1. Power Magnetic Devices: A Multi-Objective Design Approach
Chapter 10: Introduction to Thermal Equivalent Circuits

2. 10.1 Heat Thermal energy density and thermal energy
Notation: spatial average

3. 10.1 Heat
Mean temperature and thermal capacitance of a region

4. 10.1 Heat Heat flux and Fourier’s Law
Note: comparison of heat flux and magnetic flux density

5. 10.1 Heat
Heat transfer rate
Heat transfer

6. 10.1 Heat
The heat equation

7. 10.1 Heat
Derivation of heat equation

8. 10.1 Heat
Derivation of heat equation (continued 1/4)

9. 10.1 Heat
Derivation of heat equation (continued 2/4)

10. 10.1 Heat
Derivation of heat equation (continued 3/4)

11. 10.1 Heat
Derivation of heat equation (continued 4/4)

12. 10.2 TEC of One-Dimensional Heat Flow
Consider a region W
For steady state conditions, we can show

13. 10.2 TEC of One-Dimensional Heat Flow
Derivation

14. 10.2 TEC of One-Dimensional Heat Flow
Derivation (continued)

15. 10.2 TEC of One-Dimensional Heat Flow
Mean temperature
Derivation

16. 10.2 TEC of One-Dimensional Heat Flow
Derivation (continued)

17. 10.2 TEC of One-Dimensional Heat Flow
Boundary conditions
Derivation

18. 10.2 TEC of One-Dimensional Heat Flow
Derivation (continued)

19. 10.2 TEC of One-Dimensional Heat Flow
Static TEC

20. 10.2 TEC of One-Dimensional Heat Flow
Derivation (1/3)

21. 10.2 TEC of One-Dimensional Heat Flow
Derivation (2/3)

22. 10.2 TEC of One-Dimensional Heat Flow
Derivation (3/3)

23. 10.2 TEC of One-Dimensional Heat Flow
Dynamic portion of TEC
Derivation

24. 10.2 TEC of One-Dimensional Heat Flow
Derivation (continued)

25. 10.2 TEC of One-Dimensional Heat Flow
Example 10.2B. Consider a bar as follows:
Dimensions: 10 cm (length) by 2 cm by 2 cm
Specific heat capacity: 469 J/kg·K
Mass density 7500 kg/m3
Thermal conductivity: 15.7 W/m·K
Ends of bar at 26 oC
Power dissipation in the bar is 10 W
Compare transient response of TEC and direct solution of PDF

26. 10.2 TEC of One-Dimensional Heat Flow
Mean temperature versus time

27. 10.2 TEC of One-Dimensional Heat Flow
Heat transfer rate versus time

28. 10.2 TEC of One-Dimensional Heat Flow
Temperature profile versus time

29. 10.2 TEC of One-Dimensional Heat Flow
Peak temperature in region

30. 10.2 TEC of One-Dimensional Heat Flow
Derivation

31. 10.2 TEC of One-Dimensional Heat Flow
Example 10.2B. Let us find the peak temperature for steady state conditions for Example 10.2A. We obtain:
TWcx: 63.4 oC
< TW >: 38.5 oC
TW,pk: 44.7 oC
Important -

32. 10.2 TEC of One-Dimensional Heat Flow
TEC for one dimensional element w/o heat production or substantial mass

33. 10.3 Thermal Equivalent Circuit of Cuboid
Assumptions for cuboid region
Assumed temperature distribution

34. 10.3 Thermal Equivalent Circuit of Cuboid
Mean temperatures

35. 10.3 Thermal Equivalent Circuit of Cuboid
Terms of interest

36. 10.3 Thermal Equivalent Circuit of Cuboid
Derivation

37. 10.3 Thermal Equivalent Circuit of Cuboid
Static TEC

38. 10.3 Thermal Equivalent Circuit of Cuboid
Derivation (1/3)

39. 10.3 Thermal Equivalent Circuit of Cuboid
Derivation (2/3)

40. 10.3 Thermal Equivalent Circuit of Cuboid
Derivation (3/3)

41. 10.3 Thermal Equivalent Circuit of Cuboid
Dynamic portion of TEC

42. 10.3 Thermal Equivalent Circuit of Cuboid
Derivation

43. 10.3 Thermal Equivalent Circuit of Cuboid
Peak temperature
Derivation

44. 10.3 Thermal Equivalent Circuit of Cuboid
Derivation

45. 10.4 Thermal Equivalent Circuit of Cyld. Region
Consider a cylindrical region

46. 10.4 Thermal Equivalent Circuit of Cyld. Region
Conversion of heat equation to cylindrical coordinates

47. 10.4 Thermal Equivalent Circuit of Cyld. Region
Neglecting tangential heat flow
Assumed temperature profile

48. 10.4 Thermal Equivalent Circuit of Cyld. Region
Radial portion of heat equation. We can show that
Derivation

49. 10.4 Thermal Equivalent Circuit of Cyld. Region
Derivation (continued)

50. 10.4 Thermal Equivalent Circuit of Cyld. Region
Spatial average
Mean temperature

51. 10.4 Thermal Equivalent Circuit of Cyld. Region
Axial heat flow

52. 10.4 Thermal Equivalent Circuit of Cyld. Region
Radial heat flow. First, we can show

53. 10.4 Thermal Equivalent Circuit of Cyld. Region
Derivation

54. 10.4 Thermal Equivalent Circuit of Cyld. Region
We can also show

55. 10.4 Thermal Equivalent Circuit of Cyld. Region
Derivation

56. 10.4 Thermal Equivalent Circuit of Cyld. Region
Finally we obtain

57. 10.4 Thermal Equivalent Circuit of Cyld. Region
Derivation (1/3)

58. 10.4 Thermal Equivalent Circuit of Cyld. Region
Derivation (2/3)

59. 10.4 Thermal Equivalent Circuit of Cyld. Region
Derivation (3/3)

60. 10.4 Thermal Equivalent Circuit of Cyld. Region
Dynamic portion of TEC of cylindrical region

61. 10.4 Thermal Equivalent Circuit of Cyld. Region
Derivation

62. 10.4 Thermal Equivalent Circuit of Cyld. Region
Final TEC of cylindrical region

63. 10.4 Thermal Equivalent Circuit of Cyld. Region
Peak temperature

64. 10.4 Thermal Equivalent Circuit of Cyld. Region
Derivation (1/2)

65. 10.4 Thermal Equivalent Circuit of Cyld. Region
Derivation (2/2)

66. 10.5 Inhomogeneous Regions
It is often the case that we wish to develop a homogenized representation of a material

67. 10.5 Inhomogeneous Regions
Effective thermal properties in xy-direction
Effective properties in z-direction
Effective density and heat capacity

68. 10.5 Inhomogeneous Regions
Derivation in xy-direction (1/3)

69. 10.5 Inhomogeneous Regions
Derivation in xy-direction (2/3)

70. 10.5 Inhomogeneous Regions
Derivation in xy-direction (3/3)

71. 10.5 Inhomogeneous Regions
Derivation in z-direction (1/2)

72. 10.5 Inhomogeneous Regions
Derivation in z-direction (2/2)

73. 10.5 Inhomogeneous Regions
Derivation of effective density

74. 10.5 Inhomogeneous Regions
Derivation of effective heat capacity

75. 10.5 Inhomogeneous Regions
Example 10.5A. Let us compute the peak temperature in a slot of conductors.
Slot is 2.4 cm wide, 2.8 cm deep, 7.1 cm long.
Slot contains 50 conductors of 11 gauge copper wire with radius of 1.15 mm and insulation of 34.3 mm.
Current density is 7.5 A/mm2.
Thermal conductivity of copper, insulation, and air are 400 W/K·m, W/K·m and W/K·m.
Walls of slot are 50 oC, ends of slot are 52 oC, and top of slot is 53 oC.

76. 10.5 Inhomogeneous Regions
Example 10.5A continued (1/2)

77. 10.5 Inhomogeneous Regions
Example 10.5A continued (2/2)

78. 10.5 Inhomogeneous Regions
Example 10.5A continued (2/2)

79. 10.5 Inhomogeneous Regions
Example 10.5A temperature profile

80. 10.6 Material Boundaries Contact resistance
For steel to steel: hcv ~ 1 kW/K·m2 at 5 MPa

81. 10.6 Material Boundaries Convective heat transfer
Natural convection in air: hcv ~ 2-10 W/k·m2
Natural convection in water: hcf: ~ 200 W/k·m2

82. 10.6 Material Boundaries Radiation emission from object
Radiation absorption by object
Stefan-Boltzmann constant s: 56.7 nW/K4·m2
Emissivity es is between 0 and 1

83. 10.6 Material Boundaries
Net power due to radiation

84. 10.7 Thermal Equivalent Circuit Networks
Thermal equivalent circuit laws: The sum of the changes in temperature around a closed loop must be zero

85. 10.7 Thermal Equivalent Circuit Networks
Thermal equivalent circuit laws: the sum of the heat transfer rates into a node must be zero
Mathematically constructed nodes
Physical nodes

86. 10.7 Thermal Equivalent Circuit Networks
Standard branch

87. 10.7 Thermal Equivalent Circuit Networks
Origin of dependent source

88. 10.7 Thermal Equivalent Circuit Networks
Nodal analysis forms system of equations of form
Reasons to use nodal analysis (instead of mesh)

89. 10.7 Thermal Equivalent Circuit Networks
Nodal analysis formulation algorithm

90. 10.7 Thermal Equivalent Circuit Networks
Derivation (1/3)

91. 10.7 Thermal Equivalent Circuit Networks
Derivation (2/3)

92. 10.7 Thermal Equivalent Circuit Networks
Derivation (3/3)

93. 10.7 Thermal Equivalent Circuit Networks
Graphical shorthand for describing elements

94. 10.7 Thermal Equivalent Circuit Networks
More graphical shorthand

95. 10.8 Case Study: Electromagnet
Cuboids of electromagnet

96. 10.8 Case Study: Electromagnet
Representing a rounded corner

97. 10.8 Case Study: Electromagnet
Winding to core resistance

98. 10.8 Case Study: Electromagnet
Derivation (1/2)

99. 10.8 Case Study: Electromagnet
Derivation (2/2)

100. 10.8 Case Study: Electromagnet
Airgap thermal resistance

101. 10.8 Case Study: Electromagnet
Thermal equivalent circuit

102. 10.8 Case Study: Electromagnet
Thermal resistances to ambient

103. 10.8 Case Study: Electromagnet
Electro-thermal analysis (1/2)

104. 10.8 Case Study: Electromagnet
Electro-thermal analysis (2/2)

105. 10.8 Case Study: Electromagnet
Solution algorithm

106. 10.8 Case Study: Electromagnet
Solution algorithm (continued)

107. 10.8 Case Study: Electromagnet
Example 10.8A. Revisit electromagnet design

108. 10.8 Case Study: Electromagnet

109. 10.8 Case Study: Electromagnet
Perform thermal analysis for Design of Section 5.4.

110. 10.8 Case Study: Electromagnet
Thermal data

111. 10.8 Case Study: Electromagnet
Convergence: e = 150 K, 37.1 K, 6.42 K, 1.16 K, K, 37.5 mK, and 6.7 mK
Maximum node temperature: 417 K or 144 oC
Maximum device temp (element K): 379 K or 106oC
Maximum surface temp: 338 K or 65 oC

112. 10.8 Case Study: Electromagnet
Impact of temperature rise
Coil resistance increases from 4.82 W to 5.99 W
Coil current drops from 2.49 A to 2.00 A

113. 10.8 Case Study: Electromagnet
Example 10.8B. Revise design process to include thermal analysis
Steps:
Remove current density constraint
Add peak temperature constraint on winding

114. 10.8 Case Study: Electromagnet
Recall
Constraints
1. Conductors fit
2. Packing factor
3. Current density
4. Aspect ratio
5. Volume
6. Loss
7. Force
Metrics
Volume and loss

115. 10.8 Case Study: Electromagnet
Pseudo-code

116. 10.8 Case Study: Electromagnet
Pseudo-code (continued)

117. 10.8 Case Study: Electromagnet
Pareto-optimal front

118. 10.8 Case Study: Electromagnet
Gene distribution

119. 10.8 Case Study: Electromagnet
Current density

120. 10.8 Case Study: Electromagnet
Conductor counts

121. 10.8 Case Study: Electromagnet
Widths

122. 10.8 Case Study: Electromagnet
Assorted dimensions

123. 10.8 Case Study: Electromagnet
Peak winding temperature

124. 10.8 Case Study: Electromagnet
Design 65 (with thermal analysis)
Design 250 (without thermal analysis)

125. 10.8 Case Study: Electromagnet
Improvements