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Identification of hysteresis Preisach model using magneto-optic microscopy
ARTICLE IN PRESS
Physica B 403 (2008) 376–378 www.elsevier.com/locate/physb
Identification of hysteresis Preisach model using magneto-optic microscopy Valentin Ionita, Emil Cazacu Department of Electrical Engineering, University Politehnica of Bucharest, Spl. Independentei 313, RO-060042, Bucharest, Romania
Abstract The paper presents a new identification method for the Preisach models. It uses the digital images of the computer-based magnetooptic microscopy, the applied magnetic field being sinusoidal. The identification of the switching fields for each pixel of the image allows to build the statistical Preisach function. r 2007 Elsevier B.V. All rights reserved. Keywords: Hysteresis; Preisach model; Magneto-optic microscopy
1. Introduction Nowadays, the hysteresis modeling remains a challenge for scientists, due to the phenomenon complexity and the computation efficiency. The Preisach model is one of the most developed models, generating new and better versions. The classical theory was extended to cover dynamic processes or vector configuration of the applied magnetic field (e.g. [1,2]). The Preisach function, having predefined or arbitrary shapes, is identified starting from a set of experimental data (first magnetization curve, major hysteresis cycle or first order reversal curves— FORC), obtained for example with a vibrating sample magnetometer (VSM). The model accuracy depends on its intrinsic limitations and on all the errors accumulated during the identification procedure, including the experimental errors [3]. Our study starts from the statistical nature of the classical Preisach model and proposes a new identification procedure, which is based on the microscopic behavior of the material. The magneto-optic effects, especially the Kerr effect, are very useful for the visualization of the magnetic microstructure [4] in magnetic thin films or for the quantitative characterization of magnetic materials (FeSi sheets [5], Finemet [6] or Permalloy [7]). This experimental Corresponding author. Fax: +40 21 3181016.
E-mail address: [email protected] (V. Ionita). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.08.053
method allows the detection of the magnetization direction for each magnetic domain from the sample surface, using the change of polarization angle between the incident and the reflected light ray. If a variable magnetic field is applied, the dynamics of the micromagnetic structure show the switching of the local magnetization direction. In our method, these switching magnetic fields are used by assuming that each pixel of the magneto-optic image corresponds to a hysteron of the Preisach model. In this manner, the macroscopic behavior of the magnetic material results from the statistic sum of all hysterons (pixels). 2. Identification method Using a polarized light microscope and a periodic (sinusoidal) magnetic field applied to the sample, one can obtain the movie of the magnetic microstructure evolution for the considered direction. If the Kerr effects (polar, transversal and longitudinal) are combined, the method could be used for the identification of a vector function. The digital acquisition and processing of the obtained frames (images) allow the separation of the magnetic information, by subtracting a reference image, which corresponds to the saturation state, from each frame. The image contrast and brightness are also improved [8]. The analysis of the gray-level evolution for each pixel indicates the frames where the pixel changes its color (between dark and bright) so, when the pixel magnetization
ARTICLE IN PRESS V. Ionita, E. Cazacu / Physica B 403 (2008) 376–378
changes its direction. Fig. 1 shows how the values a and b of the switching magnetic fields (corresponding to the frames) could be determined for each pixel. If one assumes that a pixel corresponds to a Preisach hysteron, the value of the statistic Preisach function P(a, b) for a cell of the Preisach triangle (see Fig. 2) will result by counting all the pixels having the same pair (a, b). The obtained numerical function can be directly used in any electromagnetic field computation that involves the Preisach hysteresis model. The gray-level analysis of the digital images allows estimating the correctness of the main hypothesis of the identification procedure—a pixel corresponds to a hysteron. Indeed, if the image has a good resolution, then the pixels number is big enough to satisfy the statistic base of Preisach theory. Supplementary, the pixels must have a gray-level evolution closed to a rectangular shape, like a Preisach hysteron (see Fig. 3).
This condition allows to establish if the Preisach model is suitable for the analyzed magnetic material. 3. Experimental and numerical tests Our experimental setup is based on the polarized light microscope Axiolab, video-camera AxioCam Hsm and AxioVision software—all from Zeisss. Applying a sinusoidal magnetic field (frequency ¼ 0.1 Hz), perpendicular to the YIG garnet sample, the corresponding magnetization dynamics, at microscopic scale, can be recorded in a movie file; for thin films, the other two components of the magnetization can be ignored. The acquired magneto-optic frames (images) have (588 792) pixels, allowing a statistical study; the mean gray-level evolution is presented in Fig. 4 and the resulted threshold level is around of 2.2E+4. For this threshold, the magneto-optic hysteresis cycles obtained by the superposition of 1, 10 and 100 pixels
Fig. 3. Hysteresis cycle for a pixel. Fig. 1. Gray-level evolution during an alternating applied magnetic field.
Fig. 2. Preisach triangle.
377
Fig. 4. Mean gray level of the frames.
ARTICLE IN PRESS 378
V. Ionita, E. Cazacu / Physica B 403 (2008) 376–378
are presented in Fig. 5; the cycle rectangularity for 1 pixel confirms the equivalence pixel—hysteron. The analysis of the gray-level evolution for each pixel gives the switching fields (a, b) for each hysteron. By counting the pixels with the same (a, b) values, one computes the probability of a hysteron to have a desired pair (a, b), so the statistic Preisach function will be computed. In our case, one has (588 792) ¼ 465696 pixels (hysterons) to be distributed on ((200 frames per cycles)/2)2/2 ¼ 5000 values of the discrete Preisach function from the triangle cells. The jumps in the saturation zone (see Fig. 4) are critical for the identification precision. Indeed, Fig. 6 shows the difference between the hysteresis cycles obtained by averaging the gray-level of all pixels (cycle extracted from movie) and by numerical simulation using the identified
Preisach model (simulated cycle). The staircase shape of the simulated cycle results from the fact that the computed function has many peaks. The hysteresis cycle from Fig. 6 present an asymmetry because the sample was not enough saturated. Indeed, the shape of the magnetization curve presented in Fig. 4 shows narrow and asymmetric peaks, so the magnetization vector rotation was not yet finished. The gray-level analysis reveals that many pixels (23%) remain unchanged during a cycle and they are not taken into account for the statistic Preisach function. Consequently, the asymmetry is more increased for the simulated loop than for the experimental one. 4. Conclusions The new identification method allows a direct computation of the discrete values of the Preisach function. The procedure, including the experimental data measurement, is very fast and promises an efficient way for the identification of the hysteresis vector models. The accuracy is better for thin magnetic films, because the used experimental information is given by the magnetic microstructure of the sample surface. The method allows the computation of the specific Preisach function for each desired area (e.g. finite elements) that may be an advantage in the study of fast dynamic magnetization processes in technical devices. Despite the accuracy dependency on the experimental setup and the model constraint to be used for thin films, the identification method could be very useful to test new dynamic or vector hysteresis models. Acknowledgments
Fig. 5. Magneto-optic hysteresis cycles
The authors would like to thank to Dr. F. Fiorillo and Dr. A. Magni (INRIM Torino, Italy) for the helpful discussions. The study was supported by CEEX 33/2005 Contract (ANCS-AMCSIT) and by AC-Gr 188/2006 Grant (CNCSIS). References [1] I.D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Academic Press, New York, 2003. [2] E. Cardelli, et al., IEEE Trans. Magn. 42 (2006) 527. [3] V. Ionita, et al., Proc. IGTE-2006 Graz (2006) 87. [4] A. Hubert, et al., Magnetic Domains, Springer, Berlin, 1998. [5] S. Defoug, et al., J. Appl. Phys. 79 (1996) 6036. [6] L. Santi, et al., IEEE Trans. Magn. 39 (2003) 2666. [7] A. Neudert, et al., Phys. Rev. B 71 (2005) 134405. [8] V. Ionita, J. Optoelectron. Adv. Mater. 9 (2007) 1176.
Fig. 6. Hysteresis cycles obtained by Preisach modeling and by direct extraction from magneto-optic images.
Physica B 403 (2008) 376–378 www.elsevier.com/locate/physb
Identification of hysteresis Preisach model using magneto-optic microscopy Valentin Ionita, Emil Cazacu Department of Electrical Engineering, University Politehnica of Bucharest, Spl. Independentei 313, RO-060042, Bucharest, Romania
Abstract The paper presents a new identification method for the Preisach models. It uses the digital images of the computer-based magnetooptic microscopy, the applied magnetic field being sinusoidal. The identification of the switching fields for each pixel of the image allows to build the statistical Preisach function. r 2007 Elsevier B.V. All rights reserved. Keywords: Hysteresis; Preisach model; Magneto-optic microscopy
1. Introduction Nowadays, the hysteresis modeling remains a challenge for scientists, due to the phenomenon complexity and the computation efficiency. The Preisach model is one of the most developed models, generating new and better versions. The classical theory was extended to cover dynamic processes or vector configuration of the applied magnetic field (e.g. [1,2]). The Preisach function, having predefined or arbitrary shapes, is identified starting from a set of experimental data (first magnetization curve, major hysteresis cycle or first order reversal curves— FORC), obtained for example with a vibrating sample magnetometer (VSM). The model accuracy depends on its intrinsic limitations and on all the errors accumulated during the identification procedure, including the experimental errors [3]. Our study starts from the statistical nature of the classical Preisach model and proposes a new identification procedure, which is based on the microscopic behavior of the material. The magneto-optic effects, especially the Kerr effect, are very useful for the visualization of the magnetic microstructure [4] in magnetic thin films or for the quantitative characterization of magnetic materials (FeSi sheets [5], Finemet [6] or Permalloy [7]). This experimental Corresponding author. Fax: +40 21 3181016.
E-mail address: [email protected] (V. Ionita). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.08.053
method allows the detection of the magnetization direction for each magnetic domain from the sample surface, using the change of polarization angle between the incident and the reflected light ray. If a variable magnetic field is applied, the dynamics of the micromagnetic structure show the switching of the local magnetization direction. In our method, these switching magnetic fields are used by assuming that each pixel of the magneto-optic image corresponds to a hysteron of the Preisach model. In this manner, the macroscopic behavior of the magnetic material results from the statistic sum of all hysterons (pixels). 2. Identification method Using a polarized light microscope and a periodic (sinusoidal) magnetic field applied to the sample, one can obtain the movie of the magnetic microstructure evolution for the considered direction. If the Kerr effects (polar, transversal and longitudinal) are combined, the method could be used for the identification of a vector function. The digital acquisition and processing of the obtained frames (images) allow the separation of the magnetic information, by subtracting a reference image, which corresponds to the saturation state, from each frame. The image contrast and brightness are also improved [8]. The analysis of the gray-level evolution for each pixel indicates the frames where the pixel changes its color (between dark and bright) so, when the pixel magnetization
ARTICLE IN PRESS V. Ionita, E. Cazacu / Physica B 403 (2008) 376–378
changes its direction. Fig. 1 shows how the values a and b of the switching magnetic fields (corresponding to the frames) could be determined for each pixel. If one assumes that a pixel corresponds to a Preisach hysteron, the value of the statistic Preisach function P(a, b) for a cell of the Preisach triangle (see Fig. 2) will result by counting all the pixels having the same pair (a, b). The obtained numerical function can be directly used in any electromagnetic field computation that involves the Preisach hysteresis model. The gray-level analysis of the digital images allows estimating the correctness of the main hypothesis of the identification procedure—a pixel corresponds to a hysteron. Indeed, if the image has a good resolution, then the pixels number is big enough to satisfy the statistic base of Preisach theory. Supplementary, the pixels must have a gray-level evolution closed to a rectangular shape, like a Preisach hysteron (see Fig. 3).
This condition allows to establish if the Preisach model is suitable for the analyzed magnetic material. 3. Experimental and numerical tests Our experimental setup is based on the polarized light microscope Axiolab, video-camera AxioCam Hsm and AxioVision software—all from Zeisss. Applying a sinusoidal magnetic field (frequency ¼ 0.1 Hz), perpendicular to the YIG garnet sample, the corresponding magnetization dynamics, at microscopic scale, can be recorded in a movie file; for thin films, the other two components of the magnetization can be ignored. The acquired magneto-optic frames (images) have (588 792) pixels, allowing a statistical study; the mean gray-level evolution is presented in Fig. 4 and the resulted threshold level is around of 2.2E+4. For this threshold, the magneto-optic hysteresis cycles obtained by the superposition of 1, 10 and 100 pixels
Fig. 3. Hysteresis cycle for a pixel. Fig. 1. Gray-level evolution during an alternating applied magnetic field.
Fig. 2. Preisach triangle.
377
Fig. 4. Mean gray level of the frames.
ARTICLE IN PRESS 378
V. Ionita, E. Cazacu / Physica B 403 (2008) 376–378
are presented in Fig. 5; the cycle rectangularity for 1 pixel confirms the equivalence pixel—hysteron. The analysis of the gray-level evolution for each pixel gives the switching fields (a, b) for each hysteron. By counting the pixels with the same (a, b) values, one computes the probability of a hysteron to have a desired pair (a, b), so the statistic Preisach function will be computed. In our case, one has (588 792) ¼ 465696 pixels (hysterons) to be distributed on ((200 frames per cycles)/2)2/2 ¼ 5000 values of the discrete Preisach function from the triangle cells. The jumps in the saturation zone (see Fig. 4) are critical for the identification precision. Indeed, Fig. 6 shows the difference between the hysteresis cycles obtained by averaging the gray-level of all pixels (cycle extracted from movie) and by numerical simulation using the identified
Preisach model (simulated cycle). The staircase shape of the simulated cycle results from the fact that the computed function has many peaks. The hysteresis cycle from Fig. 6 present an asymmetry because the sample was not enough saturated. Indeed, the shape of the magnetization curve presented in Fig. 4 shows narrow and asymmetric peaks, so the magnetization vector rotation was not yet finished. The gray-level analysis reveals that many pixels (23%) remain unchanged during a cycle and they are not taken into account for the statistic Preisach function. Consequently, the asymmetry is more increased for the simulated loop than for the experimental one. 4. Conclusions The new identification method allows a direct computation of the discrete values of the Preisach function. The procedure, including the experimental data measurement, is very fast and promises an efficient way for the identification of the hysteresis vector models. The accuracy is better for thin magnetic films, because the used experimental information is given by the magnetic microstructure of the sample surface. The method allows the computation of the specific Preisach function for each desired area (e.g. finite elements) that may be an advantage in the study of fast dynamic magnetization processes in technical devices. Despite the accuracy dependency on the experimental setup and the model constraint to be used for thin films, the identification method could be very useful to test new dynamic or vector hysteresis models. Acknowledgments
Fig. 5. Magneto-optic hysteresis cycles
The authors would like to thank to Dr. F. Fiorillo and Dr. A. Magni (INRIM Torino, Italy) for the helpful discussions. The study was supported by CEEX 33/2005 Contract (ANCS-AMCSIT) and by AC-Gr 188/2006 Grant (CNCSIS). References [1] I.D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Academic Press, New York, 2003. [2] E. Cardelli, et al., IEEE Trans. Magn. 42 (2006) 527. [3] V. Ionita, et al., Proc. IGTE-2006 Graz (2006) 87. [4] A. Hubert, et al., Magnetic Domains, Springer, Berlin, 1998. [5] S. Defoug, et al., J. Appl. Phys. 79 (1996) 6036. [6] L. Santi, et al., IEEE Trans. Magn. 39 (2003) 2666. [7] A. Neudert, et al., Phys. Rev. B 71 (2005) 134405. [8] V. Ionita, J. Optoelectron. Adv. Mater. 9 (2007) 1176.
Fig. 6. Hysteresis cycles obtained by Preisach modeling and by direct extraction from magneto-optic images.