ARTICLE IN PRESS. Physica B 343 (2004) A comparative study of Preisach scalar hysteresis models

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1 Physica B 343 (2004) A comparative study of Preisach scalar hysteresis models B. Azzerboni a, *, E. Cardelli b, G. Finocchio a a Dipartimento di Fisica della Materia e Tecnologie Fisiche Avanzante, University of Messina, Salita Sperone 31, Messina, Italy b Dipartimento di Ingegneria Industriale, University of Perugia, Via G. Duranti 1, Perugia, Italy Abstract In this paper different Preisach-relays-based models are taken into account for the modeling of scalar hysteresis. The main features of such models are presented and discussed, and identification procedures are illustrated, either using complete numerical approaches, or using mixed analytical numerical approaches. The results obtained for a nonoriented grain steel suggest that the mathematical models of scalar hysteresis taken into account have comparable predictive capabilities and accuracy. r 2003 Elsevier B.V. All rights reserved. Keywords: Preisach model; Scalar hysteresis; Soft steels; Symmetric loops 1. Introduction Several Preisach-relays based models have been proposed and their accuracy and usefulness have been proved in a quite large number of different applications, ranging from magnetic recording to electrical machines. All these models are based on the formulation we will call here classical scalar Preisach model (CSPM) as described in detail in Refs. [1, pp ], [2, pp. 1 63], and [3, pp ]. CSPM has two mathematical properties: wiping-out and congruency: however, many experiments show that some magnetic materials may substantially deviate from those properties. In addition, the general procedure to determine the Preisach function for the CSPM, as presented in Ref. [2, *Corresponding author. Tel.: ; fax: addresses: azzerboni@ingegneria.unime.it (B. Azzerboni), ermanno2@unipg.it (E. Cardelli). pp ] needs the knowledge of the first-order reversal curves. This is a critical fact, because soft magnetic materials manufacturers and users usually refer to the set of symmetric loops and to the first magnetization (virgin) curve, as indicated by the Specific Standards. For the reasons mentioned above but not only for that, different Preisach-relays based approaches have been proposed for modeling with accuracy the behavior of magnetic materials. In this paper we will take into account some of these different formulations, namely the so-called modified scalar Preisach model (MSPM) [4 6], and the DOK (Della Torre, Oti, and Kadar) model [3, pp ], [7], using either numerical or analytical (with Gaussian or Lorentzian approximation functions) identification procedures. We will show a comparative study between the different formulations, and the theoretical predictions will be compared with the measurements performed on non-oriented grain magnetic steels. The main goal of the paper is to prove that /$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi: /j.physb

2 B. Azzerboni et al. / Physica B 343 (2004) different Preisach-relays-based models can reproduce with comparable accuracy symmetric loops and virgin curve in soft magnetic irons, although they use different mathematical functions and procedures. In other words, the Preisach-relaysbased approach to analyze hysteresis in a soft material is not unique, and different formulations are probably different views of the same coin Identification of PðU; VÞ for the MSPM Apart from the identification procedure of the Preisach probability distribution function, the complete procedure allows to identify the Preisach function for the CSPM using a family of symmetric loops with amplitude max i for i ¼ 1; y; n: To do that, we approximate the Preisach function with the following stepwise function: 2. MSPM formulation CSPM can be roughly described in mathematical form by the following differential equations: ¼ 2 Pð; VÞ dv ð1þ 0 for increasing applied field, and PðU; VÞ ¼P ij ; i pup iþ1 ; j pvp jþ1 ; where i ¼ max i ; 0 ¼ 0 for i ¼ 1; y; n; i ¼ max i : ð5þ ð6þ 0 ¼ 2 PðU; Þ du ð2þ for decreasing applied field; where PðU; VÞ is the so-called Preisach probability distribution function, shortly Preisach function, U and V are the switching-on and switching-off magnetic fields, respectively, and 0 is the field of the starting point on the plane B: 2.1. General formulation of MSPM Referring to the CSPM, the mathematical expression of the MSPM can be written as follows: ¼ KðMÞ WðÞþ2 Pð; VÞ dv ð3þ for increasing applied field, and 0 ¼ KðMÞ WðÞþ2 PðU; Þ du 0 ð4þ for decreasing applied field. W ðþ is an additional reversible component of magnetization, and KðMÞ describes the noncongruency property of minor loops. For more details see Refs. [4 6]. So a linear system for the unknown P ij is obtained. The solution of this system is (for i ¼ 1; y; n): P i;i ¼ð i i 1 Þ 2 ðm i ð i Þ M i 1 ð i 1 ÞÞ; P i;j ¼ð i i 1 Þ 1 ð j j 1 Þ 1 M i ð j Þ M i ð j 1 Þ Xj k¼ jþ1 P ik ð i i 1 Þð k k 1 Þ 2P ik ð i i 1 Þ 2 ; ð7þ ð8þ where M i is the magnetization for the symmetric loop between 7max i : It is easy to extend this approach to the MSPM in the hypothesis that the function WðÞ is a single-value function of the applied magnetic field. In this case we obtain P i;i ¼ð i i 1 Þ 2 DM ii ; P i; j ¼ð i i 1 Þ 1 ð j j 1 Þ 1 DM ij Xj k¼ jþ1 P ik ð i i 1 Þð k k 1 Þ 2P ik ð i i 1 Þ 2 ; ð9þ ð10þ

3 166 B. Azzerboni et al. / Physica B 343 (2004) where M i ð j Þ M i 1 ð j 1 Þ DM ij ¼ KððM i ð j Þ M i 1 ð j 1 ÞÞ=2Þ Wð jþþwð j 1 Þ : ð11þ Reduced formulation of MSPM for soft materials In consideration that magnetization of soft magnetic materials is determined by wall motion principally, and that it has been observed that wall motion is usually unidirectional when a monotonic increasing or decreasing input magnetic field are applied, we can model this fact assuming that the probability distribution of the switching fields U and V is statistically independent. Then, we can assume that the Preisach function can be written as a product of two identical single variable functions PðU; VÞ ¼P s ðuþp s ð VÞ: ð12þ So, MSPM can be rewritten in the following useful mathematical form: ¼ KðMÞ WðÞþ 2P s ðþ P s ð VÞ dv ð13þ for increasing applied field, and ¼ KðMÞ 0 WðÞþ 2P s ð Þ P s ðuþ du 0 for decreasing applied field Identification of the Preisach function for reduced MSPM ð14þ Assumption (12) allows to rewrite Eqs. (13) and (14) in the following discrete form: X n P s ð i Þ¼ ð=þð jþ 2P i¼n jþ1 s ð j ÞD ; j ¼ 1; y; n: ð15þ The numerical solution of Eq. (15) can be found with the recursive relationships below P s ð n iþ1 Þ ¼ k n i P s ð n i Þ; i ¼ 1; y; n=2 1; ð16þ P s ð i Þ¼ where ð=þð iþ ; i ¼ 1; y; n=2; ð17þ 2k i P s ð n iþ1 ÞD k n i ¼ ð=þð n iþ ð=þð n i 1 Þ ð=þð iþ1þ k iþ1 ð=þð n i 1 Þ ; i ¼ 1; y; n=2 1; ð18þ k i ¼ 1 þ k n iþ2 K i 1 k 1 ¼ 1; where n must be even. 3. DOK formulation for i ¼ 2; y; n=2; ð19þ Under the hypothesis of rate-independent hysteresis, i.e. in the plane ; M the derivative db= is independent upon the time, the total magnetization, using the DOK model has the following analytical expression: M ¼ M irr þð1 SÞða þ FðÞ a Fð ÞÞ; ð20þ where M irr is the irreversible part of magnetization computed using expressions in analogy with Eqs. (1) and (2) S is the squareness, a þ and a are magnetization-dependent functions, and F ðþ is a monotonic function related to the reversible magnetization. For more details see Refs. ½3; pp: Š and [8]. The Preisach function can be identified in complete numerical way, using the techniques presented in the previous section, or alternatively using a Gaussian approximation [3]: " P sg ðu; VÞ ¼SA 2 exp 1 U V 2 # 2 c s 2 c " exp 1 U þ V # 2 s ; ð21þ 2 c

4 B. Azzerboni et al. / Physica B 343 (2004) where c ; and s are the identification parameters, and A is a normalized constant depending on the value of magnetization at saturation state. Another possible approximation function is the Lorentzian one (see Refs. ½1; pp: 445Š [8 10]): P SL ðu; VÞ ¼SA þððv þ c = c ÞsÞ þððu c = c ÞsÞ 2; ð22þ where the identification parameters have the same meaning of the ones used in the Gaussian approximation and for this reason we have used the same symbols, even if they assume different values depending on the approximation function used. The use of approximation functions has two main practical advantages: (a) reduced computer time in the identification procedure; (b) possibility to treat partially or totally the Everett s integral in closed analytical form, so additional computer time saving (see also Refs. [8], and[3, pp ]). It must be underlined that analytical approximation function can be used also for other Preisach-relays-based formulations, for instance the same MPSM. 4. Identification of analytical approximations of the Preisach function As stated in the previous section, the identification problem for the Preisach function using the approximation functions proposed is, in fact, reduced to the determination of the parameters: A; c ; S and s: The squareness can be computed by means of the following expression: S ¼ M R ; ð23þ M S where M R is the remanence of the major loop. A simple analytical identification procedure for the Lorentzian function is presented in Ref. [9]; c can be calculated by means of ¼ max ¼ c a ; ð24þ exp where the index a means ascending branch of the major loop and exp means measured data, while ð=þ ¼c is the expression of the = computed using Eqs. (1), (12), and (22). s is computed by means of numerical solution of the equation: ð=þ a exp ¼ c ð=þ d exp ¼ c ¼ a tanð2sþ atanðð s þ c = c ÞsÞ a tanðð s c = c ÞsÞ ½1þð2sÞ 2 Š; ð25þ where the index d means the descending (upper) branch major loop. The term on the right is obtained using the expressions of = for ¼ c computed using the Lorentzian function. The same identification procedure can be used for the Gaussian function approximation: c is determined again by means of Eq. (24) where the expression of = is derived using Eq. (21) instead of Eq. (22). s is computed by means of the numerical solution of the following equation: ð=þ a exp ¼ c ð=þ d exp ¼ c ¼ erfð2sþ erfðð s þ c = c ÞsÞ ; ð26þ e 4s2 erfðð s c = c ÞsÞ where = is computed using the Gaussian approximation for ¼ c : A can be computed by means of the following equation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M S A ¼ t R s s P S ðuþ R ; ð27þ U s P S ð VÞ dv du where S and M S are the field and the magnetization at saturation state, respectively. It must be underlined that the identification procedure above is correct if the reversible magnetization is negligible with respect to the irreversible one, and/or if it is valid the state-independent hypothesis, i.e., the reversible magnetization is nonzero only in the line U ¼ V: See for more details Refs. [9] and [10].

5 168 B. Azzerboni et al. / Physica B 343 (2004) Table 1 Values of the identification parameters for MSPM Lorentzian approximation Gaussian approximation c s S A Table 2 Values of the identification parameters for DOK modeling Lorentzian approximation Gaussian approximation c s S A Fig. 2. Measured and computed symmetric loop for the same sample of Fig. 1. Exciting magnetic field AC 0:1 zwith amplitude 40 A=m: Fig. 1. Measured and computed symmetric loop for an AC 0:1 zexciting magnetic field with amplitude 100 A=m: Material: non-oriented grain Fe Si 3.2%wg thickness 0:65 mm: Samples cut at 90 : 5. Results Fig. 3. Measured and computed symmetric loop for the same sample of Fig. 1. Exciting magnetic field AC 0:1 zwith amplitude 25 A=m: In this section we show some results obtained by means of MSPM and DOK model, using different identification techniques. The computed values have been attained using dedicated software codes, working in MATLAB r environment. The complete identification procedure leads in general to a more accurate reproduction of the magnetic hysteresis in the material, but the use of analytical approximation functions allows a reduction of the computer time of more than 30 times. The magnetic material taken into account here is a commercial laminated non-oriented grain magnetic steel, having thickness ranging from 0.35 to 0:65 mm; and cut at different angles with respect to the rolling direction. The values of the parameters computed in case of analytical approximation of

6 B. Azzerboni et al. / Physica B 343 (2004) Fig. 4. Measured and computed symmetric loop for an AC 0:1 zexciting magnetic field with amplitude 100 A=m: Material: non-oriented grain Fe Si 3.2%wg thickness 0:6 mm: Samples cut at 45 : Fig. 6. Measured and computed symmetric loop for the same sample of Fig. 4. Exciting magnetic field AC 0:1 zwith amplitude 30 A=m: Fig. 5. Measured and computed symmetric loop for the same sample of Fig. 4. Exciting magnetic field AC 0:1 zwith amplitude 30 A=m: Fig. 7. Measured and computed symmetric loop for an AC 0:1 zexciting magnetic field with amplitude 190 A=m: Material: non-oriented grain Fe Si 3.2%wg thickness 0:6 mm: Samples cut at 0 : PðU; VÞ are reported in Tables 1 and 2, respectively for MSPM and for the DOK modeling. In Figs are reported some symmetric loops and the first magnetization curves predicted using the different Preisach-relays based approaches described in the previous sections. The graphs in the figures are also referred to the measured values at 0:1 z; using a special Epstein frame and a four quadrant digital-controlled amplifier. 6. Conclusions In this paper we have examined different Preisach-relays-based models for the analysis of scalar hysteresis in soft magnetic materials. We have shown in the first part of the paper the principal features of each model and we have discussed the differences either in the formulation, or in the identification procedure.

7 170 B. Azzerboni et al. / Physica B 343 (2004) Fig. 8. Measured and computed first magnetization curves for an AC 0:1 zexciting magnetic field. Material: non-oriented grain Fe Si 3.2%wg thickness 0:35 mm: Samples cut at 90 : Fig. 10. Measured and computed first magnetization curves for an AC 0:1 zexciting magnetic field. Material: non-oriented grain Fe Si 3.2%wg thickness 0:65 mm: Samples cut at 0 : predictive accuracy. This is anyway a not completely solved question and deserves further investigations. The general conclusion we have drawn is that the observed results is that Preisach-relays-based models are mathematical modeling of hysteresis rather than phenomenological, or physical ones. References Fig. 9. Measured and computed first magnetization curves for an AC 0:1 zexciting magnetic field. Material: non-oriented grain Fe Si 3.2%wg thickness 0:65 mm: Samples cut at 45 : As known, identification procedures based on approximation functions allows a remarkable reduction in computational time. We have found in our analysis a substantial confirm of this fact. In addition we have found that all the different Preisach-relays-based approaches, tested with measured data of magnetic laminated steels, gave almost identical results, even if every material has a more accurate model. We have found no relation between physical parameters of the material and the Preisach-relays based model with the more [1] G. Bertotti, ysteresis in Magnetism, Academic Press, New York, USA, [2] I.D. Mayergoyz, Mathematical Models of ysteresis, Springer, Berlin, [3] E. Della Torre, Magnetic ysteresis, IEEE Press, Piscataway, NY, [4] E. Cardelli, E. Della Torre, G. Ban, Physica B 275 (2000) 262. [5] M. Angeli, E. Cardelli, E. Della Torre, Physica B 275 (2000) 154. [6] E. Cardelli, L. Fiorucci, E. Della Torre, IEEE Trans. Magn. 37 (4) (2001) [7] E. Della Torre, F. Vajda, IEEE Trans. Magn. 30 (6) (1994). [8] B. Azzerboni, E. Cardelli, E. Della Torre, G. Finocchio, J. Appl. Phys. 93 (2003) [9] B. Azzerboni, E. Cardelli, G. Finocchio, F. La Foresta, IEEE Trans. Magn. 39 (5) (2003) [10] B. Azzerboni, E. Carpentieri, G. Finocchio, F. La Foresta, Proc. Int. Symp. on Theoretical Electrical Engineering (ISTET 03), pp