IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 6, JUNE 2017
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Measurement and Modeling of 3-D Rotating Anomalous Loss Considering Harmonics and Skin Effect of Soft Magnetic Materials Jingsong Li1 , Qingxin Yang1,2 , Yongjian Li1 , Changgeng Zhang1 , and Baojun Qu1 1 Province-Ministry 2 Tianjin
Joint Key Laboratory of EFEAR, Hebei University of Technology, Tianjin 300130, China Key Laboratory of ATEEE, Tianjin Polytechnic University, Tianjin 300387, China
By using a 3-D magnetic properties testing system, 3-D rotating core losses considering harmonics and skin effect of the bulk soft magnetic composites are measured and modeled. On the basis of the rotating core losses calculation approach, the transformed coefficients and parameters of rotating losses are determined. Then, a variable loss coefficient model and indirect orthogonal decomposition calculation model of the rotating anomalous loss are thus built in this paper. Meanwhile, a separation of the total 3-D rotating experimental core losses is worked out. In addition, measurement and separation of the total experimental core losses for typical electrical steel sheets are carried out under 3-D rotating and alternating excitations, respectively. The three different methods are analyzed and compared qualitatively. Be noted that the two novel calculation models can be more effectively presented the anomalous loss features. Moreover, the quantitative comparisons between 3-D rotating and alternating core losses have also achieved some beneficial conclusions. Index Terms— 3-D magnetic properties, core losses separation, electrical steel sheets, rotating anomalous loss, soft magnetic composites (SMCs).
I. I NTRODUCTION ARMONICS in electrical equipments mainly include the time harmonics originated from nonsinusoidal power supply and a series of spatial high-order harmonics due to nonsinusoidal distribution of air-gap magnetic potential. The impact of harmonics to electrical equipments are significant, even if their cores are made by soft magnetic composites (SMCs), not to mention the cores made of electrical steel sheets [1], [2]. Due to the existence of harmonics and skin effect in the condition of rotating excitation, especially for high-frequency motors and power transformers that run under the frequency of hundreds or even thousands of Hertz, core losses will increase significantly, which may cause local overheating damage, and the efficiency and longevity of these equipments will be decreased [3], [4]. Most previous researches of electrical steel sheets are carried out under the assumption of alternating sine-wave excitation; however, the rotating excitation, complicated higher-order harmonics, high-frequency skin effect, and anomalous loss should be considered in practical core losses calculations. Therefore, these elements would add to the complexity of the core losses model. de la Barrière [5] discussed a general method to calculate core losses in SMCs; however, the induction waveform is mostly nonsinusoidal, and the influences of harmonics and skin effect to core losses have not been considered. Appino et al. [6] measured the rotating core losses in low-carbon steels up to 1 kHz (the peak induction is about to 1.7 T) and brought to light the emergence of the skin effect under rotating excitation. Rotating eddy current loss, largely prevalent with respect to the hysteresis and anomalous loss components, is then calculated in the presence
H
Manuscript received November 19, 2016; revised January 9, 2017 and January 29, 2017; accepted January 30, 2017. Date of publication February 7, 2017; date of current version May 26, 2017. Corresponding author: J. Li (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2017.2665491
Fig. 1. 3-D rotating excitation finite element model of the bulk SMCs. (a) Diagram of overall structure. (b) Diagram of bulk SMCs under 3-D rotating excitation.
of skin effect. And yet, the harmonic effect is not considered, even at high frequency. There are few literatures involved in the deep research of anomalous loss, due to its complicated physical mechanism, its small share of core losses, etc. In addition, the investigations and modeling on 3-D rotating anomalous loss properties are the key to promote the engineering applications of soft magnetic materials. II. I NFLUENCE AND A NALYSIS OF H ARMONICS AND S KIN E FFECT A. Harmonics Analysis of 3-D Rotating Magnetic Field A finite element model of the bulk SMCs subjected to 3-D rotating excitation is modeled, as shown in Fig. 1. In the model, the flux density waveform of any point can be decomposed into a series of harmonics components by means of Fourier analysis. Thus, the loss of every magnetic field waveform is equal to the sum of its fundamental wave loss and each harmonics component loss [7] Bn sin(n · 2π fr t + ϕn ) (1) B(t) = B1 sin(2π f r t) + n
0018-9464 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 6, JUNE 2017
Fig. 2. Ellipsoidal nth harmonic flux density vector waveform at position A in the finite element model.
where B1 and f r are peak rotating flux density and excitation frequency of fundamental wave; Bn and ϕn represent peak flux density and initial phase angle of the nth harmonics component, respectively. As a result, a 3-D rotating magnetic field with nth harmonics components can be decomposed into three orthogonal alternating magnetic fields [8], of which the flux density of equatorial radius is Bnx and Bnz , the flux density of polar radius is Bny , as shown in Fig. 2. B. Influence of Skin Effect to Rotating Loss Coefficients Based on core losses calculation approach, the rotating core losses Prt of soft magnetic materials are as follows [9]: Prt = Prh + Pre + Pra = Prh + kre ( f r Bmr )
2
3 + kra ( f r Bmr ) 2
[W/kg]
(2)
where Pre and kre , and Pra and kra are the rotating eddy current and anomalous losses and coefficients, respectively; Bmr is peak rotating flux density. The rotating hysteresis loss Prh can only be expressed as [9], [10] Prh 1/(2 − s) 1/s = a1 − fr (a2 + 1/s)2 + a32 [a2 + 1/(2 − s)]2 + a32 Bmr 1 s = 1− 1− 2 . (3) Bsatr a2 + a32 Here, a1 , a2 , a3 , and s are all parameters; Bsatr is saturation rotating flux density of soft magnetic materials. Soft magnetic materials, including SMCs and electrical steel sheets, can also induce eddy current to make its current density distributed nonuniformly under rotating excitation, which is also called the skin effect [11], [12]. So when determining the rotating loss coefficients, especially the coefficient of eddy current loss, it is necessary to consider the influence of skin effect. III. E XPERIMENTAL M EASUREMENT OF ROTATING C ORE L OSSES AND C ALCULATION OF A NOMALOUS L OSS A. Determination of Transformed Coefficients and Parameters of the Rotating Losses Taking an example of SMCs (SOMALOYTM 500), the total 3-D rotating experimental core losses are fulfilled by using a 3-D magnetic properties testing system [2], [13]. (The measured Bsatr is about 1.50 T.)
Fig. 3. Rotating eddy current and anomalous loss coefficients versus rotating excitation frequency at a given peak rotating flux density Bmr = 1.40 T. (a) kre versus f r (b) kra versus f r . (Mathematical Fitting).
Fig. 3 shows that for given peak rotating flux densities, the presence of skin effect would decrease the conducting area and in turn decrease the rotating eddy current loss; thus, the rotating eddy current loss coefficient is decreasing gradually with increasing of rotating excitation frequency. The rotating anomalous loss coefficient increases with increasing of rotating excitation frequency at first and then decreases when the materials magnetization is almost saturated. This is because the rotating anomalous loss coefficient is also generally a function of peak rotating flux density and excitation frequency and eventually reduces to 0 when the materials are saturation magnetization and all domain walls disappear. Consequently, the coefficients of the rotating eddy current and anomalous losses should change with the variations of peak rotating flux density and excitation frequency kre ( f r , Bmr ) = χ(Bmr ) (4) kra ( f r , Bmr ) = λ(Bmr ) ln( f r ) + μ(Bmr ). Here, χ (Bmr ), λ (Bmr ), and μ (Bmr ) are polynomials of Bmr , and their constant coefficients are all related to f r . By uniting out (2)–(4) and being divided by f r Prt / f r = Prh / fr + Pre / f r + Pra / f r 3 2 = Prh / fr + [λ(Bmr ) ln( f r ) + μ(Bmr )]Bmr 2 2 + χ(Bmr )Bmr ( f r ) = Q + S f r + R( f r )2 .
fr (5)
Here, the experimental data of rotating core losses are used to plot curves √ of Prt / f r versus square root of rotating excitation frequency f r for different values of peak rotating flux density Bmr from the lowest to the highest frequency. These curves
LI et al.: MEASUREMENT AND MODELING OF 3-D ROTATING ANOMALOUS LOSS
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TABLE I ROTATING L OSS C OEFFICIENTS AND PARAMETERS OF SOMALOY TM 500
should be parabolas, and the coefficients Q, R, S can be obtained √ by way of parabola fitting, with Prt / f r on the vertical axis and f r on the horizontal axis. More details are given in our previous work [13]. According to the separation processes of rotating core losses, the determination of the rotating loss coefficients and parameters is shown in Table I. B. Variable Loss Coefficient and Indirect Orthogonal Decomposition Calculation Models According to the above analysis, the total 3-D rotating core losses are equivalent to three orthogonal alternating core losses. Thus, the rotating anomalous loss per unit mass of soft magnetic materials is 1 Tr Pra = kra ( f r , Bmr ) Tr 0 3 d B (t) 2 d B (t) 2 d B (t) 2 4 y x z × + + dt [W/kg] dt dt dt (6) where Tr is the rotating excitation cycle. Equation (6) is just the variable loss coefficient model of rotating anomalous loss in soft magnetic materials when considering harmonics and skin effect. According to Poynting’s theorem, the indirect orthogonal decomposition calculation model of rotating anomalous loss in soft magnetic materials can be obtained Pra = Prt − (Prh + Pre )
Tr d B y (t) 1 d Bx (t) d Bz (t) = + H y (t) + Hz (t) Hx (t) dt Tr ρ 0 dt dt dt N 2 Prh 2 2 n 2 f r2 Bnx + Bny + Bnz f r +kre ( f r , Bmr ) − fr n=1 1 Bmr 1− 2 [W/kg] (7) s = 1− Bsatr a2 + a32 where ρ is the mass density of soft magnetic materials; Bx (t) and Hx (t), B y (t) and H y (t), Bz (t) and Hz (t) are field intensities and rotating flux densities along transverse, laminated, and rolling directions in soft magnetic materials, respectively. IV. E XPERIMENTAL S EPARATION AND VALIDATION A separation of total 3-D rotating experimental core losses of the bulk SMCs specimen is worked out for given rotating
Fig. 4. Comparison between the two calculation models and experimental separation for 3-D rotating anomalous loss at Bmr = 1 T.
Fig. 5. 3-D rotating versus alternating core losses for SOMALOYTM 500: (a) under the excitation frequency form 5–50Hz and (b) under the excitation frequency form 100–1 kHz.
peak flux density in order to validate the two novel calculation models, based on (5). (The seventh harmonic component is considered for the rotating flux density, that is n = 7.) In Fig. 4, the 3-D rotating anomalous loss obtained by the separation presents the biggest discrepancy among the three different methods. Since the rotating core losses calculation approach is a kind of empirical formula with a certain range of peak rotating flux density and excitation frequency, harmonics and skin effect under high rotating excitation frequency are not considered. There are small differences between variable loss coefficient and indirect orthogonal decomposition calculation models, which reflect that the dynamic and static hysteresis loops are different [14]. Meanwhile, this paper does not take into account such case that the rotating hysteresis loss also changes with peak rotating flux density and excitation frequency under high frequency and high flux density, which will lead to smaller calculation result of rotating hysteresis loss and larger results from indirect orthogonal decomposition calculation model. In a word, the variable loss coefficient and the indirect orthogonal decomposition calculation models can be more effectively to present the anomalous loss features. Meanwhile, the measurement and separation of the total experimental alternating core losses are carried out for SOMALOYTM 500 simultaneously [2], [13]. Figs. 5 and 6 illustrate that, for SMCs, the 3-D rotating core losses, hysteresis, eddy current, and anomalous loss components are larger than the corresponding alternating losses. And then the differences between the two excitation modes are growing progressively with increasing of frequency and flux density [2]. The two novel calculation models can present
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Fig. 6. Experimental separation of total 3-D rotating versus alternating core losses at Bm = 1 T.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 6, JUNE 2017
V. C ONCLUSION Transformed rotating loss coefficients and parameters of soft magnetic materials considering harmonics and skin effect are determined in this paper. Two models evaluating 3-D rotating anomalous loss of soft magnetic materials are presented and analyzed. The variable loss coefficient and the indirect orthogonal decomposition calculation models are presented on the basis of the experimental measurement and the separation of the total 3-D rotating core losses. Accuracy and validity of the two models are verified by the separated 3-D rotating anomalous loss. The proposed separation method of 3-D rotating core losses and the quantitative comparison with alternating core losses are helpful to engineering application. ACKNOWLEDGMENT This work was supported in part by the Key Project of National Natural Science Foundation of China, under Grant 51237005 and in part by the Natural Science Foundation of Hebei Province (Youth Foundation), China, under Grant E2014202137. R EFERENCES
Fig. 7. Comparisons among the four different methods to obtain anomalous loss for 50WW800 at Bm = 1 T.
the rules as well. The fundamental reason is that the rotating magnetic field is formed by the interactions among three alternating magnetic fields. Moreover, the broadband (100 Hz–1 kHz) 3-D rotating experimental core losses of the cold-rolled NO electrical steel 50WW800 are carried out by using the same 3-D magnetic properties testing system. (The measured Bsatr is about 1.87 T.) For given peak rotating flux density, a separation of the total 3-D rotating core losses for 50WW800 is worked out at the same time. In addition, the measurement and separation of the total alternating core losses are realized on 50WW800 [15]. The four different methods to obtain anomalous loss are compared and analyzed qualitatively, as shown in Fig. 7. (The seventh harmonic component is also considered.) In Fig. 7, the rotating anomalous loss obtained by the separation based on rotating core losses calculation approach presents a large discrepancy than the calculation values of the variable loss coefficient model, as just shown in Fig. 4. The 3-D rotating anomalous loss is greater (about twice) than that of alternating anomalous loss [13], after all. As well as the effect of the high-frequency eddy current skin effect, the anomalous loss itself is a kind of special classical eddy current loss. Above all, when calculating the core losses and their components of soft magnetic materials, the influences of harmonics, skin effect, rotating excitation, etc., to materials properties should be seriously considered, especially cores under high frequency and high flux density. The traditional methods of losses calculation and separation must be restricted.
[1] Q. Yang, Y. Li, Z. Zhao, L. Zhu, Y. Luo, and J. Zhu, “Design of a 3-D rotational magnetic properties measurement structure for soft magnetic materials,” IEEE Trans. Appl. Supercond., vol. 24, no. 3, Jun. 2014, Art. no. 8200804. [2] Y. Li, Q. Yang, J. Zhu, and Y. Guo, “Magnetic properties measurement of soft magnetic composite materials over wide range of excitation frequency,” IEEE Trans. Ind. Appl., vol. 48, no. 1, pp. 88–97, Jan./Feb. 2012. [3] D. M. Said, K. M. Nor, and M. S. Majid, “Analysis of distribution transformer losses and life expectancy using measured harmonic data,” in Proc. Int. Conf. HQP, 2010, pp. 1–6. [4] S. Jiang, J. Zou, Y. Xu, and W. Liang, “Variable coefficient iron loss calculating model considering rotational flux and skin effect,” Proc. CSEE, vol. 31, no. 3, pp. 104–110, Jan. 2011. [5] O. de la Barrière et al., “Characterization and prediction of magnetic losses in soft magnetic composites under distorted induction waveform,” IEEE Trans. Magn., vol. 49, no. 4, pp. 1318–1326, Apr. 2013. [6] C. Appino et al., “Skin effect in steel sheets under rotating induction,” Int. J. Appl. Electromagn. Mech., vol. 48, nos. 2–3, pp. 247–254, 2015. [7] Y. Guo, J. Zhu, H. Lu, Z. Lin, and Y. Li, “Core loss calculation for soft magnetic composite electrical machines,” IEEE Trans. Magn., vol. 48, no. 11, pp. 3112–3115, Nov. 2012. [8] Y. Guo, J. Zhu, H. Lu, Y. Li, and J. Jin, “Core loss computation in a permanent magnet transverse flux motor with rotating fluxes,” IEEE Trans. Magn., vol. 50, no. 11, Nov. 2014, Art. no. 6301004. [9] J. G. Zhu and V. S. Ramsden, “Improved formulations for rotational core losses in rotating electrical machines,” IEEE Trans. Magn., vol. 34, no. 4, pp. 2234–2242, Jul. 1998. [10] Y. Huang, Q. Hu, and J. Zhu, “Magneto-thermal analysis of a highspeed claw pole motor considering rotational core loss,” Trans. China Electrotech. Soc., vol. 25, no. 5, pp. 54–60, May 2010. [11] O. de la Barrière et al., “High-frequency rotational losses in different soft magnetic composites,” J. Appl. Phys., vol. 115, p. 17A331, Nov. 2014. [12] C. Appino, M. Khan, O. de la Barrière, C. Ragusa, and F. Fiorillo, “Alternating and rotational losses up to magnetic saturation in nonoriented steel sheets,” IEEE Trans. Magn., vol. 52, no. 5, May 2016, Art. no. 6301204. [13] J. Li, Q. Yang, Y. Li, C. Zhang, B. Qu, and L. Cao, “Anomalous loss modeling and validation of magnetic materials in electrical engineering,” IEEE Trans. Appl. Supercond., vol. 26, no. 4, Jun. 2016, Art. no. 8800105. [14] O. de la Barrière et al., “Loss separation in soft magnetic composites,” J. Appl. Phys., vol. 109, no. 7, p. 07A317, 2011. [15] J. Li, Q. Yang, Y. Li, and C. Zhang, “Measurement and calculation of rotating anomalous loss in electrical steel sheets considering harmonics and skin effect,” Electr. Mach. Control, vol. 20, no. 6, pp. 42–49, Jun. 2016.
