2 3 4 2 3 4 2 3 4

2

3

4

1.1INTRODUCTION ... 9

1.2PRESENT AND FUTURE OF EV’S ... 10

1.3ENGINEER PHILOSOPHY ... 11

1.4FINAL CONSIDERATION ... 13

2.1INTRODUCTION ... 15

2.2ALGEBRAIC EQUATIONS,ORDINARY DIFFERENTIAL EQUATIONS,PARTIAL DIFFERENTIAL EQUATIONS, AND THE LAWS OF PHYSICS ... 18

2.3THE FINITE ELEMENT METHOD FROM THE WEAK FORMULATION:BASIS FUNCTIONS AND TEST FUNCTIONS ... 21

2.4TIME-DEPENDENT PROBLEMS ... 28

2.5DIFFERENT ELEMENTS ... 30

2.6ERROR ESTIMATION ... 31

2.7MESH CONVERGENCE ... 32

3.1INTRODUCTION ... 35

3.2MAGNETIC MATERIALS ... 35

3.3SOFT MAGNETIC MATERIALS... 36

3.4HARD MAGNETIC MATERIALS ... 44

3.5CHARACTERISTICS COMPARISON OF SOFT AND HARD MAGNETIC MATERIALS ... 48

3.6JILES-ATHERTON MODEL ... 49

3.7CONDUCTORS ... 51

3.8MODEL MATERIALS ... 54

4.1INTRODUCTION ... 57

4.2KINETIC ENERGY HARVESTING ... 58

4.3TRANSDUCTOR GENERATOR ... 66

4.4ENERGY HARVESTING BRAKES ... 72

4.5ENERGY HARVESTING DAMPERS ... 75

4.6LINEAR MACHINE DESIGN ... 78

4.7ECDOVERVIEW ... 83

4.8FINAL CONSIDERATION ... 83

5.1INTRODUCTION ... 85

5.2EDDY CURRENT SPRING-DAMPER DESIGN... 85

5.3EDDY CURRENT DAMPER DESIGN ... 92

5.4EDDY CURRENT REGENERATIVE DAMPER ... 97

5.5FINAL CONSIDERATIONS ... 100

6.1INTRODUCTION ... 102

6.2PROJECT ... 102

6.7RESULTS ... 108

6.8OPTIMIZED MODEL SIMULATION ... 116

5

7.1FUTURE WORKS ... 122

A.1AIR POLLUTION ... 126

A.2GLOBAL WARMING ... 126

B.1MAXWELL’S EQUATIONS ... 128

B.2POTENTIALS ... 129

B.3MAGNETIC FIELD ANALYSIS ... 131

B.4THE CURRENT MODEL (MAGNETIC FLUX CALCULATION) ... 132

C.1INTRODUCTION ... 134

C.2THE B-HCURVE,PERMEABILITY, AND DIFFERENTIAL PERMEABILITY ... 134

C.3HOW THE SMOOTHNESS OF THE B-HCURVE AFFECTS THE SIMULATION ... 137

6 𝐻

𝐵⃗

𝜇

𝜇

𝜇

𝑀

𝐵⃗

𝐵⃗

𝜒

𝐵⃗

𝐵⃗

𝐻

7 𝐵⃗𝐻

𝐵⃗

𝐵⃗

𝐵⃗

𝜀

𝜙

𝐸

𝐸

𝐽

𝜎

𝑣

𝐹

𝜏

𝜏

8

𝑜

𝐸 𝑘

𝐾 𝑘

𝛿

𝜔

𝜌

𝜔

9

Figure 1: Growth of population and vehicles ¹

10

11

Figure 2: Development trends of EV's and HEV's (courtesy of EVAA)

12

13

Figure 3: Interactions among EV subsystems ¹

14

15 𝑢 ≈ 𝑢

𝑢

= ∑ 𝑢

𝜑

𝑖

16

Figure 4: The function u (solid blue line) is approximated with uh (dashed red line), which is a linear combination of linear basis functions (ψi is represented by the solid black lines). The coefficients are denoted by u0 through u7 ⁴

17

Figure 5: The function u (solid blue line) is approximated with uh (dashed red line), which is a linear combination of linear basis functions (ψi is represented by the solid black lines). The coefficients are denoted by u0 through u7.⁴

18

Figure 6: Finite element discretization, stresses, and deformations of a wheel rim in a structural analysis.⁴

partial differential equations

19 ρ𝐶

𝑑𝑇

𝑑𝑡 = 𝑔(𝑇, 𝑡)

𝑇 = 𝑓(𝑡)

4

ρ𝐶

𝛿𝑇

𝛿𝑡 + ∇ ∙ 𝒒 = 𝑔(𝑇, 𝑡, 𝑥)

20

∇ ∙ 𝒒 = 𝜕𝑞

𝜕𝑥 + 𝜕𝑞

𝜕𝑦 + 𝜕𝑞

𝜕𝑧

𝑞 = −𝑘∇𝑇 → 𝑞 = (−𝑘 𝜕𝑇

𝜕𝑥 , −𝑘 𝜕𝑇

𝜕𝑦 − 𝑘 𝜕𝑇

𝜕𝑧 )

𝜌𝐶

𝛿𝑇

𝛿𝑡 + ∇ ∙ (−𝑘∇𝑇) = 𝑔(𝑇, 𝑡, 𝑥)

21

𝑇 = 𝑓(𝑡, 𝑥)

22

∇ ∙ (−𝑘∇𝑇) = 𝑔(𝑇, 𝑥) 𝑖𝑛 Ω

𝑇 = 𝑇

𝑜𝑛 δΩ

(−𝑘∇𝑇) ∙ 𝒏 = ℎ(𝑇 − 𝑇

) 𝑜𝑛 δΩ

(−𝑘∇𝑇) ∙ 𝒏 = 0 𝑜𝑛 δΩ

Figure 7: The domain equation and boundary conditions for a mathematical model of a heat sink⁴

23

∫ ∇ ∙ (−𝑘∇𝑇)𝜑𝑑𝑉

Ω

= ∫ 𝑔𝜑𝑑𝑉

Ω

ϵ ϵ

24

∫ 𝑘∇𝑇 ∙ ∇𝜑𝑑𝑉 +

Ω

∫ (−𝑘∇𝑇) ∙ 𝒏𝜑𝑑𝑆 = ∫ g𝜑𝑑𝑉

Ω δΩ

25

𝑇

(𝑥) = ∑ 𝑇

𝜓

(𝑥)

𝑖

∑ 𝑇

∫ 𝑘∇𝜓

𝑖 Ω

∙ ∇𝜓

𝑑𝑉 + ∑ ∫ (−𝑘𝑇

∇𝜓

) ∙ 𝑛𝜓

𝑑𝑆 = ∫ 𝑔 (∑ 𝑇

𝜓

𝑖

)

Ω 𝑖 𝛿Ω

𝜓

Figure 8: Finite element discretization of the heat sink model from the earlier figure ⁴

26

𝐴𝑇

= 𝑏

27

Figure 9: Tent-shaped linear basis functions that have a value of 1 at the corresponding node and zero on all other nodes. Two base functions that share an element have a basis function overlap ⁴

28

Figure 10: Two basis functions that share one element vertex but do not overlap in a 2D domain ⁴

ψ

29 𝑇

δ𝑇

𝛿𝑡 ≈ 𝑇

− 𝑇

∆𝑡

𝑇

𝑡 + ∆𝑡

𝑇

𝑡 + ∆𝑡(𝑇

)

30

Figure 11: Node placement and geometry for 2D and 3D linear elements ⁴

31

Figure 12: Second-order elements. If the gray nodes are removed, we get the corresponding serendipity elements.

The black, white, and gray nodes are all present in the Lagrangian elements ⁴

𝑒 = 𝑢 − 𝑢

𝑂(ℎ

) 𝛼

32 𝑒 = 𝑢

− 𝑢

𝑢

𝑢

Figure 13: Benchmark model of an elliptic membrane. The load is applied at the outer edge while symmetry is assumed at the edges positioned along the x- and y-axis (roller support) ⁴

33

Figure 14: Rectangular elements used for the quadratic base functions ⁴

34

Figure 15: Relative error in σx at the evaluation point in the previous figure for different elements and element sizes (element size = h). Quad refers to rectangular elements, which can be linear or with quadratic basis functions ⁴

35

Figure 16: Magnetic domains before and after alignment with the external magnetic field ⁵

36 𝜇

= 𝜇

𝜇

𝜇 𝜇

= 4𝜋 × 10

𝑚

𝜇

𝐵⃗

𝐵⃗

= 𝐵⃗

+ 𝐵⃗

= 𝜇

𝐻 + 𝜇

𝑀 = 𝜇

(𝐻 + 𝑀)

𝐵⃗

𝐵⃗

37

𝐻

𝜇

< 1

Figure 17: Field lines distribution of diamagnetic materials placed in a uniform magnetic field H ⁶

𝜇

= 1

38

Figure 18: Field lines distribution of paramagnetic materials placed in a uniform magnetic field H ⁶

Figure 19: Field lines distribution of ferromagnetic materials placed in a uniform magnetic field H ⁶

𝜇

≫ 1

39

Figure 20: Ferromagnetic material in a magnetic field. Flux lines pass through the mate- rial and the material is attracted to the source of the field ⁵

40

Figure 21: Magnetic circuit used to obtain the magnetic characteristics of a ferromagnetic sample ⁵

𝐻 = 𝑁𝐼/𝑙 𝐵⃗ = Φ/𝑆 𝐼

Φ

𝐻 𝐵⃗ 𝐼 Φ

Figure 22: A typical B(H) for a ferromagnetic material ⁵

𝐵⃗(𝐻)

𝛷(𝐼)

41

Figure 23: Permeability curve ⁸

µ

42

Figure 24: Hysteresis Loop curve ⁸

43 𝐵⃗ ≠ 0

−𝐵⃗

Figure 25: Comparison between hysteresis loop of a hard material (a) and of a soft material (b) ⁷

44

𝐻

Figure 26: Demagnetization curve, recoil loop, energy of a PM, and recoil magnetic ¹¹

45

46 𝐵⃗

= 𝜇

𝐻

= √ 𝜇

𝜎

𝑘

𝑉

𝑉

𝐵⃗

𝐻

𝐵⃗

𝐻

𝐵⃗

= 𝐵⃗

[1 + 𝛼

100 (𝜗

− 20)]

𝐻

= 𝐻

[1 + 𝛼

100 (𝜗

− 20)]

𝜗

𝐵⃗

𝐻

𝛼

< 0 𝛼

< 0

𝐵⃗

𝐻

47

Figure 27: Descending branches of hysteresis loops determined at various temperatures ¹³

48

Figure 28: BH curve comparison on Soft and Hard magnetic materials ⁸

𝐵⃗

𝐻

49

50 𝑀

𝑀

𝑑𝑀

𝑑𝐻 = 𝑀

− 𝑀

𝑘𝛿 − 𝛼(𝑀

− 𝑀

)

𝑀

= 𝑐(𝑀

− 𝑀

)

𝑀

(𝐻) = 𝑀

[coth ( 𝐻 + 𝛼𝑀

𝛼 ) − 𝛼

𝐻 + 𝛼𝑀 ]

𝑎, 𝛼, 𝑐, 𝑘 𝑀

𝛿

±1 𝑓𝑜𝑟 𝑑𝐻/𝑑𝑡 ≷ 0

51

52

◦

◦

Table 3.1: Electric resistivity, conductivity, and temperature coefficient at 20◦C ¹²

° °

𝑅(𝜗)𝑅

[1 + 𝛼

(𝜃 − 20)]

𝜌(𝜗) = 𝜌

[1 + 𝛼

(𝜃 − 20)]

𝜎(𝜗) = 𝜎

1 + 𝛼

(𝜃 − 20)

𝑅

, 𝜌

, 𝜎

, 𝛼

◦

53

° β

𝑅(𝜗) = 𝑅

[1 + 𝛼

(𝜗 − 20) + 𝛽

(𝜗 − 20)

]

°

Figure 29: Variation of resistivity ρ of metals with temperature: 1,2,3| various mild steels, 4|stainless and acid resistant steels, 5|brass, 6|aluminium, 7|copper. ¹²

54

Figure 30: HIGH INDUCTION MAGNETIZATION CURVE FOR ARMCO® PURE IRON

55

Figure 31: BH-curve of NdFeB N-50H magnet ¹⁶

56

57

Figure 32: Energy consumption distribution of an ordinary automobile in the urban transport

58

59

Figure 33: Generic model of kinetic energy harvesters

𝑦(𝑡) =

𝑌 sin 𝜔𝑡,

60 𝑚 ∙ 𝑑

𝑧(𝑡)

𝑑𝑡

+ 𝑏 ∙ 𝑑𝑧(𝑡)

𝑑𝑡 + 𝑘 ∙ 𝑧(𝑡) = −𝑚 ∙ 𝑑

𝑦(𝑡) 𝑑𝑡

𝑚 ∙ 𝑠

∙ 𝑧(𝑠) + 𝑏 ∙ 𝑠 ∙ 𝑧(𝑠) + 𝑘 ∙ 𝑠 ∙ 𝑧(𝑠) = −𝑚 ∙ 𝑎(𝑠)

𝑎(𝑡) = 𝑑

𝑦(𝑡) 𝑑𝑡

𝑧(𝑠)

𝑎(𝑠) = 1

𝑠

+ 𝑏 𝑚 𝑠 +

𝑘 𝑚

= 1

𝑠

+ 𝜔

𝑄 𝑠 + 𝜔

𝑄 =

𝑏

𝜔

= √𝑘/𝑚

−𝑚 ∙ 𝑎(𝑠) = 𝑠 ∙ 𝑍(𝑠) (𝑚𝑠 + 𝑏 + 𝑘

𝑠 )

61

−𝐼(𝑠) = 𝐸(𝑠) (𝑠𝐶 + 1 𝑅 + 1

𝑠𝐿 )

𝐼(𝑠) = 𝑚 ∙ 𝑎(𝑠), 𝐸(𝑠) = 𝑠 · 𝑍(𝑠), 𝐶 = 𝑚, 𝑅 =

𝑏

, 𝐿 =

𝑘

Figure 34: Equivalent circuit of a kinetic energy harvester¹¹

𝜁

= 𝑏

2𝑚𝜔

= 𝑏

+ 𝑏

2𝑚𝜔

= 𝜁

+ 𝜁

𝜁

=

2𝑚𝜔𝑟

𝜁

=

2𝑚𝜔𝑟

𝑄

= 1

2𝜁

62 1

𝑄

= 1 𝑄

+ 1

𝑄

𝑄

=

2𝜁𝑚

𝑄

=

1 2𝜁𝑒

𝑦(𝑡) = sin 𝜔𝑡

𝑧(𝑡) = 𝑚𝜔

𝑌

𝑘 − 𝑚𝜔

+ 𝑗𝜔𝑏 ∙ sin 𝜔𝑡

𝑧(𝑡) = 𝜔

√(𝜔

− 𝜔

)

+ ( 𝑏𝜔 𝑚 )

2

∙ 𝑌 sin(𝜔𝑡 + 𝜑)

𝜑 = tan

( 𝑏𝜔

𝑘 − 𝜔

𝑚 )

63 𝑃 = 𝑏 ( 𝑑𝑧(𝑡)

𝑑𝑡 )

2

𝑃(𝜔) =

𝑚𝜁

𝑌

( 𝜔 𝜔

)

𝜔

[1 − ( 𝜔

𝜔

)

]

2

+ (2𝜁

𝜔 𝜔

)

𝜔 = 𝜔

𝑃 = 𝑚𝑌

𝜔

4𝜁

𝑃 = 𝑚𝑌

𝜔

4(𝜁

+ 𝜁

)

𝑃

𝑃

𝑃

𝑃

𝑃

= 𝜁

𝑚𝑌

𝜔

4(𝜁

+ 𝜁

)

64 𝑃

= 𝜁

𝑚𝑌

𝜔

4(𝜁

+ 𝜁

)

𝜁

= 𝜁

𝑃

= 𝑃

2 = 𝑚𝑌

𝜔

16𝜁

𝑎 = 𝑌𝜔

𝑄

=

2𝜁_𝑚

𝑃

= 𝑚𝑎

8𝜔

65

Figure 35: Power spectrum of a Kinetic energy harvester with various Q-factor

66

67

Figure 36: Piezoelectric generators: (a)33mode and(b) 31 mode

68

Figure 37: Electrostatic generators: (a) in-plane overlap; (b) in-plane gap closing; and (c) out-of-

plane gap closing

69

Figure 38: Electromagnetic generators

𝑒. 𝑚. 𝑓. = −𝑁 ∙ 𝑙 ∙ 𝐵⃗ ∙ 𝑑𝑧

𝑑𝑡

70 𝑒. 𝑚. 𝑓. = −𝑁 ∙ 𝑆 ∙ 𝑑𝐵⃗

𝑑𝑧 ∙ 𝑑𝑧 𝑑𝑡

𝑑𝐵⃗/𝑑𝑧

𝑒. 𝑚. 𝑓. = 𝑘 ∙ 𝑑𝑧 𝑑𝑡

−𝑁 · 𝑙 · 𝐵⃗ −𝑁 ∙ 𝑆 ∙

𝑑𝑧

71

Figure 39: Comparison of different transduction mechanism of kinetic energy harvesters

72

73

Figure 40: Total traction energy and energies consumed by drags and braking in an FTP 75 urban driving cycle

Figure 41: Maximum Speed, Average Speed, Total Traction Energy, and Energies Consumed by

Drags and Braking per 100 km Traveling Distance in Different Drive Cycles

74

75

Figure 42: A typical road vehicle front suspension system

76

77

78

79

Figure 43: Basic Geometric Arrangements of Linear Electric Machines. Left: Planar, Right: Tubular

80

Figure 44: Permanent-Magnet Linear Machine Configurations.

From left to right: Moving Coil, Moving Iron, Moving Magnet

𝐶

81

Figure 45: Different PM configurations in the mover. (a) Axially, (b) axially and radially,(c) radially (inward and outward), and (d) radially (outward) magnetized PMs in the mover ⁴⁵

82

Figure 46: Tubular Machine Slot-less (left) and Slotted (right) Stator Illustrations.

83

84

85

86

Figure 47: Schematic view of the proposed eddy current spring–damper

87

𝐽

= ∇ × 𝑀 𝑗

= 𝑀 × 𝑛̂

𝑀 𝑛̂

𝑀 = 𝑀𝑧̂)

𝐽

= 𝑀𝜑̂

𝐽

= 0

𝜑

(𝑟, 𝑧)

𝐵⃗

= 𝜇

𝐼

2𝜋𝐿 ∫ (𝑧 − 𝑧

)

𝑟[(𝑅 + 𝑟)

+ (𝑧 − 𝑧

)

]

[−𝐾(𝑘) + 𝑅

+ 𝑟

+ (𝑧 − 𝑧

)

(𝑅 − 𝑟)

+ (𝑧 − 𝑧

)

] 𝑑𝑧′

𝐿/2

−𝐿/2

𝐵⃗

= 𝜇

𝐼

2𝜋𝐿 ∫ 1

𝑟[(𝑅 + 𝑟)

+ (𝑧 − 𝑧

)

]

[−𝐾(𝑘) + 𝑅

− 𝑟

− (𝑧 − 𝑧

)

(𝑅 − 𝑟)

+ (𝑧 − 𝑧

)

] 𝑑𝑧′

𝐿/2

−𝐿/2

𝐾(𝑘) 𝐸(𝑘)

88

𝐾(𝑘) = ∫ 𝑑𝜃

√1 − 𝑘

𝑠𝑖𝑛

𝜃

𝜋/2 0

𝐸(𝑘) = ∫ √1 − 𝑘

𝑠𝑖𝑛

𝜃

𝜋/2 0

𝑑𝜃

𝑘

= 4𝑅𝑟[(𝑅 + 𝑟

) + (𝑧 − 𝑧

)

]

𝐼 = 𝑀𝐿

𝐼

𝑑𝐿

𝐼

𝑑𝐿

𝑑𝐹

= 𝜇

𝐼

𝑑𝐿

4𝜋 × [ 𝐼

𝑑𝐿

× 𝑅

𝑅

]

89

𝑅

𝑅

𝑅

𝐹

= 𝜇

4𝜋 ∫ 𝐼

𝑑𝐿

× ∫ 𝐼

𝑑𝐿

× 𝑅

𝑅

𝑐1

𝑐2

(5.6)

Figure 48: Schematic of two current-carrying loops, illustrating the variables used in Eq. ⁵⁸

𝐹 = ∫ 𝐼

𝑐

𝑑𝐿 × 𝐵⃗

𝑅

= 𝑅

= 𝑅 𝐹

(𝑧

)

90 𝐹

(𝑧

) = 𝜇

𝐼

𝐼

𝑅

4𝜋 ∫ ∫ −𝑧

cos(𝜃

− 𝜃

) 𝑑𝜃

𝑑𝜃

(2𝑅

− 2𝑅

𝑐𝑜𝑠(𝜃

− 𝜃

) + 𝑧

)

2𝜋 0 2𝜋 0

𝑧

𝐹

= 𝜇

𝐼

𝐼

𝑅

4𝜋𝐿

∫ ∫ ∫ ∫ −[𝑧

− 𝑧

− 𝑧′′] 𝑐𝑜𝑠(𝜃

− 𝜃

) 𝑑𝜃

𝑑𝜃

𝑑𝑧′𝑑𝑑𝑧′′

(2𝑅

− 2𝑅

𝑐𝑜𝑠(𝜃

− 𝜃

) + [𝑧

− 𝑧

− 𝑧′′]

)

2𝜋 0 2𝜋 0

−𝑔/2

−𝐿−𝑔/2 𝑔/2+𝐿

𝑔/2

Figure 49: Geometric definition of the eddy current spring–damper

91 𝑉 = 𝑉

+ 𝑉

= − ∫ 𝛿𝑩

𝛿𝑡 ∙ 𝑑𝒔 + ∫ (𝜈 × 𝑩) ∙ 𝑑𝒍

𝑠 𝑠

𝜈 𝜈

𝒗 × 𝑩

= 0

𝑱 = 𝜎(𝑣 × 𝑩)

𝑭 = ∫ 𝑱 ×

𝛤

𝑩𝑑Γ = −𝒌 ̂𝜎𝛿𝑣 ∫ ∫ 𝑟𝐵⃗

(𝑟, 𝑧

)𝑑𝑟𝑑𝜃

𝑟_𝑜𝑢𝑡 𝑟_𝑖𝑛 2𝜋 0

Γ

𝐵⃗

= 𝐵⃗

(𝑧

) + 𝐵⃗

(𝑧

)

92

𝑧

= | 𝐿 + 𝑔 − 𝑧

𝑖𝑓 𝑧

< 𝐿 + 𝑔

−𝐿 − 𝑔 + 𝑧

𝑖𝑓 𝑧

> 𝐿 + 𝑔

93

Figure 50: Schematic view of the proposed ECD

94

Figure 51: Configuration of the proposed ECD

𝐸 = 𝐸

+ 𝐸

= ∫ (𝑣 × 𝑩) ∙ 𝑑𝑙

𝑐

𝐽 = 𝜎(𝑣 × 𝑩)

𝜎

𝐹 = ∫ 𝐽 × 𝐵⃗𝑑Γ = −𝑘̂𝜎(𝜏 − 𝜏

)𝑣

× ∫ ∫ 𝑟𝐵⃗

(𝑟, 𝑧

)𝑑𝑟𝑑𝜃

𝑟𝑜𝑢𝑡 𝑟_𝑖𝑛 2𝜋 0 Γ

Γ 𝑟

𝑟

𝜏 𝜏

95 𝐶 = 𝜎 ∫ 𝐵⃗

𝑑Γ

Γ

Figure 52: Schematic view of two PMs with like poles in close proximity

𝐵⃗

(𝑟, 𝑧)|

= 𝜇

𝐼

2𝜋𝜏

∫ (𝑧 − 𝑧′)

[(𝑅 + 𝑟)

+ (𝑧 − 𝑧

)

]

𝜏_𝑚/2

−𝜏𝑚/2

× [−𝐾(𝑘) + 𝑅

+ 𝑟

+ (𝑧 − 𝑧

)

(𝑅 − 𝑟)

+ (𝑧 − 𝑧

)

𝐸(𝑘)] 𝑑𝑧′

𝐾(𝑘) 𝐸(𝑘)

𝐾(𝑘) = ∫ 𝑑𝜃

√1 − 𝑘

sin

𝜃

2𝜋 0

96

𝐸(𝑘) = ∫ √1 − 𝑘

𝑠𝑖𝑛

𝜃

2𝜋 0

𝑑𝜃

𝑘

= 4𝑅𝑟[(𝑅 + 𝑟)

+ (𝑧 − 𝑧

)

]

= 𝑀𝜏

𝐵⃗

𝐵⃗

= 2(𝐵⃗

(𝑟, 𝑧)|

− 𝐵⃗

(𝑟, 𝑧)|

97

Figure 53: Diagram of the linear regenerative electromagnetic shock absorber.⁶⁰

Figure 54: Equivalent circuit model of the energy harvester.⁶¹

98

𝑅

Figure 55: Linear regenerative electromagnetic shock absorber dimensions and parameters⁶⁰

99 𝑃 = 𝑉𝐼

𝑅

𝑒. 𝑚. 𝑓.

𝑙 (𝑚) 𝐵⃗

𝑣

𝑉 = 𝐵⃗

𝑣 𝑙

𝐼 = 𝑉

𝑅 = 𝜎𝐵⃗

𝑣

𝐴

𝜎 𝐵⃗

𝑣

𝐴

𝑃 = 𝑉𝐼 = 𝐵⃗

𝑣

𝜎𝑙𝐴

𝐵⃗

𝐵⃗

100 𝑉 = 𝜋𝑁𝐵⃗

𝑣

𝐷

𝑃 = 𝜋𝑁𝜎𝐵⃗

𝑣

𝐷

𝐴

𝐷

𝐴

𝑁 = 𝐹 ∙

𝐴𝑊

𝐴

𝐴

𝐵⃗

𝐷

𝐴

𝐵⃗

101

102

103

Figure 56: Initial model geometry

𝑅

𝑅𝑜𝑑

𝐷

=

104

𝑅

𝑅𝑜𝑑

𝐵⃗𝐼

)

𝐴

= 0.5 𝑚𝑚

𝜏

𝐷

= 2 ∙ 𝜏

𝜏 𝑀

𝐼𝑃

𝜏

105

𝜏

/𝜏 = 0.55

𝐶

= 𝐷

/3

𝑅𝑜𝑑

𝑆𝑡

𝑅𝑜𝑑

= 𝑆𝑡

= 𝐷

Figure 57: Optimal damping characteristics of the passive damper

106

Figure 58: Displacements as function of frequency: 1 – relative displacement of shock absorber between the two mounting points, 2 – relative displacement between the piston rod and cylinder tube ⁶²

107

Figure 59: Input function of the system

108

109

𝐶

= 𝐷

/3 𝐶

𝐶

110

Figure 60: How Voltage Induced changes with Coil Internal radius, 0.5mm resolution

0.95 < 𝐶

(𝑛𝑜𝑟𝑚) < 1

111

Figure 61: How Voltage Induced changes with Coil Internal Radius, 0.1mm resolution

𝐶

(𝑛𝑜𝑟𝑚) ≈ 0.99

𝐷

= 2 ∙ 𝜏 = 2 ∙ (𝑀

+ 𝐼𝑃

)

𝜏

/𝜏 𝜏

𝜏

)

112

Figure 62: How Voltage Induced changes with taum/tau, resolution of 0.5 mm

0.6 < 𝜏

/𝜏 < 0.8

113

Figure 63: How Voltage Induced changes with taum/tau, resolution of 0.1mm

𝜏

/𝜏 = 0.69

𝑁 = 𝐹 𝐴

𝐴

= 𝐹 4𝐴

𝜋𝜙

114

𝐴

𝜙

𝜙

𝑁 = 𝐹 4𝐴

𝜋𝜙

≈ 274

𝑅

= 𝜌 𝐿

𝐴 = 𝜌 𝜋 (𝐶

+ 𝐶

) 2

𝐴

= 86.055 ∙ 10

Ω

𝐶

𝐶

𝐴

𝑅

𝑅

115 𝑟 =

𝑅_{𝐶𝑜𝑖𝑙}

𝑟 = 0.5 𝑟 = 1 𝑟 = 2

Figure 64: How power out change with load resistance

𝑟 = 1 → 𝑅

= 𝑅

116

𝑓 = 1 𝐻𝑧

Figure 65: Three-phase coil voltage at 1Hz

117

Figure 66: Three-phase coil power at 1Hz

20 𝐻𝑧 < 𝑓 < 80 𝐻𝑧

118

Figure 67: Three-phase power output for different frequencies

𝐵⃗

𝐵⃗

= 2.15𝑇

119

Figure 68: Magnetic flux density with optimized geometry model

120

Figure 69: Magnetic flux density with optimized geometry model and material feasibility

Figure 70: Magnetic flux density with optimized geometry model and material feasibility

121

Figure 71: Three-phase power output for different frequencies

122

123

Figure 72: How power changes with Coil Phases⁶⁰

Figure 73: Coil design. (a) Two-phase coil. (b) Three-phase coil. (c) Four-phase coil⁶⁰

124

125 𝐵⃗

𝐻

𝐵⃗

= 𝐵⃗

[1 + 𝛼

100 (𝜗

− 20)]

𝐻

= 𝐻

[1 + 𝛼

100 (𝜗

− 20)]

𝜗

𝐵⃗

𝐻

𝛼

< 0 𝛼

< 0

𝐵⃗

𝐻

126

127

128

𝜌

∇ × 𝑯 = 𝑱 + 𝜕𝑫

𝜕𝑡

∇ ∙ 𝑩 = 0

∇ × 𝑬 = − 𝜕𝑩

𝜕𝑡

∇ ∙ 𝑫 = 0

129 𝑩 = 𝜇

(𝑯 + 𝑴)

𝐷 = 𝜀

𝑬 + 𝑷

𝑱 = 𝜎𝑬

𝜇

, ε

𝜎



∇ ∙ 𝑩 = 0 → 𝐵⃗ = ∇ × 𝑨

130



∇ × (𝑬 + 𝜕𝑨

𝜕𝑡 ) = 0 → 𝑬 + 𝜕𝑨

𝜕𝑡 = −∇𝜑

∇

𝐴 − 𝜇𝜀 𝜕

𝐴

𝜕𝑡 = −𝜇𝐽

∇

𝜑 − 𝜇𝜀 𝜕

𝜑

𝜕𝑡

= − 𝜌 𝜀

∇ ∙ 𝐴 = −𝜇𝜀

𝛿𝑡

∇

𝐴 − 𝜇𝜀 𝛿

𝑨

𝛿𝑡

= −𝜇𝑱

∇

𝜑 − 𝜇𝜀 𝛿

𝜑 𝛿𝑡

= − 𝜌

𝜀

 

131

∇

𝑨 = −𝜇𝑱

∇

𝜑 = − 𝜌 𝜀

∇

𝐴(𝑥, 𝑡) = 𝜇

4𝜋 ∫ 𝑱(𝑥

, 𝑡)

|𝒙 − 𝒙

| 𝑑Γ′

Γ

𝜑(𝑥, 𝑡) = 1

4𝜋𝜀

∫ 𝜌(𝒙

, 𝑡)

|𝒙 − 𝒙

| 𝑑Γ′

Γ



𝛿𝑫 𝛿𝑡

=

𝛿𝑡

= 0

𝑩 = ∇ × 𝑨

132 𝐵⃗(𝑥, 𝑡) = 𝜇

4𝜋 ∫ 𝒋(𝒙

, 𝑡) × (𝒙 − 𝒙

)

|𝒙 − 𝒙

|

𝑑𝑠′

∇𝑨 = 0

∇

𝑨 = −𝜇

(𝑱 + ∇ × 𝑴)

𝑱

= ∇ × 𝑴

 

𝑩(𝑥) = 𝜇

4𝜋 ∫ 𝑱

(𝒙

) × (𝒙 − 𝒙

)

|𝒙 − 𝒙

|

𝑑Γ + 𝜇

4𝜋 ∫ 𝒋

(𝒙

) × (𝒙 − 𝒙

)

|𝒙 − 𝒙

|

𝑑𝑠

𝐽

𝑗

133 𝑱

= ∇ × 𝑴 ( 𝐴

𝑚

) 𝑉𝑜𝑙𝑢𝑚𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑒𝑛𝑠𝑖𝑡𝑦

𝒋

= 𝐌 × 𝒏 ̂ ( 𝐴

𝑚 ) 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑒𝑛𝑠𝑖𝑡𝑦

𝒏

̂

𝑴 = 𝑀

𝒛̂

𝑱

= 0

𝒋

= M𝝋 ̂

Figure B-74: Permanent magnet and its equivalent surface current density⁴⁰

134

135

Ref. 1

136

Figure 75: How the Extrapolation of the B-H Curve Affects the Simulation

137

138

139

140

141

142

143

144

145

146