Application of simplified vector Preisach model to vector magnetizing process

Physica B 275 (2000) 114}119

Application of simpli"ed vector Preisach model to vector magnetizing process Gary Kahler*, Edward Della Torre The Institute of Magnetics Research, The George Washington University, Washington, DC 20052, USA

Abstract A series of measurements has shown that when a material is increasingly magnetized in a given direction, the magnetization perpendicular to this direction decreases to zero. A simpli"ed vector Preisach model, an example of a coupled-hysteron model, has been introduced that incorporates this property into the vector model. In addition, this model satis"es the saturation property and the loss property. It can correctly compute scalar loops, since it is based upon the moving model, which properly corrects the congruency property; and it can incorporate the accommodation model and the aftere!ect model to correct the deletion property. In three dimensions, the vector magnetization is computed from the integration of the product of a state vector, Q, and a Preisach function, p, both of which are de"ned in a six-dimensional Preisach space. The Q's are compared following selection rules determined by the applied "eld. This paper illustrates the calculation procedures for a complex magnetizing process. A material sample is subjected to a saturating "eld that is then reduced to zero. The "eld is then increased in an arbitrary direction while the vector magnetization is measured. Since the model parameters are only determined along the principal axes, the predictive capability of the model is thereby demonstrated. The predicted results of the model compare favorably to the results of experimental measurements. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Preisach model; Vector modeling; Hysteresis modeling

1. Introduction Scalar models of the magnetization process consider "eld variations along the axis of the applied "eld and the magnetization along that axis. For real magnetizing process, the applied "eld not only can change in magnitude, but it can also change in direction. In particular, the applied "eld can rotate resulting in a magnetization that is not always in the direction of the applied "eld. Whereas, the scalar models characterize a magnetic material in one given direction, a model must be able to characterize this material in two or more direction if the applied "eld varies in direction.

* Correspondence address: 14804 Peachwood Drive, Silver Spring, MD 20905-5646, USA. Fax: 301-989-0980. E-mail address: [email protected] (G. Kahler)

The Mayergoyz vector and the pseudo-particle vector models characterize materials in two or more direction. The Mayergoyz model consists of a continuum of scalar Preisach transducers [1], which are incrementally rotated from one model to the next model. The input to each of these transducers is the component of the applied "eld in the direction of each transducer. The pseudoparticle vector models simulate a medium with the use of a small number of pseudo-particles. One of these models uses the Stoner}Wohlfarth model of a particle as the basic particles [2]. Another of these models uses micromagnetic calculations to de"ne the basic particles [3]. The process utilizing these particles consists of applying Preisach identi"cation techniques to each of the particles, computing their individual magnetization, and then using the vector sum of each particle's magnetization as the magnetization of the pseudo-particle. The coupled-hysteron model [4] is another category of vectors models. This category of vector models identi"es

0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 7 2 7 - 9

G. Kahler, E. Della Torre / Physica B 275 (2000) 114}119

a Preisach model along each principal axis of the system. A series of measurements [5] has shown that when a material is increasingly magnetized in a given direction, the magnetization perpendicular to this direction decreases to zero. The simpli"ed vector Preisach (SVP) model [6] is a coupled-hysteron model. The SVP model can be either two-dimensional or three-dimensional. The two dimensional SVP model "rst requires the identi"cation of a scalar Preisach model the magnetically easy axis and along the magnetically hard axis of a material. After this identi"cation process is completed, the SVP methodology [6] may be implemented.

2. Scalar Preisach model identi5cation For a two-dimensional SVP model, a scalar Preisach model must be identi"ed along the magnetically easy and magnetically hard axes of the material being modeled. This identi"cation process identi"es the following seven Preisach parameters along each of these two axes: 1. 2. 3. 4. 5. 6. 7.

M , saturation magnetization; 4 S, squareness; hM , critical "eld expectation; k a, moving constant; s , susceptibility at applied "eld equal to zero; 0 p , standard deviation of critical "eld; and k p , standard deviation of interaction "eld. i

A vibrating sample magnetometer (VSM) will acquire the data needed to identify the seven Preisach parameters and implemented the SVP model. Fig. 1 shows the placement and orientation of a material sample ion a VSM. The applied "eld in the plane of the sample. The easy and hard axes of the sample are "rst determined. With the sample oriented such that the easy axis is parallel with

115

the applied "eld, major hysteresis loop and major remanence curve data are measured. Rotating the sample 903 places the hard axis of the sample parallel with the applied "eld. With this orientation, major hysteresis loop and major remanence curve data are again measured. In addition to these measurements, the sample is placed in positive magnetic saturation, the applied "eld is reduced to H"!hM , the magnetization is measured, the k applied "eld is increased to H"hM , where the magnetizk ation is again measured. These data will be used to separate the switching "eld, p into the standard deviation of critical "eld, p and the standard deviation of interack tion "eld, p . The sample used to evaluate the SVD model i is a metal particle tape (MPT). The measured ascending hysteresis curve for the easy and hard axes of MPT is shown in Fig. 2. The measured ascending remanence curves for the easy and the hard axes of MPT is shown in Fig. 3. The seven Preisach parameters and the switching "eld, p, are shown in Table 1. Using Preisach parameters, the MPT recording material magnetization is computed for the easy and the hard axes of the material. The comparison of these computed magnetizations with measured magnetizations is shown in Figs. 4 and 5, respectively.

3. SVP model implementation Using the Preisach parameters identi"ed in Table 1, the SVP model [6] was implemented for MPT recording material with and applied "eld angle of 453, using the Preisach function,

A

1 p" j 4pp p ij kj

B

A

B

p2 (u !v !2hM )2#p2 (u #v )2 j kj kj j j , (1) ]exp ! ij j 8p2 p2 ij kj where j"x and y for the two-dimensional model. The variables, h (h and h ) are the x- and y-components of j x y Table 1 Preisach parameters for MPT recording material

Fig. 1. Placement and orientation of sample in VSM. Major hystersis loop and major remanence curve data are measured along easy and hard axes in order to identify Preisach parameters along these axes.

Parameter

Easy axis

Hard axis

M (emu) 4 S hM (Oe) k a (Oe/emu) s (emu/Oe) 0 p (emu) k p (emu) i p (emu)

0.025 0.83 1480 380900 2.49e-6 6352 2614 6869

0.025 0.39 2210 192000 9.98e-6 3502 3231 4765

116

G. Kahler, E. Della Torre / Physica B 275 (2000) 114}119

Fig. 2. Measured ascending major hysteresis curves for the easy and hard axes of MPT recording material.

Fig. 5. Computed Preisach modeled ascending major hysteresis curve along the hard axis compared to the measured curve.

Fig. 6. Two orthogonal Preisach planes showing the projection of a Preisach function on each plane. The variables h are the j components of the applied "eld, u are the up-switching "elds, j and v are the down-switching "elds. j

Fig. 3. Measured ascending major remanence curves for easy and hard axes of MPT recording material.

the applied "eld. The variables, u and v are the upj j switching "elds and the down-switching "elds, respectively, in the X and > Preisach planes, shown in Fig. 6. First the basis Preisach integrals,

P

I" j

XR

Q (u , v )p (u , v ) du dv , j x x j x x x x

(2)

where X is the region where u 'v and u 'v to R x x y y ensure physical realizability. Q is a state vector that must follow the selection rules of Tables 2 and 3. The evaluation of I and I is shown in Fig. 7. x y For applied "elds impressed on a material between the easy and hard axes of the material, the SVP model uses a rotation correction, R, DI D#DI D y . R(I , I )" x x y JI2#I2 x y Fig. 4. Computed Preisach modeled ascending major hysteresis curve along the easy axis compared to the measured curve.

(3)

For MPT recording material with an applied "eld angle of 453, the rotational correction is shown in Fig. 8.

G. Kahler, E. Della Torre / Physica B 275 (2000) 114}119 Table 2 Selection values for Q

117

x

Q x

v 'h x x

v (h (u x x x

h 'u x x

v 'h y y

h !v x x Dh !v D#Dh !v D x x y y

0

h !v x x Dh !v D#Dh !v D x x y y

v (h (u y y y

!1

No change

1

h 'u y y

h !v x x Dh !v D#Dh !v D x x y y

0

h !v x x Dh !v D#Dh !v D x x y y

Q y

v 'h x x

v (h (u x x x

h 'u x x

v 'h y y

h !v y y Dh !v D#Dh !v D x x y y

!1

h !v y y Dh !v D#Dh !v D x x y y

v (h (u y y y

0

No change

0

Table 3 Selection values for Q

y

Fig. 7. Computed evaluation of I and I for MPT recording x y material with an applied "eld angle of 453.

Fig. 8. Computed rotational correction, R(I , I ) for MPT rex y cording material.

The SVP model gives the normalization irreversible magnetization as the product of the rotational correct, R, and the basic Preisach integrals, I , j

computed from

m "R(I , I )I Ix x y x

where

(4a)

1#m Ij a " ` 2

and m "R(I , I )I . Iy x y y

m "a f (H)!a f (!H), R ` ~

(5)

(6a)

(4b)

The results of this computation are shown in Fig. 9. For a magnetization-dependent model or for a statedependent model, the reversible magnetization, m , is R

and 1!m Ij . a " ~ 2

(6b)

118

G. Kahler, E. Della Torre / Physica B 275 (2000) 114}119

Fig. 9. Computed irreversible magnetization components along the easy and hard axis of MPT recording material with an applied "eld angle of 453.

Fig. 11. Computed total magnetization components along the easy and hard axis of MPT recording material with an applied "eld angle of 453.

Fig. 10. Computed reversible magnetization components along the easy and hard axis of MPT recording material with an applied "eld angle of 453.

Fig. 12. Measured orthogonal magnetization components for MPT recording material with an applied "eld angle of 453.

and the reversible magnetization components, Eq. (8); that is,

The function f (H) is chosen to be f (H)"(1!S)M (1!e~xo H@(1~S)M4 ), S

(7)

where S, M , and s are Preisach parameters identi"ed in S 0 Table 1. Using Eqs. (5), (6a), (6b) and (7), the normalized reversible components of the magnetization is computed with m "DI DR(I , I ) [a #f (H )!a !f (!H )], Rj j x y j j j j

(8)

where j"x and y. The computation of Eq. (8) is shown in Fig. 10. The total magnetization components is the sum of the irreversible magnetization components, Eq. (4a) and (4b),

m "m #m . Tj Ij Rj

(9)

The computation of Eq. (9) for MPT recording material for an applied "eld angle of 453 is shown in Fig. 11. The VSM measures the orthogonal components of magnetization for the material attached to the VSM samples holder. For the MPT recording material, the magnetization components measured are along the direction of the applied "eld, 453, from the easy axis, and along the direction perpendicular to the applied "eld direction, 1353, or !453, from the easy axis. Fig. 12 show the

G. Kahler, E. Della Torre / Physica B 275 (2000) 114}119

Fig. 13. Comparison of computed and measured normalized total magnetization for MPT recording material with an applied "eld angle of 453.

normalized measured components for MPT recording material with an applied "eld angle of 453. The total magnetization is the vector of some of the components. Fig. 13 shows the normalized vector sum of the measured magnetization and the normalized vector sum of the SVP computed magnetization. 4. Conclusions The application of the SVP model consists of straightforward computational steps. The "rst step to implemen-

119

ting the SVP model for a given materials is the scalar identi"cation of two Preisach models } one parallel to the magnetically easy axis of the material and one along the magnetically hard axis of the material. The second step is the evaluation of the basic Preisach integrals, which utilizes a set of selection rules to ensure physical realizability. The third step is the computation of the irreversible magnetization by making use of the rotational correction and the basic Preisach integral. The fourth step calculates the reversible magnetization from a reversible function, the rotational correction, and the basic Preisach integral. The total magnetization is then the sum of the irreversible and the reversible magnetization. The total magnetization computed from measured total orthogonal magnetization components do not compare well with the total magnetization computed using the SVP vector model. This poor performance may be due to errors in the implementation of the SVP model. This performance may also be duet to some questions concerning the saturation of the measured data perpendicular to the applied "eld. The computed rotational correction may be improved by reducing the applied "eld increment.

References [1] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991. [2] J. Oti, E. Della Torre, J. Appl. Phys. 67 (9) (1990) 5364. [3] J. Oti, E. Della Torre, IEEE Trans. Mag. 26 (1990) 2116. [4] E. Della Torre, F. Vajda, IEEE Trans. Mag. 32 (1996) 1116. [5] K. Wiesen, S.H. Charap, J. Appl. Phys. 61 (1987) 4019. [6] E. Della Torre, IEEE Trans. Magn. 34 (1998) 495.