9/18/2019 Schedule • Mon 9/16: Magnetic circuits • Wed 9/18: Inductors ECE330: Power Circuits & Electromechanics • Fri 9/20: Review + Quiz 3 Lecture 8. Inductors • Mon 9/23: Transformers Prof. Richard Y. Zhang Univ. of Illinois at Urbana-Champaign [email protected] 1 2 1 2 Inductance as energy storage Wireless power: Just a big inductor Kinetic energy Centrifugal & gyroscopic forces High conversion loss Low storage loss Magnetic energy + - EM compatibility / safety No conversion loss High storage loss Ability to share energy (next week) + - https://youtu.be/pIQz8uts88o 3 4 3 4 Induction heater: Just a big inductor Mechanics of an inductor Magnetic flux (ϕ) - mmf + + emf - Electric + current (I) Applied voltage (V) https://youtu.be/d7DBS2Is0ws 5 6 5 6 1 9/18/2019 Today Magnets, how do they work? • Review: Magnetomotive force and reluctance • Electromotive force and inductance • Practical considerations Image: http://www.ukm.my/rahim/magnetic%20lecture.htm ECE330: It’s all about currents and fluxes 7 8 7 8 Magnetomotive force via Ampère’s law Magnetic circuits & Gauss’s law Goal: Compute the flux magneto- motive + Reluctance flux density force = 1 / Permeance flux in – flux out = 0 Magnetic field current density Implies flux superposition Magnetic flux (ϕ) mmf net current = enclosed by loop around loop + current All reluctances are nonlinear Interior Alignment to flux lines except for freespace (i.e. air) Boundary force ∝ flux density mmf 9 ECE 330: Assume linear reluctances 10 9 10 Linear reluctance via Ohm’s law Some material properties (Will be provided) length depth S = Siemens = Ohms-1 width At = Ampere-turns Wb= Webers 2 = Tesla-meter 6 3 + H = Henries ~10 ~10 Resistance [Ω] = 1 / (Conductance [S] ) 250 voltage [V] 1 length [m] = current [A] conductivity [S/m] width [m] x depth [m] (Material property) (Dimensional scaling) Reluctance [At / Wb] = 1 / (Permeance [H] ) Freespace mmf [At] 1 length [m] -7 = μ0 = 4π x 10 [H/m] flux [Wb] permeability [H/m] width [m] x depth [m] 11 12 Source: MIT OpenCourseWare, 6.013 Electromagnetics and Applications Spring 2009 11 12 2 9/18/2019 Leakage and fringing Aside: Magnetic materials e- Freespace is highly permeable • Inherent atomic currents mmf • Most materials: cancelation • Magnetic materials: summation mmf = 0 + mmf > 0 Saturation & Remanence © Gorchy / Wikimedia Commons 13 Magnetic behavior explained using magnetic domains 14 13 14 Today Mechanics of an inductor Magnetic flux (ϕ) • Review: Magnetomotive force and reluctance • Electromotive force and inductance - mmf + • Practical considerations + emf - Electric + current (I) Applied voltage (V) 15 16 15 16 Maxwell’s Equations Faraday’s experiment Magnetoquasistatics (MQS) = no capacitors, no electric charge Magnetic flux Voltage Current Electric circuit - mmf + Reaction force dx/dt I → mmf dϕ/dt → emf + emf - (Ampère’s law) (Faraday’s law) No emf for static system: emf requires changing flux. Magn. Pot. Magnetic circuit Flux Lenz’s law: Induced emf tries to oppose action 17 18 17 18 3 9/18/2019 Lenz’s law Faraday’s law Minus sign due to Lenz’s law -d/dt (flux) emf = - d net flux around loop dt enclosed by loop emf Almost Ampère’s law, except -d/dt electro- motive + Resistance = 1 / Conductance force Each 1 of N turns is one unit of emf Flux linkage https://youtu.be/qLCpe5P0byI 19 20 19 20 Inductor equation Aside: Energy storage i(t) Rwire Magnetic flux - mmf + + - emf Equiv. ( turns )2 v(t) - mmf + L [H] = + - reluctance Equiv. reluctance [H-1] + emf - ϕ Electric Equiv. current mmf + 1 2 + - reluctance Energy [J] = 2 L [H] ( current [A] ) Applied voltage ( mmf [At] )2 = 1 ( turns )2 2 Equiv. reluctance [H-1] L [H] = -1 Equiv. reluctance [H ] Energy is stored across reluctance (not dissipated) Magnetic materials: low reluctance, finite energy storage Trick: Getting the reluctance correct 21 Freespace: high reluctance, unlimited energy storage 22 21 22 Today Uniform flux density assumption a a mmf = 10 [At], a • Review: Magnetic circuits Rcore = 5 [At / Wb], A = 1 [m2] • Mechanics of inductors 2 + B = ? [Wb / m ] g b • Practical considerations A) 1 B) 2 a C) 5 c D) 10 Cross-sectional area What is the average A [m2] length of the core? A) 2b + 2c B) 2b + 2c – g Flux density Flux C) 2b + 2c – g – 2a B [Wb / m2] ϕ = BA 23 D) 2b + 2c – g – 4a 24 23 24 4 9/18/2019 Fringing correction Equation will be provided Schedule • Mon 9/16: Magnetic circuits depth • Wed 9/18: Inductors • Fri 9/20: Review + Quiz 3 width • Mon 9/23: Transformers gap Increase or decrease reluctance? Adjusted cross-sectional area Reluctance of air gap with fringing 1 gap μ0 (width + gap )( depth + gap) This is a “fudge factor” correction. 25 26 25 26 5.