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Magnetic field calculation considering the measured hysteresis
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 320 (2008) e988–e991 www.elsevier.com/locate/jmmm
Magnetic field calculation considering the measured hysteresis Marko Jesenik, Viktor Goricˇan, Mladen Trlep, Anton Hamler, Bojan Sˇtumberger Faculty of Electrical Engineering and Computer Science, Smetanova ul. 17, 2000 Maribor, Slovenia Available online 18 April 2008
Abstract The calculation of magnetic fields is and important part of the electromagnetic device design process. The numerical approach to the consideration of the material hysteresis is shown in the paper. Some problems with shapes of the new magnetization curves are pointed out. The results of the calculation are compared with the measurement results made on the magnetization setup for the characterization of semi- and hard-magnetic materials. r 2008 Elsevier B.V. All rights reserved. PACS: 85.70; 41.20; 81.40R Keywords: Magnetic field; Finite element method; Magnetic measurement
1. Introduction The programme for the 3D finite element magnetic field calculations considers the hysteresis of the material. The material input data are the measured major hysteresis loop and many first-order reversal curves (FORCs) for the increase and for the decrease of the excitation current. The data obtained with the measurement of the hysteresis loops are included in the calculation [1]. In each of the finite elements, the new magnetic induction B is calculated on the basis of the nonlinear finite element method calculation. The magnetic induction in each of the finite elements calculated from the previous time step, the history of the magnetic density and the excitation current are the basis for the evaluation of the new magnetization curve which will be used for nonlinear calculation in the current time step. 2. Evaluation of the new magnetization curves Hysteresis curves shown in Figs. 1, 4–9 and 12 are measured curves, obtained from the measurements made in our research group. Measurements are explained in Section 3. The calculation starts with the virgin magnetizaCorresponding author. Tel.: +386 2 220 7045; fax: +386 2 220 7272.
E-mail address: [email protected] (M. Jesenik). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.04.093
tion curve, which is shown in Fig. 1. The virgin magnetization curve is used for the calculation as long as the excitation current increases. At the moment when the excitation current changes its direction (it starts to decrease after the increasing or it starts to increase after the decreasing), we have to evaluate the new magnetization curve. Calculation procedure can be explained with the fallowing algorithm shown in Fig. 2. The algorithm which shows the evaluation of the new magnetization curves is shown in Fig. 3. If the calculated B is on the major hysteresis loop and the excitation current starts to increase, the FORCs are used for the definition of the new magnetization curve. Measured FORCs for the increase of the current are shown in Fig. 4. If the number of the measured FORCs is big, we can use the nearest FORC. Otherwise we have to make some kind of approximation between two FORCs to evaluate the necessary magnetization curve. The similar procedure is used for the decrease of the excitation current. If the calculated B is inside the major hysteresis loop, the new magnetization curve is evaluated with mirroring. The mirroring is made over the line connecting two points in the hysteresis loop where the change in the direction of the excitation current takes place. The mirrored part of the curve is shown in Fig. 5.
ARTICLE IN PRESS M. Jesenik et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e988–e991
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Fig. 1. Virgin magnetization curve.
Fig. 3. Algorithm for the evaluation of the new magnetization curves.
Fig. 2. Algorithm of the calculation procedure.
Fig. 4. FORCs.
The problems can appear if the mirrored part of the curve is outside the major hysteresis loop (in reality it is not possible), shown in Fig. 6. If the mirrored curve is outside
of the major hysteresis loop we have to evaluate the new magnetization curve as FORC, which goes through the point of the last calculated B, as it is shown in Fig. 7.
ARTICLE IN PRESS e990
M. Jesenik et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e988–e991
Fig. 5. Mirrored part of the magnetization curve.
Fig. 8. New magnetization curve made of FORC and virgin curve.
Fig. 6. Incorrect mirroring (part of the mirrored curve is outside of the major hysteresis loop).
Fig. 9. Extended FORC.
Fig. 10. Extended mirrored magnetization curve. Fig. 7. FORC, which goes through the point of the last calculated B.
FORC must be smooth to be appropriate for the Newton–Raphson calculation procedure. Fig. 8 shows the case where the calculation is made with virgin magnetization curve till the point A. In point A, the excitation current starts to decrease. The new curve should be evaluated for higher magnetic inductions as the induction in point A, to be able to perform the Newton–Raphson nonlinear calculation procedure. In Fig. 8, we can see that the virgin magnetization curve is used to calculate the magnetic inductions higher than BA. New magnetization curve is broken in point A. This break can cause numerical problems. Therefore, the extended magnetization curve is used for the calculation. It is shown in Fig. 9 as curve C3.
In Fig. 9, the curve C1 is the virgin curve, the curve C2 is the curve with the break and the curve C3 is the curve appropriate for the calculation. We obtained the curve C3 from the FORC. The last two points near the point A are linearly extended. In each calculation step, we have to check if the calculated magnetic induction B is smaller than BA. If the calculated magnetic induction B is bigger than BA, we have to repeat the calculation with the curve C1—the virgin magnetization curve. In point E in Fig. 10, excitation current starts to increase. If the calculated B from the previous step is inside the major hysteresis loop, the new magnetization curve is obtained with mirroring. The extended curve is shown in Fig. 10. The last two points of the mirrored part near the points A and E are linearly extended. In each calculation
ARTICLE IN PRESS M. Jesenik et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e988–e991
Fig. 11. The magnetization setup.
step we have to check if the calculated magnetic induction B is between BA and BE. If the calculated magnetic induction B is bigger than BA we have to repeat the calculation with the entire curve C2—the virgin curve, instead of the curve C7 shown in Fig. 10. The further calculation is made on the virgin magnetization curve until we reach the major hysteresis curve. The calculation procedure for the minor loop on the virgin magnetization curve was explained. The procedure is the same for the calculation of the minor loop on the major hysteresis. 3. B inside the sample of the magnetization setup The magnetization setup for the characterization of semi- and hard-magnetic materials was used to measure the hysteresis. It is shown in Fig. 11. The measurement is made with the frequency of 10 mHz. B is measured by induction in the measuring coil coiled around the centre of the sample. H is measured by Hall sensor that is placed near the sample (in the middle). The shape of the sample is cylinder. The same magnetization setup was modelled with threedimensional finite elements. The calculation was made as a three-dimensional nonlinear transient calculation [2]. We made the measurements and they can be compared with calculation results. To simplify the calculation, we only treated the hysteresis in the sample placed between the excitation coils. In the yoke, we used the nonlinear calculation where only the virgin magnetization curve is taken into account. Almost in the whole sample the direction of B is axial, except at the parts which are surrounded with yoke. The comparison between the calculation and the measurement of the minor loop made on the virgin magnetization curve together with used magnetization curves is shown in Fig. 12. The comparison starts with the point on the virgin magnetization curve, which is marked P1 in Fig. 12. The curve C1 is used for the calculation. The excitation current increases until the point P2 is reached.
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Fig. 12. Comparison between the calculation and the measurement of the minor hysteresis loop made on the virgin magnetization curve together with used magnetization curves.
At the point P2, the excitation current starts to decrease and we have to use the new magnetization curve C2, which is the FORC. The extended curve, obtained from the curve C2 is used for all calculation points until the point P3 is reached. In the point P3, the excitation current starts to increase. The calculation point is in the middle of the major hysteresis loop. The new magnetization curve is calculated as a mirror curve between the points P2 and P3. The new magnetization curve is shown in Fig. 12 as the curve C3. The curve C3 should be appropriately extended. The curve C3 is used for all calculation points of the increasing current until the point P2 is reached. After that point, the virgin magnetization curve is used until the point P4 is reached. In each calculation step, we have to check if curve C3 or virgin curve must be used, dependent on the B in the point P2. 4. Conclusions In Fig. 12, we can see some difference between the calculated and measured values. In spite of the seen differences, we must say that the purposed numerical method can be not very accurate in some cases and it can lead us away from the correct material behaviour. The weakness of the method is that we must measure as many of FORCs as possible to get enough data for the definition of the new magnetization curves. The advantage of the method is that the difficult mathematics of the analytical models are avoided in the numerical treatment of the material hysteresis. The method gives good results if we start the calculation in some given point at the major hysteresis loop. References [1] Prof. A. Ivanyi (Ed.), Preisach Memorial Book, Hysteresis Models in Mathematics, Physics and Engineering, Akademiai Kiado, Budapest, 2005. [2] P.P. Silvester, R.L. Ferrari, Finite Elements for Electrical Engineers, second ed., Cambridge University Press, 1983, 1990.
Journal of Magnetism and Magnetic Materials 320 (2008) e988–e991 www.elsevier.com/locate/jmmm
Magnetic field calculation considering the measured hysteresis Marko Jesenik, Viktor Goricˇan, Mladen Trlep, Anton Hamler, Bojan Sˇtumberger Faculty of Electrical Engineering and Computer Science, Smetanova ul. 17, 2000 Maribor, Slovenia Available online 18 April 2008
Abstract The calculation of magnetic fields is and important part of the electromagnetic device design process. The numerical approach to the consideration of the material hysteresis is shown in the paper. Some problems with shapes of the new magnetization curves are pointed out. The results of the calculation are compared with the measurement results made on the magnetization setup for the characterization of semi- and hard-magnetic materials. r 2008 Elsevier B.V. All rights reserved. PACS: 85.70; 41.20; 81.40R Keywords: Magnetic field; Finite element method; Magnetic measurement
1. Introduction The programme for the 3D finite element magnetic field calculations considers the hysteresis of the material. The material input data are the measured major hysteresis loop and many first-order reversal curves (FORCs) for the increase and for the decrease of the excitation current. The data obtained with the measurement of the hysteresis loops are included in the calculation [1]. In each of the finite elements, the new magnetic induction B is calculated on the basis of the nonlinear finite element method calculation. The magnetic induction in each of the finite elements calculated from the previous time step, the history of the magnetic density and the excitation current are the basis for the evaluation of the new magnetization curve which will be used for nonlinear calculation in the current time step. 2. Evaluation of the new magnetization curves Hysteresis curves shown in Figs. 1, 4–9 and 12 are measured curves, obtained from the measurements made in our research group. Measurements are explained in Section 3. The calculation starts with the virgin magnetizaCorresponding author. Tel.: +386 2 220 7045; fax: +386 2 220 7272.
E-mail address: [email protected] (M. Jesenik). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.04.093
tion curve, which is shown in Fig. 1. The virgin magnetization curve is used for the calculation as long as the excitation current increases. At the moment when the excitation current changes its direction (it starts to decrease after the increasing or it starts to increase after the decreasing), we have to evaluate the new magnetization curve. Calculation procedure can be explained with the fallowing algorithm shown in Fig. 2. The algorithm which shows the evaluation of the new magnetization curves is shown in Fig. 3. If the calculated B is on the major hysteresis loop and the excitation current starts to increase, the FORCs are used for the definition of the new magnetization curve. Measured FORCs for the increase of the current are shown in Fig. 4. If the number of the measured FORCs is big, we can use the nearest FORC. Otherwise we have to make some kind of approximation between two FORCs to evaluate the necessary magnetization curve. The similar procedure is used for the decrease of the excitation current. If the calculated B is inside the major hysteresis loop, the new magnetization curve is evaluated with mirroring. The mirroring is made over the line connecting two points in the hysteresis loop where the change in the direction of the excitation current takes place. The mirrored part of the curve is shown in Fig. 5.
ARTICLE IN PRESS M. Jesenik et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e988–e991
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Fig. 1. Virgin magnetization curve.
Fig. 3. Algorithm for the evaluation of the new magnetization curves.
Fig. 2. Algorithm of the calculation procedure.
Fig. 4. FORCs.
The problems can appear if the mirrored part of the curve is outside the major hysteresis loop (in reality it is not possible), shown in Fig. 6. If the mirrored curve is outside
of the major hysteresis loop we have to evaluate the new magnetization curve as FORC, which goes through the point of the last calculated B, as it is shown in Fig. 7.
ARTICLE IN PRESS e990
M. Jesenik et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e988–e991
Fig. 5. Mirrored part of the magnetization curve.
Fig. 8. New magnetization curve made of FORC and virgin curve.
Fig. 6. Incorrect mirroring (part of the mirrored curve is outside of the major hysteresis loop).
Fig. 9. Extended FORC.
Fig. 10. Extended mirrored magnetization curve. Fig. 7. FORC, which goes through the point of the last calculated B.
FORC must be smooth to be appropriate for the Newton–Raphson calculation procedure. Fig. 8 shows the case where the calculation is made with virgin magnetization curve till the point A. In point A, the excitation current starts to decrease. The new curve should be evaluated for higher magnetic inductions as the induction in point A, to be able to perform the Newton–Raphson nonlinear calculation procedure. In Fig. 8, we can see that the virgin magnetization curve is used to calculate the magnetic inductions higher than BA. New magnetization curve is broken in point A. This break can cause numerical problems. Therefore, the extended magnetization curve is used for the calculation. It is shown in Fig. 9 as curve C3.
In Fig. 9, the curve C1 is the virgin curve, the curve C2 is the curve with the break and the curve C3 is the curve appropriate for the calculation. We obtained the curve C3 from the FORC. The last two points near the point A are linearly extended. In each calculation step, we have to check if the calculated magnetic induction B is smaller than BA. If the calculated magnetic induction B is bigger than BA, we have to repeat the calculation with the curve C1—the virgin magnetization curve. In point E in Fig. 10, excitation current starts to increase. If the calculated B from the previous step is inside the major hysteresis loop, the new magnetization curve is obtained with mirroring. The extended curve is shown in Fig. 10. The last two points of the mirrored part near the points A and E are linearly extended. In each calculation
ARTICLE IN PRESS M. Jesenik et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e988–e991
Fig. 11. The magnetization setup.
step we have to check if the calculated magnetic induction B is between BA and BE. If the calculated magnetic induction B is bigger than BA we have to repeat the calculation with the entire curve C2—the virgin curve, instead of the curve C7 shown in Fig. 10. The further calculation is made on the virgin magnetization curve until we reach the major hysteresis curve. The calculation procedure for the minor loop on the virgin magnetization curve was explained. The procedure is the same for the calculation of the minor loop on the major hysteresis. 3. B inside the sample of the magnetization setup The magnetization setup for the characterization of semi- and hard-magnetic materials was used to measure the hysteresis. It is shown in Fig. 11. The measurement is made with the frequency of 10 mHz. B is measured by induction in the measuring coil coiled around the centre of the sample. H is measured by Hall sensor that is placed near the sample (in the middle). The shape of the sample is cylinder. The same magnetization setup was modelled with threedimensional finite elements. The calculation was made as a three-dimensional nonlinear transient calculation [2]. We made the measurements and they can be compared with calculation results. To simplify the calculation, we only treated the hysteresis in the sample placed between the excitation coils. In the yoke, we used the nonlinear calculation where only the virgin magnetization curve is taken into account. Almost in the whole sample the direction of B is axial, except at the parts which are surrounded with yoke. The comparison between the calculation and the measurement of the minor loop made on the virgin magnetization curve together with used magnetization curves is shown in Fig. 12. The comparison starts with the point on the virgin magnetization curve, which is marked P1 in Fig. 12. The curve C1 is used for the calculation. The excitation current increases until the point P2 is reached.
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Fig. 12. Comparison between the calculation and the measurement of the minor hysteresis loop made on the virgin magnetization curve together with used magnetization curves.
At the point P2, the excitation current starts to decrease and we have to use the new magnetization curve C2, which is the FORC. The extended curve, obtained from the curve C2 is used for all calculation points until the point P3 is reached. In the point P3, the excitation current starts to increase. The calculation point is in the middle of the major hysteresis loop. The new magnetization curve is calculated as a mirror curve between the points P2 and P3. The new magnetization curve is shown in Fig. 12 as the curve C3. The curve C3 should be appropriately extended. The curve C3 is used for all calculation points of the increasing current until the point P2 is reached. After that point, the virgin magnetization curve is used until the point P4 is reached. In each calculation step, we have to check if curve C3 or virgin curve must be used, dependent on the B in the point P2. 4. Conclusions In Fig. 12, we can see some difference between the calculated and measured values. In spite of the seen differences, we must say that the purposed numerical method can be not very accurate in some cases and it can lead us away from the correct material behaviour. The weakness of the method is that we must measure as many of FORCs as possible to get enough data for the definition of the new magnetization curves. The advantage of the method is that the difficult mathematics of the analytical models are avoided in the numerical treatment of the material hysteresis. The method gives good results if we start the calculation in some given point at the major hysteresis loop. References [1] Prof. A. Ivanyi (Ed.), Preisach Memorial Book, Hysteresis Models in Mathematics, Physics and Engineering, Akademiai Kiado, Budapest, 2005. [2] P.P. Silvester, R.L. Ferrari, Finite Elements for Electrical Engineers, second ed., Cambridge University Press, 1983, 1990.