Induction Motors
EE 340
Where does the power generated go?
• The electric energy generated purchased by
consumers for different needs. This energy is
converted to different forms:
– Lighting (indoor/outdoor – CFL, incandescent,
LED, Halogen…)
– Heating (electric water heaters, clothes dryers,
electric stoves and ovens)
– power supply of electronic devices (computers,
TV, DVD, battery chargers, home automation,
etc…)
– Industrial (arc furnaces, welders, manufacturing
processes….)
– Conversion to mechanical power by motors
(pumps, fans, HVAC, refrigeration – compressors,
power tools, food processors, escalators,
elevators, ….)
Types of Electric Motors
3-Phase induction machine construction
• 3 stator windings (uniformly
distributed as in a synchronous
generator)
• Two types of rotor:
– Squirrel cage
– Wound rotor (with slip rings)
The rotating magnetic field
• The basic idea of an electric motor is to generate two magnetic
fields: rotor magnetic field and stator magnetic field. The rotor
will constantly be turning to align its magnetic field with the
stator field.
• The 3-phase set of currents, each of equal magnitude and with a
phase difference of 120o, flow in the stator windings and generate
a rotating field will constant magnitude.
The rotating magnetic field
• Consider a simple 3-phase stator containing three coils, each
120o apart. Such a winding will produce only one north and
one south magnetic pole; therefore, this motor would be
called a two pole motor.
• Assume that the currents in three coils are:
• The magnetic flux density in the stator at any arbitrary
moment is given by
The rotating magnetic field
• The net magnetic field has a constant magnitude and rotates
counterclockwise at the angular velocity ω.
• The stator rotating magnetic field can be represented as a north
pole and a south pole.
• For a two pole machine,
• For a p-pole machine,
f e ( Hz)  f m (rps) 
f e ( Hz) 
1
ns (rpm)
60
p
p
f m (rps) 
ns (rpm)
2
120
Principle of operation
• This rotating magnetic field cuts the rotor windings and
produces an induced voltage in the rotor windings
• Due to the fact that the rotor windings are short circuited, for
both squirrel cage and wound-rotor, and induced current flows
in the rotor windings
• The rotor current produces another magnetic field
• A torque is produced as a result of the interaction of those two
magnetic fields
 ind  kBR  Bs
Where ind is the induced torque and BR and BS are the magnetic
flux densities of the rotor and the stator respectively
Induction motor speed
• At what speed will the induction motor run?
– Can the induction motor run at the synchronous speed,
why?
– If rotor runs at the synchronous speed, then it will appear
stationary to the rotating magnetic field and the rotating
magnetic field will not cut the rotor. So, no induced
current will flow in the rotor and no rotor magnetic flux
will be produced so no torque is generated and the rotor
speed will fall below the synchronous speed.
– When the speed falls, the rotating magnetic field will cut
the rotor windings and a torque is produced.
Induction motor speed
• So, the induction motor will always run at a
speed lower than the synchronous speed
• The difference between the motor speed and
the synchronous speed is called the slip speed
nslip  nsync  nm
Where nslip= slip speed
nsync= speed of the magnetic field
nm = mechanical shaft speed of the motor
The Slip
s
nsync  nm
nsync
Where s is the slip
Notice that : if the rotor runs at synchronous speed
s=0
if the rotor is stationary
s=1
Slip may be expressed as a percentage by multiplying the
above by 100. Notice that the slip is a ratio and doesn’t
have units.
Induction Motors and Transformers
• Both induction motor and transformer works on
the principle of induced voltage
– Transformer: voltage applied to the primary windings
produce an induced voltage in the secondary windings
– Induction motor: voltage applied to the stator windings
produce an induced voltage in the rotor windings
– The difference is that, in the case of the induction
motor, the secondary windings can move
– Due to the rotation of the rotor, the induced voltage in
it does not have the same frequency of the stator
voltage.
Rotor Frequency
• The frequency of the voltage induced in the
rotor is given by
fr 
p.nslip
120
Where fr = the rotor frequency (Hz)
p = number of stator poles
nslip = slip speed (rpm)
fr 
p(nsyn  nm )
120

p.s.nsyn
120
 sfe
Rotor Frequency
• What would be the frequency of the rotor’s
induced voltage at any speed nm?
fr  s fe
• When the rotor is blocked (s=1) , the
frequency of the induced voltage is equal to
the supply frequency.
• On the other hand, if the rotor runs at
synchronous speed (s = 0), the frequency will
be zero.
Torque
• While the input to the induction motor is electrical
power, its output is mechanical power and for that
we should know some terms and quantities related
to mechanical power.
• Any mechanical load applied to the motor shaft will
introduce a torque on the motor shaft. This torque
is related to the motor output power and the rotor
speed
 load 
Pout
m
N .m
and
2 nm
m 
60
rad / s
Horse power
• Another unit used to measure mechanical power
is the horse power.
• It is used to refer to the mechanical output power
of the motor.
• Since we, as an electrical engineers, deal with
watts as a unit to measure electrical power, there
is a relation between horse power and watts:
hp  746 watts
Example
A 208 V, 10 hp, four pole, 60 Hz, Y-connected
induction motor has a full-load slip of 5 percent
1. What is the synchronous speed of this motor?
2. What is the rotor speed of this motor at rated
load?
3. What is the rotor frequency of this motor at rated
load?
4. What is the shaft torque of this motor at rated
load?
Solution
1.
nsync 
2.
nm  (1  s)ns
3.
f r  sfe  0.05  60  3Hz
120 f e 120(60)

 1800 rpm
P
4
 (1  0.05) 1800  1710 rpm
4.  load
Pout
Pout


m 2 nm
60
10 hp  746 watt / hp

 41.7 N .m
1710  2  (1/ 60)
Equivalent Circuit
• The induction motor is similar to the transformer
with the exception that its secondary windings
are free to rotate
It is easier if we can combine these two circuits in one
circuit but there are some difficulties:
Equivalent Circuit
• When the rotor is locked (or blocked), i.e. s =1,
the largest voltage and rotor frequency are
induced in the rotor, Why?
• On the other side, if the rotor rotates at
synchronous speed, i.e. s = 0, the induced
voltage and frequency in the rotor will be equal
to zero, Why?
ER  sER 0
Where ER0 is the largest value of the rotor’s induced
voltage obtained at s = 1(locked rotor)
Equivalent Circuit
• The same is true for the frequency, i.e.
fr  s fe
• It is known that
X   L  2 f L
• So, as the frequency of the induced voltage in
the rotor changes, the reactance of the rotor
circuit also changes
X r  r Lr  2 f r Lr
Where Xr0 is the rotor reactance
 2 sf e Lr
at the supply frequency .
 sX r 0
Equivalent Circuit
• Then, we can draw the rotor equivalent circuit
as follows
Where ER is the induced voltage in the rotor and RR is
the rotor resistance
Equivalent Circuit
• Now we can calculate the rotor current as
ER
IR 
( RR  jX R )
sER 0

( RR  jsX R 0 )
• Dividing both the numerator and denominator
by s so nothing changes we get
IR 
ER 0
RR
(  jX R 0 )
s
Where ER0 is the induced voltage and XR0 is the rotor
reactance at blocked rotor condition (s = 1)
Equivalent Circuit
• Now we can have the rotor equivalent circuit
Equivalent Circuit
• Now as we managed to solve the induced voltage and
different frequency problems, we can combine the
stator and rotor circuits in one equivalent circuit
Where
2
X 2  aeff
X R0
2
R2  aeff
RR
I2 
IR
aeff
E1  aeff ER 0
aeff
NS

NR
Power losses in Induction machines
• Copper losses
– Copper loss in the stator (PSCL) = I12R1
– Copper loss in the rotor (PRCL) = I22R2
• Core loss (Pcore)
• Mechanical power loss due to friction and
windage
• How this power flow in the motor?
Power flow in induction motor
Power relations
Pin  3 VL I L cos   3 Vph I ph cos 
PSCL  3 I12 R1
PAG  Pin  ( PSCL  Pcore )
PRCL  3I 22 R2
Pconv  PAG  PRCL
Pout  Pconv  ( Pf  w  Pstray )
 ind 
Pconv
m
Equivalent Circuit
• We can rearrange the equivalent circuit as
follows
Actual rotor
resistance
Resistance
equivalent to
mechanical load
Power relations
Pin  3 VL I L cos   3 Vph I ph cos 
PSCL  3 I12 R1
PAG  Pin  ( PSCL  Pcore )  Pconv  PRCL
R2
 3I
s
2
2
PRCL  3I 22 R2
Pconv  PAG  PRCL  3I 22 R2 (1  s)
s
Pconv  (1  s) PAG
Pout  Pconv  ( Pf  w  Pstray )
 ind 
PRCL

s
PRCL (1  s)

s
Pconv
m
(1  s) PAG

(1  s)s
Example
A 480-V, 60 Hz, 50-hp, three phase induction motor
is drawing 60 A at 0.85 PF lagging. The stator copper
losses are 2 kW, and the rotor copper losses are 700
W. The friction and windage losses are 600 W, the
core losses are 1800 W, and the stray losses are
negligible. Find the following quantities:
1. The air-gap power PAG.
2. The power converted Pconv.
3. The output power Pout.
4. The efficiency of the motor.
Solution
1.
Pin  3VL I L cos 
 3  480  60  0.85  42.4 kW
PAG  Pin  PSCL  Pcore
 42.4  2  1.8  38.6 kW
2.
3.
Pconv  PAG  PRCL
 38.6 
700
 37.9 kW
1000
Pout  Pconv  PF &W
600
 37.9 
 37.3 kW
1000
4.
Pout 
37.3
 50 hp
0.746


Pout
100%
Pin
37.3
100  88%
42.4
Example
A 460-V, 25-hp, 60 Hz, four-pole, Y-connected induction motor
has the following impedances in ohms per phase referred to the
stator circuit:
R1= 0.641 R2= 0.332
X1= 1.106  X2= 0.464  XM= 26.3 
The total rotational losses are 1100 W and are assumed to be
constant. The core loss is lumped in with the rotational losses.
For a rotor slip of 2.2 percent at the rated voltage and rated
frequency, find the motor’s
1. Speed
2. Stator current
3. Power factor
4. Pconv and Pout
5. ind and load
6. Efficiency
Solution
120 f e 120  60
nsync 

 1800 rpm
P
4
1.
nm  (1  s)nsync  (1  0.022) 1800  1760 rpm
R2
0.332
Z2 
 jX 2 
 j 0.464
s
0.022
2.
 15.09  j 0.464  15.11.76 
1
1
Zf 

1/ jX M  1/ Z 2  j 0.038  0.0662  1.76
1

 12.9431.1 
0.0773  31.1
Solution
Ztot  Z stat  Z f
 0.641  j1.106  12.9431.1 
 11.72  j 7.79  14.0733.6 
4600
V
3
I1 

 18.88  33.6 A
Ztot 14.0733.6
PF  cos33.6  0.833 lagging
Pin  3VL I L cos   3  460 18.88  0.833  12530 W
2
2
3. PSCL  3I1 R1  3(18.88)  0.641  685 W
4. PAG  Pin  PSCL  12530  685  11845 W
Solution
Pconv  (1  s) PAG  (1  0.022)(11845)  11585 W
Pout  Pconv  PF &W  11585  1100  10485 W
5.  ind
 load
6.
10485
=
 14.1 hp
746
PAG
11845


sync 2 1800
 62.8 N.m
60
Pout
10485


m 2 1760
 56.9 N.m
60
Pout
10485

100% 
100  83.7%
Pin
12530
Torque, power and Thevenin’s Theorem
• Thevenin’s theorem can be used to transform
the network to the left of points ‘a’ and ‘b’ into
an equivalent voltage source VTH in series with
equivalent impedance RTH+jXTH
Torque, power and Thevenin’s Theorem
VTH
jX M
 V
R1  j ( X 1  X M )
| VTH || V |
RTH  jX TH  ( R1  jX1 ) // jX M
XM
R12  ( X 1  X M ) 2
Torque, power and Thevenin’s Theorem
• Since XM>>X1 and XM>>R1
VTH
XM
 V
X1  X M
• Because XM>>X1 and XM+X1>>R1
RTH
 XM 
 R1 

X

X
 1
M 
X TH  X 1
2
Torque, power and Thevenin’s Theorem
VTH
VTH
I2 

2
ZT
R2 

2
R


(
X

X
)
 TH

TH
2
s 

Then the power converted to mechanical (Pconv)
Pconv
R2 (1  s)
 3I
s
2
2
And the internal mechanical torque (Tconv)
 ind 
Pconv
m
Pconv


(1  s)s
3I 22
R2
s  PAG
s
s
Torque, power and Thevenin’s Theorem
 ind


VTH
3 
 
2
s 
 R  R2   ( X  X ) 2
TH
2
  TH s 

 ind 
s 
 RTH



  R2 
  s 
  


 R2 
3V  
 s 
2
R2 
   ( X TH  X 2 ) 2
s 
2
TH
1
2
Torque-speed characteristics
Typical torque-speed characteristics of induction motor
Comments
1. The induced torque is zero at synchronous speed.
Discussed earlier.
2. The curve is nearly linear between no-load and full
load. In this range, the rotor resistance is much
greater than the reactance, so the rotor current,
torque increase linearly with the slip.
3. There is a maximum possible torque that can’t be
exceeded. This torque is called pullout torque and is
2 to 3 times the rated full-load torque.
Comments
4. The starting torque of the motor is slightly higher
than its full-load torque, so the motor will start
carrying any load it can supply at full load.
5. The torque of the motor for a given slip varies as
the square of the applied voltage.
6. If the rotor is driven faster than synchronous speed
it will run as a generator, converting mechanical
power to electric power.
Complete Speed-torque Curve
Maximum torque
• Maximum torque occurs when the power
transferred to R2/s is maximum.
• This condition occurs when R2/s equals the
magnitude of the impedance RTH + j (XTH + X2)
R2
2
 RTH
 ( X TH  X 2 )2
sTmax
sTmax 
R2
2
RTH
 ( X TH  X 2 )2
The slip at maximum torque is directly proportional to the
rotor resistance R2
Maximum torque
• The corresponding maximum torque of an
induction motor equals
 max
1

2s

3VTH2

 R  R 2  ( X  X )2
TH
TH
2
 TH




The maximum torque is independent of R2
Maximum torque
• Rotor resistance can be increased by inserting
external resistance in the rotor of a woundrotor induction motor.
• The value of the maximum torque remains
unaffected but the speed at which it occurs
can be controlled.
Maximum torque
Effect of rotor resistance on torque-speed characteristic
Motor speed control by variable frequency (VFD)
Example
A two-pole, 50-Hz induction motor supplies 15kW to a
load at a speed of 2950 rpm.
1. What is the motor’s slip?
2. What is the induced torque in the motor in N.m
under these conditions?
3. What will be the operating speed of the motor if
its torque is doubled?
4. How much power will be supplied by the motor
when the torque is doubled?
Solution
120 f e 120  50
nsync 

 3000 rpm
P
2
1.
nsync  nm 3000  2950
s

 0.0167 or 1.67%
nsync
3000
no Pf W given
2.  assume Pconv  Pload and  ind   load
 ind 
Pconv
m
15 103

 48.6 N.m
2
2950 
60
Solution
3. In the low-slip region, the torque-speed curve is
linear and the induced torque is direct
proportional to slip. So, if the torque is doubled
the new slip will be 3.33% and the motor speed
will be
nm  (1  s)nsync  (1  0.0333)  3000  2900 rpm
4.
Pconv   ind m
2
 (2  48.6)  (2900  )  29.5 kW
60
Determining motor circuit parameters
• DC Test (or use Ohm meter)
Measuring V, I → R1
• Locked-rotor Test
Measuring Vφ, I1, P,
Q → R1+R2, X1+X2
• No-load Test
Measuring Vφ, I1, P,
Q → Friction +
windage , X1+Xm
Rules of thumb for dividing stator and rotor leakage
reactances
• Cross section of squirrel cage rotor bars (NEMA Class A,B,C,D)
Motor Specifications
Problems
•
•
•
•
•
•
7.4
7.5
7.7*, 7.8, 7.10
7.14, 7.15
7.18
7.19