R. W. Erickson Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder
Part III. Magnetics 13 Basic Magnetics Theory 14 Inductor Design 15 Transformer Design 1
Chapter 13 Basic Magnetics Theory 13.1 Review of Basic Magnetics 13.1.1 Basic relationships 13.1.2 Magnetic circuits 13.2 Transformer Modeling 13.2.1 The ideal transformer 13.2.3 Leakage inductances 13.2.2 The magnetizing inductance 13.3 Loss Mechanisms in Magnetic Devices 13.3.1 Core loss 13.3.2 Low-frequency copper loss 13.4 Eddy Currents in Winding Conductors 13.4.1 Skin and proximity effects 13.4.4 Power loss in a layer 13.4.2 Leakage flux in windings 13.4.5 Example: power loss in a transformer winding 13.4.3 Foil windings and layers 13.4.6 Interleaving the windings 13.4.7 PWM waveform harmonics 2
Chapter 13 Basic Magnetics Theory 13.5 Several Types of Magnetic Devices, Their BH Loops, and Core vs. Copper Loss 13.5.1 Filter inductor 13.5.4 Coupled inductor 13.5.2 AC inductor 13.5.5 Flyback transformer 13.5.3 Transformer 13.6 Summary of Key Points 3
13.1 Review of Basic Magnetics 13.1.1 Basic relationships v(t) Faraday s law B(t), Φ(t) Terminal characteristics Core characteristics i(t) Ampere s law H(t), F(t) 4
Basic quantities Magnetic quantities Electrical quantities Length l Magnetic field H x 1 x 2 MMF F = Hl Length l Electric field E x 1 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 1 and x 2 is related to the magnetic field H according to F = x 2 x 1 H dl Example: uniform magnetic field of magnitude H Length l Magnetic field H x 1 x 2 MMF F = Hl Analogous to electric field of strength E, which induces voltage (EMF) V: Length l Electric field E x 1 x 2 Voltage V = El 6
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 i(t) Changing flux (t) induces a voltage v(t) around the loop This voltage, divided by the impedance of the loop conductor, leads to current i(t) This current induces a flux (t), which tends to oppose changes in (t) Flux Φ(t) Induced flux Φ (t) Shorted loop 9
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 i(t) 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 = i(t) i(t) H Magnetic path length l m 10
Ampere s law: discussion Relates magnetic field strength H(t) to winding current i(t) 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 i(t) The total MMF around a closed loop, accounting for winding current MMF s, is zero 11
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 = 4 10 7 Henries per meter 12 Highly nonlinear, with hysteresis and saturation
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 = 10 3 to 10 5 13 Typical B sat = 0.3 to 0.5T, ferrite 0.5 to 1T, powdered iron 1 to 2T, iron laminations
Units Table 12.1. 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 1T = 10 4 G H Ampere / meter Oersted 1A/m = 4π 10-3 Oe Weber Maxwell 1Wb = 10 8 Mx 1T = 1Wb / m 2 14
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) i(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 15
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. i(t) 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 i(t). The net current passing through the path interior (i.e., through the core window) is ni(t). From Ampere s law, we have H(t) l m = n i(t) 16
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 17
Electrical terminal characteristics We have: v(t)=na c db(t) dt H(t) l m = n i(t) 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 di(t) l m dt which is of the form v(t)=l di(t) with dt 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 v(t)=na c db sat dt =0 saturation leads to short circuit 18
13.1.2 Magnetic circuits Uniform flux and magnetic field inside a rectangular element: Flux Φ { Length l MMF F Area A c MMF between ends of element is F = Hl Since H = B / and = / A c, we can express F as F = ΦR with R = µa l c A corresponding model: F H Core permeability µ R = µa l c R = reluctance of element Φ R 19
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 1 Φ 2 Node Φ 3 Magnetic circuit Node Φ 1 = Φ 2 Φ 3 Φ 1 Φ 3 Φ 2 21
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 i(t) is modeled as an MMF (voltage) source, of value ni(t). Total MMF s around the closed path add up to zero. 22
Example: inductor with air gap i(t) 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 i(t) v(t) n turns Core permeability µ Φ Cross-sectional area A c Air gap l g ni(t) 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 R g = l g µ 0 A c 24
Solution of model i(t) v(t) n turns Core permeability µ Φ Cross-sectional area A c Air gap l g ni(t) F c R c Φ(t) R g F g Magnetic path length l m Faraday s law: Substitute for : Hence inductance is v(t)=n dφ(t) dt v(t)= L = n 2 di(t) R c R g dt n 2 R c R g ni = Φ R c R g l c R c = µa c l g R g = µ 0 A c 25
Effect of air gap ni = Φ R c R g L = n 2 R c R g sat = B sat A c 1 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 B sat A c 1 R c ni sat1 B sat A c ni sat2 ni H c 26
13.2 Transformer modeling Two windings, no air gap: R = l m µa c F c = n 1 i 1 n 2 i 2 i 1 (t) n v 1 1 (t) turns Φ n 2 turns i 2 (t) v 2 (t) ΦR = n 1 i 1 n 2 i 2 Core Magnetic circuit model: Φ R c F c n 1 i 1 n 2 i 2 27
13.2.1 The ideal transformer In the ideal transformer, the core reluctance R approaches zero. Φ R c F c MMF F c = R also approaches zero. We then obtain n 1 i 1 n 2 i 2 0=n 1 i 1 n 2 i 2 Also, by Faraday s law, v 1 = n dφ 1 dt v 2 = n dφ 2 dt Eliminate : dφ = v 1 = v 2 dt n 1 n 2 Ideal transformer equations: v 1 = v 2 and n n 1 n 1 i 1 n 2 i 2 =0 2 v 1 i 1 n 1 : n 2 i 2 Ideal v 2 28
13.2.2 The magnetizing inductance For nonzero core reluctance, we obtain ΦR = n 1 i 1 n 2 i 2 with v 1 = n 1 dφ dt Φ R c F c Eliminate : n 1 i 1 n 2 i 2 v 1 = n 2 1 R dt d i 1 n 2 i n 2 1 This equation is of the form with v 1 = L M di M dt L M = n 1 2 R i M = i 1 n 2 n 1 i 2 v 1 i 1 n i 2 1 n 1 : n 2 i 2 L M = n 2 1 R 29 n 2 i 1 n 2 n 1 i 2 Ideal v 2
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
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 1 (t) and i 2 (t) do not necessarily lead to saturation. If 0=n 1 i 1 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 31
Saturation vs. applied volt-seconds Magnetizing current depends on the integral of the applied winding voltage: i M (t)= 1 L M Flux density is proportional: B(t)= 1 n 1 A c v 1 (t)dt v 1 (t)dt v 1 i 1 n i 2 1 n 1 : n 2 i 2 L M = n 2 1 R Ideal Flux density becomes large, and core saturates, when the applied volt-seconds 1 are too large, where λ 1 = t 2 t 1 n 2 i 1 n 2 n 1 i 2 v 1 (t)dt limits of integration chosen to coincide with positive portion of applied voltage waveform v 2 32
13.2.3 Leakage inductances v 1 (t) i 1 (t) Φ l1 Φ M Φ l2 i 2 (t) v 2 (t) Φ l1 Φ l2 i 1 (t) v 1 (t) Φ M i 2 (t) v 2 (t) 33
Transformer model, including leakage inductance Terminal equations can be written in the form v 1 (t) v 2 (t) = L 11 L 12 L 12 L 22 d dt i 1 (t) i 2 (t) v 1 L l1 i 1 n 1 : n 2 i 2 L M = n 1 n 2 L 12 i M L l2 v 2 mutual inductance: Ideal L 12 = n 1n 2 R = n 2 n 1 L M primary and secondary self-inductances: effective turns ratio n e = L 22 L 11 L 11 = L l1 n 1 n 2 L 12 L 22 = L l2 n 2 n 1 L 12 coupling coefficient k = L 12 L 11 L 22 34