1. Power Magnetic Devices: A Multi-Objective Design Approach
Chapter 2 Magnetics and Magnetic Equivalent Circuits
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

2. 2.1 Ampere’s Law, MMF, Kirchoff’s MMF Law
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

3. 2.1 Ampere’s Law, MMF, Kirchoff’s MMF Law
Enclosed current and MMF sources
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

4. 2.1 Ampere’s Law, MMF, Kirchoff’s MMF Law
MMF drops
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

5. 2.1 Ampere’s Law, MMF, Kirchoff’s MMF Law
We have
Thus
This is Kirchoff’s MMF Law: The sum of the MMF drops around a closed loop is equal to the sum of the MMF sources for that loop
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

6. 2.1 Ampere’s Law, MMF, Kirchoff’s MMF Law
Example 2.1A. Suppose
what is the MMF drop going from point (1,1) directly to point (5,2)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

7. 2.1 Ampere’s Law, MMF, Kirchoff’s MMF Law
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

8. 2.2 Magnetic Flux Flux through open surface Sm
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

9. 2.2 Magnetic Flux Example 2.2. Find the flux through the surface if
and
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

10. 2.2 Magnetic Flux Example 2.2 (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

11. 2.2 Magnetic Flux Example 2.2 (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

12. 2.2 Magnetic Flux Gauss’s Law Thus And
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

13. 2.2 Magnetic Flux
Kirchoff’s flux law: The sum of the fluxes into (or out of) any node must be zero
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

14. 2.3 Magnetically Conductive Materials
Diamagnetic materials
Examples include hydrogen, helium, copper, gold, and silicon
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

15. 2.3 Magnetically Conductive Materials
Paramagnetic materials
Examples include oxygen and tungsten
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

16. 2.3 Magnetically Conductive Materials
Ferromagnetic, anti-ferromagnetic, ferrimagnetic materials
Ferromagnetic materials include iron, nickel, cobalt
Ferrimagnetic materials include iron oxide magnetite (Fe3O4), nickel-zinc ferrite (Ni1/2,Zn12Fe2O4), and nickel ferrite (NiFe2O4).
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

17. 2.3 Magnetically Conductive Materials
B-H characteristic of ferromagnetic and ferrimagnetic materials
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

18. 2.3 Magnetically Conductive Materials
In free space
In a magnetic material (anhysteretic) OR
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

19. 2.3 Magnetically Conductive Materials
Relative permeability
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

20. 2.3 Magnetically Conductive Materials
Nonlinear anhysteretic relationships
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

21. 2.3 Magnetically Conductive Materials
Consider M47 silicon steel
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

22. 2.3 Magnetically Conductive Materials
Permeability functions for M47 silicon steel
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

23. 2.3 Magnetically Conductive Materials
Permeability as a function of H
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

24. 2.3 Magnetically Conductive Materials
Permeability as a function of H
Derivative of permeability as function of H
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

25. 2.3 Magnetically Conductive Materials
Derivative of permeability as a function of H
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

26. 2.3 Magnetically Conductive Materials
Permeability as a function of B
Let
r(B) for M47 silicon steel sample
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

27. 2.3 Magnetically Conductive Materials
Permeability in terms of flux density to magnetization ratio
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

28. 2.3 Magnetically Conductive Materials
Representing the flux density to magnetization ratio
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

29. 2.3 Magnetically Conductive Materials
Now consider g(B). It looks like
Thus
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

30. 2.3 Magnetically Conductive Materials
Integrating, we have
Also
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

31. 2.3 Magnetically Conductive Materials
Derivative of permeability
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

32. 2.3 Magnetically Conductive Materials
Ohm’s “Law”
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

33. 2.3 Magnetically Conductive Materials
Ohm’s ‘Law’ (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

34. 2.3 Magnetically Conductive Materials
Ohm’s ‘Law’ (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

35. 2.3 Magnetically Conductive Materials
Ohm’s ‘Law’ summarized
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

36. 2.4 Construction of the MEC
Let’s consider a UI core inductor
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

37. 2.4 Construction of the MEC
Nodes and a possible MEC
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

38. 2.4 Construction of the MEC
Reluctances
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

39. 2.4 Construction of the MEC
Simplified MEC
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

40. 2.4 Construction of the MEC
Example 2.4A. Suppose wi = 25.1 mm, wb = we=25.3 mm, ws=51.2 mm, ds=31.7 mm, lc = mm, g = 1.00 mm, ww = 38.1 mm and dw=31.7 mm. There are 35 turns with a current of 25 A. The relative permeabilty is Compute the flux and flux density in the I-core.
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

41. 2.4 Construction of the MEC
Example 2.4A (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

42. 2.4 Construction of the MEC
Example 2.4A (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

43. 2.5 Translation to Electric Circuits
Flux Linkage
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

44. 2.5 Translation to Electric Circuits
Inductance
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

45. 2.5 Translation to Electric Circuits
Consider our simplified MEC
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

46. 2.5 Translation to Electric Circuits
Faraday’s law
If inductance is constant
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

47. 2.5 Translation to Electric Circuits
Example 2.5A. Consider our UI core. Suppose the material is non-linear (permeability parameters given in Table 2.5A1). Predict the l-i characteristic.
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

48. 2.5 Translation to Electric Circuits
Example 2.5A (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

49. 2.5 Translation to Electric Circuits
Example 2.5A (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

50. 2.5 Translation to Electric Circuits
Example 2.5B. Measure the l-i characteristic. Idea and procedure.
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

51. 2.5 Translation to Electric Circuits
Example 2.5B (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

52. 2.5 Translation to Electric Circuits
Example 2.5B (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

53. 2.6 Fringing Flux Fringing flux paths
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

54. 2.6 Fringing Flux Field components at an air core interface
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

55. 2.6 Fringing Flux Normal component
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

56. 2.6 Fringing Flux Tangential component Conclusion:
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

57. 2.6 Fringing Flux Consider geometries (a) and (b)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

58. 2.6 Fringing Flux Fringing permeance for (a)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

59. 2.6 Fringing Flux Fringing permeance for (a) (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

60. 2.6 Fringing Flux Fringing permeance for (b)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

61. 2.6 Fringing Flux Fringing permeance for (b)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

62. 2.6 Fringing Flux For our UI core geometry
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

63. 2.6 Fringing Flux Example 2.6A (repeating Example 2.5A)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

64. 2.7 Leakage Flux Leakage flux paths in a UI core inductor
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

65. 2.7 Leakage Flux
Energy stored in a magnetic linear system in terms of permeance:
Energy stored in a magnetically linear system in terms of fields:
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

66. 2.7 Leakage Flux Energy in terms of permeance
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

67. 2.7 Leakage Flux Energy in terms of permeance (cont)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

68. 2.7 Leakage Flux Energy in terms of field
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

69. 2.7 Leakage Flux Energy in terms of field (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

70. 2.7 Leakage Flux Energy in terms of field (continued again)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

71. 2.7 Leakage Flux Horizontal slot leakage
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

72. 2.7 Leakage Flux Horizontal slot leakage (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

73. 2.7 Leakage Flux Horizontal slot leakage (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

74. 2.7 Leakage Flux
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

75. 2.7 Leakage Flux Other expressions for horizontal slot leakage
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

76. 2.7 Leakage Flux
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

77. 2.7 Leakage Flux Vertical slot leakage
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

78. 2.7 Leakage Flux Vertical slot leakage (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

79. 2.7 Leakage Flux Vertical slot leakage (continued again)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

80. 2.7 Leakage Flux Vertical slot leakage (continued yet again)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

81. 2.7 Leakage Flux Exterior adjacent leakage flux (approach 1)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

82. 2.7 Leakage Flux Exterior adjacent leakage flux (approach 2)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

83. 2.7 Leakage Flux Exterior term Interior term
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

84. 2.7 Leakage Flux Finding the exterior term
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

85. 2.7 Leakage Flux Finding the exterior term (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

86. 2.7 Leakage Flux Finding the interior term
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

87. 2.7 Leakage Flux Alternate orientation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

88. 2.7 Leakage Flux Finding the interior term (continued) This gets us to
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

89. 2.7 Leakage Flux Exterior isolated leakage flux
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

90. 2.7 Leakage Flux Exterior isolated leakage flux
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

91. 2.7 Leakage Flux Incorporating leakage flux into UI MEC
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

92. 2.7 Leakage Flux
Example 2.8D (Example 2.6A with leakage and fringing flux)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

93. 2.8 Numerical Solution of Nonlinear MECs
Standard branch
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

94. 2.8 Numerical Solution of Nonlinear MECs
Goal of nodal analysis
Basis of j’th row
Next
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

95. 2.8 Numerical Solution of Nonlinear MECs
Example
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

96. 2.8 Numerical Solution of Nonlinear MECs
Nodal analysis – case where branch is in positive node list
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

97. 2.8 Numerical Solution of Nonlinear MECs
Nodal analysis – case where branch is in positive node list (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

98. 2.8 Numerical Solution of Nonlinear MECs
Nodal analysis – case where branch is in negative node list
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

99. 2.8 Numerical Solution of Nonlinear MECs
Nodal analysis – case where branch is in negative node list (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

100. 2.8 Numerical Solution of Nonlinear MECs
Nodal analysis formulation algorithm (NAFA)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

101. 2.8 Numerical Solution of Nonlinear MECs
Example 2.8A
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

102. 2.8 Numerical Solution of Nonlinear MECs
Example 2.8A (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

103. 2.8 Numerical Solution of Nonlinear MECs
Example 2.8A (continued again)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

104. 2.8 Numerical Solution of Nonlinear MECs
Example 2.8A (continued yet again)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

105. 2.8 Numerical Solution of Nonlinear MECs
Goal of mesh analysis
Basis of j’th row
Flux going into i’th branch
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

106. 2.8 Numerical Solution of Nonlinear MECs
Example
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

107. 2.8 Numerical Solution of Nonlinear MECs
Example (Continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

108. 2.8 Numerical Solution of Nonlinear MECs
Thus
Suppose k’th branch is in Lp,j
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

109. 2.8 Numerical Solution of Nonlinear MECs
Mesh analysis formulation algorithm (MAFA)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

110. 2.8 Numerical Solution of Nonlinear MECs
Example 2.8B
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

111. 2.8 Numerical Solution of Nonlinear MECs
Example 2.8B (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

112. 2.8 Numerical Solution of Nonlinear MECs
Example 2.8B (continued again)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

113. 2.8 Numerical Solution of Nonlinear MECs
Example 2.8B (still continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

114. 2.8 Numerical Solution of Nonlinear MECs
Example 2.8C. Comparison of mesh and nodal analysis for nonlinear case
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

115. 2.8 Numerical Solution of Nonlinear MECs
Permeability as a function of H – given H find B then
Permeability as a function of B. Given B we can show that the following yields H whereupon permeability may be found
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

116. 2.8 Numerical Solution of Nonlinear MECs
Nodal analysis and formation of residual
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

117. 2.8 Numerical Solution of Nonlinear MECs
Nodal analysis and formation of residual (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

118. 2.8 Numerical Solution of Nonlinear MECs
Iteration of nodal equation using Newton’s method
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

119. 2.8 Numerical Solution of Nonlinear MECs
Mesh analysis and formation of residual
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

120. 2.8 Numerical Solution of Nonlinear MECs
Mesh analysis and formation of residual (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

121. 2.8 Numerical Solution of Nonlinear MECs
Iteration of mesh equation using Newton’s method
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

122. 2.8 Numerical Solution of Nonlinear MECs
Solving systems of nonlinear equations
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

123. 2.8 Numerical Solution of Nonlinear MECs
Approach for MEC analysis will be to linearize each branch equation and apply MAFA
Linearized branch equation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

124. 2.8 Numerical Solution of Nonlinear MECs
Derivation of linearized branch equation
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

125. 2.8 Numerical Solution of Nonlinear MECs
Derivation of linearized branch equation (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

126. 2.8 Numerical Solution of Nonlinear MECs
Derivation of linearized branch equation (cont. 2)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

127. 2.8 Numerical Solution of Nonlinear MECs
Derivation of linearized branch equation (cont. 3)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

128. 2.8 Numerical Solution of Nonlinear MECs
Error criteria
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

129. 2.8 Numerical Solution of Nonlinear MECs
Nonlinear MEC solution algorithm
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

130. 2.9 Permanent magnet materials
B-H and M-H characteristics
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

131. 2.9 Permanent magnet materials
To avoid demagnetization
H > Hlim (positively magnetized)
H < -Hlim (negatively magnetized)
T < Tcur
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

132. 2.9 Permanent magnet materials
Common materials
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

133. 2.9 Permanent magnet materials
MEC representation of PM materials
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

134. 2.9 Permanent magnet materials
Derivation of MEC PM model
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

135. 2.9 Permanent magnet materials
Example 2.9A. Derive an expression for the field intensity in the magnet
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

136. 2.9 Permanent magnet materials
Example 2.9A
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach

137. 2.9 Permanent magnet materials
Example 2.9A (continued)
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach