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Influence of electrical sheet width on dynamic magnetic properties
Journal of Magnetism and Magnetic Materials 215}216 (2000) 623}625
In#uence of electrical sheet width on dynamic magnetic properties T. Chevalier *, A. Kedous-Lebouc, B. Cornut Moteurs Leroy-Somer, F-16015 AngouleL me Cedex, France Laboratoire d+Electrotechnique de Grenoble, INPG/UJF-CNRS UMR 5529, Domaine Universitaire, B.P. 46, F-38402 Saint Martin d+He% res, France
Abstract E!ects of the width of electrical steel sheets on dynamic magnetic properties are investigated by solving di!usion equation on the cross-section of the sheet. Linear and non-linear cases are studied, and are compared with measurement on Epstein frame. For the "rst one an analytical solution is found, while for the second, a 2D "nite element simulation is achieved. The in#uence of width is highlighted for a width thickness ratio lower than 10. It is shown that the behaviour modi"cation in such cases is conditioned by the excitation signal waveform, amplitude and also frequency. 2000 Elsevier Science B.V. All rights reserved. Keywords: Iron loss; Electrical sheet width; Eddy currents
1. Introduction To design electrical machines, engineers use 2D "nite element software which requires a knowledge of electrical steel sheets magnetic behaviour. Such a behaviour is generally determined under standard magnetic characterisation carried out at 50 Hz on an Epstein frame or a single sheet tester. The tested samples are then large (width'30 mm) and the e!ects of the sheet edges are negligible. However, the magnetic circuit of electrical machines presents a complex geometry with large and small parts. Teeth, for example, can be 2 or 3 mm wide. In addition, their ends are usually submitted to high harmonic #ux variations. So one can wonders if the onedimensional assumption conventionally used remains true in that case. This paper presents a complete study of the width e!ects on magnetic properties of electrical steel sheets. To enable frequency in#uence analysis, sinusoidal waveform is considered. But it can be imposed either on
* Corresponding author. Tel.: #33-4-76-82-62-99; fax: #334-76-82-63-00. E-mail address: [email protected] (T. Chevalier).
the "eld H applied on the surface of the sheet or on the mean #ux density B considered in the cross-section of
the sheet. The theoretical study is based on the solution of Maxwell's equations in the cross-section of a sheet, including more or less complex local magnetic behaviour. The local magnetic law considered in this paper is either linear or non-linear. Theoretical analyses are compared with measurements in the case of 0.65 mm thick electrical steel sheets whose width can vary from 30 to 3 mm.
2. Linear study In all the studies, the "elds are always perpendicular to the cross-section of the sheet . The local "elds are denoted by B (x, y, t) and H (x, y, t) and are linked in the linear X X case by B (x, y)"kH (x, y) where k is the local permeabX X ility. The conductivity of the material is denoted by p. The boundary condition imposed is a uniform sinusoidal magnetic "eld of amplitude H and frequency u. The equation to be solved is then RHz(x, y, t) RHz(x, y, t) RHz(x, y, t) # "pk , Ry Rt Rx
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 2 4 4 - 4
624
T. Chevalier et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 623}625
Fig. 1. Pro"le of "eld H in the cross-section of the sheet obtained under a 1500 Hz sinusoidal excitation, linear case.
Hz(0, x, t)"Hz(x, b, t)"H sin(ut), Hz(0, y, t)"Hz(a, y, t)"H sin(ut). (1) The resolution is made analytically using a decomposition in Fourier series [1]. Only permanent state is considered. The mean induction B is then obtained by
integration of kH (x, y) upon the whole cross-section and X is expressed by : B (t)"kH sin(ut)
upk ! cos(ut)# sin(ut) a(m, n) K L
16kH u a(m, n)/pk 4 Q ; mnp (a(m, n)/pk)#u mnp
Fig. 2. Losses as a function of sheet width. Plain marks: H am plitude constant, linear case; empty marks: B amplitude con stant.
ratio of the steel sheet becomes lower than 10, the response of the system is sensibly modi"ed. What is more, the behaviour modi"cation depends on the excitation conditions, that is to say whether H or B is imposed.
The linear case gives a qualitative information, in order to obtain quantitative results the nonlinear case is studied and presented in the next section.
3. Non-linear study
(2)
with mp np a(m, n)" # . a b The pro"le of the magnetic "eld in the whole section of the sheet highlights the delay undergone in the middle of the sheet due to induced currents (Fig. 1). The behaviour of the material is completely characterised by H (t) and B (t). To analyse it, two magnitudes can be calculated,
one is the apparent permeability k , and the other is the losses Loss. They are, respectively, de"ned by k " Max(B (t))/Max(H (t)) and Loss"(1/o¹)2H (t) dB (t)
in which o corresponds to the mass density of the material. The Loss is expressed by Eq. (3) and can be compared as a function of k and p to the classical 1D hypothesis: 1 4uH 16kH u a(m, n)/pk Loss" (3) 2 pmn mnp (a(m, n)/pk)#u. K L
Evolution of k and Loss are calculated as a function of the width a in the case of a 0.65 mm thick N.O. iron sheet M1000-65D and for di!erent frequencies. Fig. 2 represents the variation of Loss for 50, 500 and 1500 Hz. It is observed that when the width thickness
The non-linear case is not solvable analytically except for idealised magnetic law like the step-function [1], a numerical "nite element method is therefore used [2]. The equation to be solved depends on whether H or B is imposed and is, respectively;
d(kH ) Curl(Curl(H ))#p "0, (4a) dt
Curl
1 dA Curl(A ) #p "0. k dt
(4b)
The boundary conditions are then H for Eq. (4a) and the potential vector A for Eq. (4b) whose circulation on the border of the problem allows the #ux to be imposed. The non-linearity is treated using a Newton}Raphson process. The numerical simulation requires a minimum number of signal periods in order for it to reach the permanent state. In practice about 10 periods are computed. The meshing of the sheet cross-section is composed of regular rectangles. The magnetic local law is a non-linear B (H ) extracted from measurements made on standard Epstein samples under quasi-static conditions. Simulations are performed on variable width geometries and for di!erent excitation frequencies and levels. Hysteresis cycle B (H ) undergone by each sheet is then
calculated.
T. Chevalier et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 623}625
Fig. 3. Calculated B (H ) cycles for a sinusoidal H (t) of
400 A/m, non-linear case. Dashed line: calculations, continuous line: measurements.
Table 1 Losses induced by a sinusoidal H (t) of 400 A/m amplitude Width (mm) Losses estimated (W/kg) Losses measured (W/kg)
30 29.1 33.5
6.5 29.9 33.8
3 31.8 35.4
0.65 44.2 n.d.
Table 2 Losses induced by a sinusoidal H (t) of 6000 A/m amplitude Width (mm) Losses (W/kg)
30 1249
6.5 1230
3 1170
0.65 780
Fig. 3 shows simulated hysteresis cycles at 500 Hz with a sinusoidal signal H (t) of amplitude 400 A/m for three di!erent widths. Under such an excitation, measurements are also carried out for three Epstein samples of 30, 6.5 and 3 mm width. The corresponding losses are compared to the simulation ones in Table 1. In that case the material is essentially working in the linear part and the losses follow the increase predicted by the linear study. Simulation results agree with experiment as shown in Table 1 for losses and in Fig. 3 for hysteresis loops. The same study is also made for higher excitation level. Simulation results given in Table 2 and Fig. 4 show a decrease of losses with the sample width, contrary to the low-level excitation case. This can be explained by the saturation of the material. In fact, the decrease of permeability at high "eld level tends to reduce the eddy currents e!ects. As a consequence, the induction is more uniform in the cross-section of the sheet and losses are
625
Fig. 4. Calculated B (H ) cycles for a sinusoidal H (t) of 6000
A/m, non-linear case.
reduced with width. No measurement has been performed in this case because so high a sinusoidal magnetic "eld cannot be controlled on the measurement equipment. However, the same kinds of simulation and measurements are carried out with an imposed sinusoidal induction, a decrease of losses with the width is observed. In that case the non-linearity of the material acts in the same way.
4. Conclusion This paper highlights the importance of taking into account the width of electrical steel sheet when frequency increases and geometrical width thickness ratio becomes low. The geometrical ratio limit for the studied material is evaluated as 10. It was shown that not only the hysteresis loop B (H ) but also the losses are a!ected. The
observed modi"cations are conditioned by the excitation signal waveform. Such a phenomenon should be introduced in electrical machine calculation in order to evaluate its real in#uence. However, its implementation in classical 2D "nite element software is not easy to achieve and gives rise to several problems.
References [1] R.L. Stoll, The Analysis of Eddy Curents, ISBN : 0198593112, Monographs in Electrical and Electronic engineering, Claredon press, Oxford, 1974, pp. 28}33 and 40}45. [2] H. Kardestuncer, and D. H. Norrie, Finite Element Handbook, ISBN: 0-07-033305-X, McGraw-Hill, New York, 1987.
In#uence of electrical sheet width on dynamic magnetic properties T. Chevalier *, A. Kedous-Lebouc, B. Cornut Moteurs Leroy-Somer, F-16015 AngouleL me Cedex, France Laboratoire d+Electrotechnique de Grenoble, INPG/UJF-CNRS UMR 5529, Domaine Universitaire, B.P. 46, F-38402 Saint Martin d+He% res, France
Abstract E!ects of the width of electrical steel sheets on dynamic magnetic properties are investigated by solving di!usion equation on the cross-section of the sheet. Linear and non-linear cases are studied, and are compared with measurement on Epstein frame. For the "rst one an analytical solution is found, while for the second, a 2D "nite element simulation is achieved. The in#uence of width is highlighted for a width thickness ratio lower than 10. It is shown that the behaviour modi"cation in such cases is conditioned by the excitation signal waveform, amplitude and also frequency. 2000 Elsevier Science B.V. All rights reserved. Keywords: Iron loss; Electrical sheet width; Eddy currents
1. Introduction To design electrical machines, engineers use 2D "nite element software which requires a knowledge of electrical steel sheets magnetic behaviour. Such a behaviour is generally determined under standard magnetic characterisation carried out at 50 Hz on an Epstein frame or a single sheet tester. The tested samples are then large (width'30 mm) and the e!ects of the sheet edges are negligible. However, the magnetic circuit of electrical machines presents a complex geometry with large and small parts. Teeth, for example, can be 2 or 3 mm wide. In addition, their ends are usually submitted to high harmonic #ux variations. So one can wonders if the onedimensional assumption conventionally used remains true in that case. This paper presents a complete study of the width e!ects on magnetic properties of electrical steel sheets. To enable frequency in#uence analysis, sinusoidal waveform is considered. But it can be imposed either on
* Corresponding author. Tel.: #33-4-76-82-62-99; fax: #334-76-82-63-00. E-mail address: [email protected] (T. Chevalier).
the "eld H applied on the surface of the sheet or on the mean #ux density B considered in the cross-section of
the sheet. The theoretical study is based on the solution of Maxwell's equations in the cross-section of a sheet, including more or less complex local magnetic behaviour. The local magnetic law considered in this paper is either linear or non-linear. Theoretical analyses are compared with measurements in the case of 0.65 mm thick electrical steel sheets whose width can vary from 30 to 3 mm.
2. Linear study In all the studies, the "elds are always perpendicular to the cross-section of the sheet . The local "elds are denoted by B (x, y, t) and H (x, y, t) and are linked in the linear X X case by B (x, y)"kH (x, y) where k is the local permeabX X ility. The conductivity of the material is denoted by p. The boundary condition imposed is a uniform sinusoidal magnetic "eld of amplitude H and frequency u. The equation to be solved is then RHz(x, y, t) RHz(x, y, t) RHz(x, y, t) # "pk , Ry Rt Rx
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 2 4 4 - 4
624
T. Chevalier et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 623}625
Fig. 1. Pro"le of "eld H in the cross-section of the sheet obtained under a 1500 Hz sinusoidal excitation, linear case.
Hz(0, x, t)"Hz(x, b, t)"H sin(ut), Hz(0, y, t)"Hz(a, y, t)"H sin(ut). (1) The resolution is made analytically using a decomposition in Fourier series [1]. Only permanent state is considered. The mean induction B is then obtained by
integration of kH (x, y) upon the whole cross-section and X is expressed by : B (t)"kH sin(ut)
upk ! cos(ut)# sin(ut) a(m, n) K L
16kH u a(m, n)/pk 4 Q ; mnp (a(m, n)/pk)#u mnp
Fig. 2. Losses as a function of sheet width. Plain marks: H am plitude constant, linear case; empty marks: B amplitude con stant.
ratio of the steel sheet becomes lower than 10, the response of the system is sensibly modi"ed. What is more, the behaviour modi"cation depends on the excitation conditions, that is to say whether H or B is imposed.
The linear case gives a qualitative information, in order to obtain quantitative results the nonlinear case is studied and presented in the next section.
3. Non-linear study
(2)
with mp np a(m, n)" # . a b The pro"le of the magnetic "eld in the whole section of the sheet highlights the delay undergone in the middle of the sheet due to induced currents (Fig. 1). The behaviour of the material is completely characterised by H (t) and B (t). To analyse it, two magnitudes can be calculated,
one is the apparent permeability k , and the other is the losses Loss. They are, respectively, de"ned by k " Max(B (t))/Max(H (t)) and Loss"(1/o¹)2H (t) dB (t)
in which o corresponds to the mass density of the material. The Loss is expressed by Eq. (3) and can be compared as a function of k and p to the classical 1D hypothesis: 1 4uH 16kH u a(m, n)/pk Loss" (3) 2 pmn mnp (a(m, n)/pk)#u. K L
Evolution of k and Loss are calculated as a function of the width a in the case of a 0.65 mm thick N.O. iron sheet M1000-65D and for di!erent frequencies. Fig. 2 represents the variation of Loss for 50, 500 and 1500 Hz. It is observed that when the width thickness
The non-linear case is not solvable analytically except for idealised magnetic law like the step-function [1], a numerical "nite element method is therefore used [2]. The equation to be solved depends on whether H or B is imposed and is, respectively;
d(kH ) Curl(Curl(H ))#p "0, (4a) dt
Curl
1 dA Curl(A ) #p "0. k dt
(4b)
The boundary conditions are then H for Eq. (4a) and the potential vector A for Eq. (4b) whose circulation on the border of the problem allows the #ux to be imposed. The non-linearity is treated using a Newton}Raphson process. The numerical simulation requires a minimum number of signal periods in order for it to reach the permanent state. In practice about 10 periods are computed. The meshing of the sheet cross-section is composed of regular rectangles. The magnetic local law is a non-linear B (H ) extracted from measurements made on standard Epstein samples under quasi-static conditions. Simulations are performed on variable width geometries and for di!erent excitation frequencies and levels. Hysteresis cycle B (H ) undergone by each sheet is then
calculated.
T. Chevalier et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 623}625
Fig. 3. Calculated B (H ) cycles for a sinusoidal H (t) of
400 A/m, non-linear case. Dashed line: calculations, continuous line: measurements.
Table 1 Losses induced by a sinusoidal H (t) of 400 A/m amplitude Width (mm) Losses estimated (W/kg) Losses measured (W/kg)
30 29.1 33.5
6.5 29.9 33.8
3 31.8 35.4
0.65 44.2 n.d.
Table 2 Losses induced by a sinusoidal H (t) of 6000 A/m amplitude Width (mm) Losses (W/kg)
30 1249
6.5 1230
3 1170
0.65 780
Fig. 3 shows simulated hysteresis cycles at 500 Hz with a sinusoidal signal H (t) of amplitude 400 A/m for three di!erent widths. Under such an excitation, measurements are also carried out for three Epstein samples of 30, 6.5 and 3 mm width. The corresponding losses are compared to the simulation ones in Table 1. In that case the material is essentially working in the linear part and the losses follow the increase predicted by the linear study. Simulation results agree with experiment as shown in Table 1 for losses and in Fig. 3 for hysteresis loops. The same study is also made for higher excitation level. Simulation results given in Table 2 and Fig. 4 show a decrease of losses with the sample width, contrary to the low-level excitation case. This can be explained by the saturation of the material. In fact, the decrease of permeability at high "eld level tends to reduce the eddy currents e!ects. As a consequence, the induction is more uniform in the cross-section of the sheet and losses are
625
Fig. 4. Calculated B (H ) cycles for a sinusoidal H (t) of 6000
A/m, non-linear case.
reduced with width. No measurement has been performed in this case because so high a sinusoidal magnetic "eld cannot be controlled on the measurement equipment. However, the same kinds of simulation and measurements are carried out with an imposed sinusoidal induction, a decrease of losses with the width is observed. In that case the non-linearity of the material acts in the same way.
4. Conclusion This paper highlights the importance of taking into account the width of electrical steel sheet when frequency increases and geometrical width thickness ratio becomes low. The geometrical ratio limit for the studied material is evaluated as 10. It was shown that not only the hysteresis loop B (H ) but also the losses are a!ected. The
observed modi"cations are conditioned by the excitation signal waveform. Such a phenomenon should be introduced in electrical machine calculation in order to evaluate its real in#uence. However, its implementation in classical 2D "nite element software is not easy to achieve and gives rise to several problems.
References [1] R.L. Stoll, The Analysis of Eddy Curents, ISBN : 0198593112, Monographs in Electrical and Electronic engineering, Claredon press, Oxford, 1974, pp. 28}33 and 40}45. [2] H. Kardestuncer, and D. H. Norrie, Finite Element Handbook, ISBN: 0-07-033305-X, McGraw-Hill, New York, 1987.