Lecture 1 - EE743

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SYNCHRONOUS MACHINES
Two-pole,3-phase,wye-connected,salient-pole
synchronous machine
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SYNCHRONOUS MACHINES

• In abc reference frame, voltage equations can be
  written as

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SYNCHRONOUS MACHINES

• flux linkage equations

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SYNCHRONOUS MACHINES

• flux linkage equations

• Referring all rotor variables to the stator
  windings

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SYNCHRONOUS MACHINES

• Referring all rotor variables to the stator
  windings

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SYNCHRONOUS MACHINES

• Referring all rotor variables to the stator
  windings

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SYNCHRONOUS MACHINES

• TORQUE EQUATION IN MACHINE VARIABLES

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SYNCHRONOUS MACHINES

• SWING EQUATION

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SYNCHRONOUS MACHINES

• Stator Voltage Equations in Arbitrary
  Reference-frame Variables

• The rotor voltage equations are expressed only in
  the rotor reference frame

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SYNCHRONOUS MACHINES

• The flux linkage equations may be expressed as

• The sinusoidal terms are constant, independent
  of ? and ?r only if ? ?r

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SYNCHRONOUS MACHINES

• Therefore, the time-varying inductances are
  eliminated from the voltage equations only if the
  reference frame is fixed in the rotor.

• Voltage Equations In Rotor Reference-frame
  variables park's Equations

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SYNCHRONOUS MACHINES
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SYNCHRONOUS MACHINES

• Park's voltage equations are often written in
  expanded form

• Flux linkages in expanded form

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SYNCHRONOUS MACHINES

• The Equivalent q-axis Circuits

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SYNCHRONOUS MACHINES

• The Equivalent d-axis Circuits

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SYNCHRONOUS MACHINES

• The Equivalent 0-axis Circuits

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SYNCHRONOUS MACHINES

• It is often convenient to express the voltage
  and flux linkage equations in terms of reactances
  rather than inductances

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SYNCHRONOUS MACHINES

• Also, it is convenient to define

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SYNCHRONOUS MACHINES

• If we select the currents as independent
  variables

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SYNCHRONOUS MACHINES

• If we select the flux linkages per second as
  independent variables

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SYNCHRONOUS MACHINES

• Torque Equations in Substitute Variables

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SYNCHRONOUS MACHINES

• Rotor Angle

it is convenient to relate the position of the
rotor of a synchronous machine to a voltage or to
the rotor of another machine.
The electrical angular displacement of the rotor
relative to its terminal voltage is defined as
the rotor angle,
The rotor angle is the displacement of the rotor
generally referenced to the maximum positive
value of the fundamental component of the
terminal voltage of phase a
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SYNCHRONOUS MACHINES

• Rotor Angle

It is important to note that the rotor angle is
often used as the argument in the transformation
between the rotor and synchronously rotating
reference frames
The rotor angle is often used in relating torque
and rotor speed (if ?e is constant)
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SYNCHRONOUS MACHINES
PER UNIT SYSTEM

• Base voltage
• the rms value ofthe rated phase voltage for
  the abc variables
• the peak value for the qd0 variables.

• Base power
• When considering the machine separately, the
  power base is selected as its volt-ampere rating.
• When considering power systems, a system
  power base (system base) is selected

• Once the base quantities are established, the
  corresponding base current and base impedance may
  be calculated.

• Base torque is the base power divided by the
  synchronous speed of the rotor

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SYNCHRONOUS MACHINES

• The torque expressed in per unit

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SYNCHRONOUS MACHINES

• ANALYSIS OF STEADY-STATE OPERATION

For balanced conditions the 0s quantities are
zero. ?r is constant and equal to ?e the rotor
windings do not experience a change of flux
linkages the current is not flowing in the
short-circuited damper windings the time rate of
change of all flux linkages neglected
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SYNCHRONOUS MACHINES

• ANALYSIS OF STEADY-STATE OPERATION

For balanced conditions
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SYNCHRONOUS MACHINES

• ANALYSIS OF STEADY-STATE OPERATION

Hence
and if it is noted that
Then
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• ANALYSIS OF STEADY-STATE OPERATION

It is convenient to define the last term on the
right-hand side as (excitation voltage)
if rs is neglected, the expression for the
balanced steady-state electromagnetic torque in
per unit can be written as
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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN
INPUT TORQUE
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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN
INPUT TORQUE
Dynamic performance of a hydro turbine generator
during a step increase in input torque from zero
to rated
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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN
INPUT TORQUE
Torque versus rotor angle characteristics
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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN
INPUT TORQUE
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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN
INPUT TORQUE
Dynamic performance of a steam turbine generator
during a step increase in input torque from zero
to 50 rated.
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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN
INPUT TORQUE
Torque versus rotor angle characteristics
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DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE
MACHINE TERMINALS
a hydro turbine generator
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DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE
MACHINE TERMINALS
Torque versus rotor angle characteristics
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DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE
MACHINE TERMINALS
a steam turbine generator
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DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE
MACHINE TERMINALS
Torque versus rotor angle characteristics
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COMPUTER SIMULATION
Simulation in Rotor Reference Frame
Where
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COMPUTER SIMULATION
Simulation in Rotor Reference Frame
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COMPUTER SIMULATION
Simulation of Saturation
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COMPUTER SIMULATION
Simulation of Saturation