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1 Few words and mini CV of my self 1990 - 1995 M.sc.EE Student at IET, Aalborg University. 1995 - 1999 Ph.D. Student at IET, Aalborg University. 1999 - 2002 Assistant Professor at IET, Aalborg University. 2002 - Associate Professor at IET, Aalborg University.

Content of the presentation

- Maxwell's general equations on magnetism - Hysteresis curve - - modelling - Eddy currents and hysteresis losses - Force / torque / power Maxwell 2

James Clerk Maxwell 1831 - 1879

Unified theory of the connection between and magnetism Maxwells equations in free space

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

1 c  00 Ørsted

4

H.C Ørsted 1777-1851 1820

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

{f = Bxil / 90° : f=Bil} Amperé

5

André-Marie Ampére 1775 – 1836

Just one week after Ørsteds discovery showed that two current carrying wires attract or repulse each other, and was able to show that : F  II·  0 12 lr2

I1 r I2

l Thus, for two 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 × 10-7 N/m. It was first in 1826 he published the circuit law in Maxwell’s equation Faraday

6 The Induction law 1821 1791-1867

{e = Bxlv / 90° : e =Blv}

Induction by movement ”generator”

Induction by ” Gauss

7

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 Relations in Maxwell’s equations

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

9

Return

μ = μr μ0 Where μr typically is 1000-100000 in magnetic cores Analogies

10 Faraday’s law

11 Faraday’s law (again) Some of the first electrical machines 12 Alternator by Pixii 1832

Not useful because it makes AC

Improved to a DC machine where Ampere proposed Pixii to use a mechanical commutator

Note it is made with ( 1828) Letz’s law 13 Letz’s law (consequence of Maxwell’s equation) Amperé’s law 14 Ampere’s law (again) example

15 Example : Inductor Inductor example

16 Inductor example

17

No and hysteresis Magnetic circuit model

18 Magnetic circuit model

Ohm’s law in magnetic circuits Magnetic circuit model

19 What about Kirchoff’s current law (KCL)

Gauss law Magnetic circuit model

20 What about Kirchoff’s law (KVL)

Ampere law is similar to Kirchoff’s voltage law Magnetic circuit model (Steel and air-gap)

21 Magnetic circuit model (Steel and air-gap) Magnetic circuit model (Steel and air-gap)

22 Core losses

23 Hysteresis losses Hysteresis losses

24 Hysteresis losses Hysteresis losses

25 Hysteresis curve at various H-fields 26 Hysteresis losses losses

27 Eddy current losses Eddy current losses

28 How do we reduce Eddy current losses

SOLID

LAMINATED Eddy current losses

29 Eddy current losses Eddy current losses in windings

30 Eddy current losses in windings

Can be a problem with thick wires - Low voltage machines - High speed machines Force, torque and power

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

32 Force, torque and power

33 Force, torque and power

34 Force, torque and power

35 Force, torque and power

36 Force, torque and power

37 Force, torque and power 38 Force, torque and power 39 Force, torque and power 40 Simplified machines

41 Simplified machines

42 Electromagnetic gearing

43

2 0 A 1 2 LNA  L larger than L ≈ ∞ EL  LI{ Inductor Energy  gu g 2 lg 2 0 Au 1 12 1 2 1 2 LNU  E CU2{ Energy  ELILILI    l C 2 2AUA 2 2 gu 1 E J 2{ Rotational Energy J  Δθ=90 2 E 1 LA 2 1   I E Mv2{ translational Energy  2 m 2 Electromagnetic gearing

44

Same winding but now enclosing two poles : LA is the same (if end winding is neglected the Resistance is also the same) Δθ=45

Torque doubling Normal/radial forces

45 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.