INDUCTION MOTORsteady-state model

SEE 3433

MESIN ELEKTRIK

Construction

Stator – 3-phase windingRotor – squirrel cage / wound

a

b

b’

c’

c

a’

120o120o

120o

Stator windings of practical machines are distributed

Coil sides span can be less than 180o – short-pitch or fractional-pitch or chorded winding

If rotor is wound, its winding the same as stator

Construction

a

a’

Single N turn coil carrying current iSpans 180o elec

Permeability of iron >> o

→ all MMF drop appear in airgap

/2-/2-

Ni / 2

-Ni / 2

Construction Distributed winding – coils are distributed in several slots

Nc for each slot

/2-/2-

(3Nci)/2

(Nci)/2

MMF closer to sinusoidal - less harmonic contents

Construction

The harmonics in the mmf can be further reduced by increasing the number of slots: e.g. winding of a phase are placed in 12 slots:

Construction

In order to obtain a truly sinusoidal mmf in the airgap:

• the number of slots has to infinitely large

• conductors in slots are sinusoidally distributed

In practice, the number of slots are limited & it is a lot easier to place the same number of conductors in a slot

Phase a – sinusoidal distributed winding

Air–gap mmf

F()

2

• Sinusoidal winding for each phase produces space sinusoidal MMF and flux

• Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF

F()

t

i(t)This is the excitation current which is sinusoidal with time

• Sinusoidal winding for each phase produces space sinusoidal MMF and flux

• Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF

F()

t

i(t)

t = 0

0

• Sinusoidal winding for each phase produces space sinusoidal MMF and flux

• Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF

F()

t

i(t)

2t = t1

t1

• Sinusoidal winding for each phase produces space sinusoidal MMF and flux

• Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF

F()

t

i(t)

2t = t2

t2

• Sinusoidal winding for each phase produces space sinusoidal MMF and flux

• Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF

F()

t

i(t)

2t = t3

t3

• Sinusoidal winding for each phase produces space sinusoidal MMF and flux

• Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF

F()

t

i(t)

2t = t4

t4

• Sinusoidal winding for each phase produces space sinusoidal MMF and flux

• Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF

F()

t

i(t)

2t = t5

t5

• Sinusoidal winding for each phase produces space sinusoidal MMF and flux

• Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF

F()

t

i(t)

2t = t6

t6

• Sinusoidal winding for each phase produces space sinusoidal MMF and flux

• Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF

F()

t

i(t)

2t = t7

t7

• Sinusoidal winding for each phase produces space sinusoidal MMF and flux

• Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF

F()

t

i(t)

2t = t8

t8

Combination of 3 standing waves resulted in ROTATING MMF wave

f2p2

s p – number of polesf – supply frequency

Frequency of rotation is given by:

known as synchronous frequency

• Rotating flux induced:

Rotor current interact with flux to produce torque

s

rss

Emf in stator winding (known as back emf)Emf in rotor winding Rotor flux rotating at synchronous frequency

Rotor ALWAYS rotate at frequency less than synchronous, i.e. at slip speed:

sl = s – r

Ratio between slip speed and synchronous speed known as slip

Induced voltage

Maximum flux links phase a when t = 0. No flux links phase a when t = 90o

Flux density distribution in airgap: Bmaxcos

Flux per pole:

2/

2/ maxp drlcosB = 2 Bmaxl r

Sinusoidally distributed flux rotates at s and induced voltage in the phase coils

Induced voltage

Maximum flux links phase a when t = 0. No flux links phase a when t = 90o

a flux linkage of phase a

a = N p cos(t)

By Faraday’s law, induced voltage in a phase coil aa’ is

tsinNdtd

e pa

pp

rms Nf44.42

NE

Induced voltage

pp

rms Nf44.42

NE

In actual machine with distributed and short-pitch windinds induced voltage is LESS than this by a winding factor Kw

wpp

rms KNf44.42

NE

Stator phase voltage equation:

Vs = Rs Is + j(2f)LlsIs + Eag

Eag – airgap voltage or back emf (Erms derive previously)

Eag = k f ag

Rotor phase voltage equation: Er = Rr Ir + js(2f)Llr

Er – induced emf in rotor circuit

Er /s = (Rr / s) Ir + j(2f)Llr

Per–phase equivalent circuit

Rr/s

+

Vs

–

RsLls

Llr

+

Eag

–

Is

Ir

Im

Lm

Rs – stator winding resistanceRr – rotor winding resistanceLls – stator leakage inductanceLlr – rotor leakage inductanceLm – mutual inductances – slip

+

Er/s

–

We know Eg and Er related by

rotor voltage equation becomes

Eag = (Rr’ / s) Ir’ + j(2f)Llr’ Ir’

as

EE

ag

r Where a is the winding turn ratio = N1/N2

The rotor parameters referred to stator are:

lr2

lrr2

rr

r La'L,Ra'R,aI

'I

Per–phase equivalent circuit

Rr’/s+

Vs

–

RsLls Llr’

+

Eag

–

Is Ir’

Im

Lm

Rs – stator winding resistanceRr’ – rotor winding resistance referred to statorLls – stator leakage inductanceLlr’ – rotor leakage inductance referred to statorLm – mutual inductanceIr’ – rotor current referred to stator

Power and Torque

Power is transferred from stator to rotor via air–gap, known as airgap power

s1s

'RI3'RI3

s'R

I3P r2'rr

2'r

r2'rag

Lost in rotor winding

Converted to mechanical power = (1–s)Pag= Pm

Power and Torque

Mechanical power, Pm = Tem r

But, ss = s - r r = (1-s)s

Pag = Tem s

s

r2'r

s

agem s

'RI3PT

Therefore torque is given by:

2lrls

2r

s

2s

s

rem

'XXs

'RR

Vs

'R3T

Power and Torque

2lrls

2r

s

2s

s

rem

'XXs

'RR

Vs

'R3T

This torque expression is derived based on approximate equivalent circuit

A more accurate method is to use Thevenin equivalent circuit:

2lrTh

2r

Th

2Th

s

rem

'XXs

'RR

Vs

'R3T

Power and Torque

1 0

r

s

Trated

Pull out Torque(Tmax)

Tem

0 rated syn

2lrls2

s

rTm

XXR

Rs

2lrls

2ss

2s

smax

XXRR

Vs3

T

sTm

Steady state performance

The steady state performance can be calculated from equivalent circuit, e.g. using Matlab

Rr’/s+

Vs

–

RsLls Llr’

+

Eag

–

Is Ir’

Im

Lm

Steady state performance

Rr’/s+

Vs

–

RsLls Llr’

+

Eag

–

Is Ir’

Im

Lm

e.g. 3–phase squirrel cage IM

V = 460 V Rs= 0.25 Rr=0.2

Lr = Ls = 0.5/(2*pi*50) Lm=30/(2*pi*50)

f = 50Hz p = 4

Steady state performance

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

Torque

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

Is

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

Ir

Steady state performance

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-800

-600

-400

-200

0

200

400

600

Torque

Steady state performance

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Efficiency

(1-s)