The Use of Hypergeometric Functions in Hysteresis Modeling

energies

Article The Use of Hypergeometric Functions in Hysteresis Modeling

Dejana Herceg 1,*, Krzysztof Chwastek 2 and Ðorde¯ Herceg 3 1 Department of Power, Electronic and Telecommunication Engineering, Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovi´ca6, 21000 Novi Sad, Serbia 2 Faculty of Electrical Engineering, Cz˛estochowaUniversity of Technology, Al. Armii Krajowej 17, 42-201 Cz˛estochowa,Poland; [email protected] 3 Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovi´ca4, 21000 Novi Sad, Serbia; [email protected] * Correspondence: [email protected]

Received: 25 October 2020; Accepted: 7 December 2020; Published: 9 December 2020

Abstract: Accurate hysteresis models are necessary for modeling of magnetic components of devices such as transformers and motors. This study presents a hysteresis model with a convenient analytical form, based on hypergeometric functions with one free parameter, built upon a class of parameterized curves. The aim of this work is to explore suitability of the presented model for describing major and minor loops, as well as to demonstrate improved agreement between experimental and modeled hysteresis loops. The procedure for generating first order reversal curves is also discussed. The added parameter, introduced into the model, controls the shape of the model curve, especially near saturation. It can be adjusted to provide better agreement between measured and model curves. The model parameters are nonlinearly dependent; therefore, they are determined in a nonlinear curve fitting procedure. The choice of the initial approximation and a suitable set of constraints for the optimization procedure are discussed. The inverse of the model function, required to generate first order reversal curves, cannot be obtained in analytical form. The procedure to calculate the inverse numerically is presented. Performance of the model is demonstrated and verified on experimental data obtained from measurements on construction steel sheets and grain-oriented electrical steel samples.

Keywords: hysteresis loop; ferromagnetic materials; modeling

1. Introduction Whenever a cyclic variation of magnetic field is applied to a ferromagnetic material, the so-called hysteresis loop is obtained. Usually, symmetric closed curves are considered, cf. Figure1, yet it should be mentioned that in some applications it is crucial to predict the shapes of magnetization curves for arbitrary input signals. The practical importance of hysteresis loop stems from the fact that it allows one to determine the values of some significant properties of magnetic materials like remanence induction and coercive field strength (shown in the figure for the limiting (saturating) loop). Moreover, integration of loop area allows one to determine energy dissipated per cycle and per unit volume [1]. Therefore, the study of hysteresis phenomenon still attracts the attention of the engineering community. The choice of material for a specific application may be facilitated using databases containing measurement results for different materials. However, the abundance of possible excitation scenarios makes the use of such libraries prohibitive from the practical point of view; therefore, much attention is paid to the development of mathematical descriptions of hysteresis loops. Advanced analyses and

Energies 2020, 13, 6500; doi:10.3390/en13246500 www.mdpi.com/journal/energies Energies 2020, 13, 6500 2 of 14 computations concerning e.g., distribution of magnetic ﬁelds and losses in cores of electric devices are

Energiesoften built 2020, upon13, x FOR these PEER data. REVIEW 2 of 13

Figure 1. A family of hysteresis loops loops with with different diﬀerent amplitudes for for a a grain-oriented electrical electrical steel. steel. Source: Wikimedia Wikimedia Commons, author: Stan Stan Zurek. Zurek.

TheA convenient choice of modelmaterial of for hysteresis a specific should application be simple may to usebe facilitated in calculations, using described databases with containing a small measurementset of parameters, results and provide for different a good materials. fit to the experimental However, data.the abundance Various mathematical of possible models excitation have scenariosbeen developed makes to the describe use of hysteresis such libraries loops, prohibitive such as the from wide-spread the practical Jiles–Atherton point of view; (JA) therefore, [2–4] and muchPreisach attention formalisms is paid [5–8 ]. to However, the development parameter of determination mathematical for descriptions the JA model of can hysteresis be complicated, loops. Advancedwhereas the analyses Preisach and model computations can be computationally concerning e.g., demanding. distribution Significant of magnetic effort fields is being and losses invested in coresin improving of electric the devices accuracy are often of hysteresis built upon models. these Thereforedata. of special interest are models based on convenientA convenient choice model of analytical of hysteresis functions, should whose be simple parameters to use in are calculations, easily determined. described For with example, a small setin [9of– 12parameters,], a hysteresis and model provide based a good on the fit hyperbolic to the experimental tangent function data. isVarious discussed, mathematical whereas, in models [13,14], havesimilar been descriptions, developed involving to describe the hysteresis arctan function, loops, such are presented. as the wide-spread Jiles–Atherton (JA) [2–4] and In Preisach this paper, formalisms we present [5 an–8]. improved However, hysteresis parameter model, determination based on hypergeometric for the JA functions, model can which be complicated,defines a class whereas of parameterized the Preisach curves model suitable can be for computationally modeling of major demanding. and minor Significant hysteresis effort loops, is beingas well invested as first order in improving reversal curves the accuracy (FORCs). of Thehysteresis method models. for generating Therefore FORCs, of special presented interest in this are modelspaper, is based based on on convenient methods described choice of in analytical [15–17], with functions, a modification whose parameters based on numerical are easily calculation determined. of Forthe inverseexample, model in [9 function.–12], a hysteresis Many authors model have based explored on the translationhyperbolic andtangent scaling function to generate is discussed, FORCs whereas,and minor in loops,[13,14], to similar mention descriptions, as representative involving examples the arctan Refs. function, [18–21 are]. Numerical presented. procedures for parameterIn this determinationpaper, we present and FORCan improved construction, hysteresis specific model, to ourbased model, on hypergeometric are presented. functions, The good whichaccuracy defines of the a model class of is verifiedparameterized on experimental curves suitable data obtained for modeling from measurementsof major and minor on commercially hysteresis loops,available as well steel as sheets first order and grain-oriented reversal curves electrical (FORCs). steel. The method for generating FORCs, presented in this paper, is based on methods described in [15–17], with a modification based on numerical 2. Model Description calculation of the inverse model function. Many authors have explored translation and scaling to generateAn essentialFORCs and factor minor in choosing loops, ato good mention analytical as representative model is how examples accurately Refs. it describes [18–21]. the Numerical physical proceduresbehavior under for the parameter applied magnetic determination field, when and FORC the domain construction, structure specific changes todue our to domain model, wall are presented.motion, pinning, The goodand wall accuracy rotation. of Starting the model from a is simple verified model on for experimental the magnetic data moment obtained of the fromatom measurementsand the quantum on interpretation commercially of available spin, the steel energy sheets of the and magnetic grain-oriented dipole is electrical expressed steel. via probabilities; see, for example [9,22,23]. This approach leads to the model of an anhysteretic magnetization curve 2.expressed Model Description with the hyperbolic tangent function An essential factor in choosing a good analytical model is how accurately it describes the mH e2x 1 2 physical behavior underM the= M appliedS tanh magnetic= M S field,− when= M theS 1 domain structure, changes due (1) to k T e2x + 1 − e2x + 1 domain wall motion, pinning, and wallB rotation. Starting from a simple model for the magnetic momentwhere M Sofis the saturation atom and magnetization, the quantumm interpretationis magnetic moment, of spin, ktheB is energy the Boltzmann of the magnetic constant, dipole and T is expressedtemperature. viax probabilities;= mH/kBT is thesee, reduced for example field strength [9,22,23]. expressed This approach in dimensionless leads to the units. model of an anhysteretic magnetization curve expressed with the hyperbolic tangent function

mH e2x 12 MM S tanhMMS 22xx S 1  , (1) kTB ee11 Energies 2020, 13, 6500 3 of 14

Hysteresis loop branches can be constructed by shifting the anhysteretic curve (1) horizontally by the value of hc, dependent on the coercive field. In this way, only irreversible magnetization occurring due to domain wall motion is included in the model. Reversible magnetization due to domain rotation is represented by the term qH, which exhibits an approximately linear dependence on the applied magnetic field. It is worth mentioning that the well-known Landau–Lifshitz–Gilbert (LLG) equation can be used to describe the hysteresis phenomenon with a simplified micromagnetic model that can be combined with a magnetic circuit simulation [24], yet it requires a relatively long computation time. Reference [25] points out that some solutions to the LLG equation can be explicitly expressed with confluent hypergeometric functions, which are also included in the present model. Analytical hysteresis models based on hyperbolic functions [10] usually depend on physical parameters such as magnetization at saturation, coercive force, and temperature. Hysteresis loops are thus mathematically expressed as B = B(H) or M = M(H). We refer the readers to the book [9] for further details. Consider the confluent hypergeometric function [26–28]

X∞ (a) xn F (a; b; x) = n (2) 1 1 (b) n! n=0 n where (a) = a(a + 1) (a + n 1) for n > 0 and (a) = 1, with (b) defined in the same way. n ··· − 0 n For certain combinations of a, b, and x, the hypergeometric function reduces to some of the special x functions. One such case is e = 1F1(a; a; x) for any arbitrary a. Taking tanh from (1) and replacing e2x with X∞ (a) F (a; 1; 2x) = n (2x)n 1 1 2 n=0 (n!) we get 2 f (a, x) = 1 (3) − 1 +1 F1(a; 1; 2x) Let us now define 1 Bupper(H) = Bs f a, (H + hc)τ− + qH, 1 (4) B (H) = Bs f a, ( H + hc)τ qH, lower − − − − to describe the descending and ascending branches of hysteresis, respectively, where hc is coercive force, Bs is saturation magnetic flux density, and τ is the scaling factor. For simplicity, we denote the model (4) with F(a, H) unless a specific branch is addressed, and, for a = 1, we use the shorthand notation T(H) = F(1, H). The term Bs f (a, (H + hc)/τ) in (4) is related to irreversible magnetization, and qH to reversible magnetization. This form is well suited for symmetric major and minor loops. The constant d can be added to (4) to account for vertical displacement occurring in asymmetric curves and FORCs, and also to compensate for adjustments needed to make the model curves coincide at loop tips. The hysteresis model based on the hyperbolic tangent has a physical explanation, but it does not include all physical effects occurring in materials. By adjusting the parameter a in (3), the model curve is adjusted to better represent measured data when the material approaches saturation under magnetization. As the distance between a and 1 increases, the difference between f (a, x) and the hyperbolic tangent becomes more prominent, cf. Figure2. This property of (3) has been shown to lead to improved agreement between the model and measured data. However, the difference shown in Figure2 must be accounted for in the parameter determination procedure. Energies 2020, 13, x FOR PEER REVIEW 4 of 13 shownEnergies 2020 to , lead13, 6500 to improved agreement between the model and measured data. However,4 ofthe 14 difference shown in Figure 2 must be accounted for in the parameter determination procedure.

FigureFigure 2. The 2. The difference diﬀerence between between f(,) af (a x, x )andand tanh( tanh(xx )) forfor aavalues values from from 0.6 0.6 to to 1.4. 1.4.

3. Parameter Determination 3. Parameter Determination Estimates of the model parameters τ, B , h , q, d can be determined from the measured data; Estimates of the model parameters  ,Bs , hc , q ,d can be determined from the measured data; this method is usually supplemented with a numericalsc fitting procedure. The quality of the fit can be this method is usually supplemented with a numerical fitting procedure. The quality of the fit can be assessed by examining numerical indicators such as the relative error of the residual vector expressed assessed by examining numerical indicators such as the relative error of the residual vector via the Euclidean norm tv tv expressed via the Euclidean norm Xn Xn R = (F(a h ) b )2 b 2 V nn, i i / i (5) − 2 i=1 i=1 2 RV F a, h i b i b i (5) or graphically, by plotting the error expressedii11 as a normalized difference between the model and orexperimental graphically, data.by plotting Here, (thehi, berrori), i = expressed1, 2, ... , n asdenotes a normalized an array difference of measured between data. the Curve model fitting and involves iterative solving of a non-linear least squares problem using methods such as Newton or experimental data. Here, hbii,  , in1,2, , denotes an array of measured data. Curve fitting Levenberg–Marquardt. As the process involves complex calculations, i.e., evaluating the Jacobian at involves iterative solving of a non-linear least squares problem using methods such as Newton or each iteration, a good choice of initial values is important. Wolfram Mathematica software was used Levenberg–Marquardt. As the process involves complex calculations, i.e., evaluating the Jacobian at for numerical calculations. each iteration, a good choice of initial values is important. Wolfram Mathematica software was used for3.1. numerical Choice of thecalculations. Initial Values

3.1. ChoiceInitial of values the Initial for theValues optimization procedure are determined from the physical properties of the curve in a manner similar to the one described in [10]. Model parameters can be obtained directly Initial values for the optimization procedure are determined from the physical properties of the for F(1, H) and then used as initial values in the fitting procedure with the additional parameter a in curve in a manner similar to the one described in [10]. Model parameters can be obtained directly for F(a, H). Depending on the shape of the curve and the choice of initial parameters, the optimization FH1, and then used as initial values in the fitting procedure with the additional parameter a in procedure  may converge to a less than satisfactory stationary point, or not converge at all. If necessary, theF a initial, H  . Depending values can beon adjustedthe shape using of the interactive curve and calculation the choice possibilitiesof initial paramete of the Mathematicars, the optimization GUI by procedurethe Manipulate may function.converge In to this a less way, than it is satisfactory possible to visually stationary track point, the eorff ect not of converge parameters at on all. the If necessary,model curve the and initial determine values constraints can be adjusted for the fitting using procedure. interactive From calculation our experience, possibilities a reasonable of the Mathematicaconstraint on GUI the parameterby the Manipulatea prevents function excess. distortionIn this way of, it the is modelpossible curve, to visually for example: track the effect of parameters on the model curve and determine constraints for the fitting procedure. From our 0.5 a 1.4 (6) experience, a reasonable constraint on the parameter≤ ≤ a prevents excess distortion of the model curve, for example: 3.2. Coincidence Point Adjustment 0.5a 1.4 (6) The symmetrical tanh function alone cannot represent asymmetry in the magnetization and demagnetization parts of hysteresis. To resolve this, a displacement parameter d is usually added 3.2.to theCoincidence model T P(ointH) to Adjustment shift the upper and lower branches. An example is presented in Figure3, 4 with model parameters hc = 382.4, τ = 272.9, Bs = 0.554, q = 2.17 10 , and d = 0.117. This simple The symmetrical tanh function alone cannot represent asymmetry× − in the magnetization− and demagnetizationapproach cannot parts be applied of hysteresis. to F(a, H To) due resolve to the this, nonlinear a displacement dependence parameter of parameters. d is usually added

Energies 2020, 13, x FOR PEER REVIEW 5 of 13 Energies 2020, 13, x FOR PEER REVIEW 5 of 13 to the model TH  to shift the upper and lower branches. An example is presented in Figure 3, to the model TH to shift the upper and lower branches. An example 4is presented in Figure 3, with model parameters  hc  382.4 ,   272.9 , Bs  0.554 , q 2.17 10 , and d 0.117 . This h  382.4   272.9 B  0.554 q 2.17 104 d 0.117 withsimpleEnergies model 2020approach, 13 parameters, 6500 cannot bec applied to, F a, H  , dues to the nonlinear, dependence, and of parameters.. 5This of 14 simple approach cannot be applied to F a, H  due to the nonlinear dependence of parameters.

Figure 3. The effect of the displacement parameter d in TH  . FigureFigure 3. The 3. The effect effect of the of the displacement displacement parameter parameter d d inin TTH(H). . Let us observe the example in Figure 4, where the measured data is approximated with TH  Let us observe the example in Figure4, where the measured data is approximated with T(H) Let us observe the example in Figure 4, where5 the measured data5 is approximated with TH for hc  518.1,  1017.1 , Bs  1.72 , q 3.5 10 5, and d  8.6  10 . 5 Let us note that the model  for hc = 518.1, τ = 1017.1, Bs = 1.72, q = 3.5 10− , and d = 8.6 10− . Let us note that the model h  518.1  1017.1 B  1.72 q 3.5× 105 d  −8.6 × 105 forT( H H)c+ d providesprovides, a a good good, fit fit fors for the the loop, loop in Figurein Figure3., However, and 3. However, for the for loop the. Let loop in Figure us in note Figure4 the that best4 thethe fit best model value fit Tvalueof Hd is o practicallyf d dprovides is practically zero, a good i.e., zero, fitd has for i.e. nothe, d favorableloop has inno Figure favorable influence 3. However, influence on the model for on thethe curve, loopmodel in which curve,Figure visibly which 4 the deviates bestvisibly fit valuedeviatesfrom theof fromd measured is practicallythe measured data. zero, This data. isi.e. a ,This knownd hasis a limitation noknown favorable limitation of the influence model of the Ton (modelH the). model TH curve, . which visibly deviates from the measured data. This is a known limitation of the model TH  .

FigureFigure 4. 4. MeasuredMeasured data data modeled modeled with with T(H).. Figure 4. Measured data modeled with . OnOn the the other hand, the presented model (4) is based on an inherently asymmetrical function f (a, H) which better describes asymmetric branches. Let us remark that parameter values for F(a, H), f a,On H  the which other betterhand, describesthe presented asymmetric model (4) branches. is based onLet an us inherently remark thatasymmetrical parameter function values a , 1 cannot simply be carried over from T(H). Instead, they should be used as initial values in a new f aF, H a, H whicha  1 better describes asymmetric branches.TH Let us remark that parameter values foroptimization  , procedure, cannot which simply should be carried be performed over from on all parameters . Instead, they simultaneously, should be used including as initiala. F a, H a 1 TH forvaluesPlotting in , a Fnew(a, H cannot) optimizationwith simply the same be procedure carried set of parameters over, which from asshould in Figure. Instead, be 4 performed and they some should ona , be all1 used, the parameters as situation initial valuessimulsimilartaneously, into the a new one including in optimization Figure a5 .is obtained, procedure where, which the upper should and lowerbe performed branches are on apart all parameters by ∆B and simuldo notPlottingtaneously, coincide including at Htip with, B tipa the. , Figure same5 set. To of enforceparameters coincidence, as in FigureBs 4is and adjusted some bya  introducing1, the situation the additionalPlotting constraint with the same set of parameters as in Figure 4 and some a 1, the situation similar to the one in Figure 5 is obtained, where the upper and lower branches are apart by B and similar to the one in Figure 5 is obtained,Bupper whereHtip = theBlower upperHtip and= lowerBtip branches are apart by B and(7)

which causes Bs to deviate somewhat from its original physical interpretation. Furthermore, additional

constraint is required to limit the deviation, for example

0.8bmax Bs 1.2bmax (8) ≤ ≤ Energies 2020, 13, x FOR PEER REVIEW 6 of 13

do not coincide at HBtip, tip , Figure 5. To enforce coincidence, Bs is adjusted by introducing the additional constraint

BHBHBupper tip  lower tip tip (7) which causes Bs to deviate somewhat from its original physical interpretation. Furthermore, additionalEnergies 2020 ,constraint 13, x FOR PEER is required REVIEW to limit the deviation, for example 6 of 13

0.8bmax Bs 1.2 b max (8) do not coincide at HBtip, tip , Figure 5. To enforce coincidence, Bs is adjusted by introducing the Although at the first sight it may seem that this situation could have been resolved by simply additional constraint scaling the saturation value or by shifting the branches by B , the effects of the parameters a , Bs and d are nonlinearly dependentBHBHB upper and astip  such lower must tip be tip treated simultaneously. Thus, ( the7) constraints (6), (7) and (8) must be introduced into the fitting problem. B whichWith causes the introduceds to deviate constraints, somewhat a new from fitting its originalproblem physicalwith one interpretation.additional parameter Furthermore,, , is formedadditional and constraint solved. is The required resulting to limit model the deviation, curve for fora example0.65 , hc  540.1 ,   489.8 , Bs  1.53 , q 8.7 105 d  0.63 , and shows good conformance0.8bmax Bs 1.2to experimental b max data (Figure 6). (8) The comparison of relative errors and mean relative errors between models is shown in Figure 7 Although at the first sight it may seem that this situation could have been resolved by simply for the upper branch of the loop; the results are symmetrical in H for the lower branch. The model scaling the saturation value or by shifting the branches by B , the effects of the parameters a , Bs EnergiesF a, H2020 yields, 13, 6500 a lower maximum relative error of 4.5% versus 13% for TH  and a lower mean6 of 14 and d are nonlinearly dependent and as such must be treated simultaneously. Thus, the error of 2% versus 4.5%. constraints (6), (7) and (8) must be introduced into the fitting problem. With the introduced constraints, a new fitting problem with one additional parameter, , is formed and solved. The resulting model curve for a  0.65 , hc  540.1 ,   489.8 , Bs  1.53 , q 8.7 105 , and d  0.63 shows good conformance to experimental data (Figure 6). The comparison of relative errors and mean relative errors between models is shown in Figure 7 for the upper branch of the loop; the results are symmetrical in H for the lower branch. The model F a, H  yields a lower maximum relative error of 4.5% versus 13% for TH  and a lower mean error of 2% versus 4.5%.

FigureFigure 5. 5.FF a(a, H, H) for aa=0.715before before the the adjustment adjustment of of coincidence coincidence point point is is applied. applied.

Although at the first sight it may seem that this situation could have been resolved by simply scaling the saturation value or by shifting the branches by ∆B, the effects of the parameters a, Bs and d are nonlinearly dependent and as such must be treated simultaneously. Thus, the constraints (6), (7) and (8) must be introduced into the fitting problem. With the introduced constraints, a new fitting problem with one additional parameter, a, is formed 5 and solved. The resulting model curve for a = 0.65, hc = 540.1, τ = 489.8, Bs = 1.53 , q = 8.7 10 , × − and d = 0.63Figure shows 5. F good a, H conformance for a  0.715 to before experimental the adjustment data(Figure of coincidence6). point is applied.

Figure 6. Final result of the ﬁtting procedure with F(a, H), a = 0.65.

The comparison of relative errors and mean relative errors between models is shown in Figure7 for the upper branch of the loop; the results are symmetrical in H for the lower branch. The model F(a, H) yields a lower maximum relative error of 4.5% versus 13% for T(H) and a lower mean error of 2% versus 4.5%. Energies 2020, 13, x FOR PEER REVIEW 7 of 13 Energies 2020, 13, x FOR PEER REVIEW 7 of 13 Energies 2020, 13, 6500 Figure 6. Final result of the fitting procedure with F a, H  , a  0.65 . 7 of 14 Figure 6. Final result of the fitting procedure with F a, H  , a  0.65 .

Figure 7. Comparison of errors between from Figure 6 and TH  from Figure 4. FigureFigure 7. Comparison 7. Comparison of errors of errors between between F(a, H )fromfrom Figure Figure 6 and TTH(H)from from Figure Figure4. 4.

4.4. Measurements 4. Measurements ExperimentalExperimental hysteresis data was obtained in a laboratory, laboratory, at at room room temperature, temperature, using using ring ring Experimental hysteresis data was obtained in a laboratory, at room temperature, using ring specimensspecimens equippedequipped withwith primaryprimary andand secondarysecondary windings.windings. A principal scheme of the measurement specimenssetup is shown equipped in Figure with primary8. The value and ofsecondary magnetic windings. field strength A principal H is proportional scheme of the to currentmeasurement i and setup is shown in Figure8. The value of magnetic field strength H is proportional to current i1 and setup is shown in Figure 8. The value of magnetic field strength H is proportional to current i1 and thethe voltage voltage drop dropu 1 uon1 resistoron resistorR1. TheR1 . value The value of magnetic of magnetic flux density flux B denis proportionalsity B is proportional to integrated to the voltage drop u1 on resistor R1 . The value of magnetic flux density B is proportional to voltageintegrated induced voltage in induced the secondary in the windingsecondary of winding the sample. of the The sample. values The of voltage values dropof voltageu1 and drop output integrated voltage induced in the secondary winding of the sample. The values of voltage drop voltageand output of the voltage integrator of the are integrator simultaneously are simultaneously sampled on sampled the data acquisitionon the data equipmentacquisition (DAQ)equipment and andstored.(DAQ) output Theand voltage setupstored. excites of The the setup theintegrator sample excites are with the simultaneously magneticsample wit fieldh magneticsampled strength onHfield, generatedthe strength data acquisition by H the, generated programmable equipment by the (DAQ)generatorprogrammable and of stored. the generator control The setup signal. of the excites control Measurements the signal.sample Measurements maywith bemagnetic carried may field out be using strength carried sinusoidal Hout, generated using excitation sinusoidal by the for programmablefrequenciesexcitation for from frequencies generator 10 Hz to of from 1 kHz.the 10control Hz to signal. 1 kHz. Measurements may be carried out using sinusoidal excitation for frequencies from 10 Hz to 1 kHz.

FigureFigure 8.8. Schematic of the measurement setup.setup. Figure 8. Schematic of the measurement setup. MeasurementsMeasurements werewere conductedconducted usingusing thethe measurementmeasurement setupsetup shownshown inin FigureFigure9 9.. ExcitationExcitation forfor Measurements were conducted using the measurement setup shown in Figure 9. Excitation for thethe primary primary coil coil was was supplied supplied from from the programmable the programmable GF-1 function GF-1 function generator. generator. A Tektronix A Tektronix TDS5032 the primary coil was supplied from the programmable GF-1 function generator. A Tektronix digitalTDS5032 oscilloscope digital oscilloscope was used for was simultaneous used for simultaneous data acquisition data on acquisition two channels. on two Voltage channels. on a precision Voltage TDS5032 digital oscilloscope was used for simultaneous data acquisition on two channels. Voltage 1onΩ ashunt precision of negligible 1 shunt inductance, of negligible which inductance, was connected which in was series connected with the primaryin series coilwith of the the primary toroidal on a precision 1 shunt of negligible inductance, which was connected in series with the primary sample,coil of the was toroidal sampled sample, on the was ﬁrst sampled channel. on Output the first from channel. the electronic Output integrator from the electronic was sampled integrator on the coil of the toroidal sample, was sampled on the first channel. Output from the electronic integrator secondwas sampled channel. on the Measurements second channe werel. Measurements performed for were excitation performed frequencies for excitation from 10Hz, frequencies 50Hz up from to was sampled on the second channel. Measurements were performed for excitation frequencies from 450Hz10Hz, 50Hz and for up current to 450Hz intensities and for from current tens intensities to hundreds from of milliamperes.tens to hundreds For theof milliamperes. purpose of modeling, For the 10Hz, 50Hz up to 450Hz and for current intensities from tens to hundreds of milliamperes. For the wepurpose have usedof modeling, the lowest we possible have used frequency the lowest in order possible to avoid frequency the distorting in order etoﬀ ectavoid of eddythe distortin currentsg purpose of modeling, we have used the lowest possible frequency in order to avoid the distorting inducedeffect of ineddy the currents conductive induced core material in the conductive on the shape core of material hysteresis on loop.the shape of hysteresis loop. effect of eddy currents induced in the conductive core material on the shape of hysteresis loop.

Energies 2020, 13, 6500 8 of 14 Energies 2020, 13, x FOR PEER REVIEW 8 of 13 Energies 2020, 13, x FOR PEER REVIEW 8 of 13

Figure 9. Measurement setup (A—ring sample; B—integrator; C—resistor; D—generator; Figure 9. 9. MeasurementMeasurement setup setup (A(A—ring—ring sample;sample; B—integrator; B—integrator; CC—resistor;—resistor; DD—generator;—generator; E—oscilloscope; F, G—voltmeter). EE—oscilloscope;—oscilloscope; F, G—voltmeter).G—voltmeter).

Sample 1 1 was was a a specimen specimen of of commercially commercially available available 0.4mm 0.4mm steel steel sheets sheets used used for for manufacture manufacture of deviceof device cases, cases, whereas whereas Sample Sample 2 was2 was a a specimen specimen of of grain grain oriented oriented electrical electrical steel steel (M103 (M103-27p),-27p), cf. cf. Figure 10.

Figure 10. Two ring samples used in measurements, with and without windings. Figure 10. TwoTwo ring samples used in measurements, with and without windings.windings.

55.. Numerical Numerical R Resultsesults The accuracy and practical suitability of the presented model was tested on experimental data obtained from laboratory measurements presented in the the previous previous section. section. Hysteresis data was obtained by measuring primary current and integrated output vol voltagetage from the secondary coil of the test core, core, with with di ﬀ differenterent excitation excitation currents currents at a low at excitationa low excitation frequency. frequency Calculations. Calculations were performed were performedin Mathematica, in Mathematica, using the NonlinearModelFit using the NonlinearModelFit function. function.

55.1..1. Major Major L Loopsoops TH( ) F( a, H ) The major major loop loop for for Sample Sample 1 1is isshown shown in inFigu Figuresres 4 4and and 66, ,modeled modeled with with TH T H andand FF aa,, HH respectively. Model errors are compared in Figure7. This example was used to discuss the respectively. Model errors are compared in Figure 7. This example was used to discuss the parameter determination procedure in Section3, and the advantages of the presented model are also parameter determination procedure in Section 3, and the advantages of the presented model are also discussed there. discussed there. The major major loop loop for for Sample Sample 2 2is isshown shown in inFigure Figuress 11 11and and 12, 12modeled, modeled with with TH T(H and) and FF( aa,, HH) The major loop for Sample 2 is shown in Figures 11 and 12, modeled with 4TH  and F a, H  respectively. Parameter values are hc = 68.75, τ = 5.66, Bs = 1.44, q = 2.5 10− ,4 d 0 for T(H) and h  68.75   5.66 B  1.44 q 2.5 104 d  0 respectively. Parameter values are hc  68.75 , 4 , Bs  1.44 , × , ≈d  0 for respectively.a = 0.52, hc = Parameter70.78, τ = 6.29 values, Bs are= 1.53 c , q = 1.86,  105.66− , ,d =s 0.055 for, qF(a2.5, H). 10 It can, be seen for that the × 4 and a  0.52 , hc  70.78 ,   6.29 , Bs  1.53 , q 1.86 104 , d  0.055 for . It can be seen presentedand a  0.52 model, hc follows 70.78 , the   measured6.29 , Bs data 1.53 exceptionally, q 1.86 10 well, d during 0.055 the for magnetization . It can stage be of seen the that the presented model follows the measured data exceptionally well during the magnetization stage of the hysteretic cycle. The comparison of relative errors and mean relative errors between the

Energies 2020, 13, 6500 9 of 14 Energies 2020, 13, x FOR PEER REVIEW 9 of 13 Energies 2020, 13, x FOR PEER REVIEW 9 of 13 hysteretic cycle. The comparison of relative errors and mean relative errors between the two models two models (Figures 11 and 12) is shown in Figure 13. With the mean error of 10%, F a, H  is two(Figures models 11 and (Figure 12) iss 11 shown and in12 ) Figure is shown 13. With in Figure the mean 13. With error the of 10%, meanF ( errora, H) is of clearly 10%, F better a, H than is clearly better than TH  yielding 21%. clearlyT(H) yielding better than 21%. TH  yielding 21%.

Figure 11. Major loop for Sample 2 modeled with TH  . FigureFigure 11. 11. MajorMajor loop loop for for Sample Sample 2 2modeled modeled wit withh THT(H)..

Figure 12. Major loop for Sample 2 modeled with F a, H  , a  0.52 . FigureFigure 12. 12. MajorMajor loop loop for for Sample Sample 2 modeled 2 modeled with with FF( aa, H ),, aa=0.52.0.52 .

FigureFigure 13. Comparison 13. Comparison of errors of errors for Sample for Sample 2 between 2 between F(a, Hand) and T(H )frfromom Figures Figures 1 111 and and 1 122. . Figure 13. Comparison of errors for Sample 2 between and from Figures 11 and 12.

5.2. First Order Reversal Curves 5.2. First Order Reversal Curves

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First order reversal curves can be calculated in various ways from the model for the major loop. 5.2. First Order Reversal Curves We present a technique based on the method established in [17], which in turn is based on earlier papersFirst [15 order] and reversal[16]. In [ curves17], hysteresis can be calculated loops are inexpressed various waysin terms from of themagnetizing model for current the major versus loop. We present a technique based on the method established in [17], which in turn is based on earlier flux linkage, im  in the i plane. A first order reversal curve is defined as im  f  F  x , papers [15] and [16]. In [17], hysteresis loops are expressed in terms of magnetizing current versus flux where iFm   is the equation of ascending branch of major hysteresis loop and xX  is a linkage, im(λ) in the iλ plane. A first order reversal curve is defined as im = f (λ) = F(λ + x), where function describing the displacement of the major branch along the  axis. To apply the considered im = F(λ) is the equation of ascending branch of major hysteresis loop and x = X(λ) is a function methdescribingod to the the model displacement (4), formulas of the must major be branch applicable along in the theλ HBaxis. plane. To apply the considered method to the modelA first (4), order formulas reversal must curve be applicableis generated in between the HB plane. an arbitrary starting point on the descending majorA branch first order and reversalthe positive curve loop is generated tip (Figure between 14). The an curve arbitrary is obtained starting by point displacing on the descending the major branchmajor branchby an amo andunt the expressed positive loopwith tipthe (Figurefunction 14 ). The curve is obtained by displacing the major branch by an amount expressed with the function pw 1 bb  Bb  X b  x 1 1   tip  . 1 !p   !w! (9) 2 Btip b11B tip bb 1 b b1  tip  X(b) = x1 1 − + − . (9) 2 − Btip b1 Btip b1 − 2 − where pk1  , wk1  , k Btip  b B tip  b1  B tip . Parameters  and  depend on 2 ferromagneticwhere p = 1 α materialk, w = 1 + andβk, k can= beBtip adjusted+ b Btip + to b influence1 B− . Parameters the slopeα and of theβ depend magnetic on ferromagnetic path. Here − tip material 0.1 and and can  be0.3 adjusted. to influence the slope of the magnetic path. Here α = 0.1 and β = 0.3.

FigureFigure 14. 14. GeneratedGenerated first ﬁrst order order reversal reversal curve curve on on model model (4) (4)..

For a chosen starting point 1 with coordinates (h , b ), the value of the parameter x is calculated as the 1 1 hb, 1 x For a chosen starting point 1 with coordinates  11 , the value of the parameter 1 is difference x1 = B (h1) Bupper(h1) . It represents the initial displacement between points 10 and 1. lower − calculatedAn arbitrary as the difference point P on x the1 reversal Blower h 1 curve B upper is calculatedh 1  . It represents as the initial displacement between points 1′ and 1. 1 P = B− (b X(b)), b An arbitrary point P on the reversal curvelower is calculated− as h i P B1 b X b, b for b b1, Btip . Since the ascending branch inlower the model (4) is expressed with Blower(H), the inverse ∈ 1 function H = B− (B) is needed for computation. Analytical form of the inverse hypergeometric for b b, B .lower Since the ascending branch in the model (4) is expressed with BH, the inverse function is1 nottip feasible, therefore one must resort to numerical computation. lower   1 functionThe valueHBB oflower the inverse is needed function for computation. is computed by Analytical numerically form solving of the the inverse equation hypergeometricBlower(h) = b0 for any given value b = b X(b). The initial value h for the root-finding method is chosen as function is not feasible,0 therefore− one must resort to numerical0 computation. follows. The hysteresis branch is approximated with several linear segments, constructed by a linear The value of the inverse function is computed by numerically solving the equation Blower  h  b0 fit. The value h0 is obtained from the inverse of the closest linear segment (Figure 15). The root-finding for any given value b0  b X b. The initial value h0 for the root-finding method is chosen as method then calculates the numerical solution hn in n iterations, which is taken as the value of the follows. The1 hysteresis branch is approximated with several linear segments, constructed by a linear inverse B− . Thanks to the shape of hysteresis branches, convergence conditions for the iterative fit. The valuelower is obtained from the inverse of the closest linear segment (Figure 15). The method are satisfied for all values of b0. In our experiments, using five linear segments (Figure 15), 1 root-finding method then calculates the numerical solution hn in n iterations, which is taken as the iterative method took an average of 3.6 iterations to calculate and plot Blower− . 1 the value of the inverse Blower . Thanks to the shape of hysteresis branches, convergence conditions

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forEnergies the 20 iterative20, 13, x FOR method PEER REVIEW are satisfied for all values of b0 . In our experiments, using five 11 linear of 13 segments (Figure 15), the iterative method took an average of 3.6 iterations to calculate and plot 1 for the iterative method are satisfied for all values of b0 . In our experiments, using five linear Blower . segmentsEnergies 2020 (Figure, 13, 6500 15), the iterative method took an average of 3.6 iterations to calculate and11 of plot 14 1 Blower .

Figure 15. Obtaining initial values for numerical calculation of the inverse model function. Convergence conditions are satisfied due to the shape of the model curve. FigureFigure 15. 15. ObtainingObtaining initial initial values values for numerical for numerical calculation calculation of the inverse of the model inverse function. model Convergence function. 5.3. MinorConvergenceconditions Loops areconditions satisﬁed are due satisfied to the shape due ofto thethe modelshape curve.of the model curve. 5.3.One Minor of the Loops important properties of hysteresis models is the ability of the model to represent 5.3. Minor Loops minor loops.One of There the important are various properties approaches of hysteresis to minor models loop is construction, the ability of the e.g. model, by shifting to represent and minorscaling majorloops.One loop Thereof branches the areimportant various and adjusting approachesproperties the of toloop minorhysteresis tips. loop Several models construction, minor is theloops e.g.,ability for by Sample shiftingof the model1, and approximated scaling to represent major by minorTHloop loops.and branches F There a, and H are adjustingare various shown the approaches in loop Figure tips. 1 6 Severalto and minor Figure minor loop 1 loops 7construction,, respectively. for Sample e.g. Comparing 1,, approximatedby shifting relative and by scaling Terrors(H) major loop branches and adjusting the loop tips. Several minor loops for Sample 1, approximated by ( RVand) asF ( thea, H ) minorare shown loops in Figures approach 16 and the 17 ma, respectively.jor loop, advantage Comparing of relative the model errors ( (4R)V becomes) as the minor more TH and F a, H are shown in Figure 16 and Figure 17, respectively. Comparing relative errors pronouloops nced approach (Figure the 1 major7). loop, advantage of the model (4) becomes more pronounced (Figure 17).

( RV ) as the minor loops approach the major loop, advantage of the model (4) becomes more pronounced (Figure 17).

FigureFigure 16. 16. MinorMinor loops loops for for Sample Sample 1 1approximated approximated with with TTH(H ). .

Figure 16. Minor loops for Sample 1 approximated with TH  .

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FigureFigure 17. 17.MinorMinor loops loops for for Sample Sample 1 approximated with withF (aF, H a),. H Comparison . Comparison of errors of errors obtained obtained with withT (HTH) and andF(a, H). Labels. A,Labels B, C andA, B D, denoteC and theD correspondingdenote the corresponding loops with diff loopserent amplitudes. with different 6.amplitudes Conclusions. By employing model curves based on a family of parameterized hypergeometric functions, 6. Conclusions a more accurate hysteresis model with one free parameter is constructed. The presented model has aBy convenient employing analytical model formcurves that based makes on ita suitablefamily of for parameterized practical applications. hypergeometric The conditions functions, for a moreparameter accurate determination hysteresis model using with a nonlinear one free fittingparameter procedure is constructed. are discussed, The presented as well as themodel applied has a convenientmodification analytical of the method form that for generating makes it suitablefirst order for reversal practical curves. application Using experimentals. The conditions data, it is for parametershown that determination the model can using be used a nonlinear with good fitting accuracy procedure to represent are discussed, major and minoras well loops, as the as wellapplied as modificationsimulate first of the order method reversal for curves. generating first order reversal curves. Using experimental data, it is shown thatThe the ease model of use can and be the used flexibility with good of simple accuracy analytical to represent T(x) model major is and retained minor in loops, the case as ofwell the as simulateextended first description, order reversal based curves. on hypergeometric functions with a single free parameter. The generalized modelThe ease is able of use to describeand the bothflexibility major of and simple minor analytical loops of T(x) representative model is retained soft magnetic in the case materials of the extendedaccurately. description, This fact is based important on hypergeometric from the practical functions point of view, with since a single the examined free parameter. description The generalizedis much simplermodel inis implementationable to describe thanboth the major Jiles-Atherton and minor or loops Preisach of representative models. The application soft magnetic of hypergeometric functions allows one to consider different classes of magnetic materials. Since the materials accurately. This fact is important from the practical point of view, since the examined hypergeometric function may be reduced to hyperbolic tangent in the limiting case and the description description is much simpler in implementation than the Jiles-Atherton or Preisach models. The based on hyperbolic tangent i.e., the T(x) model is particularly suited for materials with strong uni-axial application of hypergeometric functions allows one to consider different classes of magnetic anisotropy, we believe that our generalized model may represent soft magnetic materials with differing materials. Since the hypergeometric function may be reduced to hyperbolic tangent in the limiting anisotropy level. The examination of physical meaning of hypergeometric series shall be the subject of caseforthcoming and the description research. based on hyperbolic tangent i.e., the T(x) model is particularly suited for materials with strong uni-axial anisotropy, we believe that our generalized model may represent soft magneticAuthor Contributions: materials withConceptualization, differing anisotropy D.H. and level. Ð.H. methodology,The examination D.H., K.C.; of physical software, meaning D.H., Ð.H.; of validation, D.H., Ð.H.; formal analysis, D.H., Ð.H.; investigation, D.H., Ð.H.; resources, D.H., Ð.H.; data curation, hypergeometricÐ.H.; writing—original series shall draft be preparation, the subject D.H., of forthcoming Ð.H.; writing—review research. and editing, K.C.; visualization, D.H.; Authorsupervision, Contributions: D.H., K.C.; Conceptualization, project administration, D.H. D.H.; and funding Đ.H. methodology, acquisition, K.C. D.H., All authors K.C.; software,have read and D.H., agreed Đ.H.; to the published version of the manuscript. validation, D.H., Đ.H.; formal analysis, D.H., Đ.H.; investigation, D.H., Đ.H.; resources, D.H., Đ.H.; data curation,Funding: Đ.H.;This writing research— receivedoriginal no draft external preparation, funding. It D.H., was supported Đ.H.; writing by the— Deanreview of Faculty and editing, of Electrical K.C.; Engineering, Cz˛estochowaUniversity of Technology from the statutory funds for research purposes. visualization, D.H.; supervision, D.H., K.C.; project administration, D.H.; funding acquisition, K.C. All authors haveAcknowledgments: read and agreed toThis the researchpublished was version carried of out the by m Dejanaanuscript. and Ðor de¯ Herceg within the framework of research grants TR 32019 and 174030 from Ministry of Education, Science and Technological Development of the Republic Funding:of Serbia. This research received no external funding. It was supported by the Dean of Faculty of Electrical Engineering,Conflicts ofCzęstochowa Interest: The University authors declare of Technology no conflict from of interest. the statutory funds for research purposes. Acknowledgments: This research was carried out by Dejana and Đorđe Herceg within the framework of research grants TR 32019 and 174030 from Ministry of Education, Science and Technological Development of the Republic of Serbia.

Conflicts of Interest: The authors declare no conflict of interest.

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