This document provides an overview of modeling the dynamics of electric machines using a dq-winding analysis approach. It introduces the need for controlling the electromagnetic torque in electric machines. It describes representing the stator and rotor fields using equivalent d- and q-axis windings and developing the mathematical relationships between the winding currents and fluxes. The voltage equations for the d- and q-axis windings of the stator and rotor are presented. Methods for calculating the winding fluxes and currents given the voltages as inputs are discussed. The choice of reference frame speed for the dq analysis is also addressed.

1. ECOLE NATIONALE SUPÉRIEURE DES MINES DE RABAT
Support de cours - Modélisation
dynamique des machines électriques
Dr. Abdelbari Redouane
mdme.minesrabat@gmail.com
2019

2. References

3. Need for controlling the electromagnetic torque Tem
.

4. Speed control of Shunt motors by variation of
field current

6. Stator windings

7. Field distribution in the air gap; (b) with ia positive at
t1 and t2 and negative at t3 and t4

8. Example 1

9. Example 2

10. Solution

11. Example 2

12. Three-phase windings

13. Inductances
Stator Rotor

14. rotor circuit represented by three-phase windings

15. Space vector representation of mmf

16. Physical interpretation of stator current space vector

17. relationship between space vector and phasor in sinusoidal steady
state

18. all stator space vectors are collinear

19. all rotor space vectors are collinear (stator open-circuited)

20. Stator and rotor voltage and flux linkeage equations in terms of space
vectors

21. Making the case for a dq-winding analysis

22. Representation of stator mmf by equivalent dq windings

23. Per-Phase Magnetizing-Inductance Lm

24. Rotor dq Windings

25. Rotor dq Windings
(a) Per phase, steady state equivalent circuit including the stator leakage; (b) phasor diagram
Similar to the stator case, each of the dq windings on the rotor has rtsq(3/2) Ns turns,
and a magnetizing inductance Lm, which is the same as that for the stator dq windings
because of the same number of turns (by choice) and the same magnetic path for flux
lines. Each of these rotor equivalent windings has a resistance Rr anda leakage
inductance Llr (equal to R’r and Llr’, respectively, in the perphase equivalent Circuit of
induction machines)

26. Mutual Inductance between dq Windings on the Stator and
the Rotor

27. Mathematical relationships of the dq windings (at an arbitrary speed
ωd)

28. Flux Linkages of dq Windings in Terms of Their Currents

29. dq Winding Voltage Equations (stator)

30. dq Winding Voltage Equations (rotor)

31. Obtaining Fluxes and Currents with Voltages as Inputs

32. Calculating dq winding flux linkages and currents

33. Choice of the dq winding speed ωd
It is possible to assume any arbitrary value for the dq winding speed ωd . However, there
is one value (out of three) that usually makes sense: ωsyn, 0 or ωm , where ωsyn
is the synchronous speed in electrical radians per second. The corresponding values for
ωdA equal ωslip , -ωm or 0 respectively,

34. Torque On The Rotor

35. d- and q- axis equivalent circuits

36. d- and q- axis equivalent circuits

37. d- and q- axis equivalent circuits

38. d- and q- axis equivalent circuits

39. Computer simulation

40. Computer simulation

41. Calculation of Initial Conditions