1. INDUCTION MOTOR steady-state model
SEE 3433
MESIN ELEKTRIK
INDUCTION MOTOR steady-state model

2. Stator – 3-phase winding Rotor – squirrel cage / wound
Construction a b b’ c’ c a’
120o
Stator windings of practical machines are distributed
Coil sides span can be less than 180o – short-pitch or fractional-pitch or chorded winding
If rotor is wound, its winding the same as stator
Stator – 3-phase winding
Rotor – squirrel cage / wound

4. Construction Single N turn coil carrying current i
Spans 180o elec
Permeability of iron >> o
→ all MMF drop appear in airgap a  a’  
/2
-/2 -
Ni / 2
-Ni / 2

5. Construction Distributed winding
– coils are distributed in several slots
Nc for each slot 
MMF closer to sinusoidal
- less harmonic contents  
/2
-/2 -
(3Nci)/2
(Nci)/2

6. Construction
The harmonics in the mmf can be further reduced by increasing the number of slots: e.g. winding of a phase are placed in 12 slots:

7. Construction
In order to obtain a truly sinusoidal mmf in the airgap:
the number of slots has to infinitely large
conductors in slots are sinusoidally distributed
In practice, the number of slots are limited & it is a lot easier to place the same number of conductors in a slot

8. Phase a – sinusoidal distributed winding
Air–gap mmf
F()   2

9. Sinusoidal winding for each phase produces space sinusoidal MMF and flux
Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF t
i(t)
This is the excitation current which is sinusoidal with time
F() 

10. Sinusoidal winding for each phase produces space sinusoidal MMF and flux
Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF t
i(t)
F()
t = 0 

11. Sinusoidal winding for each phase produces space sinusoidal MMF and flux
Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF
i(t) t t1
F()
t = t1  2 

12. Sinusoidal winding for each phase produces space sinusoidal MMF and flux
Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF
i(t) t t2
F()
t = t2  2 

13. Sinusoidal winding for each phase produces space sinusoidal MMF and flux
Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF
i(t) t t3
F()
t = t3  2 

14. Sinusoidal winding for each phase produces space sinusoidal MMF and flux
Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF
i(t) t t4
F()
t = t4  2 

15. Sinusoidal winding for each phase produces space sinusoidal MMF and flux
Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF
i(t) t t5
F()
t = t5  2 

16. Sinusoidal winding for each phase produces space sinusoidal MMF and flux
Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF
i(t) t t6
F()
t = t6  2 

17. Sinusoidal winding for each phase produces space sinusoidal MMF and flux
Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF
i(t) t t7
F()
t = t7  2 

18. Sinusoidal winding for each phase produces space sinusoidal MMF and flux
Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF
i(t) t t8
F()
t = t8  2 

19. Combination of 3 standing waves resulted in ROTATING MMF wave

21. Frequency of rotation is given by:
p – number of poles
f – supply frequency
known as synchronous frequency

22. Rotating flux induced:
Emf in stator winding (known as back emf)
Emf in rotor winding
Rotor flux rotating at synchronous frequency
Rotor current interact with flux to produce torque
Rotor ALWAYS rotate at frequency less than synchronous, i.e. at slip speed: sl = s – r
Ratio between slip speed and synchronous speed known as slip

23. Induced voltage Flux density distribution in airgap: Bmaxcos 
Flux per pole:
= 2 Bmaxl r
Sinusoidally distributed flux rotates at s and induced voltage in the phase coils
Maximum flux links phase a when t = 0. No flux links phase a when t = 90o

24. Induced voltage a  flux linkage of phase a a = N p cos(t)
By Faraday’s law, induced voltage in a phase coil aa’ is
Maximum flux links phase a when t = 0. No flux links phase a when t = 90o

25. Induced voltage
In actual machine with distributed and short-pitch windinds induced voltage is LESS than this by a winding factor Kw

26. Stator phase voltage equation:
Vs = Rs Is + j(2f)LlsIs + Eag
Eag – airgap voltage or back emf (Erms derive previously)
Eag = k f ag
Rotor phase voltage equation:
Er = Rr Ir + js(2f)Llr
Er – induced emf in rotor circuit
Er /s = (Rr / s) Ir + j(2f)Llr

27. Per–phase equivalent circuit
Llr
Lls Ir Rs + Vs – +
Eag – +
Er/s – Is Lm
Rr/s Im
Rs – stator winding resistance
Rr – rotor winding resistance
Lls – stator leakage inductance
Llr – rotor leakage inductance
Lm – mutual inductance
s – slip

28. We know Eg and Er related by
Where a is the winding turn ratio = N1/N2
The rotor parameters referred to stator are:
rotor voltage equation becomes
Eag = (Rr’ / s) Ir’ + j(2f)Llr’ Ir’

29. Per–phase equivalent circuit
Rr’/s + Vs – Rs
Lls
Llr’
Eag Is
Ir’ Im Lm
Rs – stator winding resistance
Rr’ – rotor winding resistance referred to stator
Lls – stator leakage inductance
Llr’ – rotor leakage inductance referred to stator
Lm – mutual inductance
Ir’ – rotor current referred to stator

30. Power and Torque
Power is transferred from stator to rotor via air–gap, known as airgap power
Lost in rotor winding
Converted to mechanical power = (1–s)Pag= Pm

31. Mechanical power, Pm = Tem r
Power and Torque
Mechanical power, Pm = Tem r
But, ss = s - r  r = (1-s)s
 Pag = Tem s
Therefore torque is given by:

32. Power and Torque
This torque expression is derived based on approximate equivalent circuit
A more accurate method is to use Thevenin equivalent circuit:

33. Power and Torque sTm Tem Pull out Torque (Tmax) Trated r
rated syn
sTm s

34. Steady state performance
The steady state performance can be calculated from equivalent circuit, e.g. using Matlab
Rr’/s + Vs – Rs
Lls
Llr’
Eag Is
Ir’ Im Lm

35. Steady state performance
Rr’/s + Vs – Rs
Lls
Llr’
Eag Is
Ir’ Im Lm
e.g. 3–phase squirrel cage IM
V = 460 V Rs= 0.25  Rr=0.2 
Lr = Ls = 0.5/(2*pi*50) Lm=30/(2*pi*50)
f = 50Hz p = 4

36. Steady state performance

37. Steady state performance

38. Steady state performance