PHENOMENOLOGY OF MAGNETIC LOSSES

Transcription

1 PHENOMENOLOGY OF MAGNETIC LOSSES Any variation of the flux density in a magnetic material is associated with dissiation of energy. The main hysical mechanisms for dissiation are: 1) Eddy currents and the related Joule effect, associated with the scattering of moving charges with honons and various lattice defects. The Joule effect is dominant in conducting and semiconducting materials. ) Sin daming, related to the transfer of energy from the recessing sins to the lattice, either directly or via magnon-honon interaction. This is the favorite mechanism in insulating magnets. 1 Euroean School on Magnetism ESM13

2 Magnetic losses are conveniently defined under given time eriodic law for the rate of change of the magnetization. Tyically, one has to deal with either sinusoidal or triangular B(t) waveform. The viscous character of the magnetization rocess makes attaining a given eak olarization under defined B(t) law increasingly difficult with increasing the magnetizing frequency. There is an obvious ractical aeal to the study of magnetic losses, because there is a stringent need for ther rediction in most electrical/electronic alications. Besides the unnecessary waste of energy, the fast increase of the ower loss with the magnetizing frequency (somewhat f ) can make overheating the main obstacle to high-frequency alications (P c v dt/dt). For samle of volume V subjected to eriodic excitation at the frequency f 1/T, we can exress the eddy-current energy loss er unit volume on a eriod as W d 3 r T j ( r, t ) V σ V dt The sace-time behavior of the current density j(r, t) is hoelessly comlicated. We need to focus on the general features of the loss behavior, to be treated from a statistical viewoint. Euroean School on Magnetism ESM13

3 Grain-oriented Fe-(3wt%)Si d.3 mm J 1.7 T.5 Hz W HdB µ o H dh + HdJ H and X conjugate work variables du HdX + HdJ δ Q J (T) Hz Isothermal cycle: HdX du δ Q Hz H m (A/m) W W hyst + W cl + W exc Energy loss (J/m 3 ) G rain-oriented Fe-(3wt%)Si d.3 mm J 1.7 T W exc W cl W ( J hyst + k f + k f, f ) W ( J ) cl exc 1/ 1 W h Frequency (Hz) 3 Euroean School on Magnetism ESM13

4 Amorhous ribbon Soft ferrite ring 4 Euroean School on Magnetism ESM13

5 Soft ferrite ring kw/kg Nanocrystalline ribbon 5 Euroean School on Magnetism ESM13

6 1 nonoriented Fe-Si J 1.5 T, d.35 mm Excess energy loss under generic olarization rate dj/dt W (mj/kg) 5 W exc W cl T Wexc exc ( t) dt σ GSVo J ( t) 8 T 3 / dt nonoriented Fe-Si J 1.5 T, d.35 mm W W hyst f (Hz) Sinusoidal induction W 1/ ( π / 6) σ d J f + 8. σ GSV J 3 / f h + 76 o (W - W cl ) (mj/kg) 6 4 W exc W hyst f 1/ (Hz) 6 Euroean School on Magnetism ESM13

7 The hysteresis (quasi-static) energy loss cannot be generally redicted from knowledge of the structural roerties and the intrinsic magnetic arameters. At low inductions, however, magnetic hysteresis can usually be described by the analytical Rayleigh law. J (T),6,4,, -, -,4 Pure Ni foil (H, J ) J ( H ) ( a + bh ) H J ah + b ( H bh H ) -, H (A/m) a initial ermeability, resulting from domain wall bending and moment rotations 7 Euroean School on Magnetism ESM13

8 The dynamic loss can be formally assessed in a linear (or, in ractice, quasilinear) material, with defined DC ermeability. Dissiative henomena bring about a time delay of the B(t) waveform with resect to the H(t) waveform and the resulting hysteresis loo takes an ellitical shae.1 Co 67 Fe 4 B 14.5 Si 14.5 J 1 mt J (T) H (A/m) Bm Hz B khz B khz B 5kHz B 1MHz B MHz B 5MHz H( t ) H cos ω B( t) B B cos t ( ω t δ ) cosδ cosω t + B sin δ The associated energy loss (area of the ellitical loos) is T db( t) W H ( t) dt dt sin ω t 8 Euroean School on Magnetism ESM13 π H B sinδ

9 When dealing with a linear system we can attack in a simle way the henomenology of energy loss and the related concets of comlex ermeability, quality factor, and equivalent L - R circuit i s R s δ jωl s i s R L L s R s i s 9 Euroean School on Magnetism ESM13

10 The concet of loss searation can be given a solid hysical justification in terms of characteristic sace-time scales of the magnetization rocess (G. Bertotti, Hysteresis in Magnetism). It is valid in both conducting and nonconducting magnetic materials. W W hyst + W cl + W exc W ( J hyst + k f + k f, f ) W ( J ) cl exc 1/ y z x B(t) W cl is easily calculated in the simle case of a thin lamination (width >> thickness) under time-varying induction (e. g., B(t) B sin ωt) uniform across the sheet cross-section. 1 Euroean School on Magnetism ESM13

11 The full aroach to the classical eddy current losses, valid for whatever lamination thickness and material conductivity is based on the diffusion equation for the internal magnetic field. y E dy E µ db dt dh dt H dy H j j Eσ z x H (t) H a (t)-h eddy (t) H a (t) H σ µ dy dh dt and the boundary conditions H ( t) H a ( t) for y ± d / H / y at y 11 Euroean School on Magnetism ESM13

12 The concet of magnetic losses and hysteresis are associated with that of lag in time of induction with resect to the field. We can therefore talk aboy a time constant (or a distribution of time constants). The simlest case of relaxation is one where the change of the magnetization with time is roortional to its deviation from the equilibrium value dm M / dt k( M M ( t)) t / τ ( t) (1 e ) M with τ 1/k M / M t / τ If relaxation is due to eddy currents, the time constant will deend on conductivity, ermeability, and samle size. 1 Euroean School on Magnetism ESM13

13 Bibliograhy G. Bertotti, Hysteresis in Magnetism, Academic Press, G. Bertotti, IEEE Trans. Magn. 4 (1988) 61. G. Bertotti, J. Al. Phys. 57 (1985) 11. J.B. Goodenough, IEEE Trans. Magn. 38 () F. Fiorillo, C. Aino, M. Pasquale, in The Science of Hysteresis (G. Bertotti and I. Mayergoyz, eds., Academic Press, 6), vol III,.1. R.H. Pry and C.P. Bean, J. Al. Phys. 9 (1958) 53. H.J. Williams, W. Shockley, C. Kittel, Phys. Rev. 8 (195) 19. B.D. Cullity and C.D. Graham, Introduction to magnetic materials, Wiley, 11. F. Fiorillo and A. Novikov, IEEE Trans. Magn. 6 (199) 94. G. Bertotti and I. Mayergoyz, eds., The Science of Hysteresis, Academic Press, Euroean School on Magnetism ESM13