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1 Magnetism Content of the presentation - Maxwell's general equations on magnetism - Hysteresis curve - Inductance - Magnetic circuit modelling - Eddy currents and hysteresis losses - Force / torque / power M.sc.EE Student at IET, Aalborg University Ph.D. Student at IET, Aalborg University Assistant Professor at IET, Aalborg University Associate Professor at IET, Aalborg University. Few words and mini CV of my self

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2 Maxwell James Clerk Maxwell Unified theory of the connection between electricity and magnetism

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3 Maxwells equations in free space Not easy to understand in this form. Practical engineers tend to forget the definition of divergence, curl, surface and line integral

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4 Ørsted H.C Ørsted Ørsteds discovery was a deflection of a Compass needle when it is close to a current carrying conductor. But he was not able to describe the physics or mathematics behind the phenomena 1820 {f = Bxil / 90° : f=Bil}

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5 Amperé André-Marie Ampére 1775 – 1836 Just one week after Ørsteds discovery Ampere showed that two current carrying wires attract or repulse each other, and was able to show that : Thus, for two parallel wires carrying a current of 1 A, and spaced apart by 1 m in vacuum, the force on each wire per unit length is exactly 2 × N/m. I2I2 I1I1 r l It was first in 1826 he published the circuit law in Maxwell’s equation

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6 Faraday Michael Faraday {e = Bxlv / 90° : e =Blv} Induction by movement ”generator” Induction by alternating current ”transformer” The Induction law 1821

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7 Gauss Carl Friedrich Gauss 1777 – 1855 Basically the magnetic induction can’t escape. The equation says that we don’t have a monopole. There will always be a north and a south pole

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8 Relations in Maxwell’s equations Simplified analogies and the relations in Maxwell’s equations

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9 Permeability Return μ = μ r μ 0 Where μ r typically is in magnetic cores

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10 Analogies

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11 Faraday’s law Faraday’s law (again)

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12 Some of the first electrical machines Not useful because it makes AC Improved to a DC machine where Ampere proposed Pixii to use a mechanical commutator Alternator by Pixii 1832 Note it is made with electromagnets (Henry 1828)

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13 Letz’s law Letz’s law (consequence of Maxwell’s equation)

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14 Amperé’s law Ampere’s law (again)

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15 Inductor example Example : Inductor

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16 Inductor example

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17 Inductor example No saturation and hysteresis

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18 Magnetic circuit model Ohm’s law in magnetic circuits

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19 Magnetic circuit model What about Kirchoff’s current law (KCL) Gauss law

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20 Magnetic circuit model What about Kirchoff’s voltage law (KVL) Ampere law is similar to Kirchoff’s voltage law

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21 Magnetic circuit model (Steel and air-gap)

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22 Magnetic circuit model (Steel and air-gap)

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23 Core losses Hysteresis losses

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24 Hysteresis losses

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25 Hysteresis losses Hysteresis curve at various H-fields

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26 Hysteresis losses

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27 Eddy current losses

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28 Eddy current losses How do we reduce Eddy current losses LAMINATED SOLID

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29 Eddy current losses

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30 Eddy current losses in windings Can be a problem with thick wires - Low voltage machines - High speed machines

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31 Force, torque and power Universal modeling of terminal characteristic of electro-magnetic devices based on energy balance

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32 Force, torque and power

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33 Force, torque and power

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34 Force, torque and power

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35 Force, torque and power

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36 Force, torque and power

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37 Force, torque and power

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38 Force, torque and power

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39 Force, torque and power

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40 Force, torque and power

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41 Simplified machines

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42 Simplified machines

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43 Electromagnetic gearing L gu larger than L g ≈ ∞ Δθ=90

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44 Electromagnetic gearing Same winding but now enclosing two poles : L A is the same (if end winding is neglected the Resistance is also the same) Δθ=45 Torque doubling

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45 Normal/radial forces Normal forces Ex. Changing the air gap from 0.3 mm to 0.15 mm Gives a doubling of inductance The Δx is very small i.e a large Force In electrical machines the normal forces versus the tangential force (torque) is typically a factor 10.

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