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Lecture 7: Induction machines
Instructor:
Dr. Gleb V. Tcheslavski
Contact:
gt.lamar@gmail.com
Office Hours:
TBD; Room 2030
Class web site:
http://www.ee.lamar.edu
/gleb/power/Index.htm
Image from http://electrical-engineering-portal.com/
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Introduction
Induction motors account for most of three-phase rotating machines…
The damper windings (that were previously discussed in Synchronous
machines) can be used to produce a starting torque in an AC machine.
The machine with only the amortisseur (damping) winding is called the
induction (or asynchronous) machine, since the rotor voltage is induced
in the rotor windings instead of being supplied from an external source.
An induction motor has the same stator
as a synchronous machine, with a
different rotor design.
A typical two-pole stator
that looks (and is) the same as a
synchronous machine stator.
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Introduction
There are two different types of induction motor rotors that can be placed
inside the stator: a squirrel-cage (or just cage) rotor and wound rotor.
Image from
www.ewh.i
eee.org
The squirrel-cage rotor consists of a series of conductive bars laid
into slots carved in the face of the rotor and shorted at either end by large
shorting rings. The conductors would look like an exercise wheel that
squirrels run on, therefore, the name.
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Introduction
The wound rotor has a complete set of three-phase set of windings that
are mirror images of the windings on the stator. These windings are
usually Y-connected and their ends are tied to slip rings on the rotor’s
shaft.
The rotor’s windings are shorted
through brushes riding on the slip rings.
Therefore, wound rotor induction motors
have their rotor currents accessible at
the stator brushes, where they may be
modified (insert resistance, for instance)
to adjust the torque-speed characteristic
of the motor.
Image from www.ewh.ieee.org
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Introduction
Squirrel cage induction motor
Wound rotor induction motor
Wound rotor induction motors are more expensive than squirrel cage
motors and require much more maintenance due to brushes wear. As a
consequence, wound rotor motors are rarely used.
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Basic induction motor concepts
1. Development of the induced torque
Considering a squirrel cage induction motor with a three-phase set of
voltages applied to the stator. Thus a three-phase set of stator currents
flows. These currents produce a magnetic field Bs that rotates counterclockwise. The speed of magnetic field’s rotation (synchronous speed) is
(8.6.1)
where fe is the electrical frequency, Hz;
P is the number of poles.
This rotating magnetic field passes over the rotor bars and induces a
voltage in them.
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Basic induction motor concepts
The voltage induced in a given rotor bar is
(8.7.1)
where v is the velocity of the bar relative to the magnetic field;
B is the magnetic flux density vector;
l is the length of conductor in the magnetic field.
Therefore, it is the relative motion of the rotor
compared to the stator magnetic field that produces the
induced voltage in a rotor bar. The velocity of the upper
rotor bars relative to the magnetic field is to the right,
so the induced voltage in the upper bars is out of the
page, while the induced voltage in the lower bars is
into the page.
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Basic induction motor concepts
This induced voltage results in a current flow
out of the upper bars and into the lower bars.
However, since the rotor is inductive, the peak
rotor current lags behind the peak rotor voltage.
This rotor current
produces a rotor
magnetic field BR that
lags 90 behind the
current.
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Basic induction motor concepts
The induced torque in the machine is counter-clockwise:
(8.9.1)
Therefore, the rotor accelerates in the counter-clockwise direction.
However, there is a finite upper limit to the motor speed. If the induction
motor’s rotor were rotating at synchronous speed, then the rotor bars
would be stationary relative to the magnetic field and no voltage would be
induced. Zero induced voltage would produce no rotor current and,
therefore, no rotor magnetic field. The induced torque would be zero, and
the rotor would slow down due to friction losses. The induction motor can
thus speed up to near-synchronous speed, but can never exactly reach
synchronous speed.
In normal operation, both rotor and stator magnetic fields rotate together
at synchronous speed, while the rotor itself turns at a slower speed.
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Basic induction motor concepts
2. The concept of rotor slip
The voltage induced in a rotor bar of an induction motor depends on the
speed of the rotor relative to the magnetic fields. Since the behavior of an
induction motor depends on the rotor’s voltage and current, it is often
more logical to discuss this relative speed. Two terms are commonly
used to define the relative motion of the rotor and the magnetic fields.
One term is slip speed:
(8.10.1)
where nm is the mechanical speed of the motor shaft.
Another term is slip, the relative speed in per-unit (or sometimes percent):
(8.10.2)
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Basic induction motor concepts
(7.10.2) can be rewritten as
(8.11.1)
or in terms of angular velocity:
(8.11.1)
If the rotor spins at synchronous speed, s = 0; while if the rotor is
stationary, s = 1. All normal motor speeds fall between these limits.
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Basic induction motor concepts
Mechanical speed of the rotor shaft can be expressed in terms of
synchronous speed and slip:
(8.12.1)
or
(8.12.2)
These equations are useful in the derivation of induction motor torque
and power relationships.
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Basic induction motor concepts
3. The electrical frequency of the rotor
The induction motor works by inducing voltages and currents in the rotor;
for this reason, it is sometimes called a rotating transformer. Like a
transformer, the primary (stator) induces a voltage in the secondary
(rotor). However, unlike the transformer, the secondary frequency is not
necessary the same as the primary frequency.
If the rotor is stationary and cannot move, it will have the same frequency
as the stator. If the rotor is at synchronous speed, its frequency is zero.
For any speed in between, the rotor frequency is directly proportional to
the slip:
(8.13.1)
(8.13.2)
Using (8.6.1):
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Basic induction motor concepts
Example 8.1: A 208 V, 10-hp, four-pole, 60 Hz, Y-connected induction motor has a
full-load slip of 5%.
a)
b)
c)
d)
What is the synchronous speed of this motor?
What is the rotor speed at the rated load?
What is the rotor frequency at the rated load?
What is the shaft torque at the rated load?
a) The synchronous speed is
b) The rotor speed is
c) The rotor frequency is
d) The shaft load torque is
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The equivalent circuit of an
induction motor
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An induction motor relies for its operation on the transformer action:
induction of voltages in its rotor circuits. Therefore, its equivalent circuit is
similar to one of the transformer. Induction machines are also called
single excited (as opposed to a doubly excited synchronous machine).
Since induction motors do not have field circuits, their model will not
contain an internal generated voltage EA.
The induction motor model will be developed by starting with the
transformer model and then incorporating the variable rotor frequency
and other effects.
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The equivalent circuit of an
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1. The transformer model of an induction motor
A transformer per-phase equivalent circuit that represents the operation
of an induction motor:
Like in any transformer, the primary (stator) and secondary (rotor) include
certain resistance (R1 for stator and RR for rotor) and self-inductance.
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The equivalent circuit of an
induction motor
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Also, like any transformer with an iron core, the flux in the machine is
related to the integral of the applied voltage E1.
Typical magnetization curves for an
induction machine and for a power
transformer:
Induction motor’s magnetization curve
has lower slope, since motors include an
air gap that increases the reluctance.
Therefore, a higher magnetizing current
is needed to obtain the same flux. Thus
the magnetizing reactance XM in the
equivalent circuit will have a much smaller value that in the transformer.
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The equivalent circuit of an
induction motor
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The primary internal stator voltage E1 is coupled to the secondary voltage
ER by an ideal transformer with an effective turn ratio aeff that can be
determined (for the wound-rotor machine) as the ratio of the conductors
per phase on the stator to the conductors per phase on the rotor, modified
by any pitch and distribution factor differences. The effective turn ratio of
a squirrel-cage rotor motor is more difficult to determine.
The rotor voltage ER produces a current flow in the shorted rotor.
The primary impedances and the magnetization current of the induction
motor are very similar to the corresponding components in a transformer
equivalent circuit.
A noticeable difference between the equivalent circuits of transformers
and induction machines is the effects of varying rotor frequency on the
rotor voltage ER and rotor impedances RR and jXR.
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The equivalent circuit of an
induction motor
2. The rotor circuit model of an induction motor
In induction motors, when a voltage is applied to the stator windings, a
voltage is induced in the rotor windings of the machine. In general, the
greater the relative motion between the rotor and the stator magnetic
fields, the greater the resulting rotor voltage and rotor frequency.
The largest relative motion occurs when the rotor is stationary (lockedrotor or blocked-rotor condition), resulting in the largest voltage and
frequency induced.
The smallest voltage (0 V) and frequency (0 Hz) occur when the rotor
moves at the same speed as the stator magnetic field, resulting in no
relative motion.
The magnitude and frequency of the induced voltage at any speed is
directly proportional to the slip of the rotor.
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The equivalent circuit of an
induction motor
The magnitude of the induced voltage:
(8.20.1)
where ELR is the magnitude of the induced voltage at the locked-rotor
conditions.
The frequency is
(8.20.2)
This voltage is induced in a rotor that contains both resistance and
reactance. The resistance RR is a constant (ignoring the skin effect) that
is independent of slip.
The rotor reactance XR depends on the rotor inductance and the
frequency and, therefore, is affected by slip.
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The equivalent circuit of an
induction motor
(8.21.1)
Here XLR is the locked-rotor reactance.
The rotor equivalent circuit:
Therefore, the rotor current is
(8.21.2)
(8.21.2) can be rewritten as follows:
(8.21.2)
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The equivalent circuit of an
induction motor
According to the last equation, it is possible to analyze the rotor effects of
varying rotor speed by a varying impedance with the constant source
voltage ELR. The equivalent rotor impedance in this case is
(8.22.1)
The modified rotor equivalent circuit:
The rotor voltage is assumed
constant and the rotor impedance
accounts for the effects of varying
rotor slip.
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The equivalent circuit of an
induction motor
Dependence of the rotor
current on the rotor speed:
At very low slips, the
resistive term RR/s is much
larger than XLR, so the rotor
resistance predominates and
the rotor current varies
linearly with the slip.
At high slips, reactive term
is much larger, and the rotor
current approaches a steadystate value as the slip
becomes very large.
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The equivalent circuit of an
induction motor
3. The final equivalent circuit model of an induction motor
To produce the final per-phase equivalent circuit for an induction machine,
it is needed to refer the rotor part of the model over to the stator side. We
will use the rotor model in slide 22.
Referring will be performed similarly to what we would do for transformers.
The rotor voltage will be:
(8.24.1)
The rotor current:
(8.24.2)
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The equivalent circuit of an
induction motor
The rotor impedance:
(8.25.1)
(8.25.2)
Defining
(8.25.3)
The final per-phase
equivalent circuit of
the induction motor:
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The equivalent circuit of an
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Notice:
The rotor resistance RR, the locked-rotor rotor reactance XLR, and the
effective turn ratio aeff are very difficult (if not impossible) to determine
directly for squirrel-cage rotors. However, it is possible to make
measurements that will directly provide the referred resistance and
reactance R2 and X2. The measurement of the motor’s parameters will be
discussed later.
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Power and torque in an induction
motor
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Since induction motors are singly excited machines, their power and
torque relationships are different than for synchronous machines.
1. Losses and power-flow diagram
An induction motor can be described as a rotating transformer. Its input is
a three-phase set of voltages and currents. For an ordinary transformer,
the output is electric power from the secondary windings. In an induction
motor, the secondary windings (the rotor) are shorted out, so no electrical
output exists from
regular induction
motors. The output is
mechanical.
Power-flow diagram
for an induction motor
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Power and torque in an induction
motor
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The input power Pin is in the form
of three-phase voltages and
currents.
The first losses are I2R losses in
the stator windings, the stator
copper losses PSCL.
Some power will be lost to hysteresis and eddy currents (core losses
Pcore). The power that remains will be transferred to the rotor across the air
gap between the stator and the rotor: the air-gap power PAG. After this,
some power is lost to heat as the rotor copper losses PRCL. The rest of the
power is converted from electrical to mechanical form (Pconv). Finally,
friction and windage losses PF&W and stray losses Pmisc are subtracted.
The remaining power is the output of the motor Pout.
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Power and torque in an induction
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The core losses do not always appear in the power-flow diagram at the
point shown above. Due to their nature, these losses can be accounted in
a somewhat arbitrary place. The core losses of an induction machine
come partially from the stator circuit and partially from the rotor circuit.
Since these machines normally operate at speeds near synchronous
speed, the relative motion of the magnetic fields over the rotor surface is
quite slow; therefore, rotor core losses are much smaller than the stator
core losses. Since the largest fraction of core losses comes from the
stator circuit, all core losses are lumped together at that point in the
diagram.
Core losses are represented by the resistor RC (or the conductance GC) in
the equivalent circuit. Sometimes, core losses are specified in watts. In
this case, core losses are often lumped together with the mechanical
losses.
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The higher the sped of an induction motor, the higher its friction, windage,
and stray losses. On the other hand, the higher the peed (up to the
synchronous speed), the lower its core losses. These four categories are
sometimes lumped together and called rotational losses. The total
rotational losses are often considered as constant with changing speed,
since their components change oppositely with speed.
Example 8.2: A 480 V, 60 Hz, three-phase induction motor consumes 60 A at 0.85
PF lagging. The stator and rotor copper losses are 2 kW and 700 W; the friction and
windage losses are 600 W; the core loses are 1.8 kW; stray losses are negligible.
Find:
a) The air-gap power PAG;
b) The converted power Pconv;
c) The output power Pout;
d) The efficiency of the motor.
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Power and torque in an induction
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Solution: referring to the power-flow diagram in slide 28:
a) The air-gap power:
b) The converted power:
c) The output power:
d) The motor’s efficiency:
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Power and torque in an induction
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2. Power and Torque in an Induction Motor
Consider the per-phase equivalent circuit of an induction motor:
The input current to a phase of the motor is:
(8.32.1)
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Power and torque in an induction
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where
(8.33.1)
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Power and torque in an induction
motor
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Power and torque in an induction
motor
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