Electric Machines I
Prof. Ramzy R. Obaid
With many thanks
and appreciation to
Professor Mohamed A. El-Sharkawi
Mechanical
Power
Electrical
Power
Generator
Motor
Electrical
Power
Mechanical
Power
3
DC Machine
Source: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/motdc.html
Source: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/motdc.html
Source: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/motdc.html
Facts about DC Machines
• Most portable systems depend on the dc motor.
• Speed control is easy through terminal voltage
control
• Speed can also be controlled by changing the
armature resistance or field flux, which way is
best?
• Direction of rotation can easily be reversed, how?
Induction Motor
Facts about IM
• About 65% of the electric energy in the
United States is consumed by electric
motors.
• In the industrial sector alone, about 75% is
consumed by motors and over 90% of them
are induction machines.
• Most wind turbines use induction machines
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Why IMs are popular
•
•
•
•
•
They are rugged
Reliable
Easy to maintain
Relatively inexpensive.
Their power density (output power to
weight) is higher than some other motors.
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Squirrel Cage IM
No access to rotor windings
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Rotating Field
• The speed of the airgap flux is one revolution per
one ac cycle.
1
• The time of one ac cycle

f
• The speed of the airgap flux (Synchronous
Speed)
ns  f rev / sec
ns  60 f
rev / min
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Number of Poles
ib
ia
ic
ic
ia
ib
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Number of Poles
• If each phase has one coil, it is a two-pole
machine
– The rotor moves 360 degrees for one complete
ac cycle.
• If each phase has two coils, it becomes 4pole machine
– The rotor moves 1800 mechanical for every one
complete ac cycle.
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Number of Poles
Synchronous Speed
60 f
f
ns 
 120
pp
p
Number of pole pairs
rev / min
Number of poles
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Faraday’s Rules
e Bl v
F  Bli
e: Voltage across a conductor
B: Flux density seen by the conductor
l: Length of the conductor
v: Relative velocity of the conductor
wrt field velocity
F: Force exerted on conductor
i: Current through conductor
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Faraday law of Motion
e Bl v
F  Bli
e
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Faraday law for Rotating Motion
e Bl v
F  Bli
Linear Motion
e  f (  , n )
T  f (  ,i )
Angular Motion
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Apply 3-phase
3-phase
to Stator
to Stator
Field rotates in
airgap
i
e
Voltage induced
in rotor circuit
e = B l n
e
i
z
Torque is
developed in
rotor wdgs
f=Bli
Motor rotates
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Apply 3-phase
to Stator
n = 0
What if
motor
speed is
n s?
Voltage induced
in rotor circuit
e = B ln = 0
e
i  0
z
Torque is
developed in
rotor wdgs
f=Bli = 0
Motor
slows down
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What is the motor speed?
 n  ns  n  f ( Tload )
Speed drop is a function of load torque
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Slip (s)
 ns  n  s  
s



ns
s
ns
s
n
n
  2
60
n is in rev/min (r/min)
 is in rad/sec
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Example
• A 2-pole, 60 Hz induction motor operates at
a slip of 0.02. Compute its rotor speed.
• Solution
f
60
ns 120  120
 3600 rpm
p
2
ns  n
S
ns
n  ns 1  S   3600 1  0.02   3528 rpm
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Rotation of Induction Motor (IM)
 n  ns  n
stator
rotor
n n
s
n ns  n
s

ns
ns
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Equivalent Circuit of Induction Motor:
Rotor at Standstill (ss)
e  B l n
e  f ( ns )
N2
X2
R2
Ir I2
E2
E2 is the induced voltage in the rotor circuit at standstill – zero speed
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Equivalent Circuit of Induction Motor:
Stator circuit
R1
X1
Im
I1
V
I
Rm
Xm
E1
N1
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Equivalent Circuit of Induction Motor
at
Stand
still
X
R
R
X
1
Rm
2
I
Ir I2
Im
I1
V
2
1
Xm
E2
N2

E1
N1
E1
N1
N2
E2
I2
N1
I2


I
N2
I1
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Induced Voltage in Rotor Circuit at any
speed
X
R
2
E2 : Induced voltage at standstill
Er : Induced voltage at any speed
e  B l n
e  f (  , n )
E 2 ~ ns
E r ~ ns  n
2
Ir I2
N2
E2
ns  n
Er

E2
ns
Er  s E 2
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Frequency of Rotor Current (fr) at any
speed
X
R

r
Items change
By frequency
2
Ir I2
N2
Er
Rotor frequency at standstill n=0
f ss ~ n  ns
f ss  f
At any speed n
f is frequency
of stator voltage
f r ~ n  ns  n
ns  n
fr
fr


f ss
f
ns
fr  s f
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Equivalent Circuit of Rotor at any
speed
X r  2  f r L2  2  s f L2  s ( 2  f L2 )  s X 2
Xr = s X2
R2
Ir I2
N2
Er= s E2
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Modified equivalent circuit of rotor
Xr = s X2
I2
Ir I2
N2
Er= s E2
s E2
Ir  I2 
R2  j s X 2
R2 / s
X2
R2
N2
E2
I2 
E2
R2
 j X2
s
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Combined Equivalent Circuit
R1
X1
X2
I
V
N1 : N2
Im
I1
Rm
R2 /s
I2
Xm
E1
E2
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R1
X1
X2
Im
I1
V
Rm
R'2
 N1 

 R2 
 N2 
’
R2 /s
I2
Xm
E1 E2
2
'
I2
’
 N2 

 I 2 
 N1 
’
’
 N1
'
X 2  X 2 
 N2



2
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R1
X1
X2
V
Rm
I2
Xm
E1 E2
X1
Rm
'
R'2
R
 R'2  2 ( 1  s )
s
s
’
R2
I2
Xm
E1 E2
’
’
’
X2
Im
I1
V
’
R2 /s
Im
I1
R1
’
’
’
R'2
( 1 s )
s
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R1
X1
Im
I1
V
X2
Rm
’
R2
I2
Xm
’
’
’
E1 E2
R'2
( 1 s )
s
Equivalent to transformer’s
secondary windings
Equivalent to transformer’s
primary windings
Equivalent to transformer’s
load
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R1
X1
X2
R2
Im
I1
V
’
I2
R1
’
R'2
( 1 s )
s
’
E1 E2
Xm
Rm
’
X1
X2
’
R2
’
I1
I2
Im
V
Rm
Xm
’
E1 E2
’
R'2
( 1 s )
s
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Xeq
Req
I1
Im
V
Rm
I2
’
R'2
( 1 s )
s
Xm
'
Req R1  R2
X eq  X 1 
'
X2
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Input Power (Pin)
Stator Losses:
Copper losses (Pcu 1)
Core losses (Piron)
Airgap Power (Pg)
Rotor Copper Losses (Pcu 2)
Developed Power (Pd)
Rotational Losses (Protational)
Output Power (Pout)
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R1
Pcu 1  3
Piron  3
R1
V
Rm
'
' 2 R2
Pg  3 ( I 2 )
 Td
s
2
X2
’
Im
I1
Pin  3 V I 1 cos 1
I 12
X1
s
R2
I2
E1 E2
Xm
’
’
R'2
( 1 s )
s
’
V is a phase voltage
V
Rm
 
Pcu 2  3 I
' 2
2
R2'  s Pg
 
Pd  3 I
Protational
' 2
2
R2'
(1  s )  Pg (1  s )  Td 
s
Pout  T 
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Torque Characteristics
Xeq
Req
I1
Im
I '2 
V
V
Rm
I2
’
R'2
( 1 s )
s
Xm
' 2
R2 

 R1 
 X2

eq
s 

'
R
Td 
( I '2 )2 2 ( 1  s ) 

s


Pd
3
3 V 2 R'2 ( 1  s )

s  R1 



2 
 X eq

s 

' 2
R2 
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Td 
s
0
n
n
Pd


3 V 2 R'2

s s  R1 



' 2
R2 
2 
 X eq

s 


ns  n  s  
s

ns
s
s
s max
1
Tst
T
max
Torque
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Maximum Torque
Td 
Set
Pd


 Td
0
s
smax 
R'2
2
R12  X eq
Tmax 
3 V 2 R'2

s s  R1 



2 
 X eq

s 

' 2
R2 
3V
2

2
2 
2 s R1  R1  X eq


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Linear Induction Motor
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3-phase supply
Stator
Rotor
Stator windings
Supply
Primary
Secondary
Rotor coil
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n
n
ns
ns
Stationary Stator
Stationary Rotor
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Pole Pitch (p)
ns
vs   s r  2
r
60
2 r 120 f 2 r


2f
60
p
p
vs  2 p f
a
p
ns
r
a'
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dc or ac line
Power stick
Direction of Motion
v
Converter
Primary
a
c'
b
c
a'
b'
vs
Secondary
p
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Direction
of
magnetic
field
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Torque Equation
Pd  Td  
Rotating IM
Linear IM
3 V 2 R2' 
' 2


R2 
s s  R1    X eq2 
s 


Pd 
3 V 2 R 2' v
2
'


R2 
  X eq2 
s v s  R1 
s 


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Developed Thrust of Linear IM
vs  v
s
vs
Pd  Fd v
Pd
Fd 

v
Pd 
3 V 2 R 2' v
2
'


R2 
  X eq2 
s v s  R1 
s 


2
'
2
2
3V R


R2' 
2



s v s  R1    X eq 
s 


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Speed –Thrust Characteristics
s
s=0
v
vs
smax
Fst
Fmax
Thrust
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Maximum Thrust
s max 
R2'
R12  X eq2
Fmax 
If
s  s max

3V
2 v s R1 
Fd
s
s max
Fmax
2
R X
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1
2
eq
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
Magnetically levitated Linear Motor
(Maglev)
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