Study on the Static Load Capacity and Synthetic Vector Direct Torque Control of Brushless Doubly Fed Machines

Transcription

1 ngis Aticl Study on Static Load Capacity Syntic cto Dict Toqu Contol Bushlss Doubly Fd Machins Chaoying Xia * Xiaoxin Hou School Elctical Engining Automation, Tianjin Univsity, No. 9 Wijin Road, Tianjin, * Cospondnc: xiachaoying@.com; Tl.: Acadmic Edito:.T. Chau Rcivd: Jun ; Accptd: Novmb ; Publhd: Novmb Abstact: Compad to doubly fd machin, bushlss doubly fd machin ) has high liability low maintnanc quimnts. Ft, taking ngativ conjugation contol moto vaiabls in oto fnc fam, a stat-spac modl divd. It n tansfomd into synchonous fnc fam, calld synchonous fnc fam stat-spac modl SSSM). In th way, all vaiabls SSSM a DC und static stat. Scond, on bas analys static quations, possibl toqu limits a obtaind. Thid, causs losing contol a analyzd toqu divativs. A nw contol statgy calld syntic vcto dict toqu contol SDTC) poposd to solv losing contol poblms convntional dict toqu contol DTC). Finally, coctnss sults th pap vifid calculation xampls simulation sults, losing contol poblms can b solvd, otical capacity limits can b achd using SDTC. ywods: bushlss doubly fd machin ); stat-spac modl; load capacity; losing contol; syntic vcto dict toqu contol SDTC). Intoduction A bushlss doubly fd machin ) a nw typ AC vaiabl spd moto that has chaactti both asynchonous synchonous motos. has good application pospcts in AC diving systms, wind pow gnation bcaus it fs high liability low-maintnanc quimnts moving bush ga slip ing [ 9]. Fumo, shows commcial potntial bcaus its low convt capacity quimnt. Howv, a high-od, multivaiabl, stong coupling nonlina systm. Compad with asynchonous synchonous motos, has a complicatd stuctu a complicatd unning mchanm, thus stablhmnt a mamatical modl, analys stability, dsign a contol statgy a difficult., which oiginats fom a slf-cascadd induction machin, has on spcial oto two sts stato windings with diffnt pol pa. Among m, moto, which stato winding connctd dictly to pow gids, calld pow moto PM), o moto, which stato winding connctd to a fquncy convt, calld contol moto CM) []. Many scholas hav poposd vaious mamatical modls fo s to analyz i pfomanc study its contol mthods [ ]. Robts t al. dvlopd a ntwok modl in a th phas static coodinat []. Wallac t al. poposd a modl cascad wound oto doubly fd moto in a oto fnc fam. In s modls, all vaiabls PM CM a AC und static stat []. In od to obtain a modl which all vaiabls PM CM a Engis, 9, 9; doi:.9/n99

2 Engis, 9, 9 DC und static stat, so-calld doubl synchonous fam modl was psntd [,]. In th doubl synchonous modl, was dcomposd into two subsystms, a pow winding subsystm a contol winding subsystm, lctomagntic toqu dpnds on cunt both subsystms, so contol systm complx, contols toqu a not dcoupld. Fo a cag oto, so-calld unifid fnc fam modl was poposd [,]. Th modl most advancd widly usd on at psnt. But unifid fnc fam modl adopts mixing dsciption diffntial quations algbaic quations, so it not convnint to analyz static dynamic chaactti s. In addition, supply a dual souc, moto toqu sum asynchonous toqu synchonous toqu. Du to intaction btwn two typs toqus, moto toqu contol pocss fom a stting valu to ano stting valu a complx, oscillation loss contol appa asily [,]. To solv s poblms, sachs hav bn mad a lot ffots, a mainly th diffnt mthods, mthod woking point linaizd small signal modl, mthod fdback linaization, mthod DTC obustnss contol. In mthod woking point linaization, stability was analyzd woking point linaizd small signal modl, which pointd out that stabl opating ang was naow und opn loop contol. Expimntal otical sults showd that a small otay intia o a high CM ld to a wid stabl opating ang [ ]. woking point linaizd small signal modl latd to moto paamts, otay intia paamts contoll, changs with woking point systm, so th mthod can only study local stability systm. Fdback linaization was ft applid to basd on a stat-spac modl in oto fnc fam. modl slctd cunts, oto angl, angula spd as stats. Taking only spd as, input- fdback linaization mthod was usd to solv poblm spd contol, wh in addition to all moto paamts, load toqu otay intia must also b known []. Whn an inn cunt loop CM adoptd, contol moto can b sn as supplid a cunt souc, a stat-spac modl CM synchonous fnc fam was obtaind. modl slctd stato PM oto CM as stats. Taking toqu oto CM as, input- fdback linaization was dvlopd oto CM ointd, dcoupling contol toqu was achivd, wh all moto paamts nd to know, toqu nd to obsv, contol statgy complx []. Aft dvlopmnt vcto contol, dict toqu contol DTC) an ano high-pfomanc AC moto contol mthod. Sinc Takahashi psntd DTC fo an induction machin in 9 [], DTC tchniqu has bn widly usd in AC machin contol bcaus its simpl stuctu, high dynamic pfomanc, obustnss [,]. convntional DTC was intoducd dictly into vaiabl spd systm. Though analys lationshi among convt vctos divativs toqu, losing contol poblms a invstigatd, pioity toqu pioity statgis a poposd []. Howv, s statgis cannot liminat toqu ippls. Fo toqu ippl poblms s DTC, som sachs hav put fowad many solutions, such as fuzzy logic dict toqu contol pdictiv dict toqu contol, tc., but sults s ffots a not obvious [,]. maind th pap oganizd as follows: Sction intoducs divation pocss a synchonous fnc fam stat-spac modl SSSM). Sction obtains possibl static opation ang. Sction psnts Syntic cto Dict Toqu Contol SDTC) to solv toqu losing contol poblms. In Sction, compaativ simulation xpimnts convntional DTC SDTC a pfomd sults confim good pfomanc SDTC. Finally, conclusions a summaizd in Sction.

3 Engis, 9, 9. Synchonous Fam Stat-Spac Modl Bushlss Doubly Fd Machin ) Fo a wound oto, in oto fnc fam fnc fam), stato oto quations s pow moto PM) a: Engis, 9, 9. Synchonous Fam Stat-Spac Modl d Bushlss Doubly Fd Machin ) u i jpp Fo a wound oto, in oto fnc fam fnc fam), stato oto ) quations s pow moto PM) d p a: u p pi p u = i + dψ stato oto quations s contol + jp p ω moto ψ CM) a: ) u p = p i p + dψ p d u i +jpc stato oto quations s contol moto CM) a: ) d c u u c c ic = i + dψ + jp c ω ψ ) u c = c i c wh j = psnts unit imaginay, oto machiny angula spd, p c + dψ wh j = d q p c a PM CM pol pa spctivly, u psnts unit imaginay, ω u ju oto, u d j q machiny p up u angula p, i d j q spd, i i p p, pi c a i d jipm q a CMstato pol pa, spctivly, oto, u = stato u d + cunt ju q, u p = uoto d p + jucunt q p, i = i d + PM ji q, p p p i p = i d p + ji q p a d stato q spctivly, u, oto, stato cunt oto cunt PM spctivly, u u ju, u d j q = u d + ju q, u c uc uc, i d j q c = u d c + ju q i c, i i i d j q = i d + ji q c i ic c ic a stato, oto, stato cunt oto cunt CM spctivly,: = ic d + jic q a stato, oto, stato cunt oto cunt CM spctivly, : li l pmip ψ = l + l pm i p ψ li = l i lcmic + cm c ) ψ p lpmi p = l pm i l pip + l p i ) p ψc = l cm c lcmi lcic + l c ic a stato PM, stato CM, oto PM, oto CM, spctivly. a stato All vaiabls PM, stato s, cunts CM, oto s a complx PM, numbs. oto complx CM, vaiabl spctivly. a vcto All on vaiabls plan, so it s, also calld cunts as vcto s in a complx following. numbs. complx vaiabl Bcaus a vcto on coss connction plan, so it also oto calld windings as vcto in PM following. CM, phas squnc oto Bcaus windings a coss opposit, connction as shown in oto Figu windings. Btwn PM oto th-phas CM, phas s squnc oto windings a opposit, as shown a in a Figu b. cbtwn c b oto th-phas a a bs c cunts, a lationshi up cunts, a lationshi u a uc, p = uc, a up u b uc p = u c, up c, c uc p = u b i c i a p ic p = ic, a, i b ip ic p = ic, c, ), ) c b = ic, b d d = u d q c, q = u q d c d p = ic, d q i q q p = ic, q o p = u, i c i p = ic p ic, i.., up uc, up uc ip, ip ic, o up uc, i * p ic, wh wh supscipt * xpsss * xpsss conjugat conjugat opation. p pow gid PM A u A u CM convt stato winding oto winding B u b u p a u p b i p C u c u p a i p c ip b i c a i c B u b u c a u c c i c C u c u c coss connctd b q c d t a A Figu Figu.. aiabls aiabls stato stato oto oto windings, windings, stato stato ABC ABC fnc fnc fam, fam, static static αβ αβ fnc fnc fam, fam, oto oto abc abc fnc fnc fam fam oto oto fnc fnc fam. fam.

4 Engis, 9, 9 i Tak oto ) cunts PM as fnc, dfin oto cunt complx vaiabl, = i p = ic combin two oto quations Equations ) ), n w hav: u = i u = i + dψ + dψ = p + c ) i + jp p ω ψ + jp c ω ψ + dψ p d ψc ) ) Du to spcial stuctu oto winding, whn a stady sin cunt flows in oto winding, two magntic filds a gnatd in stato oto windings PM CM, y otat with qual lctical angula spd in opposit dictions lativ to oto. Thus, in oto fnc fam, und static stat, complx vaiabls PM CM a two sts otation vctos, i spds a sam, i dictions a opposit. complx vaiabls PM main unchangd, tak ngativ conjugat opation in both sids contol moto quation, w hav: : wh u = u = i u = i = i + dψ + dψ + dψ ψ = l i + l pm i p = l i + l pm i ) ψ = ψ ) = l i lcm ψ u = ψ p ), i ψ c = ) = ψ i p + ), ψ ψ c = + jp p ω ψ jp c ω ψ ic ) = l i + l cm i ) = lpm i + l i + l cm i ) ) ψ ) a dfind complx vaiabls, = p + c, l = l p + l c a oto stanc oto inductanc lctomagntic toqu can b xpssd as: T = p p Im ψ i + p c Im ψ i = p p Im ψ i p c Im ψ i Fo wound oto dcussd abov, xcpt that mthods conjugat tansfomation a diffnt complx vaiabls CM tak ngativ conjugation o conjugation), obtaind Equations ) ) hav sam fom as unifid fnc fam modl a cag oto []. fo, following obtaind sults a univsally suitabl fo a wound oto cag oto. In th way, und static stat, complx vaiabls PM complx vaiabls CM aft taking ngativ conjugation) otat in sam diction hav sam spd lativ to oto. Th will bing gat convninc fo analys. motion quation : ) J dω = T T l ) wh T T l a lctomagntic toqu load toqu spctivly, J total moto shaft otay intia. Equations ) ) adopt mixing dsciption diffntial quations algbaic quations. Th will lad to som difficultis in analys dsign. Th study will giv a stat-spac dsciption, which mo ffctiv fom fo contol systm analys dsign. Tak stato PM ψ, stato CM ψ, oto ψ as stats, stato PM u stato CM u as inputs, substituting Equation )

5 Engis, 9, 9 into Equation ), to liminat cunts i, i, i, w hav stat-spac modl as follows in oto fnc fam:. x = A x + B u 9) l l lcm) jp p ω l pm l cm l l pm wh A = [ x = ψ ψ ψ l pm l cm l l pm ] T [ u = l l l pm) u u + jp c ω l l cm l l cm Similaly, lctomagntic toqu can b again xpssd as: T = p pl l pm Im ψ ψ p cl l cm Im ψ l l, B = ] T, = l l l l l cm l l pm. ψ + p p+p c)l pm l cm Im ψ ψ In th way, diffnt fom Equation ), lctomagntic toqu psntd as coss poduct s. As w s, a th od nonlina systm complx numb stat-spac modl Equation 9) a = th od divativ quation, motion quation Equation ) a st od), in oto fnc fam, its complx vaiabls a still AC vaiabl. fo, modl should b tansfomd into CM synchonous fnc fam mt fnc fam). In synchonous fnc fam, vctos complx vaiabls PM ngativ conjugation complx vaiabls CM will otat synchonously und static stat. As shown in Figu, λ c angl btwn mt fnc fam oto fnc fam. Using following tansfomation: wh T = jλ c jλ c jλ c x mt = T x, u mt = T u, T = stat-spac modl SSSM) obtaind, that : wh A mt = B mt =, x mt = l l l cm) [ ψ mt [ jλ c jλ c. x mt = A mt x mt + B mt u mt j p p ω + λ. ) c l pm l cm l l pm ] l pm l cm l l l pm) + j l l cm ] T, [ u mt = B mt = T B T. lctomagntic toqu can b -xpssd as: T = p pl l pm Im ψ mt p cl l cm Im u mt, ) ), synchonous fnc fam u mt p c ω. λ c ) l l pm l l cm l l j. λ c ] T, A mt = T A T. T + p p+p c)l pm l cm Im ψ mt ψ mt ), T, motion quation sam as Equation ). Bcaus otation tansfomation dos not affct coss-poduct opation two vctos, toqu xpssions bfo aft otation tansfomation, Equations ) ), a sam fom. In mt fnc fam, all vaiabls stat quations a DC valus und static stat. static woking point asily obtaind. )

6 Im Im Im T ) motion quation sam as Equation ). Bcaus otation tansfomation dos not affct coss-poduct opation two vctos, toqu xpssions bfo aft otation tansfomation, Equations ) ), a sam fom. In mt fnc fam, all vaiabls Engis, 9, 9 stat quations a DC valus und static stat. static woking point asily obtaind. t q m o c d Figu. Rlationshi btwn CM synchonous fnc fam oto fnc fam.. Static Load Capacity Bushlss Doubly Fd Machin Using SSSM pvious sction, lationshi among, toqu spd und statitat statcan canb b asily asily obtaind. obtaind. load load capacity capacity analyzd analyzd applying applying s s lationshi. lationshi. mt Whn mt fnc fam oints on stato CM,, : : mt m t j j ) = ψm + jψ t = ψ + j ) wh stato amplitud CM. To analyz static load capacity, lt wh ψ stato amplitud CM. To analyz static load capacity, lt divativs divativs stats stats a a zo zo in Equation in Equation ), ), static static quations quations a a obtaind obtaind as as follows: follows: ) ll lpmlcm ll pm mt mt l l lcm) + jω j p u p ψ mt l pm l cm ψ + l l pm + u mt = ) ) l pml cm mt ll llcm mt mt l pm l cm ψ j c u mt l l l pm jω c ψ + l l cm ψ mt + u mt = ) l l l pm l pm mt llcm ll ψ mt mt + l l l l ψ + j j. ) λ c c ) = ) wh λ. c slip vlocity stato CM lativ to d ax oto fnc wh c slip vlocity stato CM lativ to ax oto fnc fam. In pactic, CM supplid though a convt stato amplitud CM will b a fam. constant In pactic, a stato CM supplid CM fdback though contol), a convt stato stato amplitud PM connctd CM dictly will to b a constant pow gid. a fo, stato CM amplitud fdback contol), fquncy stato PM a constant, PM connctd a dictly following to pow constaint gid. fo, conditions: amplitud fquncy PM a constant, a following constaint conditions: u u= u m u + u t,, ωp = p p ω + λ. c = π f f p ) ) m t p p c p wh wh f p f p ω p a p a fquncy fquncy lctical lctical angula angula vlocity vlocity PM pow PM supply, pow spctivly. supply, has opation chaactti synchonous moto, whn fquncy PM spctivly. moto spd a constant, has opation slip constant, chaactti fom Equation synchonous ), toqu moto, whn adjustd fquncy angl btwn PM stato moto s spd a CM constant, PM. slip Evn constant, though fom amplituds Equation ), PM toqu CM can kp adjustd constant, angl btwn toqu a stato limitd. s Ow, CM with PM. incas Evn though stato cunt, amplituds PM dop in CM stato can stanc kp constant, PM will lad toqu to a duction limitd. Ow, PM with xciting lvl, incas stato cunt, toqu fu limitd. dop in In stato following stanc dcussion, PM will limitation lad to duction stato PM xciting CM will lvl, not b takn into account. toqu That fu to limitd. say, fo In any ψ following mt dcussion,, ψ, mt always limitation a supply stato CM u mt CM will not b takn into account. That to say, fo any, that mts Equation ), Equation ) nd not b considd. Nvlss, fo a ctain stato CM ψ, spd ω, Equations ), ), ) ), solutions ψ mt its componnts m ax t ax a al numb) a only obtaind in a ctain ang toqu. n w can gt upp low limits s load capacity.

7 Engis, 9, 9 Fo gtting solutions stato PM ψ mt oto, ftly, l multiplying conjugat l j λ. ) c in both sids Equation ), lationship btwn oto, stato s PM CM, slip vlocity givn as: = l l pm. λ c + l l l l j λ. ) c ψ mt + l l cm. λ c + l l l l j λ. ) c ψ 9) Substituting Equation 9) into Equation ), to liminat oto, lctomagntic toqu can b xpssd as: wh ψ = ψ m + ψ t, a = a = } T = a ψ + R a + ja ) ψ mt ψ a ψ ) p p ll. pmλ c l l +. λ c ), a. = p p p c) l l l pml cm λ c. λ c l l + λ. ), c l l +. λ c ), a = p p+p c)l pml cm p c l lcm. λ c l l + λ. ), a c i i =,,, ) a only functions moto paamts slip vlocity, y a constant und static stat. In lctomagntic toqu xpssion Equation ), a fou tms. ft tm a ψ asynchonous toqu PM, fouth tm a ψ asynchonous toqu } CM. scond thid tms R a + ja ) ψ mt ψ a synchonous toqu. It can b sn that, fo a ctain slip vlocity stato amplitud, two asynchonous toqus a constant, always hav opposit dictions. Whn slip vlocity λ. c lativly lag th majoity woking condition ), oto impdanc lativly small lativly small), a, a, a will b small, asynchonous toqu can b nglctd. fo, lctomagntic toqu contolld synchonous toqu, it only dtmind angl btwn two stato s. Scondly, substituting Equation 9) into Equation ), to liminat oto, lationship btwn stato PM stato s obtaind as: u mt = b + jb ) ψ mt + b + jb ) ψ ) wh b =. λ c l l lcm)+ l l l l+ λ. ), b = ωp l l +. ) λ c + ll. pmλ c c l l + λ. ), b c = l pm l cm. λ c l l +. λ c ), b =. l l l pm l cm λ c l l + λ. ), b c i i =,,, ) a only functions moto paamts slip vlocity. Fumo, substituting Equation ) into constaint condition Equation ), lationship btwn stato amplitud PM stato s obtaind as: u = ) b + b ψ + R [b b + b b ) j b b b b )] ψ mt } ) ψ + b + b ψ ) Combining Equations ) ) ltting T = T l und static stat, lctomagntic toqu T quals load toqu T l ), following quation obtaind: a k + k ) k ψ m a k + ψ ) a k k a k k ψ k ψ m + a k k + a k ψ a k ψ T l = ) wh k = a b + b ) a b b + b b ) ψ, k = a b + b ) a b b b b ) ψ, k = b + b ) Tl a u + a b + b ) + a b + b ) ψ, k i i =,, ) a not only functions

8 Engis, 9, 9 moto paamts slip vlocity, but also latd to stato amplitud CM, among m, k also latd to amplitud PM load toqu. Equation ) complicatd difficult to obtain analytical solutions. Howv, fo a givn u, stato CM ψ, spd ω so slip vlocity λ. c also givn) toqu T l, numical solution ψ m can b solvd. n, substituting solvd ψ m into Equation ), ψ t obtaind, substituting solvd ψ m ψ t into Equation 9), obtaind. s solutions mt conditions Equations ), ) ). All al numb solutions constitut possibl static opation ang Stability tmpoaily not considd). Whn toqu aivs at a ctain valu, solvd ψ m not a al numb, that, toqu achs upp limit o low limit static solution ang, cosponding to maximum o minimum toqu. In th way, looking fo static woking ang tansfomd to find al solutions Equation ).. Losing Contol Poblms Convntional Dict Toqu Contol DTC) fo Its Impovd Statgy As mntiond abov, modl complx, in pactic its paamts a difficult to masu. DTC has lss dpndnc on moto paamts, spcifically, it not dpndnt on oto paamts. Th spcially impotant fo contol systm dsign. Howv, compad to induction moto, toqu losing contol poblms s convntional DTC a mo vidnt may xt som tim intvals, wh toqu cannot b contolld simultanously, toqu ippls xcd hysts ing). In [], losing contol poblms a invstigatd, pioity toqu pioity statgis a poposd. Howv, s statgis cannot liminat toqu ippls. In th pap, causs losing contol poblms a analyzd dply, an impovd contol statgy DTC poposd to solv s poblms. In mt fnc fam, CM stato amplitud toqu divativ quations about tim a as follows: dt = T ψ mt dψ mt p cl l cm Im + T d dψ + T = ψ ). d d = p pl pm l Im ) }. = ψ R ) }. ). ψ mt + p p+p c)l pm l cm Im ψ mt ψ mt Substituting Equation ) into Equations ) ), w hav that: dψ = [ α ψ R α R ) dt = α T + α [Im u mt } ) α Im ψ mt } [ α Im +α 9 [Im Im ψ mt u mt wh α = l l cm, α = l pm l cm ψ mt p c ω R u mt u mt, α = l pm l l ) ) }. + α ψ + R ] + p p ω R ψ mt + ) p p + p c ω R ). u mt + α Im ) }] ψ mt ) ) }. ) ψ mt ] ] ) ), α = l l lcm)+ l l l pm)+ l l, α = p cl l cm, α = p c l cm, α = p p l pm, α = p pl l pm, α 9 = p p+p c)l pm l cm, α i i =,,, 9) a only functions moto paamts, indpndnt moto woking points. Obviously, CM stato amplitud toqu divativs hav no lationship with slction coodinat, thus, divativs

9 Engis, 9, 9 9 Equations ) ) can b tansfomd into two-phas static fnc systm αβ fnc fam), so w hav: dψ = [ α ψ R ψ αβ ψ αβ α R ) dt = α T + α [Im u αβ ψ αβ } ) α Im ψ αβ ψ αβ } [ α Im +α 9 [Im Im ψ αβ u αβ ψ αβ ψ αβ p c ω R u αβ ψ αβ ψ αβ u αβ ψ αβ + α ψ + R u αβ ψ αβ ] ) ψ αβ }] ) + α Im ψ αβ ψ αβ } ) + p p ω R ψ αβ ψ αβ }] ] + p p + p c ) ω R Fo static woking points dcussd in pvious sction), toqu moto spd a constant. Using stato CM as fnc initial phas angl stato CM zo), amplituds all vctos, angls among s vctos a const, y can b xpssd as: ψ αβ ψ αβ ) 9) ψ αβ = ψ jωct+δ), ψ αβ = ψ jωct, ψ αβ = ψ jωct+θ), u αβ = u jω ct+δ+γ) ) wh ω c synchonous angula spd stato CM, δ angl btwn stato PM stato CM, θ angl btwn stato CM oto, γ angl btwn stato stato PM. stato CM supplid though a convt, convt vctos a as follows: u αβ = bus jπ/n )) ) wh bus DC bus, n =,,, cospond to six fundamntal nonzo vctos th-phas two lvl convt, six fundamntal vctos two zo vctos). Substituting Equations ) ) into Equations ) 9), w obtain stato amplitud CM toqu divativ quations aound static woking points a as follows: dψ = buscos ω c t π/ n )) + A ) dt wh: = α bus ψ sin π/ n ) ω c t θ) + α 9 bus ψ sin ω c t + δ π/ n )) + B ) A = α ψ cosθ α ψ cosδ + α ψ B = α T + α 9 ψ ψ [ pp ω + p c ω ) cosδ + ωp sin δ + γ) ] α p c ω ψ ψ cosθ α ψ ψ sinθ +α ψ ψ [ ωp sin θ δ γ) p p ω cos θ δ) ] + α ψ ψ sin θ δ) a constant but chang with woking points), y a divativ valus stato amplitud CM toqu whn convt two zo vctos, spctivly. As mntiond abov, if asynchonous toqu nglctd, w hav: dψ = buscos ω c t π/ n )) + A ) wh: dt = α 9 bus ψ sin ω c t + δ π/ n )) + B ) A = α ψ cosδ + α ψ

10 9bus sin ct n B ) wh: A cos Engis, 9, 9 B T 9 pp pc cos p sin [ ) B = α T + α 9 ψ ψ pp ω + p c ω cosδ + ωp sin δ + γ) ] Fo a lag pow moto, stato stanc CM small. Bcaus, a popotional Fo a lag to pow stato moto, stanc stato stanc CM stato CM amplitud small. Bcaus will dcas α, α whn α a convt popotional to stato stanc zo, A always CM small ngativ. stato amplitud stato will amplitud dcas whn CM convt toqu divativs a zo, sinusoidal A always with small synchonous ngativ. angl stato t c, amplitud whn convt CM toqu divativs u a sinusoidal with synchonous angl ω c t, whn convt u diffnt fundamntal vctos n,,, ), lctic angls αβ diffnt fundamntal vctos n =,,, ), lctic angls s divativ cuvs a s divativ cuvs a apat in tun. Fo a givn vcto u apat in tun. Fo a givn vcto u αβ, angl btwn, angl btwn stato amplitud CM divativ cuvs toqu divativ cuvs changs with stato amplitud CM divativ cuvs toqu divativ cuvs changs with vaiation woking points. vaiation woking points. Fo convntional DTC, a six nonzo vctos,. As shown in Figu, Fo convntional DTC, a six nonzo vctos,. As shown in Figu, two-phas two-phas static static fnc fnc fam fam dividd dividd into into six six sctos sctos scto scto I I), I I), ach ach scto scto occupis occupis,, angl φ btwn btwn bounday scto scto I I αα ax ax.. III II I I I Figu.. Six Six sctosin in two-phas static fnc fam. In follow dcussion, atd pow. kw usd as an xampl, paamts In follow dcussion, a atd pow. kw usd as an xampl, paamts which a shown in Appndix A, supply fquncy PM a / which a shown in Appndix A, supply fquncy PM a / Hz. Hz. Using Equations ) ), stato amplitud CM toqu divativ Using Equations ) ), stato amplitud CM toqu divativ cuvs cuvs a shown in Figu a,b.wh stato CM. Wb, moto spd. a shown in Figu a,b.wh stato CM. Wb, moto spd. ad/s ad/s subsynchonous), toqus a Nm motoing mod) Nm gnating subsynchonous), toqus a Nm motoing mod) Nm gnating mod). mod). It cla that divativ cuvs two nonzo vctos with phas It cla that divativ cuvs two nonzo vctos with phas diffnc diffnc a symmtical about lin d A. In ach scto, must b two a symmtical about lin dψ / = A. In ach scto, must b two divativ divativ cuvs, i cuvs, signsi will signs b changd, will b changd, two cosponding two cosponding vctos cannot vctos bcannot slctd, b slctd, bcaus y bcaus cannot y povid cannot a fixd povid spons a fixd diction spons. diction Similaly,. toqu Similaly, divativ toqu cuvs divativ two cuvs fundamntal nonzo two fundamntal vctos nonzo with phas vctos diffnc with a phas symmtical diffnc about a symmtical lin dtabout / = B, lin must dt b two B, toqu must divativ b two cuvs toqu in divativ ach scto, cuvs i in signs ach will scto, b i changd, signs will b two changd, cosponding two cosponding vctos cannot b slctd. vctos cannot In achb scto, slctd. if In abov ach scto, fou if abov vctosfou a not sam, vctos asa shown not in Figu sam, a,b, as shown in will Figu b fou a,b, vctos, will b fou y cannot b vctos, slctd. y Obviously, cannot b slctd. st Obviously, two vctos st cannot two satfyvctos fou cannot kindssatfy contol fou quimnts: kinds contol dcasing quimnts: in dcasing toqu, dcasing in toqu, dcasing incasing in toqu, incasing in in toqu, dcasing incasing in toqu, dcasing incasing toqu, in incasing toqu. in slction toqu. slction vcto can only makvcto losing can contol only mak tim as shot losing ascontol possibl. tim In Figu as shot c,d, as possibl. divativ In Figu cuvs c,d, scto I a nlagd to show gions losing contol mo claly.

11 Engis, 9, 9 Engis, 9, 9 Engis divativ, 9, 9cuvs scto a nlagd to show gions losing contol mo divativ cuvs scto I a nlagd to show gions losing contol mo claly. claly. d d d d, t c t c,, t c t c, dt dt dt dt,, t c t c, t c t c, I II III I I I II III I I d d a) a) I II III I I I II III I I d d b) b) losing contol losing gion contol I gion I losing contol losing gion contol II gion II t c t c,, losing contol losing gion contol I gion I losing contol losing gion contol II gion II t c t c,, c) d) c) d) Figu. Stato CM toqu divativ cuvs light load). a) Motoing mod T = Figu.. Stato Stato CM CM toqu toqu divativ divativ cuvs cuvs light light load). load). a) Motoing a) Motoing mod T mod = Nm); b) Gnating mod T = Nm); c) Patial nlagd T = Nm); d) Patial nlagd T T = Nm); = b) Nm); Gnating b) Gnating mod T mod = Nm); T = c) Patial Nm); nlagd c) Patial T nlagd = Nm); Td) = Patial Nm); nlagd d) Patial T = nlagd Nm). Nm). T = Nm). Thus, vcto switch tabls can b obtaind und motoing mod gnating Thus, vcto switch tabls can b obtaind und motoing mod gnating mod, as shown in Tabls, calld convntional DTC. Fom scto to scto I, mod, as shown in intabls,, calld convntional DTC. DTC. Fom Fom scto scto I to I to scto scto I, I, lctic angls vctos slctd fo sam contol quimnts a apat in tun. lctic lctic angls angls vctos vctos slctd slctd fo fo sam sam contol quimnts a a apat in tun. contol systm shown in Figu. Und motoing mod, will los contol whn contol systm shown in Figu. Und motoing mod, will los contol whn an incas in dcas in toqu, o an incas in an incas in toqu a an incas in a dcas in toqu, o an incas in an incas in toqu a dsid, cosponding to thid fouth ows in Tabl that cicula gion in Figu dsid, cosponding to thid fouth ows in in Tabl that that cicula gion gion in Figu in Figu c). c). Und gnating mod, will los contol whn an incas in dcas in c). Und Und gnating gnating mod, mod, will los will contol los contol whn whn an incas an incas in a dcas a dcas in toqu, in toqu, o an incas in an incas in toqu a dsid, cosponding to thid toqu, an incas o an incas in an incas an incas toqu a in toqu dsid, a cosponding dsid, cosponding to thid to fouth thid ows fouth ows in Tabl that cicula gion in Figu d). fouth in Tabl ows that in Tabl cicula that gion cicula in Figu gion d). in Figu d). *. * Choos Choos * T * T - T - T - - oltag oltag vcto vcto tabl motoing tabl stat) motoing stat) oltag oltag vcto vcto tabl gnating tabl stat) gnating stat) actan actan Flux Flux toqu obsv toqu obsv u, i u, i Figu. Contol systm convntional DTC. Figu Figu.. Contol Contol systm systm convntional DTC. DTC. /Hz /Hz Convt Convt

12 Engis, 9, 9 Tabl. oltag vcto switch tabl motoing mod). Output Hysts Compaato Six Sctos Flux Toqu I II III I I Not: indicat incasd dcasd o toqu, spctivly. Tabl. oltag vcto switch tabl gnating mod). Output Hysts Compaato Six Sctos Flux Toqu I II III I I Not: indicat incasd dcasd o toqu, spctivly. Fom Figu a,b, it can b also sn that if angl φ incasd, losing contol can b liminatd und action vcto, but losing contol will b sious und action vcto. On contay, if angl φ dcasd, losing contol can b liminatd und action vcto, but losing contol will b sious und action vcto. fo, it uslss to solv losing contol changing angl φ. Bcaus DC componnt A amplitud divativ lativly small in gnal situations, tim intval losing contol shot, ang fluctuation small. Whn load light, toqu will not los contol. With incas in toqu, DC componnts A B a incas, phas angls divativ cuvs main unchangd, toqu divativ cuvs mov ight o lft motoing o gnating mods) lativ to divativ cuvs. In Figu a,b, stato CM toqu divativ cuvs a shown, wh stato CM. Wb, moto spd. ad/s subsynchonous), toqus a Nm motoing mod) 99 Nm gnating mod). vcto switch tabls can b obtaind, which a sam as Tabls. Whn load havy, toqu will all los contol. Und motoing mod, toqu will los contol whn a dcas in an incas in toqu, o an incas in a dcas in toqu a dsid, cosponding to scond thid ows in Tabl that losing contol gions I II in Figu a). Und gnating mod, toqu will los contol whn a dcas in a dcas in toqu, o an incas in an incas in toqu a dsid, cosponding to ft fouth ows in Tabl that losing contol gions II I in Figu b). Similaly, fom Figu a,b, it can b also sn that if angl φ incasd, losing contol can b liminatd und action vcto, but losing contol will b sious und action vcto, toqu losing contol will also b sious und action vctos. On contay, if angl φ dcasd, losing contol can b liminatd und action vcto, toqu losing contol also can b liminatd und action vcto, but losing contol will b mo sious und action vcto. fo, losing contol poblms convntional DTC cannot b solvd changing angl φ.

13 Engis, 9, 9 Engis Engis,, 9, 9, 9 9 d d, t c, t c, t c, t c losing contol losing contol gion II gion II losing contol losing contol gion I gion I dt dt I II III I I I II III I I a) a),, t c t c losing contol losing contol gion II gion II losing contol losing contol gion I gion I dt dt I II III I I I II III I I b) b), t c, t c Figu. Stato CM toqu divativ cuvs havy load). a) Motoing mod T Figu.. Stato CM CM toqu toqu divativ divativ cuvs cuvs havy havy load). load). a) Motoing a) Motoing mod 9 T mod = 9 Nm); b) Gnating mod T Nm); T 99 Nm). = 9 b) Nm); Gnating b) Gnating mod T mod = 99 T Nm). = 99 Nm). Fom Fom it can b sn that DTC loss Fom analys analys abov, abov, it it can can b b sn sn that that s s convntional convntional DTC DTC asily asily loss loss contol, contol, bcaus a wid. bcaus slctd slctd vcto vcto a a fw fw scto scto lativly lativly wid. wid. losing losing contol contol poblms b o angl poblms cannot cannot b b impovd impovd incasing incasing switch switch sctos, sctos, o o changing changing angl angl φ btwn btwn ax. And DC bus tim bounday bounday scto scto I α ax. ax. And And incasing incasing DC DC bus bus,, tim tim losing losing contol contol will will but not fo will dcas, dcas, but but losing losing contol contol phnomna phnomna would would not not dappa. dappa. fo, fo, fo fo s s DTC DTC DTC xt a contol contol systm, systm, toqu toqu losing losing contol contol phnomnon phnomnon always always xt xt a invitabl. a invitabl. Without nw a six Without changing changing convt convt topology, topology, nw nw vctos vctos a a synsizd synsizd six six As in,, fundamntal fundamntal vctos. vctos. As As shown shown in in Figu Figu,,, synsizd synsizd,,, synsizd synsizd,,, synsizd synsizd,,, synsizd synsizd. duty, atio two, synsizd synsizd, to vy, synsizd synsizd. a duty. atio duty two atio fundamntal two fundamntal vctos latd vctosto latd vy tosynsizd vy synsizd vcto a vcto % a. %. six six six a fundamntal six fundamntal vctos vctos six six synsizd synsizd vctos, vctos, a atwlv twlv vctos. At sam tim, also to ach At At sam tim, switch sctos also alsoincas to totwlv, ach achscto occupis, angl angl φ. following will will point point out out that that losing contol poblms can can b b solvd incasing numb vctos switch sctos simultanously, combind with with appopiat choic angl angl φ. φ. I I III III I I II II II II III I III XII XII IX IX XI X XI Figu. Twlv sctos in two-phas static fnc fam. Figu.. Twlv sctosin two-phas static fnc fam. Aft Aft adding synsizd vctos, stato amplitud CM CM toqu Aft addinga synsizd in a,b, vctos, stato amplitud CM. Wb, CM toqu divativ cuvs a shown in Figu a,b, wh stato CM. Wb, moto spd divativ. ad/s cuvs a shown in Figu. ad/s subsynchonous), toqus wh a a stato Nm Nm 9 9 Nm Nm CM light light. load load moto havy spd load load und. ad/s motoing subsynchonous), mod), angl angl toqus σ a = Nm has has bn bn 9 chosn Nm light poply. load havy load und motoing mod), angl φ = σ = has bn chosn poply.

14 Engis, 9, 9 Engis, 9, 9 Engis, 9, 9 d d d d t c, t c,,, c t c t dt dt,, t c t c dt dt,, t c t c I II III I I II III IX X XI XII I II III I I II III IX X XI XII I II III I I II III IX X XI XII I II III I I II III IX X XI XII a) b) a) b) Figu. Stato CM toqu divativ cuvs. a) Motoing mod T = Nm); b) Figu. Stato CM toqu divativ cuvs. a) Motoing mod T = Nm); b) Motoing Figu. mod StatoT = 9 Nm). CM toqu divativ cuvs. a) Motoing mod T = Nm); Motoing mod T = 9 Nm). b) Motoing mod T = 9 Nm). Similaly, in ach scto, a also two amplitud two toqu divativ cuvs, i Similaly, in ach scto, a also two amplitud two toqu divativ cuvs, i signs signs Similaly, chang invitably, chang invitably, ach scto, cosponding cosponding a also two fou fou amplitud vctos vctos cannot cannot two toqu b slctd. b slctd. divativ Among Among cuvs, st i st signs ight ight chang invitably, vctos, vctos, vctos, cosponding y can vctos, y can fou satfy satfy vctos contol contol cannot quimnts quimnts b slctd. Among toqu toqu st i ight i divativ divativ vctos, cuvs cuvs a a vctos, fa fa y fom fom can satfy hoizontal hoizontal contol ax, ax, will will quimnts b b slctd. slctd. In In th th way, way, toqu slction slction i divativ cuvs vcto vcto a maks maks fa that that fom contol contol hoizontal ffct ffct ax, mo mo willobvious b slctd. good good Inffctivnss th way, slction quick quick adjustmnt), adjustmnt), vcto maks that ffctivnss contol ffct can can b b mo maintaind obvious stong good ffctivnss obustnss) whn whn quick adjustmnt), opating opating conditions ffctivnss load load toqu, toqu, can spd, spd, b maintaind tc.) tc.) chang. stong n twlv obustnss) sctos whn vcto opating switch conditions tabls tabls can can load b b obtaind, toqu, obtaind, spd, as as shown tc.) shown chang. in in Tabls n twlv,, sctos calld SDTC vcto contol switch statgy. tabls can Fom b obtaind, scto I asto I shown to scto in Tabls scto XII, XII, lctic, angls calld SDTC slctd contol statgy. vctos Fom fo scto sam contol I to quimnts scto XII, a a lctic apat angls apat in tun. in slctd tun. If If angl vctos φ φ still still fo, sam, scto contol I as quimnts an xampl, a slctd apat in tun. Ifvctos angl can can φ satfy still satfy contol, scto quimnts I as an xampl, toqu, but slctd will los vctos contol can satfy und action contol quimnts vcto vcto toqu,.. Bcaus but DC will DC componnt los contol A und divativ action lativly vcto small in in gnal. Bcaus situations, DC only componnt only a small a small Aadjustmnt divativ φ φ ndd lativly in in bas small in gnal. situations, analys only a gnating small adjustmnt mod simila φ imila ndd to to in motoingmod, bas. figus analys its its divativ gnating cuvs mod a omittd simila h. to motoing contol systm systm mod, SDTC figus SDTC shown itsshown divativ in in Figu Figu cuvs a omittd h. contol systm SDTC shown in Figu 9. * * * T * T T T Twlv sctos Twlv sctos vcto vcto tabl motoing tabl stat) Choos Tiangula wav motoing stat) Choos Tiangula wav Twlv sctos Twlv sctos Tiangula wav vcto Tiangula wav vcto tabl gnating tabl stat) gnating stat) actan actan u, Flux i toqu u, Flux obsv i toqu u, i obsv Convt Convt /Hz /Hz Figu 9. Contol systm SDTC. Figu Figu9. 9. Contol Contolsystm systm SDTC. following simulation will illustat that SDTC not only can solv toqu losing following contol poblms simulation will convntional illustat that DTC, but SDTC also can not mak only can solv toqu ach toqu losing otical following contol poblms simulation capacity will convntional limits. illustat that SDTC not only can solv toqu DTC, but also can mak toqu ach losing contol poblms convntional DTC, but also can mak toqu ach otical capacity limits. otical capacity limits.

15 Engis, 9, 9 Engis, 9, 9 Tabl. Twlv sctos vcto switch tabl motoing mod). Tabl. Twlv sctos vcto switch tabl motoing mod). Output Hysts Compaato Twlv Sctos Output Flux Hysts Toqu Compaato I II III I Twlv I Sctos II III IX X XI XII Flux Toqu I II III I I II III IX X XI XII Tabl. Twlv sctos vcto switch tabl gnating mod). Output Output Hysts Hysts Compaato Twlv Sctos Flux Flux Toqu Toqu I I II II III III I I I II II III IX X XI XI XII XII. Simulation Expimnts..Numical Calculation Static Opation Rangs Appndix A still still usd usd as as an an xampl, supply supply fquncy fquncy PM a PM a / / Hz, ahz, spcific a spcific calculation calculation sult sult shown shown in Figu in. Figu It can. bit sn can that b sn ang that possibl ang maximum possibl maximum toqu dcass toqu gadually dcass with gadually incas with in moto incas spdin whn moto spd stato whn amplitud stato amplitud CM constant. CM ang constant. possibl ang maximum possibl maximum toqu ft incass toqu ft n incass dcass n gadually dcass withgadually incas with in stato incas in amplitud stato amplitud CM whn moto CM spd whn constant. moto spd Whn constant. stato Whn amplitud stato CM amplitud incass to. CM Wb, incass maximum to. Wb, toqu maximum toqu achs pak valuachs Nm. Compad pak valu with maximum Nm. Compad toqu, with ang maximum possibl toqu, minimum ang toqu possibl incass minimum gadually with toqu incass in gadually stato with amplitud incas in stato CM, dcass amplitud gadually CM, with dcass incas gadually in spd. with incas in spd. T l Nm Wb ad s Figu. Maximum minimum toqu cuv sufacs a. Fo lag pow moto, situation abov somwhat diffnt. In th cas, stato Fo lag pow moto, situation abov somwhat diffnt. In th cas, stato oto stancs a vy small, angs possibl maximum minimum toqu will oto stancs a vy small, angs possibl maximum minimum toqu will tnd to b qual, indpndnt moto spd, incas with incas in stato tnd to b qual, indpndnt moto spd, incas with incas in stato amplitud CM. amplitud CM. Obviously, stabl woking points must locat in intio maximum Obviously, stabl woking points must locat in intio maximum minimum toqu cuvs, pcntag stabl opation ang latd to contol minimum toqu cuvs, pcntag stabl opation ang latd to contol mthod. Whn stato amplitud CM. Wb, maximum minimum toqu mthod. Whn stato amplitud CM. Wb, maximum minimum toqu cuvs, stability opation angs undconstant /f opn loop contol und convntional DTC a shown in Figu. As w sn that, using constant /f opn loop contol, stability opation ang na synchonous spd, occupis a small pat nti

16 Engis, 9, 9 Engis, 9, Tl Nm Tl Nm cuvs, stability opation angs und constant /f opn loop contol und static solution ang, using convntional DTC, toqu will los contol bfo it achs limits convntional DTC a shown in Figu. As w sn that, using constant /f opn loop contol, Engis static, solution 9, 9 gion. stability opation ang na synchonous spd, occupis a small pat nti static static solution ang, using convntional toqu willlos loscontol contol bfo it achs limits using solution ang, convntionaldtc, DTC, toqu will bfo it achs limits stability opation ang maximum toqu lin upp limit static static solution gion. und /f contol static solution gion. solution gion) stability opation ang toqu wold not los und contol /f contol maximum toqu lin upp limit static solution gion) und convntional DTC minimum toqu lin low limit static solution gion) ad s minimum toqu lin toqu wold not los contol und convntional DTC low limit static solution gion) Figu. Stability opation angs undconstant /f contol und convntional DTC. ad s Figu.Syntic Stability opation angs undconstant /f contol und convntional DTC... Simulation TstStability cto Dict Toqu Contol Figu. opation angs und constant /fsdtc) contol und convntional DTC...dplay Simulation Syntic cto Dict Toqu Contol SDTC) To Tst pfomanc poposd nw contol statgy SDTC, simulation tsts hav.. Simulation Tst Syntic cto Dict Toqu Contol SDTC) bn caid contol pfomancs convntional DTC SDTC a compad und To out. dplay pfomanc poposd nw contol statgy SDTC, simulation tsts hav vaious kinds opating conditions. pow moto connctd to a / Hz pow supply, To dplay pfomanc poposd nw contol statgy SDTC, simulation tsts bn caid out. contol pfomancs convntional DTC SDTC a compad undhav contol moto contol supplid though bidg convt, vaious kinds opating conditions. a pow moto connctd to adc / Hz pow supply, bn caid out. pfomancs convntional DTC SDTC aconvt compad contol moto supplid though a bidg Hz stato obsvs PMconvt, moto CM us to cunt modls, und. vaious kinds opating conditions. pow DC connctd aconvt / stato. acm usconvt, cunt modls, pow supply, contolobsvs motou supplid though i PM. In od tobidg ovcom poblm DC dc difts u i u obsvs i. stato In od to ovcom poblm dc difts convt PM CM us cunt modls, u i. s a stimatd a αβ pu intgal, stato a adaptiv compnsation intgato, wh R αβ αβ R αβ αβ αβ intgal, stato a stimatd ia adaptiv compnsation intgato, wh dc ψ =a puu i s ψ = u. In od to ovcom poblm cutf angula vlocity low-pass filt CF ad/s, PI paamts gulato a CF ad/s, angula vlocity low-pass PIcompnsation paamts intgato, gulato a difts cutf a pu intgal, stato sfilt a stimatd a adaptiv wh kp. k, [9]. toqu calculatd coss poduct s cunts, I kp. [9]., ki vlocity toqu filt calculatd coss poduct s cunts, cutf angula low-pass ωcf ad/s, PI paamts gulato * * = a k =., k [9]. toqu calculatd coss poduct s cunts, * * T pptpim I i n i i. loop adoptd, adoptd, paamts o spd i opc Im pn pc Im. spd out out loop PI PI paamts αβ p Im αβ αβ αβ T = p p Im ψ i + pc Im ψ i. spd out loop adoptd, PI paamts kp k, k,i k. upp spd loopa ak a low limits toqu hysts loop. upp upp limits low limits toqu toqu hysts I spdspd loop low hysts compaatos p =, kp I =. compaatosa paamts paamts paamts shown in in Appndix A. A. compaatosa...,.. shown a a shown Appndix a..,.., a in Appndix A. DTC SDTC......SimulationWavfoms Simulation Wavfoms DTC DTC SDTC...SimulationWavfoms SDTC i /A i /A i /A b) - b) Figu. Cont. i /A ψ /Wbψ /Wb i /A i /A i /A a) i /A i /A ψ /Wb i /A i /A i /A ω /ad/s) ω /ad/s) ω /ad/s) a) - ψ /Wb ω /ad/s) Figus dplay simulation simulation sults convntional DTCDTC SDTC, spctivly. In Figus dplay dplay sults convntional SDTC, spctivly. Figus simulation sults convntional DTC SDTC, spctivly. In th simulation, fnc valu. Wb, fnc valu. ad/s o ψ ω. ad/s o In th simulation, fnc valu. Wb, fnc valu.. ad/s o th simulation, fnc valu Wb, fnc valu ad/s subsynchonous o supsynchonous) und diffnt load toqu. ad/s subsynchonous o supsynchonous) diffnt load toqu. ad/s subsynchonous o supsynchonous) undund diffnt load toqu.

17 Engis, 9, 9 Engis, 9, c) c) ) ) f). i /Ai /A i /Ai /A.. f) d) d) g) i /Ai /A ψ /Wb ψ /Wb i /Ai /A ψ /Wb ψ /Wb i /Ai /A. ω /ad/s) ω /ad/s) i /Ai /A ω /ad/s) ω /ad/s) Engis, 9, 9 h) g) h) a) a). c) c) i /Ai /A Figu. Cont i /Ai /A i /Ai /A i /Ai /A b) b) i /Ai /A i /Ai /A. 9 9 i /Ai /A i /Ai /A ω /ad/s) ω /ad/s) ψ /Wb ψ /Wb.. ω /ad/s) ω /ad/s) ψ /Wb ψ /Wb i /Ai /A ψ /Wb ψ /Wb ω /ad/s) ω /ad/s) i /Ai /A. ψ /Wb ψ /Wb i /Ai /A i /Ai /A ω /ad/s) ω /ad/s) Figu. Simulation wavfoms convntional DTC. a). ad/s motoing b) Figu. Simulation wavfoms convntional DTC. a). ad/s motoing Figu.ad/s Simulation wavfoms DTC. a)gnating. ad/s motoing motoing c)convntional. ad/s d) b) ad/s b) ad/s motoing c). ad/s gnating d) ad/s motoing f). ad/s gnating h) ad/s gnating ).c) ad/s; ad/s; g).d) ad/s; ad/s gnating ). ad/s; f) ad/s; g). ad/s; gnating ). ad/s; f) ad/s; g). ad/s; h) ad/s. h) ad/s. ad/s. d) d)

18 . x Engis, 9, 9 Engis, 9, 9 /Nm T l /Nm T l /Nm T l ) f) g) h) Figu Figu.. Simulation Simulation wavfoms wavfoms SDTC. SDTC. a) a).. ad/s ad/s motoing motoing b) b) ad/s ad/s motoing motoing c) c).. ad/s ad/s gnating gnating d) d) ad/s ad/s gnating gnating ) ).. ad/s; ad/s; f) f) ad/s; ad/s; g) g).. ad/s; ad/s; h) h) ad/s. ad/s. Und fou conditions diffnt toqus diffnt spds, patial nlagd Und fou conditions diffnt toqus diffnt spds, patial nlagd simulation simulation wavfoms convntional DTC a illustatd in Figu a d. It can b sn that, wavfoms convntional DTC a illustatd in Figu a d. It can b sn that, dviations dviations toquwill xcd upp low limits hysts compaato, spd toqu will xcd upp low limits hysts compaato, spd fluctuation appas fluctuation appaswhn toqu loss contol. And toqu losing contol has two foms, fom whn toqu loss contol. And toqu losing contol has two foms, fom littl to big, littl to big, fom big to littl, which associatd with diffnt chang tnds toqu fom big to littl, which associatd with diffnt chang tnds toqu divativ cuvs in divativ cuvs in losing contol gion, as shown in Figu a,b. Figu h show toqu losing contol gion, as shown in Figu a,b. Figu h show toqu wavfoms in a wavfoms in a lctic angls und fou wok conditions, a six toqu ippls lctic angls und fou wok conditions, a six toqu ippls in ach m. With in ach m. With incas in toqu, toqu losing contol bcoms sious incas in toqu, toqu losing contol bcoms sious continus fo a long tim, continus fo a long tim, maximum toqu ippl achs 9 maximum popotion maximum toqu ippl achs 9 maximum popotion losing contol tim ov %. losing contol tim ov %. patial nlagd simulation wavfoms SDTC a illustatd in Figu a d. It can b patial nlagd simulation wavfoms SDTC a illustatd in Figu a d. It can sn that SDTC can ffctivly solv losing contol poblms, wh moto opats in b sn that SDTC can ffctivly solv losing contol poblms, wh moto opats motoing o gnating mods, it maks dviations stato CM toqu a in motoing o gnating mods, it maks dviations stato CM toqu stictd in hysts ing wih simultanously. moto spd stato amplitud a stictd in hysts ing wih simultanously. moto spd stato amplitud CM poply follow fnc valu. Whn a fundamntal vcto slctd, until CM poply follow fnc valu. Whn a fundamntal vcto slctd, until dviations o toqu ach limits hysts compaatos, vcto will switch dviations o toqu ach limits hysts compaatos, vcto onc. In th cas, switching fquncy latd to hysts ing wih. Whn a synsizd will switch onc. In th cas, switching fquncy latd to hysts ing wih. Whn a vcto slctd, two latd fundamntal vctos a modulatd, duty atio synsizd vcto slctd, two latd fundamntal vctosa modulatd, two fundamntal vctos a %, modulation fquncy Hz. In th duty atio two fundamntal vctos a %, modulation fquncy Hz. cas, switching fquncy fixd, until hysts compaato stat changs o In th cas, switching fquncy fixd, until hysts compaato stat changs o vcto slctd. Figu h also show toqu wavfoms sval piods und fou vcto slctd. Figu h also show toqu wavfoms sval piods und conditions. Th dmonstats that ffctivnss can b maintaind in a wid ang SDTC fou conditions. Th dmonstats that ffctivnss can b maintaind in a wid ang contol statgy whn opating conditions load toqu, spd) chang. poposd SDTC SDTC contol statgy whn opating conditions load toqu, spd) chang. poposd ffctiv in coping with losing contol poblms. stability obustnss contol systm SDTC ffctiv in coping with losing contol poblms. stability obustnss contol a impovd. systm a impovd.... Spd Comm Load Toqu Dtubanc Rsponss SDTC... Spd Comm Load Toqu Dtubanc Rsponss SDTC Figu shows dynamic spons to a stp spd comm. In th simulation, fnc Figu valu shows ψ dynamic spons to a stp spd comm. In th simulation,. Wb, fnc valu ω changs fom. ad/s to ad/s. fnc s, valu load toqu. Wb, PI fnc gulato valu limiting changs valus fom a. ± ad/s to maximum ad/s toqu at. s, Nm at load toqu ad/s whn PI gulato. Wb). It can blimiting sn that, valus whn a ± spd comm maximum changs toqu fom Nm subsynchonous at ad/s whn to supsynchonous,. Wb). It can actual b sn spd that, apidly whn follows spd comm fnc valu changs without fom ovshoots, subsynchonous to supsynchonous, cunt phas squnc actual CM spd changs. apidly follows moto acclats fnc at valu without n ovshoots, toqu tuns cunt backphas Nm squnc aft appoximatly CM changs.. moto acclats at Figu shows n toqu gulating tuns pfomanc back Nm aft appoximatly to a stp. chang s. in load toqu. In th simulation, Figu shows fnc gulating valu ψ pfomanc. Wb, fnc valu to a stp ωchang.in ad/s, load toqu. In load th toqu simulation, changs fom fnc Nmvalu to Nm at.. s. Wb, systm fnc stability valu validatd und. ad/s, a stp load toqu load dtubanc. toqu changs fom spd Nm tuns to tonm at fnc. s. spd systm aft stability sval dynamic validatd fluctuations, und a stp load toqu spd gulato dtubanc. abl tospd compnsat tuns fo to load fnc toquspd dtubanc aft sval in a timdynamic piod fluctuations, appoximatly. s. spd gulato abl to compnsat fo load toqu dtubanc in a tim piod appoximatly. s. - - /Nm T l - -

19 i i/a /A TTl /Nm l /Nm ψψ/wb /Wb i i /A/A ii/a/a /ad/s) ωω /ad/s) Engis, 9, 9 Engis, 9, 9 Engis, 9, TTl /Nm l /Nm i i/a /A ψψ /Wb /Wb i i /A/A ii/a/a /ad/s) ωω /ad/s) Figu Figu.. Spd Spd changs changs fom fom.. ad/s ad/sto to ad/s ad/sat Nmload. load. Figu. Spd changs fom. ad/s to ad/s atat Nm Nm load Figu. Load toqu changs fom Nm to Nm at. ad/s. Figu Figu.. Load Load toqu toqu changs changs fom fom Nm Nm to to Nm Nm at at.. ad/s. ad/s. Figus also show stato cunts PM CM, oto cunt. Figus Figus also show stato statocunts cunts PM PM CM, CM, oto cunt. Figus Figus also show oto cunt. Figus dmonstat that systm SDTC has a good spons chaactti fo stp spd dmonstat that systm systmsdtc SDTChas hasa agood goodspons sponschaactti chaactti fo stp spd dmonstat that fo stp spd comm, aa good gulating pfomanc und stp load toqu dtubanc. stato comm, good gulating pfomanc und stp load toqu dtubanc. stato comm, a good gulating pfomanc und stp load toqu dtubanc. stato amplitud CM follows fnc valu wll, it almost unaffctd vaiations amplitud CM follows fnc valu wll, it almost unaffctd vaiations amplitud CM follows fnc valu wll, it almost unaffctd vaiations spd toqu. spd spd toqu. toqu....output Capacity SDTC Systm...Output... OutputCapacity Capacity SDTC SDTC Systm Systm In addition to losing contol poblm, systms convntional DTC CM constant /f In Inaddition addition to to losing losing contol contol poblm, poblm, systms systms convntional convntional DTC CM constant /f /f opn loop contol will los stability whn load toqu incass to aa ctain valu. Fo xampl, opn opnloop loopcontol contolwill willlos losstability stabilitywhn whn load loaoqu toquincass incassto to actain ctainvalu. valu.fo Foxampl, xampl, und DTC, whn.. if load. und convntional DTC,whn whn ψ Wb, spd spd. ad/s, ad/s, if toqu load und convntional convntional DTC, Wb,Wb, spd ω. ad/s, if load. toqu xcds toqu will divg. toqu xcds toqu will divg. xcds toqu will divg. SDTC SDTCnot not only only can can solv solv toqu toqu losing losing contol contol poblms poblms convntional convntional SDTC not only can solv toqu losing contol poblms convntional DTC, but also can mak toqu ach otical capacity limits. In DTC,but butalso alsocan canmak mak toqu toqu ach otical capacity limits. DTC, ach otical capacity limits. In od to validat th conclusion, simulation tsts hav bn caid out. In od to validat conclusion, simulation caid od to validat thth conclusion, simulation tstststs havhav bnbn caid out. out. In.. ad/s, Inth thsimulation, simulation, fnc fncvalu valu.. Wb, Wb, fnc fncvalu valu ad/s, ψ ω In th simulation, fnc valu. Wb, fnc valu ad/s, load toqu changd gadually. Accoding to Figu, maximum toqu load load toqu toqu changd changd gadually. gadually. Accoding Accodingto tofigu Figu,, maximum maximum toqu toqu 9 Nm und woking conditions. Figu shows that actual toqu 9 Nm und abov woking conditions. Figu shows that actual toqu 9 Nm und abov woking conditions. Figu shows that actual toqu always abl follow fnc toqu whn load toqu changs fom Nm Nm. always alwaysabl abltoto tollow followfnc fnctoqu toquwhn whn load loaoqu toquchangs changsfom fom Nm Nmtoto to Nm Nmatat at..s, s, n changs fom Nm to Nm at s. intio maximum minimum n changs fom Nm to Nm at s. In maximum minimum s, n changs fom Nm to Nm at s. In intio maximum minimum toqucuvs cuvs Figu Figu,, stting sttingsom somtst tstpoints pointson on boundais, boundais,w wfind finhat that sults sultsa a toqu toqu cuvs Figu, stting som tst points on boundais, w find that sults a sam, wavfoms wavfomsa a omittd omittd h. h. SDTC SDTCcan canmak mak toqu toqu sam, sam, wavfoms a omittd h. SDTC can mak toqu ach otical otical capacity capacity limits, limits, but but convntional convntional DTC DTC cannot. cannot. ach ach otical capacity limits, but convntional DTC cannot.

20 Engis, 9, 9 Engis, 9, 9 /ad/s) ω /Wb ψ... /Nm T l /A i i /A i /A Conclusions Figu. Simulation wavfoms SDTC at limit toqu. Fo diffnt oto stuctus, mthod taking conjugation tansfomation complx vaiabls CM o PM has univsality. Using th mthod, synchonous fnc fam stat-spac modl, calld SSSM, obtaind in th pap. By dsciption SSSM, analys mthod capacity limits givn usd to tst pfomanc contol statgy. impovd DTC statgy, calld SDTC, can solv losing poblms toqu, kp advantag convntional DTC, such as simpl stuctu, lss dpndnc on moto paamts, stong obustnss. dynamic pfomanc SDTC tstd ov a wid ang spds fom subsynchonous to supsynchonous, also und conditions suddn load chang. otical capacity limitscan canb b achd SDTC contol statgy. s sults hav significant maning fo s dsign dvlopmnt also fo contol statgy study. Autho Contibutions: mthods conjugation conjugation tansfomation tansfomation synchonous synchonous fnc fam fnc stat-spac fam modl, statspac analys modl, mthod analys mthod capacity limitscapacity limits SDTC SDTC blong to Chaoying blong Xia. to Chaoying divation Xia. divation capacity latd capacity sultslatd sults all simulations all a simulations compltd a compltd Xiaoxin Hou. Xiaoxin Both Hou. authos wot vd manuscipt. Both authos wot vd manuscipt. Conflicts Intst: authos dcla no conflict intst. Conflicts Intst: authos dcla no conflict intst. Appndix A Appndix A paamts: paamts: p p =, p c =, =. Ω, =. Ω, l =. H, l =. H, l pm =. H, l cm =. H, pp, pc =,. Ω,. Ω, l =. H, l =. H, lpm =. H, lcm =. H, =. =. Ω, l l =.9 H, J =. kg m. Ω, ll =.9 H, J =. kg m. s.. Zhang, Zhang, A.L.; A.L.; Wang, Wang, X.; X.; Jia, Jia, W.X; W.X.; Ma, Ma, Y. Y. Indict Indict stato-quantitis stato-quantitis contol contol fo fo bushlss bushlss doubly doubly fd fd induction induction machin. machin. IEEE IEEE Tans. Tans. Pow Pow Elcton. Elcton.,,,, [CossRf]. Abdi, S.; Abdi, E.; Oa, A.; McMahon, R. Equivalnt cicuit paamts fo lag bushlss doubly fd. Abdi, S.; Abdi, E.; Oa, A.; McMahon, R. Equivalnt cicuit paamts fo lag bushlss doubly fd machins bdfms). IEEE Tans. Engy Convs.,,. [CossRf] machins bdfms). IEEE Tans. Engy Convs.,,.. Hashmnia, M.N.; Tahami, F.; Oyabid, E. Invstigation co loss ffct on stady-stat chaactti. Hashmnia, M.N.; Tahami, F.; Oyabid, E. Invstigation co loss ffct on stady-stat chaactti invt fd bushlss doubly fd machins. IEEE Tans. Engy Convs.,,. [CossRf] invt fd bushlss doubly fd machins. IEEE Tans. Engy Convs.,,.. Zhao, R.L.; Zhang, A.L.; Ma, Y.; Wang, X.; Yan, J.; Ma, Z.Z. dynamic contol activ pow fo. Zhao, R.L.; Zhang, A.L.; Ma, Y.; Wang, X.; Yan, J.; Ma, Z.Z. dynamic contol activ pow fo bushlss doubly fd induction machin with indict stato-quantitis contol schm. IEEE Tans. bushlss doubly fd induction machin with indict stato-quantitis contol schm. IEEE Tans. Pow Pow Elcton., 9,. [CossRf] Elcton., 9,.. Xiong, F.; Wang, X.F. Dsign a low-hamonic-contnt wound oto fo bushlss doubly fd gnato.. Xiong, F.; Wang, X.F. Dsign a low-hamonic-contnt wound oto fo bushlss doubly fd IEEE Tans. Engy Convs.,,. [CossRf] gnato. IEEE Tans. Engy Convs.,,.. McMahon, R.; Tavn, P.; Abdi, E.; Mallib, P.; Bak, D. Chaacting bushlss doubly fd machin otos.. McMahon, R.; Tavn, P.; Abdi, E.; Mallib, P.; Bak, D. Chaactingbushlss doubly fd machin IET Elct. Pow Appl.,,. [CossRf] otos. IET Elct. Pow Appl.,,.. Shao, S.Y.; Abdi, E.; Baati, F.; McMahon, R. Stato--ointd vcto contol fo bushlss doubly fd induction gnato. IEEE Tans. Ind. Elcton. 9,,.

21 Engis, 9, 9. Shao, S.Y.; Abdi, E.; Baati, F.; McMahon, R. Stato--ointd vcto contol fo bushlss doubly fd induction gnato. IEEE Tans. Ind. Elcton. 9,,. [CossRf]. Shao, S.Y.; Abdi, E.; McMahon, R. Opation bushlss doubly-fd machin fo div applications. In Pocdings Fouth IET Confnc on Pow Elctoni, Machins Divs, Yok, U, Apil ; pp.. 9. Jallali, F.; Bouchhima, A.; Masmoudi, A. Stady-stat opation bushlss cascadd doubly fd machin: A suvy. COMPEL Int. J. Comput. Math. Elct. Elcton. Eng.,,. [CossRf]. Robts, P.C.; McMahon, R.A.; Tavn, P.J.; Macijowski, J.M.; Flack, T.J. Equivalnt cicuit fo bushlss doubly fd machin ) including paamt stimation xpimntal vification. IEE Poc. Elct. Pow Appl.,, 9 9. [CossRf]. Li, R.; Wallac, A.; Sp, R.; Wang, Y. Two-ax modl dvlopmnt cag-oto bushlss doubly-fd machins. IEEE Tans. Engy Convs. 99,,. [CossRf]. Zhou, D.S.; Sp, R. Synchonous fam modl dcoupld contol dvlopmnt fo doubly-fd machins. In Pocdings th Annual IEEE Pow Elctoni Spcialts Confnc, Taipi, China, Jun 99.. Zhou, D.; Sp, R.; Alx, G.C.; Wallac, A.. A simplifid mthod fo dynamic contol bushlss doubl-fd machins. In Pocdings 99 IEEE IECON nd Intnational Confnc on Industial Elctoni, Contol, Instumntation, Taipi, China, August 99; pp Poza, J.; Oyabid, E.; Roy, D. Nw vcto contol algoithm fo bushlss doubly-fd machins. In Pocdings th Annual Confnc IEEE Industial Elctoni Socity, Svill, Spain, Novmb ; pp... Poza, J.; Oyabid, E.; Roy, D.; Rodiguz, M. Unifid fnc fam modl bushlss doubly fd machin. IEE Poc. Elct. Pow Appl.,,. [CossRf]. Li, R.; Wallac, A.; Sp, R. Dtmination convt contol algoithms fo bushlss doubly-fd induction moto divs using Floqut Lyapunov tchniqus. IEEE Tans. Pow Elcton. 99,,.. Saasola, I.; Poza, J.; Oyabid, E.; Rodiguz, M.A. Stability analys a bushlss doubly-fd machin und closd loop scala cunt contol. In Pocdings nd Annual Confnc on IEEE Industial Elctoni, Pa, Fanc, Octob.. Poza, J.; Oabid, E.; Roy, D.; Saasola, I. Stability analys a und opn-loop contol. In Pocdings Euopan Confnc on Pow Elctoni Applications, Dsdn, Gmany, Sptmb. 9. Williamson, S.; Fia, A. Gnald oy bushlss doubly-fd machin. Pat : Modl vification pfomanc. IEE Poc. Elct. Pow Appl. 99,, 9. [CossRf]. McMahon, R.A.; Robts, P.C.; Wang, X.; Tavn, P.J. Pfomanc as gnato moto. IEE Poc. Elct. Pow Appl.,, [CossRf]. Robts, P.C.; Flack, T.J.; Macijowski, J.M.; McMahon, R.A. Two stabiling contol statgis fo bushlss doubly-fd machin ). In Pocdings Intnational Confnc on Pow Elctoni, Machins Divs, Bath, Bitain, Apil ; pp... Xia, C.Y.; Guo, H.Y. Fdback linaization contol appoach fo bushlss doubl-fd machin. Int. J. Pc. Eng. Manuf.,, [CossRf]. Takahashi, I.; Noguchi, T. A nw quick-spons high-fficincy contol statgy an induction moto. IEEE Tans. Ind. Appl. 9,,. [CossRf]. Bassfild, W.R.; Sp, R.; Habtl, T.G. Dict toqu contol fo bushlss doubly-fd machins. IEEE Tans. Ind. Appl. 99,, 9. [CossRf]. Abad, G.; Rodíguz, M.Á.; Poza, J. Two-lvl SC basd pdictiv dict toqu contol doubly fd induction machin with ducd toqu ippls at low constant switching fquncy. IEEE Tans. Pow Elcton.,,. [CossRf]. Saasola, I.; Poza, J.; Rodíguz, M.Á.; Abad, G. Dict toqu contol dsign xpimntal valuation fo bushlss doubly fd machin. Engy Convs. Manag.,,. [CossRf]. Fattahi, S.J.; hayyat, A.A. Dict toqu contol bushlss doubly-fd induction machins using fuzzy logic. In Pocdings IEEE Ninth Intnational Confnc on Pow Elctoni Div Systms PEDS), Singapo, Dcmb.

22 Engis, 9, 9. Saasola, I.; Poza, J.; Rodíguz, M.Á. Pdictiv dict toqu contol fo bushlss doubly fd machin with ducd toqu ippl at constant switching fquncy. In Pocdings IEEE Intnational Symposium on Industial Elctoni, igo, Spain, Jun. 9. Hu, J.; Wu, B. Nw intgation algoithms fo stimating moto ov a wid spd ang. IEEE Tans. Pow Elcton. 99,, authos; licns MDPI, Basl, Switzl. Th aticl an opn accss aticl dtibutd und tms conditions Cativ Commons Attibution CC-BY) licns