(2.11)
Then,.the.resultant.mmf.distribution.of.the.winding.can.be.written.as
. A
(2.12)
Let.us.define,.on.the.analogy.of.(2.9bis),.the.following.equivalent.turn.number.for.the.hth.harmonic:
1 3 5 7 … . (2.13)
If. the. phase. winding. is. symmetrical,. in. the. mmf.
distribution. spectrum. there. are. odd. spatial.
harmonics.only.
As.an.example,.Figure.2.6.shows.the.mmf.spatial.distribution.for.a.symmetrical,.shortened-pitch,.
double-layer.phase.winding.with.q.=.3,.nr.=.1,.and.β.=.30°.
In.Figure.2.7,.a.comparison.of.the.harmonic.winding.coefficients.for.a.full-pitch.winding.and.a.one-
slot.shortened-pitch.winding.(both.windings.with.q.=.3.and.β.=.30°).is.shown.
Figure.2.7.highlights.that.the.pitch.shortening.affects,.in.a.modest.way,.the.fundamental.mmf.har-
monic.(h.=.1),.and.in.a.more.sensible.way,.the.amplitude.of.some.spatial.harmonics..In.particular,.the.
amplitudes.of.the.5th.and.7th.harmonics.are.considerably.reduced..As.a.consequence,.the.pitch.short-
ening.can.be.considered.as.a.simple.method.to.reduce.the.amplitude.of.some.spatial.harmonics.in.the.
mmf.distribution.
Full-pitch Shortened-pitch
Harmonic order, h
FIGURE.2.7.
Harmonic.winding.coefficient.comparison.between.a.full-pitch.winding.with.q.=.3.and.β.=.30°,.and.
a.shortened-pitch.winding.with.q.=.3,.β.=.30°,.and.nr.=.1.
0 1 2 3 4 5 76 8 9 10 11
FIGURE.2.6. Typical.mmf.waveform.for.shortened-pitch.winding.
In.the.analyses.reported.so.far,.windings.with.diametrical.or.quasi-diametrical.turns.have.been.con-
sidered..For.this.type.of.structures,.the.airgap.mmf.waveform.has.a.unique.sign.alternation.along.the.
airgap.circumference.(see.Figure.2.6)..In.other.words,.the.windings.create.two.magnetic.polarities.or.
poles.(north.and.south)..Usually,.
these.windings.are.called.two-pole.windings.or.windings.with.one.
pole.pair.(P.=.1).
In.rotating-field.electric.machines,.phase.windings.with.the.number.of.pole.pairs.greater.than.one.(P >.1).
are.often.adopted..In.these.cases,.the.airgap.mmf.distribution.has.more.sign.alternations.along.the.whole.
circumference,.and.more.magnetic.polarities.are.produced.in.the.airgap..The.easiest.method.to.produce.a.
pole.pair.number.greater.than.one.is.to.repeat.the.disposition.of.the.active.winding.lengths.of.an.elemen-
tary.two-pole.winding,.in.a.cyclic.way,.along.the.airgap.circumference,.as.shown.graphically.in.Figure.2.8.
Zf I/8
FIGURE.2.9.
MMF.waveform.and.field.lines.for.a.four-pole.(P.=.2).winding.
β/2 β/3
. A K Z I
e e
e , cos
sin( / ) sin( / )
, π . (2.18bis)
The.use.of.the.electric.angle.is.very.important.because.it.allows.to.study.a.2P-pole.winding.as.a.simple.
two-pole.winding..In.fact,.all.the.relations.involving.an.angular.airgap.coordinate.(α,.β,.etc.).written.
for.a.two-pole.winding.are.still.valid.for.a.2P-pole.winding.if.the.electric.angle.is.used.instead.of.the.
geometrical.angle.
Example 2.1
Let us consider the winding layouts reported in Figure E.2.1. The
winding polarity and the winding coefficient of these windings
are
Winding A P = 1 q = 4 Nr = 0 βe = 20° Ka = 0.925 Winding B P = 3 q
= 1 Nr = 0 βe = 20° Ka = 1.000 Winding C P = 2 q = 2 Nr = 1 βe =
30° Ka = 0.933 Winding D P = 1 q = 4 Nr = 2 βe = 15° Ka =
0.925
2.2.6 airgap MMF Waveform Produced by a Single Conductor
In.this.section,.the.mmf.distribution.produced.by.a.single.conductor.is.analyzed..This.particular.wind-
ing.structure.can.be.considered.as.a.theoretical.case.and.it.can.be.used.as.a.starting.point.to.develop.
360°
1 2 3 4 7 8 9 10 13 14 15 163 4 5 6 7 8 9 10 11 5 6 11 12 17
1812
5 11
360°
(c)
17 231 2 3 4 6 7 8 9 10 12 13 14 15 16 18 19 20 21 22 24
360°
(d)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
24
FIGURE.E.2.1
a.general.theory.of.a.nonconventional.winding.structure,.such.as.the.squirrel.cage.winding,.typically.
used.in.induction.machines.
saw . (2.20)
2 saw . (2.20bis)
0
A
When.the.airgap.mmf.distribution.produced.by.a.system.of.active.windings.is.known,.it.is.possible.to.
estimate.the.magnetic.field.waveform,.H(α),.or.the.magnetic.flux.density.waveform,.B(α).=.μ0.H(α)..This.
can.be.easily.done.if.the.following.simplifications.are.adopted:
From.this.point.of.view,.(2.23).has.to.be.considered.inadequate.for.a.point-to-point.description.of.the.
airgap.flux.density.distribution.
Supposing.that.just.one.of.the.airgap.surfaces.has.the.slots.and.the.other.one.is.smooth,.it.is.possible.
to.quantify.in.an.analytical.way.the.flux.weakening.near.to.the.slot.opening,.if.the.following.assump-
tions.are.made:
Furthermore,.let.us.define.the.parameter.ξa.=.ac/2lt..The.normal.component.of.the.airgap.flux.den-
sity,.Btn(x),.on.a.smooth.surface.can.be.evaluated.by.a.Schwarz–Christoffel.conformal.transformation..
The.result.is.expressed.in.(2.24)..In.this.equation,.the.intermediate.variable.w,.related.to.the.conformal.
transformation,.is.in.the.range.of.0–1.when.the.coordinate.x.changes.from.0.to.∞.
B A l
1 2
FIGURE.2.12. Slot-opening.effect.on.airgap.flux.density.
, arctan lnmax π ξ ξ ξ
. (2.25)
As.a.consequence,.the.magnetic.flux.in.a.slot.pitch.produced.by.the.magnetic.potential.difference,.A,.
can.be.written.as
. Φ Φd t c c t c t a a aB B l= − = − ⋅ − +( )( )
,max ,max arctan lnτ τ
π ξ ξ ξ2 2 12
=with B A lt t
,max µ0
ξ ( / ) ln
b(x)
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2x/ac
ξa = 0.5
ξa = 2.0
ξa = 5.0
ξa = 10
ξa= 1.0
Summarizing.the.considerations.taken.into.account.so.far,.the.following.conclusions.can.be.drawn:
. .
where.b(α).is.the.airgap.flux.density.weakening.function.due.to.the.slot.opening.stated.in.(2.24).and.
shown.in.Figure.2.13..The.effect.of.the.slots.on.the.actual.flux.density.waveform.is.shown.in.Figure.2.15.
1.30
1.25
1.20
1.15
1.10
1.05
1.00 0 1 2 3 4 5 6 7 8 9 10
a c /τ
MMF fundamental component
Example 2.2
Let us consider the double-layer stator winding of a three-phase,
two-pole rotating-field machine (i.e., induction motor) with 18
slots, as shown in Figure E.2.2. The winding pitch is diametrical
(full-pitch wind- ing) and there are five conductors in series per
slot per layer. Each phase winding structure uses three slots for
the outgoing active conductors and three diametrical slots for the
backward conductors, as shown in the figure.
Determine the maximum value of the mmf distribution and the
amplitude of the mmf fundamen- tal component when a phase current,
I, equal to 8 A (instantaneous value) is supplied in the phase
winding.
Number of slots per pole per phase q = 3 Number of conductors in
series per slot Zc = 10 MMF amplitude (maximum value) Amax = q · Zc
· I/2 = 3 × 10 × 8/2 = 120 A Slot angular pitch β = 360°/18 = 20°
Distribution coefficient (= winding coefficient) Kd = sen(3 ×
10°)/(3 · sen(10°)) = 0.960 Number of conductors in series per
phase Zf = 5 · 12 = 60 Amplitude of the mmf fundamental component
Afund = 0.960 · 60 · 8/3.14 = 146.6 A
Example 2.3
Determine the phase current value that produces the same amplitude
of the mmf fundamental compo- nent as calculated in the previous
example, when a pitch shortening of two slots is adopted. In this
case, evaluate the new maximum value of the mmf distribution
produced by the winding too.
Pitch shortening (in number of slots) nr = 2 Shortening coefficient
Kr = cos (2 · 10°) = 0.940 Phase current (to get Afund = 146.6 A)
I′ = 8.0/0.940 = 8.5 A MMF amplitude (maximum value) Amax = q · Zc
· I′/2 = 3 × 10 × 8.5/2 = 127.5 A
Example 2.4
In Figure E.2.4, a single-layer stator winding of a three-phase
rotating-field machine with 24 slots is shown. Using Zf = 96 active
conductors in series per phase, two winding structures, with
different pole numbers, have to be realized: a two-pole winding (P
= 1, layout (a) in the figure) and a four-pole winding (P = 2,
layout (b) in the figure), respectively.
For the two structures, determine the amplitude of the mmf
fundamental component if the phase current is I = 7 A.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Five conductors/
slot/layer
FIGURE.E.2.2
2-18 Power Electronics and Motor Drives
Number of pole pairs P = 1 P = 2 Slot angular pitch βe = 1 ·
360°/24 = 15° βe = 2 · 360°/24 = 30° Number of slots per pole per
phase q = 4 q = 2 Winding coefficient (nr = 0) Ka = 0.958 Ka =
0.966 MMF fundamental component amplitude Afund = 204.8 A Afund =
103.3 A
Example 2.5
For a stator winding, the following data are known: 18 slots, Zf =
96 conductors in series per phase, q = 3 slots/pole/phase, and nr =
2 slots. As shown in the Figure E.2.5 the airgap radius is Rt = 45
mm, the slot- opening width is ac = 2.5 mm, and the airgap
thickness is lt = 0.5 mm.
Determine the phase current value that produces an airgap
fundamental flux density amplitude equal to Bt,max = 0.857 T.
Slot angular pitch β = 360°/18 = 20° Slot pitch (linear) τc = 2π ·
45/18 = 15.7 mm Half slot-opening width/airgap thickness ratio ξa =
2.5/(2 · 0.5) = 2.5 Carter coefficient KC = 15.7/{15.7 − 2 · 0.5[2
· 2.5 · atn(2.5)−ln(1 + 2.52)]/π} = 1.087 Equivalent airgap
thickness lt′ = 1.087 · 0.5 = 0.543 mm Distribution coefficient Kd
= sin(3 · 20°/2)/(3 · sin(20°/2) = 0.960 Shortening coefficient Kr
= cos(2 · 20°/2) = 0.940 Winding coefficient Ka = 0.960 · 0.940 =
0.902 Equivalent turn number N′ = 0.902 · 96/3.14 = 27.6 Phase
currenta I = 0.857 · 0.543 · 10−3/(1.256 · 10−6 · 27.6) = 13.4
A
aBy the equation of the airgap fundamental flux density amplitude,
it is possible to evaluate the phase current, as follows:
. B
l N
t t
t t=
In.AC.electric.motors,.such.as.in.AC.generators,.the.winding.positioned.in.the.stator.is,.in.the.majority.
of.cases,.a.three-phase.winding..For.this.reason,.the.attention.is.focused.on.three-phase.winding.struc-
tures,.considering.them.as.a.special.case.of.more.generic.polyphase.windings.
τc ac = 2.5 mm
9 10 11 12 13 14 15 16 17 18
FIGURE.E.2.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 (a)
(b)
FIGURE.E.2.4
(2.29)
In. this. case,. the. presence. of. the. spatial. mmf. harmonics.
is. neglected. and. only. the. fundamental.
component.is.taken.into.account.
By.opportunely.choosing.the.origin.of.the.angular.coordinate.α,.for.the.kth.phase,.the.following.
relation.can.be.written:
K Z k k
= ′ −
t k ,
′ −
⋅ ⋅ −
=
phase1+phase0– phase2+phase1–
Obtaining.the.sum.and.after.simple.calculations,.this.relation.can.be.reformulated.as.follows:
K Z Pk k
= ′ ⋅ −
t , ( , ) sin( )3
•.
A.three-phase.winding,.with.P.pole.pairs,.produces.in.the.airgap.a.magnetic.field.with.the.same.
number.of.magnetic.polarity.of.the.winding.
The.squirrel.cage.can.be.considered.as.an.atypical.case.of.poly-
phase.windings,.and.it.is.frequently.used.as.a.rotor.winding.in.
induction. machines.. In. fact,. in. each. conductor. (or. bar).
of. the.
cage,.the.current.is.different.from.the.currents.in.the.other.bars,.
and. as. a. consequence,. each. bar. can. be. considered. as. a.
phase. winding.
From. this. point. of. view,. the. squirrel. cage. is. a.
polyphase.
winding.with.a.phase.number.m.equal.to.the.number.of.bars,.
NR,.and.each.phase.is.constituted.by.a.unique.conductor.(Zf.=.1).
(Figure.2.18).
In.addition,.the.cage.winding.does.not.have.its.own.magnetic.
pole.number,.as. is. the.case. in. the.
traditional.distributed.winding..The.current. system. in. the.cage.
is.
induced.by.the.airgap.rotating.field.produced.by.another.distributed.winding.with.P.pole.pairs..This.
induced.current.system,.flowing.in.the.cage,.automatically.generates.an.mmf.distribution.with.the.same.
pole.pair.number,.P.
. i t I t k N
k Nk R
As. stated. in. Section. 2.2.6,. the. mmf. fundamental. waveform.
due. to. each. bar. can. be. calculated. as. follows,.where.N′.
is.the.equivalent.turn.number.of.one.bar,.from.the.fundamental.mmf.distribution.
production.point.of.view:
. A t N k
N i t Nk
α π ω π= ′
π α ω=
If.the.pole.pair.number.of.the.inducing.rotating.field,.through.which.the.bar.current.system.origi-
nates,.is.equal.to.P,.then.the.bar.current.system.is.given.by
N k Nk
R R( ) cos ; , , , , ,= ⋅ − ⋅
(2.32bis)
In.this.case,.the.rotating.flux.density.produced.by.the.cage.is.given.by
R
π α ω=
The.expressions.of.the.fundamental.distribution.of.the.airgap.rotating.magnetic.field.for.three-phase.
winding.(2.31bis).and.for.the.polyphase.cage.winding.(2.34bis).are.quite.similar..For.convenience,.these.
equations.are.stated.here.again:
K Z Pt
= ′
Pt N R
π =
K Z Pt m
π =
( )S .conductors.in.series.per.phase,.and.a.symmetrical.set.of.
sinusoidal.currents.of.amplitude.I(S);
t
S
t
f S
= ′
′ − ⋅ ′ = ⋅
t
R
t
f R
S
R
Examples 2.6
For a two-pole, full-pitch, three-phase winding, the following data
are known: 12 slots and Zf = 132 conductors in series per phase.
The average airgap radius is Rt = 20 mm, the slot-opening width is
ac = 2.5 mm, and the airgap thickness is lt = 0.5 mm.
Determine the rms value of the symmetrical three-phase current set
that produces an airgap funda- mental flux density amplitude equal
to Bt,max = 1 T.
Slot pitch (linear) τc = 2π · 20/12 = 10.5 mm Half slot-opening
width/airgap thickness ratio ξa = 2.5/(2 · 0.5) = 2.5 Carter
coefficient KC = 10.5/{10.5 − 2 · 0.5[2 · 2.5 · atn(2.5) − ln(1 +
2.52)]/π} = 1.137 Equivalent airgap thickness lt′ = 1.137 · 0.5 =
0.568 mm Slot angular pitch β = 360°/12 = 30° Winding coefficient
Ka = sin(2 · 15°)/(2 · sin 15°) = 0.966 Equivalent turn number N′ =
0.966 · 132/3.14 = 40.6 rms phase currenta I
~ = 1.0 · 1.414 · 0.568 · 10−3/(3 · 1.256 · 10−6 · 40.6) = 5.2
A
aDefining I ~
as the rms value of the three-phase current system, from (2.31bis)
it is possible to write the following relation:
. I B
l N
t t= ′
′ ˆ 2
3 0µ
Example 2.7
A rotating-field machine consists of the following windings:
(a) Three-phase winding with P = 1, Zf = 234, q = 3, and nr = 0 (b)
Squirrel cage winding with 48 slots (bars)
If the bar current is equal to 150 Arms, calculate the phase
current rms value of the three-phase winding that should generate
the same airgap fundamental flux density waveform produced by the
cage winding.
Calculation of the winding coefficient for the three-phase winding
(a)
Slot angular pitch β(a) = 360°/(6 · 3) = 20° Winding coefficient K
a
a( ) ) ) .= ⋅ ° ⋅ ° =sen(3 1 /(3 sen1 960 0 0 0 Equivalent current
I
~ (a) = KI · I ~
2-24 Power Electronics and Motor Drives
2.3.4 Vectorial representation of airgap Distributions
The.fundamental.flux.density.waveform.produced.by.the.polyphase.winding,.such.as.any.sinusoidal.
distribution.along.the.airgap,.can.be.symbolically.represented.by.means.of.a.vector.
. A ˆ= ′m N
2 π
P
= ⋅ ⋅ = ⋅∫ B Bt tsin ;α α π
0
2
ˆ
2.3.6 Harmonic rotating Fields
m N K Z
= ′ ⋅ − ⋅
′ =
m k mk( ) cos ; , , , ,= ⋅ −
(2.42)
The.resultant.mmf.distribution.due.to.the.excited.m-phase.winding.system.can.be.calculated.as
m t k
⋅ − ⋅
m h t hm
k
m
h
⋅0
It.is.possible.to.conclude.that.the.flux.density.spatial.harmonics,.produced.by.the.polyphase.winding,.
can.be.grouped.in.two.sets.in.accordance.with.the.following.conditions:
Case.1. h = nm.+.1.(n.integer.≥ 0)
. B t B h th h( , ) sin( )α α ω= ⋅ − ⋅ˆ
. (2.44)
Case.2. h = nm.−.1.(n.integer.>.0)
. B t B h th h( , ) sin( )α α ω= ⋅ + ⋅ˆ . (2.45)
In.both.the.cases,.the.results.obtained.is. B m N I K lh
h
In.symmetrical.three-phase.windings,.the.phases.produce.harmonics.with.odd.integer.values.for.the.
order.h..For.these.windings,.(2.44).and.(2.45).can.be.written.as.given.in.(2.44bis).and.(2.45bis),.respec-
tively,.as.follows: Case.1. h.=.6n.+.1.(n.integer.≥ 0)
. B t B h th h( , ) sin( )α α ω= ⋅ − ⋅ˆ . (2.44bis)
Case.2. h.=.6n.−.1.(n.integer.>.0)
. B t B h th h( , ) sin( )α α ω= ⋅ + ⋅ˆ . (2.45bis)
2-28 Power Electronics and Motor Drives
Since. the.harmonic.waves. rotate.along. the.airgap.at.different.
speeds,. the. resultant.waveform.will.
change.its.shape.during.the.rotation,.as.shown.in.Figure.2.22..Figure.2.22.highlights.that.the.waveform.
distortion.is.bigger.for.the.three-phase.winding.with.respect.to.the.20-bars.cage.winding..In.fact,.as.
previously.discussed,.the.cage.winding.can.be.considered.as.a.20-phase.winding.
Linear.AC.machines.depict.a.special.case.of.the.traditional.rotating.ones..The.distributed.windings.used.
in.the.linear.machines.can.be.analyzed.as.the.traditional.ones.so.far.described.
FIGURE.2.22.
Rotating-field.waveform.at.different.time.instants:.(a).three-phase.winding.with.3.slots.per.pole.per.
phase,.(b).squirrel.cage.winding.with.20.bars.
β
FIGURE. 2.25. 24-slot,. 28-pole,. three-phase. fractional-slot.
concentrated. winding. (q. =. 0.2857. slots/pole/phase,.
Ka.=.0.9659).
2-30 Power Electronics and Motor Drives
value.of.q,.a. large.number.of.pole.pairs,.P,. restricts.
the.number.of.slots.per.pole.and.per.phase,.q,.
and.this.contributes.to.worsen.the.form.of.the.induced.emf..Winding.arrangements.with.a.number.
of.slots.per.pole.and.per.phase.lower.than.unity.become.sometimes.mandatory.for.the.construction.
of.AC.machines.with.a.
large.number.of.pole.pairs..Fractional-slot.windings.with.q.
less.than.unity.
may.indeed.yield.a.larger.number.of.poles.at.a.fixed.number.of.slots.by.placing.less.than.one.slot.per.
phase.within.each.pole..In.other.terms,.in.each.pole,.conductors.pertaining.to.one.or.more.phases.
may.be.missing..In.some.cases,.adopting.a.layout.of.this.kind.makes.it.possible.to.realize.concentrated.
nonoverlapping.windings.that,.at.the.same.time,.yield.high.values.of.the.fundamental.winding.factor.
(Figure.2.25).
r S
As.reported.in.Section.2.2.1,.in.order.to.analyze.the.airgap.mmf.produced.by.a.distributed.winding,.it.
is.not.important.to.know.if.the.active.conductors.are.interconnected..Anyway,.the.ways.to.connect.the.
active.conductors.can.be.related.to.constructive.opportunities,.
the.spatial. localization.of.the.ends.of.
the.phase.windings,.and.the.necessity.to.avoid.the.presence.of.shaft.currents..For.these.reasons,.some.
aspects.related.to.the.winding.realization.are.briefly.described.[8].
In.general,.the.following.classifications.of.the.interconnection.solutions.are.possible:
. a.. Concentric. winding:. in. this. solution,. the. endwindings.
are. different. from. each. other. (Figure.2.26a).
. b.. Crossed. winding:. in. this. case,. the. endwindings. are.
all. equal. and. overlapped. (Figure. 2.26b).
. 2..
With.respect.to.the.connections.between.a.pole.and.the.adjacent.poles:
In.general,.considering.the.whole.winding.structure,.a.double-layer.winding.can.be.considered.as.a.
type.B.winding..As.shown.in.Figure.2.30,.the.shape.of.the.endwindings,.in.an.axial.direction,.is.quite.
different.for.single-.and.double-layer.windings.
For.concentric.windings,.the.endwindings.of.each.phase.have.to.be.positioned.on.different.planes..
With.reference.to.the.three-phase.case,.the.following.situations.are.possible:
. 3.. The. crooked. coil. suggests. the. so-called. American.
winding. type,. where. all. the. coils. have.
the.same.crooked.shape..The.American.winding.structure.can.be.realizable.for.any.crossed.
winding.type.
. 4.. Type. B. winding. (Figure. 2.33):. in. this. case,. the.
endwindings. are. positioned. on. three. different.
planes.(one.plane.for.each.phase).
(a) (b)
. 3.. W..Schuisky,.Berechnung Elektrischer
Maschinen,.1st.edn.,.Springer-Verlag.Publishers,.Weinheim,.
Germany,.1960.
. 4.. H.. Sequez,. The. windings. of. electrical. machines,. A.C.
Machines,. vol.. 3,. Springer. Verlag,. Vienna,.
Austria,.1950.(In.German).
. 5.. E.. Levi,. Polyphase Motors: A Direct Approach to Their
Design,. John. Wiley. &. Sons,. New. York,.
February.1984,.ISBN-13:.978-0471898665.
. 6.. P..L..Alger,.Induction Machines—Their Behavior and
Uses,.Gordon.and.Breach.Science.Publishers.
SA,.Basel,.Switzerland,.1970,.ISBN.2-88449-199-6.
. 7.. N.. Bianchi,. M.. Dai. Prè,. L.. Alberti,. and. E..
Fornasiero,. Theory. and. design. of. fractional-slot. PM.
machines,.IEEE IAS Tutorial Course Notes, Editorial CLEUP
Editore,.Seattle,.WA,.September.2007,.
ISBN.978-88-6129-122-5.
. 8.. G.. Crisci,. Costruzione, schemi e calcolo degli avvolgimenti
delle machine rotanti,. Editorial. STEM.
Mucchi,.Modena,.Italy,.1977.(in.Italian).
3-1
*.
What.is.customarily.known.as.a.two-phase.winding.is.in.essence.a.four-phase.structure,.since.the.spatial.shift.between.
magnetic.axes.of.the.phases,.as.well.as.the.phase.shift.between.phase.currents,.is.equal.to.π/2.
was. developed. in. [4].. Its. principal. advantage,. when.
compared. to. the. matrix. method,. is. a. more.
compact.form.of.the.resulting.model.(that.is.otherwise.the.same),.which.is.also.easier.to.relate.to.the.
physics.of.the.machinery.
Following. the. extensive. work,. conducted. in. relation. to.
multiphase. machine. modeling. in. the. beginning. of. the. last.
century,. numerous. textbooks. have. been. published,. which.
detail. the. model.
transformation.procedures.for.induction.and.synchronous.machines,.as.well.as.the.applications.of.
the.models.in.analysis.of.ac.machine.transients.[5–23]..The.principles.of.multiphase.machine.mod-
eling,.model. transformations,.and.resulting.models. for.both.
induction.and.synchronous.machine. (including. machines. with. an.
excitation. winding,. permanent. magnet. synchronous. machines,.
and. synchronous. reluctance. machines). are. presented. here. in.
a. compact. and. easy-to-follow. manner.. Although. most. of. the.
industrial. machines. are. with. three. phases,. the. general.
case. of. an. n-phase.
machine.is.considered.throughout,.with.subsequent.discussion.of.the.required.particularization.to.
different.phase.numbers.
Modeling.of.multiphase.ac.machines.is.customarily.subject.to.a.number.of.simplifying.assump-
tions..In.particular,.it.is.assumed.that.all.individual.phase.windings.are.identical.and.that.the.mul-
tiphase.winding.is.symmetrical..This.means.that.the.spatial.displacement.between.magnetic.axes.
of.any.two.consecutive.phases.is.exactly.equal.to.α.=.2π/n.electrical.degrees.*.Further,.the.winding.
is.distributed.across.the.circumference.of.the.stator.(rotor).and.is.designed.in.such.a.way.that.the.
magneto-motive.force.(mmf).and,.consequently,.f
lux.have.a.distribution.around.the.air-gap,.which. can.be.
regarded.as. sinusoidal..This.means. that. all. the.
spatial.harmonics.of. the.mmf,. except. for.
the.fundamental,.are.neglected..Next,.the.impact.of.slotting.of.stator.(rotor).is.neglected,.so.that.
the.air-gap.is.regarded.as.uniform.in.machines.with.circular.cross.section.of.both.stator.and.rotor.
(induction.machines.and.certain.types.of.synchronous.machines)..If.there.is.a.winding.on.the.rotor,.
which.is.of.a.squirrel-cage.type.(as.the.case.is.in.the.most.frequently.used.induction.machines.and.
in.certain.synchronous.machines),.bars.of.such.a.rotor.winding.are.distributed.in.such.a.manner.
that.the.mmf.of.this.winding.has.the.same.pole.pair.number.as.the.stator.winding.and.the.complete.
winding.can.be.regarded.as.equivalent.to.a.winding.with.the.same.number.of.phases.as.the.stator.
winding.
The. assumptions. listed. in. the. preceding. two. paragraphs.
enable. formulation. of. the. mathematical.
model.of.a.multiphase.machine.in.terms.of.phase.variables..Of.particular.importance.is.the.assump-
tion.on.sinusoidal.mmf.distribution,.which,.combined.with.the.assumed.linearity.of.the.ferromagnetic.
material,.leads.to.constant.inductance.coefficients.within.a.multiphase.(stator.or.rotor).winding.in.all.
machines.with.uniform.air-gap..In.machines.with.nonuniform.air-gap,.however,.inductance.coefficients.
within.a.multiphase.winding.are.governed.by.a.sum.of.a.constant.term.and.the.second.harmonic,.which.
imposes.certain.restrictions.in.the.process.of.the.model.transformation..Hence,.a.machine.with.uniform.
air-gap. is. selected. for. the. discussions. of. the. modeling.
procedure. and. subsequent. model. derivation..
The.machine.is.a.multiphase.induction.machine,.since.obtained.dynamic.models.can.easily.be.accom-
modated. to.various. types.of.
synchronous.machines..Motoring.convention. for.positive.power.flow.
is.
*. In. certain. multiphase. ac. machines. this. condition. is. not.
satisfied.. The. discussion. of. such. machines. is. covered. in.
Section.3.7.
Multiphase AC Machines 3-3
3.2 Mathematical Model of a Multiphase Induction Machine in
Original Phase-Variable Domain
Consider.an.n-phase.induction.machine..Let.the.phases.of.both.stator.and.rotor.be.denoted.with.indices.
1.to.n,.according.to.the.spatial.distribution.of.the.windings,.and.let.additional.indices.s.and.r.identify.
the.stator.and.the.rotor,.respectively..Schematic.representation.of.the.machine.is.shown.in.Figure.3.1,.
where.magnetic.axes.of.the.stator.winding.are.illustrated..The.machine’s.phase.windings.are.assumed.to.
be.connected.in.star,.with.a.single.isolated.neutral.point.
s s s s
r r r r
.
[ ] [ ]
[
v v v v v v v v v vs s s s ns t
r r r r nr t
= = 1 2 3 1 2 3… …
ii i i i i i i i i is s s s ns t
= =
s s s s ns t
r r r r nr t
[ ] [ ]ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ1 2 3 1 2 3… …
. (3.2)
1s
2s
3s
ns
and.[Rs].and.[Rr].are.diagonal.n × n.matrices,. [Rs].=.diag(Rs),.
[Rr].=.diag[Rr]..Since.rotor.winding. in. squirrel-cage. induction.
machines. and. in. synchronous. machines. (where. it. exists). is.
short-circuited,. rotor.voltages. in.(3.2).are.zero..The.exception.
is.a.slip-ring.(wound.rotor). induction.machine,.where.
rotor.windings.can.be.accessed.
from.the.stationary.outside.world.and.rotor.voltages.may.
thus.be.of. nonzero.value.
Connection.between. stator. (rotor).phase.flux. linkages.and.
stator/rotor. currents. can.be.given. in.a.
compact.matrix.form.as
. [ ] [ ][ ] [ ][ ]
[ ] [ ][ ] [ ] [ ]
s
. [ ]L
L L L L L L L L L L L Ls
s s s ns
s s s ns
s s s ns=
11 12 13 1
21 22 23 2
31 32 33 3
…
…
…
… … …… … …
. (3.4a)
. [ ]L
L L L L L L L L L L L Lr
r r r nr
r r r nr
r r r nr=
11 12 13 1
21 22 23 2
31 32 33 3
…
…
…
… … …… … …
L M
n n
θ α …
− − − − − −
…
…
… … …
ke L m
Equation.of.mechanical.motion.(3.7).is.always.of.the.same.form,.regardless.of.whether.original.vari-
ables.or.some.new.variables.are.used..Symbol.Te.stands.for.the.electromagnetic.torque,.developed.by.
the.machine..It.in.essence.links.the.electromagnetic.subsystem.with.the.mechanical.subsystem.and.is.
responsible. for. the. electromechanical. energy. conversion.. In.
general,. electromagnetic. torque. is. gov- erned.with
ie t= 1
s sr
rs r =
As.stator.and.rotor.winding.inductance.matrices,.given.with.(3.4),.do.not.contain.rotor-position-depen-
dent.coefficients,.Equation.3.8.reduces.for.smooth.air-gap.multiphase.machines.to
ie s t sr
Variables.of.an.n-phase.symmetrical.
induction.machine.can.be.viewed.as.belonging.to.an.n-dimen-
sional.space..Since.the.stator.winding.is.star.connected.and.the.neutral.point.is.isolated,.the.effective.
number.of.the.degrees.of.freedom.is.(n−1);.this.applies.to.the.rotor.winding.also..The.machine.model.in.
the.original.phase-variable.form.can.be.transformed.using.decoupling.(Clarke’s).transformation.matrix,.
which.replaces.the.original.sets.of.n.variables.with.new.sets.of.n.variables..This.transformation.decom-
poses.the.original.n-dimensional.vector.space.into.n/2.two-dimensional.subspaces.(planes).if.the.phase.
number.is.an.even.number..If.the.phase.number.is.an.odd.number,.the.original.space.is.decomposed.
into.(n−1)/2.planes.plus.one.single-dimensional.quantity..The.main.property.of.the.transformation.is.
. [ ] [ ][ , ,f C f nαβ = 1 2 … ] . (3.11)
where [f ]αβ. stands. for. voltage,. current,. or. flux. linkage.
column. matrix. of. either. stator. or. rotor. after.
transformation [
f1,2,…n].is.the.corresponding.column.matrix.in.terms.of.phase.variables
[C].is.the.decoupling.transformation.matrix
1
1
2
2
…
…cos cos cos cos cos cos αα α α α α α α α α α α
0 2 3 3 2 1 2 4 6 6
…
…
sin sin sin sin sin sin cos cos co
4 2 0 2 4 6 6 4 2 1 3 6
α α α α α α α α α α
… − − − ss cos cos cos
sin sin sin sin sin sin 9 9 6 3
0 3 6 9 9 6 3 α α α α
α α α α α α …
…
… … …
− − − …… … … … …
−
−
2 2 2
−
− −
− −
− − − −
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
assuming.without.any.loss.of.generality.that.the.phase.number.n.is.an.odd.number.and.that.rotor.n-phase.
winding.is.short-circuited,.one.gets.the.following.new.model.equations:
R i L L di dt
L d dt
α α
ψ θβ β
R i L L di dt
L d dt
ψ
ψ
v R i d
s x s ls x s
y s s y s y
1 1 1
y s
d dt
R ix n s s x n s x n s
s x n(( ) ) (( ) ) (( ) )
−− −
− − −
+
= +
y n s s y n s y n
L di
3 2 3 2
y n s
R
R i L L di dt
L d dt
α α
( ) ( sin
θ
dt R i L L di
dt L d
r m s= = + = + + + −0 θθ θ
ψ
β+
v R
r x r lr x r
y r
r y r lr y ri d
dt R i L di
dt1 1
1 1+ = +
x n r (( ) ) (( ) )
y n r r y n
= +
= =
− −
− −
(( ) ) (( ) )
(( ) ) ((
r
r r r lr
. (3.14)
. T PL i i i i i i i ie m r s r s r s r s= − − + cos ( ) sin ( )θ
θα β β α α α β β . (3.15)
Per-phase. equivalent. circuit. magnetizing. inductance. is.
introduced. in. (3.13). through. (3.15). as. Lm =
(n/2)M. and.symbols.Lls. and.Llr. stand. for. leakage.
inductances.of. the. stator.and.rotor.windings,.
respectively..These.are.in.essence.the.same.parameters.that.appear.in.the.well-known.equivalent.steady-
state.circuit.of.an. induction.machine.and.which.can.be.obtained.
from.standard.no-load.and. locked.
rotor.tests.on.the.machine..Subscript.+.in.designation.of.the.zero-sequence.component.of.(3.12).is.omit-
ted.since.there.is.a.single.such.component.when.the.phase.number.is.an.odd.number.
Multiphase AC Machines 3-9
3-10 Power Electronics and Motor Drives
which.will.be.used.in.the.rotational.transformation.for.stator.quantities..Considering.that.rotor.rotates,.
and.therefore.phase.1.of.rotor.has.an.instantaneous.position.θ.with.respect.to.stator.phase.1,.the.angle.
between.d-axis.of.the.common.reference.frame.and.rotor.phase.1.axis,.which.will.be.used.in.transfor-
mation.of.the.rotor.quantities,.is.determined.with
The.second.axis.of.the.common.reference.frame,.which.is.perpendicular.to.the.d-axis,.is.customarily.
labeled.as.q-axis..The.correlation.between.variables.obtained.upon.application.of.the.decoupling.trans-
formation.and.new.d
−.q.variables.is.defined.similarly.to.(3.11):
However,.rotational.transformation.matrix.[D].is.now.different.for.the.stator.and.rotor.variables:
0 0 1 0 0 0 0 0 1
…
…
…
…
…
0
1
1
0 0 1 0 0 0 0 0 1 0
0 0 0 0 0 1
…
…
…
…
… … … … …
…
−
v R i d
v R i d
v
1 1 1
1 1 1
2 2 2
s s s s
2 2 2
0 0 0
dr lr m dr
L L i
m ds
. (3.21a)
s
Since.rotor.winding.
is.regarded.as.short-circuited,.zero-sequence.and.x–y.component.equations.of.
the.
rotor.have.been.omitted.from.(3.20).and.(3.21)..If.there.is.a.need.to.consider.these.equations.(as.the.case.
may.be.if.the.rotor.winding.has.more.than.three.phases.and.is.supplied.from.a.power.electronic.converter.
in.a.slip-ring.machine),.one.only.needs.to.add.to.the.model.(3.20).and.(3.21).rotor.x–y.equations.of.(3.14),.
which.are.of.identical.form.as.in.(3.20b).and.(3.21b).and.only.index.s.needs.to.be.replaced.with.index.r.
Upon.application.of.the.rotational.transformation.torque.expression.(3.15).becomes
. T PL i i i ie m dr qs ds qr= − . (3.22)
Model. (3.20). through. (3.22). fully. describes. a. general.
n-phase. induction. machine,. of. any. odd. phase.
number..If.the.number.of.phases.is.even,.it.is.only.necessary.to.add.the.equations.for.the.second.zero-
sequence.component,.which.are.of.the.identical.form.as.in.(3.20).and.(3.21).for.the.first.zero-sequence.
component..However,.the.complete.model.needs.to.be.considered.only.if.the.supply.of.the.machine.con-
tains.components.that.give.rise.to.the.stator.voltage.x–y.components..If.the.machine.is.considered.to.be.
supplied.with.a.set.of.symmetrical.balanced.sinusoidal.n-phase.voltages.(of.equal.rms.value.and.phase.
shift.of.exactly.2π/n.between.any.two.consecutive.voltages),.then.stator.voltage.x–y.components.are.all.
zero,.regardless.of.the.phase.number..This.means.that.analysis.of.an.n-phase.machine.can.be.conducted.
under.these.conditions.by.using.only.stator.and.rotor.d.−
q.pairs.of.equations,.in.exactly.the.same.man-
ner.as.for.a.three-phase.machine.
r dr qs qr ds= − = −(ψ ψ (ψ ψ) ) . (3.23)
As.noted.already,.angular.speed.of.the.common.reference.frame.can.be.selected.freely.in.an.induction.
machine..However,.some.selections.are.more.favorable.than.the.others.
transformation.matrix.will.also.be.different..Taking.as.an.example.a.three-phase.machine,.the.combined.
decoupling/rotational.transformation.matrix.for.stator.variables.will.be
ds qs
s s
x
y
s
s
s
s
s
θ θ α ss( ) cos( ) cos( ) cos( ) cos( ) sin si
θ α θ α θ α θ α θ α θ
s s s s s
s
− − − −
2 3 3 2… + + + nn( ) sin( ) sin( ) sin( ) sin( ) sin(θ α θ α θ α θ
α θ αs s s s s− − − − −2 3 3 2… − + − + − θθ α
α α α α α α α α α
…
… − nn sin sin cos cos cos cos cos cos sin sin
6 4 2 1 3 6 9 9 6 3 0 3 6
− − …
α
3 2 2
2 2 2
α
α α αsin sin sin sn n n … iin sin sin3 2
2 2 2
+ −
+ −
1
2
π
f f jf n
( )
++ + + +( ) = + = + + + +
−
−
f f jf n
n n
2 6
3 2
n f a f a f( )/ [( )/ ] aa fn
n ( ) /−( )1 22
It.is.again.assumed.that.the.phase.number.is.an.odd.number.and.neutral.point.is.isolated,.so.that.zero.
sequence.cannot.be.excited..It.is.therefore.not.included.here,.but.it.in.general.remains.to.be.governed.
with.the.corresponding.penultimate.row.of.the.decoupling.transformation.matrix.(3.12).
n f a f a f a
d q s ds qs s s j
s s s ns
f a f a f
ns j
r r
. (3.29)
. v R i
j
L L i L i
L L i L i
= + +
= + +
( )
where *.stands.for.complex.conjugate
Im.denotes.the.imaginary.part.of.the.complex.number
x y s s x y s x y s
x y s ls x y s
− − −
− −
= +
=
( ) ( ) ( )
( ) ( )
operation.with.symmetrical.balanced.sinusoidal.supply..Regardless.of.the.selected.common.reference.
frame,.model.(3.30).and.(3.31).under.these.conditions.reduces.to.the.well-known.equivalent.circuit.of.
an.induction.machine,.described.with
. v R i j L i L i R i j L i L i i
R i j
s s s s s s m r s s s ls s m s r
r r
( )
(0 ss r r m s r r s lr r m s rL i L i R i j L i L i i− +( ) = + − +
+ω ω ω) ( )( ( )) . (3.34)
where.ωs.stands.for.angular.frequency.of.the.stator.supply..By.defining.slip.s.in.the.standard.manner.as.
(ωs.−.ω)/ωs,.introducing.reactances.as.products.of.stator.angular.frequency.and.inductances,.and.defin-
ing.magnetizing.current.space.vector.as.im.=.is.+.ir
,.these.equations.reduce.to.the.standard.form
R s
s s s ls s m s r
r r lr r m s r
= + + +
=
.
3.8 Modeling of Synchronous Machines
3.8.1 General Considerations
. L L L L L s
sd sq sd sq 11 2 2
2= +
ie s t sr
r s t s
rotor.(using.either.permanent.magnets.or.an.excitation.winding)..The.second.component.is,.however,.
purely.produced.due.to.the.variable.air-gap.and.is.called.reluctance.torque.component..In.synchronous.
reluctance.machines,.where.there.is.no.excitation.on.rotor,.this.torque.component.is.the.only.one.avail-
able.if.squirrel-cage.rotor.winding.does.not.exist.
Stator. voltage. equilibrium. equations. (3.20a). are. in.
principle. identical. as. for. an. induction. machine,.
except.that.now.ωa.=.ω..Rotor.short-circuited.winding.(damper.winding).voltage.equations.are.also.the.
same.as.in.(3.20a).with.the.last.term.set.to.zero,.since.ωa.=.ω..Hence,
. v R i d
ds s ds ds
f= + ψ . (3.41c)
.
qs ls mq qs mq qr
dr
L L i L i
L
qr lrq mq qr mq qs
f lf
L L i L i
L L
ψ
ψ mmd f md dr md dsi L i L i) + +
. (3.42)
Electromagnetic.torque.equation.(3.40).upon.transformation.reduces.in.the.rotor.reference.frame.to.
a.simple.form,
. T P i ie ds qs qs ds= −(ψ ψ ) . (3.43a)
which.is.exactly.the.same.as.for.an.induction.machine.(see.(3.23))..However,.if.the.stator.flux.d
− q.axis.
flux.linkage.components.are.eliminated.using.(3.42),.the.resulting.equation.differs.from.the.correspond-
ing.one.for. induction.machines.(3.22).due.to.the.existence.of.
the.excitation.winding.and.due.to.two.
different.values.of.the.magnetizing.inductances.along.two.axes:
. T P L i i i i L i i ie md ds f dr qs mq qs qr ds= + + − + ( ) ( )
. (3.43b)
The.form.of.(3.43b).can.be.re-arranged.so.that.the.fundamental.torque.component.is.separated.from.the.
reluctance.torque.component,
. T P L i i i L i i P L L i ie md f dr qs mq qr ds md mq ds qs= + −
+ −( ) ( ) . (3.43c)
which. is. convenient. for. subsequent. discussions. of. permanent.
magnet. and. synchronous. reluctance. machine.types.
3-26 Power Electronics and Motor Drives
Mechanical.equation.of.motion.of.(3.7).is.of.course.the.same.as.for.an.induction.machine..Relationship.
between.original.stator.phase.variables.and.transformed.stator.d.−
q.axis.quantities.is.in.the.general.case.
and.in.the.three-phase.case.governed.with.(3.25).and.(3.24),.respectively,.where.θ
θ ωs dt≡ = ∫ .
Since.in.permanent.magnet.synchronous.machines.field.winding.does.not.exist,.the.field.winding.equa-
tions.((3.41c).and.the.last.of.(3.42)).are.omitted.from.the.model..It.is.also.observed.that.the.permanent.
magnet.flux.ψm.now.replaces.term.Lmdif.in.the.flux.linkage.equations.of.the.d-axis..If.the.machine.has.
a.damper.winding,.it.can.again.be.represented.with.an.equivalent.dr–qr.winding..Hence,.voltage,.flux,.
and.torque.equations.of.a.permanent.magnet.machine.can.be.given.as
ds s ds ds
qs ls mq qs mq qr
L L i L i
L L i L i
= + + +
= + +
( )
qr lrq mq qr mq qs
L L i L i
L L i L i
= + + +
= + +
( )
( ) . (3.45b)
. T P i L i i L i i P L L i ie m qs md dr qs mq qr ds md mq ds qs=
+ − + −ψ ( ) ( ) . (3.46)
In.torque.equation.(3.46),.the.first.and.the.third.component.are.the.synchronous.torques.produced.by.
the.interaction.of.the.stator.and.the.rotor.and.due.to.uneven.magnetic.reluctance,.respectively,.while.
the.second.component. is.
the.asynchronous.torque.(the.same.conclusions.apply.to.(3.43c),.valid.for.a.
synchronous.machine.with.a.field.winding)..This.component.exists.only.when.the.speed.is.not.synchro-
nous,.since.at.synchronous.speed.there.is.no.electromagnetic.induction.in.the.short-circuited.damper.
windings.
ds s ds ds
qs ls m qs
The.electrical.part.of.the.model.(3.47).and.(3.48).is.usually.written.with.eliminated.stator.d.−
q.axis.flux. linkages,.as
dt L i
L i
s qs
m s ds
T P i
qs s qs m s ds
e m qs
remove.from.the.IPMSM.model.terms.related.to.the.permanent.magnet.flux.linkage..Hence,.from.(3.44).
through.(3.46),.one.now.gets
ds s ds ds
L L i L i
L L i L i
= + +
= + +
( )
L L i L i
L L i L i
= + +
= + +
( )
( ) . (3.53b)
. T P L i i L i i L L i ie md dr qs mq qr ds md mq ds qs= − + − ( )
( ) . (3.54)
where.the.first.component.is.the.asynchronous.torque,.while.the.second.component.is.the.synchronous.
torque.
L i
L i
q qs
d ds
e md
. (3.55)
The.form.of.the.d.−
q.axis.equivalent.circuits.is.the.same.as.in.Figure.3.10,.provided.that.the.electromo-
tive.force.term.ωψm.is.set.to.zero.
By.far.the.most.frequently.inadequate.assumption.is.the.one.related.to.the.linearity.of.the.magne-
tizing. characteristic,. which. has. made. the. magnetizing.
(mutual). inductance. (or. inductances. in. syn- chronous.
machines). constant.. This. applies. to. both. induction. and.
synchronous. machines.. There. are.
even.situations.where.this.assumption.essentially.means.that.a.certain.operating.condition.cannot.be.
simulated.at.all;.for.example,.self-excitation.of.a.stand-alone.squirrel-cage.induction.generator..It.is.for.
this.reason.that.huge.amount.of.work.has.been.devoted.during.the.last.30.years.or.so.to.the.ways.in.
which.main.flux.saturation.can.be.in
