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Lecture 2A

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... the direction of maximum positive mmf produced by each winding taken individually. Positive mmf is assumed to be directed across the gap from rotor to stator. ... – PowerPoint PPT presentation

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Title: Lecture 2A


1
Lecture 2A
Synchronous Machine Modeling
  • Derivation of the dq0 Equations of an Idealized
    Three-Phase Synchronous Machine

Professor Ali Keyhani
2
Derivation of the dq0 Equations of an Idealized
Three-Phase Synchronous Machine
3
Derivation of the dq0 Equations of an Idealized
Three-Phase Synchronous Machine
  • Assumptions
  • 1)      A stator inner periphery has uniform
    radius (Not a function of position around the
    gap)
  • 2)      A rotor outer periphery has non-uniform
    radius. The rotor, specifically, is shaped such
    that when the motor is excited by a single, full
    pitched, concentrated winding located on the
    stator, then the flux per unit length which
    exists from the stator into the air gap is a
    sinusoidal function of position around the gap.
  • 3)      Two sets of shorted rotor coils or bars
    (amortisseur windings) are located on the two
    axes of rotor, which, although not sinusoidally
    distributed, have induced in them sinusoidal mmf
    (current) distribution due to the coupling with
    the stator circuits.

4
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • A third winding is located on one axis of rotor
    (d-axis, see Figure 1). Although this winding is
    generally concentrated, it is assumed that the
    winding produces same fundamental component of
    mmf around the gap. This is permissible since it
    was shown for two-phase case that the harmonic
    components of mmf give rise merely to a
    differential leakage flux component.
  • Linear magnetic current (no saturation). This
    assumption will be relaxed later. Since the
    circuit is linear, it is assumed that the stator
    and rotor iron (finite) can be replaced by
    material having infinite permeability. It is
    assumed that the gap can be increased to account
    for this effect.
  • Constant electrical parameters (i.e. R, L, C)
    independent of temperature or frequency.
  • The stator is connected as a 4-wire system

5
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Axes of Reference
  •  
  • Again axes of reference are chosen for each of
    the six windings. An axis of reference id
    developed along the direction of maximum positive
    mmf produced by each winding taken individually.
    Positive mmf is assumed to be directed across the
    gap from rotor to stator.
  •  
  • Machine Equations- Phase Variables
  •  
  • The stator and rotor voltages can be expressed in
    vector-matrix form as
  • Stator

6
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Rotor
  • where,

7
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • In general, the flux linkages for any orientation
    of the rotor are

8
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Stator Inductances
  • The expressions for the inductances of the
    machine can be written down by plausibility
    arguments similar to those given in the book by
    Majmudar - pages 224- 234 and 408 415. Also, in
    the book by B. Adkins pages 58- 61.

9
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Stator Rotor Mutual Inductances
  • Stator Field Windings

10
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Stator Windings and dr windings

11
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Stator Windings and qr windings
  • Rotor Inductances
  •  It may be noted that both the stator and rotor
    self and mutual inductances are constants
    independent of .

12
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Self Inductances
  • Mutual Inductances

13
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Matrix Notation
  •  The following inductance matrices may be
    defined
  • Note that represents the leakage inductance
    due to the leakage flux.

14
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Transformation from phase quantities to dq0 axes
  • Field on the rotor
  • The machine dynamic equations are
  • where,

15
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Matrices and are defined by
    (26), (27), (28) and (29). The matrix is
  • Observations
  •  The two circuits on the rotor, that is dr and
    qr, have resistances and self-inductances which
    are not necessarily equal.
  • The three stator circuits have identical
    resistances and self-inductances which are
    symmetrical. Hence, it may be possible to
    simplify the stator equations by a change in
    variable.

16
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • It may be apparent that a transformation of
    variables to any reference frame other than a
    reference frame fixed on the rotor will result in
    the differential equations associated with these
    circuits to become more complicated rather than
    simplified.
  •  
  • Transformation of Stator Variables to the dq0
    axes fixed on the rotor
  •  The dq0 axes are fixed on the rotor (see Fig.
    2) and they are rotating at angular velocity of
    . The stator variables as seen from dq0
    axes are

17
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • where, for , the is
  • Since the rotor windings are rotating at rotor
    speed, the rotor variables do not need to be
    transformed.
  •  To transform equation (30) to the dq0s reference
    frame, first multiply this equation by
    .
  •  

18
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Using (35) and (36) in (39), we will have
  • From equation (37),
  • Note that
  •  

19
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • The equation (42) can be rewritten as
  • Since the rotor rotates at the same speed as the
    reference frame, the rotor equation remains
    unchanged.
  •  
  • Transformation of the Stator Flux linkage
    Equation
  •  Multiplying (32) by ,

20
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Note that
  •  
  • The term can
    be written as
  • The second term can be written
    as

21
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  •  
  • Homework problem show that equation (48) and (49)
    are correct.
  • The stator flux equation is
  • which results in

22
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  •  
  • Transformation of the Rotor Flux linkage Equation
    to the d-q axes
  •  The rotor flux linkage equation is
  • Rewrite the above as
  • Equation (55) can be written as

23
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • After hard labor, you may obtain the following
  •  
  • Using (56) and (29), is

24
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • The scalar forms of (57) are
  •  

25
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Summary of the Synchronous Machine Equations in
    the dq0 reference frame rotating at rotor speed
  •  

26
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • where,
  •  

27
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  • Note that the flux linking the qr circuit from
    a unit current in the qs circuit is not the
    same as the flux linking the qs circuit due to
    unit current in the qr circuit.
  •  Equations (61) (72) suggest the equivalent
    circuit shown below
  •  

  • d- axis circuit

28
Derivation of the dq0 Equations of an Idealized
Three-phase Synchronous Machine
  •  
  • q- axis circuit
  • 0-axis circuit
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