1. Power Magnet Devices: A Multi-Objective Design Approach
Chapter 8: Distributed Windings and Rotating Electric Machinery
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

2. 8.1 Describing Distributed Windings
Stator construction
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

3. 8.1 Describing Distributed Windings
Position measurements
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

4. 8.1 Describing Distributed Windings
Poles
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

5. 8.1 Describing Distributed Windings
Electrical angles and positions
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

6. 8.1 Describing Distributed Windings
Discrete description of distributed windings
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

7. 8.1 Describing Distributed Windings
Developed diagram
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8. 8.1 Describing Distributed Windings
Continuous description of distributed windings
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

9. 8.1 Describing Distributed Windings
Symmetry conditions
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

10. 8.1 Describing Distributed Windings
Converting discrete to continuous
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

11. 8.1 Describing Distributed Windings
Converting discrete to continuous
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

12. 8.1 Describing Distributed Windings
Converting discrete to continuous
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

13. 8.1 Describing Distributed Windings
Continuous to discrete
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

14. 8.1 Describing Distributed Windings
End conductors
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

15. 8.1 Describing Distributed Windings
End conductors
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

16. 8.1 Describing Distributed Windings
Winding arrangements
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

17. 8.1 Describing Distributed Windings
Winding arrangements
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

18. 8.2 Winding Functions Winding functions
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

19. 8.2 Winding Functions Calculating the discrete winding function
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

20. 8.2 Winding Functions Derivation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

21. 8.2 Winding Functions Continuous winding function
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

22. 8.2 Winding Functions Derivation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

23. 8.2 Winding Functions Derivation (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

24. 8.2 Winding Functions
Example 8.2A (part 1). Finding winding function for
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

25. 8.2 Winding Functions Example 8.2A (part 2). Find winding function for
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

26. 8.2 Winding Functions Continued…
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

27. 8.2 Winding Functions Continued some more …
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

28. 8.2 Winding Functions Comparison of winding functions
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

29. 8.3 Air Gap Magneto Motive Force
Consider Ampere’s law on path below
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

30. 8.3 Air Gap Magneto Motive Force
MMF drop definitions
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

31. 8.3 Air Gap Magneto Motive Force
Defining stator, rotor, and total MMF source as
It can be shown that
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

32. 8.3 Air Gap Magneto Motive Force
Derivation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

33. 8.3 Air Gap Magneto Motive Force
Derivation (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

34. 8.3 Air Gap Magneto Motive Force
Derivation (continued some more)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

35. 8.3 Air Gap Magneto Motive Force
Derivation (and some more)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

36. 8.3 Air Gap Magneto Motive Force
A simpler result and airgap fields
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

37. 8.4 Rotating MMF
Suppose
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

38. 8.4 Rotating MMF
Then
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

39. 8.4 Rotating MMF Now suppose The stator MMF is given by
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

40. 8.4 Rotating MMF This reduces to Meaning
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

41. 8.4 Rotating MMF Meaning (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

42. 8.5 Flux Linkage and Inductance
Leakage and magnetizing flux linkage
Leakage and magnetizing inductance
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

43. 8.5 Flux Linkage and Inductance
Calculating magnetizing flux linkage
Calculating magnetizing inductance
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

44. 8.5 Flux Linkage and Inductance
Derivation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

45. 8.5 Flux Linkage and Inductance
Derivation (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

46. 8.5 Flux Linkage and Inductance
Derivation (still continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

47. 8.6 Slot Effects and Carter’s Coefficient
Carter’s coefficient specifies an effective airgap to account for slot effects
For slotted stator
For slotted stator and rotor
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

48. 8.6 Slot Effects and Carter’s Coefficient
Derivation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

49. 8.6 Slot Effects and Carter’s Coefficient
Derivation (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

50. 8.6 Slot Effects and Carter’s Coefficient
Derivation (continued again)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

51. 8.6 Slot Effects and Carter’s Coefficient
Derivation (and again)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

52. 8.7 Leakage Inductance Leakage inductance matrix
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

53. 8.7 Leakage Inductance Slot leakage
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

54. 8.7 Leakage Inductance End leakage
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

55. 8.7 Leakage Inductance Total leakage inductance
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

56. 8.7 Leakage Inductance Slot leakage paths 1-4
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

57. 8.7 Leakage Inductance Derivation of Ps1,1
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

58. 8.7 Leakage Inductance Derivation of Ps1,1
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

59. 8.7 Leakage Inductance Slot leakage paths 5-7
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

60. 8.7 Leakage Inductance Derivation of Psl,5
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

61. 8.7 Leakage Inductance Derivation of Psl,5
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

62. 8.7 Leakage Inductance End leakage permeance
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

63. 8.7 Leakage Inductance End leakage permeance Pel,1
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

64. 8.7 Leakage Inductance End leakage permeance Pel,2
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

65. 8.8 Resistance Volume of conductor in slots:
Volume of conductor in end-turns:
Total volume:
Total length:
Resistance:
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

66. 8.9 Introduction to Reference Frame Theory
Park’s transformation to the rotor reference frame
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

67. 8.9 Introduction to Reference Frame Theory
Park’s transformation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

68. 8.9 Introduction to Reference Frame Theory
Transformation of a balanced set
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

69. 8.9 Introduction to Reference Frame Theory
Transformation of voltage equations. Suppose we have
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

70. 8.9 Introduction to Reference Frame Theory
Transformation of voltage equations. In qd0 variables, we have
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

71. 8.9 Introduction to Reference Frame Theory
Derivation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

72. 8.9 Introduction to Reference Frame Theory
Derivation (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

73. 8.9 Introduction to Reference Frame Theory
Derivation (continued again)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

74. 8.9 Introduction to Reference Frame Theory
Transformation of flux linkage equations. Suppose we have
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

75. 8.9 Introduction to Reference Frame Theory
Transformation of flux linkage equations. In qd0 variables we obtain
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

76. 8.9 Introduction to Reference Frame Theory
Partial derivation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

77. 8.9 Introduction to Reference Frame Theory
Partial derivation (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

78. 8.9 Introduction to Reference Frame Theory
Transformation of power. In abc variables we have
In qd0 variables power may be expressed
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

79. 8.9 Introduction to Reference Frame Theory
Derivation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

80. 8.10 Expressions for Torque
Using an energy based approach it can be shown that
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

81. 8.10 Expressions for Torque
Derivation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

82. 8.10 Expressions for Torque
Derivation (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

83. 8.10 Expressions for Torque
Derivation (continued some more)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

84. 8.10 Expressions for Torque
A field approach to calculating torque. Using field analysis, it can be shown that
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

85. 8.10 Expressions for Torque
Derivation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

86. 8.10 Expressions for Torque
Derivation (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

87. 8.10 Expressions for Torque
Derivation (continued some more)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach