Lecture 30 ECE743

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Lecture 30 - ECE743
3-Phase Induction Machines Summary of Dynamic
Equations
Professor Ali Keyhani
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Dynamic Equations in abc Reference Frame

• Machine dynamic equations in abc can be written
  as
• Flux linkage equations are

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Dynamic Equations in abc Reference Frame
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Dynamic Equations in abc Reference Frame
Fig. 1. A 2-pole 3-phase symmetrical
induction machine.
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Dynamic Equations in abc Reference Frame
Fig. 1. A 2-pole 3-phase symmetrical
induction machine.
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Dynamic Equations in abc Reference Frame
Fig.2 Axis of 2-pole, 3-phase
symmetrical induction.
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Dynamic Equations in Arbitrary Reference Frame

• Transformation equations are

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Dynamic Equations in Arbitrary Reference Frame

• Machine dynamic equations from a reference frame
  with an angular speed ? of rotating q-d axis can
  be written as

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Dynamic Equations in Arbitrary Reference Frame

• Flux linkage equations can be written as

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Dynamic Equations in Arbitrary Reference Frame
Fig. 3. Equivalent circuits of a
3-phase, symmetrical induction machine with
rotating q-d axis at speed of ?.
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Dynamic Equations in Arbitrary Reference Frame
Fig. 3. Equivalent circuits of a
3-phase, symmetrical induction machine with
rotating q-d axis at speed of ?.
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Dynamic Equations in Arbitrary Reference Frame
Fig. 3. Equivalent circuits of a
3-phase, symmetrical induction machine with
rotating q-d axis at speed of ?.
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Dynamic Equations in Arbitrary Reference Frame

• When ? is equal to zero, the reference frame is
  fixed in the stator (qs-ds ref. frame).
• When ? is equal to ?e, the reference frame is
  fixed on the synchronously rotating reference
  frame.
• When ? is equal to ?r, the reference frame is
  fixed in the rotor. That is, the reference frame
  is rotating at speed of ?r.

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Axes Transformation

• Consider the stationary reference where the
  qs-axis ic coincident with the as-axis. That is
  ?0.
• Fig. 4. Stationary as-bs-cs to qs-ds axes
  transformation

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Axes Transformation

• From the transformation equation with ?0, we
  will have

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Axes Transformation

• Voltages in the stationary qs-ds frame can be
  converted to the synchronously rotating qe-de
  frame using the Fig. 5,
• Fig. 5. stationary qs-ds axes to synchronously
  rotating qe-de axes transformation.

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Axes Transformation

• For example, assume

or
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Axes Transformation

• The above relations verify that the sinusoidal
  variables appear as dc quantities in a
  synchronously rotating reference frame.

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Steady-State Equivalent Circuit

• Consider the stationary reference frame ?0.

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Steady-State Equivalent Circuit

• Assume Vas, Vbs, and Vcs are sinusoidal voltages,
  
• then p j?e.
• Recall

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Steady-State Equivalent Circuit

• We can use either the qss and qrs or dss and drs
  equations to obtain the steady-state phase
  equation and equivalent circuit of induction
  machine. We will use qss and qrs voltage
  equations
• Rewrite

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Steady-State Equivalent Circuit

• We can rewrite V?sqr as
• Recall
• Replace
• Term A

A
B
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Steady-State Equivalent Circuit

• Term B
• Then,

or
or
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Steady-State Equivalent Circuit

• Let
• Recall
• Then Vsqs can be written as

or
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Steady-State Equivalent Circuit

• Divide
• Recall
• Then

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Steady-State Equivalent Circuit
where,
Equivalent circuit of an inducton machine.
For short-circuited rotor V?ar0.