three phase induction machines feature of three-phase
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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
Energy Conversion Lab
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 −
=−
=
Energy Conversion Lab
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
Energy Conversion Lab
CIRCUIT MODEL OF 3φ INDUCTION MACHINE
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
Energy Conversion Lab
CIRCUIT MODEL OF 3φ INDUCTION MACHINE
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
−+
+
+−
−
−−=
Energy Conversion Lab
CIRCUIT MODEL OF 3φ INDUCTION MACHINE
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
Energy Conversion Lab
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
Energy Conversion Lab
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
Energy Conversion Lab
qd0 EQUIVALENT CIRCUIT IN ARBITRARY FRAME
ωλds
ωλqs
(ω-ωr)λdr
(ω-ωr)λqr
L
L
L
L
L
L
Energy Conversion Lab
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 == ,
Energy Conversion Lab
qd0 EQUIVALENT CIRCUIT IN ARBITRARY FRAME
Energy Conversion Lab
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 −−=−
Energy Conversion Lab
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
Energy Conversion Lab
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
Energy Conversion Lab
qd0 STATIONARY REFERENCE FRAME
Induction machine equations in stationary frame used in transient studies
Energy Conversion Lab
qd0 STATIONERY REFERENCE FRAME
Induction machine equations in stationary frame table 6.3, voltage equations
Energy Conversion Lab
qd0 STATIONERY REFERENCE FRAME
Induction machine equations in stationary frame table 6.3, flux and torque equations
Energy Conversion Lab
qd0 SYNCHRONOUS REFERENCE FRAME
Induction machine equations in synchronous frame used in steady state stability studies
Energy Conversion Lab
qd0 SYNCHRONOUS REFERENCE FRAME
Induction machine equations in synchronous frame table 6.4, voltage equations
Energy Conversion Lab
qd0 SYNCHRONOUS REFERENCE FRAME
Induction machine equations in synchronous frame table 6.4, flux and torque equations
Energy Conversion Lab
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
Energy Conversion Lab
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
Energy Conversion Lab
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
+=
−++=
−=
−−=
Energy Conversion Lab
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)
Energy Conversion Lab
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)
Energy Conversion Lab
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)
Energy Conversion Lab
SIMULATION OF INDUCTION MACHINE Steady-state curve of the induction machine: simulate from ωr=0 to ωr=ωe
Energy Conversion Lab
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
Energy Conversion Lab
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
Energy Conversion Lab
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
Energy Conversion Lab
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
Energy Conversion Lab
STEADY STATE MODEL Phasor equivalent circuit representations
developed mechanicalpower
developed powerthrough airgap
developed mechanicalpower
Energy Conversion Lab
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
++=
Energy Conversion Lab
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
Energy Conversion Lab
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’
Energy Conversion Lab
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
Energy Conversion Lab
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
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