1. ECE 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 (cont’d)
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 (cont’d)
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!
Park’s transform to the rescue!
[first row of matrix]

9. Park’s Transformation
The Park’s 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. Park’s Transformation (cont’d)
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. Park’s Transformation (cont’d)
The Park’s transformation equation is of the form:
where f can be i, v, or .

12. Park’s Transformation (cont’d)

13. Park’s Transformation (cont’d)
The inverse transform is given by:
Of course, [T][T]-1=[I]

14. Park’s Transformation (cont’d)
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 (cont’d)
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 (cont’d)
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 (cont’d)
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 (cont’d)
Now:
Just constants!!
Our double reference frame transformation eliminates the trig. terms found in our original equations.

20. Induction Motor Model in qd0 (cont’d)
We know what  and r must be to make the transformation work but we still have not determined what to set  to. We’ll 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 (cont’d)
Using the differentiation product rule:

22. Induction Motor Model in qd0 (cont’d)
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 (cont’d)
Simulation model; Stator Equations:

24. Induction Motor Model in qd0 (cont’d)
Simulation model; Rotor Equations:

25. Induction Motor Model in qd0 (cont’d)
Zero-sequence equations (v0s and v0r) may be ignored for balanced operation.
For a squirrel cage rotor machine,
vdr=vqr=0.

26. Induction Motor Model in qd0 (cont’d)
We can also write down the flux linkages:

27. Induction Motor Model in qd0 (cont’d)
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 (cont’d)
The torque of the motor in qd0 space is given by:
where P= # of poles
F=ma, so:
where = load torque

29. Induction Motor Model in qd0 (cont’d)
Example: The equations for a balanced 3, squirrel cage, 2-pole rotor induction motor:
Constitutive Laws:

30. Induction Motor Model in qd0 (cont’d)
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 (cont’d)

33. qd0 Induction Motor Model in Stationary Reference Frame (cont’d)
Stator and Rotor Voltage Equations:

34. qd0 Induction Motor Model in Stationary Reference Frame (cont’d)
Flux Linkage Equations:

35. qd0 Induction Motor Model in Stationary Reference Frame (cont’d)
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 (cont’d)

39. qd0 Induction Motor Model in Synchronous Reference Frame (cont’d)
Stator and Rotor Voltage Equations:

40. qd0 Induction Motor Model in Synchronous Reference Frame (cont’d)
Flux Linkage Equations:

41. qd0 Induction Motor Model in Synchronous Reference Frame (cont’d)
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 (cont’d)
Similarly, the rotor voltages and currents with the rotor rotating at a slip s are given by:

45. Steady State Model of Induction Motor (cont’d)
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 (cont’d)
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 (cont’d)
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 (cont’d)
and

49. Steady State Model of Induction Motor (cont’d)
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 (cont’d)
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 (cont’d)
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 =>

52. Steady State Model of Induction Motor (cont’d)
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 (cont’d)
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 (cont’d)
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 (cont’d)
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 (cont’d)
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 (cont’d)
In steady state:
The average power developed is given by:
The average torque developed is given by:

58. Steady State Model of Induction Motor (cont’d)
The operating characteristics are quite different if the induction motor is operated at constant voltage or constant current.
Constant voltage -> stator series impedance drop is small => airgap voltage close to supply voltage over wide range of loading.
Constant current -> 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 =  xls = 
rr’ =  xlr’ = 
xm =  Jrotor= 2.8 kgm2
A comparison of the performance under constant voltage and constant current is shown in the accompanying handout.