Lecture 29 ECE743

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Lecture 29 - ECE743
3-Phase Induction Machines Reference Frame Theory
Part IV
Professor Ali Keyhani
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q-d Axis Voltage Equations

• q-d axis voltage equations can be written by
  inspection of Fig.1
• And flux linkage equations are

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q-d Axis Voltage Equations
Fig.1 Axis of 2-pole, 3-phase
symmetrical induction.
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q-d Axis Voltage Equations
Fig. 2. A 2-pole 3-phase symmetrical
induction machine.
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q-d Axis Voltage Equations
Fig. 2. A 2-pole 3-phase symmetrical
induction machine.
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q-d Axis Voltage Equations
Fig. 3. Equivalent circuits of a
3-phase, symmetrical induction machine with
rotating q-d axis at speed of ?.
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q-d Axis Voltage Equations
Fig. 3. Equivalent circuits of a
3-phase, symmetrical induction machine with
rotating q-d axis at speed of ?.
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q-d Axis Voltage Equations
Fig. 3. Equivalent circuits of a
3-phase, symmetrical induction machine with
rotating q-d axis at speed of ?.
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q-d Axis Voltage Equations

• When ?0 (stationary reference frame), voltage
  equations become,
• When ? ?r (rotor reference frame) which is also
  referred to as Parks transformation, voltage
  equations become,

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q-d Axis Voltage Equations

• When ? ?e (synchronously rotating reference
  frame), the voltage equations become,
• Synchronously rotating reference frame is used
  when incorporating the dynamic characteristics od
  an induction machine into a digital computer
  program used to study the transient and dynamic
  stability of large power systems.
• Synchronously rotating reference frame is also
  used in variable frequency study of induction
  machines.

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Per Unit System

• Machine data,
• - Stator line frequency f (Hz)
• - Output horse power HP (hp)
• - Line to line voltage V (rms V)
• - Pole number P
• Base values,

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Per Unit System

• Base electrical angular velocity ?b2?fb
  rad/sec,
• Base mechanical angular velocity
  ?brm2?fb/(P/2)

Let
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Per Unit System

• Define
• Machine voltage equations in p.u can be written
  as

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Per Unit System

• Torque base

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Per Unit System

• Per unit equation of motion is
• Define quantity

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Per Unit System
Therefore,
J kg.m2 Pb watts ?b rad/sec
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Per Unit System

• The quantity H is called the inertia constant and
  has units of seconds.

n rpm KVA base KVA J kg.m2 Wk2 Lb-ft2
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Per Unit System

• Example

rs 0.016 ? r?r 0.001 ? Xls 0.0706 ? X?lr
0.0903 ? X?m 2.8413 ?
HP 115 Poles 4 Vrated 210rms/phase f 50
Hz (rated) J 100 Lb-ft2
- Base values
Pb (746)(115) 85,800 watts Vb (210)?2 297
Volts peak Ib (2/3)(Pb/Vb) 193 Amp.peak Zb
(Vb/Ib) (297/193) 1.54 ?
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Per Unit System
- Base angular frequency ?b 2?(50) 314
rad/sec
- Torque base Tb Pb(P/2)/?b 547 N-m
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Per Unit System