video

Home
Search
Collections
Journals
About
Contact us
My IOPscience
The Bilinear Product Model of Hysteresis Phenomena
This content has been downloaded from IOPscience. Please scroll down to see the full text.
1989 Phys. Scr. 1989 161
(http://iopscience.iop.org/1402-4896/1989/T25/029)
View the table of contents for this issue, or go to the journal homepage for more
Download details:
IP Address: 137.99.31.134
This content was downloaded on 01/06/2015 at 08:28
Please note that terms and conditions apply.
Physica Scripta. Vol. T25, 161-164, 1989.
The Bilinear Product Model of Hysteresis Phenomena
Gyorgy Kadhr
Central Research Institute for Physics, H- I525 Budapest, P.O.B. 49, Hungary
Received April 6, 1988; accepted June 22, 1988
Abstract
In ferromagnetic materials non-reversible magnetization processes are
represented by rather complex hysteresis curves. The phenomenological
description of such curves needs the use of multi-valued, yet unambiguous,
deterministic functions. The history dependent calculation of consecutive
Everett-integrals of the two-variable Preisach-function can account for the
main features of hysteresis curves in uniaxial magnetic materials. The
traditional Preisach model has recently been modified on the basis of
population dynamics considerations, removing the non-real congruency
property of the model. The Preisach-function was proposed to be a product
of two factors of distinct physical significance: a magnetization dependent
function taking into account the overall magnetization state of the body and
a bilinear form of a single variable, magnetic field dependent, switching
probability function. The most important statement of the bilinear product
model is, that the switching process of individual particles is to be separated
from the book-keeping procedure of their states. This empirical model of
hysteresis can easily be extended to other irreversible physical processes, such
as first order phase transitions.
1. Introduction
The behaviour of a system of magnetic units (individual
spins, clusters, fine particles, grains, domains, etc.) is complex
enough to provide a model for many types of collective
phenomena in complicated, interactive systems. The magnetization curve of a ferromagnetic body, as an example, can
be representative of hysteresis phenomena in general and can
reflect most of the characteristic features of a whole class of
static, macroscopic, non-reversible transition processes. In
such cases the deviation from reversibility does not involve
any dynamic irreversible time evolution and the macroscopic
parameters are not sensitive to microscopic details. The nonreversible behaviour is confined to a finite transition region of
the relevant parameters between two distinct states of the
system.
Outside the transition region the process is reversible, the
state variable is a single-valued function on both sides of the
transformation interval. Inside the non-reversible transition
region the function values are limited by an extrema1 major
hysteresis loop and are dependent on the direction and on the
previous history of the change in the control parameter. The
actual state of the system is described by a branching, multivalued function, which is non-reversible but still deterministic
and unambiguous if the minute details of previous history are
properly taken into account.
The branching takes place at the turning points when the
direction of the change in the parameters is switched to
opposite. Apparently the rate of change of the state variable
with respect to the control variable (e.g., the differential
susceptibility in a magnetization process) gradually increases
by moving forward along one direction and drops to a
smaller value by changing direction at any turning point.
Thus the state variable cannot follow the same route back:
branching occurs. This branching behaviour of the external
magnetic field dependence of the magnetization in a ferromagnet is characteristic of many other transition processes,
e.g., the temperature dependence of extensive parameters in
a first order phase transformation, voltage dependence
of deformation in a piezoelectric material, electric field
dependence of polarization in ferroelectrics, etc. Beyond the
transition interval the loop will be closed and the process will
be reversible.
A method of calculating some details of magnetic hysteresis
loops in ferromagnetic materials has been introduced over
fifty years ago in 1935 by Preisach [l]. The parabolic change
of the magnetization with the external field in the vicinity of
the origin of the H-M plane (Raleigh law) could be explained
by a double integral of a two-variable function. A similar
integration procedure was described for general hysteretic
processes by Everett et al. [2]. The Preisach function and
integration has been widely discussed [3-51 and. used [6] for
the calculation of magnetization curves.
In the Preisach model minor hysteresis loops calculated
between the same external field values but with different
magnetic history are congruent and this congruency property
- although not justified by the experimental data - has been
shown to be intrinsic feature of the model [7]. The “moving
Preisach function” model of Della Torre [8] could correct this
shortcoming of the traditional Preisach model. Recently a
modification of the traditional model has been proposed on
the basis of population dynamics considerations [9] in which
the switching process of magnetic units and the book-keeping
of the overall magnetic state of the system are considered
independent and are expressed as multiplicative factors in the
integrand of the Preisach model. This “product Preisach
function” model could also rectify the non-real congruency
property. The Preisach model has been scalar in character,
i.e., the external magnetic field varies along a fixed direction.
Recently, however, the application of the Preisach-function
for two- or three-dimensional vector-models of magnetic
hysteresis phenomena has become of interest [IO, 111.
In this paper the bilinear product model of magnetic
hysteresis and some of its consequences are discussed.
1. The traditional Preisach model
In the scalar Preisach model of magnetic hysteresis a change
in the magnetization of a ferromagnet along the external
magnetic field can be calculated by integrating a two-variable
function, the Preisach function, over a two-dimensional area
[I, 41. Both variables of the Preisach-function are of the
dimension of a magnetic field and the integration area is
determined by the range of the varying external field as well
as by the preceding magnetic history of the material. Thus a
change in the external field from H , to a higher value, H2,
Physica Scripta T25
162
Gyorgy Kadar
brings about a change in the magnetization [ 5 ] :
M ( H 2 ) - M ( H , ) = E ( H , , H,)
=
jz dh,:I
dh’P(h, h’)
(1)
where E ( H , , H 2 ) is called the Everett-integral, P ( h , h’) is the
Preisach-function. The integral is interpreted as a weighted
sum of infinitesimal jumps of the magnetization along the
upswitching branch of rectangular elementary hysteresis
loops. The value of the Preisach-function in a material
represents the density function of such an elementary loop
described by its upper and lower switching fields h and h’,
respectively. In a jump at h all loops have to be considered
whose upswitching field value is h, and downswitching field
values are smaller than h. Thus h’ is always smaller than h,
that is the Preisach-function is defined only on the half-plane
extending to the right of the h’ = h straight line. Outside the
transition interval, that is when the extrema1 loop has been
closed, P(h, h’) is zero by definition.
In the integral the magnetic field is supposed to be
increasing from H I to H 2 , otherwise in the case of decreasing
magnetic field we define:
E ( H 2 ,H I )
=
::j
-
dh’
f’dhP(h’, h)
the Everett-integral E ( H , , H , ) always gives the same difference irrespective of both the value of M and the previous
history of field changes. In other words all minor loops
calculated in the Preisach-model between the same limits are
congruent. This property, however, can not be justified by the
experimental curves of magnetization. Measured minor loops
of the same width in H show a tendency [I21 of being higher
in M in the vicinity of te H-axis ( M = 0) than close to the
saturated state. On the other hand calculated minor loops [5]
not touching the H-axis were found systematically higher
than the measured ones. The main weakness of the traditional
Preisach model has been its congruency property, incompatible with experimental facts.
The usual methods of the measurement of the Preisachfunction involve the mapping of the entire H-M plane, but at
least the first order reversal curves on the main loop. The
numerical values for P(h, h’) can be obtained from the
measured data by the discretization of the (h, h’) plane and
the solution of an algebraic system of equations.
The non-hysteretic reversible part of the magnetization as
a function of the external magnetic field and its relation to the
irreversible hysteresis loop has not been discussed in the
literature with satisfactory and sufficient detail.
H2
-E(-H,, -HI)
( 2 ) 2. The proposed modification of the Preisach model
From the symmetrical character of magnetic hysteresis loops The numerical values of the Everett-integrals can be most
one can show that P(h, h’) has a mirror symmetry about the conveniently calculated by the continuous summation of the
h’ = - h line in the (h, h’) plane:
derivative of the magnetization curve. For the ascending case
P(h, h’) = P(-h’, - h )
(3) this derivative that is the differential susceptibility can be
directly written as:
To calculate the actual magnetization by using the Everettintegrals, the starting point will be the demagnetized state dm(h)/dh = :jo dh’p(h, h’)
(5)
with M = 0 at H = 0, and each unidirectional cycle of field
change contributes to the magnetization change. Thus, any where m(h) = M ( h ) / M , is the reduced magnetization, M , is
complex magnetization process with any number of field the saturation magnetization, h,, is the last point of field
reversals can be followed by summing the consecutive reversal and p(h, h’) = P(h, h’)/M,. Written in this form it is
obviously seen, that the curves described by this expression
Everett-integrals:
possess the undesirable congruency property: the susceptibility
M ( H ) = E(-H,, Ho)/2
is dependent only on the external field limits of the integral.
E(&, H I ) . . . E(H,-,, H,)
E m , H ) Moreover this susceptibility will be always zero at the turning
point, when H , = h.
(4)
The congruency property can be eliminated by assuming
In this sum the so called wiping-out property of the calculation that the field derivative of the magnetization depends on the
procedure has to be taken into account. The wiping-out magnetization itself: an obvious non-linearity to be built into
property is consistent with and related to the experimental the expression. We may try to separate the magnetization
fact, that all the properties of the material connected to its dependence and the field dependence of the susceptibility into
magnetic history (e.g., domain structure) can be eliminated independent multiplicative factors. The non-zero value of the
by applying a high enough magnetic field. In case of the turning point susceptibility will be taken care of by a conPreisach-model it means that in the above natural sequence of stant, and we assume the following form:
Everett-integrals the conditions: lHol > lHzl > IH41 > . . .
and /H,I > /H,I > lH,I > . . . must be obeyed. Any time dm(h)/dh = R(m) ( P
dh’Q(h, h’)}
when an upper limit of field reversal: H k + 2 exceeds the
previous upper limit: H k , then the Everett-integrals: It is assumed that Q ( h , h’) does not depend on m any more.
E(Hk-I, H k ) + E(Hk, Hk+l)+ E(Hk+I, Hk+2) are to be /3 is the initial susceptibility fo the virgin state in the origin.
eliminated (wiped out) and replaced by E(H,_,, H k + > )in Experimental data suggest that the magnetization dependent
order to arrive at M ( H k + , ) properly. Similar procedure factor, R(m), should be an even function of m due to symapplies to the lower limits. Of course, any sequence of field metry, have a maximum at m = 0 and become zero when
reversals can be calculated with the Preisach-model, but in approach to saturation. The simplest form of R(m) with these
cases of wiping-out special care has to be taken.
properties can be given as:
Another interesting property of the Preisach-model: the so
(7)
called congruency property is related to the minor hysteresis R(m) = 1 - m2
With
the
residual
part,
Q
(
h
,
h’),
we
wish
to
satisfy
the
well
loops. It is easy to prove that for the same limits H I and H ,
=
+
+
+
+
+ jl0
Physica Scripta T25
163
The Bilinear Product Model of Hysteresis Phenomena
known symmetry property of the Preisach-function, as given
in eq. (3) and we also try to separate its two variables into
multiplicative factors. These requirements lead to the bilinear
form:
Q(h, h’) = Q(-h’, - h ) = f ( h ) f ( - h ’ )
(8)
Summarizing the assumptions of formalism of the bilinear
product model, the Preisach function takes the form:
pth, h’) = R(m) Q(h, h’) = (1 - m ’ > f ( h ) f ( - h ’ )
(9)
and the differential susceptibility, depending on the magnetic
field, on the magnetization and on the history of the actual
cycle of field variation, can be written as:
With this simplified form of R(m) we can calculate the
actual magnetization value starting from the virgin state
with an integration procedure similar to that of eq. (4).Now
the Everett integrals, E ( H , , H2), calculated with the kernel
p(h, h’) will be replaced by F ( H ,, If’), calculated similarly
with Q(h, h’), and since
( l / ( l - m 2 ) } dm(h)/dh = d{Arth (m)}/dh
(1 1)
we can write [13]:
m(H)
=
tanh {PH
+ F( - H,,H0)/2
+ F(H,, H , ) + . . .}
(12)
The wiping out property has to be taken into account, if
necessary.
3. Discussion
The modifications of the Preisach model, found reasonable
so far on grounds of simplicity of functional form, offer
some surprising opportunities of fitting them into a similarly
reasonable physical picture.
In cases of the description of saturation phenomena, like
in the theory of chemical reactions or simple demographic
processes, the logistic equation provides a satisfactory description of the phenomenon. Its simplest mathematical form
says that the rate of change of the quantity in question
(concentration, population, probability, etc.) is proportional
to the value of the quantity itself and to its distance from
the saturation value, that is with reduced values: dp/dt =
cp( 1 - p ) . In other words in a finite system the driving force
of the transition between the two possible states is proportional both to the amount of material in the new state as seeds
of further transitions, and to the amount in the old state as
room left until saturation for further transitions.
Applying this concept to the magnetization process, the
magnetic units already switched to the direction of the external field are in the new state, and those still pointing to the
opposite direction are in the old state. It is to be noted, that
the independent variable here is the magnetic field and not
the time as in the mentioned analogous examples. The time
dependence of the magnetization after a field change might
also be a logistic process, but it would involve entirely different transition states and processes and is not a subject of
this discussion. The portion of the magnetic units in the new
and old states can be given by m+ and m- , respectively, for
a uniaxial ferromagnetic material. Since m+ - m- = m
and m+ + m- = 1, one finds that the product correspond-
ing to p(1 - p ) in the logistic equation will be: m+m- =
(1 - m2)/4. Here is the reason, why the specific form of
R(m) = (1 - m’) was selected.
In a realistic case the magnetic material may have more
complicated than uniaxial symmetry. Then the magnetization
dependent factor, R(m)can be a rather complicated function
of book-keeping of the magnetization state. It will keep track
of the distribution of the magnetic units among the possible
states determined by the actual symmetry.
The next factor in the differential susceptibility is the
non-negative function f ( h ) , that can be interpreted as the
measure of the readiness of the magnetic units for switching
from the old, antiparallel to the new, parallel direction with
respect to the external field, h. It will be the probability of
switching when it is normalized so that its integral is unity.
Thusf (h)characterizes the coercive properties of the material
an can be given the name: coercivity function.
The integral off ( - h’) from the last point of field reversal
to the actual field value can be interpreted as the shortterm history of the material. This integral represents the
“mobilized and remembering” magnetic moments that play
decisive role in the magnetization process in the specific
model described here.
Recalling, that the Preisach function was the density function of elementary rectangular hysteresis loops with switching
field limits of h’ and h, the product off (- h’)f(h)has just that
meaning, since f ( - h ’ ) is the probability of switching to
antiparallel at h’ and f ( h ) is that of switching to parallel
direction at h with respect to the direction of the applied
magnetic field.
Equation (12) can describe the entire magnetic history
with the wiping-out property, however, without the congruency property.
The effect of the magnetization dependent function, R(m)
is rather obvious in eq. (12). The argument of the hyperbolic
tangent function is an unbounded expression of the applied
magnetic field, that may vary without limits. The introduction of the factor R(m) into the differential susceptibility
brings about a mathematical transformation and confines the
1.O TM
H
-1 .o
1.O
-1.01
Fig. I . Major and minor hysteresis loops calculated numerically by using
eq. (10) and a Gaussian form of f ( h ) with the values: H, = 0.25 and
H, = 0.25 in eq. (13).
Physica Scripta T25
164
Gyijrgy Kadar
values of the normalized magnetization between - 1 and + 1,
describing magnetic saturation in a natural way. Without
coercivity in the case of a paramagnetic, reversible (f(h) = 0)
magnetization curve will be described by the simple M / M s =
tanh ( P H ) function in a uniaxial material. In cases of more
complicated symmetry R(m) and the functional form of
approaching magnetic saturation will be different, but still
the magnetization curve will be confined by the introduction
of book-keeping of the magnetization state of the system.
In the argument of the saturating transformation function
the reversible part of the magnetization process is represented
by the additional term PH in addition to the non-reversible,
history dependent coercive terms.
Having an analytical or numerical form of the coercivity
function f ( h ) , any hysteresis loop can easily be calculated
with simple step by step numerical integration of eq. (6). The
plotted result of such a calculation is shown in Fig. 1 with an
analytical coercivity function of a simple Gaussian form:
The inverse problem of determiningf(h) from experimental
magnetization curves can be performed by discretizing the
magnetic field axis and finding the positive solutions of a
system of bilinear algebraic equations. With out specific
assumptions the process is straightforward and needs only
measured data of one branch of the main hysteresis loop [14].
The coercivity function determined from the experimental
data can be further analyzed and correlated to the microscopic
properties of the material. As the switching probability of
magnetic units it has to be connected with the magnetic and
geometrical structure of those units, i.e., particles, grains,
domains etc. With the understanding of such connections the
empirical data contained in a hysteresis curve may find their
basic meaning and application in a better established physical
picture of non-reversible transition processes.
4. Conclusion
The bilinear product model, a modification of the traditional
scalar Preisach model has been proposed for a two-state
Physica Scripra T25
material, offering a better physical understanding of saturating hysteresis phenomena.
The traditional, magnetic field dependent two-variable
Preisach-function has been replaced by a product of singlevariable functions. One of the functions of the product
explicitly depend on the magnetization itself showing a manifestation of the intrinsic non-linearity of the magnetization
curves. The proposed specific form, that is (1 - m’), suggests
a logistic type process of the magnetization change and determines the character of the non-linearity. The other factor of
the product, the field dependent residual part of the Preisachfunction can be represented by a bilinear form of a singlevariable, the coercivity function. The density function of
the rectangular elementary hysteresis loops, postulated by
Preisach finds its meaning in the product of probabilities of
magnetization switching as given by the coercivity function.
With this factorization the book-keeping of the actual
macroscopic magnetization state has been separated from the
statistical switching process of the magnetic units of a lower
microscopic level.
Acknowledgements
The author is indebted to Edward Della Torre for a long series of helpful
discussions, ideas and encouragement.
References
I . Preisach, F., Zeitschrift fur Physik 94, 277 (1935).
2. Everett, D. H., Trans. Faraday Soc. 51, 1551 (1955); Enderby, J. A.,
Trans. Faraday Soc. 51, 835 (1955).
3. Neel, L., Cahier de Physique, No. 12 (1942).
4. Biorci, G . and Pescetti, D., II Nuovo Cimento 7, 829 (1958).
5. Del Vecchio, R. M., IEEE Trans. Magn. MAG-16, 809 (1980).
6. Woodward, J. G. and Della Torre, E., J. Appl. Physics 31, 56 (1960);
J. Appl. Physics 32. 126 (1961).
7. Mayergoyz, 1. DD., J. A p d . Physics 57, 3803 (1985).
8. Della Torre, E., IEEE Trans. Audio Electroacoust. AU-14, 86 (1966).
9. K i d i r , G . , J. Appl. Physics 61, 4013 (1987).
10. Mayergoyz, I. D., IEEE Trans. Magn. MAG-22, 603 (1986).
11. Della Torre, E., J. Appl. Physics 61, 4016 (1987).
12. Bozorth. R. M.. “Ferromagnetism”, p. 549, D . van Nostrand Co.
Inc., New York, (1951).
13, KBdir, G, and Della
E,, IEEE Trans, Magn, MAG..23, 2820
(1987).
14. Kadar. G. and Della Torre, E., J. Appl. Physics 63, 3001 (1988).