INTRODUCTION TO ELECTRIC MACHINES 1 Magnetic circuits theory 2 Transformers Most pictures come from Jesús Fraile Mora “Máquinas Eléctricas” 5ª ed. 2003, McGraw Hill 1 MAGNETIC CIRCUITS 1 THEORY • Magnetic materials • Magnetic circuits • Magnetic energy • Alternative current and magnetism 2 MAGNETIC MATERIALS Remember: • Magnetic field • Magnetized materials • H : magnetic field intensity (A/m) • B : magnetic induction (T Tesla) • µ : Magnetic permeability (H/m) • Diamagnetic 0 ≤ µr < 1 • Paramagnetic µr ≈1000 • Ferromagnetic µr ≫1000 3 MAGNETIC MATERIALS Magnetization curves of different ferromagnetic materials 4 MAGNETIC MATERIALS Magnetic hysteresis 5 MAGNETIC CIRCUITS 1 THEORY • Magnetic materials • Magnetic circuits • Magnetic energy • Alternative current and magnetism 6 MAGNETIC CIRCUITS Ampere’s law: Every electric current I creates a magnetic field B 7 MAGNETIC CIRCUITS Ampere’s law: Magnetic field created by a loop wired current 8 MAGNETIC CIRCUITS Ampere’s law: S H N Magnetic field created by a solenoid (coil) 9 MAGNETIC CIRCUITS Ampere’s law: i H [A· v] i i γ Circulation of magnetic field S H along a curve 10 MAGNETIC CIRCUITS Ampere’s law: i B i i γ Magnetomotive force (mmf) S 11 MAGNETIC CIRCUITS Magnetic flux through S: Number of magnetic force lines through any perpendicular surface S Weber Hopkinson’s coefficient: Magnetic flux along a ferromagnetic core 12 MAGNETIC CIRCUITS Combining equations: [A· v] Magnetic reluctance Permeability Hopkinson’s law 13 MAGNETIC CIRCUITS 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) R : electric resistance (Ω) R : magnetic reluctance (H-1) E : electric field (V/m) H : magnetic field (A/m) 14 MAGNETIC CIRCUITS Magnitude Units H : intensidad de campo magnético A/m Φ : flujo magnético o número de líneas Φ = B S cos de fuerza que atraviesan cada unidad Weber Wb de superficie perpendicular a ellas µ : permeabilidad magnética H/m = (Wb/A) / m B = µ H B : inducción magnética o densidad de Tesla T = H/m· A/m = flujo magnético = (Wb/A)/m· A/m = Wb/m2 1 T = 10 000 Gauss 15 MAGNETIC CIRCUITS 16 MAGNETIC CIRCUITS Faraday’s law: 17 MAGNETIC CIRCUITS Faraday’s law: 18 MAGNETIC CIRCUITS Faraday’s law: 19 MAGNETIC CIRCUITS 1 THEORY • Magnetic materials • Magnetic circuits • Magnetic energy • Alternative current and magnetism 20 MAGNETIC ENERGY Energy: Steady state ( no movement) 21 MAGNETIC ENERGY Hysteresys: 22 MAGNETIC ENERGY Other core losses: eddy currents (Foucault) 23 MAGNETIC ENERGY Eddy current looses: 24 MAGNETIC ENERGY Eddy current looses: 25 MAGNETIC CIRCUITS 1 THEORY • Magnetic materials • Magnetic circuits • Magnetic energy • Alternative current & magnetism 26 AC & MAGNETISM R L i v Self-inductance: Magnetic flux: 27 AC & MAGNETISM Alternative voltage: R L i v 2 V N i = Φ R ; N i = R m m m N ω c.a. 28 AC & MAGNETISM Core with losses: IFe ϕ V Iµ Iexc Φ 29.