- Email: [email protected]
Statistical approach of hysteresis
ARTICLE IN PRESS
Physica B 372 (2006) 45–48 www.elsevier.com/locate/physb
Statistical approach of hysteresis Z. Sari, A. Ivanyi Pollack Miha´ly Faculty of Engineering, Department of Informatics, University of Pe´cs, Ro´kus 2. Pe´cs H7624, Hungary
Abstract In this paper a statistical approach of hysteresis phenomenon is presented. Under the assumption that each magnetic domain can be represented by a random variable a very simple model of hysteresis can be built on. In spite of the simplicity of the model it possesses many important properties of hysteretic behaviour. r 2005 Elsevier B.V. All rights reserved. Keywords: Hysteresis model; Statistics
1. Introduction In this paper a simple statistical model of hysteresis is presented. The statistical approach is not new in hysteresis modelling. There are several models based on statistical distributions to characterize the shape of the hysteresis loops [1–3]. The recent model is new in the sense that the whole hysteresis phenomenon and the resulted hysteresis curves are modelled by statistical distribution functions, based on the assumption that magnetic domains can be represented by random variables. 2. The main concept of the model A simple magnetic domain can be handled as a delayed relay operator [4] (elementary hysteron) with two states corresponding to the magnetized and the demagnetized state. The value of the switching field H þ ; H (up/down) is statistically distributed among domains of the material and the expected value of the switching field is the same for all domains. Under the above assumptions the switching field can be represented by a random variable from an arbitrary distribution with expected value m~ and standard deviation ~ Since the process of magnetization affects many domains s. with the same expected value of the switching field the net effect can be represented by a Gaussian random variable, because central limit theorem states that the distribution of Corresponding author. Tel./fax: +36 72 501 534.
E-mail address: [email protected] (Z. Sari). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.015
the average of the independent, identically distributed (IID) random variables is the normal or Gaussian distribution. 2.1. The distribution of switching field ~ sÞ ~ be a random variable from an arbitrary Let X i Dðm; distribution D corresponding to the switching field H þ . Then X i 2 ½0; H s , where H s is the saturation field strength ~ 8i. Assuming that X i ’s are IID the average and E½X i ¼ m; of them is a Gaussian random variable X Nðm; sÞ. It means that the net effect of the magnetization process can be described by the cumulative distribution function (CDF) of the normal distribution. Concentric hysteresis loops created by the model for decreasing amplitude of the exciting field can be seen in Fig. 1. The parameters of the model at this level are m corresponding to the coercive field H c , and s which depends on the maximal susceptibility wmax . 2.2. Behaviour of the model The above pair of parameters (m; s) seems to be enough to determine the major hysteresis loop with a strong saturation effect. The reversal curves can be determined in the following way. Magnetization from negative to positive saturation means no more than switching the states of all the domains. According to the concepts above, this process can be described in a purely statistical way, and from this
ARTICLE IN PRESS Z. Sari, A. Ivanyi / Physica B 372 (2006) 45–48
46 x 106
x 106 2 1.5
1
1
0.5
0.5
M [A/m]
1.5
0 5000
-0.5
H (t) [A/m]
M [A/m]
2
-1
-1.5 0
-2 -8000 -6000 -4000 -2000
-0.5 -1
0
-5000
-1.5
0
0.5 t
1
-2 -8000-6000-4000-2000 0 2000 4000 6000 8000 H [A/m]
0 2000 4000 6000 8000 H [A/m]
x 10-4
Fig. 1. Concentric loops.
point of view the starting point of the magnetization determines the whole magnetization process to the saturation. It means that a certain hysteresis curve is uniquely characterized by the last reversal point ðH r ; M r Þ, where the direction of the applied field H changes. For example, if the magnetization process starts from negative saturation (H s ; M s ) and the direction d 2 f1; 1g of the applied field is positive, accessing a certain level of M by applying a given H means switching the state of a certain percent of domains. Thus the actual M is proportional to the value of cumulative distribution function of X for the given H. If the direction of the applied field changes at this level of magnetization, the process is very similar except that demagnetization corresponds only to those domains which have been magnetized before, because the ones that have not been magnetized are in the desired state. It follows that a given (H r ; M r ) reversal point and the direction of the actual field, d, seem to determine the cumulative distribution function (CDF) which describes the hysteresis curve from (H r ; M r ) to (dH s ; dM s ) saturation point. 2.3. Formulating the model Let the cumulative distribution function of normal distribution with parameters (m; s) be the following: Z x 2 1 2 ffiffiffiffiffiffiffiffiffiffi p F ðxÞ ¼ eðtmÞ =2s dt, (1) 2 2ps 1 where s ¼ oðwmax Þ and m ¼ H c . The dependence of s on wmax can be formulated according to the fact that the maximum of F 0 ðxÞ is wmax , and from this it follows 1 pffiffiffiffiffiffi . s¼ wmax 2p
(2)
Substituting the actual applied field into the F ðxÞ function in Eq. (1) gives the percent of the domains that changed their states under the applied field H, thus F ðHÞ 2 ½0; 1 is the ‘‘magnetization ratio’’. From this function the
Probability density function
8 7 6 5 4 3 2 1 0 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 H [A/m] Fig. 2. The field reversal.
actual value of M can be calculated in view of M s saturation magnetization.1 In the case of field direction reversal, the area under the probability density function of X on the interval ½1; H r gives the ratio of the magnetized domains (M r 2 ½0; 1) 1 M r ðH r Þ ¼ pffiffiffiffiffiffiffiffiffiffi 2ps2
Z
Hr
eðtmÞ
2
=2s2
dt.
(3)
1
Because of H field reversal the expected value of X changes to m, and Eq. (3) takes the form x M r ðH r Þ ¼ pffiffiffiffiffiffiffiffiffiffi 2ps2
Z
Hr
eðtþmÞ
2
=2s2
dt,
(4)
1
where x is needed to maintain the level of magnetization at the reversal point. After calculating x the hysteresis curve can be described by x M H r ;M r ;d ðHÞ ¼ pffiffiffiffiffiffiffiffiffiffi 2ps2
Z
H
eðtmÞ
2
=2s2
dt.
(5)
1
Fig. 2 shows the hysteresis curves and the corresponding probability density functions. 1
In the case of simulated hysteresis loops M s ¼ 1:6 106 ½A=m.
ARTICLE IN PRESS Z. Sari, A. Ivanyi / Physica B 372 (2006) 45–48
47
2.4. Testing the model
3. Extending the behaviour of the model
In this section some test results are presented to demonstrate the behaviour of the introduced hysteresis model. The results presented here are derived from the simple two-parameter (m; s) model. Fig. 3 shows simulated first order reversal loops, while the accommodation property of the model can be seen in Fig. 4. Accommodation comes into effect when the excitation field is smaller than the width of the major loop. The larger the amplitude of the excitation, the faster the hysteresis loop becomes stable. As it can be seen in Fig. 4 the minor loop is shifted upwards and becomes stable after a few cycles. Since the hysteresis loop is derived from the CDF of the normal distribution the model exhibits strong saturation and accommodation properties as the figures show. The simulation results obtained by the above model seem to be appropriate; however, the model is not versatile enough since all the magnetic domains are supposed to be uniform.
Assuming that the magnetic moments of the domains are not the same for all domains, the distribution of moments can also be described by some kind of statistical distribution. The most straightforward approach is to apply the normal distribution. Let Y Nðm2 ; s2 Þ be a random variable corresponding to the magnetic moment of the domains. The statistical approach can be further extended by considering the consequence of the distribution of moments, thus the applied magnetic field will force the ‘‘weakest’’2 domains first to change their states and as the magnetic field strength increases the domains will change their states according to their moments.
2
x 106
1.5 1 M [A/m]
0.5 0
Since the switching fields and magnetic moments are described by distribution functions the magnetization process can be modelled by embedding these distributions into each other. The magnetization ratio in the case of uniform magnetic moments can be calculated by Eq. (6), and this value can be fed into the second distribution function corresponding to Y representing the distribution of the magnetic moments. The construction can be formulated as Z x 2 1 2 F ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffi eðtm1 Þ =2s1 dt, (6) 2 2ps1 1 F^ ðxÞ ¼ cotðpF ðxÞÞ,
(7)
-0.5
1 GðxÞ ¼ qffiffiffiffiffiffiffiffiffiffi 2ps22
-1 -1.5 -2 -8000 -6000 -4000 -2000
0
2000 4000 6000 8000
H [A/m]
Fig. 3. First order reversal loops ðm ¼ 1000; s ¼ 1000Þ.
14
x 105
12 10 8 2000
6
H (t) [A/m]
M [A/m]
3.1. Nested distributions
4
Z
F^ ðxÞ
2
2
eðtm2 Þ =2s2 dt,
(8)
1
where the intermediary Eq. (7) is required to map F ðxÞ 2 ½0; 1 to F^ ðxÞ 2 ½1; 1 and avoid unwanted ‘‘jumps’’ in the hysteresis curves. Substituting the actual applied field H into the GðxÞ function gives the magnetization ratio GðHÞ 2 ½0; 1 based on the assumption that the distribution of the magnetic moments is the normal distribution. At this point the actual value of magnetization can be calculated from the magnetization ratio GðHÞ in view of the M s saturation magnetization. Since the resulting distribution can be asymmetric, in the case of field reversal the probability density function has to be flipped from left to right in addition to the change of the expected value.
1000
4. Parameter identification 0
2 0 -500
0
0
500
1000 1500 H [A/m]
0.5 t
2000
1
2500
Fig. 4. Accommodation property of the model ðm ¼ 1000; s ¼ 1500Þ.
Identification of the model parameters is not too complicated. In the case of the simple model the determination of the values of parameters (m; s) goes in a 2
‘‘Weakest’’ means that least energy needed to change its state.
ARTICLE IN PRESS Z. Sari, A. Ivanyi / Physica B 372 (2006) 45–48
48
2
x 106
1.5
1.5
x 105
1
1
M [A/m]
0.5 0 -0.5
5000 H (t) [A/m]
M [A/m]
Measured Simulated
0.5
-1
-0.5
0
-1
-5000
-1.5
0
-2 -8000 -6000 -4000 -2000
0
0
0.5 t
1
-1.5 5
2000 4000 6000 8000
H [A/m]
0 H [A/m]
5 x 104
Fig. 5. Simulated concentric hysteresis loops.
Fig. 6. Measured loop compared to simulated.
straightforward manner because m ¼ H c and s ¼ oðwmax Þ as it was described before. The extended model has four parameters and the values of parameters are not so closely related to the shape of the hysteresis curve as in the former model. Let g ¼ ½m1 ; s1 ; m2 ; s2 be a vector containing the four parameters of the model. The identification procedure can be carried out in view of coercive field H c , remanent magnetization M rem , maximum susceptibility wmax , and susceptibility at point (H ¼ 0; M ¼ M rem ) denoted by wrem . The following non-linear system of equations can be formed for these values and vector g
versatile and a wider variety of hysteresis loops can be modelled.
Gðx ¼ H c ; gÞ ¼ 12, G 0 ðx ¼ H c ; gÞ ¼
wmax , 2M s
Gðx ¼ 0; gÞ ¼
M rem , 2M s
G 0 ðx ¼ 0; gÞ ¼
wrem . 2M s
A ; m
wmax ¼ 1500;
M rem ¼ 1:325 106
Since the model simply applies a distribution function to form the major hysteresis loop, the model can be generalized by substituting the cumulative distribution function by a properly scaled measured major hysteresis loop. This generalization will allow to simulate many kinds of hysteresis loops, and makes the measurement-based identification extremely easy. Fig. 6 shows a simulated hysteresis loop compared to a measured loop. 6. Conclusion
(9)
This non-linear system of equations (9) can be handled as a non-linear minimization problem and can be solved by trust region methods. The solution of the system is the g parameter vector which contains the parameters of the hysteresis model. Fig. 5 shows the behaviour of the model for the following set of properties of the major hysteresis loop: H c ¼ 1500
5. Generalization of the model
A , m
The presented model is based on a simple statistical approach of hysteresis phenomenon. Although the model is simple, it provides a good approximation to simulate the hysteresis characteristics. It is important to notice that the shape of the minor loops is completely determined by the major hysteresis loop; thus introducing a further parameter (or function) to define the shape of the minor loops is a major task in the future study of this approach. Acknowledgements The research work has been developed in the frame of the project Utilization of the infocommunication technologies in the region No. GVOP-3.1.1.-2004-05-0125/3.0.
wrem ¼ 500,
and resulting parameter vector
References
g ¼ ½1411:7; 3660:5; 1:2629; 0:5511.
[1] A. Iva´nyi, Hysteresis Models in Electromagnetic Field Computation, Akade´miai Kiado´, Budapest, 1997. [2] J. Taka´cs, Mathematics of Hysteretic Phenomena, Wiley-VCH, Weinheim, 2003. [3] G. Bertotti, Hysteresis in Magnetism, Academic Press, 1998. [4] A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994.
From the figure it can be seen that applying the modified distribution enables that the shape of the hysteresis loop can be different from the ones created by the CDF of the normal distribution. In this way the model becomes more
Physica B 372 (2006) 45–48 www.elsevier.com/locate/physb
Statistical approach of hysteresis Z. Sari, A. Ivanyi Pollack Miha´ly Faculty of Engineering, Department of Informatics, University of Pe´cs, Ro´kus 2. Pe´cs H7624, Hungary
Abstract In this paper a statistical approach of hysteresis phenomenon is presented. Under the assumption that each magnetic domain can be represented by a random variable a very simple model of hysteresis can be built on. In spite of the simplicity of the model it possesses many important properties of hysteretic behaviour. r 2005 Elsevier B.V. All rights reserved. Keywords: Hysteresis model; Statistics
1. Introduction In this paper a simple statistical model of hysteresis is presented. The statistical approach is not new in hysteresis modelling. There are several models based on statistical distributions to characterize the shape of the hysteresis loops [1–3]. The recent model is new in the sense that the whole hysteresis phenomenon and the resulted hysteresis curves are modelled by statistical distribution functions, based on the assumption that magnetic domains can be represented by random variables. 2. The main concept of the model A simple magnetic domain can be handled as a delayed relay operator [4] (elementary hysteron) with two states corresponding to the magnetized and the demagnetized state. The value of the switching field H þ ; H (up/down) is statistically distributed among domains of the material and the expected value of the switching field is the same for all domains. Under the above assumptions the switching field can be represented by a random variable from an arbitrary distribution with expected value m~ and standard deviation ~ Since the process of magnetization affects many domains s. with the same expected value of the switching field the net effect can be represented by a Gaussian random variable, because central limit theorem states that the distribution of Corresponding author. Tel./fax: +36 72 501 534.
E-mail address: [email protected] (Z. Sari). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.015
the average of the independent, identically distributed (IID) random variables is the normal or Gaussian distribution. 2.1. The distribution of switching field ~ sÞ ~ be a random variable from an arbitrary Let X i Dðm; distribution D corresponding to the switching field H þ . Then X i 2 ½0; H s , where H s is the saturation field strength ~ 8i. Assuming that X i ’s are IID the average and E½X i ¼ m; of them is a Gaussian random variable X Nðm; sÞ. It means that the net effect of the magnetization process can be described by the cumulative distribution function (CDF) of the normal distribution. Concentric hysteresis loops created by the model for decreasing amplitude of the exciting field can be seen in Fig. 1. The parameters of the model at this level are m corresponding to the coercive field H c , and s which depends on the maximal susceptibility wmax . 2.2. Behaviour of the model The above pair of parameters (m; s) seems to be enough to determine the major hysteresis loop with a strong saturation effect. The reversal curves can be determined in the following way. Magnetization from negative to positive saturation means no more than switching the states of all the domains. According to the concepts above, this process can be described in a purely statistical way, and from this
ARTICLE IN PRESS Z. Sari, A. Ivanyi / Physica B 372 (2006) 45–48
46 x 106
x 106 2 1.5
1
1
0.5
0.5
M [A/m]
1.5
0 5000
-0.5
H (t) [A/m]
M [A/m]
2
-1
-1.5 0
-2 -8000 -6000 -4000 -2000
-0.5 -1
0
-5000
-1.5
0
0.5 t
1
-2 -8000-6000-4000-2000 0 2000 4000 6000 8000 H [A/m]
0 2000 4000 6000 8000 H [A/m]
x 10-4
Fig. 1. Concentric loops.
point of view the starting point of the magnetization determines the whole magnetization process to the saturation. It means that a certain hysteresis curve is uniquely characterized by the last reversal point ðH r ; M r Þ, where the direction of the applied field H changes. For example, if the magnetization process starts from negative saturation (H s ; M s ) and the direction d 2 f1; 1g of the applied field is positive, accessing a certain level of M by applying a given H means switching the state of a certain percent of domains. Thus the actual M is proportional to the value of cumulative distribution function of X for the given H. If the direction of the applied field changes at this level of magnetization, the process is very similar except that demagnetization corresponds only to those domains which have been magnetized before, because the ones that have not been magnetized are in the desired state. It follows that a given (H r ; M r ) reversal point and the direction of the actual field, d, seem to determine the cumulative distribution function (CDF) which describes the hysteresis curve from (H r ; M r ) to (dH s ; dM s ) saturation point. 2.3. Formulating the model Let the cumulative distribution function of normal distribution with parameters (m; s) be the following: Z x 2 1 2 ffiffiffiffiffiffiffiffiffiffi p F ðxÞ ¼ eðtmÞ =2s dt, (1) 2 2ps 1 where s ¼ oðwmax Þ and m ¼ H c . The dependence of s on wmax can be formulated according to the fact that the maximum of F 0 ðxÞ is wmax , and from this it follows 1 pffiffiffiffiffiffi . s¼ wmax 2p
(2)
Substituting the actual applied field into the F ðxÞ function in Eq. (1) gives the percent of the domains that changed their states under the applied field H, thus F ðHÞ 2 ½0; 1 is the ‘‘magnetization ratio’’. From this function the
Probability density function
8 7 6 5 4 3 2 1 0 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 H [A/m] Fig. 2. The field reversal.
actual value of M can be calculated in view of M s saturation magnetization.1 In the case of field direction reversal, the area under the probability density function of X on the interval ½1; H r gives the ratio of the magnetized domains (M r 2 ½0; 1) 1 M r ðH r Þ ¼ pffiffiffiffiffiffiffiffiffiffi 2ps2
Z
Hr
eðtmÞ
2
=2s2
dt.
(3)
1
Because of H field reversal the expected value of X changes to m, and Eq. (3) takes the form x M r ðH r Þ ¼ pffiffiffiffiffiffiffiffiffiffi 2ps2
Z
Hr
eðtþmÞ
2
=2s2
dt,
(4)
1
where x is needed to maintain the level of magnetization at the reversal point. After calculating x the hysteresis curve can be described by x M H r ;M r ;d ðHÞ ¼ pffiffiffiffiffiffiffiffiffiffi 2ps2
Z
H
eðtmÞ
2
=2s2
dt.
(5)
1
Fig. 2 shows the hysteresis curves and the corresponding probability density functions. 1
In the case of simulated hysteresis loops M s ¼ 1:6 106 ½A=m.
ARTICLE IN PRESS Z. Sari, A. Ivanyi / Physica B 372 (2006) 45–48
47
2.4. Testing the model
3. Extending the behaviour of the model
In this section some test results are presented to demonstrate the behaviour of the introduced hysteresis model. The results presented here are derived from the simple two-parameter (m; s) model. Fig. 3 shows simulated first order reversal loops, while the accommodation property of the model can be seen in Fig. 4. Accommodation comes into effect when the excitation field is smaller than the width of the major loop. The larger the amplitude of the excitation, the faster the hysteresis loop becomes stable. As it can be seen in Fig. 4 the minor loop is shifted upwards and becomes stable after a few cycles. Since the hysteresis loop is derived from the CDF of the normal distribution the model exhibits strong saturation and accommodation properties as the figures show. The simulation results obtained by the above model seem to be appropriate; however, the model is not versatile enough since all the magnetic domains are supposed to be uniform.
Assuming that the magnetic moments of the domains are not the same for all domains, the distribution of moments can also be described by some kind of statistical distribution. The most straightforward approach is to apply the normal distribution. Let Y Nðm2 ; s2 Þ be a random variable corresponding to the magnetic moment of the domains. The statistical approach can be further extended by considering the consequence of the distribution of moments, thus the applied magnetic field will force the ‘‘weakest’’2 domains first to change their states and as the magnetic field strength increases the domains will change their states according to their moments.
2
x 106
1.5 1 M [A/m]
0.5 0
Since the switching fields and magnetic moments are described by distribution functions the magnetization process can be modelled by embedding these distributions into each other. The magnetization ratio in the case of uniform magnetic moments can be calculated by Eq. (6), and this value can be fed into the second distribution function corresponding to Y representing the distribution of the magnetic moments. The construction can be formulated as Z x 2 1 2 F ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffi eðtm1 Þ =2s1 dt, (6) 2 2ps1 1 F^ ðxÞ ¼ cotðpF ðxÞÞ,
(7)
-0.5
1 GðxÞ ¼ qffiffiffiffiffiffiffiffiffiffi 2ps22
-1 -1.5 -2 -8000 -6000 -4000 -2000
0
2000 4000 6000 8000
H [A/m]
Fig. 3. First order reversal loops ðm ¼ 1000; s ¼ 1000Þ.
14
x 105
12 10 8 2000
6
H (t) [A/m]
M [A/m]
3.1. Nested distributions
4
Z
F^ ðxÞ
2
2
eðtm2 Þ =2s2 dt,
(8)
1
where the intermediary Eq. (7) is required to map F ðxÞ 2 ½0; 1 to F^ ðxÞ 2 ½1; 1 and avoid unwanted ‘‘jumps’’ in the hysteresis curves. Substituting the actual applied field H into the GðxÞ function gives the magnetization ratio GðHÞ 2 ½0; 1 based on the assumption that the distribution of the magnetic moments is the normal distribution. At this point the actual value of magnetization can be calculated from the magnetization ratio GðHÞ in view of the M s saturation magnetization. Since the resulting distribution can be asymmetric, in the case of field reversal the probability density function has to be flipped from left to right in addition to the change of the expected value.
1000
4. Parameter identification 0
2 0 -500
0
0
500
1000 1500 H [A/m]
0.5 t
2000
1
2500
Fig. 4. Accommodation property of the model ðm ¼ 1000; s ¼ 1500Þ.
Identification of the model parameters is not too complicated. In the case of the simple model the determination of the values of parameters (m; s) goes in a 2
‘‘Weakest’’ means that least energy needed to change its state.
ARTICLE IN PRESS Z. Sari, A. Ivanyi / Physica B 372 (2006) 45–48
48
2
x 106
1.5
1.5
x 105
1
1
M [A/m]
0.5 0 -0.5
5000 H (t) [A/m]
M [A/m]
Measured Simulated
0.5
-1
-0.5
0
-1
-5000
-1.5
0
-2 -8000 -6000 -4000 -2000
0
0
0.5 t
1
-1.5 5
2000 4000 6000 8000
H [A/m]
0 H [A/m]
5 x 104
Fig. 5. Simulated concentric hysteresis loops.
Fig. 6. Measured loop compared to simulated.
straightforward manner because m ¼ H c and s ¼ oðwmax Þ as it was described before. The extended model has four parameters and the values of parameters are not so closely related to the shape of the hysteresis curve as in the former model. Let g ¼ ½m1 ; s1 ; m2 ; s2 be a vector containing the four parameters of the model. The identification procedure can be carried out in view of coercive field H c , remanent magnetization M rem , maximum susceptibility wmax , and susceptibility at point (H ¼ 0; M ¼ M rem ) denoted by wrem . The following non-linear system of equations can be formed for these values and vector g
versatile and a wider variety of hysteresis loops can be modelled.
Gðx ¼ H c ; gÞ ¼ 12, G 0 ðx ¼ H c ; gÞ ¼
wmax , 2M s
Gðx ¼ 0; gÞ ¼
M rem , 2M s
G 0 ðx ¼ 0; gÞ ¼
wrem . 2M s
A ; m
wmax ¼ 1500;
M rem ¼ 1:325 106
Since the model simply applies a distribution function to form the major hysteresis loop, the model can be generalized by substituting the cumulative distribution function by a properly scaled measured major hysteresis loop. This generalization will allow to simulate many kinds of hysteresis loops, and makes the measurement-based identification extremely easy. Fig. 6 shows a simulated hysteresis loop compared to a measured loop. 6. Conclusion
(9)
This non-linear system of equations (9) can be handled as a non-linear minimization problem and can be solved by trust region methods. The solution of the system is the g parameter vector which contains the parameters of the hysteresis model. Fig. 5 shows the behaviour of the model for the following set of properties of the major hysteresis loop: H c ¼ 1500
5. Generalization of the model
A , m
The presented model is based on a simple statistical approach of hysteresis phenomenon. Although the model is simple, it provides a good approximation to simulate the hysteresis characteristics. It is important to notice that the shape of the minor loops is completely determined by the major hysteresis loop; thus introducing a further parameter (or function) to define the shape of the minor loops is a major task in the future study of this approach. Acknowledgements The research work has been developed in the frame of the project Utilization of the infocommunication technologies in the region No. GVOP-3.1.1.-2004-05-0125/3.0.
wrem ¼ 500,
and resulting parameter vector
References
g ¼ ½1411:7; 3660:5; 1:2629; 0:5511.
[1] A. Iva´nyi, Hysteresis Models in Electromagnetic Field Computation, Akade´miai Kiado´, Budapest, 1997. [2] J. Taka´cs, Mathematics of Hysteretic Phenomena, Wiley-VCH, Weinheim, 2003. [3] G. Bertotti, Hysteresis in Magnetism, Academic Press, 1998. [4] A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994.
From the figure it can be seen that applying the modified distribution enables that the shape of the hysteresis loop can be different from the ones created by the CDF of the normal distribution. In this way the model becomes more