Part III. Magnetics 12. Basic Magnetics Theory 13. Filter - PowerPoint PPT Presentation

Part III. Magnetics 12. Basic Magnetics Theory 13. Filter Inductor Design 14. Transformer Design 1 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

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 2 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

12.1. Review of basic magnetics 12.1.1. Basic relations Faraday's law B(t), Φ (t) v(t) terminal core characteristics characteristics i(t) H(t), F (t) Ampere's law 3 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Basic quantities Magnetic quantities Electrical quantities length l length l magnetic field H electric field E x 1 x 2 x 1 x 2 + – + – MMF voltage V = El F = Hl surface S surface S with area A c with area A c flux density B { current density J { total flux Φ total current I 4 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

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 x 2 H ⋅ dl F = x 1 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 x 1 x 2 x 1 x 2 + – + – MMF voltage F = Hl V = El 5 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Flux density B and total flux Φ The total magnetic flux Φ passing through a surface of area A c 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 , which leads to total conductor Φ = B A c current I : surface S surface S with area A c with area A c current density J { flux density B { total flux Φ total current I 6 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Faraday’s law Voltage v(t) is induced in a area A c loop of wire by change in the total flux Φ (t) passing { through the interior of the loop, according to v ( t ) = d Φ ( t ) flux Φ (t) dt For uniform flux distribution, – Φ (t) = B(t)A c and hence v(t) + v ( t ) = A c dB ( t ) dt 7 Fundamentals of Power Electronics 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 shorted flux Φ (t) impedance of the loop loop conductor, leads to current i(t) • This current induces a flux Φ ’(t), which tends to oppose changes in Φ (t) induced flux Φ ' (t) 8 Fundamentals of Power Electronics 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 H carrying current i(t) passes through core window. i(t) • Illustrated path follows magnetic path length l m 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 ) 9 Fundamentals of Power Electronics 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)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 10 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Core material characteristics: the relation between B and H Free space A magnetic core material B B B = µ 0 H µ H H µ 0 µ 0 = permeability of free space Highly nonlinear, with hysteresis = 4 π · 10 -7 Henries per meter and saturation 11 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Piecewise-linear modeling of core material characteristics No hysteresis or saturation Saturation, no hysteresis B B B sat B = µ H µ = µ r µ 0 µ µ = µ r µ 0 H H – B sat Typical B sat = 0.3-0.5T, ferrite Typical µ r = 10 3 - 10 5 0.5-1T, powdered iron 1-2T, iron laminations 12 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Units Table 12.1. Units for magnetic quantities quantity MKS unrationalized cgs conversions B = µ 0 µ r H B = µ r H core material equation 4 G B Tesla Gauss 1T = 10 1A/m = 4 π⋅ 10 -3 Oe H Ampere / meter Oersted Φ 8 Mx Weber Maxwell 1Wb = 10 2 1T = 1Wb / m 13 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Example: a simple inductor Faraday’s law: Φ For each turn of core area i(t) A c wire, we can write + n v turn ( t ) = d Φ ( t ) v(t) turns dt core – permeability µ Total winding voltage is core v ( t ) = n v turn ( t ) = n d Φ ( t ) dt Express in terms of the average flux density B(t) = Φ (t)/A c v ( t ) = n A c dB ( t ) dt 14 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Inductor example: Ampere’s law Choose a closed path H which follows the average i(t) magnetic field line around magnetic path the interior of the core. n length l m Length of this path is turns called the mean 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) 15 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Inductor example: core material model B B sat for H ≥ B sat / µ B sat µ µ H for H < B sat / µ B = – B sat for H ≤ B sat / µ H – B sat Find winding current at onset of saturation: substitute i = I sat and H = B sat / µ into equation previously derived via Ampere’s law. Result is I sat = B sat l m µ n 16 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Electrical terminal characteristics We have: for H ≥ B sat / µ B sat v ( t ) = n A c dB ( t ) µ H for H < B sat / µ B = H(t) l m = n i(t) dt – B sat for H ≤ B sat / µ Eliminate B and H , and solve for relation between v and i . For | i | < I sat , v ( t ) = µ n 2 A c v ( t ) = µ n A c dH ( t ) di ( t ) dt l m dt which is of the form L = µ n 2 A c v ( t ) = L di ( t ) with dt l m —an inductor For | i | > I sat the flux density is constant and equal to B sat . Faraday’s law then predicts dB sat —saturation leads to short circuit v ( t ) = n A c = 0 dt 17 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

12.1.2. Magnetic circuits length l Uniform flux and + – area MMF F magnetic field inside A c a rectangular Φ { element: flux MMF between ends of core permeability µ element is F = H l l H R = µ A c Since H = B / µ and Β = Φ / Ac , we can express F as l µ A c Φ l F = with R = µ A c A corresponding model: + – F R = reluctance of element Φ R 18 Fundamentals of Power Electronics 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. 19 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Magnetic analog of Kirchoff’s current law node Physical structure Divergence of B = 0 Φ 1 Φ 3 Flux lines are continuous and cannot end Φ 2 Total flux entering a node must be zero Magnetic circuit Φ 1 = Φ 2 + Φ 3 node Φ 1 Φ 3 Φ 2 20 Fundamentals of Power Electronics 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. 21 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory

Example: inductor with air gap core permeability µ c Φ cross-sectional i(t) area A c + n v(t) air gap turns l g – magnetic path length l m Ampere’s law: F c + F g = n i 22 Fundamentals of Power Electronics Chapter 12: Basic Magnetics Theory