Part III. Magnetics. Chapter 13: Basic Magnetics Theory. Chapter 13 Basic Magnetics Theory

Transcription

1 Part III. Magnetics 3 Basic Magnetics Theory Inductor Design 5 Transformer Design Chapter 3 Basic Magnetics Theory 3. Review of Basic Magnetics 3.. Basic relationships 3..2 Magnetic circuits 3.2 Transformer Modeling 3.2. The ideal transformer Leakage inductances The magnetizing inductance 3.3 Loss Mechanisms in Magnetic Devices 3.3. Core loss Low-frequency copper loss 3. Eddy Currents in Winding Conductors 3.. Skin and proximity effects 3.. Power loss in a layer 3..2 Leakage flux in windings 3..5 Example: power loss in a transformer winding 3..3 Foil windings and layers 3..6 Interleaving the windings 3..7 PWM waveform harmonics 2 Chapter 3 Basic Magnetics Theory 3.5 Several Types of Magnetic Devices, Their BH Loops, and Core vs. Copper Loss 3.5. Filter inductor 3.5. Coupled inductor AC inductor Flyback transformer Transformer 3.6 Summary of Key Points 3

2 3. Review of Basic Magnetics 3.. Basic relationships v(t) Faraday s law B(t), Φ(t) Terminal characteristics Core characteristics Ampere s law H(t), F(t) Basic quantities Magnetic quantities Electrical quantities Length l Magnetic field H x x 2 MMF F = Hl Length l Electric field E x x 2 Voltage V = El Total flux Φ Flux density B { Surface S with area A c Total current I Current density J { Surface S with area A c 5 Magnetic field H and magnetomotive force F Magnetomotive force (MMF) F between points x and x 2 is related to the magnetic field H according to F = x 2 H dl x Example: uniform magnetic field of magnitude H Length l Magnetic field H x x 2 MMF F = Hl Analogous to electric field of strength E, which induces voltage (EMF) V: Length l Electric field E x x 2 Voltage V = El 6

3 Flux density B and total flux The total magnetic flux passing through a surface of area A c is related to the flux density B according to Φ = surface S B da Example: uniform flux density of magnitude B Total flux Φ Flux density B { Surface S with area A c Analogous to electrical conductor current density of magnitude J, which leads to total conductor current I: Total current I Current density J { Surface S with area A c 7 Faraday s law Voltage v(t) is induced in a loop of wire by change in the total flux (t) passing through the interior of the { loop, according to v(t)= dφ(t) dt Flux Φ(t) For uniform flux distribution, (t) = B(t)A c and hence v(t)=a c db(t) dt Area A c v(t) 8 Lenz s law The voltage v(t) induced by the changing flux (t) is of the polarity that tends to drive a current through the loop to counteract the flux change. Example: a shorted loop of wire Induced current Changing flux (t) induces a voltage v(t) around the loop This voltage, divided by the impedance of the loop conductor, leads to current This current induces a flux (t), which tends to oppose changes in (t) Flux Φ(t) Induced flux Φ (t) Shorted loop 9

4 Ampere s law The net MMF around a closed path is equal to the total current passing through the interior of the path: closed path H dl = total current passing through interior of path Example: magnetic core. Wire carrying current passes through core window. Illustrated path follows magnetic flux lines around interior of core For uniform magnetic field strength H(t), the integral (MMF) is H(t)l m. So F(t)=H(t)l m = H Magnetic path length l m 0 Ampere s law: discussion Relates magnetic field strength H(t) to winding current We can view winding currents as sources of MMF Previous example: total MMF around core, F(t) = H(t)l m, is equal to the winding current MMF The total MMF around a closed loop, accounting for winding current MMF s, is zero Core material characteristics: the relation between B and H Free space B B = µ 0 H A magnetic core material B µ µ 0 H H 0 = permeability of free space = 0 7 Henries per meter 2 Highly nonlinear, with hysteresis and saturation

5 Piecewise-linear modeling of core material characteristics No hysteresis or saturation B B = µh µ = µ r µ 0 Saturation, no hysteresis B B sat µ = µ r µ 0 µ H H B sat Typical r = 0 3 to Typical B sat = 0.3 to 0.5T, ferrite 0.5 to T, powdered iron to 2T, iron laminations Units Table 2.. Units for magnetic quantities quantity MKS unrationalized cgs conversions core material equation B = µ 0 µ r H B = µ r H B Tesla Gauss T = 0 G H Ampere / meter Oersted A/m = s 0-3 Oe Weber Maxwell Wb = 0 8 Mx T = Wb / m 2 Example: a simple inductor Faraday s law: For each turn of wire, we can write v turn (t)= dφ(t) dt Total winding voltage is v(t)=nv turn (t)=n dφ(t) dt v(t) n turns Φ core Core area A c Core permeability µ Express in terms of the average flux density B(t) = F(t)/A c v(t)=na c db(t) dt 5

6 Inductor example: Ampere s law Choose a closed path which follows the average magnetic field line around the interior of the core. Length of this path is called the mean magnetic path length l m. n turns H Magnetic path length l m For uniform field strength H(t), the core MMF around the path is H l m. Winding contains n turns of wire, each carrying current. The net current passing through the path interior (i.e., through the core window) is n. From Ampere s law, we have H(t) l m = n 6 Inductor example: core material model B B sat B sat for H B sat /µ B = µh for H < B sat /µ B sat for H B sat /µ µ H Find winding current at onset of saturation: substitute i = I sat and H = B sat / into equation previously derived via Ampere s law. Result is B sat I sat = B satl m µn 7 Electrical terminal characteristics We have: v(t)=na c db(t) dt H(t) l m = n B = B sat for H B sat /µ µh for H < B sat /µ B sat for H B sat /µ Eliminate B and H, and solve for relation between v and i. For i < I sat, dh(t) v(t)=µna c dt v(t)= µn2 A c d l m dt which is of the form v(t)=l d dt with L = µn2 A c l m an inductor For i > I sat the flux density is constant and equal to B sat. Faraday s law then predicts db v(t)=na sat c =0 dt saturation leads to short circuit 8

7 3..2 Magnetic circuits Uniform flux and magnetic field inside a rectangular element: Length l MMF F Area A c Flux Φ { MMF between ends of Core permeability µ element is F = Hl H R = µa l c Since H = B / and = / A c, we can express F as F = ΦR with R = µa l c A corresponding model: F R = reluctance of element Φ R 9 Magnetic circuits: magnetic structures composed of multiple windings and heterogeneous elements Represent each element with reluctance Windings are sources of MMF MMF voltage, flux current Solve magnetic circuit using Kirchoff s laws, etc. 20 Magnetic analog of Kirchoff s current law Divergence of B = 0 Flux lines are continuous and cannot end Total flux entering a node must be zero Physical structure Φ Φ 2 Node Φ 3 Magnetic circuit Node Φ = Φ 2 Φ 3 Φ Φ 3 Φ 2 2

8 Magnetic analog of Kirchoff s voltage law Follows from Ampere s law: closed path H dl = total current passing through interior of path Left-hand side: sum of MMF s across the reluctances around the closed path Right-hand side: currents in windings are sources of MMF s. An n-turn winding carrying current is modeled as an MMF (voltage) source, of value n. Total MMF s around the closed path add up to zero. 22 Example: inductor with air gap v(t) n turns Core permeability µ Φ Cross-sectional area A c Air gap l g Magnetic path length l m 23 Magnetic circuit model v(t) n turns Core permeability µ Φ Cross-sectional area A c Air gap l g n F c R c Φ(t) R g F g Magnetic path length l m F c F g = ni R c = l c µa c ni = Φ R c R g l g R g = µ 0 A c 2

9 Solution of model n v(t) turns Core permeability µ Φ Cross-sectional area A c Air gap l g n F c R c Φ(t) R g F g Magnetic path length l m Faraday s law: dφ(t) v(t)=n dt Substitute for : v(t)= n 2 d R c R g dt Hence inductance is L = n 2 R c R g ni = Φ R c R g R c = l c µa c R g = l g µ 0 A c 25 Effect of air gap ni = Φ R c R g B sat A L = n 2 c R c R g R c sat = B sat A c R c R g I sat = B sat A c n R c R g Effect of air gap: decrease inductance increase saturation current inductance is less dependent on core permeability Φ = BA c ni sat B sat A c ni sat2 ni H c Transformer modeling Two windings, no air gap: R = l m µa c F c = n i n 2 i 2 i (t) n v (t) turns Φ n 2 turns i 2 (t) v 2 (t) ΦR = n i n 2 i 2 Core Magnetic circuit model: Φ R c F c n i n 2 i 2 27

10 3.2. The ideal transformer In the ideal transformer, the core reluctance R approaches zero. MMF F c = R also approaches zero. We then obtain n i Φ R c F c n 2 i 2 0=n i n 2 i 2 Also, by Faraday s law, v = n dφ dt v 2 = n dφ 2 dt Eliminate : dφ = v = v 2 dt n n 2 Ideal transformer equations: v = v 2 and n n n i n 2 i 2 =0 2 v i n : n 2 i 2 Ideal v The magnetizing inductance For nonzero core reluctance, we obtain ΦR = n i n 2 i 2 with v = n dφ dt Eliminate : v = n 2 R dt d i n 2 i n 2 n i Φ R c F c n 2 i 2 This equation is of the form v = L M di M dt with L M = n 2 R i M = i n 2 i n 2 v n 2 i n i 2 n : n 2 i 2 i n 2 n i 2 L v 2 M = n 2 R Ideal 29 Magnetizing inductance: discussion Models magnetization of core material A real, physical inductor, that exhibits saturation and hysteresis If the secondary winding is disconnected: we are left with the primary winding on the core primary winding then behaves as an inductor the resulting inductor is the magnetizing inductance, referred to the primary winding Magnetizing current causes the ratio of winding currents to differ from the turns ratio 30

11 Transformer saturation Saturation occurs when core flux density B(t) exceeds saturation flux density B sat. When core saturates, the magnetizing current becomes large, the impedance of the magnetizing inductance becomes small, and the windings are effectively shorted out. Large winding currents i (t) and i 2 (t) do not necessarily lead to saturation. If 0=n i n 2 i 2 then the magnetizing current is zero, and there is no net magnetization of the core. Saturation is caused by excessive applied volt-seconds 3 Saturation vs. applied volt-seconds Magnetizing current depends on the integral of the applied winding voltage: i M (t)= L M Flux density is proportional: B(t)= n A c v (t)dt v (t)dt n 2 i n i 2 n : n 2 i 2 i n 2 n i 2 v L v 2 M = n 2 R Ideal Flux density becomes large, and core saturates, when the applied volt-seconds are too large, where t 2 λ = v (t)dt t limits of integration chosen to coincide with positive portion of applied voltage waveform Leakage inductances i (t) v (t) Φ l Φ M i 2 (t) Φ l2 v 2 (t) Φ l Φ l2 i (t) v (t) Φ M i 2 (t) v 2 (t) 33

12 Transformer model, including leakage inductance Terminal equations can be written in the form v (t) = L L 2 v 2 (t) L 2 L 22 d dt i (t) i 2 (t) v i L l n : n L l2 2 i 2 L M = n n 2 L 2 i M v 2 mutual inductance: Ideal L 2 = n n 2 R = n 2 n L M primary and secondary self-inductances: effective turns ratio n e = L 22 L L = L l n n 2 L 2 L 22 = L l2 n 2 n L 2 coupling coefficient k = L 2 L L Loss mechanisms in magnetic devices Low-frequency losses: Dc copper loss Core loss: hysteresis loss High-frequency losses: the skin effect Core loss: classical eddy current losses Eddy current losses in ferrite cores High frequency copper loss: the proximity effect Proximity effect: high frequency limit MMF diagrams, losses in a layer, and losses in basic multilayer windings Effect of PWM waveform harmonics Core loss Energy per cycle W flowing into n- turn winding of an inductor, excited by periodic waveforms of frequency f: W = one cycle v(t)dt v(t) n turns Φ core Core area A c Core permeability µ Relate winding voltage and current to core B and H via Faraday s law and Ampere s law: db(t) v(t)=na c dt H(t)l m = n Substitute into integral: db(t) H(t)l W = na m c dt n dt one cycle = A c l m HdB one cycle 36

13 Core loss: Hysteresis loss B W = A c l m one cycle HdB The term A c l m is the volume of the core, while the integral is the area of the BH loop. Area HdB one cycle H (energy lost per cycle) = (core volume) (area of BH loop) P H = f A c l m HdB one cycle Hysteresis loss is directly proportional to applied frequency 37 Modeling hysteresis loss Hysteresis loss varies directly with applied frequency Dependence on maximum flux density: how does area of BH loop depend on maximum flux density (and on applied waveforms)? Empirical equation (Steinmetz equation): P H = K H fb α max (core volume) The parameters K H and are determined experimentally. Dependence of P H on B max is predicted by the theory of magnetic domains. 38 Core loss: eddy current loss Magnetic core materials are reasonably good conductors of electric current. Hence, according to Lenz s law, magnetic fields within the core induce currents ( eddy currents ) to flow within the core. The eddy currents flow such that they tend to generate a flux which opposes changes in the core flux (t). The eddy currents tend to prevent flux from penetrating the core. Flux Φ(t) Eddy current Core Eddy current loss i 2 (t)r 39

14 Modeling eddy current loss Ac flux (t) induces voltage v(t) in core, according to Faraday s law. Induced voltage is proportional to derivative of (t). In consequence, magnitude of induced voltage is directly proportional to excitation frequency f. If core material impedance Z is purely resistive and independent of frequency, Z = R, then eddy current magnitude is proportional to voltage: = v(t)/r. Hence magnitude of is directly proportional to excitation frequency f. Eddy current power loss i 2 (t)r then varies with square of excitation frequency f. Classical Steinmetz equation for eddy current loss: P E = K E f 2 2 B max (core volume) Ferrite core material impedance is capacitive. This causes eddy current power loss to increase as f. 0 Total core loss: manufacturer s data Ferrite core material MHz 500kHz 200kHz Empirical equation, at a fixed frequency: Power loss density, Watts/cm kHz 50kHz 20kHz P fe = K fe ( B) β A c l m B, Tesla Core materials Core type B sat Relative core loss Applications Laminations iron, silicon steel Powdered cores powdered iron, molypermalloy Ferrite Manganese-zinc, Nickel-zinc T high Hz transformers, inductors T medium khz transformers, 00 khz filter inductors T low 20 khz - MHz transformers, ac inductors 2

15 3.3.2 Low-frequency copper loss DC resistance of wire R = ρ l b A w where A w is the wire bare cross-sectional area, and l b is the length of the wire. The resistivity is equal to cm for soft-annealed copper at room temperature. This resistivity increases to cm at 00 C. R The wire resistance leads to a power loss of 2 P cu = I rms R 3 3. Eddy currents in winding conductors 3.. Intro to the skin and proximity effects Wire Φ(t) Eddy currents Wire Current density δ Eddy currents Penetration depth For sinusoidal currents: current density is an exponentially decaying function of distance into the conductor, with characteristic length known as the penetration depth or skin depth. Penetration depth δ, cm 0. Wire diameter #20 AWG δ = ρ πµ f For copper at room temperature: δ = 7.5 cm f C 25 C khz 00 khz MHz Frequency #30 AWG #0 AWG 5

16 The proximity effect Ac current in a conductor induces eddy currents in adjacent conductors by a process called the proximity effect. This causes significant power loss in the windings of high-frequency transformers and ac inductors. A multi-layer foil winding, with h >. Each layer carries net current. Current density J iconductor h Area Φ Conductor 2 i i i Area i Area i 6 Example: a two-winding transformer Cross-sectional view of two-winding transformer example. Primary turns are wound in three layers. For this example, let s assume that each layer is one turn of a flat foil conductor. The secondary is a similar three-layer winding. Each layer carries net current. Portions of the windings that lie outside of the core window are not illustrated. Each layer has thickness h >. Core i i i i i i Layer Layer 2 Layer 3 Layer 3 Layer 2 Layer { { Primary winding Secondary winding 7 Distribution of currents on surfaces of conductors: two-winding example Skin effect causes currents to concentrate on surfaces of conductors Surface current induces equal and opposite current on adjacent conductor This induced current returns on opposite side of conductor Net conductor current is equal to for each layer, since layers are connected in series Circulating currents within layers increase with the numbers of layers Current density J Core 8 Layer h Layer i i i Layer 2 Layer 3 i i i Layer 3 Layer 2 Φ 2Φ 3Φ 2Φ Φ i i 2i 2i 3i 3i 2i 2i i i Layer 2 Primary winding Layer 3 Layer 3 Layer 2 Layer Layer Secondary winding

17 Estimating proximity loss: high-frequency limit The current having rms value I is confined to thickness d on the surface of layer. Hence the effective ac resistance of layer is: R ac = h δ R dc This induces copper loss P in layer : P = I 2 R ac Power loss P 2 in layer 2 is: P 2 = P P =5P Power loss P 3 in layer 3 is: Current density J h Layer Φ 2Φ 3Φ 2Φ Φ i i 2i 2i 3i 3i 2i 2i i i Layer 2 Primary winding Layer 3 Layer 3 Layer 2 Secondary winding Power loss P m in layer m is: P 3 = P =3P P m = I 2 m 2 m 2 h δ R dc Layer 9 Total loss in M-layer winding: high-frequency limit Add up losses in each layer: Σ P = I 2 h δ R Μ dc m 2 m 2 m = = I 2 h δ R M dc 2M 2 3 Compare with dc copper loss: If foil thickness were H =, then at dc each layer would produce copper loss P. The copper loss of the M-layer winding would be P dc = I 2 MR dc So the proximity effect increases the copper loss by a factor of F R = P P dc = 3 h δ 2M Leakage flux in windings A simple two-winding transformer example: core and winding geometry Each turn carries net current in direction shown Primary winding Secondary winding { { y x Core µ > µ 0 5

18 Flux distribution Mutual flux M is large and is mostly confined to the core Leakage flux is present, which does not completely link both windings Because of symmetry of winding geometry, leakage flux runs approximately vertically through the windings Leakage flux Mutual flux Φ M 52 Analysis of leakage flux using Ampere s law Ampere s law, for the closed path taken by the leakage flux line illustrated: Enclosed current = F(x)=H(x)l w Leakage path (note that MMF around core is small compared to MMF through the air inside the winding, because of high permeability of core) x Enclosed current H(x) F(x) l w 53 Ampere s law for the transformer example For the innermost leakage path, enclosing the first layer of the primary: This path encloses four turns, so the total enclosed current is. For the next leakage path, enclosing both layers of the primary: This path encloses eight turns, so the total enclosed current is 8. The next leakage path encloses the primary plus four turns of the secondary. The total enclosed current is 8 =. Leakage flux Mutual flux Φ M 5

19 MMF diagram, transformer example MMF F(x) across the core window, as a function of position x Enclosed current = F(x)=H(x)l w Primary winding Secondary winding { { Layer Layer 2 Layer 2 Layer Leakage path Enclosed current F(x) l w F(x) 8i H(x) i x 0 x 55 Two-winding transformer example Winding layout Core i i i i i i Layer Layer 2 Layer 3 Layer 3 Layer 2 Layer MMF diagram Use Ampere s law around a closed path taken by a leakage flux line: m p m s i = F(x) m p = number of primary layers enclosed by path m s = number of secondary layers enclosed by path i i i i i i MMF F(x) 3i 2i i 0 x 56 Two-winding transformer example with proximity effect Flux does not penetrate conductors Surface currents cause net current enclosed by leakage path to be zero when path runs down interior of a conductor Magnetic field strength H(x) within the winding is given by H(x)= F(x) l w MMF F(x) 3i 2i i 0 i i 2i 2i 3i 3i 2i 2i i i x 57

20 Interleaving the windings: MMF diagram pri sec pri sec pri sec MMF F(x) i i i i i i 3i 2i i 0 x Greatly reduces the peak MMF, leakage flux, and proximity losses 58 A partially-interleaved transformer For this example, there are three primary layers and four secondary layers. The MMF diagram contains fractional values. MMF F(x).5 i i 0.5 i i i.5 i Secondary 3i m = 3i m = 2 Primary i i i 3i m =.5 m = 0.5 m =.5 Secondary m = 2 3i m = x Foil windings and layers Approximating a layer of round conductors as an effective single foil conductor: d (a) (b) (c) (d) h h h l w Square conductors (b) have same crosssectional area as round conductors (a) if h = π d Eliminate space between square conductors: push together into a single foil turn (c) (d) Stretch foil so its width is l w. The adjust conductivity so its dc resistance is unchanged 60

21 Winding porosity Stretching the conductor increases its area. Compensate by increasing the effective resistivity, to maintain the same dc resistance. Define winding porosity as the ratio of cross-sectional areas. If layer of width l w contains n l turns of round wire having diameter d, then the porosity is (c) h (d) h η = π d n l l w Typical for full-width round conductors is = 0.8. The increased effective resistivity increases the effective skin depth: δ = δ η l w Define = h/d. The effective value for a layer of round conductors is ϕ = h δ = η π d δ Power loss in a layer Approximate computation of copper loss in one layer Assume uniform magnetic fields at surfaces of layer, of strengths H(0) and H(h). Assume that these fields are parallel to layer surface (i.e., neglect fringing and assume field normal component is zero). The magnetic fields H(0) and H(h) are driven by the MMFs F(0) and F(h). Sinusoidal waveforms are assumed, and rms values are used. It is assumed that H(0) and H(h) are in phase. H(0) F(x) F(0) Layer H(h) F(h) 0 h 62 Solution for layer copper loss P Solve Maxwell s equations to find current density distribution within layer. Then integrate to find total copper loss P in layer. Result is ϕ P = R dc 2 F 2 (h)f 2 (0) G n (ϕ)f(h)f(0)g 2 (ϕ) l where R dc = ρ l b = ρ (MLT)n 3 l A 2 w ηl w sinh(2ϕ) sin (2ϕ) G (ϕ)= cosh (2ϕ)cos (2ϕ) n l = number of turns in layer, R dc = dc resistance of layer, (MLT) = mean-length-per-turn, or circumference, of layer. sinh (ϕ) cos (ϕ)cosh (ϕ) sin (ϕ) G 2 (ϕ)= cosh (2ϕ)cos (2ϕ) ϕ = h δ = η π d δ η = π d n l l w 63

22 Winding carrying current I, with n l turns per layer If winding carries current of rms magnitude I, then F(h)F(0) = n l I Express F(h) in terms of the winding current I, as F(h)=mn l I The quantity m is the ratio of the MMF F(h) to the layer ampere-turns n l I. Then, F(0) F(h) = m m Power dissipated in the layer can now be written P = I 2 R dc ϕq (ϕ, m) Q (ϕ, m)= 2m 2 2m G (ϕ)mm G 2 (ϕ) H(0) F(x) F(0) Layer 0 h H(h) F(h) 6 Increased copper loss in layer P = I 2 R dc ϕq (ϕ, m) 00 m = P I 2 R dc m = ϕ 65 Layer copper loss vs. layer thickness P P dc ϕ = = Q (ϕ, m) P 00 m = P dc ϕ = 0 3 Relative to copper loss when h = 2.5 m = ϕ

23 3..5 Example: Power loss in a transformer winding Two winding transformer Each winding consists of M layers Proximity effect increases copper loss in layer m by the factor Q (,m) Sum losses over all primary layers: F Mn p i 2n p i n p i 0 F R = P pri = P M ϕq (ϕ, m) pri,dc M Σ m = Primary layers { Secondary layers { n p i n p i n p i n p i n p i n p i m = m = 2 m = M m = M m = 2 m = x 67 Express summation in closed form: Increased total winding loss F R = ϕ G (ϕ) 2 3 M 2 G (ϕ)2g 2 (ϕ) 00 Number of layers M = F R = P pri P pri,dc ϕ 0 68 Total winding loss P pri = G (ϕ) 2 P pri,dc 3 M 2 G (ϕ)2g 2 (ϕ) ϕ = P pri P pri,dc ϕ = 00 0 Number of layers M = ϕ 69

24 3..6 Interleaving the windings Same transformer example, but with primary and secondary layers alternated Each layer operates with F = 0 on one side, and F = i on the other side Proximity loss of entire winding follows M = curve MMF F(x) 3i 2i i 0 pri sec pri sec pri sec i i i i i i For M = : minimum loss occurs with = /2, although copper loss is nearly constant for any, and is approximately equal to the dc copper loss obtained when =. x 70 Partial interleaving Partially-interleaved example with 3 primary and secondary layers Each primary layer carries current i while each secondary layer carries 0.75i. Total winding currents add to zero. Peak MMF occurs in space between windings, but has value.5i. 3i MMF F(x).5 i i 0.5 i i i.5 i Secondary m = 3i m = 2 Primary i i i 3i m =.5 m = 0.5 m =.5 Secondary We can apply the previous solution for the copper loss in each layer, and add the results to find the total winding losses. To determine the value of m to use for a given layer, evaluate F(h) m = F(h) F(0) m = 2 3i m = x 7 Determination of m Leftmost secondary layer: F(h) m = F(h)F(0) = 0.75i 0.75i 0 = Next secondary layer: F(h) m = F(h)F(0) =.5i.5i (0.75i) = 2 Next layer (primary): F(0) m = F(0) F(h) =.5i.5i (0.5i) =.5 3i MMF F(x).5 i i 0.5 i i i.5 i Secondary m = 3i m = 2 Primary i i i 3i m =.5 m = 0.5 m =.5 Secondary Center layer (primary): F(h) m = F(h)F(0) = 0.5i = 0.5 Use the plot for layer loss (repeated on 0.5i (0.5i) next slide) to find loss for each layer, according to its value of m. Add results to find total loss. m = 2 3i m = x 72

25 Layer copper loss vs. layer thickness P P dc ϕ = = Q (ϕ, m) P 00 m = P dc ϕ = 0 3 Relative to copper loss when h = 2.5 m = ϕ Discussion: design of winding geometry to minimize proximity loss Interleaving windings can significantly reduce the proximity loss when the winding currents are in phase, such as in the transformers of buckderived converters or other converters In some converters (such as flyback or SEPIC) the winding currents are out of phase. Interleaving then does little to reduce the peak MMF and proximity loss. See Vandelac and Ziogas [0]. For sinusoidal winding currents, there is an optimal conductor thickness near = that minimizes copper loss. Minimize the number of layers. Use a core geometry that maximizes the width l w of windings. Minimize the amount of copper in vicinity of high MMF portions of the windings 7 Litz wire A way to increase conductor area while maintaining low proximity losses Many strands of small-gauge wire are bundled together and are externally connected in parallel Strands are twisted, or transposed, so that each strand passes equally through each position on inside and outside of bundle. This prevents circulation of currents between strands. Strand diameter should be sufficiently smaller than skin depth The Litz wire bundle itself is composed of multiple layers Advantage: when properly sized, can significantly reduce proximity loss Disadvantage: increased cost and decreased amount of copper within core window 75

26 3..7 PWM waveform harmonics Fourier series: =I 0 Σ j = with 2 I j cos (jωt) I pk I j = 2 I pk sin (jπd) jπ I 0 = DI pk Copper loss: Dc P dc = I 2 0 R dc 0 DT s T s t Ac P j = I j 2 R dc j ϕ G ( j ϕ ) 2 3 M 2 G ( j ϕ )2G 2 ( j ϕ ) Total, relative to value predicted by low-frequency analysis: P cu DI 2 = D 2ϕ sin 2 (jπd) Σ pk R dc Dπ 2 j = j j G ( j ϕ ) 2 3 M 2 G ( j ϕ )2G 2 ( j ϕ ) 76 Harmonic loss factor F H Effect of harmonics: F H = ratio of total ac copper loss to fundamental copper loss Σ P j j = F H = P The total winding copper loss can then be written P cu = I 0 2 R dc F H F R I 2 R dc 77 Increased proximity losses induced by PWM waveform harmonics: D = D = 0.5 M = 0 F H M = ϕ

27 Increased proximity losses induced by PWM waveform harmonics: D = 0.3 F H 00 D = M = M = ϕ 79 Increased proximity losses induced by PWM waveform harmonics: D = 0. F H 00 0 M = 0 D = M = ϕ 80 Discussion: waveform harmonics Harmonic factor F H accounts for effects of harmonics Harmonics are most significant for in the vicinity of Harmonics can radically alter the conclusion regarding optimal wire gauge A substantial dc component can drive the design towards larger wire gauge Harmonics can increase proximity losses by orders of magnitude, when there are many layers and when lies in the vicinity of For sufficiently small, F H tends to the value (THD) 2, where the total harmonic distortion of the current is THD = Σ I j 2 j =2 I 8

28 3.5. Several types of magnetic devices, their BH loops, and core vs. copper loss A key design decision: the choice of maximum operating flux density B max Choose B max to avoid saturation of core, or Further reduce B max, to reduce core losses Different design procedures are employed in the two cases. Types of magnetic devices: Filter inductor AC inductor Conventional transformer Coupled inductor Flyback transformer SEPIC transformer Magnetic amplifier Saturable reactor 82 Filter inductor CCM buck example L B Minor BH loop, filter inductor B sat B H c H c0 H c I i L BH loop, large excitation 0 DT s T s t H c (t)= n l c R c R c R g 83 Filter inductor, cont. Negligible core loss, negligible proximity loss Loss dominated by dc copper loss Flux density chosen simply to avoid saturation Air gap is employed Could use core materials having high saturation flux density (and relatively high core loss), even though converter switching frequency is high v(t) n turns n Core reluctance R c Φ F c R c Φ(t) R g Air gap reluctance R g 8

29 AC inductor L B i B sat BH loop, for B operation as ac inductor H c i t H c H c B Core BH loop 85 AC inductor, cont. Core loss, copper loss, proximity loss are all significant An air gap is employed Flux density is chosen to reduce core loss A high-frequency material (ferrite) must be employed 86 Conventional transformer i (t) i M (t) n : n 2 i 2 (t) B v (t) L M v (t) Area λ v 2 (t) BH loop, for operation as conventional transformer λ 2n A c n i mp l m H c Core BH loop i M (t) i M t H(t)= ni M(t) l m 87

30 Conventional transformer, cont. Core loss, copper loss, and proximity loss are usually significant No air gap is employed Flux density is chosen to reduce core loss A high frequency material (ferrite) must be employed 88 Coupled inductor Two-output forward converter example n i v i (t) I i i 2 (t) I i 2 2 v g n 2 turns i 2 v 2 Minor BH loop, coupled inductor B B t H c H c (t)= n i (t)n 2 i 2 (t) l c R c R c R g H c0 BH loop, large excitation H c 89 Coupled inductor, cont. A filter inductor having multiple windings Air gap is employed Core loss and proximity loss usually not significant Flux density chosen to avoid saturation Low-frequency core material can be employed 90

31 DCM flyback transformer n : n 2 i (t) i i M i 2 i,pk v g L M v i 2 (t) t B i M (t) i,pk t BH loop, for operation in DCM flyback converter B B n i,pk R c H c l m R cr g Core BH loop 9 DCM flyback transformer, cont. Core loss, copper loss, proximity loss are significant Flux density is chosen to reduce core loss Air gap is employed A high-frequency core material (ferrite) must be used 92 Summary of Key Points. Magnetic devices can be modeled using lumped-element magnetic circuits, in a manner similar to that commonly used to model electrical circuits. The magnetic analogs of electrical voltage V, current I, and resistance R, are magnetomotive force (MMF) F, flux, and reluctance R respectively. 2. Faraday s law relates the voltage induced in a loop of wire to the derivative of flux passing through the interior of the loop. 3. Ampere s law relates the total MMF around a loop to the total current passing through the center of the loop. Ampere s law implies that winding currents are sources of MMF, and that when these sources are included, then the net MMF around a closed path is equal to zero.. Magnetic core materials exhibit hysteresis and saturation. A core material saturates when the flux density B reaches the saturation flux density B sat. 93

32 Summary of key points 5. Air gaps are employed in inductors to prevent saturation when a given maximum current flows in the winding, and to stabilize the value of inductance. The inductor with air gap can be analyzed using a simple magnetic equivalent circuit, containing core and air gap reluctances and a source representing the winding MMF. 6. Conventional transformers can be modeled using sources representing the MMFs of each winding, and the core MMF. The core reluctance approaches zero in an ideal transformer. Nonzero core reluctance leads to an electrical transformer model containing a magnetizing inductance, effectively in parallel with the ideal transformer. Flux that does not link both windings, or leakage flux, can be modeled using series inductors. 7. The conventional transformer saturates when the applied winding voltseconds are too large. Addition of an air gap has no effect on saturation. Saturation can be prevented by increasing the core cross-sectional area, or by increasing the number of primary turns. 9 Summary of key points 8. Magnetic materials exhibit core loss, due to hysteresis of the BH loop and to induced eddy currents flowing in the core material. In available core materials, there is a tradeoff between high saturation flux density B sat and high core loss P fe. Laminated iron alloy cores exhibit the highest B sat but also the highest P fe, while ferrite cores exhibit the lowest P fe but also the lowest B sat. Between these two extremes are powdered iron alloy and amorphous alloy materials. 9. The skin and proximity effects lead to eddy currents in winding conductors, which increase the copper loss P cu in high-current high-frequency magnetic devices. When a conductor has thickness approaching or larger than the penetration depth, magnetic fields in the vicinity of the conductor induce eddy currents in the conductor. According to Lenz s law, these eddy currents flow in paths that tend to oppose the applied magnetic fields. 95 Summary of key points 0. The magnetic field strengths in the vicinity of the winding conductors can be determined by use of MMF diagrams. These diagrams are constructed by application of Ampere s law, following the closed paths of the magnetic field lines which pass near the winding conductors. Multiple-layer noninterleaved windings can exhibit high maximum MMFs, with resulting high eddy currents and high copper loss.. An expression for the copper loss in a layer, as a function of the magnetic field strengths or MMFs surrounding the layer, is given in Section 3... This expression can be used in conjunction with the MMF diagram, to compute the copper loss in each layer of a winding. The results can then be summed, yielding the total winding copper loss. When the effective layer thickness is near to or greater than one skin depth, the copper losses of multiple-layer noninterleaved windings are greatly increased. 96

33 Summary of key points 2. Pulse-width-modulated winding currents of contain significant total harmonic distortion, which can lead to a further increase of copper loss. The increase in proximity loss caused by current harmonics is most pronounced in multiple-layer non-interleaved windings, with an effective layer thickness near one skin depth. 3. A variety of magnetic devices are commonly used in switching converters. These devices differ in their core flux density variations, as well as in the magnitudes of the ac winding currents. When the flux density variations are small, core loss can be neglected. Alternatively, a low-frequency material can be used, having higher saturation flux density. 97