Fabien Sixdenier

Oualid Messal

Alaa Hilal

Christian Martin

Marie-Ange Raulet

Riccardo Scorretti

Temperature dependent extension of a static hysteresis model

Keywords: Temperature dependance, magnetic hysteresis, thermal stresses,

Some soft magnetic materials are strongly dependent of the temperature, because of their low Curie temperature. In order to predict their behavior in electrical devices, engineers need hysteresis models able to take into account the temperature. This paper is an attempt to take into account the temperature in an existing model of static hysteresis through its parameters. Variations of some parameters with temperature are issued or build thanks to the literature. At the end, all needed parameters have an analytical law versus temperature. Simulation results are compared to measurements and discussed.

I. INTRODUCTION

I N order to reduce the volume/mass of soft magnetic materials in electrical devices, there are most likely two main solutions: increase the "working" induction level B or the frequency f . The both solutions lead to increase the magnetic losses p mag of the device and consequently increase the temperature T , due to self heating. Moreover, soft magnetic materials that have a low Curie temperature T c are strongly dependent on the temperature and this behavior is highly non-linear. Thus, designers need models able to predict the magnetic behavior in a "large" temperature range (from 25 • C to T c ). In this aim, the authors of [START_REF] Raghunathan | Theoretical model of dependence of hysteresis based on mean field theory[END_REF], [START_REF] Messal | Temperature dependent extension of the jiles-atherton model: Study of the variation of microstructural hysteresis parameters[END_REF], used the static Jiles-Atherthon (JA) model with different parameter variation laws (analytical and/or numerical). Thanks to analytical laws [START_REF] Weiss | La constante du champ moléculaire. equation d'état magnétique et calorimétrie[END_REF] and optimization methods, a good compromise between the error levels and the monotony of the parameters is achieved in [START_REF] Messal | Thermal behavior of iron-nickel-chromium alloys and correlation with magnetic and physical properties -part a: Static effects modeling[END_REF]. Nevertheless, for complex induction b(t) waveforms, JA model generally fails to predict excitation field h(t) with accuracy due to several intrinsic, well detailed in [START_REF] Zirka | On physical aspects of the jiles-atherton hysteresis models[END_REF]. Engineers need, then, a reliable static hysteresis model able to take into account both complex inductions waveforms and the temperature influence on the magnetic material properties. To achieve this goal, an adaptation of an existing hysteresis model called Vector-Play Model (VPM) [START_REF] Bergqvist | Magnetic vector hysteresis model with dry friction-like pinning[END_REF] is proposed in this paper. We chose to test this approach on an alloy, named Phyterm260 commercialized by the APERAM company [7]. This alloy is normally dedicated to induction cookwares. It is a N i 50 F e bal Cr 9 alloy with a Curie temperature around 260 • C. The VPM has been tested and implemented in finite elements by many authors [START_REF] Franc ¸ois-Lavet | Vectorial incremental nonconservative consistent hysteresis model[END_REF], [START_REF] Henrotte | An energy-based vector hysteresis model for ferromagnetic materials[END_REF] and recently been improved in [START_REF] Bergqvist | Temperature-dependent vector hysteresis model for permanent magnets[END_REF] in the case of rotating fields. The same authors also proposed, in [START_REF] Lin | Improved vector play model and parameter identification for magnetic hysteresis materials[END_REF], a method where they considered, linear variations of saturation magnetization and coercive field to take into account the temperature for permanent magnets. In this paper, a brief reminder of the VPM is first presented in order to define some variables that will be used in the temperature dependent Paper sent on June 2015 for peer review Fabien Sixdenier, Alaa Hilal, Christian Martin, Marie-Ange Raulet and Riccardo Scorretti are working at Université de Lyon, Université Lyon 1, CNRS UMR5005 AMPERE, France, emails : fabien.sixdenier@univ-lyon1.fr, alaa.hilal@univ-lyon1.fr, christian.martin@univ-lyon1.fr, marieange.raulet@univ-lyon1.fr, Riccardo.Scorretti@univ-lyon1.fr Oualid Messal is working at Grenoble INP -UJF Grenoble 1 -CNRS UMR 5269 (G2Elab), France, email : oualid.messal@g2elab.grenoble-inp.fr corresponding author: Fabien Sixdenier extension. Then the temperature dependent extension is introduced associated to its parameters identification protocol for soft magnetic materials. Finally, the robustness of the model is tested with complex signals and temperatures that are not part of the identification protocol. The ability of the developed model to qualitatively retrieve the Hopkinson effect [START_REF] Hopkinson | Magnetic and other physical properties of iron at a high temperature[END_REF] is also tested. Simulations are compared to measurements that were performed on ring samples wound with high temperature enameled wires (placed in an oven) following the fluxmetric method with respect to the IEC 60404-4 standard [START_REF] Fausto | Characterization and Measurement of Magnetic Materials[END_REF].

II. VECTOR PLAY MODEL

A. Original Model

The evaluation of magnetization M from the applied field h is performed in two steps. First, the vector play operator h re has to be evaluated with [START_REF] Raghunathan | Theoretical model of dependence of hysteresis based on mean field theory[END_REF], with χ linked to the coercivity, h the applied field and h re0 the initial value of h re . In (1), the applied field h is decomposed into a reversible part h re and an irreversible part h ir . Fig 1 shows the vector diagram in each case. Then the magnetization M is calculated thanks to [START_REF] Messal | Temperature dependent extension of the jiles-atherton model: Study of the variation of microstructural hysteresis parameters[END_REF], where M an (h re ), is the anhysteretic curve and h re is the absolute value of h re .

h re = h re0 if|h -h re0 |< χ h -χ • h-hre0 |h-hre0| if|h -h re0 |≥ χ (1)
χ h re = h re0 h h ir h re0 h re h χ |h ir | = χ h ir
M = M an (h re ) • h re /h re (2)
Finally, the induction b is obtained by solving (3)

b = µ 0 • (h + M) (3)
Qualitative interesting results for major loops are given by VPM, in this way. In order to improve the model accuracy, the magnetization M can be separated into several parts M k that have their own χ k . Each M k associated to χ k will now be called a cell. If ω k are the the fraction coefficients, then N k=1 ω k = 1 and ω k M ≡ M k are properties that have to be respected. For each cell, the first step is still valid and implies [START_REF] Messal | Thermal behavior of iron-nickel-chromium alloys and correlation with magnetic and physical properties -part a: Static effects modeling[END_REF].

h = h k re -h k ir ( 4
)
Each cell is tested independently and each h k re depending on each χ k is calculated. Then the global magnetization M is calculated by [START_REF] Messal | Temperature dependent extension of the jiles-atherton model: Study of the variation of microstructural hysteresis parameters[END_REF] with

h re = N k=1 ω k h k re .

B. Parameters identification 1) Anhysteretic function

The model is vectorial and so able to deal with rotating fields or ellipsoidal fields, but it is generally easier to identify the parameters for alternating fields. In this paper, all the applied fields h are unidirectional, so the magnetic magnetization M is automatically in the same direction. The magnetization M is calculated thanks to [START_REF] Messal | Temperature dependent extension of the jiles-atherton model: Study of the variation of microstructural hysteresis parameters[END_REF], where M an (h re ), is the anhysteretic curve. Even if the model is phenomenological, it makes sense to use some functions and parameters from physics of ferromagnetism. In paramagnetic materials, M is generally estimated by the "Langevin" function [START_REF] Zirka | On physical aspects of the jiles-atherton hysteresis models[END_REF].

M = M s coth |H| a - a |H| • H |H| (5)
M s is the saturation magnetization at a given temperature, H is a local magnetic field and a is a shape parameter. Let's also remember that a = k B T /(µ 0 m), with k B ≈ 1.38•10 -23 J/K, the Boltzmann constant, µ 0 = 4π •10 -7 H/m, the vacuum permeability and m the effective atomic magnetic moment. For ferromagnetic materials, the local magnetic field is assumed as H = h + αM, with α, the molecular field constant in the Weiss mean field theory [START_REF] Weiss | La constante du champ moléculaire. equation d'état magnétique et calorimétrie[END_REF]. It means that the applied field h in (1) has to be replaced by H. For a given temperature, three parameters (M s , α and m) have to be identified. This will be discussed in another section.

2) Cell separation As said earlier, the model accuracy can be improved if the magnetization M is separated into several parts that have their own χ k . The authors of [START_REF] Franc ¸ois-Lavet | Vectorial incremental nonconservative consistent hysteresis model[END_REF] have shown that the accuracy is improved by increasing the number of cells. However, a number of cells superior to 8 did not improve the accuracy significantly anymore. For these reasons, we chose 7 as a reasonable number of cells. The last cell should be reserved to represent an image of the coercive field H c (obtained on a major loop). In this case, χ last = H c . One can write M = N k=1 ω k M and χ k = p k H c (with 0 < p k < 1, a weight parameter). It makes sense to assume that χ 1 = 0 associated to ω 1 M, would represent the reversible magnetization (bending of Bloch walls). The different weights (ω k and p k ) can be identified thanks to the h coer (h peak ) [START_REF] Henrotte | An energy-based vector hysteresis model for ferromagnetic materials[END_REF] and M peak (h peak ) curves. These curves are traced by measuring symmetrical loops with increasing values of h peak (amplitude of applied field). The relative h coer (h peak ) and M peak (h peak ) curves look approximately the same (see Fig. 2 (left)) in the case of this material. The weights ω k and p k have to be adjusted in order to fit as best as possible the M peak (h coer ) curve (see Fig. 2

III. TEMPERATURE DEPENDENT EXTENSION

All equations and principles of the model seen before remain valid, only some parameters will have their value changed. This section is devoted to find analytical/numerical relations of some parameters with temperature. All results of identification are reported in the table II.

A. Saturation Magnetization M s

As the saturation magnetization M s of a ferromagnetic material can be assumed to be function of the temperature T following the law (6) [START_REF] Weiss | La constante du champ moléculaire. equation d'état magnétique et calorimétrie[END_REF], with M s0 the magnetization at T = 0 K and T c the Curie temperature.

M s M s0 = tanh M s /M s0 T /T c (6)
The value of M s , at a given temperature, is calculated with (6) and its value is used in [START_REF] Zirka | On physical aspects of the jiles-atherton hysteresis models[END_REF]. A few measured major loops at different temperatures in order to identify M s0 and T c with accuracy were necessary.

B. Molecular field constant α and effective atomic magnetic moment m

Anhysteretic curves extracted from measured major loops can be compared to those calculated by [START_REF] Zirka | On physical aspects of the jiles-atherton hysteresis models[END_REF] where both m and α are present in the equation. For a given temperature, the previous unknown M s in ( 5) is now calculated by [START_REF] Bergqvist | Magnetic vector hysteresis model with dry friction-like pinning[END_REF]. One has to identify m and α to minimize the error between the measured and simulated anhysteretic curves. Weiss showed, in [START_REF] Weiss | La constante du champ moléculaire. equation d'état magnétique et calorimétrie[END_REF], that the calculation of the energy of magnetization at T ≈ 0 K and at high temperatures with α = cte give values that agree with those deduced from measurements of specific heats of pure nickel element. He concluded that α was independent:

• of the applied field • of the temperature Thus, we assume in the following that α can be considered as constant in the whole range of temperature (0 < T < T c ). In [START_REF] Gersdorf | Ferromagnetic properties of fe and ni in relation to their band structure[END_REF] the effective atomic magnetic moment m is assumed to depend on temperature, volume and external magnetic field for temperatures close to T c . In the empirical model developed by the author, a constant of the main equation depends on the density of states of the 3d-electron bands. For engineering purposes, the needed data of this approach are mostly unavailable. Nevertheless, the author mentioned that far above the Curie point, m was "stable" with temperature but that the change in volume near the Curie point (spontaneous volume magnetostriction) was sufficiently enough to modify the "apparent" atomic magnetic moment. In [START_REF] Raghunathan | Theoretical model of dependence of hysteresis based on mean field theory[END_REF], the authors made the assumption that a showed an exponential decay with temperature such as a = a 0 exp -T /(γTc) (with a 0 the value of a at T = 0 K). With this assumption and knowing that a = k B T /(µ 0 m), one can identify m as (7).

m = k B T µ 0 a 0 exp T /(γTc) (7)

C. Coercive field H c

As said earlier, χ k = p k H c , and the weight parameters p k are kept constant with temperature. One has just to identify the evolution of the coercive field H c with temperature and find an analytical law to represent its behavior. In [START_REF] Bergqvist | Temperature-dependent vector hysteresis model for permanent magnets[END_REF], the authors assumed linear dependance of H c versus temperature because the maximum temperature tested in their case was far above the Curie temperature of the permanent magnet tested (Sm 2 Co 17 , T max = 150 • C, T c ≈ 800 • C). As our need is to simulate the behavior of our magnetic material from 25 • C to Curie temperature , this approach can not be applied. In [START_REF] Raghunathan | Theoretical model of dependence of hysteresis based on mean field theory[END_REF], the authors chose an exponential decay with temperature of the parameter representing the coercive field [START_REF] Franc ¸ois-Lavet | Vectorial incremental nonconservative consistent hysteresis model[END_REF], with H c0 , the coercive field at T = 0 K and β the exponential decay coefficient.

H c = H c0 exp -T /(βTc) (8)

D. Parameters evolution with temperature

This subsection resumes the parameters M s , m and α, evolution with temperature obtained with the protocol mentioned above. All the needed values mentioned in ( 6), ( 7) and ( 8) are reported in the table II. Fig. 3 shows the per unit variations of M s , m and H c versus temperature. One can see that the measured values of M s and H c are in good agreement with those calculated by ( 6) and ( 8) respectively. Fig. 3 also shows that the calculated atomic magnetic moment slowly increases for very low temperatures (linear dependance k B T /(µ 0 a 0 ) part) showing a "stable" value [START_REF] Gersdorf | Ferromagnetic properties of fe and ni in relation to their band structure[END_REF], then increases very fast when T > 200 K. We observed that simulated anhysteretic curves were very sensitive to m and α. We checked that, when m and α were fitted as free parameters, the obtained values were very close to the ones computed analytically, and the shape of the anhysteretic curves were not modified significantly: this justifies the use of the analytical expressions instead of a "brute force" numerical fit. IV. RESULTS

A. Major loops

Fig. 4 shows the measured and simulated major loops at different temperatures. The simulated hysteresis loops are globally in good agreement with the the measured ones. The L1 norm between the measured and simulated loops was 4.11 < < 8.73 % within the temperature range.

B. "Complex" signals and temperature

In order to test the model in "complex" conditions, the applied field waveform is as h(t) = 4 (sin (ωt) + sin (3ωt)). The model had no problem to simulate harmonics (closed small loops at the tips of each loop at a given temperature in Fig. 5) as it has already been shown in [START_REF] Franc ¸ois-Lavet | Vectorial incremental nonconservative consistent hysteresis model[END_REF] . However, the simulated levels of b, for the same h input, are not respected. First (2) is not a good approximation of the measured anhysteretic curve (for this material). Second, m, γ and α parameters have been identified thanks to major loops with an amplitude much higher than the test signals (ratio of 66). In Fig. 5

C. Hopkinson effect

Fig. 6 shows the measured (m) and simulated (s) per unit magnetic permeability versus temperature. It can be seen that the shapes of the two curves are similar, but the measured curve is shifted on the high temperatures. The same behavior has been observed in Fig. 5, where after T = 160 • C, the loops start to shrink but slower in the case of the measured loops.

V. CONCLUSION

In this paper, a first attempt to take into account the temperature in an existing hysteresis model has been made. The purely magnetic behavior b(h) showed some errors for "complex" low amplitude signals. These errors could be explained by (5) used to approximate measured anhysteretic curves, which is known to be not so accurate. Accuracy might be improved by interpolating measured anhysteretic curves at different temperatures M an (h, T ), but this "black-box solution" will hide the physics of the model. However the Vector Play Model (VPM) showed no difficulties to adapt itself to the temperature dependence, by allowing some parameters to change their values. The thermal behavior is qualitatively well retrieved, loops expand or shrink ), have to be tested, in order to know if this approach can be generalized.

Fig. 1 .

 1 Fig. 1. Vector diagram for vector play operator in 2 cases: |h -h re0 |< χ (left) ; |h -h re0 |≥ χ (right)

Fig. 3 .

 3 Fig. 3. Per unit variations of Ms, m and Hc versus temperature ((m): measured values), (s): simulated values)

Fig. 4 .Fig. 5 .

 45 Fig. 4. Measured (m) and simulated (s) major loops at different temperatures

Fig. 6 .

 6 Fig.[START_REF] Bergqvist | Magnetic vector hysteresis model with dry friction-like pinning[END_REF]. Measured (m) and simulated (s) per unit magnetic permeability versus temperature

  (right)). Fig. 2. left figure: Measured per unit hcoer(h peak ) and M peak (h peak ) curves ; right figure: M peak (hcoer) measured (Mea) and simulated (Sim) curves After identification, the ω k and p k coefficients are shown in tableI. It is assumed, after this section, that all ω k and p k are independent of the temperature.

		1			1
	M peak ) h coer ) coer /max(h peak /max(M	0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9	h coer M peak	peak ) M peak /max(M	0.9 0.7 0.8 0.3 0.4 0.5 0.6 0.2 0.1	Sim Mea
			0 0.2 0.4 0.6 0.8 1 1.2			0 0.2 0.4 0.6 0.8 1 1.2
			h peak /max(h peak )			h coer /max(h coer )

TABLE I ω

 I k AND p k VALUES

	k	1	2	3	4	5	6	7
	ω k	0.1	0.2	0.15	0.1 0.15	0.15	0.15
	p k	0	0.1	0.3	0.5	0.6	0.7	1


Alaa Hilal

Fabien Sixdenier

Atef Lekdim

Christian Martin

Marie-Ange Raulet

A Single-Phase Two-Winding Transformer Dynamic Model for Circuit Simulators

Keywords: Magnetic hysteresis, transformers, energy efficiency, Circuit simulation,

Transformers are magnetic components widely used in switched-mode power electronics systems. The non-linear hysteresis behavior of the magnetic material and the high frequency effects in both core and windings have significant effects on system's efficiency, reliability and power losses. This behavior can be modeled using simple fast models or complex accurate models in order to predict and improve the transformer behavior before realization. This paper is summarized by proposing a non-linear dynamic model of transformers for use in circuit simulators. This model allows winding and core modeling including the material's accurate nonlinear dynamic hysteresis behavior. The magnetic component model is implemented in the circuit simulation software "Simplorer" using VHDL-AMS modeling language. It is validated for a medium-frequency nanocrystalline core transformer. Effects of frequency and waveform on computed efficiencies are discussed and validated thanks to experimental measurements.

I. INTRODUCTION

V IRTUAL prototyping is essential in the development of new power electronics devices like power converters and power transformers [START_REF] Ou | Analysis and design of a novel single-stage switching power supply with half-bridge topology[END_REF], [START_REF] Biela | Towards virtual prototyping and comprehensive multi-objective optimisation in power electronics[END_REF]. Circuit simulators allow the optimization of complex circuit prototypes by associating models of passive and active components [START_REF] Gruosso | Magnetic core model for circuit simulations including losses and hysteresis[END_REF]. Indeed modeling and simulation allow the prior knowledge of the system's behavior. Magnetic components constitute a major part of electronic devices. Therefore accurate modeling of magnetic materials is mandatory to predict their behavior under realistic operating conditions. Several efforts have contributed to the modeling of magnetic components such as single and three phase transformers. Transformer models included: Analytical models [START_REF] Seunghun | Analytical modeling of a mediumvoltage and high-frequency resonant coaxial-type power transformer for a solid state transformer application[END_REF], equivalent circuit models [START_REF] Pham | Dualitybased lumped transformer equivalent circuit at low frequencies under single-phase excitation[END_REF], and finite elements models [START_REF] Abed | Physics-based high-frequency transformer modeling by finite elements[END_REF]. Unlike most of state-of-the-art models which are limited either to, static conditions [START_REF] Nakmahachalasint | A static hysteresis model for power ferrites[END_REF], sinusoidal waveforms [START_REF] Wilson | Simulation of magnetic component models in electric circuits including dynamic thermal effects[END_REF], or non-conducting media [START_REF] Jiles | Theory of ferromagnetic hysteresis[END_REF], the model presented in this paper overcomes some of these limitations. The study in this paper is based on the magnetic modeling approach presented in [START_REF] Hilal | Magnetic components dynamic modeling with thermal coupling for circuit simulators[END_REF], where the authors have presented a non-linear dynamic inductor model for circuit simulators. As the model is generic and modular this paper presents its extension to a transformer model. The effects of transformer's loading on losses and thus on efficiency are taken into account by the proposed model. High frequencies up to 100 kHz and waveform influence on losses and efficiency are also studied and discussed in this paper.

II. TRANSFORMER MODELING

A transformer is a static device used in the generation, transmission and distribution of electric power. They generally consist of two or more insulated windings, and a magnetic core to insure good coupling between windings [START_REF]Power Transformers: Principles and Applications[END_REF]. The Paper sent on September 2015 for peer review All authors are working at Université de Lyon, Université Lyon 1, CNRS UMR5005 AMPERE, France, emails : alaa.hilal@univ-lyon1.fr, fabien.sixdenier@univ-lyon1.fr, atef.lekdim@univ-lyon1.fr, christian.martin@univ-lyon1.fr, marie-ange.raulet@univ-lyon1.fr corresponding author: Fabien Sixdenier model must be able to take into account the magnetomotive force (M M F ) induced in the secondary winding and thus increase the primary current based on the value of secondary current (transformer's loading). The model consists of different blocks: windings blocks, a static core block and a dynamic core block (last two blocks represent the magnetic core) as shown in Fig 1 . VHDL-AMS language [START_REF]An Introduction to the VHDL-AMS Modeling Language[END_REF] is chosen to describe each block behavior due to its multi-domain modeling feature and since it is supported by several circuit simulators.

A. Winding block

The winding block represents a coupling element between electric and magnetic circuits. It converts electrical energy into magnetic energy and vice versa (bidirectional energy flow). The winding block has two electrical pins, and two magnetic pins. It has a global parameter, the number of turns (N ) for coupling quantities between magnetic and electric domains. These domains are related by Ampere's law linking the magnetic field H to the magnetizing electric current i 0 (1) and by Faraday's law linking induced voltage e to magnetic induction B (2). A c and are the cross section and the mean path length of the core respectively. Consequently, the transformer model has two winding blocks representing the primary and secondary windings having respectively R w1 and R w2 winding resistances.

H • = N • i 0 (1) e(t) = -N A c • dB dt (2)

B. Dynamic block

The dynamic block is the center of the transformer model representing the overall behavior of the magnetic material core. This block contains a dynamic behavioral model which relates the applied magnetic field H to the flux density B. The magnetic core model is based on the principle of separation of losses into static and dynamic contributions, as well as Bertotti's theory [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF]. The applied magnetic field is the sum of static field H s and dynamic fields due to eddy currents H ec and wall motion effects H wm (3) and ( 4). As γ is related to the electrical resistivity ρ and on the ribbon thickness d, it can be identified analytically [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF]. The remaining parameter α is identified thanks to dynamic measurements.

H(t) = H s (B) + H ec + H wm (3)
H = H s (B) + γ dB dt + sign( dB dt ) • α • dB dt 1/2 (4)

C. Static Block

The dynamic magnetic model requires a static model to calculate H s (B). This is the role of the static block. It may be composed by one of the existing physical or mathematical static magnetic models such as Polynomial model [START_REF] Chailloux | Magnetic behavior representation taking into account the temperature of a magnetic nanocrystalline material[END_REF], Jiles-Atherton model [START_REF] Jiles | Theory of ferromagnetic hysteresis[END_REF], or Preisach-Néel model [START_REF] Courtay | The preisach model and its characterization[END_REF]. In this work, the polynomial model, detailed in [START_REF] Chailloux | Magnetic behavior representation taking into account the temperature of a magnetic nanocrystalline material[END_REF], is used to describe the static behavior of the materials. This mathematical model ( 5) is based on a piecewise polynomial function of order n . It is reversible and limited to materials with very low coercive field.

B =      µ 0 (H -H b ) + P (H b ) ifH ≥ H b P (H) if|H|< H b µ 0 (H + H b ) -P (H b ) ifH ≤ -H b (5)
where H b is the field needed to reach saturation and P is a polynomial function such that P (X) = n i=0 p i X i .

III. MAGNETIC MATERIAL CHARACTERIZATION

Among available materials for medium frequency transformers, the nanocrystalline Nanophy R cores N14E1, provided by Mecagis R , are chosen due to their interesting properties including high saturation flux density (B s ), low coercive field (H c ) and high relative permeability (µ r ). Material characterization is mandatory to identify the parameters of the static and dynamic models. The table I resumes some of of the magnetic material physical properties and geometrical dimensions.

A. Static measurements and modeling

A simple B(H) major loop measurement (see Fig. 2) and basic mathematical fitting is necessary to identify the needed static parameters that are resumed in the table II. For this material a polynomial model of order 5 is sufficient to represent its static behavior, where the even coefficients are almost zero due to the odd parity of the B(H) curve.

B. Dynamic measurements and modeling

Several dynamic measurements are performed on the magnetic core to identify the remaining dynamic parameter α (γ is directly calculated). Identified values of α and γ are reported in table II. Fig. 3 shows measured B(H) loops under a 280 mT sinusoidal applied flux density for 3 different frequencies (60, 80 and 100 kHz) and an example of a simulated loop.

C. Loss measurements

Fig. 4 shows the core losses versus frequency for different sinusoidal applied flux densities. These measurements will be useful later to analyze the behavior of the studied transformer.

IV. SIMULATIONS AND MODEL VALIDATION

A. Transformer losses and efficiency determination

In order to validate the developed model, a 10 W-100 kHz transformer is realized. The circuit design allows variable voltage, frequency and load. The same magnetic material characterized is used as the transformer's core with five turns for primary and secondary windings. To measure precisely primary and secondary currents, a shunt resistor (500 mΩ) is added to the primary and secondary sides of the transformer. Set of resistive loads between 5 Ω and 50 Ω are available to vary the load percentage (from 15% to 100%). The transformer is operated so as to have a desired flux density peak value in the core (example B = 280 mT) while measuring the primary and secondary voltages and currents for variable loads. From the primary and secondary currents the no-load current i o related to the applied excitation field H (1) is calculated. From v 2 and i 2 , the induced voltage e then B (2) inside the magnetic circuit can be retrieved. With H and B, the core losses p c can be evaluated precisely for each transformer test conditions. The winding losses p w are AC only and are calculated using [START_REF] Abed | Physics-based high-frequency transformer modeling by finite elements[END_REF]. R AC is the AC resistance due to skin and proximity effects and I AC is the RMS current in a single winding. The AC resistance has been determined using an Agilent 4294A impedance analyzer [START_REF]4294a precision impedance analyzer[END_REF]. Choosing a copper wire of 0.5 mm diameter, R AC is about 20 mΩ in the 10 kHz-100 kHz frequency range (no skin effects up to 100 kHz).

Finally, the efficiency η can be calculated using [START_REF] Nakmahachalasint | A static hysteresis model for power ferrites[END_REF] with P 2 the secondary power.

p w = R 1AC I 2 1AC + R 2AC I 2 2AC (6) η = P 2 P 2 + p c + p w (7)
B. Circuit simulation: Results and discussion

1) Results

A transformer circuit identical to the one used for measurements was simulated using Simplorer R software. The static and dynamic parameters were inserted in the static and dynamic model respectively in addition to the core dimensions. Winding resistances and number of turns were inserted to the primary and secondary winding blocks. Simulations were carried out for the same input voltage and frequency and for load variation between 15% and 100%. Primary and secondary voltages and currents were available from simulations. Core and winding losses were then calculated and the transformer's efficiency was deduced. Simulation results were compared to measured ones. The efficiency for three different frequencies up to 100 kHz at constant flux density level ( B = 280 mT) are shown in Fig. 5. From Fig. 5, taking the efficiency curves at a constant frequency (60 kHz for example), we can notice that the efficiency increases with load. As the flux density is almost constant, core losses are not affected by load variation. However, as copper losses are proportional to the square of the RMS current, they become more and more significant in the calculation of [START_REF] Nakmahachalasint | A static hysteresis model for power ferrites[END_REF]. On the other hand, as the frequency varies at a constant flux density, the core losses increase with frequency (see Fig. 4). As a result the efficiency decreases with increasing frequency. At any given load, the efficiency difference with frequency is related to core losses only since copper losses are the same in each case. In any case, the model is able to predict the efficiency for different proposed loads, voltage levels and frequencies. Results are in good agreement with a mean square error of about 4e -4 between measured and simulated efficiencies. In many power electronics applications, magnetic materials are excited with non-sinusoidal waveforms. This induces different losses compared to those provided by manufacturers based on sinusoidal flux density [START_REF] Boglietti | Predicting iron losses in soft magnetic materials with arbitrary voltage supply: An engineering approach[END_REF]. To test the model under non-sinusoidal excitation, a square voltage (thus a triangular flux density) at a frequency of 100 kHz is applied to the transformer. The voltage level is adapted in order to have the same applied flux density peak value as in the sinusoidal case. Resulting measured and simulated efficiencies are compared in Fig. 6. In this case core losses under square voltage are lower than those under sinusoidal voltage (under the condition of 50% duty cycle). This results in a higher efficiency value for square voltage.

2) discussion

In view of all previous results, a complete magnetic model has many advantages. Voltages, currents, magnetic fields, flux densities, losses, hysteresis loops and efficiencies are available from simulation and are in good agreement with measured ones. Besides its precision and its adaptability to different operating conditions, the real benefits of the modeling approach herein presented are its real-time aspect (the power balance is respected at any time of the simulation), ease of implantation and rapidity. However, since the model reproduces core losses accurately, we believe that the major part of the remaining error between simulated and measured efficiency (mostly visible for high relative loads) is related to flux leakages which are not considered in the current model.

V. CONCLUSION

In this paper the development of a middle frequency transformer made with a nanocrystalline core model for use in circuit simulation is presented. Model's parameters are extracted from magnetic material characterization under various conditions. Simulations of a 10 W-100 kHz transformer including the proposed model are compared to measurements performed at variable voltages, frequencies and loads. The developed model allows precise prediction of core and winding losses. As a consequence the transformer's efficiency is directly available at the end of the simulation (no post correction calculations needed). Experiments confirm model's loss prediction accuracy with a mean square error of about 4e -4 between simulated and measured efficiencies. Adding leakage elements (electric inductances or magnetic reluctances) to the model is of our future interests in the aim of even more accurate virtual prototyping. Simulation of other circuits or materials, including other or adapted static and dynamic laws is also of great interest for future work to help engineers to design innovative and effective new products.

Fig. 1 .

 1 Fig. 1. Transformer Model structure

Fig. 2 .

 2 Fig. 2. Measured (solid line) and simulated (dashed line) static loops

Fig. 3 .Fig. 4 .

 34 Fig. 3. Measured (solid lines) and simulated (dashed lines) dynamic loops at different frequencies for B = 280 mT

Fig. 5 .

 5 Fig. 5. Measured (solid lines) and simulated (dashed lines) efficiencies versus relative load for different frequencies at B = 280 mT

Fig. 6 .

 6 Fig. 6. Measured (solid lines) and simulated (dashed lines) efficiencies versus relative load for different voltage waveforms at B = 280 mT and f = 100 kHz