Power Electronics Notes 30A Review of Magnetics

Transcription

1 Power Electronics Notes 30A Review of Magnetics Marc T. Thompson, Ph.D. Thompson Consulting, Inc. 9 Jacob Gates Road Harvard, MA Phone: (978) Fax: (240) marctt@thompsonrd.com Web: Portions of these notes excerpted from the CD ROM accompanying Fitzgerald, Kingsley and Umans, Electric Machinery, 6th edition, McGraw Hill, 2003 and from the CD ROM accompanying Mohan, Undeland and Robbins, Power Electronics Converters, Applications and Design, 3d edition, John Wiley Other notes Marc Thompson,

2 Overview Review of Maxwell s equations Magnetic circuits Flux, flux linkage, inductance and energy Properties of magnetic materials 2

3 Review of Maxwell s Equations James Clerk Maxwell (13 June N November 1879) was a Scottish mathematician and theoretical physicist. His most significant achievement was the development of the classical electromagnetic theory, synthesizing all previous unrelated observations, experiments and equations of electricity, magnetism and even optics into a consistent theory. His set of equations Maxwell's equations demonstrated that electricity, magnetism and even light are all manifestations of the same phenomenon: the electromagnetic field. From that moment on, all other classical laws or equations of these disciplines became simplified cases of Maxwell's equations. Maxwell's work in electromagnetism has been called the "second great unification in physics", after the first one carried out by Newton. Reference: 3

4 Review of Maxwell s Equations First published by James Clerk Maxwell in 1864 Maxwell s equations couple electric fields to magnetic fields, and describe: Magnetic fields Electric fields Wave propagation p (through the wave equation) There are 4 Maxwell s equations, but in magnetics we generally only need 3: Ampere s Law Faraday s Law Gauss Magnetic Law 4

5 Ampere s Law Flowing current creates a magnetic field André-Marie Ampère ( ) 5

6 Ampere s Law Flowing current creates a magnetic field r v v v d r v H dl = J da + ε oe da dt C S S In magnetic systems, generally there is high current and low voltage (and hence low electric field) and we can approximate for low d/dt: C r v H dl S v v J da The magnetic flux density integrated around a closed contour equals the net current flowing through the surface bounded by the contour 6

7 Field From Current Loop, NI = 500 A-turns Coil radius R = 1 7

8 Faraday s Law A changing magnetic flux impinging on a conductor creates an electric field and hence a current (eddy current) Michael Faraday ( ) 8

9 Faraday s Law A changing magnetic flux impinging on a conductor creates an electric field and hence a current (eddy current) r v d r v E dl = B da dt C S The electric field integrated around a closed contour equals the net timevarying magnetic flux density flowing through the surface bound by the contour In a conductor, this electric field creates r r a current by: J = σe Induction motors, brakes, etc. 9

10 Circular Coil Above Conducting Aluminum Plate Flux density plots at DC and 60 Hz At 60 Hz, currents induced in plate via magnetic induction create lift force DC 60 Hz 10

11 Gauss Magnetic Law Johann Carl lfi Friedrich di hg Gauss (30 April February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. Sometimes known as the princeps p mathematicorum (Latin, usually translated as "the Prince of Mathematicians", although Latin princeps also can simply mean "the foremost") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's s most influential mathematicians. Reference: 11

12 Gauss Magnetic Law Gauss' magnetic law says that t the integral of the magnetic flux density over any closed surface is zero, or: r r B da = 0 S This law implies that magnetic fields are due to electric currents and that magnetic charges ( monopoles ) do not exist. B A B A 1, 1 2, 2 B 3, A 3 B + 3A3 = B1 A1 B2 A2 Note: similar form to KCL in circuits! (We ll use this analogy later ) 12

13 Gauss Law --- Continuity of Flux Lines φ + φ + φ =

14 Lorentz Force Law Experimentally derived rule: r r r F = J B dv For a wire of length l carrying current I perpendicular to a magnetic flux density B, this reduces to: r F = IBl 14

15 Lorentz Force Law and the Right Hand Rule r F r r = J BdV Reference: 15

16 Intuitive Thinking about Magnetics By Ampere s Law, the current J and the magnetic field H are generally at right angles to one another By Gauss law, magnetic field lines loop around on themselves No magnetic monopole You can think of high- μ magnetic materials such as steel as an easy conduit for magnetic flux. i.e. the flux easily flows thru the high- μ material 16

17 Relationship of B and H H is the magnetic field (A/m in SI units) and B is the magnetic flux density (Weber/m 2, or Tesla, in SI units) B and H are related by the magnetic permeability μ by B = μh Magnetic permeability μ has units of Henry/meter In free space μ =4π 10-7 o H/m 17

18 Right Hand Rule for B and I Reference: 18

19 Forces Between Current Loops Reference: 19

20 Inductor Without Airgap Constitutive relationships In free space: r r B = μ H o Magnetic permeability of free space μo = 4π 10-7 Henry/meter. In magnetic material: r r B = μh 20

21 Inductor with Airgap r H r dl Ampere's law: = S Let's use constitutive relationships: Bc l μ c c Bg + μ o r r J da H c lc + H g g = g = NI NI 21

22 Inductor with Airgap c A B Φ = g c A B A Φ = A c g l Put this into previous expression: + = Φ g o c c A g A l NI μ μ Solve for flux: + Φ = c A g A l NI If g/μ o A g >> l c /μa c A g NI μ Φ 22 g o c A A μ μ g μ o A

23 Magnetic-Electric Circuit Analogy Use Ohm s law analogy to model magnetic circuits V NI I Φ R R Use magnetic reluctance instead of resistance R = l σa R = l μa Magnetic reluctance has units of 1/Henry in the SI system 23

24 Magnetic-Electric Circuit Analogy 24

25 C-Core with Gap --- Using Magnetic Circuits Flux in the core is easily found by: Φ = R core NI + R gap = l c p μ A c NI + g μ A o c N turns Average magnetic path length l p inside core, cross-sectional area A Airgap, g Now, note what happens if g/μ o >> l p /μ c : the flux in the core is now approximately independent of the core permeability, as: NI NI Φ R g gap μ o A c NI + - Φ R core R gap Inductance: L = NΦ I N R 2 gap 2 N g μ A o c 25

26 C-Core with Gap --- Finite Element Analysis (FEA) 26

27 Fringing Fields in Airgap If fringing i is negligible, ibl A c = A g 27

28 Summary of Magnetic Quantities Power Electronics 28

29 Example 1: C-Core with Airgap Fitzgerald, Example 1.1: 1: Find current I necessary to generate B = 0.8T in airgap 29

30 Example 1: C-Core with Airgap 30

31 Example 2: Simple Synchronous Machine Problem: Assuming μ, find airgap flux and flux density. I = 10A, N = 1000, g = 1 cm and A g = 2000 cm 2. Solution: NI Φ = 2g μo A Φ Bg = = A g g (1000)(10) = = 2(0.01) 7 (4 10 )(0.2) π 0.13 Wb = 0.65 = 0.65T m 0.13 Wb 31

32 What Does Infinite μ Imply? 32

33 Flux Linkage and Energy By Faraday s law, changing magnetic flux density creates an electric field (and a voltage) r v d r v E dl = B da dt Induced voltage: v = N dφ dt C = dλ dt λ = "flux linkage" = NΦ Inductance relates flux linkage to current S L = λ I 33

34 Inductor with Airgap Units of inductance are Henrys, or Weber-turns per Ampere Φ λ NI g μ A o g N Φ = λ L = I μ A o μ A o g g g g N N 2 2 I Flux Flux linkage Inductance 34

35 Magnetic Circuit with Two Airgaps 35

36 Magnetic Circuit with Two Airgaps 36

37 Inductance vs. Relative Permeability 37

38 Inductance and Energy Magnetic stored energy (Joules) is: W = LI 2 38

39 Magnetic Circuit with Two Windings Note that flux is the sum of flux due to i 1 and that due to i 2 39

40 Magnetic Circuit with Two Windings 40

41 Magnetic Circuit with Two Windings 41

42 B-H Curve and Saturation Definition of magnetic permeability: slope of B-H curve 42

43 B-H Loop for M-5 Grain-Oriented Steel Only the top half of the loops shown for steel thick 43

44 DC Magnetization Curve for M-5 44

45 BH Curves for Various Soft Magnetic Materials Reference: E. Furlani, Permanent Magnet and Electromechanical Devices, Academic Press, 2001, pp

46 Relationship Between Voltage, Flux and Current 46

47 Exciting RMS VA per kg at 60 Hz 47

48 Hysteresis Loop Hysteresis loss (Watts) is proportional to shaded d area 48

49 Hysteresis Loop Size Increases with Frequency Hysteresis loss increases as frequency increases Reference: Siemens, Soft Magnetic Materials (Vacuumschmelze Handbook), pp

50 Total Core Loss Core loss (Watts/kg) in M-5 steel at 60 Hz vs. Bmax 50

51 Total Core Loss vs. Frequency and Bmax Core loss depends d on peak flux density and excitation ti frequency High frequency core material Reference: 51

52 Example 3: Core Calculations Find maximum B From this B and switching frequency, find core loss per kg Total loss is power density x mass of core 52

53 Example 3: Find V, I and Core Loss 53

54 Example 3: Excitation Voltage 54

55 Example 3: Peak Current in Winding 55

56 Example 3: RMS Winding Current 56

57 Example 3: Core Loss 57

58 Concept of Magnetic Reluctance Flux is related to ampere-turns by reluctance 58

59 Analogy between Electrical and Magnetic Worlds 59

60 Analogy between Equations in Electrical and Magnetic Circuits it 60

61 Magnetic Circuit and its Electrical Analog 61

62 Faraday s Law and Lenz s Law 62

63 Inductance L Inductance relates flux-linkage λ to current I 63

64 Analysis of a Transformer 64

65 Transformer Equivalent Circuit 65

66 Including the Core Losses 66

67 Transformer Core Characteristic 67

68 Magnetic Design Issues Ratings for inductors and transformers in power electronic circuits vary too much for commercial vendors to stock full range of standard parts. Instead only magnetic cores are available in a wide range of sizes, geometries, and materials as standard parts. Circuit designer must design the inductor/transformer for the particular application. Design consists of: 1.Selecting appropriate core material, geometry, and size 2.Selecting appropriate copper winding parameters: wire type, size, and number of turns. Core (double E) Winding Bobbin Assembled core and winding 68

69 Inductor Fundamentals As sump t ion s No co re losse s o r cop per w in ding losse s Line arize d B-H cu rv e fo r co re w it h μ m >> μ o l 2 m >> g and A >> g Magn et ic circ uit app roxi ma t ion s ( fl ux unif or m o v er co re cross -s ect ion, no fringing flux) l m = mean path length N 1 i 1 Air gap: H g Cross-sectional area of core = A g Starting equations H m l m + H g g = N I ( A m pere s Law ) B m A = B g A = φ ( Con t inuit y of f lux assum ing no l ea kag e f lux ) μ m H m = B m ( line arize d B-H cu rv e) ; Core: H m B s B μ o H g = B g Res ults ΔH NI B s > B m = B g = = φ/ A l m / μ m + g / μ o A N 2 LI = Nφ ; L = lm / μ m + g / μ o ΔB linear region μ = ΔB B = ΔH H H 69

70 Assumptions same as for inductor Transformer Fundamentals Starting equations H 1 L m = N 1 I 1 ; H 2 L m = N 2 I 2 ( A m pere 's Law ) H m L m ( H 1 H 2 ) L m N 1 I 1 N 2 I 2 µ m H m = B m ( linea rized B-H cu rv e) l m = mean path length H m L m = ( H 1 - H 2 ) L m = N 1 I 1 - N 2 I 2 + N d φ 1 d φ 2 v 1 = N 1 ; v dt 2 = N 2 dt (Faraday's Law) Ne t f lux φ = φ 1 - φ 2 = µ m H m A = µ m A(N 1 I 1 - N 2 I 2 ) L m Results assum ing µ m, i.e. ideal co re o r ide al t ransfor m er app roxi ma t ion. φ = 0 and t hus N µ 1 I 1 = N 2 I 2 m d(φ 1 - φ 2 ) v 1 v 2 v 1 v 2 = 0 = - ; = d t N 1 N 2 N 1 N 2 i 1 Cross-sectional area of core = A v v 1 1 N φ 1 Magnetic flux B s B φ ΔH ΔB linear region μ = ΔB B = ΔH H φ 2 70 H i 2 +

71 Scaling Issues Larger electrical ratings require larger current I and larger flux density B. Core loss (hysteresis, eddy currents) increase as B 2 (or greater) Winding (ohmic) loss increases as I 2 and increases at high frequencies (skin effect, proximity effect) To control component temperature, surface area of component and thus size of component must be increased to reject increased heat to ambient. At constant winding current density J and core flux density B, heat generation increases with volume V but surface area only increases as V 2/3. Maximum J and B must be reduced as electrical ratings increase. Flux density B must be < B s Higher electrical ratings larger total flux larger component size Flux leakage, nonuniform flux distribution complicate design Minor hystersis loop core g B s B - B s H fringing flux 71

72 Overview of the Magnetic Design Problem Challenge - conversion of component operating specs in converter circuit into component design parameters. Goal - simple, easy-to-use procedure that produces component design specs that result in an acceptable design having a minimum size, weight, and cost. Inductor electrical (e.g.converter circuit) specifications. Inductance value L Inductor currents rated peak current I, rated rms current I rms, and rated dc current (if any) I dc Operating frequency f. Allowable power dissipation in inductor or equivalently maximum surface temperature of the inductor T s and maximum ambient temperature T a. s p a Transformer electrical (converter circuit) specifications. Rated rms primary voltage V pri Rated rms primary current I pri Turns ratio N pri /N sec Operating frequency f Allowable power dissipation in transformer or equivalently maximum temperatures T s and T a Design procedure outputs. Core geometry and material. Core size (A core, A w ) Number of turns in windings. Conductor type and area A cu. Air gap size (if needed). Three impediments to a simple design procedure. 1. Dependence e eof J rms and db on core size.. 2. How to chose a core from a wide range of materials and geometries. 3. How to design low loss windings at high operating frequencies. Detailed consideration of core losses, winding losses, high frequency effects (skin and proximity effects), heat transfer mechanisms required dfor good design procedures. 72

73 Magnetic Core Shapes and Sizes Magnetic cores available in a wide variety of sizes and shapes. Ferrite cores available as U, E, and I shapes as well as pot cores and toroids. Laminated (conducting) materials available in E, U, and I shapes as well as tape wound toroids and C-shapes. Open geometries such as E-core make for easier fabrication but more stray flux and hence potentially more severe EMI problems. Closed geometries such as pot cores make for more difficult fabrication but much less stray flux and hence EMI problems. Bobbin or coil former provided with most cores. Dimensions of core are optimized by the manufacturer so that for a given rating (i.e. stored magnetic energy for an inductor or V-I rating for a transformer), the volume or weight of the core plus winding is minimized or the total cost is minimized. Larger ratings require larger cores and windings. Optimization requires experience and computerized optimization algorithm. Vendors usually are in much better position to do the optimization than the core user. insulating layer magnetic steel lamination 73

74 Double-E Core Example Characteristic Relative Size Absolute Size for a= 1cm Core area Acore 1.5 a cm 2 Winding area Aw 1.4 a cm 2 Area product AP = AwAc 2.1 a cm 4 Core volume Vcore 13.5 a cm 3 Winding volume Vw 12.3a cm 3 Total surface area of assembled core and winding 59.6 a cm 2 74

75 High Frequency Effects At DC, current flow in a wire spreads out uniformly across cross section of wire For high frequencies, skin effect and proximity effect affect current flow patterns For an isolated wire at high frequency, current is concentrated on the surface of the wire in a thin layer approximately one skin depth thick Proximity effect may be important if you have adjacent windings 75

76 Types of Core Materials Iron-based alloys Ferrite cores Various compositions Fe-Si (few percent Si) Fe-Cr-Mn METGLASS (Fe-B, Fe-B-Si, plus many other compositions) Important properties Resistivity = (10-100) ρ Cu B s = T (T = tesla = 10 4 Gauss) Various compositions - iron oxides, Fe-Ni-Mn oxides Important t properties Resistivity ρ very large (insulator) - no ohmic losses and hence skin effect problems at high frequencies. B s = 0.3 T METGLASS materials available only as tapes of various widths and thickness. Other iron alloys available as laminations of various shapes. Powdered iron can be sintered into various core shapes. Powdered iron cores have larger effective resistivities. 76

77 Hysteresis Loss B Minor hystersis loop B s 0 B ac t H B ac B dc B(t) - B s 0 t Area encompassed by hysteresis Typical waveforms of flux density, loop equals work done on material B(t) versus time, in an inductor during one cycle of applied ac Only B ac contributes to hysteresis magnetic field. Area times frequency loss equals power dissipated per unit volume 77

78 Total Core Loss Eddy current loss plus hysteresis loss = core loss. 3F3 core losses in graphical form. Empirical equation - P m,sp = k f a [B ac ] d. f = frequency of applied field. B ac = baseto-peak value of applied ac field. k, a, and d are constants which vary from material to material P m,sp = 1.5x10-6 f 1.3 [B ac ] 2.5 mw/cm 3 for 3F3 ferrite. (f in khz and B in mt) P m,sp = 3.2x10-6 f 1.8 [B ac ] 2 mw/cm 3 METGLAS 2705M (f in khz and B in mt) Example: 3F3 ferrite with f = 100 khz and B ac = 100 mt, P m,sp = 60 mw/cm 3 78

79 Core Material Performance Factor Volt-amp (V-A) rating of transformers is proportional to f B ac Core materials have different allowable values of B ac at a specific frequency. B ac limited by allowable P m,sp. Most desirable material is one with largest B ac. Choosing best material aided by defining an empirical performance factor PF = f B ac. Plots of PF versus frequency for a specified value of P m,sp permit rapid selection of best material for an application. Plot of PF versus frequency at P m,sp = 100 mw/cm 3 for several different ferrites shown below. For instance, 3F3 is best material in 40 khz to 420 khz range 79

80 Eddy Current Losses in Magnetic Cores B i (r) B o = exp({r - a}/δ) δ = skin depth = 2 ωμσ ω = 2π f, f = frequency μ = magnetic permeability ; μ > for magnetic materials. σ = conductivity of material. AC magnetic fields generate eddy currents in conducting magnetic materials. Eddy currents dissipate power. Shield interior of material from magnetic field. Numerical examp le σ =0.05 σ cu ; μ = 10 3 μ o f = 100 Hz δ = 1 mm 80

81 Laminated Core Cores made from conductive magnetic materials must be made of many thin laminations. Lamination thickness < skin depth. 0.5 t t (typically 0.3 mm) Insulator Magnetic steel lamination 81

82 Eddy Current Loss in Lamination Flux φ(t ) int ercept ed by c urrent loop Average power P ec dissipat ed in laminat ion of area 2xw given b y φ(t) = 2xw B(t) Volt age in current loop v(t ) = 2xw db(t ) given by P ec = < δp( t )dv > = w L d 3 ω 2 B 2 24 ρ dt core = 2wxωBcos(ωt) P ec w L d 3 ω 2 B 2 1 d2 ω 2 B 2 2w ρ P core ec,sp = = V Curr ent loop resist ance r = L dx ; w >> d 24 ρ core dwl = 24 ρ core Inst ant aneous power dissipat ed in th in loop [v(t) ]2 δp( t ) = r Average power P ec dissipat ed in lamination given by P ec = < δp( t )dv w L d 3 ω 2 B 2 > = 24 ρ core P ec P ec,sp = V = w L d 3 ω 2 B ρ core dwl = d 2 ω 2 B 2 24 ρ core x z y L w d dx x B sin( ωt) -x Eddy current flow path 82

83 Power Dissipation in Windings Average power per unit volume of copper dissipated i d in copper winding = P cu,sp = cu (J rms ) 2 where J rms = I rms /A cu and cu = copper resistivity. Average power dissipated per unit volume of winding = P w,sp = k cu cu (J rms ) 2 ; V cu = k cu V w where V cu = total volume of copper in the winding and V w =totalvolumeofthe of the winding. Copper fill factor k cu = N A cu A < 1 w N = number of turns; A cu = cross-sectional area of copper conductor from which winding is made; ; A w = b w l w = area of winding window. k cu = 0.3 for Litz wire; k cu = 0.6 for round conductors; k cu for rectangular conductors. k cu < 1 because: Insulation on wire to avoid shorting out adjacent turns in winding. Geometric restrictions. (e.g. tightpacked circles cannot cover 100% of asquarearea) area.) 83

84 Eddy Currents Increase Winding Loss AC currents in conductors generate ac magnetic fields which in turn generate eddy currents that cause a nonuniform current density in the conductor. Effective resistance of conductor increased over dc value. P w,sp > k cu ρ cu (J rms ) 2 if conductor dimensions greater than a skin depth. J(r) J = exp({r - a}/δ) o 2 δ = skin depth = ωμσ ω = 2π f, f = frequency of ac current μ = magnetic permeability of conductor; μ = μ o for nonmagnetic conductors. σ = conductivity of conductor material. Numerical example e using copper at 100 CC Frequency 50 Hz 5 khz 20 khz 500 khz Skin Depth mm mm mm mm Eddy currents H(t) I(t) J(t) J(t) I(t) r a a 0 B sin(ωt) B sin(ωt) Mnimize eddy currents using Litz wire bundle. Each conductor in bundle has a diameter less than a skin depth. Twisting of paralleled wires causes effects of intercepted flux to be canceled out between adjacent twists of the conductors. Hence little l if any eddy currents. 84

85 Proximity Effect Can Increase Winding Loss ( a) δ > d (b) δ < d x x x A A x x x B MMF x B x x Proximity effect - losses due to eddy current generated by the magnetic field experienced by a particular conductor section but generated by the current flowing in the rest of the winding. x x 10 3 Eddy Current Losses Design methods for minimizing proximity effect losses discussed later. 3 1 x 1 x 85

86 Transformer Winding Loss --- AC Effects π d d ϕ = η 4 δ δ Reference: R. W. Erickson et. al., Fundamentals of Power Electronics, pp

87 Minimum Winding Loss P w = P dc + P ec ; P ec = eddy current loss. Optimum conductor size P w = { R dc + R ec } [I rms ] 2 = R ac [I rms ] 2 Resistance R dc R ac = F R R dc = [1 + R ec /R dc ] R dc R ec Minimum winding loss at optimum conductor size. P w 1.5 P dc P ec = 0.5 P dc d opt - δ d = conductor diameter or thickness High frequencies require small conductor sizes minimize loss. P dc kept small by putting may small-size conductors in parallel using Litz wire or thin but wide foil conductors. 87

88 Losses (winding and core) raise core temperature. Common design practice to limit maximum interior temperature to C. Core losses (at constant flux density) increase with temperature above 100 C Saturation flux density B s decreases with temp. Nearby components such as power semi-conductor devices, integrated circuits, capacitors have similar limits. Temperature limitations in copper windings Copper resistivity increases with temperature increases. Losses, at constant current density increase with temperature. Reliability of insulating materials degrades with temperature increase. Thermal Considerations Surface temperature of component nearly equal to interior temperature. Minimal temperature gradient between interior and exterior surface. Power dissipated uniformly in component volume. Large cross-sectional area and short path lengths to surface of components. Core and winding materials have large thermal conductivity.. Thermal resistance (surface to ambient) of magnetic component determines its temperature. P sp = T s - T a R θsa (V w + V c ) ; R θsa = h A s h = convective heat transfer coefficient i = 10 C-m 2 /W A s = surface area of inductor (core + winding). Estimate using core dimensions i and s imple geometric considerations. Uncertain accuracy in h and other heat transfer parameters do not justify more accurate thermal modeling of inductor. 88

89 Core and Winding Scaling Power per unit volume, l P sp, dissipated i t d in magnetic component is P sp = k 1 /a ; k 1 = constant and a = core scaling dimension. J rms = P sp k cu r cu = k 2 1 k cu a ; k 2 = constant T P msp = P = k f b [B d Hence s - T a m,sp sp ac ] ; P w,sp V w + P m,sp V m = R : θsa d P sp k 3 T a = ambient temperature and R θsa = B ac = kf b = where k 3 = constant d surface-to-ambient thermal resistance of component. f b a For optimal design P w,sp = P c,sp = P sp : T s - T a Hence P sp = R θsa (V w +V c ) Plots of J rms, B ac, and P sp versus core size (scale factor a) for a specific core material, geometry, frequency, and T s - T a value very useful for picking appropriate core size and winding conductor size. R θsa proportional to a 2 and (V w + V c ) proportional p to a 3 89

90 Causes of Skin Effect Self-field field of wire causes current to crowd on the surface of the wire Method: Current i(t) changing magnetic flux density B(t) reaction currents Reference: 90

91 Effects of Skin Effect For high frequencies, current is concentrated in a layer approximately one skin depth δ thick Skin depth varies with frequency 2 1 f δ = = ωμσ π μσ σ = electrical l conductivity it = Ω -1 m -1 for copper at 300K μ = magnetic permeability of material = 4π 10-7 H/m in free space Skin depth in copper at 300K cond 5.90E+07 mu 1.26E-06 f skin depth (meter) E E E E E E E E E E-05 91

92 Skin Effect --- Increase in Wire Resistance For high frequencies, resistance of wire increases DC resistance of wire: l R DC = 2 σ πr ( 2πr δ ) w rw ( ) For frequencies above critical frequency where skin depth equals wire radius r w : R(f) l 2δ R = = AC RDC σ 2πr δ f = crit πr 1μσ 2 w R DC Result: for high frequency operation, don t bother making wire radius > δ Skin depth in copper at 60 Hz is approximately 8 mm w f crit f 92

93 Proximity Effect In multiple-layer windings in inductors and transformers, the proximity effect can also greatly increase the winding resistance Field from one wire affects the current profile in another CURRENTS IN OPPOSITE DIRECTION CURRENTS IN SAME DIRECTION 93

94 Losses vs. Winding Layer Thickness There is an optimal wire diameter to minimize losses in your magnetics Optimal wire thickness is generally somewhat smaller than a skin depth If wire is thick, DC losses are low but skin and proximity losses are high If wire is skinny, DC losses are high but skin and proximity effects are minimized Reference: 94

95 References Vacuumschmelze, Soft Magnetic Materials, 1979 Magnetic Materials Producers Association, Soft Ferrites Guide, Standard SFG-98 95