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Elec467 Power Machines &
Transformers
Electric Machines by Hubert, Chapter 4
Topics: Introduction, Induction-motor action,
Reversal of rotation, Construction,
Synchronous speed, Slip, Equivalent circuit,
locus of current, Air-gap power, Mechanical
power, Torque, and Efficiency
How is an Induction Motor wired?
•On the left is the wiring for a 3-phase induction motor
•On the right is the voltage graph showing the phase
relationships between the different coils of the stator
windings.
What happens when
the motor is turned on?
Two flux fields are created between three points. These
fields are dynamic and rotating because of the varying
voltage of the line current. The flux changes direction as
the fields cycle through the voltage oscillations positive
to negative and back. By carefully placing the stators in
3 equal-distance locations 120 apart, the field will push
and pull a metal squirrel cage around in a circle.
Rotor (aka squirrel cage)

The rotor is moving and crossing lines of flux thus generating a
current in the rotor bars. This current creates a field around the
bars and end ring of the of the squirrel cage. This field is what the
stator flux lines grab ahold of to push and pull the squirrel cage in
one direction.
Cutaway view of a motor
Examples of rotors
Speed control of motor
Number of poles affect
placement of windings
Dividing 360° by the number of poles reveal how wide an angle
the winding for one of the phases can be laid over the stator. Also
shown is the overlap for the next two phase windings.
Synchronous Speed
Synchronous speed ns is the speed of the
rotating flux from the stator coil.
 Thus it is directly proportional to the
frequency of the supply voltage, fs.
 It is also indirectly proportional to the
number of poles, P.

fs
ns 
P2
rev/sec
120 f s
ns 
P
fs = frequency of 3Ø voltage supply
ns = synchronous speed
P = number of poles
rev/min or rpm
Slip
The difference between the speed of the
rotating field (ns)and the rotor speed (nr)
in rpm is called slip speed, n = ns – nr
 The ratio of slip speed to synchronous
speed is called slip, s.
 s = n/ns also s = (ns – nr)/ns (It’s usually a fraction or 1)
 Another useful formula: nr = ns(1-s) using
the slip and the input voltage frequency
you can calculate the rotor speed.

Slip and Rotor Frequency (fr)

The voltage induced in the rotor itself has a frequency.
The current path for this voltage consist of the rotor
bars and end connections.
Pns  nr 
Pn n=slip speed
fr 
fr 
in rpm
120
120
More nifty equations with slip inserted into them:
ns  nr  sns
n  sns
sPns
fr 
120
A test condition when the rotor can’t move is called Blocked Rotor
f BR
Pns

120
f r  sf BR
Since s = 1 at BR … fr
= fBR
Rotor Voltage

We’ve learned how to compute the rotor
frequency, now we can use equation 1-25
to calculate the voltage across the rotor.
Erms  4.44 Nf max
Er  4.44 Nf r  max
Er  4.44 N (sf BR ) max
At BR, s = 1
Equation 1-25
Insert in rotor frequency for f
Adding slip and BR data
Er  4.44 Nf BR max
Equivalent circuit for the rotor
of an induction motor
This diagram represents one stator, a 3Ø motor develops
3 times the power and torque. The rotor above is an
electrically isolated closed circuit with resistive copper
loss Rr and leakage reactance jXr. Also,
X r  2f r Lr
(4-13)
Adding slip
To add slip to previous slide and incorporate BR
X r  2 (sf BR ) Lr  s2f BR Lr  X r  sX BR
Zr  Rr  jsX BR
(4-15)
I r  sE BR
Z r s 
Zr
 I r  EBR
Block Rotor equivalent circuit
Rotor
current
I r  EBR0
Z r s  r
 X BR 

 Rr s 
 r  tan 1 

EBR
 r
Z r s 
Relationship between rotor current
and rotor impedance angle
Can you tell where Blocked Rotor and starting current values lay?
What happens when you get to synchronous speed?
Locus of rotor-current phasor
If you
increase
the load
on the
shaft
connected
to the
rotor, the
current
phasor
rotates
clockwise
until a
value of
IrcosØ° can
carry the
load.
If we consider slip (s)
the value of Ø° this
graph’s lower half
shows the magnitude
of the rotor current
and the phase angle.
Another handy
equation:
E BR
Ir 
sin  r
X BR
Air-gap power
Air-gap power is the power transferred across the gap between
the stator and the rotor.
BIG POINT:
Active component Pgap is the shaft power. Including the losses
due to friction, windage, and heat losses in the rotor.
Considering mechanical forces

Mechanical force that are part of a motor
include slip, mechanical power developed,
rotor power losses, shaft power out and
developed torque.
Mechanical Power formulas


Pgap  Pmech  Prcl
Prcl  3I Rr
2
r
  Pmech


Where Prcl is rotor conductor losses (heat)
3I r2 Rr
Pgap 
s
(4-32)
3I r2 Rr (1  s)

s
(4-34) and,
Pmech  Pgap 1  s 
(4-35)
Steinmetz equivalent circuit:
Rr Rr 1  s 

R
Rr/s of Fig. 4.7(d)
s
(4-30)
s
Equivalent resistance = to mech. power
r
(4-36)
Actual winding resistance
Result of all Power formula calculations are in Watts
Torque formulas
Pmech
3I r2 Rr nr

sns
Pmech
3I r2 Rr nr

746sns
hp
(4-38) Converts from watts to horsepower
Pmech
TD nr

5252
hp
(4-39) inserts torque TD into formula for hp
21.21I r2 Rr
TD 
sns
TD 
7.04 Pgap
ns
Watts (4-37) mech. power in terms of rotor speed
lb-ft
(4-40) Converts torque into foot-pounds
Here TD is a function of two variables:
rotor current and slip
lb-ft
(4-40) Inserts Pgap into formula for lb-ft
Torque speed characteristics
with locus of rotor current
Parasitic torques
Pull-up torque of an induction motor is the minimum torque developed
by the motor from rest to the speed at which breakdown occurs. This
is the initial value for ideal motors seen if Fig. 4-11 or the pull-point in
motors with imperfections as seen above. Parasitic torques are
caused by non-sinusoidal space distribution of the rotating flux.
Loss Accounting formulas
Pgap = Pin – Pcore – Pscl Watts (4-44)
 Total power loss is:
Ploss = Pscl + Pcore + Prcl + Pf,w + Pstray
 Theoretical diagram:
Efficiency and Power Factor
Useful shaft output power: Pshaft  P mech  Pf , w  Pstray or Pshaft 
Efficiency η = Pshaft/Pin (4-49)
Power Factor:
FP = Pin/Sin
where Sin = √(3) VlineIline
Tshaftnr
5252
Example 4.6 with numbers
Please open your books to page 159, break out a sheet of paper
with pencil, and grab your calculators…
Formulas—part 1
Formulas—part 2
Formula—part 3