Induction Motor
4.2 torque Characteristic Determination
On.the.basis.of.the.equivalent.circuit.previously.defined,.it.is.possible.to.obtain.the.mechanical.torque.
produced.by.the.machine.and.the.torque–speed.characteristic..The.electromagnetic.torque.active.in.the.
air.gap.TT.is.the.ratio.between.the.transmitted.power.and.the.angular.speed.of.the.rotating.magnetic.
field,.as.shown.in.the.following.equation: power.to.the.rotor..The.mechanical.torque.is.defined.by.the.ratio.between.the.electric.power.of.the.resis-tance.((1.−.s)/s)Rr.and.the.rotor.angular.speed.as.reported.in.the.relation
. T P s s R I
ωr.=.(1.−.s)ωs,.it.is.possible.to.obtain.the.following.relation:
. T P s s R I s s R I
parameters,.Rir.and.Xm.up.to.the.stator.parameters,.because.of.the.high.value.of.the.magnetizing.current..
Xlr
FIGURE.4.5. Induction.motor.equivalent.circuit.after.the.separation.between.the.rotor.resistance,.Rr,.and.the.
equivalent.resistance.to.the.mechanical.power.Rr((1.−.s)/s).
In.fact,.in.the.induction.motors.the.presence.of.the.air.gap.requires.a.magnetizing.current,.which.can.be.
the.40%–60%.of.the.rated.current,.depending.on.the.motor.size..Using.the.previous.equivalent.circuit,.it.
is.possible.to.define.in.an.analytical.way,.the.induction.motor.torque.characteristic..In.order.to.simplify.
the.circuit.under.analysis,.it.is.possible.to.determine.the.Thevenin-equivalent.circuit.at.the.rotor.con-nections..In.addition,.to.make.the.equation.writing.easier,.all.the.apex.will.not.be.reported.anymore,.
remembering.that.the.rotor.parameter.value.has.been.reported.to.the.stator..The.Thevenin.rotor.equiva-lent.voltage.can.be.written.as
.
V V
R jX Z z
eq s
s ls p p
=( + )+ 0 0
. (4.26)
Where.Zp0.is.the.parallel.between.the.resistance.equivalent.to.the.iron.losses,.Rir,.and.the.magnetizing.
reactance,.Xm..The.Thevenin-equivalent.impedance.results
. Z R jX Z
R jX Z R X
eq s ls p
s ls p eq eq
= + ×
+ + = +
( )
( )
0
0 . (4.27)
The.new.simplified.circuit.is.reported.in.Figure.4.7.
Xls s X΄dr
Xm R΄r Rs
Rir Vs
FIGURE.4.6. Induction.motor.equivalent.circuit.with.the.rotor.quantities.reported.to.the.stator.side.
Req Xeq
Veq
Xlr
–––R΄sr
FIGURE.4.7. Thevenin-equivalent.circuit.of.Figure.4.6.
The.amplitude.of.the.rotor.current.phasor.can.be.easily.computed.by
The. quadratic. relationship. between. the. torque. and. the. supply. voltage. is. immediately. evident..
Consequently,.this.means.a.high.sensitivity.of.the.torque.with.the.voltage.variation..It.is.now.possible.
to.determine.the.characteristic.between.the.torque.and.the.slip.from.the.graphical.point.of.view.using.
some.considerations.on.the.torque-slip.function.limit,.for.slip.equal.to.zero.and.slip.equal.to.infin-ity..With.the.slips.leaning.toward.zero,.the.approximation.Req.<<.Rr/s.and.R Sr2/ >>2 (Xeq+Xlr) .can.be.2
assumed;.the.torque.relation.for.small.slips.can.be.written.as
infinity,.the.inequality.Req.>>.Rr/s.can.be.assumed.
As.a.consequence,.the.torque.relation.can.be.written.as
Remembering.the.relation.between.the.slip.and.the.mechanical.rotor.speed.ωm.=.ωs (1.−.s),.it.is.pos- sible.to.get.immediately.the.speed.vs..mechanical.rotor.speed.as.reported.in.Figure.4.9..The.two.charac-teristics.are.mirrored,.in.fact.at.slip.equal.to.one.the.rotor.speed.is.zero,.while.with.slip.equal.to.zero,.the.
rotor.speed.is.equal.to.the.rotating.magnetic.field.
In.the.mechanical.characteristic,.the.starting.torque.(torque.at.speed.equal.to.zero).and.the.peak.
torque.are.very.evident.
The.stable.part.of.the.torque–speed.characteristic.is.delimited.by.the.peak.torque.and.the.speed.ωs..
On.the.basis.of.the.torque.characteristic,.it.is.possible.to.demonstrate.the.rotor.power.balance.as.shown.
in.Figure.4.10,.where.PT.is.the.rectangular.area.TLωs,.Pmech.is.the.rectangular.area.TLωmech.and.Pjr.is.
the.rectangular.area.PT.−.Pmech..As.a.consequence,.it.is.very.evident,.as.previously.discussed,.that.the.
Slip
Starting torque
Brake or generator Motor
–1 –0.5 0 0.5 1
Brake peak torque
Motor peak torque Peak torque slip
FIGURE.4.8. Induction.motor.torque.vs..slip.characteristic.
Peak torque
ωs ωr
0 Starting torque
T
FIGURE.4.9. Induction.motor.torque.vs..mechanical.speed.characteristic.
T
ωs
ωs ωmech
PT Pjr
Load torque TL
Pmech
FIGURE.4.10. Induction.motor.rotor.power.balance.
function.of.the.slip.is.as.a.power.splitter.of.the.transmitted.power,.PT,.between.the.rotor.joule.losses,.Pjr,.
and.the.converted.mechanical.power,.Pmech.
4.2.1 Starting torque and Current
Imposing.in.the.torque.relation.a.slip.equal.to.one,.the.starting.torque.value.can.be.obtained.as
.
T p V R
R R X X
start
s eq r
eq r eq lr
=3 2 + 2+ + 2
ω ( ) ( ) . (4.33)
In.the.same.way,.the.starting.current.can.be.determined.by
.
I V
R R X X
start eq
eq r eq lr
= ( + ) (2+ + )2 .
(4.34)
The.denominator.of.the.previous.relation.is.the.short.circuit.impedance,.often.defined.as.locked.rotor.
impedance..It.is.evident.that.the.starting.current.corresponds.to.the.locked.rotor.current,.also.called.
short.circuit.current..The.starting.current.assumes.a.very.high.value.with.respect.to.the.rated.one,.and.it.
represents.a.serious.problem.for.the.motor.itself.and.the.motor.supply.source..Different.techniques.can.
be.adopted.for.limiting.the.starting.current..In.particular,.the.following.techniques.are.the.most.used:
•. Connection.of.starting.reactances.between.the.supply.source.and.the.motor..These.reactances.
have.to.be.short-circuited.after.the.motor.is.started.
•. Start.to.delta.connection.modification.during.the.starting.transient..The.motor.is.started.with.
the.motor.windings.connected.in.star.and.after.a.defined.time.interval.(depending.on.the.motor.
size.and.the.motor.and.load.inertia),.the.windings.are.switched.to.delta.connection..Obviously,.
the.procedure.requires.a.delta.connection.motor.during.normal.work.conditions..During.the.
starting.condition,.the.star.connection.reduces.the.voltage.applied.to.each.phase.of.a.factor.equal.
to. 3..Consequently,.the.line.starting.current.is.reduced.by.a.factor.equal.to.3.such.as.the.torque.
capability.
•. Soft.started.devices.based.on.solid.state.power.electronic.components..Presently,.this.technique.
is.the.most.used.for.its.high.efficiency.and.its.capability.to.control.the.starting.current.during.the.
starting.transient.
Obviously,.all.previous.methods.involve.a.reduction.of.the.supply.voltage.and.correspond.to.a.quadratic.
reduction.of.the.available.torque..For.this.reason,.in.order.to.guarantee.a.correct.starting.of.the.motor,.
it.is.very.important.to.check.that.the.actual.starting.torque.of.the.motor.is.still.higher.than.the.load.
starting.torque.
4.2.2 Peak torque
Using.the.analytical.expression.for.torque,.it.is.possible.to.determine.the.peak.torque.relationship..Due.
to.the.linear.relation.between.torque.and.transmitted.power,.the.peak.torque.condition.corresponds.to.
the.peak.of.the.transmitted.power..In.sinusoidal.supply,.the.peak.of.the.active.power.transfer.is.obtained.
when.the.equivalent.impedance.value.of.the.supply.source.is.equal.to.the.load.resistance..As.a.conse-quence,.the.peak.transmitted.power.will.be.obtained.when.the.following.condition.is.verified:
. R X X R
eq eq lr sr
2 +( + )2 =
. (4.35)
The.slip.corresponding.to.the.peak.of.the.transmitted.power,.called.peak.torque.slip,.sTx,.is.defined.by
.
s R
R X X
Tx r
eq eq lr
= 2 +( + )2 .
(4.36)
Since.in.the.induction.motors.the.term.Xeq.+.Xlr.is.practically.equal.to.the.total.motor.leakage.reactance.
Xlt.and.the.total.leakage.reactance.is.greater.than.the.equivalent.resistance.(Xlt.>>.Req),.the.peak.torque.
slip.can.be.simplified.as.shown.in.the.following.relation:
. s R
Tx Xr lt
≅ .
Including.the.simplified.peak.torque.slip,.sx,.in.the.torque.equation,.it.is.possible.to.obtain.the.value.of.
the.motor.peak.torque.as.follows:
.
T p V X
X X
x s eq lt
lt eq lt
≅3ω 2 2+(R + )2 . (4.37)
In.addition,.since.(Xlt.>>.Req),.the.final.simplified.equation.of.the.peak.torque.can.be.written.as
. T p V
x X
s eq
≅ 3 1lt
2
ω 2 . (4.38)
It.is.very.evident.that.the.peak.torque.is.inversely.proportional.to.the.total.leakage.reactance..In.other.
words,.the.motor.leakage.reactance.is.a.key.parameter.during.the.design.of.the.induction.motor,.because.
it.sets.the.motor.capability.to.produce.high.peak.torque..Induction.motors.for.industrial.applications.
have.a.ratio.between.peak.torque.and.rated.torque.in.the.range.1.5–2.5..As.a.consequence,.the.induction.
motors.have.a.good.torque.overload.capability.