Chapter 12. Basic Magnetics Theory 12.1. Review of basic magnetics 12.1.1. Basic relations 12.1.2. Magnetic circuits 12.2. Transformer modeling 12.2.1. The ideal transformer 12.2.3. Leakage inductances 12.2.2. The magnetizing inductance 12.3. Loss mechanisms in magnetic devices 12.3.1. Core loss 12.3.2. Low-frequency copper loss 12.4. Eddy currents in winding conductors 12.4.1. The skin effect 12.4.4. Power loss in a layer 12.4.2. The proximity effect 12.4.5. Example: power loss in a transformer winding 12.4.3. MMF diagrams 12.4.6. PWM waveform harmonics Fundamentals of Power Electronics2 Chapter 12: Basic Magnetics Theory 12.1. Review of basic magnetics 12.1.1. Basic relations Faraday's law v(t) B(t), Φ(t) terminal core characteristics characteristics i(t) H(t), F(t) Ampere's law Fundamentals of Power Electronics3 Chapter 12: Basic Magnetics Theory Basic quantities Magnetic quantities Electrical quantities length l length l magnetic field H electric field E x1 x2 x1 x2 +–+– MMF voltage F = Hl V = El surface S surface S with area Ac with area Ac total flux Φ total current I flux density B { current density J { Fundamentals of Power Electronics4 Chapter 12: Basic Magnetics Theory Magnetic field H and magnetomotive force F Magnetomotive force (MMF) F between points x1 and x2 is related to the magnetic field H according to x2 F = H ⋅ dl x1 Example: uniform magnetic Analogous to electric field of field of magnitude H strength E, which induces voltage (EMF) V: length l length l magnetic field H electric field E x1 x2 x1 x2 +– +– MMF voltage F = Hl V = El Fundamentals of Power Electronics5 Chapter 12: Basic Magnetics Theory Flux density B and total flux Φ Φ The total magnetic flux passing through a surface of area Ac is related to the flux density B according to Φ = B ⋅ dA surface S Example: uniform flux density of Analogous to electrical conductor magnitude B current density of magnitude J, Φ = BA which leads to total conductor c current I: surface S surface S with area A with area Ac c total flux Φ total current I flux density B { current density J { Fundamentals of Power Electronics6 Chapter 12: Basic Magnetics Theory Faraday’s law Voltage v(t) is induced in a loop of wire by change in area Ac the total flux Φ(t) passing through the interior of the loop, according to dΦ(t) v(t)= flux Φ(t) dt For uniform flux distribution, – Φ { (t) = B(t)Ac and hence v(t) + dB(t) v(t)=A c dt Fundamentals of Power Electronics7 Chapter 12: Basic Magnetics Theory Lenz’s law The voltage v(t) induced by the changing flux Φ(t) is of the polarity that tends to drive a current through the loop to counteract the flux change. induced current Example: a shorted loop of wire i(t) •Changing flux Φ(t) induces a voltage v(t) around the loop •This voltage, divided by the impedance of the loop flux Φ(t) shorted conductor, leads to current i(t) loop •This current induces a flux Φ’(t), which tends to oppose changes in Φ(t) induced flux Φ'(t) Fundamentals of Power Electronics 8 Chapter 12: Basic Magnetics Theory Ampere’s law The net MMF around a closed path is equal to the total current passing through the interior of the path: H ⋅ dl = total current passing through interior of path closed path Example: magnetic core. Wire carrying current i(t) passes H through core window. i(t) • Illustrated path follows magnetic path magnetic flux lines length lm around interior of core • For uniform magnetic field strength H(t), the integral (MMF) is H(t)lm. So F (t)=H(t)lm =i(t) Fundamentals of Power Electronics9 Chapter 12: Basic Magnetics Theory 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)lm, is equal to the winding current MMF i(t) • The total MMF around a closed loop, accounting for winding current MMF’s, is zero Fundamentals of Power Electronics10 Chapter 12: Basic Magnetics Theory Core material characteristics: the relation between B and H Free space A magnetic core material B B µ B = 0 H µ µ H H 0 µ 0 = permeability of free space Highly nonlinear, with hysteresis = 4π · 10-7 Henries per meter and saturation Fundamentals of Power Electronics11 Chapter 12: Basic Magnetics Theory Piecewise-linear modeling of core material characteristics No hysteresis or saturation Saturation, no hysteresis B B Bsat B = µ H µ µ µ = r 0 µ µ µ µ = r 0 H H – Bsat µ 3 5 Typical Bsat = 0.3-0.5T, ferrite Typical r = 10 - 10 0.5-1T, powdered iron 1-2T, iron laminations Fundamentals of Power Electronics12 Chapter 12: Basic Magnetics Theory Units Table 12.1. Units for magnetic quantities quantity MKS unrationalized cgs conversions µ µ µ core material equation B = 0 r HB = r H B Tesla Gauss 1T = 104G H Ampere / meter Oersted 1A/m = 4π⋅10-3 Oe Φ Weber Maxwell 1Wb = 108 Mx 1T = 1Wb / m2 Fundamentals of Power Electronics13 Chapter 12: Basic Magnetics Theory Example: a simple inductor Faraday’s law: Φ For each turn of core area wire, we can write i(t) Ac + Φ n d (t) v(t) vturn(t)= turns dt – core permeability µ Total winding voltage is core dΦ(t) v(t)=nv (t)=n turn dt Φ Express in terms of the average flux density B(t) = (t)/Ac dB(t) v(t)=nA c dt Fundamentals of Power Electronics 14 Chapter 12: Basic Magnetics Theory Inductor example: Ampere’s law Choose a closed path H which follows the average magnetic field line around i(t) the interior of the core. magnetic path n length lm Length of this path is turns called the mean magnetic path length lm. For uniform field strength H(t), the core MMF around the path is H lm. 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) lm = n i(t) Fundamentals of Power Electronics15 Chapter 12: Basic Magnetics Theory Inductor example: core material model B Bsat ≥ µ Bsat for H Bsat / µ µ µ B = H for H < Bsat / ≤ µ ± Bsat for H Bsat / H – B Find winding current at onset of saturation: sat µ substitute i = Isat and H = Bsat/ into equation previously derived via Ampere’s law. Result is Bsat lm Isat = µ n Fundamentals of Power Electronics16 Chapter 12: Basic Magnetics Theory Electrical terminal characteristics We have: B for H ≥ B / µ dB(t) sat sat v(t)=nA µ µ c dt H(t) lm = n i(t) B = H for H < Bsat / ≤ µ ± Bsat for H Bsat / Eliminate B and H, and solve for relation between v and i. For | i | < Isat, dH(t) µ 2 v(t)=µnA n Ac di(t) c dt v(t)= lm dt which is of the form di(t) µ 2 v(t)=L n A c dt with L = lm —an inductor For | i | > Isat the flux density is constant and equal to Bsat. Faraday’s law then predicts dB v(t)=nA sat =0 —saturation leads to short circuit c dt Fundamentals of Power Electronics17 Chapter 12: Basic Magnetics Theory 12.1.2. Magnetic circuits length l Uniform flux and +–MMF F area magnetic field inside Ac a rectangular element: flux Φ { MMF between ends of core permeability µ element is F=Hl l H R=µ Ac Since H = B / µ and Β = Φ / Ac, we can express F as l Φ F= µ with R= l A c µ A c A corresponding model: +–F R = reluctance of element Φ R Fundamentals of Power Electronics 18 Chapter 12: Basic Magnetics Theory 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. Fundamentals of Power Electronics19 Chapter 12: Basic Magnetics Theory Magnetic analog of Kirchoff’s current law Physical structure node Divergence of B = 0 Φ Φ 1 3 Flux lines are continuous and cannot end Φ Total flux entering a node 2 must be zero Magnetic circuit Φ Φ Φ node 1 = 2 + 3 Φ Φ 1 3 Φ 2 Fundamentals of Power Electronics20 Chapter 12: Basic Magnetics Theory Magnetic analog of Kirchoff’s voltage law Follows from Ampere’s law: H ⋅ dl = total current passing through interior of path closed 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. Fundamentals of Power Electronics21 Chapter 12: Basic Magnetics Theory Example: inductor with air gap core µ permeability c Φ cross-sectional i(t) area Ac + n v(t) turns air gap – lg magnetic path length lm Ampere’s law: F c + F g = n i Fundamentals of Power Electronics22 Chapter 12: Basic Magnetics Theory Magnetic circuit model F core + c – µ permeability c Φ R c cross-sectional i(t) area Ac + + n v(t) turns air gap n i(t) + Φ(t) R F – lg – g g magnetic path – length lm l F + F = n i c c g R c = µ A c l Φ R = g ni= Rc +Rg g µ 0 A c Fundamentals of Power Electronics23 Chapter 12: Basic Magnetics Theory Solution of model core µ permeability c Fc Φ + – cross-sectional R i(t) c area Ac + n + v(t) turns air gap – lg n i(t) + Φ(t) – R g Fg magnetic path length lm – dΦ(t) Φ R R Faraday’s law: v(t)=n ni= c + g dt l 2 di(t) c n R c = µ Substitute for Φ: v(t)= A c Rc +Rg dt lg R g = µ Hence inductance is 0 A c 2 L = n R c + R g Fundamentals of Power Electronics24 Chapter 12: Basic Magnetics Theory Effect of air gap Φ ni= Rc +Rg Φ = BA 2 c L = n R + R B A c g sat c 1 R Φ c sat = BsatAc 1 R + R BsatA c c g Isat = n R c + R g Effect of air gap: nI nI ∝ sat1 sat2 ni Hc • decrease inductance • increase saturation current • inductance is less dependent on core – BsatAc permeability Fundamentals of Power Electronics25 Chapter 12: Basic Magnetics Theory 12.2.