Lecture 28 ECE743

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Lecture 28 - ECE743
3-Phase Induction Machines Reference Frame Theory
Part III
Professor Ali Keyhani
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Voltage Equations in Arbitrary Reference Frame
Variables

• For two-pole, 3-phase symmetrical induction,

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Voltage Equations in Arbitrary Reference Frame
Variables
Fig.1 Axis of 2-pole, 3-phase
symmetrical induction.
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Voltage Equations in Arbitrary Reference Frame
Variables

• Using the above transformation equations, we can
  transform the voltage equations to an arbitrary
  reference frame rotating at speed of ?.
• Flux linkage equations in abc reference frame can
  be transformed to qd axes using Ks and Kr
  transformation matrices.

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Voltage Equations in Arbitrary Reference Frame
Variables
where,
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Voltage Equations in Arbitrary Reference Frame
Variables

• Voltage equations written in expanded form can be
  expressed as

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Voltage Equations in Arbitrary Reference Frame
Variables

• Flux linkage equations are
• Since machine and power system parameters are
  nearly always given in ohms or percent or per
  unit of a base impedance, it is convenient to
  express the voltage and flux linkage equations in
  terms of reactances rather than inductances.

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Voltage Equations in Arbitrary Reference Frame
Variables

• Let
• Then,

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Voltage Equations in Arbitrary Reference Frame
Variables

• And flux linkages become flux linkages per second
  with the units of volts.
• Fig. 2 presents the arbitrary reference frame
  equivalent circuits for a 3-phase symmetrical
  induction machine.

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Voltage Equations in Arbitrary Reference Frame
Variables
Fig. 2. A 2-pole 3-phase symmetrical
induction machine.
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Voltage Equations in Arbitrary Reference Frame
Variables
Fig. 2. A 2-pole 3-phase symmetrical
induction machine.
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Voltage Equations in Arbitrary Reference Frame
Variables
Fig. 3. Equivalent circuits of a
3-phase, symmetrical induction machine with
rotating q-d axis at speed of ?.
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Voltage Equations in Arbitrary Reference Frame
Variables
Fig. 3. Equivalent circuits of a
3-phase, symmetrical induction machine with
rotating q-d axis at speed of ?.
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Voltage Equations in Arbitrary Reference Frame
Variables
Fig. 3. Equivalent circuits of a
3-phase, symmetrical induction machine with
rotating q-d axis at speed of ?.
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Torque Equation in Arbitrary Reference Frame
Variables

• Electromagnetic torque in terms of arbitrary
  reference frame variables may be obtained by
  substituting the equations of transformation in
• After some work, we will have the following

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Torque Equation in Arbitrary Reference Frame
Variables

• Where, Te is positive for motor action. Other
  expressions for the electromagnetic torque of an
  induction machine are