IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014 7009404

Improvement and Application of the Viscous-TypeFrequency-Dependent Preisach Model

Miklós Kuczmann and Gergely Kovács

Department of Automation, Széchenyi István University, Gyor H 9026, Hungary

Iron parts of electrical machines are made of nonoriented isotropic ferromagnetic materials. The finite element method (FEM)is usually applied in the numerical field analysis and design of this equipment. The scalar Preisach hysteresis model has beenimplemented for the simulation of static and dynamic magnetic effects inside the ferromagnetic parts of motors. The dynamic modelis an extension of the static one; an extra magnetic field intensity term is added to the output of the static inverse model. This is aviscosity-type dynamic model. The fixed point method with stable scheme has been realized to take frequency-dependent anomalouslosses into account in FEM. This scheme can be used efficiently in the frame of any potential formulations of Maxwell’s equations.The comparison between measured and simulated data using a toroidal core shows a good agreement. A modified nonlinear versionof T.E.A.M. Problem No. 30.a is also shown to test the hysteresis model in the FEM procedure.

Index Terms— Dynamic hysteresis, dynamic Preisach model, finite element method (FEM), fixed point method.

I. INTRODUCTION

S IZE reduction of electromechanical devices in the newelectric cars and hybrid vehicles is a new trend in the

vehicle industry. The requirement of increasing the torqueof these drives and the reduction of losses are contradictoryconditions; this is why the design of these motors is agreat challenge of electrical engineering. The more accuratesimulation of losses issued inside the ferromagnetic parts ofthe motors is one of the main goals of basic research in thenumerical field analysis of electromechanical devices.

This paper presents a measurement system to obtain hystere-sis characteristics measured on a simple geometry, applying atransformer core with toroidal shape. In the first study, thescalar hysteresis is analyzed; vector modeling of this isotropicphenomenon is a future task of the research. The aim is torealize a macroscopic model of hysteresis that can be appliedin finite element method (FEM) simulations efficiently. Thescalar Preisach hysteresis model has been implemented andidentified from measured data. A model based on viscosity hasbeen realized that can be used in the numerical field analysis.The model has been inserted into finite element simulationvia the fixed point iteration scheme; finally the generalizeddynamic Preisach hysteresis model has been identified andapplied in the numerical analysis. The realized model in theframe of the implemented FEM application can be used tocalculate losses, and other quantities of electromechanicaldevices. Here, comparisons between measured and simulatedhysteresis curves are presented, and a modified nonlinearversion of T.E.A.M. Problem No. 30.a, containing a threephase motor, is shown.

II. MEASUREMENT OF HYSTERESIS

Hysteresis characteristics for identification and testing ofPreisach model have been measured applying a transformer

Manuscript received June 21, 2013; revised August 24, 2013; acceptedSeptember 15, 2013. Date of current version February 21, 2014.Corresponding author: M. Kuczmann (e-mail: kuczmann@sze.hu).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2013.2283398

Fig. 1. Measuring scalar hysteresis.

with toroidal shape. All the measurement tasks are controlledby the functions implemented in LabVIEW software envi-ronment. The communication between the computer and thedevice under test has been realized by National InstrumentData Acquisition (NI-DAQ) card inserted into the computer.The block diagram of the measurement setup can be seen inFig. 1.

The magnetic field intensity H (t) can be calculated from theexcitation current i(t) flowing through the primary coil withNp turns, by H (t) = Npi(t)/ l, where l is the mean lengthof the core. The magnetic flux density B(t) can be obtainedfrom the induced voltage u(t) of the secondary coil with Ns

turns, using u(t) = −Ns S dB/dt , and S is the cross sectionof the core material.

The excitation current is generated by a current supplycontrolled by a waveform generated in the implemented Lab-VIEW environment. The primary current and the secondaryvoltage have been measured by NI hardware. The waveformof the magnetic flux density can be controlled by a feedbackcontroller. Detailed presentation of the measurement systemcan be found in [1] and [2].

First of all, static concentric minor loops have been mea-sured for the identification of static Preisach hysteresis model.

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7009404 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014

Fig. 2. Measured concentric minor loops, f = 5 Hz.

Fig. 3. Measured concentric minor loops, f = 200 Hz.

Controlled, sinusoidal magnetic flux density has been pre-scribed, and the frequency has been selected to be smallenough to ignore eddy current field effects. After some exper-iments, f = 5 Hz has been used. The results can be seenin Fig. 2. Second, dynamic concentric minor loops have beenmeasured for the identification of the dynamic extension of thePreisach model. The waveform of the magnetic flux densityhas been controlled, sinusoidal variation has been prescribed.Measurements have been performed at several frequencies;Fig. 3 shows the dynamic concentric loops at f = 200 Hz.

III. HYSTERESIS MODEL

The Preisach model [3], [4] of scalar hysteresis has beenimplemented in its inverse form, i.e., the magnetic flux densityis the input, and the magnetic field intensity is the output ofthe model.

The well-known staircase line is used to store turningpoints of the magnetization process. This is the memory ofthe model. The Everett function E(α, β) has been identifiedfrom measured concentric minor loops shown in Fig. 2. Thecorresponding Everett function can be seen in Fig. 4. TheEverett function is approximated by 2-D, third-order splinetechnique. In the argument of E , α, and β are the switchingfields of a hysteron [3], [4].

Fig. 4. Everett function.

The static curves can be simulated very accurately in thisway.

The developed static hysteresis model can be applied asbulk model [3], [4], e.g., in circuit simulators or in FEMsimulations as a macroscopic model [4], [5]. The accuracyof the model can be very poor, if anomalous losses accordingto domain wall movements have significant effect [6]. Theexcess field term is taken into consideration as extra magneticfield intensity, depending on the actual value of the magneticflux density and its variation dB/dt , which is depending onthe frequency of supplied sources [6]. The dynamic model ofhysteresis can be identified by the use of concentric minorloops, too, but higher frequency must be supplied (e.g., loopsshown in Fig. 3).

The following viscous-type dynamic model has been studiedin this paper. The time lag of B behind H can be describedby the following equation [11]:

dB

dt= r(B)(H − Hst)

a (1)

from which

H = Hst ±∣∣∣∣

1

r B

dB

dt

∣∣∣∣

1/a

(2)

where r(B) is the dynamic magnetic resistivity, and theparameter controls the model dynamics. In (2), the first termis obtained by the static Preisach model, and the secondterm is the excess magnetic field term, Hexc. This equation isimplemented in finite element simulations, as it is presentedin the next two sections with the following function of r(B):

r(B) = R

1 −(

BBs

)2 (3)

where R = 23.79, and α = 1.0, Bs is the value of magneticflux density in saturation. Identification of R and α has beenperformed by a 1-D FEM model of the core with toroidal shape(Fig. 5), implemented in an optimization algorithm applyingsimplex search method [2]. The waveform of the magneticflux density is prescribed on the Neumann boundary �N . The1-D simplification results in a fast identification process.

IV. FEM MODEL OF THE MEASUREMENT SYSTEM

In the simulation of the measurement system, a 2-D modelhas been realized taking the rotational symmetry of the

KUCZMANN AND KOVÁCS: IMPROVEMENT AND APPLICATION OF VISCOUS-TYPE FREQUENCY-DEPENDENT PREISACH MODEL 7009404

Fig. 5. 1-D and 2-D modeling of the core with toroidal shape.

core into account, as it is shown in Fig. 5. The followingMaxwell’s equations must be solved to simulate the core withtoroidal shape taking eddy currents into account [5], [12], [13]:∇ × H = σE,∇ × E = −B, where H, B, E, and σ are themagnetic field intensity, the magnetic flux density, the electricfield intensity, and the conductivity, respectively.

The nonlinear constitutive relationship with static hysteresisis decomposed into a linear term and a nonlinear residual term,as follows [5]: B = μHst + R. Here, μ is the optimal valueof permeability, to reach global convergence, selected as [5]μ = 2/vmax + vmin, where vmax and vmin are the maximumand the minimum slope of the inverse static hysteresis char-acteristics, i.e. the maximum and the minimum reluctivity.

The index st in the polarization formulation is for theword static, because the static hysteresis model represents therelationship between B and Hst , i.e. Hst = H{B}. Excess lossterm is ignored by this representation, only the nonlinearity,i.e. hysteresis losses, and eddy current losses are present,Hst = Hh + Heddy. These terms are calculated by theFEM procedure. Decreasing the frequency of excitation, theterm Heddy is decreasing automatically, Hh is the frequencyindependent term of the magnetic field intensity.

The above equations are nonexpansive, and the fixed pointiteration scheme through the polarization formulation resultsin a contraction mapping, meaning convergent iterative process[5]. However, excess losses are not present. Excess losses canbe represented by the following scheme.

The magnetic field intensity is decomposed into three partsin the more complex model, H = Hh + Heddy + Hexc. Thelast term is responsible for the anomalous or excess losses[see the second term in (2)] [12], [13]. The advantages andconvergence properties of the fixed point technique can be holdby introducing the excess field term Hexc in the Maxwell’sequations as follows:

∇ × H = σE,∇ × E = −B, B = μ(H − Hexc) + R (4)

i.e., the decomposition of the nonlinear constitutive relation-ship must be rewritten.

The following nonlinear partial differential equation can beobtained:

∇ × ∇ × H + μσ H = μσ Hexc − σ R. (5)

This problem can be solved numerically; here the FEM hasbeen used with the weighted residual method and Galerkin’stechnique [5]. It is noted that the excess field term is presenton the righthand side of (5), and the solution of the problemis the total magnetic field intensity, containing the excess fieldterm, too.

Fig. 6. Measured and simulated hysteresis curves.

The results of simulations can be seen in Fig. 6,f = 200 Hz. The results of eddy current simulations are poor,but the inclusion of excess field term in FEM results in muchmore accurate approximations.

V. FEM MODEL OF THE TEST MOTOR

In this paper, the three-phase induction motor of T.E.A.M.Problem No. 30a [14] has been modified. The original problemconsists of induction motors with linear B–H relationship.Here, the above-mentioned Preisach model represents theconstitutive relationship between the magnetic flux density andthe magnetic field intensity.

The arrangement of the three-phase induction motor prob-lem can be found, e.g., in [14]. The phase groups of the threephases lag each other in phase by 120°. The angular velocityof the rotor is ranging from 0 to 1200 rad/s. The value ofthe upper bound is roughly three times faster than the angularvelocity of the stator field, f = 60 Hz.

The problem has been solved by the magnetic vector poten-tial formulation, B = ∇×A [5], [14]. After some mathematicalformulations, the following partial differential equation can beobtained:

∇ × (v∇ × A) + σ[

A − v × (∇ × A)]

= J0 − ∇ × I − ∇ × Hexc. (6)

Here v = 1/μ, J0 is the source current density, v is thevelocity, and

H − Hexc = vB + I (7)

is the decomposition of the magnetic field intensity.The Preisach model represents the nonlinear and hysteretic

relationship between the orthogonal components of H and Bin the rotor steel and in the stator steel, i.e., two independentscalar models are running in every finite element of some partsof the mesh.

Fig. 7 shows the torque as a function of the angular velocity,it is similar to the results of linear simulations [14]. The lossesinside steel is increasing when increasing the angular velocityas it can be seen in Fig. 7, too, but the results at smallerangular velocity obtained from static and dynamic simulations

7009404 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014

Fig. 7. Torque and steel loss of the motor as a function of the angularvelocity.

are more or less the same. The simulated results are in a verygood agreement with the results in [14], and in the literatureof induction motors, see [15]–[17].

VI. CONCLUSION

A viscous-type dynamic scalar hysteresis model has beenimplemented and identified; finally, the model has beeninserted into a 2-D FEM code to simulate the measurementsystem containing a transformer core with toroidal shape.A modified, nonlinear version of the T.E.A.M. Problem No.30.a has also been solved and analyzed, too. The FEM envi-ronment as well as the hysteresis model has been implementedusing the C programming language. The tool GMSH [18] isused as a frame for preprocessing and postprocessing, but theFEM routines have been implemented applying the functionsof PETSc [19].

The formulation and the iterative procedure mentioned inthis paper have all the advantages of the fixed point iterationscheme with global convergence and easy implementation,but, unfortunately, the convergence speed of the modifiedformulations is a bit slow.

The vector Preisach model of dynamic magnetic vectorhysteresis is the next goal of research. The measurementsystem has been developed and the measured data have beenobtained [5]; moreover the static vector model has beendeveloped. More accurate results can be obtained by a vectormodel than by the two independent scalar models applying theorthogonal field variables.

One of the aims of the current research is to develop apermanent magnet synchronous motor for electric vehicles.The size reduction and decreasing of losses are the mostimportant tasks of nonlinear FEM simulations.

ACKNOWLEDGMENT

This work was sponsored by TÁMOP-4.2.2.A-11/1/KONV-2012-0012: basic research for the development of hybrid andelectric vehicles. This work was supported by the HungarianGovernment and co-financed by the European Social Fund.The authors would like to thank D. Marcsa for the preparationof geometry, mesh, and the program of linear version ofT.E.A.M. 30.a. The authors would also like to thank to T.Budai for the development of the simulation environmentunder Linux operating system and to T. Unger for developingthe toroidal specimen.

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