Mathematical Models of Electric Machines - I. P. Kopylov by Blog da Engenharia de Produção

I. P. Kopylov

Mathematical Models of Electric Machines

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Mathematical Modds of E1edric Machines

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I. P.Kopylov

Mathematical Models of Electric Machines Translat ed f ro m the Russ ian

by P . S . IVAN OV

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n e vised [rom lh ll 1980 Russian l'diUOll

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IIU ,Il, ell bCTIlO , B loIc w lln w Ko4a _, 1980

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PubHsl,l'r5, 198<1

Contents

Pr. l,ce

• , , • • • . • • • • • • • .

I . Intr odul'llon 10 Elecl rl.>ll\eeh M I<:$ 1.1. H i ~ l orka l DevelopllH1ll1 • • • . . . • ••• • • . 1. 2. Tho La"s " f E Il'c, tro llle t h all lc~ l, Energy Donveeston . . . . 1.3. AppHcatio ll of f iclrl £ 4" 1111005 10 Ih e Scluue n of Probll'mt In El~ot to rnc<: l"u) ,c. . I. ~ . The l' r i m;tLv" t'oll.-Wlnclin ll Mach inl' . 1.5. Applica tion of C", n plll( '~ to t he SoluUon of Pro ble ms in EIK:trll ",~ h/lnlcs

. . . . . . . . . . . . . . • . , . . . . . .

A.. 2. l: len 'IlP ;n'han lt:lIl f M t'fY Cll n,'••dlllO I n ul " ln, II (iulilu Flehl 2. 1. The EquatiuWi o{ I~ O" n('t'ali ud E I~l roe Ml\('h ,llf' . . . . . 2.2 . S Il'a. ly-Sti l" Equ at ion, . . . . . . . . . . . . . 2.3 . '\ppl lu tlon of An,.l", Coll1 putel'1l to t he A na l y~i, uf EIK lric Mach!nl-s • . . . • . . . • . . . . . . •• . . • . • . . 2 .4. T ra ndnl P nxellSf't III Electrie )t'Kh in u • • • • . . • • • •

2.5. The EHl"d .. I Plra 'l1l'l l"ra OQ l b. D)'Ol mle Chllrac leris\lcs 01 In d" d iOIl M ~eh i n" . . . . . . . . . . . . . . • . . CIa. 3. C""l"•• I1.~..l ... 11 W i nd ing (' llnYfor!e' • • • • • • • • • • S.1. The In riDit. Arb it ra ry Spf'l".tn' m 01 f'ield. In tbft Air C.p 3.2. The Ge U('rllll~cd Ene rgy Coover11ir . • . • . • . • • . 3.3 . T he E q uali on~ of t he Cl"lIen liu d Enl>fg y c c nv• • ter . .

• • • • • • • •

'sae"s

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01. G. Mull iwlMlilll M-eb lnn . . . . . . . . . . . G.l. The EllullUom or M" h h..judi nt Ab ebines . . G.2. Thfl t:qu . IiQllI 01 Synehr (lllou. Muhl~ • . 11.3. T~ Equatioos of DirKl Cun fQ t MiKblntS . II... . The Dou ble 8<,,,l rTel.ca ge Inducl ion 1I010r. ThO' El fel'l of Edd y •

th , 5. Ent'f!y Cn..vtBlon Inv"l vl..g :\onslnUS/' idnl a oW '\ 1)'nlQ1 cl rlc: Supl' y V" l tllgK . . 6.1. Th e EqllltionJl 01 EIK tr ie Mach t.." , . 6.2. The Soluti on of Ellu allon l l nvo lvlng '\ ' )' nlJuetr ic: Supply V OI 1~t.. 6.3. Th e T lo l r l" ot \"nlt kge Rtg\lb IOl'- l nd"eUoo )Jolor S~tem 5.' . Pul E eelrnm6C h~ n ie.1 En" 'g)l Con vm , .

•• •• •• ••

... " ". 71

Cl,. , . Typical E'IUaliollft ur I:!l r cl .lc M,u:b ll\('l • •• •• • . 4.1. TrUM iti on from Si mpll' 10 More d oml' lex Eqllal ions 4.2 . J:; nergy Ccnveeeton I n\'ol vin g un Elh plle Field . . 4.S. IW ipli c,-Field Sl u ll>,·S t a t~ Cond itlonl . .• • • •

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s 6.5. The Induc tion Machine \Iodel l DCl udillJ( Sla l.,.- and Rotor Edd y Cum ulI . . . • . • • . • • . . . . . . • • . • • . . .• 6.6. Tb, IEffect or MilD ulaU lln "" Pac~ on t: leel.rir M ~h ;ne P... . fo r.....nce

• • • • • • • . _ • • • • • • • • • • • • .• •

00.. 7. Mp ls oJ t:lfdrl~ .\ta ebi_ with N.... liltnf "'ra~ • . _ 7.1. 1'lle Anal ysis of EIKtn c l£acbUltli wil h ~oD line ... P....mulers 7.2. The Effeel of S.l u.a tHlQ _ • • . • • • • • • . . . _ . • • 7.3. The Effoct of Cu~~ OispllliCl'm"llt ill l.lw Slot • • . . •• • 7.4. ElN'rgy eo...y.,I<OlI Prob lol1l5 In " o;Il" ,ng h ld. pell<l... ~ Vari. lII.. 7.5. The Aalill p is nr Opera lio ll of • n.a l ElllC U"k: M, dline • • • •

Cb . 8. AJ)'mll.ttric EIll:flfY C/ln Yl:I'len;

• •• • • • 8.t . Tvpe! of Asymmetry In t:lectr,r M.,h ull'S

8.2. Electr ical I nd M a g"~ l lc ,uyUl'lIelt y . . . 8.S. Spacia l .'Iaymrnel, )' . . , • • . . . 8. ~. SlRgle· Ph.ve MolOI'I . .. . . . . .. •.. . 8.5. Th, Elee\ r ie M" eltilla u an e lt'n1t'ltl (If t ho S y~ l e m

e h . 9. Tho E'luatlu,,~ fur F:Jl!Clrle M'"'h lnes or Vari ous Dcslg", 9.1. TI\II Mathematieal ~I()/Iel! of En erW'}' '.Qnverlllrs wit h a Few Degre" of Freedom , . . . . • . . • . . . . . . . 9.2. LineII' Il: llel"ll:Y Collver ten • . . . . . . . . . . . . . . . . 9.3. Il: nllll:Y Coo .trle n Wilh U 'lui d ~ ll,l Gaseou.s Rolo<'o1 • • • • 9 .4. Othu T ypes or Bnt'rtY Con. erters . . . . . . . • • . . • .

0.. 10. Eieelr l... Fk ld anll Eleetrom~oael ic- Fleld E.-gr Coll..vters to.1. Pr loc: iples of DUll·lIl,,_ ElceU"OdYl\l llI.i.:a; . . . . • 10.2. Tilt Equa liOlU for f:loo; lr;G-Fi.hl KG8I'f1 Conu cw n t o.$. PlramlllMe. Eleeu iq-F' eld E~y Cc)n ...,ters . . . . 10.<1;. Piezoelect ric E ~ c.o...e rte-s . . . _ . . . . . . . r c.e. Electro magnetlc- Pleld EMl'Il'Y Conv«U fS . • _ . . .

tho 1I . '\ pplintlon Dr tape.-llnenta l Iksigll to F. leelr k: 1l1 lm l.....,. A...lysh • • • •• •• . •• • •• • . • • - •• • •• • - tl.l. Gt<l&ral Ill.(or ,nallo.;\ on I h. 'rh~y of Ex per imen tal Deaign l U !. Th, TllChn illu, 01 ElCpt'l"lllll!nta l O('<l ,go '\ pplled ill Eleetro· lTlIlChan ics • • • • • • • . • • •. • • • • • • • • _ • •

11.3. 'T'flllu iti op 'ro m

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Ch. 12. Sr nth<:t;i! 01 Eleclrl e Itlnehl lW!l .

12.I . Opl i lu it ati on 0' ~ ~rgy Conver ters. Oplllll,u lion Methoda 12.2. Geomet riC Progra mrninJ . . . . . . . . . . . . . • 12.3. ~ig n of g l~ trl e ~ aeh lllu b}' ~me l rie PrDgfa rcu ning;

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Ch. 13. Autom.ted

t3 .1. Genua l D..ign

Deilg n of ElllCtric Maeblnes .

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13 .2. So rlware of Au toml~ DNi gu Sy ste rus 13.3. H. rd""llrfl of Aulom. led ~i gn ~ fstt lM 13.4 . Cooclll!i<ln

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Preface

T he level of a dv ance me nt. i ll t ochnol cg tca t culture prima rily depends on t he de velo pment of allergy so urces for the nee ds of man. T he use of steam and p ai-ti cul ar l. y eleo t.r toit y over LIm l ast one h u ndred yea rs br oug ht i nd us t ri al revo luti on an d ga ve a t re mend ous i mpetu s to the de vel opm ent o f so r,te ty. In the l ast few decad es wor-ldwi de peod uct ton of electri c ene rgy has incre ese d a hu ndredf old . The electric power of gene rat ing plan ts bas grown to 2 700 min kW . If the eeiee of growt h of ge nera te d ene rg'y rem ai n t he same , i n 50 ye arS from now the out put. of en 0rg y will reech 0.2% of t he to ta l OJl el-g ~' 't he onrth receiv es (ro m tho sun. Elec tric gene ra tors produce a lmo st a ll th e e lectric ene rgy used, two- th i rds of whic h is fori to electric moto rs to be conves-ted to meoha uical energ y. Elich yea r indus t ry turns out tens of millions of elec tric machines a nd tl',ll.ltsfor mers. In s erial pro d uc tion now a re t ur bi ne-driven genera tors lof 500, BOO. a nd 12000 I\1W, h ydr oel ec tr ic gene ra to rs of 700 lIliW , a nd t ran sformers ra te d at 1 000 MVA. Tod ay , mo tors an d gene r -a to rs are an esse nti al p arl of t he Ia br te of liv ing, se rv ing diverae purp oses i n industr)', agricul fure, lind in the home, Elec lric machine en gine eri ng nlltu rllil y owes Its advan ces to t he de velopmen t of t he t hem)' of elec t rd mechantcs-c- a branch of ph ys ics dealing wit h t he processes of etecmomecn entce ! ene rgy conversio n. Electric mac hines incl ud e All Y etecirnmecbaulcnl energ y conver ters (E('..s) des ti ned for va rious pur poses. El ect romechan ic al co nve rters come in a grea t va rie t y of designi an d can co nce ntra te e ne rgy in magnetic, elec tric, a nd elect r-ornag net.lc fie lds , T he equntlona of electric machin es a re written proceeding from th e th eory of elec t ric circuits , keeping in mind t hat energ y co nv erts in t he air ga p end the magn etic field is known, T he m at hemati cal model for a n Infin it e spec tr u m of fiel rls and any n um ber o r loops on HIe rotor a nd s tato r is the model of, a ge nernfized ele ctr ome chanical co nverter- a n electric ma chi ne with In an d I l wind ings on t he s ta tor and rotor . T he equ atio ns for the ge neral ts ed co nver-ter offer t he possi bil ity of wor ki ng out II ma thema t.ioal mode l for practica lly any proble m encnuntered in mod ern electr-ic ma chtne engi ne erin g, T ho pr esent Look deal s wi t h a rne them aucal t heo r y c f e lect ric machi nes that uses di ffe ren ti al eq ua tio ns as i ts base, It co vers t he

mathematical models of .eJect ri c machin es havi ng a ci rcul ar fi eld Bud a n i nfinite s pec trum of fields in tho a ir g ap . Analysie is gi ven of th o equations invo lv in g n onstnusotda l asy mmet ric s u pp ly vcneges a nd nonlinea r pa rameters and !llso to mu ltiwinding ma ch ines and mac hines wit h several degrees of fr eedo m. An attempt is ma de to ad ap t tho ach ievements i n the aren of me gn etic-field conv er ters for us e in t h e ana lysis of electric -fie ld and elec t romagnet ic- li t Jd conv ert ers. T he bo ok CO\'8I':; t opi cs devote d to t he application tll electronic com put ers to t he soluti9Jl of pro blems in ele ctromechanics. It ill expected that t he rend er i~ alread y Iaml liue with eo m pu te rs. program. m ing, and algori thmic la,n gu agos. The au t hor 'S obj ective is t o t each the s tud en t how t o form ula t e equa t ions for mos t of t he p rob lems i n the ana lysis o f t hll energ y conve rs ion proces ses in electric ma chines and rtJd,, ~e t hem to a convenient form for t heir so lu t i on b y compu ters . Much coustdoreucn is give n to an al ysi s of the obta ine d solu t ions. T hree chap t ers are devoted to th e ~yo th6l! ill of ele ctr! c ma chines an d t he com put er-aided design system ; t he hitler be in g t he hi ghes t actue vemonts in el('Clro~e ch lln i c;s. Pri ma ry a t ten t ion is focuse d on differential equati ons of electromech an ica l energ y ce n verston, wh ich form th e most general an d rl gorou 5 mauic maucet mo del f OJ' descrfhing both trans ier u and s teady-state mo des of operat ion . Po l ynomi a l mod els are al so gi ve n due trea t ment. The te x tb ook 11Iid i ~ origi n ill a series of lec t ures, "A ppl ic;ation of Compu te rs for Eng i llee ~i llg an d Econom ic Calc u lations", and in II s pecia l course, "T he Mathem atica l T heo ry of E lectric Mach in es", taug h t b y t ile author (It. t be Mosco w P ower E ngineerin g Institute. In or ga nizing the book , lllle au thor h as also us ed t he res ults of rese ar ch co nduc ted at t he !Elect ric Madlirlery Chair and in the Lahora tory for t he an alys is of prob lems in electric dri ve, electric m ach ines and ap paratus lit! the same ins ti t u te . T he presen t hook is de~i gn ed for stude nts and pos tg ra d ua tes s tudying elect r ic m ach ines en d als o for el ect romechurri ca ! an d power eng ineers engaged ill the:desl gn and serv ice of elec tr ic, m ac h iner y.

J , P, Kopy loQ

Chaple r

Introduction to Eled romechanics

1.1. Historical Development T he date th at lnlUJ,;l\ t he be!:'i UlJinÂĽ of the age of elec tric m adli nes is cons id ered to be t ho yCflr J82 1 when i\I. Faraday cons t ruc t ed II mo to r- in w h ich Il conduc t or Z re"o lvcd abo ut a peemauen t maglIet il (pig. 1.1). i\ [en: ury 3 lind u PP1r s upport J perform ed tho IU ll Ct.lon 01 a el id ing eomec t. P are d ey'aimcte r fed wi t h a de \'olta (l:o U 1(1 pro vide Hcf d exci tatio n WIIS t he first mng n I1Uc-fi(' Jd electromechn nlce l energy C.Ullvl:'rll'r l . In 1824 1-'. Barlow desc r ibed a motor 'colJsis ting of t wo co pper aear whe-els Ies t cn cd on one s ha fl lind loc at ed be twe en t he poll'S fl f pe rD'lanent magnets . B nrl ow 's wheel wns in eo n t lloU wi th merc ury lind rota ted fnst with t ile PMSIIlJ8 o f cu rren t. In t 83 1 M. F lIrfld ll}' di 5Co ver ed the law of e lf'Ct rom ilgnclir. indu eucn-ccnc of th e most hn porl an l pben om('J11I of eJeetro medu llinwhich made possibl e th e dev el c pmen t of lIew t y pes of eteetric mechill 8'!l, T ho esse nce of t he phe nom er\on d isclosed b y Farllday CO II.'lJS t5 in the ' oll owin g. If II magnetic fl u x li nk ing It con du cting l oop is mad ~ t o v ary, ele<'lromo t ive forces ;lppear ill t he loo p lind 1111 ete eu-le curren t ShHts d rt ul8li na over l he d osed loo p. In J832 P iled s ugges ted 811 ae gCIlll-fllto r with a revol vi ng hOI'SC6hoe perm an en t 1ll3!:"cl 1 a nd s t a t ion ary coli s 2 wound on s t eel eorce (F ig. i .2). I n 18 34 A. S. YaKobi de veloped a motor wor k ing Oil t ho p rin ci ple Olll lt"IIClioll and re p uls ion of pe rmaeent magllot s and elec,l romnltn ets (Fig, 1.:'1). Sw i tc hi ng on nnd off Ih e etecr eomagn ets provided ec n un uous ci rn,Ja r mot.len. I n 1 s..~ 8 Ru ssi an engin ecl'lI inSlall ed fOJ<ty molol'8 com bined in to units Oil a boat whic h coul d ru n u ps tr eam the Neva r iver with t welv e passen gers on boud . Tha t was t he (i 1'St a Uempl In harness elec t ri c motors ,for prllr t ir lll purposes. JIl t860 A. Peci nctt! /lO d , lew, in 1870 Z. Gn mme su gg:(I8ted 1II ring arma t ure (Fig , '1 .4 ). T he Gramme r ing e rmat ure conllh' t ed ~..1lI U~. ~..hPn! '~l't Mr}' . thl' geMra l tl'rm "n HJ:Y coeeertc r to dftlol" th" c1l ss of mac hil\t'S tha t ce""" Ind'i.eliou (u)'ocb rono \ls), ,,.l1dI ronll~. I lld dl ~w"rrt"111 IHItS. AIm. " f .. Ill usl,n th " rUJlO'Ctiv... ulm ~ t"lecl rk-U"ld ~onVl' n"r DIHl .......~ lfQ magnt' \k- ri<.' l d (OM"ff lt r III l'Jeetros tu ie eon'Âˇ...ru rs . 1Id eenveetees in " 'bl<:h tbe ""OI'king rid" Ih lt ce ne.. u t ra l~'l\ <'1Mll'Jr il I n f lt"t \ rom . g nd ic r;"ld._ T.....I. ' ....' . ....,...

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of lil rlllll:-t )Âˇpe mago etic eore 4 (made up from steer wi re in ear ly m achi lles) c8rT)'i ng an armet uee wi nd ing .z i n t ho form of a ec nt tClllOU5 spital. I n early mach ines, brus hes 3 d irectly s lided over tb. cce ue uocs windi ng '''Ill onered comm u tation by closing t he t u rns.

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Fig. \. <l. Paclnct tt-Grnmuie's macblot

TI le mllf,plClir field was prod uced b y m o~n els / or elee trcm ag nets. W Ill ie \"Ilkobi 's machine hed i Ll! win ding open. in th e l alter ma ch ine t he " 'Iodi nI,' WllS r.o nti nUOllS (clcse d]. T ile Gr am me m achine made â&#x20AC;˘ s t nrl for t he e,'olutlo ll ()f cc m merc lal etecrere rnachtues. It had all t he bllSic ele ments of modern elect ric machi nes. In 1873 F. Hd ncr-All eneek and W. Siemons rep laced t he ri nl arm.l ute br rhe lInn'tu ro of t he d ru m ty pe. Since 1878 man ufact uten! 11" ' 0 beg-Ill! lo prod uce drum arma t ures with eloes, and , s ince 1880. Ilrm atun'S from lam inat ions foll owinJ: the sugge! t ion put b y ~1, n nl llS A. Edi snn.

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10 1885 t he Hun gari an l'Jectriea;J engi neers pro posed II si ng lephase, s hel f-type an d core-ty pe . t eans forrner wi th II d osed mag net ic <lir eui l . In 1889 M. O. Dollvo-Dobrcvo lsky deve loped II t hree-phase esyn cheo nous motor and n t h ree-ph ase t cans lcr r ner. From 1890 t he

rbrne-phase s ys tem recetved genera l rccng lliUor' li nd marked t he begin nin g of wide a pp lication (If alte ruati ng curre nt , At the end of th e last cen t ury. firH dis tri ct and c it y power p tnnte ca me in to bein g. In 1913 tho Lo tn! power or ('1('I' t,ric s ta t.ious ill Russia amounted to t mi n kw. In 1075 t he tot lll out pu t of elect r ic ene rg y peoduced i n tl le Soviet Unio n exceeded 1 000 bill ion kW h . Electr-ic mach ines of a n elflc ieacy of 99 % are now built wi Lh

'the amount ot ecerve mater-iels s pent on t heir cous t ruc uon not exceed ing 0.5kg pel' kw. F.ig . 1.5 illus t rat es a 1.2 mi n k W t urbi ne gene ra tor op era t ing at f ill' t .e. An l'uhllnllth'c u:cnCl'lItQr t he Kost rom a power plilJll. Electric mach ine indust ry produces ac aud de mach ines t o S6l'V6 II great va r iet y of runcuone in all hr anchcs of Iud us tr y. Electric machines d o job s i n s pac e, u nder wa t er and u nder (he grou nd . S pecial machines nrc a vui lab le for work ill at omic rencrers and s panncraft. Millio ns of elect ric m achines find usc in household appl lanoes, automob iles , tra ct ors . end ot her k i ~ ds of tro ns port . Fig. 1.6 s hows lin au t omot ive ge ner a to r. T he Sovi et indu stry t urns (lilt some d ozlms of aut omot fve gene rlltors of t hi s and ot her t ypes eve ry minute.

Ch, 1. ItIl,od uction 10 El.d'o"" •• h.nic.

El ect ri c machlll " en joy wi de a pplication in timiDg aevreee, nnillll.tioll s ys tems , and ,n , di Herent t rllDWUcers. T he powe r of eteet etc mach in es va ries from fr ftGLions of II wa tt t o • m illiun k ilowllt 1.5, t he ,'oltage from fucUons of • vo lt to II mllJi on \' 011.5, t he ro tlllional speed l rom Il fow revclu uce s a d lly to 500 000 rpm , a nd t he 'I'olLage Irequency from uru to JO'l _lOlt H I. ~pile. t he gnoat prog ~ made in elec tro mec han tcs . ver y m any pro bl em' sti ll rem ai n t o tle sol ved . E lect ro mech anic al energy eon'l'e~i oo occ u ~ ill fusio n rreaerors lin d in biological species too . Furt her ad va nces in t he cre ati on 01 new t ypes of electri c macntuee w ill in t he main be de ter mined b)· th o level of deval op meu t 01 t het heeu- y of c l ec trom ec h al1 ic~ 1 energy ee nv crs tc n. A ll lloollh t he d evolop ment of el ect rc mecha nles is cons idere d to bt!gill with Pa eadny'a d l~co \' e ry , electric machines existed long bclcee hl ~ dis covery, Much errort was spent on evol vfng eleoretc-rtel d energy conv erters. t.e. :Clec l ro~ tll li c (frl ct ion ol) mac hines wit h elec crtc-netd ("lIcrgy s torage. In 165O' 0 1to '1'0 11 Guer tcke d escri bed th e fil'llt eleererc m ach ine t hllt represented II rotlll t ing ball from s uHlIr, : ..\ t th e bq in lliog or th r; ' 8 ~ h c l' n t ~l ry Franci s H aw ksbce re pleced t he s uHur ball b y /I holl ~ \\' g l as~ ball fiut>d on t he shaft. I n 1743 fritl ion al me ch tn ee with IHl iso la ted me t:i1 elee uede a ppe a red . T he elee o-o ee eo flec red eledrll:. ch argee Ilnd so the m nchin e could ccminu ously feed powe r to th e eS loJ'n al cteeun. T he s ubseq uen t ye al'1! of t b.. t Bth cent ur y Sll ~' fuct llet a rlld u:l1 im provements ill t he d llllitfD a nd perfo rm once of fri ctiona l eoeebrees. r\ mong R u ~i ll n 5(;len t ist.s, M. V. Lomon oso v, G . V, R lk hm all . _.t, . T , Bol ot ov, an d ot he rs were cn gaq:ed in th e wor kll)n t hes e maeh i nea. Dy t he en d of t he 18tb Cl' ll luQ' one. mo re t yp e (If (riction al ma ch ine Cli me in to being i n whic h th e rotor ..... (15 mild, Irom gl.'\..<6 di.!;ks up t o tw o m eters ill d iameter. The mach ine could prod u ce sp ar ks over 11 me ter long. I n th e 19th cent ury WOr k on t he Im pro ve me nt of lrictioo al m ecn tnee conti nued and resu l ed ill t he evolution of un ique ele et etc-Iteld tlnCl'IlY conveners. I n I!J::JOlhe va n de G rn alf generat or (el ect rosl llllc nccelerntor) WfIS built wbjieh deltvered II power of 6 kW at ~ voHllge of 0 min V. Bueh se tu p! lire deslgne d t o t l!:!l t ele c u-ica l equ ipm ent. Eiect rtc-Iteld energ )' convert ers wlli eh ilppe ilref! mu ch ea r li er t han me gnet.i c-I iel d ene"iY co nve r ters d id not l ind wid e p rae tl c al ap pli ca t ions . The em er gence o f m aJrlleli c.fjeld t y pe ECs in t he Hlth ce ntu r y was II new s:t.nge i n the h isLo ry of eteer ete mM-hi ne enal n",r iug nnd brough t abu u l II screeune lind t eeh utc al re\' olu\loo in t h is fi el d. T he h i!ltfJr)' of developmen t of t he t hoor) ' 01 el&ctri(' me ebmes may eolldit ionllll y be d i\'i del\ int o t hree s ta ges, Tbe (irst s t age ind udes t he per iod of Cl'@ a ti on of earl)' mach ines a nd d eve lop m ent of t he e1 aMin l t heor-y of t lech 'om eellan l(:al energy eo nveret cn . Tho seco nd

ra I lage embraces the period or oJa bohliOft lind Int rcd uetton of t he thtof)' of s tel d r- s l a le processes, ce mptex eq ua tion s, eq uival e nt creeuus. ae d ph asoe d iag rlllUS. Th o third s tage berlin in t he la te 1920s with the form ul at ion of diff~re n tiBI equ a tions a nd deve lopment of Lna t heory of t ra DSion t prec esses in elec tric machtnce. T he t heory of elecrerc m acbtnos W8..~ Sh 'e n const deeauon in t ho works of A. .M. Ampere. G. Ohm, J. P. J qule, Heinri ch F. E. Lena, Hermao L. F. HolmhoLt1., M, V. Lc mcnoecv and ot her promine nt phY.'l ictsts of tbe 18 t h and t ho 10th cen t ury, T he wor ks of J ames C. Maxwel l who ~enel'A1i~ed t he nchlcvemen te of elec t ric a l engineeri ng in hb r f eoti$e 0/ etÂŤtricu y aM M agnetUni, 187 3 , hol d a particular pla ce. Mu well introd uced the new elccl(O magnetie t heory an d pos( ula l.8d equa Uons whi ch ea ma to fonn t he th llO rel ic al base f)f electeomechentee. I Of much Impoetan ce are also t h6 wo rb of N. A. UOlO'" 11 874) l od Jo hn H . Poy nti ng (I &M, j 0 1\ tho t ransfer lll\d ee nveeeton of eoergr . T he first theoretical work conCll rning eloct ri<l machin es mll Y be considered the work of E . Arnold on th e t heor y no d desig n of wi nd ings of ele cu-rc ma chin es, issu ed in 181lt. In l bD '1 8 903 ~ l. o. Dolh 'o- Dobro';ols ky, Gis bert Kli PP, end ot her scie ntists ee r fort h t ho f,uldnmenta1s of t he t hoory lind dosian of t ransfo rmers. tn '(894 A. Hoyl a.nd l heoretica lly l!uhs tanti l ted the circle di agram of 3 D indu elion m1l4hine , a nd i n 1907 K. A. Krug offered an accura te proof of t he ci rcle diagra m. In the t 920s B. Fortesc ue suggested the method of sy d'l met ric co mpenems. In t he t 93Qs E. Arnold , R. Ri lfhtor. A. B1ond el, L. Dreyfus, FIf. Vidmllr , Charles P. Stein me l". K. A. Krug, K. I. Shenree. V. A. Tol vln s ky, and M. P. Koseenko eonsidorll. bl y exten ded and adva nced tho t heory of s tead y-st atol opera tio n of elect ric mac hines. R. Rudenberg's wor k WM ono of t ho fil'llt cont rtbut tona 10 the t heory of n-analen t processes. This theo ry , whose orig in dat es fro m lh e bogin ni ug of t hi.!l cen t ury. mad e n t re mendous ste p for ward in the t 950s Ilnd 19705 OWing to tho lWide appliea tion of co mputers. Th e fi rs ~ pa pers concerned wit h t he mathomatical th oory of electr ic mach ines ap peare d in t he middle of t he 19205, in t he 19308 and t 94O!. Among t he eutbcrs. mentio n s bc uld be lD l do of R . Park. A. A. Ocrev . G. Kron , a nd G. N. r etro v, The fund a mental works of G. Kron : rea tl y cont ribute d to the developm ent of th e mathematical l heory. He s llgallllwd tho model of e nd deri ved eqoll.tloQs for t he gooer/lli7.ed (prim iti ve) elelftric mac hin e. In t be l Ast yea rs the m athematl ~1l.1 th eor y of ele ctrtc machine!! (magnetic-field energy eo nve eters] ha s developed to a noticeab le utent owing lO t he efforta of man y' allt ho rs, fit!lt of a ll. B. Adkins. L. N. Gl'lIl0 V, A. G. IMEfyan , E;. Ya. Knovsky , K. Kovach. V. V. Khrushc hev , I. R aez, S. V. S trak hc v, D. Wltite , and J{ . WoodSOil. The use of electronic co mput ers has ena bled rese a rchers to

II nol )'", l\ lelldY-l; ~ ll te pr ocqsses n a. pllrti c;;ubr C/IS(' of teenstems l\nd I ppro . cll t he problem of dev elo ping tom pu terÂˇai dt'd desig n system! . The t heor y or eleetrosiat ic mach ine! , however. sli pped beh ind desptte t he pooling of error-ta of s uch p romi nent scientists a! A. P. Iotfe, N. D. Pa pal ekai , L. I. Mande)~h\8m. A . E. K llpl)'sn sk }' . A. A. Vorobyev. and a l6ers. largel y beceose t hei r i llvest.igllLi ons ra il ed t o creete t ho prod llcljon prot ot ypes of t hese ce e vert c-e. At preM'n t one of t he ittaporhnt t J..!lks of t he w :lthem at ieaJ t heor y of elec tric: machi nes is to tde\' e)o p t he general t heory of aU t he th ree classes of energy ceuvertera, T he ehaptera below cQllllidcl' energy ecnvertee equ atf ona, t heir tr a nd ormat ion end use Ic r most of t he basi c problema dealt wit h in the analysis lind ~ yn th hsitl of elect ric machines: presont.equm.lona .IIJH! t hei r eoluucns lly eomp cters Cor machin es (';\hibiting a rlrcul ar field a nd 1111 infi nite s peet mm of fields in t ho ai r gnp: exe mtne con\'e rt t'u in volv ing nonatuua oidal esv mcict rtc v oJt ll.g ~ , changes in t he freq uency and a mplitud e 'of su pply vollage, machines wit h nonli near plll'lu;netera, as ymmet rlc fnachinQ5; etc. Th e coverage also incl udes convert ers with a In .' d~rees of freed om , li near machines. eleetncfield elect romechanical ene"y converters , aud ot her elec:tr ic: machines. Th e t heor y of energy converters is set forth on t he bllSis (If diflerenlial equa lio n!J ",'~ie b descri be t he dyn am ic behavio r and , as a particular case , t he, s te ad ~ t ue behavi or. The COUB& in t he mathematical t heor y of electri c mac hin es gives t he base for t he mat he matical deseri pl ioll of t he process" of energy ec nv ererc n takinlf into accoun t Jlon)inear, nonsi o usoidal. as)'mmelcic as pects and manub.ctllring fac lo rs. Such an anal ysis is impm,sible to do e mploying stead r-stllle , qu e Lions. equ iv alen t cl rtu.iU5. and phasor di agr ams. T he electro mllljba nical energ y conversi on th eory presente d ill th is book enables t he :engineer to use t he eq ua tio ns for th e gener ali zed electromechanical energy convert er lIS t he base froIII wh tch he c an set up eq uations !(or s ol ving any prob lem met wit h in the pract tce of electric m a c~ino engi neeri ng.

1.1, The Law s ot'Elec:tromechanical Energy Conversion Altbough tile t heor y and pract ice of elec tro mechenieal ellerg y een versrc n have I Ion&, his to ry and achieved grea t !IuCGeSS(>.II, tllQ bas ic eneray ccn vere ten ~ ..ws hive been stated onl y q ui te reeenrly. Let us form ul a te these In .'S. F l flit law. ThI: t ffkit mv of tltctromtdlollko l tnergy ro' fIJt rl!o n (Ol1 not tqtUJl 100 " , All energy con verters ca n be di vided into simple and co mplex ont'S, In si mpl e convertees, w e energy of one for m is converted to t he energy of enc t ber for m. An example ill t he Conversion of elect ric

energy to hea l ill an elect ric hea ler. In eemp jex ecuvenere. wl,ich «I nsti tute t he majori t y of mach ines . t he e nell:Y of one form Is een "'rled to tbe energ y of t wo for ms ( II.IU' . ra rer. tu t hree or more form s). These II Nl con vert ers of energ y f Nlr:p luminous to elecrrtca l Icrm, chemical to mec ha nicel form . n uc.l\!'lIr to e.Iee t ri cnl for m. t \.l'. In «I rapleJ: conv erters t here co mmonly oecurs ell a tte nda nt ce nve eetee of energ y t o he a t. Eleer romech e nicel ellergy co nvert ers belo ng 10 th e group of complex converters beca use t he processes of energy conve reton !.lera elwllys go wi t h IJ,,,, con version or doc.Lrk e nerg y P ~ or rneolmnical enoTiY P", to t h ermal energy 1'1 ~ ' E,Cs ex hibl t th e flows of etecr roml gnel le, mechentcel , Mild t hermal en ergi~ (fig. 1 .7). The objl'Cli ve plIl'!l ued in 6\'olvin1: an EC ts to red uce t he l(Lq t hermal energ:)' (lo w:I- an d rh us to decr ease t he o verAll di menSlo n&

Fig. 1.1. Tbe energy llow dist ribution In &0 eleen lc machine

lDecbine

&/1

•

~ :~:c:~~~ ':.~:::lg ~~~~D"':J":,'I~5 and d ..b Il Ml , .. ,,~dl "d J

and cas t 01 the m:'lcblne. The efficien cy of so me co nverters IVlillbl. todey re aches 98 %. lind l hat of t rl ns! nr me l"S fUllS as high 8! 99.8% . 1l'hich is in d k lU v$ of exce pt io nal technical echteve ments. It is to be bo rne i n mind t ha t h;igh efficiencies are ac hie" ab le In high -power co nverters. In tow-power ECs t he efficie ncy niches merely III few percen t amcc th e major a.mount of mechanical o r electric energy evolv es as heat. It Is impcsefb!e to produ ce an eJe(;tr ic machine in which conver sion of ene rgy l.6 heat wou ld be ue nex tstenr: e t berwlee it must be Inrnlahed wi th superconduc tlng wlnd.logs. As w ill be s hown be low, elect romech anical energy cc nvereton eq uations have no soluttoae III

zero reststeneee.

We ce n vis uMIit.e .. lossless mach i ne (wi t hout Iro n and hav ing eu pereonducting windi ngs). but to en~bl e eueb • machine to convert. etIerg)', we need to inset1. • reetatenee in to t he CUrrflJl1. net<tI,'ork e::rterDaJ to tbe meehiae, I n t his a rr-a.ngement . it u t he eleet rome ch allieal s )·st.e.m be)'ond the ma chi ne th.. t develo ps lossl!ll. An electric

ma chine ca n be t N a Led :wilholiL reg ard to t he exte rnal eteetre e ec haniCAl 5)'stem o nly undee def inite conditions, whe n, for ex a mp le , t he line eeeis tance is equal lo zero. i .e. t he m achl ne operates fro m or in k> tile bus of infi nit e pewee. The processes of electromechanical energy co nve rsion ffiWlt. be studied WiLli due regar d for all eledrieal an d mechanical loo ps. An EC t hat does not develo p lessee becomes a sto ra~ or tank of energ y rat her t han t he energy con ver te r. E oerg y s to rage dev ices are e lec trh~1l 1 e ngineering_" rrnnge menu fe.Sll mb li nt in des ign elec t ric m acllinel . En orgy s tora ge devtcee een be b uil l a! both s t ati c dev ices and rolat ing machtnes, fo r ex emp te, as II gyr o wi th su perco nd uct tng windilillS. T his is IIlI elect ric machine t hat c en rotate per manent l y s in ce t he re is no loss i lj 'it. Bu t eu a ntilorq ue moment ap plied to ne s ha fl will brin g t he machlnc to II. s to p. T his ma chine cannot ac t as eu c uergy couveetc r. An etecteomeche ntce l ecu ve-tee ca n ue rep resented as a t wo port (Fig. 1.8) acee pUog, for ,u am ple. stim uli (yo ltag<.' U and elac t r lcoJ frequen cy f) at it pair of lel&ctric.nl te rm in als (a n elect rica l po rt) and p roduci ng responses (a tJ rque!of on t he shaft and roLatio na l Ieeq uenGY II) at a pai r of lDlX;f\ani &al t erm in als (a mecha nica l po rt). T he t ..... opor t re presen t a ti on of an elect ric mll&h ine applies to aoh' ing problems in ereereoreeenentcs where t he pl'OGeMe5 of en ergy ee n vers ion in side t he machine do nol have a dom in ont s ig n j(jcaDce. Second la w. A U , /.ectromedulIIlcal conwrw$ are revenlble, t.e . the" call act IU mo ton' lJ.Ild IU generators. The revcrs ibility is an importan t adv en lltge of ECs over othe r e nergy conv er ter.< s u&h 88 Itea m tu rb in M, di esel en~i n 9S. jet engine!, ete. T he energy-<.onyers\on mode of o perati on of a n elec t ric m. chioe de pen ds on t he m ement of resis tance (to rq ue or amuoe que) on its shalt, M r. If t he electric energy is dra wn fro m t he power li ne , the Ee o pera tes in t he m otQring mode. If t he f1 0lY of mechanical energ y dc tt veeed to rne EC s hnft trtJn s fo rm.~ to t he flow of el ect.rom eg net !c e ne rllY, t he mach ine operates in the gellllro Ung mode. Th e active power re verses iUl dire cti on with a ch ange of the operat ional fun ct ion from getierat ion to mot orin g, but the f10lY of t herm al energy does not ge ner ally Gh ange ns dtrec uo n . Losses in ee m me n Ee s lIt'e irre versible. Th ere is a great va ri ety of ec., i ncl uding electelc raechtuee whi ch w nvert neat to el eet r~& or mecha nical ene rg y. Th e di S&ussion 01 such ECs is given in eb. 9. T o provide li nka~e between win di ngs (loops) a nd cu r~n LS i t is necess ary to prod uce AI! el ectromagneti c fie ld . The rotating fiel d ill eteewtc m ach ines is se t u p by alternat ing or di rect currents. T be react ive power m ay Ilow in an EC operati ng in t h e s tead )' I t ate f('(lm eithe r th e s t a te e or rotor, or from beth s imultan eous l y.

1.2 , ... L. w. 01 Elo clrom . c t.' IJ, lu l E.... rg y Co .......lo..

One 0/ the ~lla rlU O/ liat l irlt alld I/u sÂŤond laID is tha t an Be allO rtp~n" all tMrgy conttnlr4tat. The oloct ro magoe tic ener:y. being dis t rib utod at. i nfini t.y along a,n electric powe r line, i.II s to red in mag-oetie-field ene~y conver ters f it-hin th o ai r g a p bet"""n t he nator and ro tor. In t.r ansfor me rs , t h e energy is s to red in t.he m"inoti c core end i ll the s pace b etween t hll prim ary end sec on da ry , whore leakage fluxes close 0 11 themsel ves , faili ng to btl com mon ro be t h windings . Tho nlr ga p of a co mparattvcly amall vol u me C8 11 cc neen rr ete hull'O powers . h is of Im pcrtencc to note t.h at. in t u rbi ne generators of max im um powers and in in d uctioh machln es of t he s ing le ser ies , t he power d ensity ( W/m m3 ) in t he ai r ga p is eq ua l to a pp roximn tel y 0.5. In view of this ract , dnignin r of.el ectric mach ines con be beg un with t he estimat ion or t.lte ga p vol u me and t hell proceeded wi th tlao ealcul ettcn of wl nd ings aod geo metrical pa ra me ters of t he m agn eti c I )"!tem. Activ e a nd react ive rJo_ 6r en0'1rY ca n be coinci de nt or oppotlite in di rectioll irre spect iv e of ,,\,laether tna EC Mlns as II genere tor or m ot or . This moans t nal th o eeuve po wor ma y co me from t ho st ator an d tb o re act ive pew er frolD 't he ro t or, lind v ice versa . EC5 a lso o perate in the no-loa d ~Il d i tion lit wh ich t he y COII\'ort electr ic or m ec bn n lcal po wer intO! hea t . Sync.llrono us Illoch ino.s connec t ed i n pa ra llel with the li ne ,lI. nd wn /It no lo ad lire ca ll ed $gm:}l ro Il OI/.ll captu:Uur l.

During its op era ti on , a n ollll;t ri c 1t\ll.Ch ino tel eos t'S t her mal enClrgy. It is possible to produ ce all electr!c ;rnnchi ue fur nis hed with II. ~ he rÂ moplje in order to ab sorb heat ins id o the macu tne at th e cold [une t tons 115 a resul t of tho Peltier ellect [t he reby preventing i t from heat ing) and t o evolve th ermll l enore y at t h~ hot jll nct ions outs ide th o machine. However, th e availablo se m icon ductor co up les offer cooli ng at low cn rre ot densi t ies. so the i O(o. resultin/l fr om t he i m prov ed ccolme can on l)' be brought abou t .u the eoee or an inc~ ue in l he o\'era ll d imensIo ns of the machine and a wot&elling of its COCf'gy cn llra ctem li cs . This a ltes lJ t.hnt. t he t hermAl one rg y flux es .. well 85 t he mech an ica l e nergy an d e lect ric enN.,y f1uxe.s in an EC must be regarded as closed energ y loops . Th e cendtucn or reson ance e xi s~s ill electric machines jllst II! it docs in moet enorgy converters. E Joc-tr iell.l and mech a nical phenomenu t h at occur In ECs lire reso~ a n t. El ect rtc machines oxhl bit elec tro mec h an i ca l reso nan ce pt whrc h urc rot euone t s peed of 111(l fiel d , t.. is related to t he meeh ar ucal ro t ationa l s peed of the eot oe, n , ruaasured in revol utt ons per second , by the ex press te n

/1 - pn (1.1) where p is th e number of polo pal rs. In ,.. t wo- pol e m l chlne, t he power li ne freq ue ncy and t he sync hronous s pood of the rotor aNI th e S IUD e- Electrie maeh ine.s are built : - 0 1111

C h. I. lrilro</u.:fio n to EI"<1. ol'ft/l'C honk'

in such II man ner t hat the wave of a megneuaing force in tho air gap distributes it self i n ~ eg ra ll y among the poles, so th o processes of energ y conversion In two -pole and Multipol ar machines are eseen-' tinil y id entical, t he only d)fferenc-o being that in t he latter machi nes tho synchrono us speed o~ t he field end t he mecha ni cal spee d of t he rot or are 0 factor of p lower. Th ird law. El edromlXhanic.a1 enrrgy conoersion 1$ due /0 the fields

thot are stotiOll ery with respect to tach other. T ho rotor and stator fi el d~ in the air gap of a machine, which nro stationary with respect to each other, prod uce 8 resulta nt field and electromagne tic torque; (1.2)

where 00. is tho angular veloci ty (s peed) of the field: and p . ", is the electromagnetic power. Th e fields displacing ln t he air- ga p with res pect to each ot her produce a Ilux of thermal energy, thus indi roctl y affecti ng the dist rib ut ion of tho flu xes of mecha nical lind elect ric. energies. Th e wi ndings of elect ric machine s must carr y pol yphase cu rrents flnd show a proper arra nge ment to produce 1\ rotating field ill the ai r gllp. A rotat ing fiel d (jail he set up by II two-phase current syste m, wHh t he wind ings disp laceu OO~ in s pace from one another and t he currents ahitte d in ti mo by 90°; b y a th ree-ph ase current s yste m, with th e Windin gs 120· upert in sp ace lind 120· in tim e; lind , in t he gener al case. by an In-p hase curren t system, wit h the win dings dis placed 3600 /m in space and currents shifted 36Ct'lm i n ti me. Direct cu rren t can also produce II r ot llting fiold, In which case the de windi ng must rotate. The win ding ca rr yi ng altema t lng cur rents to produce a ro tating Held er e us uedly st ationllr y. In 0 sync hronous machine, the rot l\ti ng field is hlrgel y SE'C up by the curren ts i n t he windings dis posed on t he s tator. The field rota tes at a speed 10)•• The rot or runs lit t ho s ame s peed , (0 , = 10) " thllN'forfi the frequency of the r ot or curre n t is t , "'" 0, i.e. d irect current flows th rough t he ro to r" ~ i nd i ng . In a de mach in e, t he ;field (excitati on) wind ing is on t he s t at or, and t he excita tion field is stationAry . Rnta ~i ng tb e armat ure. whic h is t he rot or here , pro duces t he rot ating arma tu re field , wh ich revol ves at the sa me spee d as t he rotor but t n t he opposite di rect ion . I n induction me chtnee, the frequ ency of curr ent in the rot or Is

f,s

(l .3)

where th e sl ip (speed differential that is a fract ion of s ynchronous speed) (1.3a ) oS "" (e, ± w ,) /w ,

Theffiore, t he s peed (angular ve locity) of t he r ot or Cllr pl us the speed wi th which the rotor field tn\'015 wit h res pect t o t he roto r Itrucloro ill , lw8ya equ al to the speed of t be field (ll •• If t he rotor t urns at a s peed hi gher th an "'. in t he sa me d ire cti on as t he field excite d by s tator eu rrenlll. th e rotor fiel d travels in the opposi te dire<: tloD to t he rotor, 50 t he Slaw' and ro to r fiel ds a re ag ain eteullOar)' wi th respect to each ot her. In tr«osfor mers the windinp are s l lt-Ioo lry, an d thus tbe ft . queociu In t he primary end seco nd ary aro lbe same. It C80 th en be assumed tha t t he fi eld s of the pr imuy and t hesecoodary lJ'a,·el . t tbo sa me apeed . The «I ncept of s ta tionarity of fields in transfo rmers is of little consequence for the InlllYllis of the proc:esses of wergy trans forma ti on . The th ir d l aw facilitates tho ana l y~is of eD81i Y cc nve rescn pI'OeeS-

in electe tc m ach i oes an d forms the b as is for t he re present att cn oJ tlnerry cenvcrsrce equ ations. For elect ric-fi eld and el ootrOmllglllltic-fiei d energy con ver ters l.ha field stationar ity conc ept dees Dot ha ve such a g reat s ig nificance as it does for magne tic-fiel d energy conv erters . T hese converters are most vivid l y represen ted IS en ergy COncen trators exhibiting elece omeclUUllcal res onan ce . Since eteceecmee hentce is pa rt of p:byslcs, aU basi c ph ysical laws are a pplicable to electric machin es . To t hese belong first of all t he law of llnerg y ccne crv e u on . Ampere 's la w (circu it al l aw). Obm 's Jaw, et c. At t he toot of t he eq u ations descr ibing energy CQ nversion in elect ric ma ch ines are Mnxwell 's equatio ns an d K irch hoff' s Iawa,

SE'S

tol . .Application of Field Equations t e the Solution of Problems in Electromechanlc:s Electrom echan ica l enorgy ce nveeeree in magn eti c-fiel d ty pe machines ()(curs in t he s pace where the machine concenl.:tates t be energ y of a mag neti c fie ld . Kn owin l tb' fi , Id, we can esumale vo l tages. curren..,. mechanic al to rques, losses, electricAl paramelan:. and other q uant ities of interes t un dor t h o st ead y-s1.lte an d t r ansien t cODdl t loJ\l. T he calc ulation of tbe el ectrom agne ti c f ield in an y ecergy converter. be it II si mple or an ip t ricate t y pe . present" a com plica ted problem, and iLs eetn uen Involves diffi cul ties even wit h t he use of modem means and mOlI t ad...an ced methods availab le. The electromagnetic field a nal }"ll15 is one of t he main aspecla t h.t a1 -wa)'. attra cts atten tio n of researchers. T he requ ireme nLs for t he lCCur. cy of elee tromlg'neUc fie ld calcu lat ions become in creasingly !tring8Qt b ecause of th e growth of the s pecific an d t otal powers of eDetVy conv er te rtl an d more sev ere t emperature CO.IIditio ns i.n wh ich they have to operate at hi gh efliCliency and Improv ed reli ability.

Ch. 1. lnlroch,dion to f loctrQm.dlonlco ,

Over tho past few decades a lorve nuwber of bot h special a nd universal met hods have ap peared for t he anal ysis a nd calcu lati on of elect romegneric fields. Maxwe ll's equ at ions are t he bun fro m which one s tarts with th e calcula tion of a n electromag neti c field. T ho)" are usuall y give n in d iUer6llt ia! for m. One of t he eq ua tions establishes t he rel at ion bet..-ee n t be vect or of magnetic field st rengt h jj and t he vee aer of curre nt density T curl fi _ T (t.4) Integratin g bot h aides of the eq ueuen over t he aRI 5 1 tee, S I )' , th o simple t wo-dimllfUllonai case of I magneti c. field

(curl 1l)" d~ _ )

<7>"tiS

( I ." )

and a pplying Stokes t heorem

~ (cur l ill" JS _ ~ H dl we arrive at t he welt-known ci rcujtal l aw [Am pere's l aw)

~ H, d'-_

m . dS

(1.6)

whore t ho area of t he surface under considera tion Is S, inside which th ere n OW! t he current f of den!!t y 7 i n t he direct ion of vector H. t he current being eonrreed within th e closed l oop I. For loops t completel y encireHng the current-cllrrying cross-section S , t he riGht-hand !Ilde of Eq. (t .6) repeeeenta tbe total CUlT6nt

J, (!>. Js_ t.

( 1.1)

The ml8netlc fiald vector B, also referre d to 13 th e magnetic tnd ucli on, or the magnet ic fl ux den!lit y, is defi ned in teems of t he permeabili t y tl of 11 medium &nd t he magne tic fi eld s t reng t h H produced in t be medium; (1.8)

where d iv O ... O

(1.9)

The di vefieQ C6 of t he fiel d is th us zero . T his means t ha t t here is no ~cu rren t~ flowing In and out of II magnetic fiel d (magnetic li nes never end h ut ol cee on themselves), t.e. free mag netic charges (mo n ~

poles) do not exi st. i n n a ture. The ma gnet ic fi el d com pon en ts n- an d li can be fou nd if we sol ve the field eq u a t tons for various p arts of a conver t er of defin ite con figu r at ion by observ ing tho boundary cond it ions of co nt inuity for the no rmal com ponen ts of the B fi eld vect ors lit tho lnterf ace b etween t\Jo med in 1 and 2 (wh ich differ in per meab tlity ) 1 B in -

.8~n

(1. 10)

and l or t he tang ential compon en ts pf t he fie ld strength 81/ = I1: l pr ov iding t hat

CU IT l."n !

(1.11)

s heets o n th e bou ndary surfaces do nut exis t.

As shown by exper imen t, Eqs. (1.7) through (1.11) permit defining th o m agn et ic field nnaly t icnll)' on ly lor a rather limited range of prchl ema wi th t he si mpl es t boundar}' condi t ions . In consi dering th e real parts of electr-ic m achi n es with rathe r compli cated sh apes of ma gn eti c cores an d.current-curryi ng elem en ts , a n umb er of ass u m p ti ons hove t o he made to obta in even an approx tmale solution . Sim plifyin g assa m puoos may a pp ly to s urf ace shapes , curren t distr ib utions, th e pro perties of me d ia, and laws of t hui r motion. In cases where t ile fie ld sources lie fa irly far awa y from t he Held reg io n un dor con s iderati on [L e. T"'" 0) , it is sometimes adva nta geous t o introd uce the n ot ion of a ma gne t ic s cala r po tenti al Illm' Because of t he curl-free ch ar ac te r of a scalar field (c url I f - Ol, the m agn eti c fie ld stren gt b jj ca n be ex pressed as it = -gr ad 'llm (i.1 2) For II scala r fi el d. L ap lace's equa t ion hold s: 'V 2 q:>m ... iPfJ'm/8:t' {f'q:>m/{)1I2 i) 21{1m/8z 2 = 0

(1. 1 3)

The fie ld li nes here prove d iscontinuous. Tho source s lind sinks olthe fie ld wfll be the surfaces havi ng differ en t m agn e t ic po tentials. The di stributio n of potentials depeiu)s Oil the d is t ri bution of cu rren ts in t he Windings of a con ver t er and is defined up to II. constant in an y lo ca l r egion. 1\lost boundar y co nd itions lor a sc alar magnetic fie ld il l el ecr rl c machines are D ir ich let con dn tons , T his is com mc nly found t o be a Iav or ahl e fac tor for t he solu t ion of a problem part jcul ar fy w hon uslug approx im at e methods. The f.id d celcu latton a Dim a t defini ng tile componen ts of th e mng neti c fi eld s t re ng t h Dlong the three axes H :< = - 8r9m/OJ;, HII = -lJ ~7Il/0y, Hz "'" - 8fpml{); (1.14) K n owi ng t hese com pone n ts and us ing (1.8 ), we can find t he B fi eld vector com pone nts an d tben 0l8 8n:etic fl uxes lind fl ux lin kages. The unit of measure of a magnetic pctcntd al is the am pere, t herefore this qu an tity corr esponds 10 t he v-agnetomotive for ce (m mf) :15

regards proves The widely b y t he

Ch. l. Introducti on to EleciromecMnin

its mea ning. The ~u uc tiou of t he fl ux in a potential field q'", to correspond to th e megnet.ic fl ux. calculation precttce of rot ational elect romagn eti c fields uses t he n oti on of a magn et ic vect or po ten t ial A- defined relatio n (1.15) B '"'" curl it

Solv ing simultanoously (t "f) , (1. 8), and (1.9), an d t hen (1.15) gi ves

Poisson's equation (1.16)

in which t he mag net ic vector potential calcul a ted up to a constant acquires a definite physical meaning. The circulation of t he vector potentia! over the loop ill found to be eq ual t o tile magnetic flux throug h the sur face boun ded b y t his l oop . W het is i mporta n t is t hat t he sha pe of the sur face is of no consequence and t h us can be arb itrary. In the three-dimensional CMe, Eq. (1.16) is written for each of t he t hree components given as the projection s on to th e corres ponding coord ina te axes. It is often permiss ible to consider t he fi eld of an olectrlc machine as a flat , two-di mensional, pattern with one curre nt compon en t , for examplo, alon g t he z axis:

o!A.'ox! + oA.loy! =

-~7z

(1.17)

Tn t his case tho mag netic vector potential takes on the mea ning of the magnetic flu x per un it lengt h in the z direction. T he B field vec tors along the x and y ax es are given by

8 x = {}A .loy,

-oA.tox

(i .18)

The solution to t he prpblem inv ulvi ng t he dete rmina ti on of t he magneti c field in eleorrte machines is most comm only s ought under t he boundary conditions of the second ki nd (Neum an n cond it ions). The function of tile [l ux in the vector field A- cor resp onds to the mag netomotlve force, i.e t he func tion of t he potent ial is proportional to t he magnetic j lux. For defining the magnetic field , it is usual to employ s l milit ude met hc ds and the methods of physi cal and ma thema tlcel modeli ng, Experie nce at tests t hat th.e not io ns of scalar and vector magnetic poten ti als equa lly well hold in modeli ng of magne ti c fields , al t hough t he realization of houndaey cond it ions when usi ng eit her of t hese two noti ons is sub srant tully d ifferent. Wher e t here is a need to s olve th e problem with ccnstde rstion for induced cur rents , t he nct tcn of th e magneti c vector potenti al is t he onl y one accep mble. 'i n which case Poisson 's equa ti on must be replaced by the s o-eatle;d heat-eond uction eq uation,

1.3. App li,otio n 01 Fie ld EquMion,

Most di verse mothods apply to sol ve t he obt ained eq uati ons for a ffillguetic field under tl lO cc ud it lons (1. 10) and (1.11) at t he boundari es be tween d ifferent med ia . His torically th e me t hods o r di r-ect solution have developed most inten,s h'ely, which commonl y give an accurate or a ppr oximate analY~ical resu l t. Among t hose, we should note the meth od of images and t he meth od of sepa ration of vari abl es. Conform al t ransformationJ of the regions of interest , by which comple x bou ndary ccndtttons undergo' s ubst en u al cha nges and become ap precia bly simpler, Hla y a not iceable pa rt in tile developm en t of the methods fo r 't he s ol uti on to mag ne tic field proble ms. Th e sol ution to L a pl ace '~ equeuo u is worked out for relarlvely simple are as and then ap plied to th e initial regio n. Th e Ineananta, Le . q ua nti ties inv a elabl a.In t ransformatio ns, a rc ma gnetic potentials , magne ti c fl uxe s, and t he moduli of the magnetic n ux density vec t ors and field s t rengt h vecto rs. 'I'he solution ue ert in the transfo rmed pl ane is found accurately, whenever posslhl a, or approxi mately usi ng an ana ly tical or numorlcal method. 'l'he metho ds of conformal trans formations mai nl y a ppl y to trrotauo na l fields. The methods of in tegral oq ua t ions are su it able for t he s olution of a num ber of ro tati onal field prob le ms. T he l ast l ew decades have seen an exceptionally ra pid developm ent of the ap proximate numeri cal tech niqu es based on the me t hods of fin ite dif ferences a nd finite elements. ; The progress in computer ongi neering an d t he creation of fas t compute rs wi t h a l arge memor y capac itY have enabled t he effect ive introduction of t.hese approximate methods. T hey per mit obtain ing the solu tio n of a desired fu nction (p9te ntial ) in the fiel d region for each particular caae. A substantial disadvemege of t hese me t hods is t ha t t hey do not aUow for de ri vi ng t he general expressi on for t he sol ution, so it becomes necessar y to obtain a new solution with any change of the parameters nffecting the field . Howe ver, t he po ten tia lities of co mputer engineering g rea tl y offse t t his inc on venience, Elect romechan ical energy conversion is t he result of interaction of el ectromagnetic forces a ppearjngt i n an energ y co nve rte r. T he determ ina ti on of t hese forces is t he most impor ta nt s tage in designing a converter. There are a fey." ap proechea to a ttacking th is probl em. A mechan ic al interaction at cu rrents, or whe t is s ome times calle d pc ndar mot.ive interaction, 6b~ys Ampere's la w. For a co nductor c Qrr ying cur rent t an d pl aced in an ex ter na l magn et ic fiel d ii, t he emf f Is gfven b)' t he vector prod uct:

1 = HBft

(U9)

whore l rs the uni t vector alollg t he ' wire ca rry ing curren t t. Where t he magn etic fiol d is knowc.: from t he sol utio n of Maawell -s equ a tions, it is convenie nt to expr ess em fs in ter ms of t he cur rent

ell. 1. InhQd.. c ~on 10 EICl Cll'Oll1ecll,niu

call ed the tensor of to t he form

t ensi ~ll ,

t he ex press ion for which can he reduced

~ It,,H,,iT-

J.lG

(nfl~/2)

(1.20)

where JI" is tile vector component of t he magnet ic fiel d stre ngt h II in tI,e dir~tion of the un'it vector it norm al to t he eur reee reglun under s lIldy. Upo n ln lc$'r atlng t he ten si on te nsor over the en tire s urf ace where tho megnebic field is slILsta nt illlly high i n magn itude, '1'0 can t hen go to t he .eom pu t at rc n of emrs aud electro magne tic torques. J t is somet imes exp e~i ent 10 deter min e elecu-omagnc ttc forces and tor qu es from t he o>.;prlc'S sion of mutu al s pecific l'nergy JW/lIV I'(Ilorroo to IIllit volu me, ,w'hich is equal to Iho scalar prod uct ot t he curre nt denstt y end lho' vect or potential of an ex t er na l magnetic field : i

aw/av _ A i

- oil

(1.21)

The "ex t MlUwelrs eque tlo n. which is of mu ch Im pcetsnce. rela tes t he vect or of electric fieitf s tNlng th 72 to th e magne t jc flux denslt.y: jcu1'l jJ_ - dJJidt

11 .22)

In it s in tegl'al form , the exp ression al l ows us t o pose t o t he ex pressi on for the IIDl f E of It'lo('p (d isregarding the grlldie111 of II sc al ar electr ic pote nt ial ): ~ E=

~ :"' (djjfdt) dS +~ I V X Bl dl

(1.23)'

- ,

The vcctcr e of B lind If gfve us ample i nl(wmal ioll on the mag_ notic fieJd end hence on all inte gr al qua ntt tf es suc h as currents, ('mfs, voltogcs , forces , aud t orqu es. The classtcal Iheory ql electric machi nes re lies I)U t he equ at ions of circuit th eor y whic h ,d efl nes th e pe re rnorers in in teg ra l no tation. T he most i mpo rta nt parameter of all ent'rgy con ver te r is its induet illite L defined all th cf rat io of tho lnstantencoua val ues of flux Iink age 'I' produced Il)' the current i to tbre curr ent : ! L = \jilt (1.24) ;

If th e Ilu x d ue to cur rent in II wind ing Of COllduc.tol' links on ly t his wind ing , we can l lt.1 k about self-lnductence: where th e flui links one wind ing due 'to curr ent in t he other, we call ta lk s bout mutual in duct ance. T o define t he Ilu x li nkltge to r the field desorilmble by Lapl ace's equa t ion , ItIsnecessary to apply Eqs . (1.14) an d (L i B) in ord er to go to t he cf l?ressi on for the ma gneti c flux densit y and th en integra to t he mllgnetfc Iluxes for a conductor onÂť' its entire ercse-sccuon S. T he nux li nk age, when ex pressed in te rms of t he

1.3. Apptlc ftlJon 01

~i.ld

Equ fttio nl

mogn<'tic v ecto r poten tial , is defined wit h respe ct t o A D taken as. th e reference for coun t ing off t he ru iming val ues of vec tor potent fak A ! exis ting ill the cross-sccuon S!

'Y=

,J(A I-~o) dSI!S

(1.25 )-

T ho pro blom of determini ng t he flux lt nka ge p racticall y re du ces.

to sim pl e arithme t ic operations if 'the conductor is b rok en down iut o a fini t e nu mb er of elementary areas each of whi ch ha s n definiteval ue of A I foun d from th e caloul atlion of tho fie ld. For th e CMe when th e n ux lor all points i n t he c ross-se ction of t h oconduct or of a wind ing (with a ce t'ln.'i n n umber of turns) is constan t.. the flu x li n k age ca n he ex pressed : as

L = If!l ",,; w(\)!t

(t. 26) Âˇ

Introduce the no t iou of permeence ' A

<!i! F

(1. 27}o

where F Is the mag ne t omo uve for ce (mm !) of a cond uctor (wind ing), The indu ct ance n ow be come s Independent of t ho curr en t alld flux and is on ly a fun cti on of pcrmcance. L ... (CilF A) / ÂŁ .... (WllWA )/l

wt ..\

(1.28) '

In 8 p art icular cnse wh en ai r gap s are t aken into cons ider a ti on. L "'" w2A i W111of.. (1.29"

where J, = ,V Vo is t he coeffi cient fJ ~ perm eance for fl uxes produ ced by the mmt. T ho resul t an t rela ~i on (1.2 9) m ask s somewh at t h e nature of orig in of Inductance and njakes th.ill parame ter appnren tdy dependent onl y on t he geomet r ical 'd imensions an d t ypes of m eteri al. H oweve r , we should rec all t he' Initf al rel ati on (1.24 ) from wh ich it un am bfg uo usfy follows t.hat such 0 pa rame ter ns ind uct an ce is not lit all m t rt ns tc in any con ducto r or wind in g but is in d ic ati v e of th o cond i ti ons of existe nce of a magnet ic field in an onergy convertee. Induc tances do uot remain consta n t but vary q ui te appreciably wh en (a) fl uxes chan ge sl owh-. (b) short-ch ouned conto urs He on t he paths of m agnetic uuxee r alying in Lim e ana amplitnde, (c) h ysteresis m akes itself fel t, an a (d ) the portion s or conver ter magn et.lc ci rc ui ts d isp lay uonlln uar cb erecteetsuce of megucuaenon. In t he th eor y of el ectri c machines, for exnmple, t his Inc t is taken into consi der at io n ill a n umber of ways, bu t a suffici en tl y ' conststeu t, a p proa oh does not cx tst.. T ho rea son is t ha t t he ta sk of quan titat iv el y oonst d erfn g all t he iulluences is extremel y com p lex , I t is eaay t o calc ula te th o omf ta,i t ... con stan t ) in t erOls of t h eself- and mu t ual Induorences procee d ing from ure changes ill the>

Ch. l. Inl',o d uctio n 10 Eledro medwlnicJ

i nt rinsic energy of t he Held in moti on I = - ow /ax = - ( i ~l2) ({)L!{)x)

(1.30)

"T his for mul a shows tha t a change tu inducta nce is t he req uisite -coudiucn for t he electromechanic al co nversion of energy . Despite a rel ati vely simple form of Hold equations (Laplace 's and Poisson' s eq uatio ns) an d a s imp le cha racter of bounda ry cond iti ons, abe solut ion for t he field of all anergy converter ha ying various [boun dar ies, a l arge nu mber of s patiall y a rra nged coils, Jet alone th e e oul.ineari t. y phenomena 'a nd hys te resis th at muse be t aken in to -accoun r, has been found only i ll t he l ast years by use of th e numeric al meth ods proceedi ng f rom yet rat her numerous assumptions . With t he use of ana l yti c~l and sa mlg ra phlcal methods of calc ula'l io n t he numb er of assum;ptio ns grows still more. In par ti cula r, we -can en umera te t he following assumptions. 1. T h to ma in field which determines eue rgy convers io n in electric m achines and gives rtse to tho main self- lind mut ual Ind uct ances -or wind ings Is pl ane- pneallel. 2. v eercus leakage tndu ouve eeacr enees ere Independent of each -ot her a od of t he main fl'e{d. It is comm on to isol ate per meances -cor reapondlng to s lot. odd , and d iffer enti nl lea kag e flu xes of so windings. 3, Th e s urfaces of s tator an d rot or cores of elec t ric machmes are ,s mooth; the ac t ual saliency Is giv en due cons idera tion by ln t rod uc- , Jug air-ga p coefficients , 4. Th e perme ab ility of fer romagneti c aecttou is taken in finite ' .e t t ho preliminary calcu la tion erege. 5. T he use of the s upetposition princi ple is per missible, 6. Th e processes of e ne('gy ec nverstcn are dependent on t he l und a.zaentel har monics of currents and magnetic fluxes . 7. The effect 01 eddy tcur rante ind uced in mag netic circuits ie ,u eg lig ible. We have ci ted but n few tnetences of all of t he possibl e .c cnstra tnts. , In t he list of approximate met hods, th e n umeri cal met hods used for t he solution of Held eq uatio ns occupy distinct posi tions and have -great s ignif ican ce, Th e f; nite difference meth od (FOM) is part.icu,lart y popula r. T his method was in ex tensive me well before t he ' m trnductio n of digital compute rs to t he calc ul ati on practice. The devel opment of hlgb-speed la rge-ca pacit y mem ory comput ers wi th '8xtBnsive g Bnerali zed prog ram li brarillS and the i nt rod uction Of T efficient algorithmic la nguages has made pop ular t he met hods of ..cal cula tion of elect romag !Il;ltiG fields on t he basi."! of finite difference , a pproxima tions of GOiltin,uity eq ua ti ons of t he most div erse forms . Th e main i dea under-l ying th e ap plicatlon of t ho FDM in elect ro\IJlagneti e cale ul at lona comes to the replacemen t of t he conti nuc us . -dts tr jbut ton of a sc al a r or vector magnetic pote nt ial by a d iscrete '

1.]. I'Ip p Uca lio n of Fjo ld Equ atio n.

dis teibut.iou of t he same fu nction i o a um ned n u mber of potnta within l he reg i on beio g stu died . The: poi nts a t whic h t ho v al ues of the run cuon n e ve to be found a re dia trfbuted o ver t ho region of Inter es t : i n other wor ds, n r,oord i nil.~ grid is d rawn Oil th e region. l u th o FDM, this g riu s hows a regu.lar pa tte r- n. In mos t ex te nsive use :1I"e the rec ta ngu lar (or qu ad rattc, in a partfc ulnr cese) system A, and the polar svstem o f ccor dtnates. Fig lll'e 1.9 shows ho w t he rocA tan gul ar gri d (ne twor k) d ivides a A, salien t-p ol e syn chrououa ma c hine region one pole pit ch ill length into a few mes hes . T he coo rdl na te system and t he {a rm of mesh es 01 the gr id ar e so chosen as t o A. approximate mos t nccura tel y th e bound aries of t he reg ion and t o P'jJ. 1.9. SUPl'ri mposing til.. aqua...... Intr od uce t.h e minlu nnn pesalhle m~b grid on lile "rea under ann(ya.i$ errors in t o t he config u ra t io ns of neighb oring regi ons . The g rid pl otting at th is stago is oft en dona by the tr ial an d e rror metho d. and d ep en d.'! o n t he exp erien ce and s k iU of t he lu veatf g ator ; t he pro ced u re hinds itself t o automation only for local zones. In acc ord ance with t he F DI\J, th o f iel d equut.ion s wnue n as partJal derivatives are re ar ra nged t o the rtnne d iffe ren ce form us ing th e expressions toe II T a ylor se ries. In t he case of a qu a dr a ti c g rid. wit h II pitch h, t he La placia n ass umes a si mplo form

\71A ~ (t l h 1)

â&#x20AC;˘

L.j (14/- A o) . .. ,-,

(1.31)

for a poin t rep rese nti ng a va l ue of 'the Iu nc t fon A D a nd s u rrou nded by po in ts AI "'" A . (see F ig . l. U). ',I.' ho erro r of digitiu llioll heee depend s on t he Iouet h-crde r de r iva.l ives In th e so ug h t-fo r Iu nc t to n and CR.n be reduced by decreas in g uie grid pi tch h. The so lut io n t o L apl aca 's equation in t he Hnlte-dtff'erenc e Cor m amoun ts t o perf orming elemen tary a ri tJlm eti c, oporatio ns. 'I'he nu m ber o f t he nodes of t he solu uo n ma y in prncuce be ve ry high lind usually rang es i nt o Il few t ho us ands . 'I'her efoee the solutt on to t be o btained system of high -orde r equations requ ires the use of i lora t ive or s tatis ti c,al methods . T he d irect so luti on to t ho s yste m of eq ua no na using , for ar ample, Gauss" method p ro ves t nilpossiblo. With th e i te ra tive meth od of ca lou la tt ou. t he v al ues o~ t he fu nction s ought are prese t at. th e Hrs t s t ag<lS ei ther ar bi t rari ly or ion t he basi s of certai n physica l consid erat ions wh ic h s u hasq uont ly improve I-he co n vergence nf t .he

C h, 1. l~lfoduetio " to Elccl,omoeh" "in

so l ut io n. By por fol-ming t he multip lo eaqueuua l t raci ng of ail nodes of the gri d a nd 501v.ilig the ttn ne-dt treren ee r el ation, it beco mes possibl e to decre ase the rema i nder of t he fie ld equ atio n t o t he max imum perm issible v alue. ' T he num ber of uee auons [repet.it fve t racings) can run into ai Caw l ens , hu n d reds , fi nd even t ho usa n ds . One COil not IIlways be cO!llidenl t ha t t he solution t ends to the ideal va lue , t hereby ensuring (t h,a COi l verg ence, T he Iterati ve method is ra ther rou t ine , is easy t o Iormahae for so lvi ng problems Oil digital compu ters, an d is se re Irom calcul atio n er ro rs s ince possibl e errors are recove rab le at subsequent s to ps. T he effective v ersion!' of the FD M lire availa ble at presen t, whi ch gi \'e good conv erg ence at a high accu ra cy of th e eesul ts. A for m of gri d ma rked out on t ho reg ion of interes t aff ects Lbe accur-acy of t he so lu tion. T his ci rcu mstance h as rece n tly stim ula te d th e search for the bes t fo rms of l ayou t of regi()ns. I t is possibl e to opti mize sequentiall y the grid structure iJy ca lNIJating the deri v ati v es of highe r ardor at' a defi nite s t age of t he eolu tlc n wi th a vi ew t o ra ise tho mesh densit)! of the grid at the n ext s tage in t he regio n of higher va lues of these denvat tvcs. T he met hod of nnrtc ~I emell t.s (FEM) develo ped in t ho last years di spl ays excep t ion al fleXibility in br ea king down t be s pace of an electr cm ag ne t lc fi eld d~st ined for eulc nlatton . Worked out til 'St for t he needs of at ruc t ur al mech en tcs, this me th od ha s t urned out to be ruther convenien t for 't he calculation of electromagneti c fiel ds in electric machi nes which ih'a ve boun d aetos co mp lex in M nfigu r aUon and ex h ibit nonliOOllri t ies no d in duced cur ren ts, The reg ion for t he fu nction sought is hrok eu, do wn into 0 Itni te n um ber of ele ments m ostl y in t he form of tria ng les wi th straight or r urvili nenr ei dee.: The dim ens ions of ell'Jrlcn ts may differ s ubst an ti All y depending on t he ex pect ed intens ity of changes of the field . The desired func t i on ins ide the ele ments i ~ assumed to obey a ce r t nl n la w. In a sim ple cas e , firs t -power s pl ine (unctions are ap plicabl e, Thus in t he tw odimens ion al case, the Iuncuion A (x, y) COr a tri angul ar elem en t with coordinates at t ho vertices , X I and Yh Z'" and Y"" Xn an d /In, c an h e wr itten lIS wh ere

+ N",A", + "'nA.. N l .", 1(1/2..1) lal + bIZ + CI Y ]

A (.r, /I)

C:.

NIA I

a l = X", Y n ~ x ng",

I-J

(1.32)

/II

b l = /I", - (lin 6. .... 1/2 1 x'" II", { CI = z ", -jr.. 1 Zn lin A sim il ar ap proach ap pl ies to determ ine t he val ues of N", an d N n _ Th us each ele ment is 'descr ibable by its own p oly nomial, which is s o chosen !IS to preserve tho con tinu ity of t h e function al ong t he

elemen t bou nd ar ies. T he values lit g rid nodes ere found by ~ I ng t he " ariat lonal pri nci ples, an u in t his ~ pect ~ he FEM is enen s t a t ed in t he con tex t o f the R ih and Gal llrki n me t hods. Wi t h t he vaneuc net for m ul ati on, th e so lu t io n w it he pro blem in vol vi ng n tw odimensional magnetic fi eld d efin ed. by Poisson's e qu ation (1.16) is equivale nt to the een d tucn of m lnlr:nl za Uon of a certa in energy fum;:LlonaJ

JHI(11. 1BdB ]:"'dv-l l/Ad%d B

itl!ido the regio n of in teg r a ti on R h The Iunc t tenet dis pl a ys s uch • pro perL)' that a ny fun ction ....·hieb mln i mi:tes it eeusnes bo..h differential eq ua t ions and houndu ry ~ndl tio ns. I n t.be CllSO oC 1101111near de pen dences ..h e process of m in~m i za t i on involves the so l ut io n of t he system of no n li near a lgebr.~c eq uatio ns. co m mo nly us ing tllo Nowton ·Raphso n method which g ives good convergence. The ca lc ul at io n of magnotic fiel ds i ll elccrete D'I11chines witll t he aid of t he fi nite-d ifference a nd f1n i~e -ol eme nt me t hods enebles a more accu rate evaluutfon of t he charac te ris t ics and pa ra met e rs or elec tr ic m eeb tucs. H oweve r, t he fID .\ { a nrl F EM call for re tn in ing a number or ess um pt tons t he l egj~lm acy of whic h are not alw ay s unques tton ahle . One of t hese 1I~ !Il lD p ti o n s made in eva l ua t ing meg net fo Iicl ds is t hat t he toothed; s to t or end roto r COl'cS oro i ll fixed mu t ua l poattlo n. Th e p osl ~'id n it llelf is mos t of ton c hosen lIrbit rQr[ly wit ho ut s uffi ci en t re as o,, ~ and t he res ults or field c alculations ar e t a ken va li d for o t her po~i.blo m u t ual positioTls , \Vhenevor t he at tempt" are m nd e t o calc ulate q:t ag not lc fie ld s wit h t he t oothod cores in motion . t he comp u t e r- aid$d ca lc ul a t io ns prove so t im eto nsum ing that the y beco me i m pr ac ti cable. It t ok es an es pec ially long time and large mom ory s i1.e to ea lcu hltll t he nir-ga p ba nd no ted for t he most in tens ive oU'Ign e t ic fi old . On t ho othe r hand . t he i nho mo· geneity of med ia In t h is ba nd s hows a ra t her reg ular chara ct er. for .....h ich reaso n II. l arge s hare of repeated ealcul atic ns can be do ne away wit h. Owi nV to t he efforts of a num bee ,of set en..is LS It bas beco me possible to evolv e t he ca lcul ation method hlllled o n tho ee peesenra uon of the fiel ds of real wlndlnga as a s qt of fiel ds of the si m ple3t loops disposed on core teet h (Fi g. 1. t O). Every loop encircles ana t oot h, The pla nes of t he c ross-sec t ion o ( loo p wires coincide with t he pia nos of t he cross·u cti on of real winding wires placed in the sl ots. The loop may abo enclose tI fe w tee th, o r extend along th e e ntire ga p. Al50, loops mtly ha ve diUore nt si deS l ocated in I lots of va rious sha pes and sizes, The essential point of th is me th od cn lled t he me t hod of per mea ncea " t ha t t he loop field must be defi ned no t for real b ut for specific

Ch. 1. Injrod ucHon to ElllCI.omcche niu

b oun dary cond i ti ons whie'h ca n be ob t ained only ortifici ally. Be yonp t he conf in es of t he loop, .r he perma en ce of tbe a ir gop be t ween t h e ro tor and s tc tor is assumed; to be infi nite. Un dor su ch boundary conditions t he fi eld tr av ers fn g t he ga p ex te n ds onl y in one d ir ect io n an d gets conce nc-e t cd in 'ihe area t h at d iffers insign ific antl y fr om t he area b ounded by the Jdo p itself. T he loop mmf here corres p onds t o t he m mf in the g ap. ,I h go ing awa y from tho l oop in opposi t e di rection s, t he loop fiel d docoys fas t. T he loop Ii old un der arti fici al bounda ry co nd it ions ol h\ bi ls a n in t er est ing Iea ture. The m agn etic

'" ,, ,, , Fig.

1. 10~ '1'b ~

Held of

.0.

&Imp lu t loop

fl ux t hl'oug- h t he g ap due'tc t he loop cu rrent is t ho ea me 35 l he umpol ar fl ux li nki ng t he l oo p when tho diffe ren ce of sca lar magne t ic potentta ts between t he cores is equ al to tho lo op c ur ren t . Als o , t he permeen ce for th o loop fl ux ~ hroug h t ho su rfa ce of an u n exci ted core cor res pon ds to t he perme ance for th e loop 1I11.x li nkage in unipol a r ffia/:Jleth:a t ion. Th i; 'is tho cas e for a ny for m of t he t wo-s id ed sa liency and f OI" OilY arrllf gemen t of loop co n d uctors in t he s lc te or In t he grup. T his fu n d ame nt al pro per ty of fl ux es an d fl ux Im k nges of t ooth-I nd uced loops oqe ns t he way of ev ol ving a ne w me t hod to ena ble th e developm ent 9f' methemau ca t mod els for descri bing the fields in el ect r ic m ac,hint'$ With du e regard roe t he tw o-s ide d s al toncy of cores . ! T1lie ma th em aticnl mo ~el t hou g h resem bl fng to II d efinite extent th e mo del ap p li ed in t he 'F EM conta ins A speci al feature. A portion of mode l eleme nts repres nnrin g th o IIi ,' ll:flp pe rme ance RfH fire not pe rmane nt and c alc ul at ed ,beforehan d eithe r wi th the ai d of rather cl rcumat nntia] ne twor ks ~ m pl o ~'e d in t h e FDM and FE t.l or an nlyt ically b y use of t he methods of conform al weneto rrnanons. T he d et e of this cetcut eucn are ent ere d in to t he com pu te r memory i n th e form~ of ap prox im at ion $J r ves o r ta bl es. T he t eeth an d y okes of cores are broken down into II n umber of elemen ts wh ose di men sions ca ll b e t aken a pp r ccia b'ly, i 3rgor (with ou t t he in troducti on of no ticeab le err ors) t ha n is t ho c ase wit h t he F DM or F EM. T he n on li nea r cha ra cte r is t ics of th ese elements are defin ed s t ar t i ng fr om the EHÂˇ

ae curves for corres pond in!; ma torials . As is done in t he frameworu of ot her methods , here too t he pe rmeabiltty ins id e a n indiv id ual element ill eeusldered cons to n t. T he mljgDetie s~a te of core etemen ts i5first se~ roughl y and t hen s pecifi ed mor e aceu re tely afte r so h'ingthe system of t he no nline.r equll t ions by th e Newto n-Rnp h80D' it er u ive meurod. Th e ma thema tj cal model bnsed on the permeanee me thod usesII relati vel y IlIrgc-'!li...e mesh patl8rJ1 and gives II Iligh acrur. cy of field I1l prOO U(".t iOIl, eapec ia ll)' iii th e gap b and . This ope ns u p poesibili ties for t he c.â&#x20AC;˘ Icu ta uon of fie ld'.!! 10 t he t ran si on t operatio n of clO<'trie> mll ch in ~ with co nsi de r ati on for th e effec ts of s aliency, Iliscr elene.ss of th e windi ng s tructu re, s al u ra t ion, an d in d uce d! currsn Ul, The equaucue Cu r all loo ps in rho per mean ce meth od repr esenreuon do not neceesnatc IIdd itiplInl coor di na te tra nsfor mations, Alt hough t he pl,.,gress in th l.' de volo pmout of etee rrto mach i ne models on th e b llsi ~ of rieh! eq ua tfons jlZ a pp reel ahle, t he meet meteria l ad vancements life mad e by use of the equ ation!' wr itten i n uie notati on adop ted i n elce t rtc circuit &hOOl)" T her efore in t he fu rthe rpreeeuta u c n of th e lcxt we will bas ic all )' e m ploy the eq uations of tbe ge nera lizocl etcc t rc meehanica l energ y ccnve etee.

1.4. The Primitive Fo ur ..W in dJng Machine All electric mac hin l.'S ere identic al in t ho 5ef15ll t hat th oy ecnvere, eOf:'rgy from elecret eel to medulII ica.\ fo rm or Ieom Ull'(;halliul to cleet rietll form . Bu t e loctr ic mar hili es even or the sa me se ries d iHe r from one anothe r in perfor m ance. Th e hosie t y pes of eteer elc mach i nes c>.n be red uced to II ge nerahaed , or prfml t.tve , me del re presenti ng :I set of two pairs of wi nd ings. moving with respec t t o eac h ethe r, I n Fi~ , 1.11 is shown t he ide ali zed model of a sy mme tri c, mach in o 118\'i ng II smoo th nir-g llp s rruc l ure lind s inusoid III wind ings . wi th. -t h'e perm ea nce eq ual t o zero. A sinusoid llll y varying ve ltuge a pplied to th e winding prod ucesa etrcu te e fio l d in t he air gap. W ith t he wind ings being sym met r ic. I ~i n Wl o i d n l s y mmet ric vo l Lllge sets up II ~ i nusoidal fi eld in th e ge p. Th e term ' pri miti ve machtne ' sta nds for an Id~aliKd lu;o-pol~ ttro-fAQ$~ I/lmmei rlc (bal anttd) machLne haVing on, pair of /rim/ings Dn th~ TOtor and th~ olher fXJir on Ou: stator l'S s hown in F ig. 1. 21. Here wU' w~ a re t he U nto r wind ings. el ollg t he cz and ~ axes ; w;. ~ are t he rotor wind inas 810ng t he IX a Dd exes; uj, u: . are voltag es al ong the cz ea d ~ u:l.'li on th e ! t a to r and ro tor reepeeth'ely; end CIl. is t he a ngu l8r s peed or t he roto r. Th e Ilnillysis of th e twopole machine M a model en1Ulles us to exte nd the fl.'S ults lind desui bc t he ' pr ocesses or.curri ng in a real multi pol ar machin e. T he t wo-p hase' marhine bes {ou r wi ndi ngs

u:..

.and is descri bable by four vol t age equa t ions (8 mi nimu m nu mber -of eq ulltions in eompeeisen wi Lh t hos8 used for describing Si llj'le-phese, t hree-phase. an d m--p hase machi nes). Conside r an idealtzed 'lIoiCorm-lIir ifap machi ne ' whose wind ings 11I"O t aken lo be in the 'form of cu rrenl ' hools ":bera t he mod distri but ion Is s lnU50idal. 'Our ide., lind m ncblne hllS:nO sa tu ration, no r non lin ear resetances, .and t herefore el:h lbits a ~i nusoid al field in t he alr gllp ....hen t he w ind ings ore fed wi t h sinuso idal voltage. T he Ideali zed ma.cbin e mooel is t he analog 01 an i nd ucti on mac hi ne when t hestah.r windi ngs", and Ultl aceept sin usoidal Yollaaes

:nt frequenc y f" 00" a part in t hn D. T he rotor wi ndiogt ca rry euerenta produced b y th e voltage ap plied to t he rotor or indueed by t ho e:urre nts in the stator win d ings. In an Ind uetion machine, tho roto r ang ular spelld is 11)• .p 11). (w . is t he synchro;II OIU speed of t he fiel d), t nd the rctce and stator fiel ds are st..atio narr. "Wi th respeet to eac h oLhh sinc e Lhe mech a nical roto r speed 11). plu sl mlllU5 t he rotor field sP,oed relative to (0), is equal to til • • T lte idea lized machin e model filpres&nls a sYllchronoU5 machinl if an ee voltage Is put 'Cl"Oi'lS the s retcr windi ngs and a dt. voltag, .aCf0S5 the rotor w ind ints, and vice vcrsa. Here (0) . = ClJ. , t.e. t he :stalor lind ro tor ftelds '8l'8 stational')' with mpeel to eac h other. If II de voh ai e d rives eqrfilnl t hrough the s tator windi ngs. t he rolor fi el d t ravels in t he dirK t ion opposlte to t hat of t he rotor, 11(1 the ~tato r an d rot or field! are s tatio nary relative to t be sla t ionar y refer. enee fra mo. With de ~u p pl y l o all t he wind ings, it is e nough t o

-01 fr equency I, = I,., 8il her

l A. Th. PrimitivD Foue-Wil1c1iI19 M . " hlnD

ha ve one field win di ng i ll which t he; ms ul tanl. m agnetizing force 13 equd to t be geo metric s um of th e magne thiog forces of each winding. 11l de mac.hines, t be arma ture winding ca nies a mu!ti pb l3e eneeIll ling curren t reetili&cl meehll.nicall ~ h y means of a commutaLor • frequency ee c verter (Fe). By red uci PI: a pol y phllSe s}'Stem to a rwoph ese 006. we o bt ain l.he model of iii de mach ine (FIg. 1.12). As in II synchronous machine, t he a rma tu re fi eld of t he de mac hi ne rotates ) «

, ", . j

.'•

Pia. 1. 12. Th e mooe l of. de IJIxhlM Ilod an 8C colltmn l.alor macbiof

in t he op posite s ense wi t h respec t to th e ar ma t ure. ' Vhen CIlr ... fit. the ar mat ure field is lIta tio no ry rela five to t he fiel d wlndi n: a nd to th e !ll ation nry referen ce Ir nme. I t. llllouid be note d t hat t ho sl i p In s ync hrcnoua mac hines and de ma cbfnes equ als taro . A com mutator can be repl aced by a aumi conductoe Ieeq uency convertor, re ed rola y conv erter, etc. T he processes of energ y conversion in t he air go p do not cha nge wit h t he repla ceme nt (If one t ype of Fe by t ho o the r. However, a co nventi onal com mutator holds a fixed ti e bet wee n t he frequ ency a nd t he ro tor speed (o)r• .wh ile a se mtcond ucre e Fe may afford the possi.bi li ~)' of eecurtng contro ll abl e (eedback rc regu ll'lte I; aecording to CIlr. A!I regards its power s u pply, a semicond uctorcommut ator mach i ne i.! Il. de mac hine. Historically. this l.yP'l of de ) _, I li a

"machine received several, names-c-eectif tee-rype mechtn e, se mtecedl,u::tor-.commutl tor machine. ee nreeness mac hi ne, etc. I n an lie comm ut ato r machine, al te rn at iog curre nts 61i.s t in t he st ator a nd rotor .....indl ngs. a Dd th e frequ enc Y converter t rallsfor ms t b o al te rn _Li ng eureem at lh . h us frequ ency in t o l h_ t of sl i p Ireq ueney (see Fig. 1. 12). .A! in other electric. machi nes, bere th o sta tor field iS lltAtio nary re lattv e t o t he ro tor fiel d . These machines.. een bfl of t he l ingle--phaso, th ree- phase., or mul tt pbese t )' pe!; th e s t at or lind ro tor windin p can be connected in series or paraUel . or ean have magne\ic coup ling. TIle primitive machine with a rotor s peed (I), - 0 can rep resent a n electeome gneuc eou veuer-c-e tra nsform er. I n t his COlle i t is s ufficient to cons ider se paratel y th e pair of wiudlnga on t he sta tor ond rot or olong the a axis or '" axis because wit h tb e roto r a t stands t tl! t here is no cou pli ng bet ween t he windings llhih ed 9O~ llpart in s pace, Although t ransformers perform electr omagnet.io ec nvaeslon of energy. t he )' belong to electric machines because of th e gene rali t}' of eq uations and for historical reasons. Tile classifica ti on of oloct ric mee hiu es hlto i ndi vidual types i.!J lare e.l y conventi onal. O ~e and t he s ame machine can cpe eete M e s ynchrcnous and lUi an ¥ ynclu'oDous mac hine. In etecter c mac hines t here occurs elC!(;tro mt'Chan ical and eledromaane tic e-ue!"ln'ec nverslc n simultaneously. Tho processes of elec1romechlUlicai e nerg y conversio n in the prjmilivo ma chine are described by vol tage equatioos (1.34) and equatioll 01 motion (1.35) ,.~ + (dldt )

(d ld t) M

•

L:..

(d /dt )J11

,.:' +(dldt ) L~

L,fIl ,

M (fJ.

'-' L~(fJ,

,.' + (d ldt )LD

(dld t) iU

(dl d t ) M

,.~

"• X

+ (dldt) L~

( 1.34)

(lIp ) ,J d fll , ldt ± M, - Jot .

(i .35)

Bqs. (1.34) and (1.35) t9gethor wit h t he equa ti on lo r an elect romagnetic torque lorm the ifundll. ment ll1 system of e-qu ations of elec tromecho nical energy ccnve rslen. In Eqll. (f .M ). uj-. u:.. loll . t:.. 4 . ':.. I~ are th e " oltages a nd curr ents in t he sta to~ a nd rotor Vo'indl op o n th e 0: and II axes respecti vely ; ~ . ,.j . "0. Ii lire th e re:sistlnces of s tator and rotor win dinp rospecLh'ely; mutual inductance ; and ~ . LA. L:.. Li I re total i nd uctances of t,he l tll.tor an d rot or 'W Indi ngs a lool t he a. and tJ Uet respectively.

u.:..

."rb

Wi ndi ng in ductances ar e d efi ned by t ile kno wn relati on s L".1

- M+ "...

L/ - M +',.

;" = M + I'..

, ~

t, ~ M+'!

('-36)

whOlO 1:.. 1~. [~ . 1~ a NI leakage iDd~ctances DC the sta to r and rot or wind ings alon g th o a and p ax es ees pee uvely. Th e mu tu al i ndu ct a nce lind le ak age inductance.. are Cou nd by th o known me t hod s In vol ving tho (' a1 (';U~lI. ti o n s or ex per imen tal nnal ys is, r.e. using eq uiv a leu t circ uits a nd desi g n formu las. T ho ass u m pucu is t hat t here is (I w orkin g tlu x 'i'h iclL li n ks t he s t a t or and rotor wind i ngs and e lso lea kage fl u xes li n ki ng o nly OM wind ing. E quation s (1.34) desc ribe a hy pp t be tica l ma ch in e h ay ln!: t he !l3me n umber of luTJUI on the stator ll.p d o n t he rot or, with th o windings beiog pse u d ostaU ona ry. T o pres er ve t h e po wer In v aeiance in 1111 ac l u al m achi ne and i n t he IIH\<: ~ne wit h s t a t ion ary wi nd in gs , tbe eq ua tions have to conlai.D tho ; mfs of !'Oh ti o", ex pressed a.s ~U) , i& MfiJ , l& for t ho roto r ""in,d ing alo ng t he a ax is and all - L;,m7 t:.. - ,4f U) ~ for the JJ-axis wi n d ing, K irch ho ff' s ec ue ucoe (CV.) includ e voltages. vehege d rops 1I1:r0Sll resistances, omls of rot ation t ha t eJ"illt on ly in ro ta ti n ~ wi ndings , And t r llOsfor mer emrs : L:' (dldt) t!, + !If (dldt ) ;M (dldt) i:,. + L~ (dld l) I~ T hE! transformer em rs for th o ~ .ax l s ~j ndi llg8 are written In 11 s i milar form. In th e eq ua ti on of motion (1.35), P Atfltlds f or t he n u m be r of pole pai l'll, and J for the mom ent of In ert l,a , If Lbe analysis is m ad e of an electric msehl ne to get he r wit h n s d elve mech en tsm , t he q uan tity J mllet re presen t t he ro to r moment Of Ine rt ia an d t he D orm aJi~ mome nt of Ine rti a of the mecha nism, In the an al ysis of electric mach i nes. the m om en t of reetste nce M . f1 urq u o) is usulIlI y t aken co ns tan t, I n tbe ana lysis of el ec rre merJJanical systems. lof . ca n be a Iuuc t tc n of (,)7 or time. The el ectro m ag ne ti c torq ue M . - t he torqu e pr odu ced b)' a con " o rtl~r- is gi vcm h y ti le pr odu c ta ot cu rrenls flowing in t he winding!!:

M . = (mI 2) M (i~ i~ - i~ / ~)

( 1.37)

wher e III is t he nu m ber of ph as es, The elec rr omec ha nt eal energ y co nv eral on eq ueuoee su ggested by Gah rie l Kron in th e 19308 co m pelae t he syste m of rtve equations (1. 34 ) an d (1, 35) i nvolving five indep endent variables (vol t ages and M 7) and fi n dependent variali les (curren t an d angul ar s peed ), The coef fici en ts . h ea d of t he d epe ndent. va rill:bJes , nam el y, lUistaocee. ind uc t ances, mu tuAl in du ctaDCM, and t.ho mo ment oJ inerti a, are the pa rameters of a n e nergy; co nv erter. }

eh. l.

I nl~od udlo n

to Electromechanics

Jn the mathematical t.heory, the coefficients at var iables may vary with the form of equauoas used, therefore it is of importance to have a clear idea of t he parameters and mathematical descri ptio n of t he processes of energy Âˇ conversion. T-he parame ter s of a ma chine een be const ant , peri odic, and nonlinear. The analyti cal sol ution of the equ ations for ulecrr omechan tcal conversion does not exist because the eq ua tions contai n product terms. The eq uat ions are solvable with the aid of computing dev lcss , tho soluti ons bei ng- appro:Ximate. This approach also permits ha ndling equa ti ons with nonJjnear coetnctents. The accuracy of soluti:on of equations dep ends on t he class of comp uters used . Compu ters can solve a simple pr oblem oven to a higher accuracy than le necessary for t he engineering purp ose. On the other hand , man y fa;ctors which affect the processes of energy convers ion in a real machine cannot be take n into account. Even the energy cen vecsion equ ations wit h consta nt coefficients are nonl inear sin ce t he torque equati on contains tho pr oducts of v ariab!1:lS. The addition of nonlinear terms onl y makes the problem more difficult. l Independent and depende n t variables in (1.34) an d (1.35) may vary in value, and t1lell:1hey describe wha t is ca lled the current dr ive. The sy stem of equ/lti0fls (1.34) consist1ng of four voltage eq uatio ns and th e equation of moti on (1.35) describes transient and ste ad y-state modes of operation. To obtain the steady-etata equation s, we should repla ce the differen ti al operator a/dt by / ID and work with complex equatJoM. In the st ea dy cond it ions the volt age equations can he dea lt wtfh inde pendentl y of t he equation of mot ion. The courses in electric mecbln eey commonly cover voltage equatio ns, and th e course in electric dri ve mainly considers the equa tion of motion. , The electromagneti c torq ue Af ~ is equal to the prod uct of curre nts In all of the lour windings. The torque (1.37) is set up by the currents in the stator and rotor windi ngs disposed on differe ut OXOS, with th e stator current Shifted in phase with res pect to the rotor cu rrent . If the rot or and -stator wind ings of the prim itive machine carry only active ac components. t he tor que M . is zero since the coup ling between the win Jli.ngs due to reactive currents that produ ce the magneti c field is absent . . The solutions to t he equ atio ns of electromecha nica l energy conversion do not exist if any ;ol the parameters ent ering into the equatio ns is zero or goes to ;infinity. If the resistan ces and in duct ive react ances are at infin ity the currents are equal to zero and the machine does not develop. t he torque M ~. At J .= 00 , t he ener gy converter picks up spee d jnfinitel y long. At J = 0, the machine cannot come up to its steady-state velocity because the rotor res-

1. ~. The Prim it ive FO\Ir_W ind ing M oehl...

pnnses t o all changes in th e p rodu cts of currants: con ti nu ousl y . If the mu t ua l induct ance is zero, tho magnetic l inkage be t ween the windings is none xis ten t and M . = (} (1.5). If t here is no resistance in t he loops th rou gh w hich t he cltrr,enls complete t hei r paths, the device will act as a s to r age of energy. T he t ime constants lire at infin ity, th e shift between curreuta Is zero , a nd M . = O. It is possible to obtain op timu m rOiatio ns bet ween t he pa ra me ters at which an elec tric mac h ine m il)'. have a maxi m u m efficiency, hig her cos Ql, a min imum mass or a des ira hle form of output characteris t ics. It s hou ld be noted . however . t hat Eqs . (1.34) and (1.35) are u nsuitable 'for use ill op tim ization at udtea because th o minimum va lu es of curren ts (dependent Io'ariflblos in t hese equ atIons) aro not ye t i ndic at ivo of an op timum mechtne. Consi der ing voltage equ a tions (1.34), we s hould pctnt. out that t he terms defi ni ng the transfermer emf Include t he induct ances and currents under t he deri vative lligll. In most elec tric machines currants are va rying quanti ties , s but the con version of energy from etectr tcet t o mechanical for m is g• 1.13. Tho principle of an MHD possible it cu rrents are constant , ge n ~rato r and inducta nces un dergo varia· ';;: And if a n mng " ," lc f Ju:r dON lty veeuc ns in a sinus oid al man ne r. T he 8, te e. 00<16 "" 111141 , .. Or li quid v el ocit y , ond mach ines perfor min g ene rgy con - emt IMp,,"'" 011" lIanne l WRI II r• • p""t1 ve\y version i ll t his manner com prise the class of param e tr ic dev ices , among w hich ind uctor ma chi nes are mos t popular. In th e gen eral: case, both in d ucta nces and currents in electric mac hines va ry etnusotd ally. Given t he mathema t ical descrip't(oD o f the processes in elect rtc mach i nes, l et us Inqutea in t o t he nature of ene rgy convers ion in t he machi nes. The gener al concl usion t pa t can be d rawn from t he cons iderati on of t he l aws an d equ atfons of e nerg y conversion comes t o t he foll owing : electromechanical en(!rgy cOllverlficJrt is possible i/ any 0/ the quantities entering the energy' eonoeaton equa ua as undergoes

iIl

variations,

Most of el ectric macbincs are sa td t o ope rate in one mode or anothe r it their windin gs carr-y af t arnat.ing currents . In these machines th e pa ramet ers ma y va ry too , Energy COil versi on is posal ble at cons ta nt vo ltages and currents bu t I\t varying parameters . Energy con vers io n can occu r when inductive re ac t an ces and res teta nces

se e lliOt i"g t he eq ua tions u nde rgo vllrilltions . With It cha nge in t he momen t of inertia , II machine arores k.ioetie enerll'Y and gives it. up to t he li ne. Far aday's mac.hi ne (&e$ Fia . lo t) llnd u re magoe lo hydrodymunle ("fliD ) I:ODera lor (Fig. 1.13) Ire t he most com plicated case for t he explanatioD of e leelro me~l1an ieal energ r c.onvcrs ion. I n Farad ay ', mach ioo w ith I permane}ll magnet , l he de circuit. cha nges etate. I t has II portion t ha t il s latlo nllry and a port ion t hat moves a bout tho mag nel. The slidi ng ee nrecr is obli lilltOry. If t ho loop is mnde u niform withou t the s lidj ng contact, the molar will not r un eve n whe n t he cu rren t source is mado t o ro ra te toge t her wit h th e loop. In t he MH O generator; the VAr )·j ll g . pa rameter is t he velocity of p lasma in th e noaalu and .orue tde i t. I n Farad ay' s meter , tra nsit ion from t he rota tin g pa rt of ,t l\ ll eu reent- carrylng loop to tho st lltiOOll ry part c ecues i n n jumplikema nner wit hi n the sl id ing contA Ct rog ton, wh ile in the MHO genera tor the vel ocit y of the act ivo mod ium cha nges smoot h ly.

t .5. Application of Computers to the Solufion

of Probl,ms In Eledromech anlc:s S ince t he eq uaUons o ~ elecl.(omcch anica l onerry conversion are llOnli near , the a nal)'lLClI ,solution exists only under certain assum pti ons. whe n w, = 0 or th e speed vartes li nearl y , in which U1!Je vo ltage equ a ti ons (1.34) a re solvable in dependent of t ho equation or mo t ion (1.35). T he llOll lysia of tr-ansient processes wit h a va rying rotat ional speed is possiblo only with t he a id o[ com put i ng devtces becaUllC t he eq ua ti ons conta in Ib\! prod ucts of vaeia bles. E lectro nic compute rs CJ)n he cl assified un der t hree mai n head ings : analog computers. dlglttJl computers, an d hybrid computus. An Ilnlliog comput er represents all variabl es by oontl nuously va ry ing phys icnl qu a ntities (eurre nta nnd vo lt ages) whose cha nge gives the solution to the pro blem bein g !Ilvestlgllto.d. An)· dy nami c cne rectc r ts uc is re produced by II recorder, for exa mple , on t he scree n of II ca tho de-ra y oscillograph. J n sol vin g prob lems on a n onal oi comp ut er , i t is well to s.ta te t he proble m firs t i n li n Incomple rcly def inite form and t he n refi ne t he slalement in tho proc:ess of t be a nalysis of t ho pr oblem j A disa d van tage of II l1alog com pu ters is t ha i t hey have low accura cy ne d limiled vetSlltility. a u t the accuracy up to a few percent is ofte n q uite suf ficient for m an ~' en gil'let'r inr studies because th e epce fjted accuracy of initial ual a is )"et lo.....or. Wh at makes a n lInalog compu ter illSll rric ie nt lr un !\'etSlll is t hat tn nsition from the solu tion or o ne problem to thlt of t he ot her nq u irel changin. th e flow dilgl'a m of the machine . Analog ecm pulers av aila ble toda y h a n~i1 e pro ble ms irn ·olYing t he in tegra ti on of

1,S. Al"Âť llcaiio n 0( Compu t. ..

OrdiIlRf)' d ifferent ia l eq u ntione, a lge;hraic a nd tra nscen de nta l eq nnt iOIlS, And pA r tial d iUere ntia l Oq UIIPOIl!. T hey a re co nvenient lor use in the a nlll~'sis of d p l.m le o~ rA tio n 01 e nergy eoe vc rte rs. Di ~ i LaI com puters fi nd use where lit is necessity to 8Oh'& ma t hem ~ ti cal pro blerms t o a h igh IIceuneY. Th e inp ut lind output i nform . ~ i on he re is in t he dtscret e form~ so t heM JnA<;b ines realiu- tbe numer ical methods lor t he solutiorl of probl ems . T he cn le ulnt ion ll(;cura C~' a rtat na ble on II di g it a l co mpu ter doponds Oil lile qUllnlit)Âˇ of bils , t he li m its being se t by u,e si1:8 of com pu ter fa clliti es em plo)'ed. Moder n d i~ ita l computers ca n a n;tomllt ica ll y perlorm II co m plete comp u tllt ion with a s peed 100 000 tt mes as rlUlt es 0 h uman bein S does and thus offer t he g r en les~ possibili ti es fOI' car ry i ng out eatenla tions . A ll dis t ingu islled from a:nalllg computers. di g it a l 01l1lS hand le pro bl ems wit h a def in it olY.s Ultod solut io n a lgorf t bm , for which th e i nst r ucti on ( pro~fIl m ) is wr it te n end give n t'O t he mechlne . Dlgit nl co mput e rs of toda y are ca pa ble of sol\' lng a wide ra nge of prchle ms . I n go ing Im m t he solut.io jt of o ne probl em to t hat of t he c rher , it is on ly necessa ry t o chAngelt he program withou t mod ifying the co mpu ter flow di AgraDl . LBr~ d ig ital co mputers are cMlt l y, 1I0phis tica\.8d . and highl)' .. ni versal i nsta lla t ions ma inly ser up at ro m pu ti nl: ec ntc l"!I whose person nel se rvice t he machines. prepare the problems to be solved am i p rogram t hem, Le , wri te t he p robl ems in a mach in a la ng u a!;e. or eode i n !II su ifa ble fonn rcq uirid for the a u to ma ted so lu t ion. A dig ital co mp ute r llpe rates wi t~ d iscrete qua nli Lies-numhera rep resented in a defi n ite notation. T;he mai n a dv antages of B d igit al co mputer Are a h igh acc uracy of co mp uta t ion, u p t o 20 d e~ i nl a l d ig its a nd over , a nd i nhere n t versatilil)' wh ich allllws fllr tile so lut ion of " w ide cl ass 01 prob lems. ?\ d isad va nt age of t h is t y pe of eom puter is t ha t p rogra mmi ng. de bul:ll'ing , fwd decod i ng of t he resu lt s ob tained i n d iscrete form -~ nsu me II- great dea l llf t tme . TIu: f, rsl a/gllal CQmpul ers were built aro und ale c trem achu ni ual rO IIl)'8 a nd t he n, la te r, a ro und vac uu m t ubes. T he on-line me mor )' of t he mach ines rel fed 0 11 t u be t ri ggers, mercur y del a)' HnM, cathoderay tubes. and , l at er , fer rite eore~ . Vacuu m t ube- based machines wi tll l\ speed in t he ord er of io.ao th ousand ope rnt ions in a seco nd' belong t o t ho fir s t ge neration of co mput ers. T hoy appeared i n 1946 aud wer e buil t up t o t he ear l)' t 96Os. 1'114 .second.generation d lgtt al CQrnpuurl t hat bega n to appea r in t 960s are the ma chinos based o n sem ico nd uctor d iscrete e teeseuts u s i n ~ ml'lgneti c-rore memoetes . The'maeh lnea llf t he second ge nerati on occupy a hundre d th t he S p:l.~ of t he f irsl -ge nera t io n cc m p utees . consu me a hun d red t h ti le amou n t of ene rg y, and ca n perf or m ll-.bout a million ope ra t illns pe r second . Th~ lhird-~n4rQtion compuUn ~ :oe m icon d uctor sma l l-sca le tnreg rille d etre u us (on the a ve roge 10 gates in II ch ip), magne t ic-core

Ch. L Intro d ucti o n to EIec1,o mechonlu

mem orie s e nd ,

p ltr ~i o. ll )' ,:

mag uetlc-disk memor ies . The computi ng

syste m of t he t h l,Âˇ;1 genc re tio n dis pla ys t hree cha r aoter ts t ic fea tures

as foll ows: emplo ys ju tegra t ed c ircu its ; ha s Input-output channels a nd the deve loped net wor k .o f pe rtphc eal un tts: a nd is ma de co m plete w i ll, so ft ware wht ch form ~ ilfl integl"lll p llr t of the com pu ti ng system . T he cost of sof tware systems grow s SI<38CUly w ith e ach p ass ing yea r . \Vbilo at th e beg inn ing of t he 19GOs t he cost of pro gra ma was 30 % lin d t h nt of equ tpmeht 70% , lit pI'()SQ n L th e cos t of sof tware re aches one ha lf the t otal cost of hnrdwo l'e. E x amp les of t h e third-ge nerat ion com pu ters inclu de tI'l! IDM36{l lind t he Soviet-ma de EC m'nch ino thll t closely resemb les tho for mer in parame te rs (EC is thl$ ab breviation of R u ss ian words meani ng th o un ifi ed system) . T ho IBM 360 system re presents n family of the third-ge nerat.iun ma chines deve lop ed by 't he world 's la rgest Ame ri ca n com p uterh uil di ng corpora t io n. Th a IBM360 displays ! 8, numbe r of d tst.inguishiug features . of whi ch th e most im portant are the foll ow ing ; the program compa t ib ili ty of var iou s t y pes o f com puters e u tcel ug tu to t he foroily, whi ch p r ovt dos for the ap pli cability of programs in go ing fro m one model of the m achine to another; tho possibil ity or con necuon of a lar go num ber of In put-output dev ice s find standard Iut.e grat. Ion of in p uto ut put devices wit h i npll t[-ou tpu t cha n nels; the ca pa bil it y to opereto i n real ti me in control systems; and the possibili t y of combining s ma ll comp u t ing-power m achines In to a s ingle evetc m . The IBM360 com puter is, 1I un iversal s ystem designed for servtng eco nom ic (buej ncss. com rneo rce) and ectcnu rtc p urposes nud also for solving t he pr oble ms i?f d ntn transfe r a nd con trol. T he ata ud a rd sys tem of prog rams offots! th e basi c com p ut ation cepebt uues nf the runc h i na . T h is com ma nd .s v srem may i ncl ud e m eans for pr ocess i ng da t a in de cimal notation . The addltton of floati ng- po in t fea t ures g rvea a scien ti fic command sys te m , an d the audit io n of secu ri ty fac ili ti es to the eco nomic 1la u d eetenurtc command sys tems providee II u niversal comma nd sys am . A few t y pes of t he I BM360 machine ca n be combi ne d w it h th(,l a id of ce nt r al prO(,'-4!SSOl'S t o Ic em 0 COIUp u ting com p lex . T he I B-"f360 system 1181'S so lid inte grated ci rcu it s ue tud lo r h igh s peed nnd ,s mtiH s b c. whi ch e ns ures h igh relia bility of t hll co m puta ra . , Th e ui ecut oes of the .E!C t ypo emp loy t he s t a nd ard network of in terconnecti on or per-iphe r-a l unua, t he so-ca ll ed i np ut-output in ter face ba sed o n t he program co nt ro l of t.hcsc uni ts. T he ha rdw are of t he EO ma chine ca n be d ivided in to fo ur grou ps (F ig . 1. (4). Group I Inc lude s pr ocess ors toget her wilh the registerbased work ing storage . ar ith me ti c and logic elemen ts, and cont ro l de vi ces. The d ev ices of Groups II a nd I II li nk pr ocessors to" pe rtphe r nl u nit s wh ich fnrm Grou p J:V. To gro up II be long se lec to r and m ul -

loS. Ap pllcet lo n of .Compute "

tiplox channels. T he selector cha nnel opera tes in t he burs~ modeto provid e for a hlgh-speod da ta inpu], to an d re adou t from onl y one peripherA l un i t for II ooct a in length [of t ime [ner u-l y a second en d oYl.'r). Th e multiplex ch an nel perm i l ~ II s imulta neous da t n t r a nsfer for II la rge number of Input -out.put,' units. TIle lin k between th eunits of Gro ups Jl a nd III for ms wha t is calle d t he in put- ou tput, ioterfaco of the sta ndard des ign , wh ich is Q de ta chable 3O-w ireconnoct tc n e nsuri ng t he tr an sfer of, contr ol signa ls a nd dcta .

0'"

Fig. i.14. The ha rdware of all. EC com puter MC - n,u lli plu cJlan1l~ ' ;8C - se l~ , d18.n.. ol. ; GD - ;-'\l Up dHlce. ; lJ /II - don><lllulal.Pn \: 111_ t.po . !Ot' l:"0 un l ll: CC.... - C lUl.n nel -t<HJ,~n o ~ DT"f _ d U , \ '" ~t., ,",u ll lplese ,"

''''1''''';

T he dev ices of grou p III inte nded to li nk t he Interface to vaeiousperipheral uni ts i nclude ind ivid ual. interface devices , grou p arrangementa M>r vici ng a few peri pheral uo tts, ehe nnel-tc-che n uel ad a ptersprovi d ing d ir ect con ne ction betweenthe selec t or cha nnels of pr ocessors, a nd multipl exors for da ta trll ll~fe r over a fow chan nels. Per tphe ral units inc lude magnet ic-tape , ma gnetic-d r um. or d tode st orages; in pu t -ou t pu t. dev ices for Pillich cards an d ta pes ; pri nters ; dal,n terminats and consoles; tbe means [mo dems} for data t eleprocessi ng a nd commun ication witll ,cont rol units . The mut hemn tfc a! program pac kage (soft ware) for th e EC com puter includes t he programs of t hree ce tegceres : opcraucg sys tems ensuejug t he Hnk between t hll o pera tor audmser a nd di s ~ri b u tillg the jobs. lin d system resources: ma intona nce programs or lest rout ines (de b ugging. checklng , a nd d iag nol:ti c rou t ines): nut! t ho packs of a ppttcnt lon ,

prograJrul , w hich a rc fu net ion :lll y co mp le te se UI a r ra nged for t l18 so lutio n o f a dofi nite cla ss 'of prob lems. T he EC u n ita r y sys te m te pres oo ts a fa m ily of progr am-co mpati ble comput in g se tcpa of the foll owi ng I)"pea : EC- t OID. ECÂˇl020,

EC-l02 t ,

EC-t 03O, EC-tQ410. EC-I05O . EC-IOOO. The workin g

s t orage capacit y ranees from 8 t o 10' ki ]ob)' les. T h e b llBlc featu re s 01 the EC ce m puter aro it s uni versal ity . eda pl abili t y for V::lriow a ppJica l io ns. an d th e possib ility of II. grA d ua l bu il d u p of tho com puting poWer over II. wi do fan go. Th e versat il i t y Is d uo to t ho in struction se t i nvo l v ing fi xed-po in t and lJo!lling. po in t ce m pu tet tons . logic And d ccl ma l ope ra t iOIl!!, ope ra tio ns w it h varia ble-length words, a nd e teo due t o vario us d ol ll Ierm m s , multiprogra mm ing possihilities . and the ad vanced sys tem of soft ware . T he ado pta bil ity for use! "ste m!! from the chnngeahle s t r uctu re of the EC eyete m (roplneeabiJity of mem ori es , ehnn uels. pe riphera l e quipment). A gradUAl Incre ase in tli e comp uting power ce ll be a chie ved by seve ra l method, . name ly, by i ncreasing t he number 01 peripheta l UDit.'l a nd t he work ing s torage cApac it y, produ c ing multim achine cc m p uting com plexes, replacing tbe processor by 1\ futer.s peed t ype, etc. The prOgn\m co mpa t ib il it y of th e EC co m r uter co m ~s f rom the unUled 10giCflI s tructu re (s ta oda rd il.a t io n 0 t he in st n lclle n se t, da ta re presen ta t ion form , And add ress system) . D urini the las t 25 ,.~a rs ;t ho com puter speed. s to rage ca pac ity, and reliAbili ty h a ve i ncre ased m llDY ti mes . Th e o1:u a ll di mens ions , th o e nerg )' co nsumed , a nd the specific cost of co mp ute rs decrell!M! very fAst eoncun-enl wil h th r improvem e nt of t he ir pa tllmeterS an d c haracteri s tics. At Lhe ato r t of the 19705, the firs t fourt h-I en.u lHio1/. computers appea red , wht eh began to \l.se medium-scale ICs (a bout 100 gates o n a chip) a nd large-scale rqs (t housands of gates o n a ch ip). Wh at ,d iat inl:ulshos t he Iour t h-genem t .icn computers is t ha t th ay widel y -e mploy se miconductor storages, en lorged Iust.ruct lon se ts , mi cr opro grommi ng , bu il t- in s ubrout ines , au tomat ed program de buggin g, peri phera l uni t e a nd cho nnols of d ivorsed t ype s a nd im proved qu ailt y, in tl"rfoces , specia li>torl .processora . T hose computers exh ibit .e nhe need reliabili ty a nd [oFm t he bas is lor the cc cstrucuon o[ mul Umachi no a nd multiprocessor comp uli ng complexes. The em erge nce of an au to ma ti c uni versa l d lgi tllli com p uter t ha t perform s Ilr it hmetic a nd logical o pera t io ns with a hi gb s peed opens up new q uali ta t ive po!ISibil it ias l or cond uct in: t he t heo reti cal i nvosti ga ti ons in co njunctio n with ebeck ex per iments. Hybri d co mpu ters which Itomprise d ig ital mach ines a nd a nalog -ue vtees hold mu ch promlso for the e ffic ien t combina t ion of the 010me e ts of & h ybrid iD.!lt.Ualio D to enable t he mcst ratiooa l solution -cr prob lems. TIle d igital computer in a hy brid complex ill a co ntro l machine

l.S" Ap p liu tio n 0 1 Co mf>\l'."

wbich s imu lta ne ously gives t he ecurce i Mor m1lt ion for t he Iur thar M1l utioa of pro blema on a nalog dev ices. T h ls complex , when een neeled to t he 5}'.!1le m of ap propr-iate lrl\nstlueorl, ca n contro l an expeellllent a nd keep t he li nk from t he mdme nt of dll. a oalY.!lis 1.0 t ho moment of obtlinint t ho resu lt. T h lS 1150 offor1l Del!.' poS!ibili ties of the search for an c pt lmnl mode of 0~ r8 tion on the pr inci ple of .self- ins t ruction of t he .!IYlIt em. At present the ways are sough t for ihe COll$t ruel iOll of h igll-performloce com puting S)'"Sle ms by us ing , ca&CArled se tu p ~ ... nposed of l llrge nu mber of iden tical u niverse] digl hll computers prog rflmOrlZllQi l l'd for t he n:!a Hntio n of a spet ified al gorithm. The devele pmen t of such s tr uct ures Involves t he ljC-finement of tile reqlli re lJlf' nlS for \'el'Slllilil)', pertoema ueo. t.omput&lioll IlU Ural )'. a lld d iroctl y depends on t he etess of pro blem s t o be so lved . B )' th e ir slruel m'lil, t he fourt h-goner. t ion mac h ines are mu lti proeeaoe se t ups de voted t o t he ro mman mem ory blo ck and t he common ex te nt of peri pherlll devtce e. An to.ggregal e of compu t ioi facillt ie3 forms a cen te r connec te d to nu merou s s ubscet bers by commumeauon linea. Such II. Ile ~wo rk orfeI1J t Ile pDMlb ili t y Jor th e communal U50 of compu ters 1.lr lin indiv id ua l or " group of reseerehers who mill' contac t th e coote r by tele graph or tel ephone from lilly region of the countr y, re la y the message Ior t he solu tton of n pro blem and recei ve the anawer on tl le giv e n do te . Both llllOloll and d igital compu loq fi nd use for t he sot uuo n of problems in ele ctro mechanics . If soma ten eq uat io ns describing t ho transienl processes In electric machtues lire eno ugh an d the par ameters ent eri ng lute t he equa t ions {\~ const llnt , it is well to solve the pro blem s 0 11 lin IInalog compu ter. Wh ore tho numbe r of equ ations i~ IlIqer _ t ile pa ra me ters ar e nonli nM l"lInd t hott) is a need for solVi ng problems for th e optimiza t ion of lin el)erg }" eon ver ter, it is nece.s.!lU'y 10 enocse tI dlll'itni cc mputer , 1/1 1M /1l1l1ly,u oj electric 'rlQChIM', It is tZ pedUflt t o tm pWII both 1If1l1llllit'G1 nuthods and arnllog and d.lgttlll computu,. The tzptr tence in chOOflng Iht combinot ton of mt thbds of anolyfi, determ Ines th, tkgru 01 w:curM!I and prolourulnu, oj the .tOlwloA of the probum. In soh -ing II problem in e leetromeplillnics, tbe ~areher should prima r il y formu lllIc t he equatio ns for t Ilt prceesses under s tu d)' \0 â&#x20AC;˘ $Offieien t degree of acc.llr.lcy flIld t llen choose a eem pute e to form I mat hematical med ol . :'Ie xt he !Ihould refine th e medel wt th the a im 10 estimlto t bi! time it woul d ta ke ;to 5OIv8 t he pro blem nnd t he ClXpef.le d .ccuracy of t he solution. T he fi nltl step involves draWing tbe plan of t he ex per inlents \.0 be r UD. Despi te t he ir great oppo rtu nities, .ee mputer facilities can soh-e a rath er Iimite.1 r l ngo of pro blom.!l in eloetro meeha nic.s. T aki ng into accou nt. eve n two or t hreo hllrmon ics i o t he ai r gllp lind lWO or tlu'e6 1oops 011 t he- s tate r and ro to r neeeeenet ee so lv ing a few tens or

Ch. 2. EI_d'OI'I'Wld'la nlo;ll1 Energy COnYllnion

equat ioD' . Conseque ntly , th o researcher s ho ul d thoroughl y choose the mathema tical mo de l, keeping in mi nd the pOWeT of co m pu te r teen u see. an d estimate the tim e re q uired for t he solutio n of th e problem a nd the poss ib le solutio n ace u roc)'.

Chapte r

Electromechanic;al Energy C onversion Involv ing h' C ircula r Field

, ,,

2.1. The Equations of the Generali zed Ele ctric Machine Coca tde r a t wo-phase t wJ.pole e lectric machine (F ig . 2 .1). It. has t wo orthogo nal sy llte ms of s ta te r a nd rol or windings uf.. w1

e, Fir.

z.t.

T he m.chia. mock-I

and an d

ID~ .

w,.

II ..

'rnll ~

In. o on U"fU)a!ormed eoordiJu te ay!tem

raspec U\'el)' , lyiz>1: on ui e ata te r a od Totoor a xel 0 â&#x20AC;˘. b. b, . T he rec llm gula r ' coordi na te rrllimu of t be s ta tor a nd

2.1. The EqueHont of lhe Gene,allzed Eleel'l( Meehlne

rotor move with respect to .each other', and t be angle 0 between th e

exes determines th e rela ti ve ro tational vel ocity. Wit h t he st ato r being st atio nary, (2.1) til , = d af~t The d ifferential equati ons of voltages in na tura l or phase (nont ransformed) coord ina tes have t he form

= i ~r~ + d'l'~ {dt ut = itrt + dlfVdt u~

- u:= i :r~ +dlJ': ldt - u~ =

(2.2)

i;;r;; + dr~ /dt

In Eqs. (2.2) , t ile fre qu encies of cur rs uta in t he s tator and ro t or are differe nt , and the 'minus' signs before t he rotor volt ngea denote that tho active power fl ows from th e~ sta to r t o t he shaft (moto r ing act ioo). The flux li nkages of th e wind ings are

'1'; = L; i~ +1H cos Oi: + M si n ei~ lJft .... L~i~

+ Iff coo M -

M s in OJ:

lJf: = L~l:

+ M cos Oi~ -

M s in M

(2.3)

lJf~ = Lb l~ + M cos etj + M s in el~ Hero t he coeffi cients ahea d .of the; currents vary with the same rate as the curre nts . If we substi t u te express tc ns (2.3) (u to (2.2), t he resultant equalions will be too awkw ard and conta in pe ri odic coefficie nts . To slmpltf y t ho so lut ion of equatio ns, it is necessary t o hav e t he sa me Irequeoctes i n the sta tor a nd rotor win dings a nd ensu re tile Invarteoce of power, t.e. to ena ble the pewee coming to t he shaft, the losses , an d the energy consumed t o be. the same as the y are in a rea l mach ine . Look i nt o the pr ocesse s of energy cbn vers ton in a machin e within the ai r ga p (t he s pacing between the rotor and sLato r) which concentr a tes the energy of a magne t ic fiel d. A r ot a li ng field is set up in t he a ir ga p of a real mac hi ne owing to a definite d istribution of the wind ings i n space. a nd to th o t i l!J.e sh ift betwee n cur re nts end vol tages . Wit h t he sin usoidall y vo~y [ n g voltag es impressed on ure ter-minals of a n ide al machine , n l;i~cu lar field app ear s in th e a ir ga p. B y" t ho t hird law of e lectr umechanics, there is a rig id li nk between t he freque ncies of curre nts ' i n the stator and r oto r , t he sta to r end rotor fields heing stationary wit h respect to each other . Accou nt m ust etsc btl t ak en here ofrt he mechan ical speeds of the rot or a nd st a l or. For a st a t ionary s ta to r, IJ) . = 1O, Âą lOJ r (where

fiJ,.

" is t he speed of t he rot or fie ld re la t ive to t he rotor) . II is cc n vcm e ut C/>. 2. Eledrom ec han lca l Ene rg y Conve rs io n

to represent t he circular Held, fn the ai r /lap by t he resu lta nt magnetie flux dens ity vecto r Ii' = B~ j B~, ff _ BOo jB~ (2 .4) and by t he result ant nux li nkage vector

'Y' "'" 11'i.' ill"6' 0/'" = "Yo. p¥~ (2 .5) T he state r end rotor vo lt ~ g~ s and cu rr ents can bo represe nted es uni t resultn nt vec tors V' , fJ' a nd 7 ' , 1' . respec t i vely . Si nce t he w ind ings in t he pr im itive ma ch ine li e s pace, Lhe vo l tages U~

= 0", si n

lVi ,

= U m COS wI

9lr

apArt i n (2 .6)

un pressed acr oss th e windin gs produ ce tho! res ul ta n t field 8' a nd IV' in t he Hi t gnp. For voltages (2.2) given in terms 01 t he resultant vect ors, t ho equn t tons assum e t he for m ,

+ d'fi' !dl ~ l ' RT + d'f' ldt

U· .., .T'R'

- D'

(2 .7)

H ere jl' = r:. = rjs, Ii' '"'" r~ = r~, Exn mine till' prooeseca of CIJ('I'gy con vcrst ou as viewed from t he cccrdmete reference fra mo rot.a t.ing a l an nrbi tr or)' speed w , (t he observer's s peed), For litis coorA lli ulIle s yste m the power invarlence depends on w< a nd Ire quc uey f •. I ll us trat e ;l change i n frequeuc y with tile rota ti on of l ite cc crd tnatc sy sturn by a n ex am ple of t he commu tator much ine shown >

..,

i n F ig , 2. 2.

III tbi s machine, t he s ta tor cueetee a thre e-phase Wi nd ing 1 , each p h ase w ind ing A , B, and C being fed wi t h eo voltage which prc duces in th e gop a field revofvln g at II sy nch rono us s peed 00 , . T ile roto r c 2 r uns at II s peed w ~ and the frequen c y i n the ro tor win d ing is Fig. 2.2. A commutator macblne I , = 1\$· Th e br ushes rigidly COIIwtt b Nvol vlllg bru shes . necte d to th e coord ina te axes sli do ove r the ro tor wind ing and rotate togeth er with t he br ush ri ng -3 at a s peed 00 , . Th o n umber of br us hes is eq ual to t he n umb er of phas es. As seen from t he figure, the ma-

2.1 , rh . Equelio "s oIth. G.n.reliz&d EI.d rlc Medlin..

<\7

chine h as t h ree brushes. and therefore t he s ys tem is of t he thre ephas e t y pe (a, b ; 0:) . Si nce the brus hes are co nnected to tho coord ina te axes, t he power tak en Cram th e three-ph ase s ys tem an d fr equency J~ dep end 011 th e speed w e' Because cos e + I s in 0 = exp (+/0), vec tor eq u at ions (2.7) for the coord inate a xes ro tn t ing at an arbi trar y s peed w . - dB j dr (2.8) have th e for m

U> ex p 10.

= nq· ex p 10. +

(4'dt ) (ip> up jO.)

{f~ exp 1 (0. - OJ """ R ' I ' exp j (0. _ fe) Takin g the deriva t ives in (2.9) y ields

+ (dldl ) (If'- exp j

+ do/'Idt + I.wj fl> R'I' + d\j!' ldt + j (w. -

fr ... R']'

(2.9)

(0. - O) (2.10)

D' = w.) 'P T ho ob ta in ed vol t ege equati ons for th e res ult ant ve ct or s arcdeHned with res pect to th e coord in ate refe rence fra me ro t atiug at an arbitrary speed wi t h t he rotor. These aro the simplest a nd most genera l Ki rcl lhorrs oqu e ttous Ior th s pr im iti ve m ach in e . T he equati ons ex p resse d i n t he vuet ab le-speed rotat ing re feren ce Iea me arein rare use . Of rno" t iJlt 9nult ure the eq uations wru to n i n coord inate& « au d ~ whe n w. = 0 and t he eq ua tl olls in t ho system of coord inates d and q when w . = w,. T he la t er aya t em is the rotating reference Irnme, wh ich is mos t po pul ar for th e anal ys is ot eynohronons. machines when w . _ W r - w, . The ro t or and s tato r fields nppe ur st at iona ry for t he obse rve r when hOI v iews t hem from th e r ot or. Hero the mod eli ng of e nerg y COli version pr ocesses ln vnl v es di rec t curren ts. For th e s t atio na ry refer ence fr-ame (/I). = 0) , with tho r eferen ce axes ex and ~ be in g rig idl y connected 't o t he st at or , E qa. (2.10) ta keon t h o for m fl o ... R')' + dW' /di (2.11) Dr _ RrTr + awrl d_t _ jw r'¥' r

Reso lv ing the res ult a nt vec tors alon g t ho axes ex an d ~ g lv C!s the equ ations of voltnges for t he primi ti ve m ach ine i n terms of fl ux. linkages: u~ = t~ ,.~ + d1f:., dt

u&= I&rB+ dlflptat ~ = I~ r~ +dW:Jdr

+ tiJ. '1'&

u;' = i ~ '8 + a'VDJd~ -

w. III;;'

(2. t2)

B y !!ubs Ut uti ng t he fJU l: linlC lIllllS 'i'~ = L:..I ~ T M ' 1¥i =- Lii~+Ml i 'I'~

L~t:.

(2.'3)

+ /If f:"

~ = L~j , +MJ i

into (2.t2 ) ....e arriv~ a t tbll energy eeeveeetc n equa tions in the GOOrd hl-lt e syste m a, j!" el:pr85lled in te rms of currents . In ioiag from t he oon t ranslormcd coord ina te ays te m to t he s ystem a , j!" we s hould refer to Fig . 2.1 a nd dotermi ne t he pro jectioos (If roto r vo lta,ges and cu rrents on the stator u :is from t he relatiol1ll

u;.

= ~

ui ,..

coe 6

-II~

+ "' s in 9

si n 9 _

~cos 9

t~ =

si n

u' eee 0

+ ;' sI11 9

a + t; C,O!! a

(2. 14) (2.15)

The tr ansfor ma tion matrix he re Is

G ~ I . l)~S a Sinal ....,.gm G cos O

(2.1G)

T h e iOVOl1l8 t ransfor ma tio n matr tx Is

eos o -sin 91 ... s inO c089

(2.1.7)

AJJ mentioned abo ve . th e ro(orellc8 fra mo with t he ref erence 1lXEl8 d and q rig idl y co nnec ted to th e ~otor struc t ure is very popul ar. 10 t his syste m. (I). co w ,. )~m (2. U,I) il foll ows tha t

lit "'" R'!' :+ rfift'ldt -

+ jfiJ. if·

= R' J' + d'V' ldt

(2. 18)

Resolving t he ruoultant vectors alooa !.he d Iud q DCS yieldl the equl tion!l for the primi Uve ma chine in t he coord inate sys18m d . q; ~

_ &~+ dlf;,ldl -tal,'Y;

u; =

j~~1f- d'¥~Jdt

itd-

j~r:i +d1J'";,ldt

U; =

I;r; + dll' ; ldl

(0,

lY~

(2. 19)

2.1. The

E qu a ~ o ns

01 the Gen e raliu d Electric Machine

Repl aci ng the fl ux linkages by currents . inductances, and mut ua l Inductances, we der ive th e direct- and quad rature-axis equ ations expressed i n teems of currents , in t'he sa me way a$ we did in th e a..~ coordinate s yst em: r~

u,, u'

•

1t~

+ (dl dt) Ld

(dldt ) ,If

(d fd l) M

rd + ( d.'dt) L~

MIJJ ,

L'd<u,

- L; w,

0 r~

td t~

-t- (d ldt ) L ; (d ld t ) M

st »,

(dld t).'!,!

r~ + ( dfdt)L;

.; i~

(2.20) ( U p) J (dw , ,!dt ) ± M , - 'pM ( i,; id -Idi ~)

(2.21)

The energy conversion equat ions In the system of coord ina tes u and u rotat ing at an arbitrary s peed U) e have the form t'• u~' r~ ;- (d ld l) L~ (dld t) M ;\[U)o £:000 u~

(dl dt ) Af ~

u •,

r~ 7 (d !dt) f., ~ £.; ( oo~ - oo.)

M(iJ)~ -(J), )

- M (we- w, ) -'~ ~(oo. - w r ) r:+ (d ld t) L: (dl dt ) M

- £.:'w;

_ Mw<

(d ld t ) .tf

+ (dldt ) ~

i~ i~

(2,22) ( II p) J:(dooA dt ) ± M • .", 'I'M (I ;i~ - t~i;)

(2.23)

Equations (2.22) and (2.23) are most general. Subs ti tu t ing 00 . = 0 In Bqs. (2.22) 3J)d (2 , 2 ~ ) . we can etn etn the a..~ equations (1.34) and (1.35), T he d-q equations (2.20) and (2,21) follow fro m (2,22) lind (2,23) if roo = (I) ,.

Th e coordinate reference frames 0., ll; d, q; and u , v are in most extens tve use. They per mit a ppl yin g equati ons to practi call y all problems de alt wit h i n elect ro mechunioa. Th e right cholct! of tM reference frame simpltfles equati ons, e.llables deriving equations w ith constant coefficients but MeS nat reduce the number of nnknoums. Th e system (1, ~ proves most suitable for tho analysis of ind ucti on machin es. The s yste m d, q app lies to t he descri ption of ene rgy convers ion processes in s ync hronous machines. This s yste m ill especially conv enient {or use in th e an al ysis 01 sali ent-pole machines, where the coordin ates extend alo ng t he direet and quadrat ure axes d and q of th e machin e. The ~ yste m Ilota ti ng at an arbitrar y s peed fin ds use for the an alys is.'oOf mach ines where both t he rotor and s rater are rotati ng members. In some cases use is elec ma de 'of the physical reference frame (for which tb e equati ons have per iodic coefficients) , for exam ple . when anal yzi ng IInjinducti on :moto r whose rotor and s tawr windings ~ _Q I 1 111

Ch, 2. £:Ie cl,om.ch..niul Ene,gy Conv orsion

recei ve pow er f"ODI II tJlydstor Jee que ucy con ver ter. H ere it is advis ahla to uso tllO sy s te m of s ix v ol t ego eq un uons r at h er thnn to r ed uce the machi ne und er s tu d y to the two -p ilose t y pe. T h is a p proach permi ts ep pfying t he act unl-wav e form v olteges of the fre quen cy conver ter 10 th o lU ao;hine win diugs. Th e equlI t io ns expressed in on e referen ce fram e or an oth er are broug ht t o II s ur ra b!e form in ucccrd e nc e w it l ~ t il", rules of ma thematiC8. 0 00 of t he Im po rt.aut way!! of t ra nsfo r ma t ion comes t o the re pl acemen t of varia bl es : iO<t

i:" + ~ .

io~

+ i&

wh ere l oa lind loa are th e Ins t. an t. a ue ous c ur rent com pon ent s at no l oad al ong t h e a lind ~ axes. Usin g Eq s. (2. 13) fo r Inllglld k flu x lin k ages , we determtoe currents: i~ = (L ' 1f:' - .M"'~ )/( V U - .U·), I~ = (UWfi _ l1fllf~) / ( L' U - AI!) i;' = (L"I':;' - M IV:' )/{J.i' L.' - ,IP ), iis _ ( 1.' 1j'~ - M lfW(L'L' - )\,,12) S u b5 t itu ~ i n g th o nb uv o current eqcnuons into th e tOl'IIUe equatlon gtves t he exp ress jon for th e tor q ue jn t erm s of t ile fl u x link ayCll

AI .

= (m I 2) [pM I (L ' L' -

AI!)J.( IVA1If; - IV:' We)

(2.2 4)

Volt{lgo equa t ion s (2. 12) and t orque equa tion (2.24) g h 'e II m ost stable mode l uf en erg y conversion p rocesses s imul at ed on an ana log computer. The ele ctromagne ti c t orque e llo he dl'fi ncd in t erms of t he s t a t or flux l ink ages and cu r-rents M . = (m !2) (,V:'/e- 'V~i:')

(2.25)

That equation (2.25) is valid is easy t o cc nftrm if we su bstit u te in (2.25) th a expressions 1J'~ = L:.t:' + Mi'; .

'Y ~ "" LMA + M/i

then perform st m ple transfor m at ions . p u t t ing L :,. = L~, lin d ob t ain the va l ue 01 to rq ue ex pres sed ill terms of currents. The torque is also definable in t er ms of th e roto r fl ux li nka ges and cu rr en ts M e = (m /2) ('Y';i B- '¥~ i ~) (2.26) The v efidlty of (2.26) 15 b orne out by s uhs t ftut.ln g in (2.26) th e expression s

and equllting

11';;'- L;.l; t o LJi.

+ .'111:', IJfli = Ln,iB + M ia

T he electrom ag net ic torq ue ca n also be de HIIBd in ter ms of t he field energy in t he air ga p or found [tom t he ex press ton for t he Poynting vecto r. T he des cribed a p pr oac hes t o t he mc dl flca tt on of equ ati ons for the prim itive mac hi ne do no t a t all mak e u p nil ex hausti ve lis t. We h ave dis cussed onfy t he bask ways and m any ot her courses are open IQ mak e tftlflsfol·mll.U Olls.

1.1. Steady-State Equations T he eq uation s for all energy conver ter in t he s tea d y-s t a te ccn dttiODS of operation are dor ive d from d ifferential equa ti ons by rep feelog i n th e electro mech anica l eq uations th e differenti al op era tor dldt b y [ta. T he steed v-s t e t e eq uatio ns for th e prim i t iv e m achi ne in t he coo rdin ate system cz, ~ c an be ob ta in ed Jzom (1.34 ) in t h e follOWing for m

iJ~

U~ - ci~

j~",

rl +jx, j X", - v.z",

,: -

+ jx:

vXz

\'$",

" Xl

' Z"Tjxz

j,x",

h X

i: i~

(2.27)

- Vii 0 0 j ,x", ' j +jXj i iJ Hero, " "" w,!w . is t he relativ e speed; ri an d ' : a re the resistances of stator and rotor wind ings respective ly; x, = wL 1 is t be in duc ti ve reactance of t he s ta tor whld ing ; Xl => (»£ . is t be ind uc tfve react ance of t he roto r Win di ng ; lind %", ""' roAf is t he ree ctenee of mut ua l ind ucti on . 'l'be eq ua tion of mo ti on in t he stea d y s tate (d!dt "" 0) is M . ""' pM . T he ele ctrom agneti c to rq ue eq ua tIon is given by M ~= (mp /2 ) M (I~~l~ + 1::",.I~, - 1&,' :'" - I P ':'T)

(2.28) (2.29)

where l:" ~ , l' ~ and I~ ~ , l~~ are th e active current compon en ts a long tb e a: and ~ ax es of t ho s tat or an d ru tor res pective l y ; an d I:'.. 1$. and l~ . , l~. ar e tbe rea ctt ve cu r rent comp onen ts al ong t he a: and l\ axes of t he s t a to r an d rotor res pectivel y. By .Us? .of &toa dy -s tate e q uations (2.27), (2,28) , a nd (2.29) for t be pri mitive mach tne we ca n fo rm ula te equations for as ynchronoue and s ynch rono us ma chines and 'als o for tr-ansfor mers . Derive the eq uations for a two- W~n d in g tr ansf or mer and all i nducti on machine. I t shou ld) be kept Iii) fuind t ho t t he eq uati ons of t he primi ti ve machine negl ec t the pre sence of a few cur ren t network s in actual m achin es, asymmetry , sa tu ratio n, and other rectors .

Ch. 2. Eleetfomeeh niul Ene' 9Y Conversion

'I'beretore, t he procedu re of derivi ng the s teed y-at a tn equations, wh ich ap pes ead earl ier t han the di ffer ential equations in the theo r y of electric maehm es, involves certain diffi culties, whate ver th e ty pe of machine t hese equations have to be set up for. It app ears logical t o conduct all (l OW stud ies in t he t heor y of electromcchuulcal converte rs in s uch. a ma nner that ~ he stead y-state conditions woufd be a pa rtial case of rra oerent condtuooe find the s tatic equations woul d be a part ial ease of dynamic equations. The ste ad y-sta te equatioua can be wr itten in any coordinates, jus t li ke t ho di ffere nt ial equ at ions. T he equations for a s j llg [ e- p ~ ase two-windi ng tra nsformer, deri ved from Eq. (1.34) fOf the primittvjJ mac hin e o n co nd it ion that zo, = 0 (cou pti ng bet ween the windin gs alon g d iffere nt axes is none xiste nt), have the form

u' I ~ lr, +(dldt)[,I

I-u.z

(d/ d t) ~l:f

(dldtl Jl{

'3 + (did !) La

Ix l~ ,ll 'a

(2,30)

In (2.30), the subscripts 1 and 2 ident ify t he p rim ar y and t he secondary eespect tvely , and the ' minus' sign before U 2 means that power flows to t he primary. Replacin g dldt by foo . from (2.30) we obtain the st eady-state ecue etooe for tho transformer-

l ""

- 6,

+ fooLJI + f oo Mi, j6>Mi , + i r + j{oL1 i a

i ,r,

(2.31)

2 2

Here / 6>[, ' ;1 = jOl (M

+ t" , ) i , = j6>M i, -!- jool", ;,

j6)L2 ' , = }oo (M

+I" ,)

Replacing va riables

U1 =

f ,, '" i ) +

' 2

j(jJ:11 I~ + f Oll ~2 ' 2

(2.3 2)

in (2.32) g i ves

1>,+ fwMi, + ,,;4- joo.11i , }<ol o

+ j lil!lf (i t + i a) + jWl" ji 1 = it" + jlil .\f [,, + ji, X I .., tÂť, -- E, + JiI Xt = - i.?,.J... i = ;" 1

(2.33)

1: ,

e, ...

where -fwMi". X, - ~ll1" Z, - '1 + /x\. Perform ing th e sa me transfor mati ons for t he eecc nd aey-wl ndin g equation g iven in (2.3 1), we get

- U~ =

;2'.+ jliJ.~ f (f, + i~) + j(jjl",i~ =

i~r2 -

e, + fi2X~

(2..'3 4)

2.2. SjI!8dy-St" t. Equ8t1001

Tra nsform ing (2.,34) yields the equatio n for t he secondary wind ing

rj~ = where

s, = E I

.!!...

- j o:nu i o, I ,

i tT-I

(2.35)

+ ix

t •

DJ .

..!.c.

" U U,

£1-

' .. Fig. 2.3. The equivalent transformer

•JIZI I lr,

- E,

U; ctrcun

j l Jx l

'. FIg. 2.4. The phaeor dlal;:ram 01 n trnn slorm ec

We have t hus arriv ed a t the known sys tem of equa tions describi ng t he processes of elec t roma gnett c energy conversion In a t rans former;

Vj = - .6: 1 + i 1zl• Ut Es - i rIs. i l + is = i , (2.36) Equ ations (2.36) follow from t he equfvalen t circ uit (Fig. 2. 3) and phasor diagram (Fig. 2.4) of II tr ansformer, which are dealt wit h. ill t he general course ill electrfe mach ines. For i nduc ti on machin es, t he stea dy-stale vol tage equa tions found from (1.34) have the form

Ch . 2. Electromechanica l Energ v Conv....(o n

+ j wLOh + j{J)Mi r,. ciA= R ·j~ + j CllL'i~ + iWlIlip

0:" - R'h

-u~ = Rri~ + iwLrh

(2.37)

+ jO}.1fi~ + Mi$wr+ L'j~(Or

- U8= trt; + j(~u i~ + jooM ' 6- M i:"wr-Uhw r For induc t ion m ach ines wit}l a s hor t-c rre u ued r ot or , and U& O. T ak ing in to accoun t

6;. =

}ooL' .... IwM + i ool:', !1UL' _ iwM !('J I:, X a =- w.l!, x' "'" wl; , X· "'" wl~ and also t he re la t ive s peed v ... fiJ r! w.. th e equa t ions (or an Ind ueti on machine assu me t he Cor m

o: = R' i~ + }x'h + jx,h + IZ~~ U6= wi~ + jz'i~ + }I o/6+ jx,/B (2 0 = - RTh - jIri; - jxj;' - I:r:, i:. - ~i 8v- (r + x . ) i~v 0= - ll' i B-!:r!i B- izj~ - jXai~ + zoi:.v + (z r + J1I ) i; v Cons ideriu g t ha t i, - Ii:... i; = lif. and om itti ng the inter038)

med iat e operati ons , Ior t he stat or and roto r wi nd ings lyi ng along the s ome axis we gel

iJ.

R.i. + tJ:J. + ixal.

0 = - R,i r - iXr

(1 -

i. - Ix, (1- v) i, -

}x o (1

-v) i , (2.3 9)

io= i . + i, w, in t roduce

rhe slip

!J = {w, ± wT)!w. = 1 ± v (2 .4.0) and t ransform t he rotor wind ing to the st etor wtnding : t hen Eq e. (2.39) wHl t ake on t he form

V. = R,i . + jX.i. + ix,i. . . 0=

- n~l ; --jx;l ;.s - ix~foi

(2A 1)

ia_i. +i; Because E. = -Ji~o and ': 0 = r o + jxo. we assu me t hat E, = £ 0' T he i mpeda nce :'0 or t he e xci t ation eurreat-carryt ng

= e, ""'

2.2. Sleady4 tal e Equation>

bra nch includes t he res ls ranc e rG equ'ivn leflt to th e i ron loss. AlLer t ransfo rmations , Eqs . (2.41) are wrj tten !IS

V.= R/. + ix/.-it. . . . 0 = E;-

(2.4.2)

jx;n- R;f ;l S

i o = i .+i; Here R ; fs = H; -+- n ; (1 - s)fs. Int rod uc iug the s t a te r and rotor Impe dances s , "'"

+ /z. ,

z; = R;

+ Ix;

yields th e eq ua tions for a n lnd uc t.iou ma ch lue

V.= - 80 + ;.:. 0 = Eo- i;z;- i .n; {'I - - .

(2.43)

S)iS

/0 = /'+ / ;

The equi vale nt c ircu it nnd t he phnsor diagra m for t he a bove equa ti ons appear ill Fig. 2.5 and Rig. 2.G res pectively . A cirde

I '.

.; -e-r-

R'r(l-s)j s

Fig. 2.5. The equivalent circuit of lin lndu ct tou mach ine

diag ram ex ts ts for a transform ed squlva lent circuit. T ile ee mp lax equat io ns (2.4:{). equivalent ci rcuit; and circle diag ram are t he basic el ements of t he t heor y of stead y-s tate operation of ind uc ti on mach ines. The Sleady-sla to eq uation s for ay uchronu us mncnruea without a damper windin g res ult, fro m Eqs. (h34) for t he pri miti ve mac hine. T he eq ua ti ons for a synchr o nous mecntne t hat do no t tak e Into accoun t the dum per windi ng are too a pprox imate (t hey are given below follo wing th e d iscussion of mul tiwi nd ing ma c hi nes]. It should be borne in mi nd tha t the ph aaor- d i agrams of synchro nous m achines are drawn for Iafrl y si mpli fied equations : those d i agr a ms ha ve a qu a l ft at.ive meani ng for mos t s ynchr onous ma chin es.

cs, 2. Eht ch omeche nicel Ene'GY Co n~ e,sio n

T he ph nsor diagr am for cor resp ond s t o t he equa tio n

nocs nli ent- pc!e machin e (Fig . 2 .7)

(2.44) wh ere I~ is t he ar mature current: U is vol t age; E is the upen -eh-eut t vol l llge or em f; T " is the armatu re res istan ce; x . = x" :I:Qd is th e

Fig. 2.1. T he phewr diallram vf a llon." li~ll l- r",le s~'n dl r<lno "S mach i"a

Fig. 2.G. Thl' pba. or dialj'ra m of liD Ind uetlcc ma chine

sync hrono us m achine reactance; -l'o is th e lea k age ind uct i ve react an ce of the ar ma ture: a nd X o d is ure i ndu ctive reactance of t he armature (oppos ftton to the crrn auirc mml]. For II salie n t - pole sync hro nous m ach ine, the phe sor di ogram (Fig. 2.8) is plotted by use or the equ at ion

E ... iJ + ior. +

ji<tX d

+ j iqX q

(2.45)

2.2. Steady.Slai. Eq..allo nl

where; iJ IUlrl ;9 Illoe cu rrents IIlo ng iho dtrcct end qu adratu re BXes of th e mac hin e res pectl vefy; z " nnll :1', nro t he j nd l1 e~ i ve rea ct an ces .Iong the d Rod q axcs rcspee t tvel yr For de m/lch ioes t he cl assical tht'Oi'}' \1'('5 /I )·[Ot si mple r cqua uc n U _ E ± 1. " . (2.406 ) where E - I (II • . F,,,) ; 1 , . ig th e n: eihtioo c ur rent ; F , ,, is t he

Ng. ::.8. The ph.so r diagrllu, of • !Ol lienl- poll' S)'llch . ono"s no-.: hiue

dir ed - a xis cu mpo e eu t of th e a r ma ture C1'OSS m mf ; a nd ' ,. i.ll th e in t8rn lll res istollce of the mach in e. Tile eq uations for sync hronous moohi nes e nd lie ma chine'S ceu be set u p b)' rep resen t i lig II machine fl!! II twcpert (S ll(' F ig. 1.8) II " 01\'1118 r. IInd.t. for a nons:l liC'nt.-pole mac hine: r., .rd, II nd .r ii Ior a sahent pole 1I1 /1" hille; a nd rh Iv r II de mlChiJl ~_ Complica ted nonlinear eou pli llgs ere u ken in tn acco unt by int rod uto ing nonline ae rlependencl'lS of t he pn rA Dl e ~ers on cu r ren ts . T ho solution of I1IM t problems ln elec rrc mec benlcs requires es teu erve stud iC3 i n to Ilteltdy-stole peoeesses , tue re tore t.hoory hargE-ly deal s with s t a tic ce ndn tcns. DiHere nti lll eq uot ions tl ~rJ b in g th e pr()e~eI o f etectromec hauieal e-I\ell:)' cn Jlvers ioJl enable tn vesugati ng tn llllicn l p rocesses and, liS a par t ir'tJIltr c ase, s tea dv-st nt e precesses. Co m pu ti ng Iactl ttt cs offer a fa irl y fas l, esumeuon of .steady-slo te p rtJC~es b y soh'i lll: diller eo t in! eq ueu ons. H owe ver, in h en lJling o ptim iza tio n pro ble m! or ill\'eJ;tiga till g eo m pl lcated

CII. 2. Elocl,o moch. nlcol E...,gy Con"o tl Jon

c cnuec t ic n di agrams of elec tr ic ma chines, s uch an ap proach comes out im pract icable because i t jtakes It groat dea l of ma chine t ime. T he proper es t ahllshm en t of ma th ematical models of elec tric machtues t ha t. would en a ble a ready simulation of both steady -state a nd t ransien t per formanc e 00 computers is one of the ma in t Mks to be hand led In the mathematical theory, ( II r 0 110 needs fi rst to lea rn how to lor mul at e e nerg y co nversion eq uations befon, plad ng -t he problem s on a co mp u te r'.

1.3. Application of Analog Computers fo the Analysis of Electric Machines Ana loj{ computers find wide use for t lto unafys ts of t ee us ien te in aloetrlc machines. In anal og s im ul at ion. cons ider ati on sh oul d be gll'en to the Icat ur es pecul rer Lo a simuln tie n model (anal og), and t he ways sh ou ld be soug ht 'to co mbi ne the con ven ien t notatio n o f eq uations wi th ure Ia vorebje ccn du toes for t he solutlon of t hose e q uations on t he an al og computer. Prali min a ry t o simulation ; Eqe. (1. 3';) exp ressed in terms of cu r rents need tIl be rre osrormed to bring t hem t o forms sunebte for se l t ing u p 1111 analog: dl~ 'dt

( 1 fl O) u~ -

df~ !ri t

= (1; U ) ui! -

(R' ID' ) i:" -

(il,flU)( dt~ :dt)

( W 'L ') i~ - (M IL') ( di~/dt )

di~ :dl = (11': 1.") i~ - (lI-1lU) (di~ /dt) -

(ij,

' i~

(2.47)

+ (ill i U ) t~ 1

r(i ~/dt = (WIl/ ) l ~ -P [/r.:) (dlMdt) + iii, [ I ~ + (M ;l.') l~1 The- block diugm m or a com pute r analog ror th e Soluli on of Eqs. (2.47 ) is s hown ill Pig. 2.9. The solutioÂť of (2. 47) comes to the jut eg rarto n of cu r re nts. Th e positive feed back loops i nterconnecting summtug amplifiers 1 , 7, 4, aud 10 of Fig. 2.9 caus e self-excit.ation of t he mod el since here t he nmp lffier g ,1i ns ],,2' k) , k l , and J,' a are higher than unlt.y. T o effect s tnbil Hr, we integrnte (2.4.7), omit th e d erivatives, a nd obta in the equations convenien t for si mula t ion (see Fig . .'\1 i n Appendi x I). T hese equ ations have II fina l for m = dl dl) ;

i~ ---... (Ilp ) ( alu~ - a 2i~) - aa l ~ 11=- ( I i p)(aiu~ - a51~) - a~t ~

/~ = ("1 (.0) [ - a11~ - {t)r (i~ + a.iMJ - ail ~ i ~ = ( 1 ;p)[-al ~t~ + (,), (i~ + all i~ )[ - a , 2tA M~=

au ( 1:;'1~ -I'I~).

pw,

=a l ~ (M .-11"1 , )

(2.48)

2.3 . Applk."ion of AN-log CO....p uhl"f

i lL'; a t - ~ - ROIL", a:r - a. M IL" ; a., = - 4 n - R rIL' ; at - a. - a ll = a l t - M I["r ; a u = (m12) p ll f; att "'" pl J. For t he co nvenie nce of de1Signation of q ua ntiti es on tb e di ag ram. we red uce t he ex pressions be t ween t h e ,brllCkeu In (2.4 8) to t he form ..here at = a. -

In tbe mod el of Fig. At t hlll ga in rec tors of t he sum mers (2 a nd 8 , S aa d 11), s hown b}' d ash [Inea, are c lose to unit y an d t he mod el is

u:,

_ di~

r -'",

", k,

~ ti:.,

l~"

,"\..J

-"••. ""'. ,I

_ "" ,C,

- '. d '~

+ " ,C,

-'.,.

-di

•

f ig. 2,!), T ile eompuw r e ualog for r.h. 51'stem of Eq l , (2.<1.) m of'll

k e-

s t ab le. Neaali" e feedback pa t hs of a mpli fiers wit h ga lllS

14 . let . k u m a.ke t he model mo re erahle In performance.

'" T he oot lltion of E qs . (2. 12) t hrough (2.24) for flu lI" li nkages is b rough t to t ho form oonv l!-Ilient for s imu la tion d 'l':' 'd t -. a t V;' + Il. V;" d 1Yl dt "'"u,ul -a.1VI+ .,.IJ'~ d'Y';'d l = -a,V:' + as'V:' -",,'t"i (2 .1,9) d'Vi ,dt= - a~'V~ + a l. II'i T W , '¥:" M c'" a" ('V.":' - 'V~'VU dfil, 'dt = (l IZ (M, - M , )

o,u:.-

where iii, - a t - U•• ,..II V2; 111 - u. -= n'L' /CL·t' - JP); " 1 = a. = = R'M/(Ut' - J1P ): u , = as - Fr L' !( L'L' - M Il; a. '" a ll '"" - R' ,l1/(L'U - ilP) ; 0 11 ""' fmp/2) '\/ /(L'L' .'11 ' ); a l l - pl/. Cu rre nt eq uat io ns are 1:... ... a l)'V!,, ~ at4 "l':. . t~ ... a,,'VI -o,.Wi 1:;' =G'I1' V;;'-al1lV:., I ~ - a l. \V ~ -RzalY' where II U "'" Il u w= U / (l!U - 1\} ~ ) ; ai ' _ Di g _ L " tL'L' _ .M I ); a l i - lilt II , ~ - 1110 = ,111(1.'£.> - ;t.,, ), I'h e hlock rli llgrn m for E qs. (2.49 ) nppon!'g in Ap peudi c J. T he se t up (){ Fig. A2 il' stoblo , t ho nei lll i" e reedb ack sig nnl exce eds th e pos it i ve feedba ck ~ ign ll l, 81ul !If g(>rlern l lOIl j ~ ll lJ~en l. T he uc t et.len of th e eq uat tous for rUrt'61U8 anll fl u ); Ii nkllges: ta c hos en suc h AS 10 be con venien t (01' t he aimuln tiou proce d ure:

d'l';, <it h a , u:" - a i~ , d'l';.. d l _ - ro,'l"i- a~ ~ , I:' ""- a lll'~ -

Q i'(f~ ,

, ; ~ a ,, "; - a ,, 'I'~. ,II . :. Cl' l dl l'V;" - i:''J1~ ),

15 =

dill;

f./t =a ;rl.~-aii~

d'l', :dt .:. w, 'V'"a~

Q,t'

'1'5 - l l ..we;,

" _ a,,"i_a..'I'1

(2 .50)

dw, d t = Cl'ltU" . - M.)

wl,crc Cl'J - Q, ' - 1.1,..!:. ~ V 2; "I - ". - N' : », - 4, "'" R', "7 = ~ ... L ' /(L'V - ,H') : =- a lt _ all "'" li lt .... l'lI I (U V - A/ll; Q II Q II = J II IU I./ M t); au = IIIJpI2) (AULT): " It = plJ. T his model is s tab lc ove r t be wi de r llnge of pa rame~ers and b appllca hle to t he sol u t io n 01 eo mpl lce t ed p rch lems (see Fig. A 3). The d loie. a si mu llll io n model d epend s on the ~o i l1s of co m pu t ing blocks , perm L!~5 ible time 01 integration lind th e !lability in operat ion. ' \ 11 im po rt a llt stage hI prepa rillg a problem for st rcu te u cn is t he ch otec of sc ales o f variables 1I11d t ho cal cul a tion of g ll in~ of tha model s co m pu ti ng blocks , In th e s lm ul ati on on 1111 analog co m pu tet . eech plly! ical q uan t ity of a real objec t hll3 it s ana log as a '·o ltege, Selli e flK'lol'!l e re ch ose n t o li'lItab lis h th e rel ntto u between tb e quan-

fl'

tit ies and vol tages . T he ma xi mum possi b le va luM of vlri .bl M should be t eke u so as nOl to faU ouLside t he l imil.$ of t he comput er working renge. T he " oltlge is Ilh"en by

u. = slf~' (2.5 1) where u ts tile model voltage: z is a dependen t ve rte ble : and /If ;c is t he s ~ Il Le of t he depend ent va rt eb lc. The t im e of t ran s ionts in th e model i$

t ...

Mil .

(2.52)

where :1£ , is t he t ime scale: lind t, is t he ume of t rans ien ts in 8 real obj eet.

In c hoosi ng the $Cal cs, it is ecnvemenr t o l ak e t he foll owi ng qu allti t ies Il! deersrve ra etces: the not\l in al-p ha Sf' c urrent I... .,.,. 11"2; nOMi naJ-ph ase volta:;o U",,.,. Y2; nominal moment /If ",: 'l",, -= U".rt. V 2Iw, ; (0,). = 2rmp /60. Hero w, - 2n/. lin t! (0 , ;s the angular ve loei l)' in rad i ans per second. Then , M ", _ IOOI U"." ,. Y2 is t hl' vO[lage scale; M I _ IOOIJ".,.", x X y 2" k ., is t he cUn' en t sca le : AI M " l 00/:lJ"k 'li I ~ t he elect ro mago(Ol ic torqu e scal e ; M 'V -= l 00/lJ'. k .9' is t he flu :!: Iin k llie scale , an d M .. - 100/ (&1, i.s the ...eloci t)' scale. The t i me scale Is c hosen pro«edi~g from thn d ~ lerati oD or accetcee uce or the proces.s of soluti on , a, def ined by (7.52) . In ch OCNIing t he Kales , it is nece:ssa ry to keep whh tn t he work inlr ra nge or an ana log co m pu te r (u .... _ ±tOO V). The ga in (octors of compu tin g blocks a re de fined as kz "'" ilf 0" ,4,/!of,,, for a su m ming II mpli fier ; l~i = M " "tflI IAf ,,,M , for a n integra ti ng

amp lif ier; k~ - Af" .. lo.llO.o i MII" M"I 'l for a s u mmer r9Coivi ng t he produ ct of vllr inb les ; k - M " ..,a,/O.O JM "" M' I"ftf l for an in tegraX

tor eece t... in g the prod uct of va riab les . Here j~f u I is tbe sea le for representa tion of a n Ollt put va riab le of aD a mp li fi er; At I" is t he sca le for repres en ta ti on oC an in pu t variable of 8 0 am plifie r; and 4, [5 the eoe lCide ot in the in itia l e qu.;tio n, which. etan ds before t he voriabl e arriv in.. a t t he a mplifier I np ut. T h.e ga ins of a mptifi&l'5 i n mos t a nalog com pute rs must he chos en bet ween 0.01 And 10. The successful i mp lem ent atio n of II si mula t ion model wi th the aid of o.J1 a nalog co m put er de pends OD ~ lte rati o between t he el ectromagne tic and elec tromecha nical lime eonet ente. If t hese cons renee di ffer hy a factor of l lY to to', th e s imul a ti on o n an ana log com pu ter becomes d ifficu lt (t his is th e c ase for lh e s im ul ation models of gy ro mceces. hI mach lo es , ete.}, [f t he s lm ul8 &ioo in ni s i q u an t i&ies meets wit h d ifficulties, it 13 advhab le to co nv ert to rei d i ve un hs. IThe rill'h t ch oice oC t he b as ic

e ll. 2. Eled romiKheniul Energ y

Con~ e" io n

queu t fuos enables t he ga ins of t he model t o lie b rough t close t o ur uty . T his decreases the d ynamic erro rs of the mode l's computing blocks and ra ises t he comput er ca pahili ty. 'I'h e conv eni en t bast e quen tl t lea ure it> = l ll . p ~ V 2; u ~ = Un. p ~V2; end ((I" - wQ - 2n/ o, nnd 0.1,50 thei r dcrt veuves such es Ole base power P I> ""' lm/2) it-ul> . base momen t M b """- P b1wb. fl ux li nkage 1f/> ... Uh 1lUb . tim e t b = t /(Jj b, imped ance = 1> = u />li l>. in d ucta nce L" = :f.,,/(il b. and mome nt of inerti a J b = M bIro/>. The m nt bcmeucet model of an energy conver ter. where the differ ent i al equ ations fot flux Hnkage e are wr it len in qu en u uee expres-

sed in dimenslonless units, has the Ierm d'ir;'!d't ""'~lU~ -

a21jf~ + ~3iV~ , d 'ir' idT = a1u', dlfitid't = -a,'Y!: + ~1f~ -;",1jr~

. , dlV ~ /d't = - ay\jf; + ~ ,g1f l + w .W:"

-as~rl + aBo/~

M . = ~ , ( 1jr~1jr ~ - q.~ q.~)

(2.5~

dw. idll= ~'1 (M.- M.)

where

a , = u, =

R'L'

R'M

= a. = 1; a~ = a, = LlL' M . I~; a s = a.'" L'L' ft/' fI ' L' L'L ' !It'

t ~ ; Us =

1tJ.-

R'M

L' L'

t il;

â&#x20AC;˘ .iP I ,,; a ll = (mP n/2) X

a;1 ...

IM /(L'U - M ~) I (II't /1I1 b); PAM ,,/Jw~. Th e bl ock diagr-am for (2.53) t o be solved on El com pu ter remai ns th e enme as befor e. The sol utio n of equatio ns in rela ti ve uUtts enab les handlin g t he pro bl ems which otherw ise (when using rea l units) lead to unsta ble models . T he model of an en,e;rg y conve r t er si mulated a ll Oil analog computer re pres ents a ec nt rel syst em cont ai nin g posit i ve and negative feedbac k loops. T he degr ee to wh ich aile feed back path or an ot her errecte th e st ability depen ds on t he relatio ns between t he para meters of t he mach ine bein gmod eled. A si mul ati on model Is chosen proceed ing from th e ty pe dl energy convener. i ts ch a racte ris t ics and t rans ient beh evfor. In Appe nd ix I are g iven t he equa tfc ns And t he bl ock dia gra ms of th e main t ypes of ener gy converters. X

1.4. Transient

Pro ~esses

In Electric: Mac:hlnes

Transi ent s in electric machi nes ar ise from ch anges in the voltages And Ir eq uen ctea at ma chine te rminals, the l oad 011 the shalt, meehtu e pa ram ete rs, d uring conn ection of II mach ine t o 01' its di scon nect ion

trom th e bus , e tc. III re nt cc udh tons , t rllos ien L peccesses COli natu ra l-

I)' occur during II s lmu h a llOOus venntjou of • Ie... fac tol'!!. T he combin .tions of t ile fact ors affec t ing the d )'n ltm ics t a n La mani fold . so t he rese ar d.lor must h ue ollou gll es perieuce lind kn ow led ge La ch oose tb e pre v. i1ing se t 01 JIICI Ors. l h.ere.'b y aim pJit),jng the problem. The re ia II Itre " t " Bri el y of l.rIUISicllts wh lch are mIlCh. more ('ompllU t h an s!.ead)'......t ete proc esses , Ih o IaUor bc ill!,; • particul ar c ese of t he for mer. B y t hei r im porlllllco find the tunuea ce t ho)" ha ve tlll l b, cpe-ett ou of mfl<:h in rs . th e l.r ll n~ icn \.ll c an he dh' ided in t o t he proeesees b rough t about d uri ng s t ar l ing . brak ing , re\· c~ill g . rt'Stllft ing . and lo ad ' ·o ri llt io n . These processes ca n llp JWo r e t sy m mc l rk llnd

•

3H 2 . 34 3. 31_

•

50. Y

100 V

0, 64 4S V 1, .,0.0 1

f ig_ 3-10.

T~ ~ I JJlll:t1I .,.

""" starting

tho

YAll..22

t}·~

4-W mOlOr

1lS1'mmQ\rie "ollag e'S in s ym me t ric a nd as ymmetri c mmchines, T h e dl'n am i(-s (If W1ynd l ronr,us alld ;;}'l.Iwwno u.ll machi nes h as us own {u lure'l, A co m muta to r or nil)' other [reque nc.y co nv ert er in t rod uces Iu OWJ\ s p(!{'irjr flll1 l u J«! In to t hu dy nll;m ic behavio r of th e m achine. Th e t ra ns ie-nt., in tra nsform er! lind ot her elec t romag net ic energy conve rters d iff er f r" m tho t ra n, ilm ls in ro t at i ng elec tric m achin es. Teenstc n ts ortco determin e t he c hoice of th e Inst alred p ower of equ ipm ent . ' hi' muss of elecrrtc machi nes lind elec trom agn eti c l oads they 118ve to ca r r) '. T h is is pa rtic ul a,rl y th e case for l mp oc,Ho ftd heav y-d ut y drlves , re ve rslng qll i ck-8 ~t i n g dr ives , etc.. T o 'ln a l ~'le tra nl ient p roce"Sel5, we should Iorm ul ete th e m arh em et.icel mod el of the ~ra n l!ll en ts , conv ert t ile eq uations 'to tho for ms con ven ien t for the ~ i m u l a tl on of the pr ocesses on II com put er and solve t hes e equ ations , At pres en t 8 III.Jll1! amou nt of m ater ial is av .il abl (> on t he tn vesuRat io", of dyn ami c processoa b y m ean s of com puters: many probl ems lu v(> become ctesstcat an d fe rm part of t he h borotory wor k us i&,oe d to stude n ts in mos t «llleges. Figu fTll 2.10 t hrough 2,13 ill u"tra le the start oecllJ ograms for motors u Di ing i ll PO"'''E!f fro m 4 W to 500 kW. As aeen fr om t he oScillognms. the st4rtiDi p rocedure pattem d iffers wit h t he relat io ns between the patllmeteTll, T he }' A,U-22 motor GOmes up to Its s taad y-s t ale velocity

Ch. 2, EI..cltom.. chaninl Energy Conw" "ion

for tw o or t hree peri ods, hut th e rot or yet oscillates a bout its steady angu lar vefccit y for some tilli e.The 50o-kW motor gai ns s peed very alo wly, but it does not overspee d ÂŤnee approachin g the s teady-state veloett y, T ho star tin g condi uoue for motors of th e 4A series, for exam ple. 75-kW moto rs, a re most typical (Fig. 2.12). T he process of rO VO l"!:IllJ differs f"orn the process of s tarting by t he e ffect of pe ra metera on th e -hu pact torqu e M ,m. impact current

I 100 V

1= }

f'i!l. 2. ' " The o""ill ol:"mm of start l o/l t hl! I" AA-I03,H typ e 2SO -W Ind uetion mowr

. h ~ ..Âˇ

100 V

Fig . 2. 12. The u$ mo gram of stlHl.lng the 4,1, -250504 t ype j5-kW induotioll mow r

,m.

1 and t he s tar t ing (speed up) t ime t. , (Fig. 2.14 ). For i ts reversal, a j'raotor is c ut off and then 'co n nected to t he line wi t h the reverse pha se sequence. T he t rllns iont hero depends on t he commutation t ime a nd o n wheth er or not t he fie ld In t ho a ir gap has died out. \ Vhere t he swi tching is ins tantan eous, t he processes nf field decoy and fie ld buildup go o n concurrentl y, so the im pact cu rr ents and torques grow. T he sche ma tic ct e program unit for reve rsi ng is s hown In Fig. A7 and Fig. AS. , In restarting a motor, t he 'i nrushes of current and the t hr us t pro-d uced are t he highes t. T he schema uc diag ra m for restar t and the

! - OIl 1 8

s ystem of eq u ati onjll for t his e e nd t uon a NI t he se mc as at !-l arting with th o dlHeren ce t h a l the in it i al vefoc t t.y lolr is ot h er l h lln zet O> (F igs . 2 . ( 5 , 2 .16) . T he IIH\I Y!lis of t eet lr t p receseee i n voh' ing ~ an

" Fig . 2.t4 .

1', _ :

•

s w . ... _

I'v

l'~

,..

-,.< 0;:::

•':1

nl!ye raill~'

t he '\Jl -3t1<\ lnducti",,, motor

l l.m, ' , _ 8 1 .. , "'. c " 1\ ) N

.I _ ~ Ol ~~ '

' M'

..~

•

: t ~v .

. I

I......

.J -!

,, ,

Fig. 2. Hi. Hes....rti ng \.he A]J.- Stl4 Dw' tor H t he lollls r sfl'I'ed or 0 .91ln "'jith the rll'ld dC'C lyed

Fi~. 2.1 6. n eltarting a Dlow r n~ t hl> lm t id ! peed or 0. 2111\. wi lli t ll/! field ,l(!u yt' d

u nd a m pe d fi eld requires t he :co mp u t a t ion o f i nitial cood i tiOM for Olaguetic nu x linkag es . T he t ta nsi eo Ul a t restarting in the u nd llm ped fi eld cond it ions lire m ost co m pl ex. From th e qual itative an al ySis of th e p recess es of s lar l i ng, reversi ng , an d rest arting it. toll o~ tha t. t hese pr ec esses d iffer fr om ea c h o ther by th e ch aracter of varia ti on of e u ere n ts, t or ques. and ang ular veloci t ies. T he effect ot pa r ameters o n t he COU !"Ml of vari ou!'l t-ranetents is d ifferen t.. T he lIna l ys is of t ra ns ion l s 11 made b y r.olvlng the sys te m of transi en t d y namic eq ual-iolls .

1.• .

T r. n .~ n l

Proc..... In Electric Mac h.ln e.

A diller.itv oj t,.QJI$ienti tn. dectrtl: m aMlne, .km. Jrom. the combilled ejject 0/ panuruteN , their nonlinear rt1atwnskt pa, ellett 0/ elemenu

co1Vlectea to the .rlator alld rotc,., /udbSf:k paUu, lUymm etrl.c ali a /Wn· $llllUOtdal pattern o/ lJOltagu, rlI.1VlmglcondUlonf, and design verdDm oj r nuc¥ CDfWt!,.Ur,.

In Appendu 1.1 a re liven th e data on motors of th e 4A se ries, their parameters, base qu antities, coeffici ents in equlllions written in fl ux li nkagll uora u c e , eeetes of va ria bles, and ga in factors of the amplifiers in a s imu l ati on model. Tabla 2.1 1isu the val ues of buic • • 11" Q u. ntltie. In Ni>\

'" M,.

11.0 U

/'~

' .55

J~..

3.'

...

19 .5

. ,.

33.'

T.bl . 2.1

l r. .... le ntt

88 . t

3."

I n ..

113 .2 213 .8 3 .i6 a.s

5. ' 2

,."

5. 21

' .M

5. 12

5.2 1

' .92

' .8

"'.1

U , U,

..., .., ' .08

210.4 2 .37

e.ss

' .M

qualltil ies (in rel ati ve un ital d escrib ~~ tramden ts in mc ac ra. T h.e dynamic processes are of importance in the a nalys is of a nu mbor of dri ves, t herefore it is ex pedient th at t pe guides to electric machines should include t he characteri sti cs of IlrlUlsients along wit h s tead ystate c har acteris tics. Th e th eor y of eynchroneus mecbtnes widel y us ee the Dotions of stea dy·state, sub t r-ausi ent , Bnd tran1Jient co nd itions and t he parameters descri bing t hese condit ions. 'Draus ien ta in Induc tion machines were given ins ufficie nt tr eat ment. Because of 1'I n arrow gap ill th ese mach ines, t here hl\$ been no need. until recently, to in tr od uce t he para meters ot s te ad y-stato nnd t ransient con ditions. T he developmen t of t he general theor y of elect ric machi nes and the th enr y of transie nla in Ind ucti on machi nes has necess itated t he Int roduc tion of transient pa rameters for induct ion machin es. Ind uetlve reacta nces greaUy var y until a molar aUalM its ope rat ine speed. In Ap pend b: II 11119 i iven the t rans ient par am eters for some machi nes of t he 4A se ries. T he anal Ytical evaluation of Inductive param eters W illi made for t he ate ady-stat.e C(lnditio Wi. TransIen t pau mete" cln be defined by using comput ers tor t he SO I UUOD of energy conversion eq ua ti on".

Ch. 2. El.etromed).a"l~el En.'gy Co "" e "ion

2.5. The Effect of Parameters on the Dynamic Characteristics.of Induction Machines As an example of application of com puters lor the s im ulatio n of dynami c processes, consi der the tr ansients in an induction machine with a ci rc ular fiel d in the air g ap. Such a field ca n exis t i n an ideal machine supplied from a sj~uSo i d al symmetr ic voltage source. Iu teal electr ic machines, the air gap contains a host of t he fiel ds of higher harmonics al ong wit h th e fiel d of the fu ndamental har mon ic, th erefore t he an alys is of the p rocesses in machin es th at takes into consideration only t he fund am ental harmonic ap plies to t he ideal machi ne a nd gives ap prox imate results . If t he elect romec han ical equations for an ener gy con verte r having a ci rcula r field i n the air ga p are c as t In th e for m conven ient Co r t he simulation of processes on co mp uters . It is possi ble to sol ve the equ at ions and t hus investi ga te the processes using the obtained res ults. The sol ution of equa tions for a ci rc ul ar field does no t present diff iculties. Gi ven t he si mulation mode l for the solu uoc of e lectro, mechan ical equation s, we ca n use th e results fro m t he processed oscillog rams or tak e the rea di ngs on digital measu rm g d evi ces and thu s eval uate t he dependent an d ind ep en den t va r ia bles and th e ti me from one ev ent to the nex t (sta rting, reversing, br aking, pulli ng in synchronis m, restarting. et c.). Compu ters offer the pOS!li biHty of analyzing tr ansi en ts, t .e. determinin g currents, im pact t orques , and t he dura t ion of a transien t process wit h a change in one pa ram eter, wh ich is im possi bl e t o do in in vesti ga tin g a real object, A nalyzin g t he IItart oscill ogr ams of inducti on motors of vario us powers , it is easy to re veal t hat t he cou rse of t ra nsients (v aria ti ons In cu rren ts , t orq ues , oscillat ions) va ries from mot or to mo t or beca use the se energy convert ers henv'jly diff er In para me ters. Figure 2.17 dtspl ays the plo'~ of M I "" ii m, t . r versus W at starting, ""Figs. 2. 18 , 2.1 9 and 2.20 ilhist rate how MIni, J'm and t il vary wi th mu tu al inducta nce (the si zo 'of an ai r gap), the mom ent of in er tia and rot or le akage in duc tance respectively. The plo ts are drawn for an A2-t02-8 i nduction mot or . H ere p~ = 100 kW, 2p = 8, U .... "" 220 Y, R' _ 0.03 Q, W i= 0.024 Q, M ... tOG x to-l H , L' ... = U = 151. 1 x 10-1. H. and J -= 6 k g mi. The curves of Fi ga. 2.1 7 t hro ugh 2.20 permit us to esti mate t he effoot of paramet ers on th e d yn am ic char act er ist ics of a motor at starling, However, th ese p l p~ are u nsuitable for estl ma t ing t he dyn amicall y optimum param,et el'll at which , fGr exa m ple. t he s ta rti ng time and th e impact t orque would be a t a min i mu m, T he cur ves of Figs. 2.17 thro ugh 2.2 0 ar e seen to d isplay ext reme. Let us note t hat a decrease or tncrease 0/ onlg one 0/ the parameUrit

2.S, The EU.. ct of Paramelers on Induction Mac hi.....

wmwt lead to opttmum result s.

All

analog computer wllit$ the system

of f/~'t equauo ns. T here 'are optimu m r,ela tlons bttU-'f1f111 th e equation.

coefficients at whIch the requisite quant tttes exhIbit extrema. Figure 2.21 illus trate s the plots of M I"" i,,,, , lind t. , versus Rr for the reversal process in t he A2-102-8 motor: Figs. 2.22, 2.23,

,• •, •

• •, • ,

•

." o

Fig. 2. 17. Star ti ng elUl e

I". curreu t

I/m' a nd w rq uc M I", VeB US rotor rcetelanc e (M

• •

"".., "" d. ,",) I

I I 1/

" i,,,,

•1/

i, m

•

1: 4

11 m

10 12 M

versu s mull.n] ind uel" Jl(c

, ,'st. i,", 8

• ,1\'

• ,

FIg. 2.18, The cu rves or 1./1 "11m' nnd

•

r- ~m .. t-

•

, i' lI> t-.::

-l- --4

t."

12 J

Fig. 2.19. The curv es or Al l"" and IfI>< "cr~u & moment of ' " cr UIl

V '" •

0.25 0.5 0.75 J.O 1.2 5 1.;,

H g. 2.ID. The curves "', ' .1 ' M r"" and lim verS\ls roto r leakage indu ctance

and 2.24 display the plots of tho sa me q uantities 3 S runeuons of mutu al indu ctance M , momen t of iner tia J , lind rotor leakage induct ance I~, respecti vely. T he transients in reversin g lire more complex t han t hose a t s tar ting. It has to be no ted t hat t he effect of parameters on t he processes occurring ; in reversing differs from the

eUeet th ey have on t he processes ahtarling . Op t ltrW m po.rdmeursln Jtor tf ng , n versing or other dy ntlmic condUioll8 dtller f rom one artother.

, •

" •

•

,•,

, ,• ,

1,.

, ,

Fic· 1.2 1. The eulYU of "I' I,,,, ve nll. rotor ~tln«, ..

M,,,,, aDd re~na

I'--

'" I

10M

Fir. 2. 22. I'1M wn'" of 'ot,

M .... and "'" ver$\l$ lDut ua l .IDduet.a1lCC in ",,-er· al Pg

•",

• , 8-

•• 0- 0

1/ J

," • ~ , 8

M 8

,,,

j I.

•

'.1>

00.25 0.50.15 1.0 1.25 t;,

s J

Fig. 2.23. The eu!'Vfl or M .... I.lId I,,. ...~l'SUi ml)mellt of ia ertUI te reYer·

Fir. 2.24. T he eureea of '",

f,,. vc ......

&I,,. aa d

.",..klp induel.l.QOl!in revee -

"·8 "'" is of interest to l ook al t he pro cess of reversing with t he field

h n ill und a mped. With a ra t her fas t reversal af ph ases a t ma chine term inals , t he li eld in th e ab ga p hB.!l no ti me lo give up its s tored energy. In other words, ' he fiel d bes nol yU decay ed a t tile IDUant

3.1. Infinite .... bil'e'y Spectrum 01 Field . in Ai' G a p

of cenoecuon of l ho moto r to t he bus. The anal ysis of the proces s of reversal t h us requires estimf\ting the li1i ti al values of the undam ped field by defi ning t he flux linkago tjetween t he st ator and rotor windings. Revorsin g coutecto rs o perate in tho severe conditions (an ec con rector c loses i n 0.03 to 0.05 s]. The most ellecrlve techniqu e of studying transients in t hese dev ices' is to use an ana log computer set up for prog rammable work. . One of tilt! interesti ng tr ans ient modes is the mode a t resta rting. The s ystem of equations and tho simulation mode l fo r its sol ution will be the s a me as at s tar ti ng with t:J1e excep tio n t hat the angul ar velocit y 00, will he oth er th an eero. ~ ll v es tig ati on s reve al t ha t t he restar tin g process at w; close to t he nomtnal (with shor t-time su pply interruptions ) is accompa nied by heav y surges of currents a nd torques. Th ese s urges exceed t he ma ximu m val ues of euerents an d torques in etartfng and reversi ng. The restar ting process wi th th e field undam ped len ds itself to t he anal ysis af ter estimat io n of th e initial cond itio ns for [lux li nkages. Since the rotor has a s tore of kine ,tic energy and th e Held has not decayed. rest ar ti ng mus t appear lit fi rst glance to be an easy mode for an electric macWne. However s ince restarting Involv es th e d isconnection of II machine h om and its connection to t he s u pply line, tile two attendant transient processes witb complic a ted changes in currents and to rques s uperpose one on th e ot her . Giv en the mat hematic-al model ,of a n in duction mach i ne. we can ana l yze tho opera tion of the machine bot h in t he br aking- and in t he gellOratlng modo. In inv ostig-~ ting the gene rator- act ion. i t suffices to change only the sign of t he to rqu e in ui e elect romecha nical equat ions. If t he line voltage is cons ta nt . t he ind ucti on generator operates in parall el wit h and into the infinite power li ne. Eert.aln difficulties anise in the ena lyaie of an' auto nomous Indu c t tcÂť generator when it dr aws the ronctive power ~ro m ca paci tors and t he voltage and froque ncy unde rgo c hanges.

Ch apt e r

General ized m-n W inding C onver ter 3.t. The Infinite Arb itrary Spectrum of Fields in the Air Gap As is 'known , t he cir cul ar field in t he ai r gap ca n be tho ught to exist only In an ideali ze d machine. I n real ma~htnt8, the air gap u hi bUs all in fini te spectrum of harmonics dif/trln g in amplitudt ana

Ch. 3. Gcr,.. •• /i zed Ill ·a WJndlllj Co nverll!l.

fr«JlUnclI along with the fundamental harmollU:. T hese har mo nics re volve both i n t he for wl ro and in the ba ckward directi o n wil b res pect to th e revol v ing fund amental harmonic. The angul ar veloci ti es of the harmonics can be h ig her a nd 10"".r than tba t of the fu nda· mental wav e and their amplitudes can vary in ro ta tio n. All heemoflies IU Y be d iv ided i nto two types. t i me an d s pace barmon ica. T ime Juumerna are th e ones whic h let i nt o ~he air gap of I machine fr om tho ou ts id o. Spa~ harmontc$ a ppear in th e ai.r g ap on ae-couot of the specifics of th e con verter's in t ern ol struct u re. I t sho uld be kop t In mind Iha t thi s cl assi fi ca ti on of b erm onrcs is rathe r ec ndlt ion a1; the Dames ' t ime and s pace harmon ica ' ar ose for h istori cal reasons in the course of development of t he weory of el ect r ic rnaeh i n ~ ,

Consid er ing tho Ee 8! /I t \l'o port (see }' ig. 1.8), we:sho uld not e t bot the conver t er has t wo in put s , one on t he s ide of e1et:tr ical teemino!.!! and thn othor on tlie s ide of mechant cei term in nis . T i me har monica arise from nons tnusoldel, as ymme tric Yoltag!!5 and nonlinear cha nges i n the a m plitu de and f req ue ncy of v oltag es. T ho)' also resul t f rom nonlinear changes in th e to r q ue and s peed. In the general ease , t im e ha rm on ics appea r from t he simu ltaneous aellon of nonlinear f ac tors at t wo in pu t termin als. T hese harmo nics m ay etec ieL in to th e ai r gap of an elec tric machine fro m t heMnal ter mi na ls (sec Fi g. r.t }, Heat s hocks (sharp t em pe rature nriati(lns of t he machi ne. frilme) cause u pper barm(lnics in th e a ir g llp as a reo6u lt of cha nges in tb o machi ne psramete rs . NOll5in uso idaJ YoU,ages whtch giv e r ise to ti me harmo nics ma y res ult from nonlinear e lements s uc h as sat l/ra ble reactors and semico nd uc tor el em ents d ispose d ah ea d of the mot or, a nonsmuaotdal waveform of the generat or vol t ag e or d ts tce uo n of t he wa vef or m of t he su pply vol t ag e, etc. If th e su p pl y vol tage eoetatn s a ccn etent co m pone nt , a harm on ic spec tr um emer ges , w h ich Includes a n in finite r/lnge of even har mon ics along wi t h od d harmonics. I n t he a bsen ce of urn e hai mo n ics in t he air g a p of 11 machin e, s pace harmoni cs originate fr om t he nonsi nusoid al dt et r tbu uc n of tu r n!'! an d magnethi ng rorces'. air g ap nnn -unHormi t y due t o t1le presence of teet h and sl ots in t he ro t or an d s ta tor, gap eIJipticity and C(lnic ity, and non lin ea rit y of t ho peeem etees en le ri 0il' te te el ectromech anica l eq uat ions. Consider in mor e detail s pace harmenice, in p nrtieulllr t ill.' harmonics of mag netizi ng Iorcea, The wiDdi ngs of elect ric machines a re cu rre nt l oo ps producing magnet iZing rorece . The s im ple:st windi ng (loo p) is a t urn (or II co il consisti ng of several t Unis) wh066 pi t ch II is equal to th o pole pilch 't (Fig. a. l a) . Such a l oop p rod uces a rectangu la r mag ne tom oti ve force (m mr) . A t 11 < 1: (Fig. 3.tb) t he m mf takes t he form (If II trllpezo ld. W here t wo or more ccus are Invo lv ed, Lbe mmf llSS u mes t he form of II. s lcpl iko cu rve (Fig. 3. Ie ).

73'

3.1. 11l/lIlUe Arbil rary Spect, ul'l ' 01 f ie ld . in Air Gap

Developi ng the mmf as a harmonic pr ogress ion , we s hould note th efact that where t he mm f distr ib u ti on is rectangu la r in sh a pe , u p per harmon ics have maximum em ptl t udee ] th o am pli tudes become low erwith II sho rt ened wind ing pHdl /lnd lower s ti ll fur th er where t ho

' oj

Pik. 3Âˇ 1. Curn-{\t looJlll

winding consists of II few coils. Only ill the case of the si nusoidal' distributi on of turns over a smooth cy li nd l"iCl,I1 su rface of th e ga p' upper har mon ics of the rmnf are no nexletent. F or a co il Windin g or Ior one tUPi, as s ho wn in Fig. 3.1a, tbeampli tudes of harmonics are g ivell by }\ = (4/n ) F c F~ -= (113) (4{n ) F e F~ = (1/5) t4/1t) F . (3.1 } F \" = (1Iv) (4/n) F e where F c = l wl2; I is t he cu rren t Clo'wing in a t urn : and w is the number of turns in t ho coil. In a willding comp r ising lJ coils, th e m mf epprextmetee a si nus oid and the amplitudes of har mon ics become lower. In t hree-phasa symmetr ic wi nd ings t hnre ap pear ha rm oni cs of the or der

'\'= 6cÂą1

where c = 0 , 1, 2, . . . Ha rm onics of th e order 6c 1 (1, 13. H) .. .) revolve in t he forw ard d irection with res pect to t he rotat ing iifllt har monic a t II s peed Wllich is a factor of 7, 13, ii}, . . . l ower than the sp eed of lite fi rs t ha rmon ic, ;Harm on ics of the order 6c _1 (5, 11 , 17, . . . ) revol ve nt. n speed of 1/5, 1111, 1/17 . t hespeed of the firs t ha rm onic in t he u troe uon o pposite t o that of t befirst harmon ic.

Cil. 3. G<lt>e,,,II..d 1lI-n W ind ing C a nvetler

For two -p hase s ymmetric wi nd ings , v = 4.:- ± 1 Harm onies of ·t he order 4c + 1 revolve in til ll same d irec ti on as the first ha rmonic, .an d harmon ics of t he order 4c _ 1 revo lve in t he op posite di recti on "t o t he firs t ha rmonic. Because t he phases of wind ings are aaymmet rtc, each upper hnr"monic of t he mmf ma y have a for ward and a backward wave . T hus , -oni y <l uonsinu s old a l a nd asymmet ric distribution of t be-mm f may g ive r ise to two ecta of har monics i n t he air gap. A s et of salfency-i nd uced h ar monics has a hea vy effect on th e -c ha ra cte r ist ics or an en erg Y :CQ nv er l er. 'I'he Win dings of e lect r ic

machines are distri buted in slots . Since the perm eance of t he air ;ga p is nonunifor m , th o field patte rn in th e ga p depen ds bo t h a ll t ho mID! distri bution an d on t ho go p per rnea nce:

"'6

B (x) """ (x) F (x) T he magne t ic n ux dens ity tIS a func t ion of t he coordi nate c ou n ted ~ff a rou nd th e circ umference is prop or t ional to t.he gap's s pecific peemeence "'6(x) expressed in terms of t he permeance per unit. a rea ·of th e g ap. I n el ect ric machines t he toot h pitch is usu al ly the same e round th e enure circumference a nd the g ap perrnea nca is re presen ted .in t he for m of a period ic curve which can be exp anded into a Fourl e 'ser ies. For a mach ine exhi biting s alienc y on bo t h sides of t he ga p, t he an al y tic al estimatio n ot! th e pe r-u nit-area perm eance even with >the rotor 8t s t ands t ill, let alone in mo tio n, presen ts a difficult p robl em. The qu antit y 1- 0 ca n be d efined as a p rod uct ). 6 <=

"'6.1'·t.

a nd 4 are t he per-u ni t-area per meanc es of t ile st ator and ector res pectively, ca lculated separatel y on the assu mption t hat e eltency res ults fro m t he slcte on l y on one si de of t he gap. T he equ ivale nt o pen ing of a s lot va ries as a res ult of s atu rati on , '9 0 t he am plitudes o f aalicncy-Induced ha r moni cs rna)' cha nge wi t h l oa d. Skewing of s lots on the s t a t or and r ot or tor one too th pi t ch -can redu ce t he amp li t udes of these harmoni cs. With t he slo ts ske wed, the emfs due t o sal iency-in du ced harmon ics ov er t he act ive leng t h -cf th o mac hin e are offsot and t he cu rrents caused by t he emfs are d ose t o tew. T he magnit ude o f sa li en cy- i nduced har mon ics depe nds on t he pr opor tion of th e n um ber of erots in tho s t at o r to t1lllt of s lots i n t he rotor. So me proport ions are u nfavou r a ble bec ause t hey a re eea pnnai ble for a p preci ahla vi br at ions and noise . Fo r three-p hase ,ind uct ion mac hin es, t he uueceepeabl e propor tions aec t he following: Zl - Z2 = 0, 1, 2, 3, 4; ZI - Z2 = p , p ± 1 Zl - ZI = 2p; 2p ± 1; 2p ± 2; 2p ± 3 (3.2) 2p ± 4 ; Zl - Z2 = 3p wh ere

hOI

l .t.

ln~ nit .

.... rb llr. 'Y Spadrum 'ol Fiel d. in ....i' G.p

Mnklng the r ight choic e of the proportion of t he number of slo1.3 in t he s ta to r to th a ~ in t he roto r, pro perl y s hor te n ing the wl mJ ing pit ch , and deciding on slot ope n ing end ske wing , i t becom es posaibla to eeduee t he a m plnudes of s pace hn r moniGll or make one of tbe hllrmon iCII p revale nt over th e others in the spectenm. In this cue the fund .mental . or fjrst . harmon ie .is an u pper s pace harm on ic !liDee its amplitude is th e h igh est over tIle etneee. Space harm onies r ene u ll y re vol vo at • lower veloc it.y t h-.n me first ha rmonic. The anr ul er velocity of a s pace harmonic B .. is a function of voltage u , e nd iu Frequ ency II' and tho pole pitch is a functi on of t he nu mber of slot.s and toet h. Therefore t he velo ci ty of B v is II factor of \' lowe r t ha n that of Lhe firs t harmonic. T bls 8xplai ns why mo tors oporating on upper spac e har monics a re of t he slow-s peed t ype, lind t hua can p er ro r ~ t he functions or m uWpoll' r machin es or gea emotore. Slow-s peed me te rs ex hi bit harmon ies whose freq uencies II ro lower tlUIO t hat of t ho firs t ha rme utc . These are lUbharmont£$ which arc also presen t in Ordinary mllcbio Qa. In muJtipola r machin es, subhermon tcs arise due to the dHfer el"!ce between permean cee u nder the poles and as a resul t of modulatio n of tbe firs t harmon ic. Spate harmonics a lso ar ise ' rOIn t he no nlinea rit y of machine p.umelers. What ex urta t he h ill"he'lt rntluenee on tbe s pecaru m of ha rmonics ca used by t he nonlinearity of pMa me ters is satu ration. t.e. t he nonl inear depe nden ce of mut ual lnduel.an ce on Cllr rent or t i me ain ce cur rent is a funct ion of t im e. All coeffi cien ts enterinit in to the electrolll8Chlnical equations may be nonli J;lear. I n an act ual machine, eeeu reneee vary on acco llnt of current .d isplaceme n t (CUrre nt dop.!Iity "aliiti on) , aud tnd uorancee depe nd on ut ura t ion : in ac me dri,'es, tbe mome n t of ine rtia undergoes cha nges. Nonli nea r va ri ations in the parameters tend t o d evelop certain spectra of har monics in t he air gap . T he effect or no nlineas pa r amet ers on t he machine char acte ristics will he de al t with in Ch. 7. The a ppear an ce of s pace hn rmonfcs is at t. rtbu t ahle t o manufacturing fMl ol'6, to whic h bel ong t he air g ap non unifo rmi t y d uo t.o lK:cent rlcity of th e s t at or witb resp ect t o th e ro to r, rotor conici t )' . n:ll\.I misalignme nt betwe en rot or and stator, and other fac tol1l of t h ~ t y pe tr ea t ed below in the text. Machine asym met ry ton Is responsible for the emergence of barrncnits in t he a ir gap. A field of t he back wa rd (nega ti ve) sequence and a fiel d of t he aero seq uenc e ap pear ill an My m me lric machine. A plllut ing zeee-seq uence field ma y fte th oug ht nf lIS consis tin g of a forward s.nd a backward field. Men ti on should also be mad e of hetercd yn e-Irequency har monies bKause el ectromech anical ECs are nonlinear s ynems: even t ....·o ha rmonies present in an energy con verte r are en ougb to g'iv e rise 10 an infin ite spectr a or harmon ics li t he terodyne freq uenci es.

If th e possibili ty exists for t he flow of currents in th e sta ~o r and rotor, which euable t he s ta~r and rotor fields to be s tat ionary whh respeer to each ot her, IU1 eJect romagn eti c torque res ults from the pa ir of h afTl:looies. In a I ho rt-ci rcuiled (squ l.rre l-c age) roto r ind ucti on mach ine, .11 hnmou.iCI pres ent in the ai r i llP Illay ~ive rise to torques . Upper harmonics sh ow u p mOllt viv id)' when starUng a machi ne. For each harm on ic t here comes 8 point where t he ro tor s peed Dquals th e fiel d s peed. 80 tne roto r may "ge t stu ck" at t his speed u nder th e in flue nce of t he synchronous to rque d ue t o t he s plice harm onic. Thus there is an infi nite arb it rary s pectrum of beemcntes making up th e air-ga p field. Th is spoet ru m ca n he broken down in to sou of hermonlcs accordi ng to thetr origi n. Earlier in the tex t we have IJltlfltio nod t he cllW!if iclIt ion of har mon ics int o t i me ODd sp ace types. III turn , s pace harmcnics ca rt be dtvtded i nt o ty pos aeeoct at ed with the mmf, sal iency , m anufact'ur ing Iac tc ra, non li nearity of param eters , a nd heterody ne freq uencies. OJ the Inf lnlte Stts of harmonics, only rom' al/tet tht duJracurlstlcs of an EC, since a grtGt man!f of the harmonIa halitl infl nitt ly low amplituda , T her efore on ly a ,small Dumber of ha rmo ni cs are g iven een srdeeanon in t ho analysis of energ y conversion p rocesses . To g llin an insight into C(lmplicllted interactions of harm(lnlcs , we first need to CODs t rue~ a m at hematica l model to d escribe energy oon " t'rs ion precesses in volv ine- an in finite s pec t ru m of ha rm oni a.

3.2. The Gen eralized Energy Converter In meg neuc-Ilejd energ y conve rters, t he "" ork ing field is II mag neti c field with its ene rgy concen tr at ed In t be air ga p- the s pace wher e atectrcmechan tcal onorgy conv ers ion t 8ke place. As mennoned earlillr, in t he ai r ga p oC a u elec t ric machln e t horo ex isU an in fini te ha r mc nlc epocrrum qepe ndeut a ll t he s u pply voltage, loa d on th e machine , and ma ch in e d esign. An y processes in a mach ine, of wh at ever charac ter, lead to changes in t h e air-gap fi eld . In ure m ath em at ical urede l, of a ma ch ine , each h arm on ic m ay be sot up b y a pa ir of windings on t he 8t at or and on th e ro to r if cu rrent s of respec uve ampl ttudes an d Irequen cles, sh ifted in ph aso w itt l respect 10 each othe r, flow i n t he wind ln:s. Gi ven an infin ite eet of wind ings , it is possible Ito produ ce a Cield of any s ha pe. In Section 1.4 we h ave deecetbed the gen l'ral iUld, or petm tt l ve. m achine defined as en ldealleed u uset u eat ed m ach ine with. un lCo rm a i""8p suueruee. whi ch carries t wo pa irs of wiu di ulr-' on t he s t lt(lr and rotor. Here we s hall d Clll 1\>.'i th t he gea er al t eed elect ro mec b an ical en orQ'Y ec n\'cr t er- t he m ath ematical m odel of 8 real m achine-wh ich IDll}' hav e ma ny loopl'l (phase win d ings ) on t ho sta t or alld rot or.

Such a pe neralieed co nve rto r eDables, us to wri le eq u aU Oll!l wi t h d ue rer ard lo r t he iield )~D t he a ir g ap and aU cUlT9nkarry in g l oops. Sum ming lop. th~ ~MralUed ~nugg con t~r'~r u an id=U:ed twopole tuXJ'pho..u: ( ketrle machine wUh Ili-n wind ings 0 /1 the slalor and rotor, ,.,spect llMl y, 4rrtlJ1, ed along 1M a "ami '" 41 fho lDn in Fig. $ .2.

au,

u'.....

""'i) ..... ..J) OIl•

i .. """i) -t' W ••

u:.

w;.

Usi ng t be mod el of tho gene rali zed converte r we can desc r ibe s ym me t ric m ul t ipbllSfI m ul tipole r ffiaj;;h in os on t he ass um pt ion t ha t t hey e re t rausform ab le t o a n eq uival en t for m l.o matc h t he t wo-phase two-p ole ma chine. As seen fro m Fig. 3.2, each phase wind ing: nee ne d eslg n aelom eubecelp ts Cl: lind ~ iden tif y t he ax es: a long wh ich wi ndi ngs w Ho; 1, 2 , .. . , m. n s tlUld for t he ordinal n umb or of t he wind in", 011 t he s t a tor e nd r ot or reepec rlv el y: And s u persc rip ts a a nd v den ot e ti le s ta t or an d ro to r win d i ngs . 1'eSpOctively, eu ppl led with vo l tages u . E ach p air of t he wind in gs is fe d fro m a n ind iv id u al s up pl y sou rce or a ll wi nd ings a lTllnl:ed in an y kiiub of ne t we rks draw cu rren t fro m a s ingle source. In the gener alized co nv ert er. mac net ic li nk betwoen t he groups of windings on ,the n me lids m a y DOt e:ziat.. E ach pa ir of windings (co ils) on the s ta to r prod uces a circ ul ar field i n t he a ir ga p. As is k n own, tb e generaliud convert er is an unsa t urat ed ma ch in e , whi ch allows m to use t he princip le of eu pee pcaitton. The field in the a.ir ga p ca n 'be .set u p by a ppl ying to t he win di oirS t ho voltag es of d iff ere nt a mp li t u deS and freq uencies , sh ifted i n

en. 1.

G...... . llted ....... W ind ing Converter

ph ase wr th respec t to eaen other. From the vie wpoin t of m etheDlfltieal t heory, eloctri c m achines d iffer from one anotbo r b)" t h.e {or m. of t he fiel d in th e a ir lap, n um ber of windings, an d the pan met ers of wi n dings . . Tho gt!ner ali f.ed co nver t er is a useful tool for descri bing allY elec t ric mecnrne. Fo r ex ample, I single-phase s ioQ:le. wind iog motor wi th â&#x20AC;˘ pUISllling field in th e aIr gnp ca n be r ep rQ!len led by 8 malhema ll ul m odel com pr is ing t wo pa irs of windings on th e sta te r and t wo-

Fig. 3.3. The lM del of a swgleÂˇph u o .aoto.

w:",

Ie:

pa irs on t he ro t or (F ill'. 3.3) . :W i nd ings and ~ build up n forwa rd (p os!Uve--sequeoee) fia l l'l l (lt~.. = U'" si n (,)1, ~ = U.,. c os wt) . Wi ndi ngs W;Ol a nd w:~ are t fed wilh voltages u~ .. = U", cos wt and u:JI - U... sin OJf which 8 8 t up a backwa rd (negat ive-sequence) f ield . It the rotor is of t he kquirre ), c8go t ype , u~ u~ D' an d u; ! are eq u al to zero . By chang ing t he vo hag(\(lJ across th e wiod lnp which prod uce Lhel orwu d a nd ba ckwe ed fields, i t is pceatble to e o oV H fro m t he pu15ating field eo t he elliptic. fiel d an d t hen t o t he circular fie ld if lIle blckward fi eld in t h e ai r gap does no t ex ist. II, a part from the forward and t he ba ck ward fiel d , upper h armon ics are present in the ga p, It is necess ary to a dd I req uisite nu m be r of p ai rs o f wi nd ings to t he mod el and l!illl u p t he fi:eJd b y ap plyi ng to the !il at or wi n dings t h e vo ltages of co rres ponding a m plitudes an d Ireq uences , whi ch sh ow a defi n ite ph ase sequence an d p h age s hift.

It:

..,

u;...

3.3. The Equ. llo ,," of the Ge ne ..lb ed EC

i\1 0fI ~ cteewre mac-hi nes ha ve sev eral wind ings. If eddy current loops al'6 ta ken into eecount, :III m ~ i n es may be t bought of IllJo multlwind ing mecb lnes. As menlioned earlier, the genera Ut.edconverter serves as II powerful tool for t he llIIol )"ais of act ual mach ines . wH h many win dings. Equ . t ion5 describi ng t he bohavior of th e m.jorl ty of elect ric mac.bl ncs can be s et up using t he e;quations for t he genenlbed' encl1'Y converter. For t hi! we need to. expand t he mml of th e field (whose shape i n t he gllp Ls known) in t.9 a harm onic seri es, cons t ruct t he model from 1I few pairs of wind illils and ap pl y th e vol tages or corresponding amplit udes lind fre qu enc ~es to th ese windings . Alt houg:h tho fielll pattern ill a rot a Hng m ae~ n o is pn ctl cally impossibleto defi ne, the mauiemauce t mod el efl e u oncrgy converter enebleeus to 50[V6 mllny probl ema to a sufficie nl accuracy by s.pecifyiog \' Olt flg C1l on t he input termi nals of the conver tor. Th, onaly$is of working processes In electric machines relit:! on two" sttll-tm, nts: (a) all Slfl U C and dynamic Cha racter/d iu o/a machIne are' governed bylh.e proc, sses occurring in the air gap : Ib) an eltaru: modl l n~ is represellttd as Q systtm oj li near dectr ie etrrulu 1l101'ing with rnp«t 10 on, onothf'r .

3.3. The Equations 01 the Generalized Energy Conve rter The v Oltllg1! eqn a uo ns for th e gei:l~r a.l iud eneq: y een veeter ant written ill th e form of a eomple:a: ma trj x sim ilar to t he K roo mlll rb: for t he prim il ive machin e of Fig. 1J 11:

""-

A:" A:." 0

R••

B• • D. 0 0 A&'

A ~' A ~

A,'

III,

(3.3 )

Each elemen l in tb e eb e ve mat rix is .. subm U rb. Rere and ere mnt rtx columns :

-I.

u:" = -I.

-:,.

u~~

1.1;. =

-I. -:,.

U~1l

u. =

-I,

. ;,

u:.. u~ . ",.

u~ 1l

u; =

u ;1l

":..

(3.' ),

Ch. ), Gonerllljzed lIl_n Winding Co nverter

Also, I:",

I~,

',. il

are matrix colu mns;

i~ =

Ii;. =

il,e

'" Ii .... ih

ih i& = .,

':"

~ ;a

'l"

'"'

Eqs. (3.4), It:... u:.., . . ." u:..c.: u~ .., u;".

. _, ur. ~ ;

u: D. u: e, . . "

u:.,. ~

(3.5)

'.,

u~,,; u~ e- u~-"

...

are t he s t at or and rotor volta ges alc ug thea and ti axes. In Eqs. (3.5), t:", a; I ~ ... . . _, i~ .. ; lill ' t; D' . . ., l~~ ; 1;/1, 1h . . _, i:"s are Ute currents .a long~lthe a. and ~ axes in the stator and rotor. Volta ge equa tions (3.3) may be written i n a mor e gBllCfal form

Iii] =

1z1

i;.., .. " "..

t;... . .

II)

(3.6)

The Impedance ma trix [zj incl udes 12 sub me t rtces. Four impe<d ance matn cee He along (he d iagonal:

-t, + (d ld t ) Lt", (d/dt) .Mh:. (dldt) }\'11 """ A:" = (dldt) M !I .. r1",+ (d /dt ) L\ .. . . • (d :dt ) ,'1-1'_

(3.i )

(Illdt ) M:"la. (d /ql) M:"2a. . .. r~ + (dld t ) L~ . Denote t he resist ances or stator and ro tor windings on th e Ct and ~ -3;1:6.S as ,;.., r;"..; r,<Z , .. " ~..: ~~, r:~ , r.. i\; r;~ , r. ~ , ~ ~. Next, denote the tota l Indu ct ances or stetor end rotor wlud inga 0 11 t he a. and P axes as Ll .., L~ <z, , L:"..; L;"" L;"" "', L~",; L:~'i L: ~, . . ., L~~~; L ;~ , L;~, , L~~. Finally, design ate t he mutual Indu ct ances between stator wind ings -o n t he Ct &'l is as M:,<Z , ., ~f:"I <Z ' Thus ide nt ifies th e mu tual i nduct ance betw een t he first and the seco nd s ta t or wi ndi ng on the -« axi s . In a machi ne with th e equai number oI tu rn s on the st ator and rotor t he mut ual ind uctances for all windings alo ng the sa me exls .are identi cal . The t ot al induc t ance L is defin ed as a sum of t be m utual tnduc tenec a nd t he leak age induct ance ot the giv en wind ing. is s tmtlae t o o4 ~ and comes ou t afte r the subst iT he sub mat ri x t ut ion of p for a. Sub matrlces A ;' and Ah ill (3.3) co me Irom A ~ an d A ~ by su bs tit ut ing , !or e. Subeuutces A:." , o4k', Ar, and A~' are indicative of li nkage bet ween st a tor an d roto r windi ngs. T he eubmatctx A:.'" h as the roen (dldt ) Mr,<Z (d /dt) M i'i« . . . (dldt) Ml';.,;. At[ = (d ldt) :1,g'1... (d id t) J1fl'i:~ . . (dl dt ) MftIGt. (3 .8)

r... ,

r.... . .., .. "

. . .,

ilf:.<z

(d ldt) M

::;I <J.

(dldt ) M::'2<J.

•• •

(d ldt) M1{,n"

3.3. The Equ<1tlon . of l"' IGenerll;zed EC

Here M;~ ... i5 t he mu tu al in du ct a nce between the Hrs t windi ngs on the stator and roto r al on g t he ex. ax is; JIlf-::: n.. is t he m utu al ind ucta nce bet ween t he mth s t a to r win ding and , t h e nt b ro t or wi nd ing al on g the ex. ax is . The s u b m a t ri x foll ows fr om t he ma trtx (3.8) afte r subs tituting ~ for ct . ~ T ho s u h ma t.rtx A~ results fr om A :[ hy in terch a ngi ng th e positions of superscripts sand r an d a lso m. end 1l j A ~' resu lts fro m A ;;~ .by chang ing ct for ~. The i nductance M:;..... is ge nerajly not equal to M~',.. to , t houg h i n runn y cas es these tn d uct ancea ma y be t aken equ al to each other. Subwa t rices D ... an d ~ t. an d als o B .. /I an d B lIa are rel ate d t o t he rotaUng em f. The llub mp t rices D a an d D /I co r res po nd ing to t he tct el tn du cu ve reactances o f 'r ot or wtud tnge alo og t he ex. and ~ axes have t h e form

L~ .. m,- M~ Z<llro, - !I1~I "ro. - L~« ro;

- M'i ~«Q). - ftf2n«@,

(3.9)

(3. 10 )

I n (3 .9) a nd (3.10 ), 1lf~2 « an d M~ 2 h are mutu al ind uctances b etwe e n t he firs t and t he second r ot or w;in ding a lon g t he ex. and f\ axes respect ivel y. ' The s u b matrices B« ~ a nd 8 11 « differ ill sign an d co r r es pond t o mu t ual inductances between t he stator and rot or windin gs:

B,.,, =

- ftf1lo;,W. -AI 12aOOr - Jd:l"w, - ;11 22 ct";l.

- fll ,,,,,,,Cll. - 111 2"",, 00.

(3. 11)

-lIf n ",,,,()).

where llfu o;, ", /lf l m .. are m utuelrinductanccs between t he ril'S t r otor w ind ing al ong t he f! axis a nd t he fi rst stato r win d i ng a long t he Ct exia: ftl n1..., . . , ftI"ma are mutu al ind uctances b et ween t he 'Ith r otor wi udi ng and t he filth s t a ter win di ng . The s u b ma t r ix B o; ll 1el l,l tes t he roto r Windings a lon g tllo a axte t o tho sta t or windings a long the ~ a xi s. T ile s igns in t he matrix are po sitive. Tho submetrtx B Il... fo ll o ws from (3.11) hy in t l'reh aAgi ng t he pos itio n of ex. and f!. T!I ~ electromagnetic torque In the wenerall ud energy co,werter 18 defined as m e products of all currents !l owing ill the loops of the machine . G-DI t7S

Ch. 3. G • .,.",.ui. d In.n Wi nd ing Co"".rt. r

~taT~: :~~ l\:~~~:, °rres;::f,~el~.r aio:t~hew~ t~n:~ :;~di:r~ ':~ t~~:

ill t he f or m At . = h l w, ld t

+.lL1rlC<.H",I ~ II - ~l1'{11l11 11H <I

+ M ~'2a.l"i", i; Il - M r2p il/li~+ . .. + Mi'n", i; ,,"j~ll- M'I';, llihi~ ... + l1:f\'I"ii.,J ~ Il - M!{I ~~llti. + 11l22C1i2"J Sll - M¥"ZlliS/liS", (3.12 )

+ ... + M~';.",t5"i~ll- MYnllttlll:;" " 's I'"II-:r ' u" 1-, ·r + . . . + .''f ...""'1..... ","11 """ " "

The express io n lor t he electromagnetic tor que of II s ymmet ri c m achi ne, where th e mutua l inBucta llces along the a. and ~ axes are the same for all windings arid equa l to M , is derived from (3.12) an d is written as Me =M ((l1... i1·. i!(l+ .. . + t;',,}( th+ iSll+ l 311 + . . . + i ~ll) - ( t h + i.~ ll -H; ll+ ' " +i :'Il) ( I 'i,, +f~,, + 13"' + . .. + £:;'",) ] (3. 13) F rom (3.13) it foll ows t hat th e electromagnetjc t or q ue ca n be defi ncd b y t wo pr od uc ts of l~IO fou r no nstnusoldal currents . Th e term s in (3. 12) and (3. 13) can be d iv id ed i n to two gro ups. T o t he fi rs t grou p bel on g tho t er ms' essocf eted wi\.h t he buildup of starti ng , brak ing , or geuerat.in g t orqu es , and t o t he second bel on g th e terms ha v ing t o d o 'wi t h pulsating t or qu es . T he torques in volv ed with the fi rs t g ro u p are se t up by t he s t ato r and rotor fields tha t are stn t.lcn ar-y rale t.ive to ea ch other. Pul~ atlng t or qu es are d ue to th e f ie lds mov i ng wi th res pect to each other. I The ex p ress ions for t he form er torques t a ke t he for m

+ +

M1'hi i ~1 1" - A/ i'l M ::;"""i~l:;'~ -

"iI"I~ <t

M:{;",,,t:..,.i:'na

(3 14)

P uls ating torques a re g iver- b y Ml~",i l~151l,

Mr2r.ibik (3 15)

M~" ... i:..r.! ~ ,.- M~,.,. i:..,.i ;';L

The l atter t or qu es ca use vib ra tions and d o not afford t he mo en torq ue component. Expressions (3.3) t h ro ug h ! (3.12) are the most gene ra l el ectromec hanic al eq uations for t he case wh ere 8 s tator or r ot or is statio na ry . where hotb th e sta tor and r otor ore in motion , expression (3.13) co ntai ns lo ur more sUb m at,r ices to all ow for stator rota tio n. Proceed i ng fro m expressions (3.3l th roug h (3.12 ), i t be comes poasibfe to dc etve oquatlons 'for roul tiw ind ing ma chines . m achines

4.1. Tr. n,itio n from Simp le fo Mh, e Compte. Eq uat ion.

suppl ied wit.h nonelnusclde l aSl' m m~tric vcl reg ce. nonslnusotdal IDllgnet ic- fiel d mac hines. and moot o~ t he ether electr ic mac hines . T he equations here are w ritten in t he i;t - ~ coc rdtn ate sys tem , th ough they ca n also be set u p in ot her eoordtna tes. E qua t ions (3.3) t hrough (3. 12) con tedn integral pa r ame te rs desp tta Ole fact tha t a wind ing consis ts of t urns. I n t he ana lysis of surges. the trea t men t of t he processes in II winding should be give n with regar d to voltage distr ibution among t he tur ns. Note also t hat "6lt.!lges distr ibu t e t hemselves nonuuiformly among t he t ur ns a t the slo t bottom , near the s lot wedge, a t t he t op an d in t he mid dle of the wind ing , and also at the end end sl ot tu r n port tons. It is t hu s possib le t o form ulate a marhemaucel model assuming t hat t he energy converter has distri buted par ~meters . The methernancel modo! beco mes 1 more complex iJ a mac hine has two, th ree, or II degrees of freedom. This is a machin e whe re t he st at or end r ot or are hoth rotating members , or t be ro to r lias the sha pe of a sphere with t he s pheeteal ~t at or being either stationary or in mot ion. In t ile elec t rcmecha ufcal eq ua t ions for elect ric-field eDergy conver ters, c fl pac i t a nces s1J bs~ i tu to for induc tan ces. T he conce pt of t he gener al ized converter conti nues to develop in keeping with t he ad va ncem ent of II,e theory of eleet romech enlcal energy conversion. G. Kron was th e firs t to introdu ce t he noti on 01 a pri mitive machi ne in the 1930s. TI(e r;enara lized el ecreomech en tcal energ y converter represents t he 'model for descrl hin g energy conversion processes in magnetic-fi el d machines having one degre e oJ freedo m, an infini t e s pect ru m of har monics , and an y n umber of loops on th e st ator an d rotor. T he notion of the ge neralized converter can be ex te nded t o ('0\'1)1' mach ines wjth any number of degrees of freedom, electr ic-fiel d con ve rte rs . lind electrom agnetic Held converters.

ChClpter

Typical Equat ions of Electric M9chines 4.1. Tran sition from Si,mple to More Complex Equ~tlons: In II par t icul ar case, the mod el 01 t he gene r aliz ed energ y conver ter enables us t o deri ve el ect eomeehan teal energy conversion equ ations for maeh tnes wi t.h II few harmo nics in th e nil' gap. mul tlw ind ing

Ch. •. Tl plu l Eq ",. t10 M 01 Ele drlc M.chi .....

maehln es, onorg ~' ee nveners s upplied fro m nonstn useld al yol t age sou rces, and for ot her actual mectnnes sh owln&, a co mpli ca ted in~er· acti on bet"..een harmonics. Orveu the equ at ion for the mo~ t ca mpIn ,"-II win dine machin e , ..... e can gra du all y ~ i mpli f~' the ma tllellul" t ica l d!l5Gription in t he eoU1S8 of anal ysis to transform t ile model t o the s implest one s pecific to a circul ae field in the air gop, the equ auens of which ha'"e been g h' en ea rlier In th e te xt . Bu~ • useful a pproach ts to begin with s imple eq uati ons and t hen gnd uaUy tra nsfor m t hem t o II rnol'tl ee mplae form. As is knowe , t he sya te m of equa tio ns comp rising four voltage equa ucne (1.34) and t he nssdcl lled equa t ion of moti on (1.35) desertbes t he processes of energy epo vers iol1 where t hl! fiel d in t he air gop Is olecula r in shape. T he expreeetcn for t ho elec tromag neti c t orque inclu des rwo prod uc ts of ~u rre n ts . A circ ula r fiel d ill t he air gap con be Silt up by a few wl nd iogs o n t he stat or and rot or. Firllt,cnosi de r a machine model havi ng 0 pair of windings on the stlto r, which produ ces a circular field in the air ga p and t wo paif9 of wlndlnga on t he rot or (Fig. 4.1). Th e voltage equations de rived fr om lbe equations (or lU I m- Il win ding machi ne heve tbo for m

R1.+

+ :t L;",

.••

71 M n

, Iii 111

•,. ur~

2•

7i M u

1t"'I!

II.

R~", +

:. Mh

L~ ""r

Mhlll.

M 1Ic..•

l~~

R ~ ", +

MS!w.

L;~w .

~/2I(1),

15.

R'ill +

~ M'i2

dt Mil

m~ +

7t M ! .

i;'

R1,+

+ :, L~'3 :, M;\

+ :' - lI1 u Ul .

-L~",Ill,

L2'"

- Af h w,

+ d.~ L~~

uh "I

M 2I w,

-M211ll, - L6.CIl.

71 M 21

, Iii

, ,

+ :, L u Mil

7t M u

i;~

, '" + di' L111 (4. 1

~. 1.

T. llnsil io n ' rom Simp l.. 10 M,o r.. Co mp t•• Eq...lll io n>

T he s ystem (~ . t ) co m pr ises six volt1l'Ke eqU!lot iOIlS, t wo lor t he t wo Slator wi n d illgs an d fou r for the lou r ro t or w i.odirogs . T h e vo l t age in t he roto r js zero Jr t he ro tor Is o f t he s q u irre l-cage type. Wi ll i t he ~Il erg y p u m ped to a ll t he t h ree pa irs of wi nd ing!! , t he e ircu lar fi e ld

•

II:,

·:·ti ."~' .

ll ~ ,

w'••

.,:-R '.-i' w'... w'

.'•

Fig. 4 ,1. 'T h. ch 'C'l,I lu -lIeld DJllchln c model ""ith. two pa in o r p halle win o tlint;lI. on th o rotor

u.:il ,e.....p W,•'

f' l i!" 4,2, Th e ma chino model wi ~h two pain o r ph u e wlndl ll.(t!I on 1he 11.1:>101 alld 01111. pair of ph alle wi nd ings on 1be rot or

in the air ga p be ing es Lab lis hed , t be vol tages across the Sl a to r and r ot or wi n d ings mus t p roduce m agnetic fi el ds wh ic h are sla tiona ry wit h res pect t o each oLber. T he t orq u e eq ua tio n l or this m od el is

AI. _ .111\ (i 1"lh - 11..t. , ) + M n

(li...l h

- J1... ii,>

(-'. 2 )

Eq u8U on (4. 2) in cl udes t wo com pc ne o ts Jl1 . ,.,.. k In

+ Al it

(-'.28)

The fi rs t co m pon en t hi due to t he interac t io n bet we en t-be eurren ts in the fi rs t wi ndings Oil t he stator an d rot o r, an d t he secon d is d ue t o t b e ln t eraetion be twee n t he cu r ren ts i n t he fi rs t s t a t o r wind ings a nd th e s ec ond rotor w hl ~ i np . A circu lar fi d d In the air gap of a machine hoVin g two pa in 0/ win· d Ing, on tlu stator and one pa ir of loind ing, on the rotor (F ig. 4.2) il def ined bV .ff~ w ltage equation. ; the d ectromtlgnetlc torque produced.

ee M re COfU u l f

of t wo

rompont:n~

M~ czM u (f ~"i : ,l - it"f~ ,1)

+Mil (i ~'1lb - ll"ti, )

(4.3) (4.3.) T he rnlO l!temalica l mode! desc ribi ng the processes of energy convenion in t his machi ne Includes fOllf voltage eq ua t ic ns for t he s ta t or. where t he ,o~.tion ol em f i.s equa l tn t ero . a nd two volta ge eq ual ions for t he roto r . For tI~ lM chi ne model of Fig, 3.9 M olng four pairs of windingl 0 11 ' he I l otoT Il nd rot Qf' ( th e JLeld i f!. the a tr gap being cl rru l ar) the syJLemo! ~lla ttOIl ' coru!l ls 0/ t ight ~YJllQ ge equat Lons. 1'11, equation lor Ihe torM . = Mil

+ Mil

que illeludes fou r «JmpOfUlI/$:

M . = M il ( l 'i a i1 ~ ~ i1 ai ~a) + M n ( j ~.. ib -l~" i h)

- /1".1 2/1 ) + jtf II Mu + Mil + :I f ..

+ ilf I I (I;".£h

(I ~" lb - i!a.i ~il)

(',.4)

M . = Mil + (4.5 ) A (urther increase in t he dumber of wi ntl ill~ does DOt ... iola te t he regulari ties a dop ted in tho fur lUlila ti o n of eq uet tons, The next !le p foll owi ng t he a nalys is dea lt with t he ci rcu la r fif'l d is t he tre almen t I)f t he processes assoc ia ted \\'i th an elli ptic field wh ich .co ns l:5 ts of a forward a nd II backward fie ld rot a t ing /It t lte ! 3.me velocity but i n oppo.1ile d irec ti ons and d iffer ing i n amplitude. A mach in e d isp la y ing e n elli pt ic field i n tho a ir Ir-lP ca n ha ve :!Cveeal Wind ings. The sy! tc m,of equations t he n com pr L-es a cerreepo ndln:: nu m ber of vo hal:'!!- equa ti ol\5. and t he to rq ue eq a a t lon fnchul l'S II correspo nd ing num ber of to rque com pone nts . T he mode l of a mac h ine with one windin g on t he stl1\.()r a nd o ne o n tho ro to r, t he field bei ng aS~IlJle<1 e ll ipti c. la an e igh t-win d in g lilach ine model descr ibed by Eqs , (3. 3). Th e processes of e nllrgy ccuvarsic n a! SOOinled wi th nn e lli pt ic H~ld in the ga p a rc give n de t ailed t rea tmen t la ter in the text . Here 'we note on l)· t hat t he s ys tem of equatio ns consis ts of e ight volta ge o q ue uo ne. one tor each Wi nding, a nd t he t orq ue equat ion includd four com ponen ts:

+ Mit -

lil u (4.6) T he ! igns in Eq . (4.G) depend on t he d irectio n of rolil tiun of Ihe harmonics of interest . If i n t'h.e a nal}"sis o r a mach ine with e n e ll fpti t field t here is a need t o «lrisidu ad d it iona l wlndil1g! , t. e. t he eddy eureent loops in th e st a lor iJiJd rotor, the eq ua tio ns beec me more cu mberso me. The number ot 'volt a ge equations co rrespo nds t o t be num ber of windings i n t he model , end th l' to rqu e equa ti on co nta i l'lS all the ter ms for pa il'"Wise t nteracun n. Th o nex t group of eq ua t ions is the s ysten l of eq ulll t io DS for Ihree fi elds hi t he ai r ga p. The ma. t hemalica l mod el here lnvelvea 12 ve t;\1. "'"

jlJ lI

Af u

~ . 2.

En6<gy Con ..e " jo n Invb lvlng an Ellip tic Fie ld

tago cqua ttnns . a nd th e t orque oq unt;ion con tains nine com po uenrs : M . """ M, J/ 1 M ~ ,..j-. M u + .1111 M 13 .'1111 + ;M 3 1 + M ~ ! (4.7) If t he fie ld in t he nir gnp has three ha rmo nics an d the sta to r a nd rotor cllrry severa l wind i ngs, t ile nu mbe r or equations t u be Inrmulated grow s. In the next. sto ps of t he a na lys is , the e quat ions a re set up for four . fi ve. lind 11 harmo nics. ' Th us the ty pi ca l eque ttous Inc tud o, t he equ at ions for c irc ula r an d eHipt ic fields wi t h three, four, an d more harm onics. T he eq uatlcns become more com plex wilh Itn increas ing nu mbe r of wind ings on t he s ector Mid rot or . Not e th fl. t th o solut ion of t he abov e e lect romecha nica l equat io ns for most electric machines req uires writi ng correspond ing programs wh ich w ill enter i nto t he librnry of sta nda rd progrfl. ms .

4.2 . Energy Conver's ion Invo lving an Elliptic 'Fie ld As mentione d above . t he e lli pt ic 'fIeld consis ts of a forw ard a nd II backward Hold . An exa mpl e of the motor exhib it. ing a n ell ipt ic field in the a ir gap is a singlephase motor wi th one windi ng on the s ta tor (F ig. 4.3). T he rue th ema ticn l mode l of a sins le-p hase motor is the modol of Fig. 3.3. ba Ying ~wo pa irs of windings on th e sta t or a nd t wo pai rs 0 11 t he rotor . O ne pa ir of wi ndings, w:,. and w:~ . is fed wit h poslttvosequ e nce volt ages to produce a forwar d field nnd the other pa ir , W:" a nd w; ~ , with negative-seq uence vol tag es to se t u p a backward field . I n a s ing le-p hase moto r t he pos it ive- And negc uvesequ e nce voltages a re th e sa me and tho amplit udes 01 t he forward , FIll" 4.3. A s lngle_pha. e mOIt>r a nd backward field s an> eq ual to ea ch other. A u el lt pt tc fie ld is bu il t up in sy n:llr-et l'ie lind asymme tric machtncs when sup plied fro m asy mme tr ic an d sy m metric voftage sources res-

~c liv ely .

For a n e lect r ic machi ne modeled in ter ms of t he macht nc of Fig. 3.3 t he sy ste m of equatto ns (4 18) holds . T h is set of eq uatio ns ;s deriv a ble from t he eq ua tions for t he ge uoeal tzod conve rter on coud t-

Ch. 4. Typkal Equal1 on, 01 Electric Machine,

uo n t hat t ha machi ne under: st ud y is of the unsa t urat ed type lind t he tnter ret a ucn between t he negn tive- e nd posi t ive-sequ ence vert-

ebles docs not extet :

-t, + pL1,.

lt~ ÂŤ

uL,

o o o o

pM 11<. -

o .M 1l<' W, o o o

u1~ u~ll

o o Ll lloor

r ~", + pL!"

pM u a

' 2" + pL'2c.

- Ll" w,

- L~"oo ,

o o

o LzlIlJ), o r~ 1l + prJ ~ 1I o ' 211 + pL211 o pM u" o pM n ...

pM a "

,0 -Mu " OJ r

o o

ll Q.

o r~" + p L~ " o

,Af I ISW,

o o o Âˇ ft:( 2 Z ~(oJ ,

p'M 1111

it" i ~" f~ "

pA/ 2211

rlll + pL1J1

rb + pL!1I 1

~~" ." i 211

(4 .8)

ti ll i!ll

To es tablish the equ atio n for t he torq ue, we s hould conside r 11 model comprisi ng t wo sta to rs fi nd two rotors cou pled toge ther

H I;:' 4.4. " two-eta toe two-ro tor IDachine model

Fig. 4.5. The ee mmon-ro tc r machine model Hluetrative of the illt erllcti oD between th e pcsntve- a nd uegett vesequence volt ages or eueeca te

Wig. 4.4 ) and a lso a model e9mposed of two sta tors an d a common rotor (Fi g . 4 .5).

4.2. f ne .!JY Co nveuio n

l n vo l ~ l ng

f m pl l~

Fie ld

As seen Irem t he al mp l fjfod tw o-s ta to r t wo-rot or model , t he pos tuv e- an d negn t Ive-se qu en ce st im u li ere Hh ly to g ive ri se to tw o Ietds in t ile gnp , t here h)' selling i tie, ro to rs in mot ion in op posi t e rltrectt o ne; t he in teract ion bet wee n <JIO pos itive-seq uen ce curren ts in th e st at or a n d th e nega rive-seq ue nce cur re nts i n t ho ro t or i~ nonexiste nt: I n th is ca se the to rque is M . = M (ih il ", - t1"ih) - :M (thiS" - 1!",i2') (4.9). (4 .1O} M . = M IL ...... M2 ~ I n th e commo n-ro to r mode l of F ig: 4 .5 t here exists coupling hetween th o p osi tive p hase sequence lind, t ho nega ti ve pha se seq ue nce; and the t or qu e eq ua t ton t hus has t ho for m M . = iIl l(i1 ~ I 'i u - i1",j;: ~) ....... (i hiia.- il",i26)

+ ( t 1~ 1 k. -

t1" i h) ""f'" (l hj~n - iin t'is)]

= M ll - l\{n+M,~ ""' M 2 '

(4 ,11}

The interactio n between, the positi ve- and T/egative-uquenct curre'l/.$ gives ru e 10 pulsati ng torqlUs which do nol prOVide the mean tcrqu.e component in, the steady-state operatio n -and al/ect the course 01 a tra ~ sisnt !Jecau.se tn the transient region tht currents art decaying and the mean value 01 current over a cucle varies . Whe n ~ = t and t he Itold pulsa tes , t he re!lUltant t orque is zero, so t here is no s La rtin g to r que on the mo to r sha ft (F ig. 4 .6(1). In or der t o sta rt a stugle-p ha se motor , it is necessar y to set the ro tor in mot ton by a n au x ili ar y mo tor o r reduce th e bac kward fi ol d; 1I111 motor th en acqui res II to r que wh ich keeps ii \ go ing in the d ir ect.ion of t hea ppl ied torq ue . T he most popu lar met h od of sWlr ti ng II s ingle- phasemot or is to red uce t he am pli tu de of t ho backward field , Le. t o t ra nsfor m II pu lsati ng fie ld int o a nellipticouc (Fi g . 4. Gb). t.-Ia ny ap proa ches are np pfi ca bl u for produ cin g lin ell iptic field in the a ir ga p of 1\ m ot or su p plfed Ir om a s i uglc- phese cireui t. . O ne of t ile wid es pread methods u t ili zes a n aux il iary st art ing windi ng dt epl ace d 00° i n. space fr om t he ma in w ind ing, th e excitatioD currents i ll bot h wi ndings boing brought into the d e.'l i rc~ ti me- ph ase rel a t io n (Fig, ~ . 7 ) . An ell iptic field in II ey mme u-lc mo tor is set up hy im pn>ss ing asy mme tri c vcneges ;ItrOS!! t he wind ings, \ Vith t he im press ed voltages be ing n t asy mme try, t he eq uat ions ex presse d in terms of fl ux. li nkages t a ke o n the for m d'V:"./dl =R' t:,,_ d1V ~ I\!dt = u:~ R 'i: 1I d'l' ;n1dt = R'i;" - P Wr'¥~ ... . dlJ1~ !I'ld t = R ' i~ ~ + p w.'I'; 8 (4. 12) d'V:,,Jdt = ±u:... - R ' i: n , dlf: ~ ldt =< ±1l:/1 - R't; B d'Y:"jdt = _R' f~ ~ + pm r'l' : /I' dW;/I:'dl = - R'i; /\ - p<o,'V;...

u: u-

Ch, 4.

Trek.1

~qil a tlo ...

of Elec lr lc M achi","

Resc tv tng tho flux li nkngl!.'j a nd curre nts i nto sym met r- ic ccmpououts , we can obta in t he ai-fJ equations expressed in terms of flux .ilinkages lind c urre nts. It is a dvtsa ble to re present flu x linkages as a M

M.o

•

M,'.

,/00 0 o~ o

0 0

~oo oy FIg_ 4,7. A ca paeitor-s tarl stngte-pheee

induction motor

Fig, 4.6. Th e me chanical characte r ia·

ti c oJ a alngle-phll5e me ter 'M

M . • _Io. qlle due ", pc.ol llv_ q uon • • Udd: I M • • _ 10"1"" <Iue t o n • • • Uy_ qu un<:ef lol d ; M j _ tOIU ( .....ull anl) lor q"'l

aum o r th e prod ucts o r curre nts an d Indu ctances , ex presse d in rel a't.ivo u nits . T ho procedure odsi mul;,'l tio n on a n Ilnatog computer will

uhen require fewer

I\ d de r.~ b..jtQ USC

the flu x lt nkegea

(i re

the s ums of

cu rrc uta w it h defi n ite coe fffcrent.s:

L'ii", + .~fj~" , 'Ir :~ = L'i: ~ L' I;" + Jilt;" , 'Y: ~ = Ui: ~ '¥~a. = U li" + ftli:" , 'I'~ ~ = Ui: ~

+ Mi ~ ~

'1': " = Ui~ ..

+ !l-/l:~

= 'II:" = '¥~ ..

+ "Ii:", 'Y;s

= UI ~ ~

+ M I;a + Mi: ~

(' .13)

SolVing t he algebra ic equa tlc ns, wh ich enter int o t he ge neral syst em or eq uenc ns (4. 12), for curre nts iT" . i~", r", an d i~ ~ expressed tn ter ms of flu x li ok ages '1'(.. , qr; 6' \I'~ .. , 'V~ lJ (oi.13) , we obta i n uie :$'l t of eq ua t io ns

+ (bUll), o/ ~'" -

I)" (8 + C) + ( h!M~"' ~ 8 - i tB (ll + C) _

(pw , faU ) 'VIB (pw. 'aL' ) \{rl ~ dl '2" ld t = OIL!", + {hIM ~ '1''2 « - i'2" (8 + C) + (rx».lu f.J' ) 'l'2B(4 ,1<'I ) 11l'2 I1'dt = ± auh (b' Af) '¥ ~Jl - i'2l' (8 + C) - (pw. 'oU) '¥'2" di t,,) dt = " au1"

dl 't ~/dt = l'IU~~

· .3. elltplic_fie ld St... d v~Slete Con di lion$

d lfl1 aldt = - R' tt" - pw. '¥ ~11 d\l'~a ldt = - W li" + peo , 'i' ~ 11 dlJ''i ll/dt = - Wli~ + pw. 'Y~.. (4.14 cont.) d'¥21l/dt= -R'ii/l- pw.'Pi e ."-'. = \f'~ ~i ~a - 1f"'i... i'i/l + lJ'511IS.. - qr~",j5~ + 'I' 'i /li ~ ", - 'I''i "lfi~ + 'Villi'i",- '¥5ai~1l (4.15) B = R ' /(crL' ); where II "'" M /(oL'U ): b = n'IIf/(aU U j: MR' /((JUJ1). Substi t uti ng t he expreaeious for curr ents in to the Ilux Ii nllllge eq uatio ns, we ca n es ta blis h t he s yste m of differentia'! eq ua tions ex pressed i n te rm s of flu x li nka ges . For Eqe. (40 .12) t hrough (4.1 :3), Q s i mul a t.Inn model is SCI u p to solv e t he equat ions on an a na log COn\pule r, or a program is wr itten to bo run o n a digi ta l com puter. As ill t ht! an alys ts inv olving a circular ficfd , t he objectives her e a re 10 Investigate thu t rnns leuts , t he effect of para meters , and seek t he wa ys for c pt.huiaetiun . T he ad justment of a eimulat.tnu mode! is do no 3::1 follow::l. T he steps e re fir-st t aken to adj ust 011('· part of ti re model for the Iorwa rd field equations and th e n t he ot her par t for t he back ward fie ld equations to enable t he ent ire ana log to be set up o n a computer , t he eteotl()magnetic torque an d Angula r ve\oeil}" being nssumed equa l to eero. The com putation procedure the n follow s to solve t he behav ior of a n ind uction ma chi ne in var ious r\todes of opera ti ou. S ince t he eq uations For an (lUi pt iC' tield a re more difEicull to SO lV0 th an t hose for II ci rcu lar fie ld , t he use of oc mput ere to deal wit h t he former eq uo t. Ions is e good a pproac h 'to obta i ning most a p propria te res ult s . >

4.3. Elliptic-Fi eld Stead y-State Conditions The s tea dy-s ta te equeun na fo r a symme t ric ma chi ne e xhl bttl ng bot h II forward a nd a back ward Hold tn the nlr ga p ca n be dcrtve.t Irom (4.8) a nd (4 .11) by su bs t it ut ing- j o> for p. Ap pl ylng (1.34) for the stead y s re te yields the set of eq uattons: u~

ciA - U~

- ci.

" + jx'

, ' + j%'

j:r...

-vz...

ix ... 0

I '•

;x",

"m ,' + jr ,,' I' m .,...",r ,"+ jr

(4 .lU)

Here v = J1pn/30l,o) . li nd th e vo lt a ~e~ Oil t he sta tor or ro tor pha se w tndtuge diffe r in ffill gn itu de.

Ch. 4. Typica l Equa!io n. 01 EI"'c1ric: Mochi nu

UlSi rlg tile me t hod of sym met ric co mponent s g ives

U~ =

Up'+ b~l, U~ = - IU~ + jiJ~

(4, 17)

whe re iJ~ and i;~ are t he pO-"1i ~ive- and negative-seq ueuce (forward nnd hack ward) st a tor vol t ages resp cc uvat y. T he voltages U~ and -ji.J~ produce n ci rculnr- Iorwerd Hel d , while U~ a nd jlh, se t u p II hackwnrd Hold. For the case 'under- s t udy, (4.18 )

where - ji;. and jh n~(!. t he pos iti ve- and st a t or cu rrents rcap ect.ivel y. . Foe t he rot or t he fo llowitig equations hold:

1ll!!l"3 t tve-se q lienee

U~ =U~+U~, ,

•

••

"I' + /./',.. /. 1I, = - 1p

('. 19)

.....ilere Up, U~ . J.p. l~ are t he ro t or vo lt ages end cu rr ents of the positive and t he negnt tve sequence, res pectively. Equations (4·.17) t hro ugh (4 .19) fo llow fro m t he mod el of Fig. 3.3 showi ng lour pairs of windings on the stator lind rot or . S in ce t he model re presents all u nsa t ur ated ma chine. coupflng be twee n the negative-seq uence win tli ngs does not extst.. T he - to r q ue can be delined by usc of the com mon-rotor mod el or Fig , 4. ' Of t ho two- stator two-rotor mod el of Fig. 4 .5 .; Cous tda r tng th e processe s i n a muchine Illollli' t ile Cl. ax is for t he posittve-sequence var tebles. Ir ojn (4.1i ) t hrough (4.19) we get

iJ~ - (r' +

j r ) it T

jxj~

- b ,,= l:Cmi~ + VX m ( -

jit) + (r" + jff) i p+ vzr ( - ji~)

(4.20 ) (4. 21)

I II a s imil lir manne r, for qle neganive-scquenee va r ta hlea we nave

iJ~ = (r" + Ix' ) j ~ + j X", h.

- iJ~ = jX,,.i~ + VX", (jh ) + (r' + jx r ) h + vx· Ui i,)

(1•. 22 ) (4. 23)

All we did in tle riv ing tile eq uatio ns for fl circu lar [ie ld. here we repla ce t he total i ndu cti ve reactn neea r' lind :r! by t he rae e te nce of mutual indu ct iOIl, :Em. nnd rho s t a to r and ro tor le akage rea ctances .:ta, and ~" t' res pecttve ly . Artee traosfoem nr lona the eq ua t ions be-

com o

iJ~ = (r'

+ /Z.,.) i~ + 1%... (;;+ i p)

(40 .24.)

- Up= lz",i,. - ivzlJl i~+ (r" -I- I Z<I ,) i;.

. . . + jr", lp- I ,,~ fp - !vr",1; ,="h'~

T ransform hlg Eq. (4 .2 5) for t he ro t or

'0=- Up= 1%...

(t - ",) i ~

+ / 'Zm (I -

(4 .25 )

lIS Lbe equa\ioll of t he

ip + / Zv. (I

- v ) I~ + r' ;;

S i nce t - v -I- ({J~fw. _ , •

-U p =

•

/7:",' ( /~ + 1,,) +- j r u ,IlI ,,+

•

r' {"

(/1.26)

D ivi ding bo th aides of (4. 26) by ' we get •

••

- Opl. _ Jz". ( J~ + J p) t- Jp(r' ''S+ /x...) Equa ti o ns (4 .24.) a nd (4 .27) for

lh.~

(4 .27) pos ttlve-seq uon ce vo llage" In

t he s ta tor a nd rotor , respectl vel)' , descr ibe th e precesses i n an Indu elion m achi ne fo r the posit iv e phase se qucnee . These e qll llt io ll$ fo ll ow (ro m tho oq uivalen t circu it of F ig . 2.3 . H eN r t. = r" r: (' _ .)/, (4 .28) The te ss i n " (1 - .)/11 is pro port io.nal Lo t ho useful po w er at t ile m achine s ltar t . A !imilll.r a pp roac h ts a pplica ble to tlc ri v i ng u.e ooga t iv0-! eq uenu

eq uat jena

U:' =

(r'

+ /z" .) i:. + 1z", (;:. + i~) +ira , ( 1 + \I) h + rrl~

- li~ = /z", (1 + \lHi:. + i:;) Si nce

... t

w ~/f.l) .

= 2 _

(4.29)

- iJ~ = /z". (2 - . ) (h + h ) t / z.., (2 - . ) j~ + r~h

(4 .30)

Eqllatio ns (4.29) a nd (4. 30) (0110....' from t he equ iv a le nt ctrc uu for t he negatlve pha !e se quence . Fo r co mple x eqceuone a nd per-pha se equ ivale nt c ircuit!! of th e i nd uctio n me toe, the phsac r a nd c ircle dl a gl'llD'lS may be found a ppli ca b le. H owever, t h o a nalys is o f t he wo rk ing p rece seea in m ntors h). the pha sor d iagr8JDS presents d.iHic lilties beca use t he curre n" dis lribulion among ph ases w it h l oa d variation depends on tbe anll'lil ol' " elodt y a nd torque exer ted on the shaft. Where the ana l ys is involves on ell ip ti c f ield, Il more p refe ('3. ble ,a ppro8 ch is 10 1IOIv9 Eq s. (4. 8) a nd (4 .11 ) r athe r tban t o proceed fro rb tho s ilnpl ified di llgra ms. Tho

Ch. S. Ener<;JY Co nversion al No", l.... " Asym. Vo ltage.

th ing is l l!:l.t pha!Or an d c trdle d iagra ms wh ich hel d a nd work well for a circu lar Field are hnpracticabl e for a more ae oE-rlll case. Lr . a n t'lliptic fie ld . Note t h(ll a t wo-phase motor ",-Hh Its sta tor w ind iogs supplied fronl sc ueees differi ng in Ireq ue ncy ma y h u e one of its phases co nnected \.0 a supp ly li ne And t hl\ ot her to a th~'ristor co nvert er or both couu ccte d 10 indi vid ua l converters. Und er t hese ca nd itions, considerati on shou ld be gi ve n fOJ' pul sati ng f ield.!i in ea ch ""ind ln: • • nd t ho a nalysla of e nergy co n\'ersio n processes should be done for a t leMli four fie lds in t he gap. Here . the model for tho t .....c- phase motor i. Iw ilt up of four pe tre of wind ings 0 11 t ho s llltor a nd four pairs of "' indi n"rs on Ille rot or. Th e Sl'!Ittlm of eq uut lons compr ises 16 vohsge eq ua tio ns a nd th e Associ/l.led torq ue eq ua t it>ll Includes 16 torqu e components. If t he cond itions devia te from t hose Iavorabla for ilC.'ll ing up II circula r fi eld , the problem become s more com pl ex . T he ate ndnrd prograrns of ty pica l energy convers ion equ ations lire n userul 1001 for studios of UI~ t ra nsient e nd stead y-s ta te perfor ma nce of elee tne mac h ines hllving ·ci rcula r a nd ell ip lic fields, f ield s wit h se ve ra l ha rmo nics. a nd ellrl1'lng two or more wiud ings on th e s ~ tor afld ro tor . T bese ProiumS for m the bas is for a utoma tic. systems of des igning e lec t ric ma chines .

Chapter

Energy Con.version Involving No nsinusoidal and A symmetric Supply Voltages 5.1. The Equat lqns of Eledric Machines Cons ider m fields in t he llir gap (i.e . R field co nta ining m he rmonies). A maeh tne with 1\ nonsin usoid nl volta~ li t ils ter minals may ser ve liSII olass tcal exa mple for t he case under st udy . A ge nerllll'::lIl\mple of t ho ne nstnusc tdal " ollage source for a n i nduct ion moto r is II eemtce nducscr ft'@ q uency co n{,erter . As! uming the ma chine to bf ideel (so lI.dd it ional harm onics in the lIir aap e re n-o nexLstcnq. it is sefe to pict ure th e field as th e one fllil l1ruJly reproduci ng th e hqrmonlc spec tru m of t he a pp l ted \'01· tage-. An ideal mach ine doos !lot ~eJ1(' n. le noise, and so for th e ha rmonic s pectru m of the mag Ml ie B fiol d to be defined . it is e nongh to ex pe nd t he phase volt /l.ges 'into a hA rmonic series. T he di roctl oJ1a

5.1. The EqIHtio ru of E'-d ,lc: Mach i"..

of rota t io n aDd ampli t udes of tim e Ila,rm onies de pend o n the nu mber f)f phAses i n t he mac hin e a nd t he ,ord ina l numb er of each harm onic. as is o bvio us from th o se t of eq uat tona (3, 1). Assum e a lso t ha t eac h of the m har monics making up the field il'll the a ir gnp i.s se t u p by t wo pa irs of Windings Arranged on th e s tllto r or rotor a long ti le a a nd JI a xes. 'IJle mode l of s uch a machi ne (Fig. 5. 1) has t wo se ts of m. windi ngs on t he s te tcr a rld rote r Il o~

u:..

w". ,.

u~"

...;..

u:..

"':..

I:lll' .5. 1. 'fhe model of a m3ch ine lor th e ana\ysi. 01 tmergy eGDv,rsi oG p~ a t Doud Duao lda l an d uy rn lDe~r:Ie s upply ..oll,a~s

tho a a nd JI a xes ra t her tha n m: a nd vd ndi ngs on th e s tato r nnd rolor, respect ivel y. T his model is ana logo us to t he model of t he gene-ralized eo nvertee. For t he mach ine w it h m wind ings be t h on th e sta tor an d on t he rot or , .....e deri ve t he s ys tem of e>qullt io ns from i3 .3). In these aq uaU :8 are the ste tor voh a ges At t he Iundam unta l Ireq ue ncy , \iODS, lind are th e stator \"ol tages prod uci ng t he fie ld of a thi rd har mo nic . Th e laller vo lta ges ha ve a fre q ucnc~' f . = 3/1 li nd li n ampli t ude t hllt corresponds to the IImp,li tude of t he fiel d of the t hird Rarm on ic. \ 'ol ta ges u;... U;, corrt"spo nd to tile fifth harmonic of frequeocy /. _ 5/1' a nd voltage' u:..... u:' 1\ are t M vcl tegee of th o m lh ha rmon ic., f .. =- m/I' The phazea a nd t ile dt eeeucns of ro tat io n of t he field har mon ics res ul t from th e cortE-spo ndlng vol ta ges pcross the sta tor wind i nas . If the a ir gap u hib ils s ubharmonics. t.e . theCields whose frequ enci es are betew the Iund amente l rreq uency . II pllrt of th e wi.ndi nga a re fed wit b voltages at freq uencie s lower t ha n.

U: .

u.:..,llu:.!

Ch. S. e .... gy Co nv"rslon .1 l'olonsln. .. AIyl'l: Vo lI!glll

'!Il)

1I11:~ Iunda me ntel . If the vol tage con tains even harmonics . a par t of t he windings a re euppljed wit h vol tages displaying eve n harm on ies. For t ach p... ir or 'rindi ng; on the s tato r t here is a corresponding pa ir of wi nd ing! on the rot or. IU rega rds an ind uctio n machine . lho voltages ac rOM t he rotor win d intrS are eq ua l to te ro for a shorL-<:i rcui100 rotor. nnd po......ee is fed t hrou gh t he sta re r: i~ ... 8 â&#x20AC;˘ j ~1 a re 't he statoe a nd rolo r curre nts of t he firs t har mon ic; i' l ' i~ .., I' ~ ore th e s tator an d roto r cu rre nts of the second harmo nic: a nd I~ ... i:" ~ . i:;'.. , I:' " are t he sutor nud rotor curre nts of t he mth harmo nic. T he equations for no nstnu sc tdal asymme tri c lllipply V OIt8 ~S can be derived from Eq s. (3.3) Of tlle aen e ra lize d con verter. T he volt age a nd eu rre nr matnces for t he coso a t hand have the sa me for ro as in {a.4) and (3.5), but t he imppIlllnce IOlltrix is differe nt . T1.., (our impeda nce mntr iecs arranged along the d iagonAl of t he Z ma tri.o:: have t he form

t: i:i;

r~ ...

+ (dl dt) L~ ll

rb + (dldt) Li...

A:'=

(5.1)

o r:", r;". . . .. r.....

r:"'" + (d ldt ) L~

where lire the res is ta nces th e windinl: oUllrs to the currents of t he 1st , 2nd an d mtll harm onies . D isl'('g ard illg current dt epla eemen r, t hese rt's i !ltan~s may be tak e n iden ti cal to a sufficle ut degree of accuracy. I'n (5. 1). L:", L:.... an d L:..... are th e total i nducta nces fff th e leo ps ca rry ing cur rents of the t sr. 2 nd . a nd mlh ha rmo niC!!. T he ma tri x A ~ comes froin A~ a fte r t he re placement of t he s uper.st ri pt I by r , a nd Ai from A ;' wu h IX re pla ced by ~ . T he impo<h'mco matrix A& comes out after s ubs tit ut ion of ~ for IX in A ~ . T he s uhma t rfx has t he term

A::

(d ldt ) M r Zll

o o

A " --

(5.2)

. . ,.,,/:;., ....

(dldt) M : _

Yo')wre M :~ Mr:roQ are t he mutual i nd uctances betwoen t he sta tor- and rotor loops, respecti vel y, over which t ho harm onics of t he sa me ordcr complete thei r pat hs .

S.I . The Equ atio o$ of Electtlc M&chIOI!l$

T he ma t rices A:"', A f, a nd Ab' Illl l,ie r ive fro m A:; on re pla c ing a. by ~ a nd i.nt ercl tltll::i ng $ a nd r, I n rn,ach ines su bjec t 10 sa t uravtc n and other nc nfinearit.iea, the m u tu a l ,i nduct a nces be t weeu t he s ta to r and ro t or loops . o n the one ha nd . and those be tween tlte rolo r lind ste tc r loo ps , on t llc ot her . (Da y diHer in ! i~ n - L , «(o),

- L:a

lil,

0 0 (5.3)

J) ,, =

L'il\w.

L~lIw,

- L:.o.w. 0 0

D,,=

(r..ti )

L~ lIw .

T he s ubrnarr tces 8"' 11 a nd IJ" ". d iffer in si gn a nd one follows Iruru the ot her by i nlereh l\ngi ng t he pos i tions of a lind ~ and of r arid 6. - M ~~ «6) r

-M ~kCll.

0 0 (~ , 5)

8«11 S2 0

.., :11;:".. .

- !U :;'_ tol,

where .\f~ : J1~ ...« I'tre t he m u t ua l Inducta nces be twee n th e ro tor a nd slAtor loo ps ca rr ying "'Irre nls of res pect ive h::t rmoniCll. Com bi ning th e eubma t elcea of vo ,ca!:OS. curre n ts, e nd Im peda nces rives t he oq uations of a n elee u-tc mac,,"ioe su p plied from n IIOlUlinusoida l vo lta ge source , ea ch !Set of equations bei ng wr it te n for a defi nit e harmo nic. ' Fo r a eea t mach in e hav ing on" pa ir or wl ndi ngll o n t he stato r a nd one pa ir on t ho ro to r , the res is ta nce in the ma thelll 8tiea l mode l m. y be eonaide eed equ a l to ea ch other . T he to t a l in d uc t an ces o f the stator lin d rote r wi ndi ngs along th e a IlPd II U N ma y differ fro m eltch other because of t he d ifference be twee n t he leaka ge fluxes (t he de.signatio n! of th650 in ducta nces a re th o ,SlIme /115 for rosistanees). Tho mutual i nd uc t a nce s be twee n t he s ta t or a nd rotor wi nd inp ma y be tak en equa l tc each o t her . ' _ Ut H

S i nce t he machi ne modol is 111)Ral m a ble , th ere is 1M) co u pli ng betwee n t he st a tor wi nd inG" !lIang tho sa me ax is. H y pc t he tt ce l w ind ing:! prc duci ug the fie ld in ure flir gllp hav e no co u pling bet wee n each ot her: in the ma ch ine under s t ud y t he non sm usot da l curre nts fl ow in o ne win din(: . As in t he cue of a ll o l llp~ie Hold . he ro we ca n cc nstd cr t wo mod els to de ri ve th e e q uat ion for t he e lec t ro ma gnetic torq ue . T he- fi rs t model

Fig. 5. 2. Til. m..fltllle r e -rotor maelline model for d<lle rmlnl o(r M .

(F ig . 5. 2) 11, :';/11 s tlllorsa nd m ro t ors . th e la tter bei ulI: r ie:idl y cou ple d. Ellell s tll to r is fed wit h \'o lt llgt"s of the fi rst a nd uppe r h llrmo niC$. For t ile mod el re prc!Ontilli t ho processe s o f l'ucrln' eenverstc e in

Fig. 5.3. The ...-m Ulr comm on -rotor maehlne mod.l for de term ining !J . t his f"shi o n, t lUI to rq ue eq un t.ion cou t a t ns on ly t he prod ucts of qua nt ities wlth Ihe sa me Ind exes . T h e pulsati ng com pone nts flrO abse nt. Accordi ng to ure seco nd mode l (F ig . 5. 3). buil l up of m s ta t ors a nd a co mmo n rot o r. all harmoni cs i rtte eac t n nd th o torque eq ua t io n i nclu de!' t he pro ducts of a ll et at e e cu rre n ts a nd a ll ro to r cur re nts of

S.l. Th. Sol ullo n 01 Equeti o ns

I n~olvi"9 .....ymm.t.ic VO lt~ge l

upper h ar mon ics . T he50 pr od ucts contain th e terms with differe nt indexes, wh ich produce pulsllting torques . Aft er t r ansfonna tto ns, the mu tua! ind uctances be tween the state r and rotor windin gs bei ng taken equal t o each other, we find that M . = 11f r (i~ " ~ (i:~

+ I:.. + . . . + i~a ) (i~~ + 1~ ~ + ... + 1~8) + l : 1l + . . + i:" ll) (i~a + i~" + . .. + 1:',, )1

(5.6\

T he eleclMmagneLic torque defined b)' Eq . (5.6) is th o product of nnnsm usotd a l currents in th e sta tor, au d rot or . Modeling porn uts analy t ing an aCL~181 energy conve r te r b)' use of Jour aquatjons s imil ar to (4.1) an d II' nons tnuseldnl voltago source or m equa tfoue nn d n sinuso idal \'olt agc; sou rce . The m-windi ng mode l is more comp lox, but it offers large po~ibi Ht ies for si mula t i ng the hehaviou r of t ile p hys ical sys tem 0 11 a digital comp uter. An a nal og compu ler is mora pr eferable for th e real taaticn of the modol for four e qu at ions il l conjunction wit h 0 nons jnusc tde ! voltago gene ralor as power supply . \Vi th t he nonainuso tda l aey m mo trlc su pply vo l tage used , bot h t he fir~t and upp er har monics ex l,jbit the forw ar d an d backwa rd fields . Therefore , t he mathome t tca l mode l must i nclude t he pa irs 01 Wind ings on the sta ter e nd rotor t o represent th o forwa rd fie ld and one more pair to re prese nt t he ba ckward field for each harmon ic . The nu mber of fields til the ga p then gro ws nud t he nu mber of windi ngs in the mode l grow s nccordiog' ly; the in it ial equat ions he re hold , and it rema ins to choose t he desired har mon ics and solve eppro xima te ly the s tated pro blems boca u~c cc r upute ra ce onor natural fy take in to a ccount the tntera cuo n of AU harm onics .

S.l. The Solution of Equations Involving Asymmetric Supp'ly Voltages Equa tions (5. 1) th rough (5.6) show a more gene ral character th a n the equauene for II cir cular- fie ld, \' her e.fore th e former equ ations require t he use of analog a nd rl igi ~lI compu te rs for t iteir sol uttcn. I n the s im u lation of pr ocesses on an, a na log computer, special feed crrcune are sot up. whic h are the ana lyse r netwo rks intended for t he solut-ion of e nergy co nver ter eq ua ti ons . MOIll po pu lar are then llt-works des igned to re prod uce t ho modes of opera li on of a thyr istor converte r; Lhey com b ine the pr inci ples of ma t.hema tlcal and physical modeling. The gate circuit is built up as ('l ph ysi cal model, and t he ana lyze r's el ectric equ ipme nt connellled t o the gate circu it representa t he m athcront.ieal mo del of 91;1 el ectric mach ine. Assum ing th at t he motor under ana lysis is fed from a no ns inu soidel vo ltage sup ply line of infi ni t e power- (the erreer of t he loa d on the

100

cn. S. e na' QY Conv.ulon ... N...... in.

a. Al rm .

Voltag e .

v oltago we vero em be i ng d i.erogarded ). it is po.ssi ble to realize nons i nusoida l vel tap; genera tors on e n ana log computer a nd use t hem i nste.nd of .si nuso idal vo ltage generators in the com pute r model for t he sc lut ton to t he eq uatfens of e n e nerg y converter (sao App(mdi:l III ) .

An a nalog compute r ca n sim ula te pOll..er s upply for a n elec tr ic mach ine from a magn et.ic amplifier af te r im plem onting uie foll owiog reteuo us

u j (t) = 10msin ",tl U'" si n ",t . ui (t) "'" IU'" cos ""I u... cos ", t Th e block diagram of t he model fol' a nonelnusct del voltage generat.or Ie shown i n Appendix; II I. T he har mon lc s pectru m va rie s on a ppl yi ng to power uni ts P UI a nd P U , t he vollaglls U l (t ) = := I U", sin ",CI U'" si n (<0» cp) a nd u , (t ) "'" I U.. cos wt I cos (cat + Ip) IlS a resu l t of changes i n t he phase aoSle cp (_ Fi g . A7). A nOJUinusoida l vo l ta ge geperato r illu.stra tod in F ig . AS {Ap pend ix Ill) gi ves a step wev eroem of tlte vol ta ge s pec if ic to t ha t of pulses d rawn from semicond uc tor converte rs. Fig. A.9 s hows the block d iagra m of an au to nomous Inverter wit h pulse-duee tf c u modul a ti on (P DM) And pu lse waveforms at. t he genera to r ou t put. As seen from Fig . Ag. the' a mplifi er 1 give s drlferent- polartt.y sa wtee th p ulses u, wh ich go to the i np ut. of t ho adde r 2 to bo com pare d with the si nusoida l voltage U.s' Th e co nt rol etemen t hero is II poteetzed rala y PR. whose contact s ehe pe differe nt.--polarity re c tangu lnr -pulsea Us modulat ed in wid th (du ration) . Appl y ing a ti me-v ar yin g pu lse voltage u p = ± O", sfn wt to the s ta ti ona ry co ntac ts of PR . enables th e circu it to ea rr}' ou t t he a mpli t ude mod ul a t io n too . T he pul se wa veform 0 , set u p by th e rolay t PR . t hen COlT@sponds to rhe waveform of t he pulse draw,Il from t he si ogle· phase PDM: inverter. In Fi a . 5.4 is shown t bo· re la y-diodo circuit dia gram wi t h two polerf aed relays PRJ a nd P.R, a nd two d iodes 0 , a nd D" wh ich can sim ulate t ho opera t ing modes of a n a mpl tt ude-eo nt ro lfad t.hyristo r dri ve . The block diagra ms of th e, m odel s for convert.ers , wh ich are bu ilt u p Iro m sta ndard computing blocks of 0 di (fereotial a na lyur e na ble II rather faithful reprod uct iop of th e t ypica l p U] 8Q wa ve fo rm s at. the o utp ut of a magnetic a mpli fie r , bridge inve rte r , a utonomous P DM i nverter , a nd t hyris tor conver ter. These m odels Are ra th er cosil y lIOt up on a n a nalog com putor a nd are reliable o vor the fre quen cy H~ . The mo dels allow lor a variation of tile l an ge between 0.1 a nd frequeocy , ampli tude, a nd harmoni c s pec t ru m of t he ou \put pu lses. They en a blo a (airl y a ccurate simulat.ion of tbe s tea d y-s ta te and d ynem te perf ormanC6 of a n e)ee tromeehaoical EC. Digi tal com pute rs offe r m uch greate r possi bil it ies of solv ing tb e eq ua ti o ns associ ated wit h no'nsi nuso ldal as ymme t ric vol ta ge s uppl y,

5.2. Th. So luti on o f Equ . tlon . Jnvo ,lving ..... . ymm." rc Vo ll.i!'"

t Ot'

For the solutio n of Eqs . (S. t) t hrough (5.6) , t he n umeric a l Run geK utta tech niqna is most pop u lar. In ~ an d li n g t he eq ua tions. it is of impo rtance t o ma ke the righ t choice of t he step of o perutto ns. for it determ ines t ho t imo a nd /l.CCUfIlCy of com pu ta ti on . T he programs for th e cal cula tion o ~ both s tatic a nd d yn a mi c eh nractertsucs of an energy converter supplied Irom /I. noua lnuso lda l vo lt ngo sou rce a re we fuc n by exper ienced pr ogra mmers . In wide use

Fi g. 5.4. R" lay-d lodc clrcult diagra m of a th yri stor wll

are s tanda rd progra ms for ex pa nding the c urv es into a har monic series, de te rmi ni ng t he eff.ectivo va lues of her-mo nies and their init inl phase . T he eq ua tio ns are solved for hot h tw o-phase and t hrua-ph ase machi nes wit h a nd without regard t o t he multtpor t a con nected to the stator a nd rotor . Of gre.a t s ig nifica nco is t he cetime t lon of s te uc chara cter is t ics for machines w it h ncn s inusofdal vll!tnge ·su ppl)'. By use of t he vo ltage oqua rrons expressed, for e xa mple , in a- ~ cc ord rna ees. t he cnleu.la tfon of curre nts is don e for "ariolls a ng ular vetocntes of t he rot or . For t he s teady-state ope m tion , the mean value of to rq ue over a pl? riot! is T

M . = (t fT )

J1I1.(t )dt

•

where T is t he pe riod of volta ge of th e fu ndam e nta l harmo nic .

(5.7)

Cn. 5. Energy Con~"rs!On al Non, in. " Asym. Voltage s

t0 2

If t he har monic componen ts of the stator a nd rotor currents i ( I ) = 10

+ ~ U l'main (I-lwt +q>..l]

(5.8 )

are k nown (where I.l denotes integers}, the n, afte r eubet uut ton of M. in to Eq. (5 .6) the torque d ue to each hnrmontc is round. Tile toeque is the n dall ned (IS

M. =M.o +~M ...

(5.9)

where t he torque due to tho eonste nt compo nent i$ M .o = (mp/2).M ( l;,O/~ - /~ D/~D ) "

(5.10)

The tor que d ue to tho uth l.mrmonic i n term s or t he. ecuve and reactive components is gtvon !,lJl . M

= (mp {2) M (I~/~ I'<f + J~I',1~ ..r-: I~"",I~I',, -J~.., I~ ..r)

(5. t l )

Here t he sub scripts a and r r~specliv ely identify t ho act ive and reactive cu rrent comp onents for tt~e: J.ltldlarmon ic a long t he a an d ~ axes for tile stator a nd rotor. If the positi ve- a nd nega tive-seq ue nce s tator and rotor curra nt components alo ng th e a a nd ' ~ axes are present , t he n, omit ting the . subsc ript ~I , we get

(5.12 )

= I", ,,+jI ,,r

- ii, + Il~

i~,, + ji fl'

(ip "

i n., )

+ I (in.

1I -

i p . ,, )

where the subseripts p and , n denote t he positive- a nd nega tivAsequence current components . Substit uting (5.12) ill (5.H) a nd performing tra nsforma tions , we have ' ftf~1' ~ Af. ~ p -- ftf~pn (5.13) For each harmo nic, the exp ression for torque i ncl udes the symm etric positive- a nd negative-seque nce components. The power at the shaft is P I -- P:w + ~ Pw.

(5.14)

where the power of the const a nt component , d13regarding a dd iti onal losses, is . (5.15) p! O = (Âť/9 .55) M . o

5,2. l he Sol ut io n of Eq uol; o n. IlIVo l.v ing "".ym melrlc 'I ol l4gel

tOO

and the t for the p t h com po ne nt is

PI" = (ttI9.55) 111 P" Th o activo powe r a bsor bed in the mac lune is g ive n by P l = P IO+ ~iRe S..

(.=:; .'16 )

(5. 17)

where P IO is t he d e power recei ve d fr om the bus ; t he adde nd is the ac active power: a nd S " is the l oLnl powor of t he ~ th harmonic , the ex press ion for wh ich takes th e form

(5 18) which is t he slim or complex powers' of t he et atcr lind ro to r pha ses . For eymrnetrtc machi nes t he ex pressio n for Sf/. in te r ms oI sy m me tric cur re nt and vol tage componen ts ;assumes the form

'. ,. ,. ..

S .. = m. (U~ /~+ U~ 1~ '-lr U ~J ~ + U~ /~)

(5. 19)

The power factor is

(5.20) In dellllng wit h oo netn usc id et voltagea, t he powe r res u lting from d ist or ti on ma kes the evetu a u cn of 8 1 imposs ible . Acco u nt th en shou ld 00 t akert of cos 'I' ro r ea ch ha r mo nic: cos 19.. = P "IS..

(5. 21)

The nort on of cos qJ. introduced fo r t he s teady-sta t e act io n a t a si n usoi da l vo ll llge , re lntes t he aetire po wer lO the reec uvo power , Th e a ng le 'I' defi nes th e pha se sh ift :bt twee n th e voltage l}nd c urre nt on t he p h1l S0 1Âˇ dt ag ra m of Fig. 2.4 .. ~h ll co ncep t, o f reactive powe r t hat hold s Io r t he stea dy -s tate con ditions docs not wor k when cons idering th e ucnste nts at nonsm usotdal volt ages . F ro m t he dyn a mi cs vi ewpoi nt , th,e rea ctive power is definable as t he power s pent o n pro du cing rnng net.ic fields . As regards tr a ns ie n ts , t he ac t iv e power t a ken fro m t he bua t ra nsforms i nt o t he eeacuve powe r which, i n t urn . co nve rts i nto "ll)e active powe r ge norlilly gi ven off as heat i n the macluna aud us loo ps where cureetus circula te, T ho deecrt pt to u of the se complex 'cnergy co nversio n processes by ulect.romechauicnl equntion s preSlln ~ sl d iff icl1lt ies aino o II. few r ath e rco m plicated a nd a pproxima te mathemat tca l model s m ust be se t u p ill hand li ng eac h problem of in terllst. It is t he refore im poss ible to define a nd mea su re ac curute l y th e re act iv e power , so there is more se nse to deal wit h the insta n taneous values o f voltages and cur re nts and t hei r pro d uc ts , i .e. i nsta ntaneous t ota l power s ,

t 04

Ch. S.

e"." gy Co nve ,"lon

oj No"'in. '" A' ym. VDltag..

T he eff iciency in th e mot ori ng mode is 1]

l' t1PI

(5.22)

and that ill Ole generating mode is 1]

= J\ IP I

(5.23)

In the etmul at jcn st ud ies of 'dynam ic eha ract erfa ttce, 0. d igital com puter sol ves Eqs. (5 .1) through (5 ,6), commo nly for three to five he- montes, t.e . eq uati ons of th o 13th to 21.8t order, T he sol ut ion of equ ati ons on moder n digi tal computers takes only a few mi nutes. T he programs desig ned for the purpose an d run 011 a computer ena ble t he inves ti gator to unulyze the eUect of the amp li tudes and phases of harmon ics, determine' thQ los~ due to each ha rmoni c, a nd study the effoct of param eters on th e d,)'na m ic behav ior at nons tnusoida l velta ges 0 11 the ter minals, . Compa ring the starti ng curVes for a motor run from sinusoidal and nonsinu soidal llOltage supp lies, we sJwuld point to an Increased fluctuation of Cllrrents, torques, and speeds In its operati on from the nonsm usoidal ooltage circuit and also to the dependence of the course of a traT/.k sient on the starung torque. Th e st udies of tran sients on' digit al computers reveal that inductio n motors fed from a ractangul j'lr-pulso volt age SO U1'(:O show al mos t a tw ofold increase in t he no-load cur ren t as agai nst that obs erved in opera ti on on th e sinusoidal " pl'tage. T he efficiency of con tro l mot ors may drop by 20 to 30 percen t. The energy cha racteristics of genera l. pur pose mot ors de viate from the nom ina l hy 10 to f S percent . Consider in brief the opera t ion of an energy converter whose wind.ings are fed with sinusoidal vol ta ges at different frequencies, Impress th e vol ta ge at Iraq uency f l on the a phase and th at at fl on the ~ ph ase. The oir gap will eahibt t four fields : the forward and the reverse field prod uced by. the 0'. phase and the sa me sat of fields prod uced by t he ~ phase, Fo~ tbe enetvere of energ y conv ersion processes, we can uao th e model of the four pairs of win dings dis tr-ibute d on the s ta tor and rotor an d t he sys te m of equations of t he 17th order si mil ar to (5, i ) t hrou gh' (5 .6). Note th at tbe ener gy eba eactertst ics here ar e much poorer th ai;! tl ley are in the case of a circula r- field , b ut in othe r respects the ellerg y convers ion processes Are simila r to t hose involved in nonsinusofda l sup ply , It is somet imes useful to employ the abo ve-me ntio ned s u ppl y circu it s for decreasing t he numb er of stati c frequency conv erters; a machine operates with ita one pha se con nected to the s upply line and lhe othe r to a frequency con vertor. Elec tric drives are oft en ma da t o operate i n the con di tion Of dynamic br aking, in which case one of the mot or wi ndings receives its excit at ion from the de source durin g th o tr ansiellt precess.'

S.3,

Ind uctio n Mlolor Syli.m

ios,

An energy convertor can be built so: that ita field angular velocity would vary with the amplitu des an d 'phases of field bar montce an<t Ule rotor would pass from one s ynchro nous s peed to ano the r wit b> changes i n th e harmonic s pectrum. This woul d enab le effi cien t angu lar \'elocit y cont rol. But design ing oflsu ch a machi ne tc vc tvee consldet-able diflicu !ties and necessitates cont rollable frequency conv&rters . It is theoreticall y possible to very t be amp li tudes of hereooDies in a regula r fashion to en ab le a linear v eeteuon of the field veloci ty and the desi red cont rol of EC parame ters.

5.3. The Thyristor Voltage Regulator-Induction Molor System At prese nt there are esseutt ally t wo a pproaches to sol ving t he pr oblem i nvolv ed in the deve lopment of a controllab le Ind uction motor driv e, Th e firs t to tall y ren ee on th ~ use of thyriator and traosls to r frequency converters an d the seco nd aims Ilt perfecti ng tho techntquo of in du ction moto r s peed control ,b y rcgu lati ng the voltage with. tho aid of thyris tors (silicon controlled rectillera, SCR). In ec m rc tling the angu lar speed through cha nges in the sli p, an electne driv& shows poorer energy cha rac teristics beca use the slip energy is losl 1I5 helt in th o moto r, But since t he elreut t vers ions here are si mple in design, reliab le, and feature good corjtrollible proper ties, they Hnd. use In motors driv ing hmÂˇtype l o ~d torqu es, s uch 113 driYcs for compressors, Ians, and blowers , lind IlIso In dri ves s ubject te control ove r a narrow s peed range_ Th e circuit diagram un der dlsc uss tcn is simila r to th at for aIt inducUon motor drtve complete with se l urllble recetoes since tho adjustment of the thyri s tM ee edue uo n angl e lea ds to II change and' addi ti ona l shift ill the firs t harmonic of t he me ter current wit h respect to th e line volLlIgc. In other we-d e, each pa ir of thyristo rs cenneeted i n pa rallel op position (Pig, 5,5) 'can be regarded as a rrcuuccs non lin ear reactance th at is a Iuneuion of th e th yristor condu ct ion anglo and the motor paremet era and ,sli p. Th yri stors designed for s tato r voltnge regulation offer II nu mber of lldvantages over setu re btereactllCS: t hy ris tor vol ta ge regulators rrVRs ) lire practicall)' in erti aJess, heve a la rger power gllin, higher efficiency, and are s malle r in site end mass. To mak e the uso of TVRs mere effect ive requires exrenetve s t ud ies in to the trans ients , tbe cffect of u pper harmonica 00 IOMOS, oVCrvO lt ll ~ , etc. T he theoretical analysis of the T VR-i nducUon molor sys tems preseats certain diffi culties du e to a number of factors. AmonI:' these we s ho uld men ti on th e non li nearitY !I[ curren t-voltage characteris t ics of semiconductor rectifi ers in tbe dynam ic and quasi-stelldy s ta tes, a~ which an electric dri ve is Ioun d to be in sequentiall y changing t ran sien t eendrucns rcsu lt ini from th e continuousl y Yarying s wi tch ing

~f t he motor s uppl y eteccn. T4e sys tcm.!l of nenltnear nonhomogen ... -cua dirferential eq uations deScribing an induct ion machin e In t he ·c ond iti onll of lIymmelrie aod asy mme tr ie voltllfe su pply to lIt.tor p b esee vary In form . Because of t he uDconV,oUable behav ior of diod es and ineomplete ly <ootrollibla operation of t hy.ri_s tolS, the out put ch u lleteris tiC6 oC a "TVR (the conducti on anile, Yol tag e waveform ) pro ve hea\·il y dependen t. on t he elec tromagneti c transien ts in motor. In other wcrda ,

r- »I" I

1 l \'KI

I I I

'1'"" . 5.$.

T M l'I,!vef1libl e

" ':C '

",-, e.,.-.--"," I ..,, I ~t.<';C·2 I "

,I ,, ,'

" ___J

lhrrlalOr supply cIrculi lOT

all

huh..;tion ml chlne

'he voltages su pplied to t he motor are depe nden t on t he t h yristo r -cc nc vcuc n a ngle, puameters end ang ula r vel oci t.y of the motor, A s ubs tantial dev ia tio n o( ' t he wavefor m of vcl tege at t he motor ter minals fro m th e si nusold 'l;llll588 high -freq uency fluctua ti ons of the electroml'lgnetic torque apd nonunltenn run ning of t he molor In 't ile stendy-state opera tion. . The equations of t ho ma th emeucal model for rbe s implest versions of t he s rete m und er s~u d y aro CSSeIl tia.ll y s olved on a nalo g co mputers. In solvi ni t he d illerentiol equeucne lor an in duct ion motor 00 a n ana loi computer , s pecial req uiremen ts a re pl aced on t he nc ie uc n of eq.uatlons 0'1d. t he choi ce or va rb blOll, This choice s hould be unde rtaken with due regard for bot h t he process .suh ject to tb o an al )-osis and t.he Iect oes dele rmining the accuracy , ca paci ty. and reliabili t y of t.he model being set up , T he si mul a ti on of t he TV R -indu ctl on motor system_on an ana log eo mput f!r ~ui tee s pecial units to be built i nto t he co mputer to s imulate t he d tseeete charac ter ~f TVR operation, The s imuJa tiOD of ind uction moto r perfor mance "Wit h coas tdeee u on for eha nge! in the paumoters due to sat.uralion

' 00

lind curren t displacem ent neeessi ra t es addilio nal u nits to perform mult iplicat ion and to account for non li near it y. Th e instabili t y of analog -co mpu te r solutio ns due t o me ~i m i led acc ur acy of co mp u ti ng amplifi ers and t he ex t rem e difficu lt y of se Uing u p the u uiv eraal models h ave stim ula ted t he ol'ol ul ion ;of t echniques fo r t he analysis of thy ris tor d riv e chc eectertsues on d ig ital com pute rs. Consider t he solutio n t o th ts proble m o n a di g it al co m puter in reore det ail choosing for t he pur pose 'the reversibl e t h yristor-t ype inductton motor driv e circult di ag ram of Fi g. 5.5. S pec ifying v arious momeJlts a t wh ich the t rigger pulses o'f dofi ni te wid th are t o be sen t to t he gates of t h yristors, ' we can 8\'011'0 var ious control ci rc ui ts. Thus 10 ob t ain II nonre veeaibl o circ ui t diagra m with t wo t hyristor cells, t he mome nts at wh ich t he pulses go to t hyristors Cl a nd C2 should be eq ual t o zero (t CI "" le o "'" and t he wid th of these pu lses sbc ul d Jre j nfi nit ~l y ls rge, I1C1 "'I ~C 2 = 00 . T o remove re versible th yr istor cells fro m th e circ uit" it is enoug h to kee p t he m In the off-s t at e si nc e a t hyr istor drive n i nto t he noncond uct ing coudtl ion in effee t br ea ks a subetrcutt. For this t he time t p 1 = t p â&#x20AC;˘ = t p s = = l p. should be t aken infin it o ass um ing t hat over th e t ime period of interest no tr igger pulses ar rlv e a t tllo g at es of t h yr is tors PI, P2 , pa. and P4, so t hey s tay ott. In devel oping an algor it hm , we assum e th at the eet of l lne voltages U... B , U B e , lind U C A is as ymme tric, Fo r cc nv euie uce , l abel t he Hoe voltages with ot her su bscri pts : U... 0 = U...1, U BC "'" U BI> and U c .... = Ve l' Se t t ing N = A, B, and C , we can 'proceed furth er wi t h t he generaltaat tou for t he nne voltage, U N l' Li ne vol tege a var y arbitra r ily with tim e, wh ich ena bles us to co nsi der t he cha racteristics of the mo t or run on t he nens jnusolda l vol t age su pply , to o, deli.... ered by va rtc ue freque ncy co nv er ters . For t he ease where th yri s l.Ors control ui e ete tcr voltage, we ass ume t hat th o su pply n ne ex hib its a sinus oidal asy mmetri c set of voltages:

USI

V2" U N t sia(6lt +

<PNI)

(5.24)

wh ere 'P,vl is t he in iti al phase of Hn b voltages

N= A}

'll i\' l =

fJlAI = IJl AII 'llt lJ' RI =lpB~ at N = 8 { lpC I=lpC" at

(5.25)

N=C

T o obtain t he most fle xibl e modelIcr s imul at ion, control ove r e ach th yristor mus t be separa te. Thus t he 'COnd uction engles of th yristors 1 and 2 lire in de pendent of each ot her and can v ary in differen t fas hions t o en abl e the indu ct ion motd.r to run i n any of t he s pee tned operating co nditlcns .

To effect the operation of loho TVR -induc.loion motor system. in various conditi ons, it is good practic e to lock lorigger pulses coming to t he gale of. eaeb N lth th yristo r (i - 1,2) wit h the corresponding li ne vohage (FIg. 5.6 ). To accom plis h the end. tho co uducrlc n angle of thyristor N, la coun ted orr from t he %oro through which U~I passes at II cer tain insts nl. Th ug lor t he t1lyr isl or A J (N _ A, t: = 1) tri ggered to con duct corrent Iri t he forwl rtl djrection from th e su pp ly

,,'

Fig_ S.t!. The $Cbeme o r Ipply lng trigger pul5l!S t.o driv e lbyrist.orll in to eenducti on

lino into LIle ph ese wind ing a. the phase shift aAJ between the lrlIVCr pulses is cou nted oU from tlle Zl!rO of the positive halfwave of li ne voltage u A B , end for A 2 (N _ A . l = 2) conducti ng in the re verse directi on, 0:.." ill counte d off from tho eere of t he negati ve halfwave of U A IJ' To est imate the moments at which trigger pulses arr ive at thyrista rs, look into th e process Involved in t he control teehuique based on the compar i.!lon between t!!e sawtooth pulae volt8ge U /'II proportional to the ph ase of the corresponding llne v clt ege and the vol tage proporti onal to the con duction angle 0: ,.- of N l . Tbo equ at ion of t he sa wt oot h pulse voltogc over th e holf-cyd e for the N 1th th yrlstor (t he lnt l'rme dln lo t ranSformll'tio ns boing omit te d) lakes 011 the fortn (5. 26)

For t he en ti re cycle. U.... l _ 1(l:.Jt + \1,...)12,,1 2n . Here t Is the ru nnin g value of lime; \' ,.. 1 c= tp/'ll "" 2n6 N (sill'n tp",,) + n (1 - t) X X sign lJI/'ii and 6,.. (sign lJI NI) is the un lL runctton dependent on

5.3. The 1l\du ,.tlo R Mo tor Sy",o m

ure sign

''''

of t he initi al ph ase of li ne voltages, lJ'r(/; 6« (sig n lJ'N I) =

i at (jIrn<O

0 :

l!t

fIlH !

As soon 4S th e s awtooth pulse U H I grows to a val ue proportional to a ppearing in th e real ci rcuit 01 t ho contr ol device, a ebcct nega live pulse t rigge rs t he bi ased blocking.oscill at or which gives a puls e of requisite wid th Ii N l to the gate of tbe thyristo r. Therefore, to deter mine t he moments t N I at whic h t rigger pulses go to tbe gates, we shou ld solve s imultaneousl y the equations describin g t he manne r of variat ion of the conduction angles q:.... l and also the correspo nding equations for sawtooth pulse vol tages, In the dynamic an d stntic oporat i~ conditions, t he momen ts at which t h yris tors and diodes begin to 'cond uct are defined by noolincar nonho mogen eous differential equa t iona describin g th a join t operation of the thyristor vol tage regulator end tho ind uction motor. At an arbitrary moment of t ime, the. system may be i n one of the five states: (1) t he on-state of the corresponding t hyristors connected to the three phases a, b, and c; (2) t bo on-s tate of thyristor cells connected to phases band e; (3) t he on-state ' of t hyri s tor cells for phases a and c; (4) t ho on-state of th yris tor teJ ls for phases a a nd b; (5) th e orf-sta te of nil t hyristor cells . For each time interval between t he two successi ve ac tions of switching of 't hy ristor cells, W1) th us obtain tbe solution of a par ti al peoblum: th e-sought-Icr solut ion Invol ves a successive solution of a La rge number of various partial problema. According to t he bas ic pr inciple of s~i tchin g , the in itial val ues of cur rents and fl ux lin k ages needed for the sol ut ion of t he next partial problem are taken from tb e procedin g sol utio n. For t he analysis of a cir cuit dia gram with its thyristors inser ted into the stator circuit , it is convenient to use t he o:-~ coordinate not ation of th e different i al equa tions for a n induction motor, which ccat ein s tator currents and roto r fl ux).ink ageg: Q; /.'l

, u:.. = T .i~ +0£. (d~/dt) + (L",fL . ) (dlJ1~ /dt) u~ = r .l~ + O'L. (drlJldt) + (L ",/ L . ) ( d'¥~/dt) 0= - T, (LmIL,) l~ + (r . tL .) 1¥~ + (d'f'".Jd t ) + tu, '1'& 0= - r. (£", /£ . ) i~ + (T. IL.) lY~ + (d'¥~/dt) - 00.11';

(5.27)

M .= (3/2) P (Lm/ L . ) (nj~ - 'I'~t~) (J tp)'(d(ij. /dt) = M . ..... sign(ij, M. where '¥~ and '¥O a re th e rotor fl Ul: li hk age vector eomp cnenta along t he a: and ~ axes respectively; sign OO,",M, is the load torque of t he deyfriction type; and 0' is t he leakage 'coefficient .

C h. 5. e""9)' Co .....ik>n ..

No~

.. My... VOIt.ge l

i,.

Us i ng t he relation be t ween use lnstOJlt aneo us ph850 cu rre nts j~ . vector components ~ . /11) and instan t aneous ph ase vohoges u~ . y., . (with th e to ta l vec tor components u.... lI~) . oftllr si mp le t ransform a tions we obtain

i: (with t ho total

t,:

d iu

· L....

ct. ;'

u... = r, ~ .. + oL, "'df" + ~ ~ (5.28)

Fro m th e set of eq ua tion! for t he line voltages of a motor. 91'0 get

(5.29)

The ex press ions i n terms .of Ilne voltages are co nv enient for t he analysis o f t hyrl!lors olHlrat,,:d fro m t he uymmetric vol ta ge s u pply. For exam pl e, t he equ atio ns for II t hree-ph ase s q ui rrel-cage ind uc ti on mot or with thy ris to rs oper nd ng in t he stato of t hree- pha se conduc tion A BC ha ve t he form U,• •

UA B . U ob e -

Us c .

u. u

... u e A

Noti ng t hat t he sum of li Q.e voltages is eeeo (in t he case nf asymme t ry too) lolA S

+ U s c + U OA

t ho su m of l ine currents is also eq ual to aero :

.; -f I,.

+ i ., -

5.). The Ind " djio n Motel Sptem

Omitting inter med i at e transform atio ns , we wr ite t he e xpression. for ste t.or c ircuits in t he form u." = (2U "8 u s c)/3 - ~ , t •• uL, (dt ,.ldt) + (L", IL .) (cN'~ /dt) . (5.30):-

' + u' L.---;It di,,, + -r:;:L.. u, ,,= 2hnc 3-U"1l = r,,1> t

d'l'~

d '¥~)

-T.~ +-,- " -Z"BC + UA8 t L d l • e + L. It.. = 3 = r • •e 11. --;n- --r:;X (

(

(5,3'}

_..!..

d 'l'~ _ Zdt

1/ 3 d'l'8)

(5,32)'

2 · dt

Eq . (5.32 ) c a ll be dt soarded since for the phose c we beve It.. = - (It ,. + al b), f H = + l.b)

-«...

For rot or ci rcuits th e equ ations assume the form 0 .... - r. (L rnIL , ) I." (r,I L,) iV& + do/&'dt + ltI,'P6

(5 . .'l3}-

0= - r ; (LmIL , ) [(2t.(.+ [",) (Val + (" , IL, ) '1'$ +d 'Y ~ldt _ IIJ ,'¥;

(5 .34»

The elect ro mag net ic torque eququc n is given by M

- ..!. 2 P

Lm Lr

[1f' (:<

(21.;,, + t. ~ l

'V 3

- 1f"ill."]

(5 .35),

A s imilar a p proac h ts a ppr opri a te in deriv ing t he eq ua tions for t hoind ucti on motor wil h t h~'ris tora ewtt c ned t o ot her mod es of operat ion . Figure 5.7 s hows th e sc hem atic di.a.gra m Hlustre tlv e of t he wa y of connect.ion or the ere tor wind i ngs 19 't he s up pl y l lne via se ri es-co nnected t h yristor cells . when on e of t he th yristors i n a t h yr istor coli TC ie hol d in th o cond uc ti ng s t ate (~'he s wit ch is 0 11), the thy ris to r ('.ell -osrstan ee is close to aero ; when' it goes t o t he noncond uc t ing Slate (t he switch is off), tho t hy rist(!r'cell has a resistance ex tending t o infini t y. B y Kirchhoff's second law ttllC

+ leR e + I"R A -

uo"e. U"e~ -

t"R n = 0 t.R C = 0

(5.36 ), From t he se t or eq uations (5. 36) We ca n deter mine curren ts and! t hen vol tages in t hy risto r cells: _ I R _ " ", " c (IiAIl - U. 1th) - ( UC"' - uuc) RA"n (5,37} U CA

UT A -

• -

R/iRc+ RARs + " c"s

Ch . S, Ene rg y Converllo n at No... ln. .. AIY"" V oltage,

1.t 2

S i mil arl y, we ca n define U ra a nd UTC' T he eq uation! for u r A , a nd U T C perm i ~ lIS to determine th e voltnges across t hYt'istor ce lla ope ra ti ng in au y of t he s ta tes of con d uctio n. These vo ltages are n~ ar y for formul a t in~ t he programs to be r un on di gital co m-

U r a

pulers. 10

devolo ping a ma t hem atical mod el l or t he s i mu la tion on

d igi-

tal co mput er, i t is desi rable th at t he model sho uld most closely approxi ma te in s t ruct ure t he roal

erreuu (If

Ole th yristor d ri ve und er

- , "n

TCA

" ><

~ TC,

~ TCe

u ,~ <.

0, /

U.., .

Usb

1I, . to

â&#x20AC;˘

tÂˇlg. 5. 7, The scheu tic dlqram of a T \'n-iodudioa 1Il010r SYlltem

i nves tiJ:otion. In clrcults whero thyri~to rs con nected to any of t he ph llSDS opera te i ndepen den t of the e ther t hyris tl)ts (for e.... a mple , in -tbe s ta r-con nec ted t hree-ph ase s tator-wind ing net work wi th t he nout ral ), t he following l ogic fu.nc tion co-responde to t he ope-an on of .each IV ,t h th yristor: X1'l IZ ."i1 + Y Nt

h ore t he fu nction X Nt corres ponds to t he voltage ac ross a th yri s tor ; Z /'1'1 corres ponds to t ho t ri a-flltr pulse on th e gat e; a nd Y"" cor ree,p onds to t he cur rent t hrough th e gate; t he multiplica ti on gign denot es II log ic AND (A ) c peratton, and tho sum matte n s ig n a lQi;ic "OR (V ) o perat io n. T he log ic func ti o ns X .~, Y "'j ' and Z ...., are eq unllo uni t y H the correspondf ng vol tages t.urrenu i n a. tbyrls wr a re postt tve: t hey a re eq ual to eerc if t hese voJ;t.ages end currents life nega ti ve in sign. fo t hree- phase networks wl.th t he iso la ted lleut ral wire, a ll t hyr llltors o pera te in an tn teerel eted fashi on. Assu me th e s yste m stays in tbe s tat e of eere cc nd uc ucn. Find th e IOllic functi on ind icat ive of switch ing of the t hyri stors to t he s ta le of t hree- phase co nd uction A BC. On ly def inite t r iple ts 01 thyris tors c.n hore fun c t ion s im ul t.lIft()usl y- t wo co nduc t CUrre nt in one dt reeu on an d t he t hi rd con d ucts

81IJ

in th e op posi te direct ion:

Au - X .u X lUX CtZ.uZ BIZ Ct + X "IX B. X c .Z " JZ UZ C2 XA1X B. X CIZ'I.1Z "2ZCI K" .X D1XC.Z,, :tZBIZ CI X ,uX s . XcIZ".Z s eZCI + X" .X SIXC.Z"2Z S2ZCl (5.38) U tile 5)"!ltem c peretee in t he A BC Inode and t he gates d o not change s ta te, th e l ogic f unct ion for t his conditi on of operlltion hu t he for m An = Y"IY SlY Ct Y"IY .. t Y CI Y "IY n Y c. Y .U Y ,U Y C I Y ".Y Sly Cl Y " tY SlY Ct (5.39) In Eq5. (5.38) and (5.39), t he tue cu oes X ...., are l ogic functions which corres pond t o voltages ur I'>' I lld are round from t be abo ve rormulas for voltlies across th yr is to r c ells. Tbu.s, if " T A > 0, X "I "'" ~ 1 eX A . - 0) ; if If..,." < 0, K " . = 1 (X " I"" 0); if I" > 0, Y A I = 1 (Y" t = 0 ); U i~ < O. Y " t ,.. t (Y" I "'" 0). 11 t he t hyristors enter th e ee rc-eeod ucucn l'\lllion and th e funcnon A OJ Is eq ua l to zerc , so t ha t none of t he t hyristor triple ls s wltohes on, we need to kno whe t her t ho t h)' ristor'S t ur n on pairwise:

An -

X"lX Bt Z UZBl + X AIX C .Z A1 Z C ~ + X A. X 1HZA 2Z BI + X A1X C:llZA IZ CI + X 8IX C' Z B I Z Ct + X nIX CIZ B,Z CI

(5.40)

When A u _ 0 , t he system ma y golfrom th e aero to the two- phase conduct ion mode. Assume, for example, trigger pulses ar ri ve at AI and B250 l bllt u.,." > o and U'I' B > O, but UT" B = U 'I' A - U T D > > O. Th e system t heu s witches from the oU-s t Ate to t he state of tw oph ase co nd uctio n. Consequently, 1lp.llrt l rom t he function A ot defined by Eq. (5.4.0) , we mu s t consi de r lin addnlo na l logic function

B .. = X A BZA IZ S2 +

X " CZ~ IZCt

+ X . " Z • • ZA.

+ X CAZ C1Z At + X s cZ elZ Ct + X caZ c IZ s.

f5.41)

where X " B - I if X S " <:> 1 eX " s "'" O}. At l.I'1'AS < 0, X BA DO "'" 1 All - 0) , and so on. T hill logic l un cUon describing t he condition I t whic h t he t h yristors are dri ven on for t he t.wo-p hase co nd uct io n bas t he form

!!"I!! •• + VAIVC. + !/AtVBI + "A tVCI + Ytll !JC. + " e2VCI

(5.42) T hus, from t be a bov e discussi on t he followi ng co ncl usion c an be drawn. a -O l l tl

Ch. So

e....'srr

CO...... .$lon f1 Monl on. â&#x20AC;˘

Mym..

Vo ltag e .

fo r the three-phase cee due u cn mode rbere corres ponds I l ogic functi on C~ - A u An + An (5.43) T he logi c fun cfi on for th e "tate of two-phMe conduc tio n is C ~ _ (A .. AnB.. An) C, (5.4 41 )

- stands for th e negative ' fu nct ions of C ~; at weeee C, and a t C,,., "1 we h i ve C, _ O.

... 0, C,

= 1

T he fu nction for t he s t l t e of zero conducti on is

Q. =

e,c.

(5.45)

Tho a bove l og ic al an alysis enables 119 t o pass from one set of dllfereo t ia! equa t io ns t o enctber depend ing VD the s ta te of condu ct ion of th yris tors lInc di odes. T ile algo r it hm for det ermln,i ng the state of t hy ris tors is so cacsen as to carry ou t, t he eeq ucn rtal ane lysia for t he t i me in terv als ov er whic h t he pulses on all t hyri.51ors remai n In va ria ble. The sYlt em can cha nge s tate ov er t he given lengt h of t ime if , th yr is tor tuns on (as U1' on t he gate changes sig n) or a t hy ris to r cell turns off (as: cu rren t ch anges s ign). T ho swi t ch i ng mom onta are so ugh t and the corres pond ing values of v ari ab l ~ n e found by di Vid ing tb e iotegrfltlon step in b u f down to H. 1n " . 10-1 s, th en t he Iinell r i nt erpo l atio n b perfo rmed. & th e pulses d ie out o r ne w OOCli emerge , aaothtN' criter ion is I n u p to de termino the presence of pulses on th yris t.ol1l ; the above-desc r iÂˇ bed method is t hoo used t o defi ne t he stale of t hyr istors an d t o conduct t he logiul a OI I)'sls of t'bei r opera t ion. Based on t h is calcul atio n technlque. a generali zed r ou t ine is writte n t o &nalYl e t he dyna mic an d s ta t ic perfo rma nco of t h'e g iven sy stem on a d igital computer for any character of ch Anges in the c.ondu ct ion ongl l"S of th y ristors,

5.4. Pulse

EI~dromec:hanlcal

Energy

~o n verters

E lec tromecha nical de and ac system, belong to t he continuous type. There are d iscrete electromechanical S)'s t oms where t he eonversion of energy proceeds by way of the pul ses of elec tromag neti c powe r. T he las t few decad Ol' h av e s een a cons iderab le ad van ce men t i n th e area of d iscrete elec t ro mecha nical syslf:D\I. Pulse ene rgy con verters ope rat e both in t he moto ri ng mode (ate p moto rs ) and in the ae neutin g mode (puls e ~enerators). Ste p mot ors tr enerorm vol ta ge pulses into discret e an gula r motions or s tep wise linea r d isplaeem en t and pUlse genera tors prod uce powerful curren t pu lse! fed t o el eetropb ys ical setups. Step moto rs have a s mal l power , us uall y up t o a few hu ndr eds of watL!!, and pulse gllDerat-ora are

SA. Pu lse Eleci,omed,.." ica l e ......gr Conv,,"""

ll S

gen er aJly b u .i l t t o g iv e a h ig h o ut put. u p to t ons of m ega wat ts in a. pulse. A s tep m ot or is m a de com plete wi t h a co rn mu t ating e r renueme ne and designed to c a rry a load on t h o r otor eh a tt. F or eac h mo tor t here' is a d efi nite s wi tch i ng freq uency a t w h ich its r otor- fol low s in step a v ar y in g fi eld in t h e ai l: gap . This Ir equoncy is blow n as a resp onsef requency d o termiued b y t he enure s ystem com p ris ing t ho sermcn uducto r co m m utator (pu ls e gone ralo r) an d t he step m o t or . an d a lso by t ile lo a d all t he s haft. Most step mo tors are multipolar mul ti p hase synchro nous m ech moa in which e ithe r g r ou ps of s ta t o r wi ndings or ea ch win d in g recelvoa unipol a r O f t wo- polarity puls es . T ho r otor re volves ncu u n ffor m fy wit h i n the en tire I\ngl e of ravof u rron as it follows th o step wis e d is plu ce rnunt of t h e m agne Uc fi eld. T h o m o to r lncc rp orates s pecial me a ns o r provision is m a de i n t h e m o t o r d03ig n t o fix t ho r otor in a def i nite position to rogisle t' t h e response . Cho nging t h e or der III ar r-iv ul of pul ses , it becomes po ssi b le to al t er the s ense of rota t ion of t hc ro t or Mid t llu:;l s u m \ I P the positi.... e a nd neg ati ve p ulses in t h eform of angu far dcnecuoos. Step electro mcchani ca! syst em s , l ik e all ol ec,l!'ir, m a chtn es , are con vertib l e. T ho }' can ac t us s ou rces o f low-power p UISM. A sl i lll por laut tusk is t o produce h igh-po wer pu lses u p t o 100 kJ wl th n steepload in g edge lind a lligh repeti tio n rete, or PU ISflS o f a defini te waveform . P u lse g en e ra ta l's must st ore e ne rg y i n ure form of th e k in e tic (' 11O.-r gy of ro tati ng me m bers a nd i n the form of t he energ-y of rh e m agn e ti c fi el d . O n t ile on e hond , 8 pulse ge nerator mus t. b e c a p a b le of co n t ro ll i ng t h e energy eto eed a n d , on t h o other. mus t h av e 8 sm al l time cons t an t to pr od uce d esired p ulses . These are con fl ie,u llg requ t rc me uts fr om t he ....ie wpof nt. o f elec teomochanlcs. Diffic u lties o ft e n ar-ise in d es ign ing a magnetic-Held ty pe pu lse ge ne ra t or a d equ ate to t ho r equirements i m posed o n it. It t.her-efo -e makes sense to reso r t t o p u lse g enera t ors of t ho electric-Hel d and electromagn otte-fiel d ty pes. Supercnnduett ng m ag net tc s yste ms orretmu e h p romis e as de vices for sto ring l a rg e e n cigtee in th e m agnetfc Held , Tho epproeches an d eq uat ions ap plied in t h e ano lyais of o l'd in a r y energy co nv ert ers also ho ld for t he s t u d ies of al ectrcm ech anle a! ener- . gy c on v ers io n processes ill pul se en ergy con ve r t ors . T h(\ form of e q un t tons fOf pulse vo ltll.gC.'! remains uie s a me us for s i n us o id a l v oll nges . I n t he si m u lati on of s tep moto rs it is common t o s im pJify equ atto na so as to t ak e i n to accoun t th e ch nr -ec t cr- o f l oa d a nd t ho in tcrll lli re efsr a nce of the s u p p ly sou r ce . Continuous s in usoidal s u p pl y v oltages ca n be t h ough t, o f Il ~ anin fini tel y l ong trai n of puls e.'! v or y i ng in ampl j tude. As the \' ollllgO'" deviates fr om th o s i n us oi d , t he ro t or rovo tves at n n onu n ifo r m ang us-

l e e s peed; t he motor ha vi ng I'l defin ite form of the fiel d d ispl a<:ing i n â&#x20AC;˘ d efin ite ma nner is a SlOp mot or th at trall!lal9!l co nt rol .!! ig rlab into the d esi red a ng les of re volution . So, as mon t ioned ab ove. t be .!!AIM eq uations an d the sa me la W! hold for descti bing con vent ional pulse gene ra lONi a nd moto rs. Al:ro. t he approach es dis eo..ued in t he present boo k work for t he an al ysis, of t be pe rfor man ce of t hese mach ines.

:C hep ler-

Multiwindlng Machines 6.1. The Equations , of MultlwJndlng M aehin es A ll dectriC machj"~$ ca ll be regarded as mull/windi ng (multtloop ) ma chln t$. The magn e ti c: COrell of eloetri c lnaeh illQ$ gene ra ll y riavolcp eddy currants which should be taken in to eccoc nt in the ana lys is of elOCltOllllttbanic al e nore ~ cc nve rsrco pro CC$SClI. Mos t of t he real ma ch ines ha ve a few Windings . An elee tri c m achiue ha vi ng oee wind ing (i. e. two ph as,e windings) o n t he s \a tot and one wrod i ll# [i. e. t wo ph ase wi ndi f1i5) o n th e totor is a ptil:nit ive machi ne adopt.ed i n t he a nalysis pre cr jce , t he const ruc tio n of which c alls for a rtlth er large number of ccnsrra tnrs. Sy nchronous machines ha ve a damper win ding a nd III field win' d ing on the rotor. Diuegard lnll edd y cur ren t loo ps , t he mo d.,1 of III synch ronous machi ne cerrillS on e wi nding (h -I,) phase windi ngs ) on t he s tato r a nd t ..... o wind il,gs (four ph ase win d ings) on t ho roto r. A de ma ch ine has win di p.gs of co m muta t ing poles (co romut ntin g wind ings) and a co mpon snthlg windin g; i ts fi eld Windi ng may COI\s ist of a c ot! of s hu nt (pal'nUo!) exc ita t io n and n co il of series exoltauon. With al lowenee mad e 'fQr ed dy c urren t 10(,1 1'9, t he de machin e is a t hree- to Ifve-wtndin g machine. A cross-Held con tr ol generntoe he a con trol wi nd ing eo n..., i~ ling of a few coils, Indus try produces in d uction ma ch in es wit h two windings on t he s .....to r an d t wo or t hree wi nd ings on t he roto r. These are mu l ti ple s peed mec htnes wit h pole-pllir changers , wh ich cons t it ute a wide cl ass of dOtlble squ irrel-caae mach ines. A d eep-slot macb ine can els e be cl assed with mu lth dn d ing mec;.hines H we con side r t ha t ;a few para ll el-co nnected cond uctors ov er t he slot hei ght form sev er"" l oops. In t he process of s ta rt ing of a n inducti on moto r, the cu rre nt s prea ds ou t non un lform Iy over the sl ot heig ht beca use of its displa gement. T bi! fa.ct can be taken int o atW UDt by sol vlDg equa ti ons for n p aeal lel -eena ected conduc to rs ple C8d o n t he ro to r.

A SUitor winding is oftell built u p of â&#x20AC;˘ few parallel-eou nect ed em ductoes. Solving t he equatio n for a multiwind ing mach ine. we c.an a llow (or th e non uniform cur rent dis l rib utio n over the co nducters , if thi ! is t he elise. Elect romagneti c conve r ters, t.e. transform erg, nrc built Wi llI t WO or th ree windi ngs. Th ey Me else mill tiwindl ng energy conve r ters If eddy CUrrent loops are ta ken ln te accoun t. In Sec. 3.2 were gh'en t he equ.tions for II hypoth eti cal m~n ""indiog mse hln e. As a pa rti cul ar CMe, ""'e ca n use these eqUitions to Iermula te th o equations for any muHiwlnding ffi ae.h ine mel with in practice. I n th o equl'lioos describing the prceessea of ene lll'Y co nversion In lIlul ti wind ing machines. mu lUll1 ind uctances dele rmi nl.' t he link bet ween wind ings, each wlnrllng being assu med to oxhi bit a defin ite magne tic leak age. T he eleet:romag netie torque Is defined as t he produ c,l.1 of cur rents enteri ng into t he equations. Unde r these cond iti ollS, Eqs. (3.3) t hrough (3.12~ attl th e equations of mnfti"'indlng Inae.hin es.

6.1. The Equ. tions of Synchronou s Machines SllnchronOlU m..d1i/'le~, both taltm t-'pol. and /'Ion~ttefl t-pole tllPQ, Ihould ~ treated as multiwinding machlnu havIng an arma /ure wind

...

1'11/. G. \. A IIYnt.hto nouJ m."hi ne

..~ -1>''''"\'''. .. In d ln . : wofof lin d "11'1-

ofÂˇ a nd

'I:'"~ IO

d Gm pe , w ln d ln;O l

"'I _ h lll d

"lllG H.1I"

dIng, field windin g, and dtJ.m per windi ng (IS shewn in Fig . 6.1. In t he si m plt!lSt case. 815)' l\c hrono~ mechln e is a tb re0-wind ing machi ne. its mod el h eing represented hy si x ph ase windings .

118

Ch. 6. M"tti .... lnClng ~o:hl n .. 1

Il is customary to ex press the equations for II e ync h eonous ma -ebtne in the sy stem of d-q cecrdinetea rig idl y conn ected to t he roto r s t ruct ure . For a ma chi ne with tra nsform ed windings we wri te down the foll ow ing cquaUoll!l

u " _ doF,i dt - 'I'"w,

+ r..l"

d'Vt/dt '1',,(1,). + 14./ - d"o/ldt + 71l f 0 ... dWd,idt r,U ' '' d 0 _ dll' ",/dt 701 tl ,, ,

14., '"

r.',

+ + H er e r", an d r, are th e fesis tnn,ces or the .o.rmaturo and fie ld

(6.1)

win d ings eeapect ivel y; rdd and r", are t he rostsr en cce of the d amper wind ing, r.e. t wo ph ase wind ings al ong t he d irect a nd t he q ua dr atu re a xis res pective ly ; 14." an d u q ere t he :voltagos ac ross th o a r ma t ure wind ing , t.e. t wo phase win di ngs alool t ha d a nd q exes r&pect ive ly; end Is Ula voltage i n t he fiel d win ding. T he m8.i'notic nU l: li nkages Ior the w i nd i ngs are 'given by ' .

•

If'..

c:o

•

L oI ~ oI

+ M odi, + At . di ll"

+ M.q'"q 'V, ... L Jt, + « , », + iII. II' . " 'Y"" .. L I .'•• + .+1.<tio! + j\f.d ', 'Y II" = L II" I " , + M Oqt, 'V, "'" L q' ,

(6.2 )

H ere Lo!, L" L" L .", and L II , ar e t he in du cta nces of j he ar ma ture, f ield , an d da mp er windi np res pec ti vely ; and M o." a nd AJ", q ar e the mu tual Inductan ces of the wl ndi ngs alo ng t he d end q axes . T he assum pt ion is t hat t he mutu a l '~ l!-du ctances between t he windi ngs l ying a lo ng t he same a xis tire iden tica l, whil e tb e leak age Ind uctances of windi ngs aro _d lUeron t . T he el6Ctromagn8t ic torque is defined in ter ms of curren ts and Il ux linkages (6.3) M . - 'Vdl "

- lV ,'.

or in te rms of c ur re nts ftf. _ M (IIi"

+ l,l ll"

i .l", )

(6.4)

T ho equality (6. 4) 15 .....ri tten for a nonsaHent·pole m achine. For a salient-pole ma chine. the equ ll.Hty includes e component to ellow fo t the d ifference between the permeances alona the d a nd q axes. T be preducta of damper winaing c urre nts and currents III. and t" g i ve rise 10 a n as ynchronous to rque. In lb e s teady-state cond i tions t hese products atO res pons ible for rotor hu nti ng.

tl9

T he eq uatioWl for a sync hro noua machi ne can be de rived fro m t he equa t ions for the genaul bed energy cc nve rt ee if we assu me t hat t he lewature winding ro tates and the field and d am per wi nd ings are at. lionary (Fig . 6.2) . It is adVlDh geoua t h at th e ma th emati cal model should ha ve a mioimu m n um ber of rot a ti ng wi ndi ngs s ln ee t he simula t ion will then req uir e a fewer mul t ipliers. T he t ran sform a t ion of windinge t o t he armature winding s hould be don e in tbe s lime d way -.s tor a n ind ucti on mech tue. " In modeli ng s yn chro nous ma chines i t is more judici otl! t o use per-uni t uni ts . T he besrc q uan tities are Laken t o be t he a!lauJ aI" veloc ity Wil _ (0) • • s t ator cu rren t t. _ J m ., ~ , and vol tage II. = = u......P ~ . We t hen ha ve t he flu x li nkage '¥ I> - U bl (i) b. i m ped ance : 11 - u ,,!i ,,. and ind uc t ance L b =

....

+_ =1l".: +A

= ; b! CJ) b ..... :1:.. ".

U 61i 6 flJlr

Here i}f G,, -

ill." - z .. q. 1.." - Z", L , - %}> L .... - % .. " ,

'" f"Tq

L, = and L ,,'l. In mOd eli n~ pc rm an entrmlgnet 9'1s. 5.2. The model of « 'f1Id u'OftOla _ hiDe ' ynclironou.s mach ines , t he dee-rea of exci t a t ion of a magnet is delined as t he p rodu ct of cu rren t J ... tl me.a t he mu t ua l Lo..du ctanee of t he a rma t u re windi ng and the fi ct itious curre nt of t he magn et. Pe rmanent mag nets a re re presented by an eq uiva len t inetti aless and Ioaalees loo p ellppli ed fro m a dc sou rce (see A ppend ix I , Fig. A6). W ha t co m pli cat es th e s i mu!at ion.ol sy nc h ronous machi nes la a eet urati on -ind uced chaege in the parame ters wit h load . In s el t en tpol e machines, th e d iapla cemem of th e fiel d nxls with respect to t he nu of poles a dds st ill more to t ha complex it y of t he model. T he t ran s ien ts were giv en tre at ment fi rst i n s ync hron ous mec hi nes, As fat bac k as t he l a le t920s P ark an d G ore v fonn ula t&d Eqs . (6.t ) wh ich now bear t.he names of P ark an d Oeeev. T he need for the s t ud y of t t ans ien ts in s ync hro no us machi nee a rose in th e cou rse of de velopmen t of power sup pl y s yst ems; t he aim was to in vesti g ate the effect of t he emerg en cy opera t ion of one O1ncbino on t he s tab ili ty of parall fll o per at ion 01 ot her llI,ach lnes. Since com pu ters c ame into being m uch la t er, it was Im posatble t o s elva E qs. (6.1) l or the oese of a vary ing ang ular veloc it y. Tho onl y wa y out was t o si m pli fy t he in it ial equat.ions to descri be the main e ven ts dete rmini ng t he beha v iou r of a mac h ina !in condi Uons co nside red most im port an t l rom t lla pr acti cal v ie w poin t. I n perfor ming th e anal )'Sis, it W all D&ce5!!ary t o define I set of param, rers cbarl cterb ing th e mac.hin e

%""

Ch. 6. Multiwlnd lng MachlrlO'

o pera ti on. Th e focus of att$ t.ion W/lS largely on t ransi ents in th e condi tio ns of suddenly ll p~lie d s hort ci rcuits. With th ree-ph ase or asymmcteic short circui ts: pl aced on sta tor windings , t he s urges of curre nts i n t he windings excee d 10 to i s t imes the nominal values. 10 solvi ng Eqs. (6.1) wit hout the use of a computer, the armatu re windi ng res ista nce is usu all ~ t aken equ al t o zero to m ake Ilu x li nkages cons tant . Thi s .assum p~ion facilit a tes the solu tio n to th e prcblem , but leads to an inconsis tenc y of th e analysts and to a number of contrad icti ons. ' I A s hort-ci rcui t t rans ient pj occsa includ es several s tages . If a mechtne has a damp er -winding, t he first s tage is determined by a direct-ax is subt ran slent inducti ve reactance

'r aY

x d = x Cq -Ti ,,"':;C";'-;: ;C:7: 1IZqd+1 /%c/+" I!% cdd where X C q is th e lea kage i ndu cti ve reactance of the armature windin'g; ' Z a d js' t he direr t-a xisl in ducti ve reactance of the arma tu re; xal is t he leak age inductive teacta llce of the field winding; and z"dfl is th e dir ect-a xis leak age ind ucti ve reactan ce of t he da mper winding. The peak sh ort-ci rcuit cUlt en t here is

1, ,,, .0=

E m/;r~

where E m is t he peak phase -emf. The second s tago of the shqrt -ctrcutt tran sient s ta rts at the in stant when the 8rt-latur'a flu :!: pessee th rough t he damper winding and hegins to travers e the field wind ing. Th is condition of t he mach ine is 'defined by t he direct-ax is trall s'i ent rea ct ance

x' -x <I -

a ,.

-ii-; 1!X ,l+1 1/%111 a

The s te ady-s tal e short--ei n;uit curre nt is a func tion of the directaxis inducti ve reactance ! Xci '= x,. a Xqd and is equa l to , J:; = E ",!Zd

Tile effecti ve valu e of short-circ uit curr ent is found from I::, t he tra nsient an d subtrans ient abort -ci rcui t curren t. The deca y of tho tran sienv curr ent is defin ed :by th e direct-axl e t ransient time conslant 1' ~ end t hat of t he subtr eus ien t current by th e direct-axi s subtranslent time const an t T;i; Ij.ere T d > T~ . all t he cu rve of t he short--c ~rou it current we call s ingle out a perio dic and lin ape riodic component, The ape riodic component dies aWIlY wit h the ar mat ure tim e constant T a which depends on t ho armatu re indu cti ve reacta nce end ar ma t ure restste nce,

6.2, Th_ EqulIllolU of Sync hron o uJ M IIChl nu

l2 .

T he aperiodic com ponents of armature wi nding cur rents produce & fiel d th at is s t ationar y with respect t o. t be ar matu re, t herefore quadrature-ext e loops als o take part in th e t ransient process. Consideration Is th en given for t he t ransient and eub tra nstent in ductive rceccences alon g t he q ax is, Tbe g-ax is sub t ran si ent ind uc tive .reec ta nce is

~ _x q

+ 1/ .1"0'1'+1V~o dq

where .t.'!' te t he g-ax is ar ma t ur e reactance; and leakage reactance of th e damper winaing. T he q-a" is transient reactance is z~ =

Z o"

+ Z ~ 'q

ZOdq

is th e q-axi !J

= z"

T he a pe ri od ic com ponent of t he ar mature c urrent oscillates lit do uble fre quency between t be currents E",/x~ and

e.,./x;

T hi s a nalysis of co m plex processes in a sync hrono us machine ca llsfor man y ass umptfous . Ne \'cr tbc less . H di sclos es well t ho phys ica~ pr ocesses and gives suf'Iie.lent fy acc urate resul ts. In the an alysis of the stat ic and d'yp a mic s tability of t he parll.ll el operation of ayn chro nous machines , wido use is m ad e of lin e<l riz ed> incremental equa tions. The Incre ments of variables are ta ken l ine ar and t he t rea t men t Is g i ven to t be modes of small osc illat ions, T be s t udy of sta tic s ta b ility on t he basis: of s mall harmoni c perturba Lions is ju s tifia ble ein ce ill t he pt olJlems involved here account should be t .aken of the parameters of t he su pp ly line Slid electric machines and t ran sformers ope rated ihto t ho anme network together wit h t he s y nchronous me chtnc un d er :llnllly~ i 8, w The cr ea ti on of tu rb cgenerat c rs of n'IHli t power of 1.Âˇ2 to 1. 5 m in kW in the last years and the emergence of more comple x powe r systems ha ve ra ised new problems rel arli ng to the s t udy oJ t rens tents, in sy nc h ro nous machines. There is Il. need for II more rigorous en alysis of t ransien ts i n asyn chronous ccu dr u one, a t rt's t ar tiJlg of otterna to rs, d uri ng rou gh s ynchrouizatton ; lind in ot her emergency conditions of operati on of synchronous machines ill t he power sys tem s. As regards t he anal ysis of t rllilsientp in synchronous machi nes, of much interes t is t he illves LigllLioli of t prsiona l vibr-atlons of t urbogenor a tor s ha fts und er \'8ri 008 emerge ncy con d itions with cons tdernli on for t rans tents in the power s yste~. I n the ana lys is, t ho el ect roma gne ti c torque and t he momen t of ,i lJerli a are taken to he distribut ed fllolIg t he rotor leng th . The eq ua t ion of motion is t hen solved alm ultane ously wlth the vollll.ge equatio ns, A mos t jud icious a ppro ac h to Invea tigati ng t he d ynam ics of aynchronous machin es is to lise com puters for th o soluuon of equll li otl8-

Ch. 6. Mutl iw lndinQ M-.:hl ".f

In con junct.lon with t rad itiqoall y adopted tec hn iq ues. The guideli ne for t he anal ysis of ulien t rPole and aonaal tent- pole ma chines wit h t hree rotor win dings an d t....;o stat or wind ings s hou ld envisage the U!8 of universal rout ines to ~e dll!llgned for so lving t he 8t8 ady -sta te a nd t ransie nt perform ance on digit al comp uter.l. h ill of much im.portauce lO t.e5t a nd make th e right choice of models obtai ned by t he meth od of experi men t al 'design. Th is wUl open u p new (Kmib lI n tes for st udyin g th e perf0f,mance of synchro nous machines.

ft.J. The Equ~Hons of Direct Current Machines Direct eurrmt

IJIllch int,

are mubtwindtng

rnachl lUlZ.

Most tdI Mtatle

d14fN1m1 of rtol tk motors lind generat or, I:on be trruu/orrMd to II , {napl l/u d modtlsuch as ilI listr at.kl in Fig. 6.2. In the model of Fig . 6.3, ;

-. -, q

...he a rma tu re windiog is sb;own t.o co nsis t of tw o symmetric win.<fings s ud t be field wind i ng is seen t o co m peise I lle pal'at. -eaen s ucn winding III. and ,; series exeneuen wind ing w. on t he s u o t or al ong t he d nis. Sho"m etec (I ll the figu re a M a co mpellStltl ng wind ing we and an i n te r pol&. (com mut a t ini ) wi nd ing WI on th e stator along t he q axl:5. H is or ee e ven tenee t o ex p ress th e equa t ions rOf de mach io85 in "the ~ coo rd ina tes , jus t li ke t he eq uat ions ror s ync hronous ra e-cl1ines . The IIi mu tauco O R a~ analog co mputer involves t he represem e-

III."

III.,;

6.3. The Eq\lalio ns o f ,DC Machin.. .

123

non of problem vee tabtes by dir ect currents and voltages. Nonli nea r coupli ngs make t he modellug of a de machine a ra t her complicated prob lem, T hese cou pli ngs ar ise from aa tu ranc n, q- an d d -a:d ll tra nsient arma t ur e rea ctances , ccm m u tat.in g armatur e rea ctions, a nd also as a result or eddy curren t iuflu enc ea. It is impossibl e t o allow accurately tor all th e abo ve facto rs. T herefore, in the s t ud y of de mech tnes , t.he cpen-circu jt characteris tic is taken ltneer and t he

u, Lo

FIll" 6.4 . The equivalent eln;ult

{ (l r

II de machi ne

parameters c ons tant. B ut s uch ass umpt ions do no t afford t he des ired accu racy in view of r ho as y mmet.r-y of fiel d (s t at or) wind ings. Apa rt Irom t he mut ua l in ductances! III Ie. 10[ 10' Jl1 IG' J-f. o. and M I c bet ween t he wind Ings of Fi g. 6. 3. t he clecrromechontcal eq uati ons mus t contain mut ual ind ucta nces Mg ~ ., M g~ . , and ftfo ~c due to the q-axis transi ent reactio ns of t he ser ies e xcneuon wind ing w" separ ate ex ci tation winding w â&#x20AC;˘â&#x20AC;˘ and co m pensaung wind ing w~; M o do and M Gd . due to th e di rect- axis u-ans tent armat ure reactions ; and J,f < ~ and M e. due to ti le com m uta(ing reacti on of the ar mat ure . Us ing the ci rcui t model of Fig. 6.1i a nd elect romllg noti c interactions bet wee n t be windings , for th e armature and exc f ta t tc n (field) windings of a generator we wri te down , th e following equa elona - UI = - e r e . + 26ftb + R~ l. + L~ l (dfd l ) to - M . a (dldt) t . I'. _ R ; i. + L . (dldl) t . - M . p (dldt) to (6.5) Here UI .is t he lus t.antaneous va lue of l oad voltage ; en ' is the rese tta ut emf of rotat io n: 6u/) is t ho volt.ngn dro p across a brush con tact ; R~ = R .. + R . HI + R . is t he Ilr:mot llre ci rcui t rest srance equ al to th e sum of resrerences of the llrm atjure. ee rree, tnter pcle, and co mpensating Wind ings res poctlvely: La is th e current in t he ar mature ci rcui t ; I. is t he cu rrent in th e exci~a t i on winding ; L; ls t he total

Ch. 6. MiJlIiwinding Machines

arm at ure circuit Induota nca eq uat to L~ = La L. + L, L , + 2M I t - M 1<, 2M 1 • ± ~lI-1a d ' ± 2 M , . - 2M ,,'1' + M,, ~ . The ter m M " in (6.5) is l ho!coeflicien t for t ho back mutu el induelance between the Armat u re land cscttauon ctrcu tte: .~I _0 = lllf . d - M,,'1 " Here M oJ .re t he c~effident for the di rect mu tu nl Ind uel ance bet ween t he a rm ature an d excitation olrcuits: M . d = fll • • ± ,\J ad. ± A-l •• - A-f l • + ilf, q .

For >I de gener at or t he 1I1!g-lIla r vcloc fty call be ce nstdered CO Ilstan L T his enables us t o d i.~reg i! rd t he equ a tion s of mot ion and limit the an alysis to the sBJutio n' of voltage eq ua tio ns. l it the ana lysis of t he d )' Hll m ic pcrfo r mauce of a machlne, it is of i mportan ce to determ ine correctly t he emf e, .. as a fu ncti on of cuerents an d to ta ke tntc account t he magneti1;ing and demag nf.'tizing forces. This can Il l.' do ne b)" use of t he 8- H cu rv e. t rnnsie ut resp onse, and relatto ns ((j.B)

wher e C. and em are th e coe ffidell t ~ that aceuun t (01' the cn eeecteessti c featu res of the mac hine ! under analys ts: an d (f) , ... is the result/l nt Il ux determined from tho transient respons e. Th e a na lys is or tr ansient ' ch ar acter jsuc s on an pnalng com puter req ui,-eg construc ti ng II cOffi .l{licflled ma th ernauical model or necessit ates t he si mula tion on a digit /ll co mpu ter. Al though t ile s tea d y-s"t/l t~ equa tio ns for l\ de ma chi ne ar e t he sim plest , t he stu d ies OIl t he d ynamic performance of an asy mmetric m achtne involve t he soluli qn of cu mberso me nouhuene equ ations . In Appe ndix I (see Figs. 1\:5 and A6) lice shown t he block diagram s of models for the sotu uon of equ at ions of de m achines. ..\ s mentioned earli er, dc ' mach ines diffe r from synchronous much i nos in Lha t t hey have cnm mutatcrs t o rect ify t he al te rnat.ing em f generat ed i ll ur e arma ture {\'i n dings . A mech e utce l freq uenc y (""O Ilvert er , or co mmut a tor. keeps n r igid t ie be tween th e anglll nr velo cit y and elect r ic frequency. while il semtcon duc tcr freq uency cneverlet may fl fford n flexible ti c, Despite th e fact t hat sync h ronous machin es have much in common wit h de machines, t he the ory of either of the t wo classes con t inued to develop se par ately for m a l~ y yea rs; com mutation p recesses were g iven treat ment in COHj U ll~ ti9 11 with operating p roces~es in 11 machi ne. I n consideri ng t he processes of energy COil vers ion in t he air gap of a dc rnachine, i t is q uit o jus~ jfiab le to employ phasor d ingr/lms and uquivalent circuits arte r red;uc iug t he multip hase ar ma tu re w in di ng to a two- phase win d ing . i n ~t s classica l deeign, t he de machine is II aalt ant- pole machi ne wit h a ;s lllt iollllry field winding. However , COD-

6.• . l ite Ooub le Squi ". I-C_

lnductioft M.oIo.

t roll ed- rec Ufiar com mut a lor mac hines widely use s ta tio nary ac w ludlu gs. N"o" stl li~lI t .pol e ue machines wilh a cn mpeastlt io, wind illg someti mes fi nd use i n practice. T he geuc r,Hzed 3pproa(:h to .studying sy nch ronous and de machi nes will promote (urUtcr the t helr ry of elect elc mach ines.

6.4. The Double Squirrel -C::age Induction Motor. The Effect 01 Eddy Currents WI! rest rtc t our IHUml ioll to t he eq uations or t he machine with o ne s t aLor Winding and two roto r wind ings . T he ma thema Lical modoJ or t his t ype is ap pli cable to t he ana lys is of I wido etsse of ml<l!ti'\cS wit h a double squl erel-cege (s lior t-cl rculte d) ro to r a nd t o t he ccnaidera aie n o( th e effect of edd y c~rreul5 i n t he roto r. Let us defi ne .....h.t one s hould understand under ed d y- curre nt! In • Ci ge rc tc e. I n t he prlctice or desill'U or a mot or . account is gener al ly ta ken oCthe &(ti oll of 0 00 short-circ uited wind i'lg made u p of bi tS Ind ond ri ngs . T he ca lcu la t io n t ech nique deals ....· it h A'I i dellli ~ed ph ys ic.al process s ince it d isregards t~e fact t hat t he rotor is currenteo ud ucting alld t he Win di ng is i n elect rica l con tact with t ile core. This a ppreec b. gene ra Hy perm issible. does not work where t hore is a need (or an acc ura te c elcule t icn of th e ~ t ll r t i ng cxaracccnsucs of a mot or whose roto r cu rrent Ireque ncy becomes co mparable to t he SU[)p1f Ilnc freq uenc y or ev en twice 'as high in plug rcverst ng. In high-freqllllllcy mojeee deelgned'foe "olt ag'lI.!I a t 400 to 1 UOO H~ t he effect of ed dy cu rren ts is not teenble. If we remove tbe cage wind h\~ (rom t he ro to r core. t hus IOIlv; ng the s lot.s em pt y, a nd t hen ec neecr t he s t a tor to the Hue, t he ro t or \\' i!1 r un lit II stead y s peed even wit h the ebeec-siee t la mi na tio~ insulated from one another. Strappi ng t he s hee t-s t eol lami natidns t ha t form a coro w ith co p per bi ts b y tight l y em bed din g t hem in tO t he slots . we mak e u p II ectce struct ure, yet with ou t t he end rings ,' t.e. with t he cag e open-ci rcuited. T he mc toe so bu il t den lopa I ~ ubs tant i al to rq ue. co mes u p to a s le ady speed , and can ea rry a loaa , abo u t eu e-ah trd of th e r ated load. In t he firs t cage co nerdeeed abJve. th e rot or keeps going under t he action (If ed dy cur ren ts i nd uced ~ n each s tool lami nat ion of th e to re; in t he s econd case, th e rot or t ur ns b y t ho act ion of ed d y curre nts in s loel-bar-! l eel l oops . Such lqops exter In a n y cage rotor a nd hev e II pr onounced el!t>ct since t ho rot or CONI l am inat ions are usual ly not isola ted from cue an ot her e nd all bats ar e soli dl y co nneet eil t o end ri ngs. com monly h ~' alu minum flow br u ing. T hus . a.5 regards its equi valan t cin;u it . a sina le s quirre l<(;Age me ter Is s lmil ill' to a tu nslorme r wit h one p ri ma ry and olle secondary, but on ly if the rot or C!tie windi ug has no electrical contact wi th sheets teel l a mina tions and the rotor core ~s made u p of t horough ly Ineulated s heets. Where t his is no t t he ease , t he motor circui t mod al

Ch . 6. My ltiwi nd jn g M. c hl.....

mus t be an alOious I,.() t ho c i r~ui t model of a mul tiwindi ng t ransformer, In whi ch t he n umb er cr-seeoud a rtes is eq ual to the number of l oopll un der al udy, or to the Q reui t model 01 a t ransfor mer with two seconderles if t he properties :Of All eddy cu r ren t l oops in t he rotor a re a menable to generaBulion to those of I n in tegral eddy curre nt wind ing. 1 T heoret ically, t he generalbed electromechanical energy cenven er per mits considering all th e ' adol y of ed dy curren t l oo ps if eve ey l oop, incl ud ing t he l oops in Ind ividual la minatio ns of t he stacked CONI, c.ao be lreatod WI an iDjdepe nden t ".-indioS' with iLol own par am olcts .

Consi der th e equations of 'an energy converter h. ·.jog one sllll to r 'A·jndi,,1I' and t wo rolor wind ipgs. In the trlnsforme d coord in ate syste m u. II re volv ing in s pace .at. e n arbl t rllty volocity we. the eq uations for l hc case of i nt erest ne ve the for m

u'" "", tPl' .. ,ld} - w. lf"." + Rei", ""d'r~lldf +W. If""1+ Rei"" 0 - d'V...fa, - (W e - v) 'Y.. + R lTf... o= d 'l'~l/dt + (<..Ie - v) 'I'll' + R 1.4, 0 = dlf1 o,u/dt - (0). _ v) '¥" + R" i". 0 = d'¥",td t + (co" - v) '¥.., + n2. i~,

U.,

(6 .7)

t.... -= D" ,1D .

1"1 """, D../ D ,

I.., - D..,I D

(6.8)

where D=

D..1 =

:1:,

.:r..

r. -'x.+ z')

x, (x. + z')

1lI,W",

folo;\(I...

"'"' 1 %,

(%0 + %')

<tl,1f"". (.111 +2:') x. Wo IV", r. ,"" V. x. lOolV~ (x. +%') r. ~\(I.. %,.. z.

Vq =

r.,

x. %"

x. (z.+ :r;') (6.9)

6.~ ,

The Dou bl e Squ;" el-<:ag" Ind ue/io n Mal o,

where (o;)O'l"1

DOl "'" (l)01J'""

.1:"1

(%o + x ' )

(Oo'l'.. (%o + -Z')

Do,=

z. wo'1'....

Xo (,)00/..

(rp +x') X"

%0 (,>00/..

Âť, D"l =

X' I

WU'+'"

X' I

Iw olf..

(.1'0 +;[:') (l)01Y...,

T he evsrem (6. 7) i ncludes th o voltage equations roe t he etetor ond for tho fi rst and t he second rotor cage; t he sys tems (6.8) and (6.9} Inclu de t he current equarlous fOl' windings on the u and /J axes respectivel y. In t hese eq ua tion s , 11' ~ 1 and '1'" lire the flux Hnkagll'!l of t he sta tor a long th e u a nd v exes: '1 ',1, nud \(1.. are t he flux li nknge9 oj th e fi rst rotor cago along the I' ar,d II axes : 1J'~, and Iy" are t he flux link ages of t he second rotor cage along t he and IIe xes; R . (R , ,, R ..) stands for resist ances of t he phase in volving t he etatc e and rot or wind ings ; x . (x,!, x,,) st ands for ind uctive reactances corresponding lo th e total inducta nce of t he pJ'8Se ~llv o l v i ng t he stator a nd rotor wind ings; x Q is t he reactence of m Ul u ~1 ind uct ion for a correspondtng mai n field of t he machine; ~' pair of windings, which is due to is t he reactan ce of mut uel induction lor rotor cages; and v = de /de Is t he angular velocit y of t he rotor. I T he ana lys is of the most popular t hree-phase induction machine with two roto r wind ings culls for t}\e formul at ion of t he relat ions bet ween self- and mut ual induct ances- of windings and use of the desi gn parameters ap plie d in th o thcon ; of electric machines , The relations aee obtainab le from t he compnr tsun of t he electromechanical equat ions l or t ho s tood y sta te (w, is conatan t] with t he classical equations for a t hree-phase double-cage moto r: x . = (j)~ (L1 - .;11 ) %~ = (j)~ (L . - .111. ) ~ = "' ~ (L. _ ~f.) (6.10) ~~ = (j) o (312) J1.f'o x' = (o.)~ (312) (At' u - M 0)

tJl1

whore L I is the ind uc tance of the s ta tor phase; L. is t he ind uctance of t he rotor phase (first cage); l...3 is tlje induct ance of t he rotor phase (sccond Cagll); M I is t he mutua l inductance between st ator win d ings;

128

c e. 6. Mu lliwi nding MoKhln,u

M l is t he m ut ua l ind uct ance b etween the ro tor windi ngs of the. Hrst c age: j\1 a is t he mutual Inductance between the rotor windin gs of the sec ond cage; an d M IJ ill the mutual in ductan ce be tween the rot or windings locat ed on th e'aama axis. SOl ti ng Q)e - W T • (O r c: \'lo or ro~ = 0 all ows us to ccnstder t he conve ner in the most prefdrahl e system of coordinates. The equat ions exp ressed in the o;-f} coor dinate system have the form

di ~ 1<:" R' _, Mi di~ a; M dl 2a d j =V- v t" - !7 (i/ - v --;rr ~ dl

ill l "

---

/If I ai ; ~ M dit/l Lt j~ - v"""'dt

R' .,

= "7::' - 77 111 R~ .r L; <Z

- --~ I -

d Il l!

d-~ I

(u.. . At r .r ) 1~ + t t l\ + --1 2 11 -

Ll"

L'1

(

.r) ,,+-M, - ',. L'j

1/\ M ; .. , llll + V 1.« + 11

L'i

L't

,' . _ 1!.. 2d • _ !!.: ~

•

4 12(1; dl

L'j

" 2 .r

- 12",,- "1

L2 !of

(6.11)

jdl

-- ~ -

AI di:'" M r dlfa. - - - - -L'j 41 L'j 41

dl~

(ft~T . ,

or M " ) ---.- lfl ..J- £ ~ lI + -- "

itt'

·L 2

di~",

-Lfdl-Lf~

"1' l rt o:)

d/Zfl n2 12f1+V (M • • • --- - -re- i",+ I1<:<+ dl

AIr ll ~

Mi' dl~lI

-[7'71 -/T"dI , e The equation of mo tion i ~ (dv ldt ) = [1/(J I pH [(mpI2) ~1 (iDI'ia- t~i ~ ll i~' ;' - i~i2f1) - M .J (6.12) In Eqa, (6.11) and (6.1~). M is the mutual in ductance between t he st at or and rotor windjngs , and Mr is th e mu tual in duct ance be tween ro tor w indings . T ile rud ucrances of sta tor an d rotor windings are L"=M +l~. L~=lIf + l~<1 ' Ls =M + lill (IL 13)

where 1:' , l l ~ ' and l20' are ehe leak age Inductances of stator and ro tor windings. It can be shown t hat fo~ the s te ady stat e, d ifferentia l eq uations

12.

4..4. Th. Double Squltrel=Cage Induct ion Mot",

(6,11) eOllv eI'l t o complex equati ons f or the ph ase of a do uhle-cage tnducttcn ma ch ine. Replacing in vol tag e equations (6.11) the differentfal op era t or hy j w, we obtain for th e steady-st ate performa nce

c.i~ .... R' i~ + jw[.' h + jwM i t.. + j ooMi'i,. u~ = ll' i~ + jw L, j-~ + jwM i t ll + jlil Mii ll

_U I..""' Rjh~ + j IilL i" h .. + j~M h + Mj~'Y +Lri r/lv +M,j;lI'Y + j wM ' i;;", -U'ill= Rji te+ I(o)Ll i'iIl 4- /(i)Mj~ - Mhv - Li:ii..'Y-M' i;.,.+ j w!w hll - Uk = R ;f2 .. + j(o)L 2r-2<z. + jroMi;.'+ ft1i .,v + L'2 i hv +M,jlll'Y+ foo MTji"..

(6.14)

- iJ211= Ri1zll + / ooLi i:; ~ + lwMj~ - M i,., v

- Lih,.,v- M'i i"aV+ /wM, j'fI Subs t it ute re lations (6.13) into (6.14), nex t mu lLipl y and div ide t he ter me in rotor circuil equa t ions for t ho e mf of ro t ati on by t h e angu lar (requeue}' of the s up pl y l in e foltage. We fina ll y ge t

U~ = R'h + fx J "' + jro it. + Ixoi,.. + / zoh .z , , , .! • , U~ = R'l il+ IxI I~+ ix"l~+ jx"/llI + j z,,1211 - U'i e = R'i i'i ", + 1%2il..+ j~oi'i .. + j%o ;~ + (%t + .1'0) j'ill'"

+ %oi~,,'

+ ix' Ii,., + Ix"i;,., +x' i2&'" +:1:0 [;;11.. .' (6.15)

•

- (xt + :£0) •

h.v'• + 1:1:';I ;;11• + I xoiz ,ll - :l:' h.., -e'

- U:; .. = Ri l ia + 1:I:~ I'2a + Izol it.z + j:x"l~ + orol t.v'

II- ol ua

+ (x~+:x,,) ~flV' + fxY'i..+ jxoT'i .. + x' Il liV ' + xolh . . • • • • f • • • U 211 = m l;lI + 1%~1 ;1! + fto l21l + fxo l~ - zol:"v' • E. • • . - (xs + .1'0) I ;..,,' + fx' I~B + fXO '~ll-z' I'i ev ' - xoli..v'

In Sqe. (6.tS) we pu t = wi:" x. = roll ~ ,

r2<t V '

- ZO

= rol~g, zJ = wIt!, v' = " fro • ... = de/dt

Ch. 6. !-'o."ltlwlndlnQ

M o~h IM'

where :1:' is t he reacta nce of mutual ind uc tion be twee n t he rotor windi ngs . Consi dering t ha t ;:"' = Jh , ,j ~..=Jit" , ii,. =j i;fl (6.16) th e processes in th e s ym metric mode of o perat ion can onl y be t ree, led for one phase of the msohtne, So, su bstit uting releuous (j . 16) into (6.14) for one phase YIel ds

+ j:ro ( j. + j ~ + i i ) -U~ = R~j; + Iz:j~ (1-!,,') + Ira (1 - "') Y, + jZg(1 -'0') j . o- =

R"i' + j,T,J'

+ jx' (i-v') i; + j::r.o (1 - v'J i ; - 0; = R;J; + jX3j~ (1 - .v') + jzo (1 - " ') i ; + Izo ( I -

v')

+ / ;;r!(I - -v' ) i; 4-jzo(1 - v')i ;

l(6.17)

Afte r tran sforming t he rotor win dings to the st ator win d ing and in tr odu cin g t he magnetiz in:g current i D .... 't- + i i' + i n Eqs . (n.H) , we have fJ' = n -i- + /x ,;'+ jzoi o

+ jZ;)'j's + jZDs]g+ I Z' si ;o= R~' i'2' -+ /.r,;i;' S+ iZDsi g+ ix'sir

o""'R'j' i'j'

(6 .18)

H ere s is the s lip . By perfo r ming a ppropriate t ran sro -mauons, fro m Eq s. (6.18) we obtai n the foll owin g system ;of equati ons for t he classica l ci rcui t model of a douhl e-cage ind uction motor; U ' = loZO +] ' ZI

iozo+ i 'iz.2 + hH t' (1 _ s) i.! +l x' i ;' ;oSo+ i ;Âˇ"3+ i t'Rt' (1 -8).'8 + j X'it'

(6 .19)

;o=j' + i'j" + i;;' In solv ing the eq uatio ns [or th e ma chine wi t h two rotor windings on all anal og com puter, it is advis able to use the model ex pressed in terms of currents s ince this model is most preferable to th e analys is of machin es with varying param eters . A more s t ablo mod el ex pressed in terme o r Hux 'l inkages become" impracticable for th e pur pose s ince a chang e t hat is to be made in a n ~' o ne of t ho Indueti ve reac t ances calls for rec alcu I/.I t ing al l t he coefficients in tile equations and rea rr angin g the same n um ber of gai n factors on t he model.

,,, Th e equat ions below correspond to t he mod el of an indu et ton machine with rwe wind ings on t he rotpr ( t / V) (i~

",U cO!! "¥ - \R' / L') f..

/II .,

/If

M" 1o. -V1h; .\1.,

I7, ''' -V'2 , «

JJ ..

.v ..

-q ' 1" - L; "

B _ HlI+ (Jl1 /Ll) i ~ .T (M ij A -= ira + ( M IL O lht+" (111 / L t) t:' D _ ih + (M IIJ5) fl ll+ (MIL;) 1&

c -: i ~+ ( M IL H HQ + ( M JLHi'; = (m p, 2) },f (ill~.. - J:,1, 1l) + ~mp/2) ft( (lli211 ~: =

(JU• .1. .'\1,>

i~I;,)

(6.20)

An ana log co m pu ter ts s uit ab le (qr the analysis of Il machin e wit b t wo windings on the roto r only i f th e w ind lll~ pa r am e ters d o DOL vary. I n s l ud yi n" t he process es wi thin II wide t 8Jlge of eha Dlu in the s li p, it is a dva ntageous t o ch~o the PQflImet ers for t he s i p which confor ms to th e inHill1 s t Age of [the tran!'ient. Whore the pu rPOliO is to de termin e tbe functi onal rela no ns betwe en th e s tolie and dynam ic characteristics d esGri hin~ tJ l~ transient, il Inakes sense to. apply t he experi ment pJannin l: techni que to Lho model. A dig ital com put er offors t he poI!!:libility 01 solv ing tho bohllYior of 1I machine wil h varying para meters of t he windj ngs. To sta rt ,.,-it h Lhe eoluuo n (If eq unliprul , we need fi~t La dete r mine Lho parameters of l oops. T he p(lra:mete~ of the integ-Nil eddycurrent l ec p ca n be defined proceedJng from tho iden tity of iLS p.rllmolOl'!l wi t b those of Lbo 801M rot o; of the u me s iitO- Expertmec-

'"till invest igat ions (rhe ebon-ctecun

C h. 6. ÂĽu lliw ind ing Mad.l nD'

l est I'l l differ en t freq uency of t he at ator ci rc uit s u pp ly vcftage, th e const r uc tio n of t orq ue cur ves for a motor) oUes t to t bo i ~ en t i ty of the para meters of Q rotor ba vi ng open-circ uited r age bars. e mbed ded in slots (an in t egral ed dy c ur rent loop ) wlth the para meters of a soli d rotor of t he same et ee. W ha t accounts for t hi s Iactjs t ha l th e b ars placed i nt o alo ta add t o e lectric cond uc t ion of cu rren t thro ugh indi vi du al l am itrat ions . To defin e the parnme tcrslof an Integr al eddy CUITan t loop llc ti ll g j oin ll r wit h th e ma ln loopi WII s hould a pp ly t he ex pressi on

z;- ~

z; (1;:w i"'- 1)/ 2x

(6.21 )

wh ich Is m e firs t equ atio n QS t ablis h iflg t he rel at ion between the i mpe dance of the loop and t he cur ren t t hr oug h it when thi a loop se ts se pa ra tely (ll.,' . I;) and together wi th t he me! n loo p (~-, 1;-) . H are % is t he or der 0 the B = klfl l ~ par abola used to ap proximate t he m ain B -H cu rv e for a ferrom agndtic m a ter ial. The secon d equat ion for the im ped a nce ~ . an d curren t

is wettten using t he eq u ivalent c irc u it . 'I'h e p ar ame te rs lI , a nd ;;; of th e sta to r and t ile main c age r espec t ive ly are t ak en from t ho c alculation data. The val ue of is found fro m t he Ij-s hu ped cu rve (t his v alue is cl ose t o t he c alcu ate d val ue). Expe rim ental Inv estlgattp ns and calculc tio ns confiro} th a t the r esu lts of ex periments compare more fa vo rabl y wi l h the calcu la t ed resul ts if acco unt I! tak en 01 rot or eddy c ur rents . The di sparity betw een th e elec tro roegnet! o t orq ues wit h end wi thout regard t o eddy c urre nts for the A03-24-4 mqt()r comes t o abou t 10 % a t SO Hz a nd t o 16 % at 1,00 H z. Th e calc u lat ions reve al th at n eglect.ing t he ertec t of ed d y cu rren ts Introdu ces a gl"i'l llte r oalcul at.icn er-ror for m ot ors des igne d t o operate lit h igh er fr eq uen ci es s ince 3. more t an gi bl e sh are of eddy curre nts affects t he elec l romag UGtic tor qu e. Any i ndu cti on moto r shou ld be t reated as a m ul til oo p syst em. The eddy cu rren ts in a rot o~ c nn be all owed for b y ad di ng t he int egr al eddy cu rrent loop to the uqulv alent cir cui t . T he effec t of t his loo p need be given d ue consider a t ion in t he dy na m ic a nd s t eadys tate a na lyses of moto rs op;cr ating wi th in a wid e ra nge of ch anges In t he sli p and also mot ors. s u ppli ed fr om s ources of increa sed frequ ency. E dd y cu rrent loops in a ;core stac ke d of sh eet -st eel l am in at ions affec t b ut li ttle t he elect romegne t.ic t orq u e of t he moccr . H owever, i t is t he loops for med b y t}Je c uge b ars an d rot or core t h at produce t be d riving t or qu e . An open-circui t ed cage corres pon ds t o t he in teg ral eddy curren t l oop and ~ i s i dent ical to Il so lid ro tor.

6.S. The Ind uct io n M a~h;no Mo d o l

T he a- ~ mode l expressed ill te rms of cur ren ts is most suit able fo r the analysis of all energy conve rter with two rotor windings on 1I 11 anolog co mputer. T he progra m for t ill! digilal-com puter soluuo n of differentia l equ ations of a machine w i ~h t wo ro tor windings per mits studying the dy namic beh av ior of the ~ac h.i lle havi ng both constant paramete rs t110 windi ngs lind pa ra meters fu nction ally v aryin g wit h time. I t is also advisa ble to us e "this pr ogram for t he s t udy of double-cage moto rs. T h.e equati ons o b t~ i n ed in t his case are cu mbersome and become s till more so with the add i tio n of a win di ng on

- - - - - - - - -M,i.+- - - -

M, Mno..,~Io::::::'JE:=1=====\!::====~~---' , o z Fig. 6.5. ln d uctfon-motor torque eharacteriJc1lc allowing Ior th e effect of the

eddy current loop

M ", .\Iml n' Ar",.,..

1>1 nom ond M ' â&#x20AC;˘â&#x20AC;˘ - &ta ,~lnl, '9ln l m~ lll. mulmum . nom ina l .... d . ...ul . ~n l t n r"(J llU r ,., J>t<:t1 Y ~I Y

tile stator to all ow for th e effec t of ed d y currents. Im aginll t he c omplexity of e q ua ti ons and th e transform at ions Inv ol ved for a l arg er number of wind ings . ; The an aly si.! of the equations f or an induc tioTt mach ine wtt h t wo rotor loop s leud.! us to th e conclusioTt th p.t all Ihe uarlely of rn~han lcal duxm cterlsttas, M .. .. reduce., 10 th e tuo-unmiing motor cha racteri.!tle M] dluegardlng edd y current loop~ fin/!. to t he d UJracler isti.c 111 2 of II motor w it h I t Jolld rotor (}-' ig . 6 .5 ).

6.5. 1he Induction M achi.n:e Model Including Stator and Rotor ~dy Currents As me nt.icne d ab ove , th e eUec l, of e~dY curre nt loops must be allowed for in solv ing t he rtynnmic nor! steady-e ra to beha v ior of motors ope ra ting over a wide ra nge of 's li p cha nges nnd also motors bllilt Ior- vol ta ges I'll i ncreased freq uencies, Tile mathem a ucs t modol of II macht ec cnr r ying t wo s tator wi ndings Bnrl two ro tor windings and httv inl: 11 ci rcul ar Held i n th e air ga p is t ho mode l of Fig . 3 .:'i. Consideri jng t hlll the mutual i nduc ta nce betwee-n the s tator and rotor wi n di ngs,l"('~ults fro m .~J , a nd tile mutua l tn rlucta nca between 1'0 1o" wimll ngs from At' , nller epprcprlate

,,,

Ch. 6. ,....,!tlw'n d inl:j MK hln • •

t ra nsformati ons we get

uta.

411...

Rr".

""'iit""' Lf - L; lilt,.

1 1"' -

Rt _.

al!"

il l / a

dii...

"{;f dT -Lf /iJ-LT ""tit

M I II'l'i..

dj ~ ..

/of

/of dl~ ..

7 =- Lf 1Z<:a - Lfldt - LT dT-Lf'dt , '

_1Zf" J+ Lflit

." 7t" =

4 /'"

f'III

- R{ v,1_r1<1 - V [ 'M ".

'71 =

,'II "'~"

Mdt- a. I. ; dj""" -

+ t'18 ..L. L M'f '· U' r ]

M ' dJk:

L ~ ~ - c:r

[,,If. + Lf M ", IZll +

Ri .r - L f ' Za. - \· .,~ ii' !of d!1",

- Lf

d ila

4110.

J,f '

(6.23)

"7i" -Lf dI

-;[j'" -

.ur.]

~ i211 + Lf 11il

<1';1\ uh R~"fl l . M !.1! AI 4 l h 101 d lb ""'ii'I ='"Lf- -z;r ll - 7;f II I - Zf dT - I.: dt d l!, .!!! ". Af II l ill M Ii/t il !II d ' ; 1\ ( i l = - LI ' 21l -W di"""- L; '""dt -Lf dl dl fll= - LR~f I' rHI + V [' ,.I' 7 _ 1.1" _

L dlZll

71= _

d fr$ _..!!.. d 'h _ ~ dl

L~~ d l

[

Vi ZJ + 'Y \ !II dl b, _

"If

" '" I IIp "dl

"( + 1:'; M j ' -t I' + M I e.-r- III /.r

I ..

II I

/. ~

]

II/i ll III

-rr 11..+ yr Ii.. + lio. +-Lf" i'i.. MM

.... .

!!.. J. ~

dj~/I _ M ' III

[2 "" ' 111 ( I"IlV". rla. - I I III 'III

+ lie:H... -

' 2l1

dl ~ ll ti'

+ 1".111 12 .r .. -

At'

)

1·f ,, '·'Zll

lk l'ill + ii "1{,,, - li".tb ) - M .]

(6.24)

Reptacilll: p lJy j w from <1>.23) we can o bta in th e eq ua ti o ns for t he steady·sta te opera t ion u~

_i:::+ i,..r.,.

~; = O= i;::+i. r...

- ';j _ 0_ h:{' -J»; + i~/l';' (\ - ')" + i:r'i;_ ;j ~ "'" 0 = i :'::;' + i...:", + i{Jl~' ( t - ')1' + j r! i r

i . -- i ,· +i ' +; ', · -l..i~ "

(6.25)

10 Eq . (6.25) . t he impedance! of (t l\l~ d ~ wind ing! are taken equa l. and t he curren ts a nd voltages beat Indexes to ide nt ify t he first and second lll.. tor an d rotor wi nd ings. I Solv ing (6.23) a nd (6.24) on a d igilp.\ co mpute r. we ca n esti mate the offect of ed dy cu rre nt l oo ps on the d ynamic Rnd stAt ic model! of cpera tio n DC an in ductio n ma ch ine . As found from inves t igation s. t ho effec t of a lItnt nr eddy-currant loop during th e period of s ta rli ng 11 7-!<W motor ill gr ent er th a n t hAt af 8 rotor eddy-cur rent l oop. hut both loo ps have fin eq ual ertcct Oil t he hnp nct slarli ng c urre nt i n t he eta rcr wi nd i ng. A p p l~'i n l: th e e:lpt'ri mollt plllnnlnlr techniq ue to t he nnn lysis of mo tors of vertoua p OWCll'S a nd wit h diJf eren t numbe rs of poles , we ca n e valua te t he etreet of eddy c urre nt loops in the s ta tor llil d rotor o n the dy ne mte lind 51,a l ic ehe reerertsucs of mo to rs . Whe nl a few loops lire in yoly tld in the peoeesa of en f'rg)' ec nvereion, of much importan ce is a n Acc ura te dete rmi nation or t he wi ndi nlt para meters, for which purp ose :l freq uen c)' methOd is arlvlllll allOOWJ . T he pamme tera of sta tor edd )'~urre.nl loo p", l<cl,-n be found from t he vnJue of iro n loss . T he ca lculatio n me th od t ho t more full y a llows fo r mllnufa cturi ng lectors gives te uee results. T IlE; effec t of eddy curronL.O\ o n t he chnrncte rist ics ola machin e is accounted fo r by th e interact io n ' o r 1111 eoetrtctcnts en teri ng into Eqe. (6. 23). an d (6.201). Th e sta tor a nd rotor s teel s hee t t h tck noss, atea l gra de, a nd ma nufact url n:: o pe ra tio ns lire c hosen alter lllo a na lys is of (6.23) a nd (G.24) and also after constdera uc n of ocooolll ic Iecto es. I n Eq. (IL23), th e vo ltages on th e second l oo ps in the e re tor a nd ro to r mil}' be ot he r t ha n eerc . If tb~ voltage impressed aeross th o -sta tor .....indin g is th o sa me nnd t hus the fie ld i n tile a ir gap is c ircu lar, t he pr oblem red uces to the Sl u"',. of current d is t ribu t ion a mong t he parallel branches nf wi ndi ngs . A small d i!lCrepancy be tween t he in ductive eeacieoeea a nd resi.!ltancos '01 pa ra lle l bea ucb es eauses a no nunifor:m distribution of cue reo ts lind t h us affects t he per for ma nce of t he ma ch ine i n the stea d y-s ta te a nd t ra ns (en\ conditio ns .

6.6, The Effect of MlInu'facturing Factors on Electric Machine Performance I n t he th eory of e nergy co nyorters jt is customary to co nsider the ai r ga p unifor m , t hough th is is not the case in D. rea l machfnc for a var iety 01 manufacturing reasons . 1!he ai r gap nonu niform lly ma y a rise from t be &CC8n\ricity of a rotor w~t.h res pect to a s tator (Fi g.6 .Gal, ector ellipticity (Fiff . 6.Gb), ro to r ond sta tor co nici ty (Fill. 6.lk) , a nd m isalignme nt (Fig . 6.6d). 'These Ifac \ors are respons ible for Additional losses, vi brations. an d varioUs errors, 80 t hey must be taken into acco un t ill t he ma th em a t ical a n ~)'s is of energy co nvers ion pro-

cesses.

IS'

Ch. 6. l!\ultlwl..d iOl2 M,d>l"...

For t he ease shown i n Fig: 6 .Ga, b. a multipolar machine s tr uct.ure can be brokeo down alo ng its length i nto a lew eleme nta ry maeh inel with different a ir ga ps and ~n to m ma chiou aro und the ga p ci rcumfere nce. For the ease of Fig. 6.&, d , th e struc ture ean be broke n a part

0) lb)

(.)

-i3.--d--E-tI I I

(0)

Fig_ 6.6. Cap

Donunif~lTlllhy

due to

m.n url>e~urlnll

bctore

in to n mechlneaw tth d illerej'l lairgapsa long iUl lengt h. 1.f t he m ultipola r me chtee of Fig. 6 .6a ~ b has parallel bra nches, th e curr ent dist ri butio n over the e temente ry machines becomes nonuniform. I 1111

e:t...FUt..FLF

-F - +

cflflflD

(Âť

FI(. 6.7. TJll Ica1 nonunUorm1Ue. o er the madlintl It)

la~

totor,

(~)

... Ior

,t..

ltnath

11110 POle _

For tbe t de of Fig. 6 .& , d, t he elemeDta ry IDll.ehiou a re conoected in seetes a nd the vOlt8~S are d i.!ltri buted nonuniform ly. CoM ider a machin e with a nonuniform ai r gap (Fig. 6.7 ) a nd d ivide it in to II piece, through out its leDi th . A usual a pproach to desi g ning a mach ine wit h its core d i1ded into pieces over a umrcem ai r gap is s imilar to the cue under s t udy si nce t bo ext reme core portions lind t he midd le core portions oPera te under diff eren t co nd itions.

131

M a fir st. a ppro::d ma tioD , U!WDO that. the m achi ne has n sh wrsand a common rotor . Supposing that there ill no li nk between n stators, Lbe voltage e quations thea take on the fonn

I, I,

(C.26)

I n Eqa . (6.26), lind ere t he vol tag e nnd current matrices of thy lth machi ne . I n th e impedance ma trix of (6 .26), th e sq ueree donole t he im peda nce mat ri ces of 8 ma chi ne wiLh a circ ula r fie ld. Each im pedance mat-fi x eouse tna c9rrespondiog para meters. Theelect romag net ic torque here is eq ua l t o t he sum of prcd ucta currents in each e le me nta ry m achine

(6. 27}

To rque equa tio n (6 .27) for the ec mmon-r ot c r ma ch ine mod el i neludes pair.....tse prod ucts of currents ~ n t he It slators and the rotor a par t fr om the pr od ucts of eurre m st n eac h ele mc llta ry ma chine. I n t he common-r ot or m ode l , th e rot or establishes the li nk bel-ween n mac h ines. For the ser ies- con nected elementary machines opera ti ng in the sleady-s tll.te co nd itions , the Hoe \' ol ta~ is

v, = VI + V, +1... + U.

(6. 28)

and for the para llel-connected m achIn e! , the li ne curre nt is

i, = i + 1. + . . . + j" . . . . l

(6. 20)

where UI> V " .. , U" an d II ' I lo . . '. I . Are ure vo lttges ond euerenre i n ele me nt ary machines. If, apart. from the linkage between the clementlll}' pieces d ue to the ro t or cu rre nt, we co nsi der t he li n kage resu lting fr om t he ," ach ille

en. 6. ~u'ItJwl nd;ng M ...hlne,

Mturation. the impedance matrix in (6 .26) will be fill ed ec mplete ly:

:,1D O

D O

0 0

D O

" (6.30)

â&#x20AC;˘ "

The torq ue will then co nt~ in not Dil ly tilt> products of curren ts in t he s ta tor a nd ro to r bu t a lso tho products of c u r re ll t ll w it la d iffore nt sii08:

+ J/, + . . . +!lI,,+Mu

+ . .. + J il l + . . . + !HI" + .. . + M eA_u " (6.3\ ) Eq ulllions (6. 30) lind (6.31/ll ro l' i lfl i l~ r in s tructure to lb n equaue ne for an m-n wi ndi ng mac line. The ('quIIl io nl! for the gtlncrllli tcd energy conver ter permit the ~ llilol ysis of en ergy cea versic n processes in elec t ric machines with due rel:,lrd for man ufactu ring tncccr e. Mac hin i ng t he s tator nnd rctce ca ll also !lfreet t he chn racle ris liCJI o f lin energy converte r. This factor can be all owed for br consrderi og th o di fferences bet wee n lhe pemmetere of e lemo nta r y machtnea or l he p resence of edd y curre ut Ioops , as is done in s ees. 0 .4 a nd 6. 5. A set of var io us ma nufftcl\lf ing t ecrcre eonun c uly de t erm ine the per formll nce of II mach tue , a nd coustccrau cn fOf each factor i n t he se t mek es the an a lysis a very d if Hcull prob le m. As noted above. the equeuo ns for the 171Âˇ/1 win.ding mach ine permit t bQ study of most of these fa ctors , each ISC pa ra,le ly lind a Iew si multa neo usly i ll One com bination or a nother . Equations (6.26) a nd (6.27) a pply to th e ana1rsis o f pein t- wi nding machines i n whi ch t he field d i.s tr i but ion lit the ond portions a ro und t he periplltl-ry d iUors fro m that close to t he center of t he a ir ga p.

... 7

Chap ter

Models of Electric Machines with Nonlinear Parameters 1.1. The Analysis o f Electric Machines with N onlinear Parameters , As noted cllr li er, th e eq uauo ns of e lec t romechen tcn! e nc'l:Y co nvers io n with constant eccr tteteoie nm Inonli nea r equa tions IIInco t hey contain t he prod uc ts of va r- inhic s . Th~ l\ntlly t ica l so lut ions to 1I1ll~ eq uati o ns d n not exist if (0) , u ndergoes c hnngcs. Const oee th e "Hect of no nHnu r coeffielo nu i n tIle clect.romechantcs l eq ua r toes n n tho p rec esses ('If e nergy ee nvees toe in eree rete ma r h illes. nllmely . t he coeffici e nts L , M , 1.". r " r" lind J li nd i nde pendent "aria bles u , J. an ll 111, . A ll ccetneren ts en teei ng into t he eq ua t ions can be no nIi nen r. T i,e eeststeuce of II rot or ch an~s w tth cu rre nt displllceme Dt, a nd tha t of /I sta tor wit h helll . T he in d uctive renclll llCC de pe nd s o n M t unlt inn . T he mome nt of inor Lili ac me dr tves is II Cu net ion of the a ng nln r spee d . , T he pemm e tcr a dep e nd nn vo ll tlÂĽC9 . lond , and other- facto rs . vul i n ge ne rnl t hoy lire funcUon" of t ame 119 is e1enr fro m Bqs . (7.1). (7.2), lind (7. :1) :

ii,

.~ I:'::~~(I) :, (I) At

Iil M (t )

ra (t J+ d

\ L 2 (t ) w,

+ 7I Lz(t, ]

x r z (l) +

Ill,

- L,. (I ) 10),

+ :, 7( t) I ,

I dT 1W;(l)

:, M(t)

(7. 1)

rl (t ) +

+ 71 Ld t )

lIf. _!If" ( t ) (IAli. - i~I ') dW, .'d t

- N (t)

. . '.

= ( p l J (Ill 1~1f. - M , ( t )l

'; (7. 2)

(7.3)

T he modol of a ll e ne'1!:f co nvener. ll!l sho w n in Fill:. 7.1 , cc ereepe uds t o Eqs . (7. 1). (7.2 ), nod (7.3).

Ch. 7. Mo de l. ot Elect, lc Ma ch ine .

-,-,.. • •

•

"I a

• •

+ "'iii

7f "' h Ill

L.l a

•, , • •• '-"71 .""Tt"M:•.....

i- JotUQ

··

" -a-1>ltl " '1

---.. · . .,.· -,. ·"'. -- ·"'. • • • 0

•

-" -:..

• '"•

:, )I', . .a.

n _

7<"' .11..

"'iiT.u I _ oil ","

. d

71 "' 11'1

*.• u"'_•

•

+ 71 (,'

'To" •

M :0 I ..

."""

-ar -" "b oa

71 .u"' " I..

•

..lI , r

7l M. -

: . )12_

21..

. 'a

~ .lf-'

• 1.. 7i" ·\1..

_ lI .... . .

", - .vI......'

-l.~ "".

-"'1_ ",

_M" .",

- ·\ft"' ''~'

-·'111<''',

- ).Ii ....." ,

- .&1'"" ,,",

-.,- "'. ..,

-.'1:"1..- ,

_ L~ ,

• •

•

' - +71 t _

• •

":.~ 0 0 0 0 The TlfJIIUntu ,ilg of the p aram eltn a/Ill! ener gy converter operating at dnuj(jldal voltact i, ru pon, ible jor lhe emergellce 0/ a harmonic

spectrum in the aIr gap.

• FiE_ 7.1. Th. mod fl or an ek!ctr ic mac hine . 'l th nonll nt!u

p.:U;lIIr~ lA'n

The mnt hemlll ic., l model 01 the processes of energy conversion in the ete ga p o r 0 lI)' m m el ric' machine hnvi ng l wo s la t o r wi ndinJ:s a lit.!

...

7.1. rh .... na lr.1I o f eled , 1c M.chlne.

•

I~ ..

•

Ii ..

•

I:....

• L~ ~"'r

M:;' ~"r

Mu ll".

"' I"'~'"

M211l",

.\ltm~'"

!II .I Il"' .

M. meOl r

·\I:;' I~.'

L~"',

U. 1. ..,

., .

·, · ·

')'~11l

.s: ., "';011\

To ~;",a

II a"~ 7i l woC:

,;.

"'iI !II_ ,. II -ar M •l>n.fI • ""2m!! • 7i

'l ll

I;.

•.... + 71 L-S

I:"

•

",j''"11

7lM Z~

• 71 "'2111

"7i

" 7l

.'II'"

-#;-uO... ,•

M ... I .

" "t I ll

71 >filII

"~ -n- .\Ill

,;.

·"" ·• Jf.. m. ....

dT ·'12111 "

, 111 71"''''

l ..

Nl .~

" Ill + ""ifi'" LIlI

.. . 'mll+

iiT L ...

.. ~

rtf •

~ m ill

•

' I B T 7T LIB

•

\"'•

':;'11

(7 .·n

two ro tor wi odi nlr-l and operating 'rom rhe symmet ric suppl y volt age so urce is t he m~n winding ma ch in e Ijlodel of F ig . 3.2. E ach har mo nic of the fie ld ca n be II!l t up on tho model by choollin g a pa ir of wind ings on the II t at or or r:o ~o r and apply ing to thei r terminaa approp ri a tol y phese-sblf ted s inysoida l voltage s of cor respo nd. in(r ampli t udes and fr eque nc ies . \ Voltage equa tions (7. 4) describe ~ he mod el of Fig. 3.2. The torq ue equation foll ows from (3.12) by su bsrtturlng m for n . The nonli nearit y of at least o ne o f tbe ooeffieie nta in the electrorneebllniclll equa tions gi ves rise to a n i nfinit8 spectrum of field har mon ies , and th e equati ons beco me s im ilar to these for t he goneralized m-n wi ndi ng ll Il(! rg}' conver ter. IHowever, th e model here b as the same num ber of w indi ngs bo th t he sta t or and on the rotor , aDd the dete rm i natten of li nk s betw'een harmonics {between treuucus wind ing!! ill th o model) diHers iw lth each paramete r. W ith a change of tbo load or volmge o n t he aermi nala of an energy con verter. the coupli ngs between harmonica (mut ual Indu ctances in the e quati ons) undergo changes too . ' Th us, the a nalysis of a n electric ma chine with nonlinear para meu rs is possi ble by WIe of t he two notat,lon" for the electromechan ical equ ations. Specify ing the pa rame te rs as fun ctions of currenU or

142

Cn. 7, Model. of Electric M.c hln• •

ti me , or other Fact ors , we can impl eme n t t hese Iuuct tons on th e 1I0nllne r u- u nits of a n ann log com pu ter or- realize t hem in t he for m of ta bles on a d igit a l computer and t hen solve Eqs. (7 .t) t o (7 .H), A second a pprce ch is to choose th e req uired numbe r of ha r mo nics ;11 the mode l of th e m-wlnd tng m achtno and solve Eq s. (7 ,4) O il 11I l a na log or d ig it::d comp u tar ualn g constan t coetncrsnts or coem cie nt s vary ing with loa d, tem pera t ure.. volt ago , etc. tsee Llllow) . The equatio ns for th o herm o otcs to he chosen in each cas e 81"l' cumberso me , but t he)' offer lar ge peselbilttlea for t he study 01 ssstoms , For exam pte , t lICY per m it us t o determine Ih e er toct of each harmon ic on t he t orq ue being produced, to consider t he pr oduc ts of cu rren ts due to ve rt cue lJa l',!\on ics, t o vn ry coupli ng , e tc. Eq s . (7.1) t o (7 .3) arc more rea d ily solvable b ut they do not a llow for a n easy a ssess men t of ma ny peculiart t.iea. T he oqun tl ons wi th nonlinear eoe lr tctcnts ere not a meue ble t o an nccurute so lulioll as is seen from tho ana lys is of t he sy s te m of OqU UIiOllS (7.4) . Howe ver, tn king i nto accou nt so me no n\i nearit ies an d so lvi og t he eq ua tt ous or 3 d igita l com pu te r , it is possi bl e to ob wtu t ho res u lt t o a des irl'd accu racy.

7.1. The lifte ct of Satura110n For most ro lali ng mach i nes , t he oper at ing poin t li es 00 t he no nItneae bmnch of the 8- H cu rve. T he saturati on of an e nergy convert er varies wit h vo ltage . frequ ency , and loa d , t hereby affec ti ng t he ma ctuna'a output cba ra ctertst.lca. \V il h t he satura t ion be i ng take n In to acco unt, i n a first a pproxjmat ton the magn et izati on M is t ake n 1.0 depe nd 0 11 th e mag neti zi ng ' cu rren t or ti me. If J11 = I, (I), the n L = 13 (t) since L = M ld,. We may ass ume here t hat t he leakage inductance /0 is i ndepe nden t! of euturauou because the leakage Ilux end s on itse lf in the air a nd a;CCOllllIS for a sma ll share of t he wor k ing Ilu x , "'Ve ma y a lso ma ke 0110 more assum pti on t ha t L and ill va ry io the sa me mann er

L' (t) - M (t )

+ lo ..

L' (t ) '"'" .l'f1 ( t )

0 •

(7 ,5)

Then

= L' (t) i:" + M (t) i~ , T~ = L' (t) i~ + !If (~) I~,

Il' ~ =

L' (t) i! + 1U (lJ I ~ W~ = U (I) i~ + M (t) /6

17.6)

To si mplify t ho mod e l fo r t he solutio n o f cqu a t tc na with nonlinear coefficie nts L and M and t111l!J t o cut dow n the nu m ber of p roduc ts, jilt li S In trod uce now variahlbs (7.7)

7,2. The Eff"ct of s'.tu,Mlon

At M (t ), t ile eq uarlo na t hen

u:. -

I R' [R'

u; _ r4. = IR'

8 5S U lJlO

tlte form

+ (dldl) lot ] it.. + (dldl ) sr (I > i",.. + (dldt) 1..,1 + (dldt ) M (I) i.....

17.8 )

+ (d/d t)J..,1 ~ + (dldt ) M ( t ) I..... + IV'. + !oJ (t) i", ..J ui - IR' + (dl elt) 1", 1 Ii + (dJdt) ill (I) f_~ /.I),

..., lUI;'

AI ( t) 1",.1

The t or q ue equatio n is wr tt te n .., JlJ" -

p ;\! (I) It .. ..' -

'",,,/6 1

(7.9)

Consider iltg It 1I0n li"ea r C h ll " ~ in ' the leak lll:e ind uctances ...... e shoul d trl' nl!for m t he equa r tc us in view of t ho fact t lilit [} (I) _ .11 4, (f) . L' (t) = M lor (t) (7 .10)

T he equat io n for

th e ll t ak es on t he (OMfJ

u:. _ [R' +

:1 1". (t) J1~ + M d~

1M "

(7 ,t1)

I n a :sim il a r '1'1'11 )' we t ra ns form Llle eq ua tio ns for u~. u~ , a nd u:" . Tho to rqu e eq ua t ion is g iven by (7,9). Th e lllla l ys19 of (7 .1) t h roug h (7.') . ' 7 .ti) lind (7 .11 ) o n analo g nnd digiti\l co mputers re ve afs l." lI L t he ,lea kage indu c tive react a nces ha ve It l:rea tl;l r eUee L on t he im plle t ~ u rre n t!l . Im pa ct tor q ues , lind st art ing t ime t hlln the react a nce 01 m ut ua l i nd uct ion . T he pa t ter-n of vuri a t ions of ltf II nul., has a s m a lle~ e rtect 011 the d ynam ic ehe eecte r teu ce . T ile valuee of t he pnr nme lers III t he in itia l stage of t he tra ns ien t p eoeess p Ia)' II dominont p ~ rt , In 8 firs t ap prox tm at fon, therefore , we <:81 11 d i.!lrega rd varia t iol1!! o f ftl a nd 1,." a nd solve t he equatio ns wi th eo ns tl'lnt cot' ffi~ie n ts l subs ti t uting in lo them t he sa t ura ti o n va lues of t he pllrD me t.erSj wllieh dete rm i ne t he at atic cbancterist it:$ at. t he end of t he tm nsiant.. At its s tarting. a m ech ln e fi rs t d raws power fro m t he Ii llf; (d ur ing one or t ..... o pe riods) neeessery t o Ilcee)Clrata from r est , then t he rnachine a nd th e line IUcbanco energy . Qependi l1g on t he eo m bi nllli o n of pnrn me te rs . t he rotor m ay relu::l. t ~e speed in exeese of the .syn-ehronous speed (a t a sma ll mom ent \1( inerUa) Of ere .....Jy gai n the ste a dyÂˇs t ate "'eloci t )' (a t a la rge m omeflt of ine rt ia). Mot ors s up pli ed (ro m h f vollllge so urces a nd mo tors with large mo men ts of inertin have s imila r stI rLing cl,naek-risti es . I n See . 7 . 1 we have d iscussed E qs . 1(7.4) for a sa t ur a te d me chlue , whi ch lUll 86t u p t o def ine a n in (i ni te rse r iC3 of harmoni es i n t he ai r gap . Co lls ide ra t ion of t he i ll te rre l a ti o ~ be twee n h arm on ics presents a eom p licatcd pro ble m . Le t \IS ill ustra te t ile way of deter m in ing. u~,

Ch. 7.

M~. II

of ElectrIc Muhlnu

t hese int erre lat ions by a n ~ n lJJ pl e of a t ra nsfor mer. Wi th . harmo nic volt. It" a pplied 10 t J,e transfo rmer i nput. t he iod uclanc8s of wi nd ings ca n be wr il ten as functions of t ime

+ L, (loll + a l ) + c, cos (2(o)t + a . ) + (l ) ... AT. + M. cos (" t + a,) + M. cos (2611 + a ,) + . ..

L,. ( t ) ...

ill

C().'5

(7. t 2) (7. t3)

The t ransformer e q ulllio ~ are co nside red here . s t he equat lon:J wit h per iod ic coeffic ie nts . [I'heso equa t ions do not ho....-ever re flect Ou tput

--@-< l,I,f, r.:

I n l'U~

u,r,

~ u,f,

~ 1,1,,1.. full y t ho processes i n tbe donli near tran sfor mer because i n t he eteeuue wit h ccou neer para meters mere ex ists a ll Inte ractio n bet wee n t he components of the harmonic earres. T he nonli near (sat urated)' t ra nsforme r is a ge nera tor of uppe r harmonics. Such a l ransformer !can be represented as a li near mul tl por t (Fig. 7.2) with a si n usoidal voltage of one freq uenc y (u1 ' h ) supplied to i ts input termi na ls e nd ':.speCl r Um of har monies a t its out put termina ls: I n III nonlinear transforme.r e ne or few windings th at rece ive en ergy ex hibit. s inu.tOid. 1 e mf of ene (req uency; th e tr a nsfer of this e oe rgy an d Ita t ra nsfcltma t ion then occur.!! not only at. t he fun damental, but a lso at upper, lower , and frac tional harm o nies. I n a n ideal ua osformÂˇ er t M s um of incomi ng e o~rgie.!l is equ â&#x20AC;˘.l to the sum of ou teo mio)t eoergier at all freq uencies.

7.2. Th"

E fl,,~1

0/ S.fur. tion

The out pu t pa ra meters of nn u-port. wilh har mo nic volt.a ge sources ean be defined by the Z mat ri x and co m pfex am plitudes of vo l t ages nt ope n-ci rcu it ed output terrotnets E~, £ " ' , . , E " . . . , en' T he assu mp tion is t llal atry of t ho wi nd irgs feat ures a sinus oida l e mf and ca r r ies cu rrents of o n ly one freq ue ncy s ince the idea l filt ers inser ted i nto ea ch wi nd ing b lock tho c urrents at other Ir-equenoles , A transf ormer , " 1; viewed from it s output termmnls , ca n be d escr ibed b y th e ma t rix '

'. "

r 1+ d

d (fiLl . M !1

lU Mit

df M nl

r' + 71 L l ' d

dI Min

lit M u

:1ilf,,,

dI 111 II

'\['2

d rr+j. di

ill /:

diM,,:

£/ •. ,

7t M ",

d . r" + liI L ..

dt /'111 .. /

" i. (i . I t.)

Th, im ped an ce matri x h" the Ior-m z 1l

ZII

::11

Inl 2 .. 2

Zli

","

"'".

2 .. ;

2 .. "

::/

(7.15)

T he sq u are matrix (7. 15) de scr i bi ng the internal so urces of harmoni c voltag es will be cal led th ~ no¥ e matrtz 0/ a transformer, I n matti c~s (7.11,> nnd (7.1S) t ho t erms ~hll l ha ve a ph ysi cnl meani ng ar e the eq u tva len t-mult.iport wi nding i mpeda nces z" , :lor' ' " ZU I ' , .. z" " ly i ng o n the prinl;ipal d iag ona ls, T ho remaining s-ma r rt x te r- ms frl" , .~f f l ' . . • , .~ 'l '.. '. . ., /1-1 1 " , wh ich dcecrt be tile in ter action of nerm onr cs in 'I sa tu re tod s ys tem, will ho termed th e caellicie ll.ls of cou pling between t ho h a rmoni cs of di ffcre nt Ireq uen ces. 'I'his Int ers oucn of h ar moni ca ill on l y .p rese nt in II nonli ne ar s yst em . The analytica l dotorm ination of t he coefficie n ts of cou pli ng .M lwee n har mo n ics pre sents great J iHic'ulties, for th e ca lcu la t io n pro cedure neceeeu e tes th o anal yt tca l e:tjlression of the magn etiza t io n cu rv e , It is therefore more advan tageous to resor t to th e g ra p h ical 1 0 - 0 ' 17 8

Ch . 7. Modell 01 Eled , k M.chilMll

l4'

o Fill;' 1.3. Detenn l"inll the eoerflclen13 of coupling b... t ...Âˇ~D hara'ooi es

(7.17) - ( I f ) i~J I1D1 J.OA C c: (til ) k M u " T he- ('oeUicien l o f ( OUplior bet...-een the tt h a nd t!le (f - Sjl.h harOlooic ill -'\:1( /-1) 1 = (I I) 11 (I i )l /.:M ll l U8)

T he llnnl)'sis of the ! IHurnled lra lll'former noS I li near multipor~ WUII Interna l sources e lillbl('jI us to de rermt nc th", int erch a nge power of eeen ha rmo n ic se pllfnte lr end also n.e nVl\i1a h le power of th e t rAnsformer ove r the e M ite s pectrum of hA rtn oni cs. The inuN:ll a nte pOUHr is t he peak VAlue of tile power lit the output of 8 source whatever t ho cha nges. in th o Ollt Pllt curre nt or vol tage . Co ns idc-ring eecb so ur ce as a one pcrt IJotWOI"~, we ge l t he Intercha nge power

PI -

(1(2) E /Et/(zll

+ :.i,)

(7.t 9)

whe re Et ill t he eemplex wDj uga te of t he effect i ve (r ms) " O H4~ E .: :;1 is Iho co njugat e-ma whed impedance of a load su pplie d [rom tho .!til onepor t of impedan ce "u. The A\'ail ah le powe r of II mu ltiport Cli O be foun d as the tolal cu tpu t power regarded as a fu ncti on of curren ts i ll a ll pole pal ra . Cons.ideri ng th e ISIl tUl'8ted; trensforrnee 1\$ A li nea r Doi lle-ge ner a li.ng multiport. ""0 tan rf lld il y vtsoalne th e wo rk i ng pr ocesses in freq uency mult ipl iers a ud d,hÂˇiders . T he eq ua ti ons th us derived ",re co nven ient for s inlUla t ioli 6n computers. We ha ve gi" en here a n

7.3. I .... Effect of Curre nt Oispl .... "' cnf

ia tile Slot

' 47

eu mple of t he lrnnsforme r to ill us lra te bow to de te r mi ne lh e interecucn betwee n ha rm o nics , though th e d iscll!!Sion certn inl~' re lates to rot at ing machi nes too. . T here e re il few me t hodJl for t he :onal Yllis of Ollcrg y co nve-srcu processes . T hey give n pprox lma te solutions to t he prohl eurs etat od , but on th e who le e ns ure t he desi red pecmney .

7.3. The Effect of Curr,nt Displacement in the Slot T he s t ud:!>' of t he dfec t of CIUTent d lBpll'lcem e nt (skin e ffec t) in t he slo t on t he d ynamic d llU"o.eterili,l ics of a n e llcrg)' conve rtor ill of milch practica l s ig nifica nco. A chl!- nge in t he An gu lo r vcl OCil)' of 11 rotor C./Hl80S fl ehe nge in th e roto r 'c ur re nt freque ncy. T his etrecte

Fl ~ . 7 .~.

T11usl",'iJlI" Gu . nml

di 5plu.emeb ~ , lll

th e illot 01 .n t llOri Y eoa ven er

T rw IIC'''" 1. r . ... . .. d"",te a .doct_

the curren t de ns it y d ist ribu ti o n overtthe he igh t of 11 conduc tor embed de d in t he slot (Fi g . 7.4) . Th e cu rr ent in R co nduc tor or cond uc tors in para lle l var-ies ove r t he slot heig ht. be ca use of th e utrrerenec bet wee n t he ind uc t ive reacta nces of co nductors ly ing I t th e slot bottom lind nea rer to the lIir p p . T he am pli l ude IIDd ph3lJC of clIrre ou th e n vuy too . The distributio n of tl1 ever tbe slot he igh t Is given in Pi, . 7.' . ' The ealcul eti c n practice usee the C?E'ffic ien t N, to account for an tncree so in the res istance d ue to cur r:e nl di splacement . It de pe nds on frequency , the ty pe of w ind ing , s lo t ~e ilrht h a nd width w, t he materral of e leme ntary co nd uc tors . fhej r num be r a nd di mens ions 2/1 end b . III use is 1I 1!l() th e coefficien t ~'" to acco unt fnr th o ver to uc n in l ea ka~ tnducuve reacta nce w it h c urrent dl splacc monr . Bo th N, a nd k ", vary no nl inen rl)' with ro t or a';'g ufo r speed. The va eta tic n of the se coeffi cie nts i n re lative unils for a deep s lot is sh own in FiC. 7.5.

." On defin i ng 11W! pat tern of <Il. ngt'l'l in the ala i reers te nce and illd ueuve reaeren ce. we a n soh-e the elecrrcmechauical e{(ua Ii OIl.:J o n nn :Il\lIlog or dig ita l compu ter . I II the anRlyals tnvol viog thl eq uat ions with pa ram eters dependen t on (\IC"Ilnl.s or t illle. use is m'ade Qf the following eq ua t io ns expressed in terms of curre nts Rod sol ved (or curre nt derlvativoe:

I:' _ (1(L'p)

u:, -

(H'l L"p ) I:' - ( M IL'J i~

/3- ( IIL'p~uf-(R"L~P) I~ - (MIL') i~

t; =

(R' ! p ) ai;' - ( I Ip ) pm.1 - 1JIt1/:"

15 _

(Wip) Oi5+ (I ip) pliI.q- M af!

(7.20)

M~ _ pM (t 'f~ - i:'.ii)

dril,Jd* _ (pll) (M. - !of. )

Here l iP is the q _ I:;' + J1fo~.

in tegra tion sy m bol;

0 '"'"

tlt/ ; f

+ Mai';

A computi na de vice peemlts c;ons ider ing Mparately an increase iu l he resist ance a nd deceeese In t he i nd ucti ve eeeera nce or tilt' rot or w ith. ChBligtI in its n l\~ular ve locity , lind etso the tntereeuo n betwee n t hese qua nt.itiea. T he s t udy of no nlin ear vll.J'IIlt tons in Impednuces 1I110 W! t h ll ~ changes in rotor rellisla nC$ have tile greatest arrett o n the dynemrcs of induction me rces a t s ta et ing.Theti meof starli llg, impact currents 'Hid torques deceeese "'dth e uereur disp lacement in r0PI,. 1.5. T he ~ .u ul l,nd<lCo tor slots. l lv' rn e, ance of . rotor Ver.lWI tq ,li p T he shape of s lo ls Il rrE!(ll~ t he cha ra cter of varia t ions in k. eud k z â&#x20AC;˘ wtlich III t urn affe ct 1M dy na mic ch ara<;\er istlc.s. Ho ....-ever , it i., t he i nit ial a nd Iinal val uE!.s of impeda nces t hll\. exert 1\ grellter i nfl uence on t ho processes of e nergy conversio n. T he chara cter of ch a lJge.~ in rnt or lmpednnce (t he pa tter n of ncult near impedance ver teuon with ti mo) pla ys a secondary role. Compul ing devices e nab le the sctuuc n of prob:~e nls for var-inus sets of linear an d nonlin ea r parameters a nd va riou.s patter ns of chlloges in k . a nd Irz wi th t ime. It is to be noUd tha t the p,roblem. of coruuur f1lg th e curren t i n II slot comes to tJu to lu l lDn.

mull lwtllding-rotm- machl tl t': ~Ull tions, tht':

JUbuqllcnt d m pli/lcatio n. 0/ !thlell t'(JR giw tht': t':t/lUJlions f or a dou bft':squ irrt':l (Dge mach ine . T he doubl e-squirrel cage rotor o( an le d ucue n

149

ma ch in e re qu lees more lo bor for its lIIu nu fl'd urc e nd has 1'I lallt'f dJllmt'te r i n com pa r iso n with III fo l nt w it h dee p dew-drop or te u teshnpsd s lo t , so i n des lVni ng new ve ratons of t uducu cn ma chines the p refere nce is S;\' on t o uie laUer rolo r . By choos ing 0 pr /)pe r shape of t ho s lo t . it is posslbfe t o b ~ iog t he dynnm ic rha l1lcle r isl ics oJ • dee p-eln t ffi:1l ch fno c lose to th 9se- (o r ~ d cubte-ea ge me enroe . I n n rea l Ind nc t tc u mecn t nc. Ilpa rli from c urre nt d tepl aecment , Ihe ffill g nol ic co re s.... tu ro ti nn lind erld y enrrem s r;rrc llll )' ll.ffee l \110 proeesses II I l'tllrli ng . T he llnl!l l ~'s i s ur these fa ct ors i n cumlnna ti on ea n give t he eq unt io ns for (I mull iwin d illg mechlne wil li nont tnea r pa rnme ters . An a pp roa ch at rned li t J"l.'d uciIlg the errec r. o r c ur rent d ispl aee ment 0/1 li la o per-a ti un of sy nc hr o nnus nr;ll d e ma chines in steady-state cooclitio llS is. to tr a nspose couduc tors tnnd dcorco:lo t hei r cross secuon. In inducti on m ac hi nes t he errect ot cu rre n t dtsplecement is tnken n d vs n l~ g<' of for im pf u y i n ~ rhe d yna mic C,htll'llc Lor iSlics.

7.4. Energy Conve rsion proble ms Involving

Independent Variables T he inde pe nden t " e ri a blClil in ele<:tromeel,rlllica l l'q llflt ions a re ec mmc u tv th e ,"ohll i" nnd momcn t of resiStllllCO ;1/ , (torq ue). I t sho uld be kept i n m i rlll t hal l ht'!e e qua tiolls ",Iso co nta in t he \'o llDgc frequen cy wh ich dtlle nnl ncs th e c\II'n 'lll Ieeq oe ucy . I n th e ge nerAl case. bo t h vc l rege (aull fr C'q uc ncy) and torq ue ma y cha nge s imulle· IIC01'sl y . I n m ost elISC!. bcwevor , Ihe, stu dY of tile etrecr of torq ue on the t1 p la m ic a nd s t a t ic ChArMt eri515cs in volves Iu va r-ia hl o vo lt nges w ilh t he to rq ue a t the s hAft s tsc kt'p t co nsta n t . Til rOlls idCl'ing r nJll ple ~ eloctromecbanrcn l s)'sel'm!! wh ich etlflsist or ma ny e le-c t ric nl:l.chi rtNI . it is n C ~!lSIl IJ' to reduc e tile nllmhcr of equ atlons lor c1o!:'CTihhlg the sim pli fied Crlt'f1Z Y convers ion pr ocesses. The researcher must corlo inl)' ha ve ~ th orollgl1 iJl ~lgh t into wh o~ ass um p t iOllll he mu at in lrod " ce and ",'h ll~ Iee t uros he ca ll ncg l('ct to make th o a na lys is llh nple r but ntleq ua te Gllm!!:h. Look II I t he effects of vo lt ages And Iecq ucn ccs 0 0 the p t.ocellS(lS of ('OGrgy ce nvere tc u in e ts e tere Illl\ch ilK'll. T ho i nvcst il;l:atio n of IrnnsiCIl\.$ a l ~mry i llg Ireq ueacles a nd ~·o l t. g e e o n th o te rm tnats of II mo tor is ~ rm ll ch a ig nifitl\ nce. This is pa rtic ula rly t he case for 811tonOJnOllJlcl<!e t roJ1J t'cha nicnl s ys tems, where the re is a need 10 o bta in t ho opt ima l cOllr!toe of Irnusi l!nlS b)' "eryillg t he vol tages a nd Ieeque ucjcs , a nd ats c l or motors III s tarl i n!: in t he cond iti ons li t wh ic h t he mo tor powtfS lind s u pply powers a re co mpar a b le , The pr ecesses 01 e tec t romeeha nice l e nergy co nve rs ion at "a r)'ing In-qllCnc il!s a nd "oltages e re deecr tbed by the !)"Stf lllS of e qua tt ons (4. 1) th rough (4.:\ ) And (3.3) t hroug h (3. 12).

150

Ch. , . Mode l. of Ele clrlc M.c hin• •

T he an alys is of d ynam ics, of tud ucucn mac hines on a n a na log com puter a t "o T ~' i llg Irequ ehctes uud voltages calls for a speci al s up ply net wo rk . Th e pt' r j o rl i ~ {lIn cti o " ssi n wt an d cos wt o f a Wlty i ng freque nc)' r nn be found Irom the solut io n of Iwo eq ua tl ons dx /dl = "lU, (7. 2 1) , dy ld t = - WI Fo r t he s tnhil tznt ic u of ~h(! volt age nmpfit ude propo r t ional to s tu 001 nud cos IJ>t , the co m ~u ler model shoul d have a n additlo nal c ircuit f Of t he soluuon of Lh,e eq ua tion s i n2wt +cQS2 wt_ 1 = 0 (7.22) Sho uld t he voltage a mpli t urle und ergo chan ges , tilt> Ieedb ack pa th prov ides for t he compe nse ttc n of errors , T he sup ply ne t work a t t he ;"ar y illg vo l ta ge a mpli t ude a nd co nst a nt Ireq ueucy is made a de qua te from the solu t ion of the equa t ion d1xldt' (,J X = 0 (7 .23)

T he problems be i ng s la ted .1I1a )" illvolv e vol t age am pli t ude s 1I 11d frequenc y t ha t a re Iuncu ons :of th e otfocu ve va l ues of t he m agne tlzi1ig eurrco r. fl ux ltnkagea , Ulf d rotor spee d. T he model for the solu-

tion of equ a t io ns of a n indu ctio n motor wit h cons tant pa ram eters is set lip witll co ns tde ra uo n fo} ehe a bove factors . Consi der trans ients at a 'var y ing su p ply vo l tage lind cons ta nt frequency, The pa t t ern of volt nge changes is recorded on nonli nea ri ty u nits . T he vo ltage is mad e t(~ V llr)' bet wee n t he limiting va l ues equ al to a bout 0 .8 and 1.2 of th e nomt na t valu e Un. The a nal ys is of esc ilIogre ms t ake n n t con sra r u pa ram eters ca n rovea! lha t for small power an d medi u m-p ower nrotors , t he cu rren ts and torques show maxi ma dur ing th e firs t onq or two peri ods when t he vof tag e s uill eha ngos l it.tlc. The ch ara cter of vol tage varia ti ons has t herefore R weak effect (Ill t he time of sHir ting or t hese machi nes . Tn h igh-power mo tors , th e peak curre n ts fi nd t orq ues fi re evldeu t du ri ng t he fir st ei gh t to twelv e pert ode, so t he vol rnge cha nges here ha ve a more pronou nced efrcct on the course of tr ansients , T h is is al so the case willi mo tors ope ra ted from M v '~ltagl! sou rces a nd wit h motors havi ng a Ifl rgl.' moment of iner t ia . T he l'('SUItS of s t ud ies show t hat t he impact. currents fwd torq ues deUc'n d o n th e eharaetur of a n i nit ia l cha nge tn the supply vo ltage. T h"e d cr- iva tiv c 0: ... dufdt or du fd'V t her efore cha racter izes th e course- of a t rnnsle nt , A cha nge in tho supply voft age exer ts a gr eeter effoct on t he Lime of s ta rt i ng of low-power a nd medi um- power motors . A decrease ill volta ge exer ts a great er efred t han a n i ncrease i n voltage at sta l·t i llg. It should be no ted t ha t i n mot ors where the cur rents a nd tor ques te ach pea k values in one or t,wo periods , t he uonflnea rlt.y o[ mu t ual i ndu ct a nce (snturarton] hae 'n smaller et rect Oil tr an s ients limn us decrease.

7.S. "' ....1, 010 of Op er.t1o" of ;tn l Eledr ic M. chlne

\ 51

It i& of in lerest. to st ud}' t he d yn ami cs of ind uc t ion motors wnb oonli near param eters , i o wWch t he treoque ne~' a nd ,' al tai\' chnnge sim ult:lntousl)'. Th is problem is o nly so lv a ble on d igita l computera. Prom t.he results of the I\llalog-r.om pute r . na l)'sls of a motor wlth ecn st e nt param et ers we ca n conclude. thAt l\ t rlosiont oc.c ur ring a t 11 va r )'i1lg '"olt age and eo nst all l frequency! di ffers ill cha racter fro m • l r:'l ll:'l il'nt prt\CQC(!i llg a t. n ,'Ilrr i ng! e nd co ns tan t U. Th e course of

Fi" 7.6. IWpKt . nd

~l nl8Oid al

load'

a t rlllu ie ol hea vil y depe nds 011 t he i;nitial values of freq ue nc)' an d VOJtll~. T'ransie nt a at cousl nn l L" Rud cons tllnt ! l'C'present 8. par t. i· cula r case . In wid espr ead \l .so lire eJlerl:Y o.;o ll" e r le rs desi gned t o opeente a t 0, ec astan t volta ge nnd peri'ld icllll r "!Jrr in !: load (drives of cr ushOMl, roll tng mu!s . e tc .}. T Ile cha racte r of ~h a ngcs i n /If r ca ll be me st d ir· Ierent , In a machine opera ti ng a t All hu pact load , t ho ai r ga p eo nre tns har mo nics wi th peak am pli tu des . bllt. tile l::lp ill free from t hese har mon ics at a s uulsoidal lea d IPi* . 7.6). If lin energy ec nvertee is cumpaeahl e in powe-r wit h t he Ii i"! . IJu~. impllcl load d is lor U t ile vo lt ages lind eurre nts ill t he EC. IIn(l the u pper IllIrmonies a r iSll ill t he line a Dd a ffect t he ope.l'1I li l)1I of o,,",c r dev ices. The inde pe ndent " ar iables U a ud ~l , lind dtlpe nd\llll ". ri8 bles i and Ill , can iUlE'reha llgc pl aces . For exam p le . i [ we cont ro l eu erem in • gtllM'rlIWI' ( in ;I :e-nerlllo r- motot s)'stem cal led the curren t dr i" c), t ile freq uency a nd "oltage will nn dt!~ chrlnees a t t he o ut pu t. In sources wit h se m icond uc to r oleme llj.s. the eo mroltod vlIr illblQ is cur ren t ra th er t han ,·ollll;;:t'.

7.S. The Analysis of Operation of a Real El ectr ic M,chine Lot us Rna l)'zo t he p t()('e~ ofel ee \jrQm ac han i CllI e llC!tID· ce nvers tc n in a rea l i nd uction machine wh ich Is t he mos t gene ral t )' Pl' of EC .!!i nce here (d, ' " CIl~. A sy mmet r ic i nd~lcl i o n ma ch ine 1tll5 u s windi nr,s arranged in ete te of the mag neti c core s ta cked of sheet-s tee l bUlli H,' -

li OIIS. I li a mllel,ioe ope ra ted en si mlsoida l vol t..' ge sup ply, the ai r ga p co nt a ins t he .!lJl('Clru nl

h limlOn icl! llSSOC iliLed w it h m l'l(:'Dl't iJ.in g

roreea. ~ Ii e ncy. non lin llll rit y: of res ssta eeee an d rndc eu vc ree ct a neee, mllnl1fllclu r ing fIlCto 1'3, ew . I n gEl llC f a l. it ma y ha ve allY k ind of harm onics . T he ind Peti oo fllll:ch illf- is a mllh iw imling e lectric machine . I n t he llnnl)"llis of t h is machine , a ile s hould ccostdee th at the-s ta tor and retce ha ve edd y curre nt l~ p.!l lind th e parall el bra ndl('!! of s lAtor ,.,.i" d ings mllY o jK'rn\e in d jJfcrent condi ti ons . Bes itl(>s , it is to be ke pt in m ind tha t the act itc SIMIC!" re of tho meehloe is bu ill IIp of core eceuc ns. th e opernt i,ng condi ti ons of which life different on tho ex t rcm fries a nd i n t he m iddle. Obv ious ly , t ho 3CCUflilc m~1l 1 he nHl t iea l deser i pUnn of t he peocceacs of e nergy convers io n in n real m;lchi ne co nunt LIl giv en because each ot t he sour ces of space hnrm ont ce prod uces All (uli nit o spoct eum of hnrm onlc s , nnd t he numbe r pf such sources in n eca l EC run s int o n Iew le ns. I T ilo most genera l 1001 for the descriptlen l)f eucr,;)' converslon i n lI. co nve nt iona l inducll on fII 11('h in", is the s)'s h' lll of oqoflLions (:i,3) t hrou; !l (3,12) wil h cOlist tlnl a nd non liooar ccerneems. Altllougl. t ho descr ipUon 91 pr ocesses i 'la n EC is /Ilwa~'.'1 a ppro ];tma te, comput i na: raeili t ie! ca n pr ov ide II su ffieit' lit accuracy in t he solu lion of most pro ble ma eneonnterod in elee lromee halli cs. It .'Ihotlld be remem bered th n ~pet ime nUl I invM llgntio ns too ca n give o nly llppro xima le dat il. I For a maelline wit h n sln\lJOidlll \' nl u. ~o at it s inpu t , it is possi ble 10 use Eqs . (3.3) l hro ugh (3.'12) with constl lll ccetnereu rs u ndee t he assum pt ion t hlll th o mac-hi lllf para meters a re Inde pendent of load . I n a sa tura ted mach ine t ho magn et izin" elllTl!ot cont ains od d JlllrtnOllics . B CClllISC of t ho sllift betwee n the wintlillg3 in s pace lind be t wee n cur rent s in tlm u, ~htose ltar mon i(,!1 pmducc fil.'lds in t bo a ir ga p. ~ha t lrllvol a t d incn/ Jlt sl ips wit h reepect te t he rotu r , B/lscct 011 t ho model of t he general ized e nergy conve r te r. lhe model of 0 eet ura te d mach ine con be .'\C'l up w il ll /II potn of wi l1 dill gS Oil the Sllltor nnd rotor along t he a ~ nd ~ axes, i ll wh ich t he np pl ted \'olLagell prod uce a f ield in t he ai r ga P s uch as l illtl Iound in 11 red ma ch ine (aeo See. 7.2). -"1S$um i n!: l hat l hel'6 is IU) i n tcrl'\!la Lion bet ween Ule har mo nica, we ca n tra ns rorm E qa. (3. 3) t hrO\lgb (::1.12) i nto Eqs . (5 .1) lIud (::'.2) 11111 co us t ruc~ II machlne IllM oJ (see Pig . 5 .1) with m wi ndi llp on 1110 stllto r . ",1 ro to r. Sueh II mode l cor~llpolldl 10 lin ide<ll mllchine 'Supplied Ire m â&#x20AC;˘ u(lll!li lluso idal asr mmetrie vo h llgc source (see See . 5 .1). Thus, considu ing II mllCÂĽ D(' with va r ta ble plIP1lmelers allli ainullOidlll \'oILD gt>s ot the te n ntJlals .....e ca ll rep resen t i ~ as a mac hine wit h const ant parameters l'iid noos in llsoidlll voh ages lit t ile in put . T he equa ti ons for II sa lllrtlted mach ine d iffer fro m those (or a n in-

1. S. A ne lyoi. 0 1 Ope,ol ion 0 1 Roe! El.. d .k Me ehl n..

deer ton machinc in tha t t he forme r contslu coe fficients M~,,, , M;...,,", M;. ~ , . ... M ;:" ~. M~h, ... ., M~:~ ~ . and a lso other eocrne re nts to acc o un t. for the Ier ruma gn e t.ic coopttugs between harm on ics . Le t us la ke n look a t {JIO effect o r It nOlls illllsoid nl dte tr tbut tc n 01 tIlt.. ma gJll! ti zing Iorcn o n the s pec t rum of harmonics i n t he lIir gn p lind gi ve t he mathern u tical descri pt(o n of the processes umler t he assumptto n t hllt t he ma chine of m tc rcet is unsa t ura hlc flno th e vol tn ga is s tnuso tdal . If we ass ume lbnt the number of he r mcnics Pis infi nite nnd t he IIi.. gap ill smoot h ' th~ 11 th e a tr- gap fi l'ld repe a ls t he pnt t er u of dt strrbuuou of th e m~go()ti ~"lg force. Kn owing the spectr um of hllrm Ol1 ics i n t he ni r gl' p; nam e ly , t he ir emp jt t u dea a nd phases , we II p pl~' th e eq uatio ns for tho gone re fizcd energy conv er te r and set u p t he rna the matical model Of 1110 m ach ine t o duscrtbo the onorgy eonve rs rou procossos . T he aSSlimp tion IU;ll\! is t ha t the st a t or end ro t or enn-y t ho sa me num ber ofjiCU tiOliS wi ndi ngs, wh ich corres po nds t o t he c hosen number of h .r mnn ies . T ile prob lem being sl a ted rnua t cover two to Jnu r hermouics. Im d it s solutin n ca n not corta inly be eccum te. T ho eestereuces o f fi ct itio us wfnd lngs IUII Y be t uke u eq ual j.o.Ihe restet nncee of ect ua l windings. 'I'he m u t ua l ind uct an ces associ rted w ilh upper hnrmo ntcs of th e m ngn llt izillg" Iorrc ma y be tak o rr a pproxima tely e q ua l to onet,hir d of t ho Ion da mc nta ! for t hc th ir d, lJarm on ic , to o ne-I ift. h for t he Ii fth he rmont c , etc. T ho coofft cten ta df coup l ing between harm oni cs, .U n /. c,011 001 be h iqllor th nn t ho mUllln l lnduot an ces between upper ha rm on ics , ,11/1. T he coeff icients Oljle riu l;l in t o elec tr omecha uical eq un t ions de pend o n load . T bus t he eq uati ons for tile ma ch ino wit h o »oestn usotdat mmf di st l·i bu\.jon arc ~ h lJ same R~ those for a satureted m ach ine . Th e oq ue t tons differ from each ot her hr the values of coeffi ci o ut.s ami the om p l it udcs of lutr moni cs . For II nOIlS/lt llJ" lIted m a chine , th e 'I, m f .liSlr ib ll t io tl ill ainusci dal . the "ir ga p is smc our, th ou gh oOIll,.(uifor m dill! to man ufact uring fac t ors ( m isa li~lImo ot of the rotor wi:lh respe ct \ 0 1 11l~ SIOt.Or. elflpt icity , con tc tt y . etc.) . T he ate-g a p numnuro rmt t v is rcs pouaihle for th e a p peara nce of t he s pec frum of ha rmoni cs in the air gap. If t he ga p Is no nuniform botl l in t~lC axta l find in th e rad ial di · rccuon . for li ll a lpi ng the processes tho mac htne is br oke n in to III p ieces in tlt e axia l d ire cti on. and t heimachine mode l is cons tr uc ted with In <,; t /l. IOr5 a nd a commo n rot or, s uch ns ill ust r a te d in Fig. 5 .3 . I I CA n be ass umed he re t hat th e Hne:\' oltnge di s tt-ibntes it se lf nniIorml y lI1110ng III m ecbtnee nud ench lill'!me lll llry ma chine d tt rcrs Hul e from t he othe r rn pru-amete ra. E ven if we ass ume the p nrnmetc rs to ho ide ntical , t ho probl em a t h a nd wil l bo in sup cra hl y diffi cult because each e lem entary mach in e ha s II lspeet.rum or hnrmonlc s due to eccentricity . T he h arm onic s pec t ru m oo n l uifl~ space he rm o nlcs w h ich inter act n n.l a lf..cr ouch other. As ment.iuned carhe r , the Ilnn ly sis .

.,,,

Ch. 1. Mod_I. 01 Electric Medllne•

m ust red uce to th e iovcst ilnluon or processes In...o h·ing a s pecUied mun ber of harmon ies. Consideri n::: t he ai r-ga p nhnun irOfOlit y , we aga in e retve lit Eqs . ·(3.3) through (3.12) whose solutfcn necessi ta tes I llMl we shou ld eo rretl l }' l!pec i f y parameters 1I 1l,d deter mine t he amplltudes lIud phll lK'.!I

of har monica. The n ir-ga p Ilpnu nUorm ilY d ue to snlioncy at sc g ive!! rise 10 II. defi n ite s pccreo m of harmonica. The mnthomll.lienl d<.'scti puc n here is tile sa me es for .the ot hor ty pes of ne nun ltorm ity . So . in 1\ real u nsnturatcd l unchlno the re are in fi n it e sets of s pace har mo nics a I-is ing h om II. n~lC)s in\l80 i d ll l mmf d lst r ih ut ion nnd ntrgn p uo nunifcrmlty du e to sq.Jienc)" a nd t'ue ntrlcily. l n 11 sa t urated machine there a ppeara uncthar spe ctr um d ill' to no nlinear se tt- lind mu lUlil i nd ucta uees a nd a lso he terod yne frequ en cies . Bu t a maj or portion of u.ese ha rmon ics lire praeth:-3 l1y hll.rm le!l'll s i nce t he" have i nll nilelr slUllo lI llo m pli tudes ~ end o nl y . sma ll po r t ion of h:l.rl'll ouies in th is spec trum 1'l Ue<: t the 'm a chine performllnce. It ill ellllY to see ,h. t Eqs . (3 .3) t1,rough (3 .12) describe t}IO pecces-.!Je3 of ellcrgy eouvers tcn ot III no nsiDusoidlil s u pply \'ol l.aljlc i ll I aa t llra ted nUtchine wit h dll. l'Cg1'l rd for e rhee splice harmon ics. I n t he a I131, 1Iis of the pr<feessell In a re nt ma chtn e. t ile resea rc her should. fi rst , hav e a cl ea r ioel'l of t he flle\ th M tile sol uti on 10 the p ro blem ca n bo ap pm xt mmc , seco nd. SOL 11 d('fin ite l im it 011 ure numbe r of eq ua t ions rnurnbor of lmrmon ics) to be de llIt with. lind, t hird . perform the most cilnql 1o,," proce dur e , IIl1ml' lr , uertnc t he nmptf tud os lind phlllSClI o f th e ha rmonics ill q uestion a nd e tsc the pere mct crs fo r ure 1'1cct rdmCtha niCll I equ IIllons . T his 0 00&, t he researcher CIlU solve t he 89ullt ions on II ecm pote r nnd obta in an Il p proll: i m ~ !.t' sc tnuo n. Fro m t he (> ngi necr's vi t'wpoi llt , the so lut ion -ean be eens idcred I\CCIU'a l~ teee use t he ob t ained ",sulu ca n com pare well v:i1 h th~ rt'Au lt!l of theh u: per irnont on R relll machi ne; in ot her words , the me3sllrelOcnt errtlrs ca n be of the ~ mll order es t he DOCS Int rod uced i n the res ul ts oili ai ned rrom the !Oluli o n of the equa'uous . Des pite t ho cnlll p)c;l: it~· er t he pre cesses in th~ ga p 01 lin electr ic maehlne an d t ile f1 if!crt'u t 'e~ nse! res ponsi ble for the eme rgence of lun"mu ni es , th e ml,lhemo. l icftl descr ipt io n or e nel'g)' eonve rs to n proce sscs CII n be g il' c/l b)' us ing o ne nnd t he some sot of eq unri o ns for 11 11 111- '1 win d i ng umehlne. The modt l of /1 machine wit h m milld /ngs 01£ th,. stator and II w indingl .()II Iht rotor permft J t he ffua rdur to [ormulate I1uJ equatio~ for any ClUe of t ll!('lromedlallfcal t nt rg" COIllJtUwn and 301vt thu e equatio" , on com pull'''. Al prrunl alm OtI all /Jtob ftrnl In~'Olctd in tM ollldll, i, of tnerg" tile 301ution to Q dt!ill Uea«UNJC'Ij. .comJt1','on prot:elln Qrt omtJlab/e

8.1. Types o f "'.ymmetry in, Elt el. le M edline s

Chap fer

A symm etr ic Energ y Converters B.t . Types of ASY",~try In Electric Mac hines The tbeor}" of electromecha nica.l bne!l:)' eonverstou gi" ne rally dea ls wit h eym me trf c me eht nes . H e.....-cve r, most e tecu-rc machi nes , o r e ve n a l most a ll rea l ma chi nes. are 115ymnllHric: if ffi Allu fa cl u r in i b el ors are "ken into aCCOlm t, bec,!l\l5e il is Impose tb te to a tt ai n t he same parameters Ior e ach pha se . As}'ln metrie machinl's can s how e leplriclll. s pec tal , a nd mag ne ti c asyMm etrr . E lectrica l asymme t ry res ults rrom a di rrefE.<nco bet wee n t ho re,si3la nce s or i nd uctive rea ct a nces or madl inil phases , To milc h Ines w Ith th is ty pe of asymme t r y M lnng in ductio n motors hlw i ng ve eiOlJ!l phil1!C-slli fti ng cl em e n ts 11ll,1 uqera l.ing Iro m II J'; lnc ll'- ph llS6 power l tne . S pncln l nsy m mlliry a ppears aa a rClIlll t of sll irL in spa ce or the n e i gll ~ hor in ll phase wi nd ing IU O!! th l'Ollgh All n ugle o lloer a la n nil lHlg1e o f 211 :m e lec t etce l l'IHlllln!!, 1'11(\ mac h in es wi t h t1 ti ~ l y pl' o r ilsy m mul r )' i nclude mo tors Illwi rig sha de d po les , ~ y n c h ro me ters , ole , M ~ gn el i l'; IIS~'m lll(l l ry er tsos from 0 nonunifor m n il' gHp 11 11\1. somer lmes- fr o m an as ym me tric Illllgpel ie core . Some mac h in es mn y S i lll ll l t ll n(Oo ll~ I~' di spl llr t h ree t ype s of asym metr)' , A n exa m l,le is 11 s i llg le- plla ~ und ucuon mo to r wit h l\ shnr tc ircuite-d s lia d ing loop ( tu r n , or co re on th e po le). The a bo ve-me .. ~ tio ned t hr ee l )'peS of /U1)'l n mll\ ry have tl> J o wit h I he pri nci ple 01a ct io n o f tho mac h rues . It is a lso o f In te res t 19 itwl'stigatl' sy m mer rte ma c hioos in whi ch Ilsy m melr y Il r l sn frOln th e cHe('1 of vario us manufa cl ur ing Iac to re. Of m uc h importa rn:tl in t oo t h,-or}' o f l!' lltrgy een vort ers is l he 111vcstiglltion of .!f)'m me t r ic ma cbines wil li 1I.~)'m me t r ic vo llaRCs 0 11 t hei r tcrmhlab , T he asymmetri e res po nse o f !!r ncb ro no us maeh tuee, ue usror mers . a nd indu cti on lIla ch io u In th o stea d y-s ta te a nd teensteet ('fmdi tlOlls dese rves pltrticul nr e ne nt to n for the st ndy 01 powe r sys tem peefcrmance . ' Wort h}' or nonce i~ t he mos t cc ncrlll 'cllS(l oooco rned with t he 1I11l l hemalicfll de.> 'lC:r ip t ion of e Ol'tg)' co nvers io n in eay m me t ele mnchtues opera.ti nl: a t 1Ill)'lIlmetr i(' "olttlge5 e n I"l'~ r terminllis. T'he t hpor y o f Ils)-m me lric ener gy 'conveners is g iven rrea un e ut i ll quite 8 few boo ks 011 t he s u bje ct. t' lough it s fu r the r d ovel cpman t is esse ut tal , fo r t he class or these m Ach ines ee vees a grea t va ril' t )Âˇ o f t ypes ,

.'"

Ch. 8. A. ym;". loiI; e "" lIY Conve r11t'"

The meth od of sym met ric eom po nonts lind too tbeory of roLll 1h ll: fiel ds II~ t he mai n lool s lor the s lu d )' of AS)' mmelric ECs. The IIl4lJumaUral /kÂŤrlptlon

Ee, repnu nl,

t il er,,, corlQullo/i ill 11I1rlImlr fc

part tcular (JIM 0/ Ihe a1Ully~i' 0/ (Uum~t'/f: Ees. f or th is reaso n it is not nlws;ys jud ici olls to ext end the ectdevemen te II:

in the t heor y of s ym me u-lo , m ach tncs into Ill o 1\l1!11 oj 1Il!1'1Il" lct ri <:

mechrn es. On t he whole, \\~h Rte vtJ r the comp lexit y of esvm me rr tc me clunes e nd JUJw(l\'C'r tlh'c rse the asy m motric cond itions in whi ch tIm)' operate , Lilli a nnlylli9 esse nt ia ll y involves th e st udy flf th o lIirga p field . For th e sol u ti on ,of th e proh lems s ta te d, the resea rc her mu st of CO UtsC thorOllgll l)' deFino the alr ga p i n wh ich t he m agne ti c fiel d stores l'oofgY a nd then give t he m athl.'m llli ~ 1 desc rip ti on o{ energy conversio n processes. I II An esym me u-ie m lld d lle oJl'!'raUng o n s ymmet ric "olt a~ s u p ply , the a tr gn p co nta ios bot ll $I 10r . . . .tlrtl ( posit ive--seq ucoc;c) e od It be ckwnnl (negALive-sequence) n ~ l d . In th eee- phese lind multlph nse machines . zero-seq uence fields appear under ce rt ain con d u tona. A ~ymÂ melry Is rerponslble for the butl cWp oj a zero-sequsnce f ield in llle air lJajJ . 7" ~ study of energy conberslrm in uJymmc trlc much incs in a f irst

apprQx imal Um rt duu.r 10 solvi ng the eledrom~cJUllllra t equfdloM f or the two f uM, f/1 t he aI r gap ~

8.2. Electrical

~ nd

Magnetic Asymmetry

T be to rq ue In $I sy mme lrl e mae ldne resul ts fro m t he products of st e tc r a flt! rere r curr ents olq ng di fferent refer en ce e xee. I n t his mach ine. t he st a tor arid ro tor c urreut pr oducts aklllg t he 5amll Ue5 I~i;' -1~1' (8.1) d o 1I0t give ri1SO to ure tor qul! siuce th o sum of to-m e ill (8 .1) is eero . l n an aSYlllme tr ic nlllchilk-, tbe c u rre nt prc due ta a lo ng the gamelu is do produce lim torque whic h i~ t1efi n:lble on t he assump t ion l hlll ure mu tu al induc ta nces between til e s ta tor II lld rot or phose wiudIn!:" are ide nlica l: .v ~ = (m '2) J1I (i~i~ io.i~ - IV. ) (8.2)

to.i' +

I II detcnn illillg the tor que I~ (Ill Ilsymmut ric ma chlne wi l li ecn etde r-

IIlioo for II dirrcronce \}e tweell t he mutn nt tud ue ta nces a long the m ach ine axes . i ~ is more jutl it ious t o defi ut! t hn lor'lu \l i n teous of [lu x Hnkoges. i In th e generlll CBS<l, Ille mat homutlca l descrip ti on or e llerg y conversion iii nil IIs ymmlltrlc lflu1li phllse mul tipolar machine in volves t he ee tut ton of Eqs. (:'\. 3) thro ugh (3 . 12) rce t he Kene ra li z('d energy converter. I n ioing fr om tI ; im ple to a more complex m alhll lll ll l k li l II llaly.'!i.!!. it ma kes se nse to co nsider so me par ti cul llr caSE'S tn votved i n the s tudy or u ym met r ie flllldJi lles .

Conside r l' tw o-phose m achtne in \\lhicb ~ =P IUd. t he Dumbe r of sl o ts per poLe and t he num ber or 510\5 pe r ph ase are differen t , a nd t he co nductors differ i ll cross sec t ion . l l'hcse co nd it ion s prom o te n111 gnet ic asymm etr-y which s hows up as 1 a d iHl'flllll sa t ura t ion nlong 0100:10 refere nce a x is of t he ma chine. Bo t h el ectrical find magnetic t ypes of

aey rum etr-y lire res pons ible fo r r hc

dif fer ence betwee n t he parame ters a long t he m achi ne Il I<:CS (F ig . 8.1). I f tho mecht nc ro t or b symmctr ic , we nee d t o co uver t one etn tor wi ndiug to t he ot her an d ha nd le the t ra nsform ed e qua tions tor th e ro tor w ind ing . I ntrod uce t he conversion Iecto r 11k = M Bb/Jlf M

= (Ivak B1wAkA)

(8 ..'1 ) a nd I'd A ~ ar e till) m lltu «1 i nd uc ta nces between t he s ta t or and

w here Al

, •

01>

Fig . 8. 1. The model of a u I;onl;o rgy c onV<'Tte r ""iUI cl c<:trtcal

asy nl nle ~ry

ro tor Wind ings shown in Fi g . 8. 1; ~ rl d k ll snd NA are t he Iactc re t.ha t a ccount fo r asymme t ry be tween: pha ses 8 a nd A . Define t he mu tu al in d uct a nces al o ?g t he 0: luis a nd a long t ho

IIf ,. = M

II ax is

A ..

'= M

M fj =MJH != kM

Hence . MA ~ .... M ~ A = M cose ,

M A60 = M ~ A

M Sb ""M u = kM cos O

IIf Si n 0, lIf oll<>

"'"

M .. s

(g .~ )

kfl,f sinO

This done, set up the voltage eg1j:at ions. F irst , express the fl ux linkages as

, + fr{ [cos e i .. - s in e t~ ) W = Ll lA + k M (sin G f.. + cos 0 f~) W.. = L' i~ + it! cos 0 I.. + k At sin 0 i 60 'If A = £ ;'1;' B

'If~

L'i~

_ M s in ,S i ..

+: k M

(8 .5)

cos O 160

T he alec reom agnet fe torque ca n be: defined as 11 par tin l der kvative of th e t o ta l ste rad electromaensrIc, e nergy with respect t o the geo-

Ch. a Asymm. tric Enet9 y Converters

mut r!c an gle: ,1-1 ~ ... PUIU, ,,,{lJe

(B.6)

uw.... = !.2"'"'JO {aro.'L (Âˇ'1' (.' J' lJf} 'a + <l'~~ ill) ! ~ + <l;: :I(i~)~+ {IIiJ~1 +2M (ki~l~ - I~ t~)}

(8.7)

Here

= (uf&.'a,) (83IDO)

O j~al{Je

= -

(0.1:"' /83) (a.!CIJ . w r )

(8.8)

" here a, = dw ,:dt is the a nkular acce leration 01 Ih e ro tor. S im il arly , ii'1~ 6 iUe -; - (i)[~1l 18$) (a, / (,) , (o) , ) (iJ~/iJe) '" -

( ()I~ /{}s) (0,1", ,"1,)

(8.9)

Cons ideri ng t ha t al~"J{J $ =.

the tor que equat ion

M.=t p { - a;:

0 and

fJl~~/fJs =

b('coDl~s

w:~.W;)2 +( ili)II+2M(ki61:' -!M~) }

{R IO}

Expre ssion (8 .10) c o n ttd n ~ two compo nen ts, of wh ich the first is 3 runcuou ()( t he change in tho ro tor wtn dt ng leak age Induc tance a nd t he second is II Iu nctl ou .of t he cha nge i n th e a ir gap energy . T i,e energy stored lip i ll lh~ roto r leakage field with II cha nge i lL eccelera tto n t akes pr o-t in energy con...-erst on. If it accounts lor /I la rgo shnre . t he leakage fi eld e nerg y a dds t o t he t orqu e a nd should be t ak en in to co nsideratio n i n the a nal vs ts of the transien t . 10 th estendy-stn te cond itions a t "!l ich a, d o, t h is energy componen t exe r ts no effec t on th e mach ine torq ue . If we dis regard a cha ngt\ i n t he leak age fiel d e nergy , Eq . (B.W ) will ass ume the form .", ~ = pM (kili:;' ~ l:'i~) (8. 11} An nsvmmet rtc ma ch ine is cne n made com plete wit h phasesh ifti ng clements Inserted into one of its phase s. These are commonly capacitors and re sist ors . ThJ voltage cqua tin n for t he phase incorpora ti ng a ca pac ito r of eap actt ance C has t he form

14 t=U~-(l!C)

1i j dl

(8. 1 ~ )

W il li the ee pac ttance C a nd se rie s resls t or r , added to the circu it ,

ul= uf. -( l iC)

Jitat -r,i A

(8 .13 )

' 50' On inS('fti ng the stllrti ng capacrtecce COl a nd ope r-d t illi capacl .... ance Cop in t ho ci rcu it, the venege equat io n becomes of the form

ul "", u~ -Il /(C#I+C.,.))

5fl a t

(8.t4)

T he mod els for t.l~ solu tion of eq uJi!tions of asymmetric m ll('hi~ are similllf to these .5('1 up (or or di na rj)' ma chi ne! . I n tbe IlllOlys ill of ellergy converters furni shed w ith ca p.adlorS Of ser ies resis tors. t he·

The se t up in ' "(lIvi ng the $\.ar li ng and ope fU lng cap aGi ta llf e5 for tile> ~lu tioD 01 t( luatlo rl$ on an 110110; eoOlpu\.t'r

- u:, f ir_ 8-3. The !Il!Wp for tbe tolUUOD of ~q. (8. 13) on

aD 1118101

eo.r:nputu

model mu st inco rpora te opcru Lonal ; am pliH eu . For exam ple. i ", I'oh'ini E q . (8 .'12) for II. mo tor with ·C .. or (: ",,, th e model h as to-

include an add tt tonal setu p Ot to cut! out COl (Fig . 8 .2). The model ror (8.13) wit h C and r. should be rtued with an attachme nt such as in Fi ll. 8 .3. 'Vitll th e ma t hematiclll model se t up Oil a n lloa Jog compute r . we can Jnvestignte t he effec t or th e pa rameter s or the mechl ne Allt l its phase-shifti ng elements on lh e sta rt c Rnd d}'na mie cll afllcte ri~tks .

8.3. Spedel Asymmetry !

To build u p a e ire-u lor fie ld in a n t' nl'rgy «I nverter . t ho windings must be at eert8.in angles te s pace with res pect 1.0 each olhe r . In ... t wo- phase machine th e . ogle betwee n l lle wtndt ngs is eq ual 10 90 e lcrlri ea l degrees, a nd in an m- pb ~~ mactllne it is equal lo 2..., tm . I n t he p nPflll case. t he flngles between t he wind;n gll can ta ke an y vll.luea . t bereb}' ca us ing s pocl ol a s}'l1l~le 'ry in a ma chine . Any uy m-

Ch, 8. Alyminefr;c Ene rg y Co nve rter s

' 60

millr y, th e s pacial one Incl uded , g ives ri se to a negative-sequence fiel d i ll t ho IIi.. gn p . I ma gine t ho t o ne o f th o windi ngs in /I two-phase m a chine g raduall y t urns from it s or igilln' toca u cn a t II = 90" t o a pos ition at

wh tcu 6 = 0 (F ig . 8. ~ ) . As the wind ing goes on mov ing. t he a ir ga p n olll co nver ts from t he qirculor 10 th e e lli p t ic; field a nd then 10 t he pu lsa ti ng on e a t II = O. ..0\8 the win dings move wf th -easpoct to

nf;-------z

~t---, .~

Usin. ,(

--tAw~

y:,

/ -"

\ B

\w;

n ' UCQ so,t

1'1ll' 8. 4. lll uslrating tb e motion, o l wind ings wilh respect to each other tbu !'I.'9ul15 In the ecnv erslcn 01; the circula r fiold to th e pulsating one

'e ach nth er , the am plitudes inf th e pos l tf vn-s equence a nd negativeseq ue nce fie ld compone nt s of til l.' ell ip ti c Held undergo changes. If th o w inu ing IllC E'S fi re eni ncide nt , we ca n supe r pose one w indi ng on the other an d co nstruct the ~n ode l of 11 single- ph ase m achine . A t epac ta l asy mmetry . L A A = L A A = llf cO!f 0 , LA ~ = L b A = ill si n 0 (8 .15) L

fl b

=L b S = ft{ si n(6-6) , Ls" =L,, s =M cos (1l -6)

F or asy mm etric Wind ings sh ifte d i n s pace , t he Ilux linkages of pha se ass ume tlt e for m 'I' A = L At A L A sill .it- M (cos Of" - s in 6 i lt) If' s = L iJi fI L lJ Af A kM lcoe (ll - 6) t" s in (ll - O)i bl 'I'" = L <l f" + M cos 6t;.- k M cos (ll - 6) I II (8 .1G) 'I' b """ L~ t b - At sin Oi.... kill s i n (6 - OJ i R

+ +

In tra ns form i ng the sot of eq c a ti o ns 10 the a. an d ues, we sho u ld re mem be r that t"

= t:.. / " =

i~ cos6

cooedrna re

+ iA.!!in6

R efe rr ing 10 (8. 10). t he eleetromagnetlc torque of an as )'mm e tr ic mach ine at. an ar bitrary a nl:le 01 oS is g iven by

I'tf . = pM 1- i~I, (1 kCOS=6) + 0. 5k s in26 (i~i~ _ i'i~) k si n 2 6 1~i')

(8.1 7)

T he fea t ure com mo n t o a ll as ymme tric mac h ines is t h a t t he y d ispl ay au e lli p lic field . Salting up t be models for the eq ua t tona of &sym me tric machi nes o n II co m pute r. we ca n analy,;" both t he t ra nsient an d s t ea dy-sta te per fOf'm fl llee o( th e e neray eonvenees . The ana l. YBis of s t a t ic ebe eeerenstres e na bles us to compare t he pote n t ia l ities of IUll p litude co nr ro t with t h ose o f ph ase co nt ro l, cl a rify th e e [feet of p ba se-sh if ti ng e lemllnts lind ma ch lpe param e ters, e tc . I II c.ornpwlsoll wtth symmetriC' macJtlflt:~ , tUymmelrl t mach iflCJ in dVnam lc operation show a gr tilter f1Qllu:n l/o r m fly 0/ t he ang ular velocity , hfgll er peakJo/ t he torqu e, Im/lfr n.o-load speedJ, and tonger transItnt

ti mes ,

T he pos t ttve-sequence IUld nega ttve-ecq uc nce Llclds p l'esellt in the air ga p o f a ma chl ne impai r it s s re t te a nd d yn a mic cha racter ist ics as agai osL t hose o f II ma ch ine w it h tl ci rc ular Hatd. Tile a na lysis th a t d isrega rd s the eUec t 01 uppe r h armon ics i n a n ell ip t ic fi e ld g iv&$ errors to wn htn 10-15 percent . In the presence o f t wo Itel d ccmponems com para ble in am pli t ude, these e rro1'3 a re g rea te r lor an elli p tic fie ld th an for a circu IB r fie ld if up per harmonica a nd ed dy CUITen ts are no t take n into ee nst ceeauo n. Th e d y na mic beha vi or of sa li e n t-pole s ync h ro no us ma ch in es I'tt asynchro nous s t u t ing depe nds on th e rotor pos itio n a t t he in sLll n t of sv.' itc h ing the machi ne i nto the su pply circu it .

8.4.

S in9Ie-Phas~

Moto rs

Single-phase m e ters ope ra t e f rom 1!Ii (lgle-phaSf! power SlIp p l)' sySterns llud fin d wide uso in d ome stic IIpplialltes . Rarer uses include trac t io n drives . Examine 110 id ea l s inglo-ph A8e mete r wh ose a ir gap exh ib its on l)' a poaijrve-se quence and 8 1)l>g/lLivo Âˇsequ~!nt(' fie ld . Sti ch a mete r has II u nif or m air-ga p s tructure . a d lSlr lbuled s inuso id a l Wind ing . a nd is free from satura t ion, In a relll motor , th is t y pe o f w inding is Impossi ble to bui ld u p . so th e a ir gn p e lwa ys co ntai ns a s pectru m \l Âˇ Gln.

of u pper har mo nics. Along with oNlillllr)' SP.1Cfl ha rmo nlrs. the refl ected WIl V(' l! of l he nlllglllrue fiel d np pear in l stngle-ph nse mot or because t he Sillg lo-phll5C vd nclim: eee up tee o nly l'\ port io n of the pole pit ch ill f'onl rnsl t o t wo-phase , t heee- pheae. a nd nrulcip hnse windi ngs. Ar rnnllillg tile Sillqle- phl'l se win di ng in Ill! s lot! is eco no micall y Impract fca b!e . lnsn IdCllJ lI i ll g le-p ha ~o mo tor, 110(> pusitive-sequonce an d negnt fveseq ue nce et e t cr vol tages IH tI equ a l lind tiL!'!r amphtu des come to 1}fI II t ho Ilm plitude o r t ho Im pressed vo l t n ~ . The pnmmeter s 01 t he mod el 01 F iJ:. 3 .3 for tho pcetuv e-eoque nee lind ne~t1ti \'e-seq lu.' oce eempooents ea n be lh t' same. T ill.' PN:K"('5..<\es 01 oll erll:Y co nveeeren in t he motor under st ud y UN dl",,,ail,"i1/o by E qa. (4 .R) th rough (4. 11) und er t he rOlld it iollS l'!peeiHed a bove Ir thf: poSiU\,6-seq ue llco a nd the nO',:ati,'&-!Cqu t'oce fiel d! Ate equa l in nrn pfit nde, t h~. mol ot dOll:! nol aeve to p A I'lll. t t i ng torq ue.

., t'lg. S.!>. Slngl.. phase induet lom OIDW,", w,\h a $l''''l'lt ol ,bad lng 10011 "'" We poll' (A) I ml U :(U1 ll\Ntlc nla!:"l't k ~)'~ lrm (h I

I t is t hen necessa r y t o reduce Ih t~ neeauvc-sequence Hdd and thu s to prod uce urc d iffere ncc Ixjt ween tho torq ues d ue 10 boll l l ie l,ls fit S <= i . t he d trrere nee be ing t he s tlll1 ing tor q ue . One of t l,,· s teps taken to red uce t ho nf'flI t ive-~q n(' ncc Held nnd br ing t ho Air ga p fiel d cl oser 10 the dl'Cu b r fi!ld palter n is 10 U~ a n :uld iti ou:a l u;indi og shifte d in space w ith rN'W I 10 th(' ma in one and i ll ro r p" r ll l e • dev ice t o secure the t imc f:.IJ ift be twccu t he currem s i n t ho 1\\"0 ~ indings . A r npac ttoe is best llllited for t ho pur pose. A singlephase mo tor l.a v i"ll lWo willd ings one of wh ieh l'oni llins n ~ t n r ti "2 en pnci t or i ~ k nown ns II cnpa<'- it or mot or whos e c ire ui l, <l in lll'~llJ1 is !Slluw lI in Fill. 4 .7. ' or t he A~~" " lllr l l' i , ' lIl olllrs. t ile .~ im p l (! st t u lIl.'s ir;n l<lId m 01l1 pop IIb r is a .~ i ll1:" Il.'. p h n Nl ' lI ol.o~ wit ll s hnded poles (willI 1I .~ hort-l',1 .~ h lul · ill~ loup Ull t he pole). s uch os iIIu st rnted i n F igs . 8 .5 :and 8 .6 . Des pite t he reer t het t he molor dosilln is s lm ple, the mat.hema ti ca l de,'ter ipt ion of c rk'rg)' MIl"l'N' iqll processes i n th is motor is most ru m-

ptex .

8.-4. Singl"-l'h...,, Mo lo ..

I II n s h or t -ci rc u it ed l oop 4 (SM Fig . 8 .6 ) a rrllnge ll 011 /l p ole 2, II cha nge in th e fl ux produced b)' 3 wl noing S gives use t o II curre nt sh ift ed i ll ti m e w it h r es pec t \ 0 t he fi ~,ld w ind ing curre n t. S ince uie wi nd ings are d ispl actld in splice { ~ ('O! F igs. 8 .5 and 8 .6a ) ant! tim curre nts a re sh ifted in t ime. a l rflve l i ng fi (>ld u ppeura i n ti le gnp Thi s is all ell ip t ic, fi el d wt t h n ru the r. leege ncgatt ve-eeq ue nce compone nt . T he tnt orac uoÂť be tw een th e .sre tor cur re nts ami th e cur ren ts.. in tI,C sho r t- oircui te d rotor 1 providca fo r th e sln rU " g a m] dri ving torque s . For t he mo de l of Fi g . 8. 6r., CllN'qy con vc rs te u equati ons ca n bo set u p, giv(\.1J t he pa mm ete rs of w ind illgs uIa. and w~ anrl also the

" ../

,{I

-rf:b '"

~ "4$: ~<:l. -......... :

(bl

rqI

I n duet ton m<.>WI'lI wllh one , ho rt.:od I OfIp on the l' ole

lor,ps on the pole (bl , an d nsymmer.rtc mllgncl lc

( lI ) ,

Il few shor t ed

S!"S U' DI ( c)

pc sitlve-seqoeuec lin d negntive-scquence vol t agos i n Ih o r- trcuu mod el w il_ll w illflings s pa ced 90'" u pa r! . A singlc -ph.. so motor w ith I.WH nr m OI'C ctosed l oo ps o n t l ,.. po le (see "~i g . 8 .Gb) is mo re d if ficu lt, II) fll)llly-:c IIm'l l lll'_ mo tor w illi 11 s ing lo loop , [(01' W'"' fhe n noed 10 [orm/llatc IIn,1 so lv e t he equmlons foenn Il"ym m ol " ic m ldti wind irlg ma chinc wi t h n n eHiptic field . H e ro Wt: a nrt luii lire e q uivnh-nt w ind in l;!~ sll'iI lo ,1 t hrough ra ngle s 6 ' e nd 6 " respect.i vul y.

Eddy currcurs E'.XI!1'1. II cons tde ra bl e e ffec t on tho cb a eec re r ts ues of nS)' m lllt> l r i(' s ing Je-p hosf' mot or-s . These cur ren t s cn u be put to lise so t hllt e m otor will ha ve a s u H iciell l, s l:H l i ng to rq ue . In f ig . 8 .6<:

Is [O;llown t he motor wi th lin as ymme tr ic mag net ic system. In the

la m inat ions of poles 5 there appea r edd y cu rre n ts due t o a cha nge of t he fl ux in th e .!lIngle- phase win din g place d o n the lo ng itu d i na l u t, of th e mach ine . T hese c urrents are in t ime dls pl llee ment wit h respect It) ewre nts in t he field wind ing 3 , a nd oddy c urre nt loops a re in

quad rature wun each olb er (see Fig. B.t ) . In eom peetsc n with mot ors wit h shaded pol es . s ingle-plul5e mol ol'$ w it h a n as ym met r ic magnetic syslt' m ca n have be t te r energy ch arac te r ist ics and are ....Âˇell Rda ptab lll t o dr h 'lng h:o usehold ta ns . I n sa lie nl- po le mach ines t he ma gne tic fiel d ene rgy concen t rate! wi t h in th e pole pitch. T he d ifie rem:e be tween peemea nces in the a rea under tho peres lIld in t he epaee bet.....ee n u.e poles (betwee n the ro Utt and pole pieces) causes the a ppt' llr'll ltt8 o' l't'neeled wa ves wh ich worsen t he ehe ree te etsues of si nghÂˇ. phase me tr es,

8.S. The Electric Machine as an Elemen t of the Syste m El el;lrie mec hl nes generaj ly se rve Il ! Iu ncuo uet un its of e lec t rom ecn ... ntc a r sys tems . If lin energ y co nve r ter opera t es fro m or into t he bus of Infi nit e power, we ca n t reAI t he processes d isrega rd ing

Fig . 8.1. Th' l i01ple I'llP~Il ~ li OD or 80 eleo;:t romeclulllicel .ystem

Fig. 8.8. TIu! t wo-ph .... eDl!l'gJ' Wllverte r as a t ...-o-eluo nllel lou rpo rt sho..l ~ the po!llll.....-q_ a nd aeg.U..~ueQCe Tol tagn u~ . up and re!lpl!Ctlv el y

w:.. ,,:;,

t he p~ rll me l e rs: or e le....e nts cunnee tc d to 1I,e Shllo r e nd rotor d reui l.5 ( Fig. 8.1). I n th e fi ~\I re. Z, lind Z, aee asymmet r tc mu ltipor ts represent ing t he eleme nts con nected 10 t he st nt or an d rotor c ircu its re spectivel y.

8.S. The

E ! ecl n ~ M.~h;n • ••

• n , f l..., .", of In. Syole"..

HIS

I n t he en.. l )"~i:s of a t wo-phase ma ch ine AS e n e leme nt. of the sysit is ce nve ntent to rt' presen t lit All a t wo-channel fourport (Fig . 8 .8) . A t hrr-e- ph /ll!Ml me chtne ean be t rea te d Illl a th reeport ne.... work , t Kn owing ti le parameters of th e circui t. model fo r the pos it ive und tbe negath,e seq ue nce, we ee n da ter mlne tile pos itiv&-sequ ('~e a nd negatil-e· .wq uclftt' e urre nt ll us ing the m ethod of s ym me t ric eozapc ue e te. GIVCIl t he parameters of Io ur-por fs connec ted to th e s talo r lind rotor o f no RS}"InIDl.'tfIC mach f nc, we; can est tmato th e poeruve-seque nee And nega t.ive-seq uen ee c nrre nte wlt h d ue regard for t he paramet ers of mu l ti pa r t net work s. D isreg a rd in g th e processes co nvers io n associa ted wi t h m ult t por ts , i ~ is po.~ ibl o to solve pro blems for cs t im8li lllo: t he chereoteetsu es ot a'll osymmerrte energy conver ter supp lie d from II nonslnu soid a l s ource v in osy mme t r!c ffi liitipor ts connec ted to t ho s tator a nd rotor. T hl s t ype of s t ud ies e na bles t ho l\. nBly,,,,t to co mpile ta bl es for vnrio us as ymmetric co n nect ion netwo rks en d for mu l" te ex pression s fo r de-s cri bin g VIe sh·~ d y·s t ll te pe r formaece, We ca nnot, in the space evarl eble, ec usld er t he com plex eoue ucne for t he internal impednnces of elec terc c ircu it eleme nts connected to the s ta to r R,Dd rot or e fre un. a nd Old y note in p3:.sing tlta t i n so me CII Sl!~ fairly lar gl' erro rs ma y a r tse H· Z. nnd Z. Are flo t ta kt' ll into consi dora t io n. A tt~lll i a pproa ch to t be s t ud y of 1111 as ym me tr ic rnaeh fne III to red uce t.h", m a chine 10 0 s )"mmc t r ic one wHh n tw opo rt thal wei udell an Im peda nce aZ:, or AZ~ a nd tb us a llows for as ymmc t r y . A powtr rul tool f ot tho s t ud y of t ile beha vio r of an e tect ete machi ne 011" plll t of t he e lec tr c me cha nlca l sys te m is t he ~ E'-n SO r IlJla l y! is firs \. cm p l o~'cd for the p uepose by G" lJrlel Krou . Th us , th e prob le ms involve d in th c a nnl )'s is o l as ym met l'ic mnchmes su pplied from " no nstnus c tda l vo t tngo source cnll Ior tho for mulaHan nnd aol utt on oi equ ettc ns w it h d ue regnrd for ed d y cur re nts. asymm e teic wi nd i ngs on tho stuto re ud ro tor , eLC. I n th o syste m ana l. ySLs re lyi ng on multi t.e rmlnlll re presentatic n. it is poss ible to t rent as mu ittpcets fi lly objec ts conn ec ted to t he mechan ical and tbcemu l termi nals of lin e lectric m achine lind thus 20 to a mo re deta iled de!rri pti o n of e nerg y conversion pr l.cessoi:l. In gol v ing technical pr oblem s , hew cve e, t he resesreuce must 1101. complicate t he rna thema ti cal mod el. H is o bjee uve is to ha nd le the task in t he allolted t ime an d to th e spceilil'd aCf'1lTa('~' so 10 give t he e nsw oe sa tisf~'ing th e e usr omer's req ui reme nt s . The ar l of a lt ack ing the pr obl ems in th e origi na l \n. y ov er t he s hor te s s time per jod natura ll y comes from t he sk ill a nd es pcr ren ee of t he en gil1fi'r.

rem,

C h. 9. Equatio n>

' 00

fo~

Ele ctric Mach;ne .

Chap le r

The Equations f or Electric Machines

of Various Designs 9. f. The M athematical Models ot Energy Converters w ith a -Few Deg rees of Freedom As is know n, the e tecr rc mecha ntce ! energy co nverter s wit h o ne d eg ree of free do m are e lectr ic machi nes h av ing one rot at ing member , namel y , II 1'0101'. Th e e uer g v co nver ters w it ll tw o deg rees of fr eedom lire electr-ic m ac hines i o wJli f h hot h the rotor (rotors) a nd stator nre ro lal i ng members (Fig. \l. I ,. These n re d c ubts- eora uo o ma c h ines d escr tbed hy t ilÂŁ' s q uauous

"~ u,; u~ ,,~

ra. + (d l dt ) L& (d ,'a t ) M - !I1w,

- Moo.

L~w. L ~w,

(d :at) s r

r[.

+ (d !dt)

L:"

-L[.w r

- I.,;:'w.

rli + (dl ll t ) q 1:' 16)

(/, lp) (dw ) dt) - M . (/ .' p ) (dol ,rdtl =- M ~ 00..

(d /tlt ) M

111.= pi'fl (i[.i' W _

MUl.

+ w.

M" Af, ~

M w, (d ldt ) !If (d ,' dt ) L !

(0 .1) (9.2) (9.3) (OA)

E q uarIo ns (9. 1) t hro ug h (9 .4) for a n el ectric machine wil h t wo d egr ees of Ireodo m d iffer f rom equnt.iuns fo r II conven tiuna l mac h i ne (with one degree of Ireedom] in u.n e t he vo l tage eq ua t ions contain t he term s ddin illg t ho I.,ml of r ot at ion o f stator and rot or wlndtngs . E qs . /9. 2) eud (9 .3) Incl ude mom e n ts of i ncr l ilt of t he r o tor !lod s ta tor , J, Âťud J ~. a nd rt'si sli ng to rques on the rotor and s eetoe. )11" a nd i11". T he system be co m es detc rm i na tc u nder de finite cond itions (9 .4) sot u p for Ihe rotor nnd s talo r ve tcc tues. Energ y cOtlvers ion in t he r uacumo of F ig . 9. 1 occu rs i n t he a ir gap lind the e lecr romagner tc torque on the rot or end s ta tor- cnusea these mem bers to rota te i n op pos ite di re ction s . I n t he s te ad y-s t a te o pcrnuo u. Ih lj ilistri hu lio n of velccities t.! OPlHlll" 0 11 t he loa d torques

'07

exert ed 0 11 t he rotor lind eo uoter-eotcr- . Ll uder overload condn io ns , cue of tile ro tc rs st e ps ("IlIm i" :: lind tlu! cu.e- , le ad -Ieee ro t or , llcjiins to accel era te . O n :JV\' ilc! l ing an !lllOt ::r conver te r, ill wh ich bo t h t ho rotor a n(\ s in to r 11. 1'6 able 10 rol llie . i nto t he s up ply c ircllit, 10) , nnd

FiJ · ' . 1. Th r machin e wi l l. ' '''0 d l'greet 01 r....oo_ 1 _ itl.... r ' ''' 6' ;

_ 0..", ••0\0. ; , _ I I" tor

!I), l hat se t iu II I 110 load become flillctinns (If t he mom ents or i nor l,;a ,

J , FInd J • . \Vi t ll n 1lll'gO in crease i n 0 110 or the mo ments o r i n€"r l ill. , II ro tor w il lt n 10 \\" I'f mom en t of inerl in sl a rts ecce lcraung , I n th o enr rg y couv e rtc r <lr (h~· nbove ty po. it 11 0 e5 nM 11111 11 01' whe re 1110 energ y I IlII l t he I' ; ' gnp rec e ives ec mcs ( ru in ~ i ll r l1 eil.! l()rf,r the r otat in g mcm l)()Nl o r t ho m uchiue

hAll II contac t arraugcme nt . Such II machin e h nll lim it ed app ti oanons , I,ho ugh some of its fen tn rcs deserve ccuctdeee r to u.

1AJt u,!! tur u our nt le nt.iun to all electric tlIll.c1l1ne with Ihroo .1<.> S'rel'$ (If Ireedo m (f i::. U.2). J .\ t hi s m ~l e bllHl- . t hl' rot or i n t he Inrm of Fig, 9. 2- T he cOlll'Jry conver ter ...llh • s pherical rotor a s phere is kept s us pe nded b)' ure , - ' lu ' 5l;olOf; ' _ ~ llctiufi of superconduc t.lng lou p.s I -" P~ t l ~. L rotor. . - A' . ' or atc ve t wo se m ici rc u la r s t a t ors . olle being tnrned !lO" wit ll respect tn tlio ot he r: lito m achi ne win di ngs produce t wn tra vi' li og f ic.lds . Otlpend hl~ on t he t orq ues peed uesd by the ~ t M loMi, t l u~ s phere ca n rota te in n t b. ree-dt mensioual epaee . An e loct ric m/lch ine witb uoec degreE'S of Ieeedcm is desc ribed b)' e i::h L vo hlll.'1l eq uation s wh ich (till lit' re present ed in tile te rm of a mat r ix 11.1 1 = [Z l li l T he imped an ce ma tet x for t w o immob ihl l ta tot3IHls four rows whe re t he emfs of rot a l ion arc eq ua l t o eerc. 1 f .....e lI{'gl{'('/ the ffiAl!llot ic

"" co up li ng

Ch. 9. e"".I;"... ,... IEI. d r;c MoIehl ....

be twee n the 1'111'0 s ta ters, Ute vollltll'fI equa t io ns lo r th is I'll. ch ill(' OOc:f"lOle s imil a r to t be vo 1Lal:@ equllt ions for III mach inr with a n {'lIi pli c Held . There a re th ree eq uatlo ns o f mo lio n: (J

. ,Jp) (d lt). ", ld f ) =- .1/ u:

(J ,y Ip) (d ro'lI fdt) "" M ' I/ -

(J " Ip) (dw ••.'d t ) -

H ero

J." . J.,. a nd

L , 1/. nndc a xes ;

.II u

M , ,, M ' II

M ••

(9 ,5) (9 .6) (9 .7)

J ,. oro t l,e r otor's mome nta of l ner t. in n lOl\1l uie are t he ro tor vel cc t li esa lOllg t he x . Y .

Io)n , (0) ' 11 ' llll il . til"

nn d :: ox cs: Jl1 .,.. .'l ey. I\lItl JlI n are e1ccttu llla i llc l ic t orq ue, a long u.e %. y. e nd :: R:!:C'S; and J ! " 1' 11 ' lind ,11.. li n> res i! t ill g torq ues a lo ng W (' :T. y, a nd :: ax ea. As we di d for t ile mll('hine wilh two degrees Creed om , he re .....e lloN I t o inl rol'luro o nc m e ee eq untr ou i ll i)rllt'r l iia l t ile 8ys l em of llil"il l ions for t he m nch iuo mut e r ~ ~ lId y sh nll1d be det er mjue te:

.>t.

w,x

w .~

+ 00,. =

Th us. tw el r;e equtd loWl, namely , tight /,Joll agl' equuliolls , three

(9 .8) eq U{l~

li ons oj motion , (lnd one "elocUy eqll!1.tion deu nbf< lil t. procesus of elleTgy renverdon /'1 a IIweh ine wf l h. t hree dt /lrL't1

01 ! rud/J/Il .

Por a sy m met r ic nHt ch ine li t J/, x =- ,1I., = .11., ... ,11r' Eqe , (9 ,5) t hro u2h (9 .i) become si m pler. ' Vbon .11u = .11c, = "ll . . ..nd J •• = _ J " = J " . t he vel ocit ies al anw t he a sp s a re eq ual to (lU I)

Mll.ch iue! wit h II. s pher ical rot o r Hnd oppli ca tio n in Illw ig8 t ion de vices . l f one of the 5tn lor" is mod e \0 re vol ve flOou t t he ro lor , t he m achi ue 10 des ig ned shows fou r deg rees " f Ireed orn. If two sta to rs eevclvc in d p.p6nd crl tJ ~' nbout. t ho roto r , th e mnc hi rw wiIJ ha ve tr ve dl'groCS of Ireedom . If \\ill r ig idl y CIH m OI: t t Wll s t a t ors II 1HI 1I 110w t holll t o revnlve ab ou t t he f ph el'i<.'8 1 rn lo r, t he IllAchi ne will ha ve a il[ Ileg reel of freed om . Bascd Oil tb e eq uRt ions for a mac h ine w it h thrill' degree s o f freed om , it is easy t o i ncrcA!8 t he nu mbe r of equ8 ti o ns a nd 11 l\~ desc r ioo a 1,)'poth Cl t ica l ma cl, lne "d l h II (lrgrees of r-eedcec, S ud' a meeh tue ca ll be l.hought. t o CII rry D rew wi ud ,nB!' Oil th e stAto r and r ot or, opera te fnun nons i" ul'I oi d lil .su pply, end e x h i bit non li llt" nr il les . FIJr i ll! d ellCript iOil. wo wou ld neell t o uc r tve n n i nl i nne num bor of vo lt llge equruio us nod cq c c t tone of me rto n , wh ich wou ld be I lie mos t ge ner a l oqun tions or e lootrome eha nical energy co nv ors to n. R l' ~lt rd i n G ty p tca l equa t.inns , we s!tolJJd note !lIn t the a bove u mch tnos are deecelba b!e by the e n-Ill a nd ll'e od d se t ...f oqonuo ns , wid Ie ccnvenuona t ma chlnea need th e od d se t of ('qlla tions for t he ir d escript io n.

9.2. Unee. Enet9V .C onve .tltu

16>

In an clc ctric ma chine wi th II d egrees of Ireedom , the erectromechnn ica.l e nergy co nv ersi on d egonerotll's beca use t he nng1l 1a r " c locity tends 10 zero ns II ep pronchcs inf i niI J!, so thn t th (' en('I'g y co nve r ter becomes a n al ectrom ngn ctic co nv er te r.

9.1. Linear Energy Con ve rte rs The re nre mOllY des i;lll versi oua of I1lec t rie d r ives in whlc h ec t uators exe cute e roc.iproca tiug mctio n b ~' v ir t ue of the meeha ni ca l tee nsforma t ion of ro t nt.io na l moti on of elcct r-ic m eclunes . IL is I ., hI-

~ , ,OJ

Fig. 9.;1. Ce neeme r ty pe.! (oj .,."-, . ..,,lona l;

(b j

"" , ,,,enl a l ; 1<1

'<'

lJ n ~ ~ r

thought t ha t all e ne rg y co nv erter has. good cba- ee reetsucs iJ i t displnys e laot eomechan fcal r CSOJlUO CC , i. Q. jl~ des ig n is s uch l hll l only <I st llnd ing wnve a p pears in th o a ir ga p nnrt rcflcc tc d WllVCS ere hot presen t . Li near e ner gy conveners nro low-per for m a nce devlces. so the ir USllS arc jus tifiable only whe re II r ot Atin g ma chtne wi lli a mech enlca l conve rt er is un accepta ble . A li ne-lit E C is n desi gn ox tc ne to» o f a eo n vco uo ne t co nve r te r (Fig. 0.30;). E vide mjy, if we rt rst build a n EC w it h a segmenta l stater (F ig . 9 .3b) un d the n in crease ~ he seg me nt ra d ius t o allo w it to go t o in lini t y . a Hnea r mo t or w ill res uIt , such ns show n iu Fi g. 9 .& . Li ne ar con vert ers li nd r are uaes for work i n t Ile g Clletn l ; ' I!:: mode . t houg h the re a re t he ca ses of op p jicat.ion of lincnr converters I" pract ice as ge ne ra tors 10 lra nsform th o cncrg J' of t he rerIpro cat.in gmot ion of II d iesel eJlgi ntl·s or s te nm ellgiwi's rod into etectr!c onergy . Linear m ot ors co me i n as ynch ronb ua or synchro nous ty pes (II,' · pemLi ng on whe t.her th ey o perate fro m an ec or de so urce. Tho oc sigo vers ion s are no loss d iverse t han th o.se of ordina r-y motors . v m-ious eonst rucuons arc ad a pt able to perfo r m t he func tio n o f a roroe . su ch a! a s tee l shee t of in fi nit('. le ng th, II ca r m ovi ng a lo ng t he ete t or, or II me gn et ic liq u id. Max well' s equ atio ns a nd t he mod el such a s iu F ig . !l.4a fo r-m th e basis fol' t he ma the mat.ical dese-rip l ioll of e nerg.V conv eeaion pr oces-

e ll. 9. Eq ..... tlon . fo ' ! \. CI,;C "'.chlne l

$ 5 in lt near me te rs . T he plots of per me llb ili l y ,... a nd Its deri va ti ve

versus s plice coordinate

= a ,l/iJz ar e s hown in Fig. gAb.

,m~----,i

,I'~,,--'''----1 , --' ,, '\,:.rI. ,' , I

U~Af...

Fi, . 9.1;. A l[nN f motot mood (", u d the plot }. (b)

f)f ~

nod Its dt'ri u tlve WI'S U'

The i U$lllll ln neo lls eleet rcmaguetic force Mc li n!? on the work in" m orn ber of H linclIl' moto r is g ive n by

t~", ==

JBi z a»

(9. 10)

•

Tbu illlcll'"ll l io ll o ve r t he volume comes Lo IIl u l l ip ly ing the w id t h 211 01 t he work ing mem ber by t he valu a of the normal jzed a i r ga p, <'I';

(9 . 11) where lJ iJS th o ma[!" rlOI ic. 1I11x de ns ity i n t he ", i t gnp: and I t is ClIf1'N 11 in the a rea of t ho rio hl s lrnc:t llre. T I,o inst nu lnllOO1l5 va l ue 01 the e lec t romag llcUe power is

~ o n dll r)'

p .... =

Jer, do •

ure

(9. 12)

lne

Oesi g uill g Iill000 f mot ors i nvo lvf'S man y d iffi cu lt ies because Const ruc t iOIl a long wit h the re t tected W3Vl:lS prese nt i n t he •Je gap lIIa kn t he uctcrmtna ucn of t he field »>llte rn in 8 .,.al mach ine 8 rat her cn mp lir.llte d problem . T he ,1t!5igrl proee d ure for Huear mot orll olte n ,0110....'5 t he llilma gu idl.'li lle as for ce uve nt iona l mot ors , usi ng th e coe fficie n ts t o IIC· count (or pee rer e nergy c ha rac teristics due 10 e dge effec ts e nd ot her es eetne (el\I O/ I\l S or operatic n. T he volta{:'o eq ua tions fire SOl tip i n U Il II~nll l

9.2. Li.... . Energy C.on....,len

t he eame IAAnner as Ior lin oro iUlI. r)' uyrnme l rie ma ch ine w ith due rega rd for UIO .... oorfi.. . te nts de pellding pn the d es ig n of t ho Iillellr motor . T he d r iv ing teeee is fou nd proceed ing from t he assump t te n th ol the powe rs ill Ihe rotalionl\ l llnd t rn Ml!t li olll\1 mot ion a re e qtllll: M (l ip) 2111 ( I - ' r) _ F'2 TI (1 - , ,) (1l.13) where s. n nd $, or o t he s li ps i ll re t nt.ioual lind tran!'lllt io nul -notton raspective ly . Suc h 11.1\ a p proac h eert.ni n ly g i ves \'eC)' approximate resul ts. but it en u prove ve tt d in l1)o l " t i,' o eel eula uens and a lso ill t ile enleulalion 0 1 llInlLlpo hi r mechlues. Li nea r motors ha ve recently foun d lt~ in h illltÂˇspee<! 1", 'ISpor L fliCiliLil'll r id ing o n (I m ag ~ Lic e ushte u. I n t he I rll'L~ por L lly-te ms of

, jODDu OOOO

\-1

Fir . 0.5. All eteerne c:au puh 1 _51._ .2_ .........

t his t ype . II,,, Sinlor is It l(lug line e xte nd lng int o I"'IJ> a nd h und red s of k ilo me ters a nd the ro lor is n (IIr slls pc nd lld a beve tim Hue bell . I n des ig lli ni mag netic-eus hi cn Irllllliport ve htc tes . the elillincc r lia.~ to eotve t he pro blems of cont rol lind sl ob ili za.l illn (Iev ita lion) of u peseeugcr c er, le t elo ne the problem of de crensl ng the cost " f gucl. Do t ra ns por l systarn. III t he a rea or linear mot ors t be re lire ye l ma ny co m plcx prnbJems t!la t a wa il Ihe ir solution , of which the DlOSL co mp lex o ne CO IO\lS 10 t ile fo llow ing . A5 far bec k U tbe m iddl e 1930:1 electeie ClI lop" lt.s were bu il t wit " l he e lm t o im pa rl a ll I'I dd ilional e ece te eaucÂť 10 fl)'in: vehicl es (F ig. 9 .5). While i n 19round l nll rnlport aj-stema t he ga p be tween t he bed a nd t he cor must be kep t accura te t o II h igh degree . in Cll in pu iting Ihe {l'ap is mad e 10 vory (t he rot or m e s into space lind t" o per e me te rs i ll equeuone und ergo clu. ng{,!I) . I n IIID la t ter CIIM! t be re is 8 nerd for ca tc ulnt'ing t he d r iv ing terce li nd RCCe-lerat.iou . Mot e d iffic ull pro ble ms Mise III lin nuo m pt t o brl "g the rolor he ck a nd tak e off 1.1'13 defi nito a mounts of t'nerg}' from h to ('ffeel the des ired decelerlliion . Alt houg h they lire not devoi d of s horlco m in{l's, line a r mo la rs e njoy n!le in graph plo llers. mn nipllilito rs o f me lAl p tecee. pus her" , and I" orhe e etecrne d elves . Il e\"ersi ng the. motion of a li near mOWi r g ives

Ch. 9,

~qu"'tl on.

[0'

~ [ ,,( !,it

Moch lne .

a n osc ttle tcrv-monon mot or. T he ana ly sis of Hnea r mot ors ell"lbJes us 10 e xte nd the result s t o rtet er mt ne th e rcleuon between electr ic machines nnd appara tu s in wh ich t he d r ivi ng el cm ents mni ni y exec. ute Iiuear dlsplae omonts w ith v ~ r~' i ng pa ramet ers of ctecu tc circuit s .

9.3. Energy Converter s with Liqu id and G aseou5 Roto rs TIll' st a tor of II li UNlr mo tor can be built in t he for m of II. pipe Ins ula of whic h a t raveli ng fillid ca n 'b e set lip. I n t he pipe fill ed w it h n magneti c Iiquirl or ~1' S f~ mov in g conduc tur}, l h\! mng ue tlc

)

Fig. 9.6, An ]'H! D generator

fi eld \I'm influe nce t he mot ion of th e flui d . If An iouree d gus I'll plasmu) or il lllag ne t ic li q uid i1; d r jven throu gh t he chan ncl , we ob ta in A ll energ y conver ter called II 'TllagJlo lohyllr ody,mmic (MHO) generato r (F ig. lUi ). An MH O generator conv er ts the rnecha nimli (ki netic) e nergy of pbsmi\ par ticles t o etcet ric tcne rgy M t ho conducting plasma Ilcws t h re ugh a ch an nel l III wh ich magne t fc co ils 2 pln ced nlougsid e t he p ipe pr nduco II magn eti c fi eld IJ . Th e cond uc t ivi ty of the hot gas grow9 with the ad d iti on of 11 11 easil y ion ized ~!<lo't' d~ al ka li metnl such as potassiu m T he motion of the pla sm a tit II speed u ill li lt' mng net ie field lnduccs II volLlIgc 0 11 elect rodes 9 and g ives nsc t o 11 curr e nt I t hat fl ows i ll an externa l ci rcui t R I â&#x20AC;˘ T he load curre nt comple tes its path acros s t he cha nnel and pro duces the arm at ure fpla.o-ma) ]Âˇenc l io)l. t here by di slorli ng lil" exciti ng Hold antl t he lC>U-'l'it tidinnl v o l l a ~ cem pcn unt-c-the H all voltage. T he H all emf E is i n a diroctj on normal to th e p lan e. 07. T ho pr ocess of MH"O power ge ner ntio n ca n he IJf t he open cy cle if the work in g rnedinm passes t hrough t he cbanuet onl y on ce . or of t he closed cyc lo if thu me dium is made t o fl ow t hrough t he genernt or

repeated ly. An MHO ge nera to r uses a n invertor to change direct current col.leete d on the elec trodes to a fternuting curre nt . The MHD cha nuei is bu ilt up of segme nts . each helng' insulated from t he other. Electrodes operate i n heavy cond iti o ns an d th eir Hfe dater mines th e serv ice life of the gene-n tor. Th e p ulse and sho rt- ti me mode s of opera t io n s how pro mise all regards th~ life e xpec ta ncy. M!lg netic hydrod Yllfimics that stud ies t he motion of li qui d a nd gaseous conducti ng med ia i n a magne tic fie ld belo ngs to electromechanf cs . foe t he tntera onou of II hlgh-velocity conducting Sll'eam wit h 8 magnetjc f ield ca uses t he convors ton or t ho kinet ic e nergy of th e s trea m into elec t ric e nergy . A c,onducting medium th at moves in a n e xte rnal magn0l k field in a d irect-ion nor ma l to the pla ne Bv in du ces nn e mf , so th a t. an electric tlnergy of direct co-re m an d of tow voltage ca n be take n off t he electrodes. As in co nvent. inna l energy eo nver tera, in MHO gs ncre.to re the lend field exerts an in flue nce 01\ t he externn l fiel d , with the result tha t cmrs appea r which aHec t l;ioth t he mot ion of the medium as a whole a nd t ile mot ion of individua l portions of the stream. A change in t he ex ternal 7J fie ld also causes ene rgy co nvers io n (see Fig. 1. 13). Th e la ws of alocrromocha rttcs al so hold for MHO g<lncrators , so t hese co nverte rs cer tAinly beloog to elect ric machi nes. Much effort bas boon spe nt in the 4 SSfl a nd US A {o r t he devel opme nt of MHD gener ators us ing plasma ns a movi ng eouductoe. 'rVi tll th e ndvancemeut s i n t he f ield of fusion reacto rs . cosmol ogy. a nd aatro phystcs , a furt her devel opme ot of magnetic hydro d ynamics become s yet me re urgent. So far , l\1H D genera tors arc i nferior to conve nt iona l energy converters from t he econ om ic e ud techn ica l viewpoi nts. There lUll rather ma ny modifica tio ns o r MHO pumps a nd MH D geneeatora, and more Improved designs are li kely to be devi sed i n t he fut ure. T he ma t he ma t ical descri pt ion of ene rgy conversio n in MHD genera tors comes to th e stm uu eneoue solution of Maxw ell's eq ua t io ns defining eloctromagne tic processes rind the Nav ler-Stokes s q ua t to ns defi ning tl\6 processes in liq uids. T he s imu ltaneous eoluuo n or t hese equations is only possible for sim ple cases Involving laminar flows. [f we Assume t hat 1111 pe euclea of R liquid move a t n co nsta nt speed , i. e . the li qu id beha vas Ilka It; soli d, t he problem becomes s impl er an d the eee egy conversion processe s in MH O generators CIlA be treated usi ng the equations for convenuonat ele ctric machlnes. Such an approach was put forwa rd by A . 1. Voldek i n 1957. T he equa tions for elect ric circuits sim ilar to E qs. (1.34) t hrough (1.. 37) or (3. 3) together with the equatio ns of magne ti c hy dro dynamics give a better descri ption of energy conv ers ion processes in MHD gene rators . The effediveness of an energy cOrnJrter with a liquid or gaseous rotor Mpends on the magneuc /l eld 8 and out flOW veloctt y v, T herefore , the

uS' III tlu ~ reo nd tlct i nl: mag llt't ic !yste ms which peoduee l!igh ma g net· it riulds otre-r:s co ns ide ra b le promise for improving the pe rformance of MHO generators . The kJne t ic energy of t he .!S t rea m r ises as a ca ll>l(lq lle llf c of he-u illlt of lhe- gil! 10 2 000 (n ;3 (00 K " lid it s aceeloro li l' " liS i t lell>('$ t he uoeal e . T h", !'IOl ul io li of th e above pro b lem s in e lec t remeche n ice l'f'q u il'l.'l1 ti le jo in t IJ rrOl'1 or l,herm e l phys ic is t s nud elcctro mec hn uic a l QDll'i-

nee re, For th e envenceme nta i n I I, ill oren to be 10 0 l"e ta ngi ble . tll Are is tin urgent nee d r U I" a p ro found lear n in g of t horm AI nn d e loct ro mllq. 1101Ic fi eld s a nd ti le prOCOSBClI of convoesto n of e ne rgy fr o m one form t o a no t he r , I n t he las t yea rs e tuctromecha uics ha s at arted ll!:,ing m ai:lle li zi ng Jiquitb (f l'rrOmllll oot ic Ii qlli lb ) . These arc co llo ida l Iiqu id~ whose c r i ~it'.:. 1 c ha ract(lr is t ic~ d epend 0 11 l he 8l 11 I,i1il Y nnd s i·too or put iclos. ~fla g lleli :r.illg liq l,id!l Clt n perform t he (.mdio n of seals. PCrlllllnOllt 11I aj!:ne ts produc..... tl fie ld in tlu~ rell'ioll se p:u8\i ng II rota t ing member fr um /I sLn liO Ill. r~· 011 1.'. Fe rrom lllflll.'llC pa r t icl e!' Ii",e up in the !ield df eectio u nod thll s mak e ure E<lIl J l igh t . T ld s tlp pron ch cnsblEl!" tmpro"ill~ t lll~ eee ts withollt Sll~la ll l.ill l "cs ii'll ,·llri lltiOlls . Magiteti:r.i llif li q Uids ril lt! ot her , th ough Iim iled . IIpp licn t ion1'. The reas on is lh nt l hu)' nee cOlIs il!crllbly infer ior to e lect r iCII I-shcol S1t*'ls ill n1 Agn" li c proper-ties, So me lwld a view poln t th OI M HD gc nc ro tors dircctly couv ert tlil.'r llllli Qlll:/rg y t o e lec t ri c \lllt'rgy. Whllt is m eant here is l/JIlI :'.I I-1 D

power gOIlt'I'llUOIl d ispe n!!Cs wilh n s tea m tur bi ne whtch is II co mmo n reet uec i ll t he c lnlls iclIl cycle or COlwl.' rti ng I1011 L10 e lec tric energy. E norgy cOllverter" are monifold both in desi gns n nd p rinc ipf es or IIctiofl . Electric ma ch ines lire a vaila bltl whi ch COI\ \'tlcl hea t 10 e ltlt t dc o r m cc lHHl ica l energy_ T hese CO nv e r te rs operat e 0 11 t he pr inci ple of c hnnaes in tilt' permeabilit.y of lerr om ag lle u ne ar t he Curie poi nt. A dlllo: e in li m inducta nct'll witb tem pe ra t ure ClI IlSe5 a cha nl!\! in t.ho parame ters of tile wi ndi ng. MRg not io- lh ecm ll1 (! oC'£gy eonvcr ters resem b le p arame l ri c convar tars, s ince in both t ypes enurgy eenver' io n ro.~ "lls from clta nges in Uw parame t ers of t he cooffi cien ts thllt e nte r i nlO elecrrcmechemce l equa t tons. MlIglletiCrt her mal C l1 e~'Y co nver te rs were s ugges ted by N. Te s ta and T . Ed iso n IU. ea rl y as the ond of th e 191h co muey , but t her, d id not trllin recogn iti o n for tho teehrucal a nd oconom ic rea so ns . I'he sea rc h for new so u rce s of etc cretc energy has aroused more in leres t ill ui esc ECs in t he Ills t yoor,.

9.4. Other Types 01 Energy Converters H ca t re moval fr om a n e lec t rt c mach i ne i3 as a n i mportant prcblem as tho improveme nt of i ts e nerg}' cha racteristics. 10 v enrtlated ma chines, t he fan ro ta tes togv ther with th e ro to r ood blows the CO(l Ii og ai r over or through the m achine . In lew -s pee d co nte 'l l m otors,

9 .4. O ilier Ty.... 0 1 Energy C on.........

hewevce. s uch :I coo li ng sy d c.m i" inerrCCl in !. Dc~i ll, ing II mnch in e ..it II two ro tors ca n rt'medr the "ll,ia lio n (F It:. 9 .7). All inle rUlI1 roto r I with 8 I& r~ mom en t of inert ia se rves 10 d ri ve l hl.' flln lind 11 11 u terml l rolor 2 i n the Ie rm of II h oll ow cup ac ts a! Ih e r<)lor p rope l 01 t he co nt rol moto r. Th e ma ch ine h:"SIl com mon et e t cr 3. By vir t ue-

of Hill rec t t hn t th e rot oe 1 haa » I" w res tste ncc . its .speed I' HI'dl y wit h vn ' t.I1 ~p. wht!e t he speed of t he ho llow rotor de pe nd s on the vo ltage . T ho!lt' t of cq uru tons {or II two-rotor- lllllc!,ine Includes lwn eq uat ions of mnno n , wilh J"~ (Id inn hlc by lJ,u prod uC't ll of cur n)n l.s ill Ihp 1I1810r a nd ro tor. an ll six vo ltAge e qua t io n!'. T Ill! se t o( vo lt age equ ntio ns co m pr i.'M:'J'I t"''' eq ua tio n.~ fOf th e s ltltor wi ndi ngs Oil Ihe a. lind ts Il;'( e" c ud fonr equa tio ns for t he wj nd illjr! o n t he two rct e-e, I lTiI lIg\ld n l o n/; Ih~, a IIDd ~ nx,",s ~:'I f!'e<' t i vt! l r. The sy!~ m of e lill, t eq ua t io ns descri bM 1I,e pro«'~s of ~ 'K' rgy ce nve raio n in lhi s t ype of m nchi ,X" . f or th o allal}'si !!of an t'ncr~r ce uvenee hll. v ill¥ n roto rs end III COIII + m Oll !lttllur . we need t o fo rm two voltage eq un t to na fo r 11'0 ! t llto r , 2n volt age eqll1l.tion s l or the rot ors . lind n eq un t .ie ns 01 mo t m n. In ell, th e sys tem wil l co n ta in 3", 2 e qua t ions . The eq ullt io lls for lin n-roto r lllollc h illO t'IHI Appl y t o llil e lll!cg}' conve ne r with n li q u id ro tor lI ' H I ~'r cer La i" nss u m ptin ns . D i~p ll1C' i n2' t ill! rot or wi lli re epcot to t he st ator, I'! O Lllnl in the Ii mil the n,l lJr nlm cst COIDCS i n co nta ct wil h the s te to e, gives n new f'lIcrg y converte r celicd III mot or wilh Il r oll ing ro tor (FIr:. 9 .&). In t h is muio r the ro tor ro ll s by wa y of o oe-s icled magneti c :lt lra ct io n. \ Vh ile ill ure e nerg}' con ver te r wit h II unU;Oflll gap t he torq ue ~ ('qu al t o th t' prrNluc t of s ln tor Illld rot or curren'" flo1\' ing in di ffere nt phases, I " ~ me rollin~- rotor m otor t he thrust arises fro m t be currents d ue Ch n Ol{Cll

Ch. 9 . Eq uatio no tor Elaet'lc Mac hi","

ec one phase : M. ""M"I:'Ii. -/Ifsi&il (!l.t4) Si nce M o. a nd M /I d iffer subs tantia lly from each other i n value, the th r us t moment ap pea rs. Tile point of tangency A (Fig. 9.8b) revol ves at tho Held veloeil'}'along the in lier surfa ce of t he stator. T he roto r rev olve s a t a s peed n2

...

( fl~

R, )/R , = 60/ (R . - R .)/R ,

(9,1.5)

Stnce II roll ing-rot or muter is made two- polar, the rotor speed depe nds on the s upply li no freque ncy and the d i(fC'Tenee bet ween t he

Fig. 9.S. Displacement of the rot or wilh reapect to the I t.ator { a)

<:<>/lTtntlonal _RetC'Âˇ con" . , !.r.

(b)

, .adll

rolll ngÂˇro(o. moto r; R. and R, _ . ato, . " d I'Olo< _p<clt.~ l ,

f ad ii of t he s ta tor and rotor . T he motor com bi nes, as it were , a mechanical speed red ucer w it~ un clecmc mach ine. Tlds is a lo w-speed motor haVing a large torque. The d isadvantages of such a motor inelu de enhanced vibratio ns a nd a short life of beari ngs. I t should be noted t ha t .j J! II conventiona l machi ne th e prod ucts of currents in one phase of t he stator a nd t he ro tor do not determ ino a d riving torq ue, t hough t hese currents need be take n into acco unt in the st udy of vib ra tions . In a rc lltng-ro tor motor , the current prod ucts of t ho ab ove type do' govern tho d riving torq ue. To t his t ype of onergy conver ter a lso bel ongs a motor with a deroemable rotor made from a flex ibl e fer romag netic ma terial . As uie Held rotates, the rotor 's deformation wav es t ravel i n synchroni sm wit h. the magnetic field, so the rotor rolls over the stator i nterior at a u a ngular velocit y descri bab le by t he sa me rela t ions as for t he mot or of Fig, 9,8,

' .4. Oth.r Types o f

~n. r9Y

Co ...... " ...

If th e r otor diame te r- is s u bs t a nti All y s malle r than the s lator diawcl ('r (F ig . ~1. 9) . th e s re to r field t h3t ro t'atecs a t (1). eud IlctS 0 0 t he rolor produces fortes P I a nd 1", ('j ifJeri ng in ma &,nill1de because th e field near the sta t or is mu ch s t r ongt' r. W he n FI ::> F t , the roto r tu rns ov er a nd begi ns to ro tate in a di rec tio n opposite to lbat of t he statoe field (see Fi g. 9.9). This effec t is easy t o prod uce by pllciltg a met o.l ball inside t ho Iita to r . T he i nte rac t ions are more comple.i: if we pla ee into rne stator bore t wo. t h ree . Rod 11 r ot ors (F ig. 9. 101. I n co nsl ruc l Lhe ma the matica l

~' ·0 ,

.0, ·0, Fir. i.i. A

rolll llg-rotor tlUICb loa

> 1'01'" F. M i l ot _ ....1... . - F,

O. _

O. -

lI .. ~ ly

t'lg . 0. 10. A tbtu-l'Otor machloa 0.,. 0" In¢ 0. _ nllor l;COI.len .... O. - • • 10< ""'"

...,Il1O.

mod el of suc h lin eteerro mec na n tce r !lys tem , eenst deeeuc« should be i1"iv(' n to t he ('rfC"l:t of one rotor o n t ho o the e a ud th e in le ractl on of onc h ro tor fie ld wit h lh .. s ta t e r He ld . Prec uca t ap plico t iollS of mururo tc r c n(,l"gr co uve r te -s a re yet u nk nown . T hill nbc vc-descj-ibe d mod if ications of energy conver ters nre ma de possibl t' by chAng i ng t he des ign of a ro to r. As rue nt.ioned earlier , energ ) ' convcrs ton i ll pa r am etric dev ices res ults fro m th e cha nges i n t he pa ram et ers of coe ff icie nts e uto rfng in t o vo ltago eq uat ions . I n Eqs . (1.34) a ni! (2.3 ), t he impeda nce me t rt x co nt a ins t he ter ms of th a fo rm (dld t) </:lJ.d' ).'I1l Bo t h eueee n te a nd in d uc ta nces usual 'ly vllry in a h nr mon ie manner . I n t ransfor m ing t h<> {'«\II1 Lion!!, 0 00 s t r i" es to make u p t he equa ti ons whh conslll Ols L " a d .11 a nd v&tyi ng cur rents t, It is poss ibl@ t o en su re e nergy co n voeston whe n i is const a nt a nd L . ud /If undergo v"rlllt io ns , The eq uat ions then have the t.erms of t he form i (dfdt ) L . l (dld t) M

'" T hese are a Iew ways or deslgn tng an olle rgy co n ver te r ill wllicll La nd M may vary h armol1 i cBlI ~' . A prefera ble design is t he one .mown i n Fig. 9. 11, whero th e lIir gap v aries in " harmonic ma n ne r IS l bo ro tor is tu rn ing . T hill is AD i ndu cto r ~oera tor which tnetudes • SlIlM ha ped roto r I in wllle;h rbe num ber of s lots is equal 10 h" ll the n um ber 01 slou in th e s tater. As t be ma chine kee ps ru nn ing. the fi eld se t u p by • de wi.ndi ng 4 pulsa t es , thereby in duci ng tlt e emf i n Sta tor wind inp I n orde r lh at the pll19alions may not bellv il)'

-....../ -- ....- - - - - ...

1:5k- -~-/-~ , V ...

J'ta:. 8.tI.

Ao IndUdor pof-rawr

. Hee t t he t orq ue a nd eeuse no Vib rations, th e ma chine is made ('omplet e willi two rot ors fi t te d o n to t he Mud! 2 tlod dI"posod 00 e lKl r ica degr~1 (rom each o lher. Su ch II. mllch w li! !I llS DO w i nlli n~ 0 11 Ih, rotor , which is its adv llDtage. Par ametr ic ene!'1Y con"ertbn eRn etec be ri tvtded in to S)TH.: lm'lnoll!l and asynch ro n ou~ t y pes . T he)' ca n o perme llt t he first hnr'llIHlir Or at upper ha rmonies i n Inotor ing. gene r;lti nj:t'. brakl ng. .. nil t rnns lcrme r mc des . EUN gy eo nveraton IH;C\lrS in t he n i t gnp. T he Willd lllR.' t hat prod uce th o fie ld i n t he lIir g ap cnn be e ither 011 IIH.' srntor or 0'0 t he rotor. T he c!ec tronjechllllica l aquatt ona offer u 1ll('1I1l'" (,I fi nd i ng th e weya for desi gn of new ECs liS well as (If a nnl),tl " !r EC6 t ha t are co mplex in th e pr(nc ip le of act ion fro m th e \'i ew poiJll of c1nssiCll I t heor)' . P nl'nl.llClri(: ECs nva iltlble tod a y ho ld lin Im por ta n t place ill ~he li st o f e lect ri c mac ht ncs . They tn clu de in du ctor ge ne r/O tors for /1111.0notncvs PO.....I·(' ey...terns , !If lt1ntl!rat ots, low-s peed mo tors , a nd olh~r t y pes . Among ,·"riOIIS Iluig liS of ECs. worth)' of ment.ieu are efec tr rc mach,ioos of t he com bined ~ype . The des igns thllt fea t ure tho m o.!l~ ef llcient com bin a ti on of e lGGt r.ic a nd m ail'ne~ ic c ircu its i ne lud e si nglearma ture converters (sync hronous me te r- de genoCrlllor u n tl..';). threeph ase t ransfo rm ers (uflit s comprising t hroe s in(:lephase trllR! forme rs).

n. molorS perform i.ng th e {unct ions of a lbp llfiers such as lJIognetic amplif ie rs Il.nd control eetc re. an d II Ilu p, ber of ot her designs . T he wa y of combioinR va rious 'llcclromecha nical tle-menls to form a desired aggre gate is 01')(' of t hema in trends in elootromeeh a Dits. Alth ough com bined e llef!:)' conveners were under s tu d y from tb beg inn ing of ind ust rial a ppli ca t ion of electric machines a nd unique discoveri es ha ve been made rnpl Dll y in t his branch of i ndusIry, much st ill rewa h.., to be done IQ prod uce new or iginal desians. From lh~ v uwpo fnt of matMrQotic al' I ~ , furthu ~lopm t,iI oj EC. 'IUJI/ tak t t M ~ for tht rue a/a"grt a/u number of ooltngt t qua. tlons [1.11 = IZ1II J and tqltati OnsOj mot.lon that would be mort compftz and 01 olhllr [orms , Th is d oes not ru le out th e a dd it ion of new eq uati ons to t he sys tem of electromecha nica l eq uatio ns t ha t would be compati ble wit h th o vcl tegc equations an d equa tions of mo tio n. There is (I posaibtlity of crea t ion of energy co nverte rs for tr ansformi ng the rmal. light . mi crowave, n'nd ot her fo rms 01 energ y into electromechan ical cnerg)' .

Chapte r

Elect ric-Field end Electromognetic-Field Energ y Co nverters 10.1. Principles of Dual-Inve rse Electrodynamics Electro mechanical enerJO' conver ters lIct I!IS conce nt ra tors of etect.romagn etlc energ y and hee t , T he ex preaslcu for electromag neti c fiolll e norgy de nsit y is of the form lV *=' ( 8I'.DÂŁ 1 j.lJ.l o HI)/2 (10. 1)

where E a nd H are the el ectr ic fi el d .!It rength an d magne ti c fiel d st re ng\ h t69pectively; II led j.I are (h e re lative perm it tlv n y and rel a t ive pe rme llb ility rt'IJpectiveh: e. a nd II. a re t he perm itt-ivit y a nd permea bilit y of vac uu m; a nd U, E2 a nd I1I1'JP lu e t he products tba t determi ne t he ene rgy deM it )' of the elec tric field and t he mllgnet ic fiel d res pect ive ly. Meder n e tee u c meehe ntcs mlli nly de als wit h mag netic-field ene rgy converters. t .e. conv er te rs wh ich ec e centre te e nerrr i n mai Detle Helds . The etess of ma gneti e-field ECs has rece ived most at tention , a nd t he grea t echteveme nts in thi!l orea ha ve cont r ibuted much to th e progre.s! in t he scie nce of the 20th cen tury.

'""

There are a lso eleer r tc-Itald COf' rg~' co nve ners . t.e. electrosta ti c co nver ters with electric-field energy s torllge. T he r ap peared earl ier th an magllet ic-field ECs. but di d DOt rece ive recogn it io n as commercia l pewe esources. Much still~ rll.m'in.'1 to be done to bring l bll i heory of th is eless of conver ters to t he a ppropriate leve l. It seeD~ reescnabl e to ex tend the rt>Sull.s in t he are a of magnetic-field ECs i nto rbe area o f elect ric- Hel d EC& a nd t he n. Cive n t he ID l'l th em at lCBI d GScriplio n of the processes of lIll(lfn' conversio n in electr ic-field electrome cha nica l sys temS. to design t he converters of t he dNl lred performance. . Th e eq uat ions d8.\lCri hi ng I t he processes of e nerlU' con version In mag netic-fiel d nod alec tr te-Held el ect rcmeeha nlcal s ystems rerollin t he sa me e nee the interchange of du al pairs in t ho follow ing tab le Elact d e cbarge Q. Electri c nux 1lI. Vo l l.a ~ "

- - - MagneLlc charge Q", _ _ Magnetic /l ux tIl",1I _ - _ Cur \'('n ~ I

Eleetl"O!IKllive lo('Ce .

- - - ilh gnet OCllo tlvt: force ...

Th e ad vantage of t he eeoee pr of magn e Lk ch arge is t hat we ca n est.ablish the d ual of th e equations for a n electrom agnetie field. T he mat hematical doser ip tion of t he phonomena i n elect ri c fiel d! t bm ~'ie lds the beh l v lor i!jl magne tic field s. Sued on t he t boory of;duel . irwerse S)'st.ems, we wri te down t he foll owing up~io n.s r =0

(dQ..l dt) ... - (dV../dt ) "'" _ (d ld t) ~ S;dS

- -I (.1J"t)a~+ 1(ii XB)aJ

(10.2)

h= - (dQ. /dt) .. - (d'l'.Id l ) ><>- - (dIem ~ DdS

= -

(1)1)10/ ) q$ +

I (5 xv )

HlvJ.. B J.. dT au d uJ.. DJ.,. dT,

from (10 .2) and (1.0.3) we r == Blv h -"'" D 1»

(i O.3)

,,, (10.') ( to.5)

where l is rue le/lglh of I cond uc to r in a ma(lletic-fi eld EC t hnt ill equal 10 rue wid t h of an e lectrode in I n elC(:lrie-fit"ld EC. Th e phen omenon of elel tromagnetic ind uctio n ts put to use in mag neti c-f ield ECs. Ind lilt of etee t resta ue i nductio n in etec wtefie ld ECs. S im il a rl )·. usip\:: two-phase , t hree- pb ase , al'id m.·p hl!lll s yste nlS of e lectrodes , we ~n produce rola ti rli electric fie lds .

lS.

Th e t o ta I en erg y of an e le ct ro m egnetjc Held is gi,-cn by

W _ lY.. + W.,, = 0.5' ~

(ED+/iB) du

(10. 6)

•

Here IV.. = 0 .5 ~ ED dl.' is th e ellergy of a n electr tc field; /Iud W ",

0.5

JiiTJdv

i s tho ell\"!rgr of II mngnct ic fiel d .

• p rocesses

T he of {J.n(' ri:'~· con versio n in Jllllg lle li r.·fie id el ectrome chan ica l sy s te ms rea u l t fr om t he i nter net.to n of magn eti c cha rges Imaqnc t te poles) a nd build u p of an elecrnc fie ld whose S O.lI t rE' S a re electric ch a rges . E nergy co n versio n In electr tc-He td sys te ms s te ms from th e Inte r-a ct.ion of ete cer tc cha rges lind bui ldup of a magn etic field wh ose sou rces lit e magn e t ic ch arges . T he equa t.ions tha t can adva ma geously for m t he bas i.!i of t he t heor y of elec t r ic- f ield ECs are t he sy s te m of e q ua t io ns which a re d ualinve rse wuh res pec t t o Ma xwell's equa uons for movlng medi a . T he theor y of t h ts c l as.~ of E Cs nl fl y re ly nn t he e qua t ions a nalogous to those for mn gne t.ic- ficl d E Cs .

10.1. The Equations for Elec1ric·Field Energy Converters P r oceed i ng from the t heor y of d uet-r everse el ect eodvne m tca, we form ulate t he fo llowing d ual- Inv erse e q ua t ions for th e generali eed e lectric-fi e ld energy conve r ter \ls i~ng Eqs. ('1.34) a nd (1. 35): i~

t;'

"il

g:' + (d ld t ) C:' (d /dt ) C

(dJdt ) C O

+ (d/dt) C;.

-Coo,.

- C~ oo ,

CID,. g~ + ( dl d ~ ) C~ (d /dt )C {d ldt)C d +(dldt ) q

('I Me =C (u~;' - u :,u&)

. f U~

qoo,.

"I "I .7)

(10.8)

Equat io ns (10 .7) and (10.8) follo ,v from ( 1 . ~) and (1 .35) lifter t he in terch ange of u a nti t , tnd oewnces L:',~ a nd tOl lll onpncttances G.'.'n . mu tu al Inducta nces M lind tn teroleetrode capacita nces C, e nd r esist a nces r:': ,'6 find con duct a nces .G~ti.

18 2

Ch. 10.

e l u j rle - F i e(~

end Eledromlg ne tie-Fle ld EC

T he tota l ea paeil/l.llce'jinCl udes the capacita nce C bet ween ure sta tor and rotor e lectrodes:a nd self-eapacitn nce 4:: C~ = C + c~ (10.9) The dua l of t he ge lle ra lite~ jnagnetfc-Ifeld energy converter is t he generaltzed elect ric-field EO with elec t rodes a t pote ntta la u:.:.'a instead of windi ngs IO:':~ . electri~/ield ECs , ltke magnetic-] ield ECs, indude syn chronous, asynchronous and , commutator m(l("!linl's, and a~o trans formers. I " II sy nchro nous electric-fie ld machin e t he vel ocity eo, = w., a nd the rotor ta kes c ur rent from 9 de circuit (Fig. 10 . 1). Elec t rodes A . B, and C (which replace wi ndings) prod uce a rot at i ng ele ctric fie ld , a nd inte relectro de FIg, tn.t . A eYllcbrooQus etecmc- capacn a nce a nd self-eapaetta nen prese nt in t he machine vary in a fiel d machlnl> ha rmonic manne r with field tot ntion.

T he d irect- lind quadeature-ax ts eq uatio n for a s ynchro nous machines has th e form

+ (dl dt) q (dld t ) C - Cw,

(dld l) C OO u~ (dldl) C~ C;()l, cu, u~ - C~ W r g~ +(d/dt)q (d ld t)C X u~ 0 (d ldl) C g; (d{dt ) C; u~ (10. 10)

g~ +

M . = C[u ~ uJ- u~Il ~ 1

( 10.11)

Here u~ , %, u~, uq are the d.g stator e nd rotor vol tages l n t he t wophase mac hine; i~, i; , I~, ~ lire t he cur rents in t he sta tor a nd rot or electrodes: n, g~ , g:i, g~ are .the cond uctances of th e ste toc a nd ro tor elect rodes; CJ:; represents tota l capecttnncea equid to a ~~ =c+,~~

where c;',~ re presen ts th e se lf-ell. paci~ nces 0 11 t he sta t or a nd ro tor elH'trodes ; a nd C s la nd! for ea pac itll ncell be t ween the sta t e r a nd rolo r e lec t rodes . An asy nchro nous electric-field meeh tne comes from t he sy nehro-DOW! mech t re if i n t he la ller we repla ce the roto r by III dielect ric d isk or use a rol.or th at I. u the sa tne n umbe r of phe eee as t he s ta to r .00 a pply to the ro lor II volt age at a sl ip fre q uen cy . A commuta tor maehilUl CA n a b o be bll il t lro m t he s y nchro nous IDlc,hiDe by i nser ti ng II co mmut a to r lnt o t ts lie d rc uit. In a n elecrrt c-Ital d t ra nsfor mer it is 'H I el ec tric fiel d t hat li nk. the e lectrodes. T ho eq ua t ions of this 'tra nsfor mer have t he form I,

- It

I 1"'1 u,

1 8 1+

(d ld t ) C 1 (dldt) C (d ld t ) C 8t + (d ld t ) C, X

(10 .12)

Hero subscripts 1 and 2 s tand for- ~he pri mary lind secondary respeet rvel y. T ile curre nt t rllTl.<;forma t io ll ra ti o de pends on the n u mber of efee reodes cOllneetf'd in pa ralle l or eertes . Ma themAt ica l nlode b are ev a ua ble for t he dese rip tin n of eoo rgy eenverescu processes in eleemcfield 1,Il1l.e1l i ne5_ However. commerc ial high -po wer converters of t.h is etass do not pra ct ica l I)' ex ist . Th ts ~ beca use i D ve5 t i la to~ have t ried to copy magne ti c-fie ld eo n veners i n ev cl vt eg E'J eclrie- rield t ypes . l hollgh t ho Ja iler occ upy a spec tnl place a mong ot her COllverte rs lind comp lemen t t ho for mer rather IhAn rep luee t hem. T he mos t or igi na l a nd s uccessful des iqo of an elec t r ic-field ene rgy converter i., th o convecti ve-t ype Vall de G raa rt genera t or. • • al 50 clllle d Ut(' c tec uosre ue M OOIerater (F ig . iO.2) . in wh ich A movi ng rub ber belt l rllnl;pnrl.'! cl.ul.rgn heiD g separated out by a coron a di sc hArge teem 000 term ina l 8 nd deposits them at t he other • t lk-reby produc ing A la rge po te nti al d iffe re nce . E \'CD in its a p pean nee lhl~ ge ue ra tor differs fro m co nventiona l E:C!l. lt is II 6-k W, 15-m ln V, 1 l.UJ-mA setup 15 10 20 m hi gh . pla ced in a cas ing a llli fill ed with 8 l as 8t h il h pressure . T he ge nera to rs oJ t ili! 1)-po t ind use in lest un n s .

• •

••

<8,

Ch. 10. Eledtic_Fi. ld .nd EllCh o m.-gooo lic. Fi. ld EC

10.3. Parame tric Electric-Field Energy Converters I n m;lIM t ic-.ficld Eeo; t he mllgne tie field 9 nerg~' is kept sto red in the a ir ga p t ha nks 10 t he s reet magneti c core . Because of th t! low

breakd own volt age of a ir . noar 30 kV em- ". t he vc tume te rce l\nd speci fie power of ereerne-n etd EQ, is a factor of ebou t 1(11 below t ho!e 01 mag nctic:-ri eld ECs. •'\ lt em pts were mlldt> to ecnce m re te l hl.' ene rgy of I n elrcl r ie rield i n liq uid d iC' leclrir.. ..\ ud em icia n A.F . loffe used k(! rosene fOr t he purpose '(l. = 2). On fill i ng the ga p wi t h III eompressed gall. t ho e.lcel ri r fie ld stre ng th ra n rellch 600 kV em - I, Soli d

fi g. 10.8. An t1t etrl&-li eld

~ne ra lor

\lllibi ng

11) ~

('Dl'rgy 01 &ell

"'llV~

d ieluGlries a nd Ier rcelecu-ics offer st ill grea te r poss ib iliti 6S. T h& cr ystals bertum litUlIll ttl a nd pota ssiu m n ihyrltophosphat e have fl pe rmittiv it y between 9 X 10· lind (1 X 1O~ , While mllgnet ic. liel d E~ s tore t h(> ll11tll'oetlC lipId e nergy in t he nir I:fl p, e tectric-fiald ECs must co nce uwe te the elt'ct r ie-fio Jd e net gy in li qui d or sol id dfelee tr tcs . SlJlOO t'nergy conver ters m ust have cle ments movi ng wi t h res pect 1.6 QUit a not her , eleetr-ic-Hcld EC design.'! prov ide for mecheniea l c:aps or ellVis.'\go llle use of liq u id me teete ts to ~T\'C t he p urpos e of a rotor ,

Porametr tc tllclrfr-!Uld t ntrlU eolw~rtul openue on I he pr lndple

01 a ptriodlc ~~ ngt tn c:apa~ltll~ at t Onst4llt u c:ftation: I;olta~ U•• All exam ple of I llch a n ene rgy co nverte r is II gcn etRtor uti liz i ng t be onergy of SlI rf (f ig. 10.3). Milny a tte mpts to co nve r t the (lnerl )' of chaot ic moti on of 5C1I waves to el<-cttic t'nergy by use of meeh aniea l ar rnn ae men U lind c(ln\'e-nlio.na l oloet r ic mach ines have not led to t he ACGe pUlbie e ngi noor ing so lutions. T he electr ic- field generasce of F ig. 10.3 cOllsis~ of II me-till rod 1 coa ted wuh a lfl}'er of b igb -per m iU ivi t y d ielf'cl r ic 2 a nd mad e fast 00 1\ balK! a. T he rod se r....es as one of the ca}lllcitor pla tes lind the 1;03 surface IIcls 08 t he ot her plllte . A \IIIWe ch llngc.!l t he dev ice ca pacita nce l.!I it r uns in at the [00 lind awa y fr om It: the capaci tor ('ha rge variel! lit eons tl\n t U•• lh enb)' in d ud ng a n Alte rna t ing curre nt i = dqldl at U , that flows ju t he 10AtI e trcun. Connect ing such rod"

10,~,

Piezoelectric Energy Conve, ten

in paenllnl and ro ctif)'jng t he cueren vca n gt ve a subete nua t power at the output . or cou rse. suc h a gene rator differs from the conv e nttona l ty pe, bu t it a lso belongs t o clect rcmechant cal eMrgJ" converter s . Of i nt ere st lire elect.rfc-fie.ld machines wit h II liq u id or gaseous rot or . T here ill a possibility of crellt i/"ig s uch machines th ll t wo uld ha ve hi gh outflow veloc fti es (a bo ve t he sou nd velocit)' ) and s tro eg elec t ric fi eld s . It docs no t see m right to t h ink that e lec t r ic-field Eo., rna )' replace some conve nt ton al elec t ri c ma ch ine s ou ly where t he lut te r- do not produce the des ired tech uical erreeu. Elect r- ic-Held ECs e re li kely to fi nd use in or lgina l a pplica tio ns in t he near ruture .

10.4. Piezoelectric Energy Con verters A me chnnica l stress appliod in a -oer t.a ln d irect ion t o t he cry slIl1s of quart z, bnetum tita nate , Hoch efle ;sal t, a mi ot hers p roduce s th o elec trlc ch arges of op posi te sign of the crysta l Inces . 'I'h is pbeucmenon is kn own as the piezoelectric elfelt. I n qna r ta . till !! effect shows up al ong t he elec tete axes of t he ptazccrvst n l an d nor ma l 10 l is principal op ti c a x is. A change ill l!it" (!jr~cLiOIl of stress pla ced on th e cry st al ca uses t he cha rges or re verse s ign. A reve rse pi ezoele ctric elfoc,t occurs under the tnrtuanee of a n elec tr ic l teld, whlch cha nges th e linear dtm enaions of (l crys la l. Chan ging th o elec tr ic field d trccuo n reve rses tbe sign of mach a r uenl stre ss . The stud ies Oil pie zoel ectric materlnls revea l t he li near re in Lions betwe en tho deform a t ion lind ch arges a t cer tÂŤtn me chanical st reseee. At strong st resses t he relation becomes non l inear . so that t he cr ysta l di 'lJllay s a d ieloctric hyst eres is which rese mbles the B-H hy steresis cur ve of II st eel core . P taac olectrtc ene rgy oonverters uru su it ab le for use as pu lse ge nera t ors in iglJit io'1 S)'SWJlls. Attempts a rc made to em pl oy these conv er ters a s M pul se motors . Although the strains in cr ystals Mil Il(Âˇgligible . t her e is n poss ibi lit y of trabalat.ing f.hem into the desi red li nea r d ts placeme r u.s. P teeoelccc n c ECs hove two alect rica l and two mecltanica l ter m tn als. Electromecha nicn l reso nance i ~ of pa rticular importu nce for th ese dov tces . As in other ela ctrtc-Hald EG<! . in p iezoele ct ri c de v ice, t he oncegy eo nve rs ton ta kes plac e irr a. crys t al. E xt cndfng th e findings i n ele ctromccha ufcs int o t he neea of p iezoolect r ic EC~. we can assume that the T eq uiva le nt c ircuit wh ich includes ca paci ta nces and cond uctances is a pp lic a b le for t ho lrepre sen t 3tion of th e i nt rinsic resistance of t hese co nve r te rs . T he processes occ urr ing i n piezoel ectric conver te rs ronow t he baste laws of cle ct ro mech e n tcs and are a me nab le to the deecrtpno n

18 6

C h. 10. Electric -field and EI. ,tr<;Hl1l1gnellc-Fle ld EC

in te rms of t he t heo ry t)f e.lect.rc mec ha nical energ y con vers ion. Apar t f ro m the piezoel ec tri c effe ct , e ma gne tcet r tert ve effect occurs 10 cer ta l n Ierrc magnetic metals, 'which appears as a cha nge in tile velu me ami form of a ferro magnet when placed in It ffillgn etic f iel d . It is possi ble t o crea te magueteetr icrive vi bra tors lil ili t ing t he phanom onon of mng ne toet.nc non tp COli vert the ene rgy of a mag ne t.ic field t o meche nl ca l energy. Til e :r~~ v l\rse magnet ost rf ctlve effe ct al so occurs. which s hows n eelr IIIl .. chan ge ill th e mngnat izatlo n or a fere-cmegnet whon sub jected to com p ressi on or ten sio n . B oth th o theory a nd t he pra cti ce o f imple men ling plezoe lectric

and magnetns t r fct.ive onorgy co nverters ca n not, as ye t boas t of great .ectosvemonis . T hese converters are or much prllcti r.'1 1 interest nnd e rc l ike ly to rind wide a pplfcnt.ions in t he fut ure .

to.5. Eleclrofllagnetic-Field Energy C"nverfe rs Let

us ha ve n look at olec trcmagnetic-Jield ECs wh ich s tore ene rgy

J n a n alec trom ag ne t ic field (F ig . 1.0 .4). T hese convertors are not ye t

/'01""8

ECI EI<d

ECo

!>la,nc:" e: \ flc:ld \

I Ek<l rie: I flc:1d

EIc:c:u(lInq Iln ,e: " n.l d ECo EJe<:IfOmaS...... r.. 1d

!Fig. 10,4. Basic cteeeee 0( elecrrcme-

c banlc /ll energy IlOn verteri

fig. to .S. Electromagnetic-field energy con verter eo lllpri!!.lng a m:lgnetle-i1eld system (L ) and an eleotrl c-Ileld sys.te m ( C)

cc mme rcialfy a vaifah lu, though t he)' are wides pread in nature . Btc-

logica l energy convert ers c ~n probabl y be t hough t to b elo ll~ to t h is cl ass of conver ters. A ma chine illus t rnted ill }";ig . 10. 5 ca n bo t ake n es lin exa mple or t he alactromngnutic-Ituld T he ma chi ne con s ist s or two por ts . T ho firs ~ par t 0 11 t ho lef t or F ig . 1.0.5 includes :I coi l 1 in wh ich a s tee l rod 2 rcclproca tee a " l:! a n Inductance co il 3 exc ited by a d e voltage U r ' T hi s is in esse nce a l inea r m ag net ic-f iel d EC (',0 11 p led ~' ill a n Mill d- t o the othe r- pa rt o( the machiua, which cons is ts of a capac it-

me.

181

10..5. EI"ct'omagnlltic<F'ald E....'gV Convarlars

or 5 sw itche d In to n de vol l!lglJ ci rcui l 'D e and a dielectric 6 m ovi ng between the ca pac it or pla t es . T he coil a is conn ected to tile ca paci to r via a load res tstance R I , A l a res ona nce frequ ency /!lu = 1/ V lC, when /!lo!.- = lI woC, e lecu- ome chu nlc e f reso na nce sets i ll. T he sup pt y lin e frequency nnd t he meche ntcal frequen cy are tho sa me. a nd th e energy co nv erter at electro mecha nica l reso na nce exb ib i ts t he best

eherecter rsucs. This t' lierg y conver tor [l1llc.tio mlll y end s t r ucturally co m b i nes t he magne tic-field machine wit h the eleetric-field ma ch ine. Th o rea ct ive power ma y no t flow from t he outside, Tho electroma guetic-fie ld conve rte r. just likt' nny other e tectr tc ma ch ine. is eonvcr ttblo. T he mach ine ca n operate in t he gen eff-t-l2'3>-'-.,-' --- '" ratin g mode . lak ing i n the tu ech a nica l energy PI" . find ca n a ls o act as a motor converti ng t he e tec retc energ y absor bed i n its clec t r!e etrc u tt ill to' mecha nical ene rgy. III F ig . 10.6 is s ho wn t he sene - Fig, 10.6. Ele<;l.....magnl'UeÂˇHeld enerÂˇ malic di ag rn m of another etec t rogy ce everte e rnague t.ic co n var t or w h ich lIt i l ite.~ t he eh llnges in tho li ne ar dimenetons of P core u nder the i nflue nce of lin e le ct eomeg nenc f ield . T he convert er co ns ists of a meg ne tost rt ct jve pa r t 1 nud p teeoete c retc part 2. P ermendure or pure nickel ca n 'b e chose n t oe the co ns t ruc t io n of tll" m ag noto str fenve part a nd t he enltd so lut.io n of lead -atrcontum tnaua te , wh toh Sl10WI; u.e bes L pteeoe lecr r tc pro per t ies . for th e co nstruc ti on o f t he por t 2 . Th e con verter o f Fi g. to.n Ilt ilhClI the ma gn ctoetrtct tve ef fect of changes in t he sha pe a nd volu me of a ' fer ro m ag net du r in g its ma gn etteeuou . Jo in t ing mechanicall y a ma g ne toet ric t.ive lInd a pleeoele ct ri c IDnte rial roga th er-, we ca n ad just hOUI pa r is to res ona nce. In tu r n. the el ectric clrcult w hich inc ludes a w;i nd in g for pr odu ci ng the f ield B a nd an in te rn a l resis ta nce of t he pi ezoelec tric mill eria ! comes to reso na nce w it h res pect t o t he rnech a riica l vib ra tion s of the cores . I n t hese ce nd tuous the proce ss o f e ne rgy co uv eraiu n lakes pl ace . T he energy conver tor of t h is t ype is re ve rs ible: it ca n vrunsfor m en ergy from mech entca l to cJt-' ctri~1 te em. an d vi ce versa. Alt hon gll t h is converte r- sttrers i n con s tructio n from co nve n tio na l c tecte tc m achines, t he a na lysis of it s o pera t io n is posaihlo in tho Irnm cwork of the theor y of alecwo me chn nic a l onergy con versio n. AttollU ["Ils are IJc>illg milde 10 produc e composite m lll{netoelect r ic.'l from pt ezoel ectr! c and me gnc toat r tctf ve mate rf al e. In composite magne toelact r tc ma toetols, t he meg netoclecte tc effect occurs through the moenentcet interacti on of a p tezomcg netic lin d a p iezoelect ric

188

Ch. \ 0. EIo" trk-f j"ld

a~d

Eled ,o mll\lneti c-Field EC

phase. T he mecha nical de forma t.ion estehlishes II l inea r li nk between th e mag net ic a nd t he el ectr;il; fi eld . I n ton i mague tc elec t rjc mnte rtels , which (I tO mix t ures of barium rnetarttnn nte lind coha lL ferr ite. th e rnagnetce leetric effect acc oun ts for merui y a row perce nt of the energy i npu t. T he fur ther adva nces in e lect rc rnecha ntcs hugely depe nd o n the progress in t he area of elQc,ricIl J-ongin eor i ng ma teria ls . Th us, the cree ucn of new me te rlela which could be hav e like good e lect-teetsheet steels in s tro ng ma guet !c fields a nd like di e lectr ics ill s trong electric fie lds wo uld O(KI II tho ways I for prod ucing new t ypes ol energy I co nverte rs . I cc I R ill" i p r ow t i Ill!- LyIW e I ll t.tr OIl! agI ne t.ic ma ch ines can be. mad e to I t uncuc n as rota t ing mach ines by coupling the mo. gilf, Lie-f ield s t ructU M to til e elec tr ic-fi eld s tr ucture through II common shaH. T hB convert ers of ench 11 deelg n would Fi g. 10.; . The doma ins of d lff~rt n ~ hI:'! able to offer 0 g l"(lllw r speed of lyp e1l of t'nrrgy con verters eesponse an d operrue Withou t N ecuv c power sources, III some deÂˇ sig n verslous 1l1l)3() conver te rs would prolJa bly be rnure eHic ient lind economic all y me re adva nta geous t han magnott c-It eld ECs , or pa r t icula r sig nificance a re elc ct.romag net.ic enel'g )' conveners wit h Il liqui d ro to r t ha t q.ffotd the mc tor-genere tc r o pem t ton by cnn ti nuun sly rege uerut lng th e cnergy , e ner gy conver t ors with int('lnsic Iecd hack and co m pu ter-co n trolled co nve r ters, It woul d be itn pc rtaut t o eom b mo the jn egnetic-fiel d ma chin e a nd t he ele ctrictield machine into a s ingle l un it i n su ch /I ma nner as is done i n the elnsstco l designs of mag netrc-Held en er gy converters which ha ve co mm on s t r uc t ura l pa cts fo ~ Il.hl;l flows of ncuve a nd rea ctive powers. Fi gu re 10 .7 shows (in E~H eoc rd inetcs) t he Jcasthle re gio n of elecn-Ic-Held maclnn es (Inrgt' values of E and small va l ues of H), th o Ieeslble reg ion ef mag ue tlc-Hold machines (l a rge val ues of H lind srna u va lues of E ) nnd tho fea s ib le- region of elect ro magnetlcfi ~ l d mechrnes (large va lues ()f E en d H ). Com pos ite-ty pe ECs can exh ihit stil l higller val ues of E a nd H . To ÂŤue tn a h igh Pl.rfO l'Jl~llnC'" of ele et ric orachtnss , 11.50 is mad e of copper for mag not ic-fic ld fjOJ aud gaseo us and solid d ielectr ics 01 h igh pormittivl t. y for e lectr ic-fiel d ECs. T he above objec t ive is diffic u lt to acco mplis h if t he \~..orki ng med iu m used is innrgantc matter. Tile req uirements o n a wor king medium in e lecu-omagn et !c ECs ca n he less s t r inge nt, so the li~t of acu ve ma ter ial s for etectrt c maeht ner y of t lds clsss ca n be Ieir-ly long .

-------.,

IB' The mal hBmat teal desarlp ston

at e,ltrg ll concers ton

III

ele ctromagne ti c-

fIeld machines I~w<l lves tne li se 0/ the s;rt of equations for (l magne tlcfi eld and on electr ic-f ield machine. T h is se t of eq ua t.iona ma y Io ll "e the form

2 "el_1 I to 0e ZOo' lxl'eÂŤc l M u . "" M (J'/ ' ),

lll~c

c (u' u' )

(10 . 13) (10. '\4)

Hccu U r, 811ll I L e re till' vo lt a ge sud cur re n t suhma t eine s for t he mague ti c-field machl no. which Art! sh~ iln c t o (1.34) or (3.3) thro ugh (3.12); t e a nd tt o tire t ho cur re nt an ~ vo l te ge au bme t etcea for t he

alect r fc- Iteld m uclu ue , w h ich ar e sim tlar to th ose for the ma gnet ic-

field ma chlua : Zr, is t he im pedance mn t rt x o f t he mag uet.lc-Hel d machine , s uch IlS i n (1. 34) or (3.3); to is t he impedance matri x of the e tectrtc-uctd me chtne. such AS for t ho magnotio-Held ma chi ne; M . L is t he t or qu e o f the ma qnet tc- t tel d mnchi nl.'; n nd /If e<; is t he torq ue of t he ele c tr ic-fie ld maehi ne. Torque equatlons (10 . 14) CR II neve lin)' of tho for ms, from th ose of (1.35) to t hose of (3. 12). The sot of eq uattcns (10.13) a nd (10 ,(4) toget hl'r WIth th o equa t io ns of mot ion d escribes t he be hav ior or e lt'c LrOlUagneLic E Cs in tr a nsient a nd s tea d y-s t a te con d itions , It can rendil~' be seen thet , g ive n the genera l t heory of all t he classes of ele c.tr fc mac hi nes , H will be mo re conve nien t to beg in t he ana lysi s of e ue rg v con version wuh t he so lu t io n of equatio ns [or a n elect romng net ic -f iel d machine an d then , as II pa rti cula r case , procee d with t he study by solv ing t.he equat ions for mngne tie-f ield and electr ic- field mechiuea. T he representat tcu of ale cr rome g netfc-Held ECs as a more ge neral case of elec tr ic mach ines o ffers t he possi bilit y of a more ccmprehe netve use of th e t he,o ry of.magnoti e. fiold machlnes. Fi rs t , t he m agn ct ic-field ma chine is a conce nt ra t or of energ y; second , et ecerom ech antca t re aoncnee shows i tse lf mOM v tv tdl y fo llowi ng t h(' anal ysi s of eleet.roma gncttc-It cdd mschinas: lind , t hird , e nE'rgy convert ers. particularly el ect ric -fi el d ECs, se para te out the chargos, which is al so the case for megn ortc-Iteld ECs s ince t = dQ 1dt. E lect roJJ1 t'chfln ics as a b ra nch of scie nce requ ires fu rth er devel opme nt . T hore arc th e whol e cl asses of electric machines t he s tu dy whi ch is on ly at th e sta rtin g s tnga. Research m to t he prece sses in ala ctrtc-Hald a nd el oclro magnctie.fleid meclnnes offers grea t promise fot evolv tng- ne w t y pea of energy conver ters .

' 90

Ch. 11. App llco!iDn of EKp".ime ntol

Chap le r

~ig n

Application of ' Experimenta l Design

10 Electric Machinery Analysis 11.1. General Information on the Theory of Exper 'men tal Design T ho Hnal e nd pursued b y t he Investfgat.ic ns in ure fid d of elect ri c machi nery is to golva the problem s of synt hesis of e nergy co nverters Bod t hus design optima l ma bhi nes. T he t ech n iq ue of es perirne nt al des ig n (E D) wh ich cnu drll s ~ic[l il y cut. d own the nu mber of expe r jment e a nd tho scope of cll.lcula).io ns on flnnlog anI) dig ita l computers has recentl y g'l. ined still wtde r recogn it ion. i n uie procedure of opttmraa ti on of a n etecrrtc ma cht ua, it becomes necessary to ha ndle va r iou s pro ble ms , for exa mple, t o attai n t he desire d per for ma nce of the meehlno lind en act. 3 m axi m um !W.\' i n~ in material s , embod y the des ign t hnt wou ld ha ve te ue r dynamic c ha ra cter istics a nd improve cJ1l.Irgy cbnrecte rfst tcs, ore. G iven the ma ucmeuee l mod el , t he co mp uter-a ided solu ti on of t he problem comes t o find! n.&: th e optlmnl operating fact or s lind m akin g the r ight cho ice of th e 59t of pa ra meters by use of one va -tetlo na l rnct.hcd or another . A ~ Is know n. a change in one parameter , others be ing k ept consta nt. e xclu des t hll poss ibil it y of determ ining t he dos ired relatione a nd l ak es II grea t deal 01 mac hine ti me . The ED technique ena bles the edrrect choice of t he course of the experiment , decreases the number of t ria ls by a factor of 8 to 10. a nd op ens t he wa ys for optim izatio n of e nergy converters . T he .ED tech nique hM a pa rt icular sign ifica uce in th o ana lysts of ECs on comp uting fac il it ies . I n the last veers th is e ppmech has Iound IIS O in most of t.hl'l resea rch works. 'Ve pre su me that the ED tac hntquo is generall y known , so the t e:x~ bel ow will only g iv e t he sys te ma t ic accou n t of Ihe e x per ience ga tned in a p ply ing t h is t echn ique in etecr rcmec banrce. ,

The IlU of the ED ttchniC#l e in the ana111sis procedure permits the engineer to r:hoore the stratigy of perform ing tcs4 according 10 the preliminarily drawn optimiw fon u;Jume in order to obtain the relations between the paranutu s 'of an eledrlc meckine and Us operati ng fado rs tn a simple mc.thel1l4tlcal/orm. namely, as a polynomia l (1U)

11,1. Gene, .

Info...... tion

where b• • hi . b ll , lind h ll ore polynomia l coe fficients; Z, lind -LI are ".ri.ble pnr8 mt>leN:. or Iect crs; II is th e m achine operat ing factor under s t udy: lind n is t h.e nuw ber of v a r i. bles (foc\o I'S). Th us. the ED tec h n iq ue offers t he 1II1"VIl nt agu of ( I ) hi gh offet-t iveIIeS.'l since it ca lls for II mll ll er numbef 01 tes ts to obtere the du lr ed inform a tio n; (2) s iffinlt8 nco u!i st udy of th e effeGt of a few "arillble plfllmele rs 01 t ho machiDO on i\...s operlltinl: factors . lind (3) I he POllsibili l)' 01 ca r ry illg out t ile tests so th a I t ho "'lIrJance o'l (bl ) of pol ynomia l cm,(ficients in th e case o f ea ud om er ro rs is II I a m i ni nillm . Tile E D thoory re li es 0 11 th e Iact t ha t the te s' lh l:l of a n )' uperi m ~llt s in t he n-dtmeustonal b cto t spare can bo re presented by HneariU'd eq lla lj o n~ of t he form Yl

+ .:tu bl + %,Ib, + . , . + zub~ + bo + + x. hi + . . . ... + X' ( I Hlb ~ + . . . %o,bo

11 1+1 .." "'0(4+ 1 )

X ll l +l)b l

(t 1 .2)

( I+-l)

The result s of N t('l'lS ha ve the foll ow ing maid:!: form

y = Xii

(11.3)

"'here 1" is the colum n vecto r of euserva u ons: f{ is t he inf ormation matrix : !l Ol l li is t he colum n vect or of coefficients. The- equU iODB for 11. Y , an d .Y a re of t he fo rm

b, (11.4)

bll

"" y,,", 1111

%"%01 ••• X"1I

z"J'tt ( I L a)

~ ..

( I t,6)

X OII % . .... .. •

Z~ N

Ded vill g the maf.rtx H, t.e . defi ning t he coofric iilnts of the polynomial. i nv ol ves the teens post tl cn and in versi on of t he Inlorm et fcn matrix X. T ho f ina l ex pre ss ion for lJ ha s the form

(1 t .7) wbe re C- 1 is the i nverse ma t ri x with re spect to C = .Y IX ; a nd X I i• • be transposed m alri x . T ile ter m ' des ig n o f ex puiroe n ts ' , e.s.se ntia ll}· has to do with • •pet-ia l construc tio n of th e information matrix X. The wa ys 01 bo w the Inlo rma Uon m a~ rlx is se t u p spec if y th e ty pes of design (ort horo na l desi gn . rota t a bl e de sign . ete.}, :I'1M structure of th o i nform. · uce matrix determ ine s t he d Mlgn form ul as for est imating tbe pol y nomial cce n te te nts and t he va riance 01 (base coefficients. B y specially

ell. I I. Ap p lic. tion 0 1 Elp.rlmenl.l Peo io n

forming th e ffi/ltr iJt X , one ca n im pnr t . Cor ex a m ple , a n im por ta nt pro per t )' to t he Itrs t-ordee desig n. estimate th e polynomi a l coeUic ienta wi t h tl vari a nce tha t ts a factor of III below t ile ve ete nce ebserved in con ducti ng oue test : ~ (b i } . . at {V)fN (11.8) I n pra ctice t he in forma t ion m at r i! definos t he val ues of var iable para weu-rs r l , :1', . . . . . z~ when perfor ming t he tfl Sts . If we rnpl'6' Rn ta t ...·o-d imens io nQ [ Iactce- ap aee as I. coord ina te plane (F ig . 1 l .t ). tbe 2' desirn matei! of th e complete fact ori a l ex peri ment (CF E) bat the form

V.."' blt IOC' ....

T. t No.

•

+' -,

•

+•

-, -, -, <'

Fro m rhe above matri x it foll ows t ha t t he ex per ime nt is set up o nl y a t poi nt s whoso coordtnetes rep -eseut all comb inat ions of t he upp er a nd lower va lues 01 vlI.rl n1J.. le ClltlOrs, uam ely, at poi o18 J . 2. 8 , and ~ . T he seco nd-order deSil!o ma t r ix t ha t 11/1.$ the pr oper ty 01 +< ro tllt a b; l ily req uires conducting t he ex per ime nt Il t additio nal poin t.!! 5. 6 . 7. 8 a nd 9. 09pendi ng 00 tho inCor mlll ioo ma tr ix , I.e. tho poi.nts llt wh ich t he u .peri menl is to be run. the re.selU'Cher ob t a ins t he correspo odi ng pol yn omi al rela ti o M between the var illble IeeIon a nd machine opera t ing te elon. These rel at ions have im por· t.nt pro per t ies. nam el y , tlla y eo ab le a simp le calcu lat ion of t he Fl, . !t.'. A t wo-d! tlIfll5i olUlf factoJ polyno mi al eoe[ficie nt s a nd tb~if SlI·c~ va r lence. One attache.!! m uch im porta nce to t ho ca lcu la ti on 01 polynomi al coeffic ients beca use the meth od of leest equa ree a pplie d In expe ri mental design giv6S cum bersom e expu.!I5ions for t hes( coefficients. Th U-'J in t he .!!!mplest tlI.!!e of one teeter, the , foJmul ll!

•

-, •

•

+' •,

193

for bG a nd b, ha ve the for m

(11 .9)

bl =

x ~

Y.."' ,, -

u ~t N

N II" u;.,~'V .. - I

2:; "u

~ zf.- (~

.. _ 1

(t 1.1 0)

."t

If t he lnfcrma tio n ma t r ix is orthogonal, th e p olynom ta l coe fficien ts are Iouud from s imple Iorm ulaa of t he Lirst-ord et des ign bo=b,=

â&#x20AC;˘

L: x,!uy "IN .-,

(11.11 )

an d of the second-order desi gn S

hi = ,,_I L: ZUlU) ,,_I ~ x~,

{"Ii. 118)

Th o ver teucc of t he ca lcul nted pol ynom tnl coelf tcjen ts determ ines the acc uracy of the polyn omial as fI whole if we use lilis pol ynom ial to calcula te t he ob jective funct ion (the mecht nc operat ing fact ors) with.i n the Iactor space. The eFE infor mat ion matr ix for , ~8 Y , the first-order deatgn d isplays the op tf mel prope rty of sim pl icit y of the design form ul as {or polynom ial coeffi ci en ts a nd a ls o t he, p rope r ly of th e identical a nd min imum variance of t he coefficients . I n t he second-order desi gn , t he firs t and second properties a re eo nt mdfctory. H t he ma tri x is or th ogonal. t he vart ence estimates of poly nomial coeffi c ients are not ident ical : N

(I: {b l }

= 02

.-.

(y} /rn ~ X~j

(11 .J 2)

Hen ce , the prediction acc uracy for th e objective fun cti on y v a ries wi th t he d irect ion of moti on in t he fact or s pace . If the informat io n mat r ix 'X exbi'bits t he propert y of r or at nhi flt.y, the sol ut ion accuracy does not depend o n t he d irection of motion , 13 - 0 1 17 8

,,..

C I\. II , Ap plicaHon of h pe,i.... nl.1 0. ' ;9"

And the d esi gn tonnul u

b, = ,~

have th e form ....

[ 2'-: (" + 2) ~ ~.'y .. - 2A..C ~ ~ x:,tY. _

( t U 3)

1- 1 .... .

_. N

b" = ~ {C'l ((n + 2p ., - nl ~ 2=~lJf .. + C(t - A., ) .. X

~ ~ z~.Y . -

i. ,... ,

n ,c h

( 11.1 4)

%'. ,V..}

...- 1

b. = ,;

--, CÂˇ

(11.15)

%'" .11" .'"

bu = 7i'i:'" ~ ;I:", z " JY..

( I LtO)

11 .. _ I

--,T..i

C =NI 2;

( 11.1 7)

(1U S)

( 11.19)

wh ere N .. is the number of p<l i ll l';; o n a sp here of r a d ius P.. ; a nd h ill t he number of sphere s (k '= 3). The good ness of fi t of tho pc l yno mla l t o tile object und er s tu dy denoee t he degree to whi ch t he ob jec t ive fu nction y ob t a ined from ca lc ula te d th e e x peelmont corres ponds to tlte obj ec t i ve Iu ncuco w ith t he po ly nom ia l. Til e q,uftnti t y charac te ri t i ng til e d isere pl'lncy be twee u the calcu la t ed l' ud ex pes-ime nta l values is the Ina de q uac y

vll l'ill nce def ill('d b)' t he formu la

(t 1. ;1.0 )

whe re rl is the number of s illlJiJiea ul tencs in the a pp rox im e l ing pol y nomial ;

11,2. The Tech"lqu e of e'pc.im e nta l Oe. i'] "

T he lest o f t he hypo thesis of adequac y is made b)' usc of P tshe r'e var ia nce ra ti o (P -tes t ): F = S:.,!S~ {y } (t 1.21)

H t he cal culmed value of t he ra l LQ b lower than the cri t ical ra tio Fer tll ke" from t he cor-respcndin g table (see Ta ble A IV .a) al t he given si gn ifi ca nce level gO' the descrip t io n is considered HI [it ti m objec t un der study, 'I'he desig n matr ices ap plie d t o t ile enalysls of elec t r ic machines intr od uce t he de finite Jea t.ure s i n t he tec h niq ue o f e x per ime nta l tlcsign . Th e descri pt ion of t hese fea tu res ls g ivt'1J be low.

11.1. The Technique of EJfperimental Design Appl ied In Electromechanlcs Consider th e m ai n classes of pro blem s ha nd led Ly t he E D t echnique . T he ri rst class, be ing most closely re la ted 10 tl te cllls,sicAI scheme of oxperune ntal des ign , i ncl udes the prob lem s of lesli llg ele ct r ic ma ch i nes , th e ir ele men ts. or eleetromecha nl ca l sys te ms. T he sec olld etess cov ers t he pro blem s of a nll l~'s is of ph ys 'i cal a nd mathe ma tical mod e ls nud a na logs w h ich are t oo spe cific and co mp lex lo be used direcIL~' for th e so lutio II to u. e p ro b lems of s ~' n t hosb of electrfu ma chi nes . T h e t Jli rd cla ss incl ud es a p prc xbna t lo u-t.yps proble ms for the oases where i t is possible 10 re place th e com plex m a t ha m a tie o] model of ene rgy conversion in e le ct r ic m achines b ~Âˇ a s im p le po lynom ia l no ted for lin explicit li nk between the va r iable para meters nud t ile m ach ine oporatin g t ec t o-e. The 'ED tech n iq ue ap pl ied to t he sc lutto n of ench c lass of problem s "shows dist ingl ,ish ing Ieat ur es wh ich we ca n tuustr e te by tho examp,lps t ha t foll ow . Cons ider th e posst bi l ntoe of the ED r ech nlq ue by exam ill i n ~ II si mpl e , but fa irly rnsu uou ve. exarn ple , For t he ec celemted reliabjlity test s to be r-un [ bot.h e val us tio n and co n t rol tests), t he accolam tcd test r atio k a must be know n, I.o. t he ratio between the t es t, um e unde rnorm a l eonruuons an d t h at u nd er a qecleratod te st condutons . T he reuc k a Call be det ermined fr om (III experimen t Oil fI tes t ob ject . A~ a ru le there is a nee d to k now a Iun ct.ionnl r era uon between k~ a nd acce lerated t est fac lors; rat he r t han 'the unlq ue value of k". Suc h an app roach ofr()rll t he Ilex ibiI H.)' a nd u ni versa li t y of tes t procedures, enables the usc or th e evolved technique 10 CIlCfY ou t tests not on ly at a cer t a i n pla nt lind o n a pa r t icular le Jlt bed but at various plan ts differi ng i n technical nnrl p rod uc j.ion basis. The o bjecti ve of lesling II me chme for J'ellnbi lity is t o check t he machine se rv jcea htlity over t he spec ifi ed lime pe rio d u nder given cperating coo rtt u ons . T o spee d li p I.lm les t p rocedure, t ho mach ine is run under mo re su-in e unt. cuuditions of opere non. T he teet t ime is cut dow u d ope nd in g o n tbe tost s t r in gonc)' l'\ling cho sen . For mach i ne uÂˇ

C h. 11. App lict'i o " 01

E~pe r im em. 1

Oeo lgn

wi th II mean ser vice We of 10 000 hou rs , t he duter m in a tt nn of t he set of eccolern ted t est co nditio " recto rs w hich ena b!e cuttl ng t he tes t t ime by II fac. tor of '15 1,0 20 is a n u rgen t prob lem fro m the ecoM OlY point of view . Th e sol ut ion to t he problem depe nds on the est ima ti on of the Iunc t.ional re la t io ns between t ho mn chino n pero ti n:;: re ctora a nd eeceterete d -test cond iti on fac tors . T hese eala uo ns ca n be de fi ned ux purime n te'l.ly us ing the E D te chniqu e . F igu re 11 .2 Ill ustrntas th e schomat!c d iagra m of conduc t ing accele rated t est s on (I ll ene rgy conver ter for n ine acc el era ted test rec-

" -, :;

'. Fig_ 11.2. 'he ach.emllUe diagra m lor

conduc~l ng

converter

eccetereted testa on an

~o crgy

t ors . To these Iaot oes bctcng t he emhle nt te m perature x , ; re pet it ion rate of st ar ti ng , %.; a ngu lar ve loci ty :1:. ; l oad o n t he sha ft , :z:.: v ibrat ions %5: aggress ivity of a med iu m , X e: hu midi t y ;1;" 7 : ous t conte nt :1:6: a nd r:rad e of a Iub r tcan t , r e .K now ing th e machi ne oper at ion tim e top recorded d u r ing th e tests, we ca n esteb ltsh . tho ralatio n toP = I (x " .. . , Zt) a nd t he reby solve the sta t ed problem using t he ED tech niq ue , What complicat es t he p rocedu re of performing such ' cats rs t,he nc nun lfermi ty of electric mac hi nes of even the sa me ba t ch. T he causes are th e d iffere nces i ll ct eeteteet -sbeet steel p roperties , vertstions In phase re ststoncos an d ai r-gap dtmcn s to na. d Hfere nees in the perfor mnn ee of hea ri ngs . and other causes assoc iat ed wit h m a nufact uri ng erro rs. T hese ca uses necessit ate clirry ing tes ts on a few machi nes simultnneoualy and determini ng the mea n operat io n time :top a nd varia nce S~ (t op} from th e te st res ults . A highe r va lue or \1\(1 ope raticn t ime va riance ca lls for per for mi ng m or e repe t it ion tes ts on m or e elect r ic me ch tn es. To. determ ine the depend e nce t ~I' = I (:1:1 ) requires s tag ing a set of ox per tmente at several val ues of .:c" The varte nce of t he coeffic ien t bl t ha t accounts [or the affect of teruporat ure changes d uring t he ma ch tn e o porat ion t ime. is fou nd [ro m the formula 0' {b, } = o~ {1"l,}l.2.. T h is coeffic ient calcula ted by th o ED t echnique evidc nt i y has [I l ower vart ecce o' {b, } = 01 (t op }!4.

11.2. Th' Technlquo 01 Expe, lm"n t,1 0 lO,19n

As t he nu mber of a ccef er ated test factors grows . t he advan tages of ED become sti l l gre a ter , as is clea r from T able t t .1. re ble 11.1 The Humber of TeJts te b.e Run to EJtlmat, Polyn4rtllal C",'ffldenIJ U.lrtg tile e,," u lcal Precedure th e ED Techniq ue

."d

Kumber or Hm '

, a 5

I,e- I

, " e"e

â&#x20AC;˘

8 18 32

O ne of the impo rtan t I'1.'IlS0 11S of lIsi ng UU~ ED techniq ue is t hnt smaller num be r of tesis e nables a mo re accurate ealculat.ion 01 t he coeffic ie nts t ha t oC,OOunL for t he intluen ce of test teet e rs . T he ED t echn iq ue has II de cid ed adv a ntage in ,lInt t ho se t of experi men ts ru n accordi ng to th e- des ign m a trix 'd 9~ S not ten d to accumulate tile erro r. R a th er. tho i nfluen ce of the' error decreases in det er m inin g t he polyuomia l ccoffic ieut s . For exam ple , t he Ior mula lor defi ni ng th e pol yn om ia l coefficteu t var-ia nce o' { b, } in t he eFE or F'f E [Iractio nnl fnc tor ia l ex pe r imen t) hils the form o' (bd = o~ Ii/}IN (11.22) wlll.'fl'1I' is the nu mber of ex pert m en t al po ints in the fact or spa ce . Th e ED te chniq ue ha s one mo re essen tia l poin t i ll ue fnyor . The thing is t ha t whe re ti le test inv olve s 0. lorge num ber of accelerated les t taeror e. t ilt' a nllly si s of th e per-lormu nce of th e m achi ne unde r test bec omes 9 rlilfic\1 1t probl em . It is quit e poss ibl(' thata co mbination of v enous rect ors rather then eac h foct or se para tolv nlIecls t he ma ch ine ope ratio n t.ime , T he jo int effect of a few acceler ated te st fact or s is pos sib le to estimat e w itho ut a com p lete kn owledge of t he phys ica l nat ure of t h e p roces ses by useor t he E D t ech n iq ue . Cons iderat io n here is giv en to t he limit s on the n um ber (If ma chines put to t est lind t he tot a l te st time , n la rge nu m ber of t est Iac tors , lind lllso a h igh cc et of the experi men t. T he ch c tce of the design ma tr-ix a nd t est sche d ule and also t he calculutio u of po lyn om ial coeff icien ts are mad e a ccor di ng to t he eonve nuoun! E D pro ced ure . Let us on ly no te two e leme n ts ill the org a nizatio n of t h e scud tee on etecu tc ma ch ines : one r elates to t he choice of t he lim its of v ar -ia t io n or ure fact ors , lind t he other to t he experime nta l te ch n iq ue. II

1!l8

C h, 11. Appl ica tion of e.pe , imenla l Do , ig n

The proced ure or .I('si g n ing rue experhue ut LJ egills w ith t he es um a t.ton o f t l,e va r iatio n ra nge of necel era ted tes t fac to rs. T h is is one of t.he im po r ta n t s tag es whe n olilk in g t he tust 011 electr ic m a chines. On the one hand , it is lHlvAn lllgeo ull t h at the va rta t fu n f/t llg", s hou ld be /l.S w ide liS posaib!o , for t he tas t Iactora will t hen produce a great er effe ct a nd the t est cn n be A ru n teste r On t IH.~ oLher ba nd. too wide n va rfut.i ou l'u nge l uvo l\"05 1\ r isk of qua litatively cha ngoi ng' t he physicn l essence <)( t he ubjec! unde r s t ud y . f or exa mp le, x, in l cllt.ing II mec hinc ove r n wide o intc rv n l o f varta t tou of t he amb ic llllelll pel' a ln r" (fnc l orx, ). tha Fig. 11.3Âˇ "'"ch l" c-opll'-ati on lime

versus tem l,eratllrc

l:f1l;:l\SO llIa y le ak ou t or

hear ing

subasse mbly causing it. to operate i n the llbnor mll l condft tons. LI;l . the coudutons of d r y Incuc u. A s ma ll in tCf\'<l 1 of vru- Intio n of th o rec to- Is a lso llJulesim ulo hecause of a lo nger tes t time . T o ma ke 110 es ti mate of t he num ber of machfnes 1.1) he te ste.l . TIl~ b !o 11.2 gives t he da tn OJ] de ter m ining the va ri:ltio n l'>l t,ges ~ fler t he It rst , seco nd. and t he t h inl t ri a l. i\-~) IC t ha i uicso dat a rotor only 10 til ", Ii rs t- ord or dl."s ig ll: t he seco nd -order de s ign wi ll cart Ior fI mu ch grea ler numbe r of te sts . T4blc 11.2 Vari a tio n Ra ng es lo r Va rlo u< Factor ial De. 19 M Do\.< '1:tl

Tr ,al No. 2'CP E

, a a

2 0CJ' E

:'t2

2 ' C ~- P;

2' - ' PI' E

2 0" 1'<'1:;

,)4

128

Til e prac t ica l rocommc udationa o r how 1_0 choose t he varia tion It cu ts re d uce 10 Il l horovgll u lllll r s is o f II priori Inf orm ation o n th e t es t o f IOR<,:h i I\QI:I I.R ll111og ~ ) de pending 00 ea ch of the h c ll> rs sot up sepa nucf y (F ig. l L 3). So Jong as the one -di mens io na l rel atio n t M , = t (2"\) is UO(!1l r or a pp roaches t he Hllea r form . t he ent ire runge ca n be cho se n as tho in te rv a l of rect o r var ta t tcn . If th e ran ge ex te nds beyond the limit A , tho pn ly no mi al modnl ma y become Inad eqnnt e..

, 1.2. , t.. Techmq.... of E. p., lm.nl al

o..ign

ThaL is why in PE' rfor min g leslS n " eosr ly o bjoeta :!Il.lcb as Ill.-elri c m a c h i lle ~ , it is obli£atory 10 co llect_~ m axi m ,"n amou nt of • pr ior i IMurm nli on wld ch mus t be g tve u prope r !5Crll ~iny te mllke II dee fSiOl' 0'\ th e ra nge of "ar illblt's . ' A!L'fllme the stated p roblem rllq u lrd 10 estima to k. = f (.z:,l . where %/ represe nts ~ ecl:!ll! role(\ lest fac tors. Conside r a n c:tllm ple of Lhl' 1lO1,,tlon 10 t hi s prohlem us ing II mod ifie d version o f rho ED techniq ue . Fur li m iflusvrnt lve p urp ose . w e will han dle lllll pro blem r",r lour t esl lectors (A lnrgcr number of fll c ~ur s re qutros th e use of s L A nd~ rd procedu res set rcnn in t he ED the or-y}. LI;lI u.c test rnetor>! hl' tho test clHlm bcl' tem perat ure X " lll/lchilll' vibra ti on Xz (of cour se more inte n!'ivl:! t ha n t he n",r1l1 11.1) , e ngula r vclccif y %'~ lind t ime (Bc t",r t wh ich is c.erta ird)' peese ur in t he ex per tmc nts of tllill type . T he /'t'b Uo " : > k" = f 1:%'" :%'•• I) can IK' found by IL~ of t he fonn nli red EO peocedu res .... h ich ill,·ul>·e t he fo lle .... illl:: aa t ller inf; II prillr; illfor mAt io n: dr term iu illg th e cu rvlll un! o f l hl' fael or space llt roul;:"h tbe c:mll!~rlI et io n 01 cm e-d hnt'll s illllill eceuous of ""ft = f (:%'1) ' k" == - f (.I'z)' il llt! k " - f (.z-, ): d tfJOS illg l l~ lowur d lld up per linl ll.! 01 va,i, Lion of Ihe Is etc rs: carry ing "" I t l ~ I'l:pcri me llt on a p propr illhl les t tees: gi " lug t l.... mM hemal it'.1 l rt!nt men t of t lo" rcanh a lind Inter pr\ll ill8" t he-n . Suc h I't way of the d it"('cL s ulutln n to tho s ta te d pr ulilem by ~ ll tl fOl'lllnJizl'fl ED procedures meall" t hat we havll to realize t111l matr ix: of n po wer IIr at 1f~ 1I .~ t 2 4 (Ta b le ll . ;~l' T he ED t echnlq no erHlhleli liS to perf or m t ho tes LIl ~ uc~ss rl1 l1 ~' 11 11 de te rnuue til t' rci llti ollS kft _ - f ("" ;tz· %' z. tJ U t he co ust rsl nts 0 11 the sc o pe. of the ex periment (i.e . the number of mad linI'll 10 IKl p ill 10 test ) a rc rensonahle And 11 0 a mp le a mou nt of a pr ior i tnr erm e uc » i.'!l evelle ble , I1 n m e l ~· . l loe res ul ts of pre li tn iIlRr }' er per trnent s lind Pro ll,erl)' chose n vllrh. t io n U II ~S , parti cu lar l)" th e var ia ti on ra nge of t he t ime Ieet c e. Experi l" \c:e shew! t ha t if is more a.d'·ll nt allOO".'!l to 1L."Cl not th e stAnda rd E D rne tbods I". t a some ....h.. ~ refined t« h lliq uc bll ~d on t he se mo r ••lell of t ill' E O l hen ry , Accou nt must be t ake n o f t he (ael Lhl'l Lan incrt'II!8 in t he scope ct t he experi me nt requ ires I'l corf('!Jpondi ng Inc rease i ll tile n umbe r 0' test beds a nd in the t est t ime . I II so lv ing the reli a bil ity prob lems a nd es tlm nting t he fUlieli onAI re latio ns of t he t ype k .. --! (.t ,). n li tt le-k nown bra nch of olect ric mach ine l'ng illet> r ing hal! 10 be kept' in mmd. wha t ts meant he re is the physi cs of l hl:! pre cesses of n g ~Jlg a mi wenr of elect r ic mnchlne elem en ts sud . II ~ comm uta to rs , bellr ings , sli p r illg3, and II win d ing un der t he cnmplex Ilcl ion of ele" Ilte d tem pera t ures . int e ns ified ,' 1brnlion s. a nrt increased s peeds. T ho d a ta on t hese precesses , Il;l Lal ene l he da ta ill a nnJ)'liclll Iorm , is usua ll)' not Ilvllitoble for a par li cubr mllch illoll or for II loall'h of t lK' mlld ..; n!?sof Lhe s.me ser ies. I n '\'i ew of t h ill file t , setting lip t he "ll r i... Lio n limits o n th e ti me fact or bceo rlle:5 pr oblema t ic . An u nllerestimnl ed. " p PE' r l imit does not Illlo .... for th e

r.,.

T&!:> t. 11.3

Naeblne No.

~. t

No. "' I i ll

I ""

1 "',

I ..

a a

, s e 1

, 8

a a

, s

• •a """ "" "" " 7

... ......

...

+ + + + +

e 0

+ +

+ + + +

0 0

+ + + +

0 0

, ,.

"" "" "" ,,",..... ,,.... ,

...

TOUt t _ ....

+ + + + + + +0

.....

co m plete IltiliUlion o r t ile ex pe r ime n t. Th e o btal uetl values of the aceele ral.ed te!'t coe ffi cie nts prcvelcwar thnn t he possi ble oues , which , In t he fin a l ana tvsis , longt he ns out th e acce lera ted reliabfli t y t('SI8. A n () v t! ~s IiOJllted upper linl,it of th e lime teeter ma y render the ex pe r ime n t u ns uccessfu l because a t lea st oue un reeliaa hle row of the Illllt r ix mea ns It fa il u re of th e en til't! ex pe rfme nt or. nt best . t he transi t io n to an up per le ve l of fra ction i.'I&:. ie, th e l.n ll8i t ion rro m the 2· com plete fll(.torial to th e 2'--1 fra ct iona l fa eUlr ill1. from tIlt! 2· - ' te the 2"- 1 fa ctor ia l , etc. T ho se que nce of the so l ut ion 10 t h is pr oblem is a s fo ll ows . Prior to co nd uct i ng t ile ex per tme n t, t he levels of th e t imo te eter lire left normal. T he l ower , zero, a nd uppe r lev els lire round on ly for the acce ler a ted tes t l ect ors s uch as tem pera t u re , vfbrnt.ic n, end s peed . I n do ing th is , it proves possi blo t o im p leme nt the 24 de st g u rnatr-ix ror the co m plote factor ial nx per tme nt , T he fi rst Ie ctcr is t he fac tor of t.ime , T he time rec to r pe rm its n roni ng tl"sts 1I0t on Hi bu r on 8 ma cbiues in tlte 2 6 CF E . O ne a nd th e sa me machi ne ill pu t t o test bo t h al l im lower and a t t he u ppe r level o f th e t ime (a ct or .

20' Each of the t.es ts is ru n u nt i ! 011 the elamems und e r invl'Sl_ign l io n.

[Ail 10 operate . This mea ns t bllt if, for ex a m p le , a hoar i.ng sub ess nmbIy fails . the time of failure is recorde d . t he subnasem bly is replaced a nd t he ex perimen t is contin ncd , bot t he new coun te r-pert is tn ka p ou t of co ns id era tio n . If. Iurf.her-, t he s parking III the com muta tor-

exceeds t ho perm iss ib le leve l lind it is d iffi cu lt t o rem ed y the def ect , the c om m u t a t o r is left to s ta nd a nd the m a chi ne to r u n s i nce the

operator has rece iv ed the in format ion 011 th o tor lIud exclude d i t from c o ns idera t io n.

rcuure of

th o

CO m m ll l ll~

T he measurem en ts of t ile op t tm taatfo n param et er [cc nt r nlla h le par ameter a suc h a ll no rse, I UnOIl! lime , s pa rklng, Ineula t tcn res tslance) rue t aken cont inuo usly , whene ver reos tble. ra the r thau lit ind iv id ua l po int s cor res pond i ng t o t ho vctt.ices of t hc h y percubeund er s t ud y. This cnn he th e ca se for , ~ II~' , t he com m ut ato r unit e nd s l ip r ings . Th o a bo ve mea s ure men ts ca n a lso be mndo dise retuly f Of , snv . bellr in gs nnd insula tion over ins ignif icant t.ime Iutu r vuls , from 24 to 48 hours . Th is se q uence 01 rea li za t io n 01 t he desi gn mat rlx shows irupor tMlt features which m ake t I n'! solu t io n of I he stn terl pruhlem posslb to . Tho po tÂťt is t hllt tha mat hem ati ca l mode l of t he pr OCl'5S, t.e . t hepolyn nminl de pe nden ces of th e cpt tm taa u c n pnmmetce on th e t est raet ors. incl udin g lim ti me Iactor , pr oves op ti mll l s ince t he re cord of t he levels allJ u,ÂŤ in ter va la 01 va r ia li on or t he t ime fa ctor is ke p t , afte r obt n in illg t he exp erf me ute l dnla lind pcrformt ng the ma t heum t-, Ical t eeat.men t . T he nex t ste p tn vcl ves tho ceu mo uon of t he accal era ted ll:!st teet ers fro m t he obtai ne d pcl yno m iul de pendences . A no the r a d vnntage derived from the ab se nce or fi:l"ed levels an d tut cr ve ls of va rsnt.inn i ll ti me is t he foll .o wing . If th o obt ul nad llill t hemutica l model i ~ inadcq u nte , r.e . th e m at hematica l descr ip t ion doesDot ccrres po url t o HIt! ron l process, we r a n pnss to the no ulinca r- t rnns-. for mll!,iofl of coor dlnn tes, replacing lor t l>tl p urpoi"ll t ile in de pe nde nt vari ablE'S hy fl ew o nes , " = ;r.~1 or ~, '= In ,( /> a nd t hus cc nv ert tng to the logar it hm ic or ex pone ntl n l t ime scal e . This fea tu r e does m uch, towa rd t he SU<:CCI'S of th ll ex penmo nr , II Illll tra nsform ntjon of eocrdl nntes floes not give th o desired ef feet (t he mo del rem u ins inadcquat u}, t he onurs ti me i nterva l ca n besp lit up into II few por ti on s so as to o bt atn t he adequa te modol. T he Iour-Ie ct or CFE ma t r-i x (T a b Lo 11.4 ) rellli wd by t he abovemethod and re qu trod t o est.i matc th e ecee le ea tcd te s t rnt ios is etmcl t a ncoua l'y t he tb eee-ra c toe CFE ma tr ix e mp loyed to det erminethe t ime to Iai lur c {t he four t h Inct or, i .0. tim e , un dergoes we nsro emeu cn 10 becom e t he opt im izatio n pa ram e ter) . T he dl'8ign so set up pr ov ides t ho roln t tons bet ween t ho Ia i fueet ime e ud t elll Ia ctors , ui ts bei ng a re t ner im por tnnt res u lt of the experl men t , Al so , t he scope 01 the ex perime nt is cut down pr ecttc-

Ch. 11. Ap p lica tio n 01 E.pe,imo nla l 0.';91' r .,ble 11.4

The CFE

)l~~~"f" , 3

.•, u 1

S 9

~ sl gn

""!Til 10, h t ilMl ion 01 h ilut e Time

\ '....1

No.1 .,

, 2

,• ,,," "

+ +

r-:-I

"'''1 ' '''' Inr

I x, + +

+ +0

% . ,::~; ,

'."'" + + + .;0 0

'.'.'. '.'. '.

.all y hy 00 0 ha lf , The der ived, rcln t.ions gener all y giv e dHft'fIlnl accd~ I·ll1.ed It'sl, rauos for CHell of th e elemauts such as bcnnugs. commutn tors , sli p r fngs., lind Wind ings. The sim ulta ncous so lut ion of e quo tions offers the posaihil.It y of obt"in illg a co m mo n nccelernred te st rano for t he entira it e m . Thus ti m d cs c r fbod me tho d ca n de termluc til e accnlertued les t ra t.ius Iur various e temnnrs of the mach ine u nde r 1.. lIt Ilud at eo 111 0 fa ihltO ti me as a [u nction of uic t est rectors ill l;C V(,l'O test co nd ttt ous, L('. in the cond itions of buth th e li m ited « mount of II p riori i"r ormnlinn li nd t he Hmt tcd scope: o f t he ex per ime nt . T l d ,~ method cnu II<' ex te nde d t o cover t he deSigns r'Jr va r io us numb ers of acce ler ated te st. fact ors , T ho nee of ox pertm entn l (les ig ll 10 evalua te accelerated test rnuos in ~('l<ti ng cte cu-tc ma chiner y rep resents OIlO of the com plex exam ples tuustra uve of L11!.' pote ntialit ies of t he ED technique. Consider the »ppl tcauon of th il! t echnique i n s l ll lt ~' i ng ma the mat tcaln nd physi ca l models, for e xnet pte , (Ill indu ction ma ch ine model d ev e loped fro m t he mnthe mn stcn l th eory o f e lectric ma chiucr y and set up on au :lnalo~ com pur er or an in duc tion- mnchine ma gne f ic

core model for med on \JI \Jc lrka l co nduct tvo paper . Th e in du c ti on muchinc mode l in the form JJf " com puter a na log pennilJl II flll'. L in· vps t ig!\lio n of t he m ost div erse mod es of opera t tou of t he machi ne (its ,ly ll:u n ic a nd stat tc »cuon . dlll flgCS in t ho pernmet ers wit h um c . c lc.). l lc we vet-. the attemp ts at scor ch ing for the optima l paramet er s .or II maohtnc provo imp ract tca ble because of t ho uns t nhle a na log

and 1\ long t ime re q u ired t o re a r ra nge-t he g a i n Ia cto rs o f the analog. This is also t rue of the ph ysi c a l m ode l b u il l IIp on c on d uc uve pupcr , The c a lc ul a tto n o f n e ld s ami nd m tt ta nces e nahle8 eva tua ting pracuclilly o ny pat.tern of t he magnetic clrcult., bu t t-he search for t.he optimal geom e t ric pa rameters o f the m a e hinc re q uires II. mu l l ip le refe rence to t h l! prcblem se tup. I n tlit her of t he t wo OIlSt'S i t is necessary t o fi 'lt! a GO l1ve n ic n t form o f p rcsanta t Io n of Lloll i ufo r m a t.ion tle rt ve d

u. H

tÂˇil[. 11.4. Th e system w ml'rislng a mngnetic amplUi" r (M A) kmt inlh,cUun

motor (1M )

CT _

e" . "'bt ' u n' ror .... . : I' T _

,'"It"!,,,

" A".f-" " ,,," : U a _ Ao<~l ~, IIl" d L'}I>- 1, 0001"'('k v t' lIag.. . n - f'<:$ N'''''

,. "

v"ll~" ..Âˇ

fro m UU~ m,,,l o l ~ so <111 t o lise it r Ul' tll '!I' for the aotu uou of o p t i mi 1. a~ rton P " Qhlt'nl .~. Th e po fynomiul 1l0 1ll i io n is" eo nvouicut form u]" p ros entnticu. Let us il lu stra te t he featu re s of ED lor I I,e second ctuss 01 prob lem s by rQ I ~~ i ded ng nn exa m ple or th e il lIlllog of an Induction lII ol nrm a gn e ti c nmpliFiel' s ys t eru , T Ile blo ck d illgl'am of t h is s ys te m wilh c u r re n t 11 11d volta ge Ieedba ck mechn.nl s ms is show n in F'I~ . 11 .4. T o uualyzc t he ind uc t io n m a chinc C Q lI~ l n ,e t i o l\ , wu 111'c ,1 10 thl' rua chine n nn lc g a nd ob t nin 11m Iunc t ioua l rcln tic n

s t" d ~'

uII=f( Ku. K, 1 where ~II i81.1,,,, m olor' s /tUg-H IM volOlllly d irrer e nco w rth II change in t he loa d t.crq uc ov er t he s pe clfled iimits; K lJ li nd K, lire v olte gn -lIll1l cur re nt Ieed beck rnct cre. . T he II11111 ysI 5 o f t he mncluce de-:sl~ n s ho u ld be 1l11l,lu "'i t hil1 thc prodotorm tued ran ges of var tat to n oT 'fM'!Or.l x, Ilnd XI: X, = K () = = 1.5 10 2 .1 and X 2 = K , = 2.4 10 3 .li. Tho ED technique is t,Xllcd io n l fl)r usa in d c nH ng with the giv e n problem [or the follow in g rl'aSllll. we nee d to o bt a tu 11 COl\ VC[liCIlI and co nctsc rc tnt ton ba two un tI." a nd K u- K , ill the for m of n po tv nom tal:

bl4

+ b 1x . + b, ~,X1 + b ll r. + bur; + . (11.2.3)

C h. 11. Application 01 hpa r'IM ntel 0'0$;911

\ \1'0 should decrease the effect of instabili ty 01 th e a n a l o~ on t he resu [\.s of experimenta l tnvesnga u ons. Cutt i ng dow n the numbe r of uxp crim cn t al (des ign) points in llie problem of i nt eres t is not the mal n objecti ve. T he req uireme nt for tile sto bIe limits wi th in which the fact ors undergo ch angl's br ing!! in II num ber of fea t ure s in the proced ure of p la n ning t bo expcr tmc nt: Sin ce t he ran ge of v aneuo n of t ile fac tors is fair ly la rge a nd so t ho a na logs obt a ined cnn Il,' inade qua te. we shou ld r aise the order of nxper tmont a t ion. At t he li rst s ta ge of s t urlyi ng t he an a log , it is necessary 10 115C Hrs t-orde r desi gns of t he e FE nnd FF E nnrl t hen pass to second-order designs il the model is i oatleq ulllo. Th e second -Q rder tll's ign stJ u~ ta n Li ll ll y d iffe rs from th e first-or der design in t hat it docs not in clud e t he expcrim en ta ! s l(\ges wliic h comb ine s im ple d('sign fo rmul as with n high nCCUflH"ÂˇY of poly nominla . I n solv ing t he proble ma in ete ceroroechan tcs . use is ma inl y made or t wo lyp l'tl of t1 Mign - th c orthogon al Mi d ra ta tahle t y pes. T he fi rs\. l YJll! o rre r~ t he advant ages ove r Iho r ota ta ble typ e i n t ha t it emplo ys rela ti vely simple d/!'s ign formulas and needs II smnll('r number of tests to be rim . I n t IL(' e xnrnplu under c{,nsi{terlllio n we lise tho seco nd-order ortho qcn a l desig n I{) rlefln a t he rei llt ion &/1 = f (K u - K / ) "'inre the t trs t-or der des ig n g ivcll an inadequa te pofy nnrui nl , T he ma t r ix of orthogona l cl,ntr'a l composite d('~ i ~n (OOCD) is g ive n in Tn hl c 11.5 . Tf.ble 11.S

The OCCD Mat. l.

,., )\0,

,, z

,s 7

, 6

+' +1 +' +I +' +' +I +' +'

Oblcc l[ Vr ll"'et lon " 1"'1

;;- .â&#x20AC;˘i "' O .6 6 ~

-, -, +' +1 - 1 -, -, ~I

-1 +'0 o 0

;.~ d_O . 6it

+'0

0 .333

76.3

0.333

O . 3.~

93 .5 64 .1;

0. 333

0.331

71.3

0.333

- 0.666

0 .333

_ 0 .666

"se "

- 0 ,666

0 .333 0 .333 - 0. 661S

0 0

"0 0

o\. n

0.333 0 .333 0.333

+1 - 1 +' <. , 0 0

P . ~ 1C1 [(Jn hy Jl CCD

-0 .666 - O.66fI

sa 58.' 68

61J.l

'00

85.6

73 1:>7 .3

64 .;; 10. 1

69. 1

T he polynom ia l for lin has t he form .1n = 67 . ~

+ 5 . 34~ -

6 .il:{r.t - O .!iXl.f~ -

3,,')x:

+ 12r. (11,24)

[L $ho ul d be noted t hat for the g iven rAnge of variation of K u and K r , even the secon d-orde r desig n gives a pol yno m ia l of lo w acc uracy. Th is ca n be seen fr om t he 'res ul ts cont ai ned in th e extre me d g h t colum n. which are th o va l ue.'> of Il n ca lc u la te d with t n ll pol ynom ial. T he di scr ep an c ies at s ome poi nts, es pac ia lly [or tosUl Nos . 7 and 8 a re so large tbet, th ey go 'beyo nd the s peci fie r! limits. A d iffer en t a ccuracy of pr ediction in va rio us dtrectt nns 01 t he fact or s pace , t .e. a t va r ious com b i na t io n of K u a nd K I, te nne of ti l\< grave disad vu n t ag es o f the seco nd -or der orthogona l desi gn . Suc h a poly nom ia l is imp ra cti cable Iceuse in . the ann lys is a nd sy nt hesis of the mach i ne becaus e tIle sea rc h fo.Âˇ,the optima l. geom ctric pa ra meters of the machine is dono for venous eooib rneuons ot K u nnd K I a nd t he pol y nomial e rror m ay.result in a fal se ex t remu m . Where t he a ccure ev of t he pol yno m ta l over t ho s pec ifi ed intervals of variation of t he fa ctors is of prinllll'y Im'P0rUinc,e , the inv estlgauoo of an al ogs a nd ph ysi cal. m odels shou ld rely o n ro t a ta ble ce nt ral eemposjte- d es ign (RCCD). T h is t yp e 01 des ign brings ab out an t ns ig ni.fic a n t in cre ase in th e number of t est s t o be porf orm od in comparison with t ho or t hogo na l t ype but offe rs a h ighe r acc ur ac y {T uhi e i 1.6). Yeble 11.6 Tho Numb ' . of TeslS R1Ill In aC CD _ _ __ C... : : Rc e D' Met ho<h N "",b l' rI c 10"

occo

, "as

nccc

ra eo

For comp ar-ison betwee n the ac cur acies of calculat io n of 1111 e mp luying m o aceD and nCCD. Tab le t t.s lists the valuM of 11~ obta ine d wilh the aid of r ot at able desi gn. Let. us note t hat in the a nal ysi s of p hys ical and matberna tfca l modem of e lecu-ic mac hines r ot a t a ble des.g us are ca rr ie d cut w ith the \100 of moder n digital comp u te rs. It sho uld a lso he po int ed out l h:H thero ar c ot.her ways of experim ental des ign i n dcnling wlt h t he c tass Qr proble ms u nder consi de ra t ion if tho mod el obtained with t he aid of t he fi rst- or der de..... ign turn, ou t 10 b e i nad eq ua te . The n p pmaches mnin ly i nvo lve the r epla cem ent. of varia b les a nd llpli tt ing of (he varia tion rBllge of fact or s into t wo su br egions , I n bot h cases th e objective is to Iry to r edu ce t he proh lem t o t he first-order desi gn . Howev er, t hese approache s are not a.lwuys eon venie nt lind re spons tblc for (I nu mber of diiÂˇ I tcult.ies in ha ndf i ng op ti mia a t ton prob lem s . The scoo nd-ordur des-

Ch. I I. A",,,,licati o l> 01 E.pa , ima lltal De" ,. "

ig n. t hough be ing mere complex. offers good re su lt s. given rouune progr ams ru n Oil d igital comp ute rs. One of the rollin adva ntsgoa of t h is t ype of des ign is t hot it e nables nunl y ttcs l est ima t ion of t he ext remum for a certai n class of probl ems. Th e problems of the th ird cl aea-c- ap prcxlmation pro blem a-c-are rathe r oft en dealt with in t he an alys is of d y namic an d sta t ic modes of c perauo u of e lect r ic machin es . One 01 t he spe ci fic problems of t h is class relates t o th e lnveauga t ton of tran sients in ind uct ion machines supplied from nonst nueolda l voltage sources (see Ch . 5) . The eq ua t ions for a n Induc ti on mac hine h ave t he form 01 (5 .1) and (5 .2). It t a kes t he M-220 comp u ter only se ve n mi nu tes t o so lve t lns sys te m of dif'ler en t fn l equat ions invct v lng nons tnueoldu l volt ages a nd Ic urh ar mon ics . The e pt.imize tio u prob lem s call for T'llal< in g 3{Xl 11.500 s uch so lu t io ns . T h is proced ure is not always real iza ble on d ig ital co mputers. T he way nul is 10 re place t he sys tem of d ifferen t ial eq uat ions bj' a s impler lI od m ore conci se reln tt c n r nn me ly , II pcl yn omial. T he bas ic Ient ure of ED as. regar-d s l it", above class of problems is tha t. th e equa tion for t he objec t of interest en a bles its r igorous m lt l l!emeu cs l treatment; ill ue expli ci t form. t he equa tio n is cu mber some a nd inconven ie nt. Th e eq ul> t.ion i" solva ble with th e ai d of di g ital computers or ot hl;lr mea ns . I t u niquely deter mi nes the rela ti un between the vnria ble pa rame ters a nd objec tive func ti ons (thi s mea ns th e t t ire object under s tudy is sta ble) . f or t his cl ass of p roblems tho term -experfme nt a l design' is r epl aceab le by t he t erm -compu ratio n01 r1 \;,si gll'. Specific 1.0 th e t hir d class of pr oblems is the u se of desig n matr-ices (or perf orm in g tht! com put a t io n wh ich permi ts re pl ncingt.he comp lex mathe mn tica l modo! by II sim p le polyn omia l mo del. T he accuracy of a pp roxima t ion is the ma io requtremont here: t he requirement for m inim iz ing the num ber of des ign poi nts is no t rl l;:id . ~1T>~ i de r t hl'l s t11leme nt or t he pro blem invol vin g ure ss timnt lon of a n npp roxi mill ing pol ynom ial for l ire speedup ti me t u ' of an inductlo u motu r w it h a 20 % dav iat.in n of nrc v etuo s of th e Icjlowtng qu anti tie s from th e nomi na l v al ues : rotor res ist an ce J: l â&#x20AC;˘ sta t or rest sla nce 7 2, II IH I momen t of lnor tla , .T~ , T he mnl ri x of the CFE. 2' deaign is ~i "I'l Tl i n Tnhl e H,i . U n der t he exper-iment is mea nt here t he solu t ion of d rtreeeuua l eq untto ns (:' .1) a nd (5 ,2) on a lJil{ita l com pul er n t th e Itxed va lues of (110 machine parameters in sec orda nce with t ho des ign mntt-ix . Determ ine t he pol ynom ial coefficie nt s from corre sponding des ign formu las , Since the repe at cal culn t tnns of the machine rnet ors at. t ile SlI lIle va lues of paramet ers gi ve the snme resul ts , th e ex per-iment var tanco is o' {I. p } = O. an d. hence, a l l coefficien ts of the poly nomial are s ign ificllnt. A check of, the model for ndeque cy agllimlt t he F- tes t ill vcr t tces of th e Iee tc r space can not be cer eted ou t. In snlv tng th o e pproxt mntlon p ecblc ms. t ho check of pofyuomia l ade quacy fOIÂˇ l . ,> is mad e b)' selecting- othe r poi nts in t he faclor space .

11.2. f he r echn lq",.. of fo p e,;", e nt.,1 D.,;g n f ebl. 11.7

Vod al", olll,..e'po,. ·

1~1 g'"l, 1,,0\1') .1

Jl ~e,ern:. Ul' p•• ln .\ncl~\ 1 + Lo.....

le... l

J' ..~1 8

3 . 08

13. ~71 •. 28 2.116

" . {Il,

h I m " 1 . &2 X 1 O ~ 0 1.82x lO~0

+ + +

Test No.

, s

"I'"

"''''

+ +

+ + + + + +

Ol>J.... uve

t unetl on

1. 22 >:10-0

2 3

" In

+ +

+ + +

ro 4.~

3.'

e.s

+ +

As is k now n . i n per form ing the ex peri me nt s fit ver t ices (Fig. 11.5) . t he Illfges t d iscrepa ncy be twee n t il> a nd t op occurs n t poi nts 1, 2 , 3 . 4. a nd lit c~lIl trlll point a. T heref ore , th e chec k for t he model a deq ua cy sho uld be done at these pctuts using Cochra n's test on t he homoa gene ity of inadeq uacy var ian ce

ast tm a tes:

.-.

o,na>: = max (S; .aV L: S~ "d S~

( 11.25)

ad {l op} = (t ,p q ill t ile inn rlequacy VDna nca for t he 'lth check po int; lind k is th e num ber of chec k points. ~ig. 11.:;. The cha rt for perf <Jnl'llng cbecke It ven lce$ If the foun d v a lue of lJ m u is ema tlcr t han <Jor ta ke n I rom t he table for correspo nding' deg rees of fre edom . "I e = t and "to = k , a nd for the chosen significa nce level q (com monly equ al to 5%). we l'lC,c ~ p t the hypothe sta of ve rt euec homoga net ty. Nex t we make lin where

_ t,p 9)'

•

ct>. I I.

Appl i t~li o "

of " , pllri me,,'.1 De . ill"

<'51ImRI e of t he ge ncra lteed VtlriADCe of mode l lnadeq ueey:

•

".... {t .p ) -:z S~ ....l k

( t t. 26)

~ ,

H t he ,·, Iull of S {t op} satisfies t he Accu ra c y required of t he IIp"p ro x ima ti on, th e poly nomia l 'Is adequate ; if it does no t. we should pll M t o t he des it us of higher order o r reduce the iOWl'3-!! of variation of tbe factors . S ince t he lim its o'n the numbe r of ch ecks in per-for m ing the co mp uta t io n! arc POt rig id . it Is advisable to t<lke .add itional po int s inside the- factor space us ing for n.e purposo I be t.."l blll of r nndom numbera. AppL)'ing t he ED t echni q ue 10 th e so lut io u of a pprox ima Lion pro bIema , it should be ke pt in m i nd t b",t a n)' npproximlltion, wh ile o ff<:'ri n{: t he s im pl icity of Iu nerfn nal re la t io ns, i nt rod uces a dd itiona l erro rs 10 l ilt' CA lcu la t ion results . Whe t her tho ED tec hn iqu e is experl jent for use ill t he so lution of a pproximatio n pro blems is est.inulled for OAch pa rticular case.

11.3. Transition from Experiment.1 Design to 'O pt imizati on' T he ec l utlon to Hl lISt opt im iza t ion pro blem s ill efec rr tc ma chlner y p rese nts d iff icult ies s ince the f unc t iona l rel a u one between the vartable (>l1r nme l llrs o r a ffill chi,le a nd i ts opera ting tee ters t.eve a n im plic it Icnn . Th, E D techn. iqil.t~ .fub~ t(lfI / ill lly f adlftate. th e wln/ion of oplfmtzat/lm probleme , for it enubles obtaini ng un erpticit: polynomfal " {at/oil bel/wen the JlVlrll ill/l'$ "YJr/ abl, paramtllertf and object l ~ ! U III: tiOI1I . It. th us becomes possib lll to 'sol ve op t imall y a nu mber of pro illem s by Rllll ly t ica l .me t llods . If t ile S<lIUl ion or optim itfiLion pro b lems is sough t by nu mer ical m et hods , pol y no m ial and two-dhnct l! iolla l sa cuo ns cOlls l ru ct od with the n id o f t he pol y nom ia l~ offer ~ we ll-gro unded a pp rcac h t o th n enoree Ilf th e numerica l sea rch method . T ile cho ice of t he sea rc h me t hod is all ilUporLa!lt a nd urgen t prcblem . The peeeuee of opt im um clesiqn oS electr ic ma chroes ShOl',S t hd certain 5I!-<' relt me th od s i n cur re n t use eIlnoo t be ee us tdared on tversal . R a nd om Wll ik (ra ndom sca rell) Inet hod s e re a ppl iCAble whe re tho n umber of va riAble pa ram e te rs rs large . but the y offer II low pro babil it )" of o bta in ing global opllm• . Gra d ient methods a nd alternati ng d iroet ion me lhc dli lead ra th er fas t to local optima . but they requi re a p relim inAry ex per imen ta l adjustmen t wh en so lv ing specific preblems . n eluQtio n methods of th e GauM-Seidel l yptil (eoecesstve d isplacem(' IlWl methods) show algor ithmic si mp!i cilJ' , but fBii to work in t he " va lley" condit ions lind un de r no nl tnea r eonst ra int s , LQCtl I

coor df na te lllE' thodiJ of d y nami c programm inc a re free from t ho shor lco m i ngs of th e G Oliss-Se idel me t hod . H oweve r, tile )' \l\k e a rel ati ve ly long sea rch ti me a t a hi{::h accura c y of t he soreuon. In each particu lar ease. tb e e bc tcec f th e me t hod must be ma d!' alter II, definite lIual y!is of tb e Iaeto r ! ptIce reg ion wit hin wh icb th e scluttou ill to be !Ough t. Such tID anlll}'sis ca n be per formed by con sln lct ing a series of tw~d i mens i4n al sec tio ns wi th t he a id of poly nom ials. B r ap pl y il1i l ite E D t ech niq ue it Is possible 10 obU ln Lbe re lat ions between t ho mot or opeee t tng fact ors lind VAr ia ble pnt:lnH· tcf'!, such as ro t or te .!llslllnce R' and sta tor resistance R'. T IIII .!'. for 1I.e A-42-6 mot or

'1 = 0 .851 - O .OI~ W - l).0221l' cos Ip = 0.83 + O,O:'IG,W - ,O.OI!lR· + O.02uWR' t•• _ 6.5G - l."" W - 1.09R' + 0.S IR'R· / I m = 0,38 - O,o:·t lW - O.04I/l· M I ", = O.s;-i -+- O.03GW - O . O~() W O.OMWIl·

(1 1.27 )

Til l' llbove po l)'llOnJinl eelut icns a re rather si mple s ince th e prc bkn. bC'i ng so lved invol ves t he \'a r illUo" of on ll' 10'10 va rtaule paremet ers . It eculd be lIS5um('d t ha.l t h... !iC'1I 1'1:1t for 11 0 opf im um u nder tltl.'H co nd tt toes d oes not {'neou nltlr dtrnc untes. Howeve r , th e a ne t)'!li8 of va rtan ts of the scluttons pro vides dUft>l'e-nt resul ts . T he Inost spec if ic eeses TOc t with in o p l im i~/I t io n of e le.:l r ic lillichtnes resu lt from II sha r pl>- red uced rl'~i oll of SHll'I: h . T hE' l onrli in whi ch lh ~ o bject i" e Iu netio n sa ti s.Iil!s Ihe imposed eons rre lnta a rt! smnll . so it is d ifficul t, 1.0 o bta lu s uch to nus for t he orga nization of Ilrpwi..o procedures without constroc ti ng I],Cl secuone. T he solutiun of opU mlzn t lo ll pro h l elll ~ of Ih is elMI!:I prose ut s d ifficu lti es IISN)Cilll ed wi t h t he cho ice of t he re rerenec point »ccded t o inlt la te th e di roc tod S/J llrc!, . T hi ~ is e tsc l,he CI\!IC for 11 lorge cla n of pre blom s tn volved i ll t he a na ly s ls of llle ctr lC mnehtnea. Wlu'll ncccu nts [or th is f/lct is t lllll a n ind ucno u machine liS t he ob jec t un dor n n(l lr~ is is II fllir ly wct t et ud led objec t lrom bnt h t he l \;eol~ U r'lI l s ud ti le pra ct ica l am ndpoi nt. T he mnch i nc, In se r ies pr od l(ctioll /Ire ro ~hcr etose t o o pl illlni o oos iJ II", Imp osed co nst rain ts are tak e n into accou m . H e nce. thu poss ibiUlil.'s 01 impro \'ing II.('ir CMl'llct llristks lire noL large., ,, 1111 so t bo ec ne or pcrralaaible sea rch for t he so lu t ion is small . The pol ynomial re ln l ionll a ud op ti mum sea rch ZOIl('S obla i llc{1 h)' th e ED technique u sist in choosing th e rt>fcre ncn poi nt needed for \I,e ol'lrllni l.o'ltion of the directed search. Problems lire me l w it h In pra.c ti ce in which the sellrch r,,:::-iOIl is d lsco aun ucus, III hllutlling t hese proble ms b~' nu meri ea.l Dl(l th od ~ , t l,en) Is a ri sk of m i" lJIki nJ: I lle l)Oint of to pa rt ial ex t ramum for the op t imu m peinl . The 1I 1ln lys ls I l_~ l l a

210

Ct>. 12. Syn lt>"'i, .. I Eled,i" ,....,ct>in• •

or t wo-di meus te na l sect ion zones helps t he engineer avotd th is mistake. Appo udi x IV prese nts t he ass tgnmo nt fOJ' the solu tio n of desig n opttmtam.Icn probl ems fO f an induction ma chine wit h t he a id of ED tech nique . App lying t his 'techniq ue , t he anal yst can tra nsform the gene ralieed m achlne m odel eq uauons i nto s imple polyno mi al rcteuons between t he c pera t.lng factors of II mach ine a nd its parameters . Th e ED tochnlque al se «ucwe ono to s ingle out t he ma in a nd secondar y foctors tha t affect certain ope ra ti ng cnerecteetsues. By use of t he ED technique . it beco mes possiblo to co nst ruct hi gherorder mn t hema tica l mode ls of enl.' rgy conver ters a nd pass \.0 geemutrfc progrnmmi ng .

Chap ter

Synthesis of Ele ctric Ma chines 11.1. Optimizat lo.n of Energy Converters . Optimization M ethods The objective of t he the or-y of elect r ic maeh tn erv is to lind t ho wnvs Ior the evo lutio n of now ellergy conver ters a nd o pt tm taa rton of cc nven uo na l t yp ClS. T he des ign of II ma chi ne Inolud ea th e stages or computa tio n lind enginee ri ng development . I n the gener nl case, eompu rat ton repre sen ts :I mat hemntlcafly indefin ite pr oblem wlt h many sot uuo ns beca use the number of un k nowns ill the problem is greater tha n the number of oquations. Th e wny out Is -t herefore to prese t the va lues of cer tain quantiuies reaso ning from t.he exp erience gllined in eteernc ma chino engineer ing . Of a few desig n versions, the most adva ntage ous. optimum one . is t hen chose n . A cr tter-lcn of' optim iza t io n ca n be 3 mi nimum total cos t required for t he productio n lind cpe rat.ion of a ma chtne, a minlmum mnss and toSl of tho ma ch ine, fi nd e ther- factors . An opt imi u t ion cr ttorte n depends on th e field of a pplitnt ion of II particular me chfne. t. ime, cos ts of materials a nd e eergy, etc. The se fa ctors complicate t he task of seoking a ll opt imum des ign . T here is a grea t num ber of methods of elec tri c ma chine deaig-n, Ea ch me t hod re lies on cor ta m mathemettcaf mode ls interrelating inpu t an d out put chere oterfstics of a machlne . Most me th ods usa des ign e que t.ions obtai ned both a na lyticall y a nd empir ica lly. Scmo methods employ equa cions deri ved from equi....a lent ctr cun models. In t he Iast yea rs a t tem pts hove been mad e to use d ifferent illl e quatio ns to Icem a mnth ema tica ! mode l for des igning energy co nve rt era,

12.1. Opl l",lu I;Of1 01 EC.

Oel;mj~"tion

M.n.oa.

The use oj dl//rrentilll equoUon.f 01 tlrdromn:Jl(lmcal energ!l eon ver'ion fo r dulgn purfKJU~ olfers 0 means 01 conllder/ nl! bofh :Il(d ff: un d

dynamic chQrtJCt"/slfC$ 01 ecs. Computi ng fa c;i1itiE'S m ake it possib le to establish t he chllraet",ris liCi o f stu d y-st a t e OPl'rai ion Ill> II pe- ueul er ease of t ra ns ien t operation . Th us it is posalhle to proc:eed wt t h dy na mi cs no t fro m th e eq u ivalellt etreun. t .e . not fro m t he pllrlir ula r t o t.he ge ner a l. bu t from t he ge ner,,1 equations o f c ne rgy eo nve rsfc n t o sta t icl'l. T he m od el s being se t up in thi s manner- hav e br ough t Ic r t h th e pr ob lem s for s imp Ufr i nU- the rullth p.mntica l Ana lys is of onorg)' conversf o n a nd m ade il necess ary 10 In tr od uce th o notion of specific p ower. It proves i de ntlcnl t h ro ng ho ut the air ga p in p rope rl y d esi c ne d eleelric ma ch i nes lind cnn pro ba b ly se r ve as II ba s ic qu a nti ty in desig ning ECs. T hE' p ee ble ms i!,,'ol " ed i n t he c p t lml ea tlon o f EC5 genera II)' ha ve seve r/ll so lu t io ll8. Tht end pursued by opttmLz ation is to uek tM best solution among man ll pcl~n ttallll poulN~ wlu.tlonJ. A n IlDllmbia;uous probl em doea not n~ o p ti mi zM io n . Optiln izat ion rt> li es o n m a ny met hods , from rn t he r ( oDlplit n lcd a na lytic lind numeriral llp proll chel t o t hose be sed (II I Illm d ee m p iu aUOIl. Rega rdi ng It s m athe m attcnl s t e te me a t , the c p t .i m iaa t.in n probrem red uces In th e pro blem of non l inea r progrllmmi ng . I ts Iorm u lnt ion CIl II be flS fo ll ows : find t he e x trem um (the ma xtmum or m in im um) of 1\ uennne run eucn F ix) dependent 01 1 va r ta bles X" x .' .. .. X n on co nd iti on t h al the t wo k inds of cons t ra ints a re sa t isfi ed: CI ~ z / :S;; H I. !,.., 1,2 , • . . , /l. U, <, lJ'/ (i) <. h J , j ... I , 2, . . ., m whe re ii is t he ve eter w hose cecr d tna tes II re~, x • . . . . . :I: ~ : If, tX) re presents so me co ns t ra int Ju nc tio ns ; a nd C" H " II" h J lire t he q uant.iti es defi ni ng th e bounds 0 1 11 pe rmissib le area . T he constra in ts in t he d ~i g n s t~ o f e nergy con ve ne r s an! tho ones Im posed o n overheat ing, cost. dimensi on s . ete . T he witlellpn:'od meth ods t ha t a ll ow for co nst ra int...' are t he follow ing . 1. T I,t' mctbcd for de t erm ini lli bou nd ar )' va lues of vn rre bl es , whi ch ls ec nvenle n u for us e in g ra d ie n t a nd coo rd t ne t e-see rc h op t iroi1llli o n pr oce d ures t nv o lv t ng th o c1msh·oi nts lmpoaed on i n dope ndent var iab les . 2. The pe nal t y f unction method that allows for inequa lH y CQnsrremts a nd rt' liu o n II nE'W run cu en J IX:,

u. tl "'" F (i, u ,"t) + P (T, ii. t )

(12. ( )

WMCfl F (z , U, t ) is t h e ohjeeti vo fu nct ion unde r I nll}-s is ; J (~ii. ij is I ne wly derived fUJlc t io n; a nd P (7". U. I) Is t he pe nalty ro e c uon . T be sig n of a pt'n a lt y func tion depends on t.ho COUr&> of the solo-

212

t io" : t he fu nc t io n is ' Ii.'gllt iv(l in tl'liling it Icet he max imum . 8.1,d postt ivu in l el!Une: it for lite m illim mn . F,nch ptI11 llt~, fu nct ion pro,' ides II IIUIII('I'ic,,1 np prcach 10 t he di l'ltC t scluuo n of tile prob lem , a. T he LlIgl1lll11h, n multi pli er method wh ich rat her of!i ciolltl y all, ....:! for eq uali t y ce us rrn inta. T ho meth od use s Lagrllngi lul muJ t ipli('. rs to introduce t h('. COlll! t rn ints into th o Ill'w l)' implomQn w(\ fu netio n. T ill! ext rem um of the 1lC1!' fu nct ion can be so ught b)' e mplo)'j ng stllllllllrd method:! or mll lh~m llti clli p rogTlmm i n ~ , L\\Sl! t1 ol inl: WIl ha ve defined uie pro ble m su bject to optim i1ll.tioo. we Cll 11 choose 000 of the general opti mi ut ioll methods. T hese i n. etude th e mctl.ods of in vesti gation of va etous dl'si~ n vers too s . wh ich i OlVO'VO th e u a lr s is of II few peestble S("jlut ions of th e sa me problem w it h Iho n itll of fimli n!:" the bes t: tn :pcrimonhll .ne lh odll w ith ou t Iho s i nd )' of corNSpontlillg m l< lhom lll i c ~ 1 ' n..d eb uf t he o bjec ts of i ntc~l! li "mph it'lll met hods based o n t ho I:rl'l pldCll 1 rep rese n tat ion of t ho function be ing opt imil M a od depe nden t on one or two va ri ab le!: lllllllrlic al me t hod! ruling t.ho e tessrca t pri nciples o f di fferl' nt illl o nd ,' nri llt iOll111 edclI' us; lind numer iclIl meth ods . For llu~ Solu t ion of cngineer illg pro hhlms involv ioe non li llf llr rela uo ns. il ill expedie nt t o U.IlIl' uum er ice ! llIt'l hods . TI,o fl ~ l npprMc!ll,\" to op t inll Xll l ion of uleetr-tc ma ch ines ha ve rr uuerly roli'HI 0 11 t he met hod. of seq uonttn l t rnc illl! of ."p'ld a l erld pouus, or nodes . III t lli:> met hod , Ih o per missible lIma of vm-iatlo n or each or l lll' facto rs is hrokoll down i nl') In I'IH:sIll'.'1 w it h n dcf fnite s te p 1I0d n n op t imum \·('.l'Sio n i.~ SO ull: hl by II'lIci ng Iho gr id poi uts s te p l,r s tep. T h6 me tho d , lwill:;:: \' irlllully ~ i m p lll , In-comes rat her clU !l borso m ~ "·III! nu tncrceee in tho nu mber of des i1;:"1l Vll r i;lblos lind {I dCt'l'('lIs0 In t ile slo p lengLh. T ho t ol nl num bt' r 01 LlIC sclntloua I,), 11,ls met hod is eq ual tc a pPl'o)(i1na ll,I)' thc pro duct of the ste ps mllcle for /III \'/lr illblu. t.e . III) X mot . , , tI'I" . Th o met hed co nsumes" fll ir l ~' {:roal dellI of t ime . 'I'IUl- next melhod 111.:lt ha s received wide l\ppl kn li ou ts t he GIlIlSSSc idl'l re la xl\li on (succl'SS ivl'-d isp l n c~ tne n lS) 10..11101"1. T ile idea of l ire method lt es in t ile seq uential S('u d l for til l' pa r ti nl ('~I r,.. m u m of tloo Olltp ut tueeucn for {'a d , va elable of t he sy ste m . l 'lIe l'Carr h is t erm inntcd .....hen t he 'I\;'x\ point of the s pace tu rns out to 00 the er . tr cmum in 1111 (oo r4 in/lle di 1'(>{'1 ionl!, Th o met hod is effic ie nt for IIpp licl1 t,ion 10 Ihe objects in wh ich tl,e (Om l. l io n betwee n t udept'odcl.t \' rori /l bl e~ UOl.'$ no l c~ L"'t . T IIi' sil~ l e lr:t\'er:llt' throll~b . 11 " lIrillhles ca n t hen lend t o t ~ sohu ion of t ho posed prob lcln . •' or t l'o objarts ~'i t h II l orge num ber of ,' a r i/lhiM whirh I' orrelll l(' wit h ea ch other , the met hod peev es Ince uv aut eut , T ile Ine lhod of t he s tee pest s lo pe or stecpesi. descent (hco bi's nlulh od ) ilf a versio n 01 th e grad ien t method . The d irl'Ct io n of senrd . lM.' re i:< oppos ite 10 tim di~tio n of th e grad ie 'lt . If / (.l"n"'l" ) < / Ix,,), t lil' nll nimh al ,on procedure rcuows: e ach ~ I C P is t ake n i n th e SOllla

213

direr t iQIJ unu! f (;>'·. + 1) t ur ns out. 10 be geee tce t hll 11 I (x~ ) a t :Il'~r l:lin s tep. Afttor rc gre!';!iOD 10 U..e potn t r . a nd calcu la tio n of the ~rll,jiel\ t ' PeW. UI(> search ill con ti nued M before. T h is method is m nr o r.Hidell t tha n t he gnldit' n t m<! l hod but becomes i no~ r1Ili ..· o ill lIK' pITMI nco o f inl ernal In te rfer es, T h is demerit li 'nits UIC Use" of tile IIlt'l lioct. part tcula r.ly lor o ptiro iUl tio ll of ml ilt ip arllmetrie o b jc<I.!!'. TI,,, met llod of eO llfi:~\Irll lions C'llsures II. s ueecss rul seareh a lo ll£! 1110 edge uf 1'I sha rp rid ge . Thtl lll gQril l'un of 8ell r c:l, co nsi"is i ll UIU foll owin",. At the fir3t s inge th e illi lill t v nlues of !Ill desig n vllr illbles :rJ' iud ul.l i olf the iui l inl i" c....m enl tl z J•.am .set 1I11d t he v31110 of I (%) ts Inund . ElU'-h ve r te ble is t he n mad,. 10 vnry c ycl i~I I I )' by the \'aIIlO of the a do p ted i nure ment in ord er thllt All the pnr llmlllers m ight thll llg" . H . tor cx a mp la , t ho va l ue of z ' _ ~ -t- .... .t. does nul IIl'H.l to 1I decrea se ill ti ll' ohjectivo fu nctio n (\ /1 solv inq tho mi ni Fll i~llli\lH proble m}. J'~ i ~ v/lr ied by - Ax , nml f (;,:) is ag~ i ll t es ted na te rorc. If till s "lIrill lion dOOlf not " Hll r im provement . i ~ kept conSlll llt. Th e llt' Xl si nge invo lves II v a na uc n of ~ by 4 T, . mcu 1'1 " nri:lt iOIl of t he ."til:r~l:'S "·e. "lIria l,le lind so lo ng. l'j {: rch~' d nw gin g 811 110!1i:lll Vn r illb le.:! lind com pleti rlg t he first c ye te of l ho in vesti ga t ion. At each !'ltlp mnde in the II n lll~l!- i s of 1111 in depe nd en t " , r ia bl e tile va lue of 1 12') is co mp ared wi th ils va lue at t he prece dlng poi nt . If the objee l ive funct ion dec reases a t a gi\'e n ste p. it s preced in g val ue is ehnn:"ld for. lIe w oue II lul USil'd li t Slbsell "tHll :step" for c(lm p.lIri!i(HI. To s pee d up the 5('on: h ror the optimum , the length of s tep ~ cAn be c hllng1!rl b ~· ~.rH xing II eeetatn multipl ie r ;.} > t 10 the q\,an t i Ly . .t Wh lll s hould be rt'g8 rded;os lhe m a ru d iJl::ltlvlln lagt! of lids me t hod is t lLli l /Ill te nw uve s te ps lire t ak en pnmllel 10 cec rd tna te lI:tC!1 nod no infor mnt iu n abo ut oll. er di rec t ions is ;'lvlli/oble l!O it is quite posstbl e tn mil!s t be ..idp . Tlll>r/! lire 1I 1.w me thods bnsod on qua dr a tf c co nve rge nce. Fe e qUlId rn\ ic Iunctrous , L1le.!l(l. me thods enable fin ding II minimum tud ependent of t he refere nce po int rQr a umned numbe r o! iter a t fons . For rlln ctj OI1 ~ whi ch e re smenoblc to:' higll-dl.'greo of upp ecx tmnt tcn nenr , Ill' extrem um po in t by I II"" 'l Hltd rll llt: dependence, meso metho ds ensure quick con verge nce , BUl lhe~' ca ll for th o enlc uln ti o n of «ot on ly th o fir s t derfvatives b\ll e tsc th e second doetvauves. I II !loh 'ing man~' e ng i""er ing prob lems, howev er. the cldc ula t ion of pa rti l\! derl va tiv{'s bo t h a na\ ylicn ll y an d nu merica lly presents d iffi culties . T he abo ve descrtbed nu mer iea l met hod ' o f seArch for t he ex trem um ean he pllllced in ll! th o gro " p of determi n istic melhoos because t he di rect ion of se;or<:h ""it h t hese method s is unlq cel y defined b)' t he logic o r t he seereh proce ss . In t he pr acuce of o plimum des ign of e lec t ric mach i nes Ilnot llt'r gro up of In('th Qrls filld~ wi de Ill!!!. These are s ll")(:hllst ic (rllndo m) meu.cd s in wh!..l, the d irect ion of sea rch if! Cho.i6n q uite rnno.l O'rlly .

2':

'14 If tllll ohjoctive fun ction sho ws a n i ncrease i ll value ill an y one direc tion. t he search is carr ied ou t in t hi s di rect io n Ullt il t he ex rromurn is lou no . T hese me t hod s prove ra the r effi cien t fo r t he search of 11[(' ex t remum of t he fu nct.lon whose behav ior is not k nown lit all . H owever , th e r a ndom sea rch does not use the tn ronoauon on both t he pr ev ious walk and t he beh av ior of th e h yperpla ne, whi ch leads t il n co oeidem ble loss of time . Th e me thods t hat show most pro m ise a re t ho DUel' wh ich successfll ll ~' com b ine l ite elem ent s of dete r mini st ic mot hods with thnse of stecheeuc moth cds of search . O ne of those is the com pl ex me th od . Th is method Is a mod ification of t he simplex method i nve nted b)' G . B. Dantzi g and ret ains -tlje ba sic pr tnctple of th e la tter . The com p le x me t hod uses N P vcr ucce (P ~ 0). each of wh ich must sa tisfy constra ints a t a ll K sta ges . I n t ho perm issi ble a rea of th e teet er s pace t hese ve r tices lire se t up in 0 ra ndom manner, foll owing w h ich t he va lue of t ho objecuvc fllncli oll is found lit each ve rtex o f tho complex. T ho vertex at whic h th e fu nct ion f (xl has t he worst va lue is rep laced by a now vertex l oca ted o n the line t ha t passes through th e rejected pninl. nod t he ce nter of cluster ol the re ma in ing vcr n ccs of the comple x a t a d is ta nce equ al 10 or gre eter t hft n the di st ance [rom t he reje cted point to t he ce nter 01 cl uster . If it ha ppens th a t th e new ver tex has t he worst. va l ue as aga tna t. tho se of t he venrces of t he new complex , a DCW vert ex is set (I t a hett- d tsra nee from the wors t poi nt to t he best ver te x of t he compl ex. H t he sea rch is success ful. the com plex ex pa nds a nd deforms ill the d tr ecuo n of t he ext remum . The process of S(lIH"C,h oon t fnuos \lllt il t he com plex is d raw n logeth el' a t the ca nter of clust er wi thi n th e li mi ts of t ill! s pecif ied acc ura cy. The me th od en a bles a successful! s olut io n or mul t iex tr em um pro blems. I n progress toda y is t he develo p me nt 01 new opt im iza t ion me thods . However . t he num ber of p robl ems to be solved g ro w~ laster t han the number or met hods req u ire d for t heir so tut ion. It scorns unli kely that sometime a u ntverset method wou ld a ppe al' and ed ge out all ot her mot hcd a. More probabl y, tho best opum u ntt on meth ods will be fou nd for t he def in ite cla sses or prob-

lems.

12.1. G eometric Programming Geometr ic p rogramming is a 'lew br anch of matJumat lcal, programm-

ing , u;hkh call ~ used 10 a4valll age [or the soluti on of opei mizallon problems i n electromechanics . T he geometric pro gr ammi ng t ech niq ue ca n effic i6nt l y t ackl e the m in im iza ti on prob lems , i n whi ch t he opti mlllil y lind const raint cr iter ia are ex pre ssed i n t erms of the nonlilw ar functi ons of th e dcrtnnc form . Geometr ic p rogramming i n combination wUh expertmentol de~ lgn provules a powerfu llool for the con3lr lÂŤ'Ho1l of Il t W mat./umlQtiral lTl ooeis ap plied ill t he syn thesis of t ller g y ccnoerters .

In ~peeirie cases. llOOme l rie p rogMlm m ill g ena bles t he l!lliu t io n of p roblem e IInAlrl il.'IIIl}' fo r qui le 0 d al i nitl,'l elllJl8 o f f u ne li" ns of the form

•

6 ( t ) ""' ~

•••ud' )

U,(t) ... C, t~" I:h . .. . , whe re g (t );

Ii ,. ",_ .. "... ~

t':.! ..

,.r", t ill:' posit ive cc mpone nre of t he f un ct ion

£.\. Ct . .. ", C. (lI: C Jl'fls ft ivc ee ust nnt ecer ncte nrs: 0.1/ lire er bltrllr)' rell l numbers (j _ I . 2 . .. " n: / _ I , 2 , . ", m) ; nnll I" .. _. t", are pos tt tve independe nt vnetebt es . Suc h f",w li on.~ lire knuwn es /lfMitilil' polyllnm ia/3, or posynomlo.!5 for sho r t ( nunlinear polyuomluls w ith po ~ ili ,'o coef fici e nts ), nnd form t il", bas is of th e !lr0l!'rtlm m i ng tcchnlquc , T ho bnstc requtre Dlent o f rho tcehutq ne c o mes 10 t IL" j'l'p m selll nl.io n o f t he fu nct ion s

'I'

und er IInnl ys is by t he Lin ear su ms of posi tive com ponents Uj _

I II e ng i' ll'er inJl: pro b lc m ~ t l,,, fun ctio llS lire ofte n e xpresse d in implic it for m . By per r<lrmi .,:; lIpprOprill te t rllnsform a t io ns , t he fnnr t inn ca n be C:lll t i ll t l,t' rorm of II po5yn om iol t o e na b le t he eoro(io ll of lUI tlrJillo1r~' problem of geo met ric progt"lI mming . The rUn<tion of the form g (t) _ 1 (I ) lq (tjl-h it ) (12.3) red uCf'!' to Q pnsyoom illl (12.4 ) g (1) _ I (t ) t: h (t ) w ith th e /l id of n n lIddi ti o na l ind e pe nden t vaelahl e to 011 cond itio n tll a t

t i ' q (t ) "";;; 1

I n t lte n hove Ic r mut a s , / ( t ) . q (I). lind h (ll e ee POs~' I\om i ll ' S in t ILt! in de pe nde n t \'''' ri lllhle; 1 _ (t l' I , . , ... till); a> 0 : nud T _ (In' II ' I, . . . .. t rs III mi n im izing [ t ra nsformed ) vecto r v nne b le . T he Iuuc rt cn or t he Iorm g ( t) _ f ( t) - u (t ) wh ere u (t ) is II single- term posy nomial. al so bel ongs to lI,e c1a s.~ of p r'lb le llls so lva ble b r t he l:iven (eo: h niq ue because t he lIlin im iZlll io n of t he Iune uon (12.41 is e q ui \'a le nt to t he m ini m illl t ion of t he June t ion H 2,:;) g('f)_ II. s u lljeel to the constrai nt t. fu t l) + f (t )/u (t):O; t where M )

'f _

(I ) -

(t ),

t,.

.. ..

1",1

T ho slIbjee t of iovesliga l ion in i,'t'om c-t rie prog ram ming is the ex press tc u

(12_6) wh t'l't! qu (l ) a nd P u (l ) are posynomiab; and " o nnd b u are pos il h-e co nstan t qu a n ti t ies _ Tho s ig n-cha ngi ng property of 11 poliW Domial leads 10 /I. \-nr ioty o f co ns t ra ints _ ine q unl ities. Consl ra in \.S are ure s ubjec t of i llv{'sU&,a.l ion in in verse lieomeLriG p rogr nol min g whi ch is not II PlIrt of COlll'OJ: p ro&,rllmmi ng a nd th erc!or o most of t he import a nt t heorems of t he origilllli gElvrnctrie progra mming are una ccep tahle for tid:! t~-po of p ro~rll llJ m i" g _

Th e geo metr ic p roJ~'I"ll m ,uill g teclmlque re lic>:,I c u the etessrc e t ine q uali lY accord ing to wh ich t lltl a r it hme ti c. mea n does not excecrl the ll'tlOml:llri c mean :

6,UI+6,U1 + . .. + 6" U,, > U~ I .

U1 1 - -. U~"

( 12.7)

where l·,. {/,' . __ , U" Arc ar bitra ry pos iti ve Il" m b.!..,o; (co mpo nents of the f lwc l io ns); iiI . 6 . . _. 6" an. llJ'bilr err positive weigb ~ 1I4l isl )' ing li te flOrll1 I1 Ji t )· "co nrlil io n

+ 6 1 + . . . + 6"

For ,'a r illbles III = 6.U, . u t = 6 1 U t __ metric i_qua lity ny u mes t.lte Icem

•• ,

I (12.8) u. - 6" C.., t il", ~

u , +U: + .. _ + u ,,;;a. (u . i,fi ,)6, (~, 6z}~ ... (1l ,, !6~ )6· (12 .9) There i.!l a one-t o-o ne eG"fOspo nde nce be twee n 1I1e w(Jig-IIts 6 / a nd posit ivI' ee mpooe nta of Ihl! Iune t ic n , A L t he point of optimum the .....eight! 6; lire t he re la tiv e va lues of t hese te r ms. there fore t.lle \' l,K:IOrs 6 (0.. ,fit, .. _. 6~) /lllll r( (ILl ' UI •. _ _• 14..) Me pRralleJ to ellch oHler, I n proble m! assoc iat ed wit h COllstn i n\ !i th e pcet uve compono nte of l he norm nli t y vector are non norm (lJit e t1 co mpone nts : 6, + + 6" ""'" J (12.10) No rmll1i1.a t io ll occu rs wit h t he lIid of 111(' mult iplier ,. wh ich is eq ua l tfl t he slim of uon ne rrnalizerl weig:h ls: " = 6 1 ~I + ... + 6 ~ (12_11) whe n' ~ .. Jot• . _ ., j. ~ lire nonnorma l teed weigb ts: A is the muitlpli cr ro lD ti ug each normalized llo't'igh t t o II cor res polld in g nonno nll lllized .....olgllt.; j,f = 18" l = I , 2, .. " tI T he geOI1l t'l r ic i lleq ua !i ly wilh nonllorm aJiud weights bas t ile

IOI In

(u l

+ u l +.. _+ U ~)A ,. (u . , A. ).I., (uz /li J'"

... (u."h\.)"'. A A

(12. 12)

12.2. Ge omel" c P,o g '<lmm' '' 9

217

The pro ble m of gE!QmeLl'ic progrnmm tng re duc es 1,0 th e minbufaarion o f t he obj ec t ive Iun ct.ion g:.o(rt» (u, /.6. , )<l. 1(u21.6.2P'" •. · (u n/An)lI." ,\ ~

(12.13)

StlbjecL t o constrai nts

l ~lk (u) ~ n (1t1 ' .6.,)"'"; I'i IO[~ J

It;

lkl "'" {mit;.

\A/t;

1t;_ 1 ' It;

1n1t; + 1'

• •,

nit;}

(12.14 ) (12 .15)

where k = 1, 2, . . P ill Eq . (12.11j) and k = 0 , 1 , 2, . ... p ; = I ; 111, = n o 1, . .. , n p = II in Eq . (12. 1.',). One of the va r- ablcs 8 1 corre sponds 1,0 neeh t erm Cit; [u (Ill of l ila cons t ra uu Iunc t. Ions (J~ -= L :l, . . ., I' )' Th e constra int func tions g" (t ) yiehl mutt jpl tor s Alt. . Th o c bjecuv o fun ction does no t e ntnit the e mergouce of multi pliora becnuse = 1 lor k = 0 accord ing to t he Ilorma lit y con dltio u. t.c. !J.cC3Use the nor me t tanuo n teete r is ta ken e qua l t o unlt.y. Tbe nO,I' maHt y eondltton represents th e sole PIll'L or li te op l-im lzat io n prob lem , in wh ich th e re is II tus uncl ion be t ween the ob jcctivc fu nc tion Ro (t ) nod posynominl coo srr atn ts Hit. ( t) , k = l , 2, . . .. p . If we ex press t he pos itiv e components of t he object ive functi on nud cc ner ra iut Iunct.io ns i n term s of the Independen t vnr ta bles t J , 12, • .. , t m , the relet ton wilt trik e t ho form

In o

"It;

(t 2. 18) where C 1 denotes the coefficients o n posilive componen ts U I; t 1 denot es posi t iv e independe nt vr u- iahlca : An d D j re pres ents ure une ar combina tion of ex pone nts of i nde pe nden t ve rt nblcs :

,-,

Dr = L Stil l} '

/ = 1, 2, . . . ,

(12.1i)

Hara, a t) rlonot os t he or bit rar y real lIumbu s use d t o Iorm t ho ex ponent rnatrix I au I. T he t'igltt side of (12. 16) is ter med the pre-dlluljunction nnd dl"s i g~ nated as V (6 , I). I t is a Iuncuou or tJl<.\ posi t ive weights an d illrle pe nd· ent vari abl es. T he pre-dua l IunctIon 1' (6 . l ) la kes Oil a mi nim um val ue ......hen th e Iuroa r com bina t io ns a re eq ua l to zero (12 . 18) In tlri s per t tcul ar ease 1\.11 i ndepe nde nt vnrla btes t 1 are raise d \.0 t he aero power and th e pre-dua l I unct ten is o nly depen dent o n 6 1 positi ve weights.

Ch, 12. Synlhesii 01 eled.lc Meehi"."

T he num ber n of r OI\ S of th e ex ponent me t rtx a' l (t == 1, 2. . . j = I . 2, Il ) de ter mines t he di mens ion of t he v"CIOf -0£ expone nts IIn,1 is ill st r ict corre s pondence wlth tha number of terms of All posyuomlals. T ho num ber m of colum ns dete rmines the dim ensi on of the o;!H\C O of the exponents e nd is equlIl to th e numb er -of ind epen dent vari nhlcs or t he ob jec t ive fun ction .

. ., /11;:

the nu m ber of rows is equa l 10 or sm allc r than t he nu mber of

colum ns, i.e. lite d imensi o n of th e ex punent vector is grea t er than the dimensi on of t he colu mn "CCIOf of th e e xpc nc ut ma t r ix, t he -ort hogo nuht. y cond it iun ca nno t 1Jc met a nd , he nce , t he p roblem has 110 sol ut ion. T he q uan t.if. y d =n - (m+ 1.) (12.HI) th~ degree of d ifJ irult y or a p ro ble m . If t he di me nsio n of t he e xponent vector is gre a ter th an t he dimeu$i Oll ul the column vec t or of t he ex ponent ma t ri x, t hen , in th is par tic ular case. th em is an a nal yticnl sol uti on of t he p roblem . wbe n II "'" In 1. thefe exis ts th e s ingle d ire ct ton o f th e so!uucn vector lh at is norma l to a ll col umn vec to rs of the ex pone nt m nt r !x . T he orth ogonnUt y con dition. which forms th e vector suba pace of tim n-dime us tc nal space. uniq ue ly dete rmines t ho n llfll vector 6 withu ut performlog tim normal tm ti nn proce ss . In t h is ca se t he n um ber of u nknown cq ua t lons is equal to me num ber of l tuear e qual ions which lIat isl r th e co nditio ns of th e rlual prob lem (pro gram) in t he form of equali ties . I n t Ill.' genera l caw . when n > m t , a ll th e soluti ons for or t]' I>golla.li ty co nd iti ons defi ne on l y t he d ual s pace w hose d imensio n ts eq ua l to n - til . Exp ression (I2 .Hj) then t akes th e form

'is terme d

L! (6) =- (

n(C,/6 )OI1Ilm\l{ ,\ ~Il

I_ I

(12.2 0)

w h ich is Q du al fu nctio n . and bl' 6 t • . . .. 6~ a re dual variables. Here t wo k inds of Ii lleal' co nerr arn te a ppear i n t he Ir mn of in equel it les and ill th e form of equa li t ies. T he fi rst co ns tre rnt imposed O il th e vect or 6 is th e co nd ition of ,f\o l\ne~a L i vi t y of d ua l varia hies: 6, ;;.. O. 6 2 ;;;, 0.. '. s, 0 (12.2t )

T he coost ratnt (12.21 ) impl ies t ha t none 01 t he components 01 -Ii cn n 00 »ega nve . Accor diug lo t he seco nd kind of ltnenr eo ns team ta, t oo sum of c om pon c uta co rres po nding t o t he pri mnl fu nc t ion is e qu al t o u nity ·('t he unr rnalit.y co nd iti on). Th er e is II th ird k ind of constr aints , na mely , vector orthogonalit y .eo ns tr nrn ts. T he)' ap pear as" resuft o f a pplica t ion o f t he in eq ualit y 11.0 which th ere must cor res po nd the sca la r produ ct of e - dim eneic nal

,,, vec tors. T he o rt hogo naHl-y eeedt uo n underlies t he ~ Ileoriell 1)( dua lily a nd req uires t ha t the vector 6 ~lo ng to t ho s pace ofLhogo ua l w il h respect, t o the colu mn vecrora ul til£> exponent ma tri x . Til e assu m ption is Lhll l t he column vectees of t he exponent ma t ri .:t are li nea rl y Ia dc pendeu r. For geom ct rt e programming pro blems wit h th e aero deg ree of d iffie uhy t he du nl regio n co ntliin s one point . T ho soIut.ion of t he d na! a nd the prim al pro ble m re duces to th e !Ml ution of th e sys tem of Hoellr eq ua ti ous , IIIwhich t he de te rm tne nt is diffe rent from ' .0 (0 . Th is sys te m or equa t ions has a unique solutio n. A ~ II ItoSiti ve o('gl'(\o of di ffi c u lt)' , t he rlllitl r ej;i Ull ha s a h ig h degree of nrbu rert ncss . A unrquc aetuuon of t110 sys tem of Iincnr equat.lnns docs not t hen ex ist. Th e du a l Iunct tc n ill s imp ler th l ll the prim ",1

Iuneuon. A du a l prot:Tam o t reee com pu ti ng a d" a ntat:l!s beca use t he co ns t ra ints a re li nea r; the co nstrai nl!l of a pr imlll program a rt' nonlinea r. Th a t is why th e fi rs t stage of the sol utio n to geome t ric prog ra mming prob1@llls Involves th e esn me t re n of t he du a l veet cr sa tisf~' i r)g tho rnli lli ti o ns of or l hogo nali t)' ODd oo rmali ty_ T he sol ut ion of the syste m Clf Ii llClIr equations

61 + 6s +_ .·+6,, =1

" U61 + o u6s

0",,61 + umt6,

+ , . + 0'1,.6"

= 0

+ . . . + 0'",,,6,. =

(12, 22)

f ull y de Hne! the dt reeucn of t ho du al vector . The du a l vect e r 6' (6;, 6~ , . . .. b;.) "Offers t ile possi bility of ee te emi nh1i' the dual func l ion fro m the ex press ion

v (6') .... (CI /6:t ': (C, '6; )ei ... (C" '6;,)6~ , \~a

(' 2 2.1)

By t he dualilY th eo r y, masi miz ing dual vJtria bles 6; )'iel d min im· b i ll i i nde pen den t varin ules of t he c bjee t tve fu nctio n. ti_ T he refore . Cl t~I< ' :Io

.. . t:. = 6jl' (6'),

i = I. 2, .... n

( 12.24)

Tak ing t he logs of (12. 24) gi ves t h,c s rste .n of n linear eq ua t io ns i n in dependent vllrin bles t,; t he p r ima l {unct ion . T lte sol ut io ll of t beec cqllnLions dct ur m tnos th e so lt ~ l, t- ror up t.imnl vnlue o r t he c bjeeuv e hillel ion . I n a nology to Hooar program min g , the techniq ue o{ geome t ric proa ra mm ing reli('lI o n t he du a li t y t heory. Bu t ui eee is a dl lteren cc, nam el y, in goini fro m t he pri ma llo t he dua l program , t he nonlinear funct ion (12.6) becom es l inear due te th e tra nsition from t he ngio n o f inde pende nt var iAbl('s to th e regi ~o of exponents. The lh('O r y of geomet r ie program ming is!Ot forth to dcscrtbe eo nvex funct io ns de fined 011 n convex se t. Aecord i ni to ti le ba sic t heorem

220

of (' O Il VC.:I: pr ognll ll mill g , en y poin t of II lucrll minimu m of the Iunction is a lso ure point of II globlll m in imum of th o ~iv o H Junctlon. T her efore , t here is 110 noed t o obtatn local extrema Am! com pa re them \.(1 mak e rue ebetec of t he !l'lolJal selut.io n . Tho l,:, si c cr ite r ion of tho me t hod of geome t r ic progr am mi ng is tilt' cbject.lve cons traint . T he objective fllllCli oll , or opt im a lfty erttorin n . is su bject to vario us co us tratnts (ill thc form of equulities lind tnequelttics] im posed o n , sny , t ho cost , rlfm e naicns , a nd overlimIt in dt)s ign ing e lec tr ic rnnchines. I II the tex t above (see &OC, 12.1) a Jew co ust ro m t mathods ha ve bee u brilugh t out . nllllle ir . t he me th ods of hound a ry VA lu es . pe nalt y funcj.lous. und La gr/lllg iaJi ffiu1Li pli o,'s. III geomet ric p<'ogrulJ' m ing, t.he pr nceduro of so l·, ing pra ctica l prob lems ma y inv olv e .. grea t numher of cOl, ~l r a iIl 1 5 which proson t diffic u ll-ies ill 1 1~i n g t h is tec hur q ue. T Ill! ohjo cLive const re ln r, t.e . tile Iuuct ic nal cons tr aln t equi" llie nt t o all i ndh' il l u(l l eonst reints , obvia tes these difficult ies, Tho ohjective cons t rarut is. 8 monom ial O(C I~'lf' "

t~m } ~ 1

(12 .25 )

wher e C is t he coerrrc tcnr of the objcctt ve ~onstrAi nt (C > 0); f , •. . , t", nrc i ndepende nt. vartnblcs or [he s yste m; and XI ' %" ' . %", ar c objlw tive oxp oncnts . II nv iilg q Iunc tinna l constraints, o ne hal! 10 deflue "I' At' , , '. A~ " .. Aq Lnqra ngia u mutt tpl lc- s . E ach Ln grnngtnn multip lil'l' C(,r responds to t he const ra int woil!'lil. T I' e oh jcct tvc co ns t ra fut cumhines a ll colls trui nt.s. so on c La gr uug ia n nll tl t ip li l'l' "'n Mtled An obiutt~ ~ multiplier cor responds 10 t h is cnust raint.:

ll'

(12,26 )

Where (lo t) reglou of sem'l:h for tuc o bjec li\'ll fu nc UolI ts e pofnt, the sea rch for t ho functi on ex t re mu m nnd cnrrespo nding coord in a tes co mes t o II wclt-Iuunded si ng le eom p m a uo n procedure . TI ".. ohject.lvc ennstmint \:1I.1l1l0t n ppl y 10 nlI prnblems. l t ca n u\:t all II useful 1001 in so l v ing problems or th e zero ,lel!l'l'IJ of (l im · culty. 1lI1d 11 1.~O in expc r im enta l Ile.'1ign and goOl1lfJll'ir. pr ogr Ulllln in~ .

12.3. Design of Electric Machines by Geomeirlc Programm ing The ck$tgn of a lt electri c: mndune by gCQm,etr ic; programming begin" with calculaliofl$ based on lhe ex per im ental: de.~ifPl techn ique, The tntti lll stage comes 10 working fllIt th e program by use of th e known desIgn formufos to /($/ th e fu ncti on in tne region oj interest an d dl'_l'iv l.' t he f irs L system of Huellr eqcaucne covc r i ll l{ lilt II pri or i mfurmn tln n

12.3.0.,0;9 " 01 EI., d,;c MClchino o b y Goome l,k P'09'Clmm;ng

on t he mcchtne. Such a system of uquat tons rnllY tak e cus 'll 1'j

= =

Cux , Cn ""

+ C..x . + .. + Cll~x .. + Ct~ : + .. + C 2 " ,x

u.e

221

form (1:!.27)

+ ..

It, = C e,x , -i- C" x . + C cmxm where C-QS 1'. e fficio nc)' '1, an d C n IT<:l Il ~ rnl io k j a re t h e out pu t cha ra c t erIsli (~" of the meen tnc : x" ,xl ' .. X'" a re ll'\i Qhkcli\'o ex pouerua (If uul e pend e ut varfa hl cs in the o bjeat ive cons t ra i nt : Gil' Cl ~ ' . • . ' . C ern arc t he coeff tcie nrs of mgrossi o n; u ml r. is t he ntllull er of ea nst.ralut Iuncu cns of the optimu m d es ig n . At o[Jl imtu n de.,ign bV eeomeir tc programming . the [ unrt ione M e

cast in the form of posymllniak T hus , ,I,he mass of t he sta to r wi nd ing i s fou nd frurn the Io rm ulu

Ge.tl = gS cZ j1c u (l l .) '10-' ("12. 2$ ) Gel' is t he ma ss nl th e t op per wi ndi ng . kg; C is t ho de nl'i l y (If t he st ator wfud in g m a ter-iel, g 1C1ll3 ~ S c is th e num hei- of errecuve con d uc rce s iu t il", s lo t ; z, is th e nu m be r of 5101,5 In 1. 11t~ stutoe; qC.. is th o cross sec ti o n of t he df(' cl ille conduc t or, mrn"; I is th e s t a tor l;ON; leng t h, mm; end I. is the a verage le ngll\ or th e st a tcr-winding face por ti o n . m m . On s u bsti tut i ng Ihe value of t he fll ce pe r tleu IInJ IIlllk lng s tm ple tran slonns tt on s , th e ex pressio n for th e st a to r win d ing Irl Il ~S assu mes t he for m w l ,,~ro

+ gS c'1cukn Yj rtD, + gS <I'JCUkJ' Y1M JI + 2gS J-,Qcu8

Gel> ...... gScZj 'kul

( 12 . 2~ )

wher e 1.'11 uud n a rc th e q ua ntit. ies eccc unt ing for tim geome t r ic dr mcnsto us of t he wi ncH ng Ince portion ; Y, is Ute st a tor w inding pit ch ; D , is t he inner dl mue ter of t he s lh lUr , m m; nndh' L is t ho uesi g n !Illig-lit (If t he s ta t or slo t , mill . Tho etot nreu is givcll by wher e b . is t ha s lot wid t h ; a nd k e " is t he s pa ce Ia ot o r ucc o u nt.ing for tuc dcgcee of filling t he slot wi t h coppe r. The ox pre sslcn Ior t he w indi ng mass finnLl y t a kes th e form

Gcu

gZlkc~ b .hul

+ gr{llllrfL kcub,hu D

+ gy ,:I1fJl kr;ub.h ' l + 2gz, ko:,u b •h", B

T ho effic ie nc.)' express io n (If the rorlll "l = ( I - 'f.P!P I ~O "')

t OO ~~

(12 .:W)

( 1 2 "~'

is noL a lOelUl ble to tes t ing by t he tech ni que of gee metr ic progroro· m tng. Bu t . assnm i ng t hllt I ma s jmnm effi d eTl()' deri ves [rom .. m in imu m 105& , we perfo rm tra nl\forUipl io ns a nd ob l.ll in t he ex pr essi on for t he llli l l lees in \he po!I)'IXJ mia l form

•

~ P"",P I+PJ + . . +P"

( 12.32)

••

""hue PI ' PI ' , , ., p . Ilrt> Ind iv id lla l lo!!!e!l In t he macbl ne. whi ch ca n be de fined in te rms of geom e tr ie dime llllioos . I n II s im ilar wa y we ca n deri ve eJ:prel!sions lor th e machine eosr, hellt ing , nnd ot her fae l ors . De pending on ti le st a tement of th e pm bte rn. th o unknown qu nn til. ies ca n be t he pa ra meters of t he eq uiv el1.'11 1 etrcuu . voh egc, t orqu e . eve. 'fho s ta r ti ng t orque clln be ex pre ssed in t he foll owin g posynomlal

rorm

i /M = ~ .!l + ~r C +....!!LC,r' .J,.. ~ ..t .'

"' ,~l

ri 2I.>t

"',"1

"l,~ i

C,,,,,,,

""ltl

1 I

m,lIf

+ ~ C~ (2j)1

...,lIi

rr (12.33)

T h ~ so lu tio n of Bqs. (12. 27) appl ies to t he 5OColid system 01 linen eq ua t ions whi ch describe t ho link between the e lemen t.:s of tho upcnent Ulatrix of the desi gn and t ho pos it ive com pone nts of the dua l vecto r Glib, a , t6~ al.6.. - 0 a l ,,6.. ." 0 112.34) Gull, Gn 6.

n...,61

+ 11",.6 2 + , .. + a"'ft6"

""" 0

where 0 Il' •. . • 0 ",. tire t he ex po nent s of i ndep endent "Bria hl l.'lI in t he opt imum desigll j lilt 61 , . • • , 6. lite t he Jl(Is ilive com po nen t! of t he mi n im iz ing du e l vect or. T he lIys l.(' m 01 equ a ti o ns (12.34) rep rese nts t he rela ti o n be tw een th l' expone u t, ma t ri x of t he des ign a nd t he posi tive com pone nre of t],tI m ini m i[i ng du a l ve ct or . Th e last ecl umn ve cto r of t ho <'x pon e nt mlltr iJ: resuus fr om t he selu t ton of t he first system of li ne llr equat ions. The i ndepen den t (u nk nown) variables of th e second sys te m of equations are t he pesntvc eomponerus of t he minimit in g d ual vec to r, The coeffi c ients I fli Jo: od t o va r i~ IJles cons titute th e ox penea r mat ri x (12. 34) a nd eette cr t he cnnd il ions of nonnA l it)' an d orthornn· " lity of th e pestuvc ee mpeeents of the dual vec to r. T llis S)'s tcm hilS II u niqu e so lu t io n at the tero degree of d iJll cul\)' of t he p r(lblem . In t h is CUi" t he num ber of Inde pe nde nt VfIr lllbles of the desi a n i. sma ll er t ha n t he n umber of te r ms in t he pro ble m by one.

12.3. Oesig n "f Ele cll ic M..chl .... by G.. om.. j rl " Progr..rrunlnp

223:

T he coeffici e nts on t he variables of th e third sy stem lir e equa l tot ill>' ex ponents of t he Inde pe nde n t varia bles of t he des ign:

+ a . , z. + . . . + a" "'::,,, = 0, + anz. + + a"'tZ", = 0, a l~zl + o."z. + + a",~ lI:", = b", 01lZ:!

a u z,

(12. 35)

where .::" z" . . .• Z ItI ar e th e logs of i. hu abso lu to va l ues of independen t var ia bles; 01' bt • • • • • OM e re t he logs of t he p rod uct of t he posi t ivo compone nts OJf t he mi ni mj ~ing dua l vector lind the object ive fu ncticn minus t he logs of t he cocrnc re nts a ll t he va rln bios or t he posn.i vc compon ents the objective fun ct io n, T he des ign process yie lds fi llll i resu lts by t aki ng the a ll! ilogs of t he results of th(' so lut ion t o the third sys te m of linear a lgebra ic eqn e t tc ns . As An ex ample , cc ns rdcr t he prob lem of sea rch for the geo metric dime nsio ns of a n energy convertor t hnt ensure t he s peci fied si e uc and d yn ami c out put ch aracto nstt es a nd effec·t It maximu m s/ld ng in copper, 'The objec tiv e fu nct io n is g ive n bi' (12.30) for d et ur min ing 1110 mnss of th e wi nd in g. T h is funct ion dep en ds on f Ollr var iables, D/, l, b , . and h n. Deno te the expo nents by X" x~, %• • a nd I t for t he va ria bles b•• h n, I, And D 1 respe ct ive ly. T he ex press lo u for cons tra in t func t ions will t hen taka t he for m of (12.27). The snlution of (12.27) enables us t o for m UIC o bjective censtrnint

C / ( o: 'h~;I"·.m'} ~ l

(12.36)-

The seco nd eq uatio n in (12 .34) will t he n t ake t he for m 16, ~ 16. 1- 16. 16 t ~ 065 ~ 1 161 -t- 16. 16. -i- 16t - X16 1 = 0

11\

+ 16

-i- 16.

16,

-t- 06.

0&,;

+ 26

t -

+ 06 -t- Ollt + Dill + 16a ~ 06. -

0 X;/>, = 0 xt 6. = 0 Zt65 =

(12.37)

Soh' i ng (1,2.37) wi t h co nsideratio n .Iur (12. 27) g ives an op timum. (m inim um) value of th o ob jectiv e funct ion

Geu mi n co (

kC6, "" , )' . (ZkC~I:., 8 )'. 6.

X (nkeuk"t!{,

)'. (:t~CU3l<fll:li l )' . ( .!'.. ) ~· &.ll .6, s

(12.38}

At the o pti m um po in t the i ndi vidua l t erm s ar e equal 10 th e product of the min imum va lue of t he objective t uncu on by the resp sct-.

e ll . 12. Synth,, ";. of EI4ct ti e M .ch; n~.

h e posit ive com pone n ts o f t he mi niltlizing d ua l vec to r b;

<'l ICe.. mho = k,,,Z.lgb,h'11 b , GCU m ld = lkeugz.18b.J!I'

6j:h.• ' nl n

6.G.........l ~

( 12.Y,l)

nk,,;,.kflt:l/tb .h ••D I

nkc."JIBlllbA ,

T a k i ng t ile log of l'- llch of the equauo us ill (12 .39) )' ields In (6IGeu ..,. ) = In (kc.: IC) In b, In h i, ....... In I ILl l6sGclIIlll. ) = In (2kc,of:I IJ) In b. In hI ' (12.1.0) In (liPcu "" ,,) = In (:lkc.,.k/l gg, ) In b, In hI' In D , h i (O..CCollDh. ) = In (lIkc.k/ ,gy, ) I" 11. 21n hI '

I lll ro<hlci llJ:!: th e nnlll li on b, = 11l 16, Cc • In, ,,) b. = In lbsGcll ''' ln) -

In lk..:..2,Z) Ill l 2/"c. rtI . B)

b. = ] 11 (o.Gc" "'I..) I,. = In (6. Gc,,'nLJ,) -

In (n "'":,,,~'fl K!lI )

l.L = In h I'

In It.. .

( 12 .41)

I ll l n ~·c..k l , gY l )

=. -

In l. :.. = In O J

WL' Il:d tJ,,'\ Lhh'd svs tcm o( equnnous which huvc I' uniq ue solu uo n

I:, ts,

+ h : 'r 11., + Oz.. + h • ..;- 0Z.3 + Oz. ,

(12.1;2)

+ 0 1 ...... 1:. "'" b. + 2:. + 0=. -i- 0: . = b..

h, -

I: .

..... IiI = bl

1: 2

Tll k in~ 11,0 a lltl l o~ o r th e sotut iOllS o I l \ 2 .1i2) g ivM 11.(' c.,.. rd i nlileS o f tIlt' m i ni miz ing po i nl of t he nl ljt o.:l iv(' ftl nct io n . T he fi1l8! rt'suhs lire Jd ilwd b r thl' ..x press jons o f Ih o foron

b. """e"',

h u =e" ,

I = ea>.

D, =e"

( t2 .4.3 )

T he m('thod of geome t r ic p rog n"m mi ng ca ll ~·iclu the r(')llt ions 1'>401 "1*'n t he c ut put lind in pu t d llLra c tor is lj(jIl of lilt' o bject u nder s l lld r in t he form fit th e sys te ms of Ii nt'ar a lgt'-braic equa ti ons . Thu IInnl )"si:5 or b lll.'.rg~· co nver ters w!tll the a id (If th e (';\" Il()II\' I1 IS or illll(' Plw d t' 1I1 vn rie blc s is the ~ llI!rn li znt lol\ o f II higll6t order I I,,,,, ill tho case for co nve nt iona l medcls . '{'l it' o bjec uvc exponents of ind epcn dOllt va rtah les in lil a ohjeclivo OOIl;llrll.int a nd the poetuve CO lll pol\('nlS or t he mini m izing dun! vector fpc clcetr!c mnr l' incs of t he ."l1 I\l(, se etes cha nge e ver ins ig n ificlln t in ter val s . T h is pe rm ils l'- ;\" !on d lng lh o dnl n .1/1 a t horou ghl y clcsjzllcd machine mode l 10

\ 3.1. Evo lu tio n of Syllem; o f Automatl!d Oesi g n

225

o the r dc~ig n vers tc ns or ct cct rtc machlnes of th e sa me se r ies, th ere by cultillg dc w u hoav ily t he ti me requi ted for t he comp uta tions. T Ile des ig n te ch n ique ba sed on geo me t r ic progr'a uuning follo ws fro m the geneml lae t to n of the theor-y of mathema t ica l s imilit ude, where tho desi gn procedure relies on tim element ar y laws of li nea r alge brn .

T he me t hods of gec me t ctc p ro~r:\ m rni ng t oge ther wit h ex puelm c nual ues ign me t ho ds hold mil ch prom ise for the so l u tion of op t imizatio n pro ble ms . T hese me th od s do not ce:rta ili ly ru le out t he np plica tic ns of othe r cpti mfaa t.lon me tho ds .

Chapte r

Automated Design of Electric Machines 13. f. General Points on the Evolution of the Systems ot Automated Design I n th e USS I{ t ho use of compu ters for d es ig n of e lec t r ic ma chines was begun in the lnte 1050,;. In tho mid- H170s a ten dency evolved t o illtogrRte sepa rate. desi g n ap proaches a nd rorm tl s ing le me thodolog y t o promo te opum um de si gn of etccu-re mac hines us iug the concept of (I ~ Jleril 1iwd enor!:~' con ve r te r . H owe ver. t he general d es ign proeedure li t the t ime llU'gely re mained manua l desp ite the fAeL t hat computers held 1\ defi n i w ple ce lIm!lllg o l her desi gn mea na. In the modern peri od of lIppliCjltlOll of computing fflciJi t ies t o the processes of des ign of cteet- te rnnchi ncs , the trend is to rearra nge fu Hy t he d('sign techniq ues on t ile bas is of automated desig n sys tems so as to tra nsfer the dMiqn prob lem i n t he for m of II rnuthe mauce l model on to a co mpu te r to Âˇge nernt e so luti o ns. T hus , princi pall)" new desi g n a pproaches h llvo come i nto being. Ob v iousl y t high-qnnli ty des ign work mu st be done in Lh0 shortes t t ime possi ble . c r berwrso t he ide;l,' pot in to t he project and th e e ngl net'Ci llg sotc t tcns will be o bso lescen t. even before t he ma ch ines como into service. A de.:lig n insl\[fi c,ie ntl y wor ked out a t t ho ellrl y stagea of its developm e nt en t a ils 11 ]c Dg t hy per iod of " upll:flllii ng" t ho pro to t ypes or ev en re mode li ng t hem at. t he produ cti on stage , t here by add ing to t he ex pe nses lind ~rotrnc t i ng t he d O$ig n impl ementat ion . A pr- tucip le o bst acle t o th e i mpro veme nt of the q ua lity of des ig ns an d red uct ion tu tho ti me of the ir d evolo pme nt is the inco ns istency betw een com plex moder n mach iner y a nd the ol d me t hod s an d IUea nS 1 ~ _ O I1l8

of tles ijt:nillg'. l\t lh e nge or le clm ica l progress. 11I1 Increased comp lu it)' 01 des ig ned objocLs is Inev lta b le, ",nd one (:Ol'Wlillly ca onol TIl! ! t he q uu lily 01 d ~i C D work .. nd e ecclc eatc t ile dQSit:n peoeess Ly just incre asi ng t he number of dClli¥1I offices. Th o problem is a mell. ble W the so lu tio n o illy with t he .id of mRlhe mfl ti u l lllelhoos amI eompu1IOll f~ cili ties insta lled at project and dc.sil:lI servj ces , milnu{ael Uf Ulg Ofio Diu tio ns , a nd at var lces plan lS. Th \! most eff octi ve des ign guideline is 1.0 con ve rt lrom e u tomaucn of indi v id ual desigllS to inlegrll tcd illlt(imal io n evo lv i ng f (lf tile p urpose a utoma ted deefan ! )"s tllms (A DS.~) . The AD S is lin o perat ionn l eng ineering s ystem s csocte tcd w ith deJIign c tnce su bu nits amI inten ded to fu lfi ll the nss ig nmcnts by t he availa ble e uto maue mea ns wh ich for m II s pec ific (', " lllp l" x . Tho enmplax e nsures me th odteal . program m ing , OI nl ori nl , i nlnrm ation, fw d org<l ni za Uollal s up-

pert.

T he s ubun its ct e dt'si ltl\ office do c,l)urd intlt(lll jo bs us ing harrlwnre lind re ly ing on orgn llh.nli ono l support . The bllsic fu nctio n of ure ADS is Lo ca r ry out cc r uputu r-eid ed desi gn at. all s tages or some s Lae' tIS of t he desjg n pro ced ure usi ng mat.hematica l e nd other model s . I n Lh(' $)'sl.em.s of a utoma ted des ign, the opera tor i nle ra cLs wilb a co mp ut er to tn tred uce approp r ia t e cha nges i nto t ile doscrtp tt c n of t he desien a nd represent tr In euue bte la ngu ap s . I n so me lIutoma tic procedures, the co mpu te r performs t hese opera t ions 011 a c1 o...~ sho p bas is . Str uc t ura lly , the ADS consiSlll of severa l llUbs)·slems . A ", u ~ys le m is an ADS par t h avi ng iLJ ow n s pecif ic fea t ur es a ml ca pa ble of )'leJdina ec m pte te de!lign so lu tio ns a nd o Lher \l ~fll l illrorm a t ion. T h.ere are ebject-c efe at ed (objec t) subsystems a nd eh joct-independeDt (i nvar ia nt ) subsystems. T he e bjec t subsys te m de ve lops ti lt desIgn of a certa in object ( Il c1 ~ !! of object s) a l A de/ illi le design s \. a::e , It ca n wor k 011 II dosi gn of mach ine par ts , compone nts. peorluc t to u pro cesses, etc . Th o invarian t su bsy ste m perfl'lrllls con t rol runct ions a nd p rocesses Intormat tcn Independe nt of t he Ientures or the n bjecl bei ng des ig ned . I t ca n control, t he ADS; conversa uo nal procedur es: numer ica l ana lysis ; c nu mreeuon: input, process ing , anti outp ut of gr aph ica l info rma t io n; a nd also i nform a ti on r et ri eva l procedures. An ADS subsys tem co ns ists of components t ntcnded 10 per form II com mo n object ive Iuncttcn. A n A DS eempoueut il! a n ele me nt of the mealls of s upport t h. l per forma a de fin ite fu nc t ion i n i ta ADS SU bl!ys telll . All t he subBYIIt ems stru ct u ra lly for m till integral sy:s tem by v rr t ue nf t ile Jioks be twee n t oo .W S co mpone nLs of va r ious suhsystems . Th e co m ponents of method ica l s uppo rt are t he docu me nta tio n wh ich offers the foll owing (eithe r fully or gi ves references to pr im . a ry llO urceS): th e t heo ry. met hods. a pproac hes , mat hem atic. 1 mo-

13.1. Evo lution 0 1

$y.,. ""

of Atrio", . t. d De.ign

227

dels , Illgor il hrns, Illgori thmic ll'l ngtlagns for tile tJescripl ion {II Ob]OGls, terminology , s pec: jfica l ion,. s ta ndards, a nd oth er infor m:!.lion lI~d Â eel rcr t he met hod oloJ>' o r des ig n work in the llubs)'s tc lrls. 'I'hll compone nta of prOgl'a romi ng .!tlppor t are II collec t ion of doeumc lll.! co mÂˇ prising var ioll.!! program s , prof:J'a ms complied {Ill machine d nta ClU r ters . a nd serv ice form s nnd records for proper 11lIXt io ning {If ee rrespood ing su bsystems . P rogr amming eup poet iuc illdes the a ll-systom 1)'p6 a nd tho a ppl j. catio n ly pe of sof tware. T he compol)enl!' of a ll-s ystem software a re opera l ing s yste rns, symbolic eom pl ters , ere . The components of a ppll car ton software nre progu ms a nd pack s or a pplicllt ioll peegrill ms inte nded for desig n eclut ton ~ Ilera l i o n. T he com po ne nts of mllteriol sUPPQrl a re co mputing fi nd office mecha niza tion facilitit>s , da ta t ran sfer mea ns. mCllsu rin g devic es , a nd oth er ty pes of hardwa re . T he components of Informe t.lon su pport th at form t he da ~8 bose of tho ADS arc th e fil es of documents giv i ng t he descr ipti on of I lflDda rd desig n protoedu rea, sta nda rd des ign soluti ons, sta lldu d eleme nts , ite ms of sets. ma teri a ls. a nd also o t her i nfor ma tio n, Inelu d ilijC t he files a nd daill. uni ts with th e record of tbe a bove dcc ument.! on th e ma chine med i um. Th e compo oonts of orga nilaliona l su pport are methodiCAl an d ot her ma nua ls , reg ulallona lind i nslnl~l lons ens uri ng the intera ction betwee n office su buni ts. . T ho . nal)',is of l he des ign of a n o bjec t is made by way of ai mulal ioo of th e objecl on a oomp uter. Th e syntbesis of t he design a nd its opt imi ution are put i nto effec t by relrievi ng informa t ion Irom the comp ute r me mory in t he co nvetlla l lonal mode whieh Involves l he inte r. Cl io n of t he des ig ner wn h th e computer, Th e des ig n work t h us acq ui res th e Ieaturea of what is k nown all "computer-aided desi gn". Here t ho computer act.! as II mea ns of io tegra tlo n DI t he part tal mc dela lint! tec hniq ues (dllciaioll-mfl k ing proced ures) starting from t he ge neral da ta a nd peogramm tng- taehnlcal base of t he AD S. I n t he ADS , th e design prc eosa requi res ti le i nt rica tel y ccoedl neted wor k of the technica l, program ming , and Infonn ation fac ili ti es. The computer i nteracts wit h t he e x~na l world t hrough t he means of i npllt , out put , storage. a od tre neler of i nforma t ion i n a plllnbeti r.. dig it.al e nd grfl phic forms. P rior to .....ork ing o n t he project of lin . ul o ma ted desig n .IIylll elll, the ere pa are take n \() l nalyu t he e xis t ing design sla t tlS, s ubsta nli llle tbe projec t . dl'! term ine t h.e prior ily of a ulo m.tion of individull i d~ ign stoge&, es t imate 11K u pedle n~ level of lII utom. lio n, ra t iona li ze the flows of informa tion . analY le a Dd improve t he D!C'th ods of soInti o llS to th el pre hle ms. An A DS we ll s" bsla nti n\.ed ecc ncmiealty cnn ensure t he cho ice of tIll' opt imum problem eelu uc n. desi red /I. CCUflil CY of solnl ions ,

...

228

s ho rte r li me req u ired to SOh 'Cl pro hiems, fll l,io nn i acq u is it io n lind p l"OÂŤ'!Si llg' o f dola : most effide n t lIt illUl lio ll of operlltiOllrtl potenlin li lictl of co rnplIli ng fac ili lie:S n nd o ther means . Til e le v.,. ) o f " lI t a mRt io n l or the pm jeci is c hos e n on lile basi s o! ()(:onom,' witf co ns idera t io n fo r t he in for ma t io n cil pacil)' . la bor i npu t n!tl'lired fo r t he so lutio n. co mp u te r s l ofllge ce pa e uy . eueeession o f so lu t io n run s , a nd m u hivariArlt"O of , I'I! eotuuon t o t he prob-

lema. The hOllie faelor3 used 10 csttm etc t he erree uveness of d('sig n in th o frRroell-or k n( t ill' AD S arc lite labor producttvl t y. qu nl.it y indexee of tho eer uuon to problems. an d due da le s o n the develo pment of d es ill'fl p ions and epecificatidns. T he a /l3 1)\~is of li te b..'\sir tltil~ In t he P~ e5l'l o f m om"l l d &5ign nr ind uct io n ma chi n('5 per m its es ta b l ish illll t he u pt.im um le ~'e l llnd pr ioril )' of a llto m8 tion o f va rlous de!lign st a ~s in evo lv in; elect ric m ach ines with t he nld of t he ADS . T ho Soviet ind ust r y 1 8 rgcl~, produces nu to rne t od des ig n s)'s lelTlll for ind uCli" o mnchtnes (ADS 1M) . T Ile s)"s le m of t hi s t ypc offers 8u to mn lic m M OS of optim izing t he des igns. per forming g ra phic opcrati onil. 11 0,1 mllk i ug u p drnw inlr.! in t he devefcprn e nt of gen eral purpose le w-volta ge mducuo n mach tnes. P roject.s a n! put fo r war d for t he devel opm ent of a syste m (or des ig ni ng Anti lin ing gra phle work on me n- ma c h ine basis. T he ADS 1M co nta i ns improved m/llh llmn ti ca l model s a nd graphic se ts of ind uct ion motors . includ inll ata nds rd perrs n oli su basscm hl ies. T he system c nv isa go"l t ho design o f ma chlncs wh ich are t o i nclude sta udard 5u bRSS(lm bli es a nd parts s uc h AS hen r ings . bolts , nu ta , and ke ys ma d e AS s t.lpul nt ed in pert inen l sta nd a rd s. T he mod ified vers ions of II mach-Ina de-sign ca n be b uilt up by vary i ng t he d im e ns io n! of sta nda rd units a nd parte ifl n l inOll. r Iashte n (Sh8f ts . stat or coree, etc.) a nd i n a nonli nea r fashio n (foce portio ns of w ind irqr-;, s t a tor a nd ro tee lllmi natio ns , etc.). The Ieatu res embod ied i n th e sys te m opcrared on the man-co m p uter bllsi s pe rtnit t he d l'5ig-ne r 10 evolve n ew units ADd parts of t he machi ne . A mo lar d t'Sign is made u p 01 elelDen lll obtai ned fro m t he eeteutation lC!iu ll.S or data read out fro m the co m pu ter st o rllge . At ea ch Stllp o f the design proce d ure, tho ADS 1M y ie lds t he grAphic so tu uo n 01 t he dellign s hown o n II d isp la y . prese nted as a drawing. Or In lIoy ot her fonn . All ADS em ployed for ea ch Âˇt y pe of elect ric mAch ine h as: its own fea l ures . In one ca se , t he m ach i oe ca n prese r ve i L, ce rtllin base s t ruc turn, so a s}'s le m S\Ic h as ADS 1M c a ll lie se l 10 fill in only cerla in jl:Il P:l III the des ign . In ;. nOl lter r a!!('. l he e lll ;"" co us t.ruetie u of, ~ )'. Il nl\lI:I,"1(' o( n utc nc m ous power mus t be re bu lll cnmpletely to chnnge it.ll e lomo nt base, H ere , the ADS ill cn ll t>d up on 1,0 me ke a e hctce o f 11K" op timum des ign cut o r II. variety 01 t ilt" d eaig n ve rsions,

13 ,1. E~of"ti on o f Syste m. of Au lo moh'd 0 0. 19n

229

e ffec t, o pt imulIl unifica t.inu n nd s t und a rd tza t.ion of s ubassembl tca a nd pe rt a of t he mach ine . T h is crrcum stn nce considera b ly ca mpI tcot ea th e proble m of dlls ign work. T he re ason is t h at, wh ile conv entiona l problems gc ncrlllly hnv e n clear-cu t mllLhe nl/lCic,nl s ta teme nt ilnd Me solv a bl e by form a l me tho ds , lh ~ proble m of search des ig ni ng shows u no nfor rnai cha r acter and , .hence , is not a mena ble to the solu t ion by etrecuve nurn or tca l mll-l/lo(1S. Nevart.hefess. \1\'(' 11 ill the abse nce of Ilonfom ln l ma t hcm at.lcel mot ho.ts , ure p l"ag mnt ic basis for t he solu t ion of des ign pr oblems ean be man- mechfne co nverse t toun l prn ccdurus . T it/.' d inlog is ma de poss ib le b)' t he method of search fur nlte rn anves . A s he fo llowA t he course 'lf t ho so l ut ion to t he probtern, the d~ ig Jlc r (',1111 I,nke n numb er of posafhle dectsion s Ilt urc g ivon H Il/N of the solunon. Th e desig ner m akes :) cho tec na Ito adjusts t he sv st em with t he grap h ic termina l for t he st ep- by -s te p snlut.ion . T tm eflcotIveness And f1o)(ib ility of the s ys tem de pen d 0 (1 whet he r ttl" net th e sequence of nc lio'lS suggest ed t o the designer ca n affor d H s uffic ie nl ly large number of possi bl e solu tio ns II"t ea ch step . I II t h is eonnect ton, infor ma t ion sup por t of t he evercm Acqu ire s mu ch Im port ance. One fin ds it ndv autegeo us to Accompli sh inform ntion a id s as n se t of prugraurm nble m odels whic h de scr tbe at an de ed desig n elements lind sim p lc geolllclric. fi gures an d also aa sem antic mode ls reflect in g the hiera rch y fl lI n s truct ure of cleme nts, par ts , a nd ll11 it.s of, t he machine being" dtÂˇs iglll.'d. Suc h II roprcso utnt.lon of g rup hlc nnd somant ic info rmnt.ln n hn s it s rcote i ll tho preli llliullr y n nnlys is of lh e geome t.r-ir, for ms of t he pnss tb le des igns of a mac hine find us el emen ts, a nd nls o in t he syst emat.ic 'a naly sis of th o ma chine dcsig.\ l ll ~' u u t lind h ternrchy of t he Isola te d el emen ts ( ill the for m of n gra ph of poss ible solutlcns). Ap R!"t fro m the program s 01 t he geo mot ric forms of ele me nts end par ts , th e grap h ic inform nl ion aids. of the syste m nlso i ncl ude the progr ams of the pr oj ec t ions. Gross soct.ions, re pr esentat.ivo dtmcnsions a nd di me nsional t olera nces , ra qut re ma nts t or winding sur face clellllli " ess . s t a nd ard t ex t, informnt.io n. e tc . T Ile AD S of etecu-rc machi nes (AD S E M) ha s n geue rn l pr ogra m of drnwlng . It e uvtsages work on vnrinus pr ohl ems lind collects a nd stores i nforma t ion . Th o prchl ems of et orsges, search, nud proooss ing of tla la acq uire pr imary im por t an ce j n. mode r n desig n sys tems nnd ha ve a substnnt.ie l effed Oil t he str-uct ure nnd (he pri nci ples of acti on of th e s ys te m as n wuo lo . The fil es of data nnd progra ms direct ly int e nded to peovjde for cent.rellzed st cm ge und scorch of Inrcrmauo n nnd al so to estlllllish li nk s w it h a pplica tio n program s wldcl~ pr ocess this in form at ion form a bank 01 data . l n systems using data ha nk s , up phcat.ron progr a ms rece ive da ta lor pro cessing not from external du t n cerriers uut Irom ro m rol syst ems of the deta base.

'30

Ch. 13. Avt om eled

.oâ&#x20AC;˘â&#x20AC;˘lg n 01 Eledrle

M""l\jne .

One of t ile im por ta nt- problems of ADS EM ill t o create" t he da ta bank of elecrromechnntca l sot uuons. For t h is i t is necessary th at deaign an d de velopmenf work carried 0 11 in t ilt' rte ld of en ergy COn ver ters sho u ld comply with tho requirem ent s of tho ADS. The proce dur e in t he ds valo pme nt or II new energy co nverter should fo llow th e ado pted gl. ideli nes on\' isngi ng t ho st orage of finlll reeults (II th e- dat il bonk for fur ther uso. T he bank ca n store infor mat ion i n li braries o r~ n i 1.ed in a n hiera rch ie orde r accor di ng t o t he ty pos of EC or (or each leading resea rch insti tute (F ig. 13. 1). T he lib rary ca n con t ai n dat il 011 deslgn Fig_ 13.1. The general ~ehem8 tl c of a of Ee ele me nts a nd s ta ndard data bank for energy cOll vert.er~ so lutio ns. Th e evolution of lin ADS EM is Il lengthy process a nd call s for fur t her deve lo pment in the field of electromechamcs and mom coor dine tc d work nf deSign dep a rtm ents at plants a nd research i n!<ti~ utes.

n .1. Software of 'Automated Desig n Syste ms Sot rwara of a n ADS of elect ric mach ines can be of seve ral types , such ns general , spe c lal , serv ice, a nd d ia log cont rol t ypes . General $O/tu'are. be ing inv ariant witl\. res pect to the ob ject u nde r stud y . comprtses applfcatton prog ram pa cks for th e solution to glJ nerol. l ma themat ical problems otva rio us cleseea. such as the problems of mat hema tielll progr amming. proble ms dor the sol uti on of l tnear a nd non linear Il l~braic eq uations , the syste ms of di fferential equation s , probl ems of mllthematicll l sta t is t ics , etc. Service software a nd dialog rolltrol software mus t pr ov id e for effective interact ion betwee n a compu ter a nd a use r. Sp ectal troflwar~ is a prc blem-orio nt ed pa rt 01 programm ing su ppor t , the de velop men t of which needs pa ettcu lar- enee, for it is t he qu al it y of t he evo lve d mat hcrna t.ieal model s of F.:Gs, t ha ~ det crmtnes , in t he fi na.l a nalys is , tho effect ive ness of the ADS. T he st ru ct ure of ADS EM scft wnre m ust meet t he follo wi ng bas ic require men ts: (I) hi gh modu tnelty , t.e . II hi gh de gree of a ut onomy of sub systems; (2) open-loo p nb il ity t o en ahl e tho ex to nuc n of ti le system or Its corroc uo n w ithi n the s t ructure withou t cha ngi ng other Mocks ; (3) flillcUon nl eom plateness of the set of \les ig ll opera t ions

13.2. S"ltw.., ,, "I

A ~ l om al.d Dc.i 9~

Sysl......s

23<

real tae blc in the ADS so t liat t im system con exert t he desired lntlue nce o n all dos tgn stages. st arting wlth Iii", re quest for the pr opnsa t a mi e nd ing w it h t he for ms and reco rds on t he pr odu ction or o. n e nerg>' con v arto r , : Cons ide r t hl;l sou were nil ADS EM. All A D S EM is capable of dOing both compu tatlolla l and de~'l!lopm.ental IQQrk . from the mc num t of rrcew fng the order to tht lIIom/"lIl. of issui ng the produdton documentatton and tes ti ng cLpI!rim/"ntal mode" of tht mackl nt (IvUkout menu[aeuu-tng the production prototypes), T he most ,Hfficu l l ,,"d importa nt task is t he tas k of evolv tng t he sys te ms for a utoma ted des ign of ma ch ines destin ed for work in

OptJml,.,l"" " r EC I" 01<..,,,-

och .;" l ' lÂˇlI'

T......' I' l"" f ,om

po,am" . " ,n

Tn" er EC

I ÂŤlm<rr y

Fig. 18,2, An AOS

or elect ric

machines

a n elect romechan ica l sys te m wlth con tro l clement s a nd feed back mechauisms. wha t odds to t he d ifUcu lly of t he t ask, apart from t he fact. t ha t the desig n work must be dOM in the shortes t poss ible poeted. is l ha t l he tl ~~ i g n mu st e uv isage t ile small-ba tch or p iece-work produe rion of the abo ve ma ehme s ifor opera t ion li t vary ing lre que nd es a nd vo ltages a mi u nder d ive rse env ir onme n ta l con d itio ns. kl men t ioned corlto r , the A DS EM co ns ists of a few subsy stems, each ln-Ing ca pa ble of so lvi ng Ind ividua l problems Oil the men-eo-nput er basi s (Fig. n .2) . Th o l i r!l ~ su bsystem et rects the opt tmi ea t ton of t ho pa ra meters of II ma c/ti ne in t he e fac t.rorneeh anica! syst em . G ive n the mat hemat.lca l descrf pttou of energy co nvers io n processes and UU'l equ at ions of t he system e leme nts , it is possib le to pass t o th e represent a t ion of the mnch i ne .nnd its s upply net work as a sys te m of multfport s a nd Implement. tb e s yn thea is o r t hl) mul up orts. I n el ectric dr ives a m otor is l,"ll(le ra ll y fod fro m II ge nera tor or s rauc [semicondnctor) co nt ro l cteceote whic h can be con nected to the s ta to r or rotor . T he supp ly sys te m ca n be sp lit li p i nto on in fini te

poWt' r li ne e nd a fetid m ulr tpon. T h is m u ltipo rt Is i ll t um broken apnr-t. In to II pass ive mu h iport an d a n emf wid elL a lso repn!sellt an in fhlite power li ne . T he ge lltril lizcd rou li ne en ables calclll lll lllg the ch:U"act~r il! li Cl of II machi ne with ils sla to r or rot or windillg supp lied ÂŁrom th ., li nt of Dousi nu:lOidnl a symmet ri c \' olt llge III rough I Hooor mu lt iport of a rhit nlr )' form h 3v i ng lin t mcme! ene rgy SOurce. The e fectrtc ma chine eau "" of t he t\Io'o- p ha9 as ym mel ric l.i' p<" or of the m.ptlasc symmelr ie t ype w it h n de or IIC sllpp l)' SOUl't'B. Fo ll ow ing t he ex pauatou Illto a h""noll ie se ries , we dele rm ina tile erreeuvc va l ue of a harme uie or !llI initio! pha se. Ne:r t ce lc")lIt e t he t hl l'/lCteris tiu tor ee en harm onte a nd ovnh ulle t he hn peda uces of t he ma chine a nd ge- nel'3H1erl mu ll.iporl . T ho ex per un ent e! des ig n te chniq ue permi!.s us 10 sirnplify li re doscl'iptio ll of t ile .Ilys lClrl, .!Ii llg le Dill s ig lllf k a nt a nd in s ig nlficunt fucl on , And re prese n t the a ppropr ta te rc ln ucns AS pol yn cmlals. Us i ng' eouune progrllm8 t o e:lli m a to all o pti m um , \VEl l:llli preh miua ri1y d e te.rmi nc t he pore merc rs of li m mlu :hilill 11 1111 t ile con t rol lJI ultipor t . T ho seco nd sub sys te m IWSUre5 tho eosvereto u from Ih(' pa ram eter s t o t Ill! ,,\lOnle ll')' o f t ho nla cldn o, T ho prohle m of tra ns iti o n to lllll opti mu m l{eolllel r)' :It the g ive ll opt im um IUIl'11 mo te~ of the e loc1ric m ach i lle ca n be st a t ed AS t ho pr ob lem of ncnlinear prog rlllllmi uiC. T he lise of r O\l t ino opl imiution p ror:rom s e nA bles fjlldill~ t he conditio na l ex trem um of t h\' o b}el:li ve fllnet io tl ill the :;h ort l'st t hut:' possib le. T he t rans tuo u from t he pAram c lc r>< t o t he ge oml'l.ry of the mllchi llU ean be s peed ed up lilling the me t hod of geo me tric pro gramm in i , Alt er dc terrn inina t hu geome t ric d imen s iuIIlI of t im ma chine, the 1011l' l si llge foll ows Wh ich In volves th e ~t i ",a li o n of paramet ers (cooUic io" ls 0 11 t he vurtn hle s i.n t he illitl,,1 eq uatio us) a nd t be chock of th e peeume tc rs to r th e op li mum in t ho sy" te m . If uro di scre pancy be t wee n tho op t imu rn pll.ram e,ter s in th e fi rst 30 l\ t he seco nd sub s vs te m is nuncceptnbl e . BOW eom pu tnt fons ore per for med . T bu th ird subsys t em )' ieJd:l s t r uc tu rn l uml produ ct ion dr nwings. It ca n s ll bS l:l ll li a ll ~' cu t dow n load times , T ile ADS CII /I Incorp orate A subsys turn fo r rorm i ll~ lin imA ge of t he machi ne u nde r d eve lopnr e nt on a T\" 8<.:1'ÂŤ! 1l give n tho mnchme l,,'tlo motl1' rla tA stored in t ho co mp u te r, S Ul'l1 Il so bsyst em toget he r w it h the a ut-0 013 te d ("on t rol aYbtem ca n s peed up tl' ll rnach i lle Ile:;ll:ll proc ed ure . Moder n com pu t ing fac ilities a llow for t he crea t io n of a subsystem for testi ng th e C:l:per ime nla l models of a li etecmc mach ine pri or to IlIAlI lI(a("tur iIlK i ts prod u ction pro totype. Alter t rl'ul.Sla ti llg t ll\, data 0 11 the mae hlne ;come t r )' in t o the com wlllld la ug llhge , we ca n llSCl Il p rOl:rtlm""'CODtrolJe d OI nelliuc too l t o prod uce l'I mod el oi II mneh ine sli lmuem b ly. for on mple. II fuJl ~! I IC or geome t ri ea ll )' llim ib. r eorc r.

w.,

1) .2. Softw.,e of Automalei:l De,llI n Sy,iem.

aaa

At th e mode rn sr nts of th e a rt , it is poss ib le to collect e nough useful datn for construc t fng 0. geueral- Lv pe rot or mnde l for most oner g y conver t ers . T hu s , for in d uct ion m aehlnca, II so lid ro tor "wiles for the m oda l. 'rests ru n o n the roto r m odel unde r cc nditions s tm fla r 10felll CO!lfli t iOlli:! e nable us to pr ed ict i lle mIlSS, e ller~y chnractertetlcs, reliabili t y , a nd o t he r teete rs of t ho machine w rt h t he lIi;1 o r II d igi t nl com pu te r lind to pass to ti le stnge of ljvolv ing a produ c li on prot oty pe, One of the im por tant prob lems of com pnt er-aided des ign of e tocrr rc mach i nes is the pro blem of evcl vlng n system of optimu m desi!:ll' ccm puta t.ion th at y ie lds the bes t mnc hiue-des ign ver s ton wh ich AAlisf i e~ t he restri cti ons Impose d by spectrrceuc ne aud eaq ul ram eu ts . I n it" general st at emen t . t he pr oblem of S)'n lhe sis of eleen-ic runchines red uce s In the o ptlrniznt.ion ( nlinim i~llliol'l or ma x imi z" UolI} oI a ce l'lai " fun ct inll ll] : {? ) ....... m in 4 s·,

it -

x'E D

s E- S ,

( 13 .1)

wil li D s ub jec t to the co ns tr a in t

filt(.?) =lfHx:, i = 1. 2, . , ., m ,

.t;, ... , x:')~O

= {t , 2 , . , " s. . , ., I, }

{13 .2}

is a se t Qf st r oc m res (design versio ns) of e lec tr ic mac hlues: (If n ptim um pa ram e ters of t he op tfmum des ign. T he pr oblem d efined h~' (Uti ) an d GI0.2) cov ers bo th at ruc t. ur nl a nd parame tr-ic o pumuaucu. The cond utons (J:i.Z, ex press t ho reswrcl ions i m posed on th e ma chine design by t he spe cificatlons a nd dcecribo the pe l'm iSl!ib/11 reg ion D . The reg io n for de finiu g- 1I11l Iune t.I nns. P' (? ) 1IJl(1 'r1 {? ) 11;1 a cer ta in region D l subject to t im const rahu

whe re

$0'" is the vec to r

'~l t?);<a- 0 ,

1, 2, • • ., I

0 3.3}

If s = (1 }. I.o . whe n th er e is a need t o opt tmiae the give n t ype oC ma chine , the opti m um design pro ced ure COUles t o t he so lu tio n uf the pr ob lem Fj(x) -I- mi n e>;'(13.4}

?tE D wit h D s ubjec t to UlC ilLeqll llli Ly cpustraints I ... I, 2 , . . "

II ~

(13 .5»

Op timu m d esig uing redueee tu the sol uti on of n nonl inear progl'am m illg p roblem oC th o genera l t yp e t ha t SIIOWS t he smoothness of the objec tive func tio n, mulrl pa rnm e telc a nd mu hlex tr emc featu r es , e nd "va lleys" ill t he hyper spn oe cr perrniss thle solu tions . i n

en. 13. Aulomated Oetign of

E l e~ j ,l<; Ma~h i"' ,

'mos t ex tre mum -seekl ng pro blems, a " vel le y" associa ted wit h a poor eo ndltional fw of t he matrix oJ seco nd part ial deri va tives s hows up !IS a strong elongation (in the t,o pological represe nta tio n} of t he Hues of the objec tive functi o n level. However . in th e problems u nder d iscuss ron, the va lley sttua tl on .ruore oft e n a ppears liS a resul t 01 n s mall an gle formed b y t he objec t ive- func t ion level line wlt.h Ihe bounda r y of th e permissible reg ion . Th e proced ures of round i n!; on co ndu ctor dia mete rs acenrding to th e tabu la ted data co nsiderably compllce te l he strah'g}' of search 'for the opti mum. Opt fmtaauon met hods diff er from eac h ethor by the programming procedu res h IVO!vC'd , count t imo, rat e of conver geuee , etc, T he a bove fac tors are de pende nt on both th e effec ti veness e f t he met hods employed a nd o n t he complex i Lr of the optima lit y cr ite rion nnd geo metr v of the- perm issi ble a rea. Th ere a re two approache s to the. problem of opli miztlti o ll in tI bounded nrea , wh ich d iffer in pri nciple si nce one a pproach co netders const ra in ts in im plicit form lind t he ot her i ll ex plic it form. Tile implicit me t hods such as Lhli Lag ra ngiun multiplier me th od a nd penaltv Iunc tton me tho d presu ppose d uo. ifllplemeu t nli o n of t he genetallzed o bjecuve functio n whlc h coinc ides wi th t he ilJiLill i c ptimalu.y crite rio n wttht n tho area. bu t grows ra th er rast beyo nd the a rea or even nea r to its boundary. Tile ex pl tc u me thods (possthle .d trec u cu methods ) mekc it necessnrv to move i nto t he permissible a rea an d ta ke s teps i ns ide of it . By t he cha racte r or dat a collection, optim iza tion metho ds ca n be d tvid ed in to the me t hods of loca l lind nonl ocal search. Local search -me t hods ca ll fo r th e a nalysis of the result s of eac h exper-iment (computo t ton results for the mnthema tica l model ) a nd t he use of the informa t ion so obtained Icr t hc ne xt experimen t . The fea t ure peculiar to these me t hods is that. eech successi ve ex per imen t. U5l:\S t he informa t io n ou t he bahn vioe of 111\1 mathematica l model of the dev ice under study t hat covers only II smal l reg ion of t he va lues of t he parameters in Ule I)roced ing ex pcr unont , Loca l search ta kes a rcla tiv el y sma ll amount of machine t ime hu t yields a t best 00 1)' one loca l extremu m. Th e mos t important stage of loen l sea rch is lh e choice of t he d irocno n of searc h for an optimum. This choice ca n be made tr orc t he es tim at e of the gra die nt defi ned with the a td of fi nite diU erences. or from the s ta tistica l da ta (ra ndom sea rch), or from the es t.ima tes of PAl'ti 1l 1 de rivatives. [ 0 some cases the direct io ns can be chose n pre fim tuaril y, for exa mple. a long th e coord i na te axes or a lo ng or t bogoonI pa l hs. In solv ing prob lema for many extrema a nd pro blems inv olVing va lley cond iti ons. Lito autonomous fun ctioni ng of algo ri thms of 'loca l sea rch proves ineffect ive. :T his s it ua t ion hAS s purred ti le de velopment of nonlo cal searc h methods wh ich in flict re q uire a de finite .orgiluizaLio n of a s uccess ion of local searc h s tages . T hus th e nnnlocal

Algorit hm for t he so lu t io n to IIllllt ie :l.:tre mlil prob lem s ell\' iSllaes the cboree of i n itia l po l nLs w it hi n tl l8 g ive n field .nd pr ocessi ng of t he re.5l.J1 ts nf loca l WArt h ma de from t he se points . T ho Illgor iUull dfor ds th e Ana lysls o f ti le g iven field And esumauce of local es t roma . I n use are I\lso Spt'c iAI uo nloea t searc h a lgo r it h ms for t he so lu tio n of p roblem s i o \'olv inr vlIlIey-co nta i ni rli field s. O ptrmum dl!s igll of e lect r ic madli nes requi res II pec k of u pplicat ion program s a nd pr ogr am s re aliz in g loca l se ar ch l\lgorithms . T hl;'se aro t ht> al~rithm s fOI' opt imit ation a t t he criter ion i n the peeuuss tbl e sea rc h Held , II d ire ct ion of motion in t o t he f ie ld Irom a n IlI'bill"llry Initia l po int , n nd p reli m i llar ~' sOIIl'ch of the give n field l o determine the loca tio n of ex t re me e nd val teys a nd t o solve t ho prob le ms for t he fie ld w it h vllll ey s . T o <lClJo mp1ish ec mp u rer-a ided ll'rl'lph ic constructions l\.;o."oc il\ted wit h t he ucsreo. il is necessary 10 wo rk ou t a !Cmi gra ph ic mod el ( be i n~ a lso II mn l he m" tica l mod lll) wh ich wo u ld usc t he d/llla 0 11 t lLll m a ill d tmensions of etatee and rotor COfl'S I., e na b le t he gt'A phi c rep~ll tl'lt i on of t he basic d('si~n of Ihe mach ine . A u lo lnated str uctll ra l de sig n of lUI e le<: tr ic maeh fne iucllld ;lS t wo bllsi c s t ages : th e d evel opment And e xe cutio n of t he pri nci pIII v iew o f t he ma chine in long it udina l lind cross sec u o ne: t ho devet opmeut a nd execut ic n of vtews of assem bly u n its a nd pUl" ts in II pro pel' way for furf.her use o f the d rawin gs n ~ dll~ igll Iltlpo. r tln ollls nn d en ll'inee r-

InR plnnta .

The two S ~/lgÂŁlS o f des ig n work pr im aril y usc th e re sults of co mpute raided e Joct ro ma!:net ie lind hent-rem cval 8 1l!l1)'ses based on n de finite c rite rio n for t he au i c me t tc se a rc h of lh o o pt imu m des igu of l he ma chIne as regards Its d lua ensto ns Dnlt pa ra me te rs . T he geo me t r ic dhae osi o l~ of t he act ive pa r t of the mac h ine ( i.e . et ator a nd rotor cores ~..It h sluls a nd w indings) for m the basis for t he lrt'1\p1 Lic rep resent clion at halh sla ge!. A l 1110 fi rs ~ stage of des ig n _ t he mach ine s t r uc t ur e m us t possess ce r t ai n ge oera l fea tures spec ifi c to t he ml'leh ll\es of t he g iven t ype. Furlher wor k o n t ile ge nera l desi g n must be do ne o n t he 1J3sls of a d ial og of t he des ianer with t he co m pu ter , i n wh ich procedure t he dealguer must per form a ll ad d it ion al ca lc ul a r tou a neee saary for the s tructural deSign o f the shatt , benr iugs , a nd Inst ano rs. for th " v thrat.ton-e coust!c Ilonly si s lind Ih e c ho ice of st a ud nrd pa rts , et c . T he ~ oeral dr awi ng o f tho ma ch ine ca n be th ough t c t ee II su m of Ind lv id ulIl a le morus e ach of wh ich s ho u ld prncticllll y re prese nt a n Il ~ Âˇ som bly un it or pa r t of the cons uuc uou. The des ign er m ust assess th e COlu p\llRt io l"lS a nd e nne r acce pt l ite resultstc e further use or i nt rod uce c ha nge s in the d ra wi n!:, wh ile "IIlni n); t h e com p uter in t h.e cc:mve rsa t ion lll mod e . T he nee d (or int roducing COrrect iOllS ca n appea r IIfler the ca lcu la t io n of II sha ft a nd bearings. t he eh cc k 00 t ho de pth o f s h iel ds a nd tli e len gl h of

t he feemc for t or N'sponde nce wuh t he ove r h:m g (If th e wi llfling '.!\" (8C(> Ilor t io05. t he cho ice of fa!~c nc r.!l . etc .. n ru l »tsc i n th e eo ursc of briJlp: ing 1110 d es j ~ll to tile Il na l Jenn III com plia nce wit h t he fl'q ulI'eme nt s of the naelg nment . Afl('f co mpleLion of t he gcnc:ra l d ra win g of th e mecn tne , t he seco nd stoge o f des ign t a ll follow , wh ich invo lYf's dC'vr lopin:.: the eesc m bly u ni lS arul pArI.! of t he co nlll.rllt tion a nll de le rnl il1i' lg t h('!r maas, d imclllliolls. lind to le ra nces req uired III tho preduc rto n slagI.'. Thi s dooc , the a utoplorter ca n fi na l!y ma ke up t he work ing draw i ngs i n co mp lilUlcc wuh the roq uirem(!nts of sillo ndllrdi:r.alion . T ho s u bs}'. . t ern of d rll\\'lIlg fllc ili t ics pro v id es d es igll lind prod uct ion d ra wings a nd s ubs ta n tia lly cut! down th e lead t imes. Tho deve lo pment of auLoma l.(od lJe:li{!1l l!I ~'s lf' J1HI poses a num ber of co m plex pro ble ms . Of m uch l!i(:'n if iu nctl is th o es tn bfisiunent of i nt er na t.lo ua l p roKrllm libra r ies whi ch would pu ll in ~iC ll L jfk peIen tia l 01 e n" " Il.>e rl!' for t he so l ut io n of t he most im po r t a nt pr ob lems o f alecr rc meeha ntcs .

13.3. Hardware of Autom ated Design Systems The ctnlra l p rO<'esstng un ll (processor) iorms the btU /if l or A DS /uud. ware. Thfl I, computer,

hIgh-rap aci t y 'com p u ter with (or without) a 'a tell ite

The ti rs ~ Sovteu- made s ys tems use d conrputors of the .r.1-220 IUld other I·y pt"s nn d ver-lous se te utte com p uters. More a dva nced sy st oms of nutomc ted deSign a ppear ed as t he !:ioviel U ni on log-e U,e r wit ll the count rte a ent.el'illlo: i nto t he Councf l fur E co nom ic M u t ual A SS1.';t " IICe cre a ted And pili int o production ele et routc com puters of I litl t hird genc m t tnn w il h 0 wide m nge Qf s torego lind per iphe ra l uu ita . A t pro/i(! nl ADSs nrc b uil t arou nd th o ce mputcra of t he EC sys tem IE C is th e abbl'e vlatio n for l ilt' R \l! l!ia n words men ning un if ied syst em) Il. ud EC l!3tCl Jlitel:Om putl' rs or s mQIl-en pAci t y compu t ers of tile iu le r nll. li o lln) system . h"t: com put ers rep resent the faol ily o f prog rllm-compnl ib Je ctecIr on ic ma elnnes perform ing from II few t-h<)ustlnd:> to II few milJi ons or opern l iOllll i ll <l eeec od Rnd Il.:lv i llg II lll lifi Ni rllllg@ of pt>r ipl,ent l ullits. H llrd....·llre o r com pu ters usod in spec ifi c sy s tems m ll)' vary in s rruetuee over R wide ra nge . T ill' ADS req uires wel t-develo ped means of dll ia in put . out pu t . and search, mean s o r co pYing f!J"llph ic . nd I{'xl dQCu llle.nt ll. t ion. a nd m eQus of Oil-li nt il lll' rllc;tiQII wit h n rom p ull' r . T lir Jisl of EC co m p ute r peri pll{>ra l u n its produce d ill ll'ri~~ toda y r;\,le nds In o'\"...r 200 ut tcs . By t ile p ur pose lh o)' have Iv ~ ...rv e. lhe peri phe rtll unus of Be co mparers a rc brok en down iu t o the foll owin (:' gro ups : ex t e -ue t st e reace: ~r:, ph ic tnre rrne u on tn pu t-oc t p ut d l'v ires ; ccnveesa ucna r proceslu nll de v ices: o nd Inp u t pre pa ra tio u eq ui pmen t.

237

I n neconte ncc with t he ree tu res of the a utoma ted design process, o ut pu t de v ices fO IÂˇl:r~ phic infor mt< t;ioll can gc nera llr be d iv ided int o o n-Hue mflp ping (d is pla y) menus 111'1 d mea ns of pl o tt i ng fi Ulll g mp hs a nd for ms. T h is is because t he dectsio n o n tho c hotec of th e d c~ i gn

ver-non is commonly token in t ho course of Ute n orettve prncadu re lnvolvlug eo nstdcra t to n of n [O) W des ig ns befor e a n il' i u!:, lit t he fi nal deetsto n. I n this connecuou the dovlcos of o n-l i no ma ppi ng (dLspLay) of grilph ic informat io n mu s t mee t me re q ui re men ts of a hig h ra te of ma pp ing 1) 11(( t he plot li;ng devices must meet t he requi remonts o r 0. h igh accuracy and Ill gll qu ellt y o f graphs lind \Irnwings. At present t here nre vartous mot boda of out p.n of gl't1 ph ic In foema tion , T hes e H"C t he me t hods of n,'Hl ki ng up ima ges Oil POP01" a nd photogra ph ic pa per , display ing tillages O il (I CRT scree n, clw lIging t he cul nr of pa per by tho reaction of etecu-oty ess . etc. The sys te ms o r anto ma tcd dealg n wtrlc fy use ole!:-tl'omcch lHliclll. e lccwcntc . a nd s ca n nin g de llices for out put of gra ph ic Iulorm a tio u. Ele c l ro mochan iC<l1 nuto mru!c d i"a\lIillg ma ctnnas arc sim iln r to numerical ly controlled mille rs in design and prin ciple of ac tlou. Deawtng o n paper (or t raci ng pnpe r] ill broug ht. nhout by nn e xecu t ive unn wluc u conetste of II plo t ti ng he ard or dr um. electr!c d rive. 811d n t rac i ng IlIlH . I n nuto mnt.ic enecbtnes (gra ph plotters) of t he plott.ing boa rd t y po, t he "ra ci ng unit; moves in two mutualfy pcm e udlcula r dtrec uons x lind II whi lc t he c hart cuer toe re mai ns s ta t ionary . T he prin ciple r"Jf n dr um-type plotter d iffer s fro m t hut of n boa rdtype ptuste r. I n t he for mer t ho ste p metor-d rlvcu tra ci l1::! unt t mOV N! on ly llio ng th e ); ÂŤx is a nd t he ([rivi ng dru m shifts t ho pa per shee t a lo ng t ho y ax ls . \Villl t he paper blling ree led 011t fro m t ile roil , tile t raci ng clement draws II pat h as it euttts i n the x d irec t io n. T he tracing unit has pen holders to fasten ball- pen or pen-a nd-Ink reco rde rs: tho num ber of re corders ca n var y from one to ro ue. Elich recorder traces Hues 01" rtrnws sy mbols of n de fin ite thickness o r n defi nit e color. Aut oma ted des ig n systems witlol j' em plo y tho a bove two t y pos of nutomatic dr awiug mac hines. l n use ore also e utoma tt c draw ing ma chines of t ho Iln ifi o{1 system. T he boa rd- t y pe ma chine of t his class with u plo t r t ug board measu ring 1 200 by 1 150 mm t races a. ~ a ea te of 50 mmhl. TIl(~ machi ne hns n data convert er which performs lin ea l' a nd c ur vili nea r interpola t ion find affords a u tom atic tracing of up to 253 sy mbols a nd t hroe t y pes of li nes , name ly, soli d . dash. a nd dot-and-dash lines. There arc t wo vers io ns (If d r um-type ma ch ines w ith p ap er rolls of 1.20 X 80 000 IU m and SiS X 20 000 mm in s ize , wh ich have :'l ma xim um rate of tra c t ne of 200 mm/s a nd 150 mm /e respec tively. Electrcrnechantcal d raw ing mllcJiine-s offer a number of advantages: na mel y. they ensure a h igh accuracy lind quality of l fnes And s ymbols. can make up d rnwlngs of whatever s ize , rre co lines of a ny t y pes and colors , a TO adaptable Ior .dot ng o t her jo bs s uch as e ngra v-

238

Ch. 1),

A uto mtlt~

De5;gn of Ele ctr;c Mach;","

i og firll l IIlll. rld ng路 ol'f, and can nN aa auto nomous dcv tces . H owever . the y :; IIO \\' a low ra te of tr ac in g. do !lo t nl luw lo r CQI'l't)ct i llg A ll e rror in th e process of drnwlng , ha ve relati vel y la rge overa ll dtmonst ona, anrl a lso pre sent oth er drawback s. E lec trur-heru ica l an d olectrotherme l d('llwing ma uhiues rela te to re s te r-type devi ces . A c om b of electr ode s form s a r aster end se rves es 11 t rac ing un it t o prod uce all ima ge a ll ele errocnem tce t paper lmpre g na ted with n spocinl olectrolyltl. On o ne of j Is sides the pnpe r CODles i ll contac t with com b lllect rodilS, a nd on i ts othe r si de wiLli n me t al el ec trod e sh a ped like a cy li" dcr;. 'fh e volla ge IIpplicd t o j'l d ividual e lectrodes ind uces t he el e clrol y~i s re ecuo n tJw t chQ II!,Tf.'s th e p aper co lor at ton. Cbangi ng t he vo ltage on the comb electrodes g ives dil Ierc nt lilie s on t he pupal' euntlnuous ly uure eked (tQ lll t ilt: roll . In c om pa r iso n with d rAwing machi nes of t he cleere omec haut cal t y pe , ras ter - t ype a ut om a ti c mach in es ha ve a Ilighcl" rat e or draw ing . Bu t t huy demo ns tr ate diff ieu ltic1>l in ob t ei nin g l inea of va ri ous th ickneseea, req u tre micro fi lming and s pec ia l moistened pape r , fall t o fu netiun in the autonomou s mode , e re . El ec tr onic dev ices ll si n~ a CJil,T aa an ex ecu tive \Jllit are rathe r pr om is ing gra phi c d isp l ay se t ups. Control of an im age on th e scree n is brou gllt about t hrough a re panttve display of t he im age lit II dol路 illite regeneration froqua nc y . T he image 0 0 t he scree n becomes stable a nd fli cker s vanish a t, a fre que ncy of 40 H z. E lect ronic dis play de路 v ices wi t h II cathodo-s-e y s t orage tuoo(CRST ) enjoy lise today. The number of add ressi ng poi n ts on the screen rea ches 4096 x 4096, w it h t he worki ng fie ld meesurf ng a bout. 50 em a long t he di ago nal . This makes i t poss ib le, t o pr oduee q ualit y images on t he scr een and t hen p hotogra ph t he drawings u p \0 Ute :Mlh shee t she. The mnin ad va n tages of CBST路bllsed devices !Ire a hi gh speed of image for ma ti o n o n t he screen , pc saihilit.y of produci ng co lor lind half-tone images, on-Hue cor recti o n of errors b y era si ng the l in es and d is pla ying repetitively the Imag es , low cost lind s ma ll overall dim ensions. The li mitat ions lire a rel a ti ve ly low qua lity of l ines and symbols. low reso lu tion , necessity of microfil ming, etc. Automat.ion of the pro cess of in pu t of gr a ph ic informat ion is II ra ther urgent pro blem. 10 the ADS, gr a ph ic info rmation input de v ices (G l Ds) sup plem ent the ma in se t .o f eq u tp men t of th e computer . T iley eo n be automarl c lind se mi a u tomat ic. Automatic dev ices are of th e scanning and tracki ng t y pes . In sca nn ing devices, t he scanne r beam swee ps over the dra win g fie ld , Ii no-b y-li ne . Traeking dev ices track t he lines of the d rawing and p red ict the ir poss ib le axta n tio n where a few Lines i nt erse ct. Automatic, devices onn set onl y rather si m ple graphic data into t he com pu te r an d r eq uire drawings of en ha nced qua lity . The st or/lga ca pacity of II compu ter should be appeeci abl y large to store t hll cede obtained i n automatic rea d ou t. H owev er , reliable

algor i t hrus fur h igh- s peed recognftic n of geome tr ic pa ue rus ar c not ye t avar lnhle . Fur t h is r ea son , e u t om er rc graphic dllta inp ut devi ces ha ve net fou nd w ide ap plica tions in AD Ss, Semtautomn uc G ID s o por e te on t h!.! foll ow ing gen eral principle , Wld le a na l,\' ziug t lll,\ tlrnwing, t he 0JIOI'aLo r fi xes t he ec rue t or lit II deff nite po fnt of t he d l'awi llg a nt! tilt'.n bri ngs the n ov ice in to opera tion to ca l cula te t he po in t coo -d tne tes and r epresen t th e da tn in t he num eri cal co de. T hus t he dev ice automa t Ically ee t cu te te s the cocr d tnotes of po int s chose n by the operator. AD Ss use 1,0 ad va ntage sem ia utom at ic data inpu t dev ices h e v iug a working rillid t 000 mm by 1 OCIIJ mm ill ai ze , wh ich nU! lIS111'C coc r dmatee 10 better than 0.2,':1 mm . Seurtoutomnttc G ID~ us ing CR T s WiUl rcgonore uon , li ght; pen s . coorruue t o hells , MId ot.her ac t ua tors IJOld rathe r consfde r a hl a promise fOI' uso in ADS.'l, Al o ng with the fllcilitiC's perform i ng the funct ions of da ta i np ut or date o utp ut, de vi ce s ha ve reccn t ly fou nd use wh ich co n se-rve. both fun ct ion s. Se t u ps Ilre /lv lli lllb l~ wh fch combi ne II sem ia ut om a t ic gra ph ic d a t a input de vi ce w nh a n elec t ro mec ha n ica l auto ma ti c dr awi ng mach ine. The a p proach to in tegra tin g da ta in pu t a nd ou tpu t dev ices into single CR T- Imst' d in p ut -out put units h as led 10 the crea tion of II device (or L111l o n-li ne gra ph ic acce ss 1-0 th e comp uter sto r age. wh ich is k now n as " gra phic dis p lay . T he s ys te ms or eutoma te d des ign no w emp'loy graphrc di splnys of ver tous t ypes deve loped i n severa l cou nt r ies (USSR, USA, Fra nce , J npa n , e te. ). In evolv ing II Il auto m at ed des ig n system, the set of rnc ruu ee is ch osen in eecn pnrtt cular ca se w ith cons tdera t ton for th e probl em s t o be sol ved ; the body lind s t re a ms of infn r ma t.ion; th e time of genera ti on, process ing , a nd tre usrer of i rifor ma t ion; the fo rm and kind of input an d Olltpu t d a ta ca rr iers ; compa t ibi li ty of available equ ipment . code and p rog ram fnciJit ie.'l ; t he t ime required lor cons trueu on of the sy stem: an d t he cos t. Figuro 13, 3 shows th e block d iagram of a probl em-orie nt ed autome tod design com p lex w ith /Ii u nive rsal set of per ip heral fa cilities. Th is com p le x can for m th o basis for t he de ve lo pme nt of v a r-i ous s yste ms of a utoma ted des ign. As see n: from t he figure , the s yst em h as t hree gr oups of ha rd ware me a ns wh i ch e na b le t he compu ter t o be run ill t he i nte rac t ive mode . The fi rs t gr ou p in cludes t he facil ities hav in g a direct li nk w it h. t he cen t ra l processor, T hese are a ca r d r ea der . t ape render-, alphanumecic pr i nte r, graphi c display , and opera tor conso le com p r is ing a vi deo k eyboard a nd dev i ces of grou p con t rol aud b lock contro l. T a pe an d di sk stora ges for m ex te r na l stora ge faciU Ue.'l. The sec o nd grou p of hardware formÂˇs a sa te lltte s ys tem h a v i ng 8 li nk with t he ce n t ral processor vi a a fro nt-e nd com p ut er of th e int ernational syst em t)'pe with a small stor age capacity. The external storllge uni ts here ar e di sk s t or age and laJXl ca rtridge storage u n it s ,

D;l I11 i npu t fucil ilics are card lIun tape readers, a nn a eemi a uto ma trc de ta In pu t devi ce or the 'plo tti ng board t y pe. Othac mea ns or me teete t s up por t in clu de a n nlphanumer tc pr inter , gra ph plo tter , gra phic monito r, microfilming setup. and de v ice for nn-l i ne e xecut ion of recor ds . T he third gro up of ha rdware forms a te rm inal sys tem connected 1.0 tne cent ral processor vta i\ commumcatinn line , This sys te m whose

fig - 1:S.3, T'lie 1Iiock diagI1llJ'l of ADS

r<~ -c~'~""';'~l~~'i;/gP~~~\';~jd~rJ:.~01t ns~!!':~.'1un:z;:~ ~~i nW:"'~ZJl; t:f.,"r ~~:

FEe - r, on l"'M "" mput. ,; BP _ bO.,d Pll>t l. , ; TCS - tal'<' ea' t'~dII. ilarDi . ; GP _ i to-

p!lle

fl l ~Il . , ;

a.,r- g,u.h lC m O ll ll ,~ , ;

MS - mlcroltl><>tor;re p h r "" 1" 1t; Btl - o ,>-]In. Utc1lÂˇ

lion or l't"O,d. ; t.;G _ 1I<001t con l ro l ; Be _ lilock con' ro l

siting ean be in a des ign otttce or ;l s ubd jv teto n en sures para llel work of Il few auto mated set ups in the netw ork of t he functionally oriented com plex . The automa ted se tup for an-ope rator (desi gner) ge nerel ly comprises n smal l-cap acity computer, graph ic da ta Input- outp ut a nd display dov lcos. exter nal s torage u nit s , a nd me ans or commu n ication wil li t he cent ral processor. Soviet 'industry has sta rted t o tu rn out au tomated se tups for des ign of rud toetect ronic equi p ment , eolu uo n t o pro blema involved in too ling-up for prod uct ion , and des ign of mach tncry. The Sov iet-made st andard a uto mated set up consists of a p ro-

I).~ ,

Conclu sion

eeesc r , pr imnrl' s tora ge un it , s r mb olie da t il input a nd display uev tce , pu nched-ta pe inpu t -output dev ice. ex ternal s lo ra ge- u n it. grl\pb ic lerll iinal tl,'1'a pb ic d isp lay), analog-to-n umbet ec nver ter (ima go eode r ], t8 pe re ader, lind bo ard pl otter. T he software of tb e SEllnp pt'rmill lh e s.imu ltll Dl'O us upe ra t.ion of a ll de vt ccs . ioelu d inil' ' he in JllIl 0ut p uL device and :raPh ic ed ito r . T i,e ha rd wa re e ng inee ri lli is making progn'S9 II I 8 treme ndou:s peee nnd more ad vRllCed dev tces ta ke e ve r con5Ln ntl y. But Ibe peln liulil. ci pill arc nu ecture cr u.e A DS a ud its beetc ha rd W1l Te clUI . 4

13,4. Conclus ion O lll" Il f Ihe lmp c r tn ut t as ks of e l(!cl n)lllecJulllics is 10 ev olve alectrfe mac hi nes of c c tre met y h igh powers , m ach tnas of nov e l u nified Sl"r lNl, a nd also s peoia t m eclunes for Ylll"io1t.'! e ppllcat lona . Th o gro wing de-ma ud Ior energ y h a,' rniS(>d l ila 41 rob loln s of St.'tl.l;ing ne w e ner gy 90ur«'5 a nd de \'e lo pi ng new onerg~' co nver te rs a par t fr om t ho probIt'1lI of Im pre v ing t he e nerlD' cha rllcle r i5li cs of eouve n tiona l ele cte lc m achi nes . I II tI,ls eo n eec t tc n it is hi~ltly impor ta nl lo find wa ys of cre. ti nr eHidl' nt gcnerators Iha t woul d oon\'C!rt the solar COHIl)' Into elee tr tc form a nd 1111r n('2 t he t hermonu ctear rea ct or for g(l,.... rlil li o n of h igh - v o ltll ~ e ne rll:)· . T IH~ pro blem co nf ro nti llg e ngi neers t od lly ca ll 0 111)' btl w pcd .,,.-jl h b~' ad vl> lIc illl; the t he or )' or eteerere ma eld ne.!l Slill fu r tl,,-'r , us i ng fo r t ile pu rpo se r nmput lug fltl' m t res Rnd prun nr il y dig ita l com pu te rs . I n roce nl )'('8 rs e ngin(l('r ll ne ve mn naged 10 in \'e:;t iga le a num be r- of pro ble ms ""rlie r eunsidc red u uscl vahlc, These nre tl l(\ prc h lems for t he eet u uc n of cquationa wit h nOlll illclir pnrl\mtllcrs an d equ III,ion.'l of t ra nsi en ts in c-IIerg )' ce nv er te -s npc rl\li ng o n nons t nuso td a l a nd 4s)' m me l ri c s up pky vo ttege a. B"!I[des, iI 11M become poss ibl e 10 solve proble ma re lnt in g lo Irl\USle n ts in m uil iwind ing mllch ines . e Ll crlrY cc uvo rs to n pmcess es ill nia chillC! with 'n ail )' deg rees of

freedom , build up or the electromagne t.ic torq ue III one rc votuuoo. a nd Intcraono n o f lin eOQrg~' co nver te r w il h e lem en ts con necte d to th e stato r an d rotor . T he p rob le ms for t ho I1 na l)' s is of l"llot' rg r co nv ers ton in etecr rt e rnlldllne.s I'e ma i n t he fOC1l5 of a t teauo n a i, th e prese nt l imo tno , " " lIlog lind rligi tlll co m puters e M bll" ure a na lysl IU pass from diHcrenl illl eq u anc us desc r ib in g tra nsi e ntll t o com plex e qUAtion!' a nd in\,c" t igate st ead y-ata te processes as II spot:;ifi c e.'\SO of t he SOl1l4 lio n of com plica ted d iUel"tntial equat ions. T he n "" I)'sl. ca n proceed from t he ll'ener nl to the pArhcu la r ra the r t ha n fro m t he e ire u it models a nd equa t te ns of lIta l ic., 10 d)' Dllm ics lIS w• .e t he case before tile ndv en t o f eo mp u t i ng rnllch incs . T here is lin urge nt need for co nduct ing invc$t il.!'a t io lls w Hh t he (lim 10 generaliz e t he ap plicat ion of a na log II - UIl 13

Ch. 13. Aotornatotd

D.~;'; n

01 E!ed rk Machi ne.

a nd d igital com pu ters to the sol ut ion o f the definite tYPIlS of problems . An increase in t.he number of comple x sy st ems of d iffer enti al eq ueucne poses <In importa nt pro blem of s implify ing th e ma t hcmetteal models ami oval ua t i ng the acc uracy of sotuuons. Si mplified me thcmet tca l model s obta ined by t he ex per ime n ta l des ign technique are fi ndi ng eve r increasing a ppli cation . The deve lopm en t of poly nomial models a pplied t o t he solu t ion 01 t he probll?ms of sy nth esis of electric machines will facilitate th o evolu ti on of mo re unproved method s of geom etric p rogra mmi ng. It sho uld he recognized tha t tile adva ncemen ts in t he sy nt hesis of e nerg y con verte rs lire fa. r fro m bei ng as h igh as they 8 1'0 in the fie ld of analysi s, so much s till rem uins to be done to ra ise t he precne e el sy nt hesls 10 a h igher level. T he search tor mom improved methods of optimiu tio n will cr-ll'tnin ly con t i nue and pro bably yi el d several op t imization methods Jor the definite classes of pro blems. Aut omated desig n svs te ms re present t he h ighesl achieve me nt ill t he field of synt hesis of ener gy converters . Of paramount importance here is t he estebhshme nt of dat a ba nks a nd progra m lt brartes . To accomplish the end requ ires the pooling of efforts 1I0t onl y wlthin n pa rtic ular bra nch of i ndustey but also in the fram ework of th e I merna t iona l Electrotechutea l Comm issi on . The creat ion of a n ADS of ele ct r ic maehines tha t could do jobs stnrling from t he req uest for the pcoposa I an t! e nd ing with t he shipmeut of machi nes is au eco nomically warran ted task . t hou gh it is one of th e d iffi cu lt ta sk s and requ ires cc nstdera ble errocts. T he erea uou o f the gt'lwr a l t heory of ele ctr-ic ma ch ines is essential for t he unifi ed mathematical description of ene rgy co nversion pro-cesses in magnatic-fia lrl. electr ic-fie ld, an d elecl rom:lguct ic-fiel d energ y co nverter s . As is k nown from the h is l ory of alect rcmccha nics. many sc ie ntis ts ma de a ttem pts t o work out Lhe gl.neral t heo ry of l.'nergy conveners. T here Are f1 ~ present equa t ions describing energy eonve rsrcn pr ocesses in alcotrfc-Held an d electromagne li c-fi eld ma ch ines. How ever. man y difficu ltles ha ve yt't t o be o vercome to prod uce commercia l ver sio ns cf t hese types of machine. More rese ar ch in th e field of the gen eral theo ry of elocu- lc machi nes lind exte nsive work on the ciea u on of new e lec tr -ical engineerin g meterInls will obv iousl y Iee nne te t he d evelo pme nt of new cres ses of etecnte machines. T he equations for 8 generalize([ olac t ro meeha nical euergv converte r permit form ul at ing fI mathe matical mode l praot .ioall v Ior a ny pr oble m in mod ern e leotromecha nlcs. Th e notion of th e ge nC'rtlli zed energy conver t er will u ndergo ch a nges with tim e. Th e general model will obv ious ly be needed t o "wri le equation s for electr ic- fiel d a nd el ect romagnett c-I leld e nergy convene rs with ma ny degrees of freedom , equa t ions for rlescr ibing t ilt' conversion of energy in el ectric

ma ch ines ill e t her form s of en ergi . a nd equa tious lor tilo solu uo n of uniq ue problems in e lect rfo maclline e n: inl!'e rlng. B il L t he principa l ap proach to tile si multaneous anll.!}'s! s of fie lds /'Iud r urre nt.ll teking p.rt In ene rg)Âˇ ee nversi c n peoeeesee wl1l lArgely remain the sa me. T he notion of th e e ledric. machi ne as 011 elee tecmeche nice.I e llergy conv e rte r of Rny design veraie n wiU b.,'e 10 exte nd co ncurren t wit h t be SCl.rch for new ph y~ ical pben omen a l ha t would enable t he creat io n of an eDug}' ecnvener with unique proper ti es. TIleorctical investigat ion! into rnagn otic. electr ic. lJlenna l . a nd mec ha nica l fields find t heir compl icated i nt.eracti on in a n energy eon "ertet vdJl promote Iurthe r Ui& develo pme nt of th e t neoty of eleetrtc machi nes an d w ill th us offer innovllUOns in the fie ld of e lect ric ma ehln e ellgim"e rirlg. T he satement that a n electric mach ine eonvert a ,me rtl' from elee t rfc to mE'Chllnical for m or from mechB nlca.l to el ectne form wtth t he Ilt tendll n'- Lrfln.'4fotmfltio n of e nergy into bea t CIlJlS for ex te ll,!h'e ln vestlgat.ion on therma l- physical prob lems. T horo e re medli nl!$ in whi ch it is d ifficult to givo preference e it her to eleerre meche nlcel phenom en a or t hN mDl-physicll l phe nomena. Crvogen ic electric machi nes, MGD e nerg y converters, a nd energy s t ~r8ge devices may serve as a n example . ':\o dou bt rllrther advancemen u in t ilt< theory of ele ctromecha nica l e ne-l'gy convers ion will grelltl y pr omote olccete machine enii neer ing.

Appe nd ice s

Appendix I. Tho Equation s 01 thCl " sic Types ot Electric Mach ihe . Blcx k Diagrams lo r Solution o f lhe Equat ions on Co mp utClrs T he equa t io ns {in curre nts} of .•

" " ;

IHI

indu ct ion mo tor ha ve the form

( I llQ• R'.) 17 - -ro '" -

M .,

77 1"

. =-;I [ - 77 R' "1'+ Q), ({,.. + 77 M{')J M '..lJ " - 77

whe re P = dldl; a nti p is t he number of polo pairs . T he bloc k dtegr3m for the so lut ion of these eq uatlons te she wn i n r ig . A1. T he equat ions (in flu ): lin k ll.gc ~l of no induction mot or e re

d'f'; rtr:: -

n'u

LoL'

R' L'

ift'"= - L'L '

R 'Itf

1,. .. + L'L' M'

W_+

R oAf

L' L'

"'. ,," ) It• , . = "'1' z I." .'M "'" ("'''''' IP .. Q. '1 ~ .

.'l!'

Ip"

.. -

'l.r' l'

,u f V. + liJ,'I".. .~o -T P (' f ""iii'""" " . - 'If r )

2<'

A.ppe ndlce.

i'~

- '''t

- C,

:E:r""

,- II

", ~,C, ~, c, ~ .,. MU, "l U, -"IU.

'.',

"I V.

i~~~' -~~ I~,~-@J V

:.I,

u-:

Fig_ AI. The block dl l&n.m for the Iilliut ioo of equations of I D induc tion mo tor J t o If _

(i n eurnm l.!ll l !l' _ lDUll l phu ,,"' b

.m l'll t l~n ;

T be block dtagra m {or Lite sol uu cn of equ auo ne (in flu x linh ges and currents) of a n indoct tcn motor is shown in Fig. A3. T ile equa-

Appoondke â&#x20AC;˘

~~~

" ,~L_/ _ \~ _r-,

... '~

\ ,'

\ ",

);

.' ~ '. ,

. ~ 14

Fig. 1\2. The Mock d iagram For the solution or eqcauocs 01 an lndw:t i"n moto r {ln nux link ages) .m l,lIf lt ..: ,\I V -

1- 11 _

lll ~ l tl P l\ or

u n it:/.

l ions have the form do/'

d de 'l'~ ._,

U m COl! et - H' i:", -

lJl r ' l'rs

- RT/:;' ,

d'P\ dt~

U m Sln Âˇ U)t - J' "i"

do/'

-_". = w 'Y.' _ Rrl',

"•

-l~

'J , "

-i[,

-<»''' ,

'. ~

- I~

(1,) ,,...,,

'",_ r - , ~' ~L_/ i~ ., ~-L_./

- i~ _.r-,

Fig. 1\3. The blo ck d b lf ram lor th o sol ut ion of ecca ucca of on luduction ml, l" .. (In flux li nkages and ClIrre llts) l oU _ " n' plit lu 5: stu _ mu ltiplier unl b

it = "

i~ =

iii""

U lfl L", L fl.' L~ PL 'L' L f" L' m e L", 1,' 1.." L ~ T « - t 'L ' Lf"

'" 1.'Lt

L ;"

,It. rll -

LM L' L ' L i,.

'V,'

\Jf:"

If.

T he blo ck ui a gr:l," for th e so humn of eq ua ti o ns of n de motor i'$ given i n Fig . 1\4. T he eq unuo ns ar l' defi ned as dl a c. <D It" t : 1 d/lJu , I' I r" liI = - 7:;; " '.' - 7; " 7:; U' - d-' - = - 2fX1 u:" ' ;- ZPlJlj·. U

(])u , = CDm - Q), ,,•. G '

Ii " il7,', (11/ "1 - 1If' rl, "'"'Ji"" = J;;

Itl "" C",l])rr.r:,

:HS

whe re I ~ and I ; are lI,(· ( ur re n t" ill t he nr ma t uee c ltctlit lind escitali e n ci rc ui t r espoc l i" oly; 0 i.!. t he loa kagt clX'f ficie nt of mai n poles ; C~ a nd e... nro oJes i2 n eo nsra ms o f ll le mot ue: I II li1l R. Me lbe i" d w:.ln nce n nd re s ist all ce of 1111.' nrme t ure t.irellit rcs JI'{'cl ive ly; J ..

" ~"L_J

, 'i!C- .\ 40. 1'he bind , '{i"griln, lor tht' ~'l hn l " l1 of e\ l\lll H ollf ur " de moto r 1'6 - ~" W h ne ..; ,Vl/ _ nOlllll,.a" l \ " "n • • -'lU _ ,",,1111'11('1 unit •

.\1... nn d ,11r 11 ~ thl' m o l or torque a nd Ioa d t e r que ( 1lIOJlICllt o f restelAnce ) respective ly ; <1>. .. _ <!Jill - II>. ..... is t he n'-sult nlll ma :O:llette flll x dU8 In ex (";llI.l ioli Wi lld ll l~ S ; II i.!l t l",~ ro t at io na l s pee d: a od J .. is t l", mome nt of ille rt ill of tbe mc t e r., T he bl Ol:'k Iljll ~ril m fGr t he sol u tion flf e q l1lllio liS flf til dc gOJlel'1l tor I::> i/l us t ra ll',1 i ll F i ~ . A5 . The eq ue tious arc dl.

II"I

= - 7;; I<I - r;-

I ")A

" - T; - "' /.I ~

.lf b dl . + L;;I em", L;; dl

dl.

r dw ti l

L . die

d"l = 7i; U .

.11,1 d l "

til "7J - 7i;" dI

= J ...+J J , (.l1. - JI s J, .I/ .. = i o C . (J) r ...

Appe"d ices

where 11- / is th e Instautanecus value of vQ ltn gu JlCl'QSS tho lond ; e".. is th e tns umm nec ua (r6J>ulla nl) value of em f or generator rota tio n ; :11" a nd ,11 " II l"O rospe ctlvul'y u.o ccorrtctcnt s Ior th e direct a nd ba ck-

,.o-@-t!

HI." AS. T ile block ui agraol lor 11Ie- so!ulloll of equ ations of a de ge-Il cre lttf I-If _

~ m l! I ' lJ e ,"~

N IJ _

M" I 'ne~ " ll

" OliO, M V - n-,.. ' tJ p ll .'r lin " .

wa rd mut ua l inducta nces hut woen u.e excl tn tie n c trc un and ermnturn ctrcu it ; 2 .:..\u ~ is t he voltage dro p IIcr OO8 t he bru sh co nta ct ; /n fwd t , a re the in ~ t lln tant'ou s va l ues in tho ar mat ure an d excna uo n

Append, c.. .

etreena res peettvely: !II". a nd fIl, are the to rques on Lhe shafts of t he moto r a nd of t be gellC'ra lor; a nd J ". an d J1Il re lh emoments of inertia of ti,e mete r And geeer atce n,spec t ively.

- s".\-..l

- .m

~u "'D

' . , 'l'q

- '.

. ,.

_..

<O$ t ~ l

~~~

- ' '\ ,

~ ..

'-<,j

-,-

- \'~ '4

Y4iq

-'I'~1jj

- M,

. - vc ~ .. 1 - 1 ,

- e"

~. I., II '.

"~ " -;,

"'U

>, *,,"

Fi ll· A6.

l'h~

black di agtllnJ ror tho ~hl\J on of eqll.UOIU of • 9)1 nchrlln<l1l1 p<! r. m anen t-D1' l/n ~ t motor

l _l l _ . m p l ltltr• .

"R_

poll ,lI"

.. l ~ y ; ..11/ _ rtlull lll l l<'r u nU I: ,,·U - nonl ll\l!....llr un l"

Tbe block d iaaralll for the $Olutlo n of eq ua tiona of a llynchronous perm .Ilt>IIt-ma:::nel utotor ts shown In Fi g. A6. The equat ionl lin': of

Appe oo lces

25\

the form

d~d = -u",sin O+wr'l'q - i/iR"

d;q ..... u ntc os6-W r IY d-l qlf .

+ ;e<). Ro dqlQ _ R tJ%d<i i """"ifI --:T'Q'Q q -

dll'D RD ~M' Rf) 1p dt"='7DD~d-; D

i = d

"' DD :Z:d%DD j

d~r = ~

":d If

I: M! z dz!.JD

xQQ "If .2.qZQQ :Z::q q-

'Â¥

rDD - :r~ d

0x~ q

%ttl"QQ

('l' diq - 'l'q id)-M"

:r<t:1'DD

1p "'~q

'.!

= l - Wr

:Z::d

Ro If

rQO e

E E

-c

Hl (o ~ ,H I I"

'l'l O ~ ~· '('

~ I~~n-n

" l(> o ~ · n

IWh l 'V ~

' I~H l ''' '

~ a \lU l l -V!'

' \' ~ B -Yl

'''' IL-V' H it....'·""

, y ,'1-V\"

-, 0 0

-. <,'

;;, ~

:£

s ~ ~

...

•

, i

•--

•

lPl O ~i"- llV l

t " U~;; -Y '

' S O ~ ~-\'l

, 0

u' 9U -v,

1100l: - Vt

W,O'1-'"

I N n l -Y'

-, ~i <.

: ; 0

tI1 UH:'I'(l"

n rOf ;:· V,

• !'SGU' yt

t lf ~ ~ t · vt

" ~ U · vt

"JI'51 -V'

•

• •

;

,,•

•• ~

uno-v, H<Zll- Vt

\ VOli- V,

II o

,VU'VV t

,VU-VV"

'Y&~- Vvt

tI" (\~V\l '

•• •• I

~ v

.. ,. , .. . . · 0" J

;

·0

••

· 0

•

•. ••• -.r. .",

Tab le ,1,11.4 Seili el of VilFillb leli Mowr

Stlll._

•< .•, <

. .• .

. .,

roo

100

·••,

< -e

••

•<

M,~

100

1tXl

M M~

AI, Ml

< <

.?i i! •• . " roo " lO so •• so

m eo

50 25

o.tun eo

?6 ,G

2(1 2

21> I

:le

. ."

.,.

.•

•• ~

' flO 50 2U

' 00 50

teo

0 .1&9 HL 2

1I .1&!J 14 .r.

(J. Hi\)

o I [,!J

fU 5(1

O. t S!!

11 .2

14. 2

1-1 .2

1;-1.4

•

t n> ~

,•a

,-•...

••

.,:Q

-e

•

1((1

'00

HiJ

O.WJ 101.1

o.isc

100 50

15.4 Tdbl.. AIl.S

Gainli of Amp Uflers of the Comp lrtet Analog

1r1 =k l Irs = k.

0.5 u.55

0.5 088

0.5 0. 67

0 .5

c...js.j

0 .477 0 .7S 0 .6 0 . 714 o.en 0 .53 I). S!1.1 f). 57!! O.I,M

0.43

~~ = k.l ~

~1 ~ - ~I ~

7 . t 2G 9.8

~ n -~ l '

fl.!)?

~·a .... ~,

~': = ~IO

~3 =~ 1l

k ,, - k 1t

k u ""' kr,o

LSt 2.15

~, ~ = kzo -~' ~ s "": ~-H

, ,

6.4

3. t4

4.23

1. 31 1. 56

1.t 9

0 ."" 0 ,86

4 .07 0 .421 2 .41, O . ~ )()

:{ . t !,

3 . t <\.

2.~

, .4

1 . 1~

0.28

u. ~

:!BAli 8 .1

7.38 0 .21)2 0. 31/1 0 .249 I).·m 2 ,(l~ 2 ,53 j!.05 2.3 2 .388 2.2\, l.St 2 .22 2. 56 a.ac r .eas 2 .26

O .~

1.7 I .M 1M

:i .H

3 .H

1. 06 1.06 0 .55

0.(142 O. 7S 0 .004 0. 75

3 .14

0 .458 o .37 0.1,2 0.30 (; .28 6.28 6.28 O.2S 7 .70 7 .12 9.49 6 .78 0 .628 a .ll l l, 0 ." 0 .220 t.t,3 L 3 0. 807 L 2~ 1. 37 1.29 1. 2 \ 119 1. 42 1.<12 1.2', 1. 22 0 ,5\

3. 14

3 .B o.us 0. 8\ o.szs 0.59 0.369 0. 3S8 O.3Q 0.35 6 .28 6 .28 8 .&1

' .M

3 . 11,

0 .6

0.56 0 .33

c.aa 6 .28 9 .8~

0.143 G.tHS O.t3 5 t .tae 1 .\ tu t.tM l. 158 t.B r .ron U8 t. I i) I 1 ~1

' 51 .appendix III . Block Diagrit~ 01 the Models o f NonsinuJoid al Voltag e G enefaton D, v. ~

UJiIl'"

U. - U.,.sin....

u.Â·u..~o_

" "

Inn-

U. 0<4U..<05,"1

Fill'_ A7. Th. bloo::k diagr:lln

or Lh~

model of " noalinuaoldal "olu ge itllentor

, ... - ."1)111141: ...; JoJV - mu lllpilu _nl'-

R" "'1 11,

ID,

R. R"

Fig_ AS. The bl ock dl"lJ1''-Dl of " tt<: tani"l. r puJ.. gen HaUlr

u..,..

, ... - . ",p ll ll..,. . : MlJ - .,,,m pn......11\ P H - -..olarl Ull ....

,/, I T_ Oll n

25'

Append1c".

nDn

"T U,

DUD

~U,

>.l''1hU. - <>-<> Us 1.I

U , R,

I PR

u, -tJ:J-.t'l7"t;fI,,<1

block ,l inllr.1m or a n Inverter model w i th nulee- d"r~ llo n modulariou .:I ud wa n -f"rms 8~ gener ato r o\l~pu l u cut - c lll-oll vonall": PR _

p ~ l ar 'red

lela,

". App endix IV. An Example ,o f th e Eledr ie M ach in e

De l ig n Assignment

Que sti on s to be Treated 1. fih kc up lh (' oqun tions lor- a n lri\llIctiofl s q uirrel-cage mot or in s la l iO nllry cocrd fna tos t;L Rn~l ~ and redu ce t hem to t he form conven-

ie nt for tbeir so lu tion

0 11

nn a nrllog compu ter. Genera te the solu-

l ion s 10 th e equ ntio us ;11 tt,rms uf ellt r'HI' =< find fl ux li nk ages. 2 . Cons t r uct t he bl oc k ttill g ril. m for the sulutton of the syste m of

eq nu ti ous a utl ca lcu la te tho ga in for Q(lI:h inp ut of t he amplifier in t h e chose n llnolog of t he mot or.

Desi gn o f tile Expe rime nt a nd th e Solution

to the Dptt mi aetton P ro blem t . Choose lltQ vnria hlc pnr nmot c ra of t he mctor Mill form the ex perirne nta l de st gn m atr ix .

2 . Sim ula te t ho experimen t on o n nnnJo{!' ( om put ...," find doi tuo th e po fync miu l rula tio us betwee n t he chllToc l.(lIÂˇil,tir,s a nd param eters of t he mo lor. 3. De tcemi uo 11m lIp l lmum c nrn mcrers ,)f 1110 i udu et.inn motor under til(' s pecHie d eoudit.lo ns of nplim i ~l1 tl o n. q . E v nlua tc t il l' accurncy of

InreeuonÂť for Ihe

~l'n rc h

for 1111) o ptimum parame ters;

SlI /tt:tjOfl I II

the Problem

1. For the dortvn tt on 01 mduarlon mnchine eq ua t ions in terms of currc nts n ul! Ilux l ;Jllow gc~ , St.' 1) ClIn,pte r 2. These equ a ti on s fi nd t he bl ock il ill f;T~1ll (or t heir solutio n /Ire gi ve n in Appond !x T. T o Hlust r a te, tho way of how to reduce th o cq ue t Ion e to tho form cou vc mc ut for th eir so fut.ton 011 /Iu -aj)/I log compute r. we cons ider- a n exa m ple of the A42-lj t hroe-phase sq tur re l-ca ga motor Ilsinj;t its equn t icn s expre sse d i n term s of current s . Tho A42-6 hllS t he foll owin g nom tna l ChnI'D Cl"tist ir,s: p ~ ~ "'... 1.7 kW, U~ = 220/380 V, {~ = 7 .5 '4 .5 A. and n" ... 930 rpm . T ile mot or para me ters ot (I wor kl ng tem per ature of 75 (Ire as foll ows : r;:" = r~ _ 3.$Q r.~ = r~ =' 3.5/ Q L~ ~ r.J -= 0. 27\J H t; = L~ = MO . 28~ .H it! = 0 .263 1'1 J = 152. X IO -~ eu m p = 3 m =3 T ill' "qua t ion for C~ has th l')o rm U:. = i~ [ r& + (di d ! ) L:.l + i;:' (d l dl) M QC

200

Ap pe nd ieeJ

U:. = i!.r.

+ Ii. (dl dt ) L:,. + Ii. (d ldt ) AI

whence

i:'" (dld t) r..:, = U:'- r:'I:"- (dld t) AHi.

I ntroduce the des igna t ion dldt ...

Then,

. . i:"", U!.r. /Ll.r.p-,r!.r.i:"/L:"p- Mi;'ILI.r.

Substitut e her e the moto r pa rameters

i:" = 220V2 eos OJt/O.279p - 3.57i:"/O,279p - (0.263/0.279) Ii. i:" ... 116 COB

fl)tlp -

O .935~

12.8t:.fp -

This for m of the eq uation is more prefe rab le for it s solution on an ana lo g computer stnca t he analog sot up on t he com puter proves more

s table. T he equations for t he ro to r voltage a eo brought to t ne destred form in a s imilar manoer wit h conatderatte n for th. fact tba t 0 & = = U~ ... 0 because the rotor is of the cage typ o:

0= pM il.r. +[ri. +(d/dt ) Li.l ~ +L ~w~ i r. +Mwri&

pLi.ii.= - pMi:" - ii.r:.. - w, {L~i~ + Mi') 'r

~a=

r;' i~

<dr{l.r. I~+m i V

L:,.p

L i. p

-, ta - ~ L",

At ter su bstitution of t he numerical values, Lhl) Hnal expression becomes

i&= - 13,141&IP - O.91ic.- wc (O . 9 1i~ + l &)Ip The same approach works for redu cing the romllini ng sq uations to t he form conve nient for setting up t he comp uter Il;nIl log. T he ays(em of equat ions for the A4.2-6 motor has th e foll owing fi nal for m

i:'= t 116 cos (j) t l p -12.8i:,/p - O . 935i~ i ~ - 1 1t6 sjn wt/p - 1 2 .8i ~/p - O.9:-15i~ 1&= -

i~ =

t 3.t 4i&fp - O.9ti:. -w, (O.91t~ + 1 ~)lp t3 . 14i~ /p -O . 9 IiA + 6), (O.91/c. + i;')/p M . =O .121 6(/ ~ /& -i;'!;) dw. ldt = 1 Ui5 (AI . - M,)

'" To clea r up poi nt 1 of t he opt.imiaa t.ion pr oblem . Wll should take t ho mo tor para meters fr om 'I'a hle Ai V. 1 ill accorda nce with t he n um ~ ber o f the ass ign me nt va r -iant and reduce eac h i rntla l oq ue.uo n ex p ressed ill curre n ts or fl ux lin ka ges t o t he desi re d form. T he bl ock di a gra m for t he solut ion of t he syste m of equn uous of t he i nd uct io n m otor is built up i t). s tages . 1'110 first s tage invol ves

Pig. AtO. 'fhe block

dl~gram

for determining Slut.or eu r",n~

the cho ice of a pp ropr ia te C]ClIl\ln15 and calculn tt on o f t he ga in for each e qua tion and tJIC seco nd stage i uvolves lin.' connectio n an d switchi og of th e en t ire anal og networ k . As lin exa mp le . con sid er t he conswuc uo n of a com pu t.iug network li nd t he calcu la t ion of ga i n Ia cto rs for a ile of t he eq uat tc ns:

if. = 1 H 6 cos wtlp - 1 2.8 i~ /p - 0 .\l35i;' Th e computer a na log for t h is oq !lllti nn is shown i n F ig'. AtO. l t conta ins an Int eg ra tor wi t h t wo in puts t o in tegrute t he firs t an d t he seco nd ter m of the equa tio n a nn ,01/10 a s um mer t o ad d t oget her t he ln t cgr aj.ic n results lind m e va lues of tho subseque nt ter m. -0.935 ii;.. To der ive from the ou t put the vnlue of -i~ wh ich is th en se nt t o t he i npu t. of an amplifier 5 lind t o other computi ng ele me nts, t h{' net wor k in cor pora l es a n inve rter b nilt around nn nmplifie r 1.'1. Aft er co nst r uct i ng t ho block d iagram . we need to ca lcula te I,he am plifier ga i ns e quival ent 1,0 t he co nst a nts in the equnt.lo na. T he calcula rio n for th e summ ing lind th e {nlegr<ll i ng u n i(,g i~ mnde by use of the follow ing for mul as: k t = 111 ou,a /M i n '

k = M ou,alll! ,,,,!rT ,

whe re j'T j ", . ,'Vlo u l ' a nd M, are t he sca les of t he in put q ua ntHy, ou tpu t q ua nt ity . and lime respect iv ely. S inc.(l!ll, = 4 V IA, M (} = 100 vo lts per un it. an d M , = 50. we beve k~"'1 = 4 X 12.8/4 X 50 = = 0.256. k a_, = 4 x 1 116 /100 x 5 = 0 ./)93 , k l>-' = 414 = 1, and k'H = 4.0 X 03511, ...... 0.935 . l n II si milar' way t h o cal cu latio n is per for med for other un its lind t he ga in fa ctors are Ioun d , T he entire block d iagram o f t he com pu te r setup for th e so l utio n of the system of e quauo na is s he w n in F ig . Al i.

.... p p., ~d i c.,

I n till' ex peri me n1 be in g d es igned . th e va r ia hl e pa eamecers \ fnct ors) of the mot or am chosen [rum Ta b le AI V.2 ( Ill eccorua ece w u h the a!llticnmcll t VIlriant n um ber ) and the ve r te ucn f IU}t!;!:! of the fllr loTS li te preset , A l y p lCld dcs il.t" proce dure comet< to dl'si !::ll ing t he CPE of t he ~ t y pe (see Ta b le Al\T. 3) . I n each /I.~iio meot va r jant tK-ing

'-'

•

r:2l ., -,.-=>,13:>-' ".'~

l'i So! ' ,\ 11. The block dl;:ll: r~ Ol fur \.he ~ I " tilln or ( 'l uiltiom' of an Indnrllon

nHJt or

WH' " 1..,01 , t hI' moto r pll rll l ne~rs pre !!Cnh' d ill T il hi t< A.I \ '. 1 Il rl;' I ;,1..(' II

ba:oi,' va lues , The upper 11I{d lewee Iim il b of t l,l' ~' a ri n bl l' pa ranjete r a ro dd i'l£!(\ in Fll'w r da nec wi t l, th(' sp('t:Uied interva ls 110,1 me d••!,l IHl' put dow!> in the w hle (m a t r ix) . Til l! experimon t IX'in!: desi g n('.t1 <:il ll!! for th o varm ticn of a lew vlIriallle rect ors at d in t' re nt re vet s . Tho 2 1 e F'E tnvorv es oight oom hilllltio ns in the var tat lcu of v a. i:lb lo fact ors. t bc ecrore for ea ch row of t he rll.'$ ign IT\lltri.l" ""'0 need Il ~

"ppo ndiu '

t o oa rry OUI tho a pprop ria te recalculnt.io u of th e g" ius of lim a nal og, Fo r exam ple , if the vnria Lh.· fnctor is t ake n to be r', t he n it s value de turm i ncs lhn cocffi cien t affixed to I~.!p , wh ich il> C(llI:11 to 1,2.8 a t II ba ste v alue of f' ('(\I ~'\I to :~ .[,7Q. I n prcpor nou to Ih is vlllo O) we cn lculnre t he gnln "'~ ~ L = O.2!ii;. III SlJlling IIp lh e ox pcriment., th o vullle o f f ' is mud u t o va ry n \ tw,) levels: th e upp er le vel , + 1 nr 4.28Q ; th e lower lev el. -1 or 2.86n . I II aec ordn ncc wi t h t hQSO ve toes . the gain lor tl 'e nct wor'k of Fi g . AI1 need he rcenlcula te d . FI,r the love! wit h II "minus' sig n

k~_t

ll1" ~ I" !ilfin ·l{ t =

4 X 11.6[,.'1, X .10 ".0.2:13

<l nd {or th e le ve l with a ' plul>' s igo n k}-l ",.. .:, X 14.65M X 50 = 0 .293

I II cl fl r Hyi ug- pllilll 1 of '·ho llrn b l ~m. we sho uld lir e 10 it , ~o ll s-i ,Il'T_ i ng the specrl red varin bLo par cme t urs , what coelricicll t-s in the sysrem of equa t.lnns rHO vnrtable , the n rucalc.ulutc th o cOl'rellpo ud illl; gain recrors nnd s umma r-ize the cto;lu]ls in Ilw tn ble . _,, ~ 1' " r-xamplo, T nbfe A IV.::I g i v e~ t he ros uns or Ihl1 ce lcu lutlu n of ~u i1J fatl ol''! <I I mQrte l i ng the A42-B mo tor by so lv ing it s eq uatio ns ex pre ssed in Il ux liuknges. :\ s tm llcr teble I>h01l 1l1 be milde u p for t ho llss ig nml' nt vlll'innl iuv elviug l.JH~ sol ut io n o f ~h l:l mo t or equat lous e xp ress ed in tenas of curre nts . It IS now necess a ry to a nalyze We ob tnined design nmtrr x lind det cemine t he expedie nt seque nce o( it s iOlph.• men t a l.Inn on a ny nnnl og comput er SODS to reca lculate u sma l lae num ber of t ho ga i n fact ors in go ing from on ", t r ial 10 t he o tJ,er , T he t r iol i n expe r imenta l des ign me a ns t ile calc u lat.io n on a n a nalog computer of th e ebsrecr ertsucs o f the tra nsien t and slea dy~stll te pro ceseee ill t he i nduction motor at t l\e fi xed vnl uos of its parnmetor a. T ill" u s e d va hles of the motor parameters are ch ose n for eac h assignmon t. v nriant from T u ble AI V.2. Since in t ho co m puter all a los;: tho ampfifieea s how zero d r ift. t ho exper iment m ns r he re ru n 11 1· leas t th ree tun es and th e results entered, in t he tab le. 2. The e xper-i me nta l des ign e nables us t o obta in a sim ple rclall o n betwe e n t he va r iab le pa rame ters of t he m acbjnc and u s cbe rac toeistics. T h is re la tion has the form of a po lynomial

!I "" bo+~ bl2'l + I_ I

I<J

Il 1i%12)+.il bu x1+ I_ L

...

where b., b/o bjJ , bll e r e poly nom ial coarr tcte nrs: X" XJ a re the ve rt sMe pa ra met er s of t he i ndu ct.in n motor ; y is the operati ng fa ct or or faet ors (o bjlwttvo fun ctio ns) of tho mot or in its s ta u c nnd dynamic opeeat ton: a nd n is t h e num ber of vnri a ble fact or s [pnrc mc tees).

""The po ly nom ia l coeff icie nts in the CFE Ap p.",d lce.

" 3./y. /N, bl} = ti, = l}

lH O

found b)' t he formula

(1 = 0, 1, ... ,n)

Al ter det erm inin g tho poly nom ial coerncrent s hI> bi}. wo shou ld check t he tr s ignifi cance (tost the n ull hypothesi s) . A check of th o hypothesis is dOM by t he St ude nt t-ta st formulated in th is case as t , = Ib, l/S {b,}, S {b/}= Y S Z{bt } where S ~ (b,) ];1 t he err or varia nce i ll oSli mllt ing t he coof.liciont bj â&#x20AC;˘ I n th o CFE tllis ve rtanee for al l va lues of b, is -'l:ive n by S~ fbi } = 8 2 {y }/Nm where N is the numJx.r of points o[ the fa cluT s pace i n wh ich t he

des igned ex per im ent is

Sl' t

â&#x20AC;˘

UP i 8 2 {y } It /(m- t )) ): (YDI ",",

th e ex pcr lmeut v aria nce obta iu ed in cond uct ing 01 d Ill p..,inls

YV)l is

tr ia ls for one

.,t lh tl Iaetoe SPIiCtl u nd er s t udy : and fl.

",",

Yol/m.

I f t he found value of t he z-teat exceeds t he value t a ke n fr om 1'1lble A lVA for t he numbe r of degrees of freed om , v, = N (m - 1) , at t he s pecifled sign ificlln l level g. (% ), th e h y pot hesis is re ject ed und th o eccrnc tent Ql is 115SUml:ld sig nifica nt: otherwise , the hy po thellis is acce pte d and th e coeff lc ien t 01 is considered i osign if ie(lnt , I.e . equa l t o zero. The che ek of t he hypo thesis for the adequate re preaenta l.lnn of t he ex per-iment results by t he rou nd pol yn om ial te done Ieom lha es timate of the d iscrep ancy be t wee n the output va lu e of Yo an d t he exper imenta l va lu es o f Yo a t all po ints of the facto r space . Th e d is pers io n of the ex per-i men t re sults on t he a ppro xi matfug pol yn om ia l is describable by th e inadequa cy ve rten ce 0;" whoso es ti ma te S;" is found from tbe formula ~

,.,

S=.t= [ t f( N -d)]}:1 (y.-y. )' wh er e d is the numb er of the si gnifica nt t er ms i n t he ap pro xi ma ti ng

po lyn omia l . The inadequacy varia nce is dependent. o n t he number of t he degrees of freed om v g<l = N - d The check on ade qu acy consists in est.imat.ing the dev iation of th e inadeq uacy variance 0: <1 fro m the re prod ucibility variance o' {y }.

ae If a: d does no t e xcee d the ex per tme n t v ariaIl(;(l . tl, e ob t e ine d ffi l'llhema 'iul mod el adequat e ly r eprese nts t he res u ne of the expertment: if 0: " > a' {II} . t he d l!scr ip t io n is cons idered Ina deq ua te 10 t he o bj ect u nder analysis . T he chec k of t he hy po thes is for adequa cy is ca rr ied o ut usi ng F isher ' s v ar iance rat io tes t . T he F- Iest perm lU chec k i ng the Dull h )'pulhe.sis for the eq ua li ty of tw o IlC ne rali zed Vll.ri~ nce!l an d gt {y } i n th e c.nse w he re 5aDlpli ng ",'or ia ntes S: d > S: {II} = st. T ho F-lellt is defi ned as th e va r ia nce rat io

S:.

F = S:"IS'

H t he cefcule ted va l ue o f t he F- te st is s mAller t ha n its c ritlce l va lue fou nd from T a bl e A IV .5 f (}l' the cor respo nd ing deg rees of Ir eedorn , " a d = " Qd - N - d nn d "tool = " s = N (In - 1). at Hie s pee tfied s ig n ifica n t lev el qnd 1% ). lh e null hyp ot hesi s is accepted. Ot hcrw ise t he h y poth es is is rejected an d t he d escripti on Is eo natdc rod lnadequata t o t ilt! o bje ct unde r st" d ~' . If t he hypolhllSis of ad eq ullC)' is rlljec ted , Il sm alle r s to p of v e r tauc n is til ke n lind t h\:l e xperime nt i.:! ru n a new. :~. O pti mum param e ters o f t he mo lor I\r8 fou nd us ing the pol ynom h Ils defined in t he tex t. of point 2 of lhe pro ble m . Tho opti m i2 ~ tio n c rit er ion an d co nst.NIinl run cuc ns lire c hose n for eac h asslgnInenl vuia ut. from T able A IY.6. The po lyn om ial s for lJ lnd cos « in l\lIll.!Sigmne n t variants llN! t ho s s me and equa l to

= 0.8t5 - O.OIV - 0.0221"" fJl = 0.83 + 0.036"" - 0.018r' + O.026r' r'

cos

I n the typ ica l dea lgn proce du re. the c p tlm um p:lrllma lers are d et er m tued by t he se m igra ph ica l ml'tll od in wh ich tw o pa rame t ers are va ned withi n the limit s set up in t he ass lgn me nt . Consi de r the WI )" of es l.imoLi ng t h o op ti mu m parameters . ASIIl1nle that t he des ign e d ex por tmon t co nd uc ted for a cc et a!n t yp e of i nd uc t io n motor ha s g ive n lh o poly nomi nl re ln ttc us (or the Im pact current . im pact lorqu e , a nd sta r ti ng (s peedu p) ume 1lS Iu nctfo ns of t wo peeama ters. T hcStl re la t te ns nre of th e rcem

I ,... = 4 .5 - 0.31"" - OAr' + 0.1r''''= 3.5 + 0.5"'- - 0.5r' '.1 = 150 - 12,... + 2r' + 2r"r' ] ( to-I)

M,.

where r' lind r art' th e s ta to r lind rotor wind in s res ist a nces , respec th·o) y, ex pre ssed in ret euve u n us . Wh llt we ueed to det e rurinc are the va lues of ,... an d r' I t wh ich the s tM ti ng time is 1'1 mi n imu m , with Ih e oonstrlli nts tM- ing imposed Oil a nd M I•. 10 Its mAthematica l form , t he o ptim iza tio n prob-

I,,,,

I I -Oli n

,,, le m ea n bo wr it ten

;a:;1

'. , =

50 - 121"'

+ 1r + 2r' r' ...... mi n +

, , ... ..... 4 .!i _ 0 .31'" - O.4r' O. t r"r' + O.'>r'" _ O.Y ...... "

~ 4.~,

l,/ ,.. _ :S .!> H _ 1 "i; ,.. 0;;;;

I~ ' I

1 li n d _ M

....

i " ' ,.....,; .... t {t il. Yl'lr l " Uon r ang<-lI are """'ro>IllCU in rol Ative u ni ts i n an'" l ogy ",,11·h t h e d ""lg n m (lt d x }. (" ,'. or "r Ill' . ...·n _. l i ' n " n " io " " l "pllC(' fu r I ho ""rl abl,;,s r' ",,0.1 ,.. ' s flhnw n i " P ll,!. "12 . Th " v crtiee>l o f ti,,, H-qUII"-, n r c I I", p o ln l ~ r....

"--....:. »: ' In-· J.. ,, -,~ P A

A A

...'L

&> t tl ll ~

up t h e ! 1,cl or;,, 1 " "-lJ'Cr i·

m c n l .''l' hfl SP"C\' ClluHnod wi t!lin

t ho sq ua re is t he inirinl [ichl of t ho perm tss tbte ~ (/l ll ~iO rt>'! w i th t he cn nsl.rll i ll t.! d isf('!S'lI r dl'd .

Till.'

ri ri< ~

f ,,.. .... 4.5 -

co ns t rai nt 0.3 r: - 0 .4 r'

+O. Ir'"r" = 4.5 represents an tu toeeept of th e hyperbola brnncll .lf LV which p ll S!l',; t h rullj;l, fl (',cIIIrn 1 poin t Il n~1 0 110 of th" veeu ees . This see uon ea n be pl ntl l~1 U)' bri n:;:in~ t ill' equation for in to Cll nonic,,1 Ioem or by .!i u lJ.~ t il ll l ing ( t'rt lli" ( h ell values of rand r . Tho cnnslrni nl -l1,.. = 3.5 0.5"" - O.!".r' ~ 4 Pig. A12. 1'IM! h\l ..dilJll'U5ionaJ raetc r s~ for ~rillbles r' ulul ,.

I,.

represents au intercept of liltl li lle PLQ passi ng through the points with coordi nates (- I; 0) lind (0: ..L.l). 1'110 cons t ra i nts so impOlWd li mi t the Held 'If p~ r m is~i lJ ltl sohnioos Lo n polygon PLNG. T I ll) rel" l io n 1,1 - 1,jO - 12"" -"-- 2r + 2r""J (10- 1 )

defi nes the fnm ily of C.lI rn'!' rl tlJ,. A : H: , A : IJ, . , ., et c. 0 11 ench of these ('lIr\"l'~ III(' \'lI lu0 of t . 1 is cousrnnt , I n rno\·ing from. My. AIB t to AI8~. thl' value of t une t . 1 d~ )"! . T he point L is tile ectulio n Lo 111(' problem s ineii' al t h is poi nt th e um e t .. ta kes 11 minimum value upon !ati!fying the to lt !ill mi nl1l .w l lip (III / I.. I nu .11/.. . Tllll coe edinntcs of the ,"tl'iIl Hlln point or the value s of t he op timu m parameters of lhe Illari d ne ca n 00 found dirl'clly on t ho gr it! of Fi g. A12: r:1'1 ..., _ 0.45 . r.P/ = 0.58, .1 . 1 = 4.16 9

Append ices

4 . T o osti mat e tho accurac y u f t he svl nUoJl, we ~h oll hJ co m pare t he cnlcula ted d ntn with the simu latio n result s for th e u ptim iza tio n crit er ion IUld con stretn t runcucns a t t he opti m u m poin L. F UI' thts it is necessar y to conver t {rom th e rel ati ve u nits used in .les ig lling the ex perfme n t to t ho real uni ts. T rw eo nversl o n is acccmp l tshc . l . hy li lt .' Iormuln

::.;;, =

X i ",) IX j

( XI -

X j m in

where :1:-; is Ihe f lllln in .!!: v alue of ti ll: mot or pnr nme t nr- in relnti ve uo its: Xi m is I hi' meA n basic vnluo of t he vo rtehle pnrlUncu,r of the motor: ;1'1 q'ln is 11m lowe r li mit of the Vllrillbll.â&#x20AC;˘ par a me te r: nod Xj is t he runnhll( value of the mo tor para mete r in rue l unil.s , T h us, for r T vru-ied over t he ra ngl' of IIp t o 20 % (sec 'fa bl e A IV .:5). Xl

= r' =

a.s.

ZI ,Jl.ln

,. ;" IJl.

= :'l .<\

The o pt im um va lue u l Iln- rotor res ts tunee r~l)/ in rel a t ive units is to /p ' lll t o 0 .;18 . So . O. 5~ = (I'~ 11! - ;L~) !(3 .8 - 3.0-'.) or r; I" =

4.v.n .

In ' I s im ilflr wa y we cat culatc 11m SIlllor rcxis t.ancc, r~ " f ~ ., 3.2:JQ , an d then suhst.it uta the fou nd v a lues of th e opt i mu m pa rame ters i'lto thc. s ystem or eq lla li " " s " f' t he lnduc t.ion moto r whh thu simulutiuu in ter-ms of currents ur Il ux Itukages . I n accord anc e wit.h the op t imu m parcme tors of the motor . we recal cula te tho gllin teeter s of t he a na log IHhl set up t he pro ble m 0 11 a ll nmll og eo mputer [.0 measure th o opt tmiza t. ic n cri teri on find co natraiut Iu net to ns for LI,e op t imu m po int. hble " IV .I

Au11lnrncnt Vati.l'Ils

Var b M

I 2 3

, s

"7 8 Q

...

'" "

:>;".

' ... -'/\

, , , , ,s.e 3. 8

3.6

8.(, 3 .6

:> .1:1

MGI".. Hram. l t .

':.- "6

j~- I~

I" I

l~ _ j~

3 .5 7

0 . 2711

0 .2li3

0. 2119

a. 7

0.2.85

0. 20

0. 28 11 .28 0 .28 0 .28 0 ,28

(1.285

0 .2G

3 .~

o.285

3 .' 3.' 3. 7 3.5

0 .28$ 0 .285 0 .285

0. 26 0 .2(; 0 .25 0.25

'1. 5

0 .185

3. 7 3. 7

O. 27 ~

3 .7

O .Z&~

0.275

0 .25 0 .2 5 O.2G

o.ac

" 3 2

3 3

O.Z8

2 3

0. 28

0 .28

1).28

o.ae

Appond;cOI Tioble " IV.1 (<'OlIl lllr.td) ...to r p........ I ...

\'L lU t S D.

""" co 19

~:.

,,,.

a.7 a .7 3. 5 3. ' 3.7

4 4

s.e

'.G

' .5 3.7 3 .7 3 .5

a.s

a.e

,a.e

3.'

3. i

a.'

z.:. -t;

0. 2i 5 0 .215

0 .26 0 .26

0 .20

0 .%75 0. 273 0 .275 0. 285

U, 0 ." 0 .25 0.25

0 .26 0 .26

0.285

0 .26

0 .285 0.285 0.285 0 .285 0 .285 0. 285

0 .26 0.26

~ -Ji

I~ , 3

0 .:28

, I

0 .28

0.28 0 .27 0 .27

0 .26 0 .25

0. 27

0 .27

0 .211 0 .25

0 .27

o.n

, I

0 .21

NOll, . I . For all vari.nts l he loJlowl!1& ,·_. I ue ~ are 10 be set: I ... l52x fO· ' (I'll m=l. ", _3 . end (,',,- 22:0/3S0 v. 2. t wo ~ ria l ll U lD be l'$ of tbe .... rl.ot.5 IDYoh~ simulat ion t n le no s of ( utl"!'nl.5. . .... odd se rial J)ul'llbi'~ 10 te rms of fh u; linklll{l!s

~ho

\" rlant No.

, I

fo, tM ElperlNent to '" D05l9'"'"

V"ri a ble p........ 1U

" "I

" " "0 ' "

o' J I

• 0

10 II U 13

AT AT J M

J I J

AT AT J AT M M

MachIn, t ac to, (obje<: Uye r " I'ltIlOfl)

'" '" 1,. 1,. '" '"1,. M ,... I,. M ,. M,. '" '" I, .

AI ,,,,

1,.

'" I'" "

""'"

I i ni

M'm ,I( fm

1V..rl lt l"" -,

ranu ,

eo eo

"'ao20

'eoso" 20

eo 20

eo ao

Append ices

269 Tab le A IV.2 (ccllrt ,u l ed)

l\I"' IIlM r. CIO< (<>bJ«t"·. f~ncll on )

J I

" " " "r

""'

" " "

" "M M

" " ."

'., ,,"

M M

M I J I

Vat lOl lon r a nte.

M / n,

1,"

.1' 1...

',j

,ll l'" '" M,. ',.

I J I

J,f 1m

,," '., I'"i'" Table A IV.]

The G. ins of Ampl1flers 01 ihe Computer Analog

V. rloble ~o" ""'e"

Gai n of

Rel.r""ce

13 8 IJ 5 1 ~

~ " 2' 28 1 . 82

Upper 1" ' .1

~~~' I - I~ ~+.8+ .Z2 ff ~ ~ ~Nt~ 11~t I I ,.[ •. [··1..1·· <,

+ 3 4 + 3 e + 1 8 +

+ +

+ + + + +

·.• • to

+ + + + - - + + _+ _ + + - -+ +- - + - + - + - - + + + + + +

•, ;'

1).•

•, •

•..•, •., •,

•

1.252 1.252 2 .33 2 .33 1. 252

•

" nlo,

.•,•• •,,,. •,". ,

", ;.

1.138 1.2S5 1. 211 1.138 2 .39 2.2 5 2 . 11 L 2S5 1. 211 a.u 2 .39 2.35 1. 128 1.285 1.211 1 .~S2 1.138 2 .39 2 .25 2 .33 2 .11 1 . 2.8~ 1 211 2 .33 2 .U 2 .39 2 . 25

2 .77 2. 17 2 .17 2.n

1.1 95 I.l 9S 1.1 95 1.195 I. ~ O.6U 1 _8:'> O.&H _ 1 .8~ 1) .644 t .~ 0_6-\.i _

_ _ _

27Q

Appe ndlce. Tab le AII/.4 I h.e

, \

,a 5

V .l ~ • •

e l ille

12 .71

4. W

ia 13

3. 18

2.;8 V,7

2 .31

2 . 210

2 .23

'S~, a

Confidenc e l e ul

2 .20 2 .t 8

a.re

:l .13

Z.08 2 .07 2. 01 2.06 2.V<

2. 12

26 27

2. 0:;

2.'14

16 t7 18

23<;

'_le.t al

2.45

SI~denl

a.u

2 .W .2.UlI 2 .09

2.0!!

2.0~

2 .0'

l. 96

hble "I V.S FiJhe"s VarIance Ratios . 1 'S"f" Con lld e nce l e ve l

" I

IM .4 18. 5 10. 1 7 .7

e.o

5.S

a 9 10

"13

12 14

re 16 t7 18 19

21f>.7

224 .ij

230 .2

19.2

1!l .2

l!l.3

9 .0 6. 9

e.s

19 .J 9. \ 6.·4 5 .2

"3 .'

S., ,. e.e '" c.a

199.5

5.8

5 .~

5.1 5.0 ' .8 4 .8 4 .7

, .c 4 .5 4. '

,., '" 4 ..~

' .1

1, .0 3.' 3 .8 3 .7 3 .7 3 .1i 3 .' 3 .6 3 .5

e.e , .;

4 .' 4 .' 4 .1 3 .9 3 .7 3.' 3 .' 3.4

a.a 3. 3 3. 2 3.2 3.2 :U

4.1

3 n

3 _~

:i.4

3.3 3.2 3.1 3 .1 30 3.0 J..9 2 .'

c 23o\. .() \9 .3

o.c

8.'

6. 3 5. \

G. 2

4 .4 4 .0 3 .7 3.5 3 .3 3 .2 3.\ <1. 0 3 .0 2.'

4.3 3 .9 3.6

~ .O

3.2 3. 1 3 .' 2' 2 .8

2.8

2 .9

2 .7

2.8 2. ' 2. 1

2 .7 2 .8

2 .7

" I~ 24t, .~

HI.4 ' .7 5 .9 4 .7 4.0 3 .' 3.3 a.t 2 .~ I

2.8

249. 0 19 .4

2M .3

e.e

8.'

5 .8 4.' 3.8 3. ; 3. 1 2 .9

2.7 2 ,U

HI .S

s.n

4.4 3. 7 3 .2 2.9 2.7 2 .:' 2. 4 2 .3

2 .7

2 .5

2.'

2 .2

2.3 2.2

2 .1 2.1 Z.O

2.5 2 .5 2.4 2 .4 2 .3 2 .3

a.a

a.z

Z.O

2 .' 2 .1

I. ~ l

..,

2i l

Apptt ndic"s

l l"<!l1 l , ~,, ~,i)

Jabl e AIV .5

"aa

ae ac

'"uo 120

1, . 4 Ii .J

:'1 .5 3 .4

a.t

2 . fI

3. 1

4.3 '-2

3 .4

a.e

'.0

4.3

2.8 2. 8 2 .1 2 .'

.'1.2 , .1 4.Q

;l .;~

.'1 .2 3 .2

3 .!J

a. t

.~. 8

a.o

3. " 30 2 .9

2.9 2 .8 2 .1 2.6

2 .< 2. 6 2 .5 2.5 2.'

-l' .r-,

2.6 :U;

2 .3 2 .2

2 .6

2 .~

Z. 2

2 .11 2 .('

2.6 2 .11 2 .5

2."

2.~

2 .5 2 .5

2.2 2t

,.r.

2.5

2.~

2.4

2 .3

2.\) 1. 9

2 ." \. () 1. 8

1. 7

l.t,

2. 3

2 .2 2.£

I.r.

1. 3

1. 5

1.0

2 .2

:.l. 1

'-' I. ,

1 8 1

U L1 I .1

1. .5

r " b l,. AIV.6 Optirnila!i" n Crite rion .. ~ C on ,I.ai nt Fundio ns V ~ r l ~" t

1'\.,.

r..... ' ~\ .a ln \

"I'T in '\ 'M 100l or lt Pr i"lt

r ~ " tll O"S

1 2 3

r.. ",h,

1 /", ';;;: O.i1!>

1 / ", ", . ~ ' Imn l i m m tn

' _/ <0; 6 1'1"';; 6 '1 ~0 . 8J

Jf l '" -c 1).8 MI", ';';;; O.8 1 )", "'; 0 ..'17 I., ';;;5

I_I ml"

'1> 0.8.'1

l ' m ~ O . 3;

u 1 8

" "

' I'" ml" 'l"'a~

Il mu COSq'm u

'1,,,.,,,

11m mln I " ,n il. 'lm ..,

"" ""

"ar

2')

::!::!

,M,, ,,, e:

0.85 ..;;; 0..35

'I" o.as

M I ," ~ 'i. 85

M I '" "' 0 .8 l:oS 'f'> \l .8

M I II , ~ I I. ~

' 1 ~ ') _8"

CQ ~

'I' > 0.8

l i '~ ';;;

11 .37

'I ~ U. 83

C{l ~ 'f ~ O.8

'1 :':!> l".8;{

cO~'l'~ U , S2

·l ", . ~

CW 'l' >- 0.8 M I", ';':;; 0 .85

1., ,,, r.

I., ."1"

'1>0 .8:1

M lm ... 0 •8

1_/ 0;;;

1 ,m ", In '1 ",."

c()5 q·~ t'.8

'1> 0 .85

11m ';;; 0. 35

C<''''f ~ O. S

1 1m ", t"

M ' m <:;.0 .75 'l , 0 .82

cos e ze o.a

t., m i ll 'l'u.' x 'l nlu <:<lS 'Pmu 'l,n-x I / m ,n l"

11 ~ O,82

Al lr,, ';;; O . i~

1'1 ':;;;1• .35

' ! m ~ f) · 35 I .! .:;;; .... JI ~

M l m ";; 0 .75 V.S2 I lm ";;;O :1"

'I ~ U . .%

t.. ~ ...

.1f t ... 0;;;0.:5

'1 "

.as

Bibliog rophy

l. B. ,\ . A o:l k i n~. The Gelltro' Til"" , . , Elt cl,lc Ah. ll ll't•. Lond on , Chopmln a nd HIlII, 19,,9. 2. D. A. A"e ti~)·an . V. S. Sckol nv, lind V. KII. XII. n. Cotnp u tn - A ld"d Dn /,It Optlmiu//olt 0/ Elu/ rlc /lI ach /un . MOllCQ w, E lll'rg iYlI. 1076 (i n RlIl'!lilln j. 3. K. J . BllmB and P. J. Lnwrc nscn, A " " I,.sis "" d Compul"U6fl cf £ l' <lrl~ ond M It" u tl. F l. 1<1 }> rllblr Il'I . Oxford . P<!rgn mo n Pree s, 1963. <I . B. Broe hnll. S"p u<oll dll,r lll, "hr."" S y. u m,. Bt.l;n, Spri nif'r Vt rl .• 1073. 5. Drsll " 01 Blntru. .4112rhlnu. Edllt'd by I. P. Kopyl o'f . ?otOK OW, £ Mrr' Y' , 198<1 (in Rossl:)n ). G. DI":rYl , El« II'U Drl"" 1611.\ Sup M "lor,. E d ittd by M. G . Chilik in. Mosco_, Enttfiylt. l!111 (in Ru ssi an ). 1. fl . J . Duffin . E. L. Pel.l!nIOIl, and C. z eeer. C,.,nodrl c Pro, . OJ..... / II' . r~

N_ - V",*. Wilo~T, 1987 . I. La Wl'ftl5Qn. I nd M. SLcl'l'naon . Pet- U Nit S lid e", . "'flit. II> F.1n /ric M.clll" n . Camb r idlfl . Unl•. Pn.-ss . ll r; o. V. 1.... ,.o" ·Smo"',,!k y_ F./H ,......." ;lIc F lcldt. Prw- . I" F.l~lrlc c.'ll ..... • nll 1'lIy"nl M IHI; j("• • M ~O'ili, EMtriYI, 196'01 {Ill R' l.55bnl . A. hobolr-nJu~. N. r _ IIinMc y. and " p. K OI,)'JO' " e~~rl.WJl I.l Orllr " F.~'ro",-~ .,,'c•. MOlICO...... Enecg iYI . li15 (i" "'I$$i, ,.!. V• • I\ . l o' ''l!ky. T..... I. " '. '''' A C 1\1111"" ' 11", Mli!"Co " . USSR .... ndr-my o r

.. "" A p p l ko t ip ll .

S. AI . n .

Huri~. P. 1l~/~"JI'u

Srff''''

(I. A. .lh 10. U. I.. E.

...

~ itll(:e.o P" h l i~hi n~ H OUR , 1962 (i n R" uillu l. E. V. K onOllc'f1kn, G. .... . Slp_y ]6v . Bud 1\. A. "Khorkov . j;' /« I rl c J,h rh,,,u. MMeU"", "ysh.:I~·o d l1<oh, \ \115 (In Il uss h"'l. 13. I. P. l( oV)·l ov. E kclttJl ", crll.niu l P./U'1lI Co nI:t' Jl lu• . "'1 """0,,", En l'l'lilyl , 1'373 (i n Ih l,"ion) . 1. 1'. Kopyl()V. 1". A. ~lamed o", on d V. Va . 8e~ pn l<l v . M " tJ"'''' /tf l r QI M od.l· ' nr hI Jm/"m "" M Q M ln~· Mok ow. En~ "lt l y", 19119 (Ill ll.uOlS; nn-l. 15. /. P . Ko py ]ov 11m] O. P. S ht /'led rin . D l rllill Ch", ,.,,/;r- A l d. d D cll~ " III Jm/u rl /QIt "Hlldl l,u •. l\!OK Ow, E n,'rg lyn, n '13 l in ll u s.~i lln l . t e. C;. Kro n. Tm ra. A n" I, ,,:. hI N ;IWhrh. New Yor k, Lond on, """ Cd CIl. ld.

t a.

...

l OO!l. 1; . G. K ron . E If"l ". ',,'" Cfrr ,<lu al E lecl .,c ,lh clluu r, .

N~w

Yor k , Wil l Y.

100 7. \' . V. I\hn,~hc;~v. F,..cfl.~.l J10rl.... P_~ A1 O"<: ow, Ellt'l"l(iya , 19. & (i n Ru:!!:!!ian).

ce.

,\I e G r. .,· H,Il. 19611. \". ".. ti l.lImo'\'". TIl~ TMoIl' QJ F.~f';riltU"'. )f ~'. Nauh, 19;1 (lp R u!-a il.DI.

i u. E. Ltv , a nd U.

". ".

"'.cAI";' I~' A ot" .... Uc f)cli'C~J. POIlIPl"_ EIt<-I. . ...I'CIl. /lICJJII l'_e~ e""""""n. New Yo rk ,

C . N. Peue v. ez..c,. fc ." .dIM', M ~._, E ...·"lliy•• Pa rt I : 1(1,4, I' . rt 2; 19&3, Part 3; 19Gd (i n ",,~i an1 · . K. lot. Poli ~.no ... F.1Jci:1 rt>d' JIfu . lc. of .11".,,", .U;di• . MO!ll:ont, £nereo lIllat. 19l52 (I u RW;oSill1ll'

273'-

23. I. M. Pcatnikov. G~,..,.rall"d Th~ory ~n d T .a nd~"'s In EI«irle M achln n •. Moscow, Vyssha ya shkola , 1\175 (in , B II~; a ll) . 24. Spula J El «/rie Machlnos. S o" t.e' a nd Con~rlerr 01 energy . Edi ted by ' A. J. Hert tnuv. Mo.se"w, Energob da l, 1982 (in RusSianl' 25. I. E. Tamol . F llnda ment al, of ~ Theory 0/ Eleclrlcltll' r. 05(:OW, Mir PllbUshers, 1979. 26. o. V. TOl OpL AII/omattc Compillation 01 E kclroma r""'l1~ FIeld Problems_ Kiev, Tek bnik a , IOB7 (Ill KllSll ian). 27. V. A. Venlk ov. Th e ThelJrV of S imili tude and Modelln ,. Moscow, Vyssl,ayaehkoia , 1!J 7G (In Ru ssian ). 28. O. N . Vu e lovsky and Ya . A. Sbnei be'1l' P l1Wa E" g/ n ~ rl n , u nu 11& D ev< l ~ opm enl. MOo'lCow. Vys::;baya shk ol a, 11170 (In Russian ). 29. A. l, Voldek. li k clric }.fad./nes. Mosecw, Epc rglyo . 1974 (in RUMinn]. 30. A. I. Voldck . ]n d" cllon M agnelM ydrady nam lc 1I1 ac/llnee Willi Ll 2111dMetall IC Wa rk lng M edia. Mosec w, En erg lya . 1070 (in RUS>l inn ). 31. }'. M. Y uferov. E k elrlC Ma chfnrr 01 A ll /o nl /llle D n l ee8. Moscow, Vysshaya. shkola, 1\176 (in RUSlli an) .

Index

.Adk i ll _~ ,

ap pl lcatlon of. 38-<14 h",ic f('IHure'< elf. <12 ,li gil ,,!. 3~·"4 fil'>lI_lll'nc rH;oll, 39 rourus-ge nere uc n. 42·43

13.. 1:1

A IllP~ ("(',

A , ~1. . 13

ArnpCl·e'.• tR W, 20, 23 Am pli ficr Il ~in , . f.l -1'2 AM log eomputces , ;;8, (;2 appllcauou " f , 5>1-(;2 block di Qgr"m "I. ss .Angul,, , ~ {lt:e d lit f ield , 18 Asyrn m ,' lr~', .;!4",tr ic'll , 15l\· 1~9 magl\l'l ic , 1 ~1 5 !l

spuctcf.

socond-gene eauen, 39

1 5!l-1l~1

All l o m ~ l.(>d

l"' ,ig n eystcme (A DS), 2211-2/1\ block lli,, !o! ra m 01, 240 f", ~ 1o'l\lri" mu h i n~'!!, 229-2.1 0 lor i nd "c~ i o ll mn C h ; ll<'~ , l 'l B-'l'l9 llud wnl"e elf. 231...21, \ , ult '\'a re or. 230-230 t yl'~'!'

-r.

2:10

Barl ow. P " 9 Blomlel. A .. 13 Bol" lo\', A. T . , 12

'Choieo

"r , r.a l ~ , .

I~J

Coerf k icnl,; lIf rOllplillg, 1/,5-146 C.o lnnlUtoh)T IIwc.llillC "'il h ircvolviug

brushes. 40 1 !lZ-1!I ~

.c"mpu l"r~.

" ..,101;,

:-18-,,'"

~-3!l

"x l' ~ri m cll t

l}~tJl

h'lIlk . 22~ f..Z:W

Deg 1'{'l' u r diHicu ltr of problcm, ZUI DiffN'(.'Ilt' " 1 o,w mto,', 36 llirichlct condit ions , 21 IH~pl~y. grap hic, 239 trohvo-Dcbrc vclsky. M. 0 " 11, 13 nrc)'fus . L. , 13 Umd Iuuc t i r m. 218 Dual v"l'i " bJc, 218

Edison , T . A., 10. l H "'free_I of sat uratt on . 142-147

Coenran's t~'St. 207

Complete r" d oc'i" l

t hircl""/l'{'ner;ll ion, 39-<12 £C rnaehtne, 40-42 I UM360, 4(' hard wll'" or. 41 h~' br i d , <12-43 sof twoec 01. 41 Curre n t loops . 73

(en,j,

£1<,.:1"; <: Call1 llUlt . 17 1 r.kr. l. l·"m~'ch""ir~,l

r ne'IU'

14·1 9 c rri ci" Ilc)' of, H · 15

first law of. 14·\1; """:'U1l<l Jaw of. 16- 18 ~h l rd l" w of, HI-HI

con ~rr;i'ltl ,

Inclu

'E!e.:IrOllltd,a niea l t nt 'll'Y con verte r, comr~ll.

u -re

eleclromag" ellc-fie ld. 18().1811 lfeDl'raliled, 16-33 lilll"" r. Hoi-In mall'oell~fj ellt , t 19 mode- " f " 1'E'r l l ion or. If, plItluoclric . 178 I'In"M!tric eleel ric- f1eld . 184· HI5 pi. lOl'l l'C l ric. 18S-186

pul.:Je. 114·111> sim pk-. 14-1S E1et ln"lnll!C hln lul _11lCt'. t1 Eq WltiOl\5 .. r ll_... lited elec tric. Inac hi nc. 44· 51

li me, 11-n. upper. 13 U.,,·kfhec . t' . • 12 Hcfnl'r-A!t etll'( k , .' o, t o IIclmholl ll. H. L.v. . 13 Her lan d , A. , 13 H i ~lo;Ki n l d" \',,I" I"'>(' 1I1 of Ihl't>rr of l'lftlric ,,,,,ch ine:5. 9-1'

fndllCt lnn

mkh i ~.

n',

ph msor

loll•. A.

Pqtl,v:alO"f\t

c i~u i l

,j~

d l~l'lIm

l>f, 5G

s., t4

losi f)'ID , A, G. , 13

Itereuve 'lIl'\ hod, 21·28 " I rillb y . M•• !I F.... d~ r ' llllolor. 0·10. 38

J OIIl':!. I. Po, 13

FIeld pqullt lnn 5. \ 9-31 .'i n ;l e d ilf cn:n C\! methOlI (F OMI. :!t\-31 .'i ~h e r· ~ va rillncll . nlio (F-t f"lt). 195.

K~ I) lY ~ Il~k)' ...\

206, 21(1 1'Iow

di ~ l rl b\lH on

In eler.trlc machin e ,

rs r cr te eue. D., 13 I'ollric PIl rlu. 14 Fre'lucnc)' cOIl'·crler.

. £o, 101 Go, \3 i:\~ ~ov$k )', E, Ya., I ~ Kir chhoff's aecollil Jaw , II \ f(~ P I',

KOlI \{'n ko, M.P.. HI KO"llc h, K .. \3 KhnJ$hchev , V. V., 13

KI'Oll . G. , \;1, 35. 83 Co p " "",," ifon llil)', 136 Centro !Or, auto mot ive . 11 eO"w . t in .. I)·Jl(' \'00 de Gral ff. t:l ,

1S3 Indlld o e, 178

rMo!lWloh)'drodsnam Le (MHO), 3 138 . 112· 174. oOll:"inll5O\i dl l 'l'olu.,.ee, 100 GI'Oll>l' l r ic pr"Gu mmin;;. 214·:220 GOl"o!v . A• •\ .. U Gra"unoe. 1.:.. 9 Gnuu1", L. N., 13 G""'rickl. ,'on 0 . • 12 J-1 arm Oll 'e:I. hel('f'lldyne-lre'l u<:!fIcl", 7S

spact' , 11-16

K mg . 1\. A .. \ 3

' .:Iir.nqil ll I\\uIU,' h er:;. 212. 220 Lllpl:lC1'" " '1\1. 1'0" . 2 1. ea. 24.. [.cPt , H. •'• •; •• 13 lAl'Ill>l105O v. )I . v.. 12· 13

Midline. COlIIDlUllllllf. 33 de, 122-125 donble-rulltiO". 100 fdu lhcd , 32 ;Ildue lion . 32 JIlul hw indl nr . 116-111 prl m;t ive. 31· 38 sync hronoU!l . 55-5 1, 117-122

-a

It>d.. wi t h with w ilh

t breoe t1 eg l'('t'S

06' fou r t1 ('gT~l~ fl ve

06' dl'greeli 01 Ir l"E!dom, 06' degrees of free dom , bypot he· " i;>e

wH h

wi t h

of frCCd\>lll, 108 t1 egre e.!l of fT e(!dom,

t ica l. 168 Magn et ic field strengt h, 2 1 Ma gneHc IJUJI d ensit y , dc fin iti on of,

20 Mal)ll elshto m , L. I. , 14 r.lat rh , d('Sign . In illveT5e, 4 8 , 19 1 noise , H ~ of com ple te fn<: to ria l

c~pHi m ent

(er E), r na ort hogoll l\\ cen t ra l cc moceu c lIesigll (OCeD ), 2{J4 ro tata ble C M ~ra l co m posi t,· d,-slgn

Ne " m pnn Gond i t ioml, 22 New t.on-Il aphll()n me~h od , 29, 39 Nonl tneee \ u nsf\>r " ,er. t44

Objlletl ve m ult iplier , 220 Ohm, G . • 13 Opli mintion cri te rion , 210

Pae fno t t t, ,\ " 9 PpcJo oU I-Gramme'.i machine , tl-I O p " p;,lebsl , N, D., 14 Park , R" 13 Pe t ro\' , G . N" 13 i' iezoeloctri c effee t, 185 P lx:tj' ! j:lenc ra l llr , 9- 10 P oiSllOn' s eq llnt ion. 22 PosHi v,", pnl}'l\ornia )g ( I'Q!~· nom ials) . 2 1S

Max well . I. C " ~ 13 Max wl'll's equat tcns , 2(1 , 23·2 4 Meth od , bn~(l 011 qU311 rnt ic COllver genr e . ~ I J complex , 2 14 dcte ron i n i~t ic . 2 13 Ga uss-Sei.lcl I"I.'lax nl ion , 203-209 ,

P owec, ,I<:: \ i".... 'W , 103 in te rc han ge . 14 (; reacti ve . ro. 1 1 s pecific , 2W l oh l. 103 Pow er i"v a riance , 4(1 Po ynti ng , I. II . , 1::1 Po y n linll ' ~ vector , SI P re- dua l r,,,, ~ ,;o" , 2 17

L"g r,lIIg, n!l muh.rpllcr . 212of tol\fjR\lr.. t i nn~ , 2 13 of fi nit e eh'm clIlS, 2.8·3 t of I "a~t squares. 192 of perm eanc es, s o-a r of s teepcst slope. 2 12 pen al t y fund lon. ZI t ran dom search , 2U'l stm ple x . 214 stOChast ic, ~1 3 M odlll ~ of el('C t l'ic ruochln ee. J <l. 4~ , 7;. 78 , 8:>. 87. 1>8, 115, !)8.

Ra el , I. . ; 3 R espon,;o f~ q(j(>ncy. l i S llel!ta,·ti ng II motor, 64 , 11 (;, a R e ve r~; og " m" L:!r, f>4 . Il.... , t\3-iO l! iehler , R ., 13 Itikhman , G. V.. '12 Illl( lenhe rg , R.. 13 A unge- K " t ~3 lechniqut , l ut

l\ l om eo ~,

She nlc r . K . I. , 13 Sie mens . \V • 10 Skin ef lect . j!,i - I!'!l

cnccr».

205

a-a

I I!), 14 0 . 1s 7

cllJet rm fl8g ll('tk•• :IS. 8 t 01 h:lUrti a, 3.... of roais l.;) nce . 35

Slip. Iii

277

Ind e1

Sl4Irli 0l: a me te r, 63-65 , 68-&l S t4'ilrl)'4 Ia lf cq UaliOllS, SI-58. 9 1·V4 Stelnll)('ll. Po, 13 Sl(l k~ Ibtorrm . 20 Shakbo .., S. V•• 13 Sludell1 f-1e51, 210 S ubhumooiC!l, 75 Sy nch ronous ca paci tor! , 17

T ensor of h llSion , 24 T e.sJa. N. , \ 74 " ol \' iQ~ k y . v. A.. 13 To rque, braking , 8:1. generat illg, 82 mU im um , 133 min imum , 133

nominal , 133 puw.l il\lf, 82 l'e!ull4lnt. 133 st n 1lng , 82, 133 in elKtric machiT nm. lfIll proc:_ nes, 6%-68 ban c q\lilllt ities In. Gi Th yrl Jl"lor vollage nogulato l'l (T VIt ), 10~t 14

Ulno>', N. A. , 13

Vari.DU. expe rl meot, 20Ii

Inl'dequa<:,.

Vidmar . M.. 13 Voldd:, A. I. , 113 \ ' ollag n {KlrebhnU'a}I!([Wlliona , 34-31 Vo robrev , A, A.. 14

Whil e, D. , '13 WoodiiOn, n.. 13 Wi oding , a rlDlllU~ , 123 damper , tiS dLslribuled s inUJIoidaJ. \ 81 cx:el L.a llon (1ll!ld ), 118, 123 th ree-ph.ll5C sym,oetri e, 73 t wo-pbaM lymllll' lrie. 72

Yak obi, B. So, 9 YakobI's machi ne. 9-10

T O TH E: T\E ADEI\

Mir Publi sh('rll welcome lou r commen ts OD It,,, COll te n ts, tl'lln:;lat IQn, a n d dl'l!igll of the hoo k.

W" would '1 h o be p lc/l""d 10 receive auy you ClI~ 10 mak e ab ou t o ur Iu ture

~ul:'!::i':; I I"II:;

j>" L'l ica l loHs. Our lul dn'S!I Is : USS R, 12 'J(l20, MQ."1::Ow, 1- 110, C S I' ,

I' " rv >Â· Hh hsky Pereulck, 2 , Al i l" I' u b l; sh('l'll

Other Books f or Your Library

C YB EJ\ Nt::T ICS IN E LECTnlC POW EH S YSTE :'IS

P rof. V. A. v ent kov (General edito r) 'MI' boo k cov('n r.la ti vcl )' ntlW maU-ri:ll not )'ft full y wo rked ou t ! ei t'lIt Uical l y . kcllnologlea.lly ud melllodo logiulJ)·. l>ll rticul-

arly

,)(I

reg. rd& rOl'l'easti!ll:', pl'lRDinll and contml or la rgE' p..wer

'!l)1' l.t'IIl.ll _ Becauso 01 Its novelt y and some cODt rovt'uiA! vic",lI. the .!IlIbjec l- In,a Uc r lark.!! un iform ity o f p ~n l .t i on , g elK'r3 1 W !M:l' PU Rnd ten lll n()l~y , " hesll li m ita li ons e an onl y hI' rem oved b)" furt M r c:olll'C-livt effm't, and Ihe a ulhur;r wlll IIpprec-ia lll a n)' sU.!igot'l;l ion.!l Irom n·alh:r>j _~tuden t.', lect urers a nd ~'nll' i n~,\..ra,

T his book hllS.been written U I' stlldy a Id lor st udents at power and ek c trieal engi neeri ng eoJl~d lind depart ments , especially whe re t he c..rrleulo includ e ecue es 011 va rlou s aspectll of PQwersystem eybemet tce. Apart rrum ~ ~lId e n l$ , t he book may be c value tn J'{'l!l.'nreh work",rs , p' lllt ~r;uh'lI! e:s and enginl.'Crs euncernctl wit h powc r-~y~ lem con tml nud alJi('d tJe](l., .

A N I N'I' HOD UCT IO N TO COM J' UT EI\S

N . Sergeev , N. Vashk avleh Thi .s book coven! the oire ui tr y an rl opeta llnll pri nciple!! 01a na log a nd d lg ild COl.uputer!!, spectal - purpes e computing dev ices. ma.-hl ue!! end sys tem s. a mple s paceis devo ted Io a basic the ory of s imili t ude and slmulatiO Jl, and a la i rl }' detailed deserip rlcn is givon o f basic Juac t tona l elemerua , assem bhea and un its wh ich make up I)' piea! a nalog an d d igil al romp uteTll. in ro nj unct ion with a n in troduction 10 the mat hl"matieaJ and logicnl basis or elec tron ic .lig itaJ ecm nurees. Se parnll" S/!'Ct'OM deRi wit h biock-d tag mm sy utbesl$ or a nalog computeI'!! to sol ve a lgl"l" roie, trensecodentul , ord inary a nd pa rt ial di fferen t ial (!<jua ti ons and lh eir s)"'te ms. T he read er will find a suevey o r me thods liSl!rl to prog ram p rohJemsolv ing on an aiog and d igita l computers a long wil l. /I brief out li ull of d ig ilal differen tia l analrtNS and h )'b rld srs tem.'J. Th e book will be of pri ma ry value to college st uden ts and facult y me Ul b1lrs. and abo to researchers a nd engi nee rs concerned witb t he de velop ment 8nd ap plica ti on of com p ute rs.