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Use of quasi-static loops of magnetic hysteresis in loss prediction in non-oriented electrical steels
Physica B xxx (xxxx) xxx
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Physica B: Physics of Condensed Matter journal homepage: http://www.elsevier.com/locate/physb
Use of quasi-static loops of magnetic hysteresis in loss prediction in non-oriented electrical steels Jan Szczyglowski Czestochowa University of Technology, Faculty of Electrical Engineering, Al. Armii Krajowej 17, 42-200 Częstochowa, Poland
A R T I C L E I N F O
A B S T R A C T
Keywords: Eddy current losses Non-oriented electric steel Bertotti’s model
One of the most important factors taken into account in the design and optimization of magnetic circuits of electronic and electric devices are energy losses. Therefore, it is necessary to have accurate models of energy losses in magnetic materials. At present, the most popular model describing excess losses in material Pexc is the semi-empirical model proposed by Bertotti. An important source of error in its application is the methodology of determining hysteresis losses Ph , that are part of total losses expressed in the form of the following formula Ptot ¼ Ph þ Pc þ Pexc . The second component of this expression Pc is defined as classical eddy-current losses. The paper proposes a new approach to calculate hysteresis losses by using the Warburg theorem in which B(H) is expressed in the form of parametric equations. The proposed solution allows one to reduce the loss prediction error significantly.
1. Introduction One of the most important factors taken into account in the design and optimization of magnetic circuits of electrical devices are energy losses. In the model proposed by Bertotti [1,2], energy losses are the result of eddy currents generated by the movement of various “magnetic objects” such as: parts of domain walls, single domain walls, and rota tion of the magnetization vector. Eddy currents can be generated in various space-time scales. On the micro scale, the source of eddy cur rents are Barkhausen jumps responsible for hysteresis losses Ph . On the scale of the entire volume of material responsible for classic losses Pc . Changes in the magnetic field due to interactions between “magnetic objects” generate eddy currents responsible for excess losses Pexc . Total losses according to Bertotti are the sum of these three components. The Bertotti model is currently the most used model to describe energy losses in electrical steel [3–7]. Resulting from the Bertotti model, the formula describing the dependence of energy loss vs. magnetic induction and frequency for non-oriented steel sheet takes the following form [6]: 2
2
1:5
Ptot ¼ Ph þ Pc þ Pexc ¼ ch Bm f þ ce Bm f 2 þ cexc Bm f 1:5
(1)
where: Ptot – total power loss per unit of mass (W/kg), Bm – mean magnetic flux density throughout the sample (T), f – field frequency, (Hz), ch , ce and cexc are material-dependent coefficients. Coefficients
take the following forms: ce ¼ π2 σd2 =6ρ, where ρ is mass density, σ is electrical conductivity and d is the sample thickness of the tested ma pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi terial cut as a strip, cexc ¼ 8 σGSV0 , where G is the damping parameter equal to 0.1356, S is the sheet cross-section area and V0 is model parameter that characterizes the distribution of the internal co ercive fields, which permits one to address quite effectively the rela tionship between dynamic losses and microstructure of the non-oriented electrical alloy microstructure [8,9]. The hysteresis losses component for calculations based on the above formula is usually obtained by extrapolation the measured data of Ptot =f as a function of frequency and taking its limit when f→0 . The second component of the total losses is separately calculated based on the knowledge of the material’s con ductivity and density. The third component of the total losses is calculated after deter mining the value of its coefficients. The coefficient V0 is determined according to the slope of the excess losses chart in coordinate system ðPtot Pc Þ=f vs: f 0:5 based on experimental data. Correct determi nation of the value of individual components of losses in expression (1) allows one to calculate the total losses, the knowledge of which is necessary at the stage of designing the magnetic circuits of electrical devices. The maximum errors between the values of the total losses calculated based on the dependence (1) and the values of losses obtained from measurements can reach 40% [7]. It was shown that the value of this error depends strongly on the value of magnetic induction and frequency. The accuracy of the approximation of excess losses described
E-mail address: [email protected]. https://doi.org/10.1016/j.physb.2019.411812 Received 29 June 2019; Received in revised form 9 October 2019; Accepted 21 October 2019 Available online 24 October 2019 0921-4526/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Jan Szczyglowski, Physica B, https://doi.org/10.1016/j.physb.2019.411812
J. Szczyglowski
Physica B: Physics of Condensed Matter xxx (xxxx) xxx
by Bertotti to the excess component determined experimentally does not exceed 12% [10]. Further metrological analysis should concern the verification of the amount of excess losses by other measuring methods. The practical use of this model, however, encounters a number of con straints related in particular to the skin effect and the need to separate hysteresis losses from total losses, which is necessary to determine the excess losses underlying the model. The paper presents the possibility of limiting this error using the new method of calculating hysteresis losses based on the Warburg theorem [11] in which the dependence BðHÞ is described by means of parametric equations [12].
calculating the shape factor using the dependence of total losses on the frequency for the frequency tending to zero. At frequency tending to zero the ratio of total loss and the frequency equals the ratio of the hysteresis loss and the frequency, because other terms in formula (1) vanish. Hence, lim f →0
According to the Warburg theory [11] hysteresis losses are given by the following relation: I ZZZ 1 Ph ¼ HdBdV (2) VT
where ρ is density and V is volume of the sample. But Eq. (4) allows us to write � � W ZZZ ZZZ Ptot kg 1 1 1 B2max 1 B2 ¼ limf →0 limf →0 2 sdV ¼ limf →0 2 max sdV: f f ρV μ ρV μ VT V (7)
Calculation of the integral with respect to B in expression (2) requires knowledge of the relationship B(H), which is expressed by the hysteresis loop. In the paper, the ascending branch of the hysteresis loop, when the value B changes from Bmax to Bmax , is described by the following parametric equations: � BðτÞ ¼ Bmax cos τ; (3) HðτÞ ¼ H0 ða0 þ a1 cos τ þ a2 cos 2 τ þ …Þ;
When the frequency is nearly zero, the magnetic field has the same amplitude throughout the sample, therefore the volume integral is easy to calculate, leading to � � W Ptot kg 1 B2max B2 ¼ lim sV ¼ 2 max s: (8) 2 f →0 f ρV μf ¼0 ρμf ¼0
where H0 ¼ Bmax =μ, τ is a curve parameter changing from 0 to π, and a0 ; a1 ; a2 … are dimensionless parameters, which can be determined by curve fitting of the experimental hysteresis loop. Magnetic permeability was determined from dynamic hysteresis loops, using peak values for successive minor loops μ, according to Fig. 1 a. The integral in the expression (2) can be depicted as follows: I Z π B2 HdB ¼ 2 HðτÞBmax sin τdτ ¼ 2 max s; (4) where s ¼ 2
Hence � � s¼
μ
P
n¼0;2;4;…
an 1 n2
(5)
The area of the static hysteresis loop is proportional to the hysteresis loss so that � � � � W W ZZZ I Ptot kg Ph kg 1 1 1 limf →0 HdBdV ¼ limf →0 ¼ limf →0 f f f ρV VT ZZZ I 1 1 1 ¼ limf →0 HdBdV (6) f ρV VT
2. Description of the hysteresis losses
0
Ptot Ph ¼ lim : f →0 f f
is defined as the quasi-static loop shape factor
ρ
Ptot limμ
2B2max f →0
f
W kg
(9)
:
Since Ptot can be found in data sheet of a material, it follows that s can be easily found without using directly the definition formula. In fact, such approach was used in the numerical calculations. Expression (4) can be used in Eq. (2) which gives the following expression:
(only an terms for even n matter). The calculation of this factor requires the determination of the coefficients an , which may be calculated using the measured hysteresis loops [12]. The paper presents a simpler way of
a)
b)
Fig. 1. Family of hysteresis loops for non-oriented electrical sheet M 330–35 at a frequency of 5 Hz (a), shape of the sample under consideration (b). J in Fig. 1b denotes eddy current density vector. 2
J. Szczyglowski
Ph ¼
Physica B: Physics of Condensed Matter xxx (xxxx) xxx
ZZZ
2f
μ
V
B2max sdV:
permeability was determined from the dynamic hysteresis loops, using peak values for successive minor loops i.e. from μ ¼ Bm =μ0 Hm , cf. Fig. 1a. The frequency range of the loss measurements was chosen to limit the skin effect (δ≫d). The estimated upper frequency value for the tested samples was 400 Hz. The calculations are presented in Ref. [15]. In Table 1, the results of hysteresis loss computations from expres sion (14) (denoted as P1h ) as well as the corresponding hysteresis losses determined from the extrapolation of measurement data ptot =f for f→0 (denoted as P2h ) are shown. It should be stated that the hysteresis losses per cycle computed from the proposed expression are rate-dependent, which means that the losses per cycle are frequency dependent. This is the result of the frequency influence on the magnetic permeability value in expression (14). Similar results are presented in Refs. [16–18]. On the other hand their counterpart calculated from the expressions used so far are rate-independent, which means that the losses per cycle are fre quency independent. In Fig. 2, the comparison of approximation errors for the two ap proaches used to compute the total losses based on Eq. (1) is presented in two cases. In the first case, in the computations, the term P1h was accounted, in the second one - P2h : The relative error of the approximation was defined as it follows: δ ¼ ðPtot Pm Þ=Pm ⋅100%, where Ptot are the losses calculated with Eq. (1) and Pm are the losses taken from the measurements. From a comparison of the level of approximation errors for the considered cases, it could be stated that the application of the proposed relationship that considers rate-dependent hysteresis, makes it possible to reduce the approximation errors. The dissipation of energy in the magnetic material is mainly caused by eddy currents generated in the material in various spatial and tem poral scales [19–23], which is expressed in the multi-component form of the formula describing the total losses. Including the hysteresis component obtained from Wartburg’s the orem in the description of the total losses allows to eliminate the semiempirical character of the expression (1). The whole description be comes physical by eliminating the empirical component associated with the hysteresis losses. The introduced component is associated with physical material coefficients such as dynamic permeability and mate rial conductivity. Previously, the component of the hysteresis losses was obtained empirically and its parameters were devoid of physical meaning. The improvement of the approximation of the expression (1)
(10)
To calculate the integral in Eq. (10) it is necessary to know Bmax and s. As for the latter, it is justified to assume that s does not change in volume V due to homogeneity of the sample. Calculation of Bmax is much more complicated and requires solving Maxwell equations in the sample shown in Fig. 1b. According to them, time varying magnetic field in duces electric field ðr � EÞ ¼ ∂∂Bt , which in turn causes eddy currents in the conductive sample J ¼ σE, and they generate magnetic field ðr � HÞ ¼ J. It is practically impossible to solve the equations exactly due to hysteresis, therefore some simplifications must be assumed. Since the width w and the length l are much larger than thickness d, it is justified to assume that magnetic induction has only a y-component By , which de pends on the x coordinate. Moreover, it is assumed that magnetic in duction is a time harmonic quantity, which allows us to use the complex notation. The electromagnetic field equations lead then to the following relationship:
d2 By
γ2 By ¼ 0, in which By ðxÞ is the complex phasor of By ,
dx2
and γ 2 ¼ jωμσ , where j - imaginary unit, ω - angular frequency, μ magnetic permeability, σ - electric conductivity. The equation can be solved easily and the integration constants can be found by symmetry under the given magnitude of the total magnetic flux Φ0 . Hence, the � � local value Bmax in Eq. (10) can be taken as �By ðxÞ�, which yields � � � � �γΦ cosh γx� � � 0 Bmax ¼ � (11) �: � 2w sinh γd2 � � �
Performing the integration we get the expression for hysteresis loss in the following form: 1 0 Ph ¼
2sf Φ20 l B1 d sinh dδ þ sin dδ C A; @ wμd 2 δ cosh dδ cos dδ
(12)
pffiffiffiffiffiffiffiffiffiffi where δ ¼ 1= πf μσ is the skin depth, and f is frequency. The magnetic flux Φ0 , can be expressed as a function of the average magnetic flux density Bm , Φ0 ¼ Bm wd . In the case under consideration the sample thickness is much smaller than the skin depth (d≪ δ), which allows one to simplify Eq. (12) to the following form: Ph ¼
2slwd
μ
(13)
2
B mf :
Table 1 The dependence of the hysteresis losses ðP1h and P2h Þ on induction and frequency for M 330–35 electrical steel.
Dividing Eq. (13) by the mass of material ðlwdρÞ the expression for specific power losses will take the following form: Ph ¼
2s
μρ
2
2
B m f ¼ ch B m f :
(14)
3. Experiment The experimental measurements were made on a single sheet of nonoriented electrical steel with geometrical dimensions 500 mm � 500 mm x 0.35 mm. Data sheet: type M 330–35, σ ¼ 2.13 � 106 S/m, ρ ¼ 7650 kg/m3. Using a Single Sheet Tester Device correlated with the measurement stand MAG-RJJ-2, the sine waveform of the flux density in the sample was achieved [13]. The measurements were carried out in accordance with the international standard IEC 404 [14] and with an error of 1.5%. 4. Results and discussion The power losses Ptot (W/kg) were measured in a range of magnetic induction from 0.1 T to 1.5 T. The computations of power losses have been carried out only up to 1 T, in order to diminish the nonlinearity effect of B vs. H dependence in deep saturation. The magnetic 3
f (Hz)
B (T)
P1h (W/kg)
P1h =f(J/kg)
P2h (W/kg)
P2h =f(J/kg)
20 30 40 50 100 200 300 400
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.0060 0.0093 0.0123 0.0157 0.0339 0.0804 0.1361 0.2011
0.000301138 0.000308659 0.000307631 0.000314972 0.000339259 0.000402146 0.000453544 0.000502789
0.0059 0.0090 0.0118 0.0147 0.0294 0.0588 0.0883 0.1177
0.000294 0.000299 0.000294 0.000294 0.000294 0.000294 0.000294 0.000294
10 20 30 40 50 100 200
0.5 0.5 0.5 0.5 0.5 0.5 0.5
0.0431 0.0886 0.1365 0.1885 0.2485 0.6129 1.6199
0.004309481 0.004429625 0.004549191 0.004711825 0.004969867 0.006128713 0.008099303
0.0421 0.0844 0.1266 0.1674 0.2093 0.4186 0.8348
0.004209 0.004221 0.004221 0.004186 0.004186 0.004186 0.004174
10 20 30 40 50 100
1 1 1 1 1 1
0.1305 0.2616 0.3941 0.5297 0.6622 1.5054
0.013046909 0.01307874 0.013138319 0.013242011 0.013243111 0.015054129
0.1300 0.2596 0.3900 0.5200 0.6500 1.3025
0.013 0.012979 0.013 0.013 0.013 0.013025
J. Szczyglowski
Physica B: Physics of Condensed Matter xxx (xxxx) xxx
might allow one to include the temperature and mechanical stress ef fects by proper modifications of the value of magnetic permeability. The results obtained in the paper can be used to describe losses in magnetic materials with other structures, for example micro- and nano-crystalline alloys, where the description of dissipation phenomena is more complicated. Declaration of competing interest None. Acknowledgements The author would like to thank Dr. P. Jablonski for his help in calculating and useful discussions.
Fig. 2. The relative approximation error of the calculated losses with respect to the measured data (solid symbols-proposed method; open symbols-Bertot ti’s method).
References
to the experimental data was mainly obtained through the artificial modification of its parameters [24,25]. The observed changes in error levels are the result of changes in the flow of eddy currents generated in various spatial-time scales depending on the magnetic induction and on the frequency of the excitation mag netic field (cf. Fig. 2). The errors reach the highest values in the low frequency range, where the so-called hysteresis losses predominate, and the energy dissipation is due to eddy currents induced by Barkhausen jumps during the movement of magnetic domain walls. As frequency increases, the errors for both analysed cases tend approximately to a 20% level, where other loss components from expression (1) begin to dominate.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
5. Conclusion
[15]
From the obtained results it should be stated that the use of a new description of hysteresis losses allows one to make a significant reduc tion of the prediction error for total losses in non-oriented electrical steel. This is due to a more realistic approach to describe the hysteresis losses based on the Warburg theorem. In the proposed approach, they are calculated on the basis of material coefficients, such as the dynamic magnetic permeability expressed in the magnetic hysteresis loop and the conductivity of the material. There is no need for artificial non-physical decomposition of losses into components, which is necessary to obtain the value of excess losses. The proposed relationship for hysteresis losses
[16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
4
G. Bertotti, Hysteresis in Magnetism, 1 st edn, Academic Press, San Diego, 1998. G. Bertotti, IEEE Trans. Magn. 24 (1988) 621–630. F. Fiorillo, et al., IEEE Trans. Magn. 38 (3) (2002) 1467–1476. K. Chwastek, J. Szczygłowski, W. Wilczy� nski, COM 28 (3) (2009) 603–612. V. Manescu Paltanea, Gh Paltanea, H. Gavrila, Physica B 486 (2016) 12–16. J. Szczyglowski, J. Electr. Eng. 69 (6) (2018) 1–3. J. Szczyglowski, P. Jablonski, Computation of core losses in electrical sheets used in electrical machines, IET Electric Power Applications (2019) (in review). G. Ban, G. Bertotti, J. Appl. Phys. 64 (1988) 5361–5363. G. Bertotti, J. Phys. Colloq. 46 (1985) C6–C389. K. Chwastek, J. Szczygłowski, W. Wilczy� nski, J. Magn. Magn. Mater. 322 (2010) 799–803. E. Warburg, Ann. Physik 13 (4) (1881) 141–164. J. Barranger, NASA Technical Note, 1965. D-3114. The Computerized Measuring System MAG-RJJ-2.0, R&J Measurement Technical Information, 2015. IEC Publication 404-3, Magnetic Materials, Part 3; Methods of Measurement of Specific Total Losses of Magnetic Sheet and Strip by Means of a Single Sheet Tester. M. Bereznicki, Proc. Conf. Selected Problems of Electrical Engineering, 2015, pp. 1–4. Kielce, Poland. J. Bishop, J. Magn. Magn. Mater. 86 (1990) 341–348. D. Seagle, S. Charap, J. Appl. Phys. 53 (1982) 8299–8301. H. Winner, W. Naber, R. Sanded, W. Grosse-Nobis, Phys. Scr. 39 (1989) 666–669. D. Cullity, Introduction to Magnetic Materials, Addison-Wesley Publishing Company, 1972. K. Sokalski, B. Slusarek, J. Szczyglowski, Scaling in magnetic materials, in: K. Maaz (Ed.), Magnetic Materials, first ed., InTech. Press, 2016. K. Sokalski, J. Szczyglowski, M. Najgebauer, COM 26 (2007) 640–649. K. Sokalski, J. Szczyglowski, Acta Phys. Pol. 115 (2009) 920–924. G. Bertotti, J. Appl. Phys. 57 (1985) 2118–2126. D. Eggers, S. Steentjes, K. Hameyer, IEEE Trans. Magn. 48 (11) (2012) 3021–3024. R. Roy, A. Dalal, P. Kumar, J. Magn. Magn. Mater. 410 (2016) 248–256.
Contents lists available at ScienceDirect
Physica B: Physics of Condensed Matter journal homepage: http://www.elsevier.com/locate/physb
Use of quasi-static loops of magnetic hysteresis in loss prediction in non-oriented electrical steels Jan Szczyglowski Czestochowa University of Technology, Faculty of Electrical Engineering, Al. Armii Krajowej 17, 42-200 Częstochowa, Poland
A R T I C L E I N F O
A B S T R A C T
Keywords: Eddy current losses Non-oriented electric steel Bertotti’s model
One of the most important factors taken into account in the design and optimization of magnetic circuits of electronic and electric devices are energy losses. Therefore, it is necessary to have accurate models of energy losses in magnetic materials. At present, the most popular model describing excess losses in material Pexc is the semi-empirical model proposed by Bertotti. An important source of error in its application is the methodology of determining hysteresis losses Ph , that are part of total losses expressed in the form of the following formula Ptot ¼ Ph þ Pc þ Pexc . The second component of this expression Pc is defined as classical eddy-current losses. The paper proposes a new approach to calculate hysteresis losses by using the Warburg theorem in which B(H) is expressed in the form of parametric equations. The proposed solution allows one to reduce the loss prediction error significantly.
1. Introduction One of the most important factors taken into account in the design and optimization of magnetic circuits of electrical devices are energy losses. In the model proposed by Bertotti [1,2], energy losses are the result of eddy currents generated by the movement of various “magnetic objects” such as: parts of domain walls, single domain walls, and rota tion of the magnetization vector. Eddy currents can be generated in various space-time scales. On the micro scale, the source of eddy cur rents are Barkhausen jumps responsible for hysteresis losses Ph . On the scale of the entire volume of material responsible for classic losses Pc . Changes in the magnetic field due to interactions between “magnetic objects” generate eddy currents responsible for excess losses Pexc . Total losses according to Bertotti are the sum of these three components. The Bertotti model is currently the most used model to describe energy losses in electrical steel [3–7]. Resulting from the Bertotti model, the formula describing the dependence of energy loss vs. magnetic induction and frequency for non-oriented steel sheet takes the following form [6]: 2
2
1:5
Ptot ¼ Ph þ Pc þ Pexc ¼ ch Bm f þ ce Bm f 2 þ cexc Bm f 1:5
(1)
where: Ptot – total power loss per unit of mass (W/kg), Bm – mean magnetic flux density throughout the sample (T), f – field frequency, (Hz), ch , ce and cexc are material-dependent coefficients. Coefficients
take the following forms: ce ¼ π2 σd2 =6ρ, where ρ is mass density, σ is electrical conductivity and d is the sample thickness of the tested ma pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi terial cut as a strip, cexc ¼ 8 σGSV0 , where G is the damping parameter equal to 0.1356, S is the sheet cross-section area and V0 is model parameter that characterizes the distribution of the internal co ercive fields, which permits one to address quite effectively the rela tionship between dynamic losses and microstructure of the non-oriented electrical alloy microstructure [8,9]. The hysteresis losses component for calculations based on the above formula is usually obtained by extrapolation the measured data of Ptot =f as a function of frequency and taking its limit when f→0 . The second component of the total losses is separately calculated based on the knowledge of the material’s con ductivity and density. The third component of the total losses is calculated after deter mining the value of its coefficients. The coefficient V0 is determined according to the slope of the excess losses chart in coordinate system ðPtot Pc Þ=f vs: f 0:5 based on experimental data. Correct determi nation of the value of individual components of losses in expression (1) allows one to calculate the total losses, the knowledge of which is necessary at the stage of designing the magnetic circuits of electrical devices. The maximum errors between the values of the total losses calculated based on the dependence (1) and the values of losses obtained from measurements can reach 40% [7]. It was shown that the value of this error depends strongly on the value of magnetic induction and frequency. The accuracy of the approximation of excess losses described
E-mail address: [email protected]. https://doi.org/10.1016/j.physb.2019.411812 Received 29 June 2019; Received in revised form 9 October 2019; Accepted 21 October 2019 Available online 24 October 2019 0921-4526/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Jan Szczyglowski, Physica B, https://doi.org/10.1016/j.physb.2019.411812
J. Szczyglowski
Physica B: Physics of Condensed Matter xxx (xxxx) xxx
by Bertotti to the excess component determined experimentally does not exceed 12% [10]. Further metrological analysis should concern the verification of the amount of excess losses by other measuring methods. The practical use of this model, however, encounters a number of con straints related in particular to the skin effect and the need to separate hysteresis losses from total losses, which is necessary to determine the excess losses underlying the model. The paper presents the possibility of limiting this error using the new method of calculating hysteresis losses based on the Warburg theorem [11] in which the dependence BðHÞ is described by means of parametric equations [12].
calculating the shape factor using the dependence of total losses on the frequency for the frequency tending to zero. At frequency tending to zero the ratio of total loss and the frequency equals the ratio of the hysteresis loss and the frequency, because other terms in formula (1) vanish. Hence, lim f →0
According to the Warburg theory [11] hysteresis losses are given by the following relation: I ZZZ 1 Ph ¼ HdBdV (2) VT
where ρ is density and V is volume of the sample. But Eq. (4) allows us to write � � W ZZZ ZZZ Ptot kg 1 1 1 B2max 1 B2 ¼ limf →0 limf →0 2 sdV ¼ limf →0 2 max sdV: f f ρV μ ρV μ VT V (7)
Calculation of the integral with respect to B in expression (2) requires knowledge of the relationship B(H), which is expressed by the hysteresis loop. In the paper, the ascending branch of the hysteresis loop, when the value B changes from Bmax to Bmax , is described by the following parametric equations: � BðτÞ ¼ Bmax cos τ; (3) HðτÞ ¼ H0 ða0 þ a1 cos τ þ a2 cos 2 τ þ …Þ;
When the frequency is nearly zero, the magnetic field has the same amplitude throughout the sample, therefore the volume integral is easy to calculate, leading to � � W Ptot kg 1 B2max B2 ¼ lim sV ¼ 2 max s: (8) 2 f →0 f ρV μf ¼0 ρμf ¼0
where H0 ¼ Bmax =μ, τ is a curve parameter changing from 0 to π, and a0 ; a1 ; a2 … are dimensionless parameters, which can be determined by curve fitting of the experimental hysteresis loop. Magnetic permeability was determined from dynamic hysteresis loops, using peak values for successive minor loops μ, according to Fig. 1 a. The integral in the expression (2) can be depicted as follows: I Z π B2 HdB ¼ 2 HðτÞBmax sin τdτ ¼ 2 max s; (4) where s ¼ 2
Hence � � s¼
μ
P
n¼0;2;4;…
an 1 n2
(5)
The area of the static hysteresis loop is proportional to the hysteresis loss so that � � � � W W ZZZ I Ptot kg Ph kg 1 1 1 limf →0 HdBdV ¼ limf →0 ¼ limf →0 f f f ρV VT ZZZ I 1 1 1 ¼ limf →0 HdBdV (6) f ρV VT
2. Description of the hysteresis losses
0
Ptot Ph ¼ lim : f →0 f f
is defined as the quasi-static loop shape factor
ρ
Ptot limμ
2B2max f →0
f
W kg
(9)
:
Since Ptot can be found in data sheet of a material, it follows that s can be easily found without using directly the definition formula. In fact, such approach was used in the numerical calculations. Expression (4) can be used in Eq. (2) which gives the following expression:
(only an terms for even n matter). The calculation of this factor requires the determination of the coefficients an , which may be calculated using the measured hysteresis loops [12]. The paper presents a simpler way of
a)
b)
Fig. 1. Family of hysteresis loops for non-oriented electrical sheet M 330–35 at a frequency of 5 Hz (a), shape of the sample under consideration (b). J in Fig. 1b denotes eddy current density vector. 2
J. Szczyglowski
Ph ¼
Physica B: Physics of Condensed Matter xxx (xxxx) xxx
ZZZ
2f
μ
V
B2max sdV:
permeability was determined from the dynamic hysteresis loops, using peak values for successive minor loops i.e. from μ ¼ Bm =μ0 Hm , cf. Fig. 1a. The frequency range of the loss measurements was chosen to limit the skin effect (δ≫d). The estimated upper frequency value for the tested samples was 400 Hz. The calculations are presented in Ref. [15]. In Table 1, the results of hysteresis loss computations from expres sion (14) (denoted as P1h ) as well as the corresponding hysteresis losses determined from the extrapolation of measurement data ptot =f for f→0 (denoted as P2h ) are shown. It should be stated that the hysteresis losses per cycle computed from the proposed expression are rate-dependent, which means that the losses per cycle are frequency dependent. This is the result of the frequency influence on the magnetic permeability value in expression (14). Similar results are presented in Refs. [16–18]. On the other hand their counterpart calculated from the expressions used so far are rate-independent, which means that the losses per cycle are fre quency independent. In Fig. 2, the comparison of approximation errors for the two ap proaches used to compute the total losses based on Eq. (1) is presented in two cases. In the first case, in the computations, the term P1h was accounted, in the second one - P2h : The relative error of the approximation was defined as it follows: δ ¼ ðPtot Pm Þ=Pm ⋅100%, where Ptot are the losses calculated with Eq. (1) and Pm are the losses taken from the measurements. From a comparison of the level of approximation errors for the considered cases, it could be stated that the application of the proposed relationship that considers rate-dependent hysteresis, makes it possible to reduce the approximation errors. The dissipation of energy in the magnetic material is mainly caused by eddy currents generated in the material in various spatial and tem poral scales [19–23], which is expressed in the multi-component form of the formula describing the total losses. Including the hysteresis component obtained from Wartburg’s the orem in the description of the total losses allows to eliminate the semiempirical character of the expression (1). The whole description be comes physical by eliminating the empirical component associated with the hysteresis losses. The introduced component is associated with physical material coefficients such as dynamic permeability and mate rial conductivity. Previously, the component of the hysteresis losses was obtained empirically and its parameters were devoid of physical meaning. The improvement of the approximation of the expression (1)
(10)
To calculate the integral in Eq. (10) it is necessary to know Bmax and s. As for the latter, it is justified to assume that s does not change in volume V due to homogeneity of the sample. Calculation of Bmax is much more complicated and requires solving Maxwell equations in the sample shown in Fig. 1b. According to them, time varying magnetic field in duces electric field ðr � EÞ ¼ ∂∂Bt , which in turn causes eddy currents in the conductive sample J ¼ σE, and they generate magnetic field ðr � HÞ ¼ J. It is practically impossible to solve the equations exactly due to hysteresis, therefore some simplifications must be assumed. Since the width w and the length l are much larger than thickness d, it is justified to assume that magnetic induction has only a y-component By , which de pends on the x coordinate. Moreover, it is assumed that magnetic in duction is a time harmonic quantity, which allows us to use the complex notation. The electromagnetic field equations lead then to the following relationship:
d2 By
γ2 By ¼ 0, in which By ðxÞ is the complex phasor of By ,
dx2
and γ 2 ¼ jωμσ , where j - imaginary unit, ω - angular frequency, μ magnetic permeability, σ - electric conductivity. The equation can be solved easily and the integration constants can be found by symmetry under the given magnitude of the total magnetic flux Φ0 . Hence, the � � local value Bmax in Eq. (10) can be taken as �By ðxÞ�, which yields � � � � �γΦ cosh γx� � � 0 Bmax ¼ � (11) �: � 2w sinh γd2 � � �
Performing the integration we get the expression for hysteresis loss in the following form: 1 0 Ph ¼
2sf Φ20 l B1 d sinh dδ þ sin dδ C A; @ wμd 2 δ cosh dδ cos dδ
(12)
pffiffiffiffiffiffiffiffiffiffi where δ ¼ 1= πf μσ is the skin depth, and f is frequency. The magnetic flux Φ0 , can be expressed as a function of the average magnetic flux density Bm , Φ0 ¼ Bm wd . In the case under consideration the sample thickness is much smaller than the skin depth (d≪ δ), which allows one to simplify Eq. (12) to the following form: Ph ¼
2slwd
μ
(13)
2
B mf :
Table 1 The dependence of the hysteresis losses ðP1h and P2h Þ on induction and frequency for M 330–35 electrical steel.
Dividing Eq. (13) by the mass of material ðlwdρÞ the expression for specific power losses will take the following form: Ph ¼
2s
μρ
2
2
B m f ¼ ch B m f :
(14)
3. Experiment The experimental measurements were made on a single sheet of nonoriented electrical steel with geometrical dimensions 500 mm � 500 mm x 0.35 mm. Data sheet: type M 330–35, σ ¼ 2.13 � 106 S/m, ρ ¼ 7650 kg/m3. Using a Single Sheet Tester Device correlated with the measurement stand MAG-RJJ-2, the sine waveform of the flux density in the sample was achieved [13]. The measurements were carried out in accordance with the international standard IEC 404 [14] and with an error of 1.5%. 4. Results and discussion The power losses Ptot (W/kg) were measured in a range of magnetic induction from 0.1 T to 1.5 T. The computations of power losses have been carried out only up to 1 T, in order to diminish the nonlinearity effect of B vs. H dependence in deep saturation. The magnetic 3
f (Hz)
B (T)
P1h (W/kg)
P1h =f(J/kg)
P2h (W/kg)
P2h =f(J/kg)
20 30 40 50 100 200 300 400
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.0060 0.0093 0.0123 0.0157 0.0339 0.0804 0.1361 0.2011
0.000301138 0.000308659 0.000307631 0.000314972 0.000339259 0.000402146 0.000453544 0.000502789
0.0059 0.0090 0.0118 0.0147 0.0294 0.0588 0.0883 0.1177
0.000294 0.000299 0.000294 0.000294 0.000294 0.000294 0.000294 0.000294
10 20 30 40 50 100 200
0.5 0.5 0.5 0.5 0.5 0.5 0.5
0.0431 0.0886 0.1365 0.1885 0.2485 0.6129 1.6199
0.004309481 0.004429625 0.004549191 0.004711825 0.004969867 0.006128713 0.008099303
0.0421 0.0844 0.1266 0.1674 0.2093 0.4186 0.8348
0.004209 0.004221 0.004221 0.004186 0.004186 0.004186 0.004174
10 20 30 40 50 100
1 1 1 1 1 1
0.1305 0.2616 0.3941 0.5297 0.6622 1.5054
0.013046909 0.01307874 0.013138319 0.013242011 0.013243111 0.015054129
0.1300 0.2596 0.3900 0.5200 0.6500 1.3025
0.013 0.012979 0.013 0.013 0.013 0.013025
J. Szczyglowski
Physica B: Physics of Condensed Matter xxx (xxxx) xxx
might allow one to include the temperature and mechanical stress ef fects by proper modifications of the value of magnetic permeability. The results obtained in the paper can be used to describe losses in magnetic materials with other structures, for example micro- and nano-crystalline alloys, where the description of dissipation phenomena is more complicated. Declaration of competing interest None. Acknowledgements The author would like to thank Dr. P. Jablonski for his help in calculating and useful discussions.
Fig. 2. The relative approximation error of the calculated losses with respect to the measured data (solid symbols-proposed method; open symbols-Bertot ti’s method).
References
to the experimental data was mainly obtained through the artificial modification of its parameters [24,25]. The observed changes in error levels are the result of changes in the flow of eddy currents generated in various spatial-time scales depending on the magnetic induction and on the frequency of the excitation mag netic field (cf. Fig. 2). The errors reach the highest values in the low frequency range, where the so-called hysteresis losses predominate, and the energy dissipation is due to eddy currents induced by Barkhausen jumps during the movement of magnetic domain walls. As frequency increases, the errors for both analysed cases tend approximately to a 20% level, where other loss components from expression (1) begin to dominate.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
5. Conclusion
[15]
From the obtained results it should be stated that the use of a new description of hysteresis losses allows one to make a significant reduc tion of the prediction error for total losses in non-oriented electrical steel. This is due to a more realistic approach to describe the hysteresis losses based on the Warburg theorem. In the proposed approach, they are calculated on the basis of material coefficients, such as the dynamic magnetic permeability expressed in the magnetic hysteresis loop and the conductivity of the material. There is no need for artificial non-physical decomposition of losses into components, which is necessary to obtain the value of excess losses. The proposed relationship for hysteresis losses
[16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
4
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