Bushlss Doubly-Fd Induction Machins: Toqu Rippl Tim. D. Stous, Xuzhou Wang, Hn Polind, Snio Mmb, IEEE, and J. A. (Bam Fia, Fllow, IEEE Abstact-- Th Bushlss DFIM without its bush-ga and slip-ings loos pomising as succsso of th nomal DFIG in wind tubin divtains. Howv, th bushlss DFIM magntic fild has a ich spac-hamonic spctum, which causs additional toqu ippl. This pap focuss on toqu ippl in th bushlss DFIM. Th causs and oigin of toqu ippl a discussd and an analytical modl is dvlopd that divs th toqu ippl fom th ai-gap magntic fild distibution. Futh, a FE mthod fo toqu ippl calculation is psntd. Both mthods a usd to calculat th toqu spons of a cas study machin. Rsults a compad and th accuacy of th analytical modl is validatd with good sult. Th analytical modl is thn usd to futh analys th toqu ippl, sulting in xact toqu ippl fquncis and additional insight in th caus of th toqu ippl. Indx Tms-- Bushlss Doubly-Fd Machin (BDFM, Coss Coupling, DFIG, Spac Hamonics, Tim Hamonics, Slotting, Toqu Rippl. W I. INTRODUCTION IND ngy is gowing apidly, with a total installd capacity of 336GW halfway 4, which accumulats to appoximatly 4% of th total woldwid lcticity usag []. This apid gowth also incass th nd fo liabl and fault-tolant dsigns, spcially fo off-sho wind tubins. Th Bushlss Doubly Fd Induction Machin (DFIM is an xpimntal machin typ that sponds to that concn. This machin typ has no bush-ga and slip-ings which incass systm liability and dcass th nd fo maintnanc. Bcaus of this th bushlss DFIM is a wothy succsso of th nomal Doubly Fd Induction Gnato (DFIG, which is nowadays th most common machin topology in wind tubins []. Th bushlss DFIM has two stato-windings, th powwinding and th contol-winding, with spctivly p p and p c pol-pais. A spcial nstd-loop oto stuctu nsus coupling with both stato magntic fild componnts. By conncting th pow-winding to th gid and th contol- Th sach lading to ths sults has civd funding fom th Euopan Union's Svnth Famwo Pogamm managd by REA Rsach Excutiv Agncy (FP7/7_3 und Gant Agmnt N.35485. T. D. Stous, X. Wang, H. Polind and J. A. Fia a with th Dpatmnt of Elctical Sustainabl Engy, Dlft Univsity of Tchnology, Th Nthlands (-mail: t.d.stous@tudlft.nl winding to a patially atd Pow Elctonic (PE convt, synchonous opation is assud ov a limitd spd ang. Hnc, th opating chaactistics of th bushlss DFIM and th nomal DFIG a compaabl. As alady indicatd in [3] th Bushlss DFIM magntic fild distibution has a ich spac-hamonic spctum. This is mainly causd by th nstd-loop oto stuctu, bcaus th oto is dsignd to coupl to two main hamonic fild componnts it poducs a lag spac-hamonic distotion. This affcts th machins opating bhaviou. Th incasd oto spac-hamonic distotion incass oto laag inductivity, which could b usd bnficially to impov voltag id though pfomanc [4]. Th dawbac of incasd spac-hamonics is that it causs additional toqu ippl in th machin, which could ngativly influnc componnt lif-tim of th componnts in a wind tubin divtain. Toqu ippl is of gat impotanc fo lctical machin (and wind tubin divtain dsign. Bcaus toqu ippl can b xpctd to b mo sv in th bushlss DFIM machin typ, th main focus of this pap will b on toqu ippl modlling and analysis of th bushlss DFIM. Fist, th causs and oigin of toqu ippl in lctical machins a discussd. Thn an analytical modl is dvlopd to calculat th toqu ippl fom th magntic fild distibution of th bushlss DFIM. Th magntic fild distibution is modlld including th ffcts of oto timhamonics and spac-hamonics du to th winding distibution and slotting. Th dvlopd modl can b usd to pfom a spac- and tim-hamonic analysis of th toqu ippl in th bushlss DFIM. Futh, FE tchniqus fo calculating toqu ippl a psntd in this pap. Th dvlopd analytical modl is validatd with th us of FE calculations, by applying both mthods on a cas study machin and compaing sults. Th analytical modl is thn usd to dtmin th causs and xact fquncis of th diffnt toqu ippl componnts psnt in th bushlss DFIM lctomagntic toqu spons. II. TORQUE RIPPLE IN ELECTRICAL MACHINES Toqu ippl in lctical machins is causd by many factos. By valuating toqu with th us of Foui thoy, it is found that toqu ippl can b causd by ith spachamonic componnts in th ai-gap magntic fild o by
Fig.. Th ffcts of spac- and tim-hamonic componnts on th toqu in a bushlss DFIM. tim-hamonic componnts in th cunts flowing though th stato and oto. Th two a latd, sinc spachamonic magntic fild componnts, induc tim-hamonic componnts in th cunts, which thn poduc thi own st of spac-hamonic componnts in th magntic fild. Although, th latt has small influnc on th toqu, which will b shown in this pap. Toqu ippl aiss whn both stato and oto poducs spac-hamonic magntic fild componnts of th sam od, but with diffnt fquncis. Th following ffcts that caus toqu ippl can b distinguishd: Spac-hamonic componnts in th adial ai-gap magntic fild distibution, oiginating fom: o Th winding distibution. o Slotting. o Satuation: Satuation causs th dfomation of th magntic fild. (outsid th scop of this pap. o Mchanical constuction: Roto ccnticity o an unbalancd oto. (outsid th scop of this pap Tim-hamonic componnts in th machins cunts: o Spac-hamonic inducd tim-hamonics. o Supplid by th gid o PE convt to th statowinding cunts. (outsid th scop of this pap. Th ffcts causing toqu ippl a schmatically psntd in figu. It shows th lations of spac- and tim-hamonic componnts and thi ffct on th toqu. III. THEORETICAL DEVELOPMENT This sction psnts th dvlopd modls fo toqu calculations of th bushlss DFIM. An analytical toqu calculation mthod is psntd, which divs th calculatd toqu fom th lina sht cunt dnsity and magntic fild distibution in th ai-gap of th bushlss DFIM. Fist an analytical adial ai-gap magntic fild distibution modl is psntd, that includs slotting and winding spachamonics as wll as oto inducd tim-hamonics. Thn a mthod fo FE toqu calculations is psntd. Both mthods a applid on a cas study machin and th sults that a obtaind a dmonstatd. Thfo, fist an ovviw of th cas study machin is psntd in tabl I and figu. Th modl is dvlopd fo bushlss DFIM s with nstdloop oto stuctus, that a opatd in synchonous mod of opation. Thfo, th cunt inducd in th oto nstd-loop by th pow-winding main magntic fild componnt, matchs th cunt inducd by th contolwinding main magntic fild componnt. Both componnts hav th sam fquncy and phas-dlay btwn consqunt oto nsts [5]. Th synchonous otational spd f m is thn givn accoding: f m f p N f nst c Th numb of oto nsts N nst must b qual to th numb of pow-winding p p and contol-winding p c pol-pais addd. TABLE I: Bushlss DFIM Dsign Constuction Paamts Numb of phass N ph 3 Numb of pol-pais p p, p c 4, 6 Numb of stato slots N ss 7 Numb of oto slots N s 8 Numb of oto nsts N nst Numb of loops p nst q 4 Gomtic Paamts Axial stac lngth l st.6 m Ai-gap lngth l g.5 mm Stato out adius so.83 m Stato inn adius si.67 m Roto out adius o.67 m Roto inn adius i.58 m Evaluatd opating point Pow winding fquncy f p 5 Hz Contol winding fquncy f c - Hz Rotational spd f m 6 Hz Pow-winding slot cunt dnsity J p A/mm Contol-winding slot cunt dnsity J c A/mm Initial position shift γ shift (9/N nst (
Th tim-hamonic oto cunts, poducd by th stato winding spac-hamonic magntic fild componnts hav spctiv fquncis f (s : f ( s fp s fm s pp nnph fc s fm s pc nnph (6 Fig.. Cas study machin, including D magntic fild flux-dnsity. A. Magntic Fild Modlling In [6] th adial componnt of th magntic flux dnsity distibution B tot in th ai gap of th bushlss DFIM is xpssd as th summation of winding spac-hamonic componnts s and of spctivly th stato B s and oto B magntic filds: Btot (, t Bs B ( j( f( ts sshift s Bs Bs( s s q j( f( t s B B,( l s, s l ( is xpssd in th oto fnc fam as indicatd with th supscipt. Thfo, th stato magntic fild is found aft tansfomation to th oto fnc fam: s mt shift γ shift is a position shift of th oto with spct to th stato at tim t =. Th stato magntic fild consists of th supposition of th pow- and contol winding fild componnts. Thfo, th a also two main hamonic th fild componnts. Th p p spac-hamonic componnt cosponding to th pow-winding and th -p th c spachamonic componnt cosponding to th contol-winding. Th stato magntic fild componnts poduc tim-hamonic oto cunts, ach poducing its own st of oto spachamonic magntic fild componnts. Th st of stato s and oto winding spac-hamonics consists of: (3 s pp( nnph pc(nnph (4 nn n (5 s nst B. Modlling th Magntic Fild including Slotting Th adial ai-gap magntic fild distibution of ( nglcts th ffct of slotting. This sction intoducs a slotting function b slot to cat fo th ffct of slotting. Using th slotting function, slotting is thn intoducd in ( by a summation of slotting spac-hamonic componnts. Th total ai-gap magntic flux dnsity distibution (including slotting ffct B tot,slot can thn b psntd accoding: Btot, slot (, t B (7 tot bslotbtot sum of main+winding hamonics sum of slottinghamonics Th slotting function b slot consist of a stato b s,slot and a oto b,slot slotting pat, which a divd fom th ai-gap lngth functions l gs and l g spctivly: j mt shift bslot (, t bs, slot(, sslot b, slot ( slot, l FFT gff, s bsslot, ( sslot, sslot, nnss lgs ( l FFT gff, b slot, ( slot, slot, nns l ( j g H l g,ff is an ffctiv ai-gap lngth, aft compnsation fo th cas slotting is not tan into account [7].Th ai gap lngth functions xpss th ai-gap lngth as function of position. Sinc all quations in this pap a xpssd fom th oto fnc point of viw, th stato ai-gap lngth function changs with position and tim. Th ai-gap lngth btwn slots is dtmind using th staight lin cicula ac mthod, as psntd in figu 3 [8]. Fig. 3. Ai-gap lngth as function of position angl.
By combining (,(7 and (8, a distinction can b mad btwn 4 sts of slotting spac-hamonics, as is psntd in (9: Spac-hamonic componnts of od s ± s,slot du to stato slotting in th stato magntic fild B s,slot. Spac-hamonic componnts of od s ±,slot du to oto slotting in th stato magntic fild B s,slot. Spac-hamonic componnts of od ± s,slot du to stato slotting in th oto magntic fild B,slot. Spac-hamonic componnts of od ±,slot du to oto slotting in th stato magntic fild B,slot. b B B B B B slot tot s, sslot s, slot, sslot, slot sum of slottinghamonics bs, slotbs b, slotbs bs, slotb b, slotb s s, slot s, slot s, slot, slot Figu 4 psnts th modlld stato magntic fild including th ffct of stato slotting and th oto magntic fild including th ffct of oto slotting, modlld fo th cas study machin dscibd in tabl I. Th ffct of oto slotting in th stato magntic fild and stato slotting in th oto magntic fild a not shown, fo th sa of poviding a mo cla pictu. Th total magntic fild distibution is thn psntd in figu 5. Calculatd using th analytical mthod and validatd using FE calculations. C. Analytical Toqu Calculation Mthod Toqu is xtd on th oto sufac S oto of th bushlss DFIM whn th is a coupling btwn stato and oto magntic fild componnts of th sam spac-hamonic od. Whn ths magntic-fild componnts hav a diffnt otational spd, toqu ippl occus. A spaation can b mad btwn toqu du to winding spac-hamonics T,win (including th main toqu componnts and toqu du to slotting hamonics of both stato and oto, T,sslot and T,slot spctivly. 4 BDFIM magntic fild i p,pa =.88A, i c,pa =.88A (9 Ai gap magntic flux dnsity B g (T 3 Fig. 5. Bushlss DFIM total magntic fild distibution in th ai-gap. [6] alady psnts an analytical toqu calculation mthod that is divd fom th Lontz-foc quation. Using th poduct of lina sht cunt dnsity J and magntic fild distibution (without slotting, intgatd ov th oto sufac S oto, povids th xtd lcto-magntic toqu T,win. T ( t J B ds (, win o tot oto lst o J ( tim, spac Bs( spac d tim spac spac s, tim s s s, slot Th lina sht cunt dnsity distibution can b divd fom th magntic fild distibution accoding: J l BDFIM magntic fild distibution Analytical calculation FEM calculation 3.5pi.5pi.75pi pi Roto position angl θ (ad B gff, (, t ( o H μ is th magntic pmability of ai (4π -7 H/m. ( includs th bushlss DFIM main toqu componnts as wll as th toqu ippl componnts du to winding spachamonics and spac-hamonic inducd oto tim-hamonics. Th toqu ippl fquncis f T poducd by th magntic fild componnts of spac-hamonic od spac, a givn by: Ai gap magntic flux dnsity B g (T 3 Stato fild: B s + c b s,slot B s f f f T(, ( ( ( spac tim tim spac Tabl II povids an ovviw of winding spac-hamonic componnts causing toqu ippl, calculatd accoding (, fo th cas study machin psntd in tabl I. Also th cosponding toqu ippl fquncis f T a psntd, as calculatd in (. Roto fild: B + c b,slot B 3.5pi.5pi.75pi pi Roto position angl θ (ad Fig. 4. Bushlss DFIM stato and oto magntic fild componnts including th ffct of slotting
TABLE II: Winding toqu ippl componnts and fquncis Main tim-hamonics: f (4 = f (-6 = 6 Hz spac 4-6 44-66 -76 4-6 4 f T (Hz 34 38 38 7 7 7 Roto tim-hamonic: f (- = 7 Hz spac - - - - - - - - f T (Hz - - - - - - - - Roto tim-hamonic: f (-8 = 8 Hz spac 8-4 -5 68 78-9 - 38 f T (Hz 36 38 34 34 7 74 7 Roto tim-hamonic: f (3 = 9 Hz spac - - - - - - - - f T (Hz - - - - - - - - Roto tim-hamonic: f (-4 = 4 Hz spac 8-4 -5 68 78-9 - 38 f T (Hz 36 7 7 36 36 8 Using th sam mthodology, th toqu ippl du to stato and oto slotting hamonics can b tan into account. Howv, th Lontz-foc quation in ( uss th magntic fild of th stato and th lina sht cunt dnsity of th oto fo calculating th toqu. Th lina sht cunt dnsity obviously dos not ta into account th ffct of slotting. Thfo, fist th toqu ippl T,sslot du to th intaction of stato magntic fild slotting spac-hamonics of od s ± s,slot coupling with th oto magntic fild winding spac-hamonics is tan into account: tim spac,, T ( t J B ds (3, sslot o s, sslot oto l J B d, st o ( tim, spac s, sslot( spac spac s s slot tim s s s slot Thn th toqu ippl T,slot du to th intaction of oto magntic fild slotting spac-hamonics of od ±,slot with th stato magntic fild winding spac-hamonics is povidd: T ( t J B ds (4, slot si s, slot oto lsto si Js( spac B, slot( tim, spac d, tim spac,, spac s slot tim s s s slot Th ngativ sign in (4 is du to th calculation of toqu xtd on th stato, which has an opposit sign fom th toqu xtd on th oto. Tabl III povids an ovviw of stato and oto slotting spac-hamonic componnts causing toqu ippl and th cosponding toqu ippl fquncis fo th cas study machin psntd in tabl I. TABLE III: Slotting toqu ippl componnts and fquncis Stato slotting: f (4 = f (-6 = 6 Hz (Main tim-hamonic spac 76 66-44 -4-76 -64 s ± s,slot 4+7-6+7 8-7 -4-7 -+44 --44 f T (Hz 38 38 34 7 7 6 Roto slotting: f (4 = f (-6 = 6 Hz (Main tim-hamonic spac -76-66 4 44 64 74 ±,slot 4-8 4-8 34+8-36+8 4+6 4+6 f T (Hz 38 38 7 34 6 8 D. FE Toqu Calculation Mthod Th lcto-magntic toqu in th machin can b divd fom th magntic fild in th ai-gap of th bushlss DFIM. Applying Maxwll s stss tnso mthod, a cicula intgal of th -dimnsional magntic fild aound th ai-gap cicumfnc at adius g is calculatd to obtain th lctomagntic toqu T : l T B B d st g tan (5 H B and B tan a th adial and tangntial componnts of th flux dnsity in th ai-gap at adius g. Equation (5 can asily b applid on a FE divd magntic fild as psntd in figu. Howv a D FE pogam solvs th magntic vcto potntial A z (in axial z-diction. Th flux dnsity distibution B is divd fom A z accoding (6 and hnc on od lss accuat. B Az Az ; Btan (6 g To impov th calculations accuacy, an analytical filt is applid to Maxwll s stss tnso mthod. As is dscibd in [9]. This mthod ducs th snsitivity to FE mshing paamts. Fist th flux dnsity B is divd fom an analytical xpssion of A z in a shll in th ai-gap. This aigap shll is dfind by two concntic cicls in th ai-gap with spctiv adii and, wh ( < g <. At th boundais of this ai-gap shll A z can analytically b xpssd as a Foui sis of spac-hamonic componnts : A (, a [ a cos( b sin( ] (7 z A (, a [ a cos( b sin( ] z Fom (5-(7 th impovd computational quation fo diving th lcto-magntic toqu is divd: l T ( a b a b (8 st g
Toqu (Nm 44 43 4 4 4 39 38 37 36 35 FE toqu spons Analytical toqu spons BDFIM Toqu spons ippl componnts and thi cosponding fquncis du to tim-hamonics in th oto magntic fild. (.g. all oto magntic fild componnts not gnatd by th main oto cunt tim {4,-6}. Last, figu 9 psnts all toqu componnts and thi cosponding fquncis du to slotting hamonics, including stato and oto slotting as wll as toqu componnts du to tim-hamonics inducd by th stato slotting spac-hamonics, accoding (-(4. Toqu ippl, du to winding spac hamonics 34.5..5..5.3.35 Tim (s Fig. 6. Bushlss DFIM toqu spons duing nominal opation. IV. TORQUE EVALUATION Sctions III-C and -D psntd mthods fo calculating toqu and toqu ippl in a bushlss DFIM using analytical calculations and FE calculations spctivly. Both mthods a applid on th bushlss DFIM dsign as psntd in Tabl I. Th gnatd toqu sponss fo a calculation tim of on oto lctic piod ( /f = 38.5ms a psntd in figu 6. Th analytical calculation mthod is validatd by th FE calculation mthod. Th sulting analytical calculatd toqu spons has a good accuacy, whn compad to th FE calculatd toqu spons. With th us of (9 th toqu ippl is xpssd as a pcntag of th man toqu. This povids a good masu fo th toqu ippl in this machin at its nominal opating point. T ippl T,max T T man,,min % (9 Th sulting man toqu T man and toqu ippl T ippl fo both th analytical and th FE calculation mthod a compad in Tabl IV. Th advantag of th analytical modl ov a FE modl is that it hlps idntifying th caus of th toqu ippl and additionally to dtmin th xact toqu ippl fquncis. Figus 7 until 9 povid an valuation of toqu ippl amplituds du to diffnt spac-hamonic componnts and thi cosponding fquncis. Figu 7 fist psnts th toqu componnts du to winding spac-hamonics only. Following ( and (. Sinc th main toqu is causd by th main winding spac-hamonic componnts (.g. th 4 th and th -6 th, ths a also indicatd in figu 7 (By th uncompltd gy bas. Figu 8 thn psnts th toqu TABLE IV: Toqu ippl calculation mthod compaison T man : T ippl : Analytical calculation: 38.Nm % FE calculation: 38.Nm 3% Toqu amplitud (Nm 976 48 988 494 7 36 Fquncy f T (Hz 4 9 34 49 64 79 94 9 4 Coupling spac hamonic fild componnts spac Fig. 7. Toqu ippl componnts du to th winding distibution spachamonics, without taing into account tim-hamonics. Toqu amplitud (Nm 5 4 3 7 36 Fquncy f T (Hz Toqu ippl, du to oto tim hamonics 4 9 34 49 64 79 94 9 4 Coupling spac hamonic fild componnts spac Fig. 8. Toqu ippl componnts du to oto tim- hamonic inducd fild componnts Toqu amplitud (Nm 5 4 3 7 36 Fquncy f T (Hz Toqu ippl, du to slotting hamonics 4 9 34 49 64 79 94 9 4 Coupling spac hamonic fild componnts spac Fig. 9. Toqu ippl du to slotting hamonics and slotting hamonic inducd oto tim-hamonics.
Fom figus 7 until 9 can b concludd that fo th cas study machin as psntd in tabl I, th lagst contibution to th toqu ippl is causd by winding spac-hamonics. Slotting spac-hamonics also contibut to th toqu ippl, but to a lss xtnt. Th ffct of th oto tim-hamonics, inducd by th stato spac-hamonics, on th toqu ippl is ngligibl. Intsting is to s that th oto tim-hamonics also poduc toqu componnts at Hz. Ths a th toqu componnts sponsibl fo th cawling ffct in nomal induction machins. Th 66 th and 76 th spac-hamonic fild componnts hav th lagst contibution to th toqu ippl. Both ths spac hamonic fild componnts poduc a ippl fquncy of 38Hz at th spcifid opating point. Th pow- and th contol-winding distibution as wll as stato and oto slotting hamonics all contibut to th 66 th and 76 th spachamonic componnts, as can b sn fom tabl s II and III. This xplains why thy contibut most to th toqu ippl. V. CONCLUSION This pap fist discussd th causs of toqu ippl in lctical machins and xplaind thi oigin du to spachamonic componnts in th ai-gap magntic fild o timhamonic componnts in th cunts flowing though th machins windings. Two mthods fo toqu calculations w intoducd. An analytical mthod and a FE mthod. Th analytical mthod divs th toqu fom th adial ai-gap magntic fild distibution using th Lontz-foc quation. Th ai-gap magntic fild is modlld including th ffcts of winding spac-hamonics, slotting and oto tim-hamonics. Th FE toqu calculation on th oth hand is basd on Maxwll s stss tnso mthod, wh an analytical filt is applid fo impovd accuacy. With th us of a cas study machin, th analytical toqu calculation mthod is validatd by compaison to th FE toqu calculation mthod. Th analytical divd man toqu and toqu ippl wh calculatd with good accuacy. With th us of th analytical toqu calculation mthod it is futh possibl to dtmin th spac- and tim-hamonic componnts contibuting most to th toqu ippl and to dtmin thi oigin. Thi cosponding ippl fquncis can also b divd. Whn analyzing th cas study machin it was found that th lagst contibution to th toqu ippl was causd by th winding distibution spac-hamonics. This vifis that th xcssiv spac-hamonic spctum, psnt in th nstd-loop oto stuctu of th bushlss DFIM, also has an considabl ffct on th toqu ippl. Th ffct of tim-hamonic oto cunts was found to b ngligibl. Th ffct of slotting (half opn stato and oto slots in th cas study machin also has som contibution to th toqu ippl. VI. REFERENCES [] WWEA half-ya pot 4 [Onlin]. Availabl: http://www.wwinda.og/wbimags/wwea_half_ya_pot_4.p df [] R. A. McMahon, X. Wan, E. Abdi-Jalbi, P. Tavn, P. C. Robts, and M. Jagila, Th BDFM as a gnato in wind tubins, in th Int. Pow Elcton. Motion Contol Conf., EPE-PEMC, 6, pp. 859 865. [3] H. Goingpou, B. Jandaghi, H. Oa, Tim and Spac Hamonics in Bushlss Doubly-Fd Machin, in 9 th Ianian Conf. Elct. Eng. (ICEE,, pp. 6. [4] U. Shipua, T. D. Stous, H. Polind, and J. A. Fia, LVRT pfomanc of bushlss doubly-fd induction machins - a compaison, to b psntd at Int. Conf. Elct. Machins & Divs (IEMDC, 5. [5] S. Williamson, A. C. Fia, A. K. Wallac, Gnalisd thoy of th bushlss doubly fd machin. Pat I : Analysis, in IEE Poc.- Elct. Pow Appl., vol. 44, no., 997, pp. -. [6] T. D. Stous, N. H. van d Blij, H. Polind, and J. A. Fia, Bushlss doubly-fd induction machins: Magntic fild modlling, in Int. Conf. Elct. Machins (ICEM, Sp. 4, pp. 7 78. [7] J. Pyhönn, T. Joinn, V. Habovcová, Dsign of Rotating Elctical Machins, st d., John Wily & Sons, 8. [8] D. C. Hanslman, Bushlss pmannt-magnt moto dsign, McGaw-Hill, 994. [9] M. Popscu, D.M. Ionl, T.J.E. Mill, S.J. Dlling and M.I. McGilp, Impovd finit lmnt computations of toqu in bushlss pmannt magnt motos, in IEE Poc.-Elct. Pow Appl., vol. 5, no., 5, pp. 7-76.