three phase induction machines feature of three-phase

Energy Conversion Lab

THREE PHASE INDUCTION MACHINES

Feature of Three-Phase Induction Machine Motoring – operate below synchronous speed Generating – operate above synchronous speed Much less expensive compared to equivalent size

of synchronous or dc machine Require very little maintenance Speed is not easy to control Large starting current, six to eight times their full

load values Operate at poor power factor when lightly

loaded


ROTATING MAGNETIC FIELD AND SLIP

Rotating mmf field in stator winding a

Synchronous speed radian per second

Slip speed between rotor and synchronous rotating stator field slip speed = ωsm - ωrm Per unit slip:

Relations between rotor speed and synchronous speed ωrm + sωsm = ωsm rotor itself rotate at ωrm, rotor mmf field rotate at sωsm relative to

rotor, therefore, rotor mmf rotate at ωsm

( ) )cos(423, tI

PNtF e

eam

ea ωθ

πθ −=

esm Pωω 2

=

e

re

sm

rmsmsωωω

ωωω −

=−

=


CIRCUIT MODEL OF 3φ INDUCTION MACHINE

Stator voltage equations

Rotor voltage equations

Flux linkage equations

Winding inductances (stator-stator, rotor-rotor)

dtdriv

dtdriv

dtdriv cs

scscsbs

sbsbsas

sasasλλλ

+=+=+= , ,

dtdriv

dtdriv

dtdriv cr

rcrcrbr

rbrbrar

rararλλλ

+=+=+= , ,

166616

×××

=

abcr

abcs

abcrr

abcrs

abcsr

abcss

abcr

abcs

ii

LLLL

λ

λ

+

+

+

=

)()()()(

)()()()(

)()()()(

)(

rrssrlsrmsmrmsm

rmsmrrssrlsrmsm

rmsmrmsmrrssrlsabc

rrss

LLLLLLLLLLLL

L

[ ]13

13

×

×

=cs

bs

as

abcs

λ

λ

λ

λ [ ]13

13

×

×

=cr

br

ar

abcr

λ

λ

λ

λ



Winding inductances (stator-rotor)

Machine model in arbitrary qd0 reference frame transformation equation

qd0 transformation matrix

[ ]

−

+

+

−

−

+

==

=

rrr

rrr

rrr

srtabc

rscsr

bsr

asr

abcsr

θθθ

θθθ

θθθ

LLLLL

L

cos 3

2cos 3

2cos

32cos cos

32cos

32cos

32cos cos

ππ

ππ

ππ

( )[ ]

=

c

b

a

qdd

q

fff

Tfff

θ0

0

( )[ ]

+

−

+

−

=

21

21

21

32sin

32in sin

32cos

32cos cos

32

0πθπθθ

πθπθθ

θ sTqd( )[ ]

+

+

−

−=−

1 3

2sin 3

2cos

1 3

2in 3

2cos

1 sin cos

10

πθπθ

πθπθ

θθ

θ sTqd



Stator winding voltage equations abc frame

transform to qd0 frame

abcs

abcs

abcs

abcs irpv += λ

000100

0100

0100

0

010

010

010

)]([)]([

)]([)]([)]([)]([

)]([)]([)]([

qds

qds

qdsqdqd

qdsqd

abcsqd

qdsqdqd

qds

qdsqd

abcs

qdsqd

qdsqd

irTpT

iTrTTpTv

iTrTpvT

+=

+=

+=

−

−−

−−−

λθθ

θθλθθ

θλθθ

][)]([ 010

qdsqdTp λθ −

][)]([][

0 3

2os 3

2sin

0 3

2os 3

2sin

0 os in-

010

0 qdsqd

qds pT

dtd

c

c

cs

λθλθ

πθπθ

πθπθ

θθ

−+

+

+−

−

−−=



Stator winding voltage equations transformation to qd0 frame

000100

0 )]([)]([ qds

qds

qdsqdqd

qds irTpTv += − λθθ

( )[ ] 000100

0

000100

0

][)]([

][

0 3

2os 3

2sin

0 3

2os 3

2sin

0 os in-

21

21

21

32sin

32in sin

32cos

32cos cos

32

)]([)]([

qds

qds

qdsqdqd

qds

qds

qds

qdsqdqd

qds

irpTT

c

c

cs

s

irTpTv

++

+

+−

−

−−

+

−

+

−

=

+=

−

−

λθθ

λω

πθπθ

πθπθ

θθπθπθθ

πθπθθ

λθθ


qd0 VOLTAGE EQUATIONS

Stator voltage equations

Rotor voltage equations

00000

0 0 00 0 10 1 0

qds

qds

qds

qds

qds irpv ++

−= λλω

( ) 00000

0 0 00 0 10 1 0

qdr

qdr

qdr

qdrr

qdr irpv ++

−−= λλωω


qd0 FLUX LINKAGE EQUATIONS

Stator flux linkage equations

Rotor flux linkage equations

00

0100

0100

0

0 0 0

0 23 0

0 0 23

0 0

0 23 0

0 0 23

)]([)]([)]([)]([

qdrsr

sr

qds

ls

ssls

ssls

qdrrqd

abcsrrqd

qdsqd

abcssqd

qds

iL

L

i

L

LL

LL

iTLTiTLT

+

+

+

=

−−+= −− θθθθθθλ

00

0100

0100

0

0 0

0 23 0

0 0 23

0 0 0

0 23 0

0 0 23

)]([)]([)]([)]([

qdr

lr

rrlr

rrlr

qdssr

sr

qdrrqd

abcrrrqd

qdsrqd

abcrsrqd

qdr

i

L

LL

LL

iL

L

iTLTiTLT

+

+

+

=

−−+−−= −− θθθθθθθθλ


qd0 EQUIVALENT CIRCUIT IN ARBITRARY FRAME

ωλds

ωλqs

(ω-ωr)λdr

(ω-ωr)λqr

L

L

L

L

L

L


qd0 STATIONARY REFERENCE FRAME

Stator and rotor flux linkage relationships flux and torque equations

rrr

Ssr

r

Sssm L

NNL

NNLL

23

23

23 === Lx bb ωλωψ == ,


qd0 EQUIVALENT CIRCUIT IN ARBITRARY FRAME


qd0 TORQUE EQUATION

Sum of instantaneous input power Pin=vasias+ vbsibs+ vcsics+v’ari’ar+ v’bri’br+ v’cri’cr Pin = (3/2)(vqsiqs+ vdsids+ 2v0si0s+v’qri’qr+ v’dri’dr+ 2v’0ri’0r)

Torque equation neglect the i2r copper loss neglect ip(λ) terms of rate of exchange of magnetic field

energy between windings electromagnetic torque is developed by sum of ωλi

divided by mechanical speed

torque equations can be expressed in stator or rotor variable forms

( ) ( )( )[ ]''''

223

drqrqrdrrdsqsqsdsr

em iiiiPT λλωωλλωω

−−+−=

[ ] ( )[ ]''''

223

223

qrdrdrqrdsqsqsdsem iiPiiPT λλλλ −=−=

( ) ( )''''drqrqrdrdsqsqsds iiii λλωλλω −−=−


qd0 TORQUE EQUATION

Torque equation resistive element is with copper loss reactance is with magnetic field energy speed voltage term is with mechanical work we can express torque with speed voltage terms definition of speed voltage

Eqs = ωλds, Eds = -ωλqs

E’qr = (ω-ωr)λdr, E’dr = - (ω-ωr) λ’qr

steady-state torque is the real power absorbed by these four speed voltage sources

( )( ) ( )( )[ ]*''* ''Re22

3drqrdrqrdsqsdsqs

rem jiijEEjiijEEPT −−+−−=

ω


Project 6-1

Induction machine dq0 transformationYou are given the dq0 transformation matrix [Tdq0(θ)] as follows,

derive the dq0 voltage equations in form of (6.25) derive the dq0 flux vs. current equations in form of (6.29) plot equivalent circuit of the induction machine in

arbitrary reference frame

( )[ ]

+

−−

+

−

=

21

21

21

32sin-

32in- sin

32cos

32cos cos

32

0πθπθθ

πθπθθ

θ sT ddq


qd0 STATIONARY REFERENCE FRAME

Induction machine equations in stationary frame used in transient studies


qd0 STATIONERY REFERENCE FRAME

Induction machine equations in stationary frame table 6.3, voltage equations


qd0 STATIONERY REFERENCE FRAME

Induction machine equations in stationary frame table 6.3, flux and torque equations


qd0 SYNCHRONOUS REFERENCE FRAME

Induction machine equations in synchronous frame used in steady state stability studies



Induction machine equations in synchronous frame table 6.4, voltage equations



Induction machine equations in synchronous frame table 6.4, flux and torque equations


SIMULATION OF INDUCTION MACHINE Three phase stator voltages

vas = vag-vsg, vbs = vbg-vsg, vcs = vcg-vsg 3vsg = (vas+vbs+vcs)-(vag+vbg+vcg)

qd0 stationary frame include the voltage source

Stator and rotor connections

( ) ( )

( ) ( )

ssgsgcsbsassgcsbsassgsg

sgcgbgagcsbsass

bgcgbscssds

cgbgagcsbsassqs

idtdLRiii

dtdLiiiRv

vvvvvvvv

vvvvv

vvvvvvv

0

0

)(3)()(

31

31

31

31

31

31

32

31

31

32

+=+++++=

−++=++=

−=−=

−−=−−=

[ ]

+

−

+

−

=

21

21

21

32in

32in sin

32cos

32cos cos

32)(0

πθπθθ

πθπθθ

θ ssTqd

0=θ


SIMULATION OF INDUCTION MACHINE Overall scheme of stationary reference frame

Induction Machine Simulation in Stationary Reference Frame

i0s

vqs Tem

wr/wb

vag

vcg

vbg

v0s

psiqsiqs

psiqr

ibs

ics

vds

psidsids

psidr

ias

qds2abc

120*piomega*t

Initializeand plot

m1

abc2qds

Zero_seq

y

To Workspace

? ? ?Tmech

Term1

Term

Sum

Scope

Rotor

Qaxis

Product1

Product

Mux

Mux

Vm*cos(u[1]+2*pi/3)

Fcn2

Vm*cos(u[1]-2*pi/3)

Fcn1

Vm*cos(u[1])

Fcn

Daxis

Clock


SIMULATION OF INDUCTION MACHINE Inside the abc2qds block

vsg

3

out_v0s

2

out_vds

1

out_vqs

Sum

Mux

Mux1s

Integrator

(u[1]+u[2]+u[3])/3

Fcn2

(u[3]-u[2])/sqrt(3)

Fcn1

(2/3)*(u[1] - (u[2]+u[3])/2)

Fcn

50*Zb*wb

1/Csg

4

ias+ibs+ics

3

in_vcg

2

in_vbg

1

in_vag

(6.106)

( )

( )

ssgsgsg

sgcgbgags

bgcgsds

cgbgagsqs

idtdLRv

vvvvv

vvv

vvvv

0

0

)(3

31

31

31

31

32

+=

−++=

−=

−−=


SIMULATION OF INDUCTION MACHINE Inside the qds2abc block

3

out_ics

2

out_ibs

1

out_ias

Mux

Mux

-(u[1]-sqrt(3)*u[2])/2 + u[3]

Fcn2

-(u[1]+sqrt(3)*u[2])/2 +u[3]

Fcn1

u[1] + u[3]

Fcn

3

in_i0s

2

in_ids

1

in_iqs

(6.109)


SIMULATION OF INDUCTION MACHINE Inside the q-axis block

(6.112), (6.113), (6.117), (6.115)

iqspsiqs

psiqm

iqr'psiqr'

4

out_psiqr'

3

out_iqr'

2

out_iqs

1

out_psiqs

1s

psiqs_

1s

psiqr'_

Mux

Mux4

Mux

Mux3

Mux

Mux2

Mux

Mux1

Mux

Mux

(u[1]-u[2])/xplr

Fcn5

(u[1]-u[2])/xls

Fcn4

xM*(u[1]/xls+u[2]/xplr)

Fcn3

wb*(u[2] +(rpr/xplr)*(u[3]-u[1]))

Fcn2

wb*(u[2]+(rs/xls)*(u[1]-u[3]))

Fcn

2

in_(wr/wb)*psidr'

1

in_vqs

(6.112)

(6.113)

(6.115)

(6.117)

(6.115)


SIMULATION OF INDUCTION MACHINE Inside the rotor block

Tdamp

2

out_wr/wb

1

out_Tem

Tfactor*(u[1]*u[2]-u[3]*u[4])

Tem_

Taccl

Mux

MuxDomega

Dampingcoefficient

1s

1/s

1/(2*H)

1/2H

5

in_Tmech

4

in_ids

3

in_psiqs

2

in_iqs

1

in_psids

(6.118)(6.120)


SIMULATION OF INDUCTION MACHINE Steady-state curve of the induction machine: simulate from ωr=0 to ωr=ωe


SIMULATION OF INDUCTION MACHINE No-load startup and step load response

0-50% 50-100% 100-50%

startup

Project. 6-2 Simulation of Induction Machine

Read carefully on project 1 in 6.11.1: operating characteristics

Use the simulation model parameters as page 224 to run the simulation as follow: (steady-state operation) plot the no-load steady-state

curve as Fig. 6.27 (generating) when machine is running at steady state,

apply Tmech=Tb (numerically positive-generating), plot iqs, ids, Tem, ωr , observe your results

(motoring) when machine is running at steady state, apply Tmech=-Tb (numerically negative-motoring), plot iqs, ids, Tem, ωr, observe your results

(braking) when motor has started up to its no load speed and transients are over, reverse the sequence of the supply voltage to the stator winding. Plot iqs, ids, vas, ias, Tem, ωe , ωr and express your observation, also plot ωr vs. Tem which is similar to Figure 6.29


DERIVATION OF STEADY STATE MODEL Steady-state induction machine equations

stator voltage equations (on stator frame)

stator current equations (on stator frame)

rotor voltage equations, current equations (voltage and current rotate at a slip of s) (on rotor frame)

( )

−=

−==

34cos ,

32cos ,cos πωπωω tVvtVvtVv emscsemsbsemsas

( )

−−=

−−=−= semscssemsbssemsas tIitIitIi φπωφπωφω

34cos ,

32cos ,cos

( )( )

( )

( )

−−−=

−−−=

−−=

δθπω

δθπω

δθω

03

4cos

03

2cos

0cos

remrcr

remrbr

remrar

tsVv

tsVv

tsVv ( )( )

( )

( )

−−−−=

−−−−=

−−−=

rremrcr

rremrbr

rremrar

tsIi

tsIi

tsIi

φδθπω

φδθπω

φδθω

03

4cos

03

2cos

0cos


DERIVATION OF STEADY STATE MODEL Transform the above equations into stationary

reference frame qd0 equations q-axis aligned with a-phase stator axis

rotor voltage equations, current equations

the rms time phasor

rms space vector is rms time phaser times ejwt

( ) ( ) ( ) )())0((s)())0(()(s i ,v

i ,vtjtsj

mrrdr

rqrr

tjtsjmr

tjrdr

rqrr

tjjms

sds

sqss

tjms

sds

sqss

rrrerrer

ese

eeIjiieeVejvv

eeIjiieVjvvθφδθωθδθωθ

ωφω

−−−−−

−

=−==−=

=−==−=

( ) tjjmr

sdr

sqrr

stjmr

sdr

sqrr ere eeIjiieVjvv ωφδδω )()( i ,v +−− =−==−=

( )r

s

jmrjmr

jmsjms

eIeV

eIeV

φδδ

φ

+−−

−

==

==

2I~ ,

2V~

2I~ ,

2V~

arar

as0

as

tjsdr

sqrtj

sdr

sqr

tjsds

sqstj

sds

sqs

ee

ee

ejii

jejvv

j

ejii

jejvv

j

ωω

ωω

arsdr

sqrar

sdr

sqr

assds

sqsas

sds

sqs

I~2

II ,V~2

VV

I~2

II ,V~2

VV

=−

=−=−

=−

=−

=−=−

=−

Rotor frame

Stationary frame

)0()1()( :statesteady at rr θωθ +−= tst e


DERIVATION OF STEADY STATE MODEL

Stationery qd voltages/currents expressed in rms qd0 voltages and currents

Phasor voltage equations (see pg. 185)

Another form of phasor equations

electromagnetic torque

( ) )())0(()())0(()( i ,v

i ,vtjtsj

mrrdr

rqrr

tjtsjmr

tjrdr

rqrr

tjjms

sds

sqss

tjms

sds

sqss

rrrerrer

ese

eeIjiieeVejvv

eeIjiieVjvvθφδθωθδθωθ

ωφω

−−−−−

−

=−==−=

=−==−=

( ) ( )( ) ( )'

aras'

ar'ar

'arasasas

I~I~I~''V~I~I~I~V~

+++=

+++=

melrer

melses

LjsLjsr

LjLjr

ωω

ωω

( ) ( )( )'

aras'

ar

'ar

'arasasas

I~I~I~''V~I~I~I~V~

++

+=

+++=

melrer

melses

LjLjs

rs

LjLjr

ωω

ωω

( )rmsm

rarr

sm

arsmrar

rm

emem

rIsrIsr

ssIPT

ωωωω

ω −=

=−

−==

'2''2''2'

~3~3)1(1~3

Pem


STEADY STATE MODEL Phasor equivalent circuit representations

developed mechanicalpower

developed powerthrough airgap

developed mechanicalpower


STEADY STATE MODEL Constant voltage supply equivalent circuit

thevenin’s equivalent circuit at stator side

torque developed by constant voltage supply

maximum torque developed with constant voltage supply

where max power transfer at

max torque quantity is not dependent on the rotor resistance rr or slip s

( )

( )mlss

lssmthth

asmlss

mth

xxjrjxrjxjxr

xxjrjx

+++

=+=

++=

)(Z

VV

th

( )( ) ( )2'2'

'2

//

23

lrthrth

rth

eem

xxsrrsrVPT

+++=

ω

( )2'2

2max

43

lrththth

th

eem

xxrr

VPT+++

=ω

Vth

rth jxth jxlr’rr’/s

Vas

rs jxls

jxlr’ rr’/s

jxmequivalentcircuit

2'2

max

'

)( lrththr xxr

sr

++=


STEADY STATE MODEL

Average torque vs. slip with constant voltage supply

( )2'2

2max

43

lrththth

th

eem

xxrr

VPT+++

=ω

2'2

max

'

)( lrththr xxr

srwhen ++=

Given the Vth, f, andr (x), Tem is obtainedwith slip s

( )( ) ( )2'2'

'2

//

23

lrthrth

rth

eem

xxsrrsrVPT

+++=

ω


STEADY STATE MODEL

Constant current supply equivalent circuit Thevenin’s equivalent circuit of stator input impedance

stator input current and complex power

constant current supply is operated at constant stator current with variable stator voltage varies with Zin (varies with slip)

average torque developed with constant current supply

( )mlrr

lrrmlss xxjsr

jxsrjxjxr++

+++= ''

''

in /)/(Z

*asas

inas IV3 ,

ZVI =+== ininin

as jQPS

( ) ( )2'2'

222'

'2'

/I where,3

mlrr

asmar

rmsm

rar

rm

emem

xxsrxIrIPT

++=

−==

ωωω

Vs

rs jxls

jxlr’ rr’/s

jxm

equivalentcircuit

Vth

rth

jxthjxls’

rs’


OPERATING AT CONSTANT VOLTAGE SUPPLY Operating characteristics with constant voltage supply 20-hp, 60-Hz, 220-V three phase induction machine high starting torque and rotor current

high current

high torque

Zin


OPERATING AT CONSTANT CURRENT SUPPLY Operating characteristics with constant current supply 20-hp, 60-Hz, 220-V three phase induction machine lower starting torque because lower rotor current and airgap voltage

lower current

lower torque

Zin

Project. 6-3 Steady State Analysis

A 20-hp, 60 Hz, 220V three-phase induction machine operated with power supply with the following parameters Rs=0.1062Ω, rr’=0.0746Ω, xls=0.2145Ω, xlr’=0.2145Ω,

xm=5.834Ω, Jrotor=2.8 kgm2

Show the equivalent circuit model of constant power supply and constant current supply with associated parameters given above

Plot the operating characteristics with constant voltage supply as of Fig. 6.11 (including Vth and Ith)

Plot the operating characteristics with constant current supply as of Fig. 6.12 (including Vth and Ith)

Report your observation

three phase induction machines feature of three-phase

Documents