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9. Contents
Pr. l,ce • , , • • • . • • • • • • • .
a" I. Introdul'llon 10 Eleclrl.>lleehM 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. AppHcatioll of f iclrl £ 4" 1111005 10 Ihe Scluuen of Probll'mt
In El~ot to rnc<: l"u) ,c. .
I. ~ . The l' rim;tLv" t'oll.-Wlnclinll Machinl' .
1.5. Application of C", n plll( '~ to the SoluUon of Problems in EIK:-
trll ",~ h/lnlcs . . . . . . . . . . . . . . • . , . . . . . .
A.. 2. l: len 'IlP;n'han lt:lIl f M t'fY Clln,'••dlllO In ul" ln, II (iulilu Flehl
2.1. The EquatiuWi o{ I~ O"n('t'aliud E I~l roe Ml('h,llf' . . . . .
2.2. SIl'a.ly-Stil" Equ at ion, . . . . . . . . . . . . .
2.3. 'ppllu tlon of An,.l", Coll1putel'1l to the A na l y~i, uf EIK lric
Mach!nl-s • ... •.. . • . . .. . . •• . .• .• . .
2.4. T randnl PnxellSf't III Electrie )t'Khinu • • • • . . • • • •
2.5. The EHl"d ..I Plra'l1l'll"ra OQ lb. D)'Ol mle Chllrac lerislcs 01
Ind" d iOIl M ~eh i n" . .... . .. . . . . . .• ..
CIa. 3. C""l"••I1.~..l ... 11 Winding (' llnYfor!e' • • • • • • • • • •
S.1. The InriDit. Arb itrary Spf'l".tn'm 01 f'ield. In tbft Air C.p
3.2. The Ge U('rllll~cd Energy Coover11ir . • . • . • . • • .
3.3. The E q uali on~ of the Cl"lIen liu d Enl>fgy c cnv••ter ..
Cl,. , . Typical E'IUaliollft ur I:!lrcl.lc M,u:b ll('l • •• •• •
. 4.1. TrUMition from Si mpll' 10 More doml' lex Eqllalions
4.2. J:;nergy Ccnveeeton In'olving un Elh plle Field . .
4.S. IW iplic,-Field Sl u ll>,·S t a t~ Conditlonl . .• • • •
th , 5. Ent'f!y Cn..vtBlon Inv"l vl..g :onslnUS/'idnl aoW ' 1)'nlQ1cl rlc:
Supl' y V" ltllgK . .
6.1. The EqllltionJl 01 EIK tr ie Macht..", .
6.2. The Solution of Elluallon l l nvo lvlng ' ' )'nlJuetr ic: Supply V OI 1~t..
6.3. The Tlol rl" ot "nltkge Rtglb IOl'-l nd"eUoo )Jolor S~tem
5.' . Pul E eelrnm6C h~ n ie.1 En"'g)l Convm , .
01. G. MulliwlMlilll M-eblnn . . . . . . . . . . .
G.l. The EllullUom or M" h h..judint Ab ebines ..
G.2. Thfl t:qu. IiQllI 01 Synehr(lllou. Muhl~ •.
11.3. T~ Equatioos of DirKl Cun fQt MiKblntS .
II.... The Double 8<,,,lrTel.cage Inducl ion 1I010r. ThO' Elfel'l of Eddy
C Ul'n'n .5 _ • •• •• •• •• •• • • • • • • • • • • •
,
s,
"
""3S
....
"sa
"..71
u
re
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sa
!~;
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",
01'01'122
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10. s
6.5. The Induc tion Machine Iodel l DCludillJ( Sla l.,.- and Rotor Eddy
Cum ulI . . . • . • • . • • . . . . . . • • . • • . . .•
6.6. Tb, IEffect or MilDulaUlln "" Pac~ on t: leel.rir M ~h ;ne P....
for.....nce • • • • • • • . _ • • • • • • • • • • • • .• •
00.. 7. Mp ls oJ t:lfdrl~ .taebi_ with N....liltnf "'ra~ • . _
7.1. 1'lle Anal ysis of EIKtnc l£acbUltli wilh ~oD line... P....mulers
7.2. The Effeel of S.lu.atHlQ _ • • . • • • • • • . . . _ . • •
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 hld. pell<l...~ Vari. lII..
7.5. The Aalillp is nr Opera lioll of • n.al ElllCU"k: M, dline • • • •
Cb . 8. AJ)'mll.ttric EIll:flfY C/lnYl:I'len; • •• • • •
8.t . Tvpe! of Asymmetry In t:lectr,r M.,h ull'S
8.2. Electrical I nd Mag"~llc ,uyUl'lIelt y .. .
8.S. Spacial .'Iaymrnel,)' . . , • • . . .
8. ~. SlRgle·Ph.ve MolOI'I . .. . . . . .. •.. .
8.5. Th, Eleerie M"eltilla u an e lt'n1t'ltl (If tho S y~ l e m
eh. 9. Tho E'luatlu,,~ fur F:Jl!Clrle M'"'h lnes or Vari ous Dcslg",
9.1. TIII Mathematieal ~I()/Iel! of EnerW'}' '.Qnverlllrs with a Few
Degre" of Freedom , . . . . • . . • . . . . . . .
9.2. LineII' Il:llel"ll:Y Collverten • . . . . . . . . . . . . . . . .
9.3. Il:nllll:Y Coo.trlen Wilh U 'lui d ~ ll,l Gaseou.s Rolo<'o1 • • • •
9.4. Othu T ypes or Bnt'rtY Con. erters . . . . . . . • • . . • .
0.. 10. Eieelrl... Fk ld anll Eleetrom~oael ic- Fleld E.-gr Coll..vters
to.1. Prloc:iples of DUll·lIl,,_ ElceU"OdYll llI.i.:a; . . . . •
10.2. Tilt EqualiOlU for f:loo;lr;G-Fi.hl KG8I'f1 Conu cwn
to.$. PlramlllMe. Eleeu iq-F'eld E~y Cc)n...,ters . . . .
10.<1;. Piezoelect ric E ~ c.o...erte-s . . . _ . . . . . . .
r c.e. Electromagnetlc- Pleld EMl'Il'Y Conv«UfS . • _ . . .
tho 1I . 'pplintlon Dr tape.-llnenta l Iksigll to F.leelrk: 1l1lm l.....,. A...l-
ysh ••• •• •• . •• • •• • . • • - •• • •• • - -
tl.l. Gt<l&ral Ill.(or,nallo.; on Ih. 'rh~y of Ex perimen tal Deaign
l U !. Th, TllChnillu, 01 ElCpt'l"lllll!nta l O('<l,go ' pplled ill Eleetro·
lTlIlChanics • • • • • • • . • • •. • • • • • • • • _ • •
11.3. 'T'flllu itiop 'ro m E J: pf-ri "I&ll ~1 D~", ign lJ) Oplimi la tioo . ..
Ch. 12. Sr nth<:t;i! 01 Eleclrle ItlnehllW!l .
12.I . Opliluit ation 0' ~ ~rgy Converters. Oplllll,u lion Methoda
12.2. GeometriC ProgramrninJ ... . .... .. . ..•
12.3. ~ig n of g l~ trl e ~ aeh lllu b}' ~me l rie PrDgfarcuning;
Ch. 13. Autom.ted Deilg n of ElllCtric Maeblnes . . ... .
t3 .1. Genual Po in ~ on tllu Evolution 01 lh, SySlenu of .-ut4 mallld
D..ign • • ••• • • • • •• •
13.2. Sorlware of Au toml~ DNi gu Syste rus
13.3. H. rd""llrfl of Aulom.led ~i gn ~ fstt lM
13.4. Cooclll!i<ln
Appeodiul
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214.
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11. Preface
The level of advance ment. ill t ochnol cgtcat culture primarily
depends on the de velo pment of allergy sources for the needs of man.
T he use of steam and pai-ti cul ar l.y eleo t.rtoity over LIm last one h u nd-
red yea rs brought indus trial revo lution and gave a t remend ous
impetus to the de velopm ent of sor,tety.
In the last few decad es wor-ldwi de peoduct ton of electric energy
has increese d a hundredfold . The electric power of generat ing
plan ts bas grown to 2 700 min kW . If the eeiee of growth of ge nera ted
energ'y rem ain the same , in 50 yearS from now the out put. of en0rgy
will reech 0.2% of the total OJl el-g ~' 'the onrth receives (rom tho sun.
Electric generators produce almost a ll th e electric energy
used, two-thirds of which is fori to electric motors to be con-
ves-ted to meoha uical energy. Elich yea r indust ry turns out tens of
millions of electric machines and tl',ll.ltsfor mers. In serial production
now are tur bine-driven generators lof 500, BOO. and 12000 I1W,
h ydroelectr ic generators of 700 lIliW , and t ran sformers ra ted at
1 000 MVA. Tod ay , motors and gene r-a tors are an esse nti al parl of
t he Iabrte of living, se rv ing diverae purposes in industr)', agri-
culfure, lind in the home,
Elec lric machine engineeri ng nlltu rllily owes Its advan ces to t he
development of t he t hem)' of electrdmechantcs-c-a branch of physics
dealing with t he processes of etecmomecnentce! energy conversion.
Electric machines incl ud e All Y etecirnmecbaulcnl energ y converters
(E('..s) des ti ned for va rious pur poses. Elect romechan ical co nve rters
come in a great varie t y of designi and can concentrate energy in
magnetic, electric, and electr-ornagnet.lc fields ,
T he equntlona of electric machines are written proceeding from th e
theory of elect ric circuits , keeping in mind that energy co nverts
in the air gap end the magn etic field is known, T he mat hematical
model for an Infin it e spec tr u m of fielrls and any num ber or loops on
HIe rotor and stator is the model of, a ge nernfized electr omechanical
converter- an electric machi ne with In and I l windings on the sta tor
and rotor .
T he equ ations for the ge neraltsed conver-ter offer the possi bil ity
of working out II mathemat.ioal model for practically any problem
encnuntered in modern electr-ic machtne engi neering,
Tho present Look deal s with a rnethem aucal t heory c f electric
machi nes that uses differen ti al equa tio ns as its base, It covers t he
12. 8
mathematical models of .eJectric machin es having a circul ar field
Bud an infinite spectrum of fields in tho air gap . Analysie is given
of th o equations involving nonstnusotda l asymmetric supply vcne-
ges and nonlinear pa rameters and !llso to multiwinding ma chines
and machines with several degrees of freedom. An attempt is made
to ad apt tho achievements in the aren of megnetic-field conv er ters
for use in the analysis of electric-field and elect romagnetic-lit Jd
conv ert ers.
T he book CO'8I':; topics devoted to the application tll electronic
computers to t he soluti9Jl of problems in electromechanics. It ill
expected that the rend er i~ alread y Iaml liue with eompu ters. program.
ming, and algori thmic la,ngu agos. The author 'S obj ective is to t each
the studen t how t o formulate equa t ions for mos t of the problems in
the analysis of thll energy conversion processes in electric ma chines
and rtJd,, ~e them to a convenient form for their solu tion by compu ters.
Much coustdoreucn is given to an alysis of the obtained solu t ions.
Three chapters are devoted to th e ~yo th6l! ill of electr! c ma chines an d
the com puter-aided design system ; the hitler be ing the highest
actuevemonts in el('Clro~e ch lln i c;s.
Pri ma ry attent ion is focused on differential equations of electro-
mechanical energy cen verston, wh ich form the most general and
rlgorou5 mauicmaucet model fOJ' descrfhing both transier u and
steady-state modes of operat ion. Polynomi al models are also given
due treatment.
The textb ook 11Iid i ~ origi n ill a series of lec tures, "Applic;ation of
Compu te rs for Engi llee ~i llg and Econom ic Calc ulations", and in
II specia l course, "The Mathem atical Theory of Electric Mach ines",
taught by t ile author (It. tbe Moscow Power Engineering Institute.
In organizing the book , lllle au thor has also used the results of re-
search conducted at the !Electric Madlirlery Chair and in the Laho-
ra tory for the an alys is of problems in electric dri ve, electric ma-
chines and ap paratus lit! the same ins ti tute .
The present hook is de~i gn ed for students and postg radua tes study-
ing electric machines end also for el ect romechurrica! and power
eng ineers engaged ill the:deslgn and service of electric, machinery.
J, P, KopyloQ
13. Chaple r 1
Introduction to Eled romechanics
1.1. Historical Development
The date th at lnlUJ,;l the be!:'iUlJin¥ of the age of electric madlines
is cons idered to be tho yCflr J821 when iI. Faraday constructed
II moto r- in which Il conduc tor Z re"olvcd abo ut a peemauen t mag-
lIet il (pig. 1.1). i [en:ury 3 lind uPP1r support J performed tho IU ll C-
t.lon 01 a eliding eomect. Paredey'aimcter fed wi th a de 'olta(l:o U
1(1 provide Hcfd excitation WIIS the first mngnI1Uc-fi('Jd electromechn-
nlce l energy C.Ullvl:'rll'rl.
In 1824 1-'. Barlow described a motor 'colJsisting of two co pper aear
whe-els Iestcncd on one s hafl lind located between t he poll'S fl f per-
D'lanent magnets. B nrlow's wheel wns in eo n tlloU with mercury lind
rotated fnst with tile PMSIIlJ8 of current.
In t831 M. FlIrfldll}' di5Covered the law of elf'Ctrom ilgnclir. indu e-
ucn-ccnc of the most hn porlan l pben om('J11I of eJeetromedullin-
which made possible th e dev elcpmen t of lIew types of eteetric me-
chill8'!l, T ho esse nce of the phe nomeron disclosed by Farllday COII.'lJS t5
in the ' ollowing. If II magnetic flux link ing It con ducting loop is
mad ~ to vary, ele<'lromot ive forces ;lppear ill the loop lind 1111 eteeu-le
current ShHts d rt ul8li na over l he d osed loop.
In J832 Piled sugges ted 811 ae gCIlll-fllto r with a revol ving hOI'SC6hoe
perm an ent 1ll3!:"cl 1 and station ary colis 2 wound on steel eorce
(Fig. i .2).
In 1834 A. S. YaKobi developed a motor wor king Oil t ho prin ciple
Ollllt"IIClioll and repuls ion of permaeent magllots and elec,lromnltn ets
(Fig, 1.:'1). Switchi ng on nnd off Ihe etecr eomagn ets provided ecn u-
nuous ci rn,Jar mot.len. In 1 s..~8 Russi an enginecl'lI inSlall ed fOJ<ty
molol'8combined into units Oil a boat whic h could ru n upstr eam the
Neva river with twelve passen gers on boud. That was the (i 1'St
aUempl In harness elec tri c motors ,for prllr t ir lll purposes.
JIl t860 A. Peci nctt! /lO d, lew, in 1870 Z. Gn mme sugg:(I8ted
1II ring armat ure (Fig, '1 .4 ). The Gramme ring ermat ure conllh'ted
1 Ttuouf:houl tbi. book " ... ~..1lI U~. ~..hPn! '~l't Mr}' . thl' geMral tl'rm
ml~l.ie.fltld "n HJ:Y coeeertc r to dftlol" th" c1l ss of machilt'S that ce"""
Ind'i.eliou (u)'ocbronols), ,,.l1dIronll~. I lld dl ~w"rrt"111 IHItS. AIm. "f ..Ill
usl,n th" rUJlO'Ctiv... ulm~ t"leclrk-U"ld ~onVl' n"r DIHl .......~ lfQmagnt' k-ri<.'l d
(OM"ff ltr III l'Jeetros tuie eon'·...ru rs . 1Id eenveetees in " 'bl<:h tbe ""OI'king rid"
Ih lt cene..u t ra l~'l <'1Mll'Jr il I n f lt"t rom. g nd ic r;"ld._T.....I. '....'. ....,...
14. of lil rlllll:-t )·pe magoetic eore 4 (made up from steer wire in early
m achi lles) c8rT)'ing an armetuee wind ing .z in tho form of a ecntt-
ClllOU5 spital. In early machines, brus hes 3 directly slided over tb.
cceueuocs windi ng '''Ill onered commutation by closing the turns.
I-{I~
'N
.- / '
~
-c'::- s r- -
- '-"- -
,
I'lli" 1.3. Yakobi' s llIncb'lue Fig. . <l. Paclncttt-Grnmuie's macblot
TIle mllf,plClir field was produced by m o~nels / or elee trcm agnets.
WIllie "Ilkobi's machine hed iLl! winding open. in th e lalter ma chine
t he " 'Iodi nI,' WllS r.onti nUOllS (clcse d]. T ile Gramme m achine made
• stnrl for t he e,'olutlo ll ()f ccmmerclal etecrere rnachtues. It had all
the bllSic elements of modern electric machi nes.
In 1873 F. Hd ncr-Alleneek and W. Siemons rep laced t he rinl
arm.lute br rhe lInn'tu ro of t he drum ty pe. Since 1878 manufact u-
ten! 11" '0 beg-Ill! lo prod uce drum armatures with eloes, and , since
1880. Ilrmatun'S from lam inat ions followinJ: the sugge! tion put by
~1, n nl llS A. Edisnn.
15. 1.1. Hi . to.lul o..~..lopm" nl
"10 1885 the Hungari an l'Jectriea;J engineers pro posed II si ngle-
phase, shelf-type and core-type. teans forrner with II d osed magnetic
<lireuil.
In 1889 M. O. Dollvo-Dobrcvo lsky developed II three-phase
esyncheo nous motor and n three-ph ase tcanslcrr ner. From 1890 t he
rbrne-phase system recetved general rccnglliUor' li nd marked the
beginning of wide application (If alteruati ng current ,
At the end of th e last century. firH distri ct and city power ptnnte
came into bein g. In 1913 tho Lotn! power or ('1('I't,ric stat.ious ill Russia
amounted to t mi n kw. In 1075
t he totlll out put of electric energy
peoduced in tlle Soviet Unio n
exceeded 1 000 billion kW h.
Electr-ic mach ines of an elflc i-
eacy of 99 % are now built wiLh
'the amount ot ecerve mater-iels
spent on their coustruc uon not
exceeding 0.5kg pel' kw. F.ig. 1.5
illustrat es a 1.2 min kW
turbine genera tor opera ting at f ill' t.e. An l'uhllnllth'c u:cnCl'lItQr
the Kost rom a power plilJll.
Electric mach ine indust ry produces ac aud de mach ines to S6l'V6
II great va riety of runcuone in all hranchcs of Iud ustr y. Electric
machines do jobs in space, under water and under (he grou nd . Spe-
cial machines nrc avuilable for work ill at omic rencrers and spann-
craft. Millions of elect ric machines find usc in household appl lanoes,
automob iles , tractors. end ot her ki~ds of tro nsport . Fig. 1.6 shows
lin automot ive generator. The Soviet industry turns (lilt some dozlms
of automot fve gene rlltors of this and ot her types every minute.
16. Ch, 1. ItIl,od uction 10 El.d'o""••h.nic.
,
Electric machlll" en joy wide application in timiDg aevreee,
nnillll.tioll systems, and ,n, di Herent trllDWUcers. The power of
eteetetc machines va ries from frftGLions of II watt to • milliun kilo-
wllt1.5, the ,'oltage from fucUons of • vo lt to II mllJion ' 011.5, t he
rotlllional speed lrom Il fow revclu uces a dlly to 500 000 rpm , and
the 'I'olLage Irequency from uru to JO'l_lOlt HI.
~pile. the gnoat prog ~ made in elec tromechantcs. very many
pro blem' sti ll rem ain to tle solved . Electromech anical energy eon-
'l'e~i oo occ u ~ ill fusion rreaerors lind in biological species too.
Further ad vances in the creati on 01 new t ypes of electri c macntuee
will in the main be determined b)· tho level of devalop meut 01 the-
theeu-y of clectromec hal1 ic~ 1 energy eenv crstc n.
All lloollh the devolopment of elect rcmechanles is considere d to
bt!gill with Paeadny'a dl~co' ery, electric machines existed long
bclcee hl~ discovery, Much errort was spent on evol vfng eleoretc-rteld
energy conv erters. t.e. :Clec l ro~ tll li c (frl ct ion ol) machines with
eleccrtc-netd ("lIcrgy storage. In 165O' 0 1to '1'011 Guer tcke descri bed
th e fil'llt eleererc mach ine t hllt represented II rotlllting ball from
suHlIr, :
.. t the bq inlliog or th r; ' 8~h cl'nt ~l ry Francis Hawksbce re pleced
t he suHur ball by /I holl~' gl as~ ball fiut>d on the shaft. In 1743
fritlional mechtnee with IHl isola ted met:i1 eleeuede appeared. T he
elee o-oee eoflecred eledrll:. chargee Ilnd so the mnchine could ccmi-
nu ously feed power to the eSloJ'nal cteeun. The subseq uent yeal'1! of
t b.. t Bth centur y Sll ~' fuctllet a rlldu:l1 im provements ill the dllllitfD
and perfo rmonce of frictiona l eoeebrees. r mong R u ~i ll n 5(;lentist.s,
M. V. Lomonoso v, G. V, R lkhmall. _.t, . T , Bolotov, and others
were cngaq:ed in the workll)n these maeh inea. Dy the end of the 18tb
Cl' ll luQ' one. more t ype (If (rictional ma chine Cli me into being in
which the rotor .....(15 mild, Irom gl.'..<6 di.!;ks up t o two meters ill d ia-
meter. The machine could produce sp arks over 11 me ter long.
In th e 19th century WOr k on the Impro vement of lrictiooal me-
cntnee continued and resu led ill the evolution of unique ele et etc-Iteld
tlnCl'IlY conveners. In I!J::JOlhe van de Grnalf generator (el ect rosl llllc
nccelerntor) WfIS built wbjieh deltvered II power of 6 kW at ~ voHllge
of 0 min V. Bueh setup! lire deslgne d to tl!:!l t elecu-ical equipm ent.
Eiectrtc-Iteld energ )' convert ers wllieh ilppe ilref! mu ch earlier
than megnet.ic-Iield ene"iY converters did not lind wid e praetlcal
applicat ions. The emergence of maJrlleli c.fjeld type ECs in the
Hlth centu ry was II new s:t.nge in the hisLory of eteerete mM-hi ne enal -
n",riug nnd brought abuul II screeune lind teehutc al re'oluloo
in this field.
The hi!ltfJr)' of development of the thoor)' 01 el&ctri(' meebmes may
eolldit ionllll y be di'idel int o three stages, Tbe (irst stage indudes
the period of Cl'@ati on of earl)' machines and development of the
e1aMin l t heor-y of t lech'omeellanl(:al energy eonveret cn . Tho second
17. ra
I lage embraces the period or oJa bohliOft lind Int rcd uetton of t he
thtof)' of stel dr-slale processes, ce mptex equa tion s, equival ent
creeuus. ae d phasoe d iagrlllUS. Th o third stage berlin in the late
1920s with the form ulat ion of diff~re n tiBI equ ations and develop-
ment of Lna t heory of traDSiont precesses in electric machtnce. T he
theory of elecrerc macbtnos W8..~ Sh 'e n constdeeauon in tho works
of A. .M. Ampere. G. Ohm, J. P. Jqule, Heinrich F. E. Lena, Her-
mao L. F. HolmhoLt1., M, V. Lcmcnoecv and other prominent phY.'li-
ctsts of tbe 18t h and tho 10th cen tury, T he works of J ames C. Maxwel l
who ~enel'A1i~ed the nchlcvemen te of elect rical engineeri ng in hb
r feoti$e 0/ et«tricu y aM M agnetUni, 1873, hold a particular place.
Mu well introd uced the new elccl(O magnetie t heory and pos(ulal.8d
equaUons which eama to fonn the th llOrelical base f)f electeomeche-
ntee. I
Of much Impoetance are also t h6 worb of N. A. UOlO'" 11874)
l od John H. Poynti ng (I &M,j 0 1 tho transfer llld eenveeeton of
eoergr . T he first theoretical work conCllrning eloctri<l machin es
mll Y be considered the work of E. Arnold on the theory nod
desig n of windings of elecu-rc machin es, issu ed in 181lt.
In lbD'18903 ~l. o. Dolh'o-Dobro';olsky, Gisbert KliPP, end other
scientists eer fort h tho f,uldnmenta1s of t he thoory lind dosian of
transformers. tn '(894 A. Hoyla.nd lheoretically l!uhstanti l ted the
circle diagram of 3 D induelion m1l4hine , and in 1907 K. A. Krug
offered an accurate proof of the circle diagram. In the t 920s B. For-
tescue suggested the method of sy d'l met ric compenems.
In t he t 93Qs E. Arnold, R. Ri lfhtor. A. B1ond el, L. Dreyfus,
FIf. Vidmllr, Charles P. Stein mel". K. A. Krug, K. I. Shenree.
V. A. Tolvlnsky, and M. P. Koseenko eonsidorll.bly extended and
advanced tho t heory of steady-statol operatio n of elect ric machines.
R. Rudenberg's work WM ono of tho fil'llt cont rtbut tona 10 the
theory of n-analent processes. This theory, whose orig in dates from
lhe boginni ug of thi.!l cen tury. made n tremendous step forward in
the t950s Ilnd 19705 OWing to tho lWide appliea tion of computers.
The firs ~ papers concerned wit h the mathomatical thoory of
electric machines ap peare d in the middle of the 19205, in the 19308
and t94O!. Among the eutbcrs. mention sbculd be lDl do of R. Park.
A. A. Ocrev. G. Kron, and G. N. r etrov, The fundamental works
of G. Kron : rea tl y contributed to the development of th e mathe-
matical l heory. He sllgallllwd tho model of end deri ved eqoll.tloQs
for the gooer/lli7.ed (primitive) elelftric machine.
In t be lAst yea rs the mathematl~1l.1 theory of electrtc machine!!
(magnetic-field energy eonveeters] has developed to a noticeable
utent owing lO t he efforta of man y'allt hors, fit!lt of all. B. Adkins.
L. N. Gl'lIl0 V, A. G. IMEfyan, E;. Ya. Knovsky, K. Kovach.
V. V. Khrushc hev, I. Raez, S. V. Strakhcv, D. Wltite, and J{. Wood-
SOil. The use of electronic computers has ena bled researchers to
18. "IInol)'", l lelldY-l; ~ ll te procqsses n a. pllrtic;;ubr C/IS(' of teenstems lnd
I ppro. cll the problem of developing tomputer·aidt'd design system!.
The theory or eleetrosiatic machine! , however. sli pped behind
desptte the pooling of error-ta of such prominent scientists a! A. P. Iot-
fe, N. D. Papalekai, L. I. Mande)~h8m. A . E. Kllpl)'sn sk}'.
A. A. Vorobyev. and al6ers. largely beceose t hei r illvest.igllLi ons
railed to creete tho prod llcljon prototypes of these ceevertc-e.
At preM'nt one of the ittaporhnt t J..!lks of the w:lthematieaJ t heor y
of electric: machines is to tde'e)op the general theory of aU t he three
classes of energy ceuvertera,
The ehaptera below cQllllidcl' energy ecnvertee equ atfona, their
tr and ormat ion end use Icr most of the basi c problema dealt wit h
in the analysis lind ~yn thhsitl of electric machines: presont.equm.lona
.IIJH! their eoluucns lly eompcters Cor machines (';hibiting a rlrcular
field and 1111 infinite speetmm of fields in tho air gnp: exemtne con-
'ertt'u involv ing nonatuuaoidal esvmcict rtc v oJt ll.g ~ , changes in the
frequency and amplitude 'of su pply vollage, machines with nonli near
plll'lu;netera, asymmetrlc fnachinQ5; etc. The coverage also includes
converters with a In .' d~rees of freedom, li near machines. eleetnc-
field elect romechanical ene"y converters, aud other elec:tric: ma-
chines. The t heory of energy converters is set forth on the bllSis (If
diflerenlial equa lion!J ",'~ie b describe the dyn amic behavior and ,
as a particular case , the, stead~tue behavior. The COUB& in the
mathematical theory of electric machin es gives the base for the
mathematical deseri plioll of the process" of energy ecnv erercn
takinlf into account Jlon)inear, nonsiousoidal. as)'mmelcic aspects
and manub.ctllring faclors. Such an anal ysis is impm,sible to do
employing stead r-stllle , queLions. equivalent clrtu.iU5. and phasor
diagrams. The electro mllljbanical energ y conversi on theory presented
ill this book enables t he :engineer to use the equations for the gene-
r ali zed electromechanical energy converter lIS the base froIII whtch
he can set up equations !(or solving any problem met with in the
practtce of electric m a c~ino engineeri ng.
1.1, The Law s ot'Elec:tromechanical Energy
Conversion
Altbough tile t heor y and practice of electromechenieal ellergy
een versrcn have I Ion&, history and achieved great !IuCGeSS(>.II, tllQ
basic eneray ccnvereten ~ ..ws hive been stated only quite reeenrly.
Let us form ulate these In .'S.
F lflit law. ThI: t ffkit mv of tltctromtdlollkol tnergy ro'fIJt rl!on
(Ol1not tqtUJl 100 " ,
All energy converters can be di vided into simple and complex
ont'S, In si mple convertees, w e energy of one form is converted to
the energy of enctber form. An example ill the Conversion of electric
19. energy to hea l ill an electric hea ler. In eempjex ecuvenere. wl,ich
«Insti tute the majori ty of machines. the enell:Y of one form Is een -
"'rled to tbe energy of two forms (II.IU' . rarer. tu three or more forms).
These II Nl con vert ers of energ y f Nlr:p luminous to elecrrtcal Icrm,
chemical to mechanicel form. nuc.l!'lIr to e.Ieetricnl form. t .l'. In
«I rapleJ: converters there commonly oecurs ell attendant ce nveeetee
of energy to heat.
Eleer romechenicel ellergy converters belong 10 the group of
complex converters because the processes of energy convereton !.lera
elwllys go with IJ,,,, conversion or doc.Lrk energy P~ or rneolmnical
enoTiY P", to thermal energy 1'1 ~ ' E,Cs exhiblt th e flows of etecrro-
ml gnelle, mechentcel , Mild thermal en ergi~ (fig. 1.7).
The objl'Cli ve plIl'!lued in 6'olvin1: an EC ts to reduce t he l(Lq-
thermal energ:)' (low:I- and rhus to decrease the overAll di menSlon&
Fig. 1.1. Tbe energy llow distribution
In &0 eleen lc machine
~ :~:c:~~~ ':.~:::lg ~~~~D"':J":,'I~5
and d..b IlMl ,..,,~dl "dJ
,
p. EC
,
Pm
lDecbine &/1 •
and cast 01 the m:'lcblne. The efficien cy of some converters IVlillbl.
todey reaches 98%. lind lhat of t rl ns!nrmel"S fUllS as high 8! 99.8% .
1l'hich is indk lU v$ of exce pt ional technical echtevements.
It is to be borne in mind that h;igh efficiencies are achie" able
In high-power converters. In tow-power ECs the efficiency niches
merely III few percent amcc th e major a.mount of mechanical or electric
energy evolves as heat.
It Is impcsefb!e to produce an eJe(;tr ic machine in which conver-
sion of energy l.6 heat would be uenex tstenr: e tberwlee it must be
Inrnlahed with superconduc tlng wlnd.logs. As w ill be shown below,
electromechanical energy ccnvereton equations have no soluttoae
III zero reststeneee.
We cen visuMIit.e .. lossless machine (without Iron and having
eupereonducting windi ngs). but to en~bl e eueb • machine to convert.
etIerg)', we need to inset1. • reetatenee into the CUrrflJl1. net<tI,'ork
e::rterDaJ to tbe meehiae, In t his arr-a.ngement. it u the eleet romecha-
llieal s)·st.e.m be)'ond the machine th..t develo ps lossl!ll. An electric
20. '"
machine can be tNaLed :wilholiL regard to the external eteetre e e-
c haniCAl 5)'stem only undee def inite conditions, whe n, for examp le,
the line eeeistance is equal lo zero. i.e. the machlne operates fro m or
in k> tile bus of infi nite pewee.
The processes of electromechanical energy co nversion ffiWlt. be
studied WiLli due regard for all eledrieal an d mechanical loo ps.
An EC t hat does not develop lessee becomes a sto ra~ or tank of
energy rather than the energy converter. Eoerg y sto rage devices
are e lec trh~1l 1 engineering_" rrnngemenu fe.Sllmbli nt in design electric
macllinel .
Enorgy storage devtcee een be buil l a! both s tati c devices and
rolating machtnes, for exemp te, as II gyro with superconduct tng
windilillS. T his is IIlI electric machine that cen rotate permanently
s ince t here is no loss i lj 'it. But eu antilorque moment applied to
ne shafl will brin g the machlnc to II. sto p. T his machine cannot act
as eu cuergy couveetc r.
An etecteomechentcel ecu ve-tee can ue represented as a t woport
(Fig. 1.8) aceepUog, for ,u am ple. stim uli (yoltag<.' U and elac trlcoJ
frequency f) at it pair oflel&ctric.nl term in als (an electrical port) and
produci ng responses (a tJrque!of on t he shaft and roLational Ieequen-
GY II) at a pair of lDlX;fani&al term inals (a mecha nica l port). T he
t .....oport representation of an electric mll&hine applies to aoh'ing
problems in ereereoreeenentcs where the pl'OGeMe5 of energy eenver-
sion inside the machine do nol have a dominont sign j(jcaDce.
Second law. AU , /.ectromedulIIlcal conwrw$ are revenlble, t.e . the"
call act IU moton' lJ.Ild IU generators.
The revcrsibility is an important adv en lltge of ECs over other
energy converter.< su&h 88 Itea m tu rbinM, diesel en~i n 9S. jet engine!,
ete. T he energy-<.onyerson mode of operati on of an elec tric m. chioe
depends on the mement of resistance (torque or amuoeque) on its
shalt, M r. If the electric energy is drawn from the power line, the
Ee opera tes in the motQring mode. If the f10lY of mechanical energ y
dctt veeed to rne EC shnft trtJn s fo rm.~ to the flow of elect.romeg net!c
enerllY, the machine operates in the gellllroUng mode.
Th e active power re verses iUl directi on with a change of the opera-
t ional fun ct ion from getieration to motoring, but the f10lY of thermal
energy does not generally Ghange ns dtrecuon. Losses in ee mmen
Ee s lIt'e irre versible.
There is a great va riety of ec., incl uding electelc raechtuee which
w nvert neat to eleetr~& or mechanical energy. Th e diS&ussion 01
such ECs is given in eb. 9.
To provide linka~e between windings (loops) and cur~nLS it is
necess ary to prod uce AI! electromagneti c field . The rotating field ill
eteewtc machines is set up by alternating or di rect currents. Tbe
reactive power may Ilow in an EC operati ng in the stead)' It ate
f('(lm either the statee or rotor, or from beth simultaneously.
21. 1.2 , ... L. w. 01 Eloclrom. ct.'IJ, lu l E....rg y Co.......lo..
One 0/ the ~lla rlU O/ liat lirlt alld I/u s«ond laID is that an Be
allO rtp~n" all tMrgy conttnlr4tat. The oloctromagoetic ener:y.
being dis tributod at. infini t.y along a,n electric power line, i.II sto red
in mag-oetie-field ene~y converters f it-hin th o ai r gap bet"""n t he
nator and rotor. In t.ransformers, the energy is stored in t.he m"ino-
tic core end ill the space between thll prim ary end secondary , whore
leakage fluxes close 0 11 themsel ves , failing to btl com mon ro beth
windings.
Tho nlr ga p of a comparattvcly amall volume C811 cc neen rr ete
hull'O powers . h is of Im pcrtencc to note t.hat. in turbine generators
of max imum powers and in inductioh machlnes of the single series,
the power density (W/mm3
) in the ai r ga p is equa l to approximn tely
0.5. In view of this ract , dnigninr of.electric machines con be begun
with the estimat ion or t.lte ga p volume and thell proceeded wi th tlao
ealculettcn of wl nd ings aod geometrical pa rameters of the magneti c
I)"!tem. Active and react ive rJo_ 6r en0'1rY can be coinci dent or
oppotlite in di rectioll irrespective of ,,,laether tna EC Mlns as II genere-
tor or motor. This moans tnal tho eeuve powor may come from tho
stator an d tbo reactive pewer frolD 'the rotor, lind vice versa .
EC5 also operate in the no-loa d ~Il d i tion lit which they COII'ort
electric or mecbnn lcal power intO! heat . Sync.llronous Illochino.s
connected in pa ra llel with the li ne ,lI.nd wn /It no load lire called
$gm:}lro Il OI/.ll captu:Uurl. .
During its operati on , a n ollll;tric 1tll.Chino tel eost'S t her mal enClrgy.
It is possible to produce all electr!c ;rnnchiue furnished with II. ~ he r
moplje in order to absorb heat insido the macutne at the cold [unettons
115 a result of tho Peltier ellect [thereby preventing it from heat ing)
and to evolve th ermlll enore y at t h~ hot jllnctions outs ide th o ma-
chine. However, the availablo semiconductor coup les offer cooli ng
at low cnrre ot densi ties. so the i O(o. resultin/l from the improv ed
ccolme can onl)' be brought abou t .u the eoee or an inc~ ue in l he
o'era ll dimensIons of the machine and a wot&elling of its COCf'gy
cnllractem lics. This alteslJ t.hnt. the thermAl onergy flux es .. well
85 the mech an ical energy and elect ric enN.,y f1uxe.s in an EC must
be regarded as closed energ y loops .
Th e cendtucn or resonance exis~s ill electric machines jllst II!
it docs in moet enorgy converters. EJoc-triell.l and mechanical pheno-
menu that occur In ECs lire reso~ a n t. El ectrtc machines oxhl bit
electro mechanical resonance pt whrc h urc rot euonet s peed of 111(l
field, t.. is related to the meehar ucal rotational speed of the eotoe,
n , ruaasured in revoluttons per second, by the expressten
/1- pn (1.1)
where p is th e number of polo palrs.
In ,.. two-pole ml chlne, the power li ne frequency and the synchro-
nous spood of the rotor aNI th e SIUD e- Electrie maehine.s are built
: - 0 1111
22. " Ch. I. lrilro</u.:fio n to EI"<1.ol'ft/l'Chonk'
in such II man ner that the wave of a megneuaing force in tho air
gap distributes itself i n ~ eg ra ll y among the poles, so th o processes
of energy conversion In two-pole and Multipolar machines are eseen-'
tinily identical, the only d)fferenc-o being that in t he latter machines
tho synchronous speed o~ the field end the mechanical speed of t he
rotor are 0 factor of p lower.
Th ird law. EledromlXhanic.a1 enrrgy conoersion 1$ due /0 the fields
thot are stotiOll ery with respect to tach other.
Tho rotor and stator fi el d~ in the air gap of a machine, which
nro stationary with respect to each other, prod uce 8 resultant field
and electromagnetic torque;
(1.2)
where 00. is tho angular veloci ty (speed) of the field: and p .", is
the electromagnetic power.
The fields displacing ln the air- gap with respect to each other
produce a Ilux of thermal energy, thus indi roctly affecti ng the
dist rib ut ion of tho fluxes of mechanical lind electric. energies.
The wi ndings of elect ric machines must carry polyphase currents
flnd show a proper arra ngement to produce 1 rotating field ill the
air gllp. A rotating field (jail he set up by II two-phase current syste m,
wHh the windings disp laceu OO~ in space from one another and the
currents ahitted in timo by 90°; by a th ree-phase current system,
with th e Windin gs 120· upert in space lind 120· in tim e; lind, in
the general case. by an In-phase current system, wit h the windings
displaced 3600
/m in space and currents shifted 36Ct'lm in ti me. Direct
current can also produce II rotllting fiold, In which case the de wind-
ing must rotate. The winding carrying altematlng currents to pro-
duce a rotating Held ere usuedly stationllry.
In 0 synchronous machine, the rot lti ng field is hlrgel y SE'C up by
the currents in the windings dis posed on the stator. The field rotates
at a speed 10)•• The rotor runs lit tho same speed, (0 , = 10)" thllN'forfi
the frequency of the rotor current is t, "'"0, i.e. direct current
flows th rough the rotor" ~ i nd i ng .
In a de machine, the ;field (excitati on) wind ing is on t he stator,
and the excita tion field is stationAry. Rnta~ing tb e armature. whic h
is the rotor here, produces the rot ating armature field , which revol-
ves at the same speed as the rotor but tn the opposite di rection.
In induction mechtnee, the frequency of current in the rotor Is
I. m f,s (l .3)
where th e slip (speed differential that is a fraction of synchronous
speed)
oS "" (e, ± w,)/w , (1.3a )
23. 19
Theffiore, the s peed (angular velocity) of the rotor Cllr plus the
speed with which the rotor field tn'015 wit h respect to the rotor
Itrucloro ill , lw8ya equal to the speed of tbe field (ll •• If t he rotor
turns at a speed higher than "'. in the sa me direction as the field
excited by stator eu rrenlll. the rotor fiel d travels in the opposi te
dire<:tloD to t he rotor, 50 the Slaw' and rotor fields are again ete-
ullOar)' with respect to each other.
In tr«osformers the windinp are sllt-Ioo lry, an d thus tbe ft.
queociu In t he primary end second ary aro lbe same. It C80 then
be assumed that the fields of the pr imuy and thesecoodary lJ'a,·el . t
tbosame apeed. The «Incept of stationarity of fields in transformers
is of little consequence for the InlllYllis of the proc:esses of wergy
transforma tion .
The third law facilitates tho analy~is of eD81i Yccnverescn pI'OeeS-
SE'S in electetc machioes an d forms the basis for the re presentatt cn
oJ tlnerry cenvcrsrce equations.
For electric-field and elootrOmllglllltic-fieid energy converters l.ha
field stationar ity concept dees Dot have such a great significance as
it does for magnetic-field energy converters. These converters are
most vivid ly represented IS energy COncentrators exhibiting elece o-
meclUUllcal resonance.
Since eteceecmeehentce is part of p:byslcs, aU basic physical laws
are applicable to electric machin es. To these belong first of all t he
law of llnerg y ccnecrv eu on. Ampere's law (circu it al law). Obm 's
Jaw, etc. At the toot of t he equations descr ibing energy CQnversion
in elect ric machines are Mnxwell 's equatio ns an d K irchhoff's Iawa,
tol. .Application of Field Equations
te the Solution of Problems
in Electromechanlc:s
Electromechanical enorgy cenveeeree in magn etic-field ty pe
machines ()(curs in the space where the machine concenl.:tates tbe
energy of a magnetic field . Kn owin l tb' fi,Id, we can esumale volt-
ages. curren..,. mechanical torques, losses, electricAl paramelan:. and
other quantities of interest un dor tho steady-s1.lte and transient
cODdltloJl. The calculation of tbe electrom agne tic field in an y
ecergy converter. be it II si mple or an ip tricate type. present" a comp-
licated problem, and iLs eetn uen Involves difficul ties even with
the use of modem means and mOlIt ad...anced methods available.
The electromagnetic field anal }"ll15 is one of t he main aspecla t h.t
a1 -wa)'. attracts attention of researchers. The requ iremenLs for the
lCCur. cy of eleetromlg'neUc field calcu lations become increasingly
!tring8Qt because of the growth of the specific an d total powers of
eDetVyconv ertertl and more sev ere temperature CO.IIditions i.n which
they have to operate at high efliCliency and Improved reliability.
,.
24. 20 Ch. 1. lnlroch,dion to f loctrQm.dlonlco,
Over tho past few decades a lorve nuwber of bot h special and
universal methods have appeared for the anal ysis and calculati on
of elect romegneric fields.
Maxwell's equat ions are the bun fro m which one starts with the
calculation of an electromagneti c field. Tho)" are usuall y given
in diUer6llt ia! form. One of the equations establishes the relat ion
bet..-een tbe vector of magnetic field strengt h jj and the veeaer of
current density T
curl fi _ T (t.4)
Integrating both aides of the equeuen over the aRI 5 1 tee, S I )' , tho
simple t wo-dimllfUllonai case of I magnetic. field
J(curl 1l)" d~ _) <7>"tiS (I ." )
,
and applying Stokes theorem
~ (curl ill" JS _ ~ H dl
we arrive at the welt-known circujtal law [Ampere's law)
~ H, d'-_ 5m. dS (1.6)
,
whore tho area of the surface under consideration Is S, inside which
there n OW! the current f of den!!t y 7 in the direct ion of vector H.
the current being eonrreed within the closed loop I. For loops t
completely 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 referred to 13 the magnetic tnduc-
li on, or the magnet ic flux den!lit y, is defined in teems of t he per-
meabili ty tl of 11 medium &nd the magnetic fi eld strength H pro-
duced in t be medium;
where
div O ... O
(1.8)
(1.9)
The divefieQC6 of the field is th us zero. T his means t hat there is
no ~cu rren t~ flowing In and out of II magnetic field (magnetic lines
never end hut olcee on themselves), t.e. free magnetic charges (mon~
25. "
poles) do not exi st. in nature. The magnet ic field components n-an d
li can be found if we sol ve the field equattons for various parts of
a converter of defin ite configuration by observ ing tho boundary
conditions of cont inuity for the normal com ponents of the B field
vectors lit tho lnterface between tJo med in 1 and 2 (wh ich differ
in permeabtlity) 1
B in - .8~n (1.10)
and lor the tangential components pf the field strength
81/ = I1: l (1.11)
providing t hat CU IT l."n ! s heets on the bou ndary surfaces do nut exist.
As shown by experiment, Eqs. (1.7) through (1.11) permit defining
tho magnetic field nnalyt icnll)' only lor a rather limited range of
prchlema with the si mpl est boundar}' condi t ions.
In consi dering the real parts of electr-ic machi nes with rather
complicated sh apes of magneti c cores and.current-curryi ng elemen ts,
a number of assumpti ons hove to he made to obtain even an approxt-
male solution. Sim plifying assa mpuoos may apply to surf ace shapes,
current distrib utions, th e properties of me dia, and laws of thui r
motion. In cases where tile field sources lie fairly far awa y from the
Held region undor considerati on [Le. T"'" 0), it is sometimes advan-
tageous to introd uce the notion of a magnetic scalar potenti al Illm'
Because of the curl-free character of a scalar field (curl If - Ol,
the magneti c field strengtb jj can be expressed as
it = -grad 'llm (i.1 2)
For II scalar field. L aplace's equa t ion hold s:
'V2
q:>m ... iPfJ'm/8:t' + {f'q:>m/{)1I2
+ i)21{1m/8z 2 = 0 (1. 13)
The field li nes here prove discontinuous. Tho sources lind sinks
olthe field wfll be the surfaces having differen t m agnetic potentials.
The di stribution of potentials depeiu)s Oil the distri bution of cur-
rents in the Windings of a converter and is defined up to II. constant
in any loca l region.
1lost boundary conditions lor a scalar magnetic fie ld ill elecr rlc
machines are Dirichlet con dntons, T his is commcnly found t o be
a Iavorahl e fac tor for t he solu t ion of a problem part jcularfy whon
uslug approximate methods. The f.id d celculattona Dim at defini ng
tile componen ts of the mngnetic fi eld st rength Dlong the three axes
H :< = - 8r9m/OJ;, HII = -lJ~7Il/0y, Hz "'" - 8fpml{); (1.14)
Knowing these components and using (1.8), we can find the B field
vector components an d tben 0l88n:etic fluxes lind flux linkages.
The unit of measure of a magnetic pctcntdal is the am pere, t herefore
this qu an tity corr esponds 10 the v-agnetomotive force (mmf) :15
26. 22 Ch. l. Introducti on to EleciromecMnin
regards its mea ning. The ~uuctiou of the flux in a potential field q'",
proves to correspond to the megnet.ic flux.
The calculation precttce of rot ational electromagn etic fields
widely uses the noti on of a magnetic vector potential A- defined
by the relation
B '"'" curl it (1.15)
Solving simultanoously (t "f) , (1.8), and (1.9), and then (1.15) gives
Poisson's equation
(1.16)
in which the magnet ic vector potential calculated up to a constant
acquires a definite physical meaning. The circulation of the vector
potentia! over the loop ill found to be equal to tile magnetic flux
through the sur face boun ded by this loop . W het is important is
that the shape of the surface is of no consequence and thus can be
arbitrary. In the three-dimensional CMe, Eq. (1.16) is written for
each of t he three components given as the projections on to th e
corres ponding coordinate axes. It is often permissible to consider
the field of an olectrlc machine as a flat, two-di mensional, pattern
with one curre nt component, for examplo, along the z axis:
o!A.'ox! + oA.loy! = -~7z (1.17)
Tn this case tho magnetic vector potential takes on the meaning of
the magnetic flu x per un it lengt h in the z direction. T he B field
vectors along the x and y axes are given by
8 x = {}A .loy, 8 , = -oA.tox (i .18)
The solution to the prpblem inv ulving t he dete rmination of the
magnetic field in eleorrte machines is most comm only sought under
the boundary conditions of the second kind (Neum an n condit ions).
The function of tile [lux in the vector field A- corresponds to the
magnetomotlve force, i.e
j
the function of the potent ial is propor-
tional to the magnetic lux.
For defining the magnetic field, it is usual to employ slmili-
tude methcds and the methods of physical and mathematlcel modeli ng,
Experience attests t hat th.e not ions of scalar and vector magnetic
poten tials equally well hold in modeli ng of magnetic fields , although
the realization of houndaey condit ions when using eit her of t hese
two notions is subsranttully different.
Where there is a need to solve th e problem with ccnstderstion for
induced currents, the ncttcn of the magnetic vector potenti al is
the only one accepmble. 'in which case Poisson's equation must be
replaced by the so-eatle;d heat-eonduction equation,
27. 1.3. Appli,otio n 01 Field EquMion,
Most di verse mothods apply to solve the obt ained equations for
a ffillguetic field under tllOccudit lons (1.10) and (1.11) at the boun-
daries between different med ia. Historically the methods or di r-ect
solution have developed most inten,sh'ely, which commonly give
an accurate or approximate analY~ical result. Among those, we
should note the meth od of images and t he meth od of sepa ration of
variables. Conform al transformationJ of the regions of interest, by
which complex bou ndary ccndtttons undergo' substenu al cha nges
and become ap preciably simpler, Hla y a not iceable part in tile
developmen t of the methods for 'the solution to magnetic field
problems. The sol ution to Lapl ace '~ equeuou is worked out for
relarlvely simple areas and then ap plied to the initial regio n. The
Ineananta, Le. quantities invaelabla.In transformations, arc magne-
tic potentials , magne tic fluxes, and the moduli of the magnetic n ux
density vect ors and field strengt h vectors. 'I'he solution ue ert in
the transformed plane is found accurately, whenever posslhl a, or
approximately using an ana ly tical or numorlcal method. 'l'he
methods of conformal transformations mainly apply to trrotauonal
fields. The methods of in tegral oquat ions are su it able for the solu-
tion of a number of rotational field prob lems. T he last lew decades
have seen an exceptionally ra pid developm ent of the ap proximate
numerical tech niques based on the me thods of fin ite differences and
finite elements. ;
The progress in computer ongineering and t he creation of fast
compute rs with a large memory capac itY have enabled the effect ive
introduction of t.hese approximate methods. T hey permit obtaining
the solu tion of a desired function (p9te ntial) in the field region for
each particular caae. A substantial disadvemege of these methods
is that t hey do not aUow for deri ving the general expression for the
solution, so it becomes necessary to obtain a new solution with any
change of the parameters nffecting the field. However, t he poten -
tia lities of co mputer engineering greatly offset this incon venience,
Elect romechan ical energy conversion is t he result of interaction
of electromagnetic forces appearjngt in an energ y conve rter. T he
determination of t hese forces is the most importa nt stage in desig-
ning a converter. There are a fey." ap proechea to attacking th is
problem. A mechanical interaction at cu rrents, or whet is sometimes
called pcndarmot.ive interaction, 6b~ys Ampere's law. For a con-
ductor cQrr ying current t an d placed in an external magnet ic field ii,
the emf f Is gfven b)' the vector prod uct:
_ . 1
1 = HBft (U9)
whore l rs the unit vector alollg the ' wire carrying current t.
Where t he magnetic fiold is knowc.: from t he solutio n of Maawell -s
equations, it is convenient to express emfs in terms of t he current
28. ell. 1. InhQd..c ~on 10 EIClCll'Oll1ecll,niu
called the tensor of tensi~ll , the expression for which can he reduced
to the form ,
T" ~ It,,H,,iT- J.lG (nfl~/2) (1.20)
where JI" is tile vector component of the magnetic field stre ngth II
in tI,e dir~tion of the un'it vector it norm al to the eurreee reglun
under slIldy. Upon lnlc$'ratlng the tension tensor over the entire
surface where tho megnebic field is slILsta ntilllly high in magn itude,
'1'0 can then go to t he .eomputatrcn of emrs aud electromagnetic
torques.
Jt is sometimes expe~i ent 10 determine elecu-omagncttc forces
and torques from the o>.;prlc'Ssion of mutu al specific l'nergy JW/lIV
I'(Ilorroo to IIllit volu me, ,w'hich is equal to Iho scalar product ot the
current denstt y end lho' vector potential of an external magnetic
field : i
aw/av _ Ai = - oil (1.21)
The "ext MlUwelrs equetlo n. which is of much Impcetsnce. relates
the vector of electric fieitf stNlngth 72 to th e magne tjc flux denslt.y:
jcu1'l jJ_ - dJJidt 11 .22)
In its integl'al form , the exp ression allows us to pose to the expres-
sion for the IIDl f E of It'lo('p (disregarding the grlldie111 of II scalar
electric potential): ~
E = ~ :"' (djjfdt) dS+~ IV X Bl dl (1.23)'
e
- - ,The vcctcre of B lind If gfve us ample inl(wmalioll on the mag_
notic fieJd end hence on all integral qua ntttfes such as currents,
('mfs, voltogcs, forces, aud torques.
The classtcal Iheory ql electric machi nes relies I)U the equations
of circuit theory which ,deflnes the pere rnorers in integral notation.
The most important parameter of all ent'rgy conver ter is its indue-
tillite L defined all thcf ratio of tho lnstantencoua val ues of flux
Iinkage 'I' produced Il)' the current i to tbre current:
! L = jilt (1.24)
;
If the Ilux due to cur rent in II winding Of COllduc.tol' links only
this wind ing, we can llt.1k about self-lnductence: where the flui
links one wind ing due 'to current in the other, we call ta lk sbout
mutual inductance. To define the Ilux linkltge tor the field desorilmble
by Laplace's equation , ItIsnecessary to apply Eqs . (1.14) and (L i B)
in order to go to the cf l?ression for the magneti c flux densit y and
th en integra to the mllgnetfc Iluxes for a conductor on»' its entire
ercse-sccuon S. The nux li nkage, when expressed in terms of the
29. 1.3. ApptlcftlJon 01 ~i.ld Equfttio nl
mogn<'tic vecto r potential, is defined with respe ct to A D taken as.
the reference for count ing off t he ru iming values of vector potent fak
A ! existing ill the cross-sccuon S!
'Y= J(AI-~o) dSI!S (1.25 )-
,
Tho pro blom of determining t he flux lt nka ge practicall y redu ces.
to simple arithmetic operations if 'the conductor is broken down
iuto a fini te numb er of elementary areas each of which has n definite-
value of A I found from th e caloul atlion of tho field.
For the CMe when the n ux lor all points in the cross-section of tho-
conductor of a wind ing (with a cet'ln.'i n number of turns) is constant..
the flux linkage can he expressed :as
L = If!l ",,; w()!t (t.26)·
Introduce the no tiou of permeence' A
A '= <!i!F (1. 27}o
where F Is the mag netomo uve force (mm!) of a conductor (wind ing),
The indu ct ance now becomes Independent of tho curr en t alld flux
and is only a fun cti on of pcrmcance.
L ... (CilF A)/£ .... (WllWA )/l = wt .. (1.28)'
In 8 particular cnse when air gaps are taken into considerati on.
L "'" w2A i W111of.. (1.29"
where J, = ,VVo is the coefficient fJ ~ permeance for fluxes produced
by the mmt. Tho resul tant rela~ion (1.2 9) masks somewhat the
nature of origin of Inductance and njakes th.ill parameter appnren tdy
dependent onl y on the geometrical 'dimensions and types of mete-
rial. H oweve r, we should recall the' Initf al relation (1.24) from which
it unambfguousfy follows t.hat such 0 parameter ns induct ance is
not lit all m trtns tc in any conductor or wind ing but is in dicati ve
of th o conditi ons of existence of a magnet ic field in an onergy con-
vertee. Inductances do uot remain constant but vary quite appre-
ciably when (a) fluxes chan ge slowh-. (b) short-chouned contours
He on the paths of magnetic uuxee r alying in Lim e ana amplitnde,
(c) hysteresis makes itself felt, ana (d) the portions or conver ter
magnet.lc circuits display uonllnuar cberecteetsuce of megucuae-
non. In the th eor y of electric machines, for exnmple, this Inct is
taken into considerat ion ill a number of ways, bu t a sufficientl y '
conststeut, approa oh does not cxtst.. T ho reason is tha t the task of
quantitatively oonst derfn g all the iulluences is extremel y complex,
I t is eaay to calcula te tho omf ta,i t ... constan t) in terOls of the-
self- and mutual Induorences proceeding from ure changes ill the>
30. Ch. l. Inl',oduction 10 EledromedwlnicJ
intrinsic energy of the Held in motion
I = - ow /ax = - ( i ~l2) ({)L!{)x) (1.30)
"T his formul a shows tha t a change tu inducta nce is the requisite
-coudiucn for the electromechanical conversion of energy.
Despite a relati vely simple form of Hold equations (Laplace 's and
Poisson' s equations) and a simple character of boundary conditions,
abe solut ion for the field of all anergy converter haying various
[boundaries, a large number of spatiall y arranged coils, Jet alone the
e oul.inearit.y phenomena 'and hysteresis th at muse be t aken in to
-accoun r, has been found only ill t he last years by use of the numeri-
c al methods proceedi ng f rom yet rather numerous assumptions.
With the use of analytic~l and samlgraphlcal methods of calcula-
'lion t he numb er of assum;ptions grows still more. In particular, we
-can enumerate the following assumptions.
1. Thto main field which determines euergy conversion in electric
m achines and gives rtse to tho main self- lind mut ual Induct ances
-or windings Is plane-pneallel.
2. veercus leakage tnduouve eeacrenees ere Independent of each
-other aod of t he main fl'e{d. It is comm on to isolate per meances
-correapondlng to slot. odd , and differenti nl lea kage flu xes of so
windings.
3, The surfaces of stator and rot or cores of electric machmes are
,smooth; the actual saliency Is given due considera tion by ln troduc- ,
Jug air-ga p coefficients,
4. The permeability of ferromagnetic aecttou is taken infinite '
.e t t ho preliminary calcula tion erege.
5. The use of the supetposition principle is permissible,
6. The processes of ene('gy ecnverstcn are dependent on the lund a-
.zaentel harmonics of currents and magnetic fluxes.
7. The effect 01 eddy tcurrante induced in magnetic circuits ie
,u egligible. We have cited but n few tnetences of all of t he possible
.ccnstratnts. ,
In the list of approximate met hods, the numeri cal methods used
for the solution of Held eq uations occupy distinct posi tions and have
-great significan ce, The f;nite difference method (FOM) is part.icu-
,lart y popula r. This method was in extensive me well before t he '
mtrnduction of digital compute rs to the calculation practice. The
development of hlgb-speed large-capacit y memory computers with
'8xtBnsive gBnerali zed program li brarillS and the int roduction Of T
efficient algorithmic la nguages has made pop ular the methods of
..cal culation of elect romag!Il;ltiG fields on the basi."! of finite difference ,
approximations of GOiltin,uity equations of the most div erse forms.
The main idea under-lying the ap plicatlon of tho FDM in elect ro-
IJlagnetie cale ulat lona comes to the replacemen t of the continucus .
-dtstr jbutton of a scalar or vector magnetic pote nt ial by a discrete '
31. A,
A .. A,
A.
1.]. I'Ipp Uca lio n of Fjold Equ atio n.
dis teibut.iou of the same fu nction io a umned number of potnta
within lhe region beiog studied. The: poi nts at which tho val ues of
the runcuon n eve to be found are dia trfbuted over t ho region of
Inter est: in other words, 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 patter-n. In mos t extensive
use :1I"e the recta ngu lar (or qu ad -
rattc, in a partfculnr cese) system
and the polar svstem of ccordt-
nates.
Fig lll'e 1.9 shows how t he roc-
tan gular gri d (ne twork) d ivides a
salient-pole synchrououa machine
region one pole pit ch ill length
into a few mes hes. T he coo rdl na te
system and t he {arm of mesh es
01 the grid are so chosen as to
approximate most nccuratel y th e
bound aries of the reg ion and to P'jJ. 1.9. SUPl'rimposing til.. aqua......
Introduce t.he minlunnn pesalhle m~b grid on lile "rea under ann(ya.i$
errors in to t he configurations of
neighb oring regions. The grid plotting at th is stago is oft en dona by
the tr ial an d error method. and depen d.'! on the exp erience and skiU
of the lu veatfg ator; t he proced ure hinds itself to automation only
for local zones.
In accord ance with the FDIJ, th o fiel d equut.ion s wnuen as par-
tJal derivatives are rearranged to the rtnne d iffe rence form us ing
th e expressions toe II T aylor se ries. In t he case of a qu adr ati c grid.
wit h II pitch h, t he Laplacian assumes a si mplo form
•71A ~ (t l h1) L.j (14/- Ao) . ..
,-, (1.31)
for a poin t representi ng a value of 'the Iu nc tfon A D and surrou nded
by poin ts AI "'" A . (see Fig. l. U). ',I.'ho erro r of digitiu llioll heee
depend s on t he Iouet h-crde r de riva.l ives In th e so ught-for Iunc tton
and CR.n be reduced by decreasing uie grid pi tch h. The so lut io n t o
Laplaca's equation in t he Hnlte-dtff'erenc e Corm amounts to perf orm-
ing elementary ari tJlmeti c, oporations. 'I'he nu mber of the nodes of
t he solu uon may in prncuce be ve ry high lind usually ranges int o
Il few tho usands. 'I'her efoee the solutt on to t be obtained system
of high -orde r equations requ ires the use of ilorative or statisti c,al
methods. T he direct solution to tho s ystem of eq ua nona using , for
ar ample, Gauss" method proves tnilpossiblo. With th e i te ra tive
meth od of calou la tt ou. t he v al ues o~ the function sought are preset
at. th e Hrst st ag<lS either arbit rari ly orion the basis of certai n physica l
considerat ions which suhasquont ly improve I-he convergence nf t.he
32. where
Ch, 1. l~lfoduetio " to Elccl,omoeh""in
solution. By porfol-ming t he multip lo eaqueuual traci ng of ail
nodes of the gri d and 501v.iligthe ttnne-dt trerenee relation, it beco mes
possibl e to decre ase the rema inder of the field equ ation to the max i-
mum permissible value. ' The number of ueeauons [repet.it fve
tracings) can run into ai Caw lens, hundreds , find even thousa nds.
One COil not IIlways be cO!llidenl t ha t the solution tends to the ideal
va lue, t hereby ensuring (th,a COi lvergence, The Iterati ve method is
ra ther routine, is easy to Iormahae for solvi ng problems Oil digital
computers, and is se re Irom calculation erro rs since possible errors
are recove rab le at subsequent stops. The effective version!' of the
FDM lire availa ble at present, which gi 'e good converg ence at
a high accu racy of the eesults.
A form of gri d marked out on tho region of interest affects Lbe
accur-acy of the solu tion. T his circumstance has recently stimula ted
the search for the bes t fo rms of layout of regi()ns. It is possible to
opti mize sequentiall y the grid structure iJy ca lNIJating the deri-
vati ves of highe r ardor at' a defi nite stage of the eolutlcn with a vi ew
t o raise tho mesh densit)! of the grid at the next stage in t he regio n
of higher va lues of these denvattvcs.
The method of nnrtc ~I emell t.s (FEM) developed in tho last years
displays exceptional fleXibility in brea king down tbe space of an
electrcmagnetlc field d~stined for eulcnlatton. Worked out til'St
for the needs of at ructur al mech entcs, this meth od has turned out to
be ruther convenien t for 'the calculation of electromagnetic fields in
electric machi nes which ih'ave boun daetos complex in MnfiguraUon
and exhibit nonliOOllrities no d induced currents, The region for t he
function sought is hrok eu, down into 0 Itni te num ber of elements
mostly in the form of triangles with straight or rurvilinenr eidee.:
The dim ensions of ell'Jrlcnts may differ subst anti All y depending
on the expected intensity of changes of the field . The desired func -
tion ins ide the ele ments i~ assumed to obey a certnln la w. In a sim ple
cas e, first-power spline (unctions are ap plicable, Thus in the two-
dimensional case, the Iuncuion A (x, y) COr a tri angular element
with coordinates at tho vertices, XI and Yh Z'" and Y"" Xn and /In,
can he written lIS
A (.r, /I) C:. NIA I + N",A", + "'nA.. (1.32)
N l .", 1(1/2..1) lal + bIZ + CIY ]
{
a l = X",Yn ~ xng", I-J XI /II Ibl = /I", - (lin 6..... 1/2 1 x'" II",
CI = z",-jr.. 1 Zn lin
A simil ar ap proach ap pl ies to determ ine t he values of N", an d Nn _
Th us each element is 'describable by its own poly nomial, which
is so chosen !IS to preserve tho con tinu ity of the function along the
33. element boundaries. T he values lit grid nodes ere found by ~ I ng the
" ariatlonal pri nci ples, anu in this ~pect ~he FEM is enen stated
in the contex t of the R ih and Gal llrki n methods. Wi th the vane-
uc net formulati on, th e solu tion w it he pro blem in volvi ng n tw o-
dimensional magnetic field defined. by Poisson's equation (1.16)
is equivale nt to the eendtucn of m lnlr:nl za Uon of a certain energy
fum;:LlonaJ
B
F_ JHI(11. 1BdB]:"'dv-l l/Ad%dV
itl!ido the regio n of integ rati on R h The Iunct tenet dis plays such
• properL)' that any function ....·hieb mln imi:tes it eeusnes bo..h dif-
ferential eq ua tions and houndu ry ~ndl tions. In t.be CllSO oC 1101111-
near dependences ..h e process of min~mi za tion involves the solut ion
of the system of nonli near algebr.~c equatio ns. commo nly using
tllo Nowton·Raphson method which gives good convergence.
The calculat ion of magnotic fiel ds ill elccrete D'I11chines witll t he
aid of the finite-difference and f1n i~e -ol eme nt met hods enebles
a more accurate evaluutfon of the characte ris t ics and parameters
or electr ic meebtucs. However, t he fID.{ anrl FEM call for retn ining
a number or ess um pttons t he l egj~lm acy of whic h are not alw ays
unquesttonahle. One of t hese 1I~!Il lD pti ons made in evaluating
meg net fo Iiclds is that the toothed; sto tor end roto r COl'cS oro ill
fixed mutual poattlon. Th e posl ~'idn it llelf is most ofton chosen
lIrbitrQr[ly wit hout sufficient reaso,, ~ and t he res ults or field calcu-
lations are taken va lid for other po~i.blo mutual positioTls , Vhenevor
the attempt" are mnde to calculate q:t agnotlc field s with the toothod
cores in motion. the comp uter- aid$d calculations prove so time-
to nsum ing that they beco me impracti cable. It tokes an especially
long time and large momory si1.e to ea lcu hltll the nir-ga p band no ted
for the most in tensive oU'Ignetic fiold. On t ho othe r hand . the inhomo·
geneity of med ia In this ba nd shows a rather regular chara cter. for
.....hich reaso n II. large s hare of repeated ealcul aticns can be done
away with.
OwinV to the efforts of a num bee ,of seten..isLS It bas become pos-
sible to evolve the calculation method hlllled on tho eepeesenrauon
of the fields of real wlndlnga as a sqt of fields of the si mple3t loops
disposed on core teeth (Fig. 1. t O). Every loop encircles ana tooth,
The pla nes of the cross-sec tion o ( loo p wires coincide with the pianos
of the cross·u ction of real winding wires placed in the slots. The
loop may abo enclose tI few tee th, or extend along th e entire gap.
Al50, loops mtly have diUorent si deS located in Ilots of various shapes
and sizes,
The essential point of th is meth od cnlled t he method of permeancea
" tha t the loop field must be defi ned not for real but for specific
34. 30 Ch. 1. InjroducHon to ElllCI.omccheniu
boundary conditions whie'h can be obt ained only ortificially. Beyonp
t he confines of t he loop, .rhe perma ence of tbe air gop between the
rotor and stctor is assumed; to be infinite. Undor such boundary con-
ditions the field traversfn g the gap exte nds onl y in one direct ion
an d gets concenc-e tcd in'ihe area t hat differs insign ific antly from
t he area bounded by the Jdop itself. T he loop mmf here corres ponds
to the mmf in the gap. ,Ih going awa y from tho loop in opposite
di rections, the loop fiel d docoys fast. T he loop Iiold under arti fici al
boundary cond it ions ol hbi ls an interesting Iea ture. The m agnetic
' "
,,
,,
-,
,
Fig. 1. 10~ '1'b ~ Held of .0. &Implut loop
fl ux thl'oug-h the gap due'tc t he loop current is t ho eame 35 lhe um-
pol ar flux li nki ng the loop when tho difference of scalar magnet ic
potenttats between t he cores is equal to tho loop c urren t. Also,
the permeence for th o loop flux ~hrough tho surface of an unexcited
core corres ponds to the permeance for th e loop 1I11.x linkage in uni-
polar ffia/:Jleth:ation. Th i; 'is tho case for any form of the two-sided
sa liency and fOI" OilY arrllf gement of loop conductors in the slcte or
In the grup. T his fundament al property of fluxes and fl ux Imknges
of tooth-I nd uced loops oqens the way of evol ving a ne w me thod to
ena ble the developm ent 9f' methemau cat models for descri bing the
fields in electric mac,hint'$ With du e regard roe the tw o-sided sal toncy
of cores . !
T1lie ma th ematicnl mo~el though resemblfng to II definite extent
th e model applied in t he 'FEM contains A special feature. A portion
of model eleme nts represnnrin g tho IIi,' ll:flp permeance RfH fire not
permanent and calcul ated ,beforehan d either wi th the aid of rather
cl rcumat nntia] ne tworks ~ m pl o ~'e d in the FDM and FEt.l or an nly-
t ically by use of the methods of conform al wenetorrnanons. T he det e
of this cetcuteucn are ent ered into the compu te r memory in the
form~ of approxim ation $J rves or tables. The teeth an d yokes of
cores are broken down into II number of elemen ts whose di mensions
call be t aken apprccia b'ly, i3rgor (without t he in troduction of notice-
able errors) tha n is tho case with the FDM or FEM. The nonli near
characterist ics of these elements are defined starting from the EH·
35. ae
curves for corres pondin!; ma torials. As is done in the frameworu of
other methods , here too t he permeabiltty inside an indiv id ual ele-
ment ill eeusldered cons tont. T he mljgDetie s~a te of core etements i5-
first se~ roughl y and t hen s pecifi ed more aceu retely afte r soh'ing-
the system of the nonline.r equll tions by th e Newto n-Rnph80D'
iteru ive meurod.
The ma thema tj cal model bnsed on the permeanee me thod uses-
II relativel y IlIrgc-'!li...e mesh patl8rJ1 and gives II Iligh acrur. cy of
field I1lprOO U(".t iOIl, eapeciall)' iii th e gap b and . This opens up poe-
sibili ties for the c.• Icu tauon of field'.!! 10 the transiont operation of
clO<'trie> mll ch in ~ with considerati on for th e effects of saliency,
Iliscrelene.ss of th e windi ng structu re, salura tion, and induce d!
currsn Ul, The equaucue Cur all loops in rho per mean ce meth od repre-
senreuon do not neceesnatc IIdditiplInl coordi na te transfor mations,
Although t he pl,.,gress in th l.' devolopmout of etee rrto machine
models on th e b llsi ~ of rieh! equa tfons jlZ appreel ahle, the meet mete-
rial advancements life mad e by use of the equ ation!' written in uie
notati on adopted in elce trtc circuit &hOOl)" Therefore in the fu rther-
preeeutau cn of th e lcxt we will basicall )' employ the eq uations of
tbe genera lizocl etcc t rc meehanica l energ y ccnveetee.
1.4. The Primitive Four..W indJng Machine
All electric machin l.'S ere identical in t ho 5ef15ll that th oy ecnvere,
eOf:'rgy from elecreteel to medulII ica. form or Ieom Ull'(;halliul to
cleet rietll form . Bu t eloctric mar hilies even or the same series diHer
from one another in perfor mance.
The hosie types of eteer elc machines c>.n be reduced to II genera-
haed, or prfml t.tve, medel representi ng :I set of two pairs of wind ings.
moving with respect to each ether, I n Fi~ , 1.11 is shown the ide ali -
zed model of a sy mme tri c, machino 118'i ng II smoo th nir-g llp srruc-
lure lind sinusoidIII wind ings. wi th. -t h'e permeance equal to zero.
A sinusoidllll y varying ve ltuge applied to th e winding produces-
a etrcutee fiold in t he air gap. W ith the windings being sym met ric.
I ~inWloi dnl symmetric vo lLllge sets up II ~inusoidal field in th e ge p.
Th e term ' primitive machtne' sta nds for an Id~aliKd lu;o-pol~
ttro-fAQ$~ I/lmmeirlc (balanttd) machLne haVing on, pair of /rim/ings
Dn th~ TOtor and th~ olher fXJir on Ou: stator l'S shown in Fig. 1. 21.
Here wU' w~ are the U nto r windings. elollg t he cz and ~ axes ; w;.
~ are t he rotor wind inas 810ng the IX aDd r- exes; u:.. uj, u: . u,
are voltages al ong the cz ead ~ u:l.'li on th e ! tator and rotor reepee-
th'ely; end CIl. is t he angul8r speed or t he rotor.
Th e Ilnillysis of th e two-pole machine M a model en1Ulles us to
exte nd the fl.'Sults lind desuibc the ' processes or.curri ng in a real
multi polar machin e. The two-phase' marhine bes {our windings
36. .and is describable by four voltage equa tions (8 minimum nu mber
-of equlltions in eompeeisen wiLh thos8 used for describing Sillj'le-
-phese, t hree-phase. and m--phase machi nes). Consider an idealtzed
'lIoiCorm-lIirifap machi ne 'whose wind ings 11I"O t aken lo be in the
'form of currenl ' hools ":bera the mod distri but ion Is slnU50idal.
'Our ide., lind mncblne hllS:nO sa turation, nor nonlinear resetances,
.and therefore el:hlbits a ~i nusoid al field in t he alr gllp ....hen the
w indings ore fed with sinusoidal voltage.
The Idealized ma.cbine mooel is the analog 01 an inducti on ma-
c hine when thestah.r windi ngs", and Ultl aceept sin usoidal Yollaaes
,
:nt frequency f" 00" apart in thn D. The rotor windiogt carry euerenta
-01frequency I, = I,., 8ilher produced by th e voltage ap plied to the
rotor or indueed by tho e:urre nts in the stator windings. In an Indue-
tion machine, tho roto r angular spelld is 11)• .p 11). (w. is the synchro-
;II OIU speed of t he field), t nd the rctce and stator fields are st..atio narr.
"With respeet to each oLhh since Lhe mechanical roto r speed 11). plusl
mlllU5 the rotor field sP,oed relative to (0), is equal to til • •
T lte idea lized machine model filpres&nls a sYllchronoU5 machinl
if an ee voltage Is put 'Cl"Oi'lS the sretcr windings and a dt. voltag,
.aCf0S5 the rotor wind ints, and vice vcrsa. Here (0) . = ClJ., t.e. t he
:stalor lind rotor ftelds '8l'8 stational')' with mpeel to each other.
If II de voh ai e drives eqrfilnl through the stator windi ngs. the rolor
field travels in the dirK tion opposlte to that of the rotor, 11(1 the
~tato r and rotor field! are stationary relative to tbeslationary refer.
enee framo. With de ~u p pl y lo all the windings, it is enough to
37. l A. Th. PrimitivD Foue-Wil1c1iI19 M. " hlnD sa
have one field win di ng ill which the; msultanl. magnetizing force 13
equd to tbe geometric sum of the magnethiog forces of each winding.
11l de mac.hines, t be armature winding canies a mu!tipb l3e enee-
Ill ling current reetili&cl meehll.nicall ~ h y means of a commutaLor -
• frequency eecverter (Fe). By red uciPI: a poly phllSe s}'Stem to a rwo-
phese 006. we obt ain l.he model of iii de machine (FIg. 1.12). As in
II synchronous machine, the arma tu re fi eld of t he de machine rotates
) «
, ",
. j
,
Fe
.'•
Pia. 1.12. The mooe l of. de IJIxhlM Ilod an 8C colltmnl.alor macbiof
in the op posite sense with respect to the armature. ' Vhen CIlr ... fit.
the ar mat ure field is lItatio nory relafive to t he field wlndi n: and
to th e !ll ation nry reference Ir nme. It. llllouid be note d that tho slip
In sync hrcnoua machines and de macbfnes equ als taro. A commutator
can be replaced by a aumiconductoe Ieeq uency convertor, reed rolay
converter, etc. T he processes of energy conversion in t he air go p
do not change wit h t he repla cement (If one t ype of Fe by t ho other.
However, a conventional com mutator holds a fixed ti e between the
frequ ency and the rotor speed (o)r•.while a se mtconducree Fe may
afford the possi.bi li ~)' of eecurtng contro ll able (eedback rc regu ll'lte I;
aecording to CIlr. A!I regards its power supply, a semiconductor-
commut ator machine i.! Il. de machine. Historically. this l.yP'l of de
) _,I lia
38. "machine received several, names-c-eectiftee-rype mechtne, se mtece-
dl,u::tor-.commutl tor machine. eenreeness machine, etc.
In an lie commutato r machine, al tern atiog currents 61i.s t in the
st ator and rotor .....indlngs. aDd the frequencY converter trallsforms
tbo al tern _Ling eureem at lh. hus frequency into lh_t of sl ip Irequeney
(see Fig. 1.12). .A! in other electric. machi nes, bere tho sta tor field
iSlltAtionary relattve to t he rotor field. These machines..een bfl of the
l ingle--phaso, three- phase., or mul tt pbese t)' pe!; th e stator lind rotor
windin p can be connected in series or paraUel. or ean have magneic
coupling.
TIle primitive machine with a rotor speed (I), - 0 can represent
an electeomegneuc eou veuer-c-e tra nsformer. In this COlle i t is suf-
ficient to consider separatel y th e pair of wiudlnga on the stator ond
rot or olong the a axis or '" axis because with tbe roto r at stands ttl!
t here is no coupling between the windings llhih ed 9O~ llpart in space,
Although transformers perform electromagnet.io ecnvaeslon of energy.
the)' belong to electric machines because of the generali t}' of equa-
tions and for historical reasons.
Tile classifica tion of oloctric meehiues hlto individual types i.!J
laree.l y conventional. O ~e and t he same machine can cpeeete M
e synchrcnous and lUi an ¥ ynclu'oDous machine. In etecterc machines
there occurs elC!(;tromt'Chanical and eledromaanetic e-ue!"ln'ecnverslcn
simultaneously.
Tho processes of elec1romechlUlicai energy conversion in the prj-
milivo machine are described by voltage equatioos (1.34) and equa-
tioll 01 motion (1.35)
"•
o
(lIp) ,J d fll,ldt ± M, - Jot.
o
,.~ + (dldt) L:.. (d/dt )J11 0 0
(dldt) M ,.:' +(dldt ) L~ L,fIl, M(fJ.
X
'-' L~(fJ, ,.' + (dldt )LD (dld t) iU l~
(dldt ) M ,.~ +(dldt) L~ I~
(1.34)
(i .35)
•
'"
Bqs. (1.34) and (1.35) t9gethor wit h the equa tion lor an electro-
magnetic torque lorm theifundll.mentll1 system of e-quations of elec t-
romechonical energy ccnve rslen.
In Eqll. (f.M). u.:.. uj-. u:.. loll . t:.. 4 . ':.. I~ are the "oltages
and currents in the sta to~ and rotor Vo'indl op on the 0: and II axes
respecti vely ; ~ . ,.j . "0. Ii lire the re:sistlnces of stator and rotor
windinp rospecLh'ely; ."rb mutual inductance; and ~ . LA. L:.. Li
Ire total inductances of t,he l tll.tor and rotor 'WIndings alool the a.
and tJ Uet respectively.
39. 35
('-36)
relati onsinductances are defined by tile known
L" - M+ " ;" = M + I'.1 ... , ~ ..
Wi ndi ng
L/ - M +',. t, ~ M+'!
whOlO 1:.. 1~. [~. 1~ aNI leakage iDd~ctances DC the sta to r and rotor
windings along th o a and p axes eespeeuvely.
Th e mu tu al indu ctance lind leak age inductance.. are Cou nd by th o
known me thods Involving tho (' a1 (';U~lI. ti o n s or ex perimen tal nnaly-
sis, r.e. using eq uivaleu t circuits and design formu las. T ho ass ump-
ucu is that there is (I working tlux 'i'hiclL links t he stator and rotor
wind ings and elso leakage fluxes linking only OM wind ing.
Equations (1.34) describe a hypptbetical ma chine hay ln!: the
!l3me number of luTJUI on the stator ll.pd on the rotor, with tho wind-
ings beiog pseudostaUonary. To preserve the power Invaeiance in
1111 ac lual machi ne and in the IIH<: ~ne wit h stationary wi nd ings ,
tbe eq uations have to conlai.D tho ; mfs of !'Oh ti o", ex pressed a.s
~U) ,i& + MfiJ,l& for tho rotor ""in,ding along t he a ax is and all
- L;,m7 t:.. - ,4f U) ~ for the JJ-axis winding,
K irchhoff' s ec ue ucoe (CV.) include voltages. vehege drops
1I1:r0Sll resistances, omls of rotation tha t eJ"illt on ly in ro ta ti n ~ wi n-
dings, And t rllOsformer emrs :
L:' (dldt) t!, + !If (dldt ) e: ;M (dldt) i:,. + L~ (dldl) I~
ThE! transformer em rs for th o ~ .ax l s ~j ndi llg8 are written In 11 simi-
lar form.
In th e eq uati on of motion (1.35), P Atfltlds for t he number of pole
pai l'll, and J for the mom ent of Inertl,a, If Lbe analysis is mad e of an
electric msehl ne to gethe r wit h ns delve mech entsm , t he quan tity J
mllet re present t he roto r moment Of Inertia an d t he DormaJi~
moment of Inerti a of the mechanism,
In the an al ysis of electric mach ines. the mom ent of reetste nce M .
f1urq u o) is usulIlI y t aken constan t, In tbe ana lysis of elecrreme-
rJJanical systems. lof . can be a Iuuct tcn of (,)7 or time.
The electrom agne ti c torque M . - the torque produ ced b)' a con -
" o rtl~r- is givcm h y tile produ cta ot cu rrenls flowing in the winding!!:
M . = (mI2) M (i~i~ - i~/~) ( 1.37)
where III is t he number of phases,
The elecrromec hant eal energy converal on eq ueuoee su ggested by
Gahriel Kron in the 19308 com pelae the system of rtve equations
(1. 34) an d (1, 35) i nvolving five independent variables (vol t ages
and M 7) and fin dependent varialiles (current and angular s peed),
The coefficients . head of t he dependent. va rill:bJes, nam ely, lUis-
taocee. induct ances, mutuAl in du ctaDCM, and t.ho mo ment oJ iner-
tia, are the parameters of an energy;converter. }
.-
40. 36 eh. l. I nl~od udlo n to Electromechanics
Jn the mathematical t.heory, the coefficients at variables may
vary with the form of equauoas used, therefore it is of importance
to have a clear idea of the parameters and mathematical description
of the processes of energy·conversion. T-he parameters of a machine
een be const ant , periodic, and nonlinear.
The analytical solution of the equations for ulecrromechan tcal
conversion does not exist because the equations contain product
terms. The equations are solvable with the aid of computing devlcss,
tho solutions being- appro:Ximate. This approach also permits hand-
ling equa tions with nonJjnear coetnctents.
The accuracy of soluti:on of equations depends on the class of
computers used. Computers can solve a simple problem oven to
a higher accuracy than le necessary for the 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 taken into account. Even
the energy cenvecsion equations with constant coefficients are
nonlinear sin ce the torque equation contains tho products of variab-
!1:lS. The addition of nonlinear terms onl y makes the problem more
difficult. l
Independent and depende nt variables in (1.34) and (1.35) may
vary in value, and t1lell:1hey describe what is called the current
drive.
The system of equ/lti0fls (1.34) consist1ng of four voltage equa-
tio ns and th e equation of moti on (1.35) describes transient and
steady-state modes of operation. To obtain the steady-etata equa-
tions, we should replace the differential operator a/dt by /ID and
work with complex equatJoM. In the steady cond it ions the volt age
equations can he dealt wtfh inde pendently of the equation of motion.
The courses in electric mecblneey commonly cover voltage equa-
tio ns, and th e course in electric drive mainly considers the equa tion
of motion. ,
The electromagnetic torque Af~ is equal to the product of currents
In all of the lour windings. The torque (1.37) is set up by the cur-
rents in the stator and rotor windi ngs disposed on differe ut OXOS,
with the stator current Shifted in phase with respect to the rotor
current. If the rotor and -stator windings of the primitive machine
carry only active ac components. the tor que M . is zero since the
coupling between the winJli.ngs due to reactive currents that produce
the magnetic field is absent. .
The solutions to the equations of electromechanical energy conver-
sion do not exist if any ;ol the parameters entering into the equa-
tio ns is zero or goes to ;infinity. If the resistan ces and inductive
react ances are at infinity the currents are equal to zero and the
machine does not develop. the torque M ~. At J .= 00 , the energy
converter picks up speed jnfinitel y long. At J = 0, the machine
cannot come up to its steady-state velocity because the rotor res-
41. s
iIlg• 1.13. Tho principle of an MHD
, gen~rator
8,';;:And if an mng " ," lc fJu:r dON lty vee-
tee. 00<16"" 111141 , .. Or li quid vel ocity , ond
emt IMp,,"'" 011" lIanne l WRI II r• • p""t1 vey
1.~. The Primit ive FOIr_W inding M oehl...
pnnses to all changes in th e produ cts of currants: con ti nuously.
If the mu tual inductance is zero, tho magnetic linkage be tween the
windings is none xis ten t and M . = (} (1.5). If there is no resistance
in the loops th rough which t he cltrr,enls complete their paths, the
device will act as a s torage of energy. T he time constants lire at
infinity, th e shift between curreuta Is zero, and M . = O.
It is possible to obtain op timum rOiations between t he para meters
at which an electric mac hine m il)'. have a maxi mum efficiency,
higher cos Ql, a min imum mass or a des ira hle form of output cha-
racteristics. It should be noted .
however. that Eqs. (1.34) and
(1.35) are unsuitable 'for use ill
optim ization at udtea because th o
minimum values of currents (de-
pendent Io'ariflblos in these equ a-
tIons) aro not yet indicat ivo of
an optimum mechtne.
Consi dering voltage equ ations
(1.34), we should pctnt. out that
the terms defining the transfer-
mer emf Include t he induct ances
and currents under the derivative
lligll. In most elec tric machines
currants are va rying quanti ties,
but the conversion of energy from
etectr tcet to mechanical for m is
possible it cu rrents are constant
and inductances un dergo varia·
ucns in a sinusoid al manner. T he
mach ines performin g energy con -
version ill t his manner comprise
the class of param etr ic dev ices , among which inductor machi nes
are most popular. In th e general: case, both in ducta nces and
currents in electric mac hines va ry etnusotdally.
Given the mathematical descrip't(oD of the processes in electrtc
machines, let us Inqutea into the nature of ene rgy convers ion in the
machi nes. The general concl usion tpat can be drawn from t he cons i-
derati on of the laws and equ atfons of energ y conversion comes
to the foll owing: electromechanical en(!rgy cOllverlficJrt is possible i/ any
0/ the quantities entering the energy' eonoeaton equa ua as undergoes
variations, ,
Most of electric macbincs are satd to ope rate in one mode or
another it their windings carr-y aftarnat.ing currents . In these ma-
chines th e parameters ma y va ry too, Energy COilversi on is posal ble at
consta nt vo ltages and currents bu t It varying parameters . Energy
convers ion can occur when inductive reactances and restetances
42. se
elliOt i"g the equations undergo vllrilltions. With It change in the
momen t of inertia, II machine arores k.ioetie enerll'Y and gives it. up
to t he line.
Faraday's mac.hi ne (&e$ Fia. lot) llnd ure magoe lohydrodymunle
("fliD) I:ODera lor (Fig. 1.13) Ire the most complicated case for the
explanatioD of e leelro me~l1an ieal energr c.onvcrsion. I n Faraday',
machioo with I permane}ll magnet , lhe de circuit. cha nges etate.
It has II portion that il slatlonllry and a port ion that moves about
tho magnel. The sliding eenrecr is oblililltOry. If tho loop is mnde
uniform without the slidj ng contact, the molar will not run even
when the curren t source is mado to ro rate together with th e loop.
In the MHO generator; the VAr )·j ll g . parameter is the velocity of
plasma in th e noaalu and .oruetde it. In Faraday' s meter , tra nsition
from the rotating part of ,tl ll eureent-carrylng loop to tho stlltiOOll ry
part cecues in n jumplikemanner within the sl iding contACt rogton,
while in the MHO generator the velocit y of the activo modium
changes smoothly.
t .5. Application of Computers to the Solufion
of Probl,ms In Eledromechanlc:s
Since the equaUons o ~ elecl.(omcchanical onerry conversion are
llOnlinear , the anal)'lLClI ,solution exists only under certain assump-
tions. when w, = 0 or the speed vartes li nearly, in which U1!Je
voltage equations (1.34) are solvable independent of tho equation
or motion (1.35). The llOlllysia of tr-ansient processes with a varying
rotat ional speed is possiblo only with the aid o[ comput ing devtces
becaUllC the equations contain Ib! products of vaeiables.
Electronic computers CJ)n he classified under three mai n head ings:
analog computers. dlglttJl computers, and hybrid computus.
An Ilnlliog computer represents all variables by oontl nuously
varying physicnl quantities (eurre nta nnd volt ages) whose change
gives the solution to the problem bein g !Ilvestlgllto.d. An)· dy namic
cnerectcrtsuc is reproduced by II recorder, for example, on the screen
of II cathode-ray oscillograph. Jn solving problems on an onaloi
computer , i t is well to s.ta te the problem first i n li n Incomple rcly
definite form and then refine the slalement in tho proc:ess of tbe
analysis of tho problemj A disa dvan tage of II l1alog computers is
thai they have low accuracy ne d limiled vetSlltility. a ut the accu-
racy up to a few percent is ofte n quite sufficient for m an ~' engil'let'rinr
studies because the epcefjted accuracy of initial ual a is )"et lo.....or.
What makes an lInalog computer illSll rricient lr un!'etSlll is that
tn nsition from the solu tion or one problem to thlt of the ot her
nquirel changin. the flow dilgl'am of the machine . Analog ecm pu-
lers availa ble today h a n~i1 e problems irn·olYing the integration of
43. 1,S. Al"» llcaiion 0( Comput. .. 39
OrdiIlRf)' d ifferent ial equntione, alge;hraic a nd tra nscendenta l eqnn-
t iOIlS, And pArtial diUere ntia l OqUIIPOIl!. T hey are convenient lor
use in the a nlll~'sis of dp l.mle o~ rA tio n 01 energy eoevcrte rs.
Di ~ i LaI computers fi nd use where lit is necessity to 8Oh'& mathe-
m~ tical problerms to a h igh IIceuneY. Th e input lind output infor-
m . ~ i on here is in the dtscrete form~ so t heM JnA<;b ines realiu- tbe
numer ical methods lor t he solutiorl of probl ems. The cnleulnt ion
ll(;cura C~' artatnable on II di gital computer doponds Oil lile qUllnlit)·
of bils, the li m its being set by u,e si1:8 of computer faclliti es emplo)'-
ed. Moder n d i~ ita l computers ca n a n;tomllt ica ll y perlorm II complete
computllt ion with a speed 100 000 ttmes as rlUlt es 0 human bein S
does and thus offer the g r en les~ possibili ti es fOI' carrying out eaten-
la tions. A ll dist ingu islled from a:nalllg computers. di gita l 01l1lS
handle problems with a definitolY.sUltod solut ion algorftbm , for
which th e inst ructi on ( pro~fIl m ) is writte n end give n t'O the mechlne .
Dlgitnl computers of toda y are ca pable of sol'lng a wide range of
prchlems. I n go ing Im m the solut.iojt of one probl em to that of the
crher , it is on ly necessa ry to chAngelthe program withou t mod ifying
the computer flow di AgraDl.
LBr~ dig ital computers are cMltly, 1I0phis tica.8d . and highl)'
..niversal insta lla tions ma inly ser up at ro mputi nl: ecntc l"!I whose
person nel service the machines. prepare the problems to be solved
ami program them, Le , wri te t he probl ems in a mach ina la ngua!;e.
or eode in !II su ifa ble fonn rcq uirid for the automated solut ion.
A digital compute r llperates wit~ d iscrete quanli Lies-numhera
rep resented in a defi nite notation. T;he main adv antages of B digit al
computer Are a high accuracy of computa tion, up to 20 d e~ i nl a l
digits and over , and inherent versatilil)' which allllws fllr tile solu-
t ion of " wide class 01 prob lems. ? d isadvantage of th is type of
eomputer is that programmi ng. debul:ll'ing, fwd decod ing of the
resu lts ob tained in discrete form -~ nsu me II- great dea l llf t tme .
TIu: f, rsl a/gllal CQmpulers were built aro und alectrem achu niual
rOIIl)'8 a nd then, la te r, around vac uu m t ubes. The on-line memor)'
of the mach ines relfed 011 tube tri ggers, mercury dela)' HnM, cathode-
ray tubes. and , later, ferrite eore~. Vacuum t ube-based machines
wi tll l speed in t he order of io.ao th ousand opernt ions in a second'
belong t o tho first generation of computers. T hoy appeared in 1946
aud were buil t up to the earl)' t 96Os.
1'114 .second.generation dlgttal CQrnpuurl that began to appear
in t 960s are the machinos based on sem iconductor discrete eteeseuts
usin~ ml'lgneti c-rore memoetes . The'maeh lnea llf the second genera-
ti on occupy a hundre dth the Sp:l.~ of t he firsl-ge nera tion ccmpu-
tees . consu me a hundred th tile amou nt of energy, and ca n perform
ll-.bout a million ope ra t illns per second .
Th~ lhird-~n4rQtion compuUn ~ :oem iconductor small-scale tnre-
grilled etreuus (on the averoge 10 gates in II chip), magnet ic-core
44. Ch. L Introduction to EIec1,omechonlu
mem orie s e nd , p ltr ~i o. ll )' ,: maguetlc-disk memories. The computing
system of the th l,·;1 gencre tion displays three charaotertst ic fea tures
as foll ows: employs jutegrated circu its ; has Input-output channels
a nd the developed network .of pertphceal un tts: and is made complete
will, software wht ch form~ ilfl integl"lll pllrt of the computi ng system .
T he cost of sof tware systems grows SI<38CUly with each pass ing
year . Vbilo at the beginn ing of t he 19GOs the cost of pro gra ma was
30 % lind thnt of equ tpmeht 70% , lit pI'()SQnL th e cost of software
reaches one half the total cost of hnrdwol'e.
E xamples of the third-ge nerat ion com pu ters include tI'l! IDM36{l
lind the Soviet-made EC m'nch ino thll t closely resembles tho former
in parame ters (EC is thl$ abbreviation of Russian words meaning
th o un ifi ed system) .
T ho IBM 360 system represents n family of the third-generat.iun
ma chines developed by 'the world 's largest Ame ri can computer-
huil ding corpora t ion.
Th a IBM360 displays ! 8, number of dtst.inguishiug features . of
whi ch the most im portant are the foll owing; the program compa t i-
b ili ty of various types of computers eutcelug tu to the foroily, which
provt dos for the appli cability of programs in going from one model
of the machine to another; tho possibility or con necuon of a largo
num ber of In put-output dev ices find standard Iut.egrat.Ion of input-
out put devices with inpll t[-ou tput channels; the capabil ity to opere-
to in real ti me in control systems; and the possibili t y of combining
sma ll comput ing-power machines In to a single evetc m .
The IBM360 com puter is,1I un iversal system designed for servtng
econom ic (buej ncss. comrneorce) and ectcnurtc purposes nud also
for solving the problems i?f dntn transfe r and con trol. T he ata udard
system of programs offots! th e basic computation cepebtuues nf the
runchina. This comma nd .svsrem may incl ud e means for pr oces-
sing da ta in decimal notation. The addltton of floati ng-point feat ures
grvea a scienti fic command sys tem , an d the audit ion of secu ri ty
facili ties to the eco nomic1laud eetenurtc command sys tems providee
II universal command sys am . A few types of the IBM360 machine
ca n be combined with th(,l a id of central prO(,'-4!SSOl'S to Ic em 0 COIU-
puting complex. The IB-"f360 system 1181'S solid inte grated ci rcuit s
uetud lo r h igh speed nnd ,smtiH sb c. whi ch ensures h igh reliability
of thll com putara . ,
The ui ecut oes of the .E!C typo employ t he standard network of
in terconnection or per-ipher-a l unua, t he so-ca lled input-output
interface based on the program contro l of t.hcsc units.
T he ha rdware of the EO ma chine ca n be divided in to fo ur grou ps
(Fig . 1. (4). Group I Includes processors toget her wilh the register-
based work ing storage. arithmetic and logic elements, and cont rol
devi ces. The devices of Groups II and I II li nk processors to" pertphernl
unit s wh ich fnrm Grou p J:V. To gro up II belong selec to r and m ul-
45. loS. Ap pllcetlo n of .Compute",
tiplox channels. The selector channel opera tes in the burs~ mode-
to provide for a hlgh-speod da ta inpu], to and readout from only one
peripherAl un it for II ooctain length [of t ime [ner u-ly a second en d
oYl.'r). The multiplex channel perm i l ~ II simultaneous da tn transfer
for II large number of Input-out.put,' units. TIle lin k between the-
units of Groups Jl and III forms what is called the in put-ou tput,
ioterfaco of the standard design, wh ich is Q detachable 3O-wire-
connocttcn ensuring the transfer of, control signals and dcta.
0'"
Fig. i.14. The hardware of all. EC computer
MC- n,ulli plu cJlan1l~ ' ;8C - se l~ , d18.n...ol. ; GD - ;-'l Up dHlce. ; lJ/II - don><lllulal.Pn :
111_ t.po . !Ot' l:"0 un lll: CC.... - ClUl.nnel -t<HJ,~no ~ ''''1''''';DT"f _ d U, '"~t., ,",ull l-
plese,"
The devices of group III intended to li nk the Interface to vaeious-
peripheral uni ts include individual. interface devices , group arrange-
menta M>rvici ng a few peri pheral uotts, ehennel-tc-chenuel adapters-
providing direct connection betweenthe selector cha nnels of proces-
sors, and multiplexors for da ta trllll~fer over a fow chan nels.
Pertpheral units include magnet ic-tape, magnetic-drum. or d tode
storages; in put-outpu t. devices for Pillich cards and tapes ; pri nters;
dal,n terminats and consoles; tbe means [mo dems} for data tele-
processi ng and communication witll ,control units .
The muthemntfca! program package (soft ware) for the EC computer
includes the programs of t hree cetegceres: opcraucg systems ensue-
jug the Hnk between thll operator audmser and dis~ri bu tillg the jobs.
lind system resources: ma intonance programs or lest rout ines (debug-
ging. checklng , and d iagnol:ti c rou t ines): nut! tho packs of appttcnt lon,
46. prograJrul , w hich arc funet ion:llly complete seUI arranged for tl18
so lution of a dofi nite cla ss 'of problems.
T he EC unita ry syste m tepresoots a family of program-co mpatible
comput ing setcpa of the following I)"pea: EC-tOID. EC·l020,
EC-l02t , EC-t 03O, EC-tQ410. EC-I05O.EC-IOOO. The working
s torage capacity ranees from 8 to 10' ki ]ob)' les.
The bllBlc features 01 the EC cemputer aro it s uni versal ity. eda p-
l abili ty for V::lriow appJicalions. an d the possibility of II. grAdual
bu ildup of tho computing poWer over II. wido fan go. Th e versat ility
Is duo to tho instruction set involving fixed-po int and lJo!lling.
po int cemputettons. logic And dccl mal opera t iOIl!!, opera tions wit h
varia ble-length words, a nd eteo due to various dolll Ierm ms , multi-
programming possihilities. and the advanced sys tem of software.
The ado pta bil ity for use! "ste m!! from the chnngeahle structu re of
the EC eyetem (roplneeabiJity of memories , ehnn uels. peripheral
equipment). A gradUAl Increase in tlie computing power ce ll be
achieved by seve ral method, . namely, by increasing t he number 01
periphetal UDit.'l and the working storage cApac ity, produ cing multi-
machine ccmputing complexes, replacing tbe processor by 1 futer-
.speed type, etc. The prOgnm compat ibil ity of the EC comr uter
co m ~s from the unUled 10giCflI structu re (staodardil.at ion 0 the
inst nlclle n set, data representa t ion form , And address system) .
Durini the las t 25 ,.~a rs ;tho com puter speed. storage capacity,
and reliAbili ty have increased mllDY ti mes . The o1:ua ll di mensions,
tho energ)' consumed , and the specific cost of computers decrell!M!
very fAst eoncun-enl wilh thr improvement of their patllmeterS and
characteristics.
At Lhe atort of the 19705, the first fourth-I en.u lHio1/. computers
appeared , whteh began to l.se medium-scale ICs (about 100 gates
on a chip) a nd large-scale rqs (thousands of gates on a chip). Wh at
,d iat inl:ulshos the Iourth-genem t.icn computers is that th ay widel y
-e mploy semiconductor storages, enlorged Iust.ruct lon sets , mi cro-
progromming, built- in subrout ines , au tomat ed program debugging,
peri pheral unite and chonnols of divorsed types and im proved qu ail-
t y, in tl"rfoces , speciali>torl .processora. Those computers exhibit
.enheneed reliabili ty a nd [oFm t he basis lor the cccstrucuon o[
mulUmachino and multiprocessor compuli ng complexes.
The emergence of an au tomatic universal dlgi tllli computer that
performs Ilr ithmetic and logical operat ions with a higb speed opens
up new q uali tat ive po!ISibilit ias lor conduct in: the theoretical
invosti gati ons in conjunction with ebeck ex periments.
Hybrid computers which Itomprise digital machines a nd a nalog
-uevtees hold much promlso for the efficient combinat ion of the 010-
meets of & hybrid iD.!lt.UalioD to enable the mcst ratiooa l solution
-cr prob lems.
TIle digital computer in a hybrid complex ill a contro l machine
