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Physica B 372 (2006) 61–67

www.elsevier.com/locate/physb

Preisach functions leading to closed form permeability

Zsolt Szaboa,b,�

aDepartment of Broadband Infocommunications and Electromagnetic Theory, Budapest University of Technology and Economics,

Egry J. u. 18, H-1111 Budapest, HungarybNanophysics Group, Nanomaterials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan

Abstract

In this paper the factorization possibilities and analytic approximations of the Preisach function are investigated, so that the dynamic

permeability or the susceptibility may be expressible in closed form. The parameters of the analytical approximations can be determined

by a minimization procedure.

r 2005 Elsevier B.V. All rights reserved.

PACS: 75.60.Ej

Keywords: Magnetic hysteresis; Preisach model

1. Introduction

Many practical electromagnetic problems involve the knowledge of the dynamic permeability or susceptibility data. Forinstance the Kirchoff equations of the simplest magnetic circuit, a coil with toroidal ferromagnetic core and a resistance R

connected serially to a sinusoidal voltage source u leads to the equation

u ¼ Ri þNdfdt

, (1)

where i is the current, f is the magnetic flux, N is the number of turns. Applying the transformation

dfdt¼ S

dB

dt¼ S

dB

dH

dH

dt¼

NS

l

dB

dH

di

dt, (2)

(S is the cross section, l is the average length of the core) the current flowing in the circuit can be determined by solving thefollowing differential equation:

di

dt¼

l

N2S

u� Ri

dB=dH. (3)

As another example, consider the scattering of the electromagnetic field with components Ez and Hy propagating in thex direction incident perpendicularly on a ferromagnetic slab of thickness d. The Maxwell equations for the ferromagneticmedia are

qHy

qx¼ sEz þ �0�r

qEz

qt, (4)

e front matter r 2005 Elsevier B.V. All rights reserved.

ysb.2005.10.020

ng author at: Nanophysics Group, Nanomaterials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044,

1 29 851 3354 x8830; fax: +81 29 860 4795.

sses: szabo.zsolt@nims.go.jp, szabo@evtsz.evt.bme.hu.

ARTICLE IN PRESS

Fig. 1. Measured hysteresis curve and dynamic susceptibility computed by numerical differentiation.

Z. Szabo / Physica B 372 (2006) 61–6762

qEz

qx¼

qBy

qt. (5)

The time derivatives of the magnetic flux density can be written as

qBy

qt¼

qBy

qHy

qHy

qt, (6)

and substituting in (5) results in

qEz

qx¼

qBy

qHy

qHy

qt. (7)

Solving the partial differential equation system (4), (7) the behavior of the electromagnetic waves in the ferromagnetic slabcan be investigated.

As it can be seen knowledge of the dynamic permeability plays an important role in the above simple circuit andelectromagnetic field computation problems. Generally, the equations of the dynamic electromagnetic problems cancontain the dynamic permeability or susceptibility as a three-dimensional tensor.

In this paper the Preisach model [1,2] is applied for the description of the hysteresis in ferromagnetic media. Numericaldifferentiation can amplify the measurement errors and usually provides a randomly oscillating dB=dH function (see Fig.1) mainly when the Preisach model is identified directly from measured data [3] (e.g. the Everett function is constructedfrom first order reversal curves), numerical differentiation can amplify the measurement errors and usually provides arandomly oscillating dB=dH function (see Fig. 1). The solution of the governing nonlinear differential equations usuallyrequires iterative process and rugged dB=dH may lead to numerical instability. To avoid the numerical differentiationdifferent forms of the Preisach function are investigated which lead to closed form dynamic permeability. With thisprocedure a well-defined smooth dynamic permeability can be obtained, moreover the point-wise storage of the Everettfunction or Preisach distribution can be also avoided leading to economy in memory usage.

2. Separation possibilities of the Preisach function

To represent a ferromagnetic material with the Preisach model

BðtÞ ¼

Z ZT

mðh1; h2Þgðh1; h2;HðtÞÞdh1dh2, (8)

(where g represents a hysteron with switching down field h1 and switching up field h2, T denotes the Preisach triangle) thetwo-dimensional Preisach function mðh1; h2Þ or

Eðx; yÞ ¼

Z y

x

Z h2

x

mðh1; h2Þdh1dh2, (9)

ARTICLE IN PRESSZ. Szabo / Physica B 372 (2006) 61–67 63

the Everett function must be known. In order to simplify the formulation of the model, the Preisach function can beapproximated as a product of two one-dimensional functions. Usually two types of separations can be applied. In the firstone the switching up and down fields are considered to be uncorrelated

mðh1; h2Þ ¼ f ðh1Þgðh2Þ ¼ jð�h1Þjðh2Þ. (10)

In Eq. (10) the Preisach function is considered symmetric on h2 ¼ �h1 line. This condition can be justified by the symmetryof the magnetization process. In the second one the equivalent characterization of the hysterons by coercive force hc andinteraction field hm is exploited

mðhc; hmÞ ¼ ZðhcÞxðhmÞ ¼ Zh2 � h1

2

� �x

h2 þ h1

2

� �. (11)

In this case the coercive force and the interaction field are uncorrelated and regarded as consequences of independentphysical phenomena.

With the aid of the Preisach model the dynamic permeability for increasing magnetic field intensity can be expressed as

dB

dH¼ 2

Z H

Hr

mðh1;HÞdh1, (12)

and for decreasing magnetic field intensity

dB

dH¼ 2

Z Hr

H

mðH ; h2Þdh2. (13)

In these relations Hr is the last dominant extremum of the magnetic field intensity.

3. Uncorrelated switching up and down split of the Preisach function

First the validity of Eq. (10) is investigated. The consequence of this separation is that the Preisach function (or a part ofit) can be determined point-wise from the increasing or decreasing branch of the major (concentric) hysteresis loop.Dividing the interval between negative and positive saturation into n equal subintervals, approximating dB=dH

numerically, considering j constant in each subinterval and taking into consideration the symmetry of the one-dimensionalPreisach function jð�h1Þ ¼ jðh2Þ, from Eq. (12) the following equation system can be obtained:

Biþ1 � Bi

Hiþ1 �Hi¼ 2jðHiÞDh

Xn

j¼n�iþ1

jðHjÞ. (14)

The solution of Eq. (14) determines the two-dimensional Preisach function. The procedure has been applied to fit thehysteresis characteristic of several ferromagnetic materials. It was found that the major loop fits very well, but the interiorloops differ from the measured values. The validity of the obtained Preisach function can be tested with the firstmagnetization curve. Along the first magnetization curve the dynamic permeability can be expressed as

dB

dH¼ 2jðHÞ

Z H

0

jð�h1Þdh1. (15)

Applying the previous discretization n=2 equations with n unknowns can be obtained

dB

dH

����i

¼ 2ji

Xi

j¼1þn�i

jjDh; i4n

2. (16)

If Eq. (10) is correct the solutions of Eq. (14) must satisfy Eq. (16). However, in general this fact is not verified. Theobtained Preisach function presents biaxial symmetry. To represent simultaneously with the same accuracy the major andthe first magnetization loops a more general Preisach function is needed. This requirement cannot be satisfied by Eq. (10),however, as it is shown in the sequel this approximation can lead to relatively simple and useful mathematicalapproximations.

3.1. Gaussian type one-dimensional Preisach function

The one-dimensional Preisach function in Eq. (10) is approximated as

jðxÞ ¼Xn

i¼1

aie�ððx�biÞ=ciÞ

2

. (17)

ARTICLE IN PRESSZ. Szabo / Physica B 372 (2006) 61–6764

For increasing values of magnetic field intensity the dynamic permeability results in

dB

dH¼Xn

i¼1

aie�ððH�biÞ=ciÞ

2 Xn

j¼1

ajcj

ffiffiffipp

erfH þ bj

cj

� �� erf

Hr þ bj

cj

� �� �. (18)

For the decreasing values of magnetic field intensity similar expression can be obtained from Eq. (13).

3.2. Lorentz type one-dimensional Preisach function

In this case the one-dimensional Preisach function is approximated as

mðxÞ ¼Xn

i¼1

ai

1þ ððx� biÞ=ciÞ2, (19)

for the increasing branches of the hysteresis loop the dynamic permeability can be written as

dB

dH¼ 2

Xn

i¼1

ai

1þ ððH � biÞ=ciÞ2Xn

i¼1

ajcj atanH þ bj

cj

� �� atan

Hr þ bj

cj

� �� �. (20)

3.3. Inverse cosine hyperbolic type one-dimensional Preisach distribution

Approximating the one-dimensional Preisach distribution with

jðxÞ ¼Xn

i¼1

aie�ðx�biÞ=ci

ð1þ e�ðx�biÞ=ci Þ2¼

1

2

Xn

i¼1

ai

1þ coshððx� biÞ=ciÞ¼Xn

i¼1

aie�bix

ð1þ gie�bixÞ

2, (21)

where a ¼ aeb=c, b ¼ 1=c, g ¼ eb=c, for increasing magnetic field intensities the dynamic permeability takes the form

dB

dH¼Xn

i¼1

ai

1þ coshððH � biÞ=ciÞ

Xn

j¼1

aj

bjcj

1

1þ eðHr�bÞ=c�

1

1þ eðH�bÞ=c

� �. (22)

4. Preisach function leading to closed form Everett function

Considering the Preisach function in the following form:

mðh1; h2Þ ¼Xn

i¼1

miðh1; h2Þ ¼Xn

i¼1

aiebih1

ð1þ giebih1Þ

2

aie�bih2

ð1þ gie�bih2Þ

2, (23)

the double integration in Eq. (9) can be performed symbolically and the Everett function can be expressed in closed form

Eðx; yÞ ¼ �Xn

i¼1

a2ib2i

ðg2i � 1Þðebix � ebiyÞ þ ðgi þ ebiyÞð1þ giebixÞ lnð1þ gie

biyÞðgi þ ebixÞ=ð1þ giebixÞðgi þ ebiyÞ

ðg2i � 1Þ2ðgi þ ebiyÞð1þ giebixÞ

. (24)

When in Eq. (21) b ¼ 0, then g ¼ 1 and Eq. (24) becomes singular, however, applying the l’Hospital rule the followingrelation can be obtained:

limg!1

Eðx; yÞ ¼1

2b2ðebx � ebyÞ

2

ð1þ ebxÞ2ð1þ ebyÞ

2. (25)

In this way a completely analytical Preisach model can be introduced. The main advantage of Eq. (21) is that in Eq. (24)appear only ordinary functions. For example, the magnetic flux density along the increasing branch of concentric hysteresisloops (from �Hm! Hm) can be computed as

B ¼ � Bm þ 2Xn

i¼1

a2ib2i

�ð�ebiH � e�biHm Þðg2i � 1Þ � ðgie

�biHm þ 1Þðgi þ ebiH Þ lnð1þ giebiH Þðgi þ e�biHm Þ=ð1þ gie

�biHm Þðgi þ ebiH Þ

ðg2i � 1Þ2ðgi þ ebiH Þð1þ gie�biHmÞ

, ð26Þ

ARTICLE IN PRESSZ. Szabo / Physica B 372 (2006) 61–67 65

where Bm is the maximum value of magnetic flux density along the concentric loop (corresponding to Hm),

Bm ¼Xn

i¼1

a2ib2i

ð1� e�2biHm Þðg2i � 1Þ � ð1þ gie�biHm Þ

2 lnð1þ giebiHm Þðgi þ e�biHm Þ=ð1þ gie

�biHm Þðgi þ ebiHm Þ

ðg2i � 1Þ2ð1þ gie�biHm Þ

2. (27)

5. Separation of the Preisach function in coercive field and interaction field dependent terms

For this type of split the difficulty in the evaluation of Eq. (12) appears with the fact that the Preisach function is nolonger separable in independent h1 and h2 terms. In the following approximations the function xðhmÞ is consideredsymmetric.

5.1. Gauss–Gauss expression of the Preisach function

In this case the Preisach function is approximated as

mðhc; hmÞ ¼Xn1i¼1

aie�ðhc�bi=ciÞ

2 Xn2j¼1

ajðe�ððhm�bj Þ=cjÞ

2

þ e�ððhmþbjÞ=cjÞ2

Þ, (28)

the permeability results in the following expression:

dB

dH¼ 2

ffiffiffipp Xn1

i¼1

Xn2j¼1

Ki;j e�ðH�bi�bj Þ2=ðc2i þc2j Þ erf

c2i H þ c2j bi � c2i bj

cicj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic2i þ c2j

q0B@

1CA

264

8><>:

�erfHðc2i � c2j Þ �Hrðc

2i þ c2j Þ þ 2ðc2j bi � c2i bjÞ

2cicj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic2i þ c2j

q0B@

1CA375þ e�ðH�biþbjÞ

2=ðc2iþc2j Þ erfc2i H þ c2j bi þ c2i bj

cicj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic2i þ c2j

q0B@

1CA

264

� erfHðc2i � c2j Þ �Hrðc

2i þ c2j Þ þ 2ðc2j bi þ c2i bjÞ

2cicj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic2i þ c2j

q0B@

1CA3759>=>;,

Ki;j ¼aiajcicjffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c2i þ c2j

q . (29)

5.2. Lorentz–Lorentz approximation of the Preisach function

In this case the Preisach function has the form

mðhc; hmÞ ¼Xn1i¼1

ai

1þ ððhc � biÞ=ciÞ2

Xn2j¼1

aj

1þ ððhm � bjÞ=cjÞ2þ

aj

1þ ððhm þ bjÞ=cjÞ2

!, (30)

and the permeability can be expressed as

dB

dH¼ 4

Xn1i¼1

Xn2j¼1

aiajcicj

x1ðHÞcicjðh2 � bi � bjÞ ln

½ðH � bjÞ2þ c2j �½ðH þHr � 2biÞ

2þ 4c2i �

½b2i þ c2i �½ðH �Hr � 2bjÞ

2þ 4c2j �

þ ciððH � bi � bjÞ2þ c2i � c2j Þ

(

� arctanH � bj

cj

� arctanH �Hr � 2bj

2cj

� �þ cjððH � bi � bjÞ

2� c2i þ c2j Þ arctan

bi

ci

þ arctanH þHr � 2bi

2ci

� �)

þ 4Xn1i¼1

Xn2j¼1

aiajcicj

x2ðHÞcicjðh2 � bi þ bjÞ ln

½ðH þ bjÞ2þ c2j �½ðH þHr � 2biÞ

2þ 4c2i �

½b2i þ c2i �½ðH �Hr þ 2bjÞ

2þ 4c2j �

þ ciððH � bi þ bjÞ2þ c2i � c2j Þ

(

� arctanH þ bj

cj

� arctanH �Hr þ 2bj

2cj

� �þ cjððH � bi þ bjÞ

2� c2i þ c2j Þ arctan

bi

ci

þ arctanH þHr � 2bi

2ci

� �),

ARTICLE IN PRESSZ. Szabo / Physica B 372 (2006) 61–6766

x1ðHÞ ¼ H4 � 4ðbi þ bjÞH3 þ 2½3ðbi þ bjÞ

2þ c2i þ c2j �H

2 � 4ðbi þ bjÞ½ðbi þ bjÞ2þ c2i þ c2j �H

þ ½ðbi þ bjÞ2þ ðci � cjÞ

2�½ðbi þ bjÞ

2þ ðci þ cjÞ

2�,

x2ðHÞ ¼ H4 � 4ðbi � bjÞH3 þ 2½3ðbi � bjÞ

2þ c2i þ c2j �H

2 � 4ðbi � bjÞ½ðbi � bjÞ2þ c2i þ c2j �H

þ ½ðbi � bjÞ2þ ðci � cjÞ

2�½ðbi � bjÞ

2þ ðci þ cjÞ

2�. ð31Þ

For inverse cosine hyperbolic type approximation or mixed cases the integral in Eq. (12) has no analytical primitivefunctions.

6. Identification of the parameters

The parameters are identified by fitting to the lower branch of a symmetric hysteresis loop. To determine economicallythe required values of the magnetic flux density the interval between �Hm;Hm is divided into n intervals. Eq. (9) iscomputed on each interval. The first integral is substituted by the resulted closed form expression and the second one isperformed numerically with four point Gauss quadrature. Then a corresponding summation is applied and an errorfunction is evaluated taking the difference between the values of the measured and computed magnetic flux densities. Incase of Eq. (23) the error function is determined directly with Eq. (26). A reversible part in the form

BrevðHÞ ¼ k1H þ k2 tanhH

k3

� �(32)

is added exterior to the Preisach model. The hysteresis loops of the TEAM Problem 32 [4] were fitted to investigate theproperties of the approximations. In all cases the concentric loop can be approximated very well; however, the computedinternal loops can differ significantly from the measured one (see Fig. 2).

7. Discussion and conclusions

Generally the major loop is not enough to identify unambiguously the Preisach function and the particular choice of theanalytic function does not have big effect. With (28) and (30) very complex Preisach functions can be constructed, but theevaluation of (29) and (31) is time consuming and the minimization algorithm can often stuck in local minima. Thepresented closed form Everett function leads to completely analytic Preisach model. This approximation is the fastest andthe most applicable. To increase the accuracy, minor loops or the first magnetization curves can be used in the fittingprocedure and improved Preisach models e.g. the moving model and the product model can be applied. With adequatesuperposition of the closed form expressions, Mayergoyz type vector Preisach model can also be built efficiently.

Fig. 2. Measured and computed hysteresis loops.

ARTICLE IN PRESSZ. Szabo / Physica B 372 (2006) 61–67 67

Acknowledgements

This work was supported by the ‘‘Magyary Zoltan’’ and JSPS postdoctoral fellowships.

References

[1] E. Della Torre, Magnetic Hysteresis, IEEE Press, New York, 1999.

[2] G. Bertotti, Hysteresis in Magnetism, Academic Press, 1998, pp. 433–505.

[3] P. Kis et al., Physica B 343 (2004) 357.

[4] hhttp://www.compumag.co.uk/teamindex.htmli