INTRODUCTION TO ELECTRIC MACHINES

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Magnetic circuits theory

Transformers

Most pictures come from Jesús Fraile Mora “Máquinas Eléctricas” 5ª ed. 2003, McGraw Hill

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MAGNETIC CIRCUITS THEORY

• Magnetic materials

• Magnetic circuits

• Magnetic energy

• Alternative current and magnetism

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• Magnetic field

• Magnetized materials

• H : magnetic field intensity (A/m)

• B : magnetic induction (T Tesla)

• μ : Magnetic permeability (H/m)

Remember:

• Diamagnetic μr ≈1

• Paramagnetic μr ≈1000

• Ferromagnetic μr ≫1000

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Magnetization curves of different ferromagnetic materials

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Magnetic hysteresis

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MAGNETIC CIRCUITS THEORY

• Magnetic materials

• Magnetic circuits

• Magnetic energy

• Alternative current and magnetism

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Ampere’s law:

Circulation of magnetic field H along a curve

[A· v]

ii

i

S

ii

i

H

iγ

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Ampere’s law:

Magnetomotive force (mmf)

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B

i

S

γ

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Magnetic flux through S:

Number of magnetic force lines through any perpendicular surface S

Weber

Magnetic flux along a ferromagnetic core

Hopkinson’s coefficient:

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[A· v]

Combining equations:

Magnetic reluctance

Hopkinson’s law

Permeability

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Electric circuit Magnetic circuit

e : e.m.f. F : m.m.f.

j : current density (A/m) B : induction (T)

σ : conductivity (S/m) μ : permeability (H/m)

i : electric current (A) : magnetic flux (Wb)

V : electric potential (V) U : magnetic potential (AT)

E : electric field (V/m) H : magnetic field (AT/m)

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Electric circuit Magnetic circuit

H : intensidad de campo magnético A/m

Φ : flujo magnético o número de líneas de fuerza que atraviesan cada unidad

de superficie perpendicular a ellas

Φ = B S cosWeber Wb

μ : permeabilidad magnética H/m = (Wb/A) / m

B : inducción magnética o densidad de flujo magnético

B = μ HTesla T = H/m· A/m =

= (Wb/A)/m· A/m = Wb/m2

1 T = 10 000 Gauss

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Faraday’s law:

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MAGNETIC CIRCUITS THEORY

• Magnetic materials

• Magnetic circuits

• Magnetic energy

• Alternative current and magnetism

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Energy:

Steady state ( no movement)

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Energy:

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Energy:

Self inductance

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Hysteresys:

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Other core losses:

eddy currents (Foucault)

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Eddy current looses:

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Eddy current looses:

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MAGNETIC CIRCUITS THEORY

• Magnetic materials

• Magnetic circuits

• Magnetic energy

• Alternative current & magnetism

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v

i

R L

Self-inductance:

Magnetic flux:

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v

i

R LAlternative voltage:

c.a.

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Core with losses:

V

I

Iμ

IFe

Φ

ϕ

exc

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