Electrical Machines

Lecture 1 :Introduction of Polyphase

Induction Machines
Instructor : Dr. Eltaib Said Elmubarak
• an induction motor is one in which alternating
current is supplied to the stator directly and to
the rotor by induction or transformer action
from the stator.
• When the stator winding excited from a
balanced polyphase source, it will produce a
magnetic field in the air gap rotating at
synchronous speed as determined by the
number of stator poles and the applied
stator frequency fe

120
𝑛𝑠 = 𝑓𝑒
𝑝𝑜𝑙𝑒𝑠

Where
ns – synchronous speed [rpm]
fe - applied electrical frequency [Hz}
• The rotor of a polyphase induction machine may
be one of two types:
 A wound rotor is built with a polyphase winding
similar to, and wound with the same number of
poles as, the stator.
• The terminals of the rotor winding are connected
to insulated slip rings mounted on the shaft.
• Carbon brushes bearing on these rings make the
rotor terminals available external to the motor.
• Wound-rotor induction machines are
relatively uncommon, being found only in a
limited number of specialized applications
• a squirrel-cage rotor with a winding consisting
of conducting bars embedded in slots in the
rotor iron and short-circuited at each end by
conducting end rings.
• The simple structure of the squirrel-cage
construction are outstanding advantages of
this type of induction motor and make it by far
the most commonly used type of motor in
sizes ranging from fractional horsepower on
up
• Let us assume that the rotor is turning at the
steady speed of n r/min in the same direction
as the rotating stator field.
• Let the synchronous speed of the stator field
be ns r/min
• This difference between synchronous speed
and the rotor speed is commonly referred to
as the slip of the rotor; in this case the rotor
slip is ns - n, as measured in r/min.
• Slip is more usually expressed as a fraction of
synchronous speed.
• The fractional slip s is
𝑛𝑠−𝑛
S=
𝑛𝑠
• The rotor speed in r/min can be expressed in
terms of the slip and the synchronous speed as
n = (1 - s)ns
• Similarly, the mechanical angular velocity 𝜔𝑚 can
be expressed in terms of the synchronous
angular velocity s and the slip as

𝜔𝑚 = (1 - S) s

• The relative motion of the stator flux and the

rotor conductors induces voltages of frequency fr

fr = Sfe

called the slip frequency, in the rotor.

• Thus, the electrical behavior of an induction
machine is similar to that of a transformer but
with the additional feature of frequency
transformation produced by the relative
motion of the stator and rotor windings.
• In fact, a wound-rotor induction machine can
be used as a frequency changer.
• The rotor terminals of an induction motor are
short circuited; by construction in the case of
a squirrel-cage motor and externally in the
case of a wound-rotor motor.
• The rotating air-gap flux induces slip-
frequency voltages in the rotor windings.
• The rotor currents are then determined by the
magnitudes of the induced voltages and the
rotor impedance at slip frequency.
• At starting, the rotor is stationary (n = 0), the
slip is unity (s = 1), and the rotor frequency
equals the stator frequency fe.
• The field produced by the rotor currents
therefore revolves at the same speed as the
stator field, and a starting torque results,
tending to turn the rotor in the direction of
rotation of the stator-inducing field.
• If this torque is sufficient to overcome the
opposition to rotation created by the shaft
load, the motor will come up to its operating
speed.
• The operating speed can never equal the
synchronous speed however, since the rotor
conductors would then be stationary with
respect to the stator field no current would
be induced in them, and hence no torque
 would be produced.
• With the rotor revolving in the same direction
of rotation as the stator field, the frequency of
the rotor currents is sfe and they will produce
a rotating flux wave which will rotate at sns
r/min with respect to the rotor in the forward
direction.
• But superimposed on this rotation is the
mechanical rotation of the rotor at n r/min.
• Thus, with respect to the stator, the speed of
the flux wave produced by the rotor currents
is the sum of these two speeds and equals
sns + n = sns + ns(1 - s) = ns
• From equation we see that the rotor currents
produce an air-gap flux wave which rotates at
synchronous speed and hence in synchronism
with that produced by the stator currents.
• Because the stator and rotor fields each rotate
synchronously, they are stationary with
respect to each other and produce a steady
torque, thus maintaining rotation of the rotor.
• Such torque, which exists for any mechanical
rotor speed n other than synchronous speed,
is called an asynchronous torque
• The figure shows a typical polyphase squirrel-
cage induction motor torque-speed curve.
• The factors influencing the shape of this curve
can be found in terms of the torque equation,

2
𝜋 𝑝𝑜𝑙𝑒𝑠
𝑇= 𝑠𝑟 𝐹𝑟 𝑠𝑖𝑛𝛿𝑟
2 2

• Where 𝑠𝑟 is the resultant flux per pole

 produced by the combined effect of the
stator and rotor mmf's.
Fr – rotor mmf
𝛿𝑟 is the angle measured from the axis of the
resultant mmf wave to the axis of the rotor
mmf wave.
• Note that the resultant air-gap flux sr in this
 equation is approximately constant when the
stator-applied voltage and frequency are
constant.
• Also, recall that the rotor mmf Fr is
proportional to the rotor current Ir.
• Then the torque equation can then be
expressed in the form:
T = - K Ir sin r
• where K is a constant and r is the angle by
which the rotor mmf wave leads the resultant
air-gap mmf wave.
• The rotor current is equal to the negative of
the voltage induced by the air-gap flux divided
by the rotor impedance, both at slip
frequency.
• The minus sign is required because the
induced rotor current is in the direction
 to demagnetize the air-gap flux, whereas the
rotor current as being in the direction to
magnetize the air gap.
• Under normal running conditions the slip is
small and in this range the rotor impedance is
largely resistive and hence independent of slip
• The rotor-induced voltage, on the
• other hand, is proportional to slip and leads
 the resultant air-gap flux by 90 ° .
• Thus the rotor current is very nearly
proportional to the slip, and proportional to
and 180 ° out of phase with the rotor voltage.
• As a result, the rotor-mmf wave lags the
resultant air-gap flux by approximately 90
electrical degrees, and therefore sin gr
sin 𝛿𝑟 ≈ - 1.
• Approximate proportionality of torque with
slip is therefore to be expected in the range
where the slip is small.
• As slip increases, the rotor impedance
increases because of the increasing
contribution of the rotor leakage inductance.
Thus the rotor current is less than
proportional to slip.
• Also the rotor current lags farther behind the
induced voltage, and the magnitude of sin r
decreases.
• The result is that the torque increases with
increasing slip up to a maximum value and
then decreases, as shown in torque-speed
curve.
• The maximum torque, or breakdown torque,
• limits the short-time overload capability of the
motor.
• We will see that the slip at which the peak
torque occurs is proportional to the rotor
resistance.