考慮趨膚效應和動態磁滯效應的電機旋轉鐵芯損耗模型

宋 澤，李永建※，張長庚，劉 洋

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考慮趨膚效應和動態磁滯效應的電機旋轉鐵芯損耗模型

宋 澤1，李永建1※，張長庚1，劉 洋2

（1. 河北工業大學電氣工程學院省部共建電工裝備可靠性與智能化國家重點實驗室，天津 300130；2. 全球能源互聯網研究院有限公司，北京 102211）

實際農業用電設備鐵芯損耗的預測不準是趨膚效應造成農用電器效率低下、使用成本升高的重要原因之一。針對這一問題，該文綜合考慮了交變激勵下趨膚效應、局部磁滯作用和動態磁滯回環對經典鐵芯損耗模型系數的影響，將系數修正后的交變模型引入旋轉鐵芯損耗模型中，提出一種基于正交分解和損耗分離的改進無取向硅鋼疊片旋轉鐵芯損耗模型。利用磁特性測量裝置進行不同頻率下的鐵芯損耗測量，對所建模型驗證。結果表明：與經典鐵芯損耗模型和只考慮單一因素的改進模型相比，該模型在高頻和高磁密下的鐵芯損耗計算精度分別提高了25.32%和9.16%。研究結果可為農用電氣設備的電機設計與優化提供參考。

模型；試驗；磁損耗；趨膚效應；磁滯回環

0 引 言

如何通過提高電機的效率減少資源浪費是許多農用電機應用中亟待解決的問題。一方面可以采用新材料、提出新結構、優化舊有結構[1]；另一方面可以通過改進電機的控制方式及控制策略[2-3]使其運行在最佳效率范圍內。而上述2種途徑均基于對電機鐵芯損耗的精確估算[4-6]。

實際電機使用情況下的定子鐵芯處于旋轉磁場中，產生的旋轉損耗不同于單一方向交變激勵產生的損耗[7-10]。而硅鋼疊片生產廠家由于設備和規范問題往往只測量了疊片在單一激勵方向下的鐵芯損耗曲線，這對于電機定子的設計和控制策略的精確性產生了較大影響。關于電機鐵芯損耗的估算，19世紀S Steinmetz提出了鐵芯磁滯損耗公式[11]，該公式適用于激勵為正弦工頻的情況，20 世紀90年代Bertotti G在其基礎上增加了額外損耗[12]，隨著電力電子器件的出現和其應用技術的迅速發展，此類公式產生了多種變形。Aldo Boglietti提出了一種任意電壓波形均可適用的軟磁材料的鐵芯損耗計算方法[13]，該方法基于工程應用的數據擬合，能夠較快的計算出損耗值大小。隨后，Aldo Boglietti在對大量數據分析的基礎上提出了一種適于脈沖寬度調制（pulse width modulation，PWM）供電的異步電機鐵耗模型[14]，具有良好的可重復性。以上方法均是從磁路的角度出發研究損耗，需要進行解析，準確度不高。

目前主流的鐵芯損耗計算方法都是基于對已知測量數據進行參數識別之后采用數據擬合得出系數[15-16]，因此計算鐵芯損耗方法的準確性基于精確的鐵芯損耗測量。對于磁性材料的交流特性測試方法目前分為一、二、三維磁特性測量。一維方法比較成熟[17]、二維測量裝置國際上目前并沒有統一的標準，根據勵磁方式和被測樣品形狀的不同分為水平方形樣片旋轉測量裝置[18]、水平圓型樣片旋轉測量裝置[19]、垂直型旋轉測量裝置[20-21]等，而考慮到實際中農用電氣設備處于的磁場為三維磁場，一、二維測量裝置無法真實模擬實際磁場。隨著電子計算機技術的發展，有限元軟件的出現使得材料層面的模型可以直接應用于實際設備中。Zhu等[22]考慮了交變激勵下動態磁滯回環的情況，用變系數的方法進行改進。黃平林則在考慮磁滯回線面積和磁密關系的基礎上引入修正系數來表示局部磁滯回環對磁滯損耗的影響[23]。上述兩者雖然都包含了諧波產生的影響但并沒有考慮到實際電氣設備處于旋轉磁場的情況。江善林等[24]將交變激勵下趨膚效應對磁滯損耗的影響引入旋轉模型中，提出了在旋轉磁通條件下考慮趨膚效應的改進鐵芯損耗公式，但并沒有考慮諧波對旋轉磁化下磁滯損耗的影響。

本文考慮到電機鐵芯實際處于的局部磁場為旋轉磁場，在原有只考慮單一因素影響的交變激勵鐵芯損耗公式基礎上進行綜合考慮，并將其引入旋鐵芯損耗模型，對損耗系數進行修正。將合成的旋轉場進行分解，對2個方向的計算值進行耦合，得到旋轉磁化條件下的鐵芯損耗公式。在此基礎上運用磁特性測量裝置對樣品進行測量，將測量值進行數據擬合得出本文改進模型的計算值，并將改進模型計算值和測量值以及只考慮單一因素的鐵芯損耗公式計算值進行比對，以期驗證模型精度，為農用電機的設計和優化提供更為直接的理論參考。

1 鐵芯損耗的損耗分離

1.1 交變激勵下的鐵芯損耗

圖1 硅鋼片B35A210在磁密0.5 T時的鐵芯損耗分離結果

1.2 旋轉磁場下的鐵芯損耗

一維交變場的磁密圖像呈一條直線，實際空間磁場磁密必然是三維的，而由一維向三維過渡過程中，二維旋轉場是必不可少的。旋轉場的磁密矢量大小和方向隨時間變化，其頂點形成的軌跡為橢圓或圓形。旋轉磁場的形成必須使外加磁場的方向能與被測樣片的軋制方向成任意角度，其實現方法是產生2路相互垂直并且相差一定相位角的磁場，被測硅鋼樣片的磁密矢量值在2個相互垂直磁路的平面上會合成一個確定的圖形，當圖形為橢圓時，說明磁場既有旋轉分量，又有交變分量。根據坡印廷矢量定理[26]計算出旋轉磁場下磁性材料每周期的能量損失[27]，進而得到旋轉磁場下的總損耗P為：

2 旋轉磁化下鐵芯損耗模型的建立

2.1 磁滯損耗模型修正

注：和為鐵芯材料在2個相互垂直激勵磁路方向測得的磁密，T；，為橢圓形磁密軌跡的長軸長和短軸長，T；為旋轉磁密的幅值，數值上為和矢量和的模值，大小同，T；θ為橢圓軌跡長軸與被測鐵芯材料軋制方向的夾角，rad。

Zhu[8]給出磁性材料在旋轉磁化中全部的磁滯損耗為：

電能變換設備的大規模應用帶來的大量諧波會產生局部磁滯回環，影響經典公式中磁滯損耗部分的計算準確性?？紤]到磁滯回線面積與磁密的關系，磁滯損耗中引入表示局部磁滯回環中磁密變化與整體最大磁密的比值的修正系數[30]：

式中為鐵芯材料的磁密，T；、、為隨激磁頻率變化而變化的常系數，通過取對數擬合求出[32]。

2.2 總損耗模型的確定

當鐵芯材料置于變化磁場中時，磁疇壁會發生跳躍的、不連續的巴克豪森躍變與彎曲運動，從而在其內部產生環繞疇壁邊界的磁通，產生感應電壓或電流，進而引起鐵芯材料的歐姆損耗，即交變模型中的額外總損耗。在交變模型基礎上，考慮旋轉場中磁密運動軌跡并對其進行正交分解，得出渦流總損耗和額外總損耗分別如下所示：

3 驗證試驗

如果只考慮鐵芯材料處于低頻磁場的情況則測量條件和現實工況差距較大，為了進一步模擬現實中電氣設備的運行狀況，試驗的激磁頻率選擇50、100、200 Hz。圖4顯示了50、100、200 Hz下本文模型的計算值和測量值的比較，誤差平均值分別為0.123 1、0.147 5、0.192 W/kg，分別占測量值的11.55 %、8.03 %和6.56 %。其中絕對誤差隨頻率增大而增大主要是由于鐵芯總損耗隨著頻率的增加而增加，相對誤差的逐漸減小說明本文提出的模型計算值精度在高頻率下更加精確。

圖5為不同頻率下35WW270硅鋼片鐵芯損耗計算值和測量值的對比。圖5a、5b為50和200 Hz下本文模型、經典鐵芯損耗模型、考慮單一因素的改進模型的計算值和測量值的比較。圖5c為200 Hz時樣品在激磁方向為材料的切向、軋制方向時的交變鐵芯損耗和旋轉磁化下鐵芯損耗的對比。從圖5a、5b中可以看出，經典鐵芯損耗公式在頻率較高時的計算誤差相比本文提出的改進模型更大，50 Hz時本文模型鐵芯損耗的計算值與測量值的誤差比經典鐵芯損耗模型減小9.21%，而200 Hz時則減小39.76%。由圖5b可知，在低磁密時只考慮單一因素的改進模型與本文模型在精度方面差距不明顯，當磁密較高時本文模型的精確度更高：在磁密為0.95 T時對比只考慮趨膚效應、局部磁滯作用、動態磁滯回環的鐵芯損耗模型，本文模型的計算精度分別精確提高4.34%、15.2%、7.95%。而在磁密為0.38 T時分別提高1.94%、3.88%、7.37%。

1.三維磁特性測量裝置 2.電阻 3.電容箱 4.采集卡 5.示波器 6.功率分析儀 7.功率放大器 8.信號放大電路 9.電流表 10.Labview界面1.3D magnetic properties measurement device 2.Resistance 3.Capacitance box 4.Acquisition card 5.Oscilloscope 6.Power analyzer 7.Power amplifier 8.Signal amplification circuit 9.Ammeter 10.Labview interfacea. 三維磁特性測量系統各部分設備a. Equipment component of 3D magnetic properties measurement system1.勵磁線圈 2.鐵軛 3.內置被測鐵芯材料的傳感箱 4.勻場保護層1.Excitation coil 2.Iron yoke 3.Sensing box with built-in core material for test 4.Protective layer for uniforming magnetic fieldb. 磁特性測量裝置主磁路和被測立方體樣品b. Magnetic circuit of magnetic properties measurement device and tested cube sample 1.傳感線圈 2.傳感線圈 3.傳感線圈 4.傳感線圈 5.傳感線圈 6.傳感線圈1.sensing coil 2.sensing coil 3.sensing coil 4.sensing coil 5.sensing coil 6.sensing coilc. 傳感箱結構c. Structure of sensing box

圖4 硅鋼片35WW270鐵芯損耗測量值和計算值對比

圖5c中交變激勵下的鐵芯損耗大于旋轉磁化的鐵芯損耗，這是由于在交變激勵中，鐵芯處于不停換向的磁場中，鐵芯材料中磁疇的不可逆換向和疇壁移動一直發生，致使磁滯損耗一直隨磁密增加而增加；而在旋轉磁場磁化作用下，由于外磁場為方向固定的矢量，磁密接近飽和（0.76T）后由于材料內部磁化強度逐漸和外界磁場強度一致，疇壁逐漸消失，而由疇壁可逆和不可逆運動帶來的磁滯損耗和額外損耗也逐漸減小至0，總損耗主要表現為經典渦流損耗。圖4中50 Hz時旋轉鐵芯損耗的下降趨勢不如200 Hz時的明顯，主要是由于材料在低頻時磁化激勵的周期較長，磁滯損耗減少速度比200 Hz時的慢。在磁密逐漸飽和的過程中50 Hz激勵下的鐵芯總損耗中磁滯損耗的占比比200 Hz激勵下的高。以磁密為0.8 T時為例，50 Hz時的鐵芯總損耗比200 Hz時高了33.54%。而200 Hz時，在磁密由0.2 T增大到0.9 T過程中，磁滯損耗在鐵芯總損耗占比減少了12.33%，鐵芯總損耗中的磁滯損耗部分隨之減少，只剩經典渦流損耗。由于高頻時的渦流損耗遠高于低頻，所以對外呈現的鐵芯總損耗數值更大。

圖5 硅鋼片35WW270鐵芯損耗測量值和計算值對比 Fig.5 Core loss comparison of silicon steel sheet 35WW270 between measured value and calculated value

4 結 論

本文根據農用電器的實際使用情況，綜合考慮趨膚效應、局部磁滯作用和動態磁滯回環，構建了適用于無取向硅鋼片材料的旋轉鐵芯損耗模型，并以農用電氣設備中常用的無取向硅鋼片35WW270為對象進行試驗驗證。得到如下結論：

1）與經典鐵芯損耗模型相比，本文提出的改進模型考慮了實際應用中存在旋轉磁化的條件，綜合了高頻諧波的趨膚效應、局部磁滯作用和動態磁滯回環的情況，計算精度較高，誤差不超過11.55%。與經典鐵芯損耗模型和只考慮單一因素的改進模型相比，本文模型在高頻和高磁密下的鐵芯損耗計算精度分別提高了25.32%和9.16%。而且本文改進模型隨激磁頻率的增加而更接近于測量值。當激磁頻率為50 Hz時本文模型對比經典鐵芯損耗模型的誤差減小9.21 %，而200 Hz時誤差則減小39.76 %。

2）無取向硅鋼片的磁密尚未飽和時，交變激勵下的鐵芯損耗隨磁密增加而增加，而旋轉磁化條件下的鐵芯損耗則隨磁密增加到一定值后開始減少。本文旋轉磁化下的鐵芯損耗在磁密增加到0.76 T處開始出現減少，而交變激勵下的鐵芯損耗則一直隨磁密增加而增加。

3）交變激勵情況下和旋轉磁化時鐵芯損耗隨磁密增加的不同變化趨勢主要是由兩者磁滯損耗的不同變化趨勢造成的。而磁滯損耗的不同變化趨勢是由鐵芯材料中磁疇壁運動的不同方式所導致的。

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Rotating core loss model for motor considering skin effect and dynamic hysteresis effect

Song Ze1, Li Yongjian1※, Zhang Changgeng1, Liu Yang2

(1.300130,; 2.102211)

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 3D magnetic properties measurement system was used to measure the rotating core loss of electrical sheet steels. The 3D 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 3D 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.

models; experiments; core loss; skin effect; hysteresis loop

2018-10-26

2018-12-29

國家重點研發計劃（2017YFB0903904）；河北省百名優秀創新人才支持計劃項目（SRLC2017031）

宋 澤，主要從事工程電磁場與磁技術。Email：837892770@qq.com

李永建，教授，博士，博士生導師，主要從事工程電磁場與磁技術。Email：liyongjian@hebut.edu.cn

10.11975/j.issn.1002-6819.2019.06.009

TM15

A

1002-6819(2019)-06-0074-07

宋 澤，李永建，張長庚，劉 洋. 考慮趨膚效應和動態磁滯效應的電機旋轉鐵芯損耗模型[J]. 農業工程學報，2019，35(6)：74－80. doi：10.11975/j.issn.1002-6819.2019.06.009 http://www.tcsae.org

Song Ze, Li Yongjian, Zhang Changgeng, Liu Yang. Rotating core loss model for motor considering skin effect and dynamic hysteresis effect[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2019, 35(6): 74－80. (in Chinese with English abstract) doi：10.11975/j.issn.1002-6819.2019.06.009 http://www.tcsae.org