Model of the Magnetostrictive Hysteresis Loop with Local Maximum

materials

Article Model of the Magnetostrictive Hysteresis Loop with Local Maximum

Roman Szewczyk Institute of Metrology and Biomedical Engineering, Warsaw University of Technology, 02-525 Warsaw, Poland; [email protected]; Tel.: +48-609-464741

Received: 6 December 2018; Accepted: 26 December 2018; Published: 30 December 2018

Abstract: This paper presents a model of the magnetostrictive hysteresis loop with local maximum. The model is based on the differential equations describing magnetostriction due to the domain wall movement as well as domain magnetization rotation. The transition between these mechanisms of magnetization is quantified by the Maxwell–Boltzmann distribution. Moreover, the lift-off phenomenon in the magnetostrictive hysteresis loop is considered. The proposed model was validated on the results of measurements of magnetostrictive hysteresis loops of Mn0.70Zn0.24Fe2.06O4 ferrite for power application and 13CrMo4-5 construction steel. The results of modeling confirm that the proposed model corresponds well with experimental results. Good agreement was confirmed by 2 determination coefficient R , which exceeded 0.995 and 0.985 for Mn0.70Zn0.24Fe2.06O4 ferrite for power application and 13CrMo4-5 construction steel, respectively.

Keywords: magnetostriction model; Mn-Zn ferrites; construction steels

1. Introduction The magnetostriction phenomenon is connected with the changes of the linear dimensions of the sample during the magnetization process. The magnetostrictive phenomenon has great technical importance. It can be used in the development of specialized position [1] and level sensors [2], MEMS (Micro-Electro-Mechanical Systems) sensors [3], as well as ultrasonic transducers [4] and high accuracy actuators [5]. In spite of the fact that magnetostriction has been known for over one hundred years, as well as the great effort that has been taken to understand it, until now a complete quantitative model of the magnetostrictive hysteresis loop has not been presented. However, it is obvious that magnetostrictive characteristics of soft magnetic materials plays a key role in understanding other magnetomechanical effects, such as the Villari effect [6], the Wiedemann effect [7], or the stress dependence of giant magnetoimpedance phenomenon [8], as well as the Matteucci effect [9]. Previously presented research clearly indicates that magnetostriction (λ) is mostly determined by a parabola-shaped curve with the value of magnetization (M) of the material (which may be estimated by the ﬂux density (B)) [10]. Another important approach was the use of hyperbolic Bessel functions, proposed by Sablik [11]. However, proposed solutions are suitable for parabola-shaped λ(B) curves only. It should be stressed that alarge number of soft magnetic materials, such as some Mn-Zn ferrites [12] or constructional steels [13,14], exhibit local maxima on the λ(B) dependence, which makes parabola-shaped models unsuitable for the modeling of magnetostrictive hysteresis loops of such materials. This paper ﬁlls the gap in the state of the art, presenting a quantitative model of the magnetostrictive λ(B) hysteresis loop with local maximum. The presented model considers the transition from the domain wall movement to the domain rotations, causing the changes of the sign of magnetostriction curve derivative. As a result, the proposed model may be used for both the

Materials 2019, 12, 105; doi:10.3390/ma12010105 www.mdpi.com/journal/materials Materials 2019, 12, 105 2 of 11 technical development of magnetostrictive sensors and actuators, as well as for physical analyses of magneto-mechanical processes in soft magnetic materials.

2. Principles of Modeling the Magnetostrictive Hysteresis Loops Magnetostriction in soft magnetic materials is caused by the changes of the total free magneto-mechanical energy of the material sample due to the transition from the paramagnetic to ferromagnetic state, during the cooling of the material and overturning of the Curie temperature. On the basis of magneto-crystalline anisotropy analyses, it may be stated that the dependence of magnetostriction (λ) on the magnetization (M) of magnetic material may be described by the fourth order polynomial [15]: 2 4 λ(M) = a1 M + a2 M (1) This dependence is commonly reduced to the second order polynomial [16]:

3 λ ( ) = s 2 λ M 2 M (2) 2 Ms where λs is the saturation magnetostriction and Ms is saturation magnetization. In the case of soft magnetic materials, where relative permeability µr >> 1, Equation (2) may be presented as [17]:

3 λ ( ) = s 2 λ B 2 B (3) 2 Bs where B and Bs are flux density in the material and saturation flux density of the material, respectively. Such form of magnetostrictive characteristic model is more convenient for technical applications. For this reason, Equation (3) was successfully used for the technical modeling of magnetostrictive actuators, especially with cores made of giant magnetostrictive materials, such as Terfenol-D [18,19]. On the other hand, analysis of experimental results of measurements of λ(B) hysteresis loop clearly indicates that accurate modeling of these characteristics requires consideration of λ(B) hysteresis (which is different than B(H) hysteresis), as well as the so called “lift-off” phenomenon. The “lift-off” phenomenon is connected with the fact that, during the magnetization loop, magnetostriction never comes back to the value observed in the demagnetized state. The physical origins of hysteresis on the λ(B) relation are connected with the interaction between residual stresses and magnetostrictive strain. In previous research, it was connected with the hysteretic magnetization equal to the difference between total magnetization (M) and anhysteretic magnetization (Man)[20]. Another approach to this hysteresis was based on the hyperbolic Bessel functions [11], connecting the magnetostriction with the efficient magnetizing field (He) in the magnetic material equal:

He = H + αM (4) where H is the magnetizing ﬁeld, whereas α is the interdomain coupling accordingly to the Bloch theory. The “lift-off” phenomenon is connected with the fact that, during the magnetization process, the λ(B) characteristic never returns to zero [20]. The physical origins of this effect are not clearly explained; however, it has been observed in experimental measurements of the magnetostrictive hysteresis loops λ(B) and λ(H) of most ferromagnetic materials [11,20–24]. Known previous approaches to quantitative modeling of the “lift-off” phenomenon were focused on the reproduction of the shape of magnetostrictive hysteresis loops considering this phenomenon [20]. The most important problem connected with modeling the magnetostrictive hysteresis loops λ(B) and λ(H) is the fact, in the case of some soft magnetic materials, that local maxima occurs on these dependences. This local maxima was observed in experimental results [12,14,25–27]; however, the quantitative model of such a magnetostrictive hysteresis loop was never presented before. Lack of such a model is the signiﬁcant barrier for understanding the physical background of Materials 2019, 12, 105 3 of 11 magneto-mechanical effects, as well as for the practical description of the behavior of ferromagnetic materials required for, for example, the development of transformers or nondestructive testing of elements made of constructional steels.

3. The Proposed Model of the Magnetostrictive Hysteresis Loop with Local Maxima The proposed model of the magnetostrictive λ(B) hysteresis loop is based on the fact that the mechanism of the magnetization of ferromagnetic material changes during the magnetization process. For smaller values of magnetizing ﬁeld, magnetization is connected with the domain walls movements, whereas magnetization in the saturation region is mostly caused by the rotation of domains magnetization [15,16]. As a result, magnetostriction λmov(B) in the domain walls movement region of the magnetic hysteresis loop has a parabola shape [16], and may be described by the following differential equation: dλ mov = 2a B (5) dB 1 where a1 is the parameter determined in the same way as the parameters in Equation (3). However, for the domain rotations region, the magnetostriction λrot is connected mostly with the rotation from the easy axis to hard axis. As a result, the linear dependence of λrot(B) may be observed in this area [28], represented by the following differential equation:

dλ rot = a (6) dB 2 where a2 is the parameter describing the slope of the magnetostrictive curve in the saturation region. The transition between the magnetization mechanisms based on the domain wall movement and domain magnetization rotation may be quantiﬁed by the Maxwell–Boltzmann statistical distribution, of which the cumulative distribution function is given by the following equation [29]:

r 2 B − B 2 (B − B )e−(B−Bswitch)/(2k ) W(B) = er f √switch − switch (7) k 2 π k where Bswitch is the value of ﬂux density B when the mechanism of magnetization starts to change from domain walls movement to domain magnetization rotation, and k determines the intensity of this process. The so-called error function erf (x), necessary to determine Maxwell–Boltzmann statistical distribution, is given by the following equation [29]:

x 1 Z 2 er f (x) = √ e−t dt (8) π −x

Figure1 presents the example of the magnetic hysteresis B(H) loop together with W(B) dependence. The region of the domain wall movement, the region of the domains magnetization rotation, as well as the region of mixed mechanism can be clearly observed. Finally, the differential equation determining the ﬂux density B dependence of anhysteretic magnetostriction λanhyst is stated as:

dλanhyst dλ dλ = mov (1 − W(B)) + rot W(B) (9) dB dB dB with the initial condition λanhyst(B) = 0 for B = 0. The example of the λanhyst(B) curve stated by Equation (9) is presented in Figure2. As in the case of real samples, the maximal value of magnetostriction λmax is signiﬁcantly higher than saturation magnetostriction λs. Materials 2019, 12, x FOR PEER REVIEW 3 of 11 materials required for, for example, the development of transformers or nondestructive testing of elements made of constructional steels.

𝑑𝜆 =2𝑎 𝐵 (5) 𝑑𝐵 where a1 is the parameter determined in the same way as the parameters in Equation (3). However, for the domain rotations region, the magnetostriction λrot is connected mostly with the rotation from the easy axis to hard axis. As a result, the linear dependence of λrot(B) may be observed in this area [28], represented by the following differential equation:

𝑑𝜆 =𝑎 (6) 𝑑𝐵 where a2 is the parameter describing the slope of the magnetostrictive curve in the saturation region. The transition between the magnetization mechanisms based on the domain wall movement and domain magnetization rotation may be quantified by the Maxwell–Boltzmann statistical distribution, of which the cumulative distribution function is given by the following equation [29]:

()/( ) 𝐵−𝐵 2 (𝐵−𝐵)𝑒 𝑊(𝐵) =𝑒𝑟𝑓 − (7) 𝑘√2 𝜋 𝑘 where Bswitch is the value of flux density B when the mechanism of magnetization starts to change from domain walls movement to domain magnetization rotation, and k determines the intensity of this process. The so-called error function erf(x), necessary to determine Maxwell–Boltzmann statistical distribution, is given by the following equation [29]: 1 𝑒𝑟𝑓(𝑥) = 𝑒 𝑑𝑡 (8) √𝜋 Figure 1 presents the example of the magnetic hysteresis B(H) loop together with W(B) dependence. The region of the domain wall movement, the region of the domains magnetization Materials 2019, 12, 105 4 of 11 rotation, as well as the region of mixed mechanism can be clearly observed.

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Figure 1. Changes in the mechanism of magnetization in Mn0.70Zn0.24Fe2.06O4 ferrite for power applications: (a) Magnetic hysteresis loop B(H); (b) W(B) dependence determining the mechanism of magnetization.

Finally, the differential equation determining the flux density B dependence of anhysteretic magnetostriction λanhyst is stated as:

𝑑𝜆 𝑑𝜆 𝑑𝜆 = (1−𝑊(𝐵)) + 𝑊(𝐵) (9) 𝑑𝐵 𝑑𝐵 𝑑𝐵 with the initial condition λanhyst(B) = 0 for B = 0.

The example of the λanhyst(B) curve stated by Equation (9) is presented in Figure 2. As in the case Figure 1. Changes in the mechanism of magnetization in Mn0.70Zn0.24Fe2.06O4 ferrite for power of real samples, the maximal value of magnetostriction λmax is significantly higher than saturation applications: (a) Magnetic hysteresis loop B(H); (b) W(B) dependence determining the mechanism magnetostrictionof magnetization. λs.

Figure 2.2. The ﬂuxflux densitydensity BB dependencedependence ofof anhystereticanhysteretic magnetostrictionmagnetostriction λλanhyst,, determined by anhyst µm 1 µm 1 Equation (9). Parameters: a1 == 5 5 , a, 2a = =10 10 , Bswitch, B = 1 T,= 1k T,= 0.1. k = 0.1. 1 mT2 2 m T switch

The solution proposed to model the “lift-off” phenomenon is similar to the one proposed by Sablik et al.al. [[11,20].11,20]. ValueValue ofof the the magnetostriction magnetostriction component componentλ λlift-offlift-off describing this phenomenon is proportional to fluxflux density B, if the value of fluxflux density B is higher than the value of fluxflux density Bprev reachedreached previously previously by by the the material. material. Otherwise, Otherwise, component component λlift-offλlift-off remainsremains unchanged, unchanged, as the as last the lastreached reached value. value. Such Suchmechanism mechanism describes describes the physic the physicalal background background of “lift-off” of “lift-off” phenomenon, phenomenon, where wherethe magnetostriction the magnetostriction component component λlift-offλ lift-offis connectedis connected with with the the interaction interaction of of the the magnetostriction magnetostriction strain with the residualresidual stressesstresses inin thethe material.material. TheThe magnetostrictionmagnetostriction component componentλ λlift-offlift-off isis determined determined by the following set of differential equations [[11]:11]: 𝑑𝜆 dλ li f t−o f f =𝑎 𝑓𝑜𝑟 𝐵 >𝐵 𝑑𝐵 = a − f or B > Bprev dB li f t o f f (10) 𝑑𝜆 (10) dλli f t−o f f ==0 otherwise 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 dB𝑑𝐵 The example of flux density B dependence of anhysteretic magnetostriction λanhyst considering The example of flux density B dependence of anhysteretic magnetostriction λ considering the “lift-off” phenomenon is presented in Figure 3. anhyst the “lift-off” phenomenon is presented in Figure3.

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Figure 3. The flux density B dependence of anhysteretic magnetostriction λanhyst considering the FigureFigure 3. TheThe flux ﬂux density B dependence of anhystereticanhysteretic magnetostrictionmagnetostriction λλanhyst consideringconsidering the the “lift-off” phenomenon, calculated for the following parameters: a1 = 5 , a2 =anhyst 10 , Bswitch = 1 T, k µ µ “lift-off” phenomenon, calculated for the following parameters: a1 = 5 m 1 , a2 = 10 m ,1 Bswitch = 1 T, k “lift-off” phenomenon, calculated for the following parameters: a1 = 5 2 , a2 = 10 , Bswitch = 1 T, = 0.1, alift-off = 1 . m T m T µm 1 = 0.1, alift-off = 1 . k = 0.1, alift-off = 1 m T . For modelling the hysteretic behavior of magnetostriction λ(B) dependence, it should be ForFor modelling modelling the the hysteretic hysteretic behavior behavior of of magnetostriction λλ(B)(B) dependence, dependence, it it should should be be highlighted that hysteresis is connected with the domain walls movement magnetization highlightedhighlighted thatthat hysteresis hysteresis is connectedis connected with thewith domain the domain walls movement walls movement magnetization magnetization mechanism. mechanism. Domain magnetization rotation is anhysteretic, which is what can be observed on both mechanism.Domain magnetization Domain magnetization rotation is anhysteretic, rotation is anhy whichsteretic, is what which can be is observedwhat canon be bothobservedB(H) andon bothλ(B) B(H) and λ(B) dependencies. As a result, the hysteretic component of the magnetostriction hysteresis B(H)dependencies. and λ(B) dependencies. As a result, the As hysteretic a result, the component hysteretic of co themponent magnetostriction of the magnetostriction hysteresis loop hysteresisλhyst can loop λhyst can be described as: loopbe described λhyst can be as: described as: 𝜆λhyst(𝐵)(B) = =( 1 1− W −𝑊((𝐵B)))∙𝑎·ahystB𝐵 (11)(11) 𝜆(𝐵) = 1 −𝑊(𝐵)∙𝑎𝐵 (11) wherewherea hystahystdetermines determines the magnetostrictivethe magnetostrictive hysteresis. hysteresis. The flux densityThe fluxB dependence density B of magnetostrictivedependence of where ahyst determines the magnetostrictive hysteresis. The flux density B dependence of hysteresismagnetostrictiveλhyst is hysteresis presented inλhyst Figure is presented4. in Figure 4. magnetostrictive hysteresis λhyst is presented in Figure 4.

Figure 4. The flux density B dependence of magnetostrictive hysteresis λhyst calculated for the Figure 4. The ﬂux density B dependence of magnetostrictive hysteresis λhyst calculated for the following Figure 4. The flux density B dependenceµ of magnetostrictive hysteresis λhyst calculated for the following parameters: Bswitch = 1 T, k = 0.1, amhyst1 = 1 . parameters: Bswitch = 1 T, k = 0.1, ahyst = 1 . following parameters: Bswitch = 1 T, k = 0.1, amhyst T= 1 . Finally, thetheλ (Bλ)(B) hysteresis hysteresis loop canloop be can calculated be calculat as a sumed ofas the a componentssum of the of magnetostriction:components of Finally, the λ(B) hysteresis loop can be calculated as a sum of the components of magnetostriction: magnetostriction: λ(B) = λanhyst(B) + λli f t−o f f (B) + λhyst(B) (12) 𝜆(𝐵) =𝜆(𝐵) +𝜆(𝐵) +𝜆(𝐵) (12) 𝜆(𝐵) =𝜆(𝐵) +𝜆(𝐵) +𝜆(𝐵) (12) The example of the magnetostrictive λ(B) hysteresis loop is presented in Figure 5. It can be The example of the magnetostrictive λ(B) hysteresis loop is presented in Figure 5. It can be observed that the shape of the λ(B) hysteresis loop well reflects the shape of such loops presented in observed that the shape of the λ(B) hysteresis loop well reflects the shape of such loops presented in the literature. the literature.

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The example of the magnetostrictive λ(B) hysteresis loop is presented in Figure5. It can be observed that the shape of the λ(B) hysteresis loop well reﬂects the shape of such loops presented in theMaterials literature. 2019, 12, x FOR PEER REVIEW 6 of 11 Materials 2019, 12, x FOR PEER REVIEW 6 of 11

FigureFigure 5. 5. Magnetostrictive λλ(B)B hysteresis loop, determined by Equation (12), calculated for the Magnetostrictive ( ) hysteresis loop, determined by Equation (12), calculated for the Figure 5. Magnetostrictiveµ λ(B)µ hysteresis loop, determined by Equationµ (12), calculatedµ for the parameters: a1 = 5 m 1 , a2 = 10 m ,1 Bswitch = 1 T, k = 0.1, alift-off = 1 , ahysm = 1 . m 1 parameters: a1 = 5 m 2 , a2 = 10 m T , Bswitch = 1 T, k = 0.1, alift-off = 1 m T ,ahys = 1 m T . parameters: a1 = 5 T , a2 = 10 , Bswitch = 1 T, k = 0.1, alift-off = 1 , ahys = 1 . 4. Validation4. Validation of of the the Model Model 4. Validation of the Model 4.1.4.1. Materials Materials and and the the Method Method of of Measurements Measurements 4.1. Materials and the Method of Measurements TheThe proposed proposed model model was verifiedwas verified on the basison ofthe experimental basis of experimental measurements measurements of magnetostrictive of hysteresismagnetostrictiveThe loops proposed of Mnhysteresis 0.70modelZn0.24 loopsFewas2.06 Overifiedof4 ferriteMn0.70 Zn foron0.24 powertheFe2.06 Obasis application4 ferrite of experimentalfor [ 12power] and 13CrMo4-5application measurements construction[12] andof steelmagnetostrictive13CrMo4-5 [14]. Measurements construction hysteresis of steel the loops magnetostrictive [14] . ofMeasurements Mn0.70Zn hysteresis0.24Fe 2.06of Othe4 loopferrite magnetostr were for carriedpowerictive out hysteresisapplication on the frame-shapedloop [12] wereand samples13CrMo4-5carried [30 out] with onconstruction the strain frame-shaped gauges. steel Semiconductor[14] samples. Measurements [30] with strain strain of gauges the gauges. magnetostr and foilSemiconductor strainictive gauges hysteresis strain were gaugesloop used were forand the carried out on the frame-shaped samples [30] with strain gauges. Semiconductor strain gauges and measurementsfoil strain gauges of the were samples used madefor the of measurements Mn0.70Zn0.24 Feof 2.06theO samples4 ferrite made and 13CrMo4-5 of Mn0.70Zn0.24 constructionFe2.06O4 ferrite steel, respectively.foiland strain 13CrMo4-5 gauges Measurements wereconstruction used were for the performedsteel, measurements respective in roomly. of temperaturetheMeasurements samples made in thewere of quasi-static Mn performed0.70Zn0.24Fe mode,2.06 Oin4 withferriteroom the magnetizingandtemperature 13CrMo4-5 field in the frequency constructionquasi-static of 0.2 mode, steel, Hz. with respective the magnetizingly. Measurements field frequency were of performed0.2 Hz. in room temperature in the quasi-static mode, with the magnetizing field frequency of 0.2 Hz. A schematicA schematic block block diagram diagram of the of magnetostrictionthe magnetostricti measuringon measuring system system [12] is[12] presented is presented in Figure in 6. A schematic block diagram of the magnetostriction measuring system [12] is presented in MagnetizingFigure 6. Magnetizing winding of winding the frame-shaped of the frame-shap sampleed was sample connected was connected to the voltage-current to the voltage-current converter Figure 6. Magnetizing winding of the frame-shaped sample was connected to the voltage-current BOP36-6Mconverter producedBOP36-6M byproduced Kepco, by USA. Kepco, The USA. sensing The sensing winding winding as connected as connected to fluxmeter to fluxmeter type type 480, converter480, produced BOP36-6M by Lakeshore, produced USA, by Kepco, enabling USA. real-tim The sensinge measurements winding asof connectedflux density to in fluxmeter the core. type The produced by Lakeshore, USA, enabling real-time measurements of flux density in the core. The strain 480,strain produced gauges bywere Lakeshore, connected USA, to enablinga specialized real-tim MT-12e measurements bridge, enabling of flux strain density gauge in the sensitivity core. The gauges were connected to a specialized MT-12 bridge, enabling strain gauge sensitivity adjustments. strainadjustments. gauges Duringwere connected the measurements, to a specialized the temp MT-12erature bridge, of the enabling sample strainwas monitoredgauge sensitivity by the During the measurements, the temperature of the sample was monitored by the thermocouple-based adjustments.thermocouple-based During sensor.the measurements, The whole systemthe temp waeratures controlled of the by sample PC computer, was monitored with LabView by the sensor. The whole system was controlled by PC computer, with LabView software equipped in thermocouple-basedsoftware equipped in sensor.PCI-6221 The data whole acquisition system and wa controls controlled card. by PC computer, with LabView PCI-6221software data equipped acquisition in PCI-6221 and control data acquisition card. and control card. strain gauge strain gauge strain gauge voltage-current magnetizing frame-shaped strain gauge bridge MT-12 convertervoltage-current BOP-36 magnetizingwindings frame-shapedcore bridge MT-12 converter BOP-36 windings core sensing fluxmeter windingssensing Lakeshorefluxmeter 480 temperature windings Lakeshore 480 temperaturesensor sensor

PC computer with National Instruments PCI-6221 data acquisition card and LABview software for control PC computer with National Instruments PCI-6221 data acquisition card and LABview software for control

Figure 6. Schematic diagram of the system used for the measurements of magnetostrictive λ(B) and FigureFigureλ(H), 6. as 6.Schematic well Schematic as the diagram magnetic diagram of B(H)of the the hysteresis system system usedused loops. forfor the measurements of of magnetostrictive magnetostrictive λ(B)λ( Band) and λ(Hλ(H),), as as well well as as the the magnetic magneticB B(H)(H) hysteresis hysteresis loops.loops. 4.2. Identification of the Parameters of the Model 4.2. Identification of the Parameters of the Model

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The proposed model of the magnetostrictive λ(B) hysteresis loop, stated by Equations (5)–(12), 4.2.was Identificationimplemented of thein ParametersOctave 4.4. of the1, open-source Model Matlab alternative. For solving the ordinary differentialThe proposed equations, model stating of the the magnetostrictive model of themλagnetostrictive(B) hysteresis loop,hysteresis stated loop, by Equations the fourth (5)–(12), order wasRunge–Kutta implemented algorithm in Octave was 4.4.1, used open-source [31]. Identification Matlab alternative.of the model’s For solvingparameters the ordinarywas carried differential out on equations,the basis of stating the minimization the model of pr themagnetostrictiveocess, performed by hysteresis a derivativ loop,e-free the Nelder fourth orderand Mead Runge–Kutta simplex algorithm [32]. was usedThe target [31]. Identificationfunction G for of minimization the model’s parameterswas determine wasd carriedas the sum out onof the the squared basis of thedifferences minimization between process, the experimental performed bydata a derivative-free and the results Nelder of the andmodeling Mead given simplex by algorithm the following [32]. Theequation target [33]: function G for minimization was determined as the sum of the squared differences between the experimental data and the results of the modeling given by the following equation [33]: 𝐺=𝜆(𝐵) −𝜆(𝐵) (13) n 2 G = ∑(λmeas(Bi) − λmodel(Bi)) (13) where λmeas(Bi) is the results of measurementsi=1 of the magnetostrictive hysteresis loop for the set of given values of flux density Bi; and λmodel(Bi) is the results of the modeling of the magnetostrictive where λmeas(Bi) is the results of measurements of the magnetostrictive hysteresis loop for the set of hysteresis loop for the same set of flux density Bi values. given values of flux density Bi; and λmodel(Bi) is the results of the modeling of the magnetostrictive Parameters of the magnetostrictive hysteresis loops of Mn0.70Zn0.24Fe2.06O4 ferrite for power hysteresis loop for the same set of flux density Bi values. application and 13CrMo4-5 construction steel, identified during the optimization process, are Parameters of the magnetostrictive hysteresis loops of Mn0.70Zn0.24Fe2.06O4 ferrite for power presented in the Table 1. The results of this modeling are presented in Figures 7 and 8 for application and 13CrMo4-5 construction steel, identified during the optimization process, are presented Mn0.70Zn0.24Fe2.06O4 ferrite, for power application and 13CrMo4-5 construction steel, respectively. in the Table1. The results of this modeling are presented in Figures7 and8 for Mn 0.70Zn0.24Fe2.06O4 Figures present both λ(B) and λ(H) hysteresis loops. ferrite, for power application and 13CrMo4-5 construction steel, respectively. Figures present both λ(B) and λ(H) hysteresis loops. Table 1. Parameters of magnetostrictive λ(B) hysteresis loops determined in optimization process.

TableParameter 1. Parameters Unit of magnetostrictive Mn0.70Znλ0.24(BFe) hysteresis2.06O4 Ferrite loops determined 13CrMo4-5 in optimization Steel process. μm 1 a1 8.10 6.73 Parameterm 𝑇 Unit Mn0.70Zn0.24Fe2.06O4 Ferrite 13CrMo4-5 Steel μm 1 µm 1 a2 a 2 −10.07 8.10−25.93 6.73 1 m T m T a µm 1 −10.07 −25.93 Bswitch 2 T m T 0.312 1.338 B T 0.312 1.338 k switch - 0.039 0.101 k - 0.039 0.101 μm 1 µ alift-off m 1 0.253 0.495 alift-off m T 0.253 0.495 m T µm 1 ahystμm 1 m T 0.561 0.908 ahyst 2 0.561 0.908 R m T - 0.995 0.985 R2 - 0.995 0.985

(a)

Figure 7. Cont.

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(b)

Materials 2019, 12, x FOR PEER REVIEW 9 of 11 FigureFigure 7.7. ResultsResults ofofmodeling modeling the the magnetostrictive magnetostrictive hysteresis hysteresis loops loops of of Mn Mn0.700.70ZnZn0.240.24FeFe2.062.06O44 ferrite forfor powerpower applications:applications: (a)) λ((B)B) hysteresishysteresis loop;loop; ((b)) λ((H)H) hysteresis loop. Experimental results—red line;

resultsresults ofof modeling—blackmodeling—black line.line.

(a)

(b)

FigureFigure 8. ResultsResults of of modeling modeling the the magnetostrictive magnetostrictive hysteresis hysteresis loops loops of of 13CrMo4-5 steel: ( (aa)) λλ(B)(B) hysteresishysteresis loop;loop; ((bb))λ λ(H(H)) hysteresis hysteresis loop. loop. Experimental Experimental results—red results—red line; line; results results of modeling—black of modeling—black line. line. It should be highlighted, that the proposed model very well described the shape of both magnetostrictiveIt should beλ highlighted,(B) and λ(H) hysteresisthat the proposed loops. The model quality very of the well model described was also the conﬁrmed shape of by both the magnetostrictivevalue of the R2 determination λ(B) and λ(H) coefﬁcient, hysteresis loops. which The describes quality the of proportionthe model was of the also variance confirmed described by the value of the R2 determination coefficient, which describes the proportion of the variance described by the model and the variance of the results of measurement. For λ(B) hysteresis loops, the R2 coefficient exceeded 0.995 and 0.985, in the case of Mn0.70Zn0.24Fe2.06O4 ferrite, for power application and 13CrMo4-5 steel, respectively. Such high values of R2 determination coefficients confirms, quantitatively, that the proposed model is suitable for modeling λ(B) and λ(H) magnetostrictive hysteresis loops with local maxima. On the other hand, accuracy of the model for 13CrMo4-5 steel is worse than for Mn0.70Zn0.24Fe2.06O4 ferrite. This effect is probably connected with the fact that the presented model does not consider nucleation and annihilation mechanisms in the sample. These phenomena should be the subject of further research.

Materials 2019, 12, 105 9 of 11 by the model and the variance of the results of measurement. For λ(B) hysteresis loops, the R2 coefficient exceeded 0.995 and 0.985, in the case of Mn0.70Zn0.24Fe2.06O4 ferrite, for power application and 13CrMo4-5 steel, respectively. Such high values of R2 determination coefficients confirms, quantitatively, that the proposed model is suitable for modeling λ(B) and λ(H) magnetostrictive hysteresis loops with local maxima. On the other hand, accuracy of the model for 13CrMo4-5 steel is worse than for Mn0.70Zn0.24Fe2.06O4 ferrite. This effect is probably connected with the fact that the presented model does not consider nucleation and annihilation mechanisms in the sample. These phenomena should be the subject of further research.

5. Conclusions The presented results of the measurements of both magnetostrictive λ(B) and λ(H) hysteresis loops of Mn0.70Zn0.24Fe2.06O4 ferrite, for power application and steel 13CrMo4-5, confirm that, in the case of these materials, local maxima occur on the magnetostrictive hysteresis loop. To model such a magnetostrictive hysteresis loop the, change of magnetization and magnetostriction mechanisms should be considered. The proposed model utilizes the cumulative distribution function of Maxwell–Boltzmann statistical distribution to quantify the transition from domain walls movement to domain magnetization rotation. As a result, a new model of the magnetostrictive hysteresis loop was presented, considering previously proposed descriptions of the magnetostrictive hysteresis and magnetostrictive “lift-off” phenomenon. The proposed model very well describes experimental results, efficiently reproducing the maxima on the magnetostrictive λ(B) and λ(H) hysteresis loops. The high quality of the model was confirmed, quantitatively, by the value of the R2 determination coefficient, which exceeded 0.995 and 0.985, for Mn0.70Zn0.24Fe2.06O4 ferrite, for power application and 13CrMo4-5 steel, respectively. Due to good agreement with the experimental results, the proposed model may be used for works focused on understanding the magnetostrictive phenomena in steels and soft ferrites. In addition, the proposed model may be used for the modeling of the magnetostrictive behavior of inductive components with inductive cores made of soft magnetic materials with non-monotonous magnetostrictive characteristics. Such components are commonly used in power conversion devices, such as switching-mode power supplies.

Funding: This work was fully supported by the statutory funds of the Institute of Metrology and Biomedical Engineering, WUT. Conﬂicts of Interest: The author declares no conﬂict of interest.

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