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Wide frequency complex permeability function for linear magnetic materials
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 743–744
Wide frequency complex permeability function for linear magnetic materials Alex Van den Bosschea,*, Vencislav Valchevb, Marc De Wulfc a
Department of Electrical Energy, Systems and Automation, Ghent University, St-Pietersnieuwstraat 41, Gent B-9000, Belgium b Department of Electronics, Technical University of Varna, Bulgaria c OCAS-ARCELOR, B-9060 Zelzate, Belgium
Abstract A mathematical model is presented in which both the energy loss, the reactive power and the complex permeability of soft magnetic material cores are described in a wide frequency range. This general function is based on impedance functions and on the theory of one-dimensional homogeneous transmission lines. Besides the three material parameters (thickness, density and electrical conductivity), only four fitting parameters are used in the model. A good correspondence has been found for the description of a variety of soft magnetic materials. A comparison of the model prediction with experimental data is presented for toroidal nanocrystalline cores. r 2003 Elsevier B.V. All rights reserved. PACS: 81.70.q; 75.50.y Keywords: Energy losses; Permeability function; Wide frequency range; Nanocrystalline material; Reactive power
In the classical loss separation theory, the permeability and the losses are described in a separate way. There is however a relation between the frequency characteristic of the active and reactive power, and between the angle and frequency characteristic of the permeability [1]. Impedance functions have no poles nor zeros in the right half plane and are ‘minimum phase’. The complex permeability associated with hysteresis losses has an angle close to a constant, whereas its magnitude decreases with increasing frequency. Analytical functions compatible with this behaviour are described in [2]. A one-dimensional approach is known using complex tanh-functions [3]. The theory of impedances and the theory of onedimensional homogeneous transmission lines is used to find a general function in which energy losses (hysteresis and eddy currents) and reactive power are combined and used to derive complex analytical functions describing *Corresponding author. Tel.: +32-9-264-34-19; fax: +32-9269-35-82. E-mail address: [email protected] (A. Van den Bossche).
the losses. The proposed complex functions are easy to handle in todays mathematical software. First, we consider a constant loss angle impedance function for the description of the hysteresis losses. The hysteresis effect can be characterized by a constant loss angle dh of the complex impedance of the material over a wide frequency range. A constant loss angle impedance function, is given by zh ðsÞ ¼ smh ðsÞ ¼ mhr s12dh =p ;
ð1Þ
where dh is the loss angle (in radians) and s is the Laplace operator (s ¼ jo). The hysteresis reference permeability mhr is a real constant and is fitted such that Eq. (1) matches the data at a reference frequency, e.g. 10 kHz. For dh ¼ 0; the material is loss-less and has a constant permeability. For a low loss angle, the amplitude of the permeance is almost independent on the frequency and the losses almost proportional with frequency. High loss angles (combined with a high permeance) are present in some amorphous alloys, grain oriented FeSi steels and in Ni–Fe alloys. Using the one-dimensional transmission line theory, the following equation for a loss-less complex
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.671
ARTICLE IN PRESS 744
A. Van den Bossche et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 743–744
permeability mc can be derived rffiffiffiffiffi pffiffiffiffiffiffiffiffi d 2 sm tanh sms : zc ðsÞ ¼ smc ðsÞ ¼ d s 2
ð2Þ
in which s is the electrical conductivity and d is the lamination thickness. Similar expressions are used in Refs. [3,4]. At high frequencies, the permeance reduces with the root of the frequency, and the angle tends to p=4: Substituting the intrinsic material characteristic (1) into (2) leads to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 mhr s12dh =p d : ð3Þ tanh smhr s12dh =p zc ðsÞ ¼ d 2 s In order to be able to apply Eq. (3) at low frequencies, we place a parallel permeability mg : One could intuitively consider this as taking into account the parasitic airgap in a perfect core. This parallel permeability is defined by the reference permeability mgr and loss angle dg : zg ðsÞ ¼ smg ðsÞ ¼ mgr s12dg =p
ð4Þ
The wide frequency complex permeance is thus given as follows: 1 ; ð5Þ mw ðsÞ ¼ 1=mc ðsÞ þ 1=mg ðsÞ Eq. (5) allows to calculate the complex power SðsÞ per volume under sinusoidal flux !2 1 pffiffiffiB# s 2 ; ð6Þ SðsÞ ¼ jHj2rms jomw ðsÞ ¼ mw ðsÞ where B# is the peak induction and H is the effective value of the magnetic field. Using Eq. (6), we can calculate the apparent power |S|, the power loss Re(S) and the reactive power Im(S). As far as the permeability variation due to the change in induction is negligible, the derived equations are applicable for a wide frequency range. With increasing frequency however, the induction level at the side edge of the sheet becomes significantly higher than the average induction in the sheet. The ratio between the average induction, compared to one on the side edge, is denoted KðsÞ: For the frequency range where KðsÞ is close to unity, the linear model can be used. The proposed model is used for the description of soft magnetic nanocrystalline material (Vitroperm500F). A comparison of the model prediction with experimental data is shown in Fig. 1. A good correspondance is found. The values of the used model parameters are listed in Table 1. The advantage of the proposed model is that it explains both low- and high- frequency behaviour of the loss angle and amplitude of the permeability in one single expression. Furthermore, it explains a loss angle exceeding p=4 in the high frequency range (which is observed in most of the magnetic materials).
Fig. 1. Comparison of the model prediction with experimental data on nanocrystalline Vitroperm500F. Table 1 Material parameters in the wide frequency model for the description of soft magnetic nanocrystalline material d (mm) r (Om) Density dh ( ) (kg/m3)
mhr (/)
dg ( )
mg (/)
0.021
250e6
0.025
15560
1.15e-6 7300
50
V. Valchev wishes to thank NATO—FWO Vlaanderen to grant his stay at the Department EESA of Ghent University, Belgium.
References [1] J. Slama, R. Dosoudil, R. V!ıcen, A. Gruskov!a, V. Olah, I. Hudec, E. U$sa! k, J. Magn. Magn. Mater. 254–255 (2003) 195. [2] M. Fogiel, The Handbook of Electrical Engineering, Research and Education Assocation, USA, 1996. [3] R. Lebourgeois, S. B!erenguer, C. Ramiarinjaona, T. Waeckerl!e, J. Magn. Magn. Mater. 254–255 (2003) 191. [4] J. Lamebrain, M. Stafl, Eddy Currents, Iliffe Books, London, UK, 1966.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 743–744
Wide frequency complex permeability function for linear magnetic materials Alex Van den Bosschea,*, Vencislav Valchevb, Marc De Wulfc a
Department of Electrical Energy, Systems and Automation, Ghent University, St-Pietersnieuwstraat 41, Gent B-9000, Belgium b Department of Electronics, Technical University of Varna, Bulgaria c OCAS-ARCELOR, B-9060 Zelzate, Belgium
Abstract A mathematical model is presented in which both the energy loss, the reactive power and the complex permeability of soft magnetic material cores are described in a wide frequency range. This general function is based on impedance functions and on the theory of one-dimensional homogeneous transmission lines. Besides the three material parameters (thickness, density and electrical conductivity), only four fitting parameters are used in the model. A good correspondence has been found for the description of a variety of soft magnetic materials. A comparison of the model prediction with experimental data is presented for toroidal nanocrystalline cores. r 2003 Elsevier B.V. All rights reserved. PACS: 81.70.q; 75.50.y Keywords: Energy losses; Permeability function; Wide frequency range; Nanocrystalline material; Reactive power
In the classical loss separation theory, the permeability and the losses are described in a separate way. There is however a relation between the frequency characteristic of the active and reactive power, and between the angle and frequency characteristic of the permeability [1]. Impedance functions have no poles nor zeros in the right half plane and are ‘minimum phase’. The complex permeability associated with hysteresis losses has an angle close to a constant, whereas its magnitude decreases with increasing frequency. Analytical functions compatible with this behaviour are described in [2]. A one-dimensional approach is known using complex tanh-functions [3]. The theory of impedances and the theory of onedimensional homogeneous transmission lines is used to find a general function in which energy losses (hysteresis and eddy currents) and reactive power are combined and used to derive complex analytical functions describing *Corresponding author. Tel.: +32-9-264-34-19; fax: +32-9269-35-82. E-mail address: [email protected] (A. Van den Bossche).
the losses. The proposed complex functions are easy to handle in todays mathematical software. First, we consider a constant loss angle impedance function for the description of the hysteresis losses. The hysteresis effect can be characterized by a constant loss angle dh of the complex impedance of the material over a wide frequency range. A constant loss angle impedance function, is given by zh ðsÞ ¼ smh ðsÞ ¼ mhr s12dh =p ;
ð1Þ
where dh is the loss angle (in radians) and s is the Laplace operator (s ¼ jo). The hysteresis reference permeability mhr is a real constant and is fitted such that Eq. (1) matches the data at a reference frequency, e.g. 10 kHz. For dh ¼ 0; the material is loss-less and has a constant permeability. For a low loss angle, the amplitude of the permeance is almost independent on the frequency and the losses almost proportional with frequency. High loss angles (combined with a high permeance) are present in some amorphous alloys, grain oriented FeSi steels and in Ni–Fe alloys. Using the one-dimensional transmission line theory, the following equation for a loss-less complex
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.671
ARTICLE IN PRESS 744
A. Van den Bossche et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 743–744
permeability mc can be derived rffiffiffiffiffi pffiffiffiffiffiffiffiffi d 2 sm tanh sms : zc ðsÞ ¼ smc ðsÞ ¼ d s 2
ð2Þ
in which s is the electrical conductivity and d is the lamination thickness. Similar expressions are used in Refs. [3,4]. At high frequencies, the permeance reduces with the root of the frequency, and the angle tends to p=4: Substituting the intrinsic material characteristic (1) into (2) leads to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 mhr s12dh =p d : ð3Þ tanh smhr s12dh =p zc ðsÞ ¼ d 2 s In order to be able to apply Eq. (3) at low frequencies, we place a parallel permeability mg : One could intuitively consider this as taking into account the parasitic airgap in a perfect core. This parallel permeability is defined by the reference permeability mgr and loss angle dg : zg ðsÞ ¼ smg ðsÞ ¼ mgr s12dg =p
ð4Þ
The wide frequency complex permeance is thus given as follows: 1 ; ð5Þ mw ðsÞ ¼ 1=mc ðsÞ þ 1=mg ðsÞ Eq. (5) allows to calculate the complex power SðsÞ per volume under sinusoidal flux !2 1 pffiffiffiB# s 2 ; ð6Þ SðsÞ ¼ jHj2rms jomw ðsÞ ¼ mw ðsÞ where B# is the peak induction and H is the effective value of the magnetic field. Using Eq. (6), we can calculate the apparent power |S|, the power loss Re(S) and the reactive power Im(S). As far as the permeability variation due to the change in induction is negligible, the derived equations are applicable for a wide frequency range. With increasing frequency however, the induction level at the side edge of the sheet becomes significantly higher than the average induction in the sheet. The ratio between the average induction, compared to one on the side edge, is denoted KðsÞ: For the frequency range where KðsÞ is close to unity, the linear model can be used. The proposed model is used for the description of soft magnetic nanocrystalline material (Vitroperm500F). A comparison of the model prediction with experimental data is shown in Fig. 1. A good correspondance is found. The values of the used model parameters are listed in Table 1. The advantage of the proposed model is that it explains both low- and high- frequency behaviour of the loss angle and amplitude of the permeability in one single expression. Furthermore, it explains a loss angle exceeding p=4 in the high frequency range (which is observed in most of the magnetic materials).
Fig. 1. Comparison of the model prediction with experimental data on nanocrystalline Vitroperm500F. Table 1 Material parameters in the wide frequency model for the description of soft magnetic nanocrystalline material d (mm) r (Om) Density dh ( ) (kg/m3)
mhr (/)
dg ( )
mg (/)
0.021
250e6
0.025
15560
1.15e-6 7300
50
V. Valchev wishes to thank NATO—FWO Vlaanderen to grant his stay at the Department EESA of Ghent University, Belgium.
References [1] J. Slama, R. Dosoudil, R. V!ıcen, A. Gruskov!a, V. Olah, I. Hudec, E. U$sa! k, J. Magn. Magn. Mater. 254–255 (2003) 195. [2] M. Fogiel, The Handbook of Electrical Engineering, Research and Education Assocation, USA, 1996. [3] R. Lebourgeois, S. B!erenguer, C. Ramiarinjaona, T. Waeckerl!e, J. Magn. Magn. Mater. 254–255 (2003) 191. [4] J. Lamebrain, M. Stafl, Eddy Currents, Iliffe Books, London, UK, 1966.