考虑趋肤效应和动态磁滞效应的电机旋转铁芯损耗模型
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摘要
实际农业用电设备铁芯损耗的预测不准是趋肤效应造成农用电器效率低下、使用成本升高的重要原因之一。针对这一问题,该文综合考虑了交变激励下趋肤效应、局部磁滞作用和动态磁滞回环对经典铁芯损耗模型系数的影响,将系数修正后的交变模型引入旋转铁芯损耗模型中,提出一种基于正交分解和损耗分离的改进无取向硅钢叠片旋转铁芯损耗模型。利用磁特性测量装置进行不同频率下的铁芯损耗测量,对所建模型验证。结果表明:与经典铁芯损耗模型和只考虑单一因素的改进模型相比,该模型在高频和高磁密下的铁芯损耗计算精度分别提高了25.32%和9.16%。研究结果可为农用电气设备的电机设计与优化提供参考。
In order to improve the efficiency of agricultural electrical equipment and reduce energy consumption, many scholars attempt to estimate the iron loss accurately. The analysis of finite element showed that in rotational electric machines total core loss comprised alternating core loss and rotational core loss. Precise measured value and modeling of rotational core loss in electrical steel sheets and rotating electrical machines is very important to design and optimize the kind of agricultural motor. According to the separated core loss model under alternative excitation, the core loss can be separated into hysteresis loss, eddy current loss and excess loss. As for alternating core loss, the specific core loss with a circular magnetic flux density can also be separated into 3 portions: the rotational hysteresis loss, the rotational classical eddy-current loss and the rotational excess loss. By means of fourier analysis, the rotating core loss model which considered the influence of alternating and rotating magnetic field, skin effect, dynamic hysteresis loop and minor hysteresis loop was proposed in this paper. Actually,the static hysteresis loop and the dynamic hysteresis loop are different, when the flux density is in saturation, the hysteresis loop shape will be changed. In order to consider the complex behavior of dynamic hysteresis, variable coefficient hysteresis loss was used in the model. The classic exponential coefficient were chosen to be substituted in to the 3 parameters polynomial and fitted out the rotational hysteresis loss with the logarithm. Considering the impact of skin effect at high frequencies,eddy-current loss coefficient were corrected in the paper. Minor hysteresis loop generated by the massive harmonic components leaded the inaccurate prediction of the core loss, the influence of minor hysteresis loop was described by the modified coefficient in the improved formulations, and the modified coefficient was related to the ratio of local flux density to flux density amplitude. By applying orthogonal decomposition technology, 2 mutually orthogonal magnetic flux field was used to describe elliptical rotating magnetic field and replace the rotating loss data. Taking an object affected by the elliptical flux density as an example, the applied magnetic field intensity might not be an elliptical vector because of the nonlinear magnetic flux density-magnetic field intensity relationship and magnetic anisotropy, when it was expanded into a fourier series, however,it shown that the higher harmonics of magnetic field intensity did not contribute to the total core loss as long as magnetic flux density only contains the basic components, the total core loss under the elliptical flux was the summation of alternating core loss and rotating core loss. In order to verify the accuracy of the improved model, a new 3 D magnetic properties measurement system was used to measure the rotating core loss of electrical sheet steels. The 3 D excitation structure consisted of 3 orthogonal C-shaped cores, 6 multilayer excitation coils which were wound around core poles, a sensing box with built-in core material was placed in the center of 3 D magnetic properties measurement device, 6 thin pieces, named as protective layer of uniform magnetic field were fixed around the specimen to make the measured field more uniform at the surface of specimen.Experimental results showed that compared with the classical model and the improved model considering the single factor of skin effect or hysteresis loops only, the accuracy of core loss calculation value of the proposed model was increased by 25.32%and 9.16%, respectively, especially under the condition of high flux density and high frequency. When the frequency was 50 Hz,compared with the classical model, the accuracy of core loss calculation value of the proposed model was increased by 9.21%,and the accuracy was increased by 39.76% at 200 Hz. The comparison of measured value between alternating core loss and rotational core loss showed that the energy of magnetic domain could not accumulated to the maximum in a fixed direction that would lead to irreversible magnetic domain conversion under alternating excitation, which resulted in the increase of hysteresis loss with the increase of magnetic density. The research results can provide reference for the design and optimization of agricultural electrical equipment.
In order to improve the efficiency of agricultural electrical equipment and reduce energy consumption, many scholars attempt to estimate the iron loss accurately. The analysis of finite element showed that in rotational electric machines total core loss comprised alternating core loss and rotational core loss. Precise measured value and modeling of rotational core loss in electrical steel sheets and rotating electrical machines is very important to design and optimize the kind of agricultural motor. According to the separated core loss model under alternative excitation, the core loss can be separated into hysteresis loss, eddy current loss and excess loss. As for alternating core loss, the specific core loss with a circular magnetic flux density can also be separated into 3 portions: the rotational hysteresis loss, the rotational classical eddy-current loss and the rotational excess loss. By means of fourier analysis, the rotating core loss model which considered the influence of alternating and rotating magnetic field, skin effect, dynamic hysteresis loop and minor hysteresis loop was proposed in this paper. Actually,the static hysteresis loop and the dynamic hysteresis loop are different, when the flux density is in saturation, the hysteresis loop shape will be changed. In order to consider the complex behavior of dynamic hysteresis, variable coefficient hysteresis loss was used in the model. The classic exponential coefficient were chosen to be substituted in to the 3 parameters polynomial and fitted out the rotational hysteresis loss with the logarithm. Considering the impact of skin effect at high frequencies,eddy-current loss coefficient were corrected in the paper. Minor hysteresis loop generated by the massive harmonic components leaded the inaccurate prediction of the core loss, the influence of minor hysteresis loop was described by the modified coefficient in the improved formulations, and the modified coefficient was related to the ratio of local flux density to flux density amplitude. By applying orthogonal decomposition technology, 2 mutually orthogonal magnetic flux field was used to describe elliptical rotating magnetic field and replace the rotating loss data. Taking an object affected by the elliptical flux density as an example, the applied magnetic field intensity might not be an elliptical vector because of the nonlinear magnetic flux density-magnetic field intensity relationship and magnetic anisotropy, when it was expanded into a fourier series, however,it shown that the higher harmonics of magnetic field intensity did not contribute to the total core loss as long as magnetic flux density only contains the basic components, the total core loss under the elliptical flux was the summation of alternating core loss and rotating core loss. In order to verify the accuracy of the improved model, a new 3 D magnetic properties measurement system was used to measure the rotating core loss of electrical sheet steels. The 3 D excitation structure consisted of 3 orthogonal C-shaped cores, 6 multilayer excitation coils which were wound around core poles, a sensing box with built-in core material was placed in the center of 3 D magnetic properties measurement device, 6 thin pieces, named as protective layer of uniform magnetic field were fixed around the specimen to make the measured field more uniform at the surface of specimen.Experimental results showed that compared with the classical model and the improved model considering the single factor of skin effect or hysteresis loops only, the accuracy of core loss calculation value of the proposed model was increased by 25.32%and 9.16%, respectively, especially under the condition of high flux density and high frequency. When the frequency was 50 Hz,compared with the classical model, the accuracy of core loss calculation value of the proposed model was increased by 9.21%,and the accuracy was increased by 39.76% at 200 Hz. The comparison of measured value between alternating core loss and rotational core loss showed that the energy of magnetic domain could not accumulated to the maximum in a fixed direction that would lead to irreversible magnetic domain conversion under alternating excitation, which resulted in the increase of hysteresis loss with the increase of magnetic density. The research results can provide reference for the design and optimization of agricultural electrical equipment.
引文
[1]葛研军,袁直,贾峰,等.笼型异步磁力耦合器机械特性与试验[J].农业工程学报,2016,32(12):68-74.Ge Yanjun,Yuan Zhi,Jia Feng,et al.Mechanical properties an testing for squirrel cage asynchronous magnetic coupler[J]Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2016,32(12):68-74.(in Chinese with English abstract)
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[3]孙宇新,沈启康,叶海涵,等.基于改进UKF的无轴承异步电机无速度传感器控制[J].农业工程学报,2018,34(19):74-81.Sun Yuxin,Shen Qikang,Ye Haihan,et,al.Speed-sensorless control system of bearingless induction motor based on modified adaptive fading unscented kalman filter[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2018,34(19):74-81.(in Chinese with English abstract)
[4]Daisuke Yokoyama,Takeru Sato,Takashi Todaka,et al.Comparison of iron loss characteristics of divided cores considering vector magnetic properties[J].IEEE Transactions on Magnetics,2014,50(4):1-4.
[5]Ding Xiaofeng,Ren Suping,Xiong Yanwen,et al.2-Dmagnetic properties measurement system for electrical steel sheets considering laminated direction mechanical stress[J].IEEE Transactions on Magnetics,2017,53(10):1-11.
[6]Xu Weijie,Duan Nana,Wang Shuhong,et al.Modeling of magnetic properties in soft magnetic composite material under rotational magnetization[J].IEEE Transactions on Industrial Electronics,2017,64(3):2459-2467.
[7]Natheer Alatawneh,Pragsen Pillay.The minor hysteresis loop under rotating magnetic fields in machine laminations[J].IEEE Transactions on Industry Applications,2014,50(4):2544-2553.
[8]Zhu Jianguo,Victor Stuart Ramsden.Improved formulations for rotational core losses in rotating electrical machines[J].IEEE Transactions on Magnetics,1998,34(4):2234-2242.
[9]Ashraf Rohanim Asari,Guo Youguang,Zhu Jianguo.Core loss measurement under elliptical loci of magnetic flux density[C]//2017 20th International Conference on Electrical Machines and Systems(ICEMS).Sydney,NSW:IEEE,2017:1-4.
[10]杨娜,李丹丹,宋寅卯.软磁复合材料损耗特性的研究进展[J].电工材料,2018(4):32-35.Yang Na,Li Dandan,Song Yingmao.Review of loss characteristics of soft magnetic composites material[J].Electrical Engineering Materials,2018(4):32-35.(in Chinese with English abstract)
[11]Chas Proteus Steinmetz.On the law of hysteresis[C]//Transactions of the American Institute of Electrical Engineers,New York:IEEE,1892,4(1):3-29.
[12]Aldo Boglietti,Paolo Ferraris,Mario Lazzari,et al.About the possibility of defining a standard method for iron loss measurement in soft magnetic materials with inverter supply[J].IEEE Transactions on Industry Applications,1997,33(5):1283-1288.
[13]Aldo Boglietti,Andrea Cavagnino,Mario Lazzari,et al.Predicting iron losses in soft magnetic materials with arbitrary voltage supply:an engineering approach[J].IEEETransactions on Magnetics,2003,39(2):981-989.
[14]Aldo Boglietti,Andrea Cavagnino,Dan M Ionel,et al.Ageneral model to predict the iron losses in PWM inverter-fed induction motor[J].IEEE Transactions on Industry Applications,2010,46(5):1882-1890.
[15]迟青光,张艳丽,曹政,等.电工钢片旋转损耗特性分析与损耗模型修正[J].中国电机工程学报,2017,37(8):2418-2426.Chi Qingguang,Zhang Yanli,Cao Zheng,et al.Analysis of rotational loss property in an electrical steel sheet and correction of loss model[J].Proceedings of the CSEE,2017,37(8):2418-2425.(in Chinese with English abstract)
[16]龚文君,黄邵刚,杨玉文,等.ANSYS Maxwell硅钢片铁耗系数提取方法的实现[J].电机与控制应用,2016,43(4):82-85.Gong Wenjun,Huang Shaogaug,Yang Yuweu,et al.Realization of the method of extracting iron loss coefficient of silicon steel sheet in ANSYS Maxwell[J].Electric machines&control application,2016,43(4):82-85.(in Chinese with English abstract)
[17]李玉梅.考虑电工钢片旋转磁特性的感应电机铁心损耗问题研究[D].沈阳:沈阳工业大学,2013.Li Yumei.Study on Iron Core Loss of the Induction Motor Considering Rotating Magnetic Property of the Electrical Steel Sheets[D].Shenyang:Shenyang University of Technology,2013.(in Chinese with English abstract)
[18]Martin A Rückert,Patrick Vogel,Anna Vilter,et al.Rotational drift spectroscopy for magnetic particle ensembles[J].IEEETransactions on Magnetics,2015,52(2):1-4.
[19]Jemimah C Akiror,John Wanjiku,Pragasen Pillay,et al.Rotational core loss magnetizer:design and measurements[J].IEEE Transactions on Industry Applications,2018,54(5):4355-4364.
[20]Anouar Belahcen,Paavo Rasilo and Antero Arkkio.Segregation of iron losses from rotational field measurements and application to electrical machine[J].IEEE Transactions on Magnetics,2014,50(2):893-896.
[21]曹政.基于二维磁特性测量技术的电工钢片损耗特性研究[D].沈阳:沈阳工业大学,2016.Cao Zheng.Research on Power Loss Properties of Electrical Steel Sheet Based on Measurement Technology of Two-Dimensional Magnetic Properties[D].Shenyang:Shenyang University of Technology,2016.(in Chinese with English abstract)
[22]Zhu Lixun,Junho Park,Koh Chang-seop.A dynamic hysteresis model based on vector-play model for iron loss calculation taking the rotating magnetic fields into account[J].IEEE Transactions on Magnetics,2018,54(3):1-4.
[23]黄平林.旋转电机铁心损耗的分析与计算[D].南京:东南大学,2007.Huang Pinglin.Analysis and Calculation of Rotating Core Loss[D].Nanjing:Southeast University,2007.(in Chinese with English abstract)
[24]江善林,邹继斌,徐永向,等.考虑旋转磁通和趋肤效应的变系数铁耗计算模型[J].中国电机工程学报,2011,31(03):104-110.Jiang Shanlin,Zou Jibin,Xu Yongxiang,et al.Variable coefficient iron loss calculating model considering rotational flux and skin Effect[J].Proceedings of the CSEE,2011,31(3):104-110.(in Chinese with English abstract)
[25]李劲松,杨庆新,李永建,等.考虑谐波及集肤效应的电工钢片旋转异常损耗计算与测量[J].电机与控制学报,2016,20(6):42-49.Li Jingsong,Yang Qingxin,Li Yongjian,et al.Measurement and calculation of rotating anomalous loss in electrical steel sheets considering harmonics and skin effect[J].Electric Machines and Control,2016,20(6):42-49.(in Chinese with English Abstract)
[26]曹磊.硅钢叠片材料的损耗特性测及工程模拟研究[D].天津:河北工业大学,2016.Cao Lei.Core Loss Characteristics Measurement and Engineering Modeling of the Silicon Steel Lamination[D].Tianjin:Hebei University of Technology,2016.(in Chinese with English abstract)
[27]Wan Xuezhi,Li Yongjian,Li Jingsong,et al.Orthogonal decomposition of core loss along rolling and transverse directions of non-grain oriented silicon steels[J].AIPAdvances,2017,7(5):1-7.
[28]Zhu Jianguo.Numerical Modelling of Magnetic Materials for Computer Aided Design of Electromagnetic Devices[D].Sydney:University of Technology Sydney,1994.
[29]李永建,王利祥,张长庚,等.基于三维励磁结构的电工磁材料动态磁特性测试与分析[J].电工技术学报,2018,33(1):166-174.Li Yongjian,Wang Lixiang,Zhang Changgeng,et al.Dynamic magnetic properties testing and analysis of the electric magnetic materials based on three-dimensional excitation structure[J].Transactions of China Electrotechnical Society2018,33(1):166-174.(in Chinese with English abstract)
[30]曹诗侯.铁磁材料非正弦激励下的损耗计算方法研究[D].杭州:浙江大学,2017.Cao shihou.Predicting Iron Losses in Magnetic Materials with Non-Sinusolidal Power Supply[D].Hangzhou:Zhejiang University,2017.(in Chinese with English abstract)
[31]K Atallah,Z Q Zhu,D Howe.An improved method for predicting iron losses in brushless permanent magnet DCdrives[J].IEEE Transactions on Magnetics,1992,28(5):2997-2999.
[32]Chen Yicheng,Pragasen Pillay.An improved formula for lamination core loss calculations in machines operating with high frequency and high flux density excitation[C]//Conference record of the 2002 IEEE Industry Applications Conference.37th IAS Annual Meeting(Cat.No.02CH37344)Pittsburgh,PA,USA:IEEE,2002:759-766.
[33]Mehdi Taghizadeh Kakhki,Jér?me Cros,Philippe.Viarouge.New approach for accurate prediction of eddy current losses in laminated material in the presence of skin effect with 2-D FEA[J].IEEE Transactions on Magnetics,2016,52(3):1-4.
[2]杨泽斌,李方利,陈正,等.基于低频信号注入法的无轴承异步电机转速自检测控制[J].农业工程学报,2017,33(2):41-47.Yang Zebin,Li Fangli,Chen Zheng,et,al.Revolving speed self-detecting control base on low-frequency signal injection for bearingless induction motor[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2017,33(2):41-47.(in Chinese with English abstract)
[3]孙宇新,沈启康,叶海涵,等.基于改进UKF的无轴承异步电机无速度传感器控制[J].农业工程学报,2018,34(19):74-81.Sun Yuxin,Shen Qikang,Ye Haihan,et,al.Speed-sensorless control system of bearingless induction motor based on modified adaptive fading unscented kalman filter[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2018,34(19):74-81.(in Chinese with English abstract)
[4]Daisuke Yokoyama,Takeru Sato,Takashi Todaka,et al.Comparison of iron loss characteristics of divided cores considering vector magnetic properties[J].IEEE Transactions on Magnetics,2014,50(4):1-4.
[5]Ding Xiaofeng,Ren Suping,Xiong Yanwen,et al.2-Dmagnetic properties measurement system for electrical steel sheets considering laminated direction mechanical stress[J].IEEE Transactions on Magnetics,2017,53(10):1-11.
[6]Xu Weijie,Duan Nana,Wang Shuhong,et al.Modeling of magnetic properties in soft magnetic composite material under rotational magnetization[J].IEEE Transactions on Industrial Electronics,2017,64(3):2459-2467.
[7]Natheer Alatawneh,Pragsen Pillay.The minor hysteresis loop under rotating magnetic fields in machine laminations[J].IEEE Transactions on Industry Applications,2014,50(4):2544-2553.
[8]Zhu Jianguo,Victor Stuart Ramsden.Improved formulations for rotational core losses in rotating electrical machines[J].IEEE Transactions on Magnetics,1998,34(4):2234-2242.
[9]Ashraf Rohanim Asari,Guo Youguang,Zhu Jianguo.Core loss measurement under elliptical loci of magnetic flux density[C]//2017 20th International Conference on Electrical Machines and Systems(ICEMS).Sydney,NSW:IEEE,2017:1-4.
[10]杨娜,李丹丹,宋寅卯.软磁复合材料损耗特性的研究进展[J].电工材料,2018(4):32-35.Yang Na,Li Dandan,Song Yingmao.Review of loss characteristics of soft magnetic composites material[J].Electrical Engineering Materials,2018(4):32-35.(in Chinese with English abstract)
[11]Chas Proteus Steinmetz.On the law of hysteresis[C]//Transactions of the American Institute of Electrical Engineers,New York:IEEE,1892,4(1):3-29.
[12]Aldo Boglietti,Paolo Ferraris,Mario Lazzari,et al.About the possibility of defining a standard method for iron loss measurement in soft magnetic materials with inverter supply[J].IEEE Transactions on Industry Applications,1997,33(5):1283-1288.
[13]Aldo Boglietti,Andrea Cavagnino,Mario Lazzari,et al.Predicting iron losses in soft magnetic materials with arbitrary voltage supply:an engineering approach[J].IEEETransactions on Magnetics,2003,39(2):981-989.
[14]Aldo Boglietti,Andrea Cavagnino,Dan M Ionel,et al.Ageneral model to predict the iron losses in PWM inverter-fed induction motor[J].IEEE Transactions on Industry Applications,2010,46(5):1882-1890.
[15]迟青光,张艳丽,曹政,等.电工钢片旋转损耗特性分析与损耗模型修正[J].中国电机工程学报,2017,37(8):2418-2426.Chi Qingguang,Zhang Yanli,Cao Zheng,et al.Analysis of rotational loss property in an electrical steel sheet and correction of loss model[J].Proceedings of the CSEE,2017,37(8):2418-2425.(in Chinese with English abstract)
[16]龚文君,黄邵刚,杨玉文,等.ANSYS Maxwell硅钢片铁耗系数提取方法的实现[J].电机与控制应用,2016,43(4):82-85.Gong Wenjun,Huang Shaogaug,Yang Yuweu,et al.Realization of the method of extracting iron loss coefficient of silicon steel sheet in ANSYS Maxwell[J].Electric machines&control application,2016,43(4):82-85.(in Chinese with English abstract)
[17]李玉梅.考虑电工钢片旋转磁特性的感应电机铁心损耗问题研究[D].沈阳:沈阳工业大学,2013.Li Yumei.Study on Iron Core Loss of the Induction Motor Considering Rotating Magnetic Property of the Electrical Steel Sheets[D].Shenyang:Shenyang University of Technology,2013.(in Chinese with English abstract)
[18]Martin A Rückert,Patrick Vogel,Anna Vilter,et al.Rotational drift spectroscopy for magnetic particle ensembles[J].IEEETransactions on Magnetics,2015,52(2):1-4.
[19]Jemimah C Akiror,John Wanjiku,Pragasen Pillay,et al.Rotational core loss magnetizer:design and measurements[J].IEEE Transactions on Industry Applications,2018,54(5):4355-4364.
[20]Anouar Belahcen,Paavo Rasilo and Antero Arkkio.Segregation of iron losses from rotational field measurements and application to electrical machine[J].IEEE Transactions on Magnetics,2014,50(2):893-896.
[21]曹政.基于二维磁特性测量技术的电工钢片损耗特性研究[D].沈阳:沈阳工业大学,2016.Cao Zheng.Research on Power Loss Properties of Electrical Steel Sheet Based on Measurement Technology of Two-Dimensional Magnetic Properties[D].Shenyang:Shenyang University of Technology,2016.(in Chinese with English abstract)
[22]Zhu Lixun,Junho Park,Koh Chang-seop.A dynamic hysteresis model based on vector-play model for iron loss calculation taking the rotating magnetic fields into account[J].IEEE Transactions on Magnetics,2018,54(3):1-4.
[23]黄平林.旋转电机铁心损耗的分析与计算[D].南京:东南大学,2007.Huang Pinglin.Analysis and Calculation of Rotating Core Loss[D].Nanjing:Southeast University,2007.(in Chinese with English abstract)
[24]江善林,邹继斌,徐永向,等.考虑旋转磁通和趋肤效应的变系数铁耗计算模型[J].中国电机工程学报,2011,31(03):104-110.Jiang Shanlin,Zou Jibin,Xu Yongxiang,et al.Variable coefficient iron loss calculating model considering rotational flux and skin Effect[J].Proceedings of the CSEE,2011,31(3):104-110.(in Chinese with English abstract)
[25]李劲松,杨庆新,李永建,等.考虑谐波及集肤效应的电工钢片旋转异常损耗计算与测量[J].电机与控制学报,2016,20(6):42-49.Li Jingsong,Yang Qingxin,Li Yongjian,et al.Measurement and calculation of rotating anomalous loss in electrical steel sheets considering harmonics and skin effect[J].Electric Machines and Control,2016,20(6):42-49.(in Chinese with English Abstract)
[26]曹磊.硅钢叠片材料的损耗特性测及工程模拟研究[D].天津:河北工业大学,2016.Cao Lei.Core Loss Characteristics Measurement and Engineering Modeling of the Silicon Steel Lamination[D].Tianjin:Hebei University of Technology,2016.(in Chinese with English abstract)
[27]Wan Xuezhi,Li Yongjian,Li Jingsong,et al.Orthogonal decomposition of core loss along rolling and transverse directions of non-grain oriented silicon steels[J].AIPAdvances,2017,7(5):1-7.
[28]Zhu Jianguo.Numerical Modelling of Magnetic Materials for Computer Aided Design of Electromagnetic Devices[D].Sydney:University of Technology Sydney,1994.
[29]李永建,王利祥,张长庚,等.基于三维励磁结构的电工磁材料动态磁特性测试与分析[J].电工技术学报,2018,33(1):166-174.Li Yongjian,Wang Lixiang,Zhang Changgeng,et al.Dynamic magnetic properties testing and analysis of the electric magnetic materials based on three-dimensional excitation structure[J].Transactions of China Electrotechnical Society2018,33(1):166-174.(in Chinese with English abstract)
[30]曹诗侯.铁磁材料非正弦激励下的损耗计算方法研究[D].杭州:浙江大学,2017.Cao shihou.Predicting Iron Losses in Magnetic Materials with Non-Sinusolidal Power Supply[D].Hangzhou:Zhejiang University,2017.(in Chinese with English abstract)
[31]K Atallah,Z Q Zhu,D Howe.An improved method for predicting iron losses in brushless permanent magnet DCdrives[J].IEEE Transactions on Magnetics,1992,28(5):2997-2999.
[32]Chen Yicheng,Pragasen Pillay.An improved formula for lamination core loss calculations in machines operating with high frequency and high flux density excitation[C]//Conference record of the 2002 IEEE Industry Applications Conference.37th IAS Annual Meeting(Cat.No.02CH37344)Pittsburgh,PA,USA:IEEE,2002:759-766.
[33]Mehdi Taghizadeh Kakhki,Jér?me Cros,Philippe.Viarouge.New approach for accurate prediction of eddy current losses in laminated material in the presence of skin effect with 2-D FEA[J].IEEE Transactions on Magnetics,2016,52(3):1-4.