Multiphase AC Machines
3.6 Space Vector Modeling
Since,.upon.application.of.the.decoupling.transformation,.one.gets.pairs.of.axis.components.in.mutually.
It.is.again.assumed.that.the.phase.number.is.an.odd.number.and.neutral.point.is.isolated,.so.that.zero. current,.and.flux.linkage.space.vectors.are.obtained.in.the.common.reference.frame.by.rotating.corre-sponding.α−β.space.vector.by.an.angle,.which.is.for.stator.θs.and.for.rotor.θr..This.is.done.by.means.of.
the.vector.rotator,.exp.(−jθs).for.stator.and.exp.(−jθr).for.rotor.variables..Hence,.space.vectors.that.will.
describe.the.machine.in.an.arbitrary.common.reference.frame.are.governed.with
To.form.the.induction.machine’s.model.in.terms.of.space.vectors,.it.is.only.necessary.to.combine.d.− q.
axis.equations.of.the.real.model.(3.20).and.(3.21).as.real.and.imaginary.parts.of.the.corresponding.
Indices.d.− q,.used.in.(3.29).to.define.space.vectors,.have.been.omitted.in.(3.30).and.(3.31).for.simplicity..
In.(3.30).and.(3.31),.space.vectors.are.vs.=.vds.+.jvqs,.is.=.ids.+.jiqs,.ψ−s.=.ψds.+.jψqs.and.vr.=.vdr.+.jvqr ,.ir.=.idr.+.
jiqr ,.ψ−r.=.ψdr.+.jψqr..Torque.equation.(3.22).can.be.given,.using.space.vectors,.as
. T PLe= mIm * .
( )
i is r (3.32)space.vector.equations.that.describe.x–y.circuits.of.stator..Using.again.real.model.(3.20).and.(3.21).and.
and.there.are.(n −.3)/2.such.voltage.and.flux.linkage.equations.for.x–y.components.1.to.(n −.3)/2.
Model.(3.30).and.(3.31).is.the.dynamic.model.of.an.induction.machine..Consider.now.steady-state.
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
. also.be.of.different.time.dependence,.depending.on.the.selected.common.reference.frame..For.exam-ple,.in.the.stationary.reference.frame. vs
(
ωa=0)
= nVexp( )
j tωs ,.while.in.the.synchronous.reference.frame.in.which.d-axis.is.aligned.with.the.stator.voltage.space.vector.vs
(
ωa=ωs)
= nV.Stator.voltage.space.vector.under.symmetrical.sinusoidal.supply.conditions.is.shown.in.Figure.3.5.for.
a.three-phase.machine..It.travels.around.the.circle.of.radius.equal.to. 3V ..Instantaneous.projections.
of.the.space.vector.onto.α-.and.β-axis.represent.space.vector.real.and.imaginary.parts,.in.accordance.
with.the.definition.in.(3.28)..Upon.application.of.the.vector.rotator.of.(3.29).with.θs=
∫
ωsdt=ωst.the.Rs jXls jXlr
jXm
i s i r
i m
v s Rr/s
FIGURE.3.4. Equivalent.circuit.of.an.induction.machine.for.steady-state.operation.with.sinusoidal.supply.in.
terms.of.space.vectors.
stator.voltage.space.vector.becomes.aligned.with.the.d-axis.of.the.common.rotating.reference.frame.so.
that.the.q-component.is.zero..Since.the.d.− q.system.of.axes.rotates,.its.position.continuously.changes;.
thus,.the.illustration.in.Figure.3.5a.applies.to.one.specific.instant.in.time,.when.the.angle.is.45°..Since.the.
machine.is.in.steady.state,.the.stator.current.space.vector.is.in.essence.determined.with.the.ratio.of.the.
stator.voltage.space.vector.and.impedance..The.angle.that.appears.between.the.stator.voltage.and.stator.
current.space.vectors.is.the.power.factor.angle.ϕ.(Figure.3.5b)..Speed.of.rotation.of.the.stator.current.
space.vector.is.of.course.equal.to.the.speed.of.the.voltage.space.vector,.but.the.radius.of.the.circle.along.
which.the.stator.current.space.vector.travels.is.different.
If.the.machine.has.five.or.more.phases.and.the.stator.supply.is.either.not.balanced/symmetrical,.
or.it.contains.certain.time.harmonics.that.map.into.x−y.stator.voltage.components,.then.it.becomes.
necessary.to.use.additional.equivalent.circuits,.one.per.each.x−y.plane.(i.e.,.only.one.for.a.five-phase.
machine,.but.two.for.a.seven-phase.machine,.and.so.on)..In.principle,.the.form.of.equivalent.circuits.for.
x−y.components.is.governed.with.(3.33)..However,.since.x−y.voltages.may.contain.more.than.one.fre-quency.component,.a.separate.equivalent.circuit.is.needed.for.steady-state.representation.at.each.such.
frequency..Assuming,.for.the.sake.of.illustration,.that.stator.x−y.voltages.contain.a.single-frequency.
component,.the.equivalent.circuit.is.as.given.in.Figure.3.6.
Whether.or.not.the.stator.winding.x−y.circuits.are.excited.entirely.depends.on.the.properties.of.the.sta-tor.winding.supply..If.the.supply.is.a.power.electronic.converter,.which.produces.time.harmonics.in.the.
Rs jωx–y(s)Lls
ix–y(s)
vx–y(s)
FIGURE.3.6. Equivalent.circuit,.applicable.to.each.frequency.component.of.every.x−y.stator.voltage.space.vector.
in.machines.with.more.than.three.phases.
Im (β) Im (q)
ωa= ωs ωa= ωs
Re (d)
v s
Re (α)
√3V
(a)
θs= ωst = 45°
Im (β)
v s
i s
Re (α)
(b)
φ ωs
FIGURE.3.5. Illustration.of.the.stator.voltage.and.current.space.vectors.for.symmetrical.sinusoidal.supply.condi-tions..(a).Stator.voltage.space.vector.and.(b).stator.voltage.and.current.space.vectors.
output.phase.voltage,.then.some.of.these.harmonics.will.map.into.each.x–y.plane..As.an.example,.Table.
3.1.shows.harmonic.mapping,.characteristic.for.five-phase.and.seven-phase.stator.windings.[24]..As.can.
be.seen,.one.particular.time.harmonic.in.each.x–y.plane.for.each.phase.number.is.shown.in.bold.font..
These.are.the.time.harmonics.of.the.supply.that.can.be.used,.in.addition.to.the.fundamental,.to.produce.
an.average.torque..The.idea.is.to.increase.the.torque.density.available.from.the.machine,.and.this.applies.
equally.to.both.generating.operation.[25].and.motoring.operation.[26]..However,.for.this.to.be.possible,.
it.is.necessary.that.the.stator.winding.is.of.the.concentrated.type,.so.that,.in.addition.to.the.fundamental.
space.harmonic,.there.exist.the.corresponding.low-order.space.harmonics.of.the.mmf..In.simple.terms,.
this.means.that.the.spatial.distribution.of.the.mmf.is.not.regarded.as.sinusoidal.any.more;.it.is.quasi-rectangular.instead..Modeling.of.such.machines.is.beyond.the.scope.of.this.article..It.suffices.to.say.that,.
while.the.decoupling.transformation.matrix.remains.the.same,.rotational.transformation.changes.the.
form..Also,.the.starting.phase-variable.model.in.this.case.has.to.take.into.account.the.existence.of.the.
low-order.spatial.harmonics.through.appropriate.harmonic.inductance.terms..In.the.final.model,.d.− q.
equations.remain.the.same.but.electromagnetic.torque.equation.and.x–y.circuit.equations.change.