MESIN ELECTRIC INDUCTION MOTOR STEADY STATE

MESIN ELECTRIC INDUCTION MOTOR STEADY STATE

1. INDUCTION MOTOR
steady-state model
SEE 3433
MESIN ELEKTRIK

2. Construction
Stator – 3-phase winding
Rotor – squirrel cage / wound
a
b
b’
c’
c
a’
120
o
120
o
120
o
Stator windings of practical machines
are distributed
Coil sides span can be less than
180
o
– short-pitch or fractional-
pitch or chorded winding
If rotor is wound, its winding the
same as stator

4. Construction
a
a’
Single N turn coil carrying current i
Spans 180
o
elec
Permeability of iron >> 
o

→ all MMF drop appear in airgap

/2-/2-
Ni / 2
-Ni / 2

5. Construction
Distributed winding
– coils are distributed in several slots
N
c
for each slot

/2-/2-
(3N
ci)/2
(N
c
i)/2
MMF closer to sinusoidal
- less harmonic contents

6. 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:

7. 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

8. Phase a – sinusoidal distributed winding


Air–gap mmf
F()

2

9. •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

10. •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

11. •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)
 2
t = t1
t1

12. •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)
 2
t = t2
t2

13. •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)
 2
t = t3
t3

14. •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)
 2
t = t4
t4

15. •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)
 2
t = t5
t5

16. •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)
 2
t = t6
t6

17. •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)
 2
t = t7
t7

18. •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)
 2
t = t8
t8

19. Combination of 3 standing waves resulted in ROTATING MMF wave

21. f2
p
2
s  p – number of poles
f – supply frequency
Frequency of rotation is given by:
known as synchronous frequency

22. •Rotating flux induced:
Rotor current interact with flux to produce torque
s
rs
s



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

23. Induced voltage
Maximum flux links phase a when t = 0. No flux links phase a when t = 90
o
Flux density distribution in airgap: B
max
cos 
Flux per pole:  



2/
2/
maxp drlcosB = 2 B
max
l r
Sinusoidally distributed flux rotates at 
s
and induced voltage in the phase coils

24. Induced voltage
Maximum flux links phase a when t = 0. No flux links phase a when t = 90
o

a
 flux linkage of phase a

a = N 
p cos(t)
By Faraday’s law, induced voltage in a phase coil aa’ is
tsinN
dt
d
e
pa 


p
p
rms
Nf44.4
2
N
E 



25. Induced voltage

p
p
rms
Nf44.4
2
N
E 


In actual machine with distributed and short-pitch
windinds induced voltage is LESS than this by a
winding factor K
w
wp
p
rms
KNf44.4
2
N
E 



26. Stator phase voltage equation:
V
s
= R
s
I
s
+ j(2f)L
ls
I
s
+ E
ag
E
ag
– airgap voltage or back emf (E
rms
derive previously)

E
ag
= k f 
ag

Rotor phase voltage equation:

E
r = R
r I
r + js(2f)Ll
r
E
r – induced emf in rotor circuit
E
r /s = (R
r / s) I
r + j(2f)Ll
r

27. Per–phase equivalent circuit
R
r
/s
+
V
s
–
R
s
L
ls
L
lr
+
E
ag
–
I
s
I
r
I
m
L
m
R
s
–stator winding resistance
R
r
–rotor winding resistance
L
ls
–stator leakage inductance
L
lr
–rotor leakage inductance
L
m
–mutual inductance
s –slip
+
E
r
/s
–

28. We know E
g
and E
r
related by

 rotor voltage equation becomes
E
ag
= (R
r
’ / s) I
r
’ + j(2f)L
lr
’ I
r
’

a
s
E
E
ag
r
 Where a is the winding turn ratio = N
1/N
2
The rotor parameters referred to stator are:

lr
2
lrr
2
r
r
r
La'L,Ra'R,
a
I
'I 

29. Per–phase equivalent circuit
R
r
’/s
+
V
s
–
R
s
L
ls L
lr’
+
E
ag
–
I
s I
r’
I
m
L
m
R
s
–stator winding resistance
R
r
’ –rotor winding resistance referred to stator
L
ls
–stator leakage inductance
L
lr
’ –rotor leakage inductance referred to stator
L
m
–mutual inductance
I
r
’ –rotor current referred to stator

30. Power and Torque
Power is transferred from stator to rotor via air–gap,
known as airgap power
s1
s
'R
I3'RI3
s
'R
I3P
r2'
rr
2'
r
r2'
rag

Lost in rotor
winding
Converted to mechanical
power = (1–s)P
ag
= P
m

31. Power and Torque
Mechanical power, P
m
= T
em

r
But, s
s
= 
s
- 
r
 
r
= (1-s)
s

 P
ag
= T
em

s
s
r
2'
r
s
ag
em
s
'RI3P
T




Therefore torque is given by:
 
2
lrls
2
r
s
2
s
s
r
em
'XX
s
'R
R
V
s
'R3
T










32. Power and Torque
 
2
lrls
2
r
s
2
s
s
r
em
'XX
s
'R
R
V
s
'R3
T









This torque expression is derived based on approximate equivalent circuit
A more accurate method is to use Thevenin equivalent circuit:
 
2
lrTh
2
r
Th
2
Th
s
r
em
'XX
s
'R
R
V
s
'R3
T










33. Power and Torque
1 0

r
s
T
rated
Pull out
Torque
(T
max)
T
em
0 
rated

syn
 
2
lrls
2
s
r
Tm
XXR
R
s


 











2
lrls
2
ss
2
s
s
max
XXRR
V
s
3
T
s
Tm

34. Steady state performance
The steady state performance can be calculated from
equivalent circuit, e.g. using Matlab
R
r’/s
+
V
s
–
R
s
L
ls L
lr
’
+
E
ag
–
I
s I
r
’
I
m
L
m

35. Steady state performance
R
r’/s
+
V
s
–
R
s
L
ls L
lr
’
+
E
ag
–
I
s I
r
’
I
m
L
m
e.g. 3–phase squirrel cage IM
V = 460 V R
s
= 0.25  R
r
=0.2 
L
r = L
s = 0.5/(2*pi*50) L
m=30/(2*pi*50)
f = 50Hz p = 4

36. Steady state performance
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
100
200
300
400
500
T
o
r
q
u
e
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
50
100
150
200
250
I
s
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
50
100
150
200
250
I
r

37. Steady state performance
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-800
-600
-400
-200
0
200
400
600
T
o
r
q
u
e

38. Steady state performance
0 0.10.20.30.40.50.60.70.80.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
E
ffi
c
ie
n
c
y
(1-s)