ECE 8830 Electric Drives

1
ECE 8830 - Electric Drives
Topic 4 Modeling of Induction Motor using
qd0 Transformations Spring
2004
2
Introduction

• Steady state model developed in previous topic
  neglects electrical transients due to load
  changes and stator frequency variations. Such
  variations arise in applications involving
  variable-speed drives.
• Variable-speed drives are converter-fed from
  finite sources, which unlike the utility supply,
  are limited by switch ratings and filter sizes,
  i.e. they cannot supply large transient power.

3
Introduction (contd)

• Thus, we need to evaluate dynamics of
  converter-fed variable-speed drives to assess the
  adequacy of the converter switches and the
  converters for a given motor and their
  interaction to determine the excursions of
  currents and torque in the converter and motor.
  Thus, the dynamic model considers the
  instantaneous effects of varying
  voltages/currents, stator frequency and torque
  disturbance.

4
Circuit Model of a Three-Phase Induction Machine
(State-Space Approach)

5
Voltage Equations

• Stator Voltage Equations

6
Voltage Equations (contd)

• Rotor Voltage Equations

7
Flux Linkage Equations

8
Model of Induction Motor

• To build up our simulation equation, we could
  just differentiate each expression for ?, e.g.
• But since Lsr depends on position,
• which will generally be a function of time,
  the trig. terms will lead to a mess!
• Parks transform to the rescue!

first row of matrix
9
Parks Transformation

• The Parks transformation is a three-phase to
  two-phase transformation for synchronous machine
  analysis. It is used to transform the stator
  variables of a synchronous machine onto a dq
  reference frame that is fixed to the rotor.
• The ve q-axis is aligned with the magnetic
  axis of the field winding and the ve d-axis is
  defined as leading the ve q-axis by ?/2. (see
  Fig. 5.16c Ong on next slide).

10
Parks Transformation (contd)

• The result of this transformation is that all
  time-varying inductances in the voltage equations
  of an induction machine due to electric circuits
  in relative motion can be eliminated.

11
Parks Transformation (contd)

• The Parks transformation equation is of the
  form
• where f can be i, v, or ?.

12
Parks Transformation (contd)

13
Parks Transformation (contd)

• The inverse transform is given by
• Of course, TT-1I

14
Parks Transformation (contd)

• Thus,
• and

15
Induction Motor Model in qd0

• Acknowledgement
• The following notes covering the induction
  motor modeling in qd0 space are mostly courtesy
  of Dr. Steven Leeb of MIT.

16
Induction Motor Model in qd0 (contd)

• This transform lets us define new qd0
  variables.
• Our induction motor has two subsystems - the
  rotor and the stator - to transform to our
  orthogonal coordinates
• So, on the stator,
• where Ts T(?), (? to be
  defined)
• and on the rotor,
• where Tr T(?), (? to be defined)

17
Induction Motor Model in qd0 (contd)

• STATOR
• abc ?abcs Ls iabcs Lsr iabcr
• qd0 ?qd0s Ts ?abcs Ts Ls Ts-1 iqd0s Ts Lsr
  Ts-1 iqd0r
• ROTOR
• ?qd0r Tr ?abcr Tr LsrT Ts-1 iqd0s Tr
  Lr Tr-1 iqd0r

18
Induction Motor Model in qd0 (contd)

• After some algebra, we find
• where Lar Lr-Lab
• and similarly for .
• But what about the cross terms? They
• depend on the choice of ? and ?.
• Let ? ? - ?r , where ?r is the rotor position.

19
Induction Motor Model in qd0 (contd)

• Now
• Just constants!!
• Our double reference frame transformation
  eliminates the trig. terms found in our original
  equations.

20
Induction Motor Model in qd0 (contd)

• We know what ? and ?r must be to make the
  transformation work but we still have not
  determined what to set ? to. Well come back to
  this but let us first look at our new qd0
  constitutive law and work out simulation
  equations.

21
Induction Motor Model in qd0 (contd)

• Using the differentiation product rule

22
Induction Motor Model in qd0 (contd)

• For the stator this matrix is
• For the rotor the terminal equation is
• essentially identical but the matrix is

23
Induction Motor Model in qd0 (contd)

• Simulation model Stator Equations

24
Induction Motor Model in qd0 (contd)

• Simulation model Rotor Equations

25
Induction Motor Model in qd0 (contd)

• Zero-sequence equations (v0s and v0r) may be
  ignored for balanced operation.
• For a squirrel cage rotor machine,
• vdrvqr0.

26
Induction Motor Model in qd0 (contd)

• We can also write down the flux linkages

27
Induction Motor Model in qd0 (contd)

• How do we pick ??
• One good choice is
• where ?e is synchronous frequency.
• Remember that this choice makes a balanced 3?
  voltage set applied to the stator look like a
  constant.

28
Induction Motor Model in qd0 (contd)

• The torque of the motor in qd0 space is given
  by
• where P of poles
• Fma, so
• where load torque

29
Induction Motor Model in qd0 (contd)

• Example The equations for a balanced 3?,
  squirrel cage, 2-pole rotor induction motor
• Constitutive Laws

30
Induction Motor Model in qd0 (contd)

• State equations
• ?r rotor speed
• ? frame speed
• J
  shaft inertia
• ?l load torque

31
qd0 Induction Motor Model in Stationary Reference
Frame

• The qd0 induction motor model in the
  stationary reference frame can be obtained by
  setting ?0. This model is known as the Stanley
  model and the equivalent circuits are given on
  the next slide.

32
qd0 Induction Motor Model in Stationary Reference
Frame (contd)

33
qd0 Induction Motor Model in Stationary Reference
Frame (contd)

• Stator and Rotor Voltage Equations

34
qd0 Induction Motor Model in Stationary Reference
Frame (contd)

• Flux Linkage Equations

35
qd0 Induction Motor Model in Stationary Reference
Frame (contd)

• Torque Equation

36
Induction Motor Model in qd0 Example

• Example 5.3 Krishnan

37
qd0 Induction Motor Model in Synchronous
Reference Frame

• The qd0 induction motor model in the
  synchronous reference frame can be obtained by
  setting ? ?e . This model is known as the Kron
  model and the equivalent circuits are given on
  the next slide.

38
qd0 Induction Motor Model in Synchronous
Reference Frame (contd)

39
qd0 Induction Motor Model in Synchronous
Reference Frame (contd)

• Stator and Rotor Voltage Equations

40
qd0 Induction Motor Model in Synchronous
Reference Frame (contd)

• Flux Linkage Equations

41
qd0 Induction Motor Model in Synchronous
Reference Frame (contd)

• Torque Equation

42
Induction Motor Model in Synchronous Reference
Frame Example

• Example 5.5 Krishnan

43
Steady State Model of Induction Motor

• The stator voltages and currents for an
  induction machine at steady state with balanced
  3? phase operation are given by

44
Steady State Model of Induction Motor (contd)

• Similarly, the rotor voltages and currents
  with the rotor rotating at a slip s are given by

45
Steady State Model of Induction Motor (contd)

• Transforming these stator and rotor abc
• variables to the qd0 reference with the
  q-axis
• aligned with the a-axis of the stator gives
• where s and r qd0 components in stationary
• frame and rotating ref. frames, respectively.

46
Steady State Model of Induction Motor (contd)

• In steady state operation with the rotor
  rotating at a constant speed of ?e(1-s),
• This equation can be used to simplify the
  rotor voltage and current space vectors which
  become

47
Steady State Model of Induction Motor (contd)

• Use phasors to perform steady state analysis.
• Notation
• A - rms values of space vectors
• - rms time phasors
• Thus,

48
Steady State Model of Induction Motor (contd)

• and

49
Steady State Model of Induction Motor (contd)

• Referring the rotor voltages and currents to
  the stator side gives
• where the primed quantities indicate rotor
  quantities referred to the stator side.

50
Steady State Model of Induction Motor (contd)

• In the stationary reference frame, the qd0
  voltage and flux linkage equations can be
  rewritten in terms of the complex rms space
  voltage vectors as follows

51
Steady State Model of Induction Motor (contd)

• Using the relationships between the rms space
  vectors and rms time phasors provided earlier,
  and re-writing (?e-?r) by s?e, and dropping the
  common ej?t term, we get
• ?s gt

52
Steady State Model of Induction Motor (contd)

• The relations on the previous slide can be
  rewritten as
• where ?b is the base or rated angular freq.
• given by where frated rated
  
• frequency in Hz of the machine.

53
Steady State Model of Induction Motor (contd)

• A phasor diagram of the stator and rotor
  variables with is shown below
  together with an equivalent circuit diagram.

54
Steady State Model of Induction Motor (contd)

• By adding and subtracting rr and regrouping
  terms, we get the alternative equivalent circuit
  representation shown below

?e
55
Steady State Model of Induction Motor (contd)

• The rr (1-s)/s resistance term is associated
  with the mechanical power developed.
• The rr/s resistance term is associated with
  the power through the air gap.

56
Steady State Model of Induction Motor (contd)

• If our main interest is on the torque
  developed, the stator side can be replaced by the
  Thevenin equivalent circuit shown below

57
Steady State Model of Induction Motor (contd)

• In steady state
• The average power developed is given by
• The average torque developed is given by

58
Steady State Model of Induction Motor (contd)

• The operating characteristics are quite
  different if the induction motor is operated at
  constant voltage or constant current.
• Constant voltage -gt stator series impedance
  drop is small gt airgap voltage close to supply
  voltage over wide range of loading.
• Constant current -gt terminal and airgap
  voltage could vary significantly.

59
Steady State Model of Induction Motor- Constant
Voltage Supply

• Shorting the rotor windings and operating the
  stator windings with a constant voltage supply
  leads to the below Thevenin equivalent circuit.

60
Steady State Model of Induction Motor- Constant
Voltage Supply

• The Thevenin circuit parameters are

61
Steady State Model of Induction Motor- Constant
Voltage Supply

• The average torque developed for a P-pole
  machine with constant voltage supply is given by
• We can use this equation to generate the
  torque-slip characteristics of an induction motor
  driven by constant voltage supply.

62
Steady State Model of Induction Motor- Stator
Input Impedance

• The stator input impedance is given by
• The stator input current and complex power are
  given by

63
Steady State Model of Induction Motor- Constant
Current Supply

• With a constant current supply, the stator
  current is held fixed and the stator voltage
  varies with the input impedance given on the
  previous slide.
• The rotor current Iar can be used to
  determine the torque and is given by

64
Comparison of Constant Voltage vs. Constant
Current Operation

• Consider a 20 hp, 60Hz, 220V 3? induction
  motor with the following equivalent circuit
  parameters
• rs 0.1062? xls 0.2145 ?
• rr 0.0764? xlr 0.2145 ?
• xm 5.834 ? Jrotor 2.8 kgm2
• A comparison of the performance under constant
  voltage and constant current is shown in the
  accompanying handout.