Electrical Machines II
Week 5-6: Induction Motor Construction, theory of operation, rotating
magnetic field and equivalent circuit
Asynchronous (Induction) Motor:– industrial construction
Two types of induction motors – industrial products
Asynchronous (Induction) Motor: Introduction

Three-phase induction motors are the most
common and frequently encountered machines
in industry

simple design, rugged, low-price, easy
maintenance

wide range of power ratings: fractional
horsepower to 10 MW

run essentially as constant speed from no-load to
full load

No dc current is required (for magnetic field
production)
Asynchronous (Induction) Motor: Construction

An induction motor has two main parts: a stationary stator

consisting of a steel frame that supports a hollow, cylindrical core

core, constructed from stacked laminations (for eddy current reduction), having a number of
evenly spaced slots, providing the space for the stator winding

Two basic design types depending on the rotor design

squirrel-cage: conducting aluminum bus bars laid into slots and shorted at both ends
by two aluminum shorting rings forming a squirrel-cage shaped circuit (squirrel-cage).

Stator of IM
wound-rotor: complete set of insulated three-phase wire windings exactly as the
stator. Usually Y-connected, and the ends of the three rotor wires are tied to 3 slip
rings on the rotor shaft. The rotor windings are shorted through brushes riding on the
slip rings. Wound rotor IM have their rotor currents accessible at the stator brushes,
where they can be examined and where extra resistance can be inserted into the rotor
circuit. similar to the winding on the stator to modify the torque- speed
characteristics.
Squirrel cage rotor
Asynchronous (Induction) Motor: Stator Construction
Stator with three symetrical (balanced) distributed phases , a, b. c (windings). There are 3 isolated
balanced phase (windings), spaced with 120° (for 2-pole machine). 3-phase symmetrical stators
winding is supplied by 3-phase symmetrical voltage supply 120°
Stator windings
Rotor winding
air gap

Air gap
It must be as small as possible. Smaller air gap resulting in small magnetizing current needed for magnetic field. That field is
important for effective electromechanical energy conversion.
Asynchronous (Induction) Motor: Rotor Construction
bars
ring
rings
ring
resistors
More expensive than squirrel cage
rotor and require much maintenance
because of the wear associated with
their brushes and slip rings
Squirrel cage rotor of induction
motor
Stator and rotor connections of a slip-ring
Asynchronous (Induction) Motor: Rotating magnetic field
𝑖𝑎 𝑡 = 𝐼𝑚 𝑐𝑜𝑠 𝜔𝑡
𝑖𝑏 𝑡 = 𝐼𝑚 𝑐𝑜𝑠 𝜔𝑡 − 120°
𝑖𝑐 𝑡 = 𝐼𝑚 𝑐𝑜𝑠 𝜔𝑡 − 240°
 A balanced 3-phase set of currents is applied to the coil
 Each current will create a magnetic flux distribution, but the 3-phases are
displaced both in time and in space.
 The flux distribution due to the coil “b” is 120 degrees behind coil “a”
and the flux due to the coil “c” is 120 degrees behind coil “b”.
Asynchronous (Induction) Motor: Rotating magnetic field
At wt=0 the current in phase “a” is at a positive max.
While the other 2 phases have negative currents
With an amplitude of ½. By the right hand flux rule
the net stator flux is directed as indicated forming an S
pole to the left and N pole to the right.
At wt=90˚ the current in phase “a” is zero while phase
“b” is positive and phase “c” is negative. The right hand
flux rule will determine the net stator flux is directed as
indicated forming an S pole at the bottom and N pole at
the top.
A rotating magnetic field with constant magnitude is
produced, rotating with a speed
𝑛𝑠 =
120 × 𝑓
rpm
𝑃
𝑛𝑠 : synchronous speed
𝑓 : supply frequency
𝑃: number of stator poles
Asynchronous (Induction) Motor: Rotating magnetic field
At 𝜔𝑡 = 0°
Bnet (t )  Ba (t )  Bb (t )  Bc (t )
 BM sin(t )0  BM sin(t  120)120  BM sin(t  240)240
=0+
=
3
𝐵
2 𝑀
3
−
𝐵 ∠120° +
2 𝑀
3
𝐵 ∠240°
2 𝑀
− cos 120°𝑥 + sin 120°𝑦 + cos 240°𝑥 + sin 240°𝑦
=
3
𝐵
2 𝑀
1
3
1
3
𝑥 −
𝑦− 𝑥−
𝑦
2
2
2
2
3
=
𝐵
− 3𝑦
2 𝑀
= −1.5 𝐵𝑀 𝑦
= −𝟏. 𝟓 𝑩𝑴 ∠ − 𝟗𝟎°
𝑥: unit vector in the 𝑥 direction
𝑦: unit vector in the 𝑦 direction
0.866= 3 2
Asynchronous (Induction) Motor: Principle of Operation
When the stator winding of a three-phase induction motor is connected to a three phase power source, it produces a
magnetic field that:
(a) is constant in magnitude and
(b) revolves around the
periphery of the rotor at the synchronous speed.
If 𝑓 is the frequency of the current in the stator winding and P is the number of poles, the synchronous speed of the revolving
field is
Synchronous speed 𝑁𝑠 =
120×𝑓
𝑃
rpm
The speed with which the stator magnetic field rotates
 The revolving field induces electromotive force (emf) in the rotor winding.
𝑒𝑖𝑛𝑑 = (𝑣 × B) . l
𝑣: velocity of the bar relative to the magnetic field
𝐵: Magnetic flux density
𝑙: length of the conductor in the magnetic field
It is the relative motion of the rotor compared to the stator magnetic field that produces
voltage in the rotor bar.
 Since the rotor winding forms a closed loop, the induced emf in each coil gives rise to an
induced current in that coil.
 When a current-carrying coil is immersed in a magnetic field, it experiences a force (or
torque) that tends to rotate it. The torque thus developed is called the starting torque.
 If the load torque is less than the starting torque, the rotor starts rotating. The force
developed and thereby the rotation of the rotor are in the same direction as the revolving
field. This is in accordance with Faraday’s law of induction.
 Under no load, the rotor soon achieves a speed nearly equal to the synchronous speed.
However, the rotor can never rotate at the synchronous speed because the rotor coils
would appear stationary with respect to the revolving field and there would be no induced
emf in them. In the absence of an induced emf in the rotor coils, there would be no current
in the rotor conductors and consequently no force would be experienced by them. In the
absence of a force, the rotor would tend to slow down. As soon as the rotor slows down,
the induction process takes over again. In summary, the rotor receives its power by
induction only when there is a relative motion between the rotor speed and the revolving
field.
Asynchronous (Induction) Motor: Synchronous speed and slip
Let 𝑁𝑚 (or 𝜔𝑚 ) be the rotor speed at a certain load. With respect to the motor, the revolving field is moving ahead at a
relative speed of:
𝑁𝑟 = 𝑁𝑠 − 𝑁𝑚
𝜔𝑟 = 𝜔𝑠 − 𝜔𝑚
The relative speed is also called the ‘slip speed’. This is the speed with which the rotor is slipping behind a point on a
fictitious revolving pole in order to produce torque. However, it is a common practice to express slip speed in terms of the
slip (s), which is a ratio of the slip speed to the synchronous speed. That is,
Per unit Slip 𝑠 =
𝑁𝑟
𝑁𝑠
=
𝑁𝑠 −𝑁𝑚
𝑁𝑠
𝑁𝑚 = (1 − 𝑠)𝑁𝑠
𝜔𝑚 = (1 − 𝑠)𝜔𝑠
 When the rotor is stationary, the per-unit slip is 1 and the rotor appears exactly like a short-circuited secondary winding of
a transformer.
Asynchronous (Induction) Motor: Synchronous speed and slip
The frequency of the induced emf in the rotor winding is the same as that of the revolving field. However, when the rotor
rotates, it is the relative speed of the rotor 𝑁𝑟 (or 𝜔𝑟 ) that is responsible for the induced emf in its windings. Thus, the
frequency of the induced emf in the rotor is:
𝑃𝑁𝑟 𝑃(𝑁𝑠 − 𝑁𝑚 ) 𝑃𝑁𝑠 𝑁𝑠 − 𝑁𝑚
=
=
120
120
120
𝑁𝑠
= 𝑠𝑓𝑒
𝑓𝑟 =
rotor frequency depends upon the slip of the motor. At standstill, the slip is 1 and the rotor frequency is the same as
that of the revolving field. However, the rotor frequency decreases with the decrease in the slip. As the slip approaches
zero, so does the rotor frequency. An induction motor usually operates at low slip. Hence the frequency of the induced
emf in the rotor is low
Asynchronous (Induction) Motor: Equivalent Circuit

The induction motor is similar to the transformer with the exception that its secondary windings are free
to rotate

As we noticed in the transformer, it is easier if we can combine these two circuits in one circuit but there are some
difficulties

When a balanced three-phase induction motor is excited by a balanced three-phase source, the currents in the phase
windings must be equal in magnitude and 120˚ electrical apart in phase. The same must be true for the currents in the
rotor windings as the energy is transferred across the air-gap from the stator to the rotor by induction.
Asynchronous (Induction) Motor: Equivalent Circuit

However, the frequency of the induced emf in the rotor is proportional to its slip.

Since the stator and the rotor windings are coupled inductively, an induction motor resembles a three-phase
transformer with a rotating secondary winding. The similarity becomes even more striking when the rotor is at
rest (blocked-rotor condition, s = 1). HOW????
The greater the relative motion between the rotor and stator magnetic fields, the greater the resulting rotor voltage 𝑬𝑹
and rotor frequency 𝒇𝒓 . The largest relative motion occurs when the rotor is stationary, or so-called blocked rotor or
locked-rotor. The smallest occurs when 𝑬𝑹 is zero, when rotor moves at same speed as stator magnetic field.
Asynchronous (Induction) Motor: Equivalent Circuit
Thus, it is concluded that the voltage induced in the rotor at any instance is directly proportional to the slip of the rotor
𝐸𝑅 = 𝑠𝐸𝑅0
Where ER0 is the largest value of the rotor’s induced voltage obtained at s = 1(locked rotor)
 The rotor voltage is induced in a rotor circuit which contains both resistance and reactance. The rotor resistance 𝑅𝑅
is constant (doesn’t change with operation, except for the skin effect) and is slip independent.
 Rotor reactance is affected in a more complicated way
Asynchronous (Induction) Motor: Equivalent Circuit
It is known that:
𝑋 = 𝜔𝐿 = 2𝜋𝑓𝐿
So, as the frequency of the induced voltage in the rotor changes,
the reactance of the rotor circuit also changes:
𝑋 = 𝜔𝐿 = 2𝜋𝑓𝑟 𝐿𝑟 = 2𝜋𝑠𝑓𝑒 𝐿𝑟 = 𝑠𝑋𝑟0
Where Xr0 is the rotor reactance at the supply frequency (at
blocked rotor)
rotor equivalent circuit
Asynchronous (Induction) Motor: Equivalent Circuit

To calculate the rotor current as
IR 


ER
( RR  jX R )
sER 0
( RR  jsX R 0 )
Dividing both the numerator and denominator by s so nothing changes we get
𝐼𝑅 =
𝐸𝑅0
𝑅𝑅
𝑠 + 𝑗𝑋𝑅0
Where ER0 is the induced voltage and XR0 is the rotor reactance at blocked rotor condition (s = 1)
rotor equivalent circuit
Rotor circuit model with all frequency (slip)
effects encountered
Asynchronous (Induction) Motor: 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