Modeling the Electromagnetic Behavior of Nanocrystalline Soft Materials

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1 678 IEEE TRANSACTIONS ON MAGNETICS, VOL 45, NO 2, FEBRUARY 2009 Modeling the Electromagnetic Behavior of Nanocrystalline Soft Materials Peter Sergeant 1;2 and Luc Dupré 1 Faculty of Engineering, Department of Electrical Energy, Systems and Automation, Ghent University, B-9000 Ghent, Belgium Department of Electrotechnology, Faculty of Applied Engineering Sciences, University College Ghent, B-9000 Ghent, Belgium A model that describes the magnetic behavior of nanocrystalline ring cores is useful for simulations of electronic circuits that contain inductors or transformers using these cores A general but computationally demanding model combines a macroscopic model of the ribbon with a dynamic Preisach hysteresis model In this paper, we present two models that taking into account the principle of loss separation make it possible to avoid the use of the CPU time consuming dynamic Preisach model Both models compute the waveform of the magnetic flux density for an arbitrary waveform of the magnetic field, or vice versa The first model uses a macroscopic model based on the plane wave theory and the classical rate-independent Preisach formalism The macroscopic model operates in the frequency domain and applies the harmonic balance principle Because of nonlinearity, the model is solved iteratively by a Newton-Raphson scheme The second model starts from a single evaluation of the classical Preisach model Additionally, it uses a lookup table that is a function of the flux density and its time derivative to evaluate the classical and excess field to be added The models are validated by measurements between 2 and 100 khz on Vitroperm nanocrystalline ring cores Index Terms Hysteresis, losses, nanocrystalline material I INTRODUCTION T HE high frequency behavior of passive magnetic components is important in power electronic circuits These high frequencies originate from the fundamental frequency and harmonics of modulated power signals By fast switching between two or more voltage levels, the desired voltages and currents can be synthesized The faster the switching, the more accurate the desired waveform can be generated by, eg, pulse width modulation For electrical machines, the frequency of pulse width modulation is typically khz, with extremes between 2 and 100 khz Here, nanocrystalline materials are useful thanks to a relatively high induction (compared to ferrites), a high permeability and low electromagnetic losses up to rather high frequencies They are widely employed in power electronic circuits to build inductors or transformers that create filters of high harmonics, galvanic isolation, or energy storage For several types of soft magnetic materials, manufacturers provide data about the geometry, the magnetic permeability, the saturation and the losses as a function of the frequency [1] [4] However, the losses are usually given for sinusoidal waveforms, while many applications in power electronics use other waveforms such as square waves Based on the statistical loss theory (Section II-B), the losses can be computed to whatever supply waveform, provided that all the material coefficients are evaluated starting from a limited number of measurements under sinusoidal excitation Recently, the losses in nanocrystalline cores under square waveforms have been studied in [5] by a model based on the one-dimensional transmission line theory in combination with an impedance function to represent the hysteresis and excess losses [6] In this paper, we do not only want to compute the losses, we also want to reproduce the waveforms by using fast models so that circuit simulations of a network including nonlinear and Manuscript received August 07, 2008; revised October 16, 2008 Current version published February 11, 2009 Corresponding author: P Sergeant ( petersergeant@ugentbe) Digital Object Identifier /TMAG hysteretic magnetic components can be elaborated The considered frequency range in the simulations is khz, which is sufficient for applications of pulse width modulation in electrical machines Nevertheless, the presented models are valid also outside this frequency range II HOW TO MODEL ELECTROMAGNETIC BEHAVIOR IN A LAMINATION? A numerical model that describes the magnetic behavior in a lamination of a material, has to be a combination of a macroscopic model that implements the Maxwell equations, and a constitutive law One such model has been developed in [7] where the macroscopic model is a 1-D time domain finite element model (FEM) and the constitutive law is a rate-dependent (dynamic) Preisach model [8] In this paper, two other methods are presented, based on the statistical loss theory of Bertotti [6] In method 1 Section III-A the macroscopic model uses the one-dimensional plane wave theory in the frequency domain and the constitutive law is a rate-independent classical Preisach model (CPM) [9], [10] In method 2 Section III-B the macroscopic model is a lookup table and the constitutive law is also the CPM Before discussing the two methods, Section II-A explains the macroscopic model that will be used in method 1: the plane wave model (WM) based on the plane wave theory Section II-B describes how the macroscopic model and the constitutive law together calculate the three loss terms hysteresis loss, classical loss, and excess loss in the statistical loss theory A Plane Wave Model (WM) The plane wave theory is explained in [11] to find the field in magnetic shields in the frequency domain for one frequency component (no additional harmonics), but we extend the theory towards harmonics In this one-dimensional approach, all electromagnetic vector quantities are represented by scalars /$ IEEE

2 SERGEANT AND DUPRÉ: MODELING THE ELECTROMAGNETIC BEHAVIOR OF NANOCRYSTALLINE SOFT MATERIALS 679 Fig 1 Geometry of the lamination with thickness 2l (a) with indication of the fields and (b) divided in p fictitious sublayers for the numerical model are constant, ie, independent from the position inside the lamination A tensor independent of results in a linear plane wave theory Notice that the dependence of and on the time is automatically taken into account in the frequency domain approach Indeed, the tensor elements are found using the waveforms of and that may contain memory effects in time Combining Maxwell s equations leads to the well-known plane wave equations in the time domain as only one component of the vectors differs from zero The direction of the vectors in the lamination is shown in Fig 1(a) 1) Constitutive Law: For harmonics, the constitutive law of the material relates the frequency components of the magnetic flux density with the frequency components of the magnetic field with the conductivity, the permittivity, and the electric field In the frequency domain and for harmonics, (2) and (3) can be written as (2) (3) (4) Here, and are the real and imaginary part of the frequency component of the flux density and the magnetic field respectively For a given material and magnetic field waveform, the elements in the permeability tensor can be identified as follows The flux density waveform is determined by the Preisach model starting from With both and known, the tensor is found by applying the harmonic balance principle as explained in [12] or [13] Notice that the permeability terms in the matrix are not unique; they depend on the waveform of the magnetic quantities When converting the spectra of and from the constitutive law (1) to the time domain, the -graph of the resulting time waveforms is the original hysteresis loop Indeed, the real and imaginary components of the magnetic field and the magnetic induction introduce the phase shift essential for hysteresis phenomena 2) WM With Constant Permeability Tensor: In this paragraph, we assume that the elements in the permeability tensor (1) with the angular velocity If is independent from, substitution of the constitutive law (1) results in From (6) and (7), equations can be generated as the equality must be valid for each frequency component and for (5) (6) (7)

3 680 IEEE TRANSACTIONS ON MAGNETICS, VOL 45, NO 2, FEBRUARY 2009 both cosine and sine terms separately These equations can be written in matrix notation by introducing the matrix The matrix is The unknowns are the and the components Solving the equation requires the introduction of and : the matrix with eigenvectors and the diagonal matrix with eigenvalues on the diagonal The determination of the solution is elaborated in detail in [13] Here, we give immediately the total solution for harmonics taken into account the matrix with characteristic impedances (8) (10) 3) WM With Permeability Tensor That Is Function of the Position in the Lamination: By combining the Preisach model with a lamination model, the solution reveals a dependence of the permeability on, because both and vary with due to eddy currents However, the variation of the permeability with the position cannot be modeled by the macroscopic lamination model as described in previous Subsection II-A2, because it assumes a constant permeability tensor This problem is tackled by discretizing the material into several sublayers, each having a constant permeability, as shown in Fig 1(b) In each segment, the material is linearized and the permeability is constant (independent of ) This means that the continuously varying permeability is approximated by a step function It is emphasized that the sublayers are fictitious, they are not physical layers in the material The discretization into sublayers can be seen as a cascade in series of transmission line segments, each having different properties The permeability tensor depends on the amplitude of all field components in the frequency domain Therefore, the approach in the plane wave model is iterative and the decomposition has to be repeated in every new iteration, for all sublayers This makes sure that the tensor is at each sublayer in the lamination compatible with the corresponding field and induction waveforms in (the middle of) each sublayer The determination of the permeability tensor in each sublayer is based on the magnetic field and the corresponding induction at the midpoint of each sublayer, as described in part 1 of this section The vector of unknowns in the LHS of (9) becomes in the case of several sublayers a cascade of the vectors containing the unknowns for each sublayer This equation relates the unknowns at side source at side of the lamination [Fig 1(a)] (9) with the imposed The number of real unknowns for a problem with sublayers and harmonics is For sublayers with, the source field is imposed at the last sublayer, and not like in the case for one sublayer in (9) at sublayer The source vector in the RHS of (9) is The matrix in (9) becomes (11) with the identity matrix of size Evidently, the matrices are calculated using (10) for each sublayer The dimensions of matrix are A Newton-Raphson (NR) routine is used to solve the nonlinear problem (12)

4 SERGEANT AND DUPRÉ: MODELING THE ELECTROMAGNETIC BEHAVIOR OF NANOCRYSTALLINE SOFT MATERIALS 681 The Jacobian containing the derivatives to all -components is determined by applying perturbations in the magnetic field and by evaluating the constitutive law and the macroscopic model In case of one harmonic where the use of eigenvalues and eigenvectors is not necessary an analytical solution of the derivative can be used to reduce the number of evaluations This cascade of sections and the combination with the Preisach model result in this approach to be almost completely different from [5] In the latter, only one section is used and the constitutive law is a single valued scalar complex function of the frequency and two fitted (constant) parameters B Statistical Loss Theory For soft magnetic laminated materials under unidirectional flux, the loss theory separates the total loss into three terms: the hysteresis loss, the classical loss caused by macroscopic induced currents and the excess loss For Fe-Si soft magnetic laminations, the total loss under sinusoidal flux excitation with amplitude and with frequency is given in [6] Fig 2 Limit cycles obtained by the CPM (showing H ), by the combination of the CPM and the macroscopic lamination model (MLM) at 20 khz (showing H + H ) and by measurement at 20 khz (showing H + H + H ) with (13) wherein is the electrical conductivity, the lamination cross section and The parameters and, depending on, are defined by the microstructure of the material The statistical loss theory was initially validated on Fe-Si alloys but more recently, it has been applied to nanocrystalline materials [15] In the cited paper, the expression for the excess loss was slightly changed The classical rate-independent Preisach model accounts for the hysteresis loss only The CPM uses an Everett map [16] to reproduce hysteresis loops in a time efficient way This Everett map is generated from experimental data a number of measured hysteresis loops at sufficiently low frequency [17] To calculate next to the hysteresis loss also the classical loss inside a lamination, the hysteresis model must be solved together with macroscopic field calculations in the lamination The excess loss finally can be incorporated in the Preisach model in order to make it dynamic, thus rate dependent In [7], the time domain FEM is combined with this dynamic Preisach model to compute the field in a lamination The computation requires a rather high CPU time due to the dynamic Preisach model Moreover, a matrix equation resulting from the 1-D FEM has to be solved at every time step To avoid this numerical burden, it is desirable to develop computationally less demanding models III TWO PRACTICAL METHODS TO MODEL ELECTROMAGNETIC BEHAVIOR IN A LAMINATION In [6], it is shown that, based on physical considerations given in [18], it is possible to rewrite the (13) valid for sinusoidal excitation, to a formulation valid for any waveform of wherein is the flux density averaged over the cross section of the lamination (14) (15) The two presented methods are based on the fact that similar to the loss also the magnetic field can be split up into three contributions Fig 2 shows the three fields for a Vitroperm core [1] In [19], two alternative time domain methods are presented that calculate the average induction in a lamination much faster Analytical formulas from [6] are used to avoid the dynamic Preisach model and/or to find the classical losses without a finite element model These methods are somewhat less accurate, but much faster than the method in [7] The WM presented in this paper is an alternative in the frequency domain As there is no time-stepping, the model requires the solution of only one matrix equation in case of linear material For nonlinear materials, an iterative approach is mandatory The WM uses the classical Preisach model and consequently neglects excess loss Therefore, the approach in [13] is only valid for materials with negligible excess loss However, the two methods presented in the following sections do take into account excess loss, each in their own way The frequency domain approach has both advantages and disadvantages compared to the time domain method If the WM is used to characterize eg, inductors in an electrical circuit, the frequency domain approach has an advantage concerning the study of the complete electrical circuit The frequency domain model of the inductor is compatible with the remaining of the (linear) electrical circuit that is also solved in the frequency domain A disadvantage of the frequency domain approach is the iterative technique for which stability and convergence are not guaranteed Therefore, the WM in the frequency domain is the preferred technique for mildly nonlinear systems, while it seems that the time domain methods are better for highly nonlinear problems In literature, techniques are described for a better handling of nonlinearities [14] The WM has an advantage compared to the finite element approach in [7] in case of mildly nonlinear materials The WM

5 682 IEEE TRANSACTIONS ON MAGNETICS, VOL 45, NO 2, FEBRUARY 2009 gives an exact solution of Maxwell s equations for a material with constant permeability tensor Therefore, it needs a discretization to model the changing permeability, while the finite element approach in [7] requires discretization to model the changing fields In the ideal case of a perfectly linear material at high frequency (strong skin effect), the plane wave approach needs only one layer (no discretization), although the field is decaying quickly inside the material Here, the finite element approach needs a very fine discretization to take into account the fast changes of the field as a function of the position inside the lamination As nanocrystalline material is quite linear, not many sublayers are needed, even at frequencies where the penetration depth is small compared to the lamination thickness Therefore, the WM is faster for materials with mild nonlinearity (so that few sublayers are required) at frequencies where strong skin effect occurs TABLE I ENCLOSED AREA OF B 0 H-LOOP OBTAINED BY THE TWO METHODS AND BY MEASUREMENTS A Method 1: Model Hysteresis and Classical Field, Add Excess Field By using the classical Preisach model, method 1 describes the behavior, which is the hysteresis loop in the nanocrystalline ribbon at very low frequency (quasi-static) This Preisach model is combined with the macroscopic lamination model the WM that evaluates the field in a number of fictitious sublayers inside the lamination, where the magnetic field is changed by eddy currents If sufficient sublayers are considered in the lamination, both and are modeled correctly, as well as the corresponding losses and The only contribution missing is, corresponding to the excess loss It is shown in Fig 2 that it is not allowed to neglect this term In Method 1, is approximated by the analytical expression (15) The evaluation of the classical Preisach model and this analytical expression require much less CPU time than the dynamic Preisach model in the general approach As it is observed from (15) that this is only function of and, an alternative is to retrieve the excess field from a lookup table Then however, a major advantage of method 1 the absence of a lookup table is lost B Method 2: Model Hysteresis; Add Classical Field and Excess Field Method 2 also starts from the classical Preisach model, but not in combination any more with the WM This makes it extremely fast: not only, there is no evaluation of the WM any more, but also, only one evaluation of the Preisach model is required In method 1, the Preisach model is evaluated in every iteration of the computation process, during each iteration in several fictitious sublayers The missing contributions and are added by a lookup table It is assumed that both the classical and the excess field are function of only and, not of the history This makes it interesting to create this map once, so that afterwards hysteresis loops can be generated very fast by using a very simple method Disadvantage of Method 2 is the extra effort to generate this map of In practice, it is produced by measuring limit cycles for maximal (119 T for the core considered in the simulations) at frequencies between the quasistatic one (about 1 khz) and the maximal frequency of interest (100 khz) In nanocrystalline materials, the permeability may decrease with increasing frequency, as is stated in [15] Method 2 is able to take this effect into account Method 2 is a 0-D method that uses a static Preisach model and a lookup table for the dynamic field The latter is obtained from numerous high frequency measurements Although the static model doesn t change the permeability with the frequency, the dynamic field compensates for this Method 1 however uses the WM in combination with the static Preisach model and an analytical expression for the excess field Here, the decrease of permeability as a function of the frequency is not modeled The results in Section IV and Table I show that neglecting the changing permeability does not cause unacceptable loop shapes and loss values A possible extension of the model would be to insert a frequency dependent correction function in the static Preisach model Another possibility is to couple the WM with the dynamic Preisach model, in order to improve the capability of the model to reproduce the material behavior for higher frequency The latter approach reduces the computational efficiency C Waveform Control for Both Models Both methods start from an imposed waveform of, calculate the resulting waveform of and add a dynamic waveform to the magnetic field The final waveforms of and are a

6 SERGEANT AND DUPRÉ: MODELING THE ELECTROMAGNETIC BEHAVIOR OF NANOCRYSTALLINE SOFT MATERIALS 683 result of the calculation and cannot be imposed from the beginning Therefore, an iterative waveform control algorithm modifies the input sequence in order to obtain the loop for any arbitrary waveform of either or It is recalled that for nonlinear materials, the plane wave model used in method 1 is iterative itself Section IV-G gives an indication about the required number of iterations and the convergence IV EXPERIMENTAL VALIDATION A Experimental Setup The experimental setup is based on a computer with a data acquisition card The software controls an arbitrary waveform signal generator that produces the desired voltage waveform The signal is amplified by a linear power amplifier that is connected to the excitation winding (the primary winding) of the magnetic core under test The current in this winding is measured together with the induced voltage in a secondary winding As a result of the feedback in the software, possible dc components are eliminated and the induction is controlled to the setpoint The data acquisition should sample sufficiently fast to have a lot of sample points per period even at the highest considered frequency Indeed, to calculate the loss, accurate phase information is necessary A sample rate of 2 gigasamples per second was obtained by coupling an oscilloscope Tektronix TDS3034B with the data acquisition card of National Instruments The measurement of the current is carried out by either a shunt resistance or a current clamp that introduce almost no parasitic phase shift up to 1 MHz: an accurate loss determination requires an accurate measurement of the phases of the primary current and of the induced voltage in the secondary winding The output generation is established by a National Instruments NI5401 function generator Thanks to the sample rate of 40 megasamples per second and 12 bits amplitude resolution, the high frequency waveforms can be generated with high accuracy: up to 100 samples per period at 400 khz The linear power amplifier has a slew rate of 5000 V/ s and is stable up to a frequency of 10 MHz In order to validate the models, we compare them with measurements The considered material is a core W403 in Vitroperm from Vacuumschmelze [1] with outer diameter 16 mm, inner diameter 10 mm and height 8 mm The Everett map was created from 21 measured loops with increasing amplitude of The waveform of was sinusoidal with a sufficiently low frequency in order to avoid dynamical effects Furthermore, 11 limit cycles with sinusoidal were measured for increasing frequency, up to 100 khz Up to 25 harmonics are considered, so that the highest considered frequency is 25 MHz For method 1, the parameter in (15) was fitted at 20 khz No adjustments were done for other frequencies, and the parameter mentioned in [15] was not considered For method 2, a map was constructed from the 11 mentioned loops Finally, in order to validate the model, 30 loops were measured for sinusoidal (various frequencies and amplitudes of ), 23 additional loops were measured for sinusoidal (various frequencies up to 150 khz and several amplitudes of ), and 15 loops for square voltage waveform (equivalent to a square waveform for ) The results are discussed in the following sections Fig 3 Comparison of the measured B0H loop for 2 khz and sinusoidal H(t) of 10 A/m amplitude with the quasi-static loop and with the loops obtained by the two methods using 3 sublayers and harmonics up to order 25 B Measurements and Simulations for Sinusoidal Magnetic Field For a frequency of 2 khz, where the classical and excess loss in nanocrystalline material is rather small compared to the hysteresis loss, Fig 3 shows the loops obtained by measurements, by method 1 and by method 2 As a reference, a static loop is shown to illustrate that the hysteresis loss dominates the other two loss terms For each loop, 3 shield layers and 25 harmonics are considered Table I provides the data to study the effect of the numbers of layers and harmonics considered Using less harmonics, also an accurate prediction of the losses is obtained at 2 khz, but the shape of the loop becomes somewhat less smooth The number of shield layers was chosen equal for all figures that show loops However, for low frequencies, 1 sublayer would be sufficient: the penetration depth is much higher than half the thickness of a nanocrystalline ribbon The contribution of eddy current loss to the total loss is quite small for 2 khz: only 11% of the total measured losses at 10 A/m A further discussion about the influence of the number of shield layers and harmonics can be found in Section IV-F At 10 khz, the dynamic field is larger: see Fig 4 The loops of both methods differ from each other, but their deviations from the measured loop have a similar magnitude The loss predicted by method 1 differs from the measurements by only 2% for three or more sublayers and by 6% for method 2 The calculated eddy current loss is 40% of the total measured loss for 10 A/m At 100 khz, the waveform control for method 1 converged well (Fig 5), while the waveform control for method 2 suffered from discontinuities in the lookup table of For high, it is difficult to obtain a smooth lookup table because the measurement of the limit cycles at high frequency changed the core temperature (and also the magnetic behavior) For method 2, it can be seen in Fig 5 that the peak induction is lower, causing the waveform control to increase the magnetic field to almost 6 A/m For high frequencies, the contribution that has to be added to the quasi-static loop is very large, and here, method 1 seemed to be more robust Although the penetration depth is small at 100 khz, the number of considered sublayers can be limited to 3 or 5 (Table I) One sublayer however is unacceptable at this frequency The number of harmonics can be low for 5 A/m, but if the waveform saturates, more harmonics are needed The eddy current loss is 56% of the total loss for 5 A/m In Fig 2 of [15],

7 684 IEEE TRANSACTIONS ON MAGNETICS, VOL 45, NO 2, FEBRUARY 2009 Fig 4 Comparison of the measured B 0 H loops for 10 khz and sinusoidal H(t) of 10 A/m amplitude with the quasi-static loop and with the loops obtained by the two methods using three sublayers and harmonics up to order 25 Fig 6 Comparison of the measured B 0 H loops for 10 khz and sinusoidal B(t) of 12 T amplitude with the quasi-static loop and with the loops obtained by the two methods using three sublayers and harmonics up to order 25 Fig 5 Comparison of the measured B 0 H loops for 100 khz and sinusoidal H(t) of 5 A/m amplitude with the quasi-static loop and with the loops obtained by the two methods using three sublayers and harmonics up to order 3 the classical loss is shown as a function of the square root of the frequency for transverse field annealed ribbons It can be seen that the classical losses become larger than the sum of hysteresis and excess losses starting from about 10 khz For longitudinal field annealed ribbons, the effect of excess losses remains important up to higher frequencies C Measurements and Simulations for Sinusoidal Magnetic Induction For sinusoidal at 10 khz, the loop of method 1 shows the effect of the limited number of harmonics (see Fig 6) Near saturation, both methods predict a lower magnetic field than the measurements For method 1, this means that the analytical equation for the excess loss is here not accurate enough For method 2, the cause of the error is inaccuracy in the interpolation, because the is in this region of and very sensitive to small variations However, the losses the enclosed surface of the loops are quite accurate for both methods (Table I) The waveforms and the spectrum of and can be observed in Fig 7 Fig 7 Waveforms of H(t) in (a) and B(t) in (b) for a sinusoidal B(t) of 10 khz and 118 T, obtained by experiment, by method 1 and by method 2 The spectrum of the waveforms of method 1 is shown in (c) The contribution of eddy current loss to the total loss is quite small for 2 khz: only 12% of the total measured losses at 118 T At 10 khz, the eddy current loss is 49% of the total measured loss for 06 T For higher induction, the contribution of eddy current loss decreases: it is 39% at 1 T and 26% at 118 T At 50 khz and 1 T, it is 48% of the total losses D Measurements and Simulations for Square db/dt Waveforms In power electronics, often square voltage waveforms occur The resulting waveform for the induction is triangular Simu-

8 SERGEANT AND DUPRÉ: MODELING THE ELECTROMAGNETIC BEHAVIOR OF NANOCRYSTALLINE SOFT MATERIALS 685 Fig 9 Losses predicted by method 1 for sinusoidal H(t) at 10 khz and 10 A/m as a function of the number of harmonics n and sublayers p Fig 8 Waveforms of H(t) in (a), B(t) in (b), and error of the waveform control in (c) for a square waveform for B(t) at 10 khz, obtained by methods 1 and 2 lations show that the losses predicted by both models deviate more from the measurements than in case of sinusoidal or (Table I) This is because the waveform control was not always able to fit the waveform accurately to the desired one The loss values however are always acceptable: the deviation is maximally 20% for method 1 and 50% for method 2 For 10 khz and 1 T amplitude of, the eddy current losses are 49% of the total losses E Simulations for Square B Waveforms To illustrate the ability of both presented methods to deal with any arbitrary waveform, simulations were carried out for square waveforms of the magnetic induction These rather extreme waveforms may be caused by strong voltage spikes: very high voltages during very short time The experimental setup was not able to generate such spikes, but Fig 8 shows that the simulations are capable of approximating the square waveform for the induction The error of the waveform control the sum of the absolute values of the differences between corresponding points of the obtained and the desired waveform is also shown As a result of the limited number of 25 considered harmonics, it is not possible to reproduce an exact square waveform that corresponds with error zero The figure shows that method 2 can generate more steep edges of the waveform which results in a lower error compared to method 1 Table I reveals that the losses in case of harmonics up to order 25 still make sense for both methods, although the accuracy is definitely worse than for sinusoidal waveforms of of For 10 khz and 1 T amplitude of, the eddy current losses are 41% of the total losses F Number of Considered Harmonics and Sublayers For method 1, Fig 9 shows the loss (the surface of the enclosed area of the loops) at 10 khz and 10 A/m for several considered harmonics and numbers of sublayers The corresponding loop can be seen in Fig 4 Concerning the number of harmonics, is the highest order considered; as even harmonics are zero, for example means that frequency components 1, 3 and 5 are considered It is observed that the error of the waveform control decreases with increasing : the waveform can be modeled more accurately with more harmonics The losses with 1 frequency component differ about 5% from the ones with more harmonics for this waveform where the core is saturated (see Fig 4) If the core is not saturated, also yields accurate results This means that it is not necessary to take larger than 5 or 7 for obtaining accurate loss values The only reason to take equal to 25 may be to produce smooth loop shapes The number of sublayers has no influence on the accuracy of the waveform control However, influences the loss value by about 10% at 10 khz (see Fig 9) For this frequency, is the minimal number of sublayers At 2 khz, has almost no influence ( is sufficient) and at 100 khz, it is necessary to take at least three sublayers In general, the penetration depth is a good indicator: the thickness of the sublayers should be smaller than the penetration depth A detailed discussion about the influence of the number of shield layers and harmonics for silicon steel also showing loops with several harmonics to illustrate the effect on the loop shape can be found in [13] For method 2, the number of harmonics has effect only because the spectrum of the induction and the spectrum of the total magnetic field (hysteresis field plus dynamic field) are truncated For a sinusoidal magnetic field at low frequencies, the influence of the truncation on the loss is almost negligible (Table I) For other waveforms, the effect is more visible, especially for higher frequencies and high amplitudes wherefore the core is saturated For, some unrealistic values can be found in the table If is five or higher, the loss prediction is acceptable except for the highest amplitude 100 T at the highest considered frequency of 50 khz G Convergence of the Model and the Waveform Control For method 1, distinction should be made between convergence of the nonlinear plane wave model using a Newton-Raphson scheme on the one hand, and convergence of the waveform control on the other hand The nonlinear

9 686 IEEE TRANSACTIONS ON MAGNETICS, VOL 45, NO 2, FEBRUARY 2009 scheme of the plane wave model uses typically 2 3 iterations or 3 4 iterations if the core is not saturated In saturation, the number of iterations becomes 4 or 6 The frequency seems to have only little influence on the number of required iterations Method 2 always considers only one evaluation of the CPM and interpolation in the lookup table Both models are iteratively evaluated by the waveform control For one frequency component, the waveform control does not work well: for low amplitudes (where the nanocrystalline material is almost linear), the number of iterations is still 5 6 for method 1 and 6 8 for method 2 The remaining error is large because for one frequency component, all waveforms are sinusoidal At 1 T amplitude, the number of iterations is already 12 (method 1) or 18 (method 2) For T, the waveform control doesn t converge any more However, with 25 harmonics, convergence is observed already after 4 6 iterations (method 1) or (method 1) for T For stronger induction and higher frequencies, the convergence requires (method 1) or (method 2) iterations In general, method 2 requires some more iterations than method 1, but method 2 is faster as it doesn t iterate within one iteration of the waveform control V CONCLUSION Two methods are presented to model the electromagnetic behavior of nanocrystalline cores with a small computational effort Method one uses a plane wave model and a classical Preisach model for the hysteresis and classical losses The excess losses are approximated by an analytical formula Method two models the hysteresis loss from a classical Preisach model and determines the classical and excess loss from interpolation in a table that contains the dynamic field as a function of the induction and its time derivative Both methods are used iteratively in order to reproduce the desired waveforms Method 1 is usually slower than method 2, because it contains an iterative plane wave model However, it doesn t need the creation of a lookup table required by method 2 From viewpoint of implementation and calculation time, method 2 is preferred for intensive use, as the extra effort to create the lookup table will be compensated by faster evaluations of the model From viewpoint of accuracy, method 2 is recommended only for waveforms without extreme values of : rather low frequencies and smooth waveforms of or guarantee good interpolation results Precautions should be taken against out of range values in the interpolation table, because these may lead to completely wrong loops and loss Method 1 is in general more accurate concerning the prediction of the loss, and almost never produces completely wrong waveforms It is more robust but slower Both methods 1 and 2 are able to produce loop shapes and loss values that are acceptable for a wide range of induction waveforms ACKNOWLEDGMENT This work was supported by the FWO project G008206, by the GOA project BOF 07/GOA/006, and the IAP project P6/21 funded by the Belgian Government The first author is a postdoctoral researcher for the Fund of Scientific Research Flanders (FWO) REFERENCES [1] Vacuumschmelze, VAC-VITROPERM Nanocrystalline Tape-Wound Cores for Common Mode Chokes, 2007 [2] Ferroxcube Catalog, Soft Magnetic Products and Accessories, 2007 [3] Metglas Catalog, Magnetic Alloy 2705 M Cobalt-Based, 2006 [4] Hitachi Metals, Nanocrystalline Soft Magnetic Material, FINEMET, 2005 [5] V Valchev, A Van den Bossche, and P Sergeant, Core losses of nanocrystalline soft magnetic materials under square voltage waveforms, J Magn Magn Mater, vol 320, no 1 2, pp 53 57, Jan 2008 [6] G Bertotti, Hysteresis in Magnetism New York: Academic Press, 1998 [7] L Dupré, O Bottauscio, M Chiampi, M Repetto, and J Melkebeek, Modeling of electromagnetic phenomena in soft magnetic materials under unidirectional time periodic flux excitations, IEEE Trans Magn, vol 35, no 5, pp , 1999 [8] G Bertotti, Dynamic generalization of the scalar Preisach model of hysteresis, IEEE Trans Magn, vol 28, pp , 1992 [9] F Preisach, Uber die magnetische nachwirkung, Zeitschr Für Phys, vol 94, pp , 1935 [10] I Mayergoyz, Nonlinear Diffusion of Electromagnetic Fields San Diego, CA: Academic Press, 1998 [11] R B Schulz, V C Plantz, and D R Brush, Shielding theory and practice, IEEE Trans Electromagn Compat, vol 30, no 3, pp , 1988 [12] L Degroote, W Ryckaert, B Renders, and L Vandevelde, Harmonic analysis of distribution networks including nonlinear loads and a nonlinear transformer model, in Proc 3rd IEEE Young Researchers Symp Electrical Power Engineering, Ghent, Belgium, Apr 27 28, 2006, paper no 38 [13] P Sergeant and L Dupré, Plane wave model for the electromagnetic behavior of SiFe alloys, IEEE Trans Magn, vol 44, no 4, pp , Apr 2008 [14] J Saitz, Newton-Raphson method and fixed-point technique in finite element computation of magnetic field problems in media with hysteresis, IEEE Trans Magn, vol 35, no 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