6.11s Notes for Lecture 4

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6.11s Notes for Lecture 4 Analysis of Induction
Machines June 15, 2006 J.L. Kirtley Jr.
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Induction motor stator and rotor windings are
coupled together much like windings of a
transformer. But the coupling is dependent on
rotor position
Rotor angle
Stator Rotor
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Assume currents are of a balanced form
And that the rotor is turning at steady speed
Note the frequencies will match if
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We can calculate stator flux
Do some trig and this reduces. And the rotor flux
is similar
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Now we see that this simplifies if we use
complex notation
So that
Now make a couple of definitions
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And our flux-current relationship becomes simple
Now we can write voltage equations
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Now we need to look at the internals of the
machine winding self inductances are of the form
Note these both have fundamental inductances that
have the same permeance and then some leakage
inductance. The mutual inductance has the same
permeance as the fundamental of the self
indctances
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Those inductances can be written as
Slip is defined by
We can re-write the voltage equations
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Now refer rotor current across the transformer
ratio
Then, if we short the rotor, voltages become
Where we have made a number of definitions
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Those equations describe this equivalent circuit
Now look at the power balance
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Squirrel Cage Motor Model We want to work with
a rotor surface current
This current makes a magnetic flux density in the
gap
And that produces a flux (in the stator) of
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So that now total stator flux is
And this leads to a definition of rotor current
Now currents in the rotor bars will be like this
And now a decent description of rotor surface
current is
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So now we want to describe surface current as
And doing the Fourier analysis, the complex
amplitudes are
For these harmonic orders
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Each of these will produce a flux density
Which produce axial electric fields in the rotor
If we can neglect higher order harmonics,
voltage that drives current in the rotor slot is
And that is evaluated to be
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That can be translated into flux density (divide
by speed)
Integrate to get flux
And that yields air-gap voltage
Now the rotor looks like this with current as
indicated
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The complete picture of the machine is a bit more
complex, as the space harmonics of the stator
produce flux that interacts with the rotor too.
This diagram shows the belt leakage harmonics,
but slot order (slots per pole pair plus and
minus one) might also be shown here.
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Higher harmonic air-gap inductances are what you
might expect
And there are equivalent leakage reactances for
the harmonic orders
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Those space harmonics are often regarded as
simply leakage inductance, in which case the
two components are
And the slot order components are
To reduce noise and stray load loss due to
harmonics, the rotor is often skewed (perhaps by
about one stator slot pitch. Flux linked by a
full pitch coil is described by
So there is a skew leakage that needs to be
added
Slot and end winding leakage are also generally
included
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And the equivalent resistances are
Note that these resistances are calculated at the
right harmonic frequencies
Involved slips are
And this turns into electromagnetic energy
conversion
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Back to fundementals note that slot impedance is
most important here. If the slot is deep
Which evaluates to
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Which leads us to the arbitrary slot model (this
is current research)
What we do is to break the slot up into a
(sometimes large) number of slices vertically
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Current in one of those layers would be
Reactance of that layer is
And resistance
This leads to an equivalent circuit
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It is necessary, in most cases. To correct for
end ring resistance
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Speed Control Note that flux is the ratio of
voltage to current. We would expect constant flux
control might be a way of controlling an
induction motor. Start by ignoring stator
resistajce
This can be reduced to a simple equivalent
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Current and torque are found (quite simply) to be
Defining slip and voltage with respect to base
quantities
We find torque with respect to an absolute slip
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Above that base speed assume constant voltage
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With a more realistic motor model
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Slip Ring Motors Doubly Fed Configuration Slip
Ring Machines have wound rotor and stator, and
the rotor winding is brought out to slip rings.
Use of such machines include adjustable speed
drives with the rotor fed by an adjustable speed
drive. Here is a configuration for a possible
ship propulsion scheme
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For operation in this mode, we might be able to
neglect winding resistances (at least until we
need to calculate efficiency). The equivalent
circuit looks like this
Voltage equations are about what you would
expect speed voltage is proportional to relative
speed
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We repeat the torque expression for convenience
Developed mechanical power is
And electrical power into the stator terminals is
Rotor electrical power input is
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Current/Flux relationships are
Then rotor and stator innput power are
And they are rerlated by
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Reactive Power at the Stator terminals
Make a few definitions then reactive power is
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Now we can divide stator reactive power into
these parts The sum is
Rotor reactive power is
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This is reactive power in rotor leakage
Rotor reactive power is then
If the stator is providing reactive power
Then rotor input reactive power is
The real power relationship is
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