Transcript INDUCTION MOTOR (squirrel cage)

SEE 3433
MESIN ELEKTRIK
INDUCTION MOTOR
steady-state model
Construction
Stator windings
120o of practical machines
are distributed
a
120o
c’
Coil sides span can be less than
b’
180o – short-pitch or fractionalpitch or chorded winding
Ifc rotor is wound, its winding the
same as stator
b
a’
120o
Stator – 3-phase winding
Rotor – squirrel cage / wound
Construction
Single N turn coil carrying current i
Spans 180o elec
Permeability of iron >> o
→ all MMF drop appear in airgap
a

a’
Ni / 2
-
-Ni / 2
-/2
/2


Construction
Distributed winding
– coils are distributed in several slots
Nc for each slot

MMF closer to sinusoidal
- less harmonic contents
(3Nci)/2
(Nci)/2
-
-/2
/2


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
i(t)
t
This is the excitation
current which is sinusoidal
with time
F()

• 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
i(t)
t
0
F()
t=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
i(t)
t
t1
F()
t = t1

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
i(t)
t
t2
F()
t = t2

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
i(t)
t
t3
F()
t = t3

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
i(t)
t
t4
F()
t = t4

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
i(t)
t
t5
F()
t = t5

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
i(t)
t
t6
F()
t = t6

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
i(t)
t
t7
F()
t = t7

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
i(t)
t
t8
F()
t = t8

2

Combination of 3 standing waves resulted in ROTATING MMF wave
Frequency of rotation is given by:
2
s  2f
p
p – number of poles
f – supply frequency
known as synchronous frequency
• Rotating flux induced:
Emf in stator winding (known as back emf)
Emf in rotor winding
Rotor flux rotating at synchronous frequency
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
s  r
s
s
Induced voltage
Flux density distribution in airgap: Bmaxcos 
Flux per pole:
p  
/ 2
 / 2
Bmax cos l r d
= 2 Bmaxl r
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
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
ea  
Erms 
d
 N p sin t
dt
Np
2
 4.44f Np
Maximum flux links phase a when t = 0. No flux links phase a when t = 90o
Induced voltage
In actual machine with distributed and short-pitch
windinds induced voltage is LESS than this by a
winding factor Kw
Erms 
N p
Erms 
 4.44f N pK w
2
Np
2
 4.44f Np
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
Per–phase equivalent circuit
Rs
+
Llr
Lls
+
Is
Vs
–
Rs –
Rr –
Lls –
Llr –
Lm –
s–
Im
Lm Eag
–
stator winding resistance
rotor winding resistance
stator leakage inductance
rotor leakage inductance
mutual inductance
slip
Ir
+
Er/s
–
Rr/s
We know Eg and Er related by
Er s

Eag a
Where a is the winding turn ratio = N1/N2
The rotor parameters referred to stator are:
Ir
2
2
Ir '  , R r '  a R r , L lr '  a L lr
a
 rotor voltage equation becomes
Eag = (Rr’ / s) Ir’ + j(2f)Llr’ Ir’
Per–phase equivalent circuit
Rs
Is
Llr’
Lls
+
+
Vs
–
Rs –
Rr’ –
Lls –
Llr’ –
Lm –
Ir ’ –
Ir ’
Lm
Im
Eag
–
stator winding resistance
rotor winding resistance referred to stator
stator leakage inductance
rotor leakage inductance referred to stator
mutual inductance
rotor current referred to stator
Rr’/s
Power and Torque
Power is transferred from stator to rotor via air–gap,
known as airgap power
Pag  3Ir'2
Rr '

s
3Ir'2R r ' 
Lost in rotor
winding
3Ir'2
Rr '
1  s
s
Converted to mechanical
power = (1–s)Pag= Pm
Power and Torque
Mechanical power, Pm = Tem r
But, ss = s - r

r = (1-s)s
 Pag = Tem s
Pag
3Ir' 2R r '
Tem 

s
ss
Therefore torque is given by:
Vs2
3R r '
Tem 
2
ss 
Rr ' 
2
Rs 
  X ls  X lr ' 
s 

Power and Torque
This torque expression is derived based on approximate equivalent circuit
A more accurate method is to use Thevenin equivalent circuit:
3R r '
VTh2
Tem 
2
ss 
Rr ' 
2
2


R


X

X
'
 Vs Th
3R r 'Th
lr
Tem  
s  2
ss 
Rr ' 
2
R

 s
  X ls  X lr ' 
s 

Power and Torque
Tem
sTm  
Pull out
Torque
(Tmax)
Tmax
Trated
0
1
0
R s  X ls  X lr 
2
2

Vs2
3 

ss  R  R 2  X  X 2
s
ls
lr
 s
sTm rated syn
s
Rr
r




Steady state performance
The steady state performance can be calculated from
equivalent circuit, e.g. using Matlab
Rs
Is
Llr’
Lls
+
+
Vs
–
Ir ’
Lm
Im
Eag
–
Rr’/s
Steady state performance
Rs
Is
Llr’
Lls
Ir ’
+
+
Lm
Vs
Im
–
Eag
–
e.g. 3–phase squirrel cage IM
V = 460 V
Rs= 0.25 
Lr = Ls = 0.5/(2*pi*50)
f = 50Hz
p=4
Rr=0.2 
Lm=30/(2*pi*50)
Rr’/s
Steady state performance
500
Torque
400
300
200
100
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
250
200
Is
150
100
50
0
250
200
Ir
150
100
50
0
Steady state performance
600
400
Torque
200
0
-200
-400
-600
-800
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Steady state performance
1
0.9
0.8
Efficiency
0.7
(1-s)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1