Transcript Rotor reactive power is

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:
Pag  3 I2
2
R2
s
Ps  3 I2 R2
2
R2
1 s
s
1
p
p p
2 R
Te  Pm
 Pm
 Pag  3 I2 2
m
 1 s
 
s
Pm  Pag  Ps  3 I2

2
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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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