Multiphase AC Machines
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.
of.any.phase.of.either.stator.or.rotor.is.of.the.same.principal.form,.v = Ri + dψ/dt..Here,.v,.i,.and.ψ.stand.
for.instantaneous.values.of.the.terminal.phase.to.neutral.voltage,.phase.current,.and.phase.flux.linkage,.
FIGURE.3.1. Schematic.representation.of.an.n-phase.induction.machine,.showing.magnetic.axes.of.stator.phases.
((α.=.2π/n)).
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.
where.[Ls],.[Lr],.and.[Lsr].stand.for.inductance.matrices.of.the.stator.winding,.the.rotor.winding,.and.
mutual.stator-to-rotor.inductances,.respectively..Relationship.[Lrs].=.[Lsr]t.holds.true.and.it.has.been.
taken.into.account.in.(3.3)..Due.to.the.assumed.perfectly.cylindrical.structure.of.both.stator.and.rotor,.
Here,.for.both.stator.and.rotor,.winding.phase.self-inductances.are.governed.with.L11.=.L22.=.….= Lnn,.
while.for.mutual.inductances.within.the.stator.(rotor).winding.Lij.=.Lji.holds.true,.where.i ≠ j,.i,.j.= 1 … n..
For.example,.in.a.three-phase.winding.L12.=.L13.=.L21.=.L31.=.L23.=.L32.=.M.cos.2π/3,.since.cos.2π/3.=.cos 4π/3,.
so.that.there.is.a.single.value.of.all.the.mutual.inductances.within.a.winding..Also,.Lii.=.Ll.+.M,.where.Ll.is.
the.leakage.inductance..However,.taking.as.an.example.a.five-phase.winding,.one.has.two.different.values.
of.mutual.inductances.within.a.winding,.L12.=.L21.=.L15.=.L51.=.L23.=.L32.=.L34.=.L43.=.L45 =.L54.=.M.cos 2π/5.
and.L13.=.L31.=.L14.=.L41.=.L24.=.L42.=.L35.=.L53.=.L52.=.L25 .=.M.cos.2(2π/5)..In.general,.given.an.n-phase.wind-ing,.there.will.be,.due.to.symmetry,.(n−1)/2.different.mutual.inductance.values.within.the.winding.
Due.to.the.assumption.of.sinusoidal.mmf.distribution,.mutual.inductances.between.stator.and.rotor. TL.is.the.load.torque
ωm.is.the.mechanical.angular.speed.of.rotation
The.inductances.of.(3.6).are.functions.of.electrical.rotor.position.and,.hence,.according.to.(3.5),.electri-cal.rotor.speed.of.rotation..The.equation.of.mechanical.motion.(3.7).is,.therefore,.customarily.given.
in.terms.of.electrical.speed.of.rotation.ω,.which.is.related.to.the.mechanical.angular.speed.of.rotation.
through.the.number.of.magnetic.pole.pairs.P,.ω.=.Pωm..Hence,
. T T J 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.
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
. T P i d L
d i
e st sr
= [ ] [ ][ ]θ r . (3.10)
This.means.that,.in.machines.with.uniform.air-gap,.electromagnetic.torque.is.solely.created.due.to.the.
interaction.of.the.stator.and.rotor.windings.
Any.multiphase.induction.machine.is.completely.described,.in.terms.of.phase.variables.(or,.as.it.is.said,.
in.the.original.phase.domain).with.the.mathematical.model.given.with.(3.1).through.(3.8).(or.(3.10).instead.
of.(3.8).)..The.model.is.composed.of.a.total.of.2n.+.1.first-order.differential.equations.(3.1).and.(3.7),.where.
2n.differential.equations.are.voltage.equilibrium.equations,.while.the.(2n +.1)th.differential.equation.is.the.
mechanical.equilibrium.equation..In.addition,.there.are.2n.+.1.algebraic.equations.(3.3).and.(3.8)..The.first.
2n.algebraic.equations.provide.correlation.between.flux.linkages.and.currents.of.the.machine,.while.the.
(2n +.1)th.algebraic.equation.is.the.torque.equation..Finally,.the.model.is.completed.with.an.integral.equa-tion.(3.5),.which.relates.instantaneous.rotor.electrical.position.with.the.angular.speed.of.rotation.
Substitution.of.flux.linkages.(3.3).into.voltage.equilibrium.equations.(3.1).and.electromagnetic.torque.
(3.10).into.the.equation.of.mechanical.motion.(3.7).eliminates.algebraic.equations,.so.that.the.machine.model.
contains.2n.+.1.first-order.differential.equations.in.terms.of.winding.currents,.plus.the.integral.equation.(3.5)..
This.is.a.system.of.nonlinear.differential.equations,.with.time-varying.coefficients.due.to.variable.stator-to-rotor.mutual.inductances.of.(3.6)..While.solving.this.model.directly,.in.terms.of.phase.variables,.is.nowadays.
possible.with.the.help.of.computers,.this.was.not.the.case.100.years.ago..Hence,.a.range.of.mathematical.
transformations.of.the.basic.phase-variable.model.has.been.developed,.with.the.prime.purpose.of.simplify-ing.the.model.by.the.so-called.change.of.variables..Model.transformation.is.therefore.considered.next.
Before.proceeding.further,.one.important.remark.is.due..Since.stator.and.rotor.variables.and.parameters.in.
general.apply.to.two.different.voltage.levels,.rotor.winding.is.normally.referred.to.the.stator.winding.voltage.
level..This.is.in.principle.the.same.procedure.that.is.customarily.applied.in.conjunction.with.transformers,.
and.it.basically.brings.all.the.windings.of.the.machine.to.the.same.voltage.(and.current).base..In.all.machines.
where.the.squirrel-cage.rotor.winding.is.used.(induction.machines.and.synchronous.machines.with.damper.
winding),.the.actual.values.of.rotor.currents.and.rotor.parameters.cannot.anyway.be.measured.and,.hence,.
this.change.of.the.rotor.winding.voltage.level.has.no.consequence.on.the.subsequent.model.utilization.since.
rotor.voltages.of.(3.2).are.by.default.equal.to.zero..However,.if.there.is.excitation.at.the.rotor.winding.side,.as.the.
case.may.be.with.slip-ring.induction.machines.(and.as.the.case.is.with.the.field.winding.of.the.synchronous.
machines),.in.which.case.rotor.winding.voltages.are.not.zero,.it.is.important.to.have.in.mind.that.rotor.voltages.
and.currents.(as.well.as.parameters).will.in.what.follows.be.values.referred.to.the.stator.winding..No.distinction.
is.made.here.in.terms.of.notation.between.original.rotor.winding.variables.and.parameters,.and.correspond-ing.values.referred.to.the.stator.voltage.level..As.a.matter.of.fact,.it.has.already.been.implicitly.assumed.in.the.
development.of.the.model.(3.1).through.(3.10).that.rotor.winding.has.been.referred.to.the.stator.winding.