1. ECE 8830 - Electric Drives Topic 3: Induction Motor Modeling -
Steady State
Spring 2004

2. Introduction
Induction machines are the most widely used of all electric motors. They offer the following attractive features:
Generally easy to build and cheaper than corresponding dc or synchronous motors
Rugged and require little maintenance
Offer reasonable asynchronous performance
A manageable torque-speed curve
Stable operation under load
Generally satisfactory efficiency
Range in size from few Watts to several MW

3. Introduction (cont’d)
Some disadvantages of induction motors are:
Speeds not as easily controlled as dc motors
Draw large starting currents, typically 6-8 x their full load values
Operate with a poor lagging power factor when lightly loaded

4. Introduction (cont’d)
New designs for high performance induction machines, such as in high speed motors for gas compressors, will be required to have new characteristics from existing machines, it is important to have a good fundamental understanding of these types of machines.
Goal: To develop a “simple” model for the induction machine that is useful for control and simulation.

5. Structure of an Induction Machine
Two types of induction machine:
Wound rotor or squirrel cage rotor

6. Rotating Magnetic Field and Slip
We previously showed that a balanced set of three-phase currents flowing in a set of symmetrically placed, three-phase stator windings produces a rotating mmf given by:
[eq. (6.1) Ong, eq. (2.9) Bose]
where ae is the electrical angle measured from the a-phase axis and e is the angular speed of the stator mmf in electrical radians/second.

7. Rotating Magnetic Field and Slip (cont’d)

8. Rotating Magnetic Field and Slip (cont’d)
In mechanical radians/sec. the synchronous speed is related to the electrical speed by:
If the rotor is rotating at an angular speed rm the slip speed is simply equal to sm - rm. The “slip”,s, is the normalized slip speed and is given by:

9. Torque Production
The torque produced by an induction motor may be derived and expressed by the following equation: (see ref. [1] in Bose)
where P= # of poles
l = axial length of motor
r = radius of motor
Bp= peak air-gap flux density
Fp= peak value of rotor mmf
and

10. Per-Phase Equivalent Circuit Model
A per-phase transformer-like equivalent circuit is shown below:

11. Per-Phase Equivalent Circuit Model (cont’d)
Synchronously rotating air gap flux wave generates a counter emf Vm. This in turn is converted to a slip voltage in the rotor phase, Vr’ = nsVm, where n=rotor:stator turns ratio, and s=normalized slip.
Stator terminal voltage, Vs = Vm + VRs +VLls
where VRs=voltage drop across stator resistance (Rs) and VLls=voltage drop across stator leakage inductance (Lls).

12. Per-Phase Equivalent Circuit Model (cont’d)
Excitation current, I0 = Ic + Im
where Ic is core loss current (=Vm/Rm)
and Im is magnetizing current (=Vm/ )
Rotor-induced voltage, Vr’ = VRr’ + VLl’
where VRr’ = voltage drop across rotor resistance
and VLl’ = voltage drop across rotor leakage
inductance
The induced voltage in the rotor leads to a rotor
current Ir’ at slip frequency sl.

13. Per-Phase Equivalent Circuit Model (cont’d)
The stator current, IS = I0 + Ir
where Ir is the rotor-reflected current induced in the stator. I0

14. Per-Phase Equivalent Circuit Model (cont’d)

15. Per-Phase Equivalent Circuit Model (cont’d)

16. Per-Phase Equivalent Circuit Model (cont’d)
Torque expression can be written as:
where = peak value of air gap flux
linkage/pole
and = peak value of rotor current

17. Per-Phase Equivalent Circuit Model: Power Expressions
Input Power: where cos is input PF
Stator copper loss:
Rotor copper loss:
Core loss:
Power across air gap:
Output power:
Shaft Power: where PFw is friction and windage power loss

18. Per-Phase Equivalent Circuit Model: Torque Expression
The torque can be expressed as:
where is the rotor
mechanical speed (radians/sec.)

19. Per-Phase Equivalent Circuit Model: Torque Expression (cont’d)
Using a little algebra (see Bose) it can
be shown that the torque may be further
expressed as:
where
This torque expression is similar to that for
a dc motor, where Im = magnetizing
component of stator current and Ia =
armature component of stator current.

20. Simplified Per-Phase Equivalent Circuit
A simplified circuit dropping Rm and shifting Lm to the input is applicable to integral horsepower machines.
Performance of this equivalent circuit is typically within 5%.

21. Simplified Per-Phase Equivalent Circuit Model (cont’d)
The current Ir in this circuit is given by:
The torque of the motor using this circuit
is given by:

22. Example of Calculating Efficiency of an Induction Motor
Example 5.1 Krishnan

23. Flowchart for Computing Steady State Performance of Induction Motor
Ref: R. Krishnan,
“Electric Motor Drives”

24. Torque-Speed Curve of Induction Motor
The torque-speed curve as a function of slip can be calculated from the equation two slides back.

25. Torque-Speed Curve of Induction Motor (cont’d)
Three regions in torque-speed curve:
1) Plugging (braking) region (1<s<2)
Rotor rotates opposite to direction of air gap flux. Can happen, for example, if stator supply phase sequence reversed while rotor is moving.
2) Motoring region (0<s<1)
Te=0 at s=0. As s increases (speed decreases), Te increases until max. torque (breakdown torque) is reached. Beyond this point, Te decreases with increasing s.

26. Torque-Speed Curve of Induction Motor (cont’d)
3) Regenerating Region (s<0)
Here the induction machine acts as a generator. Rotor moves faster than air gap flux resulting in negative slip.

27. Torque-Speed Curve of Induction Motor (cont’d)
Ref: R. Krishnan,
“Electric Motor Drives”

28. Performance Characteristics of Induction Motor
Ref: R. Krishnan, “Electric Motor Drives”

29. Starting Torque of Induction Motor
The starting torque of an induction motor is given by substituting for s=1 and is given by:

30. Starting Torque of Induction Motor (cont’d)
This torque can be enhanced for line start motors (ones started directly with full line voltage) by increasing the rotor resistance. This can be achieved by connecting external resistors in the case of slip ring rotors. However, with squirrel cage rotors where the rotor is shorted, deep bar or double-cage rotors can be used to increase starting torque.

31. Characterizing Induction Motors
One way to characterize an induction motor is with the No-load/blocked rotor tests which yield the per-phase equivalent circuit model shown earlier (see figure below).
ias
iar
vas =M

32. Characterizing Induction Motors (cont’d)
We can characterize an induction motor with the variables Rs, Lls, M, Llr, Rr determined through lab tests using balanced 3 excitation. This circuit described the impedance perceived per phase on a line-neutral connected machine. Everything in the dashed box is a rotor quantity that has been “referred” to the stator by the ideal transformer in the machine model. From now on, assume that Llr, Rr and iar are referred to the stator.

33. Characterizing Induction Motors (cont’d)
No-Load Test (s=0)
Equivalent circuit:
Ref: R. Krishnan, “Electric Motor Drives”

34. Characterizing Induction Motors (cont’d)
No-Load Test (s=0) yields:
In sinusoidal steady state, ignoring resistances:
But -ias = (ibs+ics) 
From transformer model:
Las
=>

35. Characterizing Induction Motors (cont’d)
Blocked rotor test (s=1) yields estimates of Lls and Llr. Equivalent circuit at standstill is shown below:
Ref: R. Krishnan, “Electric Motor Drives”

36. Characterizing Induction Motors (cont’d)
Ohmmeter/Power loss tests give Rs
and Rr.
So, with Llr, Rr and all ir’s understood as
referred rotor quantities, the “stator-side”
tests identify all the model parameters for
the induction motor.

37. Example of Determining Induction Motor Model Parameters
Example 5.2 Krishnan

38. NEMA Classification of Induction Motors
The National Electrical Manufacturers Association (NEMA) has classified induction motors based on their torque-slip characteristics. (see text for details)

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

40. Voltage Equations
Stator Voltage Equations:

41. Voltage Equations (cont’d)
Rotor Voltage Equations:

42. Flux Linkage Equations

43. 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]