ECE 8830 Electric Drives

1
ECE 8830 - Electric Drives
Topic 14 Synchronous Motors
Spring 2004
2
Introduction

• The stator of a synchronous motor is identical
  to that of an induction motor. However, unlike an
  induction motor, a magnetic field is created by
  the rotor either through the use of permanent
  magnets or through a rotor winding with slip
  rings and brushes. The presence of the magnetic
  field on the rotor allows the rotor to move at
  synchronous speed with the stator field.

3
Introduction (contd)

• The rotor shape of a synchronous motor may be
  salient or non-salient, i.e. the airgap may be
  non-uniform or uniform, respectively.

4
Introduction (contd)

• Synchronous motors are more expensive than
  induction motors but offer the advantage of
  higher efficiency, an important advantage at high
  power.
• Thus, synchronous motors are used for power
  generation and large motor drives.
• A comparison of a 6MW induction motor and a
  wound field synchronous motor is shown on the
  next slide.

5
Introduction (contd)

6
Introduction (contd)

• Salient pole synchronous generators are used
  in low speed applications, such as in
  hydroelectric power stations where they are used
  to match the low operating speed of the hydraulic
  turbines.
• Non-salient pole synchronous generators are
  used in high-speed applications such as
  steam-power stations to match the high-speed
  steam turbines.

7
Introduction (contd)

• In addition to the field winding, the rotor of
  a wound field synchronous motor usually also
  contains a second winding. This armortisseur, or
  damper winding, is like the short-circuited
  squirrel cage bars in an induction motor.
• Also, additional damper windings in the rotor
  can be used to represent the damping effects of
  eddy currents in the solid iron of the rotor
  poles.

8
Introduction (contd)

• The de-axis is aligned with the North pole of
  the rotor and the qe-axis is aligned orthogonally
  to the de-axis.
• Note In Ongs book
• the qe-axis leads the
• de-axis whereas in
• Boses book the qe-
• axis lags the de-axis.

9
Equivalent Circuit of Non-Salient Pole Wound
Rotor Motor

• A simple per-phase equivalent circuit for a
  round rotor synchronous motor can be developed in
  a manner similar to the per-phase equivalent
  circuit of the induction motor.
• The figure on the next slide shows a
  transformer-coupled circuit linking the stator
  and the moving rotor winding.

10
Equivalent Circuit of Non-Salient Pole Wound
Rotor Motor (contd)

• The rotor is supplied with a current If
  produced by a voltage Vf.

11
Equivalent Circuit of Non-Salient Pole Wound
Rotor Motor (contd)

• The rotor circuit can be replaced by an ac
  current source whose amplitude is If(nIf) and
  frequency is ?e. That results in the equivalent
  circuit below

12
Equivalent Circuit of Non-Salient Pole Wound
Rotor Motor (contd)

• Neglecting Rm and replacing the current source
  and parallel inductance with a Thevenin
  equivalent, we get the equivalent circuit shown
  below

13
Mathematical Model of the Wound Rotor Motor
(contd)

• The winding inductances for deriving a
  mathematical model of a synchronous machine are
  shown below

14
Mathematical Model of the Wound Rotor Motor
(contd)

• Before writing down the equations for this
  particular circuit, let us examine the variation
  of inductances with rotor position.
• While the rotor mmfs will be aligned along
  the d- and q-axes, this is not necessarily true
  of the stator mmfs. We therefore resolve the
  stator mmfs in these two directions.

15
Mathematical Model of the Wound Rotor Motor
(contd)

• The resolved components of the a-phase mmf,
  Fa, results in flux components along the d- and
  q-axes given by
• where Pd and Pq are the permeances along the
  d- and q-axes.

16
Mathematical Model of the Wound Rotor Motor
(contd)

• The flux linkage of these flux components with
  the a-phase winding is
• where and

17
Mathematical Model of the Wound Rotor Motor
(contd)

• Similarly, the linkage of the flux components
  by the b-phase windings that is 2?/3 are given
  by

18
Mathematical Model of the Wound Rotor Motor
(contd)

• Based on these relationships, we can write
  expressions for the self-inductance for the
  a-phase (excluding leakage inductance)
• Similar expressions can be written for the
  self-inductances for the b- and c-phases except
  that ?r is replaced by (?r-2?/3) and (?r-4?/3),
  respectively.

19
Mathematical Model of the Wound Rotor Motor
(contd)

• Similarly, the mutual inductances between the
  a- and b- phases are given by
• Similarly, Lbc and Lca expressions are
  obtained by replacing ?r by (?r-2?/3) and
  (?r-4?/3), respectively.

20
Mathematical Model of the Wound Rotor Motor
(contd)

• The voltage equations for the seven stator and
  rotor windings can now be written as
• where
• vsva vb vcT, vrvf vkd vg vkqT,
• isia ib icT, irif ikd ig ikqT,
  rsdiagra rb rc,
• rrdiagrf rkd rg rkq, ?s?a ?b ?cT, and
  ?r?f ?kd ?g ?kqT.

21
Mathematical Model of the Wound Rotor Motor
(contd)

• We can now write down the equations for the
  flux linkages of the stator and rotor windings
  as
• where
• Lss

22
Mathematical Model of the Wound Rotor Motor
(contd)

• Lrr
• Lsr

23
Mathematical Model of the Wound Rotor Motor
(contd)

• We notice from these equations that the
  elements of Lss and Lsr are functions of the
  rotor angle ?r which is varying with rotation of
  the rotor. This is the same problem that we
  encountered with the induction motor and the
  solution was to transform to the rotor reference
  frame. We do the same thing here - Parks
  transformation to the rescue once more!

24
Mathematical Model of the Wound Rotor Motor
(contd)

• Recall the Parks transform matrix is
• and the inverse transform matrix is

25
Mathematical Model of the Wound Rotor Motor
(contd)

• Applying the Parks transform to the stator
  voltage equations, we get
• If rarbrcrs, the first term simplifies to

26
Mathematical Model of the Wound Rotor Motor
(contd)

• Now,
• It can be shown that,
• where .

27
Mathematical Model of the Wound Rotor Motor
(contd)

• and that
• Since,
• the stator voltage equations in the rotor qd
  reference frame become simply

28
Mathematical Model of the Wound Rotor Motor
(contd)

• The flux linkages can be obtained in a similar
  manner by only transforming the stator quantities
  
• The resulting equations are

29
Mathematical Model of the Wound Rotor Motor
(contd)

• The rotor winding flux linkages do not need to
  be transformed. The expressions for the rotor
  flux linkages are

30
Mathematical Model of the Wound Rotor Motor
(contd)

• The 3/2 terms in the above rotor equations
  will result in non-symmetric inductance
  coefficient matrices. Therefore we multiply the
  actual rotor current terms by 2/3 to produce
  equivalent rotor currents which result in
  symmetric inductance coefficient matrices, i.e.

31
Mathematical Model of the Wound Rotor Motor
(contd)

• The rotor quantities can be referred to the
  stator as described in the Ong text pp. 267-269.
  The resulting equations in the rotors qd0
  reference frame (with rotor quantities referred
  to the stator) can be summarized as shown on the
  next slide.

32
Mathematical Model of the Wound Rotor Motor
(contd)
33
Mathematical Model of the Wound Rotor Motor
(contd)

• The qd0 equivalent circuits from these
  equations are shown below

34
Mathematical Model of the Wound Rotor Motor
(contd)

• The power into the machine is given by
• Using the transformations of the stator
  quantities to the rotor qd0 reference frame, this
  equation becomes

35
Mathematical Model of the Wound Rotor Motor
(contd)

• With further algebraic manipulation and
  removing the ohmic loss and rate of change of
  magnetic energy terms, the electromechanical
  power developed by the motor can be expressed as
• For a P-pole motor with rotor speed ?rm
  mechanical radians/sec. we can write

36
Mathematical Model of the Wound Rotor Motor
(contd)

• Thus the electromechanical torque provided by
  the motor is given by

37
Steady State Operation

• Assuming balanced, steady state conditions
  with the rotor rotating at synchronous speed, ?e,
  and the field excitation held constant, we can
  write the stator phase voltages as

38
Steady State Operation (contd)

• The steady state stator currents flowing into
  the motor are given by

39
Steady State Operation (contd)

• At this stage, we do not know the relative
  orientation of the rotors qr axis and the
  synchronously rotating qe axis but since the qr
  axis rotates at synchronous speed, the relative
  orientation will be constant in time.
• Let us define an angle ? (known as the power
  or torque angle) which represents the phase
  difference between VA and Ef (?r(t)- ?e(t)).

40
Steady State Operation (contd)

• In steady state, the qd voltage equations of
  the stator windings in the rotor qd reference
  frame may be written as
• where Ef is the steady state field excitation
  voltage on the stator side given by

41
Steady State Operation (contd)

• These equations can be used to derive an
  expression for ? (see Ong pp. 274-276). The
  resulting expression is
• For a non-salient pole motor, the torque can be
  shown to be given by (see Bose pp. 79-80)

42
Steady State Operation (contd)

• For a salient pole motor, the torque can be
  shown to be given by (see Bose pg. 81)
• The torque-? curves for the two types of motor
  are shown below

43
Phasor Diagrams

• The phasor diagrams for motoring mode are
  shown below. See Ong pg. 278 for details.
• Leading PF Lagging PF

44
Simulation Model of Three-Phase Synchronous Motor

• The winding equations derived earlier can be
  used for a simulation model.
• The model inputs include
• 1) the stator abc phase voltages,
• 2) the excitation to the rotor field
• windings, and
• 3) the applied mechanical torque to
• the rotor.

45
Simulation Model of Three-Phase Synchronous Motor
(contd)

• The simulation model outputs are
• 1) the three stator abc phase currents,
• 2) the field current in the rotor,
• 3) the electromagnetic torque generated
  
• by the motor,
• 4) the speed of the rotor,
• and 5) the torque angle of the motor.

46
Simulation Model of Three-Phase Synchronous Motor
(contd)

• The first step in developing the simulation
  model is to transform the abc phase voltages to
  the qd reference frame attached to the rotor.
  This can be performed in two steps by first
  transforming to the stationary reference frame
  and then to the rotating reference frame of the
  rotor.

47
Simulation Model of Three-Phase Synchronous Motor
(contd)

• The transformation from the abc phase voltages
  to the stationary reference frame is given by

48
Simulation Model of Three-Phase Synchronous Motor
(contd)

• The transformation from the stationary
  reference frame to the rotor reference frame is
  given by
• where

49
Simulation Model of Three-Phase Synchronous Motor
(contd)

• For the case of a wound rotor synchronous
  motor with one field winding in the d-axis and a
  pair of damper windings in the d- and q-axes, we
  can write integral equations for the winding flux
  linkages as

50
Simulation Model of Three-Phase Synchronous Motor
(contd)

• In the previous expressions,
• Note All of the above equations assume a
  motoring current convention.

51
Simulation Model of Three-Phase Synchronous Motor
(contd)

• From the flux linkages, we can get the d- and
  q-axis stator, and d- and q-axis and field rotor
  (referred back to the stator) winding currents as
  follows

52
Simulation Model of Three-Phase Synchronous Motor
(contd)

• We can now get the abc phase winding currents
  by a two-step transformation. First we transform
  from the rotor qd0 reference frame to the
  stationary qd0 reference frame. This is
  accomplished through the below transformation

53
Simulation Model of Three-Phase Synchronous Motor
(contd)

• The second step is to transform from the
  stationary qd0 reference frame back to abc using
  the following transformation

54
Simulation Model of Three-Phase Synchronous Motor
(contd)

• For a motor,
• net acceleration torque TemTmech-Tdamp
• From Newtons 2nd law of motion applied to a
  rotating body, we have
• ?

55
Simulation Model of Three-Phase Synchronous Motor
(contd)

• In this equation, the electromagnetic torque
  produced by the motor can be calculated as shown
  earlier by
• Also, the torque angle ? can be calculated in
  simulation from the equation

56
Simulation Model of Three-Phase Synchronous Motor
(contd)

• The flow of variables for simulation of the
  synchronous motor is shown for the q-axis circuit
  here and for the d-axis circuit on the next
  slide.

57
Simulation Model of Three-Phase Synchronous Motor
(contd)

58
Per-Unit Expressions for Synchronous Motor

• Normalization of machine parameters allows for
  standardized comparison of different machines.
  The per-unit (p.u.) expressions for synchronous
  motors are as follows
• Base Power, Sb, is rated kVA of machine
• Base Voltage, Vb, is peak voltage,
• i.e.
• Base Current, Ib, is peak current 2Sb/3Vb
• Base Impedance, ZbVb/Ib
• Base Torque, Tb SbP/2?b ,where ?b is base
  electrical angular frequency.

59
Synchronous Motor Parameters

• Synchronous motor parameters from
  manufacturers are usually in the form of
  reactances, time constants, and resistances which
  are derived from stator measurements. Two time
  constants appear in the transient behavior of the
  rotor - a subtransient period (during the first
  few cycles of a short circuit of the windings)
  when the current decay is very rapid. This is
  followed by the transient period in which the
  current decay is slower. These two time constants
  are denoted as Tdo and Td0.

60
Synchronous Motor Parameters (contd)

• The reactances, resistances and time constants
  can be used to calculate the synchronous motor
  parameters as described on pp. 302-304 Ong.
• A sample calculation of synchronous motor
  parameters is also provided in this section.

61
Higher-Order Models

• Significant error is found in some cases
  between simulations using the model described
  earlier and actual synchronous motor performance,
  especially in the rotor winding components of
  motors with solid iron rotors. The main source of
  error is the shielding effects of the damper
  winding currents and the eddy currents in a solid
  rotor.

62
Higher-Order Models (contd)

• A third circuit to account for the eddy
  currents induced in the pole surface can be
  incorporated into the model. The three current
  components in the rotor slot are shown below

63
Higher-Order Models (contd)

• The field current path is deep in the slot and
  links the slot leakage flux components shown.
• The damper current path is higher up in the
  slot and links ?r1c partially and all of ?r2c
  directly above it.
• The eddy currents flow in the surface of the
  slot and partially links ?r2c but not ?r1c.

64
Higher-Order Models (contd)

• The qd0 circuits need to be modified from the
  earlier circuits to incorporate additional
  inductances Lr1c and Lr2c to account for the
  fluxes ?r1c and ?r2c in the rotor slot and L1c,
  L2c, and L3c to account for the self-leakages
  of the field, damper and eddy currents.
• The modified qd0 circuits are shown on the
  next slide.

65
Higher-Order Models (contd)
66
Higher-Order Models (contd)

• The simulation equations for the higher order
  model represented by these modified circuits are
  developed in sec. 7.8.1 (pp. 307-309) in Ongs
  book.