winding saxes

7/30/2019 Winding Saxes

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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.

winding saxes

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