Synchronous Machine Modeling Lecture 4 EE746

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Lecture 4
Synchronous Machine Modeling
Summary of Dynamic Equations of Three-Phase
Synchronous Machine
Professor Ali Keyhani
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Summary of Dynamic Equations of Three-Phase
Synchronous Machine
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Summary of Dynamic Equations of Three-Phase
Synchronous Machine

• The voltage equations are

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Summary of Dynamic Equations of Three-Phase
Synchronous Machine

• The flux linkage equations are

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Summary of Dynamic Equations of Three-Phase
Synchronous Machine

• The equations 1- 11 suggest the equivalent
  circuit shown below. Note that mutual couplings
  have been replaced by a common magnetizing
  reactance.
• The d-axis
•  
•  
•  
• d-axis Equivalent Circuit of a , 4-wire
  Synchronous Machine

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Summary of Dynamic Equations of Three-Phase
Synchronous Machine

• The q-axis
•  
•  
• q-axis Equivalent Circuit of a , 4-wire
  Synchronous Machine
•  
•  
•  
• 0-axis Equivalent Circuit of a , 4-wire
  Synchronous Machine

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Summary of Dynamic Equations of Three-Phase
Synchronous Machine

• Electromechanical Torque Equation
• The 3-phase synchronous machine has 3
  electrical inputs (outputs), one electrical input
  (field voltage), and one mechanical input
  (output). The electromagnetic torque may be
  expressed as
• The co-energy, , is expressed as

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Summary of Dynamic Equations of Three-Phase
Synchronous Machine

• where, the evaluation of the co-energy at
  particular operating point is assumed. Also, ,
  can be expressed as
• and
• where, and
• or
• Note For the derivation of (12), see the book by
  H. Majumdar Electromechanical Energy
  Converters, Allyn and Bacon Inc., Boston, 1965.
  Also, The General Theory of Alternating Current
  Machines by B. Adkins R.G. Harley.

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Summary of Dynamic Equations of Three-Phase
Synchronous Machine

• The (electrical speed) is function of
  , the mechanical speed, that is,
• Therefore,
• or
• where,

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Summary of Dynamic Equations of Three-Phase
Synchronous Machine

• The , and are defined by
  equations (26), (27) and (29) of part 13.
• After some hard labor, it can be shown that
• or
• or
• Also, it can be shown that in abcs reference
  frame

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Summary of Dynamic Equations of Three-Phase
Synchronous Machine

• Power equation
• The total power input to the machine is
• Transforming the stator variables to a d-q axis
  rotating with the rotor, the power equation
  becomes
• Equations of Motion
• If motor convention is assumed such that the
  machine is loaded with a torque which is
  independent of speed of the shaft, then

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Summary of Dynamic Equations of Three-Phase
Synchronous Machine

• where, is the total rotor inertia in
  , is the number of poles , is the
  rotor mechanical speed and is the
  equivalent electrical speed.
• Bonus Homework Problem
• Problem 1 Starting with equation (15) show that
  
• a)
• b)
• c)