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DIRECT RELATION BETWEEN HYSTERESIS AND DYNAMIC LOSSES IN SOFT MAGNETIC
MATERIALS
G. Bertotti
To cite this version:
G. Bertotti. DIRECT RELATION BETWEEN HYSTERESIS AND DYNAMIC LOSSES IN SOFT MAGNETIC MATERIALS. Journal de Physique Colloques, 1985, 46 (C6), pp.C6-389-C6-392.
�10.1051/jphyscol:1985672�. �jpa-00224927�
Colloque C6, supplCment au n09, Tome 46, septembre 1985 page C6-389
DIRECT RELATION BETWEEN HYSTERESIS AND DYNAMIC LOSSES I N SOFT MAGNETIC MATERIALS
G . B e r t o t t i
I s t i t u t o EZettroteenico NazionaZe GaZiZeo Ferraris and G.N. S.M. -C. I.S.M., 1-20125 To'orino, I t a l y
Rksumk - La perte dynamique en exchs d e s matkriaux polycristallins magnb- tiques doux est exprimCe en fonction de l a perte par hystCrCsis et l a dimen-
sion moyenne d e s grains, sans introduire aucun autre parametre a r b i t r a i r e . L'Cquation obtenue permet de calculer correctement soit l a valeur absolue, soit l a dgpendance de l a dimension des grains des pertes en exc&s mesurkes s u r des t a l e s de FeSi 3% non-orientkes.
Abstract - On the b a s i s of a statistical theory of l o s s e s recently developed by the author, the excess dynamic loss of polycrystalline soft materials i s expressed directly in t e r m s of the hysteresis l o s s and the average grain size, with no adjustable parameters. The obtained l o s s equation correctly predicts both the absolute value and the dependence on grain size of excess l o s s e s measured in nonoriented 3% SiFe laminations.
I - INTRODUCTION
In a recent s e r i e s of papers / 1-41, the author has proposed a new, general inter- pretation of eddy current losses, based on the use of statistical theoretical concepts.
It is well known that statistical methods prove particularly powerful when the inves- tigated process can be decomposed into a sum of statistically independent contribu- tions. In magnetization processes, these independent contributions cannot evidently be found at the level of single spins, and neither a t the one of single Bloch walls , which show, in general, strong mutual interactions 1 3 1 . Nevertheless, the author has proposed that such independence exists at the level of what he has called I t mag- netic objectstt (MO). A MO corresponds to a group of neighbouring walls whose evolution, just due t o the presence of local wall-wall interactions, is s o strongly correlated that can be treated in t e r m s of the dynamic properties of an equivalent, single object. The introduction of t h e concept of MO leads t o a number of important simplifications in the physical description of the magnetization process, which a r e discussed in detail in Ref. / 4 / and can be briefly summarized a s follows.
i ) - The dynamic behaviour of a single MO is, t o a good approximation, the s a m e a s the one of the plane 180' wall considered by Williams, Shockley and Kittel in t h e i r classical paper / 51. T h e r e exists a simple proportionality b?tween the excess dy- namic field Hexc acting on a MO and the magnetic flux r a t e @ correspondingly ob- tained, expressed by the equation
Hex, = OG(W) 6 (1)
where a is the electrical conductivity of the material and, according t o paper / 5 / , G(W) = 4 [ x 1 / ( 2 k + 1 ) 3 ] / ~ 3 = 0.1356 ... .
ii) - At each magnetization r a t e i = 4 ImaX f m s the domain structure dynamics can be described in t e r m s of ii simultaneously active MOs randomly placed in the sample Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985672
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c r o s s section S. A s a consequence, the dynamic l o s s splits into the classical l o s s and an e x c e s s contribution related t o the individual behaviour o f each MO / 4 /
p ( e x c ) = ij=~,,, = a G ( W ) s i2 / E , (2)
where Hex, i s the same as i n eq. ( 1 ) and it has been taken into account that the flux rate 6 provided b y each active M 0 must be a fraction l/i? o f the total flux rate S i .
i i i ) - i s expected t o be an increasing function o f the field H e x c , expressible, t o a f i r s t order approximation, i n the f o r m
Essentially, assuming this direct dependence o f ?i on He,, i s based on the idea that each MO experiences a d i f f e r e n t , local coercive field, so that, at each magnetizing frequency f m , only the MOs for which the applied field i s greater than tpe local coercive field will be active. When fm i s increased, a higher flux rate @ i s demand- ed f r o m the active MOs, and this implies an increase o f the applied field ( e q . ( 1 ) ).
T h i s increased magnetic pressure, acting uniformly in the sample c r o s s section, will overcome the local coercive fields o f further MOs, thus inevitably producing also an increase o f "n ( eq. ( 3 ) 1. Equations ( 2 ) and ( 3 ) lead t o the l o s s expression
~ ( e x c ) = i ~ o V o [ ~ 1 + 4 o G ( W ) S i / ^ n ~ v ~ - l ] / 2 = t 8 1 m a x f m ] o ~ . (4) T h e last approxiKate f o r m becomes in gef?eral quite accu'kte when f m >, 10 Hz, and points at the field V o a s the fundamental parameter o f the theory.
A systematic comparison with experiments / 4 / shows that eq. ( 4 ) has a wide range o f validity, f r o m v e r y fine grained nonoriented steels t o SiFe single crystals. In R e f . 141, however, V was essentially treated a s an arbitrary, adjustable parame- t e r , used t o fit experimental data. Actually, V o has an important physical meaning, 0 since, a s previously mentioned, it i s related t o the distribution o f the local coercive fields o f d i f f e r e n t MOs. In the present paper, the basic physical consequences o f t h i s fact will be discussed i n t h e case of fine grained materials. An equation will be derived which e x p r e s s e s the e x c e s s dynamic l o s s directly i n t e r m s o f the h y s t e r e s i s l o s s and t h e average grain s i z e , with no adjustable parameters. I t will be shown that t h i s equation correctly predicts both the absolute value and the dependence on grain size o f the e x c e s s l o s s measured in nonoriented 3% SiFe laminations.
I1 - THE MODEL
Let u s consider a polycrysta_lline magpetic lamination. W e shall assume that it con- tains, in i t s c r o s s section, No available MOs and that the distribution o f the local coercive fieids o f d i f f e r e n t MOs i s flat, with constant density l / v o . In other words, the number N o f MOs with coercive field lower than H i s simply
~ = H / v , . ( 5 )
In quasi-static conditions, t h e value 0f.H at each point o f the magnetization loop will be determined b y the requirement o f having at least one active MO ( t h a t i s
- no- 1 ) / 4 / .Conversely, i n dynamic conditions, H will be, at each point o f the loop, greater than i t s static value b y an amount AH=HeXc , which, according t o eq. (5), corresponds t o an increase AN =&Eo =H,,</ V o o f the number o f active MOs, i n agreement with eq. ( 3 ) . From t h i s point o f m e w , t h e r e f o r e , the fact that eqs. ( 2 ) - ( 4 ) are capable o f correctly predicting a great variety o f d i f f e r e n t experimental results c o n f i r m s the general existence, i n soft materials, o f a f a i r l y flat distribution o f local coercive fields. On the other hand, the approximate validity of eq. ( 5 ) i s also supported b y the investigations o f Del Vecchio on low carbon steels / 6 1 .
T h e magnetization reversal i n each MO will be described, for simplicity, a s a 180' reversal, and the spread o f the directions o f t h e e a s y magnetization a x e s i n d i f f e r e n t
Is with its weighted average <Is> = Is<cosO > over a l l the easy axes closest to the applied field direction ( < I ~ > r 0.85 Is r 1. 7 T in isotropic F e and SiFe samples 1. a Thus, when a M0 r e v e r s e s completely its flux, it gives a contribution A1 = 2 < 1,s /No to the total magnetization, and the number of MOs involved in a cycle of peak mag- netization Imax will be
On the other hand, eq. (5) implies that the maximum local coercive field reached in each magnetization half cycle will be 2 Hhyst, with Hhyst = P ( ~ Y s ~ ) / i ( P ( ~ Y s ~ ) is the hysteresis l o s s 1. We therefore obtain, taking account of eqs. 15) and (6),
Equations (4) and (7) give evidence of a direct physical connection between dynamic and hysteresis l o s s e s in soft magn;lle_tic materials.
What can we say about the number N, of available MOs appearing in eq. ( 7 ) ? Here, we shall explore the co;vnsequences o f assuming that each MO actually coincides with a single grain, that is No = S / s 2 , with s equal to the average grain size. This con- dition i s equivalent to assuming that each grain is characterized by a fairly uniform local coercive field, and that its domain structure behaves in a highly coherent fash- ion, while different grains in a same c r o s s section evolve quite independently from each other. All these conditions a r e expected t o hold in annealed samples character;
ized by a grain size much smaller than the sample thickness d. Conversely, when s + d , the inhomogeneities of coercive field inside each grain begin to be important and the number of MOs becomes definitely l a r g e r than the number of grains. In fact, the material tends to approach the limiting condition, met in monoc~ystalline Sam- ples, where each MO coincides with a single Bloch wall 141. With No = S / s 2 , we obtain from eas. (4) and (7)
which expresses the excess dynamic l o s s in t e r m s of the hysteresis l o s s and of the average grain size. We s t r e s s the fact that eq. (8) contains no adjustable parameters.
We have compared the predictions of eq. (8) with two s e t s of available experimental results, both referring to nonoriented SiFe laminations of variable grain size / 7,8/.
Figure l a reports the measured excess l o s s e s together with the predictions (broken lines) obtained from eq. (8) on the basis of the average behaviour of the measured
hysteresis l o s s ( Fig. l b ) . In spite of the simplicity of the proposed model and the absence, in eq. (a), of any a r b i t r a r y parameter, we obtain an essentially correct prediction of the absolute value of the excess loss, a s well a s of its dependence on both grain s i z e and peak magnetization. On the other side, the fact that the theoret- ical curve becomes substantially higher than the measured l o s s when s 3200 pm simply points, a s previously discussed, to a progressive splitting of each grain into several independent MOs.
I11 - CONCLUSIONS
The previous results show that the concept of MO, f a r from having a purely math- ematical character, is instead of relevant physical importance in the description of the magnetization process. The fact that the proposed model can provide a correct quantitative interpretation of the dynamic loss behaviour in c a s e s where previous models, a s the one of P r y and Bean 191, completely fail is the consequence of two basic assumptions : i ) - the single grains, rather than the single Bloch walls, a r e the fundamental objects providing coherent magnetization changes in fine grained materials ; ii) - the spatial fluctuations of local coercive field strongly reduce the
C6-392 JOURNAL DE PHYSIQUE
number of simultaneously active regions of the material c r o s s section, thus giving r i s e to a magnetization process much more inhomogeneous than expected on the ba- s i s of the geometrical arrangement of magnetic domains, and to a consequent large increase of the excess l o s s contribution.
Finally, we wish to point out that, although in this paper we have only discussed the l o s s behaviour a t fm = 50 Hz, eq. (8) provides indeed the complete dependence of
~ ( e x c ) on fm . We have verified that even the prediction of a linear dependence of the excess l o s s per cycle on rm is in excellent agreement with experiments.
Fig. 1 - Dependence of excess dynamic l o s s e s ( a ) and hysteresis l o s s e s ( b ) on av- erage grain size in nonoriented 370 SiFe laminations of thickness d = O . 35mm (open c i r c l e s after 171, I,, = 1 T ; other symbols after / 8 / , Imax = 1 . 5 T 1. The broken lines in Fig. l a represent the theoretical predictions obtained from eq. (8) on the basis of the average behaviour of the measured hysteresis loss.
REFERENCES
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/ I / Bertotti, G., J. Magn. Magn. Niater. 41 (1984) 253.
/ 2 / Bertotti, G., J. Appl. Phys. 54 (1983) 5293.
/ 3 / Bertotti, G., J. Appl. Phys. 3 (1984) 4339, 3 (1984) 4348.
/ 4 / Bertotti, G . , J. Appl. Phys. 57 (1985) 2110, 57 (1985) 2118.
/5/ Williams, H. J.. Shockley, W. and Kittel, C., Phys. Rev. 80 (1950) 1090.
/ 6 / Del Vecchio, R. M. and Charap, S. H., IEEE Trans. Magn. MAG-20 (1984) 143 7.
/7/ Matsumura, K. and Fukuda, B., IEEE Trans. Magn. MAG-20 (1984) 1533.
181 Bertotti, G . , Di Schino, G., F e r r o , A. and Fiorillo, F . , this Conference.
/ 9 / Pry, R.H. and Bean,C. P., J. Appl. Phys. 2 (1958) 532.
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