Lecture Set 1 Introduction to Magnetic Circuits

Transcription

1 Lecture Set 1 Introduction to Magnetic Circuits S.D. Sudhoff Spring

2 Goals Review physical laws pertaining to analysis of magnetic systems Introduce concept of magnetic circuit as means to obtain an electric circuit Learn how to determine the flux-linkage versus current characteristic because it will be key to understanding electromechanical energy conversion 2

3 Review of Symbols Flux Density in Tesla: B (T) Flux in Webers: Φ (Wb) Flux Linkage in Volt-seconds: λ (Vs,Wb-t) Current in Amperes: i (A) Field Intensity in Ampere/meters: H (A/m) Permeability in Henries/meter: µ (Η/m) Magneto-Motive Force: F (A,A-t) Permeance in Henries: P (H) Reluctance in inverse Henries: R (1/H) Inductance in Henries: L (H) Current (fields) into page: Current (fields) out of page: 3

4 Ampere s Law, MMF, and Kirchoff s MMF Law for Magnetic Circuits 4

5 Consider a Closed Path 5

6 Enclosed Current and MMF Sources Enclosed Current i Γ = N i + N i N i enc, a a b b c c MMF Sources F F a b = N i = N i b b a a F c = N i c c Conclusion i enc, Γ = Fs SΓ = {' a ',' b',' c '} s S Γ 6

7 MMF Drops Going Around the Loop MMF Drop F y Thus Or H d l = H d l + H d l + H d l + H d l l l l lγ l Γ = H d l l y H d l = α β δ H d l = F + F + F + F l Τ F d DΓ = {' α ',' β ',' γ ',' δ '} l d DΓ Γ α β γ δ 7

8 Ampere s Law Ampere s Law H d l = i l Γ Recall i enc, Γ Thus = s S Γ enc, Γ F F = s s S d D Γ s Γ F d H d l = l Γ d D Γ F d 8

9 Kirchoff s MMF Law Kirchoff s MagnetoMotive Force Law: The Sum of the MMF Drops Around a Closed Loop is Equal to the Sum of the MMF Sources for That Loop 9

10 Example: MMF Sources 10 conductors, 3 A = 5 conductors, 2 A = 10

11 Example: MMF Drops A quick example on MMF drop: 11

12 Another Example: MMF Drops How about F cb? 12

13 Magnetic Flux, Gauss s Law, and Kirchoff s Flux Law for Magnetic Circuits 13

14 Magnetic Flux Φ m = B S m ds 14

15 Example: Magnetic Flux Suppose there exist a uniform flux density field of B=1.5a y T where a y is a unit vector in the direction of the y-axis. We wish to calculate the flux through open surface S m with an area of 4 m 2, whose periphery is a coil of wire wound in the indicated direction. 15

16 Example: Magnetic Flux 16

17 Example: Magnetic Flux 17

18 Kirchoff s Flux Law Gauss s Law Thus Or B d S = S Γ o O Γ S o Φ o = o O Γ 0 B ds O Γ = {' α ',' β ', ' γ ', } 0 18

19 Kirchoff s Flux Law Kirchoff s Flux Law: The Sum of the Flux Leaving A Node of a Magnetic Circuit is Zero 19

20 Kirchoff s Flux Law A quick example on Kirchoff s Flux Law 20

21 Magnetically Conductive Materials and Ohm s Law for Magnetic Circuits 21

22 Ohm s Law On Ohm s Law Property for Electric Circuits On Ohm s Law Property for Magnetic Circuits 22

23 Types of Magnetic Materials Interaction of Magnetic Materials with Magnetic Fields Magnetic Moment of Electron Spin Magnetic Moment of Electron in Shell These Effects Lead to Six Material Types Diamagnetic Paramagnetic Ferromagnetic Antiferromagnetic Ferrimagnetic Superparamagnetic 23

24 Ferromagnetic Materials Iron Nickel Cobalt 24

25 Ferrimagnetic Materials Iron-oxide ferrite (Fe 3 O 4 ) Nickel-zinc ferrite (Ni 1/2 Zn 12 Fe 2 O 4 ) Nickel ferrite (NiFe 2 O 4 ) 25

26 Behavior of Ferromagnetic and Ferrimagnetic Materials 26

27 Behavior of Ferromagnetic and Ferrimagnetic Materials M19 MN80C 27

28 Modeling Magnetic Materials Free Space B = µ H 0 Magnetic Materials B = µ 0( H + M ) M = χ H B = µ H + M Either Way B Or B 0 M = µ χ H = µ 0 (1 + χ) H B = µ 0µ r H µ r = 1+ χ = µ H µ = µ 0µ r 0 28

29 Modeling Magnetic Materials B = µ H ( H ) H B = µ B ( B) H 29

30 Back to Ohm s Law 30

31 Relationship Between MMF Drop and Flux F = R ( ξ ) Φ a a a R a ( ξ ) la Form 1: ξ = Fa Aa µ H ( Fa / la ) = la Form 2 : ξ = Φ Aa µ B( Φa / Aa ) a Φ = P ( ξ ) F a a a P a ( ξ ) = A µ ( F / l ) a H a a l a A µ ( Φ / A ) a B a a l a Form 1: ξ = F Form 2 : ξ = Φ a a 31

32 Derivation 32

33 Derivation 33

34 Construction of the Magnetic Equivalent Circuit 34

35 Consider a UI Core Inductor 35

36 Inductor Applications 36

37 UI Core Inductor MEC 37

38 Reluctances 38

39 Reluctances 39

40 Reduced Equivalent Circuit 40

41 Example Let us consider a UI core inductor with the following parameters: w i = 25.1 mm, w b = w e = 25.3 mm, w s = 51.2 mm, d s = 31.7 mm, l c = mm, g =1.00 mm, w w = 38.1 mm, and d w = 31.7 mm. The winding is composed of 35 turns. Suppose the magnetic core material has a constant relative permeability of 7700, and that a current of 25.0 A is flowing. Compute the flux through the magnetic circuit and the flux density in the I-core. 41

42 Example 42

43 Example 43

44 Solving the Nonlinear Case R ( Φ ) = R ( Φ ) + R ( Φ ) + 2 R ( Φ ) + 2R eq ic buc luc g Φ = R eq Ni ( Φ) 44

45 Translation of Magnetic Circuits to Electric Circuits: Flux Linkage and Inductance 45

46 Flux Linkage λ = N Φ x x x 46

47 Inductance L x, abs = λ i x x i, λ x1 x1 L inc = λx i x i x1 47

48 Example Suppose λ = 0.05(1 e 0.2i ) i Find the absolute and incremental inductance at 5 A 48

49 Computing Inductance Φ x N x i x + - R x 49

50 Computing Inductance 50

51 Nonlinear Flux Linkage Vs. Current 51

52 Nonlinear Flux Linkage Vs. Current 52

53 Nonlinear Flux Linkage Vs. Current % Computation of lambda-i curve % for UI core inductor % using a simple but nonlinear MEC % % S.D. Sudhoff for ECE321 % January 13, 2012 % clear work space clear all close all g=1*mm; % air gap (m) N=100; % number of turns mu0=4e-7*pi; % perm of free space (H/m) Bsat=1.3; % sat flux density (T) mur=1500; % init rel permeability Am=w*d; % mag cross section (m^2) % start with flux linkage lambda=linspace(0,0.08,1000); % assign parameters cm = 1e-2; % a centimeter mm = 1e-3; % a millimiter w=1*cm; % core width (m) ws=5*cm; % slot width (m) ds=2*cm; % slot depth (m) d=5*cm; % depth (m) % find flux and B and mu phi=lambda/n; B=phi/(w*d); mu=mu0*(mur-1)./(exp(10*(b-bsat)).^2+1)+mu0; 53

54 Nonlinear Flux Linkage Vs. Current % Assign reluctances; Ric=(ws+w)./(Am*mu); Rbuc=(ws+w)./(Am*mu); Rg=g/(Am*mu0); Rluc=(ds+w/2)./(Am*mu); % compute effective reluctances Reff=Ric+Rbuc+2*Rluc+2*Rg; % compute MMF and current F=Reff.*phi; i=f/n; % find lambda-i characteristic figure(1) plot(i,lambda); xlabel('i, A'); ylabel('\lambda, Vs'); grid on; % show B-H characteristic H=B./mu; figure(2) plot(h,b) xlabel('h, A/m'); ylabel('b, T'); grid on; 54

55 Nonlinear Flux Linkage Vs. Current 55

56 Hardware Example Our MEC

57 Non-Idealities: Fringing and Leakage Flux Log10 Energy Density J/m y,m x,m 57

58 Our MEC With Leakage and Fringing 58

59 Faraday s Law Faraday s Law, voltage equation for winding vx = rxix dλ + x dt If inductance is constant λ x = L x i x Combining above yields vx = rxix + di L x x dt 59

60 Energy Stored in a Magnetically Linear Inductor We can show Proof E = x 1 2 L x i 2 x 60

61 Mutual Inductance 61

62 Mutual Inductance 62

63 Case Study: Lets Design a UI Core Inductor 63

64 Architecture Depth into page = d Node 8 Node 7 Node 1 Node 2 w Node 3 g Node 4 N Turns d s w Node 6 Node 5 w ws w 64

65 Design Requirements Current Level: 40 A Inductance: 5 mh Packing Factor: 0.7 Slot Shape η dw : 1 Maximum Resistance: 0.1Ω 65

66 Information Cost of Ferrite: 230 k$/m 3 Density of Ferrite: 4740 Kg/m 3 Saturation Flux Density: 0.5 T Cost of Copper: 556 $/m 3 Density of Copper: 8960 Kg/m 3 Resistivity of Copper: 1.7e-8 Ωm 66

67 Step 1: Design Variables Depth into page = d Node 8 Node 7 Node 1 Node 2 w Node 3 g Node 4 N Turns d s w Node 6 Node 5 w ws w 67

68 Step 2: Design Constraints Constraint 1: Flux Density 68

69 Step 2: Design Constraints Constraint 2: Inductance 69

70 Step 2: Design Constraints Constraint 3: Slot Shape 70

71 Step 2: Design Constraints Constraint 4: Packing Factor 71

72 Step 2: Design Constraints Constraint 5: Resistance 72

73 Step 2: Design Constraints 73

74 Approach (1/6) 74

75 Approach (2/6) 75

76 Approach (3/6) 76

77 Approach (4/6) 77

78 Approach (5/6) 78

79 Approach (6/6) 79

80 Design Metrics Magnetic Material Volume Depth into page = d Node 8 Node 7 Node 1 Node 2 w Node 3 g Node 4 N Turns d s w Node 6 Node 5 w ws w 80

81 Design Metrics Wire Volume 81

82 Design Metrics Circumscribing Box Depth into page = d Node 8 Node 7 Node 1 Node 2 w Node 3 g Node 4 N Turns d s w Node 6 Node 5 w ws w 82

83 Design Metrics Material Cost Mass 83

84 Circumscribing Box Volume 84

85 Material Cost 85

86 Mass 86

87 Least Cost Design N = 260 Turns d = cm g = mm w = cm a w = (mm)^2 d s = 8.94 cm w s = 8.94 cm V mm = m^3 V cu = m^3 V bx = m^3 Cost = $ Mass = Kg 87

88 Loss and Mass 88

89 Design Criticisms 89

90 Manual Design Approach Flux Density B/m Analysis y,m 0 Design Equations Numerical Analysis Design Revisions x,m Final Design 90

91 Formal Design Optimization Evolutionary Environment Detailed Analysis Fitness Function 91

92 Another Design Trade-Off 92

93 Lunar Rover Propulsion Drive Pareto-Optimal Front 93