Mathematical Models of Hysteresis

Chapter

Abstract

This chapter offers an overview of the hysteresis models that will be used throughout the book. After a short general classification of hysteresis models and parameter identification methods, the rectangular hysteresis operator is introduced. Then, the chapter focuses on summarizing the main equations, properties, and characteristics of the Preisach, energetic, Jiles-Atherton, Coleman-Hodgdon, and Bouc-Wen models. Particular attention is given to the analytical description of the general properties of hysteresis curves such as differential susceptibilities, remanence, coercivity, saturation, anhysteretic curve, energy lost, stability, accommodation, and limit cycle for each model. The second part of the chapter presents two techniques for the modeling of rate-dependent hysteresis, one based on the feedback (effective field) theory and the other one on the relaxation time approximation. Finally, a unified theory of vector models is presented; this theory can be applied to generalize any scalar model of hysteresis to vector systems.

References

  1. 1.
    Mayergoyz, I. (1991). Mathematical models of hysteresis. New York: Springer-Verlag.Google Scholar
  2. 2.
    Bouc, R. (1971). Mathematical model for hysteresis. Acustica, 24, 16–25.MATHGoogle Scholar
  3. 3.
    Wen, Y. K. (1976). Method for random vibration of hysteretic systems. Journal of the Engineering Mechanics Division-ASCE, 102, 249–263.Google Scholar
  4. 4.
    Duhem, P. (1897). Die dauernden Aenderungen und die Thermodynamik. Zeitschrift für Physikalische Chemie, 22, 543.Google Scholar
  5. 5.
    Jiles, D. C., & Atherton, D. L. (1983). Ferromagnetic hysteresis. IEEE Transactions on Magnetics, 19, 2183–2185.CrossRefGoogle Scholar
  6. 6.
    Jiles, D. C., & Atherton, D. L. (1984). Theory of ferromagnetic hysteresis. Journal of Applied Physics, 55, 2115–2120.CrossRefGoogle Scholar
  7. 7.
    SPICE. (University of California at Berkeley), The SPICE Webpage. Retrieved from http://bwrc.eecs.berkeley.edu/classes/icbook/spice/.
  8. 8.
    Coleman, B. D., & Hodgdon, M. L. (1987). On a class of constitutive relations for ferromagnetic hysteresis. Archive for Rational Mechanics and Analysis, 99, 375–396.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Coleman, B. D., & Hodgdon, M. L. (1986). A constitutive relation for rate-independent hysteresis in ferromagnetically soft materials. International Journal of Engineering Science, 24, 897–919.CrossRefMATHGoogle Scholar
  10. 10.
    Hauser, H. (1994). Energetic model of ferromagnetic hysteresis. Journal of Applied Physics, 75, 2584–2596.CrossRefGoogle Scholar
  11. 11.
    Harrison, R. G. (2009). Physical theory of ferromagnetic first-order return curves. IEEE Transactions on Magnetics, 45, 1922–1939.CrossRefGoogle Scholar
  12. 12.
    Harrison, R. G. (2011). Positive-feedback theory of hysteretic recoil loops in hard ferromagnetic materials. IEEE Transactions on Magnetics, 47, 175–191.CrossRefGoogle Scholar
  13. 13.
    Bergqvist, A., et al. (1997). Experimental testing of an anisotropic vector hysteresis model. IEEE Transactions on Magnetics, 33, 4152.CrossRefGoogle Scholar
  14. 14.
    Leite, J. V., et al. (2005). A new anisotropic vector hysteresis model based on stop hysterons. IEEE Transactions on Magnetics, 41, 1500–1503.CrossRefGoogle Scholar
  15. 15.
    Matsuo, T., et al. (2004). Stop model with input-dependent shape function and its identification methods. IEEE Transactions on Magnetics, 40, 1776–1783.CrossRefGoogle Scholar
  16. 16.
    de Almeida, L. A. L., et al. (2003). Limiting loop proximity hysteresis model. IEEE Transactions on Magnetics, 39, 523–528.CrossRefGoogle Scholar
  17. 17.
    Takács, J. (2003). Mathematics of hysteretic phenomena: The T(x) model for the description of hysteresis. Weinheim: Wiley.Google Scholar
  18. 18.
    Kucuk, I. (2006). Prediction of hysteresis loop in magnetic cores using neural network and genetic algorithm. Journal of Magnetism and Magnetic Materials, 305, 423–427.CrossRefGoogle Scholar
  19. 19.
    Cao, S. Y., et al. (2006). Modeling dynamic hysteresis for giant magnetostrictive actuator using hybrid genetic algorithm. IEEE Transactions on Magnetics, 42, 911–914.CrossRefGoogle Scholar
  20. 20.
    Rayleigh, L. (1887). On the behaviour of iron and steel under the operation of feeble magnetic forces. Philosophical Magazine, 23, 225–248.CrossRefMATHGoogle Scholar
  21. 21.
    Ikhouane, F., & Rodellar, J. (2007). Systems with hysteresis. Chichester: John Wiley & Sons, Ltd.Google Scholar
  22. 22.
    Xue, X. M., et al. (2010). Parameter estimation for the phenomenological model of hysteresis using efficient genetic algorithm. ISCM II and EPMESC XII, 1233, 713–717.Google Scholar
  23. 23.
    Sun, Q., et al. (2009). Parameter estimation and its sensitivity analysis of the MR damper hysteresis model using a modified genetic algorithm. Journal of Intelligent Material Systems and Structures, 20, 2089–2100.CrossRefGoogle Scholar
  24. 24.
    Liu, G. J., & Chan, C. H. (2007). Hysteresis identification and compensation using a genetic algorithm with adaptive search space. Mechatronics, 17, 391–402.CrossRefGoogle Scholar
  25. 25.
    Peng, L., & Wang, W. (2007). Adaptive genetic algorithm with heuristic weighted crossover operator based hysteresis identification and compensation. 2007 IEEE International Conference on Control and Automation (Vol. 1–7, pp. 3260–3264).Google Scholar
  26. 26.
    Zheng, J. J., et al. (2007). Hybrid genetic algorithms for parameter identification of a hysteresis model of magnetostrictive actuators. Neurocomputing, 70, 749–761.CrossRefGoogle Scholar
  27. 27.
    Chwastek, K., & Szczyglowski, J. (2006). Identification of a hysteresis model parameters with genetic algorithms. Mathematics and Computers in Simulation, 71, 206–211.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Zidaric, B., & Miljavec, D. (2005). Nested genetic algorithms in determination of Jiles-Atherton hysteresis model parameters for soft-magnetic composite materials. Informacije Midem-Journal of Microelectronics Electronic Components and Materials, 35, 92–96.Google Scholar
  29. 29.
    Cao, S. Y. et al. (2005) Parameter identification of strain hysteresis model for giant magnetostrictive actuators using a hybrid genetic algorithm. ICEMS 2005: Proceedings of the Eighth International Conference on Electrical Machines and Systems, 1–3, 2009–2012.Google Scholar
  30. 30.
    Chan, C. H., & Liu, G. J. (2004). Actuator hysteresis identification and compensation using an adaptive search space based genetic algorithm. Proceedings of the 2004 American Control Conference, 1–6, 5760–5765.Google Scholar
  31. 31.
    Naghizadeh, R. A., & Vajidi, B. (2011). Parameter identification of Jiles-Atherton model using SFLA. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 31, 1293–1309.Google Scholar
  32. 32.
    Ye, M. Y., & Wang, X. D. (2009). Parameter identification of hysteresis model with improved particle swarm optimization. Proceedings of the 21st Chinese Control and Decision Conference, 16, 415–419.Google Scholar
  33. 33.
    Ye, M. Y., & Wang, X. D. (2007). Parameter estimation of the Bouc-Wen hysteresis model using particle swarm optimization. Smart Materials and Structures, 16, 2341–2349.CrossRefGoogle Scholar
  34. 34.
    Fulginei, F. R., et al. (2007). Symbiotic evolutionary algorithm for the Preisach hysteresis model identification. International Journal of Applied Electromagnetics and Mechanics, 25, 681–687.Google Scholar
  35. 35.
    Andrei, P., et al. (2007). Identification techniques for phenomenological models of hysteresis based on the conjugate gradient method. Journal of Magnetism and Magnetic Materials, 316, E330–E333.CrossRefGoogle Scholar
  36. 36.
    Mayergoyz, I. (2003). Mathematical models of hysteresis and their applications: Electromagnetism. San Diego: Academic Press.Google Scholar
  37. 37.
    Pike, C. R., et al. (2005). First-order reversal curve diagram analysis of a perpendicular nickel nanopillar array. Physical Review B, 71, 134407.CrossRefGoogle Scholar
  38. 38.
    Della Torre, E., & Bennett, L. H. (1998). A Preisach model for aftereffect. IEEE Transactions on Magnetics, 34, 1276–1278.CrossRefGoogle Scholar
  39. 39.
    Cardelli, E., et al. (2000). Direct and inverse Preisach modeling of soft materials. IEEE Transactions on Magnetics, 36, 1267–1271.CrossRefGoogle Scholar
  40. 40.
    Fry, R. A., et al. (2000). Preisach modeling of aftereffect in a magneto-optical medium with perpendicular magnetization. Physica B-Condensed Matter, 275, 50–54.CrossRefGoogle Scholar
  41. 41.
    Patel, U. D., & Della Torre, E. (2001). Fast computation of the inverse CMH model. Physica B, 306, 178–184.CrossRefGoogle Scholar
  42. 42.
    Reimers, A., et al. (2001). Implementation of the preisach DOK magnetic hysteresis model in a commercial finite element package. IEEE Transactions on Magnetics, 37, 3362–3365.CrossRefGoogle Scholar
  43. 43.
    Reimers, A., & Della Torre, E. (2002). Implementation of the simplified vector model. IEEE Transactions on Magnetics, 38, 837–840.CrossRefGoogle Scholar
  44. 44.
    Cardelli, E., et al. (2004). Modeling of laminas of magnetic iron with a reduced vector Preisach model. Physica B-Condensed Matter, 343, 171–176.CrossRefGoogle Scholar
  45. 45.
    Della Torre, E., et al. (2004). Differential equation model for accommodation magnetization. IEEE Transactions on Magnetics, 40, 1499–1505.CrossRefGoogle Scholar
  46. 46.
    Burrascano, P., et al. (2006). Vector hysteresis model at micromagnetic scale. IEEE Transactions on Magnetics, 42, 3138–3140.CrossRefGoogle Scholar
  47. 47.
    Della Torre, E., et al. (2006). Vector modeling—part I: Generalized hysteresis model. Physica B-Condensed Matter, 372, 111–114.CrossRefGoogle Scholar
  48. 48.
    Della Torre, E., et al. (2006). Vector modeling—part II: Ellipsoidal vector hysteresis model. Numerical application to a 2D case. Physica B-Condensed Matter, 372, 115–119.CrossRefGoogle Scholar
  49. 49.
    Della Torre, E., & Cardelli, E. (2007). The coordinated vector model. COMPEL-the International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 26, 327–333.CrossRefMATHGoogle Scholar
  50. 50.
    Della Torre, E., et al. (2008). A model for vector accommodation. Physica B-Condensed Matter, 403, 496–499.CrossRefGoogle Scholar
  51. 51.
    Cardelli, E., et al. (2009). Analysis of a unit magnetic particle via the DPC model. IEEE Transactions on Magnetics, 45, 5192–5195.CrossRefGoogle Scholar
  52. 52.
    Stancu, A., et al. (2005). New Preisach model for structured particulate ferromagnetic media. Journal of Magnetism and Magnetic Materials, 290, 490–493.CrossRefGoogle Scholar
  53. 53.
    Mitchler, P. D., et al. (1996). Henkel plots in a temperature and time dependent Preisach model. IEEE Transactions on Magnetics, 32, 3185–3194.CrossRefGoogle Scholar
  54. 54.
    Mitchler, P. D., et al. (1999). Interactions and thermal effects in systems of fine particles: A Preisach analysis of CrO2 audio tape and magnetoferritin. IEEE Transactions on Magnetics, 35, 2029–2042.CrossRefGoogle Scholar
  55. 55.
    Spinu, L., et al. (2001). Time and temperature-dependent Preisach models. Physica B, 306, 166–171.CrossRefGoogle Scholar
  56. 56.
    Matsuo, T., et al. (2003). Application of stop and play models to the representation of magnetic characteristics of silicon steel sheet. IEEE Transactions on Magnetics, 39, 1361–1364.CrossRefGoogle Scholar
  57. 57.
    Matsuo, T., & Shimasaki, M. (2005). Representation theorems for stop and play models with input-dependent shape functions. IEEE Transactions on Magnetics, 41, 1548–1551.CrossRefGoogle Scholar
  58. 58.
    Bergqvist, A., et al. (1997). Experimental testing of an anisotropic vector hysteresis model. IEEE Transactions on Magnetics, 33, 4152–4154.CrossRefGoogle Scholar
  59. 59.
    Deane, J. H. B. (1994). Modeling the dynamics of nonlinear inductor circuits. IEEE Transactions on Magnetics, 30, 2795–2801.CrossRefGoogle Scholar
  60. 60.
    Cadence. (2005), PSPICE. Retrieved from http://www.cadence.com/.
  61. 61.
    Jiles, D. C., & Atherton, D. L. (1986). Theory of ferromagnetic hysteresis. Journal of Magnetism and Magnetic Materials, 61, 48–60.CrossRefGoogle Scholar
  62. 62.
    Sablik, M. J., & Jiles, D. C. (1988). A model for hysteresis in magnetostriction. Journal of Applied Physics, 64, 5402–5404.CrossRefGoogle Scholar
  63. 63.
    Hauser, H. et al. (Oct 2007). Including effects of microstructure and anisotropy in theoretical models describing hysteresis of ferromagnetic materials. Applied Physics Letters, 91, 172512. Google Scholar
  64. 64.
    Jiles, D. C., et al. (1992). Numerical determination of hysteresis parameters for the modeling of magnetic-properties using the theory of ferromagnetic hysteresis. IEEE Transactions on Magnetics, 28, 27–35.CrossRefGoogle Scholar
  65. 65.
    Hauser, H. (1995). Energetic model of ferromagnetic hysteresis. 2. Magnetization calculations of (110)[001] fesi sheets by statistic domain behavior. Journal of Applied Physics, 77, 2625–2633.CrossRefGoogle Scholar
  66. 66.
    Hauser, H. (2004). Energetic model of ferromagnetic hysteresis: Isotropic magnetization. Journal of Applied Physics, 96, 2753–2767.CrossRefGoogle Scholar
  67. 67.
    Andrei, P., & Adedoyin, A. (Apr 2009). Noniterative parameter identification technique for the energetic model of hysteresis. Journal of Applied Physics, 105, 07D523.Google Scholar
  68. 68.
    Baber, T. T., & Wen, Y. K. (1981). Random vibration of hysteretic, degrading systems. Journal of the Engineering Mechanics Division-ASCE, 107, 1069–1087.Google Scholar
  69. 69.
    Baber, T. T., & Noori, M. N. (1985). Random vibration of degrading, pinching systems. Journal of Engineering Mechanics-ASCE, 111, 1010–1026.CrossRefGoogle Scholar
  70. 70.
    Baber, T. T., & Noori, M. N. (1986). Modeling general hysteresis behavior and random vibration application. Journal of Vibration Acoustics Stress and Reliability in Design-Transactions of the ASME, 108, 411–420.CrossRefGoogle Scholar
  71. 71.
    Hodgdon, M. L. (1991). A constitutive relation for hysteresis in superconductors. Journal of Applied Physics, 69, 2388–2396.CrossRefGoogle Scholar
  72. 72.
    Hodgdon, M. L. (1988). Mathematical theory and calculations of magnetic hysteresis curves. IEEE Transactions on Magnetics, 24, 3120–3122.CrossRefGoogle Scholar
  73. 73.
    Stancu, A., et al. (1997). Models of hysteresis in magnetic cores. Journal de Physique IV, 7, 209–210.CrossRefGoogle Scholar
  74. 74.
    Hodgdon, M. L. (1991). Computation of superconductor critical current densities and magnetization curves. Journal of Applied Physics, 69, 4904–4906.CrossRefGoogle Scholar
  75. 75.
    Boley, C. D., & Hodgdon, M. L. (1989). Model and simulations of hysteresis in magnetic cores. IEEE Transactions on Magnetics, 25, 3922–3924.CrossRefGoogle Scholar
  76. 76.
    Andrei, P. (1997). Phenomenological models for the study of ferromagnetic and fertic materials (in Romanian). B.S., Physics Department, Alexandru Ioan Cuza University, Iasi.Google Scholar
  77. 77.
    Mayergoyz, I. D., & Friedman, G. (1987). Isotropic vector Preisach model of hysteresis. Journal of Applied Physics, 61, 4022–4024.CrossRefGoogle Scholar
  78. 78.
    Mayergoyz, I. D., & Friedman, G. (1987). On the integral-equation of the vector Preisach hysteresis model. IEEE Transactions on Magnetics, 23, 2638–2640.CrossRefGoogle Scholar
  79. 79.
    Mayergoyz, I. D. (1988). Vector Preisach hysteresis models. Journal of Applied Physics, 63, 2995–3000.CrossRefGoogle Scholar
  80. 80.
    Andrei, P., & Adedoyin, A. (Apr 2008). Phenomenological vector models of hysteresis driven by random fluctuation fields. Journal of Applied Physics, 103, 07D913.Google Scholar
  81. 81.
    Krasnosel'skii, M. A., & Pokrovskii, A. (1989). Systems with hysteresis, Nauka.Google Scholar
  82. 82.
    Visintin, A. (1994). Differential models of hysteresis. Berlin: Springer.Google Scholar
  83. 83.
    Brokate, M., & Sprekels, J. (1996). Hysteresis and phase transitions, Springer.Google Scholar
  84. 84.
    Krejčí, P. (1996) Hysteresis, convexity and dissipation in hyperbolic equations. Tokyo: Gakkotosho Co., Ltd.Google Scholar
  85. 85.
    Mayergoyz, I. D., & Bertotti G. (2006). Science of hysteresis, Academic pressGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2014

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringHoward University and Stefan cel Mare UniversityWashingtonUSA
  2. 2. Department of Electrical and Computer EngineeringFlorida State University and Florida A&M UniversityTallahasseUSA

Personalised recommendations