Proceedings

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11th International IGTE Symposium 2004

Energy Loss in Nanocrystalline Materials K. Chwastek, M. Najgebauer, J. Szczyglowski Faculty of Electrical Engineering, Al. Armii Krajowej 17, 42-200 Czestochowa, Poland E-mail: [email protected]

Abstract: The paper presents an attempt to describe the energy loss in FINEMET-type nanocrystalline materials. An extension to the quasi-static Jiles-Atherton description of magnetization process has been proposed, that makes it possible to take into account the influence of eddy currents induced in the material. The effect of eddy currents has been incorporated into the formula describing the anhysteretic magnetization. The obtained dynamic hysteresis loops have been used to predict energy loss in Fe73,5 Cu1 Nb1 Si13,5 B9 . It was stated, that the dependencies of energy loss on magnetic flux density and frequency of alternating magnetic field are within the range of measured loss. The obtained results might be applied in the design and application stages for the prediction of magnetic properties of cores used in electronic and electric appliances. Keywords: energy loss, nanocrystalline materials, effective field, Langevin function.

I. I NTRODUCTION Currently applied models for description of the energy loss in soft magnetic materials, such as Pry’s-Bean’s, Bertotti’s models reveal considerable differences between the measured values of loss and the values predicted by analysis, especially in the case of materials with microand nanocrystalline structure [1]-[2]. The paper presents an attempt to describe the energy loss in FINEMET-type nanocrystalline materials. An extension to the description of the magnetization process proposed by Jiles and Atherton [3], has been introduced in order to include the influence of eddy currents generated within the material by alternating magnetic field. The basic equation of JilesAtherton model Man − Mirr dMan dM = (1 − c) +c , dH kδ − α(Man − Mirr ) dH

(1)

where c, k, α — the model parameters, Man — the anhysteretic magnetization, Mirr — the irreversible component of magnetization M , was adapted to dynamic conditions by a modification of the formula describing the anhysteretic magnetization. The influence of eddy currents on the magnetization process was evaluated by the introduction of an additional component of the effective field in the Langevin function. The component is related to the energy loss bound with eddy currents generated within the material [4]. The equation describing the global state of thermodynamic balance in the magnetized material will take the following form  µ0 hmiH + µ0 hmiαM + µo hmiHd Man =Ms coth − kB T  kB T − , µ0 hmiH + µ0 hmiαM + µo hmiHd (2) where µ0 hmiH denotes the energy related to interaction of a magnetic moment with the external field, µ0 hmiαM —

the energy related to spontaneous magnetization, whereas the component µ0 hmiHd — the energy related to the flow of eddy currents, generating counterfield field Hd in the material. By introduction of the aforementioned equation into the Jiles-Atherton description, dynamic hysteresis loops for a material with nanocrystalline structure were obtained, which later were used (by means of numerical integration) for the prediction of energy loss during magnetization with alternating magnetic field. The counterfield coming from eddy currents may be induced by current pulses generated by Barkhausen jumps for time scale about 10−8 [s], currents generated around moving domain walls, as well as currents generated in the whole bulk material — the so-called intralaminar eddy currents. In the case of magnetic cores the field may also be induced by interlayer (interlaminar) eddy currents, when there is no sufficient electric insulation between the layers of the ribbon the core is made of [5]. Therefore the following dependence may be postulated Hd = Hexc + Hinter + Hintra ,

(3)

where Hexc — field bound with excess eddy currents, Hinter — field component of classical eddy currents bound with interlaminar eddy currents, Hintra — field component of classical eddy currents bound with intralaminar eddy currents. The change in tension force between neighbouring ribbon windings may lead to change of value of measured loss in the case of insufficient insulation of ribbon surface, even for low frequency of magnetic field the core is subjected to. It has been proved, that the increase in tension between two neighbouring uncoated ribbons windings from 5 [N] to 25 [N] results in the double increase in loss per cycle at frequency 60 [Hz] and induction 1.4 [T] [5]. In the present paper an idea how to include the component bound with interlaminar eddy currents into the analysis of classical eddy currents is presented. This is achieved by the introduction of effective (resultant) conductivity into calculations. Classical eddy currents are very often neglected in the analysis of loss, especially in the range of low frequencies of field magnetizing the core [6]. In the case of an anisotropic environment (a magnetic core is an example thereof), the electric conductivity assumed for loss calculation ought to be a tensor, whose coordinates depend on spacial variables due to a very complicated current path, which is influenced by shortings between neighbouring ribbon layers the core is made of [5], as well as to the complex structure of the material itself. Determination of the tensor coordinates in the aforementioned case is practically impossible. Taking the above into account, it was assumed in the calculations of the counter-

11th International IGTE Symposium 2004

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Proceedings

Figure 1. Geometry and characteristic dimensions of the tested core. Figure 2. Calculated /—/ and measured /- -/ hysteresis loops.

field Hd on the basis of the dependence ~ ~d = −γ ∂ B , rot rot H ∂t

(4)

that the conductivity γ is a scalar representing a certain medium value of conductivity given by γ=

ln (r2 /r1 ) , 2πRd

(5)

where r1 — the internal radius of the core, r2 — the external radius, d — the core height, while R — the resistance between internal and external core surfaces (cf. Fig. 1). For the examined cores the conductivity was about 0.7 [S/m]. II. R ESULTS AND DISCUSSION Theoretical calculations and measurements of energy loss have been carried out for a magnetic core wound into toroidal shape of uncoated FINEMET-ribbon at tension 15 [N]. The nanocrystalline state in the ribbon has been obtained by annealing the core in temperature 520 [◦ C] in vacuum within 1 hour in magnetic field ≈ 50 [kA/m], aligned transverse to the direction of ribbon casting. The anisotropy coefficient Ku ≈ -15 [J/m3 ] has been obtained. The shape and dimensions of the tested sample are depicted in Fig. 1, where r1 = 8 [mm], r2 = 12 [mm], d = 10 [mm]. The model parameters were estimated according to the algorithm published in [4] and were as follows: a = 2, 51 [A/m], α = 2, 53 · 10−6 [-], k = 1, 01 [A/m], c = 1, 56 · 10−2 [-], Ms = 1, 11 · 106 [A/m]. Figure 2 depicts hysteresis loops calculated on the basis of dynamic hysteresis model and obtained during measurements at frequency equal to 400 [Hz]. The carried out calculations of hysteresis loops are in a good accordance with the results obtained experimentally. In the proposed dynamic model of hysteresis the magnetization process occurs in a modified effective field, which includes the field component coming from eddy currents. It can be stated, that as the frequency of the magnetizing field increases, the hysteresis loops widen (cf. Fig. 3, [4]), what in the general case may be the result of increase of eddy currents around moving domain walls and the component of classical eddy currents bound with interlaminar eddy currents. The component bound with intralaminar

Figure 3. Experimental hysteresis loops for various frequencies.

eddy currents in the considered range of frequency may be neglected. In order to estimate the influence of these components on the measured loss, the loss dependence on frequency of field magnetizing the core has been determined. In the case, when the first component prevails, the dependence should obey ∝ f 3/2 [7], while for the second component the dependence should take the form ∝ f 2 . Calculating numerically the area covered by hysteresis loop, the total energy loss bound with magnetic hysteresis and eddy currents generated within the material, as well as between the layers may be obtained. For aRgiven radius of a toroidal sample this is given by w(r) = HdB, where BH

H is the external magnetic field intensity. The energy loss in the whole volume of the considered geometric structure of the sample is given by Zr2

Z W =

w(r)dV = 2πd V

w(r)rdr,

(6)

r1

where dV = 2πrddr. Figure 4 depicts modelled dependencies of the total loss on frequency for various values of flux density calculated on basis of the formula given above. The Figure also shows the results of measurements of loss density of the examined cores, obtained with the use of measurement system, whose description is given in [9].

Proceedings

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11th International IGTE Symposium 2004

eddy currents by application of ribbon surface insulation [8]. In the case when the influence of currents generated around moving domain walls prevails, the counterfield Hd may be calculated on the basis of Pry’s-Bean’s model. The obtained theoretical dependencies of energy loss on frequency and flux density may be helpful for quantitative analysis of performance of the cores working in magnetic circuits of inductive elements in electric and electronic appliances. R EFERENCES

Figure 4. Total energy loss density vs. frequency: calculated (solid line) and measured (points) for various values of flux density.

Figure 5. Component of total loss bound with eddy currents for the examined core.

On the basis of Kolmogorov’s test carried out at the significance level equal to 0.05 it was stated, that the measured values of energy loss, obtained for various samples of nominally the same material, (in the considered case 5 measurements were carried out), reveal normal distribution. Figure 4 depicts medium values of energy loss density and deviations from them. They were calculated using the three-sigma rule. The deviations may be a result of scatter of technological parameters during the manufacturing process of the material, during core forming and also during the process of inducing magnetic anisotropy in the material. On the basis of obtained results it was stated, that energy loss dependencies on magnetic flux density and frequency of alternating magnetic field calculated on the basis of the dynamic energy loss model are leveled within the range of measured loss. The component of total loss bound with eddy currents is depicted in Fig. 5. From the dependencies of loss from frequency depicted in Fig. 5 it results, that they are subjected to ∝ f 2 , what indicates an influence of interlaminar eddy currents on the value of total loss in the range of measured frequencies of magnetizing field. For commercial applications it is possible to reduce the influence of interlaminar

[1] G. E. Fish, C. F. Chang, R. Bye, ”Frequency dependence of core loss in rapidly quenched Fe-6.5 wt. %Si”, J. Appl. Phys., vol. 64, pp. 5370–5372, 1988. [2] J. Szczyglowski, P. Kopciuszewski, W. Wilczynski, A. Roman, ”Energy losses in Fe-based and Co-based amorphous materials”, Mater. Sc. Eng. B75, pp. 13–16, 2000. [3] D. C. Jiles, D. L. Atherton, ”Theory of ferromagnetic hysteresis”, Journ. Magn. Magn. Mater., vol. 61, pp. 48–60, 1986. [4] J. Szczyglowski, ”Influence of eddy currents on magnetic hysteresis loops in soft magnetic materials”, Journ. Magn. Magn. Mater., vol. 223, pp. 97–102, 2001. [5] D. M. Nathasingh, C. H. Smith, A. Datta, ”Effects of coatings on the soft magnetic properties of an iron based amorphous alloy”, IEEE Trans. Magn. 1984, vol. MAG-20, No. 5, pp. 1332–1334. [6] E. Ferrara, C. De Luigi, C. Beatrice, C. Appino, F. Fiorillo, ”Energy loss vs. magnetizing frequency in field-annealing nanocrystalline alloys”, Journ. Magn. Magn. Mater., vol. 215-216, pp. 466–468, 2000. [7] G. Bertotti, ”Hysteresis in Magnetism”, Academic Press, New York, 1998. [8] Y. Okazaki, H. Kanno, E. Sakuma, ”Improved coating for amorphous alloy with low loss deterioration”, IEEE Trans. Magn. 1989, vol. 25, No. 5, pp. 3352-3354. [9] J. Bajorek, ”Komputerowy system do pomiaru dynamicznych wlasciwosci magnetycznych materialow magnetycznie miekkich (Computer-aided system for measurement of dynamic magnetic properties of soft magnetic materials)” (in Polish), Zesz. Nauk. PSwietokrz. Elekt., pp. 33–39, 1984.