IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 2, MARCH 2000 429
Energy-Based Hysteresis Model for MagnetostrictiveTransducers
F. T. Calkins, R. C. Smith, and A. B. Flatau
Abstract—This paper addresses the modeling of hysteresis inmagnetostrictive transducers in the context of control applica-tions that require an accurate characterization of the relationbetween input currents and strains output by the transducer. Thisrelation typically exhibits significant nonlinearities and hysteresisbecause of inherent properties of magnetostrictive materials. Thecharacterization considered here is based on the Jiles–Athertonmean field model for ferromagnetic hysteresis in combinationwith a quadratic moment rotation model for magnetostriction.As demonstrated by comparison with experimental data, themagnetization model very adequately quantifies both major andminor loops under various operating conditions. The combinedmodel can then be used to accurately characterize output strainsat moderate drive levels. The advantages of this model lie in thesmall number (six) of required parameters and its flexibility undera variety of operating conditions.
Index Terms—Hysteresis model, magnetorestrictive transducer.
I. INTRODUCTION
T HIS PAPER addresses the modeling of hysteresis inmagnetostrictive transducers. The capabilities for actua-
tion and sensing in such transducers are provided by the dualmagnetostrictive effects in the core material: 1) the applicationof a magnetic field generates strains in the material and 2)material stresses yield measurable magnetic effects. One coremagnetostrictive material which has proven very effectiveat room temperatures and nominal operating conditions isTerfenol-D (see [1] and [2] for descriptions of the material andits capabilities). Due to the magnitude of the strains and forcesgenerated by the material, Terfenol-D transducers have beenemployed as ultrasonic transducers and sonar projectors, andthey provide the capability for controlling vibrations in heavystructures and industrial machinery.
Several properties inherent to magnetostrictive materialsmust be addressed when designing systems that employ them.The first concerns the hysteresis and nonlinear dynamics
Manuscript received September 20, 1997; revised August 4, 1999. The workof F. T. Calkins and A. B. Flatau was supported in part by Graduate Student Re-search Program Grant NGT-51254, NASA Langley Research Center, R. Silcox,Technical Advisor, and National Science Foundation Young Investigator AwardCMS 9457288. The work of R. C. Smith was supported in part by the Air ForceOffice of Scientific Research under Grant AFOSR F49620-95-1-0236.
F. T. Calkins is with the Boeing Company, Phantom Works Flight/Con-figuration Technology, Seattle WA 98124-2499 USA (e-mail: [email protected]).
R. C. Smith is with the Center for Research in Scientific Computation, De-partment of Mathematics, North Carolina State University, Raleigh, NC 27695USA (e-mail: [email protected]).
A. B. Flatau is with the Department of Aerospace Engineering and En-gineering Mechanics, Iowa State University, Ames, IA 50011 USA (e-mail:[email protected]).
Publisher Item Identifier S 0018-9464(00)01831-8.
exhibited by the materials. This is due to inherent magneticproperties of the materials and is particularly pronounced athigher drive levels. It is also well documented that Terfenol-Dperformance is highly sensitive to operating conditions such astemperature, mechanical prestress, magnetic excitation (biasand AC amplitude), frequency, and external load [3]–[5]. Sev-eral of these aspects (e.g., prestress and external loads) involvesystem aspects external to the core Terfenol-D material whichmakes the extrapolation of results from isolated laboratorysamples to actual transducer design difficult and motivatesconsideration of the transducer as a whole.
Accurate modeling of transducer dynamics is necessary totake advantage of the full capabilities of the materials and toprovide the ability for tailoring the performance of the trans-ducers by modifying easily adjusted operating conditions. Toattain these objectives, the model must accurately characterizeboth major (symmetric) and minor (nested and asymmetric)hysteresis loops as well as constitutive nonlinearities. Themodel must also incorporate the sensitivities with respect tooperating conditions and be in a form amenable for eventualincorporation in structural system models. Finally, the modelmust be suitable for controller design in the sense that it isefficient to implement and characterizes all dynamics whichmay be specified by the control law. For example, a model thatcharacterizes major loops but not minor ones would be lessuseful in a feedback control law which cannot differentiatebetween the two.
The model we consider is obtained through the extensionof the ferromagnetic mean field model of Jiles and Atherton[6]–[9] to magnetostrictive transducers. This provides a charac-terization for the inherent hysteresis which is based upon the an-hysteretic magnetization along with reversible and irreversibledomain wall movements in the material. When coupled withnonlinear strain/magnetization relations, this yields a model thatcharacterizes strain outputs in terms of input currents to thedriving solenoid. Minor loops are incorporated through the en-forcement of closure conditions.
With regard to design criteria, this model is currentlyconstructed for a transducer with quasi-static input and fixedtemperatures (these are commonly employed conditions forinitial transducer characterization). The capability for havingdifferent prestresses and variable input magnitudes to thedriving solenoid are included in the model and demonstratedthrough comparison with experimental data. The advantages ofthis approach lie in the accurate fits attainable in the consideredregimes with a small number (six) of physical parameters tobe identified through least squares techniques. This providesthe method with significant flexibility and low computational
430 IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 2, MARCH 2000
Fig. 1. Cross section of a typical Terfenol-D magnetostrictive transducer.
overhead. The model is also in a form which can be extendedto variable temperature and frequency regimes and can beincorporated in a large variety of structural models (e.g.,[10] and [11]). As a result, it shows great promise for use intransducer design for precision positioning as well as structuralcontrollers and structural acoustic controllers [12].
To place this modeling approach in perspective, it is usefulto summarize briefly the existing techniques for character-izing magnetostrictive transducers. For initial applications,linear field/magnetization relations were used to approximatethe transducer dynamics [1], [13]. While this approach isreasonable at low drive levels, it is inaccurate at moderateto high input levels due to inherent hysteresis and materialnonlinearities. In this latter regime, various phenomenologicalor empirical techniques, including Preisach models, havebeen employed to quantify the input–output relations [14],[15]. Phenomenological approaches circumvent unmodeledor unknown physical mechanisms and have the advantage ofgenerality. While some connections have been made betweenunderlying physical processes and Preisach models [16], thisgenre of model typically provides less insight into physicaldynamics than a model developed from physical principles.Furthermore, such empirical models generally require a largenumber of nonphysical parameters and are not easily adapted tochanging operating conditions. This increases implementationtime [17] and will limit flexibility if employed in a control law.
A typical magnetostrictive transducer is described in SectionII. This illustrates the system being modeled and indicates de-sign issues which must be incorporated in the model. The en-ergy-based model is discussed in Section III, and the appli-cability of the model in a variety of experimental settings ispresented in Section IV. These results illustrate the accuracyand flexibility of the model at fixed temperatures and low fre-quencies and indicate the extensions necessary for use in otherregimes.
We point out that the primary focus in the model develop-ment centers on the characterization of strains generated by theTerfenol-D rod in response to inputs from the remaining trans-ducer components (e.g., magnetic fields, stresses). Hence themodel is applicable for a variety of transducer configurations inwhich strains are generated by axial magnetostrictive elements.Because of the coupling between the various components of thetransducer and the magnetoelastic properties of the Terfenol-D
rod, parameters in the model reflect attributes of both the trans-ducer material and auxiliary transducer components. This im-plies that while the model can be applied in a variety of ap-plications, parameters must be identified for a given transducerconfiguration. Furthermore, this indicates that while trends maybe noted when comparing parameter values for the transducermodel with those for the isolated materials, direct comparisonsmay be difficult due to the influence of the coupled transducercomponents.
II. M AGNETOSTRICTIVETRANSDUCERS
The issues which must be addressed when developing a com-prehensive model are illustrated in the context of the transducerdepicted in Fig. 1. As detailed in [14] and [18], this constructionis typical for actuators currently employed in many structuralapplications; hence it provides a template for the development ofmodels which will ultimately enhance design and performance.Details regarding the specific experimental setup used here areprovided in Section IV.
From a design perspective, the transducer can be consideredas the entire system which facilitates the utilization of the mag-netostrictive outputs for applications. For modeling purposes,the key components are the magnetostrictive core, a dc mag-netic circuit, a driving ac circuit, and a prestress mechanism.The magnetostrictive material used in the transducer for the ex-periments reported in Section IV was comprised of Terfenol-D,Tb Dy Fe , while the driving ac magnetic field was gen-erated by a surrounding wound wire solenoid. As illustrated bythe experimental data plotted in Fig. 2, the relationship betweenthe applied field and resulting magnetization exhibits sig-nificant hysteresis while the relationship between the magneti-zation and strain is highly nonlinear. Moreover, the strains inan unbiased rod are always positive since the rotation of mag-netic moments in response to an applied field always producean increase in length. To attain bidirectional strains, a dc biasis provided by the enclosing cylindrical magnet (alternatively,a biasing dc current could be applied to the solenoid). Finally,the prestress bolt (not shown) and spring washers further alignthe orientation of magnetic moments and maintains the rod in aconstant state of compression.
To fully utilize the transducer for structural applicationsand eventual controller design, it is necessary to characterize
CALKINS et al.: ENERGY-BASED HYSTERESIS MODEL 431
(a)
(b)
Fig. 2. Relationship in experimental transducer data between: (a) themagnetic fieldH and the magnetizationM and (b) the magnetizationM andthe generated strainse.
the relationship between the currentapplied to the solenoid,the resulting field , the associated magnetization , andfinally, the generated strains. A characterization based uponthe Jiles–Atherton ferromagnetic hysteresis model is presentedin the next section.
III. D OMAIN WALL MODEL
The model described here is based on an energy formulationof the magnetoelastic properties of magnetostrictive materialswhen they are employed in a transducer. This formulation isbased on the observation that Terfenol-D is ferromagnetic attemperatures below the Curie temperature and hence exhibitsa well-defined domain structure. The application of an externalmagnetic field causes the domain magnetic moments to rotate
which produces a subsequent change in the bulk magnetizationand magnetostriction.
For a material that is defect free, the former mechanism leadsto anhysteretic (hysteresis free) behavior that is conservative andhence reversible. Such a situation is idealized, however, sincedefects are unavoidable (e.g., carbides in steel), and in manycases defects are specifically incorporated in the material to at-tain the desired stoichiometry (e.g., Dysprosium in Terfenol-D).These defects or inclusions provide pinning sites for the domainwalls due to the reduction in energy which occurs when the do-main wall intersects the site. For low magnetic field variationsabout some equilibrium value, the walls remain pinned and themagnetization is reversible. This motion becomes irreversibleat higher field levels due to wall intersections with remote in-clusions or pinning sites. Note that pinning effects lead to phe-nomena such as the Barkhausen discontinuities observed in ex-perimental magnetization data [6], [19]. The energy loss due totransition across pinning sites also provides the main mecha-nism for hysteresis in ferromagnetic materials.
A. Magnetostriction
The model presented here ultimately provides a relationshipbetween the current input to the solenoid and the strainoutput by the transducer. As a first step, we characterize themagnetostriction which results at a given magnetization level.The magnetostriction indicates the relative change inlength of the material from the ordered, but unaligned state, tothe state in which domains are aligned. While the magnetostric-tion does not quantify the effects of domain order or thermaleffects, it does provide a measure of the strains generated in aTerfenol-D transducer.
As detailed in [6], consideration of the potential energy forthe system yields
(1)
as a first approximation to the relationship between the mag-netization and magnetostriction in isotropic materials. Hereand respectively denote the saturation magnetization and sat-uration magnetostriction. For an isolated Terfenol-D sample,
represents the magnetization required to rotate all momentsand has been observed to have the approximate value
A/m [20]. This parameter has a similar interpreta-tion in the full transducer model but will be shown in the ex-amples of the next section to have the slightly smaller value of
A/m. This illustrates the necessity of esti-mating such parameters for the specific transducer under consid-eration. The value of depends upon the initial orientation ofmoments and hence upon the applied prestress. In the absence ofapplied stresses and under the assumption of a cubic anisotropymodel, can be defined in terms of the independent saturationmagnetostrictions and in the and direc-tions, respectively. As detailed in [6], under the assumption thatthe material contains a large number of domains and has no pre-ferred grain orientation, averaging of domain effects yields theexpression
432 IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 2, MARCH 2000
for the total saturation magnetostriction (typical satura-tion values for Terfenol-D are and
). As will be noted in the examples of thenext section, this saturation value is highly dependent upon theoperating conditions (e.g., applied prestress) and the parameter
in this model must be estimated through least squarestechniques for the specific conditions under consideration.
We point out that in the absence of an applied prestress,Terfenol-D is highly anisotropic and the quadratic model(1) provides a poor approximation for the magnetostrictiongenerated by an applied magnetization. At sufficiently highprestresses, however, the stress anisotropies will dominatecrystalline anisotropies and the model (1) is adequate. Thedetermination of prestress levels at which this occurs is de-pendent upon the configuration of easy axes in the material,the operating temperature and the specific stoichiometry underconsideration. For example, the computations [6, pp. 126, 410]employ the relation whereas other easy axisconfigurations yield the relation . Furthermore, valuesfor the crystal anisotropy constant are highly dependentupon the temperature and specific stoichiometry of Terfenol-Dbeing employed. Given this variability in underlying assump-tions, computed values of the stress levels at which stressanisotropies begin to dominate crystalline anisotropies rangefrom psi (6.25 MPa) [6, p. 410] to ksi (18.75MPa). For the transducer under consideration, prestresses inexcess of 1 ksi are typically employed.
Given the difficulty associated with determining the prestresslevels at which the quadratic model (1) becomes inaccurate,we considered this issue empirically through considerationof the model fit to experimental strain data measured for thetransducer. As illustrated in Section IV, the quadratic lawprovides an adequate model at low to moderate drive levelsfor the prototypical transducer with 1.0 and 1.3 ksi prestresses.While this provides an empirical motivation for the quadraticlaw at commonly employed prestresses, the material may stillexhibit significant crystalline anisotropies which are neglectedin the model. Hence for certain applications, anisotropicmagnetostriction models may be required. Furthermore, forhigher drive levels and frequencies along with variable tem-perature and stress conditions, the model must be extended toinclude mechanisms such as stress effects [3]–[5]. This can beaccomplished through the incorporation of stress dependencein and the use of higher order magnetostrictive modelsas discussed in [21]. Alternatively, higher order effects andmagnetostrictive hysteresis can be incorporated through anenergy formulation as detailed in [9]. Finally, the effects ofmagnetomechanical coupling and mechanical resonancesmust be incorporated in various operating regimes. Hence thiscomponent of the transducer model should be extended asdictated by operating conditions.
We next turn to the characterization of the magnetizationin terms of the input current. To accomplish this, it is necessaryto quantify the effective field associated with the magneticmoments in the core material, the anhysteretic magnetization
, the reversible magnetization , and the irreversiblemagnetization .
B. Effective Magnetic Field
In general, the effective field is dependent upon the magneticfield generated by the solenoid, magnetic domain interactions,crystal and stress anisotropies, and temperature. For this model,we are considering the case of fixed temperature and compres-sive prestresses in excess of 1.0 psi. Under these conditions, theeffective magnetic field is modeled by
whereis the field generated by a solenoid withturnsper unit length;quantifies the field due to magnetic interactionsbetween moments;is the stress-dependent field component.
The parameter quantifies the amount of domain interactionand must be identified for a given system. The field componentdue to the constant applied stresses can be quantified throughthermodynamic laws to obtain
(see [9] and [21] for details). Here is the free space perme-ability, and the subscript denotes constant temperature in de-grees Kelvin. Note that with the approximation (1) for, theeffective field can be expressed as
where .
C. Anhysteretic Magnetization
The anhysteretic magnetization is computed through consid-eration of the thermodynamic properties of the magnetostrictivematerial. Under the assumption of constant domain density,Boltzmann statistics can be employed to yield the expression
(2)
where coth is the Langevin function. The con-stant is given by where is Boltzmann’sconstant and represents the Boltzmann thermal energy. Wepoint out that cannot directly be computed for a transducer dueto the fact that is unknown. Hence it is treated as a parameterto be identified for the system. We also note that this expres-sion for is valid only for operating conditions under which
is valid. For example, if prestresses are sufficiently smallso that crystal anisotropies are significant, the expression mustbe modified to incorporate the differing anisotropy energies inthe different directions. One approach to modeling the effectsof anisotropy is given in [22].
D. Irreversible, Reversible, and Total Magnetization
The anhysteretic magnetization incorporates the effects ofmoment rotation within domains but does not account for do-main wall bending and translation. As noted previously, the con-sideration of domain wall energy yields additional reversible
CALKINS et al.: ENERGY-BASED HYSTERESIS MODEL 433
TABLE IPHYSICAL PROPERTIES AND EFFECTS OF
MODEL PARAMETERS�; a; k; c; M ; � ;THE PARAMETER ~� IS THEN GIVEN
BY ~� = �+ (9=2)(� � =� M ) WHERE� IS THE APPLIED PRESTRESS
and irreversible components to the magnetization. The consid-eration of energy dissipation due to pinning and unpinning ofdomain walls at inclusions yields the expression
(3)
for the differential susceptibility of the irreversible magnetiza-tion curve [7], [21]. The constant ,where is the average density of pinning sites, is the av-erage energy for 180walls, is a reversibility coefficient, and
is the magnetic moment of a typical domain, provides a mea-sure for the average energy required to break a pinning site. Theparameter is defined to have the value1 whenand 1 when to guarantee that pinning alwaysopposes changes in magnetization. In applications,can be di-rectly determined from the magnetic field data whileis iden-tified for the specific transducer and operating conditions.
The reversible magnetization quantifies the degree to whichdomain walls bulge before attaining the energy necessary tobreak the pinning sites. As derived in [7], to a first approxima-tion, the reversible magnetization is given by
(4)
The reversibility coefficient can be estimated from the ratioof the initial and anhysteretic differential susceptibilities [8] orthrough a least squares fit to data. Properties of all the modelparameters are summarized in Table I.
The total magnetization is then given by
(5)
with and defined by (3) and (4) and the anhys-teretic magnetization given by (2). The full time-dependentmodel leading from input currents to output magnetization issummarized in Algorithm 1. When combined with (1), thisprovides a characterization of the output strains in terms of thecurrent input to the solenoid. Note that this model is validfor fixed temperature and quasistatic operating conditions. The
Fig. 3. Closure requirements for minor loops.
TABLE IIESTIMATED MAGNETIZATION PARAMETERS FOR THETRANSDUCER WITH
PRESTRESSES OF1.3 AND 1.0 ksi
extension to more general operating conditions will involve thepreviously mentioned modifications to the effective field.
i)
ii)
iii)
iv)
v)
vi)
Algorithm 1: Time-dependent model quantifying the outputmagnetization in terms of the input current . The pa-rameter is given by whereis the applied prestress.
434 IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 2, MARCH 2000
(a) (b)
(c) (d)
Fig. 4. Experimental data (dashed line) and magnetization model dynamics (solid line) for multiple drive and prestress levels: (a) three drive levels with 1.0 ksiapplied stress; (b) magnified view of 1.0-ksi case; (c) two drive levels with 1.3-ksi applied stress; and (d) magnified view of 1.3-ksi case.
E. Asymmetric Minor Loops
The final aspect that we consider here concerns the modifi-cation of the model to incorporate minor (asymmetric) loops.Such loops occur when the sign of is reversed for a tra-jectory lying within the interior of the major loop. To preserveorder in the sense that forward paths do not intersect, it is nec-essary that minor loops close. The model (5) can be employedfor the first half of the minor loop but does not ensure closure.This property is incorporated in the model through the consid-eration of a working volume and volume fraction for either themagnetization or the reversible and irreversible components ofthe magnetization.
To illustrate the first case, we let and , respectively,denote the times when the minor loop starts, when it turns dueto a change in the sign of and when it closes (Fig. 3).The corresponding values of the magnetic field and magneti-zation are and .Note that in order to guarantee closure of the minor loop, it is
necessary to require that and .Direct integration of (5) yields
which in general will not be equal to . To attain closure,we define
for . The magnetization valuesand are computed using (5). Through the inclusion ofthis volume fraction
(6)
the magnetization is forced to satisfy the closure property. A similar formulation of volume fractions
CALKINS et al.: ENERGY-BASED HYSTERESIS MODEL 435
(a)
(b)
Fig. 5. Experimental data (dashed line) and quadratic magnetostriction modeldynamics (solid line). (a) Low drive level. (b) High drive level.
for the component reversible and irreversible magnetizations isgiven in [23] while extensions of the model to accommodatemore complex anhysteretic effects can be found in [24].
The viability of the model with minor loops closed via (6)is illustrated in the next section. We note that for the operatingconditions targeted in this paper, the model accurately charac-terizes the transducer response including both major loops andnested minor loops.
IV. M ODEL FITS TO EXPERIMENTAL DATA
The model fits to experimental transducer data using the rela-tions summarized in Section III are presented here. Following adescription of the experimental transducer, two cases are consid-ered. The first illustrates the performance of the magnetizationand magnetostriction models under various drive levels with a1.3 ksi prestress applied to the Terfenol-D rod. Included in these
results are model fits to data which contains minor loops. Thesecond case illustrates the performance of the model for a pre-stress of 1.0 ksi. As discussed in the last section, the stress-dom-inated anisotropy model for the magnetization is adequate inboth cases for the considered device. Taken in concert, theseexamples illustrate the accuracy and flexibility of the magne-tization model for a range of drive levels, magnetic biases, andprestresses for quasi-static operating conditions at fixed temper-ature. The quadratic magnetostriction model is also accurate atlow to moderate drive levels but must be extended to incorpo-rate the hysteresis and saturation present at high drive regimes.
A. Experimental Transducer
The experimental data reported here was collected from abroad-band Terfenol-D transducer developed at Iowa State Uni-versity. The nominal resonance range was designed for struc-tural applications (1–10 kHz). Furthermore, the transducer wasdesigned to produce an output free from spurious resonancesand to permit adjustable prestress and magnetic bias.
The Terfenol-D (Tb Dy Fe ) rod employed in the trans-ducer had a length of 115 mm and a 12.7 mm diameter. The rodwas placed inside two coils consisting of an inner single layer110-turn pickup coil and a multilayer 800-turn drive coil. A cur-rent control amplifier (Techron 7780) provided the input to thedrive coil to produce an applied ac magnetic field and dc bias asnecessary. The reference signal to this amplifier was provided bya Tektronix spectrum analyzer and the applied magnetic fieldgenerated by the drive coil had a frequency of 0.7 Hz and mag-nitude up to 5.6 kA/m (700 Oe) per ampere. The pickup coilwas used to measure the induced voltage from which the timerate change of the magnetic inductionwas computed usingthe Faraday–Lenz Law.
A cylindrical permanent magnet surrounding the coils pro-vided the capability for generating additional dc bias if neces-sary and is a component in the flux path. This permanent magnetwas constructed of Alnico V and was slit to reduce eddy currentlosses. Note that for the experiments reported here, biases gen-erated in this manner were unnecessary and the reported datais unbiased (i.e., the permanent magnet was demagnetized). Fi-nally, mechanical prestresses to the rod were generated by com-pressing Belleville washers at one end of the rod by tighteninga threaded prestress bolt at the opposite end of the transducer.
The measurable quantities from the transducer included thecurrent and voltage in the drive coil, the voltage induced in thepickup coil, and the mechanical output. To quantify the mechan-ical output, a Lucas LVM-10 linear variable differential trans-former based upon changing reluctance was used to measurethe displacement of the transformer output interface connection.Temperature was maintained within 5C of the ambient temper-ature (23 C) by monitoring two thermocouples attached to theTerfenol-D sample.
B. Parameter Estimation
The use of the magnetization and magnetostriction modelsto characterize transducer dynamics requires the estimation ofthe parameters , and summarized in Table I.The parameters and are in essence averages that arise
436 IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 2, MARCH 2000
when extending physics at a microscopic level to the macro-scopic scale necessary for control implementation. Hence, whilethey have physical interpretations and tendencies, they must beestimated for individual transducers. The parametersand
are macroscopic and have published values for Terfenol-Dunder various operating conditions. Sufficient variation occursin the values, however, that we also estimated them for the in-dividual transducer.
The full set of parameters was estimated through a leastsquare fit with experimental data from the previously describedtransducer. The optimization was performed in two steps. Inthe first, the values of were estimatedthrough minimization of the functional
(7)
where denotes the experimentally measured value of the Ter-fenol-D magnetization at time. The modeled magnetization attime for parameter valuesis denoted by [(5) or vi)of Algorithm 1]. The functional (7) was minimized using a con-strained optimization algorithm based upon sequential quadraticprogramming (SQP) updates.
With the estimated values of and , the model fitsto the experimental magnetization curves can be obtained. Thesecond step concerns the estimation ofto attain reasonablefits in the magnetostriction model (1). This was accomplishedthrough a least squares fit with displacement or strain data fromthe transducer.
Initial magnetization parameters were estimated using thistechnique for the transducer with an applied prestress of 1.3 ksi.The resulting values are summarized in Table II while modelfits are illustrated in Fig. 4. From strain data, the best fit esti-mate of the saturation magnetostriction was determined to be
for high drive levels andat low drive levels (the difference in values is further discussedin the next section).
To ascertain the robustness of the model with respect to ap-plied prestresses, we then considered the estimation of param-eters and performance of the model with a prestress of 1.0 ksi.For this case, we fixed the parameters that have theleast theoretical dependence upon prestress and estimated theparameters through a least squares fit to the data. Theestimated magnetization parameters are again summarized inTable II while the saturation magnetostriction was found to be
at high drive levels.A comparison of the estimated values ofindicates sig-
nificant changes due to the effects of stress on the pinningenergy at magnetic inclusions. The change in the satura-tion magnetostriction is due to stress-induced changesin the initial domain configuration. The stress dependencein is primarily due to mag-netomechanical stress anisotropies which are quantified bythe term . Note that for the compressiveprestress and the estimated valuesfor , the magnetic coupling parameter has thecomputed value while it has the valuefor ksi. This small variation in the values of(less
than 9%) illustrates the consistency of the model with regardto nearly constant applied stresses. Moreover, it indicates thatone has the capability for identifying and fixing the parameter
and incorporating subsequent stress effects through thecomponent . The use of this strategy hasbeen substantiated by the highly accurate model fits obtainedwith fixed .
C. Magnetization Model
We consider first the performance of the quasi-static mag-netization model summarized in Algorithm 1 under a varietyof operating conditions. The model is formulated to be flexiblewith regard to various drive levels and prestresses and it waswithin this regime that the performance was tested. Data wascollected at multiple drive levels with prestresses of 1.0 and 1.3ksi applied to the Terfenol-D rod. As detailed in [3], prestresseswithin this range yield nearly optimal magnetomechanical cou-pling and strain coefficients for the specific transducer. Param-eters for the magnetization model were estimated through thepreviously described least squares techniques and used to obtainmodel responses under the various conditions. In each case, themeasured applied field was used as input to the model.
The model fits at three drive levels for the 1.0 ksi case are il-lustrated in Fig. 4(a) and (b) while fits for two drive levels witha 1.3 ksi applied stress are illustrated in Fig. 4(c) and (d). Foreach fixed prestress, the same fixed parameters in Table II wereused to attain the model responses at the multiple drive levels.The variation in model behavior is due solely to the changesin the input fields. This illustrates the flexibility of the modelwith respect to drive levels. As noted in previous discussion andsummarized in Table II, only the parameterand stress contri-bution to must be modified to accountfor changes in prestress. Hence the model is also highly flexiblewith respect to applied prestresses.
Close examination of Fig. 4(a) and (c) indicates that one as-pect of the experimental transducer behavior that is not quanti-fied by the model is the constricted or “wasp-waisted” behaviorthat occurs at low applied fields. This behavior has been notedby other investigators [19], [25] and is hypothesized to be dueto 180 domain changes [26]. While quantification of this effectis ultimately desired, the accuracy and flexibility of the currentmagnetization model are sufficient for control applications inthis operating regime.
D. Magnetostriction Model
The second mechanism which must be modeled for the uti-lization of transducers in control design is the magnetostrictiondue to changing magnetization. Once this model is obtained, itcan be combined with the previous magnetization model to pro-vide a characterization of the strains output by the transducerin terms of currents input to the solenoid. For this investigation,we considered the quadratic model (1) as a first approximationto the relation between magnetization and magnetostriction.
The performance of this model is indicated in Fig. 5. At mod-erate drive levels, the strain data exhibits minimal hysteresisand is adequately characterized by the quadratic model. At thehigh drive levels illustrated in Fig. 5(b), the data exhibits signif-icant hysteresis and saturates from a quadratic to nearly linear
CALKINS et al.: ENERGY-BASED HYSTERESIS MODEL 437
relationship as approaches its maximum value. One com-ponent of this hysteresis is due to magnetostrictive hysteresiswhile other effects are due to mechanical hysteresis caused bythe prestress mechanism. The performance of the model is muchless accurate at high drive levels due to such unmodeled be-havior. This loss in accuracy is also reflected in the change ofthe estimated saturation value in the lowdrive regime to at the high drive level. Wenote that at levels below that depicted in Fig. 5(a), the value
provides adequate model fits.The same tendencies are apparent when the magnetization
and magnetostriction models are combined to provide a rela-tionship between input currents and output strains. As illustratedin Fig. 6(a), the combined model is accurate at moderate drivelevels and will be adequate for control design in this regime.Fig. 6(b) illustrates that at high drive levels, however, the magne-tostrictive model degenerates due to unmodeled dynamics andhysteresis.
E. Minor Loop Model
The modeling of minor asymmetric loops comprises the finalcomponent of this investigation. Accurate minor loop character-ization is important for numerous applications including controldesign for transducers in unbiased and biased states. In a generalunbiased state, it is crucial that the model be able to characterizeboth major and minor loop dynamics to attain the full range ofdynamics specified by the control law. The characterization ofminor loops in a biased state is important since it represents acommon operating condition for transducers.
For both cases, we employed the volume fraction (6) to at-tain closure in the minor loop magnetization model. The re-sulting model fit is illustrated in Fig. 7(a) and (b), where 1.0-ksidata containing a major loop and two minor loops is considered.Both the major and minor loops are resolved by the model withthe slight discrepancy in minor loop position due to differencesin experimental and model major loop magnetizations for thevalues of at which the turn points occur. We note that themodel parameters for this case are those summarized in Table IIand no parameter changes are necessary to accommodate theminor loops. As with the major loop case, turning points aredictated solely by the input magnetic field (or equivalently, theinput current ). Fig. 7(c) and (d) illustrates the performanceof the magnetization and magnetostriction models in resolvingmajor and minor loops in the 1.3-ksi data. The accuracy of thefit in Fig. 7(c) reflects the accuracy of the underlying magne-tization model while discrepancies in the major loop strain fitin Fig. 7(d) are due to the previously mentioned unmodeled be-havior in the magnetostriction at high drive levels. At moderatelevels, the minor loop model is sufficiently accurate for controlapplications.
V. CONCLUDING REMARKS
An energy-based model for characterizing magnetization andoutput strains for magnetostrictive transducers is presented. Themagnetization model, which is based upon the Jiles–Athertonmean field theory for ferromagnetic materials, provides a meansof characterizing the magnetic hysteresis inherent to the trans-ducer. Through enforcement of closure conditions, nested asym-
(a)
(b)
Fig. 6. Experimental data (dashed line) and combined magnetization andmagnetostriction model dynamics (solid line): (a) low drive level and (b) highdrive level.
metric minor loops as well as symmetric major loops are re-solved by the characterization. This magnetization model is cur-rently constructed for a transducer with quasistatic input andfixed operating temperature. Within this regime, the model pro-vides the capability for characterizing variable input levels tothe solenoid and differing applied stresses to the Terfenol-D rod.The good agreement of this theory with experimental data illus-trates the flexibility of the model under a variety of operatingconditions. Furthermore, similar model fits have been obtainedwhen the model is used to characterize additional transducerconfigurations under differing operating conditions.
A quadratic model based upon the geometry of moment rota-tions was employed to quantify the magnetostriction and strainsgenerated by the transducer. As illustrated through comparisonwith experimental data, this characterization was adequate atmoderate drive levels but degenerated at high drive levels due
438 IEEE TRANSACTIONS ON MAGNETICS, VOL. 36, NO. 2, MARCH 2000
(a) (b)
(c) (d)
Fig. 7. Experimental data (dashed line) and minor loop model dynamics (solid line): (a) two minor loops in magnetization data with 1.0-ksi applied stress; (b)magnified view of 1.0-ksi case; (c) minor loop in magnetization data with 1.3-ksi applied stress; and (d) minor loop in strain data with 1.3-ksi applied stress.
to unmodeled nonlinearities and hysteresis. Certain aspects ofthe magnetostriction hysteresis can be included through the en-ergy model of [9] but adequate quantification of the full relationhas not been attained and is under current investigation.
At moderate input levels, the combination of the magnetiza-tion and magnetostriction models provide an accurate charac-terization of output strains in terms of input currents to the sole-noid. For quasi-static applications in which temperature can beregulated, the model is sufficiently accurate for control design.The robustness of the model with regard to operating conditionsand the small number of required parameters (six) enhance itssuitability for such applications.
We point out that while the model quantifies certain lossmechanisms through the hysteresis characterization, otherpotential sources of loss can arise in certain regimes includingdynamic operating conditions. These include eddy currentand anomalous losses as well as potential mechanical lossesin the prestress mechanism. Due to the coupled nature of
the transducer components in response to varied operatingconditions, the isolation of additional magnetic and mechanicallosses may be difficult although certain aspects of the lossescan be minimized through careful construction techniques.
ACKNOWLEDGMENT
The authors would like to thank M. Dapino and D. Jiles fornumerous discussions and input regarding the modeling tech-niques employed here.
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