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Hysteresis losses in soft magnetic composite materials

Hysteresis losses

Bogomir Zidaricˇ Teces, Maribor, Slovenia

157

Mykhaylo Zagirnyak Kremenchuk State Polytechnic University, Poltava, Ukraine, and

Konrad Lenasi and Damijan Miljavec Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia Abstract Purpose – To analyze the Jiles and Atherton hysteresis model used for hysteresis losses estimation in soft magnetic composite (SMC) material. Design/methodology/approach – The Jiles and Atherton hysteresis model parameters are optimized with genetic algorithms (GAs) according to measured symmetric hysteresis loop of soft magnetic composite material. To overcome the uncertainty, finding the best-optimized parameters in a wide predefined searching area is done with the proposed new approach. These parameters are then used to calculate the hysteresis losses for the modeled hysteresis. The asymmetric hysteresis loops are also investigated. Findings – The classical GAs are good optimization methods when a pre-defined possible set of solutions is known. If no assumption on solutions is present and a wide searching area range for parameter estimation is selected then the use of the new approach with nested GAs gives good results for symmetric hysteresis loops and further for the estimation of hysteresis losses. Research limitations/implications – The use of the Jiles and Atherton hysteresis model for asymmetric hysteresis must be explored further. Only one set of optimized Jiles and Atherton hysteresis model parameters used for estimation of hysteresis losses gives good results for only symmetric hysteresis loops. These parameters have limitations for asymmetric hysteresis loops. Practical implications – Nested GAs are a useful method for optimization when a wide searching area is used. Originality/value – The originality of the paper is in the establishment of nested GAs and their application in Jiles and Atherton hysteresis model parameters optimization. Also, original is the use of the Jiles and Atherton hysteresis model for hysteresis loop description of soft-magnetic composite material. Keywords Genetic modification, Algorithmic languages, Magnetism, Composite materials, Modelling Paper type Technical paper

Feromagnetic hysteresis model The mathematical model for ferromagnetic hysteresis introduced by Jiles and Atherton ( Jiles, 1989; Jiles and Thoelke, 1989) (J-A model) is made in compliance with physical principles of ferromagnetic materials and not strictly on mathematical or experimental curve fitting. The J-A model assumes that total magnetization M (equation (4)) of ferromagnetic materials is decomposed into an irreversible Mirr (equation (2)) and a reversible Mrev (equation (3)) magnetization component ( Jiles, 1989; Jiles and Thoelke, 1989).

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 25 No. 1, 2006 pp. 157-168 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640610634416

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Anhysteretic magnetization Man (i.e. hysteresis curve, if there are no losses) follows the Langevin function (Jiles and Thoelke, 1989; Jiles et al., 1992):
   H þ aM a M an ¼ M S coth 2 : ð1Þ a H þ aM

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The irreversible magnetization component Mirr is manifested in energy losses through domain wall pinning (Jiles, 1989; Jiles et al., 1992): dM irr M an 2 M irr ¼ : dH kd 2 aðM an 2 M irr Þ

ð2Þ

The reversible magnetization component Mrev is proportional to the difference between anhysteretic Man and irreversible magnetization Mirr value. In the model it has the form (Jiles, 1989; Jiles et al., 1992): M rev ¼ cðM an 2 M irr Þ:

ð3Þ

The total magnetization M is the sum of irreversible Mirr and reversible Mrev magnetization ( Jiles, 1989): M ¼ M rev þ M irr :

ð4Þ

Substituting Mrev with (equation (3)) yields: M ¼ ð1 2 cÞM irr þ cM an ;

ð5Þ

and this leads to the total differential magnetization susceptibility dM/dH (Jiles, 1989; Jiles et al., 1992): dM M an 2 M irr dM an ¼ ð1 2 cÞ þc ; dH kd 2 aðM an 2 M irr Þ dH

ð6Þ

which with equation (1) forms the J-A hysteresis model. The five material-dependent parameters appearing in equation (1) and (6) are: . MS – magnetization saturation; . a – shape parameter; . a – main field parameter; . k – domain wall pinning constant proportional to hysteresis losses; and . c – domain flexing constant. In this paper, the parameter d has no physical meaning and is equal to þ 1 for dH =dt . 0 and to 2 1 otherwise. If the first order differential equation (6) is solved, the induction B can be calculated using: B ¼ m0 ðH þ M Þ:

ð7Þ

Identification of the five parameters included in the J-A model is obtained by using the measured results of hysteresis loops and genetic algorithms (GAs).

Hysteresis measurements To determine the five J-A model parameters (equation (1) and (6)) and hysteresis losses, the measured values of material hysteresis have to be known. Our measurements of quasi-static hysteresis loops are made on a SMC composite material Somaloy 500e ring sample (Jansson, 1999) (ring data in Table I) by using a precise ballistic galvanometer. Figure 1 shows the measured quasi-static hysteresis loops for a SMC material. The measured quasi-static hysteresis loops (Figure 1) do not include the response of the material to the alternating magnetic field excitation. They do not take into account neither eddy current losses nor anomalous losses. The measured values of the quasi-static hysteresis loops for different magnetization levels are used to define the J-A model magnetization level-dependant parameters and estimation of hysteresis losses. The hysteresis loops (Figure 2) measured at different applied frequencies and maximum induction of 0.9 T ( Jansson, 1999) include besides hysteresis losses also eddy current losses and anomalous losses. The difference in the surface between the quasi-static hysteresis loops and the loops at one of the frequencies presents eddy current losses and anomalous losses. Figure 2 shows the measured hysteresis loops for a SMC material at different applied frequencies (Jansson, 1999) for the maximum induction of 0.9 T. The small differences between the loops in Figure 2 are due to the physical structure of the SMC material. In this material, small ferromagnetic particles are insulated from each other and pressed together. In this way, eddy currents are significantly reduced.

Hysteresis losses

159

Ring SMC sample data Inner diameter Outer diameter Thickness Cross-section

45 mm 55.1 mm 5.05 mm 49.7425 mm2

Table I. SMC ring data

Figure 1. Measured quasi-static hysteresis loops at different induction levels

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160 Figure 2. Hysteresis loops measured at different applied frequencies and at maximum induction of 0.9 T

Genetic algorithms for J-A model parameters identification GAs are used to identify proper values of J-A hysteresis model parameters. These algorithms are an optimization technique based on the concept of natural selection and genetics (Wilson et al., 2001). They start with a set of possible solutions – individuals called the population (Figure 3) (Wilson et al., 2001; Leite et al., 2004). Solutions from this population are used to form new children with the hope that the new population will be better than the old one. Solutions that will form a new children set of solutions are selected according to their fitness. This is repeated until the GA stop criteria (for example, the number of generations or improvement of the best solution) are satisfied. The GAs can optimize with continuous or discrete parameters and can handle numerically generated data, experimental data or analytical functions (Leite et al., 2004). The use of GAs in this matter is further discussed below. J-A model parameters identification The five material-dependent parameters to determine the J-A model (equation (6)) are obtained for a quasi-static hysteresis loop using the measured values from Figure 1. The construction of the population (Leite et al., 2004) for the J-A model parameters used in GAs for the optimization process is shown in Figure 3. The population contains n individuals – chromosomes (n is any positive integer) where each row represents only one individual, composed with parameters of the J-A model called genes. The initial parameter values assigned to the population are selected from a predefined range (Table II). The last column (Figure 3) shows estimation of the individuals – fitness according to the chosen evaluation criteria of individuals. Wilson et al. (2001) use the least square error approach, and the mean square error between an experimental and simulated hysteresis loop is used by Leite

Figure 3. Construction of the population

et al. (2004). In our case, fitness is obtained by the absolute difference between measured and computed induction B: p X ðBm;i 2 Bc;i Þ ð8Þ fitness ¼ 2abs

Hysteresis losses

i¼1

where Bm – measured induction B, Bc – calculated induction B using the J-A model, p – number of measurements. After the initial population is established, GAs search for the best solution (individual) in the area of possible solutions (Table II) using GAs operators (Matlab toolbox, www.ie.ncsu.edu/mirage/GAToolBox/gaot): selection, crossover and mutation on the temporary best fitted individuals. The selection mechanism forms pairs, with regard to their fitness, which will create children. The most used technique by Wilson et al. (2001), Leite et al. (2004) and Riganti and Salvini (2005) is the roulette wheel rule. The selection technique (Matlab toolbox, www.ie.ncsu.edu/mirage/GAToolBox/gaot) used in these cases is a ranking selection function based on the normalized geometric distribution by probability of selecting the best individual. The chosen probability is 8 percent. Crossover is an arithmetic function that takes selected pairs and creates new children. The used crossover function creates children with a random interpolation along the line formed by two individuals. Some new children genes are then chosen to mutate using a non-uniform probability distribution. The Gaussian distribution is used for the mutation which starts widely and narrows to a point distribution when the current generation approaches the maximum generation. The area of possible solutions (searching area) is limited with predefined upper and lower limit values (Table II) individually for each J-A model parameter. After a certain predefined number of populations (stop criteria), GAs return an individual with the best fitness representing the existing optimum for the J-A model parameters in a limited searching area. The best fitness means the maximum negative fitness value found during GAs operation (ideal value is “zero”). The higher is the fitness value (equation (8)) the more accurate are the calculated hysteresis loops. Using the described population (Figure 3), estimation of fitness (equation (8)) and selected searching area (Table II) with GAs, identification of the J-A model parameters is made. Optimization results for n ¼ 40 individuals after 100 generations of GAs on a measured quasi-static hysteresis loop with an induction level of 1.3 T are shown in Figure 4 and Table III. From Figure 4 it can be seen that GAs does not find the best (global) optimum but some other local optimum. To determine the reason for this problem, convergence of GAs solutions was analyzed. Convergence of the J-A model normalized parameters is shown in Figure 5. Parameters a a k MS c

Upper area limit

Lower area limit

2,000 0.002 2,500 1.7 £ 106 0.3

100 0.0001 100 1 £ 106 0.01

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Table II. GAs parameter area limits for a measured quasi-static hysteresis loop B ¼ 1.3 T

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From convergence of parameters k, MS and a (Figure 5) we can see that their selected searching area is most likely too narrow. This means that the global optimum may not exist in a determined searching area. The next possible reason might be the small number of individuals and generations with regard to the GAs operators. When the searching area is wider, GAs must operate with a larger number of individuals and generations meaning that good results can be expected (GAs search J-A model parameters

Table III. Optimized values for the J-A model parameters for measured quasi-static hysteresis loop B ¼ 1.3 T

Figure 4. GAs optimized hysteresis and measured quasi-static hysteresis loop B ¼ 1.3 T

Figure 5. Convergence of the J-A model normalized parameters

a a k MS c

Optimized values 876.7888 0.000965 310.878 1170224.135 0.142

through the whole area and convergence is slower). On the other hand, thought narrowing the searching area may give rise to the loss of the global optimum, less individuals and generations (fast convergence) need to be used. In (Leite et al. 2004) it is proposed to calculate suitable ranges for the J-A model parameters with a trial-and-error procedure using a small number of individuals with few generations by observing the error (fitness). The previous discussion shows how to overcome these low quality results. In its continuation the paper proposes how to operate with a wide searching area. The main idea is to use GAs built in another GA called GgaA. Its flow diagram is shown in Figure 6. The main property of GgaA is that the outer GA finds the best upper and lower searching area limits and the inner GA is used to estimate the individual’s fitness for the outer GA. In fact, from a wide searching area the region containing the global optimum is detected.

Hysteresis losses

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Figure 6. Flow diagram of GgaA

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The outer GA individual of the population is composed of 11 genes, representing the upper and lower limits of the searching area (Table IV) plus fitness. For example, the first gene of an individual represents the upper boundary area limit for parameter a, the second gene represents the lower boundary area limit for parameter a, and in the same manner further for the rest of the parameters. The inner GA uses 40 individuals and 100 generations. This number of individuals and generations is selected because the searching area of the inner GA is narrow and is expected to find the best optimum for all searching areas (fast convergence) delivered from the outer GA. To determine fitness for the outer GA, the inner GA finds the J-A model parameters global minimum for the current area limits. It uses measured values of quasi-static hysteresis loops and fitness from equation (8). The inner GA is also used when GgaA stopping conditions are reached (number of generations of the outer GA). In this case, the inner GA optimizes the J-A model parameters with the upper and lower limits of the parameters searching area optimized with GgaA (Table V). The J-A model parameter identification is done using new GgaA optimization on the measured hysteresis loop with induction of 1.3 T. The upper and lower J-A model parameter boundary limit values for GgaA are in Table IV. The results are shown in Figure 7 and Table V. The outer GA is set to ten individuals and 20 generations. From Figure 7 it can be seen that the modeled hysteresis loop is well fitted to the measured one. This confirms the discussion and work done on GgaA’s. A good approximation between the modeled and measured hysteresis loops (Figure 7) is essential for proper estimation of the hysteresis losses in SMC materials. The difference between the modeled and measured hysteresis losses from Figure 7 is 5.92 percent. This quite considerable difference is due to the use of high induction level

Parameters of GgaA

Table IV. J-A parameter boundary limits for GgaA for measured quasi-static hysteresis loop B ¼ 1.3 T

a upper limit a lower limit a upper limit a lower limit k upper limit k lower limit MS upper limit MS lower limit c upper limit c lower limit

J-A model parameters Table V. Optimized values for the J-A model parameters for measured quasi-static hysteresis loop B ¼ 1.3 T using GgaA

Search area limits for GgaA

a a k MS c

4,000 4 1,950 1,950 4 100 0.004 4 0.00195 0.00195 4 0.0001 4,000 4 1,950 1,950 4 100 2.4 £ 106 4 106 106 4 0.4 £ 106 1 4 0.4975 0.4975 4 0.005

Search area limits optimized with GgaA

Optimized values with inner GA

790.23 4 327.03 0.000852 4 0.000273 2704.89 4 821.44 1.103 £ 106 4 1.08 £ 106 0.9687 4 0.71998

663.907 0.000694 2454.013 1.09538 £ 106 0.95382

Hysteresis losses

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Figure 7. Optimization result of GgaAs for measured quasi-static hysteresis loop B ¼ 1.3 T

hysteresis loops. Better results are achieved by modeling lower maximum induction level hysteresis loops. It has to be noted that the J-A model can also be used for different shapes of the quasi-static applied magnetic field in function of excitation. For example, in stator teeth of brushless direct current motors (BLDC) the total excitation (Figure 8) is the sum of the external applied field and excitation of built-in permanent magnets. Permanent magnet excitation is constant (Figure 8) and it is superimposed to motor winding excitation. For such excitation asymmetric minor hysteresis loops are obtained (Figure 9). Using the same set of optimized J-A model parameters from Table V for modeling a minor hysteresis loop with excitation from Figure 8 shown unsatisfactory results. The descending part of the hysteresis in Figure 9 is well fitted but at a lower turning, point when the hysteresis loop ascends, the difference between the modeled and measured hysteresis loop became obvious. This is the reason why the J-A model parameter

Figure 8. Sample of excitation in BLDC motors

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values were again optimized with GgaA (the same boundary limits from table IV) for asymmetric minor hysteresis loops. Results are shown in Table VI and Figure 9. Figure 10 shows that with the new optimized set of the J-A model parameters results are better. But again unsatisfactory results are obtained when hysteresis ascends and approaches the knee of the hysteresis loop. When the applied field ascends, it describes

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Figure 9. Quasi-static hysteresis appearing only in the first quadrant with excitation from Figure 8 and optimized J-A parameters

J-A model parameters Table VI. Optimized J-A model parameter values for a minor hysteresis loop

Figure 10. Quasi-static hysteresis appearing only in the first quadrant with excitation from Figure 8 and new optimized J-A parameters

a a k MS c

Optimized values with GgaA 805.206 0.001016 737.32 1.13 £ 106 0.77

the new hysteresis with a different set of the J-A model parameters. Riganti and Salvini (2005) proposed that the set of the J-A model parameters is changed when the hysteresis ascends (use of two sets of parameters). To achieve better results with asymmetric hysteresis loops models, further work must be done. The authors propose the use of neural networks and fuzzy logic to evaluate the J-A model parameters for such kind of magnetization.

Hysteresis losses

167 Conclusion Identification of the J-A hysteresis model parameters for the symmetric hysteresis (Figure 1) with GAs for SMC materials is an efficient method when the searching area is predefined with a possible set of solutions. The J-A hysteresis model parameters values can also be obtained by empirical equations (Jiles et al., 1992) in characteristic points on the hysteresis loop. If no assumption on solutions is present and a wide searching area range for parameter estimation is selected, then the use of the GgaA approach gives good results. This is clearly seen from Figure 7. The obtained results speak in favor of the use of the J-A hysteresis model optimized by GgaA for SMC materials. Best-fitted hysteresis loops are used to calculate hysteresis losses in SMC materials. It has to be understood that the physical meaning of the J-A model parameters is lost when they are optimized for AC hysteresis loops. Modeling the asymmetric hysteresis loops for SMC materials must be further exploited. References Jansson, P. (1999), Soft Magnetic Composites – A Rapidly Expanding Materials Group, Ho¨gana¨s AB, Hoganas. Jiles, D.C. (1989), Introduction to Magnetism and Magnetic Materials, Encyclopedia Britannica, 15th ed., Chapman & Hall, London. Jiles, D.C. and Thoelke, J.B. (1989), “Theory of ferromagnetic hysteresis: determination of model parameters from experimental hysteresis loops”, IEEE Trans. Magn., Vol. 25 No. 5, pp. 3928-30. Jiles, D.C., Thoelke, J.B. and Devine, M.K. (1992), “Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis”, IEEE Trans. Magn., Vol. 28 No. 1, pp. 27-35. Leite, J.V., Avila, S.L., Batistela, N.J., Carpes, W.P. Jr, Sadowski, N., Kuo-peng, P. and Bastos, J.P.A. (2004), “Real coded genetic algorithm for Jiles-Atherton model parameters identification”, IEEE Trans. Magn., Vol. 40 No. 2, pp. 888-91. Riganti, F.F. and Salvini, A. (2005), “Soft computing for the identification of the Jiles-Atherton model parameters”, IEEE Trans. Magn., Vol. 41 No. 3, pp. 1100-8. Wilson, P.R., Ross, J.N. and Brown, A.D. (2001), “Optimizing the Jiles-Atherton model of hysteresis by a genetic algorithm”, IEEE Trans. Magn., Vol. 37 No. 2, pp. 989-93. About the authors Bogomir Zidaricˇ received the BSc degree from the Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia, in 2002, respectively. Currently he is a researcher with the Faculty of Electrical Engineering, University of Ljubljana and Teces, Maribor. His interests include finite-element modeling and design and optimization of electrical machines. He is the corresponding author and can be contacted at: [email protected]

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Mykhaylo Zagirniak he graduated from Lugansk Machine-Construction Institute in 1970 and was qualified as an Electrical Engineer. He received the degree of Cand. Sc. (PhD) from Kharkov Polytechnical Institute in 1977, and the degree of DSc from Kharkov State Polytechnical University in 1996, both in Electrical Engineering. His research interests are in the area of calculation of magnetic fields for electromagnetic separators, electric machines and other electromagnetic devices. Presently he is rector of Kremenchuk State Polytechnical University and Chairman and Professor of Electrical Engineering Department. Konrad Lenasi received the BSc, MSc, and PhD degrees from the Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia. Currently he is a Professor with the Faculty of Electrical Engineering, University of Ljubljana. His interests include construction and analyzes of power transformers. Damijan Miljavec (S’93-M’99) received the BSc, MSc, and PhD degrees from the Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia, in 1993, 1996, and 1999, respectively. Currently he is an Assistant Professor with the Faculty of Electrical Engineering, University of Ljubljana. His interests include finite-element modeling and design and optimization of electrical machines together with the introduction of new magnetic materials. He has been involved in several research projects on the development of modern permanent-magnet machines with a special focus on hybrid vehicle propulsion systems.

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