lecture set 1 introduction to magnetic circuits

Lecture Set 1

Introduction to Magnetic Circuits

S.D. Sudhoff

Spring 2017

1

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

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

Ampere’s Law, MMF, and Kirchoff’sMMF Law for Magnetic Circuits

4

Consider a Closed Path …

5

Enclosed Current and MMF Sources

• Enclosed Current

• MMF Sources

• Conclusion

6

,enc a a b b c ci N i N i N iΓ = − + −

a a aF N i= −

b b bF N i=

,enc ss S

i FΓ

Γ∈

= ∑ ' ', ' ', ' 'S a b cΓ =

c c cF N i= −

MMF Drops

• Going Around the Loop

• MMF Drop

• Thus

• Or7

y

y

l

F d= ⋅∫H l

l l l l l

d d d d dα β δγΓ

⋅ = ⋅ + ⋅ + ⋅ + ⋅∫ ∫ ∫ ∫ ∫H l H l H l H l H l

dd Dl

d FΓΓ ∈

⋅ = ∑∫ H l ' ', ' ', ' ', ' 'D α β γ δΓ =

l

d F F F Fα β γ δΤ

⋅ = + + +∫ H l

Ampere’s Law

• Ampere’s Law

• Recall

• Thus

8

,enc

l

d iΓ

Γ⋅ =∫ H l

dd Dl

d FΓΓ ∈

⋅ = ∑∫ H l,enc ss S

i FΓ

Γ∈

= ∑

s ds S d D

F FΓ Γ∈ ∈

=∑ ∑

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

Example: MMF Sources

10

10 conductors, 3 A =

5 conductors, 2 A =

Example: MMF Drops

• A quick example on MMF drop:

11

Another Example: MMF Drops

• How about Fcb ?

12

Magnetic Flux, Gauss’s Law, andKirchoff’s Flux Law for Magnetic Circuits

13

Magnetic Flux

14

m

m dΦ = ⋅∫S

B S

Example: Magnetic Flux

• Suppose there exist a uniform flux density field of B=1.5ay T where ay is a unit vector in the direction of the y-axis. We wish to calculate the flux through open surface Sm with an area of 4 m2, whose periphery is a coil of wire wound in the indicated direction.

15


16


17

• Gauss’s Law

• Thus

• Or

18

Kirchoff’s Flux Law

0dΓ

⋅ =∫S

B S

oo

dΓ∈

⋅∑ ∫O S

B S ' ', ' ', ' ', α β γΓ = ⋅ ⋅ ⋅O

0oo Γ∈

Φ =∑O


• Kirchoff’s Flux Law:

The Sum of the Flux Leaving A Node of a Magnetic Circuit is Zero

19


• A quick example on Kirchoff’s Flux Law

20

Magnetically Conductive Materials and Ohm’s Law for Magnetic Circuits

21

Ohm’s Law

• On Ohm’s LawProperty for Electric Circuits

• On Ohm’s LawProperty for Magnetic Circuits

22

Types of Magnetic Materials

• Interaction of Magnetic Materials with Magnetic FieldsMagnetic Moment of Electron SpinMagnetic Moment of Electron in Shell

• These Effects Lead to Six Material TypesDiamagneticParamagnetic FerromagneticAntiferromagneticFerrimagneticSuperparamagnetic

23

Ferromagnetic Materials

• Iron

• Nickel

• Cobalt

24

FerrimagneticMaterials

• Iron-oxide ferrite (Fe3O4)

• Nickel-zinc ferrite (Ni1/2Zn12Fe2O4)

• Nickel ferrite (NiFe2O4)

25

Behavior of Ferromagnetic and FerrimagneticMaterials

26

Behavior of Ferromagnetic and FerrimagneticMaterials

27

M19 MN80C

Modeling Magnetic Materials

• Free Space

• Magnetic Materials

• Either Way

• Or28

0B Hµ=

0( )B H Mµ= +

0B H Mµ= +

M Hχ=

0M Hµ χ=

0(1 )B Hµ χ= + 0 rB Hµ µ= 1rµ χ= +

B Hµ= 0 rµ µ µ=

Modeling Magnetic Materials

29

( )HB H Hµ= ( )BB B Hµ=

Back to Ohm’s Law

30

Relationship Between MMF Drop and Flux

31

( )a a aF R ξ= Φ

B

Form 1:( / )

( )

Form 2 :( / )

aa

a H a aa

aa

a a a

lF

A F lR

l

A A

ξµ

ξξ

µ

== = Φ Φ

( )a a aP FξΦ =B

( / ) Form 1:

( )( / )

Form 2 :

a H a aa

aa

a a aa

a

A F lF

lP

A A

l

µ ξξ

µ ξ

== Φ = Φ

Derivation

32

Derivation

33

Construction of the Magnetic Equivalent Circuit

34

Consider a UI Core Inductor

35

Inductor Applications

36

UI Core Inductor MEC

37

Reluctances

38

Reluctances

39

Reduced Equivalent Circuit

40

Example

• Let us consider a UI core inductor with the following parameters: wi = 25.1 mm, wb = we = 25.3 mm, ws = 51.2 mm, ds = 31.7 mm, lc = 101.2 mm, g =1.00 mm, ww = 38.1 mm, and dw = 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

Example

42

Example

43

Solving the Nonlinear Case

44

( )eq

Ni

RΦ =

Φ

( ) ( ) ( ) 2 ( ) 2eq ic buc luc gR R R R RΦ = Φ + Φ + Φ +

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

45

Flux Linkage

46

x x xNλ = Φ

Inductance

47

1 1

,

,x x

xx abs

x i

Li λ

λ=

1x

xinc

x i

Li

λ∂=∂

Example

ie i 03.0)1(05.0 2.0 +−= −λ

48

• Suppose

• Find the absolute and incremental inductance at 5 A

Computing Inductance

49

+

-xxiN

xΦ

xR

Computing Inductance

50

Nonlinear Flux Linkage Vs. Current

51

52



% 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

% 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)

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);

% 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


% 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

Hardware Example – Our MEC

Non-Idealities: Fringing and Leakage Flux

57

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Log10 Energy Density J/m3

x,m

y,m

Our MEC – With Leakage and Fringing

58

Faraday’s Law

• Faraday’s Law, voltage equation for winding

• If inductance is constant

• Combining above yields

59

dt

dirv xxxx

λ+=

xxx iL=λ

dt

diLirv x

xxxx +=

Energy Stored in a Magnetically Linear Inductor

• We can show

• Proof

60

221

xxx iLE =

Mutual Inductance

61

Mutual Inductance

62

Case Study: Lets Design a UI Core Inductor

63

Architecture

64

Node 1

Node 3

Node 5

Node 4

Node 8

Node 7

Node 2

Node 6

Depth into page = d

g

sd

sw ww

w

w

N Turns

Design Requirements

• Current Level: 40 A

• Inductance: 5 mH

• Packing Factor: 0.7

• Slot Shape ηdw: 1

• Maximum Resistance: 0.1Ω

65

Information

• Cost of Ferrite: 230 k$/m3

• Density of Ferrite: 4740 Kg/m3

• Saturation Flux Density: 0.5 T

• Cost of Copper: 556 $/m3

• Density of Copper: 8960 Kg/m3

• Resistivity of Copper: 1.7e-8 Ωm

66

Step 1: Design Variables

Node 1

Node 3

Node 5

Node 4

Node 8

Node 7

Node 2

Node 6

Depth into page = d

g

sd

sw ww

w

w

N Turns

67

Step 2: Design Constraints

• Constraint 1: Flux Density

68


• Constraint 2: Inductance

69


• Constraint 3: Slot Shape

70


• Constraint 4: Packing Factor

71


• Constraint 5: Resistance

72


73

Approach (1/6)

74

Approach (2/6)

75

Approach (3/6)

76

Approach (4/6)

77

Approach (5/6)

78

Approach (6/6)

79

Design Metrics

• Magnetic Material Volume

80

Node 1

Node 3

Node 5

Node 4

Node 8

Node 7

Node 2

Node 6

Depth into page = d

g

sd

sw ww

w

w

N Turns

Design Metrics

• Wire Volume

81

Design Metrics

• Circumscribing Box

82

Node 1

Node 3

Node 5

Node 4

Node 8

Node 7

Node 2

Node 6

Depth into page = d

g

sd

sw ww

w

w

N Turns

Design Metrics

• Material Cost

• Mass

83

Circumscribing Box Volume

84

Material Cost

85

Mass

86

Least Cost Design

• N = 260 Turns• d = 8.4857 cm• g = 13.069 mm• w = 1.813 cm• aw = 21.5181 (mm)^2• ds = 8.94 cm• ws = 8.94 cm• Vmm = 0.00066173 m^3• Vcu = 0.0027237 m^3• Vbx = 0.0075582 m^3• Cost = 153.7112 $• Mass = 27.5408 Kg

87

Loss and Mass

88

Design Criticisms

89

90

Manual Design Approach

Analysis

Design Equations

NumericalAnalysis

DesignRevisions

FinalDesign

-0.1 -0.05 0 0.05 0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Flux Density B/m2

x,m

y,m

91

Formal Design Optimization

Evolutionary Environment

Fitness Function

DetailedAnalysis

Another Design Trade-Off

92

Lunar Rover Propulsion DrivePareto-Optimal Front

93

lecture set 1 introduction to magnetic circuits

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