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Overview
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21-Torque and Back EMF
ECEGR 450
Electromechanical Energy Conversion
Torque
Power
Angular Acceleration
Rotational Dynamics
Torque vs Speed
Torque and Machine Size
Back EMF
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Dr. Louie
Questions
Introduction
• What determines the speed an object will rotate
at?
motor
load
electrical
power
• Why do bicycles and automobiles have different
gears?
mechanical
power
shaft
prime
mover
generator
electrical
power
mechanical
power
shaft
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Torque
Torque
• Torque: tangential force times radial distance at
which it is applied measured from axis of rotation
• T = Frsin( ) (in Nm)
Which experiences greater torque?
B.
A.
rsin(180- ) = rsin( )
r
r
A
r
T
T
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B
F
T
F
F
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Torque
Alternative Forms
B. experiences greater torque
Frsin( B) > Frsin( A)
•
B.
A.
•
r
r
B
A
T
T = r x F (cross product, use right hand rule for direction)
 Thumb out of the paper CCW
 Thumb into paper: CW
Torque: tangential force multiplied by r: T rF
F
T
r
F
F
T
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

F
F
F
F
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Work
• Torque through an angle is work
• For constant torque: W = T(
F
Power
• Recall that power is rate of work
(Joules)
P
1: starting angle (radians)
2: ending angle (radians)
T
dW
dt
P
T
d
dt
T
m
(watts)
m
• where:

m:
angular velocity (rad/sec)
2
1
T
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Angular Acceleration
Example
• A shaft experiencing torque will accelerate
according to:
(T1
A large reel of paper installed at the end of a paper
machine has a diameter of 1.8m and a moment of
inertia of 4500 kgm2. It is coupled to a variable
speed dc motor revolving at 120 rpm. The paper is
kept under constant tension of 6000 N.
T2 )J
• Where
 : angular acceleration (rad/s2)
 J: mass moment of inertia (kgm2)
Compute:
the torque exerted on the reel
the power output by the motor
If T1 = T2: speed unchanged
If T1 < T2: slows down
If T2 > T2: speeds up
T1,
T2
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6000N
d
paper reel
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Example
Torque vs Speed
A large reel of paper installed at the end of a paper
machine has a diameter of 1.8m and a moment of
inertia of 4500 kgm2. It is coupled to a variable
speed dc motor revolving at 120 rpm. The paper is
kept under constant tension of 6000 N.
• The power, torque and speed relationship is of
interest to electric motors and generators
• Governs rotational dynamics of system
• Different machines have different torque vs
speed relationships for a given power level
Compute:
the torque exerted on the reel
T = Fr = 6000 x 0.9 = 5400 Nm
the power output by the motor
P = 5400 x 120 x 2 /60 = 67.85 kW
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Torque vs Speed
Torque vs Speed
• Recall: P = T
 T = P/
• For a hypothetical constant power machine:
400
Automobile
Induction motor
150
100
P = 300 W
300
250
200
150
100
50
50
0
0
0
0.2
higher speed,
lower torque
P = 200 W
0
torque developed (Nm)
torque (Nm)
higher torque,
lower speed
torque developed (Nm)
350
50
100
150
0.4
0.6
speed (p.u.)
0.8
cadence (rpm)
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Generator Torque
Generator Torque
• Magnitude of the force
experienced by the coil
is: F 2iLB
N
Fa
aB
x
 L: length of conductor
• electrical torque is
Te
b
 Torque Te in CCW
direction
rpm
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• Consider a generator
whose shaft is rotating
with torque Tm
(supplied by prime
mover) in CW direction
at speed
• Conductors experience
Lorentz Force
1
Source: http://www.gusongames.com/content/car_physics_2.html
200
Fer
Fa
x
aB
2iBLr
 r: radius where the
conductors are
located (m)
Fb
N
b
S
Fb
S
Same force and torque equations
can be applied to motors
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Generator Torque
Example
In generators, applied mechanical torque is
opposed by electromagnetic torque
Consider a generator supplying a load of 1MW at a constant
voltage at 3600 rpm. The load suddenly increases to 1.1MW.
Which of the following statements are true?
 Te > Tm: shaft decelerates
 Te < Tm: shaft accelerates
 Te = Tm: shaft continues to rotate at
A.
B.
C.
D.
E.
The
The
The
The
The
generator will begin decelerating.
generator will continue to operate at 3600 rpm.
generator will begin accelerating.
electric torque is higher at 1.1 MW than 1MW.
electric torque is lower at 1.1 MW than 1MW.
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Example
Back EMF
Consider a generator supplying a load of 1MW at a constant
voltage at 3600 rpm. The load suddenly increases to 1.1MW.
Which of the following statements are true?
A.
B.
C.
D.
E.
The
The
The
The
The
Recall that a motor with shown current polarity
will rotate in CCW direction
generator will begin decelerating
generator will continue to operate at 3600 rpm
generator will begin accelerating
electric torque is higher at 1.1 MW than 1MW
electric torque is lower at 1.1 MW than 1MW
N
ax
b
S
If voltage is constant and power increases, current must increase.
Increased current will lead to increased electric torque. Unless
mechanical torque also increases, the generator will decelerate.
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Back EMF
Back EMF
Recall that a generator rotating CCW will have
induced current with the shown polarity
• Applied current (voltage) and induced current
(voltage) oppose each other
• Induced voltage is known as back emf
• Analogous to Te opposing Tm in generators
N
a
N
N
x
b
a
ax
S
b
x
S
S
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b
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Back EMF
Torque and Back EMF
• Consider the following scenario:






Applied voltage to stator is increased
I increases (Ohm’s Law)
Te increases (Lorentz Force equation) Te
(T1 T2 )J )
increases (
P
Back emf increases Ec
P m
Current decreases
• Counter torque and back emf exist in rotating
electric machines
• Act to oppose change in operating state
• Account for torque, back emf in mechanical and
electrical models of machines
2IBLr
Rc
Vs
+
I
-
+
-
Ec
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Torque and Machine Size
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Torque and Machine Size
Which generator do you expect to be physically
larger?
Which generator do you expect to be physically
larger?
 Machine A Ratings: 100 kW, 250V, 2000 rpm
 Machine B Ratings: 100 kW, 250V, 1000 rpm
 Machine A Ratings: 100 kW, 250V, 2000 rpm
 Machine B Ratings: 100 kW, 250V, 1000 rpm
d
According to e N P sin
if speed is
dt
lower, either p or N must increase to
maintain the same voltage. This means
using larger magnets (or electromagnets) and/or increasing number of
turns. Both options increase the size.
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Torque and Machine Size
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Summary
• For equal power output, lower speed machines
are more expensive than higher speed machines
 If low rpm is needed, consider using a gear box
• Generally, size of the machine is dependent on
the torque required
 100 kW, 2000 rpm machine is similar in size to a
10 kW, 200 rpm machine
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• Torque = radius x force
• Power = torque x speed
• Machines exhibit different torque vs speed
characteristics and are important in determining
the application of the machine
• Generators producing current will experience a
counter torque in opposite direction to the
applied mechanical torque
• Motors drawing current will experience an
induced voltage that opposes the applied voltage
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