1. INDUCTION MOTOR steady-state model (squirrel cage)
MEP 1523
ELECTRIC DRIVES
INDUCTION MOTOR steady-state model (squirrel cage)

2. CONTENTS
Construction and Principle of Operation
Per-phase Equivalent Circuit
Power And Torque
Steady-State Performance

3. Stator – 3-phase winding Rotor – squirrel cage / wound
Construction and principle of operation
120o a c’ b’ c b a’
Stator – 3-phase winding
Rotor – squirrel cage / wound

4. Construction and principle of operation
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 and principle of operation
Distributed winding
– coils are distributed in several slots
Nc for each slot 
(3Nci)/2
(Nci)/2 -
-/2
/2  

6. Distributed winding (full-pitch)
Construction and principle of operation
Distributed winding (full-pitch)
The resultant MMF is the total contribution of MMF from each coil
Considering only the space-fundamental component,
Concentrated
Distributed
Concentrated space fundamental
Distributed space fundamental

7. IDEAL case: sinusoidal flux
Construction and principle of operation
Phase a – sinusoidal distributed winding
conductor density 
Turns/rad OR conductors/rad
If Ns is the number of turns per pole,
Hence for sinusoidal distributed winding,
IDEAL case: sinusoidal flux
In PRACTICE: negelect harmonics and only consider the fundamental component

8. Phase a – sinusoidal distributed winding
Construction and principle of operation
Phase a – sinusoidal distributed winding
IDEAL case: sinusoidal flux
In PRACTICE: negelect harmonics and only consider the fundamental component

9. Phase a – sinusoidal distributed winding
Construction and principle of operation
Phase a – sinusoidal distributed winding

10. Phase a – sinusoidal distributed winding
Construction and principle of operation
Phase a – sinusoidal distributed winding

11. Combination of 3 standing waves resulted in MMF wave rotating at:
Construction and principle of operation
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
Combination of 3 standing waves resulted in MMF wave rotating at:
p – number of poles
f – supply frequency

12. Construction and principle of operation

13. Rotating flux induced:
Construction and principle of operation
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 producing 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

14. ✔ CONTENTS Construction And Principle Of Operation
Per-phase Equivalent Circuit
Power And Torque
Steady-State Performance ✔

15. Per-phase equivalent circuit
Stator voltage equation:
Vs = Rs Is + j(2f)LlsIs + Eag
Eag – airgap voltage or back emf
Eag = k f ag
Rotor voltage equation:
Er = Rr Ir + js(2f)Llrlr
Er – induced emf in rotor circuit
Er /s = (Rr / s) Ir + j(2f)Llrlr

16. Rs – stator winding resistance Rr – rotor winding resistance
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

17. We know Eg and Er related by
Per-phase equivalent circuit
We know Eg and Er related by
Where a is the winding turn ratio
The rotor parameters and variable referred to stator are:
𝑅 𝑎 ′ = 𝑎 2 𝑅 𝑟 𝐿 𝑙𝑟 ′ = 𝑎 2 𝐿 𝑙𝑟 𝐼 𝑟 ′ = 𝐼 𝑟 𝑎
rotor voltage equation Er /s = (Rr / s) Ir + j(2f)Llrlr becomes
 Eg = (Rr’ / s) Ir’ + j(2f)Llr’ Ir’

18. Rs – stator winding resistance
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

19. ✔ ✔ CONTENTS Construction And Principle Of Operation
Per-phase Equivalent Circuit
Power And Torque
Steady-State Performance ✔ ✔

20. Converted to mechanical power = (1–s)Pag
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 = (1-s)Pag

21. 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 (based on approximate equivalent circuit):

22. sm Tem Pull out Torque (Tmax) Trated r 0 rated s s 1 0
Power and Torque
Tem
Pull out Torque
(Tmax)
Trated r
rated s sm s

23. ✔ ✔ ✔ CONTENTS Construction And Principle Of Operation
Per-phase Equivalent Circuit
Power And Torque
Steady-State Performance ✔ ✔ ✔

24. 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

25. e.g. 3–phase squirrel cage IM V = 415 V Rs= 0.25  Rr=0.2 
Steady state performance
Rr’/s + Vs – Rs
Lls
Llr’
Eag Is
Ir’ Im Lm
e.g. 3–phase squirrel cage IM
V = 415 V Rs= 0.25  Rr=0.2 
Llr = Lls = 1.6mH Lm= 95.5mH
f = 50Hz p = 4

26. Steady state performance : m code
Rs=0.25;
Rr=0.2;
Lr=(0.0971)-(0.0955);
Ls=Lr;
Lm=0.0955;
f=50;
p=4;
V=(415/sqrt(3));
Zs=Rs+(2*pi*f*Ls)*i;
Xm=2*pi*f*Lm;
Zth = Zs*Xm*i/(Zs + (Xm*i));
Vth = V*(Xm*i)/(Zs + (Xm*i));
s=linspace(0,1,1000);
Ir=Vth./(Zth + 2*pi*f*Lr*i + Rr./s);
Is=Ir.*(((Rr./s) + (2*pi*f*(Lr+Lm)*i))/(i*Xm));
Te= (p/2)*3*(abs(Ir).^2)*Rr./(s*2*pi*f);
wr=(1-s)*(2*pi*f)/(p/2);
figure(1);
subplot(3,1,1);
plot(wr,Te,'k','LineWidth',2);grid;
xlabel('rotor speed (rad/s)'); ylabel('Torque (Nm)');
subplot(3,1,2);
plot(wr,abs(Is),'k','LineWidth',2);grid;
ylabel('Stator current (A)');
subplot(3,1,3);
plot(wr,abs(Ir),'k','LineWidth',2);grid;
ylabel('Rotor current (A)');

27. Steady state performance

28. Steady state performance