IEEE SENSORS JOURNAL, VOL. 6, NO. 3, JUNE 2006 721
A 3-D Hybrid Jiles–Atherton/Stoner–Wohlfarth
Magnetic Hysteresis Model for Inductive
Sensors and Actuators
Panagiotis D. Dimitropoulos, Georgios I. Stamoulis, and Evangelos Hristoforou
Abstract—The Jiles–Atherton (JA) theory of hysteresis is cur-
rently used in the majority of commercial CAD tools, mainly due
to its implementation simplicity in fast and stable algorithms. The
JA model provides precise results in the case of isotropic, poly-
crystalline, multidomain magnetic devices, where flux-reversal
is governed by pinning mechanisms. Dynamic response of such
devices, including Eddy-current loss and magnetic resonance, can
also be accurately modeled. However, JA theory is not applied for
three-dimensional (3-D) magnetization simulations and does not
account for anisotropy that affects severely hysteresis curves of
single-domain, thin-film devices, which are usually incorporated
in miniature inductive sensors and actuators. In that case, the
Stoner–Wohlfarth (SW) theory can be applied, which, however,
does not account for dynamic response and incremental energy
loss. In this work, we employ a virtual 3-D anisotropy-field vector
calculated with SW theory that introduces magnetic feedback
to the classical equation of Paramagnetism, in order to derive
a proper 3-D “input” for the JA algorithm. This way, a hybrid
3-D JA/SW model is developed, which incorporates both models
into one single formulation, capable of modeling simultaneously:
1) temperature effects, 2) pinning and Eddy-current loss, 3) mag-
netic resonance, and 4) uniaxial anisotropy, the orientation of
which can be simulated to vary with time. The model that owns
a solid physical basis has been implemented in a computation-
efficient, stable algorithm capable of functioning with arbitrary
excitation-field input. The algorithm has been successfully applied
to model the behavior of a series of miniature Fluxgate mag-
netometers based on the Matteucci effect of thin glass-covered
magnetic wires.
Index Terms—Inductive sensor, magnetic hysteresis, magnetic
sensor.
I. INTRODUCTION
S
INGLE-DOMAIN magnetic devices such as ferromagnetic
films are widely used as: 1) magnetic flux concentrators
in read-write heads, in micro-actuators, and Hall devices [1],
[2], 2) flux modulators in fluxgate sensors [3]–[5], and 3) field
sensing heads in magnetoresistive and spin-valve sensors
Manuscript received April 19, 2005; revised November 11, 2005. The asso-
ciate editor coordinating the review of this paper and approving it for publication
was Prof. Fabien Josse.
P. D. Dimitropoulos is with the Institute of Microelectronics and Microsys-
tems, Swiss Federal Institute of Technology Lausanne, 1014 Lausanne, Switzer-
land (e-mail: panagiotis.dimitropoulos@epfl.ch; pdimi@pdimi.gr).
G. I. Stamoulis is with the Department of Computer and Telecommuni-
cations Engineering, University of Thessaly, 38221 Volos, Greece (e-mail:
georges@uth.gr).
E. Hristoforou is with the Department of Mining Engineering and Metal-
lurgy, National Technical University of Athens, 15777 Athens, Greece (e-mail:
eh@metal.ntua.gr).
Digital Object Identifier 10.1109/JSEN.2006.874454
[6]. Thus, the development of models with strict physical
background that enable fast and precise computation of mag-
netization of ferromagnetic devices in three dimensions is
important for sensor designers. The Jiles–Atherton (JA) theory
of hysteresis [7], [8] is known to serve well this purpose mainly
due to its implementation simplicity in fast and stable algo-
rithms. This explains why it is incorporated into the majority of
CAD software tools [9]. The theory assumes in-average con-
stant energy loss per unit magnetization increase attributed to
domain wall pinning. Thus, JA model provides accurate results
in the case of isotropic polycrystalline, multidomain magnetic
devices, where pinning mechanisms dominate during flux-re-
versal and magnetization increases in small steps. Thermal
and dynamic responses of such devices including Eddy-current
loss and magnetic resonance can also be precisely modeled
by JA theory [10], [11]. However, JA model, isotropic as it
is, fails in simulating magnetic hysteresis of single-domain
devices that show large Barkhausen jumps and is no more
governed by average pinning loss, but anisotropy. Moreover,
its classical formulation is one-dimensional (1-D). On the other
hand, Stoner–Wohlfarth (SW) theory [12] models accurately
the magnetization in three dimensions of magnetic devices
exhibiting anisotropy, such as thin ferromagnetic films. So, the
SW model is governed by a magnetostatic energy minimization
equation that is difficult to expand in order to simulate dy-
namic behavior, although not totally impossible [13]. Dynamic
behavior, however, is well treated with ordinary differential
equations (ODEs), and that is why the JA theory seems to
prevail in this aspect. To our knowledge, there are generally
three ways to introduce anisotropy effects into the JA model,
namely: 1) by using a tensor instead of a scalar quantity for
the JA pinning loss constant [14], 2) by employing additional
terms of effective field [15] apart from Weiss molecular-field
term [16], and 3) by modifying the anhysteretic magnetization
function employed in the JA model introducing the effects of
anisotropy energy [17], [18]. Particularly, in the latter case, the
modified anhysteretic magnetization function, also called the
total anhysteretic [12], the final JA simulation results make best
fit for the corresponding results of SW modeling.
In this work, we propose a novel treatment for the problem of
introducing anisotropy effects into the JA model by developing a
new iterative algorithm for fast SW hysteresis curve calculation.
Instead of using an anhysteretic magnetization function to act as
“input” for the classical JA algorithm, we let hysteresis curves
derived with SW theory to serve this purpose. The “input” of the
JA algorithm is not anhysteretic anymore, but is composed by
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