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INDUCTION MOTOR steady-state model. SEE 3433 MESIN ELEKTRIK. a. 120 o. 120 o. c’. b’. c. b. a’. 120 o. Stator windings of practical machines are distributed . Construction . Coil sides span can be less than 180 o – short-pitch or fractional-pitch or chorded winding.

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induction motor steady state model
INDUCTION MOTORsteady-state model

SEE 3433

MESIN ELEKTRIK

slide2

a

120o

120o

c’

b’

c

b

a’

120o

Stator windings of practical machines are distributed

Construction

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

Stator – 3-phase winding

Rotor – squirrel cage / wound

slide4

Ni / 2

-

/2

-/2

-Ni / 2

Single N turn coil carrying current i

Spans 180o elec

Permeability of iron >> o

→ all MMF drop appear in airgap

a

Construction

a’

slide5

(3Nci)/2

(Nci)/2

-/2

-

/2

Distributed winding

– coils are distributed in several slots

Nc for each slot

Construction

MMF closer to sinusoidal

- less harmonic contents

slide6

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

slide7

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

Construction

  • 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

slide8

Phase a – sinusoidal distributed winding

Air–gap mmf

F()

2

slide9

i(t)

t

F()

  • Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF
  • Sinusoidal winding for each phase produces space sinusoidal MMF and flux

This is the excitation current which is sinusoidal with time

slide10

i(t)

t

F()

  • Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF
  • Sinusoidal winding for each phase produces space sinusoidal MMF and flux

0

t = 0

slide11

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

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

i(t)

t

t1

F()

t = t1

2

slide12

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

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

i(t)

t

t2

F()

t = t2

2

slide13

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

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

i(t)

t

t3

F()

t = t3

2

slide14

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

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

i(t)

t

t4

F()

t = t4

2

slide15

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

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

i(t)

t

t5

F()

t = t5

2

slide16

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

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

i(t)

t

t6

F()

t = t6

2

slide17

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

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

i(t)

t

t7

F()

t = t7

2

slide18

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

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

i(t)

t

t8

F()

t = t8

2

slide21

Frequency of rotation is given by:

p – number of poles

f – supply frequency

known as synchronous frequency

slide22

Emf in stator winding (known as back emf)

Emf in rotor winding

Rotor flux rotating at synchronous frequency

  • Rotating flux induced:

Rotor current interact with flux to produce torque

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

slide23

Flux per pole:

= 2 Bmaxl r

Induced voltage

Flux density distribution in airgap: Bmaxcos 

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

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

slide24

Induced voltage

a  flux linkage of phase a

a = N p cos(t)

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

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

slide25

Induced voltage

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

slide26

Stator phase voltage equation:

Vs = Rs Is + j(2f)LlsIs + Eag

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

Eag = k f ag

Rotor phase voltage equation:

Er = Rr Ir + js(2f)Llr

Er – induced emf in rotor circuit

Er /s = (Rr / s) Ir + j(2f)Llr

slide27

Per–phase equivalent circuit

Llr

Ir

Lls

Rs

+

Vs

+

Eag

+

Er/s

Is

Rr/s

Lm

Im

Rs – stator winding resistance

Rr – rotor winding resistance

Lls – stator leakage inductance

Llr – rotor leakage inductance

Lm – mutual inductance

s – slip

slide28

We know Eg and Er related by

Where a is the winding turn ratio = N1/N2

The rotor parameters referred to stator are:

  • rotor voltage equation becomes
  • Eag = (Rr’ / s) Ir’ + j(2f)Llr’ Ir’
slide29

Is

Lls

Llr’

Ir’

Rs

+

Eag

+

Vs

Rr’/s

Lm

Im

Per–phase equivalent circuit

Rs – stator winding resistance

Rr’ – rotor winding resistance referred to stator

Lls – stator leakage inductance

Llr’ – rotor leakage inductance referred to stator

Lm – mutual inductance

Ir’ – rotor current referred to stator

slide30

Power and Torque

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

Lost in rotor winding

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

slide31

Power and Torque

Mechanical power,Pm = Temr

But, ss = s - r  r = (1-s)s

 Pag = Tems

Therefore torque is given by:

slide32

Power and Torque

This torque expression is derived based on approximate equivalent circuit

A more accurate method is to use Thevenin equivalent circuit:

slide33

Power and Torque

Tem

Pull out Torque

(Tmax)

Trated

r

0 ratedsyn

sTm

s

1 0

slide34

Is

Lls

Llr’

Ir’

Rs

+

Eag

+

Vs

Rr’/s

Lm

Im

Steady state performance

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

slide35

Is

Lls

Llr’

Ir’

Rs

+

Eag

+

Vs

Rr’/s

Lm

Im

Steady state performance

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