AN ENERGY-BASED HYSTERESIS MODEL FOR
MAGNETOSTRICTIVE TRANSDUCERS
F.T. Calkins
Department of Aerospace Engineering
and Engineering Mechanics
Iowa State University
Ames, IA 50011
tcalkins@iastate.edu
R.C. Smith
Department of Mathematics
Iowa State University
Ames, IA 50011
rsmith@iastate.edu
A.B. Flatau
Department of Aerospace Engineering
and Engineering Mechanics
Iowa State University
Ames, IA 50011
abf@iastate.edu
Abstract
This paper addresses the mo deling of hysteresis in magnetostrictive transducers. This is
considered in the context of control applications which require an accurate characterization
of the relation between input currents and strains output by the transducer. This relation
typically exhibits signicant nonlinearities and hysteresis due to inherent properties of mag-
netostrictive materials. The characterization considered here is based up on the Jiles-Atherton
mean eld model for ferromagnetic hysteresis in combination with a quadratic moment ro-
tation model for magnetostriction. As demonstrated through comparison with exp erimental
data, the magnetization model very adequately quanties b oth ma jor and minor lo ops under
various op erating conditions. The combined model can then be used to accurately characterize
output strains at moderate drive levels. The advantages to this model lie in the small number
(six) of required parameters and the exibility it exhibits in a variety of operating conditions.
i
1 Intro duction
This paper addresses the mo deling of hysteresis in magnetostrictive transducers. The capabil-
ities for actuation and sensing in such transducers are provided by the dual magnetostrictive
eects in the core material: (i) the application of a magnetic eld generates strains in the
material and (ii) material stresses yield measurable magnetic eects. One core magnetostric-
tive material which has proven very eective at room temp eratures and nominal operating
conditions is Terfenol-D (see [1, 2] for descriptions of the material and its capabilities). Due
to the magnitude of the strains and forces generated by the material, Terfenol-D transducers
have been employed as ultrasonic transducers, sonar pro jectors and provide the capability for
controlling vibrations in heavy structures and industrial machinery.
Several prop erties inherent to magnetostrictive materials must be addressed when design-
ing systems which employ them. The rst concerns the hysteresis and nonlinear dynamics
exhibited by the materials. This is due to inherent magnetic prop erties of the materials and
is particularly pronounced at higher drive levels. It is also well documented that Terfenol-D
performance is highly sensitive to op erating conditions such as temperature, mechanical pre-
stress, magnetic excitation (bias and AC amplitude), frequency and external load [3, 4, 5].
Several of these asp ects (e.g., prestress and external loads) involve system aspects external to
the core Terfenol-D material which makes the extrap olation of results from isolated laboratory
samples to actual transducer design dicult and motivates consideration of the transducer as
a whole.
Accurate modeling of transducer dynamics is necessary to take advantage of the full ca-
pabilities of the materials and to provide the ability for tailoring the performance of the
transducers by mo difying easily adjusted operating conditions. To attain these ob jectives, the
model must accurately characterize b oth ma jor (symmetric) and minor (nested and asymmet-
ric) hysteresis loops as well as constitutive nonlinearities. The mo del must also incorporate
the sensitivities with resp ect to op erating conditions and be in a form amenable for eventual
incorporation in models for underlying structural systems. Finally, the model must be suit-
able for controller design in the sense that it is ecient to implement and characterizes all
dynamics which may b e specied by the control law. For example, a mo del which charac-
terizes ma jor loops but not minor ones would b e less useful in a feedback control law which
cannot dierentiate b etween the two.
The model we consider is obtained through the extension of the ferromagnetic mean eld
model of Jiles and Atherton [6, 7, 8, 9 ] to magnetostrictive transducers. This provides a
characterization for the inherent hysteresis which is based up on the anhysteretic magnetization
along with reversible and irreversible domain wall movements in the material. When coupled
with nonlinear strain/magnetization relations, this yields a mo del which characterizes strain
outputs in terms of input currents to the driving solenoid. Minor lo ops are incorporated
through the enforcement of closure conditions.
With regard to design criteria, this mo del is currently constructed for a transducer with
quasi-static input and xed temp eratures (these are commonly employed conditions for initial
transducer characterization). The capability for having dierent prestresses and variable input
magnitudes to the driving solenoid are included in the mo del and demonstrated through
comparison with exp erimental data. The advantages of this approach lie in the accurate
ts attainable in the considered regimes with a small number (six) of physical parameters
1
to be identied through least squares techniques. This provides the metho d with signicant
exibility and low computational overhead. The mo del is also in a form which can b e extended
to variable temp erature and frequency regimes and can b e incorp orated in a large variety of
structural mo dels (e.g., see [10, 11]). As a result, it shows great promise for use in transducer
design for precision positioning and structural and structural acoustic controllers [12].
To place this modeling approach in p erspective, it is useful to briey summarize existing
techniques for characterizing magnetostrictive transducers. For initial applications, linear
eld/magnetization relations were used to approximate the transducer dynamics [1, 13]. While
this approach is reasonable at low drive levels, it is inaccurate at moderate to high input
levels due to inherent hysteresis and material nonlinearities. In this latter regime, various
phenomenological or empirical techniques, including Preisach mo dels, have b een employed
to quantify the input/output relations [14, 15 ]. Phenomenological approaches circumvent
unmodeled or unknown physical mechanisms and have the advantage of generality. While
some connections have b een made between underlying physical processes and Preisach mo dels
[16], this genre of mo del typically provides less insight into physical dynamics than a mo del
developed from physical principles. Furthermore, such empirical models generally require a
large number of nonphysical parameters and are not easily adapted to changing operating
conditions. This increases implementation time [17] and will limit exibility if employed in a
control law.
A typical magnetostrictive transducer is describ ed in Section 2. This illustrates the system
being modeled and indicates design issues which must be incorporated in the mo del. The
energy-based model is discussed in Section 3 and the applicability of the mo del in a variety
of exp erimental settings is presented in Section 4. These results illustrate the accuracy and
exibility of the model at xed temperatures and low frequencies and indicate the extensions
necessary for use in other regimes.
2 Magnetostrictive Transducers
The issues which must be addressed when developing a comprehensive model are illustrated
in the context of the transducer depicted in Figure 1. As detailed in [14 ], this construction is
typical for actuators currently employed in many structural applications; hence it provides a
template for the development of mo dels which will ultimately enhance design and p erformance.
Details regarding the sp ecic exp erimental setup used here are provided in Section 4.
From a design persp ective, the transducer can b e considered as the entire system which
facilitates the utilization of the magnetostrictive core for applications. For mo deling purp oses,
the key components are the magnetostrictive core, a DC magnetic circuit, a driving AC
circuit and a prestress mechanism. The magnetostrictive material used in the transducer for
the experiments rep orted in Section 4 was comprised of Terfenol-D, Tb
0
:
3
Dy
0
:
7
Fe
1
:
9
, while the
driving AC magnetic eld was generated by a surrounding wound wire solenoid. As illustrated
by the experimental data plotted in Figure 2, the relationship between the applied eld
H
and
resulting magnetization
M
exhibits signicant hysteresis while the relationship between the
magnetization and strain
e
is highly nonlinear. Moreover, the strains in an unbiased rod are
always p ositive since the rotation of moments in resp onse to an applied eld always pro duce
an increase in length. To attain bidirectional strains, a DC bias is provided by the enclosing
2
cylindrical magnet (alternatively, a biasing DC current could be applied to the solenoid).
Finally, the prestress b olt further aligns the orientation of magnetic moments and maintains
the rod in a constant state of compression.
To fully utilize the transducer for structural applications and eventual controller design,
it is necessary to characterize the relationship between the current
I
applied to the solenoid,
the resulting eld
H
, the asso ciated magnetization
M
and nally, the generated strains
e
. A
characterization based up on the Jiles-Atherton ferromagnetic hysteresis mo del is presented in
the next section.
Cylindrical Permanent Magnet
Wound Wire Solenoid
Spring
Terfenol-D Rod
Washer
Steel Casing
Figure 1.
Cross section of a typical Terfenol-D magnetostrictive transducer.
−6 −4 −2 0 2 4 6
x 10
4
−6
−4
−2
0
2
4
6
x 10
5
Magnetic Field (H)
Magnetization (M)
−8 −6 −4 −2 0 2 4 6 8
x 10
5
0
0.2
0.4
0.6
0.8
1
1.2
x 10
−3
Magnetization (M)
Strain (e)
(a) (b)
Figure 2.
Relationship in experimental data between (a) the magnetic eld
H
and the
magnetization
M
, and (b) the magnetization
M
and the generated strains
e
.
3
3 Domain Wall Dynamics
The transducer model described here is based upon the theory that magnetization in fer-
romagnetic materials is due to the realignment of magnetic moments within the material.
Such materials exhibit the property that at temp eratures b elow the Curie point, moments are
highly aligned in regions termed domains (the reader is referred to [6, 18] for further discussion
regarding the exp erimental verication of domain properties). The reorientation of moments
can o ccur b oth in bulk within the domains or within transition regions, termed domain walls,
between domains.
For a material which is defect free, the former mechanism leads to anhysteretic (hystere-
sis free) behavior which is conservative and hence reversible. Such a situation is idealized,
however, since defects are unavoidable (e.g., carbides in steel) and in many cases, incorpo-
rated in the material to attain the desired stoichiometry (e.g., second-phase materials such as
Dysprosium in Terfenol-D). These defects or inclusions provide pinning sites for the domain
walls due to the reduction in energy which occurs when the domain wall intersects the site.
For low magnetic eld variations about some equilibrium value, the walls remain pinned and
the magnetization is reversible. This motion becomes irreversible at higher eld levels due to
wall intersections with remote inclusions or pinning sites. Note that pinning eects lead to
phenomena such as the Barkhausen discontinuities observed in exp erimental magnetization
data [6, 18 ]. The energy loss due to transition across pinning sites also provides the main
mechanism for hysteresis in ferromagnetic materials.
Magnetostriction
The mo del presented here ultimately provides a relationship between the current
I
input
to the solenoid and the strain
e
output by the transducer. As a rst step, we characterize the
magnetostriction which results at a given magnetization level. The magnetostriction
d`
`
indicates the relative change in length of the material from the ordered but unaligned state
to the state in which domains are aligned. While the magnetostriction do es not quantify DC
eects, the eects of domain order, or thermal eects, it do es provide a measure of the strains
generated in a Terfenol transducer.
As detailed in [6], consideration of the potential energy for the system yields
=
3
2
s
M
2
s
M
2
(1)
as a rst approximation to the relationship b etween the magnetization and magnetostriction.
Here
M
s
and
s
respectively denote the saturation magnetization and saturation magnetostric-
tion. For an isolated Terfenol-D sample,
M
s
represents the magnetization required to rotate
all moments and has been observed to have the approximate value
M
s
7
:
9
10
5
A=m
[19].
This parameter has a similar interpretation in the full transducer mo del but will b e shown in
the examples of the next section to have the slightly smaller value of
M
s
= 7
:
65
10
5
A=m
.
This illustrates the necessity of estimating such parameters for the specic transducer under
consideration. The value of
s
depends upon the initial orientation of moments and hence
upon the applied prestress. In the absence of applied stresses and under the assumption of
a cubic anisotropy mo del,
s
can be dened in terms of the independent saturation mag-
netostrictions
100
and
111
in the
h
100
i
and
h
111
i
directions, resp ectively. As detailed in
4
