Limits and rules of use of a dynamic flux tube model

Limits and rules of use of a dynamic flux tube model M.A. Raulet, F. Sixdenier, B. Guinand, L. Morel and R. Goyet Université de Lyon, F69622, France ; Université Lyon 1, F69622, France; CNRS UMR5005 AMPERE, 43, Boulevard du 11 Novembre 1918, Villeurbanne, F-69622, France. E-mail: [email protected], [email protected] Abstract — The simulation of electromagnetic devices remains an essential research tool for optimization. Nowadays, the miniaturization of devices leads to increase the supply frequency, in this case the material is hardly sollicitated. An accurate description of dynamical material law must be introduced in the magnetic circuit representation. Our team has already created a dynamic behavioural magnetic model which lumps together all dynamic effects developped in the circuit. The main assumption of this model is to consider that all magnitudes are homogeneous in the cross section. The aim of this paper is to analyse this assumption and to define a validity domain and rules of use. I. INTRODUCTION Nowadays, the miniaturization of electromagnetic devices leads to increase the fundamental supply frequency, moreover, thes systems are usually fed by static converters. Thus the magnetic materials of these devices are hardly stressed due to the fast dynamic working conditions. The design of these devices requires simulation tools, which need to take into account accurately both the description of the geometry of the system and dynamical material laws. A 3D field calculation including a dynamic realistic material law would lead to a prohibitive calculation time and numerical difficulties ; at present time, it hardly seems possible with standard computers. Some authors [1] [2] consider dynamical effects due to the material in a 2D field calculation considering a modified law of the material. Our laboratory has developped a magnetic dynamical flux tube model [3]. This model points out the dynamical behaviour of the material and considers a simple geometry (flux tube with a constant cross-section). The association of different flux tubes allows to simulate a real industrial device [4]. This model has already been effectively used to represent different industrial devices. Nethertheless, the main assumption of the model which is to lump together the different dynamical effects has not yet been tested. The purpose of this paper, is to analyse in details the main assumtion of this model and to define its validity domain and its rules of use. II. DESCRIPTION OF THE DYNAMICAL FLUX TUBE MODEL The model allows to obtain the dynamic behaviour of a magnetic circuit with a constant cross-section, where the anisotropy of the material is neglected. The different dynamic effects developped in the circuit (wall motion, macroscopic eddy currents) are considered in this model by a single representation. The dynamic behaviour of the circuit is described by a first order differential equation which can be represented by a bloc-diagram in Fig. 1 The quantity Ba is the average flux density in the crosssection, Hdyn , is the excitation field applied at the surface, and Hstat (Ba ), is a fictitious static excitation field value We thank you for your contribution to the success of the EMF 2006 Symposium. Fig. 1. dynamic bloc-diagram of the flux tube model which corresponds to a given value of Ba .The magnetization history of the material can be taken into account by considering an hysteretic static model for Hstat (Ba ). The parameter γ is a dynamic behavioural parameter. Its value has to be fitted by comparing simulated and measured dynamic loops. This model presents several advantages : • it requires only one parameter γ (apart those required to model the static hysteresis), supposed to be independant of the waveform and velocity of the excitation • it is a time domain model • the calculation time is short • it is reversible (Ba (Hdyn ) or Hdyn (Ba ) • it can be easily introduced in many kinds of softwares (circuit type, design, simulation. . . The main assumption of this model is to consider the homogeneity of the phenomena in the cross-section of the flux tube. This model has been tested for different materials and devices. For the representation of ferrite components, this model allows to obtain accurate results, in that the assumption of homogeneity holds with good approximation. For other materials, where the conductivity leads to a field diffusion across the section, the model provides quite satisfying results. III. VALIDITY OF THE MAIN ASSUMPTION The dynamic flux tube model lumps together all dynamic effects developped in the circuit. The value of the dynamic parameter γ, determine the width of the dynamic loop of the material. This value depends on the thickness of the circuit, the resistivity ρ of the material, the wall motions, the static hysteresis phenomenon and the permeability of the material. In the aim to simplify the problem, we limit our study to simple geometries : we consider torus samples where the anisotropy phenomenon is negligible[1]. Consider the formula (1) associated to the dynamic flux tube model. Hdyn − Hstat (Ba ) = γ. dBa dt (1) TABLE II. S AMPLE N 1 RESULTS This formula can be compared with the expression (2) defined in the case of a magnetic lamination if the skin effect, saturation and edge effects are negligible. Htot σ.d2 ∂Ba = Hstat (Ba ) + . 12 ∂t (2) f (Hz) 25 50 100 150 δ (mm) 0.22 0.16 0.11 0.09 ε1 (%) 19 4.2 23 36.7 ε2 (%) 14.5 22.7 29.8 37.2 Where : Htot = excitation field at the lamination surface when eddy currents are induced in the cross section, d = thickness of the lamination, σ = conductivity of the material, Ba = averaged magnetic flux density over the thickness. In both cases, the homogeneity of magnetic data assumption is assumed. An analogy between (1) and (2) allows to estimate the value of the parameter γ : γ= σ.d 12 (3) In the aim to validate on the one hand the estimation of γ and on the other hand the lumped model, we carry out successively tests on four samples numbered 1 to 4. A. Sample tests n1 We consider a toroidal sample made of NiFe(50/50) main characteristics are reported in the table I. The value of ρ is given by the manufacturer and µr is estimated considering the static characteristic Hstat (Ba ) in its linear part. Due to the large thickness of the torus, the weak resistivity and its high relative permeability, eddy currents are not negligible in this sample. A sinusoidal excitation field H is imposed at the surface of the sample. Relative Amplitude of the applied excitation field (%) 100 2 90 80 70 60 50 40 30 20 25 Hz 50 Hz 100 Hz 150Hz 10 0 0 20 40 60 80 relative distance from the surface (%) 100 Fig. 2. Normalized amplitude of the excitation field H versus the relative thickness TABLE I. S AMPLE DATA Dout (mm) Din (mm) thickness (mm) ρ (Ω.m) µr 18.8 9.9 1.1 48.10−8 100000 100 The table II regroups for different working frequencies (25Hz, 50Hz, 100Hz and 150Hz) : δ= r ρ µr .f (4) • the relative error ε1 between the areas of the simulated and measured loops (which is representative of the losses) • the quadratic error ε2 defined by the formula (5). Relative Amplitude of the fluxdensity (%) • the skin depth δ : 90 80 70 60 50 40 30 10 v u n u1 X Bsimulated (i) − Bmeasured (i) ε2 = t n i=1 25 Hz 50 Hz 100 Hz 150Hz 20 (5) We observe that the quadratic error ε2 increases with the frequency. This result agrees with the indication given by 0 20 40 60 80 relative distance from the surface (%) 100 Fig. 3. Normalized amplitude of the flux density B versus the relative thickness the skin depth value : in fact when δ becomes comparable to the size of the section, the assumption of homogeneity doesn’t hold. However, we point out that the classical formula (4) has been obtained for linear materials and with a semi-infinite plane conductor : hence it provides a very approximate result with our samples (for instance see [5] for analytical computations of δ with different geometries). Local information inside the cross section is not available. So as to obtain this information, we use a numerical tool based on the magnetic field diffusion [6]. The figure (2) shows the simulated excitation field H, normalized with respect of its intensity at the surface, as a function of its relative position in the thickness of the lamination (H (0) = field at the surface, H (100%) = field in the middle of the thickness) for different frequencies. One observes that even at 25 Hz the excitation field H is not uniform ; however, the skin effect is less important than the prediction obtained by (4) In the same way, the figure (3) shows the normalized flux density as a function of the relative geometric position through the thickness. One sees that the saturation phenomenon tends to homogeneize the flux density through the thickness of the lamination. Hence, the validity of the flux tube model is enlarged. 1 0.8 simulation measure Averaged Flux density B (T) 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −50 0 Excitation field H (A/m) 50 Fig. 5. Quadratic error is 48.10−8Ω.m, its relative permeability µr estimated from the linear part of the quasi-static characteristic is about 6000. Conversely to the previous sample, few eddy currents can be induced in the circuit (thin ring) : by using (4), one sees that the skin effect can be neglected until at least 1500 Hz. First, we analyse the results provided by the model considering the value of γ = 6.94.10−3 computed by (4). We carry out some 300Hz to 2400Hz simulations with an imposed excitation field H at the surface. The figure 5 shows the quadratic error ε2 computed for each frequency. An important error is observed, and can not be explainde by the skin effect, which is negligible until at least 1500Hz frequency. Therefore, we carried out other simulations by taking the value of γ which minimizes the quadratic error ε2 at the frequency of 800 Hz. The obtained value of γ is now 0.013. The quadratic error ε2 is reduced for all the considered frequencies (not only at 800Hz), as shown in figure 5. Hence, the formula (4) is no more applicable. By considering the assumption of magnetic losses separation [7], the excitation field Hdyn at the surface of the sample can be decomposed by the sum of different terms : Hstat (Ba) due to the static law of the material, Hedd due to the eddy currents, and Hexc due to the effects of wall motions. Fig. 4. Measured and simulated loops A last observation concerns the comparison between both errors ε1 and ε2 carried out for 50Hz frequency operation where a discrepancy appears. The figure 4 shows the simulated and measured loops for this frequency. Both loops have nearly the same area but are hardly different. These different results bring out several preliminary conclusions : • If this model provides accurate results on the estimation of the area of the loop, we must be more careful concerning the estimation of the waveform when a great heterogeneity exits in the cross section. • The skin depth provides information about the validity domain of the tube flux model • The saturation phenomenon tends to homogeneize the flux density distribution, and thus enlarges the limits of use of our model. Hdyn = Hstat (Ba) + Hedd + Hexc (6) We now compare (1) to (6), by using formula (2), so as to obtain the following expression for Hexc : Hdyn =   σ.d2 dBa γ− . 12 dt (7) The dynamic flux tube model lumps different dynamic effects, which are represented by a sole formulation. The representation of dynamical effects associated with the wall motions has a similar formulation as those linked to eddy currents. This result has already been validated in previous works [8] to simulate ferrite circuits. In this kind of material, dynamic effects due to the wall motions are dominant, and the flux tube model gives accurate results. The assumption of magnetic losses separation together with the identification of the parameter γ allows to specify the different energy dissipations (W/kg) associated respectively with eddy currents and to wall motions. B. Sample test n2 The second sample is a stack of rings (thickness 0.2mm) of SiFe(3%). The material resistivity given by the manufacturer Pedd σ.d2 = 12 I dBa .dBa dt (8) TABLE IV. S AMPLE N 3 AND 4 RESULTS Pexc =  I σ.d2 dBa γ− .dBa 12 dt (9) Many simulations using the dynamical flux tube model have been carried out by using the same value of γ = 0.013. The contribution of each kind of losses cannot be measured separately. So as to obtain such a comparison, we use the results obtained with the diffusion model (diff ) [6]. Different simulations have been carried out by imposing a sinusoidal excitation field H at the surface of the sample, for the range of frequencies (800 Hz - 2400 Hz). The results are regrouped in the table III. PT are the total losses, Physt are the static losses, Pedd are the losses due to eddy currents and Pexc are the excess losses. TABLE III. S AMPLE N 2 RESULTS f (Hz) 800 1200 2400 model DFTM Diff DFTM Diff DFTM Diff PT (mW/kg) 0.63 0.58 0.75 0.7 0.96 0.89 Physt (mW/kg) 0.32 0.38 0.32 0.4 0.25 0.37 Pedd (mW/kg) 0.16 0.14 0.21 0.2 0.35 0.32 Pexc (mW/kg) 0.16 0.06 0.21 0.1 0.35 0.2 These results lead to different investigations : • Physt are important in this range of frequencies, thus an accurate static hystersis model Hstat (Ba) must be used. • By considering (8) and (9) formulas, the power losses due to eddy currents and to the wall motions are propor
tional to the coefficients σ.d2 /12 and γ − σ.d2 /12 • The dynamic flux tube model allows to obtain accurate results, until the skin effect is not dominant. C. Sample tests n3 and n4 These tests have been carried out on rings made of CrNiFe. The resistivity is ρ = 94.10−8Ω.m, the relative permeability related to the linear part of the static first magnetization is about 50000. The manufacturer produces two series of rings with different thicknesses: d = 0.2 mm and 0.6 mm. Manufacturing precautions are taken into account in the aim to ensure the same magnetic properties of both ring series. So as to avoid the static hysteresis phenomenon, we carry out simulations limited to the first magnetization. For the previous sample, we obtain the value of γ which minimizes the quadratic error ε2 for the frequency 50 Hz. The obtained results are regrouped in table IV. The optimal value of γ depends upon the thickness of the ring. This result agrees with the formula (4). Conversely, we 2 observe that the parameter γ − σ.d 12 seems to be constant. This parameter is characteristic of dynamic effects due to wall motions. This result is coherent, in that wall motions are linked to the structure of the material, and are independant of the geometry of the sample. d (mm) 0.2 0.6 ε2 (%) 1.77 4.37 γ 0.05 0.033 σ.d2 12 0.00354 0.0319 0.00146 3.08 0.0011 1 σ.d2 12 2.δ d γ− IV. CONCLUSION The limits of a dynamic flux tube model is analysed by considering tests (simulations + measurements) on 4 different samples. This model lumps dynamic effects, which are represented by a same formulation. It gives accurate results when the skin effect is negligible, and when a weak heterogeneity of magnetic data exists. Nevertheless, the saturation phenomenon enlarges the validity domain of the model. The value of γ can be decomposed into two terms : the first one is linked to eddy currents, and is given by the classic expression depending of σ and d (2). The second one seems to be a constant value, depending upon the structure of the material, and not upon the geometry. This decomposition based on the assumption of losses contributions, allows to obtain separately an estimation of the losses linked respectively to the static hysteresis, to eddy currents and to wall motion effects. The estimation of γ is still empiric; we are working on this subject. REFERENCES [1] J.J.C. Gyselinck and L. Vandevelde and D. Makaveev and J.A.A. Melkebeek, ”Calculation of no load losses in an induction motor using an inverse vector Preisach model and eddycurrent loss model”, IEEE Trans. on Magn., vol. 36, no. 4, pp 856-860, July. 2000. [2] N. Sadowski, N.J. Batistela, J.P.A Bastos and M. LajoieMazenc, ”An inverse Jiles-Atherton model to take into account hysteresis and hysteresis in time-stepping finite element calculations”, IEEE Trans. on Magn., vol. 38, no. 2, pp 797-800, March. 2002. [3] F. Marthouret and J.P. Masson and H. Fraisse, ”Modelling of a non-linear conductive magnetic circuit”, IEEE Trans. on Magn., vol. 31, no. 6, pp 4065-4067, Nov. 1995. [4] F. Sixdenier, ”Modélisation d’un capteur de courant asservi”, in Proceedings Conference, JCGE2005., vol. 1, pp. 324–327, June 2005. [5] N. Ida, ”Numerical modeling for electromagnetic nondestructive evaluation”, Chapman & Hall, London, pp 64. [6] M.A. Raulet and B. Ducharne and J.P. Masson and G. Bayada, ”The magnetic field diffusion equation including dynamic hysteresis : a linear formulation of the problem”, IEEE Trans. on Magn.,vol. 40, no. 2, pp 872-875, March. 2004. [7] G. Bertotti, ”General properties of power losses in soft ferromagnetic materials”, IEEE Trans. on Magn.,vol. 24, pp 621630, January. 1998. [8] P. Tenant, J.J. Rousseau, ”Dynamic model for soft ferrite”, Power Electronics Specialists Conference proceedings,vol. 2, pp 1070-1076, January. 1995, Atlanta.