vecto of agnetc potental A, feld ntent H, o the vecto of agnetc nducton B. It alo poble to obtan the ntegated paaete uch a co-eneg and eneg of a feld, nductance of the wndng and pondeootoc foce encounteed n a te. Fg..7 Ste of n agnetcall lned nducto In the eaned cae we have to do wth the ta concenng the tateent of genealzed fo whch enable one to detene the functon of co-eneg and eneg of a agnetc feld n the functon of genealzed coodnate and the te devatve electc cuent. The ba fo the devaton of uch foulae le n the fact that eneg and co-eneg of a feld ae the functon of the tate.4 and a uch ae not elatve to the ean ued to obtan a cetan tate tepoal functon but depend on the ntantaneou tate a detened b

97 76 Dnac of Electoechancal Ste vaable Ψ, Q. Hence fo plfcaton of the atheatcal tateent, n a te of lned nducto the patcula cuent Q hould be uccevel aed fo zeo to the fnal value, whle the eanng vaable ae beng ept at a tead level. Th conducted n ucceon fo nducto to the lat one, whch denoted a n, whle the cuent vaable denoted wth a gn placed above t nae Q ~ n ode to dtnguh t fo the eanng vaable. T e Q Q Q n ~ ~ ~ Ψ Q,,,, dq ~ ~ Ψ Q, Q,, Q,,, dq Th can be etated n a oe abbevated fo: T e n Q Ψ ~ ~ Ψ Q, Q,,, dq Q ~ ~ Q, Q,, Q,,, dq n ~ Ψ Q, Q,, Q dq n n. The calculaton of the agnetc eneg of the te of nducto a nvolve a la pocedue,.e. ntegaton of the te tpe Q d Ψ o pl the ue of foula n.7. Thu, we obtan T ΨQ. e T e Fo a te of lned nducto we defne utual nductance coeffcent M l Ψ Q. Ψl whle n the lnea cae : M l, Q l whee: Ψ l a coponent of the lnage Ψ eultng fo cuent l. The co-eneg of the feld n the lnea cae eult dectl fo elaton. and the defnton of nductance coeffcent. and cleal equal to the agnetc eneg of the feld T T e Q e Q M Q M ~ ~ M Q dq Q Q M l M Q M ~ Q dq ~ ~ ~ Q dq.

98 . Electoechancal Ste 77 B ntegaton we obtan: T T e e M Q M M Q M Q Q M QQ M Q Q Q n M M nn n n, n Q Q n M nq Q n M Q Q n Q n.4 whch can be etated oe befl a: T T e e n n M lqq l l.5 In the epeon.5 fo eneg and co-eneg of a te of lned nducto one can appl the elaton M l M l. A a eult of epanon we obtan: fo l M Q, whle fo l M lq Q l, whch lead to the coect eult uch a n Potental Eneg n Electc Feld Convete Electc potental eneg aocated wth qua-tatc dplaceent of an electc chage Q n an electc feld n the decton of the ng potental. In accodance wth.8 th eneg fo a ngle patcle equal to Q, whee, t the cala potental of a feld. Th chage condeed a a all tetng one a t telf doe not affect the cala potental. The dceet eleent elated to the accuulaton of electc eneg electc feld the capacto, whoe ablt to accuulate a chage chaactezed b paaete C called the electc capact. The capact of a capacto epeed b the elaton of the accuulated chage to the voltage between the electode n a capacto Q C.6 U whle fo a lnea capacto Q C.7 U the capact a contant value. The potental eneg aocated wth the chaged capacto can be deved fo electc powe P uq uppled dung chagng: U e t t Q ~ dq ~ ~ u dq.8 dt uqdt u ~ dt

99 78 Dnac of Electoechancal Ste Fo elaton n.8 t eult that the potental eneg elated to an electc feld n a capacto the functon of the tate and depend onl to the voltage and chage n that capacto. Fo th acocopc cae, n contat to the eleenta chage n an etenal feld, chage Q affect the potental n the chaged capacto a a eult of chagng. Hence, n.8 the elaton u Q/CQ fulflled. Q ~ Q ~ Theeb, ung.8 we obtan U e ~ dq. C Q Fo a lnea capacto th gve a fo that popula and often efeed to Q ~ Q ~ Q U e dq ~.9 C Q C..5 Magnetc and Electc Te of agange Functon: Electoechancal Couplng It now poble to detene the electoechancal te e of agange functon n an electoechancal te. It cont of the netc co-eneg T e elated to agnetc feld.6 and a te fo potental eneg U e of the electc feld.9 deduced fo th te T U. e The applcaton of agange functon n t electoechancal pat aocated wth a need of ang two ea. Fo lnea agnetc ccut wthout agnetc atuaton the co-eneg of the agnetc feld T e equal to the eneg of the feld T e ; hence, the dtncton not necea. The econd ea that n connecton wth low fequence of voltage and cuent altenaton n electoechancal te we have to do wth qua-tatona feld, whch ean that the feld ae vtuall not coupled. A a eult, we have to do wth the agnetc te of the agange functon, the electc te, whle the electoagnetc te aocated wth the coupled feld abent. Electoechancal couplng. The ablt of electoechancal convete to convet eneg occu a a eult of electoechancal couplng. Th ean that n equaton fo electcal ccut thee ae voltage eultng fo echancal oton, whle n equaton of oton fo echancal vaable dplaceent, otaton thee ae toque o foce eultng fo cuent o electc chage. The occuence of couplng aocated wth the fact that co-enege o enege of electoagnetc natue ae elatve to echancal vaable. et u conde a devce wth n coupled wndng, whoe nductance coeffcent ae elatve to a echancal vaable, fo ntance angle of otaton φ. Th the cae n electcal ult-phae achne T e n n M l l e e Q Q. l

100 . Electoechancal Ste 79 In equaton nvolvng electcal vaable th eult n the addton of anothe te n the -th equaton d dt T Q n e M d dt l l l l l n Q M n l Q l M l Q l. n whch we have to do wth two te fo voltage nduced n the wndng: the ft te coe fo vaaton of the cuent n te d l / dt and denoted a the voltage of tanfoaton, whle the othe one aocated wth angula veloct of convete oton denoted wth the te otaton nduced voltage. In the equaton fo the echancal vaable φ we have to do wth the te Te n n M l l Q Q l. It ean the toque eultng fo the nteacton of electc cuent o, accodng to a dffeent ntepetaton, fo the effect of the nteacton of agnetc feld and electc cuent. The latte ntepetaton qute elf-evdent when we tae nto condeaton a convete wth non-lnea chaactetc of agnetzaton. In th cae t necea to appl agange functon accountng fo co-eneg T e of the agnetc feld and n accodance wth. we can calculate: Te n Q Ψ ~ ~, Q, Q,, Q,,, dq.4 In a la anne, f the capact of a capacto n a convete wth electc feld elatve to a echancal vaable, fo eaple the dtance between the electode, we have alo to do wth electoechancal couplng. The electcal eneg of a capacto wth a oble electode epeed b foula Q U e.5 C whee a echancal vaable. In th cae n the equaton fo the electc vaable we have an electc te: U e Q Q C a a eult of whch the voltage elatve to the poton of an electode. In the equaton fo a echancal vaable we obtan the followng te:

101 8 Dnac of Electoechancal Ste U e Q Q C C U C C.6 whch denote the echancal foce of attacton of the oble electode of a capacto epeed n [N]. Th confed b the denonal anal of foula.6: F A VA J V V [ N] V The la pncple apple to the electotatc voltete fo eaple. Electoechancal couplng ae encounteed n all eaned convete nce the conttute the dea govenng the opeaton...6 Eaple of Applcaton of agange Equaton wth Regad to Electoechancal Ste Eaple.. Fo an electcal te fong a netwo peented n Fg..8 we wll tate dffeental equaton of oton ung agange equaton ethod. The luped paaete R,,C n th netwo ae contant. The eleent peent n the banche of the netwo n Fg..8 ae nubeed n accodance wth the nube of banche n th netwo. The nube of banche aount to g 5, whle the nube of node w. Hence, the nube of the degee of feedo equal to g w Fg..8 Electcal netwo wth luped paaete The vecto of the coodnate of the pa decpton.5,.e. chage Q n the patcula banche tae the fo T [ Q, Q,, Q5 ] and, b paallel, the vecto of electc cuent Χ.7 Q and vtual chage δq ae:

102 . Electoechancal Ste 8 T T Q Q Q Q Q Q ],,, [ ],,, [ 5 5 δ δ δ δ Χ Χ.8 The electon of the genealzed coodnate can be undetaen n vaou wa. The electon of cuent n the banche wth nductance a genealzed velocte lead to the decouplng of the equaton of oton wth epect to the devatve of the cuent and, a a conequence, we obtan a dagonal at of nductance. Gven uch electon, we have T T T Q Q Q Q Q Q Q Q Q ],, [ ],, [ ],, [ δ δ δ δ q q q.9 The contant n th te eult fo the equaton fo cuent etablhed fo the Kchhoff ft law.9. Two of the eve fo the calculaton of tanfoatonal foulae.96, whch n the eaned cae lead to the elaton 4 5 Q Q Q Q Q Q. Fo the thd node, whch wa not appled fo the devaton of tanfoatonal foulae fo cuent., t poble to undetae the vefcaton of the foulae b checng whethe the balance wll be zeo a a eult of etateent of th foula n the fo of cuent beng genealzed velocte. The equaton n. can be ntegated and dectl lead to the devaton of tanfoatonal foulae fo electc chage and vtual dplaceent of the chage: Q Q Q Q Q Q Q Q Q Q Q Q δ δ δ δ δ δ. agange functon fo the eaned te tae the fo: C Q Q C Q C Q Q Q Q Q U T e e. Th agange functon ha alead been epeed n genealzed coodnate ung.. The vtual wo n the coodnate te of the pa decpton tae the fo: Q Q R e Q Q R e Q Q R Q Q R Q R Q u A δ δ δ δ δ δ Afte the applcaton of tanfoatonal foulae.-., we obtan:

103 8 Dnac of Electoechancal Ste,, ~ P Q Q Q R Q R R e e Q Q R Q R Q R R R e Q Q R Q R R u A δ δ δ δ δ. The coectne of thu calculated vtual wo can be vefed b ung cloed loop cuent ethod. The equaton of oton fo th electcal netwo can be foulated n the followng anne: º fo ~ : P Q Q dt d Q q Q R Q R R u C Q Q Q dt d o: R R R u C Q Q dt d º Slal, fo: ~ : P Q Q dt d Q q R R R R R e C Q Q C Q C Q Q dt d º and fo: ~ : P Q Q dt d Q q R R R e e C Q Q dt d Thee equaton hould be copleented b the followng defnng equaton: 4º - 6º 4 4 ; ; Q Q Q.5 et u futhe aue that between the nducto and thee a agnetc lnage n the fo dafted n Fg..9. Th a negatve lnage decbed b utual nductance M. Th ean that the cuent n the ft and thd banch n the decton of cuent flow aed wth an aow lead to the deceae of the eneg of the agnetc feld. Accodngl, the new fo of agange functon ~ the followng ~ Q Q M

104 . Electoechancal Ste 8 whee the agange functon.. A a conequence, the equaton º and º n the te n.4 have changed: M u R R Q RQ dt dt C d M dt e R 5 d R R d d Q Q dt C 5 Q Q Q C 5 R R 4 Q Q C The eultng equaton ae lned b the devatve wheea the eanng equaton n the te.4-.5 ean unchanged. Fg.. and. peent eaple of wavefo fo th netwo uppled wth contant voltage of u 8 [ V], e 4 4 [ V] and e 5 [ V]. The value of the paaete ae a follow: R C [ Ω] R [ Ω] R 5 [ Ω] R4 [ Ω] R5 4 [ Ω] [ H ] [ H ] 4 8 [ H ] [ μf ] C 5 [ μf ] C [ μf ] 5 Fg..9 Magnetc lnage of wndng and n the netwo n Fg..8 Fg.. Cuve of electc cuent fo the netwo n Fg..8 unde contant uppl voltage, wtched unde zeo ntal condton

105 84 Dnac of Electoechancal Ste Fg.. Cuve of voltage on capacto fo netwo n Fg.8 unde contant uppl voltage Followng Fg.. and. peent the cuve fo wtchng unde altenatng voltage u wth the apltude of U 8[V], f 5 [Hz], φ.6, whle the eanng voltage ae contant e 4 4 [V], and e 5 [V] and the ccut paaete ean unchanged. Fg.. Cuve of electc cuent fo netwo n Fg.8 afte wtchng altenatng voltage u, whle the eanng voltage ae contant and ccut paaete unchanged Fg.. Cuve of voltage on capacto fo netwo n Fg..8 afte wtchng altenatng voltage u whle the eanng voltage ae contant and ccut paaete unchanged

106 . Electoechancal Ste 85 Eaple.. The eaned electoechancal te a etcal clnde of a wth a col wound aound t. Th clnde ollng down along an nclned plane wth an angle of α. The col cont of z tun and fued n contact nto the de uface of the clnde n a wa that doe not ntefee wth potental oton of the clnde. The clnde ade of non-agnetc ateal whoe agnetc peeablt equal to that of the a. The oton occu n a contant agnetc feld, whoe nducton vecto B pependcula to the bae of the plane. Ft, we wll etablh the equaton of oton fo the te, whoe neatc daga peented n Fg..4. et u aue that the oton lple and the a of the clnde fo the duaton of the oton paallel to a whch goe along the hozontal edge of the plane. Th te chaactezed b the followng paaete: - a of clnde ncludng col J - clnde oent of neta n elaton to the cental a - clnde adu l - length of clnde along the a R - etance of the wndng w - nductance coeffcent of the wndng z - nube of the col tun α - angle of plane nclnaton Fg..4 Clnde wth col wound aound t on an nclned plane unde contant agnetc feld Unde uch aupton the contant poed on the te ae holonoc and the te ha two ndependent vtual dplaceent: δφ - fo the ev olute oton of a clnde δq - fo the electc chage n an col Fo the above we can deve genealzed coodnate n the fo q, Q.7 The vtual wo of the te δ A D δ RQ δq

107 86 Dnac of Electoechancal Ste and the netc echancal eneg J T.8 Knetc co-eneg of the agnetc feld, whch an equvalent to the agnetc eneg due to lneat of the envonent, ha two te and jut a the feld a upepoton of two feld: one fo the cuent Q n the col and the etenal feld wth contant flu dent B. T T Q Q Ψ Q Q Ψ co.9 e e w z w z whee: Ψ z φ - denote the flu aocated wth the col ognatng fo the etenal feld. The potental eneg U aocated wth the a of the clnde n the gavtatonal feld gh g h nα.4 U The above elaton contan the followng te: - ntal value of the angle of evoluton h - ntal elevaton of the clnde cente of a S above the ount Ψ z Blz - au value of the etenal feld lned wth the col. Afte tanfoatonal foulae ae taen nto account,, agange functon n genealzed coodnate tae the fo: T e T U J.4 Q Q co g h nα w Ψz Hence, the eultng equaton of oton ae the followng d º q : D dt hence: J Q Ψz n g nα D Q Ψz n g nα D J d º q Q RQ :.4 dt Q Q whee: d dt d dt Q Ψ co w w Ψ z w z Q T R / w w n T w RQ w w

108 . Electoechancal Ste 87 The epeon: Μ e w Ψz n.4 denote electoagnetc toque whch bae the clnde dung t ollng oton down the plane. The ean value of th toque dffeent fo zeo nce the cuent w n the col the altenatng cuent. It poble to elect Ψ z n uch a wa that the ean veloct Ω av contant and n that cae the te functon of the cuent w peodcal, a eultng fo.4. Gven an appoated cuve of the cuent w n the tead tate n the fo w n Ωavt θ and fo the otaton angle φ Ω av tφ, t poble to detene electoagnetc toque actng upon the clnde fo the foula.4 Μ e Ψ z co Ω t θ co Ω t θ av av.44 The bac coponent of th toque fo ae the contant toque and the vaable one coepondng to the double of the ean angula veloct: θ coω θ Μ e Ψz co avt Fo the atheatcal odel of the te peented n.4 the paaete have been calculated fo a hollow clnde wth the dent of ρ. [g/dc ]. The bac denon of the tuctue ae peented n Fg..5. The paaete of the te ae followng: S Cu.5 [c ] - co-ecton of the wndng 7. [g] - a of the clnde z 5 - nube of tun n the col R 4.5 [Ω] - etance of the wndng w.4 [H] - nductance of the wndng J. [N ] - cental oent of neta of the clnde J z J [N ] - oent of neta of the clnde n epect to tangenc lne of ollng oton.75 [] - adu of the clnde cz.4 [Wb/T] - contant value whch elate flu wth flu dent α. - angle of plane nclnaton The llutaton of the eult of copute ulaton fo th te ae peented n Fg The nvolve two tpe of oton. Fo the ft et Fg the oton tae place unde a tong agnetc feld,.e. fo B.5 [T] and tatona n the ene of poeng peodc chaactetc.

109 88 Dnac of Electoechancal Ste Fg..5 Scheatc daga of clnde contanng col Howeve, t cleal non-unfo n the functon of the angle of evoluton and the cuent n the wndng dffe uch fo the nuodal hape due to the condeable effect of the angula veloct, whch peodc but fa fo tead. Unde a weae feld B. [T] the oton uch fate but alo tatona, veloct ha a vaable te wth a lowe value and the cuent peent a cuve oe endng a nuod. Fg..6 Angle of clnde evoluton fo flu dent B.5[T] Fg..7 Cuent cuve n col fo flu dent B.5[T]

110 . Electoechancal Ste 89 Fg..8 Angula veloct of clnde oton fo flu dent B.5[T] Fg..9 Electoagnetc toque deceleatng clnde oton fo flu dent B.5[T] Fg..4 Angle of clnde evoluton fo flu dent B.[T] Eaple.. Dnac of a contacto wth electoechancal dve. Fg..44 peent a odel of a contacto wth non-lnea agnetzaton chaactetc. It cont of an electoagnet, ovable jupe of a, a te of contant png wth tffne, coeffcent of vcou dapng D, and png and dape actng wthn the ange of all wdth of the gap fo deceleatng and

111 9 Dnac of Electoechancal Ste Fg..4 Cuent cuve n col fo flu dent B.[T] Fg..4 Angula veloct of clnde oton fo flu dent B.[T] Fg..4 Electoagnetc toque deceleatng the oton of clnde fo flu dent B.[T] dapng the elatc collon of the jupe agant the coe. The tffne coeffcent ae defned a and and dapng one a D and D. An adequate electon of the png and dape pla a e ole n the ecung the coect opeaton of the electoechancal te of the contacto bede the chaactetc of agnetzaton n the functon of the wdth of the gap a peented n Fg..45. Th te ha two degee of feedo: one aocated wth the echancal oton and the othe one fo the electc chage n the col of the wndng. Hence, the genealzed coodnate follow n the fo

112 . Electoechancal Ste 9 q, Q whee: - the wdth of the a gap Q- the electc chage n the wndng Fg..44 Electoechancal te contng of a contacto wth a ovable jupe Fg..45 Chaactetc of agnetzaton of contacto ccut fo vaou value of a gap fo the openng to clong of the gap agange functon fo the te nvolve the netc co-eneg of the agnetc feld, netc eneg of the oveable jupe and potental eneg of the jupe n the gavtatonal feld a well a potental eneg of the te of png gven b the functon U. Q ~ ~ Ψ Q, dq g c U.45

113 9 Dnac of Electoechancal Ste d d d d d d d d d U.46 The elaton n appl the followng bol: - a of the ovable jupe d, d, d - contant whch detene the fee length of the png c - a contant. The vtual wo of the te nvolve the echange of eneg wth the envonent and contan two te: an electcal and a echancal one Q P P A A A Q e δ δ δ δ δ.47 The echancal te of vtual wo contan vcou dapng coeffcent D eultng fo the oton of the jupe and dape wth condeable dapng coeffcent D and D, whch account fo dapng of the pact between jupe and electoagnetc coe fo all gap wdth jut a fo the cae of the png d D D D d D D d D P A δ δ δ δ δ.48 The electc te of vtual wo foulated fo the ccut of the col Q RQ u A e δ δ.49 whle the enegzng povded fo n two wa: b AC and DC wth the voltage conveted fo a doublng ectfe co co α ω α ω t U t U u.5 The equaton of oton fo the echancal vaable tae the fo: º : P dt d q and afte the ntoducton of agange functon.45: Ψ Q P U g dq Q dt d ~ ~, whch can be tanfoed to tae the fnal fo:

114 . Electoechancal Ste 9 d D D D d d d d D D d d d D d g F e.5 whee: Ψ Q e Q d Q F ~ ~,.5 - the pull foce of the electoagnet. The equaton fo vaable Q - electc chage eult fo the geneal fo: º Q P Q Q dt d Q q : o: Q R u Q dt d Ψ, Afte dffeentaton, we obtan RQ u Q Q Q Q Ψ Ψ,,.5 In the equaton of oton we have to do wth devatve of the chaactetc of agnetzaton wth epect to the cuent of the col Q Q Ψ, and wth epect to the wdth of the a gap Q Ψ,, whch afte ntegaton.5 coepond to the pull foce of the actuato. In the eaned cae of the chaactetc of agnetzaton Fg..45 appoaton wa caed out analtcall ung plne functon of the thd degee, than to whch the devatve accodng to the col cuent ae contnuou. Howeve, the devatve of the agnetzaton chaactetc wth epect to the wdth of the gap ae calculated a the dffeental value Δ ΔΨ / b lnea appoaton of the quotent between the patcula chaactetc. Fg..46 peent the foce of jupe pull calculated n accodance wth elaton.5 n the functon of the elatve wdth of the gap, epeed a the pe cent of the gap wdth n the tate of coplete openng of the contacto, fo eveal value of the cuent, [A],, 5 Q. On the ba of the atheatcal odel wth the equaton of oton.5-.5 and chaactetc of agnetzaton fo Fg..45 a nube of ulaton wa conducted fo the eaned te of the contacto wth electoechancal dve nvolvng huttng of the ovable jupe afte the voltage appled. The calculaton wee pefoed fo the uppl of both altenatng cuent a well a the voltage conveted fo a doublng ectfe, whch lead to the flow of dect cuent n the wndng wth onl a all altenatng coponent. The appopate opeaton of a contacto aocated wth the necet of an adequate electon of uppl

115 94 Dnac of Electoechancal Ste voltage and paaete egadng both png and dape. Fo the uppl of the contacto wth altenatng voltage the apltude dung wtchng t on U [V], whle afte the jupe ha hut, the value nceae to U 4 [V] n ode to antan the cloue tate. Fg..46 Chaactetc of pull foce of contacto n the functon of elatve wdth of a gap fo contant value of col cuent Q 5,,, [ A] Fo the cae of uppl wth ectfed voltage the tuaton eveed: fo the wtchng on of contacto the apltude of altenatng voltage undegong ectfcaton equal to U [V] whle afte the huttng of the jupe educed b a half. The vaou value of the uppl voltage ae necea to peeve the appopate opeaton of the contacto dung the huttng of the jupe and ubequentl to eep t n the cloed tate. The eult fo the equed flu dent n the a gap and an ndpenable value of the cuent necea to ntate the fat oton of the jupe and ubequentl to t table antenance n the cloed tate, ovecong tong png puh, wth onl all powe loe. Fg..47 Cuent wavefo fo poweng on contacto wth altenatng voltage [V] dung huttng of the jupe and wtchng voltage to 4 [V] afte huttng

116 . Electoechancal Ste 95 Fg..48 Veloct cuve of the jupe fo poweng on contacto wth altenatng voltage 4 [V] dung huttng of jupe and ubequent wtchng of voltage to 4 [V] afte huttng Fg..49 Poton of the jupe wdth of a gap fo poweng on contacto wth altenatng voltage fo condton peented n Fg..47 and.48 The peented llutaton adequatel chaacteze the opeaton of a contacto unde the uppl of altenatng voltage and ndcate a need to nceae the voltage afte huttng of the jupe n ode to enue the uffcent foce fo the antenance of the cloed poton unde all ocllaton of the poton. Fg..5 Electoagnet pull-n foce fo poweng on contacto wth altenatng voltage [V] dung jupe huttng and wtchng voltage to 4 [V] afte huttng

117 96 Dnac of Electoechancal Ste The ntant of wtchng the voltage to a hghe one ndcated b all peented cuve; howeve, t ot cleal dcenble fo the cae of the cuve of cuent, veloct of the jupe and the foce of electoagnetc pull. The cae of the applcaton of ectfed voltage to the contacto llutated n Fg Fg..5 Cuent wavefo fo the contacto to whch ectfed voltage of apltude [V] wa appled dung huttng and wtched to 6 [V] afte cloue of jupe Fg..5 Poton of the jupe fo poweng on contacto wth ectfed voltage of apltude [V] dung huttng and wtched to 6 [V] aftewad Fg..5 Jupe veloct fo enegzng contacto wth ectfed voltage of apltude [V] dung jupe huttng and wtched to 6 [V] afte huttng

118 . Electoechancal Ste 97 Fg..54 Pull-n foce of the electoagnet fo enegzng contacto wth ectfed voltage fo condton decbed a n pevou fgue Eaple.4. Paallel plate oton electotatc actuato. Fg..55 peent the daga of an electoechancal actuato wth a paallel plate oton n a capacto. In the dcued te the ovng electode of a and uface S ove n the vetcal decton and ubjected to the foce of gavt. The delectc found between the plate of the capacto ha a delectc contant ε, whch n the calculaton egadng the oton of th actuato equal to the delectc contant of vacuu ε. The ccut of the actuato uppl nclude: etance R, addtonal capacto wth the capact of C and a all nductance of the ccut. The ole of the capact C cont n the tablzaton of the oton of the electode and, oe pecel, an nceae of the ange of the tatona opeaton of the actuato. Inductance onl aue a all value nce t foed b the nductance of the enegzng we. In addton, tude have hown that the applcaton of appopate ethod of ntegaton fo tff dffeental equaton ae t poble to degad th nductance. Fg..55 Daga of electotatc actuato wth paallel plate oton n vetcal decton Unde the aupton of the full et of the te and eultng paallel oton of the ovable electode, the te ha onl two degee of feedo: one fo the echancal oton decbed b genealzed coodnate, whch denote the dtance between the electode and the othe one fo electc chage defned

119 98 Dnac of Electoechancal Ste b the genealzed coodnate Q. Hence, the vecto of genealzed coodnate tae the fo q, Q.54 agange functon fo the te decbed n the genealzed coodnate defned a: Q Q Q g.55 C C whee: - the fee length of the png Q Q Q - electc chage of each of the capacto; electcal genealzed coodnate ε S C - capact of a capacto wth oveable electode actuato ε S C C - nonal capact of the actuato fo. The vtual wo of the te, whch ealze the echange of eneg wth the envonent fo the eaned te, equal to δ A u RQ δq D δ.56 The dffeental equaton of oton of the te ae followng: fo q d P dt º d Q g D dt C A a eult, th gve: F c Q g D ε S whee F c the electotatc foce of pull of the oveable electode; fo q Q d PQ dt Q Q º F c d Q Q u RQ dt C C.57

120 . Electoechancal Ste 99 whch eld: C C Q Q u RQ C C C Followng the tateent that: C /C α and b aung that one obtan d dt C.58 C Q C α u R.59 Fo the cae when we aue that, the equaton of oton fo the electc ccut tae the followng fo: dq dt Q u T α c.6 R dq whee: T RC c dt Statona tate, tablt of the te. Unde the aupton that the tatona tate et fo cont, Q cont, fo equaton.57,.59 we can etablh the condton fo the tablt of the te: whee: F F F.6 Fc Q ; F ; ε S c g F g g UC Concuentl, Q z.6 α z The eanaton of the poble oluton to equaton.6 n the functon of the dtance between the electode and fo actuato paaete, uch a the uppl voltage U and α C /C, lead to the gaphcal epeentaton of foce F c and F n Fg..56 and.57. The pont of nteecton epeent the poble equlbu pont fo the te fo F g, whch denote the actuato n the hozontal oton. Thee ae the thee poble cae of the actuato opeaton ange < z detened b vaable z /. The ft cae elevant fo the tuaton when thee ae two pont of nteecton of cuve epeentng foce F c and F. The tatona pont of opeaton the one fo whch the elaton Fc F >.6 z z fulflled, nce n uch a cae the eultant foce etoe the te to t ntal poton afte t ha been put out of balance. Th condton atfed fo the

121 Dnac of Electoechancal Ste pont of the nteecton between cuve fo F c and F, whoe locaton cloe to the value of z. In the econd cae thee jut a ngle pont of the nteecton of the cuve wthn the aea of actuato opeaton, whch tatona, a t fulfll the condton n.6. The thd cae encounteed when thee ae no pont of nteecton between the two cuve and the eaon fo th too hgh a voltage appled to the te followed b a coplete hot ccut between the plate n the actuato. Fg..56 peent tatc chaactetc fo α. and vaou value of the voltage U 4, 5, 6, 64.5, 7 [V]. The lo of the tablt of the opeaton fo actuato tae place fo U 66 [V]. Fg..57 peent the cuve fo a ngle value of the uppl voltage U 5 [V] and ato of the capacte equal to α.8,.9,.,.,.4,.6. Fo α.4 thee ae two nteecton pont: a tatona one fo z.6 and a non-tatona tate fo z.. In ode to eane oe pecel the contol of actuato b ean of changng the uppl voltage, that, n ode to detene the chaactetc, the equaton.6 ha been tanfoed ung.6 to tae the fo β U C /.64 whee: β g - - α / - a degn contant. Th equaton.64 nvolve vaable, whch a lnea functon of the dtance between the electode of an actuato and depend alo on the ato α z α / α.65 The calculaton of chaactetc and cuve fo dnac tate that follow have been conducted fo actuato wth the followng paaete:. [N] - fo a hozontal oton actuato. [N] - fo a vetcal oton actuato.4 [g], 5 [], C.8 [pf], S [c ], ε ε 8.85 E- [C/V], g 9.8 [/ ] Fg..56 Chaactetc of electotatc foce F c and png foce F n the functon of the elatve dtance z between electode fo α. and fo uppl voltage U 4,5,6,64.5,7 [V]. The cae of the hozontal oton actuato wthout the effect of gavt

122 . Electoechancal Ste Fg..57 Chaactetc of electotatc foce F c and png foce F n the functon of the elatve dtance between the electode fo uppl voltage U 5 [V] and α.8,.9,.,.,.4,.6. The cae of the hozontal oton actuato The fee length of png ha been elected n uch a wa that n the ntal tate unde zeo voltage the wdth of the gap, hence g /.66 The vetcal oton of the actuato a dawbac nce t aocated wth the need to ntall a elatvel tff png, whoe fee length.66 hould not be dculoul lage n copaon to the tavel of the actuato,.e. n the ange <, whch coepond to 5 [] n the eaned cae. Hence, the lage tffne of the png fo a vetcal oton te eult n the need to uppl hgh voltage to contol the actuato. The hozontal oton te need a elatvel le tene png and th eult n the condeabl lowe voltage fo the contol of the ovng electode. Fg..58 peent the chaactetc of contol n the functon of voltage U fo a te n vetcal oton accountng fo gavt,. [N/] fo vaou value of paaete α C /C. The lne that goe aco the cuve the bounda lne of tablt of the contol. If the bounda eceeded b the ovng electode a hot-ccut wth the peanent electode follow. The net Fg..59 peent la chaactetc fo a hozontal oton actuato wth no gavt,. [N/]. The chaactetc ae la, howeve, fo a te n the hozontal oton of the electode the contol voltage tend to be condeabl lowe. The two chaactetc ndcate that the ue of a capacto C put n a ee wth an actuato eult n a benefcal etenon of the ange of the balanced contol and poved n te of the pecon; howeve, t aocated wth a need of applng hghe voltage. The dnac cuve have been odeled on the ba of the equaton of oton and olved ung Roenboc pocedue fo tff dffeental equaton te. Such a need eult fo the lage pan of the te contant value encounteed n the te, whch aocated wth the all value of capact C.

123 Dnac of Electoechancal Ste Fg..58 Contol chaactetc z fu fo a capact vetcal oton actuato ubjected to gavt foce, fo vaou value of paaete α.,.4,,.4. The lne llutatng the bounda of the tablt peented Fg..59 Contol chaactetc z fu fo a capact actuato a n Fg..58, but fo a hozontal oton te whch apple a le tene png The applcaton of th pocedue ae t poble to degad paaete, whch denote edual nductance n the ccut and, conequentl, calculate chage Q of the capacto fo equaton.6. Thu, one gan ve la cuve of the actuato oton n both cae. The et of cuve n Fg peent the actuato oton opeatng n a hozontal oton te fo a lowl nceang uppl voltage u U Δu t 5.5t [V], fo α.. The lo of tablt ecoded fo voltage aound 7 [V],.e. fo a value la to the one ganed fo tatc chaactetc Fg..59, whch aount to U 66 [V].

124 . Electoechancal Ste Fg..6 Wavefo fo chage Q n the functon of te fo lowl nceang voltage fo the hozontal oton actuato Fg..6 Wavefo fo cuent Q n the functon of te fo lowl nceang voltage fo the hozontal oton actuato In contat, one can copae the eult fo the ae actuato fo a fat nceang voltage,.e. u U Δu t 5.5t [V], fo whch cae the lo of the tablt occu fo the voltage of U 6 [V] afte aound 5 [], uch above the thehold of tatc tablt. Fg..6 Dtance between electode n the actuato n the functon of te fo lowl nceang voltage fo the hozontal oton actuato

125 4 Dnac of Electoechancal Ste Fg..6 Veloct of oton of ovng electode n the actuato n the functon of te fo lowl nceang voltage fo the hozontal oton actuato Fg..64 Tajecto of chage Q fo voltage u U u t 5.5 t [V] low nceae n the hozontal oton actuato Fg..65 Wavefo of chage Q n the functon of te afte wtchng on voltage U [V] n an vetcal oton actuato Fo the actuato n the te n vetcal oton the ole of the gavt pull condeable; hence, t necea to appl oe tene png to pevent the ue of too long one.66. The fgue that follow peent the cuve of the oton of the ovng electode afte an abupt applcaton of the voltage of U [V], α.. Th value eceed a lttle the value of U 9 [V] detened on the

126 . Electoechancal Ste 5 Fg..66 Poton of the ovng electode n the functon of te afte wtchng on voltage U [V] n the vetcal oton actuato Fg..67 Veloct of the ovng electode n the functon of te afte wtchng on voltage U [V] n the vetcal oton actuato Fg..68 Tajecto of electc chage afte wtchng on voltage U [V] n the vetcal oton actuato and lo of tablt of the poton of the ovng electode. The bounda of the tablt fo th cae U 9 [V] ba of equaton.64 a the value of the tatc tablt. In th cae we have to do wth a fat lo of tablt wthn 4 [] and hot-ccut between the electode of the actuato.

127 6 Dnac of Electoechancal Ste The peented chaactetc and dnac cuve of the electoechancal actuato wth vaable wdth of the a gap ndcate the etent of the dffcult aocated wth the contol of uch an actuato. It tend to opeate bette n the te wth the hozontal oton of the electode and fo table opeaton t eque the ue of a capacto n ee whch affect the ange of the contol n a ve benefcal anne. The cot to cove a a eult of the ue of uch a capacto wth a adequate capact α..4 aocated wth the need to uppl hghe value of the voltage fo the contol of the actuato dplaceent. Capact actuato pla nowada nceang ole n vaou McoElectoMechancal Ste MEMS le head potonng te o co wtche etc.[,4,7,,4,5]. Refeence [] Bahan, I.B., Geen, R.J.: Coputatonall-effectve ecuve agangan foulaton of anpulaton dnac. Int. J. Cont. 54, [] Bhuhan, B. ed.: Spnge Handboo of Nano-technolog. Spnge, Beln 4 [] Bloch, A.M., Rehanoglu, M., McClaoch, N.H.: Contol and tablzaton of nonholonoc dnac te. IEEE Tan. Aut. Cont. 7, [4] Catane,.M., Sentua, S.D.: Speed-eneg optzaton of electotatc actuato baed on pull-n. IEEE J. Mcoelectoech. S. 8, [5] Deeno, A.: Moveent ulaton n fnte eleent odel of electc achne dnac. IEEE Tan. Mag., [6] Feathetone, R.: Rgd Bod Dnac Algoth. Spnge, Hedelbeg 7 [7] Fujta, H., Suzu, K., Ataa, M., et al.: A coactuato fo head potonng te of had dc dve. IEEE Tan. Mag. 5, [8] Geenwood, D.: Pncple of Dnac. Pentce-Hall, Englewood Clff 988 [9] Hung, E.S., Sentua, S.D.: Geneatng effcent dnacal odel fo coelectoechancal te fo a fnte-eleent ulaton un. IEEE J. Mcoelectoech S. 8, [] Kobaah, M., Hoowtz, R.: Tac ee contol fo had dc dual-tage evo te. IEEE Tan. Mag. 7, [] Kozłow, K.: Modelng and Identfcaton n Robotc. Spnge, Beln 998 [] anczo, C.: The Vaatonal Pncple of Mechanc. Dove Publcaton, NewYo 986 [] andau,.d., fchc, E.M.: Mechana, Eletodnaa. Naua, Mova 969 [4], Y., Hoowtz, R.: Mechatonc of electotatc actuato fo copute dc dve dual-tage evo te. IEEE Tan. Mechat. 6, [5] McCath, B., Ada, G.G., McGue, N., et al.: A dnac odel, ncludng contact bounce, of an electotatcall actuated cowtch. IEEE J. Mcoelectoech. S., 76 8 [6] Nea, Y.I., Fufaev, N.A.: Dnaa negolononch te. Naua, Mova 967 [7] Otega, R., oa, A., Nclaon, P.J., et al.: Patvt-baed Contol of Eule- agange Ste. Spnge, ondon 998 [8] Paul, C.R., Naa, S.A.: Intoducton to electoagnetc feld. McGaw-Hll, New Yo 98 [9] Pon-nn, J., Rodgez, A., Catane,.M.: Voltage and pull-n te n cuent dve of electotatc actuato. IEEE J. Mcoelectoech. S., 96 5

128 Refeence 7 [] Sclano, B., Khatb, O. ed.: Spnge Handboo of Robotc. Spnge, Hedelbeg 8 [] Sobcz, T.J.: An eneg baed appoach to odelng the agnetc non-lneat n AC achne. Achve El. Eng. 48, [] Su, C., Stepaneno, J.: Robut oton foce contol of echancal te wth clacal nonholonoc contant. IEEE Tan. Aut. Cont. 9, [] Uhl, T., Bojo, T., Moze, Z., et al.: Rapd pototpng of echatonc te. J. Theo. Appl. Mech. 8, [4] Whte, D.C., Woodon, H.H.: Electoechancal Eneg Conveon. John Wle & Son, New Yo 959 [5] You,., Chen, B.: Tacng contol degn fo both holonoc and nonholonoc contaned echancal te. Int. J. Cont. 58,

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130 Chapte Inducton Machne n Electc Dve Abtact. Chapte deal wth electcal dve wth nducton achne. Afte hot ntoducton concenng bac contucton vaant and geneal poble of odelng nducton oto dve, atheatcal odel ae developed on a ba of pevoul ntoduced agange ethod. Model tanfoed n a clacal wa ae peented b ue of othogonal tanfoaton along wth the opton govenng fee paaete of thee tanfoaton. Subequentl, othe odel ae developed, n whch onl one de electcal vaable ae tanfoed tato o oto, whle the othe de ean untanfoed, wth natual vaable. Thee odel ae appled n ulaton and peentng vaou poble of dve te wth electonc powe convete and fo othe etenal aet cae. Soe of the ae clacal dve poble le DC beang, opeaton of Schebu dve o oft-tat te. Othe concen oden te wth electonc powe convete, oe pecel and uccefull odeled and coputed n th wa. The ae dve wth two- and thee-level voltage ouce nvete ued n PWM contol SVM, DPWM a well a cuent ouce nvete CSI. Thee nd of odel ae alo appled to peent vecto contol VC o feld oented contol FOC and dect toque contol DTC of nducton oto dve. Aboveentoned poble ae wdel llutated b eaple coputed fo vaou dnac tate, wth and wthout autoatc contol, and fo that pupoe fou nducton oto of dffeent ated powe ae peented. Fnall a poble of tuctual lneazaton of nducton oto dve coveed, bede a nube of ueful tate obeve te that ae dcued.. Matheatcal Model of Inducton Machne.. Intoducton The hto of contucton and applcaton of nducton oto n electc dve date bac a fa a ove ea and the nducton achne conttute the bac unt enegzed fo altenatng cuent n a etcal thee-phae powe dtbuton te. Bede the thee-phae laout oe all ze achne can be uppled fo a ngle phae fo houehold applcaton and two-phae achne can be ued n the dve of evoechan. The pactcal eanng of theephae nducton achne ephazed b the fact that the conue neal 7%

131 Inducton Machne n Electc Dve of the geneated electc powe. The nae gven to nducton oto ognate fo the ngle ded powe uppl to uch achne ot fequentl occung fo the decton of the tato. Inde the oto thee a fed wndng whch geneate electc powe a a eult of nduced voltage eultng fo agnetc flu that change otate n elaton to the oto wndng. The utual nteacton of agnetc feld n the a gap wth the cuent n oto wndng lead to the ogn of electoagnetc toque T e, whch et the oto n oton. In the nducton oto the a gap between the tato and oto a naow a t techncall poble fo the equeent of the echancal tuctue, nce the eneg poweng the oto b the agnetc feld ha a condeabl hgh value. Th feld hould have a hgh value of flu dent n the gap whch eque an adequate agnetzng cuent, that appoatel popotonal to the wdth of the a gap δ. Thee-phae nducton oto fnd an applcaton n all banche of ndut and n uncpal utlt anageent a well a n fang and evce wohop. Th goup of achne nvolve devce poduced wthn a wde ange of powe atng fo unde [W] and can each a uch a [MW]. The tadtonal nducton achne wee appled n dve that do not eque the contol of otatonal peed. Th wa due to the cot and poble aocated wth the ue of uch contol devce whle ecung the antenance of hgh effcenc of tanfong electc eneg nto echancal one. The coue of event ha changed condeabl ove the pat ea. The nceae of acceblt and elatve fall n the pce of powe electonc wtche uch a SCR, GTO, MOSFET and IGBT wa followed b contnuou developent of dvee electonc convete [,9,,4,5,69,8,84,86,95]. The applcaton ae t poble to tanfo electc powe wth the paaete of the uppl netwo nto vaable paaete equed at the nput of nducton oto to eet the need of the effectve contol of otatonal peed. Such contol oete nown a cala egulaton dcued n detal n Secton.. The followng tage n the developent of the contol te of nducton achne focued on the poveent of the powe electonc devce nvolved n the eecuton of coand and, n patcula, wth the developent of poceo fo tanfeng nfoaton, ncludng gnal poceo adaptable fo ndutal applcaton. A a eult, t wa poble not onl to degn and pleent dnac contol of dve contanng nducton oto but alo develop contol that tac the tajecto of the poton and oto peed. Th tpe of contol encounteed otl n two vaete [,8,74,76,96,99,] Feld Oented Contol FOC and Dect Toque Contol DTC. The ethod appled n th epect ae dcued n Secton.4... Contucton and Tpe of Inducton Moto A tpcal nducton achne a clnde haped achne whoe ato of the daete to length n the ange of.-.8. Inducton oto ae bult to eet the equeent of vaou nube of phae; howeve, ot coonl the ae thee-phae achne. The a gap between the tato and the oto a all a t achevable and wndng ae located n the lot Fg. of the tato and oto. The feoagnetc ccut ade of a lanated elatc teel agnetc heet n

132 . Matheatcal Model of Inducton Machne ode to lt eneg loe aocated wth altenatng agnetzaton of the on dung the opeaton of the achne. Ve potant ole plaed b the wndng, whch ae engneeed n eveal bac tpe. In the tato of hgh voltage achne the ae often ade n the fo of ba fo olated ectangula haped conducto foed nto col neted nto the lot of the tato. In th cae the lot ae open, ectangula and t poble to net the ead ade gd col nto the n contat to the e-cloed lot appled n wndng ade of col we. The ateal whch conduct the cuent the hgh conductvt coppe. Anothe potant clafcaton aocated wth the ngle and two lae wndng. Fo the cae of ngle lae wndng the de of the col occupe the ente pace avalable n the lot, whle n two lae wndng nde the lot thee ae two de belongng to two dffeent col, one above the othe, whle the de could belong to the ae o dffeent phae of the wndng. In achne wth hghe capact we uuall have to do wth wndng n two lae. Stll anothe clafcaton of wndng n nducton oto aocated wth ntegal and factonal lot wndng [,,,4]. Integal lot wndng the one n whch the nube of lot pe pole and phae an ntege nube. Mot nducton achne appl ntegal lot wndng nce the offe bette chaactetc of agnetc feld n the a gap. Factonal lot wndng ae ued n the cae when the achne degned n a wa that ha a lage nube of pole but t not jutfed to appl too lage a nube of lot n a all co ecton. Anothe eaon fo the applcaton of factonal lot wndng aocated wth econoc facto when the ae feoagnetc heet ae ued fo oto wth vaou nube of pole pa. In th cae fo a gven nube of lot and cetan nube of pole pa we have to do wth factonal lot wndng. Howeve, the ot potant ole of an engnee n chage of the degn of an nducton achne to focu on the developent of uch a wndng whoe agnetc feld n the a gap eultng fo the flow of cuent though a wndng follow a cloel a poble a ne cuve Fg.. The wndng n the oto of nducton oto ae encounteed n two vaou odel whoe nae ae adopted b the tpe of nducton oto: lp-ng oto and quel-cage oto. The wndng n a lp-ng oto ade of col jut a fo a tato n the fo of a thee phae wndng wth the ae nube of pole pa a a wndng n a tato and the tenal of phae ae connected to lp ng. Wth thee ng and b adequate buttng contact ung buhe lppng ove the ng t poble to connect an etenal eleent to the wndng n a oto. Th poblt ued n ode to facltate the tat-up of a oto and n an cae alo to contol t otatonal peed. The quel cage fo the othe vaet of an nducton oto oto wndng that oe coon. It ot often ade of cat ba ade of alunu o, oe ael of ba ade fo welded coppe allo placed n the lot. Such ba ae claped ung ng on both de of the oto. In th wa a cage foed Fg.; hence, the nae quel cage wa coned. The cage foed n th wa doe not enable an etenal eleent o uppl ouce to be connected. It doe not have an defnte nube of phae, o oe tctl peang: each eh n the netwo foed b two adjacent ba and connectng ng egent fo a epaate phae of the wndng. Hence, a quel

133 Inducton Machne n Electc Dve Fg.. Co ecton of an nducton achne wth e-cloed lot n the tato Fg.. Shape of a agnetc feld poduced n the a gap of a thee-phae nducton achne and t fundaental haonc fo a 4 lot tato wth the nube of pole-pa p and fo a 6 lot tato and p cage wndng wth ba n a detaled anal could be condeed a a wndng wth phae. Moeove, a quel cage wndng doe not have a defned nube of pole pa. In the ot bac anal of an nducton oto one can aue that a quel cage wndng a econda wndng that pavel adapt n epone to the agnetc feld a a eult of nduced voltage and conequentl cuent. It poble to futhe aue that the agnetc feld n an a gap wth p pole pa nduce n the ba of a cage a te of voltage and cuent wth p pole pa a well. Snce the nube of phae n the oto bacall abta a the wndng not uppled fo an etenal ouce th alo a thee-phae wndng la to the wndng n a tato. Hence, n t bac engneeng dawng along the ccufeence of the tato the agnetc feld n the a gap of the nducton oto

134 . Matheatcal Model of Inducton Machne decbed b ne cuve wth p te ecuence dung the ound of the gap ccufeence Fg.. The dffeence between the actual hape of the agnetc feld n the a gap and the fundaental haonc of the ode ρ p appoated b a et of ne cuve, fong the hghe haonc of the feld, whoe pecta and apltude can be calculated b accountng fo all contucton detal of the tato and oto of a achne. The bac eaon fo the occuence of hghe haonc of the feld n the a gap aocated wth the dceetl located conducto n the lot and the accuulaton n a all pace, the patcula pan of the col cang cuent and non-hoogenou agnetc peeance n the a gap [8]. Th a gap depte havng t contant engneeng wdth δ n the ene of the agnetc peeance elatve to the denon of the lot n the tato and oto. The hghe haonc of the agnetc feld n the a gap account fo a nube of undeable phenoena n nducton achne called paatc phenoena. The nvolve anchonou and nchonou paatc toque that defo the bac chaactetc of the electoagnetc toque [8], a well a addtonal loe eultng fo hghe haonc and pecfc fequence peent n the acoutc gnal etted b the achne. Fg.. A fequent hape of a quel-cage wndng of a oto of nducton oto In the cuentl anufactued nducton oto paatc phenoena ae encounteed on a elatvel low level and do not dtub the opeaton of the dve. Hence, n the dcuon of the dvng chaactetc the nducton oto epeented b a atheatcal odel whoe agnetc feld dpla onohaonc popete. The onl haonc the fundaental one wth the nube ρ p, whch equal to the nube of pole pa. The ltaton of paatc phenoena and contucton of a achne that vtuall onohaonc coe a a eult of a nube of engneeng pocedue, of whch the ot bac one nvolve an appopate electon of a nube of lot n the tato and oto. The nube n queton ae N and N, epectvel and the ae neve equal to each othe and

135 4 Inducton Machne n Electc Dve the electon depend lagel on the degned nube of pole pa p. Condeable poge ha been ade n the degn and engneeng of nducton oto ove the oe than ea old hto. The eaue of th poge not onl nvolve the ltaton of paatc phenoena but alo an nceae the effectvene of the tuctue n te of the toque atng pe loga of the achne a, long evce lfe, eneg effcenc, ecologcal chaactetc, poge n the ue of nulaton ateal, whch ae t poble to uppl fo convete wth hgh fequenc and apltude of voltage haonc... Fundaental of Matheatcal Modelng... Tpe of Model of Inducton Machne Matheatcal odelng pla a ve potant ole n the degn, eplotaton and contol of electc dve. Modelng and copute ulaton, whethe wth egad to electc dve o n othe banche of engneeng, that adequate and effectve educe the te needed and the cot of ganng an optu degn of a dve and t contol te. Thu, new oppotunte ae offeed n te of educng lead te n the pototpe tetng phae of the degn. The odelng of an nducton oto cople to the degee that we have to do wth an electoechancal devce wth a lage nube of degee of electcal feedo, epeented b chage and electc cuent n phae wndng and, addtonall, that can account fo agnetc lnage. The latte ae delveed b the agnetc feld n the feoagnetc ateal n whch the wndng of the tato opeate and the feoagnetc coe often n the condton of agnetc atuaton. The ultaneou and copehenve accountng fo electoagnetc and electoechancal pocee n an nducton oto that nvolve atuaton of the actve on n the tato and oto, eneg loe dung altenate agnetzaton, pece appng of lnage between the wndng, the non-tead wong ege of the oto and the potental effect of the heat geneated on the popete of the te n fact too cople and too cotl and, hence, even n the ot advanced odel of nducton achne thee pocee tend to be plfed. The bac and ot coon plfcaton cont n the dtncton ade between the agnetc and electc feld due to the all fequence of the altenaton of the feld. Fo that eaon, the feld condeed to be agnetotatc. Moeove, thee a tendenc to plf the ue aocated wth eneg loe dung the altenate agnetzaton of the on, and oete t degaded. Phae wndng n a achne ae ot coonl condeed a electc ccut wth luped paaete and the connecton wth the agnetc feld epeed b flu lnage ψ, whee ubcpt denote the nube of the adequate wndng. Oveall, the poble aocated wth the detenaton of the flu lnage a the functon of electc cuent n the patcula phae wndng of a achne [9]. The ue of the echancal oton of a oto not a cople phenoenon nce a tpcal nducton oto ha onl a ngle degee of echancal feedo angle of otaton of the oto θ. In atheatcal odelng of a an nducton achne dve we tae nto condeaton two cae: non-hoogenou oton of the oto n the dnac tate fo eaple dung

136 . Matheatcal Model of Inducton Machne 5 tat-up o bang of the oto and oton unde a contant angula peed,.e. n a tead condton of the dve n opeaton. A a conequence of not accountng fo the paatc toque wth nchonc chaactetc we do not tae nto condeaton all ocllaton of the peed aound the balance tate; th coe a a conequence of the agnal ole n a degned dve. The bac and the coon foundaton dung the developent of a atheatcal odel of an nducton achne the aupton of t geoetcal and ateal et. Th allow ve lagel to plf the odel and t ot often followed n the ue aocated wth the electc dve. Abandonng of the aupton of et dung the odelng of an nducton achne necea onl n pecal ccutance, uch a odelng of eegenc condton fo a dve and fo eaple n the tude devoted to the toleance of the engneeng tuctue of the achne to t chaactetc and potental eegence. Such an eaple encounteed dung the anal of an nducton achne the tud of the effect of the aet of the a gap between the tato and oto to the eultng foce of agnetc pull and beang wea. The aupton of the et alo enable one to lt the aea of calculaton undetaen wth an a of developng feld odel and detenaton of bounda condton fo uch calculaton. Due to the peented pedent and coplcaton the odel of nducton oto uuall account fo a nube of plfcaton whch fo an adaptaton of the eaned queton and can lead to the tateent of an anwe. In th epect we can dentf thee geneal categoe of atheatcal odelng of a dve. The categoe nclude: odel evng fo the optzaton of the contucton chaactetc of a oto, econdl, odel ued fo the detenaton of electoechancal chaactetc and, thdl, odel whoe object to appl an nducton oto dve contol. The peented thee categoe of odel can be decbed a follow: a atheatcal odel of an nducton oto aed at the optzaton of t contucton wth egad to the tuctue of a agnetc ccut, a a ule, a feld baed odel whoe oluton peented n D o D pace, wth a patcula epha on the hape of a feoagnetc coe along wth the degn of the tato and oto lot a well a patal dtbuton of the wndng. The feoagnetc ateal condeed a non-lnea tang nto account t chaactetc of agnetzaton. The condeaton tend to oe fequentl nvolve a agnetc htee loop and le often the occuence of edd cuent [7,49]. Hence, calculaton ae pefoed fo fed poton of the tato n elaton to the oto o a contant peed of the oton, whle the cuent dent n the wndng a a ule contant ove the ente co-ecton of the wndng n the lot. Fo the cae of wndng ba wth lage denon we have to account fo the non-hoogenou dtbuton of the cuent dent n the adal decton. The contucton of a tpcal nducton oto due to the plane-paallel feld epeentaton enable one to pefo feld calculaton n D pace wthout affectng the pecon of the eult. The calculaton appl pofeonal oftwae ute ung Fnte Eleent Method FEM o Edge Eleent Method EEM. Such oftwae contan pocedue ang t poble to gan vaou data and age egadng feld chaactetc n a patcula ubject, to obtan a nube of ntegated paaete uch a the value of eneg and co-eneg of the agnetc feld, electoagnetc toque, foce calculated b

137 6 Inducton Machne n Electc Dve ean of vaou ethod and nductance of the wndng n the aea of calculaton [4,48]. A one can conclude fo th decpton, feld odel ae applcable not onl wth an a of povng the engneeng and condeng detal of ateal paaete but can alo povde valuable data n the fo of luped paaete fo the calculaton of the poble encounteed n the dve. In patcula, elevant nght offeed b the data egadng the nductance of the wndng and t elaton to the agnetc atuaton. The atheatcal odel evng fo the detenaton of electoechancal chaactetc of a dve, both n tatc and dnac tate, a a ule ae foed a odel wth luped paaete. The eaon that n th cae the engneeng detal ae elated to n an ndect wa ung a all nube of paaete, whch ubequentl cobne a nube of phcal popete of a achne. Dung the detenaton of chaactetc, n patcula the echancal one, the paatc phenoena ae fequentl accounted fo n the fo of addtonal eleent of electoagnetc toque deved fo hghe haonc of the agnetc flu and haonc aocated wth vaable te epeed b othe eleent n the peeneance of the a gap. The odel whch ae applcable fo tatng the chaactetc n an cae have to be pece n te of eneg balance nce one of the applcaton n the detenaton of the loe of eneg and effcenc of the dve. The anal of luped paaete pefoed b a nube of pecalzed calculaton ethod. Th baed on feld calculaton n the electcal achne fo the pecfc condton of opeaton [7,48,9]. The atheatcal odel appled n the ue aocated wth dve contol tend to be the ot plfed odel. A a pncple, the degad the loe n the on, the phenoena of agnetc atuaton and nuance n the fo of ult-haonc pectu of the agnetc feld n the a gap. Such odel tae the fo of a te of odna dffeental equaton. The odel ae tanfoed ung the popete of the achne et nto te of equaton, n whch the fo of the equaton elatvel ple n the ene of the aupton of contant paaete of a te, the nube and tuctue of epeon. Thu, the odel coepond to the equeent of the contol te due to t nteacton wth the tanfoed eaueent gnal deved fo feedbac n the te. The atonale fo ung the pobl ot uncoplcated n te of calculaton odel n the queton of contol aocated wth the fact that the ae late ued fo the calculaton of the vecto of tate vaable of the dve n eal te. The atheatcal odel of the nducton oto fnd an nceangl wde applcaton n the oden ethod of contol concened wth lneazaton though non-lnea feedbac of the dnac odel of a dve, whch, n ealt, ha a non-lnea tuctue. A tpe of th nd of contol alo naed contol wth nvee dnac. An ang queton concened wth the pactcal applcaton of odel that do not account fo a nube of phenoena n nducton achne ncludng agnetc atuaton. The oluton popoed nvolve the contepoa contol ethod, alo appled n electc dve, whch ae oe etant to the uncetant of the paaete of the odel and dtubance along the eaueent path. Such odel nclude obut and adaptve contol [,45,5,57,65,75,5], n whch cae the atheatcal odel cobned wth etaton of the paaete n eal te. The cuentl olved ta n dve

138 . Matheatcal Model of Inducton Machne 7 contol appl the followng pocedue: ple and functonal contol odel n te of calculaton ae accopaned wth the coecton of dcepance eultng fo paaete etaton ung gnal that ae eal acceble b wa of eaueent. Fo the pont of vew of the cuent boo the pncpal nteet focue on the atheatcal odel degned fo detenaton of the chaactetc of the dve and the one appled fo the pupoe of contol.... Nube of Degee of Feedo n an Inducton Moto The queton of the nube of the degee of feedo. encounteed n te wth luped paaete whoe oton dnac decbed b a te of odna dffeental equaton. Fo the cae of an nducton achne Fg..4 th ean one degee of feedo of the echancal oton, fo vaable θ denotng the angle of oto poton and the adequate nube of the degee of feedo e fo electc ccut foed b the phae wndng. Fo the cae when both the tato and oto have thee phae and the wndng ae ndependent, n accodance wth the llutaton n Fg..4a, the nube of electc degee of feedo el 6. The aupton that electc ccut tae the fo of phae wndng wth electc chage Q a tate vaable doe not eclude the applcablt of a feld odel fo the calculaton of agnetc flue ψ lned wth the patcula wndng of the oto. Th poblt eult fo the decouplng of the agnetc and electc feld n the achne and the condeaton of electc cuent Q n the achne a ouce of agnetc vecto potental.8, functon that ae eponble fo feld geneaton. a b Fg..4 Daga wth co-ecton of nducton oto: a -phae tato and oto wndng b oto quel-cage wndng In th cae we have to do wth feld-ccut odel [48], n whch the odel wth luped paaete decbng the dnac of an electoechancal te n th tud the nducton oto accopaned b an nteactvel poduced odel of the electoagnetc feld n whch the peent flu lnage ψ ae

139 8 Inducton Machne n Electc Dve defned. Hence, the odel of an nducton oto whoe daga peented n Fg.4a ha 7. e degee of feedo. In th place one can tat to thn about the tate encounteed n the wndng of a quel cage oto Fg..4b, whch doe not contan a tandad thee-phae wndng, but ha a cage wth N nube of ba. The quel cage wndng epond to the MMF poduced b tato wndng cuent. The nduced EMF n quel cage wndng dpla the ae et popete on condton that the quel cage of the oto etc n the ange of angula pan coepondng to a ngle pole of the tato wndng o t total ultple. Hence, the eultng nube of degee of feedo q fo a etcal quel cage wndng [] epeed b the quotent p u v q u. whee: - the nube of ba n the etcal cage of a oto u,v - elatve pe ntege The nube of the degee of feedo of the electc ccut of a oto quelcage wndng q u coepond to the allet natual nube of the ba n a cage contaned n a pan of a ngle pole of the tato wndng o t ultple. Th done unde the lent aupton that the tato wndng ae etcal. If the et not actuall the cae, the au nube of the degee of feedo of a cage equal to q. whch coepond to the nube of ndependent electc ccut ehe, n accodance wth.95, n the cage of a oto Fg.4b. Fo the oto n Fg..4b, we have p, N, hence the quotent: u and, a a eult, the nube of electc degee of p 4 v feedo fo a quel cage wndng aount to q u. Th ean that n th cae the two pole ptche of the tato contan coplete lot cale o lot ptche of the oto, afte whch the tuaton ecu. The lage nube of the degee of feedo of the cage ae t poble to account n the atheatcal odel fo the paatc phenoena [8], fo eaple paatc nchonc toque. Howeve, f we degad defoaton of the agnetc feld n the a gap and aue that t a plane-paallel and onohaonc one wth the ngle and bac haonc equal to ρ p, then n ode to decbe uch a feld we ethe need onl two coodnate o two ubttutve phae wndng, n ot ple cae othogonal one. Fo uch an aupton of onohaonc feld the nube of the degee of feedo deceae to q egadle of the nube of ba n the oto cage. In the tude of nducton oto dve and t contol the pncple to aue the plana and onohaonc feld n the a gap. Nevethele, at the

140 . Matheatcal Model of Inducton Machne 9 tage when we ae tatng to develop the atheatcal odel of nducton achne, t aued fo the lp ng and quel cage achne that the oto wndng thee-phaed a n Fg..4a fo the pupoe of peevng a unfo coue of eaonng. Hence, a ndcated eale, unde the aupton of a plana and onohaonc feld n the a gap, lp-ng and quel-cage oto ae equvalent and can be decbed wth a ngle atheatcal odel wth the onl dffeence that the wndng of a quel-cage oto not acceble fo outde, n othe wod, the voltage upplng the phae of the oto ae alwa equal to zeo. Accodng to.89 and., agange functon fo a oto wth thee phae wndng n the tato and oto can tae the fo: Q 6 ~ ~ J θ ψ Q, Q,,, θ dq.4 and the vtual wo.98 epeng the echange of eneg equal to: δa T l 6 D θ δθ u R Q δq.5 whee: q Q, Q,,Q 6, θ - vecto of genealzed coodnate J - oent of neta elated to the oto haft, T l - load toque on the oto haft, D - coeffcent of vcou dapng of the evolute oton, R - etance of -th phae wndng, Q, u - electc cuent and uppl voltage of -th phae wndng, ψ - agnetc flu lned wth -th wndng. The odel n th fo alead contan two plfcaton,.e. t degad on loe aocated wth agnetzaton of the coe and change of the wndng etance followng a change n the tepeatue. Fo the above the equaton of oton fo electc vaable follow n the fo: d dt Q Q u R Q,,6.6 wth the capacto ng fo the te. Wheea, accodng to.4 and ung the degnaton fo cuent obtan Q Q, we

141 Inducton Machne n Electc Dve l l l l l d Q l ~,,,, ~,,,,,,,,, 6 θ ψ θ ψ.7 If -th wndng wee the fnal one, a fo n 6, then,,, n Q θ ψ.8 The eultng equaton tae the fo l l l l l n d Q l ~,,,, ~,,,,,,,,,,,, 6 θ ψ θ ψ θ ψ.9 Fo the copaon of.8.9 t eult that fo the plct of notaton we hould teat the equaton n.7, whch cuentl condeed, a the fnal one. In th cae the equaton fo the ccut of an nducton oto tae the fo n R u dt d,,, θ ψ. Afte the dffeentaton of the left-hand de we obtan,6, 6 R u dt d θ θ ψ ψ. The left-hand de epeon. denote electootve foce nduced n -th phae wndng a a eult of the vaaton n te of flu lnage. The ft te ae deved fo the vaaton of the cuent and ae called electootve foce of tanfoaton, whle the fnal te elated to the angula peed of the oto Ω θ and called the electootve foce of otaton. The equaton fo the toque epeed wth vaable θ tae the followng fo l D T dt d θ θ θ and, conequentl,

142 . Matheatcal Model of Inducton Machne d dt 6 J ~ ~ θ ψ θ d Tl D,,,,,, θ θ, Whch, n cae of contant oent of neta J, can be denoted altenatvel a J θ T T D θ. e l whee: T e 6 ~ ψ ~,,,,,, d θ θ. the electoagnetc toque poduced b the nducton oto. In pte of the fact that the eultng equaton of oton ae tated fo a te wth luped paaete and n th cae fo 7 vaable coepondng to 7 degee of feedo of the oto, the can fnd a ve boad applcaton. Th eult fo the geneal fo of the flu lnage aocated wth the patcula phae wndng ψ,, n, θ. It could be ganed b vaou ethod accountng fo the atuaton and vaou engneeng paaete of the agnetc ccut. Fo a quel-cage oto, fo the cae f one needed to account fo the etng paatc toque, t would be necea to abandon the tatng aupton of the onohaonc age of the feld n the a gap and, hence nceae, the nube of equaton fo the phae of the oto fo to q, a t eult fo....4 Matheatcal Model of an Inducton Moto wth nea Chaactetc of Coe Magnetzaton..4. Coeffcent of Wndng Inductance In a ajot of ue aocated wth the oton of an nducton achne, n patcula n the ue aocated wth the contol of dve wth nducton oto, we can aue a plfcaton nvolvng an appoaton of the chaactetc of oto agnetzaton ung lnea elaton. Hence, the defnton of the coeffcent of elf-nductance and utual nductance of the achne wndng follow n the fo ~ M l ψ ψ l l l.4 Fo l th coeffcent naed the elf-nductance coeffcent and nclude two te: ~ M M.5 σ

143 Inducton Machne n Electc Dve whee σ the leaage nductance, whch eult fo the agnetc flu lned olel to -th phae wndng. In contat, fo l th coeffcent denoted a utual nductance coeffcent ~ M l M l.6 The detenaton of the coeffcent of the nductance of an nducton achne could be deved fo eaueent on an etng achne o could be baed on calculaton, whch alead poble at the tage of oto degn. The epeental tude, whch can eve n ode to detene the coeffcent of wndng nductance, nvolve the eaueent of the chaactetc of the dle unnng of a oto, hot ccut fo the pupoe of tatng the nductance of the leaage and othe tet fo ntance of the epone to a voltage tep functon. On th ba t poble to etablh the appoate paaete of a achne, ncludng the nductance of the wndng a well a to appl the ethod fo the etaton of the paaete fo elected eaueent chaactetc [7]. The calculaton ethod nvolve the calculaton of the feld n the achne ung feld poga [9], whch povde nfoaton egadng ntegated paaete, ncludng nductance. Fo an nducton oto t uffcent to aue calculaton of plano-paallel feld D wth uppleenta data and coecton egadng the bounda ecton of the feld n the achne. In patcula th concen leaage nductance of the end wndng ecton of the wndng. Moeove, a nube of analtcal ethod ha been developed fo the calculaton of the feld and nductance coeffcent n an nducton oto, thu povdng valuable nfoaton fo nducton oto odel. Howeve, thee tend to be le pece than the one that eult fo feld calculaton nce the account onl fo the ajo te of the eneg of the agnetc feld,.e. the eneg of the feld n the achne a gap. In the fundaental noton Fg.., unde the aupton of onohaonc dtbuton of the feld n the gap, the coeffcent of utual nductance tae the fo: M l M co α α.7 whee: M - value of nductance coeffcent fo phae concdence α,α l - angle whch detene the poton of the ae of wndng,l In accodance wth Fg..4 thee angle ae: l α, α, α, π / p, π /p α 4, α5, α 6 θ, θ π /p, θ π / p.8 - fo the tato wndng and oto wndng, epectvel. The nube of the pole pa p eflect p-te ecuence of the te of wndng and patal age of the feld along the ccufeence of the a gap. On the ba of elaton n.5.8, the at of the nductance coeffcent of tato wndng tae the fo

144 . Matheatcal Model of Inducton Machne M ph σ M.9 and lal, fo the wndng n the oto M ph σ M. At the ae te, utual nductance atce between tato and oto wndng ae elatve to the angle of otaton θ M ph M M T ph co pθ co pθ π / co pθ π / co pθ π / co pθ co pθ π / co pθ π / co pθ π / co pθ. In the above equaton: σ, σ - ae leaage nductance coeffcent of tato and oto wndng M, M - an feld nductance coeffcent of tato and oto wndng M - utual nductance coeffcent of tato and oto wndng fo full lnage between the wndng algned poton of wndng ae - unta at wth denon...4. Model wth nea Chaactetc of Magnetzaton n Natual phae Coodnate At the begnnng t necea uaze the plfng aupton fo th odel of the nducton oto, tatng wth the ot potant one: - coplete geoetcal and ateal et of the electoagnetc tuctue of the oto - lnea chaactetc of agnetzaton of the electoagnetc ccut - plana and onohaonc dtbuton of the feld n the a gap, eultng n a ngle haonc wth the nube ρ p - degadng of the loe n the on - degadng of the etenal nfluence fo eaple tepeatue on the paaete of the oto. Snce, n accodance wth the econd aupton, we conde a lnea cae of agnetzaton, on the ba of.4 the followng elaton atfed

145 4 Inducton Machne n Electc Dve l l l l l M ψ ψ. Ung the elaton., the flu lnage of wndng of the tato and the oto can be ecoded n the at fo [ ] [ ] ph ph T ph ph ph T ph M M M M Ψ Ψ ψ ψ ψ ψ ψ ψ. [ ] [ ] T T - wth vecto of phae cuent of the tato and oto, epectvel. Wth the ad of. the equaton of the electc ccut of the nducton oto can tae the followng at fo ph ph ph ph ph ph dt d R dt d R M M U M M U θ θ.4 o ph ph ph ph ph ph dt d R R M M M M I I U U θ θ.5 whee:, - zeo at and unta at wth denon. [ ] [ ] T ph T ph u u u u u u U U On the ba of. and.5, the electoagnetc toque of the oto can tae the followng fo: [ ] ph T ph ph T ph ph ph ph T T l l l e M T M M M M M M 6 6 θ θ θ θ θ θ θ θ θ.6 Snce both te of the quadatc fo.6 ae equal, then ph T e T M θ θ.7

146 . Matheatcal Model of Inducton Machne 5 The equaton concenng the evolute oton. ean unchanged, t necea onl to appl a oe detaled epeon.7 to detene electoagnetc toque of the oto...4. Tanfoaton of Co-odnate Ste Mot of the dnac cuve and oluton wth egad to contol ae not conducted n phae coodnate, uch a the cae of the atheatcal odel n.5,.7, but the ae tated n the tanfoed coodnate. Thee ae eveal eaon fo that: an adequatel elected tanfoaton capable of tanfong the te.4 that contan peodcall vaable coeffcent tgonoetc functon nto a te wth contant paaete. Thu, thee no need to appl lage coputng powe and the cot theeof educed, whch patculal elevant n the ue of dve contol n eal te. In addton, a one can conclude fo the fo of utual nductance at of the wndng.7,., that the ode not but onl. The have one denon too an, whch can be concluded b addng up all the ow n each of the atce. The phcal eaon elf-evdent: n ode to decbe a onohaonc plana feld t uffcent to ue two vaable. Hence, the feld can be poduced b cuent n two phae wndng that ae not tuated along a ngle a pependcula ae ae ot applcable. The eal applcaton of the tanfoaton of the coodnate te ognate fo Pa and eved n ode to analze the opeaton of nchonou geneato. The geneal theo of tanfoaton of coodnate te n ult-phae electc achne baed on the Floquet theoe. Fo t eult that fo lnea te of odna dffeental equaton wth te peodc coeffcent t poble to dentf uch a tanfoaton T θ, θ fo whch n the new defned coodnate the achne equaton ae ndependent of the angle of otaton of the oto [5]. In the patcula cae onohaonc feld dtbuton t poble to gan th eult b the applcaton of tanfoaton Tθ,.e. onl elatve to the poton of the oto. The eaple of oluton and applcaton n th aea ae ultple and can be found n the bblogaph [76,8,8]. Fo the techncal pont of vew t onl enble to appl othogonal tanfoaton n electc dve. Th ean the one whoe atce fulfll the condton that T Τ Τ.8 Th the cae when the vecto whch fo the at of tanfoaton T ae othogonal and have an eleenta length, that : v v fo j and v v fo, j,,.9 j whee v the colun ow of the at T. Th popet ndpenable nce othogonal tanfoaton peeve the cala poduct and quae fo fo the tanfoed vecto. The cala poduct fo the cae of an electc achne coepond to the ntantaneou powe delveed to the clap of the achne j

147 6 Inducton Machne n Electc Dve [ ] u u u u u u p A a eult of tanfoaton of the above epeon, ung othogonal tanfoaton T, we obtan: γ γ β β α α u u u p T T T * * U T T U I. whee [ ] [ ] T T u u u γ β α γ β α T TU U * * ae tanfoaton of the vecto of the voltage and cuent of a thee-phae electc achne. Concuentl, electoagnetc toque n the ult-phae achne n the algebac ene epeed b the quadatc fo * * * M T MT T T M I I T T T T T T e T θ θ. whee M M MT T M T θ ~ ~ *. the tanfoed at of the devatve of utual nductance between the wndng. A one can conclude fo., the vaable have been tanfoed nto the fo *, whle due to the othogonalt of the at of tanfoaton T the electoagnetc toque ean unchanged. Unde the aupton adopted at the begnnng of th chapte the achne ha a onohaonc feld n the a gap, whch eult n the fact that the at of utual nductance between the tato and oto elatve onl to aguent pθ of the peodc functon. Th enable one to eal dentf the othogonal tanfoaton of T uch that ode the atheatcal odel n the ene of leadng to the contant coeffcent of the dffeental equaton. Th tud apple the followng othogonal atce of the tanfoaton: - fo the quantte elatve to -phae wndng n the tato n n n co co co a t a t t a t a t t c c c c c c ω ω ω ω ω ω T. - fo quantte elatve to -phae wndng n the oto

148 . Matheatcal Model of Inducton Machne 7 n n n co co co a p t a p t p t a p t a p t p t c c c c c c θ ω θ ω θ ω θ ω θ ω θ ω T.4 whee: ω c - an abta pulaton that conttute the degee of feedo of the planned tanfoaton a π/ - aguent of the etc phae hft. The tanfoaton of the patcula phae vaable occu n the followng wa: [ ] [ ] T v u ph uv T v u ph uv z z z z z z Z T Z T Z Z.5 Net, we wll poceed to ee how th wo fo a te of etc -phae nuodal voltage of fequenc f upplng tato wndng: [ ] T ph uv u u u T T U U.6 that : ] n[ ] co[ co co co * * n n n co co co γ ω ω γ ω ω γ ω γ ω γ ω ω ω ω ω ω ω t t U a t a t t U a t a t t a t a t t u u u c c ph ph c c c c c c v u.7 whee: ω πf. A a eult of the tanfoaton the voltage u, and voltage u u and u v fo an othogonal te. In geneal, tanfoaton of T,T.,.4 lead to the etateent of the phae vaable n the tato o oto wndng fo two pependcula ae u,v whch ae n evolute oton wth the abta angula peed of ω c - a t wa peented n Fg..5. The thd of the tanfoed ae a pependcula to ae u,v and act n the aal decton; hence, t doe not contbute to the plana feld of the achne.

149 8 Inducton Machne n Electc Dve Fg..5 Illutaton of the othogonal tanfoaton of -phae wndng of AC achne nto otatng u,v ae..5 Tanfoed Model of Inducton Moto wth nea Chaactetc of Coe Magnetzaton..5. Model n Cuent Coodnate Ftl, we tanfo equaton.4 fo electc vaable cuent,. Fo the equaton of the tato wndng T U d dt uv d T RIT TM pht T TM dt T T T T M pht T T M ph θ T T M uv M uv ph ph uv T θ T T.8 The above tanfoaton appl the popet of the othogonal at: T T T T T T I and lal fo the at of tanfoaton T. Fo equaton.8 we obtan: whee: d U uv R uv ω ca M uv uv M uv uv.9 dt uv uv T T ph ph T [ u v ] [ ] T u v.4

150 . Matheatcal Model of Inducton Machne 9 d dt M uv T M ph T T σ σ - elf-nductance of tato wndng M - nductance of agnetzaton T M uv TM ph θ T.4 ω n ω t n ω t a n ω t a ω T.4 co ωct co ωct a co ωct a T c c c c ca whle: A - a ew-etc at d and ω ca - an opeato actng on the ght-hand de epeon. dt Actng n a la anne fo the phae ccut of the oto, afte tanfoaton, and eplong we obtan: whee: d dt ωc p θ A T T d R uv ωc p θ A M uv uv M uv uv.4 dt σ T M uv T M pht.44 σ - elf-nductance of oto wndng d ω p θ A c - opeato actng on the ght-hand de epeon. dt The quadatc fo of the.7 tanfoed lewe: T T T T T e T T M θ T T p uv A uv θ uv uv

151 Inducton Machne n Electc Dve whch fnall eult n a elaton T p v.45 e v u Fo equaton afte the tanfoaton t eult that the zeo equence equaton ae not nvolved n the conveon of eneg and thee equaton ae autonoc u R R d dt d dt σ σ u.46 The aal cuent occu onl fo the cae of aet n the powe uppl.7, n addton to whch, t peent onl n the cae when the wndng of the tato ae not connected n a ta. Othewe, the equaton of contant elnate the poblt of the occuence of even n cae of the etng voltage u. Fo th eaon n futhe pat of atheatcal odelng of nducton oto we wll account onl fo two te epeentng electc quantte n the ae u,v : U uv R R uv uv d dt d dt ωca ωc p θ A and fo the equaton fo the evolute oton dω J dt vu Te uv uv uv uv.47 p uv Tl DΩ.48 whee: Ω θ - the angula peed of oto evoluton T T [ ] [ ] [ ] T uv u v uv u v U uv uu uv A - a ew-etc at of denon. The denon of the tate vecto of the odel of the nducton oto 5, whch coe a a conequence of the lac of equaton fo the zeo cuent,...5. Model n Med Coodnate The tanfoaton of flu lnage of phae wndng. n a la anne to the pocedue wth equaton.8 lead to elaton whch detene aal flue n the functon of aal cuent

152 . Matheatcal Model of Inducton Machne Ψ Ψ uv uv uv uv uv uv.49 whee: ae called coeffcent of agnetc couplng fo tato and oto wndng. Equaton.47 ung.49 can be etated and tae the fo U uv R R uv uv d dt d dt ω A c c Ψ uv ω pω A Ψ uv.5 In equaton.5 we have to do both wth aal cuent uv, uv and aal flue Ψ uv, Ψ uv - that wth a double et of vaable. In ode to elnate one of the, we need a elaton that the evee to the one n.49: uv uv σ Ψ σ Ψ uv uv.5 whee: σ called the coeffcent of wndng leaage. a Med coodnate Ψ, uv uv Fo the elnaton of vaable Ψ, fo equaton.5 we appl the followng uv uv elaton: Ψuv Ψuv σ uv uv Ψuv uv.5 Ung.5 and tanfong the eult nto the tandad fo, we obtan: Ψ uv uv U σ ω pω c uv α pω A R A uv uv ω A Ψ c U uv σ Ψ uv uv α α σ uv.5 B tatng.5 n the fo of equaton wth a ngle denon and uppleentng wth the equaton of oton we obtan the odel:

153 Inducton Machne n Electc Dve ψ ψ u v Ω u v u u α ψ σ α ψ σ u u R u R v u v ω ψ α α σ α α σ [ p ψ uv ψ vu Tl DΩ J T c ω ψ c v u pω ψ pω ψ e v u ] u v ω pω c ω pω c v u uu σ uv σ.54 The above calculaton of the toque apple followng popet of the ewetc at A : T A b Med coodnate uv, Ψ uv Th te we hould elnate vaable uv, Ψ uv fo equaton.5. Th done ung the followng elaton: uv Ψ uv uv Ψ uv Ψ uv σ uv Afte ubttuton and tanfoaton we obtan: Ψ uv uv α Ψ σ uv α α pω A uv ω pω c γ uv Ψuv uv c uv A Ψ ω A Uuv σ.55 whch eult n a ngle denonal equaton and the oveall odel contanng the equaton of oton: ψ ψ u v Ω u v α ψ α ψ σ α ψ σ α ψ u v α u v ω pω ω pω γ γ ψ ψ [ p ψ uv ψ vu Tl DΩ J T u α v pω ψ pω ψ e v u c c u v u c v c u v ω ω ] U u σ U v σ.56 Below a ua of coeffcent ued n the above odel:

154 . Matheatcal Model of Inducton Machne R α σ γ σ σ α α R α.57 The odel peented ung the te of equaton.54,.56 eve not onl fo the pupoe of the calculaton of the dnac of the dve but alo ae appled n oden ethod of nducton oto contol...5. Model n Flu Coodnate In ode to obtan a odel n flu coodnate t necea to elnate cuent uv, uv, fo equaton.5 b ung fo th pupoe elaton n.5. A a eult we obtan: Ψ Ψ uv uv U uv α σ α σ Ψ Ψ uv uv ω A uv Ψuv Ψuv ωc pω A Ψuv c Ψ.58 The coplete odel accopaned b the equaton of the evolute oton peented below: ψ ψ ψ ψ Ω u v u v uu α ψ u ψ u ωcψ v σ uv α ψ v ψ v ωcψ u σ α ψ u ψ u ωc pω ψ v σ α ψ v ψ v ωc pω ψ u σ p [ ψ vψ u ψ uψ v Tl DΩ J σ..5.4 Specal Cae of Selectng Aal Ste u,v T e ].59 Angula peed ω c encounteed n tanfoaton T, T.,.4 a a fee paaete ae t poble to cofotabl conduct calculaton and ntepet the eult fo the tateent of the peed of the evoluton of othogonal ae u,v Fg..5. Fo th pupoe we appl thee bac ubttuton encounteed n the followng tage a ω. Equaton n α,β ae c One of the coonl appled oluton egadng the electon of an aal te aocated wth oblzaton of the te u,v wth epect to the

155 4 Inducton Machne n Electc Dve oto tato. In th cae we have to do wth the te of ae denoted a α,β cobned wth the tato of the achne. Befoe ubttutng ω c the equaton of oton fo the phae ccut.47 ae peented n a at fo: Ω Ω uv uv c c c c uv uv uv uv uv p p dt d R R A A U ω ω ω ω.6 Afte the ubttuton ω c and eaangeent we obtan equaton n α,β ae n the tandad fo: Ω αβ αβ αβ αβ αβ αβ αβ αβ α α α α σ σ p dt d A A U U.6 Voltage U αβ eult fo the te of ne voltage.7 afte applng tanfoaton T wth ω c. The cuve fo cuent αβ, αβ have a pulaton ω n the tead tate, whch eult fo the voltage U αβ. The coplete atheatcal odel of the nducton oto n the aal coodnate found below: ] [ l T D T p J p u p u p u p u e Ω Ω Ω Ω Ω Ω β α α β α α β β β β β β α α α α α α β β β β β β α α α α α α σ σ α α σ σ α α σ σ α α σ σ.6 In a la anne on the ba of equaton.56,.59 followng ubttuton ω c, we a have the equaton of oton fo the oto n α,β ae fo ed and flu coodnate. b c pω ω. Equaton n d,q ae Followng the ubttuton ω c pω n the equaton of oton, the te of othogonal equvalent ae u,v tffl elated to the oto and, hence, t ha been gven the nae d,q te. Th te ot coonl appled n the decpton of the dnac of nchonou achne and a d then tuated contentl wth the longtudnal a of the achne oto. The te of ae

156 . Matheatcal Model of Inducton Machne 5 d,q n th cae can be effectvel appled wth egad to the nducton oto a well. In th cae fo poweng tato wndng wth etc nuodal te of -phae voltage.7 n tato the cuent cuve we have to do wth pulaton ω ω ω pω pω / ω Ω ω / ω f.6 whch called the lp pulaton of an nducton oto. Slp eflect a elatve dela n otaton peed Ω of the oto n epect to the agnetc feld otaton peed ω f ω /p. Slp dung the tandad opeaton of an nducton oto ange fo a facton of a pecent to a few pe cent of one. Concuentl, ω the phcal pulaton of oto cuent. Fo ntance one can note the equaton fo a oto epeed n ae d,q n ed coodnate uv, Ψ uv. On the ba of equaton.56 and followng the ubttuton ω c pω, we obtan: ψ ψ d q Ω d q α ψ α ψ σ α ψ σ α ψ d q α d q γ γ [ p ψ dq ψ qd Tl DΩ J T d α q pω ψ pω ψ e q d d q pω q pω d ] U d σ U q σ.64 c ω ω. Equaton of oton n, ae c The aal te, a te whch otate wth the peed ω pω f, that the p-ultple of the agnetc feld peed n the a gap. In cae of upplng phae wndng wth a etc nuodal te of -phae voltage, afte the applcaton of tanfoaton.7, voltage U contant. A a eult, the electc vaable, Ψ n the tead tate pla the ole of contant functon. In the tanent tate the vaablt eflectng envelope cuve of altenatng quantte n the phae wndng. The atheatcal odel baed on the te of coodnate benefcal to conduct nuecal calculaton of dnac cuve, nce the pactcal coputng concen the envelope of the phae cuve and t poble to conduct the wth a uch lage ntegaton tep. A an eaple, t poble to tae nto condeaton the atheatcal odel of a oto n flu coodnate.59 epeed n the aal te:

157 6 Inducton Machne n Electc Dve α ψ σ α ψ σ α ψ σ α ψ σ p Ω [ ψ ψ J σ ψ ψ ψ ψ ψ ψ ψ ψ ω ψ ω ψ ω ψ ω ψ ψ u u ψ T l DΩ ].65 The odel of the nducton oto peented n te ae patculal applcable n the contol, whch coe a a conequence of the plct of the equaton fo the oto n th te. Th epecall dcenble fo Feld Oented Contol FOC technque...6 Matheatcal Model of Inducton Moto wth Untanfoed Vaable of the Stato/Roto Wndng Thee a nube of pactcal eaon wh t benefcal to peeve untanfoed vaable on one of the de of an nducton oto. Th ean that the vaable whch defne the electc tate n the tato o oto wndng ean n the fo of natual vaable, whle the tate of the connecton between the wndng antaned b the ntoducton of adequate equaton wth contant eultng fo Kchhoff law.9. The peevaton of the untanfoed vaable and the eultng equaton of oton fo a ngle de enable one to deve the o-called ntenal aet wthn wndng, whch ae defned ung natual vaable. Hence, t poble ethe to ncopoate abta luped eleent n the patcula phae wndng o appl aetcal uppl voltage wth abta wavefo. In patcula, a a eult of th, t poble to pefo the calculaton fo the bang wth dect cuent fo an connecton between phae wndng, opeaton unde ngle-phae uppl, opeatng wth an aula phae fo capacto tatng a ngle-phae oto, anal of a ee of eegenc ue and elect afet eaue. Howeve, the ot potant applcaton of the atheatcal odel of th tpe n the odelng of electonc powe convete n cobnaton wth the uppled achne, n whch powe tanto o lcon contolled ectfe SCR ae degned fo the contol of voltage and cuent n the patcula wndng. Th tpe of odelng, whch ha been the object n nueou eeach, can be ot effectvel conducted n the ccutance of peevng a fed tuctue of an electc te b the ntoducton of etance wth vaable value coepondng to the tate of the eaned powe electonc wtche n the patcula banche of electc ccut. In the blocng tate the aue hgh value ltng the flow of cuent aco a cetan banch, whle n the conductng tate the value ae all,.e. one whch coepond to the paaete of conducton calculated on the ba of the data fo uch coponent

158 . Matheatcal Model of Inducton Machne 7 taen fo anufactue catalogue. Th tpe of odelng aocated wth a nube of pedent n nuecal calculaton of the cuve of eaned vaable due to te contant n the patcula electc ccut, whoe value dffe b eveal ode of agntude, a a eult of applcaton of hgh value blocng etance. Fo th eaon thee odel appl tff ethod of ntegaton fo Odna Dffeental Equaton ODF. Anothe oluton to be appled nvolve the ue of ple ngle-tep pocedue fo the oluton of tff te wth a all tep ze. In an cae, howeve, powe econducto wtche encounteed n the banche of electc oto ae oe eal odeled fo the cae when vaable fo a gven ccut ae the phae vaable. The eaon aocated wth the fact that the tate of a gven powe econducto wtch elatve to the contol gnal and fowad cuent n th eleent. In the ue of contol of quel-cage nducton oto econducto te ae ebe of the ccut of the tato n a achne, whle the vaable concenng quel-cage wndng of the oto ae tanfoed to the othogonal ae u,v. Howeve, f contol occu n the oto of a lp-ng oto and the contol eleent ae tuated thee ncludng the convete, t benefcal to have untanfoed vaable phae cuent n the oto. But then the electc vaable of the tato wndng could a well be tanfoed nto othogonal ae u,v fo the pupoe of uccnct notaton...6. Model wth Untanfoed Vaable n the Electc Ccut of the Stato We hall aue that we deal wth a -phae oto wth wndng connected n a ta. A a eult, when the cuent of the phae wndng and ae condeed a tate vaable, the te of tato wndng chaactezed wth contant Fg..6 Fg..6 Sta-connected -phae tato wndng o oe befl: W.66 W

159 8 Inducton Machne n Electc Dve o oe befl: u u u u u U W W u u U ph.67 A a eult of the ntoducton of the equaton of contant and the tanfoaton of the cuent of tato wndng, we obtan fo the tato te of equaton.4: M M d W u * U ph R ph ph θ dt Wu U ph RWu W U A a conequence: In equaton.68: R d dt Wu M phw M U R uv T Wu M pht T M uv uv d M M uv.68 dt R R R.69 R R R ae t poble to account fo non-hoogenou etance n the tato phae wndng, and fo the cae of et R R R and, hence: R R.7 M σ.7 M uv.7 Sla tanfoaton ae pefoed fo the te of equaton n the oto ccut.4: M M d T * R ph θ dt and, conequentl: ph

160 . Matheatcal Model of Inducton Machne 9 ph ph T ph T ph dt d dt d dt d dt d R uv uv uv uv uv W M T W T M T T M T T T T M T M M In the followng tanfoaton we appl two elaton: Connecton of tato wndng n a ta ae t poble to elnate the zeo equence equaton. c p dt d T A T θ ω whch fo the te, c ω β α gve p dt d T A T θ.7 A a eult, the equaton of the oto ccut tae the fo: uv uv uv uv uv uv uv p dt d R M M A M M θ.74 whee: uv uv M M.75 It potant to note that: T uv uv M M, nce T W u W. A a eult, the te of equaton fo the electc ccut of the nducton oto fo th cae can be noted a v u v u dt d R p p R p R R R R R u u θ θ θ.76

161 4 Inducton Machne n Electc Dve The electoagnetc toque epeed n thee vaable tae the fo: T e p[ u v ].77 The equaton fo an nducton oto n the fo.76 ae t poble to account fo econducto eleent of convete ung vaable value of etance R, R, R encounteed n phae wndng n a fed tuctue te...6. Model wth Untanfoed Vaable of Electc Ccut of the Roto Th odel fnd applcaton n the ue egadng ntenal aete and contol of dve wth a lp-ng nducton achne. Fo eaple, t appled n the calculaton of the tat-up of a lp-ng oto wth aetc etance dung the tat-up, atpcal te of connecton between phae wndng of the oto, anal of cacade te wth a lp-ng oto, fo ntance the Schebu cacaded te. In the dcued eaple the electc vaable of the tato ccut undego aal tanfoaton along u,v ae, whle the vaable of the oto ccut ean untanfoed. Unde the aupton of the ta connecton of oto wndng, we have to ntoduce equaton of contant. Th te to enue et of the eultng equaton the efeence phae the one denoted wth the nube and, a a conequence, the cuent contant offe the elnaton of the cuent, whle voltage contant efe nte-phae voltage to the tenal of phae wndng : u u u u u W u.78 o U Wu U ph.79 In th cae, n contat to.66,.67 the followng fulflled: T W W u.8 whch eult n the et of atce n equaton.88. The aupton of the la coue of acton a n ecton..6. lead to the tanfoaton of the te of equaton n the followng wa: TU U M M d T * U ph R ph ph θ dt d T R T TM pht T T M ph θ W dt uv Muv uv M uv ph uv

162 . Matheatcal Model of Inducton Machne 4 whee: M uv n pθ co pθ co pθ n pθ [ ] W n pθ co pθ Afte the equaton fo the zeo equence degaded, we obtan: U uv.8 d d R uv M uv uv M uv p θ AM uv.8 dt dt whee: A a ew-etc at appled wth egad to the elaton: d dt Fo oto equaton we obtan: M uv p θ A M uv M M d W u * Uph R ph dt WuU ph WuR W U d dt R WuM phw M T WuM T T M uv uv Fo an aetc at of etance of oto wndng R we obtan.8 R R R R WuR W.84 R R R whch fo the cae of the et of the etance can be educed to the fo: R R.85 In a la wa at M tae the fo: M Concuentl, at M uv due to t et.8 fulfll the elaton:

163 4 Inducton Machne n Electc Dve T uv uv M M.86 Afte abandonng the zeo equence equaton fo the tato cuent the equaton fo the electc ccut of the oto tae the fo: uv uv uv uv p dt d dt d A M M M R U θ.87 The above equaton apple the elaton A A T. A a eult of the cobnaton of.8 wth.87, we obtan a te of equaton fo electc ccut of the lp-ng oto wth untanfoed vaable of the electc ccut of the oto: v u T v u T v u dt d R v v R u u u u θ θ θ θ N N K K.88 whee: p v θ p p p p p p θ θ θ θ θ θ θ n co n co n co K p p p p p p θ θ θ θ θ θ θ co n co n co n N The epeon fo the electoechancal toque tae the fo: ] n [co v u v u u v e p p p T θ θ.89 The chaactetc popet of equaton fo a lp-ng nducton oto wth untanfoed ccut of the oto the dependence of thee equaton on the angle of the otaton of the oto θ.

164 . Dnac and Statc Chaactetc of Inducton Machne Dve 4. Dnac and Statc Chaactetc of Inducton Machne Dve The atheatcal odel of nducton achne developed n the pecedng ecton wll be appled hee n the ulaton and calculaton of tpcal dnac tate and tatc chaactetc of the dve. In the ft tage the peentaton wll nvolve dnac chaactetc calculated fo tandadzed electoechancal paaete of a dve. In the ubequent ecton the equaton of oton wll be educed to the tead tate and on th ba an equvalent ccut daga of an nducton oto wll be deved togethe wth tatc chaactetc, tpcal paaete and gaphcal age fo the chaactetc... Standadzed Equaton of Moton fo Inducton Moto Dve Depte the coon opeatng pncple and a unque decpton n the fo of a atheatcal odel, nducton achne fo a cla that condeabl dtnctve. The ange of the ated powe vae fo a facton of a [W] to the achne eceedng [MW]. Concuentl, peed atng eultng fo the nube of pole pa appled n the contucton, tpcall ange fo a achne fo p to p6 pole pa, and patcula anufactue offe achne wth a hghe nube of pole pa. Rated voltage appled to uppl the pa wndng alo tend to ve aco the achne n accodance wth the tandadzed ee of voltage, whle the ajot of the oto un off a /4 [V] uppl voltage o a hgh voltage of 6 [V]. In addton, nducton achne ae dffeentated b the tuctue of the wndng of the econda de oto and n patcula b the hape and pofle of the co-ecton of the ba n the quel-cage oto. Th pat eponble fo the nceae n the value of etance R, whch the bac paaete whch chaacteze the atheatcal odel and fo the fact that th paaete condeabl elatve to the fequenc of the cuent n the cage ba. Squel-cage achne, whch ae dtnguhed b the tall and lende hape o patcula pofle that tend to becoe thnne towad the a gap ae chaactezed b etance R, whoe value nceae fo hghe cuent fequence n the cage occung dung oto tat-up. Fo hgh fequence of the cuent n the cage the leaage eactance defntel donate n the pedance of the ba n the cage oto and the cuent dplaced towad the a gap, whch bng a educton of the actve co-ecton of the ba and nceae etance. Secton. ha dealt wth the developent of the atheatcal odel of a oto wth contant etance of oto wndng R hence, the ae elevant fo lp-ng and nglecage oto wth a wea effect of cuent dplaceent. The applcaton wth egad to quel-cage oto, fo ntance deep lot oto ntoduce a condeable eo n patcula n te of the chaactetc at the phae of tat-up and all angula velocte of the oto. In ndutal pactce, double-cage nducton oto ae appled n ode to pove the tat-up popete [6]. In the uppe cage uch oto have ba wth a alle daete, whch fo hghe cuent fequence have hghe etance level. A a eult, oe cople atheatcal odel ae necea fo the odelng.

165 44 Inducton Machne n Electc Dve In ode to atonall pefo the tandadzaton of the paaete [8] of vaou nducton oto the equaton of oton n flu coodnate.56 wll be ncondeabl tanfoed and epeed n the te α,β, ω c. It necea to ecale the vaable egadng the aal flue b dvdng the b the ated voltage U n : ψ /U n.9 A a eult, the equaton fo the aal flue tae the fo: u α σ v α σ u α σ v α σ u v u u v u u u u v / U / U pω v v v pωu n n.9 In a la anne, the equaton of oto oton tanfoed and t ultpled b p, hence tandadzed to the efeence of one pole pa achne: p pu n p Ω vu uv Tl DΩ J σ Th equaton tanfoed to tae the fo: T J D J l e ce vu uv ωe.9 ω U N σ p p p Tbω whee: ce.9 J J the electoechancal contant fo an nducton oto, whle J D T J l D Tl.94 p p p epeent, epectvel: oent of neta, coeffcent of vcou dapng and load toque deved fo the nube of pole pa p. T b - the bea toque of the oto n a tead tate. ω e pω - the angula veloct of the oto epeed n te of a oto wth a ngle pole pa, called electcal angula peed. Th veon of the equaton of oton.9 epeent tandadzed equaton fo an nducton oto fo whch the ente cla of ngle-cage oto, not accountng fo cuent dplaceent n the oto cage, ae epeed n te of a ngle nchonou veloct fo p egadle of the value of the ated N

166 . Dnac and Statc Chaactetc of Inducton Machne Dve 45 voltage and the nube of pole pa. Th acheved a a eult of dvdng aal flue ψ b the ated voltage U n.9. In th odel we have to do wth the followng ndependent paaete elatve to the degn of nducton oto:, - coeffcent of agnetc couplng of wndng.57 α, α - coeffcent of dapng.57 nvee of the te contant c e electoechancal contant fo a dve.9 The above paaete ae tandadzed, nthetc paaete of the atheatcal odel of an nducton oto n t ot ple veon that doe not account fo the agnetc atuaton of the coe and cuent dplaceent n the oto cage. The cove each ndvdual oto and ae t poble to calculate t dnac and tatc chaactetc. Othe paaete ued n the odelng of an nducton oto, fo eaple coeffcent of wndng leaage σ.57 ae elatve to the one peented n.94 o ae nvolved n the alead fo ntance nube of pole pa p. The peented ethod of tandadzaton of equaton and paaete baed on [8]. On the ba of the calculaton of paaete.94 conducted on of data ganed fo ndutal catalogue fo eveal eanngful anufactue fo a few dozen of quel-cage oto wth vaou powe atng and ated voltage, the table found below ha been developed Table.. Table. Standadzed paaete of tpcal nducton quel-cage oto Rated powe/ tandazed paaete [W] [W] [W] [W] [W] [-] α [/] α [/] c [ ] e σ [-] γ [-] It contan tandadzed paaete fo a wde ange of quel-cage nducton oto wth bac degn and a all nfluence of cuent dplaceent n the oto cage. Bede the bac paaete.94 t contan elatve paaete.57, whch ae encounteed n eveal veon of the atheatcal odel. The data n Table. gve ean value of the paaete fo the goup of eaned oto wth the nube of pole pa p 5. The eeach that follow eve fo the pupoe of ettng an eaple and llutaton and baed on fou oto fo the goup of 6 oto that wee ued fo the pepaaton of Table.. The elected oto ae epeentatve of the goup of all, edu and lage powe one, epectvel. The paaete ae peented below; one can ae an effot to copae the wth the paaete n Table..

167 46 Inducton Machne n Electc Dve Sall powe oto S P n α σ 5.5 [ W ] U 4 [ V ] I. [ A] T 55. [ N] n n n n n 95 [ / n] p n.5.4 [/ ] α 7. [/ ].67 [ H ] R.9 [ Ω] R [ H ] [ Ω] c e [ ] J.4 [ N T b / T.5 T / T.7 I / I 5.8 n t n t n ].95 P n α P n α Medu powe oto - hgh voltage SH 5 [ W ] U 6 [ V ] I 6.5 [ A] T 4 [ N] n n n n n 48 [ / n] p n.4.6 [/ ] α. [/ ].9 [ H ] R.9 [ Ω] R.85.8 [ H ] σ [ Ω] c e.5 [ ] J 5.5[ N T / T. T / T. I / I 4.8 b P n α n t n t n Medu powe oto low voltage S 56 [ W ] U 4 [ V ] I 975 [ A] T 59 [ N] n n n n n 45 [ / n] p n.4.7 [/ ] α.5 [/ ].4 [ H ] ].75 [ H ] σ.4.98 R 5.7 [ Ω] R 8.4 [ Ω] c e. [ ] J 4. [ N T b / T.8 T / T. I / I 5.7 n t n t n Hgh powe oto S 5 [ W ] U 4 [ V ] I 49 [ A] T 8 [ N] n n n n n 9 [ / n] p n.8. [/ ] α.6 [/ ].79 [ H ]..985 R.86 [ Ω] R 8 4 ].785 [ H ] σ.5 [ Ω].98 c e. [ ] J 45. [ N T / T.9 T / T. I / I 5.7 b n t n t n ]

168 . Dnac and Statc Chaactetc of Inducton Machne Dve 47.. Tpcal Dnac State of an Inducton Machne Dve Eaple of Tajectoe of Moton Equaton of oton fo an nducton achne dve peented n Secton..5 fo vaou coodnate te have 5 degee of feedo: fou fo electc vaable n ae u,v of the tanfoed te of electc vaable of a achne and one degee of feedo fo the echancal vaable n the fo of the angle of otaton of the oto. It poble to have to do wth oe than one coodnate of the echancal oton, fo ntance unde the aupton of a fleble dve haft, whee the angle of haft otaton on the de of the oto and on the de of the dve ae dffeent and th dffeence coepond to the toon angle of the haft. The patcula dnac tate foall conttute dtnct ntal condton fo a te of odna dffeental equaton that fo the atheatcal odel of a dve. The nube of thee tate can be nfnte fo vaou ntal condton; howeve, fo the pactcal pont of vew a few of the ae encounteed ot fequentl and hence the deeve a oe n-depth anal hee. The tpcal dnac tate fo an nducton achne dve nclude: tat-up fo tandtll, oto tat-up unde non-zeo otatonal peed, change of a load, eveal.e. change n the decton of otaton and electcal bang. In the ft ode we wll dcu dnac tate, whch can be eal and effectvel olved ung odel wth tanfoed vaable of the tato and oto. The peentaton wll cove, epectvel: tat-up fo a tandtll, tat-up unde angula peed dffeent fo zeo epeated tatup and eveal and dve ege of opeaton unde cclc vaable load. The followng tage wll nclude the peentaton of dnac tate, whch can be convenentl calculated on the ba of odel n whch ngle de of a oto untanfoed. Such cae nclude the ue of a oft-tat and DC bang of an nducton oto.... Stat-Up dung Dect Connecton to Netwo A copute ulaton of th dnac tate pefoed fo zeo ntal condton. It poble to conduct calculaton b applcaton of vaou veon of the atheatcal odel n a tanfoed coodnate te. It benefcal to appl the odel n cuent coodnate n α,β ae.6 o n flu coodnate.59. Both te ae n the tandad fo and the applcaton of the α,β te enable one to acheve the natual fequenc of voltage n the tanfoed te. The calculaton of the cuent cuve eque the applcaton of a evee tanfoaton T T. unde the aupton that ω c,. Fo the cae of the odel n.59 wth flu coodnate, the tanfoaton of T T, whch lead to tanfoaton of cuent n the aal α,β te ha to be peceded b the tanfoaton of.5 to convet flu vaable nto aal cuent. A a eult, we obtan:

169 48 Inducton Machne n Electc Dve σ σ ψ ψ α β ψ ψ α β.99 The eaple of tajectoe of the oton peented n th ecton appl a odel n the flu coodnate.9-.9, whle phae cuent ae obtaned ung.99. The calculaton of the tajectoe of tat-up fo a tall wa conducted fo thee nducton oto wth epectve, all, edu and lage powe, whoe paaete ae gven n.95,.97 and.98, epectvel. Fg..7 peent the cuve of the phae cuent, electoagnetc toque and angula peed fo an unloaded all powe oto whoe oent of neta on the haft J J, whee J denote the oent of neta of the oto oto. The wavefo of the ae tpe ae peented n Fg..8 and.9 fo edu and lage powe oto. In Fg..8 fo the edu powe oto the tajecto of electoagnetc toque,.e. the elaton of the toque and angula peed addtonall peented. a b c Fg..7 a phae cuent b electoagnetc toque c angula veloct dung fee acceleaton afte dect connectng to the uppl netwo, fo the all powe oto. Moto unloaded and J J

170 . Dnac and Statc Chaactetc of Inducton Machne Dve 49 a b c d Fg..8 a phae cuent b angula veloct c electoagnetc toque te hto d toque tajecto, dung fee acceleaton afte dect connectng to the uppl netwo, fo the edu powe oto, whle T l and J J a b c Fg..9 a phae cuent b electoagnetc toque c angula veloct, dung fee acceleaton afte dect connectng to the uppl netwo, fo the hgh powe oto, whle T l and J J

171 5 Inducton Machne n Electc Dve One can note the condeable ocllato change of electoagnetc toque wth a lage ntal value 4 6 T n and a fequenc la to the netwo voltage dung the dect connecton of the nducton oto to the uppl netwo. Th eult fo the occuence of an apeodc coponent of the agnetc flu geneated b the tato wndng Ψ n aocaton wth lowl nceang flu of the oto wndng Ψ. The ocllato tate of the toque occu untl the ntant when the two flue each a tead tate dung the otaton ove a ccula tajecto. Th well llutated n Fg.. b the peentaton of flue Ψ, Ψ dung the tat-up of the all and lage powe oto. Th fgue efe to the tat-up cuve peented n Fg..7 and.9. The peented toque wavefo dung dect connecton to the netwo poe a hazad to the echancal pat of the dve uch a the haft, clutch a well a the ve devce that connected. Fo th eaon the dect connecton oe and oe fequentl eplaced wth the ethod of oft-tat, whch ae oe wdel dcued n the futhe pat of th ubecton. a b c d Fg.. Magnetc flu vecto tajecto n the a gap of nducton oto dung fee acceleaton afte dect connecton to the netwo: a Ψ b Ψ fo the all powe oto c Ψ d Ψ fo the hgh powe oto

172 . Dnac and Statc Chaactetc of Inducton Machne Dve 5... Reconnecton of an Inducton Moto Reconnecton the te whch denote the dnac tate encounteed dung engagng a oto dung coatng.e. fo a non-zeo angula peed. One can note the dffeence between a econnecton: eultng fo a beadown n powe uppl fo... [], when the agnetc feld n the oto fo the weaenng cuent of the cage doe not deca copletel and a econnecton afte a beadown of powe uppl of ove [], when the agnetc feld deca copletel. Fo fat a b c d e Fg.. Reconnecton of the all powe nducton oto at the nchonou peed and zeo cuent ntal condton: a tato cuent b electoagnetc toque c angula peed d tato flu Ψ e oto flu Ψ tajecto

173 5 Inducton Machne n Electc Dve econnecton one ha to tae nto account non-zeo ntal condton fo electc vaable on the oto wndng Ψ, nce the foed electoagnetc toque then condeabl dependent on the phae of the voltage connected to the tato tenal. Jut a n the cae of nchonzng a nchonou achne wth the netwo t poble to undetae a econnecton n accodance wth the a b c d e f Fg.. Reconnecton of the hgh powe nducton oto at the nchonou peed and zeo cuent ntal condton: a tato cuent b angula peed c electoagnetc toque d toque-peed tajecto e tato flu Ψ f oto flu tajecto Ψ

174 . Dnac and Statc Chaactetc of Inducton Machne Dve 5 phae, whoe chaactetc nclude low value of the connecton cuent and electoagnetc toque. In contat, n the ot advee cae, uch econnecton can occu n the ccutance of the oppoton between the phae of netwo voltage and the one on the oto tenal. In the latte cae we have to do wth a lage connecton cuent, whch hazadou fo the dve due to a toque uge. Fo th eaon t bet to avod fat and dect econnecton of an nducton oto nto the netwo. Fo lage powe oto the duaton peod of the hazadou econnecton lat fo about.8-. [], whle fo all and edu ze one the beadown the te.5 []. The econnecton of the oto afte the peod of the voltage beadown ove [] could be condeed a the connecton fo the zeo ntal condton of electc vaable. The cuve of the cuent and toque afte uch a econnecton elatve to the angula peed Ω afte whch the econnecton ha occued; howeve, t doe not eceed the value that ae peent dung the dect connecton of a oto dung tandtll. The cuve of the cuent, electoagnetc toque and angula peed and agnetc flue n the tato and oto afte connecton to the netwo fo nchonou peed ae peented n Fg.. and. fo all and lage powe oto, epectvel.... Dve Reveal The te eveal denote tunng on a dve unde a peed that evee to the decton of the otaton eultng fo the equence of phae of the uppl netwo afte tunng on. A eveal a be aocated wth the need of a technolog o a b c d Fg.. Reveal of the edu powe nducton oto dve J J : a tato cuent b oto flu c elatve oto peed d toque-peed tajecto

175 54 Inducton Machne n Electc Dve a fo a tpe of bang eultng fo a counte cuent. In th cae the eveal hould be dcontnued when the otatonal peed of the dve cloe to zeo and befoe the dve tat t otaton n the oppote decton. The dnac cuve of the cuent and electoagnetc toque n the ntal phae of the eveal ae la to the value of the cuve fo thee value dung tat-up unde dect tunng on. The calculaton of the tajectoe of the dve oton appl zeo ntal condton fo cuent flue, b aung an adequatel long nteval n the uppl [] o oe and an angula peed la to the nchonou peed but wth a negatve gn. Eaple of cuve dung the eveal of a ddle powe nducton oto.97 ae peented n Fg Cclc oad of an Inducton Moto oad on a oto a contan a vaable te. In th cae tead opeaton tate, undetood a fed pont, not acheved b a dve on t chaactetc. In contat, the dve opeate n a cloed tajecto when the opeatng ege becoe tead. Fo hgh neta of the dve and a elatvel all vaable te of the load toque the tajecto of the oton cloe to a fed pont. In the oppote cae the tajecto of the dve oton fo a cuve that condeabl dvege fo tatc chaactetc. The tajecto elatve to the value of the vaable te, fequenc of the load vaaton and oent of neta elatve to the oto haft. The eaple of the dve ege of opeaton fo a lage coponent of vaable load ae peented n Fg..5,.6,.7. Fg..4 that pecede the peent the tepwe vaable load toque actng on the nducton achne haft. Fg..5 peent the dnac cuve fo the ean load equal to the ated toque n the ccle of the load, n whch fo τ.τ l the load toque equal to T la 4.T n, and n the eanng pat of the ccle T ln.t n wth the fequenc of the toque vaaton f l [Hz]. The tajecto of the electoagnetc toque n epect to angula peed fo a cloed cuve wth the hape of an eght. Fg..6 peent the cae of the dentcal load on a dve but fo fequenc f l 6 [Hz] and f l 5 [Hz]. A a conequence, thee a condeable educton and ltaton of the tajecto loop. Fg..7 peent a vaable load wth the fequenc of f l [Hz] and f l [Hz] at the bounda of dve bea. Fg..4 The cclc load toque of the dve

176 . Dnac and Statc Chaactetc of Inducton Machne Dve 55 a b c d Fg..5 Inducton oto dve chaactetc unde peodc load change: f l [Hz], T lav T n, T la 4. T n : a phae cuent b otatonal peed c toque te hto d toquepeed tajecto...5 Soft-Stat of an Inducton Moto fo Non-ultaneou Connecton of Stato Wndng to the Netwo A t wa entoned eale..., the dect connecton of an nducton oto to the uppl netwo eult n the hgh value of an ocllato coponent of electoagnetc toque dung tat-up. Bede, thee condeable value of the tat-up cuent. Th well llutated n Fg fo all, edu and hgh powe oto. The ocllaton of the electoagnetc toque can be condeabl lted and, hence, t vtuall poble elnate the effect a a eult of the applcaton of nchonzed connecton of phae wndng n the netwo. In the ft tage, two clap of the tato wndng ae connected wth a utable nchonzaton wth the netwo and n the econd tage the thd clap connected wth an adequate phae dela. A copute ulaton of the eaple that llutate th ue can be convenentl conducted b ue of a odel of an nducton oto wth untanfoed electc vaable of the tato wndng. Th

177 56 Inducton Machne n Electc Dve odel peented n equaton , whle the notaton ued fo tato wndng follow the one n Fg..6. Unde the aupton that the voltage of the uppl netwo U gven n the fo of the functon a b Fg..6 Inducton oto electoagnetc toque unde peodc load change fo vaou fequenc value: a f l 6[Hz] b f l 5[Hz] a b Fg..7 Inducton oto pefoance unde peodc load change cloe to the bea toque loadng: a f l [Hz], T la 5. T n, T lav.87 T n, b f l [Hz], T la.5 T n, T lav.85 T n

178 . Dnac and Statc Chaactetc of Inducton Machne Dve 57 U U n ω t γ. the oft-tat of the oto follow fo the phae angle: γ χ π χ.46.5,. whle the connecton of the eanng, thd, clap follow wth a phae dela: δ π / χ χ.4.6. a b c d e Fg..8 Dela oft-tat of the all powe nducton oto J J : a phae cuent b angula peed c electoagnetc toque d tato flu tajecto e and oto flu tajecto

179 58 Inducton Machne n Electc Dve a b c Fg..9 Dela oft-tat of the edu powe nducton oto J J : a electoagnetc toque b tato flu tajecto c oto flu tajecto The value of coeffcent χ, χ ae detened a a eult of the calculaton nvolvng ulaton fo elected quel-cage oto; the etee value ae elevant wth egad to oto fo a all to lage powe. The ntance of uch connecton, eult n a vtual lac of apeodc coponent n the geneated agnetc flu Ψ, Ψ. Th ue ha been llutated ung eaple baed on copute ulaton fo oto fo all to lage powe, and the obtaned eult ae peented n Fg..8,.9 and.. One can eal note the ooth cuve of the cuent wthout the apeodc coponent and all ocllaton of the electoagnetc toque at the ntal tage of the tat-up. Th coe a a conequence of the cuve of tajectoe of agnetc flue peented n the fgue. Th tpe of tat-up, that aocated wth the

180 . Dnac and Statc Chaactetc of Inducton Machne Dve 59 a b c d Fg.. Dela oft-tat of the hgh powe nducton oto J J : a phae cuent b electoagnetc toque c tato flu tajecto d and oto flu tajecto need to appl powe econducto wtche, ha condeabl oe advantage than dect connecton, and th can be concluded fo a copaon between the above llutaton and eult peented n Fg..7,.8,.9 fo the ae oto....6 DC Bang of an Inducton Moto Bang ung dect cuent nvolve DC uppl to the utabl connected tato wndng n uch a wa that enable the potentall hgh contant agnetc flu n whch the oto put n oton. The cuent poduced b oto wndng a a conequence of nducton cobne wth the agnetc feld thu poducng bang toque, whch appoache dle un fo a DC uppl,.e. the condton when the oto talled. Fg.. peent two tpcal laout fo aong the lt of the poble connecton between the tato wndng fo bang. The odelng baed on equaton fo an nducton oto fo untanfoed cuent of tato wndng. Fo a thee-phae te of connecton Fg..a, we dectl appl equaton b aung that: u U u. DC Fo two-phae powe uppl dung bang Fg..b the followng contant ae applcable: u U DC.4

181 6 Inducton Machne n Electc Dve Fg.. Connecton of nducton oto tato wndng fo the DC thee-phae beang and the two-phae beang a b c d e f Fg.. -phae DC beang of the all powe nducton oto wth J 5J, DC I n.: a tato cuent b tato cuent c agnetzng cuent d MMF tajecto u / v e electoagnetc toque f otatonal peed

182 . Dnac and Statc Chaactetc of Inducton Machne Dve 6 Conequentl, the tato wndng connected n a ta have a ngle degee of feedo, fo whch we aue the vaable. Ung.4 and afte elnaton of the latte of equaton.76, we obtan the odel fo th tpe of bang: v u v u DC dt d R p p p R p R R U θ θ θ θ.5 v u e p T On the ba of the obtaned veon of the atheatcal odel, ulaton wee conducted fo bang of a all and lage powe oto.95,.98 fo a bang cuent, whch n the tead tate equal to DC I n. The chaactetc wavefo ae peented n Fg.. to.5. a b c Fg.. -phae DC beang of the all powe nducton oto wth J 5J, DC I n.: a tato cuent b agnetzng cuent c MMF tajecto u / v d electoagnetc toque e otatonal peed

183 6 Inducton Machne n Electc Dve d e Fg.. contnued a b c d e f Fg..4 -phae DC beang of the hgh powe nducton oto wth J.5J, DC I n : a tato cuent b tato cuent c agnetzng cuent d MMF tajecto u / v e electoagnetc toque f otatonal peed

184 . Dnac and Statc Chaactetc of Inducton Machne Dve 6 a b c d e Fg..5 -phae DC beang of the hgh powe nducton oto wth J.5J, DC I n : a tato cuent b agnetzng cuent c MMF tajecto u / v d electoagnetc toque e otatonal peed The cloe falat wth the eult of calculaton fo DC beang lead to the followng geneal concluon: - -phae bang condeabl oe effectve than -phae bang wth dect cuent; howeve, t chaactetc nclude ocllaton of toque and peed n the fnal phae of bang. Th eult fo the lac of dapng of the claped ccut n the wndng n phae and - agnetzng cuent dung bang qute all and defntel alle than the agnetzng cuent dung etc oto ege. Afte the oto talled the agnetzng cuent eache the value of DC. Hence, the atuaton of the agnetc ccut ove the ente ange of peed dung bang la to chaactetc of oto ege of opeaton and the appled odel wth contant paaete ean n the ae pecon ange a dung oto ege. Th

185 64 Inducton Machne n Electc Dve concen the uppl of the oto wth the dect voltage, fo whch cuent DC doe not eceed eveal te the ated cuent. - condeable dffeence ae abent fo the dnac cuve of bang fo all and lage powe oto. Salle oto tend to bae oe dnacall, n accodance wth the lage value of the electoechancal contant c.9... Reducton of a Matheatcal Model to an Equvalent Ccut Daga A dnac te, uch a electc dve decbed wth odna dffeental equaton fo gven ntal condton and nput functon, chaactezed wth a pecfc tajecto of the oton. Th tajecto epeent the hto of all vaable n a te. The tead tate of uch a te occu when the tajecto epeented b a fed pont, that t { q } {} q o b a peodc functon wth the peod of T, when t T T.6 Fo an electc dve th occu when vaable n a te fong the vecto of genealzed coodnate q ae ethe contant functon o peodcall vaable one. In a nducton oto dve we can aue n an dealzed wa that the tead tate occu when the angula peed contant,.e. Ω θ cont and the electc cuent whch uppl the wndng ae peodc functon wth the peod n confot wth the voltage enfocng the flow of the cuent. One can note that the hto of both the uppl voltage and the eultng cuent elatve to the tanfoaton of the co-odnate of the te, a peented n the odel of the oto n α,β, d,q o, ae In a, te otatng wth the peed ω c ω pω f the etc te of nuodal voltage upplng phae wndng a a eult of tanfoaton.7 educed to contant voltage. In uch coodnate te the tead tate lteall ean a fed pont on the tajectoe of all vaable. The tuaton wll be dffeent fo a tead tate n the cae of aet of the uppl voltage o cclcall vaable load toque. In uch a cae the tead tate wll be chaactezed b peodcall vaable wavefo of electc cuent and angula peed, whle n the peed wavefo the contant coponent wll fo the pedonant eleent. The acquantance wth tead tate elevant fo the degn and eplotaton of a dve nce t povde nfoaton egadng t opeatng condton and, hence, fo the ba fo the developent of tatege egadng ethod of dve contol. The falat wth the tead tate ae t poble to detene the chaactetc of the dve,.e. functonal elaton between vaable that fo the et of contant pont on a tajecto and one that ae te nvaable. Fo the eaon gven hee the tead tate of the nducton oto dve can be convenentl decbed n aal coodnate,. Theefoe, we wll tae a the tatng pont the tanfoed equaton.6 n cuent coodnate, whch afte the ubttuton ω c ω gve:

186 . Dnac and Statc Chaactetc of Inducton Machne Dve 65 whee: u R R d X dt X X X A X A ω X ω X ω.7 ω pω whle.8 ω - the lp of the oto peed n elaton to the otatng agnetc feld ee.6. t We aue that the tead tate fo the fed pont of the tajecto q q, hence, t denote the contant angula peed Ω cont and the contant lp cont. Th condton poble due to the contant value of cuent,, and, a a eult, the contant electoagnetc toque T e. Th eque the contant uppl voltage afte the tanfoaton of,, whch tae the followng fo n accodance wth.7: u coγ u U ph.9 u nγ The fo of voltage.9 ugget the ntoducton of cople value: ph jγ U u ju U e U ph. whee: U ph - the RMS value of the voltage upplng the phae of the oto. Subequentl, we can ubttute: I j I j. In the followng tanfoaton of equaton.7 the latte of the equaton n each pa ultpled b the agna unt j and added to the ft of the equaton, thu gvng the equaton fo a cople vaable. Fo the tato we obtan: U R I jx I jx I. d d Hee we have appled: I, I, whch eult fo the tead tate dt dt and A j whch afte addton of ow vecto lead to: A j. j A we pefo la opeaton fo the othe pa of equaton,.e. oto equaton, and dvdng th equaton b lp, we obtan:

187 66 Inducton Machne n Electc Dve R I jx I jx I. Th ae t poble to develop an equvalent ccut fo an nducton oto n the tead tate a a eult of egng equaton.,. n the fo of a two pot, ung a coon agnetzng eactance te jx. The equvalent ccut n the fo n Fg..6, bede the voltage and cuent elaton peented n eve two pot, alo ealze n an undtubed anne the enegetc elaton occung n the tead tate. Th coe a a eult of the applcaton of othogonal tanfoaton that peeve cala poduct and quadatc fo n the tanfoaton of equaton. Fg..6 Equvalent ccut of an nducton oto fo the tead tate In th ccut we have to do wth a etance te R /, whch ealze n the enegetc ene both Joule loe n the oto wndng and the echancal output of the dve tanfeed va the achne haft a the poduct of toque T e and the angula peed of the haft Ω. Hence the etance te can be dvded nto two te: R, R -/, whch ealze the loe of the powe n the tato wndng and echancal powe P, a t peented n Fg..7. The followng coponent of the electc powe ae encounteed n the equvalent daga: Fg..7 Equvalent ccut of nducton oto wth phcal ntepetaton of electc powe coponent

188 . Dnac and Statc Chaactetc of Inducton Machne Dve 67 P U I co nput powe el I R f I R / P P Joule loe n tato wndng a gap feld powe.4 Pel I R Joule loe n oto wndng P I R echancal powe P P el P f The eneg balance fo a -phae achne peeved due to the fact that U U ph, hence, the tanfoed powe thee te hghe than the powe of a ngle phae. In the anal of the epeon fo the echancal powe output of an nducton oto dve we can dtnguh the followng aea of opeaton:. fo < < P f >, P > - oto ege. fo < P f <, P < - geneatng ege. fo > P f >, P < - bang ege 4. fo P f >, P - tall of the oto 5. fo P, P - dle un f Fo the epeon fo the echancal powe we can calculate the oto toque n the tead tate: T e P p I R.5 Ω ω The equvalent ccut can addtonall be ueful n the calculaton of the tato and oto cuent: I R jx I I U X R / jx jx R / jx.6.7 It would be valuable to peent the cuent n the tandadzed paaete.57 nce a a conequence of uch peentaton t poble to depat fo the patcula degn of an nducton oto. The tandadzed paaete aue value n the ange peented n Table.. In th cae the elaton.6-.7 tae the fo:

189 68 Inducton Machne n Electc Dve I I ω X I U jω jω jα ω σ α jω ω jα.8 Cuent I, I epeent bolc value of tato and oto cuent fo tead tate ne cuve. The electoagnetc toque n the tead tate can be deved fo elaton.45 T p p I I I.9 e Ung elaton.6,.7 peentng tato and oto cuent we obtan: whee: pu Te ω R R R X z R X X z R. σx. - a bloced-oto eactance. The electoagnetc toque can alo be peented ung tandadzed paaete.59, and tae th fo: α pu σ Te. X α α α σ ω α ω The epeon.,. epeentng electoagnetc toque elatve to uppl voltage and oto paaete ae fequentl ubjected to cetan plfcaton n ode to plf the anal of thee epeon. The bac pocedue apple degadng of the etance of the tato wndng R and, ubequentl, α n oe o all te of th epeon. A detaled anal of th tpe of plfcaton wll be conducted late on dung the detenaton of the chaactetc of the dve ege...4 Statc Chaactetc of an Inducton Moto Statc chaactetc concen the tead tate of a dve and gve n an analtc o gaphc fo the functonal elaton between the paaete chaactezng oto ege. Tpcal tatc chaactetc can fo ntance ndcate the elaton between electoagnetc toque, cuent and the capact of a oto o between effcenc and the lp, voltage uppl and the powe output of the dve and the le. One can note that tatc chaactetc conttute a et of contant pont along a

190 . Dnac and Statc Chaactetc of Inducton Machne Dve 69 tajecto { q} q fo elected vaable of the tate q o the functon, that llutate the value that ae nteetng fo the pont of vew of the pecfc ege of a achne, fo eaple electoagnetc toque T e. Statc chaactetc collect the end pont of tajectoe fo whch the te eache a tead tate. The do not povde nfoaton egadng the tanfe fo a pecfc pont on the chaactetc to anothe one, how uch te t wll tae and whethe t attanable. Hence, n tatc chaactetc we do not have to do wth uch paaete a oent of neta J, and the electoagnetc toque T e and load toque T l ae equal nce the dve n the tate of equlbu,.e. t doe not acceleate o bae ee.. Fo eaple, ve elevant chaactetc ae peented ung functon.,.. The llutate the electoagnetc toque fo an nducton achne dependng on a nube of paaete. A tpcal ta nvolve the tud of the elaton between the chaactetc of the electoagnetc toque and the lp T e fo contant eanng paaete, nce t nfo of the dvng capablte of the oto n the tead tate. The elaton between achne toque and lp T e gve the au of th functon fo two lp value called bea toque lp o pull-out lp. o n tandadzed paaete R R X b ±. σx R σ X b χ α α ω ±.4 ω σ α / σ ω The oot te χ n foulae.,.4 the facto fo coecton of the value of the bea toque lp a a eult of the of tato wndng etance R nfluence. Snce leaage coeffcent σ.8. ee Table. the followng nequalt fulflled R σx χ R >> X.5 In addton, thee elaton ae nveel popotonal to the quae oot of the fequenc of the uppl ouce. Two degee of plfcaton that ae applcable n the developent of tatc chaactetc of an nducton oto eult fo the peented etate. The ft of the not ve fa-eachng and nvolve degadng of etance R n the te denotng toque.,. and bea toque

191 7 Inducton Machne n Electc Dve lp.,.4, n whch th effect alle n accodance wth the etaton n.5. In th cae we obtan: T e b pu ω ± σx R σx R σx R R R.6 The ot etenve plfcaton concen the cae when the etance of the tato wndng copletel degaded,.e. R. In th cae we obtan: T e R pu ω R and σx b ± R σx.7 Fg..8 peent tatc chaactetc of the oto toque n the functon of the lp fo a all powe nducton oto fo the thee eaned vaant of plfcaton egadng etance R. One can note the all dffeence between the cuve fo the toque aed wth old lne.e. the one peentng elaton wthout plfcaton.,., and dotted lne.e. the one peentng the eult of calculaton on the ba of foulae.6 nvolvng the ft degee of plfcaton. Howeve, when the etance of tato wndng totall degaded R n accodance wth foulae.7, the eo n the chaactetc of toque T e condeable, a the elatve nvolveent of etance R n the tall pedance of all powe oto eanngful. Fg..8 Toque-lp chaactetc fo the all powe nducton oto llutatng plfcaton concenng tato etance R : R taen nto condeaton copletel, accodng to.,.; nto condeaton taen onl the ot gnfcant coponent contanng R, accodng to.6; R totall degaded.7

192 . Dnac and Statc Chaactetc of Inducton Machne Dve 7 It notewoth that fo R the chaactetc of oto toque becoe an odd functon of the lp, o t etcal n elaton to the pont of the dle un. Accountng fo etance R toque wavefo on the de of the oto ege > condeabl alle n te of abolute value than fo the cae of geneatng ege,.e. fo <. In addton, on the de of the geneato ege the effect of the ft degee of plfcaton accountng fo etance R oe cleal dcenble than fo the cae of the oto ege, whch can be pl ntepeted b analzng elaton.,.. The peented effect of the etance of tato wndng on the chaactetc of the toque nceae along wth the educton of the pulaton of the uppl voltage ω and becoe ve hgh fo all fequence. Th ubject wll be coveed n oe detal late. Th effect gaphcall peented n Fg..9 n the ange of the uppl fequence < f 5 [Hz]. Fg..9 Chaactetc of the bea-toque lp b veu pulaton of the uppl voltage ω fo the all powe oto The foulae fo the bea-toque lp and oto toque accountng fo plfcaton concenng the etance can be addtonall peented n foulae contanng tandadzed paaete. The equvalent of the foulae.6 tae the fo: α pu σ Te and X α σ ω α b α ω ±.8 ω σ α σ ω / Concuentl, foulae.7 ae eplaced wth the fo whch degad etance R, b ntoducng α :

193 7 Inducton Machne n Electc Dve X pu T e α σ ω α and ω σ α b ±.9 In the latte cae t ea to calculate the value of the bea-toque: X b b e U p X pu T T z z ω σ ω σ. Foula. conttute the bac ule applcable fo adjutng the RMS value of nuodal uppl voltage U of the oto to the fequenc of th voltage f n uch a anne, that guaantee a contant bea-toque value of T b. Hence, the elaton tae the fo: n n U U ω ω, whch, ubequentl gve: n n f f U U. Dung the coue of acton that follow n the dcuon of fequenc baed contol of oto otatonal peed t wll becoe evdent that th ule copletel nuffcent wthn the ange of all uppl fequence. Th o due to the ng hae of the etance R n the pedance of the oto tall along wth the deceae n the fequenc of uppl. The elaton denotng the bea-toque wthout plfcaton, n whch etance R not degaded, uch oe cople than the one n.. The geate coplet of the elaton eult fo the ubttuton of the bea-toque lp b. n the epeon denotng the electoagnetc toque of the oto. A a eult we obtan: χ ω σ α ω χ α χ σ ω b pu gn T. whee coeffcent χ eult fo. and gven b the elaton: X R X R σ χ.

194 . Dnac and Statc Chaactetc of Inducton Machne Dve 7 Unde the plfng aupton that R, we have α and χ and, a a conequence, bea-toque epeon. educed to th fo.. The elaton n. appled to ndcate the effect of etance R on the beatoque T b oe cleal. The followng llutaton n Fg.. how voltagefequenc elaton equed to povde contant value of nonal bea-toque T bn n the functon of tato voltage pulaton ω. Fo the oto ege of opeaton the equed voltage cleal hghe than fo the geneato ege. Fo Fg.. we can alo ee that alle oto, wthn low fequenc ange, eque uch hghe uppl voltage than lage oto to utan the nonal level of T bn. A cloe npecton of Fg..b ndcate that fo hghe pulaton ω the dffeence between oto dappea, but tll thee contant dcepanc between the etcal deal V-lne fo α and the cuve, fo whch tato etance R wa accounted fo. Fo the oto opeaton the equed voltage ae hghe whle fo geneato opeaton the ae lowe n copaon to the deal V- lne. One ght a that the actual V-lne fo whch etance R ncluded hfted n the decton of lowe pulaton ω n epect to the deal V-lne fo whch R copletel gnoed. a b Fg.. Voltage-pulaton cuve ndcatng the a tato voltage level equed to utan a nonal bea-toque T bn whle ω pulaton change. The cuve ae peented fo dffeent nducton oto wth α 8.8, 5.4,.7,.,. : a fo full ange of tato voltage pulaton ω, b ange of ω lted to low value Subequentl, Fg.. peent the chaactetc of the oto n the functon of the lp n two veon: copletel accountng fo paaete R - alle chaactetc n each pa, and the one totall degadng etance,.e. fo R - wth the above peented chaactetc. Fo nonal value of ω πf, the dtnctve dffeence between the two veon tae place fo the all powe oto.

195 74 Inducton Machne n Electc Dve Fg.. Toque-lp cuve elatve value fo the thee nducton oto: all, edu and hgh powe. The effect of R plfcaton llutated fo f 5 [Hz] Subequentl, Fg.. peent the chaactetc of tato cuent fo the thee oto accountng fo etance R. The elaton.4 appled n th cae, whch coe a a conequence of.8: X U I ω α α α α α ω σ α ω.4 When the etance of tato wndng degaded α, the elaton whch defne the cuent n the tato wndng tae a condeabl oe uccnct fo, whch addtonall ea to vef fo the two etee oto tate,.e. fo and. X U I α ω σ α ω.5 Self eactance of the tato wndng X encounteed n a ulttude of elaton concenng nducton oto. The value of th paaete can be eal detened fo calculaton o anufactue data fo dle un. Fo the equvalent daga Fg..6 of the oto t eult that n Z X jx R Z I U.6

196 . Method and Devce fo Fong Chaactetc of an Inducton Moto 75 Fg.. Stato cuent-lp cuve elatve value fo the thee eepla oto accodng to.4 peented n elatve value, fo f 5 [Hz] whee U, I, Z, denote voltage, cuent, pedance and phae angle fo the dle un of the oto. If the phae angle dung dle un not fala, t poble to ue aeent elevant fo the ated fequenc: R << X and calculate n an appoated wa: U X.7 I. Method and Devce fo Fong Chaactetc of an Inducton Moto B t ve natue eultng fo old oto wndng and lac of powe uppl to t wndng, an nducton oto ot utable fo opeaton unde tead condton and wth a all lp. In uch a cae the angula peed eult fo the fequenc of the uppl to the tato wndng, nube of pole pa and value of the lp. Tadtonall, t wa appled n dve n whch nethe fequent change of peed no vaable contol wee equed eaple of uch devce nclude pup, blowe, copeo, belt conveo, cane, ndutal hot. Thee wa vtuall no poblt of contollng nducton oto wthn wde ange of peed whle concuentl peevng hgh enegetc effcenc untl 97. Dve n whch the contol of peed wa necea ot fequentl appled lp ng nducton oto, n whch t poble to contol otatonal peed a a eult of ue of etenal eleent. Howeve, uch te ae ethe cople, cotl and pobleatc n contol due to the ue of cacaded te. Altenatvel, the have lowe enegetc effcenc due to addtonal etance n the oto ccut. In addton, the tat-up popete of an nducton oto unde dect connecton to the netwo ae advee due to the ntal peod of ocllaton of electoagnetc toque wth a hgh apltude and hgh value of the tat-up cuent. Depte thee dawbac the nducton oto ha becoe the ot coon achne n electc dve te due to the fundaental advantage ncludng long evce lfe and

197 76 Inducton Machne n Electc Dve elablt a well a low pce and acceble uppl ouce. Followng the developent of powe electonc and contol eleent enablng abta hapng of voltage and cuent, nducton oto becae wdel appled n cople dve due to a new angula peed contol potental and geneal obutne at heav dut. Th ecton wll be devoted to the peentaton of the ethod of fong chaactetc of nducton oto and wll cove the devce that ae t poble to ealze the equed chaactetc. The poblt of odelng chaactetc eult dectl fo the elaton defnng angula peed π f Ω ω f.8 p whee: ω f π f - nchonou angula peed of a otatng feld. Each of the p value n elaton.8 offe the poblt of odelng echancal chaactetc: nube of pole pa p, lp a well a the fequenc f of the uppl voltage. The contol of lp poble to a lage etent a a eult of the etenal ntefeence n the oto ccut and alo voltage change but wthn a all ange of otatonal peed. The peentaton of ethod ued fo odelng chaactetc aocated wth oto lp change wll follow n the ubequent ecton. Concuentl, a epaate ecton wll be devoted to an etenve peentaton of contol a a eult of odfng the fequence of the uppl voltage. The applcaton of the vaou nube of pole pa p fo changng oto peed appea to be ot taghtfowad to eplan. A ee of nchonou peed ω f fo a gven uppl fequenc cont of a dceet value. Fo the ucceve nube of pole pa p,,,4,5,6,... and fo eaple fo the fequenc of the uppl f 5 [Hz] the ae, appoatel: ω f 4.6, 57., 4.7, 78.5, 6.8, 5.4, Th fnd applcaton n ult-pole oto, n whch the wndng can be wtched to two o thee nchonou peed, whch lead to a tepwe change of oto peed. Th tpe of dve appled n cane and ndutal hot anl wth two peed tant peed wth a hghe value and a lowe appoach peed... Contol of Suppl Voltage The contol of the uppl voltage can offe onl lted poblt of adjutng otatonal peed of an nducton oto. Th eult fo the bac echancal chaactetc of the oto Fg..8 whch ndcate that the lp unde a gven load can be nceaed up to the lt of < b, whch ean t ha to eep below the bea-toque lp beond whch a lo of the tablt occu and the oto top. In addton, th tpe of contol acheved at the epene of effcenc lo nce unde a contant load the loe n the oto ae ΔP > P f. Th coe a a conequence of the nceae of the cuent and loe n the oto wndng. At the ae te, the contol of the uppl voltage cuentl ued n ode to educe the tat-up cuent and pefo a oft-tat. Th ealzed wth the ue of an electonc devce called a oft-tate. A daga of uch a devce found n Fg...

198 . Method and Devce fo Fong Chaactetc of an Inducton Moto 77 Fg.. Bac daga of a oft-tate fo an nducton oto The ntoducton of econducto eleent SCR, IGBT, GTO, MOSFET etc. fo the two decton of cuent flow fo each lne upplng the oto wndng ae t poble to eplo cuent flow wth a elected dela angle α n elaton to the zeo cong of the uppl voltage cuve. A a eult, at oe epene of alteng the cuent and voltage fo nuodal hape, t poble to contol the value of voltage and nchonze the oto wth the netwo at the ntant of connectng the patcula oto phae wndng dung tat-up. Soft-tate a, accodngl, ealze the followng functon elated to the tat-up and toppng of an nducton oto: - nchonzaton of the connecton of patcula phae wndng to the netwo and thu enablng the educton of the vaable coponent of the toque ee ltaton of the tat-up cuent n a elected ange, - bang wth the ue of dect cuent ee ecton...6 and conduct contolled top of a dve. Not all of the above functon have to be ealzed b a ngle tpe of oft tate. In the ot econoc veon degned fo alle dve, a oft tate oete contan wtche n the two uppl lne, whch onl lead to ltaton of the tat-up cuent and doe not povde et of the uppl voltage. The followng Fg peent the eaple of applcaton of a oft-tate fo an edu powe nducton oto wth a dela angle α 4º and the bac value of the oent of neta J J. The fgue peent a copaon between tatup veon wthout nchonzaton dung the connecton of phae to the netwo and the one wth nchonzaton nvolvng the connecton of lne, fo phae angle δ,.48π [ad] and a late connecton of the thd uppl lne fo angle: δ a δ, -. [ad]. A a eult, we obtan a ve oft tatng cuve dung the ntal tage of the tat-up of the oto Fg..4 accopaned b a ve favoable toque wavefo Fg..6. The nchonzed connecton fo uch a lage dela angle α 4º alo eult n the educton of the duaton of the

199 78 Inducton Machne n Electc Dve tat-up Fg..5, Fg..6 nce the value of the contant coponent of the oto toque nceae dung the ntal tage of the tat-up. The cuent wavefo n the phae wndng of the oto fo uch a uppl peented n Fg..7. The dela angle n the ange of aound 4 vtuall the hapet one fo whch t poble to conduct tat-up of the oto dung dle un wthn a enble te, due to the condeable educton of the value of electoagnetc toque of the oto. The appoate llutaton of the effect of dela angle α on chaactetc of the oto peented n Fg..8. Soft-tate fnd applcaton n dve wth an ea tat-up due to the condeable educton of the toque followng the fall of the value of the uppl voltage. a b Fg..4 ne cuent of the edu powe nducton oto dung fee acceleaton wth a oft tate α 4 : a wthout nchonzaton b wth nchonzaton: δ.48π; δ a δ, -. a b Fg..5 Relatve veloct cuve fo the edu powe oto dung the oft-tat fee acceleaton, unde condton le n Fg..4

200 . Method and Devce fo Fong Chaactetc of an Inducton Moto 79 a b Fg..6 Electoagnetc toque cuve fo the edu powe oto dung the oft-tat fee acceleaton, unde condton le n Fg..4 Fg..7 Stead-tate lne cuent fo the edu powe oto dung the oft-tat fee acceleaton, unde condton le n Fg..4 Fg..8 Inducton oto tatng chaactetc elatve value fo the edu powe oto dung the oft-tat fee acceleaton n elaton to dela angle α: a tatng cuent b bea toque c tatng toque d dle un fee-acceleaton te

201 8 Inducton Machne n Electc Dve.. Slp Contol It poble to contol lp n an nducton oto when electc powe delveed though the oto wndng to the etenal devce. Th coe a a conequence of the fact that fo a contant electoagnetc toque T e and contant uppl fequenc f the powe P ψ delveed b the otatng feld fo the tato to the oto ha to ean contant. Th o nce n the tead tate Pψ Te.9 ω f Afte oe powe etacted fo the oto wndng, the electc powe P el P Ψ nceae and the echancal powe P P Ψ - deceae, whch poble a a eult of an nceae of lp,.e. the educton of the otatonal peed of the oto. A we can ee, the contol of the lp onl poble n lp-ng oto, due to acce to the oto wndng fo outde. The othe poblt aocated wth powe uppl to the oto hadl eve pactcall encounteed. It poble fo ntance n a oto wth powe uppl fo two de and th cae wll not be dcued n th boo [5,8,6,67]. The poce of powe etacton fo wndng conducted n two wa. An ncluon of an addtonal etance R d n the oto ccut the oldet ethod of pefong oft-tat and pobl peed contol; howeve, t accopaned b huge loe aocated wth the poduced heat. Anothe ethod nvolve powe output to etenal devce whoe ole to tanfo the powe to ueful fo, fo ntance t etun to the uppl netwo. Such devce, whch ued to be electoechancal, now pedonantl ae powe electonc one ae called cacade. One of the the Schebu dve, and a ubject n the latte pat of th ecton.... Addtonal Retance n the Roto Ccut Th ethod of contol eult n change of tatc chaactetc of the toque peented n Fg..9. Accodng to.,.4,.7 the bea-toque lp b nceae popotonall to the nceae of the etance of oto wndng R, whle the bea-toque T b doe not change. Th coe a a conequence of the antenance of the contant elaton α /, whch ean that lp e popotonall to the nceae of α. Th n a wa eult n the change of the cale of the lp whch etend the chaactetc of the toque n the decton of hghe value of the lp. Th lead to an poveent of oto tat-up nce the tat-up cuent educed fo and the tatc tat-up toque nceae. Unfotunatel, the opeaton of the oto n the tead tate wth an addtonal etance n ode to educe the otatonal peed not appled nce t eult n the educton of the enegetc effcenc of the dve. Th o becaue fo an nducton oto the followng elaton antaned: η <.4 whch ean that fo ntance the educton of oto peed Ω to each the half of the nchonc peed ω f wth the ue of etance baed contol lead to the educton of the effcenc to η <. 5, whch unacceptable.

202 . Method and Devce fo Fong Chaactetc of an Inducton Moto 8 Fg..9 Toque-lp chaactetc of a wound nducton oto wth addtonal etenal etance n the oto fo α d α ; α d nclude addtonal etance R d connected to the lp-ng... Schebu Dve A a eult of the applcaton of Schebu dve t poble to contol the peed of a lp-ng nducton oto a a conequence of electc powe etacton fo oto wndng and t etun afte the deed tanfoaton nto the netwo. In t htocal odel the Schebu dve contan an electoechancal fequenc convete connected on one de to the oto of an nducton oto and the othe one to the uppl netwo nto whch the powe etun. In a oden oluton of the Schebu dve Fg..4, the cuent of the oto wndng ae ectfed n a -phae ectfng bdge and ubequentl uppl a convete whch etun the eneg nto the netwo va an adaptng tanfoe. Between the two bdge thee an nducto that oothen the flow of the cuent and whoe ole to ecue the contnut of cuent flow aco the oto even fo all echancal load of the oto haft. The contol paaete dela angle α of the thto bdge, whch fo the deed nvete ode contaned n the ange: 9 < α < 8 μ whee μ the eegenc angle whch pevent the nvete bac-feed and ha to be bgge then the au calculated coutaton angle. Such contol coepond to the feedng of voltage U d α nto the oto, whch offe a poblt of contollng the lp of an dle un. The lp dung dle un coepond to the theoetcal dle un of an nducton oto n whch thee an equlbu between the ean value of electootve foce E and voltage at the output of the nvete U d α. Snce E U U d α U coα, π π U then: coα.4 U Fo the cae of an adequatel elected tanfoe ate upplng the nvete the followng fulflled: U U and, a a conequence:

203 8 Inducton Machne n Electc Dve coα coβ whee β 8-α the nvete advancng angle. An nceae of the contol angle α of the thto, lead to an nceae of the lp of the dle un thu hftng the echancal chaactetc of the oto n the decton of lowe otatonal peed. Fg..4 Daga of a econducto Schebu dve Fg..4 Daga of the plfed Schebu dve fo atheatcal odelng The contol of the econducto cacaded dve ha been odeled n a plfed fo Fg.4, whee an nvete educed to luped eleent d, R d, U d α. An nducton oto odeled o that the electc vaable of the tato ae tanfoed nto othogonal ae u,v whle we have to do wth natual vaable,, n the oto wndng. Th odel dcued n ubecton.6. and decbed b the te of equaton n The cobnaton of the odel of a lp-ng oto wth a bdge on the de of the oto and a ccut of dect

204 . Method and Devce fo Fong Chaactetc of an Inducton Moto 8 cuent fo an nvete decbed b a te of equaton wth e 7 electcal degee of feedo and a ngle degee of feedo fo the otatonal oton of the oto. The genealzed coodnate fo the electc coponent of the odel ae the tanfoed cuent of the tato wndng u, v, phae cuent of the oto wndng, cuent of the ectfe bdge, and the cuent d n the nvete ccut. Th odel fo electc ccut can tae the fo of a at equaton: R R R R R R R R R R R R R v vc R R R R R c v c v v c v R vc c v R u u u d v u τ τ τ τ τ τ τ τ τ τ τ τ α d v u d R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R c c c c c c τ τ τ τ τ τ τ τ τ τ τ τ d v u d dt d 4 4.4

205 84 Inducton Machne n Electc Dve whee: v p θ τ n pθ cτ co pθ The te whch epee the electoagnetc toque tae the fo: T * e p * [ cτ v u u τ v u v ].4 On the ba of th odel calculaton wee conducted wth an a of fndng electc and echancal tanent of the dve and tatc chaactetc a well, fo vaou tate of contol of the nvete bdge. The calculaton wee caed out fo a edu powe lp-ng oto wth the followng paaete: U J n 5 [ N p 6 [ V ] ] I U 5[ A] n 55 [ V ] n 47[ev / n] n n P 45 [ W ] n I n 545 [ A] η.9 n T 9 [ N] n co.88 J n 5 [ N ] The tatc chaactetc of th oto ae peented n Fg..9. The paaete detened on the ba of the oto atng ae: R. [ Ω].4 [ H ] R.989 [ Ω].7 [ H ] and the paaete of the bdge ae:.[ H ].[ H ] R. [ Ω] d d The dnac calculaton wee conducted fo the dve oent of neta J J 5 [N ]. Fg.4 peent the tat-up of the dve unde a load T l.5 T n and contol angle α coepondng to Uʺd 4 [V]. Hence, the lp of the dle un, calculated accodng to.4, equal to: π U d U n.96 whee Uʺd the voltage of the nvete bdge epeed n tato wndng voltage te. In conequence the angula peed of the dle un πf Ω ω f.6 [ ad / ] p

206 . Method and Devce fo Fong Chaactetc of an Inducton Moto 85 a b c d e f Fg..4 Statng of the 45 [W] Schebu dve wth., T l.5t n, J J : a tato cuent b oto cuent c bdge cuent d DC ln cuent e angula peed f electoagnetc toque

207 86 Inducton Machne n Electc Dve Subequentl, Fg..4 peent the wavefo of electc and echancal vaable of the dve n tead tate fo Uʺd [V], whch coepond to.48 and Ω.8 [ad/], epectvel fo the load of T l.5 T n. a b c d e f Fg..4 Stead tate te-cuve of the 45 [W] Schebu dve wth.5, T l.5t n, J J : a tato cuent b oto cuent c bdge cuent d DC ln cuent e elatve peed f electoagnetc toque One can note that tat-up of th dve doe not noall occu n a cacaded te, a n the eaple n Fg..4 but n te wth an addtonal etance R d n the oto ccut, whch enue a fate tat-up wth a lage oto toque. Afte th ntal tage of tat-up oto econnected to the Schebu dve te. A ndcated b the copaon of tatc chaactetc of the toque Fg..9 wth the tat-up chaactetc n the Schebu dve te, n the latte cae the oto develop uch alle toque due to the defoaton of the oto cuent fo the ne cuve accopaned b a condeable deceae of voltage aocated wth the coponent of econducto bdge n the oto. To gve an llutaton of a tanent opeaton of the Schebu dve a tepwe change n contol of nvete voltage wa ntoduced. The output voltage of the nvete changed abuptl fo Uʺd [V] to Uʺd 4 [V], wthout a change of the load beng T l.5 T n. The dve ha to low down due to the change of the dle lp value fo. 5 to. 5. The eultng tanent ae peented n Fg..44.

208 . Method and Devce fo Fong Chaactetc of an Inducton Moto 87 a b c d e f Fg..44 Tanent chaactetc of the Schebu dve dung nvete contol change fo.5 to.5: a tato phae cuent b oto phae cuent c bdge cuent d DC ln cuent e elatve oto peed f elatve electoagnetc toque Statc echancal chaactetc of the econducto Schebu dve wee detened on the ba of a ee of calculaton of the tead tate fo vaou nvete bdge contol angle. An llutaton found n Fg..45.

209 88 Inducton Machne n Electc Dve Fg..45 Toque-peed chaactetc the Schebu dve fo U" d, 6,, 8, 4,, 4, 48 [V] o equvalent dle un lp value:.,.74,.48,.96,.7,.444,.58, Suppl Fequenc f Contol One of the fundaental ethod appled fo contol of angula peed of an nducton oto n accodance wth.8 baed on changng the fequenc f of the voltage uppl to the tato wndng. Although t wa dffcult to eecute n the pat, th ethod ha becoe wdepead a a eult of the applcaton of vaou powe electonc convete. It fnd applcaton n nducton oto dve n the ange of powe atng fo a facton of a [W] to poweful achne eceedng MW []. Dependng on the powe output and degn t poble to appl vaou oluton of nvete and vaou fequence of eneg conveon n the ange fo eveal hunded to [Hz] fo all and edu powe devce. In th chapte an epha wll be on the bac oluton appled n nducton dve wth powe nvete, ncludng: - dect fequenc convete ccloconvete, - two-level voltage ouce nvete, - thee-level voltage ouce nvete, - PWM cuent ouce nvete. Th lt of convete dve doe not fo the coplete ecod of the appled dve n patcula wth egad to lage powe dve but contan the ot coon one. Moeove, eonance baed cuent nvete and load couted nvete ae appled n addton to the lted one. Fo each one of the te t poble to appl eveal ethod of contol ealzng the vaou voltage wavefo and output cuent. The ue theeof ae ve etenve and ae wdel dcued n the efeence [,4,,5,5,97].... Dect Fequenc Convete Ccloconvete The ole of a ccloconvete the conveon of -phae altenatng voltage and cuent of the uppl netwo wth the fequenc f nto ngle-phae voltage and

210 . Method and Devce fo Fong Chaactetc of an Inducton Moto 89 cuent of a load wth the fequenc of f wthout conveon nto dect cuent. In ode to obtan a -phae te of voltage and cuent upplng the oto on the output of a convete t necea to appl thee epaate conveon unt o two unt n the econocal veon of a convete [5]. Each of uch unt cont of antpaallel goup of contolled ectfe ot coon of whch nclude 6-pule ectfe q 6. The necet of ung two antpaallel ectfng goup eult fo the need of etc conductng cuent n two oppote decton. Thto SCR ae appled n the ectfng goup of the convete and hence n a tpcal fequenc convete we have to do wth cuent coutaton. Th coe a a conequence of the fact that the tpcal applcaton of a convete the contolled lage powe nducton o nchonou oto dve wth a capact of up to a dozen MW. A SCR Slcon Contolled Rectfe a powe electonc coponent wth the hghet opeatng voltage and hgh conducton cuent; hence, t ued n lage powe convete. One of the tandad applcaton of a fequenc convete n a -phae load ac oto peented n Fg..46. Fg..46 A -phae ccloconvete wth a 6-pule ectfe bdge and epaated output. The te wth nducto ltng the cculatng cuent Th a convete wth bdge ectfng unt and epaate phae of the load, whch, howeve, uppled fo a tanfoe wth a ngle econda wndng. Between the ectfe upplng the wndng of the oto thee ae nducto

211 9 Inducton Machne n Electc Dve ltng the pact of the equalzng cuent n the ccut of the antpaallel ectfe unt nce n th oluton both of the ae contolled b the dela angle ove the ente ange of the opeatng condton of the convete. Th a oluton that doe not eque detecton of the ntant of the load cuent cong zeo and a ubequent epaate contol of the two ectfe unt. Anothe oluton of the ccloconvete te peented n Fg..47. In th cae the -phae load connected n a ta, whch lead to the uppl of patcula te of the antpaallel convete fo epaate econda wndng of the tanfoe n ode to avod hot. Fg..47 A -phae ccloconvete wth a 6-pule ectfe bdge and the Y - connected output, whch eque epaate econda wndng of the uppl tanfoe. The te wthout cculatng cuent and conequentl wthout nducto Inducto ae not appled between antpaallel te, whch ean that uch unt do not nvolve ultaneou contol of both ectfng bdge. Each of the feed the cuent nto one decton of conducted cuent that ngula fo t. Th aocated wth a need to appl oe advanced contol of the convete, whch nvolve the detecton of the ntant of a cuent flow decton change, and a hot bea dung the conducton of both bdge n th peod to etoe blocng ablt. The bac dtncton n the appled contol te nvolve electon the

212 . Method and Devce fo Fong Chaactetc of an Inducton Moto 9 contol ethod of a ccloconvete wth equalzng cuent o wthout the and t poble to ue one o the othe n eve tpe of convete. Howeve, the ltaton of the equalzng cuent aocated wth the need to ue ave and epenve nducto degned fo conductng cuent wth lage value. Th equeent lead to a tendenc to appl a te wthout equalzng cuent. The pncple fo the contol of ccloconvete eult fo the adequate contol of a -phae ectfe n uch a wa that enue an output voltage whoe bac haonc a ne wavefo wth the fequenc of f. Snce the output voltage and cuent ognate fo co-opeaton of the two ectfng unt bdge fong the nput of a ngle phae of a load, t necea to contol both of the, ultaneoul o n ucceon, n ode to enue that the uppl unfo output voltage wthn the ange of the bac haonc: U U n ω t U d n ω t U coα d coα fo the unt and unt epectvel. Hence contol angle α, α ae: α acco α acco a n ω a n ω f t f t.44 o α π α.45 whee: U a - apltude odulaton facto U d f f - fequenc odulaton facto f U - au value of the bac haonc of the output voltage of a convete U d U - ean voltage of the ectfng unt q 6 fo dela π angle α U - RMS voltage of the econda de of a tanfoe. The dela angle wavefo of the ectfe goup peented n Fg..48 n the functon of the phae angle of the output voltage. The fgue that follow peent how the output voltage foed b the unt and of antpaallel ectfe. Rectfe unt pefo the decendng ecton of the odulated voltage, whle unt eponble fo the acendng ecton of the voltage b applcaton of -phae voltage of the uppl netwo. Both voltage fo thee unt of the convete geneate an dentcal haonc of the output voltage wth the fequenc of f, whle n the ange of the hghe haonc the wavefo ae dffeent. Hence, the equalzng cuent occu n the antpaallel te of the ectfng unt fo the cae of contollng both goup ove the ente peod of the output voltage.

213 9 Inducton Machne n Electc Dve Fg..48 α contol angle of a ccloconvete anode goup a a functon of the output voltage phae angle, fo apltude odulaton facto a,.,.9,. a b c Fg..49 Ccloconvete output voltage fo a.55, f.667, f [Hz] and fo f 6 [Hz]: a unt voltage b unt voltage c output voltage fo epaate contol of ectfng unt contol wthout cculatng cuent

214 . Method and Devce fo Fong Chaactetc of an Inducton Moto 9 a b c Fg..5 Ccloconvete output voltage fo a.75, f., f [Hz] and fo f 6 [Hz]: a unt voltage b unt voltage c output voltage fo epaate contol of ectfng unt contol wthout cculatng cuent Fo the llutaton of the tanfoed voltage cuve, whoe bac fequenc f t eult that the content of hghe haonc n the output voltage hgh and nceae along wth the deceae of the apltude odulaton facto a. The anal of the output fequenc ndcate that the donant pat occuped b haonc wth the fequence of f h, qf ± f,,,.46 whch fo a low output fequenc f ean that the ajo hghe haonc have a fequenc aound qf. Th ean aound [Hz] fo a convete wth 6-pule ectfng unt and the fequenc of the uppl netwo of f 5 [Hz]. Pactcal condeaton lead to the ltaton of the uppe bounda of the output fequenc to aound.4 f and the adaptaton of voltage U of the tanfoe upplng the convete to th fequenc. Th coe a a conequence of the pncple n., whch defne the adaptaton of the value of the uppl voltage to the fequenc, whle peevng an adequate uplu of voltage. The a of th to appl a potentall hgh apltude odulaton facto a and, thu, the ltaton of the apltude of hghe haonc of the output voltage.

215 94 Inducton Machne n Electc Dve a b c Fg..5 Ccloconvete output voltage fo a.45, f., f 5 [Hz] and fo f 5 [Hz]: a unt voltage b unt voltage c output voltage fo epaate contol of ectfng unt contol wthout cculatng cuent... Two-evel Voltage Souce Invete Voltage Souce Invete VSI a powe electonc devce capable of tanfong the DC voltage and dect cuent nto voltage and altenatng cuent wth the deed chaactetc. It poble to degn ngle- and ultphae nvete. - phae nvete ae coonl appled fo the uppl of nducton oto. The bac daga of a -level voltage nvete peented n Fg..5. Th nvete coonl efeed to a VSI nvete nce t fo the ouce wth voltage chaactetc and the voltage cuve on the output load not elatve to the value of the load cuent n a wde ange of opeatng condton. Th ade poble due to the poweful voltage ouce wth all ntenal pedance addtonall booted b a capacto wth adequatel lage capact C on the nput of the nvete. A -phae nvete ha banche wth two econducto wtche and fee-wheelng dode peented n Fg..5. The contol wtche appl IGBT Iolated Gate Bpola Tanto o GCT Gate Couted Thto o MOSFET Metal-Ode Seconducto Feld Effect Tanto dependng on equed wong condton, ftl uppl voltage, load cuent and wtchng fequenc. An output voltage flte wth capacto wth the capact of

216 . Method and Devce fo Fong Chaactetc of an Inducton Moto 95 Fg..5 Daga of a -level -phae voltage ouce nvete VSI C f connected to the load. The nae fo a two-level convete coe a a conequence of the fact that voltage U G, U G, U G can onl aue two value,.e. U d o. Regadle of the pecfc anne of contol of the nvete gate at an ntant onl one of the econducto wtche can conduct cuent n a pecfc banch. The coutaton nvolve the poce n whch n a ngle banch one of the wtche tenate the conducton whle the othe wtch tat the conducton poce onl afte an adequate bea to pevent hot. At an ntant wth the onl ecepton of the coutaton bea n the nvete we have to do wth conducton of thee econducto wtche,.e. one n each banch. The deed wavefo of the output voltage ganed a a eult of an adequate contol of nvete econducto wtche. Fo the uppl of an nducton oto dve t deable to have -phae voltage wth a ne wavefo havng contollable fequenc and apltude. The bac ethod appled fo the odelng of the output voltage cont n Pule Wdth Modulaton PWM [,6,9,4,44,59,85]. Unde tandad condton t nvolve adequate wtchng of the potental U and potental at the output b econducto wtche fo the nvete banche n a hot nteval coepondng to a all facton of the peod of the output voltage. Concuentl, thee a lage nube of odulaton ethod, oe of whch wll be dcued n th ecton. Eve odulaton ethod hould eult n output voltage cloe to -phae etcal nuodal te wth all content of hghe haonc. The othe potulate egad the applcaton of pobl all nube of wtchng between the contol econducto eleent coepondng to a ngle ccle of the output voltage. Eve coutaton n an nvete banch aocated wth etve loe n the powe electonc wtche, hence the effot to ae tanton peod hot. Alo the coutaton aocated wth loe n the delectc n the wndng nulaton leadng to wea of the nulaton lae. Hence, the potulate of the ltaton of the nube of connecton follow. The dcuon hee wll focu on nuodal PWM SPWM odulaton n whch the tangula cae gnal

217 96 Inducton Machne n Electc Dve odulated wth the nuodal wavefo a well a eveal vaete of the pace vecto odulaton SVM [5,55,7,7,7,7,87]. The ft of the lted ethod ha t ogn n the analogue technque of contol and wa thu undetaen n th aea, whle the othe one coepond to the dgtal technque of contol.... Snuodal Pule Wdth Modulaton SPWM. Snuodal pule wdth odulaton SPWM Snuodal PWM nvolve appopate eploent of the cong pont between aw cae gnal and nuodal odulaton wave. When odulaton voltage u, u, u ae hghe then the voltage of the cae wave. u > u aw.47 the potental of the output econducto wtch,, aue the value of the uppl U d. In the oppote cae th potental ha the value of nce the gound econducto wtch of the adequate banch of the nvete n the ON tate. The foaton of the cae wave peented n Fg..5. The output voltage eult fo the dffeence of the potental between the appopate pa of output pont between nvete banche,, le: U U G U G.48 Fg..5 Foaton of an output voltage b the Snuodal PWM

218 . Method and Devce fo Fong Chaactetc of an Inducton Moto 97 Fg..5 contnued In the decpton of the nvete ode of opeaton we appl apltude and fequenc odulaton facto U f a f.49 U f aw When the cae fequenc f aw an ntege ultple of the output fequenc f we have to do wth nchonc odulaton. The cae when a < called pope odulaton and n that cae the fequenc of wtchng wch aw f aw f f f /.5 whle fo a > we have to do wth oveodulaton, the wtchng fequenc alle than t eult fo.5, and the voltage wavefo dtoted. The hghet attanable RMS value of the output voltage bac haonc wth the fequenc of f equal to [,5] U. 6 fo.5 U d whch ndcate a elatvel all applcaton of the uppl voltage. Concuentl, hghe haonc ode n th cuve fo the nchonc odulaton [] aount to a

219 98 Inducton Machne n Electc Dve / f ± n.5 l / f ± l n n,,, and do not contan low ode haonc beng the ultple of f. In ode to nceae the nvete cope of applcaton of the uppl voltage a ethod wth the njecton of the thd haonc of the odulatng voltage ha been developed, [4,9,6,89], whoe llutaton found n Fg..54. A a eult, the aw cae wave odulated wth the voltage of u u n ω t u nωt.5 n a anne that enue the wave u of the odulatng voltage doe not eceed the voltage of the cae wave U aw. Th condton fulflled when u. 4u u U.54 The ntoducton of the thd haonc nto the odulatng voltage eult n the dtoton of the voltage wavefo n elaton to the efeence potental U G, U G, U G. Howeve, t doe not eult n the dtoton of the output voltage n the load U, U, U nce the copenaton of the effect of the thd haonc accodng to.48. aw Fg..54 Fong an output voltage b the SPWM ethod wth a d haonc njecton

220 . Method and Devce fo Fong Chaactetc of an Inducton Moto 99 Fg..54 contnued In the eult we have to do wth an nceae of the output voltage to U U d whch ean that t ove 5% oe copang wth Space Vecto Modulaton SVM. Two-level voltage nvete ha thee banche, each of whch n one of conducton tate. The conducton tate of an nvete can be detened a: S,,,.56 Fo eaple the tate S ean that n the ft banch the uppe econducto wtch n the conducton tate and the output potental equal to U G U d. Concuentl, S ean that the gound econducto wtch n the econd banch ON and then U G. In th ethod the nvete output hot te T p aveaged voltage vecto V could be defned and effectvel contucted b ue of a concept of the pace vecto copleo V, whch detened b the bac haonc U ph of the equed output voltage and the tate of the patcula banche S, S, S : j j S Se Se Uph V.57 Snce thee ae banche, each of whch can be n ethe of two tate, the ntantaneou output V fo the nvete can aue an of 8 tate llutated gaphcall n Fg..55, n accodance wth.57.

221 Inducton Machne n Electc Dve Fg..55 Space vecto plane wth the wtchng tate <S,S,S > defned fo each vecto Along wth the gven vecto V, Fg..55 peent the tate of the nvete banche fo whch ethe one occu. Thee ae alo two zeo tate of the nvete output V, V 8 fo whch econducto wtche n all banche connect the load ethe to the gound G o potve P uppl ba. In eult the output voltage ae equal to zeo. Dectl fo the tate of the banche t poble to detene output voltage u S u U d S u S In a la anne, we can etablh phae voltage:.58 u S U d u S.59 u S The elaton.59 llutate voltage wavefo peented n Fg..56. The ft haonc ha the apltude of U π ph U d.6 whch coepond to the length of vecto V of the voltage ta peented n Fg..55. The apltude of th value achevable onl fo phae angle θ π/. The ntantaneou poton of vecto V detened b a phae angle θ. The ethod of odulaton ung pace vecto SVM wll be peented on the eaple of the nthe of vecto V tuated n the ft ecto of the voltage ta of the nvete. In the othe ecto the tuaton la, a a value of phae angle θ could be educed to the ange of the ft ecto. Th vecto nthezed b adequatel elected wtchng te of the tate that detene vecto V and V a well a zeo vecto V and V 8. Th llutated n Fg.57.

222 . Method and Devce fo Fong Chaactetc of an Inducton Moto Fg..56 Phae voltage of the two-level nvete, llutatng.59 Fg..57 A ethod of nthe of the V vecto n the ft egent of the pace vecto plane The contucton of hot te aveaged vecto V nvolve the fact that wthn uffcentl hot pulaton te T p, whch coepond to a facton of the total ccle, voltage V, V and V o V 8 ae wtched on fo the elected duaton t, t, t n. Thee nteval ae obvoul elatve to the ntant poton of vecto V detened b angle θ. A a atte of plfcaton t aued that wthn a ngle pulaton te the angle θ nvaable. The detenaton of te nteval t, t, t n pefoed ung the elaton: n p n n p t t t T t t t T V V V V.6 whee: π θ π π j d j d j e U e U e U V V V t n - a u of the nteval of the occuence of zeo vecto V, V 8. Afte olvng.6, we obtan: p n d p d p t t T t U U T t U U T t θ π θ π π n n.6

223 Inducton Machne n Electc Dve The au value of voltage U need to be elected n a wa that t n. In addton, the hghet adble value of apltude odulaton facto fo a lnea ange of nvete contol need to eult fo th condton. The au value of te nteval t t occu fo θ π/6 ee.66 and then: t t U π.6 Tp U d Defnng fo th cae the apltude odulaton facto a U π a.64 U d fo.6 we obtan the followng condton: U U d a U a a < a.65 π The te nteval.6 eflectng nuodal nvete contol ae equal to: t T t t n T p p p a T a t π n θ nθ t t T n p a π co θ 6.66 Output fequenc eult fo contol of the angle θ ω t π f t, whle the apltude of the nuodal wavefo eult fo an adequate electon of facto a. The au apltude of the nuodal voltage fo a U a accodng to.65. Altenatvel t could be calculated a: U a Uph U d U d. 55U d.67 π π - and t the value of the apltude of the phae voltage, whle U U a U d. 955U d π - apltude value of lne-to-lne voltage, and fnall U U. 675U d - the RMS value of the output voltage, whch ove % hghe than.5 fo the cae of contol ung nuodal pule wdth odulaton SPWM. Fo the above elaton t eult that fo a oto wth the ated voltage U n, the voltage U d delveed to the nvete hould be, epectvel: U n 4 [ V ] U d 6 [ V ].68 U 6 [ V ] U 9 [ V ] - that U d.5 U n. n d

224 . Method and Devce fo Fong Chaactetc of an Inducton Moto... Swtchng Sequence n the Pulaton Ccle. The bac equeent to be et b an nvete contol te ae the followng: - wde ange of lnea opeaton, - all nube of wtchng pe ccle of output voltage aed at educng eneg loe dung coutaton, - ltaton of the apltude of the haonc of voltage and cuent of the load aocated wth wtchng fequenc, - ltaton the ncdence and apltude of lowe haonc of the output voltage aocated wth load fequenc f. 6-pule wtchng equence A tandad oluton nvolve 6-pule wtchng equence fo a ngle pulaton ccle. The duaton t n of the zeo vecto V o V 8 n th ccle dvded nto thee pat: ¼ at the begnnng and the end of a ccle and ½ n the d ecton of the ccle. The chaactetc popet n th cae that the change of nvete tate occung between the ndvdual pule eque onl a ngle wtchng. An eaple of the wavefo fo a ngle pulaton ccle n 6-pule equence peented n Fg..58. A one can note, fo th nd wtchng equence the begnnng and tenaton of the ccle coe wth the V <> tate. In addton, t nvolve onl a ngle wtchng fo each ucceve pule and thee et n elaton to the d peod of pulaton. Fo the ente peod of the output voltage of the nvete the 6-pule wtchng equence peented n Table.. Fg..58 Eepla 6-pule equence n a pule ccle n the I-t ecto of the nvete opeaton

225 4 Inducton Machne n Electc Dve Table. S-pule wtchng equence wth nu nube of coutaton Swtched vecto wth the duaton of a pule Secto 4 / n t / n t 4 / n t I V V / t V / t 8 V V / t V / t V II V V / t V / t 8 V V / t V / t V III V V / t 4 V / t 8 V 4 V / t V / t V IV V 5 V / t 4 V / t 8 V 4 V / t 5 V / t V V V 5 V / t 6 V / t 8 V 6 V / t 5 V / t V VI V V / t 6 V / t 8 V 6 V / t V / t V The tanfe fo vecto V to an odd vecto V - eque jut a ngle wtchng, jut a n the cae of the tanfe fo vecto V 8 to an even vecto V. Hence, the wtchng equence fo the ethod peented n Table. eque onl one wtchng between the pule. The tanfe between the ecto occu wthout the necet of wtchng. Fo the eaned contol of the nvete the fequenc of wtchng w f N f 6.69 whee: N - nube of wtchng ccle coepondng to a peod of output voltage. The peented ethod ha onl one dadvantage, a t doe not fulfll one of the potulate peented eale. Wthn a coplete peod T the output voltage doe not fo an antetc functon,.e. one fo whch fωtt / -fωt nce n the oppote ecto e.g. I-IV, III-VI etc. the egent ae not dentcal but have oppote gn. One can note th b efeng to Table. whee fo actve vecto the elaton V V, whch ecue antet, not fulflled fo ecto I I. Th gaphcall peented n Fg..59 fo ecto IV.

226 . Method and Devce fo Fong Chaactetc of an Inducton Moto 5 Fg..59 S-pule equence n a pule ccle of the IV ecto Fo the copaon between the equence of the pule n ecto I Fg..58 and ecto IV Fg..59 and the hape of the output voltage t clea that the gnal coepondng to the voltage have equal value wth oppote gn; howeve, the phae hft ae dffeent. Th eult n ognatng even ode haonc n the output voltage wth fequence beng the ultple of f. The apltude of thee haonc ae not condeable and ae acceptable n all and edu powe dve; howeve, n hgh powe dve the cannot be accepted due to eceedng the equeent of the tandad value of cuent defoaton. Fo th eaon, thee a nube of 6-pule contol ccle enung the abence of even nubeed haonc n the voltage geneated b the nvete. One of uch ccle wll be peented n the ecton that follow. 6-pule wtchng equence elnatng even haonc In th nd of contol an nceaed nube of wtchng n a equence coe a a eult of the elnaton of the even haonc n the output voltage. Th elnaton ot eal occu a a eult of the adjutent of pule n the oppote ecto n a wa that enue that the ae ealzed b oppote vecto n the voltage ta. Th llutated n Table..

227 6 Inducton Machne n Electc Dve Table. S-pule wtchng equence wth elnaton of even haonc Swtched vecto wth the duaton of a pule Secto 4 / n t / n t 4 / n t I V V / t V / t 8 V V / t V / t V II V V / t V / t 8 V V / t V / t V III V V / t 4 V / t 8 V 4 V / t V / t V IV 8 V 4 V / t 5 V / t V 5 V / t 4 V / t 8 V V 8 V 6 V / t 5 V / t V 5 V / t 6 V / t 8 V VI 8 V 6 V / t V / t V V / t 6 V / t 8 V The opeatng pncple n th ethod peented n Fg..6 on the ba of the eaple of the equence of pule n ecto II and V. The peented coue enue that the eultng voltage wavefo an odd functon; hence, t doe not contan even haonc. An nceae of the nube of wthng occu dung the tanfe fo ecto III to IV and fo ecto VI to ecto I nce thee a change of the zeo vecto fo V to V 8 and the evee. Th lead to the nceae of the wtchng fequenc b w f f Δ pule wtchng equence DSVM Th ethod of nvete contol can lead to a futhe ltaton of the nube of wtchng. Dung a ngle ccle of wtchng one of the node of the nvete doe not change the tate and the wtchng occu n the two eanng one. Fo th eaon th ode of contol called dcontnuou SVM DSVM. We can dtnguh two tpe of wtchng ccle: tpe A when a banch that not nvolved n wtchng ean at the level of potental G and tpe B when the banch that doe not wtch connected to a hgh potental P. Th llutated n

228 . Method and Devce fo Fong Chaactetc of an Inducton Moto 7 Fg..6 S pule wtchng equence wth elnaton of even ode haonc, deontated fo the II and V ecto Fg..6. Both tpe of wtchng equence equall fo the bac haonc of the output voltage whle t hape dffeent n the ange of hghe haonc nce n each tpe of equence a dffeent ecto dvded nto two te ecton. One can alo note that the negaton of the wtchng equence tpe A n ecto I tanfoed nto the equence of wtchng DSVM tpe B n ecto IV, etc. Table.4 contan a ua of the wtchng equence fo the full peod of the output voltage.

229 8 Inducton Machne n Electc Dve Table.4 Fou-pule wtchng equence tpe A fo a nu wtchng fequenc Swtched vecto wth a duaton of a pule Secto / n t / n t I V V / t V t V / t V II V V / t V t V / t V III V V / t 4 V t V / t V IV V 5 V / t 4 V t 5 V / t V V V 5 V / t 6 V t 5 V / t V VI V V / t 6 V t V / t V The advantage of the wtchng equence peented n Table.4 the ltaton of the wtchng fequenc, whch w f N f 4.7 Howeve, the dadvantage egad the hape of the voltage, whch doe not fo an antetc functon and eult n the ogn of even haonc tpe f. The lac of the antet of the voltage cuve cleal vble n Table.4 a eult of the copaon between the voltage n the oppote ecto, e.g. I-IV, II-V, o III-VI. Th dawbac can be elnated b the applcaton of the tpe B equence e.g. n ecto IV to VI. Th uazed n Table.5.

230 . Method and Devce fo Fong Chaactetc of an Inducton Moto 9 Fg..6 Fou-pule wtchng equence of the A - and B - tpe, peented fo the I-t ecto The applcaton of the wtchng equence peented n Table.5 eult n an nceae of the nube of nvete wtchng pe peod of the output voltage. Followng the tanfe fo ecto III IV and VI I we have to do wth wtchng egadng zeo vecto,.e. V V 8 and V 8 V, epectvel. Th eult n an nceae of fequenc epeed b.7. Fg..6 llutate the applcaton of 4-pule wtchng equence wth elnaton of even haonc of the voltage, fo voltage geneated n ecto III and VI of the phae plane.

231 Inducton Machne n Electc Dve Table.5 Fou-pule wtchng equence tpe A and B cobned, fo elnatng even haonc of f ode n an output voltage Secton Swtched vecto wth a duaton of a pule I V / n t V / t V t V / t V / n t II V / n t V / t V t V / t V / n t III V / n t V / t 4 V t V / t V / n t IV 8 V / n t 4 V / t 5 V t 4 V / t 8 V / n t V 8 V / n t 6 V / t 5 V t 6 V / t 8 V / n t VI 8 V / n t 6 V / t V t 6 V / t 8 V / n t Fg..6 Fou pule wtchng equence wth elnaton of even nubeed haonc, deontated fo the oppote ecto III and VI

232 . Method and Devce fo Fong Chaactetc of an Inducton Moto Fg..6 contnued...4 Eepla Cuve of Invete Cuent. The output wavefo of a -phae load cuent uppled fo an nvete depend the contol ethod,.e. the nube N of pule equence pe peod of the cuve, apltude odulaton facto a and the load. The eaple of the wavefo unde a load tpe R, and the output fequenc of the nvete f 5 [Hz] ae peented n the fgue that follow n th wa llutatng the effect of the nube N of the equence of pule and odulaton facto a. a b Fg..6 Pefoance of a two-level VSI nvete wth DSVM and even haonc elnaton contol fo N 8, f 5 [Hz], a.85 : a DC ouce cuent b -phae load cuent

233 Inducton Machne n Electc Dve a b Fg..64 e n Fg..6, but fo N 48 a b Fg..65 e n Fg..6, but fo N a Fg..66 Pefoance of two-level VSI nvete fo the ove-odulaton contol aea. Copaon of -phae load cuent fo: a a.5 b a.5 c a.

234 . Method and Devce fo Fong Chaactetc of an Inducton Moto b c Fg..66 contnued... Inducton Moto Suppled fo -evel Voltage Invete The atheatcal odel fo the ulaton of the opeaton of a VSI nvete nducton oto dve baed on the daga peented n Fg..5 and on the te of equaton wth untanfoed vaable of the electc ccut of the tato Howeve, oe change have been pleented. In the place of the flte C f connected to the load thee ae capacto C placed paallel to eve banch of the nvete. In addton, nductance DC and etance R DC ae ntoduced to pla the ole of edual paaete of the DC voltage ouce. Moeove, edual nductance d between the banche of the nvete bdge ha been ntoduced. The contolled wtche peent n the nvete banche have been ulated wth etance wth a contolled value: a facton of an oh n the conducton tate and eveal Ω n the blocng tate, o that t appoatel coepond to the actual opeatng condton of an nvete. Th atheatcal odel of a dve te decbed ung 6 tate vaable, 5 of whch ae electcal one. Togethe, the fo a vecto of the tate vaable: [, Q, Q,,,,,,, ω, Q, Q ] T Y.7 DC u v e, 6 whee: DC - DC ouce cuent Q Q - chage of the an capacto C g 4, 5, 6, 7 - eh cuent n the nvete banche elected o, that the cuent of the oto tato ae equal to: u, v - tanfoed aal cuent of the oto

235 4 Inducton Machne n Electc Dve ω e pω - electc angula peed of the oto Q, Q 6 - capacto C chage on the banche wth nvete wtche. The odel fo electc vaable a be peented n the at fo: d U Y Z ωe Y.74 dt Snce nductance at contant, calculaton ae conducted ung the algoth U Z Y Y.75 whee: U - vecto of uppl voltage Zω e - pedance at of the te elatve to the peed of oto and etance, whoe value depend on contol of econducto wtche -contant paaete nductance at of the te. Thee equaton ae futhe uppleented wth an equaton fo the echancal oton whee: ω e ω T T Dω J.76 e e l e / u.77 Te p v - denote electoagnetc toque of the oto T l - a load toque D - a coeffcent of vcou dapng J - a oent of neta of the otatng pat of the dve. Th atheatcal odel wa appled fo ulaton calculaton of the dve n opeaton. The tetng nvolved eveal ethod of nvete contol peented eale n the boo. The eepla wavefo peented n fgue egad contol wth 4-pule wtchng equence DVSM and elnaton of haonc wth even nube Fg..6. The te N n the decpton of the fgue denote the nube of the equence of pule pe peod of the output voltage of an nvete. Dung the ulaton calculaton the edual nductance DC, d aue the value n the ange of.5% - % of the nductance of the bloc-oto tate z σ,., whle blocng etance R b fo 4 [Ω] to [Ω], dependng on the ated voltage of the oto. The voltage U DC upplng the nvete coepond to the voltage detened fo.68. Below a peentaton of oe eult of ulaton fo all and edu powe oto.

236 . Method and Devce fo Fong Chaactetc of an Inducton Moto 5 a b c d e Fg..67 Statng of a all powe S oto fed b VSI; f 5 [Hz], U DC 6 [V], T l.5 T n, J J, N : a DC ouce cuent b tato cuent c electoagnetc toque d angula peed e toque-peed tajecto

237 6 Inducton Machne n Electc Dve a b c d e Fg..68 Dect tatng of a edu powe SH oto fed b VSI; f 5 [Hz], U DC 9 [V], T l., J.5 J, N : a DC ouce cuent b tato cuent c electoagnetc toque d angula peed ω e e toque-peed tajecto

238 . Method and Devce fo Fong Chaactetc of an Inducton Moto 7 a b c d e Fg..69 Medu powe SH oto tanton to geneatng ode; f 5 [Hz], U DC 9 [V], T l.5t n -.75T n, J.5 J, N : a DC ouce cuent b tato cuent c angula peed d electoagnetc toque e toque-peed tajecto

239 8 Inducton Machne n Electc Dve a b c d e Fg..7 VSI contolled U/f tatng of the edu powe SH oto. T l T n, f 5 [Hz], u..95, J.5 J, N : a DC ouce cuent b tato cuent c electoagnetc toque d angula peed e toque-peed tajecto f tead-tate DC cuent g tead tate tato cuent

240 . Method and Devce fo Fong Chaactetc of an Inducton Moto 9 f g Fg..7 contnued a b Fg..7 VSI contolled U/f tatng of the edu powe SH oto. T l.5 T n, f 5 5 [Hz], u.5.95, J.5 J, N 6: a DC ouce cuent b tato cuent c electoagnetc toque d angula peed e toque-peed tajecto f tead-tate DC cuent g tead tate tato cuent

241 Inducton Machne n Electc Dve c d e f g Fg..7 contnued...4 Thee-evel Dode Neutal Pont Claped VSI Invete Mult-level voltage nvete [8,86,9] ae appled fo the uppl of hgh powe dve angng aound eveal MW. Bacall, thee ae two tpe of ult-level nvete. One of the a H-bdge cacaded nvete and the othe one - a dode claped nvete [,,56,64]. The advantage of the applcaton of the latte tpe aocated wth the poblt of hgh output voltage wthout eal connecton of the econducto wtche, lowe value of the voltage dv/dt devatve fo wtchng a well a lowe total haonc dtoton THD level, whch detene the degee of wavefo dtoton eultng fo hghe ode haonc. It poble to appl nvete wth dode claped neutal pont n the fo of thee

242 . Method and Devce fo Fong Chaactetc of an Inducton Moto a well a ult-level one [8,88], howeve, onl the te baed on thee-level degn have found coon applcaton n hgh-powe dve. The nae fo the thee-level nvete coe a a conequence of the fact that the output clap, o can be connected to the potve potental of the ouce P, to the neutal pont G o to the negatve potental of the uppl N. The neutal potental G ganed a a eult of capacto-baed dvon of the uppl voltage U DC and appopate contol of wtche n a wa that balance the fluctuaton of the potental at th pont Stuctue and Opeatng Pncple of a Thee-evel Invete. The daga of a thee-level nvete wth dode claped neutal pont G peented n Fg..7. Fg..7 Thee-level dode claped VSI The patcula banche contan fou IGBT o GTO Gate Ton Off wtche pe each banch connected n a ee. The neutal pont G foed b the uppe and lowe pa of wtche n banche beng connected va dode to the d pat of a capact voltage dvde. The tate of a patcula banch of an nvete detened a a eult of the algebac decpton of opeaton ung a epectve vaable S, S, S coepondng to the banch nube. Unde noal opeatng condton of an nvete thee ae poble thee tate uch that vaable S, S, S a aue the value n the et <,,->. A banch aue tate when the two uppe wtche n a banch ae n the ON tate n that cae the clap of the load connected to pont P wth the potve potental of the ouce. The tate of the banch ean that the two ddle wtche n a pecfc banch ae n the ON tate n th cae the output clap connected to the neutal G ung a uppe o lowe clapng dode dependng on the decton of the cuent flow. A banch vaable aue the value of - fo the cae when both lowe wtche n a banch ae n ON tate and, a a conequence, the load clap connected to the negatve al of the ouce wth potental N. Th llutated n Table.6.

243 Inducton Machne n Electc Dve Table.6 Poble tate of the banch n thee-level VSI nvete wth a dode claped neutal pont S S S T on off off T on on off T off on on T off off on whee: - denote an uppe de wtch - denote a lowe de wtch,, - the nube of a banch One can note that n a gven banch the wtche T, T and T, T ae n the ON tate n a copleenta anne. Snce the wtche n an nvete ae elected fo lowe blocng voltage then the total voltage of U DC, the followng ule ae obeved: - the balance between the voltage n pa T, T and T, T occu a a eult of paallel connecton of eto voltage dvde o on the ba of a pecfc electon of wtche n each banch of an nvete. Th electon nvolve the equeent that leaage cuent of the wtche T and T ha to be lowe then the leaage cuent of the wtche T and T - the dect wtchng between the tate and pohbted. Between thee tate the nvete ut pa though zeo tate. Snce the nvete ha banche and each of the can aue an of thee tate, the total nube of nvete tate equal to S 7.78 Each of thee tate can be decbed ung a vecto S, S, S followng tate: a zeo tate decbed wth vecto: V V V P N,,,,,,. Th nvolve the.79

244 . Method and Devce fo Fong Chaactetc of an Inducton Moto b 6 tate fo the ajo ta patten voltage: V V V 5,,,,,, V V V 4 6,,,,,,.8 c 6 tate fo the ddle ta patten voltage: V V V M M 5M,,,,,, V M V V 4M 6M,,,,,,.8 d 6 tate fo the no ta patten voltage tpe P: V V V SP SP 5SP,,,,,, V V SP 4SP V 6SP,,,,,,.8 e 6 tate fo the no ta patten voltage tpe N: V V V SN SN 5SN,,,,,, V SN V V 4SN 6SN,,,,,,.8 The gaphcal epeentaton of the vecto fong thee ta patten baed on the elaton.57, whee U ph the phae voltage of the no ta patten voltage. Th llutated n Fg..7. The angula and te daga fo the ntantaneou output voltage ae obtaned fo the followng elaton: u u u U DC S S S.84 Slal, fo phae voltage of the -phae load: u u u U 6 DC S S S.85

245 4 Inducton Machne n Electc Dve Fg..7 Space vecto plane fo a thee level VSI nvete The applcaton of the elaton.84 and.85 wth egad to vecto.8, whch detene ajo patten voltage lead to the voltage wavefo Fg..74 fo uch a ccle of opeaton. The wtchng equence pefoed eve ω t π/. A la coue of acton undetaen fo ddle patten voltage.8 gve voltage cuve peented n Fg..75. Fo the two et of vecto.8,.8 that detene tate of no patten voltage the voltage cuve ae dentcal n te of the hape and fo to the wavefo n ajo patten voltage Fg..74; howeve, the apltude of thee voltage ae educed b a half. The ua of the chaactetc peentng thee confguaton odel fo a -phae nvete peented n Table.7. Fg..74 Voltage cuve fo the ajo patten wtchng of a thee-level VSI: lne-to-lne voltage -tep and phae voltage 4-tep

246 . Method and Devce fo Fong Chaactetc of an Inducton Moto 5 Fg..75 Voltage cuve fo the ddle patten wtchng of a thee-level VSI: lne to lne voltage 4-tep and phae voltage -tep A t eult fo data n Table.7, the elaton between the voltage fo the patcula wtchng odel ae: U U ddle no / U / U ajo ajo h, RMS a,h, RMS SVM Contol of -evel VSI Voltage Invete. A peented n Fg..7, fo the contol of a thee-level nvete wth pace vecto odulaton SVM we have 4 actve vecto and zeo one when copaed to the total of 6 actve and zeo vecto n a two-level nvete. The geneal ule egadng the contol ae la to the pevoul condeed cae: - thee a ngle pa of wtchng at a te ee Table.6, dect tanton fo tate to - and evee fobdden - tanfe of the vecto fo a wtchng equence to anothe one, both wthn a ngle contol egon and dung the tanfe between the neghbong egon, hould not eque oe than a ngle pa of wtchng, - wtchng wthn a contol equence confgued n a wa that lt the fluctuaton of the potental of the neutal pont G to a au degee. Th ue not encounteed n two-level nvete and occu n thee-level one due to the capacto baed ettng of the neutal level of the potental of pont G.

247 6 Inducton Machne n Electc Dve Table.7 Output voltage fo two- and thee-level nvete Invete tpe -level ajo ta hape tep Output voltage U / U au h *U d h a d RMS - tep V V RMS a. π.955 π π -level edu ta 4 tep.955 π.955 π π -level π 6 no tep / ta π π level tep le fo the ajo ta Invete tpe hape Output voltage U ph / U au h *U d h a d RMS - tep V V RMS a -level ajo ta 4 tep /.67 π.55 π.47.9 π -level edu ta tep /.55 π.55 π π -level.8.6 π no 4 tep / ta π π level 4 tep / le fo the ajo ta

248 . Method and Devce fo Fong Chaactetc of an Inducton Moto 7 The tuaton of the nthe of the hot te aveaged T p output voltage V ung vecto peent n the voltage ta fo the ecto I of a cople plane peented n Fg..76. Fo the cae of the -level nvete the contol ecto n the cople plane dvded nto 4 alle tangle called egon and defned a a,b,c,d. In each of the egon thee ae conttutve vecto and the total nube of the n each ecto equal to 5. In the ecto I the ae: V S, V, V M, V S, V and n addton the zeo vecto. The ndce S,, M denote the all, lage and edu voltage ta, epectvel. The vecto n the all ta a occu ethe of two veon N o P; fo ntance: V SP <,,> ; V SN <,,->. Calculaton of te nteval fo SVM nvete contol Te nteval t, t, t z of the duaton of patcula vecto n the pule equence latng T p ae elatve to angle θ of the vecto V poton and t agntude. In ode to antan the potulate of the contol b ean of a ngle wtchng pa t poble to appl onl the vecto pontng the apee of the tangle fong a egon. The wtchng between the,- tate can occu onl n two tage b wtchng though the zeo tate. The ethod ued fo the calculaton of te nteval t, t, t z wll be peented fo vecto V tuated n egon d a n Fg..76. The te nteval t egad the vecto wth the nu value of angle θ, t wth the edu value, and t z fo the hghet value of angle θ. Fg..76 Dvon of a ecto I nto 4 egon a, b, c, d a an llutaton fo conttuton of an output voltage vecto V The u of the te nteval t t t T.87 z p

249 8 Inducton Machne n Electc Dve equal to the peod of the pulaton equence T p. Fo ntance fo egon d the followng vecto equaton vald: jθ Regadng V a a phae vecto p S j t M jπ / 6 V e T V e V e t V S e jπ / t z V S U d VS U d VM U d.88 π π π the vecto equaton aue the fo: π TpU U dt coθ j nθ U d T j U d Tz j.89 U Afte the ntoducton of apltude odulaton coeffcent.64: a U and epaaton of.89 nto the eal and agna pat, we obtan: d π atp coθ T T atp nθ T Tz Tz 9 The oluton of the te.9,.87 ae t poble to detene te nteval t, t, t z fo egon d. The ua of the te nteval fo the ente Secto I peented n Table.8. Table.8 Te nteval t, t, t z fo SVM nvete contol n a,b,c,d egon of the I-t ecto of the -level nvete Regon a b c d V S t T p w V T p V M T p w V S T p w w V t T p V V S T p S T p V M T p w w w w V S t z T p w V M T p w V V T p S T p w w foulae w π a n θ w a w a n θ π n θ

250 . Method and Devce fo Fong Chaactetc of an Inducton Moto 9 If a vecto V tuated n one of the latte egon, angle θ ha to be educed to a value wthn a ange of < θ π/ and foulae fo Table.8 hould be appled fo an appopate egon, changng the nube of ndee n V vecto, accodngl. Swtchng equence ove a pulaton peod The atte n th cae ae oe cople n copaon to the two-level nvete nce due to the necet of ltng the fluctuaton of the potental of the neutal pont G t necea to balance the te nteval of the duaton of vecto n the no ta n veon N a well a n P. Th occu dffeentl n egon b and c, whee thee a ngle conttutve vecto of the no ta than n egon a and d, whee thee ae two conttutve vecto of the no ta. An eaple of 6-pule contol n the pulaton equence fo egon b,.e. one n whch thee one conttutve vecto of the no ta, peented n Fg..77. The balanced ealzaton of vecto V S enued a a eult of pefong the wtchng equence <,,> and <,,-> wth the dentcal uazed duaton. The tuaton dffeent fo egon a and d, whee thee ae two conttutve vecto fong the no voltage ta. In th cae the egon dvde nto two etcal pat and fo each of the the ealzaton of P and N balanced fo the vecto of the no ta that donant n each of the halve n a egon. Concuentl, the othe vecto of the no ta unbalanced n the half of the egon n whch t pla a le potant ole. Th peented n Fg..78 fo Secto I, ubegon d,.e. fo the half of the egon d that le cloe to vecto V S. Fg..77 S-pule equence of pule ccle n the egon b of the ft ecto

251 Inducton Machne n Electc Dve Fg..78 S-pule equence of a pule ccle n the ubegon d of the ft ecto In th ubegon the vecto V S condeed n the ae anne a zeo vecto and the te nteval of t duaton dvded nto two ealzaton: <,,> and <,,-> wth the equal duaton. The peented ealzaton of the equence of wtchng ae not unque and can be degned n anothe anne, fo ntance b accountng fo the elnaton of even nubeed haonc of the output voltage. A oe detaled peentaton of the ue n th ecton found n efeence, e.g. [,8,9,7]....5 Cuent Souce Invete wth Pule Wdth Modulaton PWM The nvete appled n the electc dve fo the uppl of nducton oto can be geneall gouped nto voltage ouce nvete VSI ecton.. and.. and cuent ouce nvete CSI. At the ae te, CSI ae degned a load couted nvete CI and one wth poed coutaton wth pule wdth odulaton PWM [6]. Futhe on, cuent ouce nvete wth pule wdth odulaton geneate AC cuent wth the deed fequenc and apltude. The econducto wtche of a PWM cuent nvete have to povde a poblt of cuent hut-off and have to have evee voltage blocng capact. The cuent odel appl evee blocng Gate Couted Thto GCT whle foel the appled GTO thto. PWM cuent ouce nvete dpla a nube of

252 . Method and Devce fo Fong Chaactetc of an Inducton Moto advantage and ae theefoe coonl appled n ndutal dve fo eveal eaon: - n contat to VSI nvete the do not ue hgh fequenc wtched voltage to the oto wndng but tend to uppl AC wth a ooth wavefo to the wndng; hence, the poble wth the hgh value of du/dt devatve abent, - the have a condeabl ple engneeng tuctue and the wtche do not need the applcaton of feewheelng dode, - due to the necet of applng an n-ee nducto thee no hazad of hot-ccutng on nvete o oto clap. The dawbac of th oluton nclude: - necet of applng n-ee nducto degned fo a load cuent value and a capacto flte at the output, - elatvel low eacton te n epone to contol due to the peence of an nducto n the DC ccut. Th tpe of nvete CSI-PWM uccefull appled n edu and lage powe dve whee we have to do wth condeable load ove the ente ccle of opeaton and a elatvel fat contol of the oto angula peed not equed Stuctue and opeatng pncple of CSI PWM nvete. The daga of CSI cuent nvete wth PWM contol peented n Fg..79. A lcon contolled ectfe SCR wth an adequate cuent output, an nducto d wth Fg..79 Daga of a PWM cuent ouce nvete CSI: nvete oto connecton and oveall contol te

253 Inducton Machne n Electc Dve adequate nductance and a contolle that egulate the cuent fo the cuent ouce fo the nvete. PWM-CSI nvete eve eel fo the wtchng of the cuent to enegze patcula wndng whle the capacto flte aebl C f povde the cuent n the phae wndng whch ae not uppled fo the CSI nvete n a gven te peod. Th pocedue followed n uch a anne nce at an ntant ecept fo the coutaton peod, onl one wtch of the potve goup anode goup of the nvete and one n the gound goup cathode one n the ON tate. Fo the cuent ouce uppl, the tuaton when two wtche n a goup ae n the ON tate, eult n the lo of the ablt to contol n the ene of cuent dffuon nto the two wndng. In tun, f onl one wtch n the ente CSI nvete wee n the ON tate, the cuent flow foced b the ouce wll eult n a ve tong ovevoltage eultng n t falue. Fo th eaon, the adequate contol of the CSI nvete ha to ecue contnut of cuent flow though an nvete and oto n a wa that onl two wtche ae n the conducton tate, each one n a dffeent banch and anothe goup of wtche Contol of CSI PWM Invete. Mot coon ae ethod of nvete contol [4,,9]: tapezodal pule-wdth-odulaton TPWM, contol ung electve haonc elnaton SHE and pace vecto odulaton SVM. Each of the ethod ha a nube of advantage and dawbac to the. The tapezodal PWM odulaton can be ealzed n eal te o n the fo of a looup table. In addton, t dpla good dnac chaactetc; howeve, t doe not gve the poblt of cuent b-pang, whch lt the dnac of dve contol. The ethod of elected haonc elnaton offe the bet eult n te of uppeng hghe ode haonc of the cuent; although t ut opeate on the ba of pevoul pepaed loo-up table and, hence, the dnac popete deteoate. The ethod of pace vecto odulaton opeate n eal te and ha ecellent dnac popete; howeve, t doe not lead to a all level of hghe ode haonc n the cuent. Fg..8 peent the pncple govenng the contol ung tapezodal PWM. In the decpton of th contol we appl apltude and fequenc odulaton facto n the fo: U f a f.9 U aw f aw whee: U aw, f aw - denote the apltude and fequenc of the aw cae wave. A ndcated n Fg..8, the wtch n a banch of the nvete n the ON tate fo a flat ecton of the top bae of the tapezod a well a fo t the ng lope when t hghe than the gnal of the aw cae wave. A a eult, the nube N p pule coepondng to a half of the cuent peod equal to: f aw N p.9 f It benefcal when the ntege nube of peod of aw cae wave coepond to one of the lope of the tapezod,.e. when: / f 6n, n,,. Th cae llutate a etcal and nchonc wtchng and hence the ajo haonc [] of the output cuent have the fequence of

254 . Method and Devce fo Fong Chaactetc of an Inducton Moto f f h h N p ± f f aw ± f N p ± 5 f faw ± 5 f.9 Fg..8 Illutaton of a tapezodal pule-wdth odulaton PWM...5. Eepla cuve of nducton oto dve uppled fo CSI-PWM. The llutaton that follow peent the tat-up cuve of a edu-powe nducton oto S wth the applcaton of CSI cuent nvete contolled b ean of PWM odulaton. The nducton oto uppled fo CSI nvete wth all neta and low load toque a well a fo capacto wth low value of capact tend to opeate n an untable wa. An llutaton of th found n Fg..8. a b Fg..8 Statng of the CSI PWM fed nducton dve wthout an feedbac, wth J J, T/T n., C f [μf] fo edu powe oto.96: a DC cuent b electoagnetc toque c tato cuent d angula peed e toque-peed tajecto

255 4 Inducton Machne n Electc Dve c d e Fg..8 contnued The hghe the neta, capact of a flte capacto C f and the negatve feedbac n elaton to the peed, the oe tablzed the opeaton of the dve afte tat-up ee Fg..8. a b Fg..8 Statng of the edu powe.96 CSI PWM dve wth J J, T/T n., C f 5 [μf]: a DC cuent b electoagnetc toque c tato cuent d angula peed e toque-peed tajecto

256 . Method and Devce fo Fong Chaactetc of an Inducton Moto 5 c d e Fg..8 contnued The eaple peented n Fg..8 and.8 ndcate that the appopate electon of the oto paaete and feedbac lead to the tablzaton of the dve even fo elatvel all load T l /T n.5. The cuent I DC 98 [A] the nput value fo tablzaton of the nvete uppl cuent afte tat-up. a b Fg..8 Statng of the edu powe.96 CSI PWM dve wth DC cuent tablzaton afte tat-up wth J J, T l /T n., C f 5 [μf], I DC 98 [A]: a DC cuent b electoagnetc toque c tato cuent d angula peed e toque-peed tajecto

257 6 Inducton Machne n Electc Dve c d e Fg..8 contnued.4 Contol of Inducton Machne Dve.4. Vecto Contol The theoetcal fundaental of vecto contol, alo efeed to a Feld Oented Contol FOC, wee developed n 97 and 8 and ubequentl followed b attept conducted on nducton achne dve. Howeve, huge poge and wde applcaton of the te wa peceded b the developent and geate acceblt of fat powe electonc econducto wtche whch fo the ba of powe convete n engneeng. On the othe hand, th tpe of contol tctl elatve to the ue of fat copoceo to poce nuecal data of atheatcal odel and eaued data fo eno ecodng elected gnal vaable n a dve. A a conequence, the applcaton of thee ethod ha lagel epanded followng an nceae n the capact of gnal poceo. Theefoe, t wa poble to fulfll the peequte fo the pleentaton of advanced nducton achne dve contol te a late a 99 [,4,68,76,87,99]. The dea of vecto contol fo a epone to the dffculte aocated wth the contol of a -phae quel cage nducton achne, whch ha a nube of advantage depte not beng uceptble to contol b ean of eal acceble ethod uch a contol of uppl voltage and fequenc n a anne that offe

258 .4 Contol of Inducton Machne Dve 7 quc epone, ocllaton fee and ageeent wth the degned patten n te of toque, acceleaton and angula peed. The pncple govenng vecto contol the followng: fo the phcal value wth ne patal dtbuton along the a gap, whch can be peented n the fo of otatng vecto copleo, the flu lnage wth oto wndng Ψ dentfed and along th decton the a ρ of the otatng two-aal te tuated. The a ρ fo th te, whch etablhed pependcula to th decton and the angle of oentaton ρ, detene the poton of th pecfc te wth efeence to a elected two aal te u,v,. Stctl peang, th fo a patcula cae of the,, te otatng wth the peed of the agnetc feld, oented n the pace o that the oto flu lnage Ψ peented n the ρ a Fg..84, hence the nae fo Feld Oented Contol ognated. In th new coodnate te the copleo of the tato cuent ade up of the te ρ, ρ. The phcal elevance of the ente undetang the followng: the te ρ of the tato cuent fo the agnetzng cuent and dectl affect the value of the oto flu Ψ. Concuentl, the te ρ of the tato cuent that pependcula to t dectl affect the value of the oto electoagnetc toque T e. A a eult, the contol of the value of the flu n an nducton oto decoupled fo contol of toque n a wa that follow the odel of a epaatel ected DC achne. In th analog the te ρ of the tato cuent coepond to the ectaton cuent of f of the DC achne, whle the te ρ coepond to the aatue cuent a. In a DC coutato achne the contol pocedue conducted n an ea wa nce the oentaton a eult of the pecfc degn of a achne. The ae of the ectaton and aatue wndng ean pependcula a a eult of the potonng of pa of buhe collectng the aatue cuent pependcula to the a of the ectaton wndng, whch wound at the achne tato alent pole. Fg..84 Roto agnetc flu Ψ and tato cuent oentaton n the feld oented contol ethod The cot to be ncued wth the plfed contol of DC cuent achne aocated wth the ovecong poble wth coutaton, buhe, poblt of

259 8 Inducton Machne n Electc Dve pang at the coutato and necet of egula npectng uch devce. On the othe hand, n an nducton achne thee ae dffculte aocated wth eaueent and contol. The tep to follow nclude: dentfcaton, oentaton of the otatng te n pace n the ene of detenaton of the oentaton angle ρ followed b adequate decon egadng the openng of the convete wtche wth thee-phae output n a wa that th coepond to the decoupled contol b ean of ρ, ρ cuent n a two-phae ρ, ρ te. Th occu n a qute cople contol te that cuentl appled n nueou veon dffeng n te of pecfc value eaued n the te and calculated on-lne. The two bac vaete of applng feld vecto oentaton ethod called dect vecto contol and ndect vecto contol wll be dcued late n th chapte..4.. Matheatcal Model of Vecto Contol The pecfc popet of vecto contol aocated wth the oentaton of a te Fg..84 b etablhng angle ρ, whch detene the decton of the a of the oto flu Ψ. The focu of the condeaton hee a te of equaton fo a oto epeed n ed co-odnate, Ψ.56 otatng wth the peed of the feld,.e. fo ω c ω pω. It tae the followng fo: ψ ψ Ω α ψ α ψ β α ψ β α ψ α α pω ψ pω ψ pω pω γ γ Tl μ ψ ψ J Te ψ ψ u pω pω D Ω J β U β U.94 whee: - μ p / J β / σ Ω / p f / p ω π. Fo vecto contol we appl a coodnate te, whch tae nto account the angle of oentaton ρ, oto flu Ψ and, addtonall, otaton angle θ of the oto. A tanfoaton etablhed:, Ψ : ψ, ψ,,, θ, Ω, ρ, ψ, ρ, ρ Ω.95 whee: - ρ, ρ - ae the coponent of tato cuent n a feld oented te ρ, ρ. In ode to etablh the tanfoaton.95 we appl the followng elaton: ψ ρ actan.96 ψ ψ ψ ψ.97

260 .4 Contol of Inducton Machne Dve 9 and ρ ρ ρ ρ ρ ρ co n n co.98 Afte the calculaton of devatve and ubttuton, we obtan: l T p p U p U p D T J e ψ α γ ψ β σ ψ α γ βψ α σ α ψ α ψ ψ α ρ μψ θ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ Ω Ω Ω Ω Ω Ω Ω.99 Fo the ve fo of the odel.99 fo the vecto contol t poble to dectl deve the followng concluon: - contol poble onl when Ψ > - the te epeentng electoagnetc toque ρ ψ μ e T. ve ple n the ae wa a the one fo electoagnetc toque n a epaatel ected DC achne - agnetc flu of the oto Ψ elatve onl to the cuent ρ. Thee a coplete analog wth efeence to the ectaton flu and ectaton cuent n a epaatel ected DC achne. In the tead tate ρ Ψ,. - and fo ρ, achne lp equal to p Ψ Ω ρ α,. - voltage U ρ, U ρ ae tanfoed n accodance wth Realzaton of the Model of Vecto Contol Vecto contol of an nducton oto ealzed on the ba of a atheatcal odel.99 whch apple contol gnal fo eno and quantte calculated

261 4 Inducton Machne n Electc Dve on-lne n the gnal poceo. In the up-to-date oluton of vecto contol the eaued gnal nclude the cuent and voltage upplng tato wndng o, altenatvel, oto angula peed o t angle of otaton eaued b an encode, whch calculate the pule coepondng to the unt of the angle of otaton [5]. Aong the calculated quantte thee ut be the angle of oto oentaton ρ, whch ecue the oentaton of the te ρ, ρ n accodance wth the oto flu lnage Ψ. The agned value n the contol poce nclude the oto flu Ψ ef and the efeence toque T e ef, whle the ubcpt ef denote the efeence value Dect Vecto Contol Ste. The daga of uch a contol te peented n Fg..85. The contol pocedue nvolve the eaueent of the value of tato cuent,, and voltage u,,. On the ba of th and applng the elaton Ψ U R pωa Ψ Ψ σ Ψ. we can calculate the te Ψ, Ψ of the oto flu. Subequentl, we can calculate oentaton angle ρ, flu Ψ and thee-phae cuent ae tanfoed to the conttuent te ρ, ρ n a two-a w feld oented te. Th ethod called dect feld contol nce t nvolve the calculaton of the flu Ψ. Fg..85 Bloc daga of a dect vecto contol te.4... Indect Vecto Contol Ste. The daga of th contol anne peented n Fg..86. The contol pocedue nvolve the eaueent of the value of tato cuent,, and angula peed Ω. Th te ap the devatve of the oentaton angle ρ, whch ubequentl ntegated and added to the phae angle of tato voltage θ and appled fo the tanfoaton of tato

262 .4 Contol of Inducton Machne Dve 4 cuent to two-aal feld-oented coponent ρ, ρ. The te doe not nvolve the detenaton of flu Ψ and fo th eaon th veon of the feld contol ethod denoted a ndect. Fg..86 Bloc daga of an ndect vecto contol te.4.. Foalzed Model of Vecto Contol.4... Aptotc Decouplng of Flu and Speed Contol. If we nae the patcula vaable n the odel.99 n the followng wa:,,,,,,,,,, 5 4 Ψ Ω ρ θ ρ ρ.4 we can obtan the equaton of the odel n the fo: p v v a Ω α γ γ α α μ.5 whee: l p p U v p U v J D T a J p Ψ Ψ Ω Ω Ψ Ψ Ω ρ ρ ρ ρ ρ ρ ρ α β σ α β α σ μ / /

263 4 Inducton Machne n Electc Dve The atheatcal odel of the dve.5 can be boen down nto two ubte one aocated wth the contol of the flu wth the output of Ψ v γ α α.6 and the te of the contol of oto poton o angula peed: v a γ μ.7 The decouplng of the two te.6 and.7 occu n the aptotc ene,.e. fo a tctl contolled flu. A a eult, we can aue that: fe. Th tuaton full eeble the contol of ectaton flu and peed n a epaatel ected DC oto. In the cae of whch, howeve, the decouplng occu a a eult of the engneeng degn of the achne and eque a coute and buhe fo aatue. In contat, the decouplng n th cae eult fo the applcaton of the equaton fo the oentaton angle 5 ρ and tanfoaton of the vaable. One can note hee that n the peented te vaable 5 ρ not followed and uncontolled. The falat wth t necea fo the tanfoaton of phcal quantte to the tate vaable and, hence, t need to be conducted epaatel Input-Output Decouplng. In the te.99 we now wll ntoduce new vaable: ρ α α μ θ ρ ρ Ψ Ψ Ψ Ω z z z a z z z.8 Snce the flu detened n pola coodnate, t necea that Ψ >, whch ecue the etence of an nvee tanfoaton: z a z z z z z z Ω ρ μ α α θ ρ ρ.9

264 .4 Contol of Inducton Machne Dve 4 Afte the ntoducton of tanfoaton.8 we obtan: / z z p a z z v z z z v z z z z z μ α Ω. whee: e l T J a z a b J D T a b U z z p z z a z v U a z z z z pz z z v / ρ ρ β μ α α β μ γ α σ α μ α α γ β α α γ α. In the peented te. we have to do wth nput-output decouplng and ndependent contol of the two output: z θ and z 4 Ψ. In th veon of contol nput to tate decouplng doe not occu nce vaable 6 ρ ean unobeved and uncontolled. The contol eque that z 4 Ψ >. The contol gnal ν, ν ae tctl elatve to the vaable of the tate and the devatve:,,,,,,,,,,, 5 4 e e T T f b a z z z z f v v Ω Ω Ψ Ψ Such elaton tend to be ve cople and nvolve quantte that ae dffcult to eaue n the dve. The ae t poble to deve contol vaable on the ba of eaueent ung tate obeve. The cot of the ple and lnea contol of the I/O te, a peented above, aocated wth cople contol gnal and necet of the cobnng fo vaou ouce obeve/etato/eno Input to State neazaton b Dnac Feedbac. The acheveent of nput to tate lneazaton b dnac feedbac fo an nducton oto dve aocated wth the need to appl othe et of vaable than thoe appled befoe,.e. one accountng fo oentaton angle ρ. Theefoe, the cuentl appled pa vaable techncall output nclude: ρ Ω z. Thee pa vaable need to be dffeentated untl the latet of the devatve peent the contol nput, whch enue the contollablt the te. Fo the

265 44 Inducton Machne n Electc Dve adopted vaable. t wll eque the dffeentaton of each vaable thee te and condeaton of the contol voltage ν ρ a an addtonal vaable n the te: 6 v ρ. whle t te devatve v ~ 6 ρ wll be the contol value ubjected to ntegaton b dnac feedbac at the nput of the te. Th bolcall llutated n Fg..87. Fg..87 neazed and decoupled FOC te of an nducton oto wth an ntegato of the v ~ gnal ρ Fo vaable. and an addtonal θ, the decoupled and lneazed FOC te of an nducton oto tae the followng fo: 4 v [ ].4 z z z z z v [ ] whee n accodance wth the defnton of vaable. 4 d Ω d θ v 4 dt dt.5 d ρ v dt In ode to detene contol quantte ν, ν t necea to conduct ubequent dffeentaton n accodance wth.5, b applcaton of foulae.5 fo

266 .4 Contol of Inducton Machne Dve 45, 5. A a eult, we obtan: [ ] ~ ~ v v pb p v b v v v α α γ μ α α γ α γ α α γ α α γ α α α μ μα γ α μ μα γ α μ γ α μα.6 On the ba of.6 t poble to calculate the contol quantte: ] / [ ~ ] / [ ~ ] / [ 6 A d v A v A v t τ.7 The detenaton of the contol qute cople and eque the falat of Ψ, ρ, 4 ρ a well a the ft and econd devatve of the load toque,.e. b a b,. The peequte fo the contol condton Ψ >, jut a n the pevou cae. Th tpe of contol dffcult to ealze, howeve, due to the obevablt of the te and electon of the vaable z ρ t enue the oentaton of the te a a eult of t contol. It to oe degee eae to ealze the lneazed te wth educed dnac of the te afte ealzng the tablzaton of the oto ectaton cuent,.e. ρ cuent, whch eponble fo flu Ψ, a peented n the followng chapte neazaton of a Reduced Ste wth a Stablzed Ectaton Cuent. We wll eane a FOC contol te n whch a epaate contol ued fo the tablzaton of the ectaton cuent ef ρ.8 Then, the odel of the dve.5 educed to:,,,,,,,, 4 Ψ Ω ρ θ ρ.9 4 p w w a Ω α α α μ.

267 46 Inducton Machne n Electc Dve whee: - w ef, w - ae the contol value. The te. nvolve new vaable, uch a.: ρ Ω z. A a eult of calculaton of te devatve of vaable., we obtan: w w a p w w b z z z α α μ α μ μα μα. Afte the ntoducton of nput quantte n the fo: a p w w u b w w u μ α α α μα μ μα. we obtan a decoupled and lneazed dve contol te of the educed ode: ] [ ] [ u u z z z.4 The nput equaton can tae the followng fo: a p b w w u u μ α μα α α μ μα.5 Th equaton.5 offe the poblt of calculatng the contol: a p u b u W dt d w w μ α μα α μ α μα ρ ρ.6 whee: - > Ψ W α μ

268 .4 Contol of Inducton Machne Dve 47 Depte oe ple contol ethod n th equaton, t eque that the value of oto flu Ψ, tanfoed cuent ρ, load toque a T l T f /J and t devatve b a ae fala. The contol of the vaable Ω hould occu aound the natual value of the angula peed eultng fo the load and condton of powe uppl..4. Dect Toque Contol DTC.4.. Decpton of the Method Dect Toque Contol DTC fo a ve effectve and elatvel ple ethod; hence, oe and oe coonl appled n the contol of nducton oto dve [9,74,79,]. The powe devce eponble fo the contol n th te ot often a two-level voltage nvete Fg..5. The output quantte nclude: tato flu vecto Ψ and oto electoagnetc toque T e. Eght poble tate of nvete fo the pool of nput vecto to eecute contol n accodance wth elaton.57 and Fg..55. The ba of the anal of DTC contol ethod the te fo electoagnetc toque epeed n flu coodnate, whch n accodance wth odel.59 tae the fo: T pβ ψ ψ e v u ψ u ψ v pβ Ψ Ψ pβ Ψ Ψ n.7 whee: φ T φ -φ - a feld angle. Dect Toque Contol DTC nvolve the contol of the tato flu odule Ψ and t poton on the u,v plane. In th ethod t aued that the change of tato flu occu condeabl fate than the change of oto flu Ψ. On the ba of the ft of the equaton.5 and the econd one of.55 ω Ψ U R ca Ψ α α ω pω Ψ Ψ c A we can conclude that the change of tato flu ΔΨ occu dectl unde the effect of applng an adequate tato voltage ove a pecfc peod ΔtU and th an ntant effect. Concuentl, the change of oto feld vecto ΔΨ occu unde the effect of the change of the tato cuent o oto peed. The change of the oto peed n the eaned te cale of a ngle contol pule occu totall unnotceabl whle the change of the tato cuent occu wth a te contant of /α /R, whch ean a condeable dela ee Table.. Hence, dependng on the angula poton of vecto Ψ we can acheve the effect of changng tato feld Ψ and nceang feld angle φ T.7 b wtchng on of one of the ntantaneou vecto V, V, V 8 epeentng one of the output tate of the voltage nvete. Thu, the oto toque T e effected qucl and dectl. Th llutated n T Ψ

269 48 Inducton Machne n Electc Dve Fg.88 fo counteclocwe decton of otaton. Fg..88 peent alo ecto I,II,...VI nto whch the contol plane u,v can be dvded. The decon egadng the electon of a patcula vecto V fo oto contol elatve to the poton of vecto Ψ n a pecfc contol ecto and to whethe the value odule of tato feld vecto eceed one of the thehold contol value a well a whethe the value of oto toque T e eceed o below the pecbed value detenng the adble toque fluctuaton b the band-band ethod. Th done wth the ad of contolle wth a htee baed contol chaactetc, a peented n Fg..89. Fg..88 Clafcaton of the Dect Toque Contol ethod In ode to claf electon ethod we wll tae nto account the tuaton peented n Fg..88, whee the vecto of tato flu Ψ tuated n ecto IV and the decton of feld otaton counteclocwe. If the contol te equed an nceae of the flu odule Ψ and an nceae of the toque T e whch coepond to an nceae of φ T, the tate of the nvete wtche coepondng to vecto V 5 would be n ON tate. In ode to educe flu Ψ and nceae toque t necea to wtch on the tate of the nvete coepondng to vecto V 6, whle n ode to deceae flu Ψ and toque T e t would be necea to wtch on the tate of the nvete coepondng to vecto V. The ua of the nvete wtchng tate fo the patcula ecto of the contol plane and equeent egadng Ψ and T e ae found n Table.9 and. fo the counteclocwe and clocwe decton of feld otaton, accodngl.

270 .4 Contol of Inducton Machne Dve 49 a b Fg..89 Contol chaactetc: a fo a tato feld Ψ wth a two potonal htee loop b fo a toque T e wth a thee-potonal htee loop Table.9 DTC wtchng table fo the counteclocwe feld otaton Ψ T Secto e I II III IV V VI FU TU V V V 4 V 5 V 6 V FU TD V 6 V V V V 4 V 5 FU TN V 8 V V 8 V V 8 V FD TU V V 4 V 5 V 6 V V FD TD V 5 V 6 V V V V 4 FD TN V V 8 V V 8 V V 8 Inde lt: FU Flu Ψ Up; FD Flu Ψ Down TU Toque T e Up TD Toque T e Down; TN Toque T e Neutal In the DTC contol ethod the fundaental ta aocated wth the detenaton of tato flu vecto Ψ agntude and t poton on the u,v plane. Th done b the applcaton of tate obeve a the tool baed on calculaton and eaueent. Th ue wll be dealt wth n ecton.5. Fo the pupoe of atheatcal odelng and eeach of oto dve the flu Ψ can be etated n te of foulae.49 and ubequent tanfoaton. Thu, we obtan: Ψ uv uv

271 5 Inducton Machne n Electc Dve o Ψ Ψ u v u v.8 Table. DTC wtchng table fo the clocwe feld otaton Ψ T Secto e I II III IV V VI FU TU V 6 V V V V 4 V 5 FU TD V V V 4 V 5 V 6 V FU TN V 8 V V 8 V V 8 V FD TU V 5 V 6 V V V V 4 FD TD V V 4 V 5 V 6 V V FD TN V V 8 V V 8 V V 8 Th lead to the followng eult: Ψ Ψ u v u v Than to th, t poble to detene the odule of tato flu.9 Ψ Ψ u Ψ v and the ecto n whch vecto actuall located on the ba of the elaton between t coponent Ψ u, Ψ v. Slal, n ode to detene electoagnetc toque and feld angle φ T.7 t necea to detene the coponent of oto flu lnage: Ψ Ψ u v u v. Ψ Ψ u Ψ The acquantance wth the coponent.9,. of tato and oto feld ae t poble to calculate feld angle φ T Fg..9 v

272 .4 Contol of Inducton Machne Dve 5 ΨuΨv ΨvΨu T actan. Ψ Ψ Ψ Ψ u u v v Fg..9 Peentaton of a feld angle φ T.4.. Eaple of Dect Toque Contol DTC on the Ba of a Matheatcal Model DTC contol wa odeled wth the ad of the elaton.7-. and Table.9,. fo a two-level voltage nvete a the powe output devce, the one whoe opeatng pncple decbed n ecton... The odelng wa conducted on the ba of an aupton that the detenaton of flu Ψ n the ene of the length of the vecto and t poton eo fee, nce n th odel t baed on the elaton n.9. The la coue of eaonng aued fo electoagnetc toque T e, whoe calculaton baed on the atheatcal odel n.7 thu povdng eo fee eult. Th ean that the conducted calculaton and the eult conttute the llutaton of the opeatng pncple of DTC ethod but do not eflect pecel the opeaton of the dve due to the aupton of dealzed opeatng condton, n patcula n te of detenaton of contol quantte. The peented llutaton egad the tat-up, bang, contol of flu, etc. fo DTC contolled and opeated dve. The ft et of llutaton n Fg..9.9 peent the tat-up of a hgh voltage edu-powe oto SH fo a load of T l.5t n, oent of neta equal to J.5J. The band ltaton of DTC contol nvolve the value of:.5 Ψ 45 T e. 55 [ Wb] T eef 5 [ N]. Snce the te doe not appl peed contol afte the tat-up the electoagnetc toque fall to each the value eultng fo the load and the peed eache a tead value eultng fo the opeatng condton of the dve.

273 5 Inducton Machne n Electc Dve a b Fg..9 Stat-up of 4 [W], 6 [V] oto: a DC ouce cuent b tato cuent; DTC paaete gven b. a Fg..9 Stat-up of 4 [W], 6 [V] oto: a electoagnetc toque b angula peed: DTC paaete gven b.

274 .4 Contol of Inducton Machne Dve 5 b Fg..9 contnued a b Fg..9 Stat-up of 4 [W], 6 [V] oto: a tato agnetc flu Ψ te hto b u,v a tajecto; DTC paaete gven b. The followng et n Fg peent the tat-up of a all nducton oto wth the gven chaactetc of the toque. It wavefo haped n a wa that enue a fat tat-up and antenance of the gven peed afte tat fo J 4J, T l.5t n. Th te opeate wthout an feedbac, fo the followng DTC contol ltaton:.8 Ψ. [ Wb] ΔT.5 [ N]. e

275 54 Inducton Machne n Electc Dve a b Fg..94 DTC guded tat of the 5.5 [W] nducton oto: a electoagnetc toque b angula peed. DTC lt gven b. a Fg..95 a DC ouce cuent b tato cuent dung the toque guded tat up of the 5.5 [W] nducton oto. DTC lt gven b.

276 .4 Contol of Inducton Machne Dve 55 b Fg..95 contnued a b Fg..96 a tato agnetc flu Ψ n te b t u,v tajecto, dung the toque guded tat up of the 5.5 [W] nducton oto. DTC lt gven b.

277 56 Inducton Machne n Electc Dve Anothe eaple of DTC contol nvolve deceang agnetc flu of a 5.5 [W] oto b 5% dung t tead opeaton whle antanng the toque wthn the degned boundae,.e. 8.8 Te<4. [N]; T e ef.75 T n 4.5 [N]. A a eult of obevng the tanto wavefo one can note a nube of pecfc eacton of the te: ntantaneou educton of the tato flu Ψ accopanng a gadual, aptotc deceae of oto feld Ψ Fg.97. It ve pecfc to ea an nceae of feld angle φ T between feld vecto Ψ and Ψ n a anne that enue the antenance of a contant toque T e depte deceang value of the two vecto. a b c Fg..97 Stepwe educton 5% of a tato agnetc flu n tead tate of nducton 5.5 [W] oto wth DTC whle toque antaned on the contant level T ete.75 T n 4.5 [N]: a tato flu Ψ b tato flu tajecto c oto flu Ψ u,v tajecto

278 .4 Contol of Inducton Machne Dve 57 a b c d Fg..98 a DC ouce cuent b tato cuent c tato cuent u,v tajecto d oto cuent u,v tajecto - afte the 5% educton of the tato flu Ψ a Fg..99 a electoagnetc toque b feld angle φ T - afte the 5% educton of the tato flu Ψ. Condton le Fg..97

279 58 Inducton Machne n Electc Dve b Fg..99 contnued a b c d Fg.. DTC tanton fo otong to beang ege of 5.5 [W] nducton oto S: a electoagnetc toque b angula peed c tato and d oto agnetc flu tajectoe

280 .4 Contol of Inducton Machne Dve 59 a b c d Fg.. DTC tanton le n Fg.: a DC ouce cuent b tato cuent c tato cuent u,v tajecto d oto cuent u,v tajecto

281 6 Inducton Machne n Electc Dve a b c Fg.. DTC tanton to geneatng ege of 5.5 [W] nducton oto S b the change of wtchng equence fo the counteclocwe to the clocwe one: a electoagnetc toque b angula peed c feld angle φ T a b Fg.. DTC tanton to geneatng ege of 5.5 [W] nducton oto S b the change of wtchng equence fo the counteclocwe to the clocwe one: a tato flu Ψ flu tajecto b oto flu Ψ tajecto

282 .4 Contol of Inducton Machne Dve 6 a b c d Fg..4 DTC tanton to geneatng ege of 5.5 [W] nducton oto S b the change of wtchng equence fo the counteclocwe to the clocwe one: a tato cuent te cuve b tato cuent tajecto c oto cuent tajecto d DC ouce cuent a Fg..5 DTC eveng of a of 5.5 [W] oto dve: a electoagnetc toque b angula peed c feld angle

283 6 Inducton Machne n Electc Dve b c Fg..5 contnued a b c d Fg..6 DTC eveng of a of 5.5 [W] oto dve: a tato flu Ψ te cuve b oto flu Ψ te cuve c tato flu tajecto d oto feld tajecto

284 .4 Contol of Inducton Machne Dve 6 a b c d Fg..7 DTC eveng of a of 5.5 [W] oto dve: a tato cuent te cuve b tato cuent tajecto c oto cuent tajecto d DC ouce cuent Fg... llutate the tanfe of the dve fo oto ege of opeaton to bang a a eult of a change of electoagnetc toque, a peented n Fg..a. The coepondng change of angula peed peented n Fg..b. The tablzaton of the peed fo t >. [] coe a a conequence of changng load toque T l and t adaptaton to the tate of balance. Howeve, th cannot be concluded fo the efeng fgue. The tanfe to the bang ege not accopaned b an nceae of tato and oto cuent, whch deall llutated n Fg.. c,d. One can note ve good chaactetc of DTC contol n the ene of the antenance of the degned wavefo of electoagnetc toque and flu Ψ a well a a condeable ovelappng between the cuve fo the toque and feld angle φ T, whch can be ade oe copehenble b cloe eanaton of elaton.7 fo all value of angle φ T. Subequentl, Fg...4 peent the tanfe of the dve contolled b DTC ethod to the geneatng ege of opeaton. Th occu a a eult of changng the equence

285 64 Inducton Machne n Electc Dve of econducto wtchng fo the counteclocwe ode Table.9 to clocwe one Table. fo the potve decton of otaton. It accopaned b a tanfe of feld angle φ T fo the potve value fo oto opeaton to the negatve one Fg..c, negatve electoagnetc toque Fg..a and etun of eneg nto the ouce Fg..4d. Followng Fg.5.7 peent eveal of the 5.5 [W] DTC contolled dve alo b changng the equence of wtchng of a bdge tanto fo a evee otaton. The effect of changng the otaton ene qute notceable n eveal tajectoe peented. Thee eaple pove the DTC contol to be effectve, table wthn a boad ange of opeatng condton a well a deontate t pactcalt n the contol of dnac wavefo. The peequte fo th ethod concen the antenance of Ψ > and the poblt of detenng vecto Ψ on the contol plane u,v. A t wa poved b the eaple of conducted tat-up, the antenance of the degned cuve of electoagnetc toque b the DTC contol te acheved ve pecel and wthout dela wthn the ange of the capact of the oto of povdng adequate toque fo the detened condton of te uppl. DTC a nd of ldng ode contol ealzaton ee 5.6. and belong to the tpe of heutc contol ethod nce the wavefo, the nube of wtchng a well a the wtchng fequenc of the flu and toque ae elatve to the deed contol and htee loop fo toque and flu ltaton Fg..89. Concuentl, the te telf detene the ntant fo wtchng on the ba of a cuentl eecuted contol ta a well a on the ba of the pecon of the appng b the obeve of the feld Ψ, t poton and the cuent value of electoagnetc toque T e..4. Obeve n an Inducton Machne The lneazaton of a odel of a dve, feld oented vecto contol FOC, DTC and othe contol pocedue eque on-lne detenaton of the flu lnage fo the tato Ψ o fo the oto Ψ. The oentaton of the flu vecto alo equed, whch nvolve the detenaton of t poton on the u,v plane fo ntance b ean of oentaton angle ρ. Thee quantte ae vtuall nacceble on ba of eaueent and need to be calculated n the contol te. A atheatcal odel wth data lnage to the eaueent of value that ae deved ve eal and pecel can fo the ba fo calculaton of th tpe. Such quantte nclude the cuent of tato wndng, uppl voltage, angle of oto poton o otatonal peed. State obeve ae atheatcal object eultng n calculaton algoth, whch pefo calculaton of nacceble quantte that ae needed to conduct the contol poce contnuoul, on the ba of a atheatcal odel and eaueent of cetan vaable n the dve whch ae ea to obtan. Such calculaton need to be conducted on-lne dung the contol poce. The te tablt of an obeve ued hee ean the deca of etaton eo dung the detenaton of the ought quantte caued b dtubance o ncoect ntal value along wth te pang. Such tablt ethe eaned wth the ad of a ethod baed on apunov theoe [5], o etaton eo deca deontated on the ba of an adequate electon of the obeve paaete. The tablt of the obeve

286 .4 Contol of Inducton Machne Dve 65 undetood n th anne doe not ecue t coplete accuac dung calculaton but conf the tendenc of etaton eo to deca n te. The naccuac n the etaton of wavefo of the eaned vaable eult fo the eaon that ae copletel beond the tuctue of the obeve. Such eaon nclude: - naccuace eultng fo plfcaton ade dung the atheatcal odelng of the dve evng to etablh an obeve, - vaablt of paaete condeed a contant n the algoth, fo eaple coonl appled quantte α, aocated wth the nducton oto that tend to be dectl elatve to agnetc atuaton and tepeatue, - naccuace of eaueent of cuve nvolved n the obeve algoth fo the opeatng dve, - naccuace of the nuecal ntegaton of the obeve equaton. Depte thee dawbac obeve have poved the applcablt n contol even n the oe ple fo. The tude devoted to obeve, the tablt and ethod of eo educton ae wdel dcued n efeence [,4,4, 66,77,78]. Seveal ple obeve appled n nducton oto ae peented below along wth oe ue petanng to the antenance of the tablt..4.. Roto Flu Obeve n Coodnate α, β The ot ple flu obeve eult dectl fo the odel n.55. Fo ω c t a odel wth a tatona te of ae u,v n elaton to the tato,.e. α, β odel. Ψˆ Ψˆ α β α Ψˆ α Ψˆ α β pω Ψˆ pω Ψˆ β α α α α β.4 In the obeve.4 the etated value ae: Ψˆ α, Ψˆ β and the value nput fo eaueent nclude: Ω, α, β. The atheatcal odel fo the vaable Ψ α, Ψβ ha an dentcal tuctue a the one n.4, wth a note that thee vaable occu n t n the place of etated quantte Ψˆ α, Ψˆ β. The eo of the etated vaable ae defned a: e α Ψ Ψˆ e Ψ Ψˆ.5 α α The equaton of dnac of the etaton eo ae deved b ubtactng equaton fo the obeve.4 fo the equaton fo the odel: e α e β α e α α e β β pω e β pω e α β β.6

287 66 Inducton Machne n Electc Dve The tud of the behavo of the etaton eo.6 dung the dtubance nvolve the anal of the dnac of the eo fo eaple wth the ad of apunov functon. It a potvel detened functon baed on etaton eo and n the eaned cae can tae the followng fo: e α e β V B calculaton of: β β α α e e e e V, and ubequent ubttuton.6, we obtan: V e e V α α β α.7 Fo the eult.7 one can conclude about the tablt of the deca of flu etaton eo t e V t V α..4.. Roto Feld Obeve n Ψ, ρ Coodnate Th anal baed on the odel of oto.99 n Ψ, ρ coodnate: ρ ρ α α α ρ p Ψ Ψ Ω Ψ.8 B plannng the obeve n the fo: ρ ρ α α α ρ p ˆ ˆ ˆ ˆ ˆ ˆ Ψ Ψ Ω Ψ.9 and defnng the eo a: ρ ρ ρ ρ ρ ρ ρ e e e e ˆ ˆ ˆ ˆ Ψ Ψ Ψ.4 we obtan the equaton fo the dnac of obeve eo: e e e e e e α α α α ρ ρ ρ ρ ρ ρ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ ˆ ˆ ˆ ˆ ˆ.4 In equaton.4 the eo e ρ not dectl ncluded, but thee ae eo denoted a e, e, whch eult fo e ρ and ae tanfeed to cuent ρ, ρ a a eult of tanfoaton. Fg..8 peent geoetcal elaton eultng fo Ψ, ρ tanfoaton.

288 .4 Contol of Inducton Machne Dve 67 Fo Fg..8 t te that: ρ ρ ρ e ˆ, whch alo epeent the elaton n.4. Hence, ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ϑ ϑ ϑ ϑ ϑ ϑ e e e e e e e e e e n co n co co n n ˆ n co n n co co co ˆ.4 Fg..8 The nfluence of the oentaton eo e ρ on the feld oented tato cuent ρ, ρ Fo the all value of e ρ n.4, t eult that: ρ ρ ρ ρ e e e e.4 A a conequence, on the ba of.4,.4 the equaton fo the dnac of the eo can be etablhed n the fo: Ψ Ψ Ψ Ψ e Ψ e e e ρ ρ ρ ρ ρ α α α α E ˆ ˆ.44 Fo the followng we can deve the chaactetc polnoal: W Ψ Ψ Ψ ˆ ˆ ρ ρ α α α E.45 The polnoal n.45 can be tanfoed to tae the fo:

289 68 Inducton Machne n Electc Dve Ψ Ψ Ψ Ψ ˆ ˆ α α.46 wth the ad of the elaton Ψ ρ. The dcnant of the quadatc equaton.46 ˆ 4 ˆ < Ψ Ψ Ψ Ψ Δ α.47 alwa fo ρ >,.e. the cuve of the eo ha an ocllato hape. In ode to ae the value of the oot the dcnant.47 can be tanfoed to tae the fo: Ψ Ψ Ψ Ψ Ψ Ψ Δ ˆ 4 ˆ 4 ˆ ρ ρ α α Hence, j Ψ Ψ ± Ψ Ψ ˆ ˆ, ρ α and, appoatng fo Ψ Ψ ˆ :.48 ρ ρ ρ α α j j ± Ψ ±, The dapng of etaton eo cuve popotonal to α, whle the pulaton elatve to the elaton ρ / ρ,.e. the ato of the electoagnetc toque to the flu..4.. Roto Feld Obeve n, Coodnate wth Speed Meaueent Th obeve baed on the falat of the tanfoed cuent,, whch eult fo the eaueent of tato cuent and tanfoaton of,, that nvolve the need of nput of the angle of the otaton of the oto θ. Snce peed Ω eaued n the te, the detenaton of the angle of the otaton of the oto Ω θ ˆ occu wth the pecon ange to the contant,.e. the value of the

290 .4 Contol of Inducton Machne Dve 69 ntal angle Ω. The atheatcal odel.56, fo ω c pω n the ange that nteetng to u tae the fo: Ψ Ψ α Ψ α Ψ β α Ψ β α Ψ α α pω Ψ pω Ψ γ γ pω pω U U β / β /.49 The feld obeve baed on the eaued tato wndng cuent, afte the tanfoaton of, that apple the eaued peed Ω and angle of otaton θˆ, contucted [5] n the followng wa: Ψˆ Ψˆ ˆ ˆ α Ψˆ α Ψˆ β α Ψˆ β α Ψˆ e ˆ e ˆ ˆ ˆ α α pω pω Ψˆ Ψˆ μα e μα e γ γ μpω e pω pω μpω e β / U β / U e e.5 The equaton of dnac of the eo tae the followng fo: e e e e Ψ Ψ Ψ Ψ ˆ ˆ Ψˆ Ψˆ α e e e Ψ α e Ψ μα e e e ˆ ˆ μα e ˆ ˆ μpω e μpω e β β α eψ pωeψ α e pω e Ψ Ψ.5 Vefcaton of the deca of the eo nvolve apunov ethod wth potvel detened eo functon n the fo: μ μ V eψ eψ e e.5 β β Afte dffeentaton of.5, we obtan: V μ β μ β eψeψ eψeψ ee ee.5 Afte the ntoducton of devatve.5 and odeng thng, the eult :

291 7 Inducton Machne n Electc Dve ˆ ˆ e e e e V Ψ Ψ β μ α.54 The eult.54 ndcate a pogeve deca of the eo functon.5. The eo aocated wth the etaton of the flu deca wth the te contant of /α, whle the peed of the cuent etaton eo deca e, e a be contolled b an appopate electon of the value of, > Obeve of Inducton Moto Speed Baed on the Meaueent of Roto Poton Angle Inducton oto contanng the eno of oto poton, fo ntance encode, do not appl peed eno. The angula peed Ω a be calculated on the ba of dffeentaton of the poton gnal, whch fo a dceet detenaton of the poton angle eult n condeable noe wth hgh fequenc aocated wth the dffeentaton. It, howeve, poble to avod t and gan a ooth cuve of the etated peed a a eult of applcaton of a ple obeve []. Fo tandad notaton the atheatcal odel fo vaable θ and Ω,.e., oto poton and angula peed ae the followng: J T J D l Ω Ψ Ω Ω ρ μ θ.55 The popoed obeve tae the fo: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ l J T J D θ θ μ θ θ θ ρ Ω Ψ Ω Ω.56 The eo equaton ae deved a a eult of deducng the de of.55 and.56, epectvel: θ ρ ρ θ θ μ μ θ θ e e J D e e e e ˆ ˆ ˆ ˆ Ψ Ψ Ω Ω Ω Ω Ω.57 Unde the aupton that the eo n the detenaton of electoagnetc toque ˆ ˆ ρ ρ μ μ Ψ Ψ deca uch fate than fo the cuve of echancal vaable, the equaton of the dnac of eo tae the fo: Ω e Ω e J D e e θ θ /.58 The chaactetc equaton of the eo dnac tae the fo:

292 .4 Contol of Inducton Machne Dve 7 / J D J D W J D J D W.59 The oot of th equaton.59 ± J D J D J D, 4.6 a be foed abtal b electng,. The dapng of the eo -½ D/J a be adequatel lage whle the cuve of the eo deca a be ocllato o eponental dependng on the electon of Flu, Toque and oad Toque Obeve n, Coodnate Th obeve bae on the eaueent of, θ, and the etated quantte nclude: l T ˆ, ˆ, ˆ, ˆ Ω Ψ Ψ. Fo plfcaton pupoe we aue that T l contant. The coepondng odel of the te tae the fo: / Ω Ψ Ψ Ω Ω Ψ Ψ Ψ Ψ J T J T J D l l μ θ α α α α.6 The obeve degned n the followng fo [] θ θ θ μ θ α α α α e J T e J T J D e l l / ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Ω Ψ Ψ Ω Ω Ψ Ψ Ψ Ψ.6 whee: e θ θ θ ˆ The equaton of the dnac of the eo etaton tae the fo:

293 7 Inducton Machne n Electc Dve θ θ θ μ θ θ α α e e e J D e e e e e e e e e e T ˆ ˆ ˆ ˆ Ω Ω Ψ Ψ Ψ Ψ Ω Ω Ω.6 whee: J T T e e l l T / ˆ ˆ Ω Ω Ω In accodance wth the ft two equaton fo the te.6, we aue fat eo dapng t t e e e e e e α α.64 whch ae t poble to ndependentl deal wth the thee eanng equaton.6, fo a deca n the eo of toque etaton e e μ. A a eult, the dnac of the eo e θ, e Ω, e T can be etated a: Ω Ω T e T e e J D e e e θ θ /.65 The chaactetc polnoal fo th te tae the fo: / J D J D J D W.66 The polnoal.66 can have thee eal oot,, fo whch cae n ode to enue eo dapng we eque that ethe,, < o a ngle eal nube oot and two cople one that ae utuall conjugated. In th cae t equed that all thee eal pat ae negatve. Fo the oluton wth thee eal nube oot,, the gan facto of,, ae calculated on the ba of the geneal fo of the chaactetc polnoal W Hence, b copaon wth.66 we obtan:

294 .4 Contol of Inducton Machne Dve 7 / J D J D.67 The elaton.67 ae t poble to deve the gan facto of,, fo abtal elected value of dapng,,. Fo the cople oot -d±jω the equaton tae the fo: ω d d W.68 whee: - - a eal oot d, ω - coponent of cople oot the ft of whch denote the dapng coeffcent of the ocllato cuve whle the othe the pulaton of the cuve. Followng an adequate etenon: ω ω d d d d W A a eult of the copaon of the eult wth polnoal n.66 t poble to detene the gan of,, fo the deed value of, d, ω: / ω ω d J D d d J D d.69 The peented obeve.6 ha been olved n an effectve anne nvolvng the nput the nuecal value fo gauge n ode to obtan the deed cuve of eo deca functon. Th, unfotunatel, doe not nvolve the cae fo the deca of the eo of flu etaton, whch elatve to the te contant /α, whch wa the cae n the pecedng eaple Stato Flu Obeve Ψ wth Gven Rate of Eo Dapng The nowledge of tato flu lnage Ψ ndpenable fo the dect toque contol. It alo enable one to calculate oto flu Ψ, whch necea fo vecto contol.5: Ψ Ψ Ψ β.7 Concuentl, the atheatcal odel evng fo the calculaton of tato flu the followng.5: c c p p R U A Ψ A Ψ A U Ψ β ω α α σ α β ω Ω Ω.7

295 74 Inducton Machne n Electc Dve On the ba of th odel, whle the eaueent quantte nclude U,, Ω, an obeve of tato flu ha been degned [66]. The advantage of th obeve nvolve the fact that t poble to et an abta peed of etaton eo dapng, whch dtnguhe t fo the pecedng oto flu Ψ obeve whee eo dapng wa affected b the dapng of α. The degned obeve tae the fo: c c Z c p p Z c R β β ω α α σ α β ω U A Ψ A Ψ A U Ψ ˆ ˆ ˆ ˆ ˆ ˆ Ω Ω.7 whee: c, c - ae cople ultple. The equaton of the dnac of the eo n the cople vaable tae the fo: c Z c p Z c β α β ω e e A e e e A Ψ Ψ e ˆ ˆ Ω Ψ Ψ Ψ.7 In the at fo the equaton of the eo ae the followng: Ω Ψ Ψ c Z c jp Z c j e e e e β α β ω.74 Hence, the chaactetc polnoal follow: c c c p c c Z j Z c j Z c Z c jp Z c j W Ω Ω ω β α β ω β β α β ω.75 Th a polnoal wth cople paaete. Slal, Z c c,, can tae cople value. In the phcal ene t denote two-denonal contol of flu Ψ and cuent n u,v coodnate. Havng the poblt of ettng the eal contant ω c and cople contant Y R Y R c c c c c c.76 and pedance Z, thee ae condeable oppotunte fo the foaton of the cuve of the etaton eo. Aung the geneal fo of the chaactetc polnoal:

296 Refeence 75 ω ω ω ω ω ω ω ω j j j j W.77 t poble to tate the equaton fo the coeffcent: ω ω ω ω ω β α β ω ω ω β j p c c Z j Z c j j Z c c c Ω.78 A an eaple we wll peent the oluton fo: ω ω ω ω β Y c c Z.79 In th cae, on the ba of.78, we obtan: c p p p c c R Y R Ω Ω Ω ωα ω ω α α.8 Fo th oluton.8 b appopate electon of, t poble to acheve abtal the deca of the flu etaton eo Ψ. Howeve, th eult condeabl elatve to the angula peed Ω. The eult can eve fo the detenaton of flu Ψ fo the pupoe of DTC contol o help one n the detenaton of oto flu Ψ b ung.7. Refeence [] Abdn, E.S., Xu, W.: Contol degn and dnac pefoance anal of a wnd tubne-nducton geneato unt. IEEE Tan. En. Conv. 5, 9 96 [] Agawal, P.: Powe Electonc Ste. In: Agawal, P. ed. Theo and Degn, Pentce Hall, Uppe Saddle Rve [] A, T.H., Chen, J.F., ang, T.J.: A ando wtchng ethod fo HPWM full-bdge nvete. IEEE Tan. Ind. Elect. 49, [4] Ale, G.: A PWM cuent-ouce nvete fo hgh qualt dve. Eu. P. Elect. J., 99 [5] Baade, U., Depenboc, M., Gee, G.: Dect elf contol of nvete-fed nducton achne: a ba fo peed contol wthout peed eaueent. IEEE Tan. Ind. Appl. 8, [6] Bech, M., Blaabjeg, F., Pedeon, J.K.: Rando odulaton technque wth fed wtchng fequenc fo thee-phae powe convete. IEEE Tan. P. Elect. 5, [7] Belln, A., Fgall, G.: On the electon of the coutng ntant fo nducton oto dve. IEEE Tan. Ind. Appl. 5,

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302 Chapte 4 Buhle DC Moto Dve BDC Abtact. Th chapte deal wth popete and contol of buhle DC oto dve wth a peanent agnet ectaton BDC, PMDC. In the ft pat chaactetc of contepoa peanent agnet PM that ae ued n electc oto ae peented along wth the plfed wa of the odelng. A an eaple a pendulu gven, whch cont of wngng col ove a tatona PM, and the nfluence of PM odelng plfcaton upon the dnac tajectoe of oveent dcued. Futhe on, a odel of PMDC dve deved on a tanfoed d-q and alo a conta odel n whch no tanfoaton of vaable ued, wth the coutaton tang place accodng to the tate of phcal natual vaable. Howeve, the poble of nonholonoc contant not undetaen whle dealng wth PMDC odelng. In a clacal DC oto wth echancal coutato, the etence of uch contant evdent, becaue the connecton of each aatue col to the etenal ccut depend on the oto poton. In cae of electonc coutaton, a wtchng of wndng uppl contolled alo b the oto poton angle, but topogaphc tuctue of ccut ean fed and the wtchng caed out b a abupt change of pedance value of powe electonc wtche. In th chapte vaou chaactetc and tanent cuve fo BDC dve ae peented and a copaon ade between eult obtaned fo both tpe of odel: d-q tanfoed and untanfoed one. It gve the poblt of jutfng the choce egadng the nd of the odel to be ued n patcula applcaton, dependng on the denon of a whole te and equed gooune of eult. The eult peented cove the opeaton of DC dve wth and wthout contol te nteventon. The PID contol dcued n t applcaton to a gven pofle of peed and oto poton oveent and alo nvee dnac ethod ntoduced. Nueou eaple of DC dve poble ae ncluded, eplong two tpcal BDC oto wth gven data. 4. Intoducton Fo 5 ea DC dect cuent achne have plaed an potant ole n electc dve. The bac advantage aocated wth the applcaton n dve nclude: ea adjutent of otatonal peed, uncoplcated tat-up and eveal, table opeaton fo all peed a well a good dnac popete enung fat eacton to changng paaete of powe uppl. DC achne wth clacal

303 8 4 Buhle DC Moto Dve BDC degn cont of a tato wth electoagnet whch ae poweed b the ectaton cuent and whoe ole to geneate a agnetc feld of ectaton. The wndng of the otatng aatue ae connected to the echancal coutato wth the gaphte-etal buhe lppng ove t. The conttute the electcal connecton between the ovable aatue wndng and etenal ccut. The echancal coutato, whch a devce otl of htocal gnfcance, pla the ole of a echancal ectfe, whch convet AC cuent wth the fequenc coepondng to the oto otatonal peed nto the DC cuent outde the aatue. At t te, the echancal coutato wa an outtandng devce, whch wa howeve toubleoe dung eplotaton and cotl n te of cot of nvetent. It wa alo the weaet ln n the te n the ene of elablt of opeaton a well a equed fequent evce and egula ovehaul. A oden buhle DC achne BDC dpla two fundaental dffeence n contat to the echancall coutated DC achne [,5,7,8,4]. Ft of all, t doe not have a echancal coutato ove whch the buhe fong the electcal node ued to lp. A tatc electonc coutato ued n t place, whoe ole the coutaton of the cuent n aatue wndng n the functon of the angle of oto poton θ. Hence, the pncple govenng DC achne peeved,.e. the achne elf-coutatng. In the chaactetc of the achne the bac effect nvolve the fact that along wth the nceae of the load the achne tend to low down unle t uppled wth etenal peed contol fo the tablzaton of the peed. A a eult of th lowng of otatonal peed the aatue cuent tend to nceae and th lead to a new equlbu pont of the opeaton. The econd elevant dffeence between a clacal echancall coutated achne and up-to-date oto nvolve the eplaceent of electoagnet ectng the an agnetc feld wth adequatel elected peanent agnet aebl [,]. Th oluton endeed poble a a eult of agnetc paaete and othe utlt paaete and ultatel the coecal value of peanent agnet. The contan ae eath eleent, uch a neodu Nd, aau S aong othe. The applcaton of peanent agnet pove the effcenc of a achne nce thee ae no powe loe n the ectaton wndng and lead to the deceae of achne a. Howeve, n te of the contucton and theal equeent of achne opeaton thee no advanceent nce the peanent agnet ntalled n the achne and povdng the ectaton flu eque adequate opeatng condton whch do not pet the deteoaton o a deca of the agnetc feld fo the agnet. Thee equeent bacall nvolve the ltaton of the tepeatue nde the peanent agnet oto, ltaton of the nfluence of aatue eacton n a wa that enue that eveble deagnetzaton of agnet doe not occu and not etendng a gap n ode to pevent oveloadng of peanent agnet. One ha to bea n nd that the peanent agnet DC achne PMDC have to be degned n anne that enue the opeaton ove a nube of ea wthout deteoaton of the eplotaton paaete. Anothe potant dffeence between the clacal echancall coutated DC achne and a

304 4. Peanent Magnet Bac Decpton n the Matheatcal Model 8 buhle one concen the nube of wndng of the aatue and, ubequentl, the cuent wavefo on the DC de. In a coutato achne the uual nube of wndng vae aound a couple dozen, a a conequence of whch thee an adequatel lage nube of coutato egent. In connecton wth th, DC cuent contan ve all pulaton nce the coutaton occu eve couple of degee of oto angle of otaton. Conequentl, the electoagnetc toque geneated b th achne tend to deontate all pulaton. In BDC achne of the ot coon engneeng degn thee ae thee phae wndng of the aatue, whch eflected b thee banche of an electonc coutato ectfe contolled b the angle of oto poton. Th eult n condeable cuent and toque pulaton geneated b the achne nce the coutaton occu eve 6º of the angle of oto otaton, altenatvel n the anode and cathode goup of the electonc coutato. It obvoul poble to nceae the nube of the aatue wndng and nube of coutato banche thu leadng to the educton of cuent pulaton; howeve, two negatve effect follow. One of the aocated wth the need to ue a oe etenve and epenve electonc coutato, whle the othe one nvolve an nceae of coutaton loe and deceae of the effcenc of the dve. A fnal ea that can be ade at the begnnng of th ntoducton that buhle DC achne wth peanent agnet can va condeabl n te of the engneeng tuctue. Ft of all, thee can be nute achne evng a evodve n technolog, houehold applance and vehcle. Bede, thee ae lage achne, whch ae appled n electc dve of autoatcall contolled devce, ncludng dve n anpulato jont. Fnall, thee ae hgh powe achne wth the paaete of the dve ued n ndutal achne, fo eaple n teel ll o veel. BDC achne a have a vaou nube of phae, have clndcal contucton and n oe applcaton the can have a fo of a d wth oble aatue and otatng agnet. The fnal oluton can eve fo ue n low evoluton geale dve. BDC achne need not have low evoluton ange, a one dcued befoe, but alo can opeate unde otatonal peed eceedng, [ev/n]. The nube of the avalable veon lage and tll gowng. 4. Peanent Magnet Bac Decpton n the Matheatcal Model Peanent agnet have been n ue fo a long te. The have been appled a coponent of techncal devce fo neal ea. The acquantance wth the phc of agnetc ateal and pncple govenng agnetzaton on the co level ha occuped the attenton of centt fo the lat 5 ea whle the techncalte of the poce of poducton of up-to-date copote agnet ha dated nce 98. At peent we ae fala wth peanent agnet wth table agnetc popete on condton of not eceedng adble tepeatue wth hgh value of unta ntenal eneg, agnetc nducton unde the agnet n

305 84 4 Buhle DC Moto Dve BDC ece of [T] and a boad agnetzaton loop. The bac chaactetc of the peanent agnet peented n Fg 4.. Fg. 4. Geneal chaactetc of a peanent agnet wth an a gap load cuve gven b λ M Fo the decpton of the opeaton of peanent agnet we noall peent the chaactetc of agnetzaton eel n the II quadant of the coodnate te nce th the opeatng ange of a agnet. In a ple agnetc ccut contng of a peanent agnet, a gap and a all feoagnetc coe μ that ued fo clong the agnetc ccut, one can tate that Hd l H lm H δ δ 4. Unde the aupton that leaage flu neglected n uch a ccut the followng elaton fulflled B δ SM Bδ Sδ B Bδ S 4. M and nce: H δ B δ /μμ a a eult, on the ba of 4. and 4. one can tate that: In the above foulae: λ M B H lm S δ S δ M μμ S 4. - H, B, l M, S M - ae: agnetc feld tength and nducton n the agnet and, conecutvel, the length and ntenal co ecton of the peanent agnet - H δ, B δ. δ, S δ - ae: agnetc feld tength and nducton n the a gap and conecutvel, the length and co ecton of the a gap

306 4. Peanent Magnet Bac Decpton n the Matheatcal Model 85 - μ, μ - agnetc peeablt, elatve one and that of the a - λ M - denote, n accodance wth 4. the unt agnetc conductance of the agnetc ccut alo nown a the nclnaton of the taght lne of agnet load. The nclnaton of the taght wth the dectonal coeffcent λ M coepond to the conductance of the a gap n a ple agnetc ccut and the co ecton between the taght lne and chaactetc of the agnet detene the opeatng pont H,B of the peanent agnet n a gven agnetc ccut. Concuentl, the poduct of H,B detene the unta eneg of the agnet pe unt of volue at a gven opeatng pont H,B. Fo a cetan nclnaton of the taght lne λ M the ectangle wth the de aed a H,B ha the laget aea fo a gven chaactetc of opeaton and th pecfc opeatng pont detene the au opeatng eneg H B a fo the agnetc ateal fo whch the agnet foed. A gven ateal optal n te of agnetc popete when t ha concuentl a lage value of nducton of the agnetc enant B and ntent of the coecon of the agnetc feld -H c, a well a the lage value of the au opeatng eneg H B a. An deal would nvolve a agnet wth a neal ectangula agnetzaton loop fo lage value of B and -H c nce t enue a lage and neal contant nducton unde a agnet wth a wde ange of load. A a eult of the wde applcaton of ae eath eleent n agnet, the ae able to coe cloe to th pecfc equeent to a uch lage degee Fg 4.. Fg. 4. A fal of agnetzng cuve of a ae eath peanent agnet, fo dffeent tepeatue value of opeaton A one can conclude fo z Fg. 4., the nceae of tepeatue ha a condeable effect on the agnetzaton chaactetc of up-to-date peanent agnet baed on ae eath eleent. Thee a cetan, all educton of the value of the enant nducton B and ve lage deceae of the abolute value of -H c that the ntent of the agnetc coecon. Too hgh an abence tepeatue of a

307 86 4 Buhle DC Moto Dve BDC agnet eult n the deteoaton of the ange of adequate condton fo the agnet to opeate. Th coe a a conequence of the fact that followng the change of -H c the nclnaton of λ M,.e. the chaactetc of agnet loadng lted, whch ean that the adble a gap n the agnetc ccut condeabl alle. The fundaental paaete of ajo fale of peanent agnet,.e. fete agnet, allo baed one wth alunu, ncel and cobalt AlNCo and two ajo goup of agnet wth ae eath eleent,.e. one wth aau S and neodu Nd, ae peented n Table 4. and 4.. Table 4. Bac agnetc popete of the an PM ateal Fal of a PM ateal B [T] H c [A/] H B a [J/ ] Fete.4 5 Al N Co. 5 8 S Co. 75 Nd Fe B. 85 Table 4. Bac tepeatue paaete of the an PM ateal Fal of a PM ateal Mau opeatng tepeatue [ o C] Cue pont [ o C] κ B [%/ o K] κ Hc [%/ o K] Fete Al N Co S Co Nd Fe B The table contan ean and appoated value of paaete taen fo vaou efeence n a anne that doe not eflect an patcula agnetc ateal avalable n the aet. One can note that the detal of the ateal uazed n the table ae offeed coecall n vaou allo cobnaton, a copote o nte, a t the cae fo fete. The patcula ateal decbed n anufactue catalogue dpla vaou popete depte belongng to a ngle fal. Fo the data n Table 4. and 4. one can conclude that neodu agnet ae utable fo opeaton wth lowe opeatng tepeatue whle the one wth aau dpla uch bette popete n hghe tepeatue ange. Thee ae couple of ethod of odelng on the acocopc cale of PM appled n electoechancal devce. We ean hee plfed odelng, uch that ae t poble to peent the opeaton of electoechancal tanduce and enable the odelng and ulaton of opeaton n dve te. One of the ethod nvolve the eplaceent of the agnet wth a copact tun wth zeo etance and an adequatel adapted elf-nductance and cculatng cuent f n

308 4. Peanent Magnet Bac Decpton n the Matheatcal Model 87 th tun. The evee effect of the aatue on the agnet occu a a eult of the aatue cuent a va the utual nductance M. The poduct M f coepond to the agnetc flu Ψ f b ean of whch the peanent agnet affect the aatue ccut. Anothe qute ple wa nvolve the peentaton of peanent agnet flu n the atheatcal odel n the fo Ψ M f f f f M / a f 4.4 The effect of the aatue odeled ung the te a M/ f, whch educe the conventonal agnetzng cuent f ognatng fo the peanent agnet. The plet wa of odelng the cuent ognatng fo PM coupled wth a gven ccut the adopton of t value Ψ f a a contant. Th nvolve degadng aatue cuent dung the opeaton of a achne fo a all a gap n the agnetc ccut. It alo coepond to the opeaton of the agnet n the ntal ecton of agnetzaton chaactetc of a agnet poduced fo allo of ae eath eleent Fg. 4.. None of the peented hee PM odelng ethod account fo the agnetzaton chaactetc unde the effect of the tepeatue e. In ode to peent the dcued PM odelng ethod, below found an eaple of a evoechan wth a ovable col wngng above the agnet. Eaple 4. Pendulu col ove PM. A plfed odel of the electoechancal te n whch a pendulu col ove n the feld of a oble PM peented n Fg. 4.. Fg. 4. Model of a pendulu col ove PM The netc eneg of the te : T f f aa M af a f 4.5

309 88 4 Buhle DC Moto Dve BDC whle the potental eneg: g U Th te ha thee degee of feedo and afte the ntoducton of genealzed coodnate: a a f f a f Q Q Q Q,, ϑ q 4.6 Wth J l, agange functon fo th te tae the fo co ϑ ϑ l H g J M U T f a af a a f f 4.7 Aung n a plfed fo that M coϑ M af 4.8 t poble to detene the equaton of oton fo th te a agange equaton: f f f f f R Q dt d Q q 4.9 a a a a a R Q dt d Q q 4. ϑ ϑ ϑ ϑ D dt d q 4. Afte adequate tanfoaton and odeng of eleent fo the two equaton of oton egadng the electcal vaable, we obtan: a a a f f f a f a f R e R e M M ϑ ϑ co co 4. ϑ ϑ ϑ ϑ n n M e M e f f a a 4. Afte the tanfoaton to the noal fo, the te 4. n the at notaton : a a a f f f f a a f a f R e R e M M M ϑ ϑ ϑ co co co 4.4

310 4. Peanent Magnet Bac Decpton n the Matheatcal Model 89 The au value of the agnetc flu coupled wth the col : Ψ M 4.5 f Concuentl, the equaton fo the echancal oton of the pendulu fo vaable ϑ, tae the fo J ϑ Ψ n ϑ gl nϑ D ϑ 4.6 a f whee the ft ght hand de te denote electoechancal toque bang the oton of the pendulu: T f Ψ nϑ 4.7 e a The atheatcal odel peented n 4.4, 4.6 applcable wth egad to the ft, leat plfed wa of odelng PM. A oe plfed agnet odel nvolve degadng of the odelng of the agnet b ean of a epaate dffeental equaton and the peentaton of the effect of the aatue n the fo eultng fo 4.4. In th cae, agange equaton tae the fo: M co ϑ / a a J ϑ g H l coϑ f a M coϑ a f Th lead to two equaton of oton fo vaable q Q a, ϑ n the fo: M co ϑ / a J ϑ M a f a nϑ M ϑ nϑ a M a n ϑ gl nϑ D ϑ a f 4 M / coϑ R a a a a The fnal and ot plfed odel whch degad the effect of the aatue ganed fo f f, M f f Ψ f,. In th cae: ϑψ a a J ϑ M Ψ f a nϑ R f a a nϑ gl nϑ D ϑ The opeaton of the odel fo the paaete of the te wth the value of f.5.5 D 5 [ H ] a.[ H ] M.8[ H ] Ra.[ Ω] [ g] f 5.[ A] Ψ f.4[ Wb] [ N / ] l.5[ ] J,[ N ] peented n a ee of fgue Fg , whch llutate the oton of th te fo the ntal poton of the pendulu ϑ 6º.

311 9 4 Buhle DC Moto Dve BDC a b c d Fg. 4.4 Swngng oton of the pendulu col, fo ϑ 6º, coputed b the odel : a f cuent of PM b a col cuent c ϑ angula veloct of the pendulu d wa angle ϑ e electoagnetc toque T e

312 4. Peanent Magnet Bac Decpton n the Matheatcal Model 9 e Fg. 4.4 contnued a b c Fg. 4.5 Copaon of electoagnetc toque coputed wth dffeent PM odel plfcaton: a full odel b aatue eacton odel 4.4 c contant Ψ f odel

313 9 4 Buhle DC Moto Dve BDC a b c Fg. 4.6 Copaon of wngng oton - ϑ angle, fo dffeent PM odel: a full odel b aatue eacton odel 4.4 c contant Ψ f odel a Fg. 4.7 Chaactetc of oveent fo a ve tong feld lnage Ψ f. [Wb]: a f cuent of PM b a col cuent c ϑ angula peed of the pendulu d wa angle ϑ e electoagnetc toque T e

314 4. Peanent Magnet Bac Decpton n the Matheatcal Model 9 b c d e Fg. 4.7 contnued On the ba of the eult peented n Fg. 4.5 and Fg. 4.6 one can conclude that the dffeence n te of the cuve fo the vaable chaactezng the pendulu oton ganed fo vaou veon of the PM odel plfcaton ae ncondeable and the agnet odel fo Ψ f cont acceptable fo the odelng of oton paaete of the dve.

315 94 4 Buhle DC Moto Dve BDC 4. Matheatcal Model of BDC Machne wth Peanent Magnet The peentaton wll focu on the atheatcal odel of a thee-phae achne wth tpcal engneeng,.e. one wth clndcal tuctue wth aatue wndng n the tato and peanent agnet n the oto. Qute elf-evdentl, thee a wde vaet of BDC buhle dect cuent achne, both n te of the nube of phae wndng a fo ntance one baed on d tuctue, whee the ajo feld ha an aal decton [,6]. Howeve, a clndcal, thee-phae achne fo the bac engneeng oluton and occu a a all, edu and lage powe devce. A plfed co-ecton of uch a achne fo the nube of pole pa p peented n Fg Fg. 4.8 Co-ecton and cheatc vew of a BDC oto, fo p The bac plfng aupton appled dung the developent of a atheatcal odel nclude: - coplete et of the achne contucton, - degadng of facto affectng deagnetzaton of peanent agnet dung the opeaton effect of aatue, tepeatue nceae - degadng hgh ode MMF haonc of aatue wndng. The eanng, oe detaled aupton aocated wth the developent of the atheatcal odel wll be peented dung the coue of t devaton. Fo uch an electoechancal tanduce and lac of eleent evng fo the accuulaton of potental eneg, the agange functon equal to netc co-eneg:

316 4. Matheatcal Model of BDC Machne wth Peanent Magnet 95 J M M M T θ Ψ Ψ Ψ 4. whch can alo tae the fo of a at notaton: a T af a a T a J θ Ψ 4. whee: M M M M M M a at of aatue nductance Ψ Ψ Ψ Ψ f f f af - vecto of the couplng between peanent agnet flu and aatue wndng, [ ] T a vecto of aatue cuent. The patcula coponent of the nductance at of the aatue wndng account fo the vaable eluctance of the oto and dpaton flu of the aatue wndng and fo the pupoe of plfcaton can be peented a follow: co co co a M a M M a a a 4.6 co co co M M M a M M M a M M M 4.7

317 96 4 Buhle DC Moto Dve BDC Ψ Ψ n n n a a f af 4.8 whee:. / π a The angle φ account fo the nube of pole pa n the achne pθ 4.9 whle the elf-nductance of the aatue wndng aued n the fo whch dentfe the leaage nductance: a σ 4. Afte accountng fo thee ea, the nductance of the aatue wndng 4.4 can be etated n the followng fo: co co co co co co co co co a a a a a M a σ 4. The equaton of the echancal oton of the achne can be deved fo agange equaton fo a vaable denotng otaton angle θ : l D T dt d θ θ θ 4. o: l e D T T J Ω Ω 4. whee: θ θ θ af T a a a T a e T Ψ 4.4 pθ The epeon n 4.4 defne the electoagnetc toque of the achne and nvolve two te. The ft of the denote the eluctance toque of the achne, whch coe a a conequence of the eacton of aatue cuent wth the alent pole of the oto wth agnet:

318 4. Matheatcal Model of BDC Machne wth Peanent Magnet 97 T e pm T a n n a n a n a n n n a n n a a 4.5 Concuentl, the othe te of the epeon 4.4 denote the pncpal toque of the achne eultng fo the nteacton between aatue cuent wth peanent agnet ectaton flu. T ef co T a pψ f co a.6 co a In addton, BDC achne have anothe coponent of the toque,.e. coggng toque bede the eluctance elated one 4.5. It peent a a eult of the eacton of the pncpal flu wth the aatue teeth. In the peented odel 4.4 t, howeve, not encounteed nce the haonc aocated wth the tato lot ae degaded. Th oon adble nce the degne thoughout the engneeng effot [,8,9,8,,7,,5], tend to effectvel a at the nzaton of th coponent of the toque. 4.. Tanfoed Model Tpe d-q The tuctue of the nductance at of the aatue 4. ugget the applcaton of the othogonal tanfoaton, la a n the cae concenng a theephae nducton achne. In th cae we wll appl tanfoaton T.4 fo ω c. Thu, T a co co a co a 4.7 n n a n a Ta T a p θata The tanfoaton of equaton fo the electc ccut of the aatue wll be conducted n the geneal fo deved fo of agange equaton fo electc vaable: d dt Q a a U a R whch n the condeaton of Q offe the followng at equaton: a a a 4.8

319 98 4 Buhle DC Moto Dve BDC and, ubequentl, d dt d 4.9 dt a a Ψaf Ua Raa d a a dt a a Ψ f C Ua Raa 4.4 B ultplcaton of the above b T a and b applcaton of the othogonalt condton of the tanfoaton at 4.7 we can note that: T aa T T a Ψ f T a d dt a TaC T U T a a a a U a T Ta Ta T a a a 4.4 a a The epeon d a T a tanfoed n the followng anne: dt d a d d d T a Taa Ta a a A a 4.4 dt dt dt dt whee A a ew-etc at n the fo: A 4.4 Followng the tanfoaton, the at equaton 4.4 tae the fo: d dt A Ψ C U R a a f a a a 4.44 The coponent of the equaton 4.44 ae tanfoed a follow: a T a a q d U a TaUa uq 4.45 ud

320 4. Matheatcal Model of BDC Machne wth Peanent Magnet 99 d q T a a a M M σ σ σ σ T T 4.46 whee: d q M M σ σ 4.47 q d A 4.48 T a C T C 4.49 A a eult, the tanfoed equaton fo the aatue ccut 4.44 tae the fo: Ψ d a d q a q f d q q d d q d q R u R u p p θ θ o, altenatvel, q q d a d d d f d d q a q q q p R u dt d p R u dt d θ θ Ψ 4.5 The detenaton of the patcula epeon of the tanfoed voltage u q, u d 4.45 aocated wth the need to conde the poble of the coutaton of aatue cuent, whch occu n the functon of the angle of the oto poton θ. Th ue wll be dcued late. Concuentl, the quadatc fo 4.4 whch, detene the electoagnetc toque can be tanfoed n the followng anne: Ψ C a f a a e p M M p T

321 4 Buhle DC Moto Dve BDC whch offe the followng eult: T e p M Ψ 4.5 q d f The epeon 4.5 whch detene the electoagnetc toque of the oto, afte the tanfoaton, tae an uncoplcated fo: the ft te denotng the eluctance toque elatve to the poduct of aal cuent d, q and popotonal to nductance M aocated wth the bac haonc of the eluctance of the a gap, whle the othe te denote the pncpal toque popotonal to the poduct of the agnetc flu Ψ f and cuent q n the tanvee a of the achne. Thee a coplete analog hee to the coutato DC achne. 4.. Untanfoed Model of BDC Machne wth Electonc Coutaton The applcaton of the odel that doe not nvolve the tanfoaton of the coodnate te ha a nube of advantage. Fo the cae of a oto wth electonc coutaton thee a poblt of a oe ealtc odelng of coutaton and, thu, ganng eult oe pecel, ncludng the electc vaable ove te. The coutaton a well a the paaete of the wtchng tanto can be taen nto condeaton oe pecel n a anne that equed fo a pecfc poble of dve contol. Secondl, fo the lac of tanfoaton, the odelng of the achne and dve telf can account fo a nube of aete and dffeence n te of paaete, whch ende t poble to ulate the eegenc tate of the dve. In an untanfoed odel we conde that the aatue wndng ae connected n a ta Fg. 4.9, whch tae the fo of adequate contant equaton. Hee we wll appl the atce of contant W and W u.78 and.79 fo the epectve cuent and phae voltage of the oto u u W.e. a W a 4.5 u u u W u.e. U W u Ua 4.5 A a eult of the ultplcaton of the left-hand de of the equaton n 4.4 b the at of contant W u and ntoducng the vecto of aatue cuent

322 4. Matheatcal Model of BDC Machne wth Peanent Magnet we obtan: d Wua W dt Ψ f [ ] T 4.54 a a a Wu W WuC WuUa Wu RaW C U R a a 4.55 co π / co π / M M 4.56 M co π / M co co π / 6 co π / 6 6M 4.57 co π / 6 n co π / 6 C 4.58 n R Ra 4.59 Afte the tanfoaton, the equaton fo the vaable of the aatue cuent, 4.54 tae the followng fo: d dt a Ψ f C U R 4.6 a pθ a At th pont t poble to pa on to the tanfoaton of the quadatc fo of electoagnetc toque 4.4 b ntoducton of contant 4.5 and 4.5 fo the connecton of the aatue wndng nto a ta: T e T T T T a W a a a Ψaf θ W W 4.6 θ Snce W W T u, the epeon tae the fo W and the othe te tanfoed nto: a a W Wu W p θ θ T T T T T a W Ψ a W u f C f a C θ af Ψ Ψ

323 4 Buhle DC Moto Dve BDC Hence, T e Ψ C T a a f Afte adequate algebac opeaton we obtan: T pm e p 4.64 co π / 6 n co π / 6 pψ f co π / 6 n 4.65 The epeon 4.65 peent electoagnetc toque of BDC achne n the natual coodnate, wthout tanfoaton, fo the connecton of thee-phae aatue wndng n a ta. 4.. Electonc Coutaton of BDC Moto Coutaton n a buhle DC achne nvolve the wtchng of the aatue cuent to patcula phae wndng dependng on the poton of the oto angle θ. In a tadtonal buhle DC achne th occued a a eult of applcaton of a echancal coutato contng of olated coppe egent wth aatue wndng connected to the. Ove th coutato the gaphte buhe would lp thu ecevng the cuent whle the poton of the buhe wa fed n pace. In uch a anne the coutaton occued natuall dependng on the poton of the oto. The electonc coutaton enued b the convetng bdge whle the wtchng of the cuent between the wndng alo occu n the functon of the oto poton angle, and the gnal eponble fo the contol of the wtchng obtaned fo the poton eno eaung the angle of otaton θ. A a pncple, uch eno ae optcal, ncludng encode and nducton baed one,.e., eolve. Fg. 4.9 Bac chee of a bpola -phae BDC oto uppl

324 4. Matheatcal Model of BDC Machne wth Peanent Magnet In an cae, howeve, the coutaton angle need to be et at an appopate value, whch n a tadtonal DC achne wa the ole of the coect potonng of the buhe n a coutato. Fg. 4.9 peent the tandad tanto bdge fulfllng the ole of an electonc coutato fo a oto wth thee phae wndng n the aatue and bpola uppl of the wndng connected n a ta. The bpola uppl ean that n the aatue wndng the cuent flow n both decton,.e. the cuent flowng though wndng AC. The angula chee of the coutaton of BDC oto fo a potve decton of oto oton peented n Fg. 4.. a b Fg. 4. Tpcal chee of cuent coutaton n BDC oto - phae aatue n elaton to oto poton angle: a conductng of ndvdual phae wndng b conductng of pa of ta-connected wndng

325 4 4 Buhle DC Moto Dve BDC A tpcal coutaton daga fo thee-phae wndng connected n a ta nvolve ultaneou conducton of two phae belt whle the thd one ean n OFF tate. The ultaneou conducton of all thee belt occu onl n ve hot coutaton peod when we have to do wth the tanfe of the conducton fo the belt that about to tenate the opeaton to anothe phae belt, whch n accodance wth the coutaton daga tae the tun n tatng coutaton. A ngula phae belt n a thee-phae BDC achne conduct ove the peod coepondng to the angle of otaton,.e. π/ and ubequentl tae a bea ove the te coepondng to the angle of otaton,.e. π/. The ubequent conducton peod fo the angle of otaton equal to π/ occu afte th bea; howeve, fo an oppote decton of the conducton followed b anothe bea n conducton. It degned o that fo a full tun of the oto n a gven phae wndng the cuent that flow AC wth the bea n the conducton coepondng to the otaton of the oto ove π/ angle Fg 4.. In a coplete BDC achne wth thee phae wndng coutaton occu eve π/ angle of the otaton of the achne oto. The developent of a coutaton daga ae t poble to detene uppl voltage u d, u q 4.45 n the tanfoed odel of the oto and pefo detaled condeaton of the coutaton odel 4.5 to be appled fo uppl of the oto n odelng wthout tanfoaton. In both cae the value of uppl voltage contolled a a pncple b the pule wdth odulaton PWM. Due to the coue of coutaton of the cuent the patcula phae ae wtched on lghtl n advance n elaton to the theoetcal coutaton daga peented n Fg. 4.. Th advance angle denoted a δ uuall n the ange fo 5º - 5º. The peentaton that follow concened wth the detenaton of the uppl voltage u d, u q fo the tanfoed odel of the oto Suppl Voltage of BDC Moto n Tanfoed Model u, u q d The tanfoed voltage ae calculated n accodance wth the elaton n 4.45 and the coutaton daga peented n Fg. 4.. The detal of the elaton ae a follow: u u q d U u Uu c co δ c co δ a c co δ a c n δ c n δ a c n δ a 4.66 whee: U c, c, c - uppl voltage of the coutaton bdge fo the potve conducton tate - fo the evee conducton tate fo the OFF tate - conducton facto

326 4. Matheatcal Model of BDC Machne wth Peanent Magnet 5 t p u - pule wdth facto PWM contol T PWM pθ - electcal otaton angle conducton te wthn a pule peod T PWM t p t z T t - ecupeaton te wthn a pule peod T PWM PWM δ a π / p - advance angle Ove the peod of T PWM fo the duaton of the uppl t p epectve wtche ae n the ON tate and the voltage U U fed to the wndng. Concuentl, when we have to do wth contol wthout eneg ecupeaton, the cloue of the phae tae place and the cuent flow though the etun dode and one of the tanto of the bdge, and U. Fo the contol wth eneg ecupeaton all tanto ae n the OFF tate and the eneg etuned to the ouce though the two of the etun dode fo the voltage of the oto U -U. Th occu n the ecton of the contol peod t z. The above decpton of a ngle pule wth the peod of T PWM offe an eplanaton to the ue of calculaton of output voltage of the coutaton bdge 4.67 fo both tpe of bdge contol. The coeffcent u ae t poble to calculate the ean value of the voltage u d, u q. Thee ean ae detened on the ba of foulae n 4.66 whle the functon of the conducton facto c, c, c ae detened accodng to coutaton chee n Fg. 4.: u u q av d av U π U π ph ph n δ π / u co δ π / u 4.67 The eaple of the wavefo fo u d, u q ae peented n Fg. 4. and 4.. In both cae we appl an angle dependng PWM coeffcent u calculated fo the elaton: u ep / T 4.68 u u π whee: T u - the angula contant of voltage nceent. The eponental chaacte of voltage nceae u d, u q offe the poblt of the ooth oto tat-up Modelng of Coutaton n an Untanfoed Model of BDC The odelng of coutaton n an untanfoed te fo a thee-phae wndng of the aatue can have a vaou degee of detal. In th chapte we wll peent a ethod that condeabl plfed and, ubequentl, appl t n

327 6 4 Buhle DC Moto Dve BDC Fg. 4. Voltage u q and u d a a functon of otaton angle fo: δ, Tp 7.º, U 4 [V], u, T u.9 [] Fg. 4. Voltage u q and u d a a functon of otaton angle fo δ º, T p º, u 4 [V], u, Tu.8 [] eaple. It tae nto condeaton the fact that adequate uppl voltage connected to the patcula pa of phae wndng connected n a ta,.e. wndng no. -, -, - va a coutaton bdge. Dung the coutaton we wll dtnguh two tate: ft, when the coutaton begn dung the connecton of the ouce voltage to the wndng,.e. dung the actve pat t p of the uppl pule

328 4. Matheatcal Model of BDC Machne wth Peanent Magnet 7 T p and the othe tate, when the coutaton begn dung the pave pat of the pule whle the eneg etuned fo the wndng to the ouce o dung the cloue of the wndng. In both thee tate t poble to adequatel odel etance and voltage occung n the patcula electc ccut n the gven tate,.e. the etance of the oto wndng, electonc wtche a well a the blocng etance R b n the ccut of unuppled phae wndng. It wll be llutated b appopate eaple. Fg. 4. Cuent flow dung coutaton fo φ π /, and an actve pat t p of the peod T p We wll conde the coutaton occung n the actve pat of the pule t p, whch tae place fo the angle of otaton φ π /, Fg. 4., whee we have to do wth the wtchng of the cuent fo to cuent,.e. the tenaton of the conducton n the potve decton n the wndng n phae and coenceent of the conducton n the potve decton b the wndng n phae. Dung that te n the eanng wndng of phae the cuent flow contnuoul n the conventonal negatve decton. Th tuaton llutated n Fg. 4.. The taget ccut afte the coutaton uppled wth voltage U aed though tanto T, T6, -, whle the decang cuent n the wndng of phae cloed n the ccut wth the etun dode D that antpaallel to tanto T, nce the tanto T ha jut been cloed, and conductng tanto T6 connected to the phae wndng. In the tate peented n the fgue the potental of pont a aount to, potental of pont b U and the potental of pont c. Hence, the voltage U -U, U -U. Afte the coutaton,.e. afte the cuent ha decaed, the potental of the pont a wll change to ½ U and the epectve voltage wll be U -½ U, U -U, whle the etance R R b, whch ean that t wll aue the value of the blocng etance. Fo a bette llutaton of the condeaton we wll addtonall eane the coutaton fo the angle of otaton φ π. In th cae ee Fg. 4.4 the coutaton nvolve a change, whle contnung t flow. Decang cuent cloed aco the etun dode D4 of the tanto T4 and tanto T. The potental of the patcula pont

329 8 4 Buhle DC Moto Dve BDC a,b,c aocated wth the begnnng of the phae wndng n th tate aount to:,, U. A a conequence, lne-to-lne voltage upplng the wndng ae a follow: U, U U. Fg. 4.4 Cuent flow dung coutaton fo φ π, and an actve pat t p of the peod T p Concuentl, afte the tenaton of the coutaton the potental of the pont b change to ½U, whch eult n the followng value of lne-to-lne voltage upplng the wndng U -½U and U ½U. The tuaton dung the coutaton ove the pave ecton of pule T p can be peented fo two vaou altenatve of contol of the coutaton bdge,.e. fo the cae when dung the coutaton the eneg etun to the ouce and the oppote one when the eneg not etuned to the voltage ouce U. At the begnnng, we wll conde the ft of the cae, when n the pave ecton of the pule the eneg ecupeated. Fo th cae we wll conde coutaton, φ π/,.e. the ae a n Fg. 4., but fo the pave ecton of the PWM pule. Dung th coutaton tanto T jut wtched off tenatng uppl to the phae wndng, tanto not wtched on becaue of the pave peod and tanto T6 wtched off to facltate ecupeaton of eneg. The decang cuent flow though dode D and D5 agant the voltage of the ouce. In th tate the potental of pont a,b,c ae epectvel equal to, ½U, U and conequentl U - ½U, U ½U. Afte the coutaton fnhed potental of all thee a,b,c pont ae the ae and equal to ½U and nte-phae voltage ae U U. Th tate peented n Fg Concuentl, fo the othe veon of coutaton wthout eneg etun to the ouce dung the pave pat of the peod T p, the tanto T not wtched on, whle tanto T6 wtched on - contnung conducton, and the decang cuent of wndng and flow though T6 and D n a hoted ccut. At that tate the potental of a,b,c pont ae epectvel, ½U, and nte-phae voltage ae U U -½U. Afte the coutaton thee no cuent and le n

330 4.4 Chaactetc of BDC Machne Dve 9 Fg. 4.5 Cuent flow dung coutaton fo φ π / n a pave pat of the peod T p, wth eneg ecove Fg. 4.6 Cuent flow dung coutaton fo φ π /, n a pave pat of the peod T p, wthout eneg ecove the pevou cae all thee cla potental ae equal to ½U and n conequence U U. An llutaton of th found n Fg Chaactetc of BDC Machne Dve The peentaton n th chapte wll be devoted to the copute ulaton of chaactetc of buhle DC dve. Such ue nclude: tat-up, bang and dve eveal, contol of otatonal peed and tacng contol of the dve a well a t eacton to vaable paaete of the uppl and loadng. Snce thee detaled ue can be llutated wth the ad of adequatel elected eult of dnac calculaton, t potant to elect oto fo the deontaton of the dve n opeaton bede the peentaton of topc devoted to atheatcal odelng of the dve. Fo thee pupoe the paaete of two BDC oto ae peented: one wth a alle and the othe wth the hghe powe and dffeent uppl voltage. A ua of the paaete found n Table 4..

331 4 Buhle DC Moto Dve BDC Table 4. Rated data of two PMDC oto Rated Paaete Sbol Unt Moto A Moto B Powe Voltage Veloct of otaton Aatue cuent Toque Effcenc Wndng elf-nductance Mutual nductance PM ectaton flu P n W U V 4 n n ev/n 6 n I A.9.4 t n T n N. 5. η % n H.6. M H.. Ψ Wb.. f n Wndng etance R Ω.5.5 Moent of neta J N.8.5 Dapng facto D N.. Pule wdth T deg PWM Cout. advance angle δ deg Stat-Up and Reveal of a Dve Dve Stat-Up Stat-up fo the bac ue aocated wth the oton of a dve and, hence, the oto dve and the contol te have to fulfll a nube of peequte n ode to enue the appopate coue of the poce. Thee peequte nclude: poblt of tat-up fo eve ntal poton, tat-up wth a equed load a well a ltaton of the tat-up cuent to the value acceptable b the oto and the uppl te. The poce of tat-up of BDC oto futhe peded a a eult of occuence of paatc toque,.e. eluctance toque and coggng toque. The two effect ae educed dung the poce of oto degn n a anne that the ae not anfeted too tongl dung the tat-up. The ltaton of the tat-up cuent can be acheved n two wa: b nceental nceae of the voltage upplng the aatue ung PWM ethod o a a eult of contollng the tat-up cuent b ean of PWM ethod a well, elatve to the ntantaneou value of the cuent. A ooth nceae of the voltage dung the tat-up can be acheved n nueou anne. In ulaton odel appled fo the deontaton of the tat-up cuve t acheved b eponental nceae of u coeffcent 4.68,.e. the one denotng the actve pat of the pule. The fgue that follow llutate the tat-up cuve fo a oto wth hghe powe,.e. oto aed B n Table 4. a well a fo a alle oto A unde a ated load. The copaon wll nvolve the tat-up cuve fo the d-q tanfoed odel a well a fo the untanfoed oto.

332 4.4 Chaactetc of BDC Machne Dve a b c Fg. 4.7 Statng of the B 6.6 [W] oto wth a voltage egulaton: a aatue phae cuent b -phae cuent c cuent n the tead tate. The eult obtaned fo the untanfoed odel of BDC a Fg. 4.8 The ae coue a n Fg 4.7 fo: a electoagnetc toque b eluctance toque c otatonal peed

333 4 Buhle DC Moto Dve BDC b c Fg. 4.8 contnued a b Fg. 4.9 Statng of the B 6.6 [W] oto. Reult obtaned b the tanfoed odel of BDC: a tanfoed d, q cuent, b tanfoed bac aatue cuent a

334 4.4 Chaactetc of BDC Machne Dve a b Fg. 4. Statng of the B 6.6 [W] oto. Reult obtaned b the tanfoed odel of BDC: a electoagnetc toque b oto peed The two fgue that follow,.e. Fg. 4. and 4. peent the tat-up of the ae oto, howeve, fo the applcaton of a cuent delte. The opeatng pncple of the devce nvolve the dvon o ultplcaton of the pule wdth coeffcent u b a educton coeffcent ed n the ubequent pule 4.69 dependng on whethe the value of the cuent n an of the phae eceed o doe not each the value of the poed ltaton I. a Fg. 4. Statng of the B 6.6 [W] oto wth a cuent delte et fo I [A]: a ngle phae cuent b -phae cuent c U u aatue voltage

335 4 4 Buhle DC Moto Dve BDC b c Fg. 4. contnued a b Fg. 4. The ae coue a n Fg 4. but fo: a electoagnetc toque b otatonal peed

336 4.4 Chaactetc of BDC Machne Dve 5 The llutaton that follow,.e. Fg. 4. and Fg. 4.4 peent the cuve of the tat-up of the alle oto oto A, Table 4. ganed a a eult of applng untanfoed and tanfoed odel of the BDC oto. A one can conclude, the two cuve ae ve la, n patcula wth egad to the appng of electoechancal vaable. Condeable dffeence ae noted n te of the cuent cuve nce n the tanfoed odel the coutaton not a pecel odeled. In concluon, n te of the qualt of the odelng untanfoed odel a bette one, whle the bac advantage of the tanfoed odel nvolve the to te deceaed cot of ulaton. Fo thee eaon the tanfoed odel of BDC oto peent oe advantage dung ulaton of lage electoechancal te n whch a geate nube of dve peent. Concuentl, the educton of the duaton of the calculaton fo a condeable pee n favo of the eecuton of the ulaton of the opeaton of the te. a b c Fg. 4. Statng of the BDC oto A.96 [W]. The eult of ulaton b untanfoed odel: a aatue cuent b hape of cuent cuve c electoagnetc toque d eluctance toque e otatonal peed

337 6 4 Buhle DC Moto Dve BDC d e Fg. 4. contnued a b Fg. 4.4 Statng of the BDC oto A.96 [W]. The eult of ulaton b tanfoed odel: a d-q cuent b aatue cuent c hape of aatue cuent d electoagnetc toque e otatonal peed

338 4.4 Chaactetc of BDC Machne Dve 7 c d e Fg. 4.4 contnued Reveng DC Moto Buhle DC achne, jut a the coutato achne baed one can eve ve well fo opeaton n both decton of otaton. Howeve, t necea that two fundaental condton ae et: the te of the uppl and contol ha to be pepaed fo uch ccutance and a patcula contucton of the coolng te o an adequate a-flow aco the achne ha to be povded. A epaate ue aocated wth the opeaton of the dve at all otatonal peed and enung that heat caed awa n uch condton, thu, that the tepeatue nde the achne doe not eceed the petted lt. Th a be aocated wth an applcaton of a achne wth ndependentl dven fan. In ode to pefo the tat-up of a BDC oto n the evee decton, t necea to change the equence of the uppl of the oto phae and evee the value of the dela angle δ. A la coue of acton aued fo the cae when one ntend to pefo eveng of a oto dung t opeaton. The wtchng on of the evee decton of otaton n accodance wth the pecedng decpton ft eult n a peod of deceleaton, naed counte-cuent bang. Afte that, when the dve eache zeo peed of otaton the dve tat the opeaton n the evee

339 8 4 Buhle DC Moto Dve BDC decton. Such a anne adopted fo oto eveal geneall eque the ltaton of the aatue cuent due to the condton of the uppl te and the adble oto cuent. One can note at th pont that eceedng b the tat-up cuent the au adble value can caue an eveble deteoaton of the achne feld of ectaton ognatng fo peanent agnet. The eult of the ulaton tude peent the cuve of the eveng of the BDC dve wthout the ntoducton of a ltaton on the cuent Fg. 4.5 a well a dung a condeable cuent ltaton Fg 4.6. a b c Fg. 4.5 Reveng of PMDC oto A wthout cuent ltaton: a ngle phae cuent b -phae aatue cuent c electoagnetc toque d eluctance coponent of the toque e otatonal peed

340 4.4 Chaactetc of BDC Machne Dve 9 d e Fg. 4.5 contnued a b Fg. 4.6 Reveng of PMDC oto A wth cuent delte et on [A]: a ngle phae cuent b -phae aatue cuent c electoagnetc toque d eluctance coponent of the toque e angula peed

341 4 Buhle DC Moto Dve BDC c d e Fg. 4.6 contnued Fo the copaon of the cuve peented n Fg. 4.5 and 4.6 t te that the cuent delte dung the eveng of the oto opeate effectvel; but the eveng of the oto lat two te longe than n the one wthout a delte f t condeed untl the te of the tanfe of the peed aco zeo. Howeve, coplete contol ove the cuent peent, whch enue afet of peanent agnet and the electonc coutaton te upplng the dve. Bede, one can obeve that the effectve opeaton of the delte, degned a the a facton ultple ed of the PWM coeffcent, condeabl depend on t value. It follow the algoth: a a > I I n n * ed u n n / ed u u u 4.69 Fo the cuve peented n Fg. 4.6, the value of h facto et at ed..

342 4.4 Chaactetc of BDC Machne Dve 4.4. Chaactetc of BDC Machne Dve The te dve chaactetc denote the gaphcal epeentaton of a et of pont epeentng the opeaton of a dve elatve to the elected paaete chaactezng t opeaton. The paaete condeed a the ndependent vaable found on the X-a, whle the Y-a denote the value of the eaned vaable condeed a the output vaable. Often the ae fgue contan a fal of the chaactetc fo whch the patcula cuve dffe n te of anothe paaete that ve potant fo the peentaton of the opeaton of the dve,.e. one fo whoe coue t value contant. If on the X-a we fnd a paaete that not te, then uch a chaactetc can be teed a the tead-tate chaactetc cuve. Th chaactetc fo a et of pont fo whch the dnac tajecto fnd fnal tead tate, f uch a tatona tate et at all. The ente tatc chaactetc nfo at whch pont of the opeaton the dve cuentl found afte the tenaton of the dnac poce,.e. fo ntance tat-up, bang, change of the paaete of the uppl o loadng fo a gven paaete on the X- a. One hould note, howeve, that a change n the tate of the opeaton of the dve dnac tajecto doe not ovelap wth the tatc chaactetc, nce f th wee the cae, the duaton of the eecuton of the degned tajecto would be nfntel long. One can a that the tajecto begn and end at the tatc chaactetc; howeve, t cuve dffeent than the one fo the chaactetc nce t occu n a detened, fnte and ve often hot te. The hote the te, the futhe the tajecto fo the tatc chaactetc cuve. Anothe tpe of chaactetc the one n whch X-a contan te t. In uch a cae the cuve tae the fo of a te hto fo a gven vaable and geneall ha a dffeent wavefo fo othe paaete of the dve opeaton. It alo elatve to the ntal condton fo whch the cuve ognated. If fo uch a cuve thee a tead tate, the tead value of th tate fo a coponent of an coepondng tatc chaactetc. The tatc chaactetc can be deved n a nube of wa. Fo a ead dve we can ue a ethod of eaueent fo appopate poblte of vaaton of the paaete of the uppl and load of the eaned dve. If we have a atheatcal odel of a dve avalable, tatc chaactetc can be deved b defnton b conductng dnac calculaton and pefong ulaton untl the tead tate obtaned. Such calculaton have to be conducted epaatel fo each pont that detene the chaactetc. Concuentl, thee a poblt of aung adequatel favoable ntal condton, whoe dnac tajecto lead uffcentl fat to a tead tate. Depte that, t a cot and teconung entepe. Anothe effectve ethod nvolve the ubttuton of hpothetcal tead tate to the atheatcal odel n the fo of dffeental equaton of oton and convetng the odel nto a te of algebac equaton. Th pocedue ha been followed fo ntance dung the ntoducton of the equvalent daga of the nducton oto n chapte... If we ae capable of effectvel ganng uch a educton of the dffeental odel to an algebac odel, a a eult we wll obtan tatc chaactetc of the dve n the fo of functonal elaton between the paaete and vaable. Howeve, one ha to bea n nd that not all of the obtaned chaactetc have to be avalable a a

343 4 Buhle DC Moto Dve BDC eult of the tenaton of the dnac cuve o need not be avalable fo abta ntal condton,.e. fo an tatng pont. Fo that t te that, a a pncple, the atheatcal odel of electoechancal te ae non-lnea. The tatc chaactetc of the BDC can be ganed fo tanfoed odel 4.5 b algebazaton afte ubttutng contant functon fo t vaable: θ Ω q d I I q d cont cont cont dq dt dd dt 4.7 The tanfoed voltage u q av, u d av that ae peent n th odel ae decbed b elaton 4.67 and account fo the elaton wth coutaton advance angle δ. A a eult of the ubttuton of the fed vaable 4.7 n the atheatcal odel n the fo n 4.5,4.5 we obtan a te of thee algebac equaton n the fo: u u q av d av p R I a R I a Ψ f I q d q pω d I pω I M I d q I q q d T l Ψ f pω D 4.7 The non-lnea te of algebac equaton 4.7 account fo thee vaable of the tead tate of the dve I q, I d, Ω, paaete of the uppl u q av, u d av elatve to U, δ, paaete of the load T l, D and engneeng paaete of the dve, uch a p, q, d, M, R a, Ψ f. Such a tatc odel ae t poble to detene the chaactetc fo elected vaable n the ubject of the eanaton. Ipotant eaple nclude echancal chaactetc Ω, I q, I d, I a ft l,.e. chaactetc n the functon of the load toque fo the eanng paaete wth contant value, ncludng paaete of the uppl. The non-lnea te of algebac equaton of the tead tate 4.7, that ae cubc n elaton to vaable I q, I d, Ω, can be olved effectvel ung nuecal ethod, whoe applcaton ae wdel found n a nube of popula atheatcal pacage. In th cae atheatcal pacage MAPE V wa appled n ode to gan the futhe peented chaactetc. The voltage u q av, u d av of the tanfoed odel calculated n accodance wth 4.67 contan phae voltage U ph, whoe value fo a tpcal uppl of the BDC oto ae aued n the fo U ph U /, 4.7 whch fo a plfcaton b aung nuodal wavefo of the,, cuent n the patcula phae wndng of the oto. The algebac odel of the oto unde of the uppl of phae wndng n accodance wth 4.7 ha ade t poble to detene the tatc chaactetc n the both eeached BDC

344 4.4 Chaactetc of BDC Machne Dve oto. Fo the 6.6 [W] oto the chaactetc n the functon of the coutaton advance angle δ ae peented n Fg. 4.7 and 4.8. Thee ae the epectve fale of chaactetc I q, I d, I a fδ and Ω, η fδ, fo fou value of the load toque T l 5, 5, 5, 45 [N]. Moe detaled decpton ae found n the capton unde the fgue. a b c Fg. 4.7 Chaactetc of cuent of the 6.6 [W] BDC oto a a functon of advance angle δ: a I q tanfoed cuent fo T l 45, 5, 5, 5 [N], top down b I d tanfoed cuent fo T l 5, 5, 5, 45 [N] top down c I a aatue cuent fo T l 45, 5, 5, 5 [N], top down a b Fg. 4.8 Chaactetc of peed and effcenc of the 6.6 [W] BDC oto a a functon of advance angle δ: a Ω peed [ev/n] fo T l 5, 5, 5, 45 [N], top down b effcenc η fo T l 5, 5, 5, 45 [N], top-down at δ º

345 4 4 Buhle DC Moto Dve BDC Fo the peented cuve te a nube of concluon. I q tanfoed cuent doe not depend uch on advance angle δ, but anl on load toque T l, a t decde on electoagnetc toque value at the tead tate. Accodng to 4.7, t equal to: T e T DpΩ 4.7 l The cuent I d condeabl elatve to the advance angle δ nce th coepond to a change n the poton of the a of the buhe n a clacal DC achne wth a echancal coutato. The coutaton advance angle hould be et n a anne that enue that the value of I d cloe to zeo,.e. n the ange of δ 5º 5º n the chaactetc peented n Fg. 4.7b. Fo th cae the opeaton of the dve occu at a nu aatue cuent I a and au effcenc η. The electon of hghe value n th ange ae t poble to enue the opeaton of the dve fo a hghe otatonal peed Ω Fg. 4.8a at the epene of the deteoaton of effcenc. Anothe goup of chaactetc peented n Fg. 4.9 and 4. how the ae vaable a foel but the eult ae peented n the functon of the load toque T l. The othe paaete n the fgue that offe a dtncton between the patcula wavefo the advance angle δ º, º, º, 4º. a b c Fg. 4.9 Chaactetc of cuent fo the 6.6 [W] oto n a functon of load toque T l : a I q tanfoed cuent fo δ 4º, º, º, º top down b I d tanfoed cuent fo δ 4º, º, º, º top down, c I a aatue cuent fo δ 4º, º, º, º top down, at T l [N]

346 4.4 Chaactetc of BDC Machne Dve 5 a b Fg. 4. Chaactetc of oto peed and effcenc facto fo the 6.6 [W] oto n a functon of load toque T l : a otatonal peed fo δ 4º, º, º, º top down b effcenc facto fo δ º, º, º, 4º top down, at T l [N] d b c Fg. 4. Chaactetc fo the.95 [W] oto, n a functon of advance angle δ: a I a aatue cuent fo T l 6., 4.5,.,.5 [N] top down b otatonal peed fo T l.5,., 4.5, 6. [N] top down c effcenc facto fo T l.5,., 4.5, 6. [N] top down, at δ º In the coenta of the nfoaton found n the et of chaactetc n Fg. 4.9 and 4. one can conclude that the fal of the chaactetc I q ft l geneall peent the nvolveent of the eluctance toque n the total toque T e of the oto. The laget hae of the eluctance toque T e occu fo δ º and th chaactetc le the lowet n t fal. Th obevaton confed b wavefo I d ft l, whee fo δ º, I d ha negatve and deceang value, thu

347 6 4 Buhle DC Moto Dve BDC leadng to adequatel potve eluctance toque T e. The echancal chaactetc fo Ω ft l n Fg. 4.a have a hpebolc wavefo whch patculal obeved fo all load. Th coe a a conequence of the deagnetzng effect of the cuent I d n th ange, whch aue potve value thee Fg. 4.9b. The effcenc of the oto Fg. 4.b fo δ º 4º and the load that cloe to t ated value T l [N] hgh and eceed η > 9%, and eache a au of ove 9%. A one can conclude fo the hape of the chaactetc the cuve ae qute flat and even oveloadng of the oto two te doe not eult n a condeable lo of dve effcenc. The chaactetc deved n an analogcal anne fo the alle of the eaned oto wth the ated output of.95 [W] oto A, Table 4. ae peented n an abbevated fo n Fg. 4. and 4.. b b c Fg. 4. Chaactetc fo the.95 [W] oto n a functon of load toque T l : a I a aatue cuent fo δ 45º, 5º, 5º, 5º top down, at T l [N] b otatonal peed fo δ 45º, 5º, 5º, 5º top down c effcenc facto fo δ 5º, 5º, 5º, 45º top down, at T l [N] The tatc chaactetc fo an untanfoed odel of the BDC oto 4.6, 4.65 fo two-phae contol cannot be ganed pl n the algebac fo nce the ae elatve to the oto angle. Obvoul, the equaton wth the peodcall vaable coeffcent and oluton fo the tead tate n accodance wth Floquet theoe ae efeed to n lteatue wth egad to atheatcal odel of electc achne. Howeve, fo the cae of uch an abbevated atheatcal odel and low cot of calculaton, the tatc chaactetc can be deved a a

348 4.4 Chaactetc of BDC Machne Dve 7 et of tatona pont n the dnac tate. Th appoach ha anothe advantage, nael, that t peent whethe a gven tatc tate of a dve poble to acheve fo a gven ntal tate. Nueou eaple ndcate that th not alwa the cae, n patcula wth egad to BDC oto wth a hghe hae of the eluctance toque o coggng toque. The tatc chaactetc of a 6.6 [W] oto ganed a a eult of th ethod ae peented n Fg. 4. and 4.4. Fo the copaon of chaactetc deved on the ba of the tanfoed and untanfoed odel of the oto one can conclude that both of the loo ve la. The onl elevant dffeence egad the wavefo ang the cuent of the aatue I a ft l. In the tanfoed odel Fg..9c the value of the cuent fo a all load ae condeabl hghe than the one ganed on the ba of the untanfoed odel Fg. 4.a. Th coe a a conequence of the coue of the te I d n the tanfoed odel. Fo hghe load the elevance of the dffeence tat to fade. a b Fg. 4. Stead-tate chaactetc fo 6.6 [W] BDC oto n a functon of load toque, coputed b untanfoed odel: a I a aatue cuent fo δ 4º, º, º top down b Ω oto veloct fo 4º, º, º top down a b Fg. 4.4 Stead-tate chaactetc fo 6.6 [W] BDC oto n a functon of load toque, coputed b untanfoed odel: a effcenc facto fo δ º, º, 4º top down b total loe fo δ 4º, º, º, top down

349 8 4 Buhle DC Moto Dve BDC At th occaon one can note that the two odel appled n th cae,.e. the tanfoed and the untanfoed one ae not copletel equvalent. B defnton n the tanfoed odel an aupton ade that each of the thee phae of the aatue ndependentl uppled. In addton, the nfluence of coutaton degaded. The untanfoed odel account fo the contant poed b two-phae uppl and nvolve coutaton between the ta connected wndng whle the wtchng occu dung the otaton of the oto n the functon of t poton n accodance wth the daga n Fg. 4.. The advantage of the tanfoed odel that t ve ple and doe not poe an poble dung calculaton. Th pla an potant ole n a cople egulaton te n whch a ngle BDC oto fo one of an coponent of the te a a dve n one of the jont, fo ntance a an ndutal anpulato. The tatc chaactetc cope the et of the poble tead tate of the dve. Howeve, the a b c Fg. 4.5 Tanent followng tepwe change of advance angle δ: º 4º, fo 6.6 [W] oto, coputed b untanfoed odel of BDC: a I a cuent b electoagnetc toque c oto peed

350 4.4 Chaactetc of BDC Machne Dve 9 tanfe between two pont on the chaactetc occu a a eult of tanent and, hence, tanfe not alwa poble nce we have to do wth a non-lnea dnac te. Fo the pupoe of llutaton the fgue that follow peent the tanent fo the dnac tate that occu dung the change of the paaete n a te wth BDC oto. Fg. 4.5 and 4.6 peent tanent eultng fo a abupt change of the advance angle δ: º 4º fo an untanfoed and tanfoed odel of the oto, epectvel. a b c d Fg. 4.6 Sla tanent a n Fg 4.5, but coputed b tanfoed odel: a I q, I d cuent b I a aatue cuent c electoagnetc toque d oto peed

351 4 Buhle DC Moto Dve BDC Fo the cae of the eult ganed fo untanfoed odel the nceae of the peed Ω alle nce n a two-phae oto uppl the epone of the dve to the change n the advance angle δ lted n copaon to the cae of ndependent uppl of thee phae, whch addtonall confed b the tatc chaactetc of the dve peented eale n th ecton. The two fgue that follow,.e. Fg. 4.7 and 4. 8 peent tanent fo the epectve tanfoed and untanfoed odel afte a tepwe change n the advance angle δ: º º. Th a change that the oppote of the one that wa pevoul peented a t eult n the educton of the oto peed and bang n the tanent peod. a b c Fg. 4.7 Tanent followng tepwe change of advance angle δ: º º, fo 6.6 [W] oto, coputed b untanfoed odel of BDC: a I a cuent b electoagnetc toque c oto veloct

352 4.4 Chaactetc of BDC Machne Dve a b c d Fg. 4.8 Sla tanent a n Fg. 4.7, but coputed b tanfoed odel: a I q, I d tanfoed cuent b I a aatue cuent c electoagnetc toque d oto peed The copaon between the eult of calculaton fo the untanfoed and tanfoed odel ndcate a geate deceae of the oto peed Fg. 4.8d veu Fg. 4.7c accopaned b an adequatel hghe nceae of the peed Fg. 4.6d veu Fg. 4.5c fo the cae of the eult ganed ung the tanfoed odel n whch the wndng ae not connected. The ae eult ae ganed on the ba of tatc chaactetc, fo ntance fo the copaon of the eult n Fg. 4.a wth the one n Fg. 4.b.

353 4 Buhle DC Moto Dve BDC 4.4. Contol of Rotatonal Speed n BDC Moto The bac technque appled fo the contol of the DC oto, ncludng BDC oto, nvolve egulaton b alteng voltage U, whch cuentl ealzed wth the ad of the pule wdth facto u of PWM contol. Fo the te wthout eneg ecupeaton the change of the facto u, whch ealze the coplete change of the oto peed, occu appoatel n the ange: whle n the te wth eneg ecupeaton n the ange u u The dffeence fo the both tpe of the contol eult fo the fact that dung the etun of the eneg the oto ove th peod fed wth a voltage wth negatve value -U, o that fo u.5 the ean value of the uppl voltage equal to. The tatc chaactetc fo the contol of the oto eultng fo the change of the pule wdth facto wthout the ecupeaton of the eneg nto the ouce ae peented n Fg. 4.9 and 4.4 fo the adequate dffeent value of the load toque T l and vaou value of the advance angle δ. a b c d Fg. 4.9 Chaactetc fo.95 [W] oto n a functon of u facto, wthout eneg ecupeaton, fo δ 5º: a I q tanfoed cuent fo T l 6., 4.5,.,.5 [N] top down b I d tanfoed cuent fo T l.5,., 4.5, 6. [N] top down c I a aatue cuent fo T l 6., 4.5,.,.5 [N] top down d otatonal peed fo T l.5,., 4.5, 6. [N] top down

354 4.4 Chaactetc of BDC Machne Dve a b Fg. 4.4 Chaactetc fo.95 [W] oto n a functon of u facto, wthout eneg ecupeaton, fo T l [N], a I a aatue cuent fo δ 5º, 45º, 5º, 5º top down, at u b otatonal peed fo δ 45º, 5º, 5º, 5º top down Sla chaactetc ae dplaed fo the contol wth eneg ecupeaton; howeve, the change of the pule wdth facto lted n accodance wth Selected chaactetc fo th tpe of contol ae peented n Fg a b c d Fg. 4.4 Chaactetc fo.95 [W] oto n a functon of u facto, wth eneg ecupeaton: a I a aatue cuent fo δ 5º, T l 6., 4.5,.,.5 [N] top down b otatonal peed fo δ 5º, T l.5,., 4.5, 6. [N] top down c I a aatue cuent fo T l. [N], δ 5º, 45º, 5º, 5º, top down, at u d otatonal veloct fo T l [N], δ 45º, 5º, 5º, 5º top down

355 4 4 Buhle DC Moto Dve BDC The two fgue that follow peent tanent tate, whch occu afte deceang pule wdth facto fo u to u.75, change that equvalent to the educton of the uppl to the half of the ouce voltage value. a b c Fg. 4.4 Tanent followng tepwe change of u facto: u :.75, fo 6.6 [W] oto b ue of the untanfoed odel: a a aatue cuent b electoagnetc toque c otatonal peed a Fg. 4.4 Sla tanent a n Fg 4.4, but coputed b ue of the tanfoed odel of BDC: a q cuent b q, d cuent, c a aatue cuent, d electoagnetc toque, e oto peed

356 4.4 Chaactetc of BDC Machne Dve 5 b c d e Fg. 4.4 contnued The chaactetc peented n the two fgue Fg. 4.4 and 4.4 ae la wth the onl ecepton of the wavefo fo the aatue cuent. Fo the cae of the wavefo eultng fo the applcaton of the tanfoed odel n the tanent tate the aatue cuent doe not deceae dung the bang of the dve but how an nceae. Th oppote than n the cae fo tanent ganed on the

357 6 4 Buhle DC Moto Dve BDC ba of the untanfoed odel. Th coe a a conequence of the hap deceae of the cuent q and nceae of the cuent d, nce n th anne the tanfoed odel ealze the tepwe change of the oto load. Phcall th ean a dffeent tpe of oto bang fo the ndependent uppl of thee phae wndng wthout contant than t the cae n a thee-phae te connected n a ta fo the two-phae uppl. A t wa entoned befoe, obtanng tatc chaactetc n an untanfoed odel aocated wth the need of calculatng a ee of tanent that fnall gan a tead tate. In th anne the chaactetc n Fg. 4. and 4.4 wee dawn up n the functon of the achne load. The ae ethod wa followed n ode to gan the chaactetc peented below n the functon of coutaton advanceent δ fo thee dffeent pule wdth facto value u,.9,.8. Th wa conducted n a te wth eneg ecupeaton o that the antcpated value of otatonal peed ae found n the ange that n ageeent wth the foula below Ω Ωn u 4.76 that Ω n,.8 Ω n,.6 Ω n, epectvel. The chaactetc ae peented n Fg and a b c Fg Chaactetc fo 6.6 [W] oto n a functon of advance angle δ, fo u,.9,.8 and T l T n : a I a aatue cuent b total eneg loe ΣΔP c effcenc facto η

358 4.4 Chaactetc of BDC Machne Dve 7 a b Fg Chaactetc fo 6.6 [W] oto n a functon of advance angle δ, fo u,.9,.8 and T l T n : a otatonal peed b echancal powe P a b c Fg Shape of the tead-tate a cuent fo: a δ -5º b δ 5º c δ 45º; u and T l T n The chaactetc ganed n th wa ae not ooth nce the ae foed on the ba of a lted nube of pont fo the vaable δ,.e. about pont and, n addton, the fnal tead tate of the dve dffcult to detene n a copaable wa fo each fnal pont. The chaactetc peented n Fg. 4.44,.e. fo

359 8 4 Buhle DC Moto Dve BDC aatue cuent I a, total loe ΣΔP and, wthn a cetan ange, effcenc η ndcate that fo a tead load thee value ae onl lghtl elatve to facto u. Concuentl, the cuve n Fg. 4.45,.e. otatonal peed Ω and echancal powe P ae condeabl dependent on the value of the voltage and, conequentl, on the value of facto u, whch dectl affect the value of the voltage. A a eult, t the value of the voltage upplng aatue, hee epeented b pule wdth facto u, that the bac vaable eponble fo the contol of BDC dve, whle hgh eneg effcenc to be antaned and t not condeabl affected dung uch contol pocedue. In addton, Fg peent wavefo of aatue cuent a fo vaou value of the advance angle δ. 4.5 Contol of BDC Moto Dve 4.5. Contol Ung PID Regulato A t ndcated b tatc chaactetc and tanent, BDC oto dve ae contolled n a la anne to othe DC oto,.e. b changng aatue voltage. In a te wth an electonc coutato th occu a a eult of the change of the pule wdth facto u. Th a happen at eve patcula pule a a eult of odfng the gnal contollng the ee of the pule. In ot cae we have to do wth dceet contol n whch the value of the pule wdth facto u elatve to the value of vaable n the dve fo the duaton of the pule that pecede. Contol elatve to the angle of otaton θ, jut a peented eale on dung the dcuon of the opeaton of the electonc coutato. The othe value that ae eponble fo the contol nclude otatonal peed Ω and aatue cuent a. Snce fo the pupoe of the contol t necea that the angle of otaton fala, one of the coon oluton nvolve the applcaton of an encode fo detenaton of the poton of the oto and a dffeental te n ode to ndectl obtan peed Ω. It oete the cae that the aatue cuent appled Fg BDC dve contol te wth encode and PID egulato. HSG hgh de gate, SG low de gate

360 4.5 Contol of BDC Moto Dve 9 fo egulaton a the value that an aula one and can be ued fo the pupoe of ltng contol, jut a peented n the eaple devoted to tat-up of a oto. Soete the contol of BDC dve doe not appl poton eno and the necea oto angle obtaned ndectl on the ba of eaueent of voltage and cuent n aatue wndng b ue of tate obeve. Th ethod wll be peented n the futhe pat of the cuent ecton. A tpcal contol te fo a BDC oto dve wth a poton eno peented n Fg Th ecton wll be concened wth the peentaton of eaple of BDC dve contol te wth peed tablzaton fo changng load and tepwe a b c d Fg Stablzaton of oto peed b PID egulato P 6; I ; D.5 afte tepwe load toque change T l : [N] 9 [N] and conequentl 9 [N] [N]: a aatue cuent b oto peed c electoagnetc toque d peed eo [%]

361 4 4 Buhle DC Moto Dve BDC change of the nput value fo a contant oto load. Such a bac contol ta ung PID egulato eveal the ablt of BDC oto to opeate n the dve contol baed te on changng the value of the uppl voltage. Fg peent eult of ulaton conducted ung an untanfoed odel of the oto fo tablzng otatonal peed fo a tepwe change of the load toque fo the ated value of T l [N] ntall to T l 9 [N],.e. thee te oveloadng the oto followed b a tepwe change to each the value of T l [N], whch one thd of the ated load. a b c d e Fg Contol voltage coponent and aatue voltage dung tablzaton of peed ee Fg 4.48: a P coponent b I coponent c D coponent d aatue voltage e u facto

362 4.5 Contol of BDC Moto Dve 4 a b c Fg. 4.5 Stablzaton of oto peed b PID egulato ee Fg 4.48 but fo dffeent contol value: P ; I 5; D.: a aatue cuent b oto peed c peed eo Fg llutate a coect tablzaton of the oto peed fo ve dnac change of the load. Fo hgh gan value of PID egulato Fg the eo of peed egulaton n the ange ε Ω.4%, whle fo a lowe gan Fg. 4.5 th eo ange aound ε Ω %, whch denote value that ae appoatel popotonal to the gan appled fo I, P, D facto. Stablt tate acheved afte aound.5 []. Benefcal condton fo egulaton ae ecued b a uplu of the egulaton that nvolve nput of the uppl voltage U [V] to a oto whle the ated value of the voltage U [V]. A one can ee fo uch an ntenve egulaton the te of the uppl, coutato and oto telf have to be degned to wthtand the au value of the voltage [V]. The followng wavefo,.e. the one peented n Fg. 4.5 and 4.5 llutate PID egulaton fo the tepwe change of nput peed Ω d : 5 5 [ev/n] fo both untanfoed and tanfoed odel of the oto.

363 4 4 Buhle DC Moto Dve BDC a b c d Fg. 4.5 PID egulaton of oto peed of the 6.6 [W] oto, afte tepwe change of equed value of peed Ω d : 5 5 [ev/n]. Reult fo untanfoed odel and: I, P 5, D : a tato cuent b electoagnetc toque c oto peed d aatue voltage

364 4.5 Contol of BDC Moto Dve 4 a b c d Fg. 4.5 PID egulaton of oto peed tanent le n Fg 4.5, but eultng fo tanfoed d,q odel: a d,q cuent b electoagnetc toque c oto peed d aatue voltage The wavefo n Fg. 4.5 and 4.5 ndcate that the tanent eultng fo the applcaton of tanfoed and untanfoed odel ae ve la. The egulaton ealzed qucl and effectvel depte lage dffeence n te of the taget peed. Howeve, th happen unde the aupton of acceblt of the hghe value of uppl voltage U.5U n and an addtonal condton that the oto capable of the geneaton of a uge toque of about T e 5 [N],

365 44 4 Buhle DC Moto Dve BDC a b c d Fg. 4.5 PID egulaton of oto peed of the 6.6 [W] oto, wth aatue cuent ltaton to [A], afte tepwe change of equed value of peed Ω d : 5 5 [ev/n]. Reult fo untanfoed odel and I, P 5, D : a tato cuent b electoagnetc toque c oto peed d aatue voltage.e., te the value of T n. Such egulaton popete can be ganed a a eult of ung a oto and uppl te that ovezed n elaton to the ated value

366 4.5 Contol of BDC Moto Dve 45 dung opeaton. Fo alle equeent egadng the egulaton peed t poble to appl a oe econocal uppl te and ntoduce cuent delte n the oto degn. The two fgue that coe below,.e. Fg. 4.5 and 4.54 peent the egulaton of the oto fo a tepwe change of the nput peed and ultaneou applcaton of a cuent delte whch wouldn t allow aatue cuent uge that eceed et ultple of the ated value, a n the pevou eaple. Fg. 4.5 peent the eult fo the tepwe nceae of the nput value of the otatonal peed whle the cuve n Fg peent the eaple nvolvng the educton of the nput value. Fo both cae the aatue cuent lted to the value of [A], whch about fve te the value of the ated cuent. a b c Fg PID egulaton of oto peed of the 6.6 [W] oto, wth aatue cuent ltaton to [A], afte tepwe change of equed value of peed Ω d : 5 5 [ev/n]. Reult fo untanfoed odel and I, P 5, D : a tato cuent b electoagnetc toque c oto peed d aatue voltage

367 46 4 Buhle DC Moto Dve BDC d Fg contnued 4.5. Contol wth a Gven Speed Pofle BDC oto dve ae capable of pefong cople contol ta ncludng the one nvolvng a gven pofle of dve peed and a gven pofle of angula poton of the oto. Th tpe of contol naed tacng contol nce the ole of the contol te to follow the gven tajecto of the oton whle the ole of the dve egulato fo ntance, PID tpe to coect the eo of the pefoed tajecto. Th tpe of pactcal ue ae well woth degnng wth the ad of gnal poceo that have been developed pecfcall to collect gnal fo eaung convete, pefo nuecal calculaton aocated wth the contol and end coand gnal to be eecuted n ode to nfluence the opeaton of an electonc coutato. Sgnal poceo ae equpped wth ntenal coponent that eve fo the pupoe of contol fo ntance eveal geneato of PWM gnal thu eecutng the ta of the contol of an convete. An eaple of a contol te contng of a BDC oto and a gnal poceo peented n Fg The eult of a copute ulaton of a aple contol ta fo a BDC dve wth a gven peed pofle wth tapezodal hape ae peented n Fg In th ta the oto peed nceae fo to.5 [ev/n] wthn.8 [] and ean at th level fo anothe. [], afte whch wthn. [] t deceae to 5 [ev/n]. The calculaton fo the cae of th ulaton have been conducted wth the ad of untanfoed atheatcal odel of a oto Fg and wth peevng d,q tanfoaton Fg Fg contan a copaon of the eult of contol fo vaou value of gan of the PID egulato.

368 4.5 Contol of BDC Moto Dve 47 Fg Contol te of a BDC oto, wthout a oto poton gauge wth a gnal poceo a Fg Roto peed contol of the 6.6 [W] PMDC oto, accodng to tapezodal hape of peed tajecto, unde the nonal load of T l 5 [N]. PID egulato ettng ae: I, P 5, D. Untanfoed odel eploed: a aatue cuent b oto peed c electoagnetc toque d oto voltage e peed eo [ev/n]

369 48 4 Buhle DC Moto Dve BDC b c d e Fg contnued

370 4.5 Contol of BDC Moto Dve 49 a b c d Fg The ae contol poble a n Fg 4.56, but tanfoed d,q odel of BDC eploed: a d,q tanfoed aatue cuent b aatue cuent c oto voltage d oto peed e electoagnetc toque f peed eo

371 5 4 Buhle DC Moto Dve BDC e f Fg contnued a b c Fg Tacng eo fo tapezodal peed contol of the PMDC oto, le n Fg 4.56, but fo vaou PID egulato ettng: a I, P 5, D b I 4, P, D c I, P 5, D

372 4.5 Contol of BDC Moto Dve 5 On the ba of the eult peented n Fg one can conclude that the eo of the dve contol oughl popotonal to the value of egulato gan Contol fo a Gven Poton Pofle The applcaton of a gnal poceo a an electonc te that cobne the popete of a convete of eaued quantte, a dgtal calculaton poceo and a geneato of contol gnal offe the poblt of pefong cople ue egadng a BDC oto dve contol wth an altenatve of elnatng a poton eno [9,4,,8,4]. Fg peent a te ncludng a gnal poceo that collect voltage gnal fo a EMF aple and wth the ad of uch a te that capable of detenng the peod dung whch the tanto ae n the ON tate wthout dect eaueent of the angle of otaton. The queton of the enole contol fo a cople ta nce t eque the detenaton of the poton of the oto even befoe tat-up when the nduced EMF too all n ode to detene oto poton on t ba. Subequentl, afte the equed thehold eceeded, on the ba of the nduced EMF n the unuppled phae of the aatue wndng at a deed ntant the contol gnal fo the electonc coutato occu. Fg peent EMF nduced between clap, dung the tat-up of the oto. Fg EMF nduced n aatue wndng of a BDC oto hotl afte tat-up Fo the opeaton of the oto wth hghe peed the detenaton of the ntant of the coutaton on the ba of the eaued cuent and EMF n the wndng eae nce thee a condeable angula coelaton between the two value that ae elatve to the poton of the oto. Fg. 4.6 peent the coelaton between SEM and aatue cuent and Fg. 4.6 contan the coelaton of the voltage between the clap and cuent.

373 5 4 Buhle DC Moto Dve BDC a b Fg. 4.6 EMF and aatue cuent fo BDC opeaton wth a noal peed: a e, b e, a b Fg. 4.6 Phae-to phae voltage and aatue cuent fo BDC opeaton wth a noal peed: a u, b u, A t ha alead been entoned, dung the tat-up the tuaton oe cople and n ode to detene the oto poton t poble to appl the pule ethod [9,]. The applcaton of th ethod, howeve, eque that the oto ha a vaable eluctance dung the otaton,.e. a tuaton whch nvolve the alent pole oto. Fg. 4.6 peent electootve foce EMF and cuent fo the cae of the tat-up, fo whch the wavefo conf the lac of ple coelaton between the cuve.

374 4.5 Contol of BDC Moto Dve 5 a b c Fg. 4.6 EM Foce and cuent dung tatng of BDC oto: a e, b e, c e, An eaple of the contol of the eveble BDC dve ncludng the egulaton of the poton and the peed peented n the followng eaple. Eaple 4. Fg. 4.6 peent a daga of the dve of a lage, ave pendulu wng wth a contolled apltude and peod of ocllaton dvegng fo a natual peod, whch anhow dependent on dapng of the oton. The dve apple a oto wth a ated powe of.95 [W] Table 4.. Fg. 4.6 Vew of a ave pendulu dven b BDC oto

375 54 4 Buhle DC Moto Dve BDC The data fo the echancal pat of the dve ae a follow: 8 [ g] - a of the wng l.8 [ ] - length of the pendulu J J J g.8 [ N ] - oent of neta of the oto and the fat de of the gea J.[ N ] - oent of neta of the low de of the gea and the pendulu elatve to the cente of a D. [ N] - coeffcent of pendulu dapng D.5 [ N] - coeffcent of oto dapng [ ] - tanon ato g Applcaton of agange ethod eque the calculaton the netc eneg of the echancal pat of the te: T θ J l J θ J J l The vtual wo of the te equal to: δa D δ T e D θ δθ whee: T e - denote the electoagnetc toque of the oto. In t echancal pat the te ha a ngle degee of feedo. Snce the calculaton ae to be pefoed fo the pont of vew of the oton of the pendulu, n th cae q φ hould be adopted a genealzed coodnate. The contant eultng fo the tanon tae the fo: θ 4.77 g Afte the ntoducton of the contant 4.77, agange functon fo th te tae the fo: J J l co g T U gl 4.78 J whee: - J l J / l J / l p g - the total oent of neta educed to the low peed de. Concuentl, the vtual wo of the te equal to: p T D D δ Q δ δa e g g 4.79 Afte the applcaton of the agange equaton.5 fo the genealzed coodnate φ we obtan the equaton of oton fo the echancal vaable n the fo: d Q 4.8 dt

376 4.5 Contol of BDC Moto Dve 55 Afte ubttuton n the agange functon 4.78, we have: gl n Te g D D g / J p 4.8 The aued pulaton of the oton of the pendulu equal to the pulaton of the atheatcal pendulu wth oent of the neta l, whch dffe fo J p 4.78, hence: g Ω.9 [/ ] 4.8 l The aued oton of the pendulu Eaple 4.a nvolve the antenance of the undaped fluctuaton wth the apltude φ and pulaton Ω 4.8. Hence, the equed tajecto detened a follow: n Ωt π / ω Ωco Ωt π / 4.8 a ω Ω n Ωt π / The othe tpe of the gven oton Eaple 4.b the eponental tat-up of the pendulu fo the ntal angle φ to angle φ a wth the te contant of T a. t T n Ω π / a e a t ω a e T Ω a e a t T a n Ωt π / t T a co Ω π / t 4.84 a a e Ta a Ω e T a t T a t T co Ωt π / a Ω e a t T a n Ω π / t The atheatcal odel of the te alo n the echancal pat nonlnea and th o to the lage degee the geate apltude of the oton of the pendulu φ, φ a. Th could be eplaned b the lneazng appoaton of the pendulu wng n < 4.85

377 56 4 Buhle DC Moto Dve BDC that fulflled wth uch nceang eo fo hghe value of φ. The egulaton of the dve follow a a eult of the applcaton of PIDD egulato,.e. the one that eponble fo the contol of 4 tpe of eo: ε I a t a dτ ε 4.86 ε ω ω ε a ω The fgue that follow,.e. Fg to Fg 4.69 peent the eult of the egulaton fo Eaple 4.a, whch denote wngng of the pendulu wth a gven apltude. But the ft of the fgue Fg how the fee oton of the pendulu wth dapng and wthout dve egulaton. Th gve the ba fo the copaon fo eaple analzed futhe n whch the egulaton te and the actuato,.e. BDC oto and the tanon ae engaged to pefo the oton n accodance wth the gven tajecto. Fg Fee wng of the pendulu wthout an nteventon of a egulaton te, fo φ -5º, -7º, -8º The fgue that coe below peent the oton of the pendulu and egulaton cuve eultng n the acheveent of the degned tajecto 4.8. a Fg Cuve fo electoechancal vaable fo the PIDD contolled pendulu wngng, φ -7º: a φ cuent b aatue cuent c electoagnetc toque T e ft d electoagnetc toque T e fφ e u facto f U a /U oto voltage cuve

378 4.5 Contol of BDC Moto Dve 57 b c d e f Fg contnued

379 58 4 Buhle DC Moto Dve BDC a b c d e Fg Mechancal vaable cuve fo the PIDD contolled pendulu wngng, φ - 7º: a poton angle φ b angula veloct ω c angula acceleaton a d poton eo ε φ e peed eo ε ω

380 4.5 Contol of BDC Moto Dve 59 a b c d Fg Untable chaacte of pendulu egulaton fo φ -8º: a poton angle φ b angula veloct ω c angula acceleaton a d poton eo ε φ e peed eo ε ω f u facto

381 6 4 Buhle DC Moto Dve BDC e f Fg contnued a b c Fg Copaon of poton eo ε φ, fo: a φ -5º b φ -7º c φ -8º

382 4.5 Contol of BDC Moto Dve 6 a b c Fg Copaon of peed eo ε ω, fo: a φ -5º b φ -7º, c φ -8º The et of fgue that follow,.e. Fg llutate oluton to the Eaple 4.b, whch peent tat-up of a pendulu fo an ntal angle of φ º to a wng wth an apltude of φ a. The wavefo llutatng the tat-up n accodance wth 4.84 eponentall egulated wth the te contant of T a. a Fg. 4.7 Electoechancal vaable fo eponental tatng of the pendulu wng fo φ -º to φ a -7º, wth PIDD egulato: a aatue cuent b electoagnetc toque T e fω c electoagnetc toque T e ft d u facto e U a /U oto voltage

383 6 4 Buhle DC Moto Dve BDC b c d e Fg. 4.7 contnued

384 4.5 Contol of BDC Moto Dve 6 a b c d e Fg. 4.7 Mechancal vaable fo the PIDD contolled pendulu wngng, le n Fg. 4.7: a poton angle φ b angula veloct ω, c angula acceleaton a d poton eo ε φ e peed eo ε ω

385 64 4 Buhle DC Moto Dve BDC a b c d e f Fg. 4.7 Untable tatng oveent of the pendulu fo φ a -8º: a poton angle φ b angula veloct ω c angula acceleaton d poton eo ε φ e peed eo ε ω f u facto

386 4.5 Contol of BDC Moto Dve 65 a b c Fg. 4.7 Statng wng of the pendulu. Copaon of poton eo ε φ, fo: a φ a -5º b φ a -7º c φ a -8º

387 66 4 Buhle DC Moto Dve BDC a b c Fg Statng wng of the pendulu. Copaon of peed eo ε ω fo: a φ a -5º b φ a -7º c φ a -8º The peented eult of copute ulaton of the pendulu dve ndcate that uch cople egulaton ue egadng tacng contol of dve eveal can be ealzed wth the ad of a PID egulato; howeve, condeable ltaton ae poed on t. The oton of the pendulu ha to be lted to the degee that onl lghtl eceed the lnea appoaton of the odel and an adequate electon of the gan of the egulato ut be accounted fo. In addton, oton need to be uffcentl low and n th cae the peod of the oton of the pendulu T p > []. Fo the angle φ a > 7º t ve dffcult to obtan a table opeaton of the pendulu and the eo of the egulaton ae becong geate. Howeve, we can copletel conf the applcablt of BDC oto fo th tpe of dve Foal neazaton of BDC Moto Dve The tanfoed odel of the oto 4.5 fo vaable: can tae the fo: [,Ω,, ] T q θ 4.87 q d

388 4.5 Contol of BDC Moto Dve 67 q q d d d d d d d q q q q q l e p R u p R u J D T T Ω Ψ Ω Ω Ω Ω / θ 4.88 whee: Ψ Ψ B A M p T d q d q e - denote electoagnetc toque of the oto. A a eult of the ntoducton to 4.88 of new vaable : z θ, z Ω, z T e, z 4 d, we eceve: d q l v z v z J Dz T z z z z 4 / 4.89 whee: d q q d d d d d q d q q p R u v A B A v / Ω Ψ 4.9 On the ba of the elaton 4.9 and b the ue of 4.88 t poble to detene the pace fo uch lneazed contol b detenaton of the tanfoed uppl voltage u q, u d : q q d d d d d d q q d q d d q q q p R v u v A v B A p R u Ω Ψ Ω Ψ 4.9 The applcaton of th lneazed odel of BDC dve 4.89 can be the followng: the contol value ν q conttute the devatve of the gven electoagnetc toque T ed : q T ed v 4.9 Concuentl, the contol vaable ν d conttute the devatve of the gven tanent of the oto cuent d d d v 4.9 The degn of a tajecto of the oton of the dve can eal appl thee vaable obvoul n the adble aea of the contol that detened b voltage

389 68 4 Buhle DC Moto Dve BDC u d, u q. Unde the aupton that we a to educe eluctance toque of the oto that ntefee wth tanent and concuentl we need to pefo a tajecto of the oton, the coue of acton can be the followng. et u aue that appoatel d and ultaneoul on the ba of 4.67 unde tandad contol condton the elaton u d fulflled, t poble to etablh the followng equalt on the ba of 4.9: v d q q pω q pω Ted 4.94 B d d Ψ nce q B T ed Ψ, when d and the eluctance toque ng. In the ubequent tep, on the ba of 4.9 t poble to deve voltage u q : u u q d R q q Ω B Ψ B q Ψ v q ApΩ q d T B ed Ψ 4.95 Unde the aupton of a planned tajecto n the fo of ntal cuve fo the vaable P, v T, T, Ω ] 4.96 [ d q ed ed d afte the applcaton of 4.95 we wll obtan the functon fo deed contol u qd t. The contol te developed n th anne ha to enue the eecuton of the planned egulaton of u qd t and addtonall ha an adjutent te wth feedbac, whch ha been degned to enue the deed pecon of the contol depte the occuence of ntefeence. Moeove, one can note that uuall the value of A all n copaon to B Ψ, ν q nce t elated to the vaable te M of nductance and that could be pee fo degadng the econd te n the paenthee of the epeon Hence, n ot plfed fo, the eaned contol can be educed to: u q q Rqq Ω BΨ vq B 4.97 A poble aocated wth the fact that the calculated contol, fo ntance 4.95 and 4.97 have to be pefoed n an untanfoed te, whch ean that thee a necet to appl the nvee tanfoaton to obtan a te of phae voltage of the aatue. The popoed tanent fo u d ae elatvel ea n eecuton wth the ad of pule wdth facto u Anothe tpe of contol degned to atch the planned tajecto of oton wthout tanfoaton of the vaable wll be peented n the ecton that follow. Ψ

390 4.5 Contol of BDC Moto Dve Regulaton of BDC Moto wth Invee Dnac Th tpe of egulaton effectve wth egad to a te n whch cae t poble to plan the tajecto of oton, whch nvolve the detenaton of the poton, peed and acceleaton of the echancal vaable. In the eaned cae of a buhle DC oto th ean that the deed tanent θ d, Ω d, Ω d ad need to be fala. Subcpt d denote the deed value,.e. the one that a an deal tajecto of oton. The actual tajecto θ, Ω, Ω a uuall doe and wll dvege fo the deed one a a eult of the effect of a nube of facto whch ae degaded when tatng ntal aupton that tend to plf ethe the atheatcal odel of the dve, contant paaete of the uppl o othe facto that get n the wa of the poce of egulaton. The dffeence between the tanent fo the eo of the egulaton. It futhe appled fo povng the contol gnal n the addtonal coponent of the egulaton te,.e. the coecto. ε θ θ d θ ε Ω Ωd Ω 4.98 ε a a a d On the ba of the degned tajecto θ d, Ω d, a d and atheatcal odel t poble to deve the needed otatonal toque fo a dve Ted f θ d, Ωd, ad 4.99 that ubequentl pefoed b the egulato of the toque whch nvolve the calculaton and nput of a gnal, that ha the value coepondng to the pule Fg BDC dve contol b the nvee dnac ethod

391 7 4 Buhle DC Moto Dve BDC wdth facto u, nto the te of the electonc coutato. The daga of the contol te that apple the nvee dnac ethod peented n Fg Th ethod fnd a coon applcaton n a nube of dve te whoe ta nvolve tacng of a gven tajecto. The egulaton te detene the bac contol pocedue on the ba of the coputed equed tate whle the PID coecto offe an nput of an addtonal gnal thu nzng the contol eo. The beneft eultng fo the applcaton of th ethod nclude nceae of the egulaton peed, condeable educton of eo nce the pedonant pat of the contol gnal gven n advance and egadle of the eo. Othe advantage of th tpe of egulaton ae aocated wth effectve contol n non-lnea te nce n th cae the ltaton egadng PID egulaton ae no longe n foce. Th ethod fnd applcaton n a nube of ndutal dve ncludng the contol of anpulato and obot. Below a deontaton of the opeatng pncple n pactce wth efeence to an object fo Eaple 4. contol of a pendulu. Eaple 4. Th ta nvolve the contol of a lage, ave pendulu to atch a gven tajecto, whch condeabl dvege fo t natual oton. The detal ae dentcal a n Eaple 4.; bede, the llutaton n Fg. 4.6 eve fo the anal. The contol of the oton, whch fo an eaple of tacng contol can be baed on the ethod of nvee dnac. The tajectoe ae gven b the functon 4.8, 4.84 fo the vaable of the pendulu n oton and not b the actual oto oveent. The detenaton of the deed toque n accodance wth 4.8 offe the followng eult: T ed d g θ d p / g d d g J gl n D D d 4. On the ba of the copaon of the deed toque T ed and the toque actuall geneated b the oto T e t poble to detene the egulaton u, whch addtonall adjuted b the gnal fo PID coecto n the functon of the egulaton eo. Th pefoed n accodance wth the daga peented n Fg. 4.75, howeve, the te ulated. Th pactcall ean that the value θ, Ω, a a well a T e ae detened on the ba of calculaton on the atheatcal odel and ae not eaued. Slal a n the cae of the Eaple 4. the eanaton wll nvolve two veon of the deed tajecto: oton of the pendulu wth a contant apltude tatng fo the au deflecton of the pendulu and the othe veon n whch the tajecto of the pendulu begn wth tat-up fo a all angle of deflecton, equal to º, and achevng au dplaceent 8º, º at a te contant of T 5 []. Fg peent the eult of the ta n the ft veon and the oton of the pendulu wth an apltude of 8º and the followng Fg fo an apltude of º. Fg. 4.76a peent the gven toque T ed g calculated n accodance wth 4. fo the planned tajecto.

392 4.5 Contol of BDC Moto Dve 7 a b c d e Fg Electoechancal vaable fo nvee dnac contol of wngng ave pendulu, fo φ -8º: a pe-coputed deed toque T ed g b poton angle φ c angula peed, d angula acceleaton e poton eo f peed eo ε ω g u facto

393 7 4 Buhle DC Moto Dve BDC f g Fg contnued a b c Fg Electoechancal vaable fo nvee dnac contol of wngng ave pendulu, fo φ -º: a deed toque T ed g b poton angle φ c angula peed d angula acceleaton e poton eo ε φ f peed eo ε ω g oto voltage h electoagnetc toque T e g

394 4.5 Contol of BDC Moto Dve 7 d e f g h Fg contnued Fo the copaon of the tanent n Fg wth the one n Fg φ -7º and the tanent n Fg wth the one peented n Fg. 4.77

395 74 4 Buhle DC Moto Dve BDC φ -8º and φ -º lead to the concluon that thee a fundaental dffeence between the two. One can cleal conclude that the latte goup.e. eult that ae obtaned b applng egulaton wth nvee dnac, ha condeable advantage ove the one peented eale. The dffeence nvolve the tablzaton of the tanent to obtan lage value of apltude of the wng of the pendulu and alo ve elevant poveent of the accuac of the egulaton a a eult of applng nvee dnac. The educton of the potonal eo fo tablzed tanent ove 5 te, not to enton the poblt of tead opeaton fo lage dplaceent. A et of fgue that follow peent the eult of the egulaton fo the altenatve veon of the ta,.e. fo the tat-up of the pendulu. Soe aple of tanent ganed ae peented n Fg fo a ng apltude of the oton to each the value φ a 8º and φ a º n the table tate of the pendulu. a b c Fg Stang coue of the pendulu fo φ a 8º, wth nvee dnac contol: a pe-coputed deed toque T ed b poton angle φ, c angula peed d angula acceleaton e poton eo ε φ f peed eo ε ω g u facto

396 4.5 Contol of BDC Moto Dve 75 d e f g Fg contnued

397 76 4 Buhle DC Moto Dve BDC a b c d e Fg Copaon of tatng eult of the pendulu fo φ a 8º veu φ a º, wth nvee dnac contol: a deed toque T ed g fo φ a 8º b deed toque T ed g fo φ a º c poton eo ε φ fo φ a 8º d poton eo ε φ fo φ a º e peed eo ε ω fo φ a 8º f peed eo ε ω fo φ a º g electoagnetc toque fo φ a 8º h electoagnetc toque fo φ a º

398 4.5 Contol of BDC Moto Dve 77 f g h Fg contnued In the coenta of the eult that wee obtaned one can copae the eult peented n Fg. 4.7 wth the one n Fg nce both of the efe to the ae gven tajecto,.e. the tat-up of a pendulu to each φ a 8º fo vaou contol pocedue. Fo the cae of PIDD egulaton the tat-up dectl lead to non-tablt Fg. 4.7 and the eo of the angula poton nceae and even eceed alead. In contat, n the cae of the applcaton of contol ung nvee dnac Fg. 4.78, the eo ε φ dung the tat - up tend to tablze and eache the ange of ±º fo a nu value of the peed eo. The eult of the ubequent Fg ndcate that nceang the apltude to φ a º offe the poblt of effectve egulaton of the poton n the tacng oton whle the poton eo tablze n the ange of ±5º. The eaple peented hee ndcate that the ethod ve effectve depte the fact that t ha been appled wthout the pece electon of adequate egulaton paaete.

399 78 4 Buhle DC Moto Dve BDC Refeence [] Benal, F.F., Ceada, A.G., Faue, R.: Model-Baed o Mnzaton fo DC and AC Vecto-contolled Moto Includng Coe Satuaton. IEEE Tan. Ind. Appl. 6, [] Betn, F., Pnchon, D., Capolno, A.: A te-vang ldng uface fo obut poton contol of DC oto dve. IEEE Tan. Ind. Elect. 49, [] Banch, P., Bolognan, S.: Degn Technque fo Reducng the Coggng Toque n Suface-Mounted PM Moto. IEEE Tan. Ind. Appl. 8, [4] Boge, M., Wallace, A., Spee, R., et al.: Geneal pole nube odel of the buhle doubl-fed achne. IEEE Tan. Ind. Appl., [5] Bub, J.R., Matn, R.: Aal-flu peanent agnet a-coed geneato fo allcale wnd tubne. In: Poc. IEE, El. P. Appl., vol. 5, pp [6] Cnchlla, M., Analte, S., Bugo, J.C.: Contol of peanent-agnet geneato appled to vaable-peed wnd-eneg te connected to gd. IEEE Tan. En. Con., 5 6 [7] Daano, A., Gatto, G.., Maongu, I., et al.: Second-ode ldng-ode contol of DC dve. IEEE Tan. Ind. Elect. 5, [8] Doe,., Plla, P.: Coggng toque educton n peanent agnet achne. IEEE Tan. Ind. Appl. 4, [9] Flho, E.R., a, A.M., Aaujo, T.S.: Reducton Coggng Toque n Inteo Peanent Magnet Machne wthout Sewng. IEEE Tan. Mag. 4, [] Fulan, E.P.: Feld Anal and Optzaton on NdFeB Aal Feld Peanent Magnet Moto. IEEE Tan. Mag., [] Fuuhah, T., Sangwongwanch, S., Oua, S.: A poton-veloct enole contol fo buhle DC oto ung an adaptve ldng ode obeve. IEEE Tan. Ind. Elect. 9, [] Gea, J.F., Wng, M.: Peanent Magnet Moto Technolog: Degn and Applcaton, nd edn. Macel Dee Inc., New Yo [] Glna, T., Kulez, B.: Copaon of ated powe and effcenc of dc oto, nducton oto and buhle oto wth electonc coutato. In: Conf. Poc. 6th Int. Conf. on Elect. Mach. ICEM 4 4 [4] Ha, I., Kang, C.: Eplct chaactezaton of all feedbac-lneazng contolle fo a geneal tpe buhle DC oto. IEEE Tan. Aut. Cont. 9, [5] Had, E.S.: Degn of Sall Electcal Machne. John Wle & Son, Chchete 994 [6] Heat, N., e, M.C.: Coplete odel chaactezaton of buhle DC oto. IEEE Tan. Ind. Appl. 8, [7] Hendehot, J.R., Mlle, T.J.E.: Degn of Buhle Peanent Magnet Moto. Magna Phc and Claendon Pe, Ofod 994 [8] Holtz, J., Spngob,.: Identfcaton and copenaton of toque pple n hghpecon peanent oto dve. IEEE Tan. Ind. Elect. 4, [9] Hu, J., Dawon, D.M., Andeon, K.: Poton contol of a buhle DC oto wthout veloct eaueent. IEE Poc. El. P. Appl. 4, 99 [] Hung, J., Dng, Z.: Degn of cuent to educe toque pple n buhle DC oto. IEEE. Poc. El. P. Appl. 4, [] Hwang, S.M., eu, D.K.: Reducton of toque pple n buhle DC oto. IEEE Tan. Mag.,

400 Refeence 79 [] Ito, M., Kawabata, K., Taja, F., et al.: Coupled Magnetc Feld Anal wth Ccut and Kneatc Modelng of Buhle Moto. IEEE Tan. Mag., [] Jabba, M.A., Phu, H.N., u, Z.: Anal of the tatng poce of a dc dve pndle oto b te teppng fnte eleent ethod. IEEE Tan. Mag. 4, [4] Jang, G.H., K, M.G.: Optal Coutaton of a BDC Moto b Utlzng the Setc Tenal Voltage. IEEE Tan. Mag. 4, [5] Kabaha, T., Kawahaa, A., Goto, T.: Foce Calculaton Ung Magnetzng Cuent. IEEE Tan. Mag. 4, [6] Kawae, Y., Haah, Y., Yaaguch, T.: D fnte eleent anal of peanent agnet oto ected fo quae pule voltage ouce. IEEE Tan. Mag., [7] Kawae, Y., Yaaguch, T., Haah, Y.: Anal of coggng toque of peanent agnet oto b -d fnte eleent ethod. IEEE Tan. Mag., [8] Kenjo, T., Nagao, S.: Peanent Magnet and Buhle DC Moto. Claendon Pe, Ofod 985 [9] K, D.K., ee, K.W., Kwon, B.I.: Coutaton Toque Rpple Reducton n a Poton Senole Buhle DC Moto Dve. IEEE Tan. P. Elect., [] K, T.H., Ehan, M.: Senole contol of BDC oto fo nea-zeo to hgh peed. IEEE Tan. P. Elect. 9, [] ateb, R., Taoabet, N., Mebod-Taba, F.: Effect of agnet egentaton on the coggng toque n uface-ounted peanent agnet oto. IEEE Tan. Mag. 4, [] edee, D.H., Kot, N.: Modelng of non-lnea agnetc ateal ung a cople effectve eluctvt. IEEE Tan. Mag. 5, [] ee, W.J., Sul, S.K.: A New Statng Method of BDC Moto wthout Poton Seno. IEEE Tan. Ind. Appl. 4, [4] ow, T., Teng, K., ee, T., et al.: Stateg fo the ntantaneou toque contol of peanent agnet buhle DC dve. IEEE Poc. El. P. Appl. 7, [5] Łuanzn, M., Jageła, M., Wóbel, R.: Optzaton of peanent agnet hape fo nu coggng toque ung a genetc algoth. IEEE Tan. Mag. 4, 8 4 [6] Łuanzn, M., Wobel, R.: A tud on the nfluence of peanent agnet denon and tato coe tuctue on the toque of the dc-tpe buhle DC oto. El. Eng. 8, [7] Magnuen, F., endenann, H.: Paatc Effect n PM Machne wth Concentated Wndng. IEEE Tan. Ind. Appl. 4, 7 [8] Matu, N.: Senole PM buhle DC oto dve. IEEE Tan. Ind. Elect. 4, [9] Melote, H., Khoa, F.: Nonlnea adaptve contol of dect-dve buhle DC oto and applcaton to obotc anpulato. IEEE/ASME Tan. Mechat. 4, [4] Shao, J., Nolan, D., Tee, M., et al.: A novel cocontolle-baed enole buhle BDC oto dve fo autootve fuel pup. IEEE Tan. Appl. 9, [4] Wang, K., Atallah, K., Howe, D.: Optal toque contol of fault-toleant peanent agnet buhle achne. IEEE Tanacton on Mag. 9,

401 8 4 Buhle DC Moto Dve BDC [4] Zeoug, H., Boua, B., Sahaou, H.: Modelng and Anal of a Buhle DC Moto Dve. In: Conf. Poc. of th Int. Conf. El Mach. ICEM 998, pp [4] Zhu, Z.Q., Howe, D., Chan, C.C.: An poved analtcal odel fo pedctng the agnetc feld dtbuton n buhle peanent agnet oto. IEEE Tan. Mag. 8, 9 8 [44] Zhu, Z.Q., Howe, D., Mtchell, J.K.: Magnetc Feld Anal and Inductance of Buhle DC Machne wth Suface-Mounted Magnet and Non-Ovelappng Stato Wndng. IEEE Tan. Mag.,

402 Chapte 5 Swtched Reluctance Moto Dve Abtact. The chapte devoted to Swtched Reluctance Moto SRM dve. Ftl nonlnea agnetzng cuve of SRM ae peented and the potance fo oto opeaton dcued. The odel peented n the chapte tae nto account thee nonlnea chaactetc dependng on phae cuent and oto poton angle, but gnoe utual agnetzaton of phae. The agnetzaton cuve ae egaded n a pecfc and ognal wa a a poduct of nonlnea functon dependng on agnetc atuaton and oto poton angle. Th appoach ee to be ueful a t enable one to analze the nfluence of patcula contucton eleent on chaactetc of a oto. In conequence utual nductance of phae ae degaded, but the actual nfluence peented fo two tpcal SRM, and poved to be agnal. Seveal poble of SRM opeaton and contol ae peented baed on atheatcal odel and eult of copute ulaton. Aong othe the ae: detenng a pule equence fo tatng, dect tat up wth cuent ltaton, beang and a copehenve dcuon of geneato opeaton condton. The poble of egulaton paaete fttng alo peented, condeed fo a pont of vew of ganng pobl hgh effcenc and low toque pple level. A fa a contol of SRM concened thee ldng ode contol dcued a well a cuent contol and DTC wth an a to nze toque pulaton at vaou tate of opeaton. Bede, thee the poble of a contol wth and wthout poton/peed eno peented and tate obeve applcaton dcued that enable th nd of contol. 5. Intoducton Swtched eluctance oto SRM a an engneeng oluton to the degn of the electc oto n otatonal oton ealze one of the ealet dea of opeaton pncple fo the electc oto, whch ognate fo the ft half of the 9 th centu. It eplo the ple concept of an electoechancal te baed on the attacton of a feoagnetc eleent b an electoagnet. In ode to ae th dea vable fo a otatonal oto altenatel enegzed col ae tuated on Z tato teeth, whle Z alent oto teeth wthout wndng ae attacted b adequatel enegzed tato wndng. The at n the degn a well a the fundaental technologcal poble aocated wth adequate enegzng and de-enegzng tato wndng n a pope phae equence n ode to enue a ooth otaton of the oto whoe teeth ae attacted b geneated agnetc feld. Depte ple opeatng pncple the pactcal opeaton of SRM oto aocated wth a need

403 8 5 Swtched Reluctance Moto Dve to appl fat and effcent electonc wtche n ode to ealze the wtchng equence. Bede, eno of oto poton wth uffcent pecon ae needed to ecue coutaton of the cuent n the wndng. Fo thee eaon the developent and wde applcaton of the degn tated a late a 98 when adequate powe electonc coponent wee avalable to ealze uch wtchng functon [4,,]. In ode to enue a tatng toque and pobl ooth otatonal oton t necea that oto teeth ae not algned n epect to all coepondng tato teeth concuentl. Fo a oto wth a ngle pole pa p, th ultaneou algnent occu onl fo the oppote teeth,.e. oe geneall at an angula nteval of π/p. Fo th eaon the nube of tato and oto teeth uuall dffeent. The ot coon oluton appl the followng equence of teeth nube: Z /Z 6/4 and Z /Z 8/6. Self-evdentl, othe teeth nube equence encounteed, e.g. Z /Z 4/ fo a two-phae oto, o Z /Z /8 fo a oto wth fve phae wndng n the tato. The nube of phae fo oto wth a ngle pole pa equal to Z /. Th eult fo the fact that the oppote tato teeth ae enegzed ultaneoul wth an equal cuent dung the connecton of the col n ee. But paallel connecton of wndng on oppote tato teeth alo poble. A a eult, two col of oppote tato teeth fo a ngle phae wndng of the oto. Fo hghe pole pa nube, the nube of pa of tato and oto teeth, whoe ae ovelap, epectvel hghe and aount to p > whle the tato wndng belongng to the ae phae ae enegzed n the ae equence; hence, the nube of phae equal to Z /p. Th affect the epectvel hghe electoagnetc toque of the oto. Fg 5. peent a co-ecton of a Z /Z 6/4 SRM oto,.e. a eluctance thee-phae oto. Fg. 5. Co-ecton of a Z /Z 6/4 SRM oto, wth the ndcaton of oto poton angle θ, and wtch angle α on and α off

404 5. Intoducton 8 The applcaton of SRM oto ha opened new boundae n the pactce of the degn of electc oto dve. In connecton wth th, t would be of value to chaacteze t bac paaete and defne the cope of t applcaton. In te of the ated powe SRM oto ae anufactued n a ange fo ve all unt wth the output of eveal watt to enoou dve wth the powe of eveal hunded W and can even each MW level. The copaon of the eneg effcenc, tat-up toque and toque oveload between the tpcal SRM oto and the coepondng nducton oto lead to the concluon that the SRM offe oe advantage; howeve, the advantage of the SRM oto not gnfcant. A copaatve anal of a SRM oto wth a coepondng nducton oto accountng alo fo noe eon level baed on the eaple of [5]. One of the chaactetc of SRM oto nclude the poblt of ganng hgh otatonal peed, a hgh a, [ev/n] wthout pecal engneeng change n the oto. At the ae te, SMR oto whoe pecal degn ae the hghpeed can each a hgh a, [ev/n]. One ha to note that uch achne have a ooth clndcal oto whle the effect of vaable eluctance obtaned a a eult of the applcaton of ateal that va n te of agnetc peeablt along the ccufeence of the oto. Anothe pecal featue of SRM oto nvolve t echancal chaactetc Ω ft l whoe wavefo la to the ee wound DC oto. Th ean that dung an nceaed load th oto condeabl low down t otaton and when the load educed t acceleate. The negatve chaactetc of SRM oto nclude hgh toque pple and hghe noe level n copaon to e.g. nducton oto [5,8,6,7,8]. The counteeaue nclude the pope agnetc ccut contucton and applcaton of adequate contol te thu educng toque pule geneated b the oto. A fa a the applcaton of SRM oto concened, the ae la to the ue of nducton oto and ee wound DC oto. In patcula, the fnd applcaton n tacton dve and ca dve [,7] due to fleble echancal chaactetc, lage toque oveload capablt, ple contucton and hgh level of elablt. SRM oto can be uccefull appled n evooto and actuato. SRM achne can alo pla the ole of geneato; howeve, due to the pave ole of the oto the agnetc ectaton ha to occu a a eult of cuent pang though tato wndng, whch aocated wth pecfc equeent egadng contol and affect the effcenc of the achne a a geneato. Bblogaph devoted to th poble nueou [8,9,] alo n the contet of the contucton of wnd powe taton [4,6] and th aea the ubject n chapte In te of the nvetent a SRM oto dve cheape n copaon to an nducton oto dve and o convete a a eult of the uch oe plfed contucton of the tato wndng, lac of oto wndng and le coplcated te needed to uppl the achne. At peent, the lowe populat of the oto and alle offe on the pat of anufactue eult n the fact that SRM oto have not et been able to deontate all the advantage the have ove nducton oto.

405 84 5 Swtched Reluctance Moto Dve 5. Opeatng Pncple and Suppl Ste of SRM Moto The ot fundaental ue n the contol of SRM oto dve aocated wth adequate equental voltage feedng to and dconnectng fo oto wndng. Fg. 5. peent the co-ecton of a oto wth Z /Z 8/6 teeth and angula poton α on and α off of the oto fo whch epectve phae wndng ae uppled and ubequentl dconnected fo an etenal ouce. An ntutve undetandng of the opeatng pncple of uch a oto ugget that the potve toque T e >,.e. one n accodance wth decton of otaton of the oto encounteed when the oto tanfe fo the poton of hghet eluctance to the poton wth allet eluctance wth epect to the teeth of the tato whoe wndng uppled. The chaactetc poton of the oto ae denoted a unalgned poton and algned poton, the latte of whch efe to the ovelappng of the a of tato teeth and the one on oto teeth. The wndng of a gven phae hould be uppled wth a cuent of an adequate value n the ange of th otaton angle,.e. α on < θ α off. Th ean that the uppl hould be wtched on a defned b angle α on lghtl befoe the unalgned poton, whle the uppl dconnected a decbed b angle α off lghtl n advance n elaton to the ntant when a tooth eache an algned poton. The dffeence between the angle of the wtchng on of a powe uppl to the phae wndng and the angle when t dconnected nown a conducton angle, whch equal to: α α α 5. z on A one can conclude, the above entoned advance of wtchng on and off of the voltage fo a pecfc phae of the oto eult fo the dnac chaactetc of cuent nceae n the phae wndng followng an ntant the uppl wtched on and ubequent deca of the cuent afte the uppl dconnected. Moeove, t elatve to the oto peed, nductance of the wndng a well a the contol of convete wtche. Fo all otatonal peed the cuent n the wndng nceae elatvel fat wth egad to the ente conducton angle α z whle the value of the cuent contolled a a eult of ung PWM ethod. In contat, fo hgh otatonal peed the nceae afte wtchng on and deca of the cuent afte dconnectng the uppl coe elatvel lowl n the te peod whch detened b the conducton angle α z nce th angula ange of the oto oton coveed n a ve hot peod of te. Addtonall, elatvel lage bac EMF nduced n the wndng and the nceae of the cuent enfoced onl b the dffeence n voltage between the uppl voltage u fo a gven phae and the bac EMF e b. A one can ee, the contol of the wtch on angle α on and the conducton angle α z fo the bac ethod appled n the contol and adaptaton of chaactetc of SRM oto, bede the poblt of contollng uppl voltage uuall acheved wth the ad of PWM ethod. A a conequence of analzng the opeaton of SRM oto one can detene the theoetcal conducton angle fo a ngle phae ε - denoted a toe angle, whch eult fo the nube of phae and teeth of the oto unde the aupton of a epaate conducton of the wndng. In a tpcal SRM the dffeence n nube of a tato and oto teeth pe pole pa,.e. Z Z p. Hence the toe angle ε, a the allet off

406 5. Opeatng Pncple and Suppl Ste of SRM Moto 85 angula dtance between cloet tato and oto teeth, n a tate of algnent occung fo oe othe pa of teeth, π π p π ε p Z Z 5. p ZZ Z Fo a oto wth Z /Z 6/4 teeth th gve an angle ε 6 / 4, whle fo one wth Z /Z 8/6 teeth toe angle equal to ε 6 / The actual value of the conducton angle α z alwa geate than the toe angle ε a a eult of the pocee of cuent nceae and deca n the phae wndng and, hence, dung the opeaton of the SRM oto thee ae peod when o even phae wndng ae n conductng ON tate. Snce uuall the a to gan hgh value of electoagnetc toque, the conducton peod etended wthn the ange of tong attacton of the oto tooth b the electoagnet ade up b a pa of tato teeth o that the wtch off angle α off onl lghtl pecede the algned poton. Fo th eaon the poce of cuent deca n a gven phae acceleated a uch a poble to avod negatve toque value. Th occu afte the oto eache the poton detened b the angle α off a a eult of enegzng th wndng wth evee voltage u that upple the phae and, thu, caung the eneg etun to the ouce. Such a capablt ha to be ecued though the coutaton te of the phae of SRM oto. The bac te of the powe uppl and coutaton of a ngle phae wndng of SRM nvolve an aetc tanto /dode H bdge hown n Fg. 5.. Snce SRM a eluctance oto and the decton of the toque not elatve to Fg. 5. Co-ecton of a Z /Z 8/6 SRM oto, wth the ndcaton of oto poton angle θ, wtchng angle α on and α off, a well a conducton angle α z

407 86 5 Swtched Reluctance Moto Dve the decton of the cuent flow though the phae wndng, the coutato bdge doe not need to facltate cuent flow though the wndng n both decton and t uffcent to appl two powe tanto and two dode to enue eneg uppl and eneg etun to the ouce. Fg. 5. A tpcal coutaton H-bdge ccut fo wtchng cuent of a ngle phae of a SRM Ove the peod when the wndng n uppl tate fo the ouce both tanto T and T ae n the ON tate. Th occu fo the angle of the otaton n the ange fo α on to α off and when ultaneoul thee a nceae of the cuent n the ccle of the PWM egulaton of the voltage. Howeve, n the ecton of PWM ccle when phae cuent deceae, onl one tanto and one dode n H bdge ae n ON tate. Th could be tanto T and dode D of the bdge and n th cae we have to do wth a deca of the cuent n the ccut n whch thee eleent hot a phae wndng. In contat, afte the oto eache poton α off, both tanto ae wtched off and the cuent n the wndng cloed n the ccut foed b two dode D and D and the powe ouce. Th decton of the cuent flow detened b the dode eult n the etun of the eneg toed n the electoagnetc feld nto the ouce accopaned wth a apd deca of the cuent n the wndng. We can alo conde an opton of echancal eneg conveon ove th peod and t onl elatve to the actual ene of electoagnetc toque and t value. Howeve, fo the pupoe of ough eplanaton of the opeatng pncple of the oto we can aue that n the vcnt of the algned poton of the oto the electoagnetc toque all and, a a eult, the etun of the eneg cont onl n the etun of the eneg toed n the electoagnetc feld. The followng fgue Fg 5.4 Fg 5.6 contan an llutaton of the opeaton of the coutaton te fo a ngle phae wndng, fo the epectve low otatonal peed,.e. fo n 6 [ev/n], fo the d peed ange,.e. 6 [ev/n] Fg. 5.5 and fo hghe peed, e.g. fo n [ev/n] Fg The vaable decdng on the wtchng equence the value of the cuent n th

408 5. Opeatng Pncple and Suppl Ste of SRM Moto 87 wndng and t eponble fo voltage wtchng between the value of u and -.e. pefong o-called oft-choppng. Afte the copleton of the wtchng ccle n a gven phae,.e. afte the oto angle eceed poton α off, both tanto ae wtched off and the cuent n the wndng deca qucl and ecupeaton of eneg, due to the uppl wth -u voltage, tae place. Fg. 5.4 Cuent coutaton dung a ngle conducton ccle of a phae wndng wth a cuent ltaton, fo a low peed ange of SRM Fg. 5.5 Sla coutaton a peented n Fg 5.4, but fo a edu peed ange of SRM

409 88 5 Swtched Reluctance Moto Dve Fg. 5.6 Cuent coutaton a n Fg 5.4 and Fg 5.5, but fo a hgh peed ange. Cuent egulaton not poble n th cae The daga of the coutaton te peented n Fg. 5. enable one to contol the cuent n each of the phae epaatel. It poble to appl a oe econocal te,.e. one enablng the applcaton of a alle nube of powe electonc wtche n the degn of the coutaton te. One of uch eaple found n Fg. 5.7 fo a oto wth 4 phae o a geate even nube of phae. The contol of the wtchng equence of the phae baed on an aupton that dung the opeaton of the oto we don t have to do wth ultaneou conducton n oe than two phae wndng. Fg. 5.7 Banch-avng coutaton te fo 4 phae SRM achne In the peented eaple we eclude one banch of the convete pe fou banche n two coplete H bdge, whch ean that the nube of coponent deceae b 5%.

410 5. Opeatng Pncple and Suppl Ste of SRM Moto 89 Fg. 5.8 Swtch avng coutaton chee wth one geneal T wtch and ngle wtche fo each of phae wndng A te that goe even futhe n te of the econocal ue of electonc coponent peented n Fg Each of the phae appl a ngle tanto and thee one tanto T that coon to the all. Th te offe the poblt of educng coutaton loe; howeve, pactcall t doe not pet ecupeaton of eneg of the phae that beng n the wtchng off tate, becaue t eque the tanto T to be wtched off. In th cae we have to do wth the deca of the cuent n the cloed ccut of th phae wndng aco phae dode and tanto T. Condeable oppotunte n te of the poveent of the opeaton n the ange of hgh otatonal peed offeed b the te [47], n whch phae wndng contanng col tuated n the oppote tato teeth have avalable clap to uppl each col epaatel. It ean that the phae wndng can be effectvel dvded nto two equal pat. In th cae t poble to appl an altenatve, n-ee o paallel uppl of the two pat of the phae wndng and, a a eult, condeabl acceleate the nceae and deca of the cuent n the wndng and, addtonall, nceae the ange of the peed fo whch t poble to contol the cuent. Such te confguaton nvolvng dvon of a wndng and enablng ee powe uppl fo lowe peed and, concuentl, paallel connecton at hghe peed peented n Fg Fg. 5.9 Coutaton chee of a dvded SRM phae wndng fo changeable uppl confguaton of both pat

411 9 5 Swtched Reluctance Moto Dve Th te fo a ngle phae of the oto cont of two H bdge that uppl the halve of the wndng and an addtonal tanto T5 and dode D5 that connect thee bdge. Dung the peod when cuent nceae afte the oto gan α on poton and afte t obtan the poton α off aocated wth the tenaton of the phae uppl, tanto T5 wtched off and the two halve of the phae wndng b and b ae connected to full uppl voltage u. Th fact cobned wth two te lowe nductance eult n condeabl fate cuent nceae afte t fed a well a fate deca afte the phae dconnected a a eult of wtchng off all fve tanto T...T5. At the begnnng of a ccle, afte the equed value of the cuent eached, tanto T5 connectng the two bdge wtched on and the wndng halve ae connected n a ee to fo a ngle phae wndng. Wthn a ngle ccle of the uppl the wtchng of the pat phae to paallel and, ubequentl, to ee uppl can be pefoed eveal te thu nceang the ange n whch t poble to contol the cuent n the oto, a n Fg Fo the cae when afte the wtchng fo the paallel to ee connecton the cuent n the two halve ae not equal the cuent n the have to be balanced and the uplu of the cuent n one ecton of the wndng etun though one of the dode D o D4. A t wa ndcated b copute ulaton and the opeaton of uch epeental et-up [47], the applcaton of ee-paallel wtchng of the wndng halve n a two-phae oto ha led to an nceae of the otatonal peed b 8% unde ated loadng and the ated cuent ha not been eceeded. 5. Magnetzaton Chaactetc and Toque Poducng n SRM Moto In te of the contucton the capablt of a oto to tanfo eneg and poduce a toque detened b the agnetzaton chaactetc, whoe wavefo eult fo the engneeng detal and popete of the feoagnetc ateal appled. What eant hee the fal of agnetzaton chaactetc n the functon of the poton of oto tooth n elaton to tato tooth n the ange fo the unalgned poton to the copletel algned one. An eaple of agnetzaton chaactetc peented n Fg. 5. togethe wth a ngle ccle of convetng the eneg of the agnetc feld nto echancal eneg n otatonal oton of the oto. Fo the cheatc daga n Fg. 5. one can conclude about the elaton: the oe non-lnea agnetzaton chaactetc ae fo the algned poton the geate the co-eneg of the agnetc feld T that conveted nto echancal wo of the dve wthn a ccle of the powe uppl. At the ae te, le eneg of the agnetc feld T f etuned to the ouce dung the powe dode

412 5. Magnetzaton Chaactetc and Toque Poducng n SRM Moto 9 Fg. 5. A ngle ccle of eneg conveon n SRM. T co-eneg conveed fo agnetc to echancal fo, T f agnetc eneg ecupeated to a ouce n a dode conducton peod of the cuent deca conducton dut peod. On the ba of the ccle of eneg conveon peented n Fg. 5. t poble to ae the oto toque: echancal powe/ccle P ZT Te [ N] 5. otaton angle/ccle θ π The elaton peented hee 5. and the ntepetaton of the ccle of eneg conveon n Fg. 5. ae elevant fo a plfed cae when the patcula achne dut ccle ae epaate and thee n t a peod of a coon conducton. In the futhe pat of th chapte the eaple and llutaton wll be baed on two tpcal laout of SRM oto: - oto A wth ated value: Z /Z 8/6; P n 9 [W]; U n [V] - oto B, fo whch: Z /Z 6/4; P n 9 [W]; U n [V]. Detal of the two achne ae uazed n Table 5. n Chapte 5.4. Below a ua of the chaactetc of agnetzaton and eveal othe chaactetc aocated wth toque geneaton fo oto A.

413 9 5 Swtched Reluctance Moto Dve Fg. 5. A fal of agnetzng chaactetc fo oto A Table 5., fo 5 conecutve oto poton angle fo unalgned poton º to algned poton º Fg. 5. Satuated nductance cuve of oto A fo nceang tato phae cuent value 4,6,8,, [A] Fg. 5. Electoagnetc toque cuve, fo an ndvdual uppl of conecutve phae wndng of oto A wth [A] a a functon of a oto poton θ

414 5.4 Matheatcal Model of SRM Moto 9 Fg. 5.4 Electoagnetc toque cuve dung a ngle phae uppl wth nceang cuent 8,6,4, 6 [A] a a functon of a oto poton θ 5.4 Matheatcal Model of SRM Moto 5.4. Foundaton and Aupton of the Matheatcal Model The atheatcal odel fo the bac tool fo conductng ulaton of dnac coue and fndng chaactetc of the oto a well a a whole dve. It alo ndpenable n the eeach of dve contol te. The degee of coplcaton and the pecon of a gven atheatcal odel elatve to the pactcal applcaton of a odel,.e. the nube, tpe and elevance ange of the chaactetc that ae detened b t ue. The fo of the atheatcal odel cleal elatve to the plfng aupton adopted dung the tateent of the odel. It alo potant to note the ouce of nfoaton evng fo the pupoe of developng a odel,.e. whethe the ouce ognate n engneeng data o data ganed on the ba of eaung and tetng etng object, o data ha been ganed n oe othe wa. The odel baed on eaueent can ethe be detentc n natue o ognate on the ba of atfcal ntellgence ethod. The latte, howeve, have a lted cope of applcaton nce the eve n ode to eane object the nfoaton of whch often appoated [,,9,49]. Th boo baed on the developent and applcaton of detentc odel n the fo of odna dffeental equaton deved a agange equaton, n accodance wth the pocedue decbed n Chapte. The bac applcaton of uch a odel n the eeach of dnac and tatc chaactetc, electon of contol ethod and pocedue a well a n the ulaton of dve opeaton n deed ccutance. In the bblogaph n th ubject thee a condeable nube of atheatcal odel of th nd, whch, howeve, dffe n detal. Fo ntance, atheatcal odel of SRM, whch appl engneeng detal and lnage between the wndng ae peented n [], and the one wthout lnage ae found n [45]. In tun, anothe odel degned fo dnac calculaton and ulaton of wavefo peented n [9] unde the aupton of falat of functon of

415 94 5 Swtched Reluctance Moto Dve wndng nductance and the devatve. Bacall, the ajot of odel account fo non-lneat of agnetzaton chaactetc n SRM oto. Th o becaue t pla gnfcant ole wth epect to geneaton of the electoagnetc toque. Whle oe of the atheatcal odel found n bblogaph account fo the agnetc lnage between the adjacent phae wndng of the oto, a condeable nube of tude copletel degad the. Th fact of gnong the lnage between phae wndng n a SRM oto jutfed b two detal. Ftl, the value of the lnage all, nce n the ene of an uppe lt t doe not eceed % of the elf-nductance of the wndng and the tandad that the value fo 5% to 6% of th nductance. Secondl, the peod of the ultaneou conducton of adjacent phae of a oto lted and dung the te when n one of the phae the cuent gan t au value, n the othe one t alead decang. The peented atheatcal odel degned to conduct wft and ultple dnac calculaton of the wavefo of SRM; hence, t condeabl plfed. It account fo non-lneat of the chaactetc of agnetzaton of the oto, whch ndpenable, but degad agnetc lnage between the wndng. It lead to the plfcaton of the odel concuentl not caung elevant eo a a eult of the applcaton of the odel. Th confed b the chaactetc and wavefo ganed on the ba of eaueent and, n patcula, th petan to chaactetc ganed fo the cae of eveng the cuent n the adjacent phae wndng dung copaatve anal and eaueent of nductance n tandad SRM oto. The ncondeable dffeence between the chaactetc ganed on the ba of uch eaueent ndcate that the lnage between phae can be degaded wthout the deteoaton of the pecon of the eult collected b the applcaton of th odel. The atheatcal odel of the oto deved on the ba of agange equaton fo an electoechancal te whle peevng the notaton ued n Fg. 5.. The nube of the degee of feedo n the te equal to: 5.4 whle denote the nube of the electc degee of feedo of the te and the electc chage aocated wth phae wndng fo the epectve vaable q Q q Q q Q and the te devatve have the eanng of phae cuent of the oto: Q 5.5 Concuentl, the eanng degee of feedo eeved fo the echancal vaable of the te q θ and denote the angle of otaton of the oto. Unde the aupton of the lac of agnetc lnage between phae wndng, agange functon fo th te tae the fo:

416 5.4 Matheatcal Model of SRM Moto 95 ~ ~ θ d J Ψ θ, 5.6 The vtual wo of the te coepondng to the echange of eneg between the te and the uoundng envonent equal to: δa u R δq Tl Dθ δθ n whch the patcula te efe to: θ, M θ, 5.7 Ψ - agnetc flu aocated wth -th phae wndng, R - etance aocated wth cuent flow though -th phae wndng, accountng fo etance of electonc eleent and etance of the uppl ouce u - voltage appled to -th phae wndng, J - oent of neta aocated wth oto haft, D - vcou dapng coeffcent aocated wth dapng of the oton, T l - haft load toque Equaton of Moton fo the Moto The genealzed fo of equaton of oton n accodance wth agange odel.5 n the fo d dt q whee genealzed foce.7 j q j P j j δa Pj 5.8 δq the foce actng along t j-th genealzed coodnate calculated a the epectve patal devatve of vtual wo δa 5.7. Fo the electc ccut of the oto n accodance wth the aupton of degadng utual phae lnage, we obtan the total of equaton n the fo: d dt M, u R Afte calculaton of the te devatve, t : d M dt j θ 5.9 θ, M θ, M θ, u R Ω 5. θ e, θ, Ω

417 96 5 Swtched Reluctance Moto Dve whee Ω θ - the peed of the otatonal oton of the oto. The te e n equaton 5. denote bac EMF of otaton: e Ω M θ θ, 5. whch popotonal to the cuent n the wndng and the angula peed of the oto. Th te detene the lat between SRM achne and ee wound DC oto nce the phae cuent pla hee the ae ole a the ectaton cuent n the DC achne. The equaton 5., n the condeaton of the lac of lnage, can be aanged n the tandad fo: θ, d M u R e, θ, Ω / M, θ dt 5. θ The equaton of oton fo the echancal vaable the followng: d Tl D θ dt θ θ Aung a contant value J fo the neta, a a eult we obtan: The electoagnetc toque n th equaton equal to: Te θ e l 5. J θ T T D θ 5.4 M θ ~ θ, ~ ~ d Ung 5., the foula fo the toque can be etated a: Te Ω e ~,, Ω 5.4. Functon of Wndng Inductance 5.5 ~ θ d 5.6 A oe detaled nght nto the atheatcal odel, gven b equaton aocated wth the need of notng the functon of wndng nductance, whch pla a e ole n th odel. In the eaned odel the followng fo of the nductance functon ha been povded [4,4] M M ϑ λ ϑ, θ 5.7, Such nductance functon notaton n the fo of a poduct a ee coplcated; howeve, t ha a nube of advantage. The te Mϑ peent Fg 5.5 the unatuated nductance of the wndng n the fo of the functon ϑ, whch the oto angula poton educed to the ptch of the teeth τ π/z :

418 5.4 Matheatcal Model of SRM Moto 97 [ θ ε τ ] ϑ τ fac / 5.8 whee the te fac denote the factonal pat of aguent. Concuentl, the econd te n 5.7, peented n the gaphcal fo n Fg. 5.6, eponble fo the agnetc atuaton and ntoduce the adequate functonal elaton fo oto angula poton and cuent n the wndng. Such a peentaton of nductance coeffcent ae t poble to tud the effect of atuaton a well a engneeng change on nductance a well a etate the paaete of the oto on the ba of eaued chaactetc. The epectve patal devatve of the nductance 5.7 tae the fo: M θ, M ϑ λ ϑ, λ ϑ, M ϑ θ ϑ ϑ M θ, λ ϑ, M ϑ 5.9 Fg. 5.5 Unatuated nductance coeffcent of a phae wndng fo SRM oto A n a functon oto poton angle θ Fg. 5.6 Satuaton facto λθ, of a phae wndng nductance, fo poton angle value θ,.5,,, a a functon of wndng cuent

419 98 5 Swtched Reluctance Moto Dve Th wa of epeentng the nductance of the wndng and t devatve 5.9 affect the detaled fo of the epeon of electoagnetc toque 5.5, whch tae the followng fo T e M ϑ cla ϑ, ϑ M ϑ cdlaf ϑ, 5. Fg. 5.7 Phae wndng nductance coeffcent, fo 5,,5, 5 [A], a a oto poton functon Thee ae ued two ntegal wth epect to cuent n the wndng cla ϑ, cdlaf ϑ, ~ λ ϑ, ~ d λ ϑ, ~ ~ ~ d ϑ 5. The ft ntegal claϑ, peented n the gaphcal fo n Fg. 5.8 n the functon of the cuent fo vaou value of the ϑ angle. Evaluaton of th ntegal ndcate that, cla ϑ < 5. The ft te of the epeon 5., whch defne electoagnetc toque of the oto, end one of the clacal epeon denotng toque of eluctance ogn dnhed b the nfluence of atuaton, whch contat wth unatuated nductance functon, whch elatve onl to the angle of otaton. Th te denote the bac coponent of the toque. Concuentl, the othe te of the toque n the fo of an ntegal clafϑ, peented n Fg. 5.9 fo potve value of the achne angle of otaton. Snce t an odd functon and the

420 5.4 Matheatcal Model of SRM Moto 99 oto ode of opeaton tae place fo negatve value of angle ϑ, the econd te n the epeon 5. detene deceae of the bac toque coputed b ue of the ft te of 5.. Howeve, one can note that the change of the toque aocated wth non-lneat of agnetzaton chaactetc pla a oe potant ole fo lage value of the cuent. Fo the cae of oto A, whoe chaactetc ae peented hee, t deontate fo > 6 [A],.e. the value of the cuent that hghe than the ated cuent. Fg. 5.8 The ntegal claϑ, accodng to 5. a a functon of phae cuent, fo a poton angle ϑ,8.5,7,, top down Fg. 5.9 The ntegal clafϑ, accodng to 5. a a functon of phae cuent, fo a poton angle,8.5,7,, top down Th llutated b Fg. 5., whch peent both coponent of the toque wth epect to the angle of otaton,.e. the bac te elatve to the devatve of the nductance a coponent I and the othe te that elatve to the change of the atuaton a coponent II. Th decopoton of the electoechancal toque of a oto peented fo hgh value of the phae cuent n the oto [A] n Fg. 5.a and fo the cuent 5 [A] n Fg. 5.b,.e. fo all agnetc atuaton. A one can ee fo the llutaton, the le potant coponent II of the toque the alle wth the alle atuaton of the

421 4 5 Swtched Reluctance Moto Dve a b Fg. 5. Patcula coponent of SRM electoagnetc toque 5.. Coponent I bac toque ognated fo nductance devatve on a poton angle; coponent II eflectng a change n atuaton: a fo a hgh atuaton [A] b fo a low atuaton 5 [A] agnetc ccut. Fo the phae cuent [A] t equal to aound % of the value of the bac toque, whle fo 5 [A] t coepond onl to 5% of th toque. 5.5 Dnac Chaactetc of SRM Dve 5.5. Eepla Moto fo Sulaton and Tet Fo the pupoe of llutatng the chaactetc and dnac coue of SRM oto dve a electon of two low powe oto wa ade. The two oto ae found n the catalogue and wee the ubject of the eeach and eaueent n laboato condton [4]. A ua of the data found n Table 5..

422 5.5 Dnac Chaactetc of SRM Dve 4 Table 5. Selected data fo two tpcal SRM Paaete Moto A Moto B Z - tato teeth nube 8 6 Z - oto teeth nube 6 4 P - ated powe [W].9.9 n n - oto peed [ev/n] 6 n U - ated voltage [V] n n I - ated ouce cuent [A] 5.. T - ated toque value [N].6.4 n R - phae wndng etance [Ω].45.9 ph a - algned nductance [H]. 4 - unalgned nductance [H].46 7 n M ;,4 - utual nductance [% a M ;4 - utual nductance [% a ] ].9 - η - oto effcenc [%] α on; α off - wtchng angle [deg] 8 ; 5 ; Statng of SRM Dve A condeable poble encounteed dung the tat-up of SRM oto nce the toque elatve to the ntal poton of the oto, whch unfala, a a ule. Inceental encode ae ued n ode to contol the oto and detene an ntantaneou poton of the oto. Howeve, the latte do not povde nfoaton egadng the poton when t talled. Anothe poble wth the tat-up, epecall wth egad to a oto wth a all nube of teeth e.g. Z /Z 4/ but alo Z /Z 6/4 to a cetan etent, aocated wth the fact that the do not develop equed tat-up toque n eve poton of the oto fo both decton of otaton. Theefoe, fo the oto, whch fo engneeng pupoe ae ncapable of tatng n eve ntal poton of the oto, the tat-up poce ethe occu fo a all load o the oto potoned po to the tatng pocedue. Concuentl, the poble aocated wth detenng the ntal poton of the oto befoe tatng the oto can be olved b: - applcaton of abolute encode, whch povde a eadng of the ntal poton, - applcaton of eolve, whch a etenall enegzed nductance devce that ecue a pece eaueent of the poton n eve tuaton. The two oluton ae, howeve, athe epenve and the ae not appled n coonl ued dve. Soe othe poblte of pepang the dve to the pefo the tat-up nclude: - potonng a a eult of focng adequatel tong cuent flow though elected wndng,.e. t algnent po to the tat-up,

423 4 5 Swtched Reluctance Moto Dve - detenaton of the oto poton and electon of a tatng equence ung tet pule appled to the wndng po to t tatng, when the oto talled. The latte opton wll be eaned n oe detal below Stat-Up Contol fo Swtched Reluctance Moto b Pule Sequence The applcaton of tatng pule equence to detene the poton of the oto wll be peented wth egad to both oto fo Table 5. oto A and B. Fg. 5. contan a ua of the tet eult fo oto B Z /Z 6/4 wthn the angula ange of the otaton of the oto -6. a b Fg. 5. Geneal pctue of: a tet cuent pule and b epectve toque epone fo the Z /Z 6/4 oto, n the oto poton ange -6 On the ba of Fg. 5. one can conclude that the bet condton fo the tatup of th oto ae encounteed n the ange fo θ 6, whle the leat optu one ae fo the angle cloe to θ 5. Th o becaue n th poton the teeth on the oto ae dplaced n elaton to the tato teeth b ±5,±45,.e. the poton of the oto n whch the devatve of the oto wndng nductance aue a all value. In contat, fo θ the eanng teeth on the oto ae n the poton ± fo the ae of the phae ecept fo the pa n the algned poton. Th, n tun, offe optu condton fo the tat-up. The net fgue Fg. 5. peent a oe detaled epone of the oto to tetng pule fo elected oto angle.

424 5.5 Dnac Chaactetc of SRM Dve 4 a b Fg. 5. SRM oto B, epone to tet voltage pule n vaou oto poton: a θ b θ 9 c θ 5 d θ

425 44 5 Swtched Reluctance Moto Dve c d Fg. 5. contnued On the ba of Fg. 5. t poble to eane eveal tpcal tat-up tuaton fo elected oto poton angle. Fo the poton angle θ thee a all cuent pule n the phae and two lage one n phae and, whch coepond to the educed angle ±, gven b the elaton: ϑ θ ± ε ϑ τ / 5... Enegzng phae dung tat-up wll eult n the oton n the potve decton, whle enegzng phae n oton n the evee decton. Fg. 5.c llutate the tuaton occung fo θ 5. We have to do wth all cuent pule n the phae wndng and and a tong pule n phae. Th coepond to the epectve educed oto angle of ±5 and ±45. The lage cuent pule coepond to angle θ 45, whch occu between the a of phae and the a of the cloet tooth on the oto, but the eultng toque vtuall nonetent. The uppl of the phae lead to the otaton of the oto n the potve decton, whle of the phae n the negatve one. In thee cae the toque, howeve, thee te alle than fo the angle θ and, hence, the tat-up can be peded. The ot pobleatc tatng condton ae encounteed fo θ 9,

426 5.5 Dnac Chaactetc of SRM Dve 45.e. fo ϑ,, 9,-,9 - n Fg. 5.b, whle a phae o ae enegzed the negatve toque ve all a t eache aound % of the value fo angle θ. Th eult fo the fact that the poton angle 9 qute all n elaton to the half of the ptch of the tooth τ / 45, t too cloe to the algned poton. Concuentl, fo angle 9 the cuent alead condeabl lage nce nductance all but t too cloe to the lt of 45, when we have to do wth a change of the toque ene. So n conequence thee not a good opton fo the negatve decton tat-up. The fnal llutaton n Fg 5.d peent the effect of a cclc powe uppl the tuaton uch le fo θ, onl phae eplace phae, whle phae eplace phae and, n tun phae n the place of phae. The oe geneal concluon fo the tet that n ode to conduct the tatup one hould enegze the phae wndng fo whch thee a cuent pule wth the ean value o a wndng fo whch the pule epone a cloe to the ean value a poble. Dung enegzng of the wndng, whoe pule pecede the phae n whch thee wa a pule wth the hghet value, the tat-up toque potve and f we uppl the phae that follow the one wth the hghet pule epone, the tat-up toque negatve. One ha to bea n nd that all epone n the fo of cuent pule ae potve a we appl potve voltage pule, whch n the peented eaple ae equal to % of the ated voltage and ae.5 [] n duaton. The negatve value of a cuent n a phae wndng doe not change the ene of the eluctance toque ee The followng fgue llutate the tuaton aocated wth pule detenaton of the oto poton and ettng the tat-up equence of oto A Z /Z 8/6 n the ange of the otaton angle 5. a b Fg. 5. Repone fo voltage tet pule n SRM oto A 4 fo oto poton θ : a cuent pea b coepondng toque je

427 46 5 Swtched Reluctance Moto Dve Fo Fg. 5. t poble to conclude that the bet tat-up condton ae encounteed fo the oto angle θ 6 9. Howeve, fo 4 phae wndng n eve poton t poble to conduct the tat-up even unde condeable oto load. The followng Fg. 5.4 llutate the eult of the pule tat-up tet fo a nube of elected oto poton. a b Fg. 5.4 SRM epone oto A, 4 fo tet voltage pule appled to the conecutve wndng n eveal oto poton: a θ b θ 6 c θ d θ 5

428 5.5 Dnac Chaactetc of SRM Dve 47 c d Fg. 5.4 contnued

429 48 5 Swtched Reluctance Moto Dve Fo the oto wth the phae nube 4, the ubequent poton of the oto teeth n elaton to the ae of the tato teeth,.e. the ubequent value of the educed angle eult fo the elaton: ϑ θ ± ε ± ε ϑ τ / 5.4,,,4 Fg. 5.4a contan tet eult fo θ and fo th cae the ubequent value of the educed angle ae equal to ϑ,,,4,-5,-,5, epectvel, and the lage tat-up toque encounteed fo the uppl of phae and 4,.e. fo ϑ ±5. The hghet cuent pule n the wndng coepond to angle ϑ ± nce t coepond to the poton wth the nu nductance n the wndng. Fg. 5.4b peent the eult of th tet fo angle θ 6. The followng value of the educed angle: ϑ,,,4 6,-9,-4, coepond to th poton of the oto teeth n elaton to epectve tato teeth. The hghet cuent pule coepond to the angle,-4, fo phae and 4, togethe wth the hghet value of the tat-up toque. Anothe eaple found n Fg. 5.4c, fo the angle θ. The value of the educed angle of otaton fo the followng phae ae equal to: ϑ,,,4,-,-8,7, and the hghet cuent pule encounteed whle enegzng phae 4,.e. fo ϑ 4 7. Concuentl, the potve tat-up toque encounteed fo phae ϑ -8 whle the negatve one fo phae,.e. fo the angle ϑ. The tuaton peented n Fg. 5.4d egad angle θ 5,.e. the one that equal to the toe angle. Th well llutate cclc chaactetc of the tet pule nce, a one can ee, the ole of the ubequent phae ae hfted. The ule egadng the tat-up equence fo the cae of the oto wth 4 phae ae la to the one peented pevoul. In the eaned cae the tat-up toque wth a condeable value poduced dung enegzng the phae whoe pule epone the econd of the fou n te of t value along wth the one whoe pule epone ot cloe to the pevoul elected one. In the eaple n Fg. 5.4c thee ae, epectvel, phae and, and fo the cae n Fg. 5.4b phae and 4. The toque wth the negatve value poduced fo the uppl of the wndng wth the ucceve nube odulo n elaton to the phae wth the hghet epone of the cuent pule and the potve toque geneated dung enegzng of the phae wth the pecedng nube of the one wth the laget pule f t qualfe a the one that cloet to the pevoul elected one n te of the value. Th a ound cople; thu, t peented n the fo of an algoth n Fg Th algoth ha been pepaed fo the oto wth 4 phae, howeve, fo the oto wth thee phae the algoth tae the ae fo ecept z,,. Th algoth n- fo the ltaton of the et of phae nube to { } volve detenaton of the phae nube {,,,4 } z that to be enegzed n the ft ode dung oto tat-up dependng on the elected decton of the otatonal peed: n > T e > z o n < T e < z - and on the ba of the

430 5.5 Dnac Chaactetc of SRM Dve 49 Fg. 5.5 The outlne of the algoth fo a tatng equence of SRM baed on tet pule befoe tatng pule tet fo all 4 phae: I { I, I, I I }, 4. One hould note that n the algoth the pule c detened a the one that cloet n te of the apltude to the econd hghet called b could be the one wth the laget value,.e. the one decbed a a.

431 4 5 Swtched Reluctance Moto Dve Cuent Deltaton dung Dect Moto Statng The dect tat-up of SRM oto fo the cae of conductng tatng equence fo an adequate phae occu effectvel. Concuentl, thee a need of ltng the tat-up cuent n the ntal peod of tatng equence. In cae the dve ha eno of the phae cuent th ltaton can be eal pleented a a eult of ntoducton of uppe and lowe boundae of cuent fluctuaton. Soft-choppng ued wthn th ange of egulaton and nvolve de-enegzng of one of the tanto of the H bdge afte eceedng the uppe cuent lt followed b a natual deca of the cuent untl t eache the lowe bounda. Subequentl, the pevoul de-enegzed tanto wtche on, the powe etoed and the cuent n the phae wndng e agan. Th wa of egulaton accoplhe t ole fo a all nube of the wtchng equence n a pule, whch otl elatve to the boundae of cuent change poed b thee ltaton. Fg. 5.6 peent the ntal peod of the tat-up fo oto A, and Fg. 5.8 fo oto B, fo the cae of poed tat-up cuent ltaton and ated loadng. Fg. 5.7 peent the coplete tatng ange of oto A. On th eaple one can ee ee ectaton effect of the SRM achne. It deontate n a low long latng peed nceae n the fnal pat of the tatng coue. Th aocated wth the gadual deceae of the cuent and de-ectaton of the oto, thu caung the nceang peed. a b Fg. 5.6 Intal pat of tatng cuent of SRM oto A, wth cuent ltaton wthn the ange of [A] 9 [A]: a phae cuent b electoagnetc toque

432 5.5 Dnac Chaactetc of SRM Dve 4 a b c Fg. 5.7 The whole tatng coue of SRM oto A wth a cuent ltaton and a nonal load: a phae cuent b electoagnetc toque c otatonal peed 5.5. Bang and Geneatng b SRM Geneato ege of opeaton of SRM achne not povded fo ethe n the engneeng tuctue of the achne telf no due to the tuctue of the econtolled H bdge fo whch t contolled Fg. 5.. The oto of the eluctance SRM oto not enegzed and the ectaton flu of the achne coe fo the cuent n the tato wndng. Hence, thee a lac of epaatel egulated ectaton cuent that tpcal fo geneato o peanent agnet that offe an ectaton flu of the achne pactcall egadle of the achne load.

433 4 5 Swtched Reluctance Moto Dve a b Fg. 5.8 Statng of SRM oto B wth cuent ltaton n the ange of 5 [A] [A]: a phae cuent b electoagnetc toque a b Fg. 5.9 Ectaton peod and geneatng peod of SRG achne dung one wtchng ccle: a cuent ccle b cuent flow n a H bdge fo one phae wngng; e - bang cuent flow, g - geneatng cuent flow

434 5.5 Dnac Chaactetc of SRM Dve 4 a b Fg. 5. Stead-tate chaactetc fo a geneatng ode of SRM opeaton oto A a a functon of oto peed, fo u U n, α on 6, α off -,- : a phae and ouce cuent b toque and oveall effcenc The tpcal tanto-dode H bdge that enegze the achne wndng fo the DC ouce ae t poble to contol the achne cuent onl fo the cae of the oto ege of opeaton. The etun of powe fo phae wndng nto the ouce can onl occu afte wtchng off both tanto and tae place n an uncontolled anne untl the agnetc feld aocated wth the wndng deca. Moeove, the geneaton and bang of SRM achne not tead, whch eult fo the cuve of the tatc chaactetc peented n Fg. 5. and confed b the wavefo peentng the untead opeaton n the vcnt of the equlbu pont n the te wthout feedbac, a hown n Fg. 5.. Howeve, fo adequate contol ung angle α on, α off and n the te wth feedbac fo the egulaton of the output powe the SRM achne capable of pefong the dute of the geneato [9,8,4,6,4,4]. In the geneaton ege of opeaton each pule of the achne cuent cont of two pat. In the ft pat, fo two tanto n the ON tate thee an ectaton of the SR geneato, oete denoted a SRG, nce the achne daw cuent fo the ouce and t opeate a a bae o a oto dependng on the ene of the toque poduced n th peod. The tanfoaton of the echancal powe n th peod ncondeable nce the angle α on fo whch the achne opeate a a geneato

435 44 5 Swtched Reluctance Moto Dve pecede the algned poton b a few degee and the ectaton occu fo the all value of electoagnetc toque whch actuall change the ene fo the potve to the negatve one. Afte the ectaton peod the two tanto ae wtched off fo the angle of the oto of α off, whch happen eveal degee afte the algned poton of the oto and tato teeth eached. Followng we have to do wth the geneaton ege of opeaton untl the deca of the cuent n the a Fg. 5. Untable opeaton of SR achne a a geneato: EMF, phae cuent, toque and oto peed coue: a unde an equlbu pont of balance b above an equlbu pont of balance

436 5.5 Dnac Chaactetc of SRM Dve 45 b Fg. 5. contnued wndng, whch n th cae cloe though two dode of the H bdge and thu eneg etun nto the ouce. In th ange of opeaton the eneg conveon pefoed at the epene of echancal eneg, nce the electoagnetc toque negatve and to oe etent due to agnetc feld eneg aocated wth wndng cuent. The llutaton of the ectaton and ubequentl, geneato ege opeaton of SRM peented n Fg Fo adequatel elected contol angle α on, α off the ean value of the geneated cuent condeabl hghe than the ean value of ectaton cuent and the

437 46 5 Swtched Reluctance Moto Dve a b c d Fg. 5. Stead tate chaactetc of SRG A fo dffeent ectaton level ealzed b u facto: a otatonal peed b phae and ouce cuent a a toque functon c toque and effcenc a u functon d hape of cuent pule achne opeate a a geneato wth a decent eneg effcenc Fg. 5., whch, howeve, lowe than fo t oto opeaton. Fo the chaactetc n Fg. 5. one can ee that the electoagnetc toque and phae cuent deceae along wth an nceae of the oto peed, whch foecat an untead chaactetc of the achne opeaton wthn th ange. Th confed b t tanent, whch peent the behavo of the geneato afte the balance dtubed Fg. 5..

438 5.5 Dnac Chaactetc of SRM Dve 47 a b c Fg. 5. nea tanton fo oto to geneato ege of wo of SRM A, afte lnea change of: α on 5 and α off -, n 85 [ev/n]: a phae cuent b electoagnetc toque T e c patal toque conttutng total T e toque The untable equlbu n the condton peented n Fg. 5. occu fo the peed n [ev/n] and the load T l -.[N]. The geneaton ege can be tablzed a a eult of ncludng adequate feedbac elatve to the peed and actng upon the pule wdth odulaton coeffcent u defned fo the PWM contol, whch n cae of contant ouce voltage U egulate a level of an ectaton. Th o becaue the PWM voltage contol b the u facto actng dung tanto opeaton of the H bdge and th wa t effect the ectaton level. A a conequence, u facto egulaton a an tool to contol ndectl electoagnetc toque dung the geneato opeaton wth a contant ouce voltage. The chaactetc that ndcate th poblt fo oto A ae peented n Fg. 5.. A ooth tanfe fo the oto to geneato ege of opeaton llutated n Fg. 5.. In th cae we have to do wth lnea change of the contol angle α on 5 and α off - a well a a change of the toque on the

439 48 5 Swtched Reluctance Moto Dve a b c d Fg. 5.4 Fat change fo oto to geneato opeaton fo: α on 5 6, α off -, T l. -. [N]: a phae cuent b EMF n a wndng c electoagnetc toque d otatonal peed achne haft fo the load of T l. [N] to the toque dvng the geneato T l -.8 [N]. The eneg effcenc fo th tate of the geneato ege equal to η g 68%, n contat to η 8.5% fo the oto ege pecedng the change of the opeatng ege. The tanfe fo the oto to geneato ege can alo occu fat and doe not poe an poble to the tablt of the dve. An eaple of uch fat change llutated n Fg In the peented eaple we have to do wth a popt

440 5.5 Dnac Chaactetc of SRM Dve 49 a b Fg. 5.5 Detal hape of a phae cuent and nduced EMF n a phae wndng fo: a oto b geneato ode of SRM opeaton, contol paaete le n Fg 5.4 wtchng of the contol: α on 5 6 and α off - and fo the load toque of T l. [N] -. [N]. Fg. 5.5 peent the detaled te wavefo of the phae cuent and nduced EMF n the SRM achne fo oto and geneato ege n the condton defned n the llutaton n Fg One can note the change n the hape of the wavefo fo phae cuent and EMF, whch end of a evee otaton of the achne fo the oto ege. The effectvene of opeaton and eneg effcenc of the SRM achne dung geneato ege ae condeabl elatve to the contol angle α on, α off and u facto contollng the ectaton level. Unde the aupton that eneg uppled n an optu wa dung ectaton ot of t can be etuned nto the ouce dung geneato ege and, thu, the effcenc qute hgh. Unde the aupton of a contant otatonal peed and contant ouce voltage U of the dve th effcenc can be epeed b the elaton: P Pec Pgen U Iec, av I gen, av η g 5.5 P P, av Tl Ω, av whee the patcula bol denote: P - echancal powe output of the dve P - electc powe ued b the dve P ec - electc powe dawn fo the ouce dung ectaton pat of the ccle P gen - electc powe etuned nto the ouce dung geneaton pat of the ccle T l - load toque I ec,av - ean value of ouce cuent dung ectaton pat of a ccle I gen,av - ean value of ouce cuent dung geneaton pat of a ccle Ω,av - ean otatonal peed aveagng the effect of pulaton.

441 4 5 Swtched Reluctance Moto Dve The effcenc defned n th wa, whch potve fo geneato ege, ha a negatve nueato nce the ean cuent n the geneato ccle condeabl hghe than the ean ectaton cuent a well a a negatve denonato of the epeon 5.5 nce the achne actuall dven fo outde, whch ean that the load toque alo negatve. Eaple of the chaactetc of the effcenc of the achne A ae peented n Fg. 5. and Fg. 5.c the llutate cleal the effect of the contol angle and u facto epectvel on the wavefo. The effect of the contol angle on the opeaton and effcenc of SRG geneato ae peented n a nube of bblogaph te [4,4] teng the coplet of the ue. Th coplet due to the fact that dung the geneato ege the ae pont on the echancal chaactetc of SR geneato.e. oto peed and toque can be obtaned fo vaou value of contol vaable α on, α off and u howeve, the cuent and effcenc dffe condeabl. Moeove, oe bblogaph te n th ubject dcu the ectaton and elf-ectaton of SR geneato [4,6] a well a geneaton fo a hgh and low otatonal peed of the achne. The latte eult fo the potental applcaton of SRG n wnd powe taton n the engneeng odel nvolvng echancal gea and one wthout t. 5.6 Chaactetc of SRM Machne 5.6. Contol Sgnal and Tpcal Stead-State Chaactetc Although SRM oto a eluctance achne and ha a copletel pave oto, thee ae thee contol quantte decdng about the chaactetc of the opeaton of the dve. The ae: uppl voltage u, ntal angle of enegzng the phae wndng wtch on angle α on and ntal angle of de-enegzng powe uppl fo the wndng wtch off angle α off. Altenatve to the wtch off angle, the et of the contol gnal can appl the conducton angle α z 5.. The above contol angle α on, α off Fg. 5., Fg. 5. ae eant to be the angle that pecede the algned poton of tato and oto teeth. Fo a tpcal uppl of the SRM dve the followng condton fulflled: the conducton angle α z > ε, whee ε the toe angle 5.. Uuall n ode to appl the poblte of dvng the oto the contol pefoed n the ange: ε α z < ε 5.6 At the ae te, wtchng on occu fo αon α z αoff.5ε. ε 5.7 whch aed at obtanng a lage value of the cuent n the adequate ange of the educed otaton angle ϑ 5.8. The hghe value of α on concen SR oto wth hghe nube of phae wndng. Snce the nube of tead-tate chaactetc that can be peented fo thee contol vaable lage, the peentaton hee

442 5.6 Chaactetc of SRM Machne 4 wll focu onl on elected chaactetc calculated fo oto B Z /Z 6/4. The ft goup of chaactetc Fg. 5.6 pefoed n the functon of the load toque T l fo thee value of the uppl voltage: U.,,75,.5 U n. a b c Fg. 5.6 Stead-tate chaactetc of 6/4 SRM dve fo U.,.75,.5 U n and α on 5, α z 6, a a functon of a load toque: a peed cuve b phae and ouce cuent c effcenc of the dve The peented chaactetc n the functon of the angle α on ndcate that thee an optu electon of the advance angle fo enegzng wndng whch occu fo the eaned achne oewhee n the ange fo α on 5 5,.e. fo α on.7ε n th cae. Concuentl, the allet pulaton of the toque ae encounteed fo α on 4 46,.e. fo α on.4ε. Hence, t eult that the late begnnng of the conducton poce, e.g. fo α on.4ε, lead to an unnteupted cuent flow n the wndng and educe pulaton. Howeve, the dadvantage theeof aocated wth negatve toque coponent ognated fo cuent flow n patcula phae wndng of the oto, whch ae anfeted afte the algned poton eceeded. The peented chaactetc do not llutate the effect of the conducton angle α z on the chaactetc, whch fo the cae n Fg. 5.6, Fg. 5.7 equal to 6,.e. α z.ε, whch coepond to a tandad value.

443 4 5 Swtched Reluctance Moto Dve a b c d Fg. 5.7 Stead-tate chaactetc of 6/4 SRM dve fo, α z 6, U U n and T l.5,.,.5 T n a a functon of a wtch angle α on : a peed cuve b phae and ouce cuent c effcenc of the dve d pple toque level T p /T av 5.6. Effcenc and Toque Rpple evel of SRM Havng thee contol paaete α on,, α off, u t poble to detene the ae pont of opeaton of the dve along the echancal chaactetc of the oto n,t l fo a ee of vaou contol paaete. Thu, at the ae opeatng pont t poble to tanfo eneg fo vaou effcence and fo vaou level of toque pple - T p. Such eeach ha been peented n [4,4], and the geneal concluon that the qua-optal electon of contol paaete of SRM oto techncall poble. The contol value dung qua-optal opeaton va along wth otatonal peed. Fo all peed the contol occu a a eult of changng u, fo a contant value of α on,, α z. In the nteedate ange of the otatonal peed the value of the voltage u ean contant, whle the wtch on angle α on and the conducton angle α z nceae. Wthn the ange of the hgh peed onl the angle α on nceae, whle the eanng paaete of contol ean contant. The cuve fo contol vaable fo a qua-optal contol of SRM oto ae peented n Fg. 5.8.

444 5.6 Chaactetc of SRM Machne 4 Fg. 5.8 Change of contol vaable fo a au effcenc of SRM dve n epect to change of otatonal peed The peevaton of the contol n accodance wth the pncple peented n Fg. 5.8 ae t poble to ecue the opeaton of the dve wth au effcenc. Concuentl, the pple coponent of the toque at a nu fo a gven load toque of a oto fo the thehold peed n a well. The thehold value n, n, fo whch thee hould be a change n the contol anne ae elatve to the oto load toque T l and the ae deved on the ba of the nu cuent. Fo the cae of the atheatcal odel of SRM oto 5. dcued hee, t poble to peent electoagnetc toque poduced b the oto a the u of the coponent toque eultng fo the flow of patcula phae cuent. Th o becaue n accodance wth the aupton ade dung the developent of th odel, the phae wndng ae not agnetcall lned and the cuent cong fo each of the phae geneate a agnetc flu fo a ngle pa of tato teeth egadle of the cuent n the eanng phae wndng. In patcula, th offe a poblt of gaphcal peentaton of how the contol angle α on, α off affect the hto of electoagnetc toque and what value of the contol angle ae benefcal fo the educton of toque pple. Fg. 5.9 peent the developent of toque n oto B Z /Z 6/4 fo the ated value of the uppl and load. Subequentl, Fg. 5.4 peent the toque of the oto fo the contol angle elected n a anne n whch the pulaton of the toque ae the allet. Th occu fo α on 4 and α off 9,.e. fo the conducton of the phae aco α z 4. In the ft of the cae, the pule coponent of the toque equal to 7% of the ean value of the toque, and n the latte cae peented n Fg. 5.4 fo T p.7 [N], whch coepond to aound 7% of the ean toque. Th done at the epene of the educton o the te effcenc b 4 pe cent pont. If the contol angle wee to be elected at the value α on 44 and α off, the pulaton level would be onl equal to %, and the effcenc lo would be two te lowe, whch ean pe cent pont Fg A a eult, a copoe wth egad to the electon of contol paaete poble wth a condeable beneft to the qualt of the dve opeaton, whch geneall chaactezed b the qua-optal cuve of the contol paaete n Fg. 5.8.

445 44 5 Swtched Reluctance Moto Dve The llutaton of the cuve n Fg Fg 5.4 peent wh fo a oto wth thee phae thee a cetan lo of eneg effcenc dung ltng pulaton. Th o becaue the flat wavefo of toque accodng to te and educton of the pulaton occu fo the contol angle dplaced n the decton of the algned poton of tato and oto teeth n copaon to the opeaton n the ated tate. In th cae we have to do wth two phenoena educng the effcenc: lage negatve toque coponent fo eceedng the algned poton wth the cuent n the gven wndng and deceae of the otatonal peed of the oto, whch eult n the alle powe output of the achne. a b Fg. 5.9 Electoagnetc toque T e of -phae SRM a a u of patal toque ognated fo ngle phae cuent fo α on 5, α off 5 : a toque-pple cuve wth toque pple level T p /T av 7% b phae cuent a b Fg. 5.4 Electoagnetc toque T e of -phae SRM a a u of patal toque fo α on 4, α off 9 : a toque-pple cuve wth toque pple level T p /T av 7% b phae cuent

446 5.6 Chaactetc of SRM Machne 45 a b Fg. 5.4 Electoagnetc toque T e of -phae SRM a a u of patal toque fo α on 44, α off, a a copoe between effcenc and toque pple level: a toquepple cuve wth T p /T av % b phae cuent a b Fg. 5.4 Electoagnetc toque and cuent fo the Z /Z 8/6 SRM - oto A, fo the nonal tate: α on 8, α off : a toque b phae cuent The data gven above and Fg concen oto B wheea fo oto A Z /Z 8/6 the effect of the paaete on the level of pulaton elatvel alle. Fo th oto the level of pulaton cloe to the nu T p / T av %...4% wthn a wde ange of the contol angle and t dffcult to obtan a level of pulaton below %. It poble to eceed th bounda; howeve, th can onl occu fo the load of the oto that ae geate than the ated load

447 46 5 Swtched Reluctance Moto Dve a b Fg. 5.4 Electoagnetc toque and cuent fo the Z /Z 8/6 SRM - oto A, fo nal toque-pple level T p /T av %, whle α on 5, α off : a toque b phae cuent a b Fg Electoagnetc toque and cuent fo the Z /Z 8/6 SRM - oto A, fo a hgh load of the oto T l 7.8 [N] and α on 8, α off : a toque, whle toque pple level T p /T av 6%, n 95 [ev/n] b phae cuent Fg Concuentl, fo the contol coepondng to the ated tate the wavefo ae peented n Fg. 5.4, fo α on 8, α off. In th cae, the epectve value of the effcenc and toque pple level ae the followng: η 8.5%, T p /T av 9%. The lowet level of toque pple, whch equal to T p /T av %

448 5.6 Chaactetc of SRM Machne 47 tae place fo the contol: α on 5, α off, and the effcenc even hghe, a t equal to η 84.8%. The two latet tate dffe n te of the value of the powe output of the oto due to the defntel dffeent value of otatonal peed, whch ae epectvel equal to [ev/n] and 78 [ev/n]. A one can ee, n SRM oto thee a poblt of educng the level of pulaton a a eult of adequate electon of the contol angle α on, α off. Th, howeve, poble wthn a lted ange and a lead to a lght deceae of the effcenc, n patcula fo low otatonal peed [,4,4] Shape of Cuent Wave of SRS The hape of phae cuent eflect the ode of the contol of SRM oto and aue pecfc wavefo dependng on the contol angle and otatonal peed of the oto. It alo poble to dtnguh the geneato opeaton of the achne fo the oto ege on the ba of t wavefo Fg. 5.9a. Fg peent the hape of the phae cuent of oto A whch dffe n te of the a b c Fg Phae cuent of the A SRM fo α on 5, T l 5. [N] and dffeent α off value: a α off 5, b α off c α off 5

449 48 5 Swtched Reluctance Moto Dve a b Fg Phae cuent of the A SRM fo a contnuou conducton eultng fo late wtch-off angle α off 5, α on 4, T l. [N], n 9 [ev/n]: a ngle phae cuent b all 4 phae cuent wtch off angle α off,.e. fo a deceang conducton angle α z, equal to α z, 5,, epectvel. One can cleal note the ntant when the uppl voltage dconnected and the tanfe of the wndng to the peod n whch t etun the eneg to the ouce though the dode of the bdge. Fo the eaple peented n Fg. 5.45a fo α z, dung the etun of eneg one can eal notce a bulge on the wavefo whch aocated wth the deceang nductance of the wndng afte the oto tooth eceed the algned poton. Fg peent the contnuou conducton of the phae cuent of the SRM oto, whch occu fo the late de-enegzng of phae, lage load and hgh otatonal peed of the oto. In thee condton the oto opeate coectl and deontate a low level of pulaton; howeve, the eneg effcenc of the oto deceae condeabl due to the lage powe loe n the wndng. Th, n tun, bng a hazad of oto falue due to oveheatng. Fg Phae cuent hape b PWM contolled voltage oto A, fo u.7, α on, α off 8, T l. [N], n [ev/n]

450 5.6 Chaactetc of SRM Machne 49 a b Fg Phae cuent hape n cae of hgh oto peed n 54 [ev/n], fo oto B, whle: α on 5, α off 5, T l.5 [N]: a ngle phae cuent b all -phae cuent a b c Fg An nfluence of a load on cuent hape oto B, α on 5, α off 5 : a T l.5 [N], n 54 [ev/n] b T l 4.5 [N] n 7 [ev/n] c T l 7.5 [N], n [ev/n]

451 4 5 Swtched Reluctance Moto Dve a b Fg. 5.5 ate wtch on and off of SRM oto B, α on 4, α off 4, T l.[n]: a ngle phae cuent b all phae cuent a b Fg. 5.5 Cuent hape fo a geneato opeaton of SR oto A: α on, α off -8 : a ngle phae cuent b all 4 phae cuent

452 5.7 Contol of SRM Dve 4 In Fg 5.49 one can ee that the load doe not have a gnfcant effect on the wavefo of the cuent, whch la fo both all T l.5 [N] a well a lage T l 7.5 [N] load toque of the oto. Howeve, the otatonal peed of the dve dffe gnfcantl n thee two cae, whch can be alo concluded fo tatc chaactetc Fg Contol of SRM Dve 5.7. Vaable Stuctue Sldng Mode Contol of SRM Sldng ode contol wdel ued n a up-to-date electc dve. Th eult fo the fact that wtchng of contant uppl voltage to electc oto wndng foall conttute of one of the poble contol ode wth vaable tuctue. Th, n tun, wa ade poble a a eult of the ajo developent n the feld of powe electonc. Sldng ode contol nvolve the contol of the oton along a ldng uface σq,t [5,44]. Th goup of ethod nvolve contol applng PWM technque Pule Wdth Modulaton, a well a othe oe advanced contol ethod, ncludng DTC Dect Toque Contol, wth egad to nducton oto dve fo ntance, a decbed n Chapte.4. The natual tablt of the te togethe wth hgh fequenc of the wtchng ae t poble n ajot of cae fo the tajecto of the dve oton to follow n a cloe vcnt of a ldng edge even wthout applcaton of pecal effot and pece electon of the paaete. Soete engnee nvolved n t pactcal applcaton do not ee t necea to bothe theelve wth povng tablt of a dve. In uch cae the epeence eultng fo laboato tet and a naow ange of equeent egadng the contol of the dve ae t poble to degn contol on the ba of one epeence. Howeve, n a wde ange of othe cae and, n patcula, n actuato ealzng cople and vaable tajectoe of oton the electon of contol paaete tend to be oe foal and ot often t baed on the dect apunov ethod fo the anal of the tablt of the te [5,4/,6/]. It fnd applcaton n evoechan wth teppe oto a well a BDC dve [5/]. Slal, n SRM dve t poble to ealze the gven tajecto of the oton b a pope wtchng the uppl voltage n the ange ±u, o ±u, wth an adequatel hgh fequenc. The ldng contol of SRM oto that appled hee ha cetan ltaton eultng fo the natue of the oto a well a the adopted aupton. One of the that the contol occu ultaneoul onl fo a ngle phae wndng of the oto, a a eult of whch t nvolve jut one denon. The econd etcton concen the fact that the contol occu onl fo the conducton of tanto,.e. fo the flow of eneg fo the ouce. In contat, dung the dode conducton and etun of the eneg nto the ouce the oto cuent and toque ae uncontolled fo the duaton of th tage of opeaton. Obvoul, thee a poblt of applcaton of tanto-dode contol ode,.e. u, dung the peodc wtchng of the wndng and oe nd of the effect of cuent contol obtaned n th wa; howeve, t wo onl n the decton of nceang the te needed fo the deca of the cuent and t wll not be

453 4 5 Swtched Reluctance Moto Dve appled hee. The peentaton futhe on wll concentate on the applcaton of ldng contol wth egad to SRM achne ealzng gven tajecto of oton and ltng toque pple,.e. ovecong one of the dawbac of th dve n te of vbaton and noe poducton. The peentaton wll nclude DTC contol and cuent contol of the oto n accodance wth the gven ldng cuve ealzng thee taget. It poble to plan othe contol technque nvolvng ldng, all of whch ealze the gven tajectoe of the oton of the dve. The peentaton of pactcal eaple of uch te wll be the ubject of the followng ecton Cuent Contol of SRM Dve The cuent contol ued hee nvolve the ldng contol of SRM dve n whch the ldng uface defned b the gven functon of the cuent of the phae wndng of a achne σ,, θ σ θ 5.8 and the contol telf defned n the tandad wa: u >, θ fo σ, θ u, θ 5.9 u <, θ fo σ, θ The ldng contol decbed b the foulae 5.8 and 5.9, whch futhe called cuent contol due to the fact that the ldng uface degned on the ba of the value of the phae cuent, not a tpcal one. Th coe a a conequence of the fact that the analtcal epeon 5.8 of ldng uface doe not nvolve te t n an eplct wa but the contol elatve to othe vaable,.e. the angle of the otaton of the oto θ, thu, t a phae uface. Secondl, a t wll be peented late, the ldng uface detened n the functon of the u of phae cuent. In the pactce of SRM oto contol th ean the dependence of the ldng uface on a ngle phae cuent that dawn fo the ouce, whch contollable nce t uppled though the tanto, and pobl on a ngle o oe phae cuent that ae n the phae of deca. Th, n tun, ean that the contol ngle-denonal dependent on the u of the cuent n wndng, whle onl one of the cuent contollable,.e. the one that enegzed fo the ouce. The electon of the ldng uface ha two ole to pla: t nze the pulaton of the toque and eecute the gven cuve of the electoagnetc toque of the oto that defned to adequatel eflect the equed tajecto of the oton of the dve. A one can ee, the ta et n th

454 5.7 Contol of SRM Dve 4 a b c Fg. 5.5 Chaactetc cuve fo Z /Z 6/4 SRM a a functon of oto poton angle: a electoagnetc toque T e fo two neghbong phae wndng uppled b.,.,.,, [A] b Σ of cuent n two neghbong phae wndng equed fo T e.,.,,6, [N] c Σ of cuent equed fo T e. [N] n detal fo wa eque that the poble that nvee to the oto toque functon ha to be olved,.e. f T e 5. Snce th poble non-lnea and concuentl the cuent noall conducted though oe than a ngle phae wndng, the above hould be etated a follow

455 44 5 Swtched Reluctance Moto Dve f T e 5. In pactce, th poble aocated wth the detenaton of the u of two cuent beng conducted though adjacent wndng,.e. one that uppled and one that etun the eneg, n a wa that enable geneaton of a deed ntantaneou value of electoagnetc toque. The oluton of th ta the evee to the elaton peented n Fg. 5.5a and Fg. 5.5a epectvel fo the SRM oto wth the teeth nube Z /Z 6/4 and Z /Z 8/6. Thee ae the achne B and A n Table 5.. a b c Fg. 5.5 Chaactetc cuve fo fo Z /Z 8/6 SRM a a functon of oto poton angle: a electoagnetc toque T e fo two neghbong phae wndng uppled b.,.,.,,5 [A] b Σ cuent n two neghbong phae wndng equed fo T e., 6., 9.,,8. [N] c Σ cuent equed fo T e 6. [N] n detaled fo

456 5.7 Contol of SRM Dve 45 a b c d e Fg Statng of oto B Z /Z 6/4 wth cuent contolle et fo egented contant toque value: a equed Σ cuent b phae cuent c electoagnetc toque d oto peed e patal and eultant toque fo a fnal peed

457 46 5 Swtched Reluctance Moto Dve a b c d e Fg Statng of oto A Z /Z 8/6 wth cuent contolle et fo egented contant toque value: a equed Σ cuent b phae cuent c electoagnetc toque d oto peed e patal and eultant toque fo a fnal peed

458 5.7 Contol of SRM Dve 47 a b c d e Fg Tajecto foed n the B SRM achne unde a cuent contolle: a equed cuent b phae cuent c electoagnetc toque d otatonal peed e patal and eultant toque fo Ω 8 [ad/n]

459 48 5 Swtched Reluctance Moto Dve a b c d e Fg Tajecto foed n A SRM achne unde a cuent contol ule: a equed cuent b phae cuent c electoagnetc toque d otatonal peed e patal and eultant toque fo Ω [ad/n]

460 5.7 Contol of SRM Dve 49 Fo the eult n Fg. 5.5 and Fg. 5.5 t te that ganng hgh value of the toque fo oto B Z /Z 6/4 poe a uch oe dffcult ta than fo the oto wth 4 phae wndng,.e. oto A Z /Z 8/6 a a eult of the lage dtance between the teeth fo the ft of the. Howeve, n both cae t poble to pefo the cuent contol b the peented ethod, whch wll be deontated on the ba of eveal eaple. Ft, we wll dcu the tat-up of oto wth the applcaton of cuent contol. Fo oto B t peented n Fg Cuent contol, a peented can be effectvel ued to fo the tajecto of the oton of a dve a a eult of applng a gven wavefo of toque poduced b a oto. Th toque fo the applcaton of the cuent contol need to be ubequentl tanfoed 5. th ganng the equed cuent wavefo necea to pefo the ta gven b Σ. Th peented on the llutaton of the opeaton of the dve fo both oto A and B. The peented eaple of the applcaton of cuent contol Fg Fg pove that the peented ethod effectve wth egad to the both oto condeed a eepla one,.e. fo the oto wth and 4 phae wndng. Th allow one to fo the wavefo of otatonal peed, toque and cuent, the latte of whch n th ethod the quantt that dectl egulated, a well a enable one to lt the pulaton of the toque. Howeve, th ethod can be effectve onl wthn the ange n whch thee an adequate uplu of the egulaton, whch n th cae ean a uffcent uplu of the uppl voltage, that wll enable one to pefo the planned cuent contol. Th lted uplu of the contol the eaon that n the tat-up of oto Fg. 5.54, Fg caed out wth a toque deceang b tepwe ecton along wth nceang peed. The fong of the tajecto of the oton occu egadng the otatonal peed that petted b the uppl voltage n ode to enue that the gven cuent hape eultng fo aued tajecto of the oton wee poble to pefo b the contol te Dect Toque Contol DTC fo SRM Dve DTC contol wth egad to SRM oto alo fo an applcaton of ldng ethod fo dve egulaton nce t occu a a eult of apd wtchng of the voltage appled to the wndng of a achne tato n a wa that enue that the gven wavefo of electoagnetc toque ealzed. Th ethod n t pactcal applcaton la to cuent contol, whch ha alead been the focu of peentaton eale n the chapte. The pecfc chaactetc of DTC contol nvolve the fact that the ldng uface contucted on the ba of the deed wavefo of the toque: Th can be etated oe dectl a: T, θ σ q 5. e σ : T θ, Ω, t T q e

461 44 5 Swtched Reluctance Moto Dve a b c d Fg Statng of B SRM achne b ue of DTC contol ethod wth equed toque, epectvel T e.,.5, 7.,.7 [N]: a phae cuent b electoagnetc toque c otatonal peed d patal and eultant toque fo T e.7 [N]