induction motor steady-state model see 3433 mesin elektrik
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INDUCTION MOTORsteady-state model
SEE 3433
MESIN ELEKTRIK
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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
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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
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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
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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:
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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
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Phase a – sinusoidal distributed winding
Air–gap mmf
F()
2
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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Combination of 3 standing waves resulted in ROTATING MMF wave
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f2p2
s p – number of polesf – supply frequency
Frequency of rotation is given by:
known as synchronous frequency
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• 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
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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
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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
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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
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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
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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
–
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)