Lecture 8. Inductors • Mon 9/23: Transformers

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]

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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

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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

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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

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Magnetomotive force via Ampère’s law Magnetic circuits & Gauss’s law Goal: Compute the flux

magnetomotive + 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

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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)

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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

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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 electromotive + Resistance = 1 / Conductance force Each 1 of N turns is one unit of emf

Flux linkage

https://youtu.be/qLCpe5P0byI 19 20

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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

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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

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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