Fundamentals of Power Electronics Chapter 13: Basic Magnetics Theory1Part III. Magnetics13 Basic Magnetics Theory14 Inductor Design15 Transformer DesignFundamentals of Power Electronics Chapter 13: Basic Magnetics Theory2Chapter 13 Basic Magnetics Theory13.1 Review of Basic Magnetics13.1.1 Basic relationships 13.1.2 Magnetic circuits13.2 Transformer Modeling13.2.1 The ideal transformer 13.2.3 Leakage inductances13.2.2 The magnetizing inductance13.3 Loss Mechanisms in Magnetic Devices13.3.1 Core loss 13.3.2 Low-frequency copper loss13.4 Eddy Currents in Winding Conductors13.4.1 Skin and proximity effects 13.4.4 Power loss in a layer13.4.2 Leakage flux in windings 13.4.5 Example: power loss in atransformer winding13.4.3 Foil windings and layers 13.4.6 Interleaving the windings13.4.7 PWM waveform harmonicsFundamentals of Power Electronics Chapter 13: Basic Magnetics Theory3Chapter 13 Basic Magnetics Theory13.5 Several Types of Magnetic Devices, Their B–H Loops, and Core vs.Copper Loss13.5.1 Filter inductor 13.5.4 Coupled inductor13.5.2 AC inductor 13.5.5 Flyback transformer13.5.3 Transformer13.6 Summary of Key PointsFundamentals of Power Electronics Chapter 13: Basic Magnetics Theory413.1 Review of Basic Magnetics13.1.1 Basic relationshipsv(t)i(t)B(t), Φ(t)H(t), F(t)TerminalcharacteristicsCorecharacteristicsFaraday’s lawAmpere’s lawFundamentals of Power Electronics Chapter 13: Basic Magnetics Theory5Basic quantitiesElectric field EVoltageV = ElTotal flux ΦFlux density B{Surface Swith area AcTotal current ICurrent density J{Surface Swith area AcLength lLength lMagnetic quantities Electrical quantitiesMagnetic field H+–x1x2MMFF = Hl+–x1x2Fundamentals of Power Electronics Chapter 13: Basic Magnetics Theory6Magnetic field H and magnetomotive force FExample: uniform magneticfield of magnitude HMagnetomotive force (MMF) F between points x1 and x2 is related tothe magnetic field H according toAnalogous to electric field ofstrength E, which inducesvoltage (EMF) V:F = H⋅⋅dllx1x2Length lMagnetic field H+–x1x2MMFF = HlElectric field EVoltageV = ElLength l+–x1x2Fundamentals of Power Electronics Chapter 13: Basic Magnetics Theory7Flux density B and total flux Example: uniform flux density ofmagnitude BThe total magnetic flux  passing through a surface of area Ac isrelated to the flux density B according toAnalogous to electrical conductorcurrent density of magnitude J,which leads to total conductorcurrent I:Φ = B⋅⋅dAsurface STotal flux ΦFlux density B{Surface Swith area AcTotal current ICurrent density J{Surface Swith area AcFundamentals of Power Electronics Chapter 13: Basic Magnetics Theory8Faraday’s lawVoltage v(t) is induced in aloop of wire by change inthe total flux (t) passingthrough the interior of theloop, according toFor uniform flux distribution,(t) = B(t)Ac and hence{Area AcFlux Φ(t)v(t)+–v(t)=dΦ(t)dtv(t)=AcdB(t)dtFundamentals of Power Electronics Chapter 13: Basic Magnetics Theory9Lenz’s lawThe voltage v(t) induced by the changing flux (t) is of the polarity thattends to drive a current through the loop to counteract the flux change.Example: a shorted loop of wire• Changing flux (t) induces avoltage v(t) around the loop• This voltage, divided by theimpedance of the loopconductor, leads to current i(t)•This current induces a flux(t), which tends to opposechanges in (t)Flux Φ(t)Induced currenti(t)ShortedloopInducedflux Φ′(t)Fundamentals of Power Electronics Chapter 13: Basic Magnetics Theory10Ampere’s lawThe net MMF around a closed path is equal to the total currentpassing through the interior of the path:Example: magnetic core. Wirecarrying current i(t) passesthrough core window.• Illustrated path followsmagnetic flux linesaround interior of core• For uniform magnetic fieldstrength H(t), the integral (MMF)is H(t)lm. SoH⋅⋅dllclosed path= total current passing through interior of pathi(t)HMagnetic pathlength lmF(t)=H(t)lm= i(t)Fundamentals of Power Electronics Chapter 13: Basic Magnetics Theory11Ampere’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 tothe winding current MMF i(t)•The total MMF around a closed loop, accounting for winding currentMMF’s, is zeroFundamentals of Power Electronics Chapter 13: Basic Magnetics Theory12Core material characteristics:the relation between B and HFree space A magnetic core material0 = permeability of free space= 4 · 10–7 Henries per meterHighly nonlinear, with hysteresisand saturationBHµ0BHµB = µ0HFundamentals of Power Electronics Chapter 13: Basic Magnetics Theory13Piecewise-linear modelingof core material characteristicsNo hysteresis or saturation Saturation, no hysteresisTypical r = 103 to 105Typical Bsat = 0.3 to 0.5T, ferrite0.5 to 1T, powdered iron1 to 2T, iron laminationsBHµ = µr µ0BHµBsat– BsatB = µHµ = µrµ0Fundamentals of Power Electronics Chapter 13: Basic Magnetics Theory14UnitsTable 12.1. Units for magnetic quantitiesqu antity MKS unrationalized cgs conversionscore material equation B = µ0 µr HB = µr HB Tesla Gauss 1T = 104GH Ampere / meter Oersted 1A/m = 4s 10-3 OeWeber Maxwel l 1Wb = 108 Mx1T = 1Wb / m2Fundamentals of Power Electronics Chapter 13: Basic Magnetics Theory15Example: a simple inductorFaraday’s law:For each turn ofwire, we can writeTotal winding voltage isExpress in terms of the average flux density B(t) = F(t)/AccorenturnsCore areaAcCorepermeabilityµ+v(t)–i(t)Φvturn(t)=dΦ(t)dtv(t)=nvturn(t)=ndΦ(t)dtv(t)=nAcdB(t)dtFundamentals of Power Electronics Chapter 13: Basic Magnetics Theory16Inductor example: Ampere’s lawChoose a closed pathwhich follows the averagemagnetic field line aroundthe interior of the core.Length of this path iscalled the mean magneticpath length lm.For uniform field strengthH(t), the core MMFaround the path is H lm.Winding contains n turns of wire, each carrying current i(t). The net currentpassing through the path interior (i.e., through the core window) is ni(t).From Ampere’s law, we haveH(t) lm = n i(t)nturnsi(t)HMagneticpathlength lmFundamentals of Power Electronics Chapter 13: Basic Magnetics Theory17Inductor example: core material modelFind winding current at onset of saturation:substitute i = Isat and H = Bsat / intoequation previously derived via Ampere’slaw. Result isB=Bsatfor H ≥ Bsat/µµH for