winding saxes
TRANSCRIPT
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WindingsandAxes1.0 IntroductionInthesenotes,wewilldescribethedifferentwindingson
a synchronousmachine.Wewillconfineouranalysis to
twopolemachinesofthesalientpolerotorconstruction.
Resultswillbegeneralizablebecause
A machine with p>2 poles will have the samephenomena,exceptptimes/cycle.
Roundrotormachinescanbewellapproximatedusinga salient pole model and proper designation of the
machineparameters.
Wewillalsodefinean importantcoordinate frame that
wewilluseheavilyinthefuture.
2.0 DefinedaxesThemagneticcircuitandallrotorwindingcircuits(which
wewilldescribeshortly)aresymmetricalwithrespectto
the polar and interpolar (betweenpoles) axes. Thisprovesconvenient,sowegivetheseaxesspecialnames:
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Polaraxis:Direct,orDaxis Interpolaraxis:Quadrature,orQaxis.
TheQaxisis90fromtheDaxis,butwhichway?
Ahead?Orbehind?
Correctmodelingcanbeachievedeitherway,andsome
booksdo itoneway,andsomeanother.Wewillremain
consistentwithyourtextandchoosetheQaxistolagthe
Daxisby90.
Fig.1isfromyourtext,andshowstheQaxislaggingthe
Daxis,consistentwithourassumption.
Fig.2 is fromKundur,andshows theQaxis leading the
Daxis,whichwewillNOTdo.
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Fig.1
Fig.2
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3.0 PhysicalwindingsTherearetypically5physicalwindingsonasynchronous
machine:
3statorwindings(aphase,bphase,andcphase) 1mainfieldwindingAmortissuerwindingsonthepolefacesThe statorwindings and the fieldwinding is familiar toyou based on the previous notes. The amortissuer
windingmightnotbe,sowewilltakesometimehereto
describeit.
Amortissuer means dead. So this winding is a dead
winding under steadystate conditions. It is alsofrequently referred toasa damperwinding,because,
as the name suggests, it provides additional damping
undertransientconditions.
Amortissuer windings are not usually used on smoot
rotormachines, but the solid steel rotor cores of suchmachines provide paths for eddy currents and thus
producethesameeffectsasamortissuerwindings.
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Amortissuer windings are often used in salientpole
machines,butevenwhennot,eddycurrents inthepole
faces contribute the same effect, although greatly
diminished.
Amortissuers have a number of other good effects, as
articulated by Kimbark in his Volume III book on
synchronousmachines.
Amortissuerwindingsareembeddedinthepoleface(orshoeof thepole)andconsistof copperorbrass rods
connectedtoendrings.Theyaresimilar inconstruction
tothesquirrelcageofaninductionmotor.
Figures 3 (from Sarma) and 4 (from Kundur) illustrate
amortissuerwindings.Notethattheymaybecontinuous(Fig.3aandFig.4)ornoncontinuous(Fig.3b).
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Fig.3
Fig.4
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4.0ModeledwindingsandcurrentsAlthough there are typically 5 physical windings on a
machine, we will model a total of 7, with associatedcurrentsasdesignatedbelow.
3statorwindings:ia,ib,ic Fieldwindings:Thereare2:onephysical;onefictitious
oMain field winding: carrying current iF and
producingfluxalongtheDaxis.
oGwinding: carrying current iG andproducing fluxalong theQaxis. This is the fictitious one, but it
serves to improve the model accuracy of the
roundrotormachine (bymodelingtheQaxis flux
producedbytheeddycurrenteffects intherotor
duringthetransientperiod),anditspresencedoes
not affect the accuracy of the salient pole
machine. NOTE: Text does not include this one
(seepg124).TheGwindingisliketheFwindingof
themain field,except ithasno sourcevoltage in
itscircuit.
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Amortissuer winding: This one represents a physicalwinding forsalientpolemachinewithdampers,anda
fictitious winding if not. Because these produce flux
along both theDaxis and theQaxis,wemodel two
windings:
oDaxis:amortissuerwindingcarryingcurrentiDoQaxis:armortissuerwindingcarryingcurrentiQ
ItisofinteresttocomparetheFandGwindingstotheD
andQwindings.
BoththeFandDproducefluxalongtheDaxis,butDisfaster(lowertimeconstant)thanF.
BoththeGandQproducefluxalongtheQaxis,butQisfasterthanG.
5.0 FluxlinkagesandcurrentsSowehavesevenwindings(circuits) inoursynchronous
machine.Thefluxlinkageseenbyanywindingiwillbea
functionof
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CurrentsinallofthewindingsandMagneticcouplingbetweenwindingiandwindingj,ascharacterizedbyLij,wherej=1,,7.
Thatis
=
=7
1
1
j
jijiL (1)
Forexample,thefluxlinkingthemainfieldwindingis:
GFGQFQDFDFFFcFcbFbaFaF iLiLiLiLiLiLiL ++++++= (2)
Repeating for allwindings results in Equation (4.11) in
your text, with exception that your text does not
representtheGwindinglikewearedoinghere.
=
G
Q
D
F
c
b
a
GGGQGDGFGcGbGa
QGQQQDQFQcQbQa
DGDQDDDFDcDbDa
FGFQFDFFFcFbFa
cGcQcDcFccbcca
bGbQbDbFbcbbba
aGaQaDaFacabaa
G
Q
D
F
c
b
a
i
i
i
i
i
i
i
LLLLLLL
LLLLLLL
LLLLLLL
LLLLLLL
LLLLLLL
LLLLLLL
LLLLLLL
(3)
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Notetheblocksoftheabovematrixcorrespondto
Lowerrighthand44arerotorrotorterms.Upperlefthand33arestatorstatorterms;Upperrighthand34arestatorrotorterms; Lowerlefthand43arerotorstatorterms;Your text summarizes theexpressions foreachof these
groups of terms on pp. 8687. I will expand on this
summaryinthenextsection.
6.0 Inductanceblocks6.1aRotorrotorterms:selfinductances
Recallthatthegeneralexpressionforselfinductancesis
iR
2
i
i
iii
N
iL ==
(4)
whereRiisthereluctanceofthepathseenby i.
Atanygivenmoment,therotorwindingsseeaconstant
reluctancepath, independentof the positionangle .
Thus, rotorwinding selfinductances are constants, and
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we define the following nomenclature, consistentwith
eq.(4.13)inyourtext.
Daxisfieldwinding FFF LL = (5)Qaxisfieldwinding GGG LL = (6)Daxisamortissuerwinding: DDD LL = (7)Qaxisamortissuerwinding: QQQ LL = (8)Note your texts convention of using only a single
subscriptforconstantterms.
6.1bRotorrotorterms:mutualinductances
Recall
ijR
ji
j
iij
NN
iL ==
(9)
where Rij is the reluctance of the path seen by i in
linkingwithcoiliorthepathseenby jinlinkingwithcoil
j (eitherway it is the same path!). Again, by similar
reasoning as in section 6.1a, these mutual terms are
constants(i.e.,independentof).
Therefore,wehavethefollowing:
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F(field)D(amort): RDFFD MLL == (10a)G(field)Q(amort): YQGGQ MLL == (10b)Butwehavefourotherpairstoaddress:
F(field)G(field): 0== GFFG LL (11a) F(field)Q(amort): 0== QFFQ LL (11b)G(field)D(amort): 0== DGGD LL (11c)D(amort)Q(amort): 0== QDDQ LL (11d)But these pairs ofwindings are each in quadrature, so
the flux fromonewindingdoesnot link thecoilsof theother winding, as illustrated in Fig. 5. Therefore the
abovefourtermsarezero,asindicatedineqs(11a11d).
Fig.5
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6.2aStatorstatorterms:selfinductances
We can derive these rigorously (see Kundur pp. 6165)
buttheinsightgainedinthiseffortdoesnotseemworthour time. Rather, we are better served by gaining a
conceptualunderstandingoffourideas,asfollows:
1.Sinusoidaldependenceonofpermeanceon:Duetosaliency of the poles (and to field winding slots in a
smoothrotormachine),thepathreluctanceseenbythestatorwindingsdependson ,asillustratedinFig.6.
N
S
aRotation
N S
aRotation
aa'
aa'
Fig.6a: =0 Fig.6a: =90
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From Fig. 6a,we observe thatwhen =0, the path of
phasea fluxcontainsmore ironthanatanyotherangle
0180, and therefore the reluctance seen by the
phaseafluxinthispathisataminimum,andpermeance
isatamaximum.
FromFig.6b,weobserve thatwhen =90, thepathof
phasea flux containsmore air that at any other angle
0180, and therefore the reluctance seen by the
phaseafluxinthispathisatamaximum,andpermance
isataminimum.This suggestsa sinusoidalvariationof
permeancewith .
2. Maximum permeance: There will be a maximumpermeanceduetotheamountofpermeanceseenbythe
phaseafluxatanyangle.Thiswillincludetheironinthe
middlepartof the rotor (indicatedbyabox in Figs.6a
and 6b), the stator iron, and the air gap. Denote the
corresponding(maximum)permeancePs.
3. Double angle dependence: Because the effectsdescribed in 1 and 2 above depend on pemeance (or
reluctance), and not on rotor polarity, the maximum
permeanceoccurstwiceeachcycle,andnotonce.
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Taking(1),(2),and(3)together,wemaywritethat
2cosms PPP += (12)
4. Inductanceapproximation:BecauseL=N2/R=N2P,theselfinductanceoftheaphasewindingcanbewrittenas
2cosms LLLaa += (13)
Likewise,wewillobtain:
( )1202cos += ms LLLbb (14)
( )2402cos += ms LLLcc (15)
Equations(13),(14),(15)aredenoted(4.12)inyourtext.
6.2bStatorstatorterms:mutualinductances
Wewill identify3 importantconceptsforunderstanding
mutualtermsofstatorstatorinductances.
1.Sign:First,weneedtoremindourselvesofapreliminaryfact:
Foranycircuitsiandj,Lijispositiveifpositivecurrents
inthetwocircuitsproducefluxesinthesamedirection.
Withthisfact,wecanstateimportantconcept1:
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As a result of defined stator current directions, the
statormutualinductanceisalwaysnegative.
To see this,we can observe that the flux produced bypositive currents of a and b phases are in opposite
directions,asindicatedinFig.7.
ObservethefollowinginFig.7:
Thecomponentof flux fromwindinga that linkswithwindingb, ab,is180from b.
Thecomponentof flux fromwindingb that linkswithwindinga, ba,is180from a.
Theimplicationsoftheabove2observationsarethat
a
aa'
Fig.7
b
b'
b
ab
ba
Observethatphysical
locationofthebphasewill
causeitsvoltagetolagthe
aphasevoltageby120,for
counterclockwiserotation.
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Mutuallyinducedvoltagesarenegativerelativetoself
inducedvoltages.
Mutualinductanceisnegative.
2.Functionofposition:2a.MaximumPermeanceforMutualFlux:
Recall that conditionswhere the amountof iron in the
path is a maximum permeance (minimum reluctance)
condition.
Thisconditionforphaseaselffluxis =0.
Thisconditionforphasebselffluxis =60.
Thereforetheconditionformaximumpermeanceforthe
mutual flux between phases a and b is halfway
inbetweenthesetwoat =30.
2b.PeriodicityofPermeanceforMutualFlux:
Starting at the maximum permeance condition, a
rotationby90to =60givesminimumpermeance.
Starting at the maximum permeance condition, a
rotation by 180 to =150 givesminimum permeance
again.
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The implication of these observations are that
permeance, and therefore inductance, is a sinusoidal
functionof2(+30).
3.Constantterm:There is an amount of permeance that is constant,
independent of rotor position. Like before, this is
composedof the stator iron, theairgap,and the inner
part of the rotor. We will denote the correspondinginductanceasMS.
From above1,2, and3,weexpressmutual inductance
betweenthea andbphasesas
[ ])30(2cos ++= absab LML (16)
One last comment: The amplitude of the permeance
variationforthemutualfluxisthesameastheamplitude
of the permeance variation for the selfflux, therefore
Lab=Lm.Andsothethreemutualexpressionsweneedare
[ ])30(2cos ++=
msab
LML (17)
[ ])90(2cos += msbc LML (18)
[ ])150(2cos ++= msca LML (19)
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6.3 Statorrotorterms
These are all mutual inductances. There are four
windingsontherotor(F,G,D,andQ)andthreewindingson the stator (a, b, c phases). Therefore there are 12
mutualtermsinall.
Centralidea:Recallthatforstatorstatormutuals,
windingswerelocationallyfixed,and thepathofmutualfluxwasfixed,but therotormoveswithinthepathofmutualflux,andforthisreason,thepathpermeancevaried.
Now,inthiscase,forstatorrotorterms(allmutuals),
therotorwindinglocationsvaryandthestatorwindinglocationsarefixed,andso
thepathofmutualfluxvaries,andforthisreason thepathpermeancevaries.
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To illustrate, consider the permeance between the a
phasewindingandthemainfieldwinding(F).
When themain fieldwinding and the statorwindingarealigned,as inFig.8a,thepermeance ismaximum,
andthereforeinductanceismaximum.
When themain fieldwinding and the aphase statorwindingare90apart,asinFig.8b,thereisnolinkage
atall,andinductanceiszero.
When the rotor winding and the aphase statorwindingare180apart,as inFig.9,thepermeance is
againmaximum,butnowpolarityisreversed.
a
aa'
Fig.8a
F
aa'
Fig.8b
F
a
N
S
SN
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This discussion results in a conclusion that themutual
inductancebetweenaphasewindingandthemainfieldwindingshouldhavetheform:
cosFaF ML = (20a)
The Daxis damper (amortissuer) winding is positioned
concentricwith themain fieldwinding,bothproducing
fluxalongtheDaxis.Therefore,thereasoningaboutthe
mutualinductancebetweentheaphasewindingandthe
Daxis damperwindingwill be similar to the reasoning
about the mutual inductance between the aphase
windingandthemainfield(F)winding,leadingto
cosDaD ML = (21a)
a
aa'
Fig.9
F
S
N
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Nowconsiderthemutualsbetweentheaphasewinding
and thewindingson theqaxis, i.e., theGwindingand
theQdamper(amortissuer)winding.
The only difference in reasoning about these mutuals
and themutualsbetween the aphasewinding and the
windingsonthedaxis(theFwindingandtheDdamper
winding)isthatthewindingsontheqaxisare90behind
the windings on the daxis. Therefore, whereas the a
phase/daxis mutuals were cosine functions, these
mutualswillbesinefunctions,i.e.,
sinQaQ ML = (22a)
sinGaG ML = (23a)
Summarizingstatorrotor terms forall threephases,we
obtaintheequationsonthenextpage.
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cosFaF ML = (20a)
)120cos( = FbF ML (20b)
)240cos( = FcF ML (20c)
cosDaD ML = (21a)
)120cos( = DbD ML (21b)
)240cos( = DcD ML (21c)
sinQaQ ML = (22a)
)120sin( = QbQ ML (22b)
)240sin( = QbQ ML (22c)
sinGaG ML = (23a)
)120sin( = GbG ML (23b)
)240sin( = GbG ML (23c)
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7.0 SummarySummarizingallofourneededequations:
Rotorrotorselfterms:5,6,7,8
Rotorrotormutuals:10a,10b,11a,11b,11c,11d
Statorstatorselfterms:13,14,15
Statorstatormutuals: 17,18,19
Rotorstatormutuals:20a,20b,20c,21a,21b,21c,22a,
22b,22c,23a,23b,23c
Countingtheaboveequations,weseethatwehave28.
Butletslookbackatouroriginalfluxlinkagerelation(3):
=
G
Q
D
F
c
b
a
GGGQGDGFGcGbGa
QGQQQDQFQcQbQa
DGDQDDDFDcDbDa
FGFQFDFFFcFbFa
cGcQcDcFccbcca
bGbQbDbFbcbbba
aGaQaDaFacabaa
G
Q
D
F
c
b
a
ii
i
i
i
ii
LLLLLLLLLLLLLL
LLLLLLL
LLLLLLL
LLLLLLL
LLLLLLLLLLLLLL
(3)
Wehave49terms!Wherearetheother21equations?
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Note because Lij=Lji, the inductance matrix will be
symmetric.Ofthe49terms,7arediagonal.Theother42
termsareoffdiagonalandarerepeatedtwice.Soweare
missing the 21 equations corresponding to the off
diagonal elements for which we did not provide
equations.Butwedonotneedto,sincethosemissing
equationsfortheoffdiagonalelementsLijareexactlythe
sameastheequationsfortheoffdiagonalelementsLji.
We will look closely at this matrix in the next set of
notes.