宋 澤，李永建※，張長(zhǎng)庚，劉 洋

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考慮趨膚效應(yīng)和動(dòng)態(tài)磁滯效應(yīng)的電機(jī)旋轉(zhuǎn)鐵芯損耗模型

宋 澤1，李永建1※，張長(zhǎng)庚1，劉 洋2

（1. 河北工業(yè)大學(xué)電氣工程學(xué)院省部共建電工裝備可靠性與智能化國(guó)家重點(diǎn)實(shí)驗(yàn)室，天津 300130；2. 全球能源互聯(lián)網(wǎng)研究院有限公司，北京 102211）

實(shí)際農(nóng)業(yè)用電設(shè)備鐵芯損耗的預(yù)測(cè)不準(zhǔn)是趨膚效應(yīng)造成農(nóng)用電器效率低下、使用成本升高的重要原因之一。針對(duì)這一問(wèn)題，該文綜合考慮了交變激勵(lì)下趨膚效應(yīng)、局部磁滯作用和動(dòng)態(tài)磁滯回環(huán)對(duì)經(jīng)典鐵芯損耗模型系數(shù)的影響，將系數(shù)修正后的交變模型引入旋轉(zhuǎn)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(jìn)無(wú)取向硅鋼疊片旋轉(zhuǎn)鐵芯損耗模型。利用磁特性測(cè)量裝置進(jìn)行不同頻率下的鐵芯損耗測(cè)量，對(duì)所建模型驗(yàn)證。結(jié)果表明：與經(jīng)典鐵芯損耗模型和只考慮單一因素的改進(jìn)模型相比，該模型在高頻和高磁密下的鐵芯損耗計(jì)算精度分別提高了25.32%和9.16%。研究結(jié)果可為農(nóng)用電氣設(shè)備的電機(jī)設(shè)計(jì)與優(yōu)化提供參考。

模型；試驗(yàn)；磁損耗；趨膚效應(yīng)；磁滯回環(huán)

0 引 言

如何通過(guò)提高電機(jī)的效率減少資源浪費(fèi)是許多農(nóng)用電機(jī)應(yīng)用中亟待解決的問(wèn)題。一方面可以采用新材料、提出新結(jié)構(gòu)、優(yōu)化舊有結(jié)構(gòu)[1]；另一方面可以通過(guò)改進(jìn)電機(jī)的控制方式及控制策略[2-3]使其運(yùn)行在最佳效率范圍內(nèi)。而上述2種途徑均基于對(duì)電機(jī)鐵芯損耗的精確估算[4-6]。

實(shí)際電機(jī)使用情況下的定子鐵芯處于旋轉(zhuǎn)磁場(chǎng)中，產(chǎn)生的旋轉(zhuǎn)損耗不同于單一方向交變激勵(lì)產(chǎn)生的損耗[7-10]。而硅鋼疊片生產(chǎn)廠家由于設(shè)備和規(guī)范問(wèn)題往往只測(cè)量了疊片在單一激勵(lì)方向下的鐵芯損耗曲線，這對(duì)于電機(jī)定子的設(shè)計(jì)和控制策略的精確性產(chǎn)生了較大影響。關(guān)于電機(jī)鐵芯損耗的估算，19世紀(jì)S Steinmetz提出了鐵芯磁滯損耗公式[11]，該公式適用于激勵(lì)為正弦工頻的情況，20 世紀(jì)90年代Bertotti G在其基礎(chǔ)上增加了額外損耗[12]，隨著電力電子器件的出現(xiàn)和其應(yīng)用技術(shù)的迅速發(fā)展，此類公式產(chǎn)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(jì)算方法[13]，該方法基于工程應(yīng)用的數(shù)據(jù)擬合，能夠較快的計(jì)算出損耗值大小。隨后，Aldo Boglietti在對(duì)大量數(shù)據(jù)分析的基礎(chǔ)上提出了一種適于脈沖寬度調(diào)制（pulse width modulation，PWM）供電的異步電機(jī)鐵耗模型[14]，具有良好的可重復(fù)性。以上方法均是從磁路的角度出發(fā)研究損耗，需要進(jìn)行解析，準(zhǔn)確度不高。

目前主流的鐵芯損耗計(jì)算方法都是基于對(duì)已知測(cè)量數(shù)據(jù)進(jìn)行參數(shù)識(shí)別之后采用數(shù)據(jù)擬合得出系數(shù)[15-16]，因此計(jì)算鐵芯損耗方法的準(zhǔn)確性基于精確的鐵芯損耗測(cè)量。對(duì)于磁性材料的交流特性測(cè)試方法目前分為一、二、三維磁特性測(cè)量。一維方法比較成熟[17]、二維測(cè)量裝置國(guó)際上目前并沒有統(tǒng)一的標(biāo)準(zhǔn)，根據(jù)勵(lì)磁方式和被測(cè)樣品形狀的不同分為水平方形樣片旋轉(zhuǎn)測(cè)量裝置[18]、水平圓型樣片旋轉(zhuǎn)測(cè)量裝置[19]、垂直型旋轉(zhuǎn)測(cè)量裝置[20-21]等，而考慮到實(shí)際中農(nóng)用電氣設(shè)備處于的磁場(chǎng)為三維磁場(chǎng)，一、二維測(cè)量裝置無(wú)法真實(shí)模擬實(shí)際磁場(chǎng)。隨著電子計(jì)算機(jī)技術(shù)的發(fā)展，有限元軟件的出現(xiàn)使得材料層面的模型可以直接應(yīng)用于實(shí)際設(shè)備中。Zhu等[22]考慮了交變激勵(lì)下動(dòng)態(tài)磁滯回環(huán)的情況，用變系數(shù)的方法進(jìn)行改進(jìn)。黃平林則在考慮磁滯回線面積和磁密關(guān)系的基礎(chǔ)上引入修正系數(shù)來(lái)表示局部磁滯回環(huán)對(duì)磁滯損耗的影響[23]。上述兩者雖然都包含了諧波產(chǎn)生的影響但并沒有考慮到實(shí)際電氣設(shè)備處于旋轉(zhuǎn)磁場(chǎng)的情況。江善林等[24]將交變激勵(lì)下趨膚效應(yīng)對(duì)磁滯損耗的影響引入旋轉(zhuǎn)模型中，提出了在旋轉(zhuǎn)磁通條件下考慮趨膚效應(yīng)的改進(jìn)鐵芯損耗公式，但并沒有考慮諧波對(duì)旋轉(zhuǎn)磁化下磁滯損耗的影響。

本文考慮到電機(jī)鐵芯實(shí)際處于的局部磁場(chǎng)為旋轉(zhuǎn)磁場(chǎng)，在原有只考慮單一因素影響的交變激勵(lì)鐵芯損耗公式基礎(chǔ)上進(jìn)行綜合考慮，并將其引入旋鐵芯損耗模型，對(duì)損耗系數(shù)進(jìn)行修正。將合成的旋轉(zhuǎn)場(chǎng)進(jìn)行分解，對(duì)2個(gè)方向的計(jì)算值進(jìn)行耦合，得到旋轉(zhuǎn)磁化條件下的鐵芯損耗公式。在此基礎(chǔ)上運(yùn)用磁特性測(cè)量裝置對(duì)樣品進(jìn)行測(cè)量，將測(cè)量值進(jìn)行數(shù)據(jù)擬合得出本文改進(jìn)模型的計(jì)算值，并將改進(jìn)模型計(jì)算值和測(cè)量值以及只考慮單一因素的鐵芯損耗公式計(jì)算值進(jìn)行比對(duì)，以期驗(yàn)證模型精度，為農(nóng)用電機(jī)的設(shè)計(jì)和優(yōu)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(lì)下的鐵芯損耗

圖1 硅鋼片B35A210在磁密0.5 T時(shí)的鐵芯損耗分離結(jié)果

1.2 旋轉(zhuǎn)磁場(chǎng)下的鐵芯損耗

一維交變場(chǎng)的磁密圖像呈一條直線，實(shí)際空間磁場(chǎng)磁密必然是三維的，而由一維向三維過(guò)渡過(guò)程中，二維旋轉(zhuǎn)場(chǎng)是必不可少的。旋轉(zhuǎn)場(chǎng)的磁密矢量大小和方向隨時(shí)間變化，其頂點(diǎn)形成的軌跡為橢圓或圓形。旋轉(zhuǎn)磁場(chǎng)的形成必須使外加磁場(chǎng)的方向能與被測(cè)樣片的軋制方向成任意角度，其實(shí)現(xiàn)方法是產(chǎn)生2路相互垂直并且相差一定相位角的磁場(chǎng)，被測(cè)硅鋼樣片的磁密矢量值在2個(gè)相互垂直磁路的平面上會(huì)合成一個(gè)確定的圖形，當(dāng)圖形為橢圓時(shí)，說(shuō)明磁場(chǎng)既有旋轉(zhuǎn)分量，又有交變分量。根據(jù)坡印廷矢量定理[26]計(jì)算出旋轉(zhuǎn)磁場(chǎng)下磁性材料每周期的能量損失[27]，進(jìn)而得到旋轉(zhuǎn)磁場(chǎng)下的總損耗P為：

2 旋轉(zhuǎn)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(gè)相互垂直激勵(lì)磁路方向測(cè)得的磁密，T；，為橢圓形磁密軌跡的長(zhǎng)軸長(zhǎng)和短軸長(zhǎng)，T；為旋轉(zhuǎn)磁密的幅值，數(shù)值上為和矢量和的模值，大小同，T；θ為橢圓軌跡長(zhǎng)軸與被測(cè)鐵芯材料軋制方向的夾角，rad。

Zhu[8]給出磁性材料在旋轉(zhuǎn)磁化中全部的磁滯損耗為：

電能變換設(shè)備的大規(guī)模應(yīng)用帶來(lái)的大量諧波會(huì)產(chǎn)生局部磁滯回環(huán)，影響經(jīng)典公式中磁滯損耗部分的計(jì)算準(zhǔn)確性?？紤]到磁滯回線面積與磁密的關(guān)系，磁滯損耗中引入表示局部磁滯回環(huán)中磁密變化與整體最大磁密的比值的修正系數(shù)[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(shù)，通過(guò)取對(duì)數(shù)擬合求出[32]。

2.2 總損耗模型的確定

當(dāng)鐵芯材料置于變化磁場(chǎng)中時(shí)，磁疇壁會(huì)發(fā)生跳躍的、不連續(xù)的巴克豪森躍變與彎曲運(yùn)動(dòng)，從而在其內(nèi)部產(chǎn)生環(huán)繞疇壁邊界的磁通，產(chǎn)生感應(yīng)電壓或電流，進(jìn)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(chǔ)上，考慮旋轉(zhuǎn)場(chǎng)中磁密運(yùn)動(dòng)軌跡并對(duì)其進(jìn)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(yàn)證試驗(yàn)

如果只考慮鐵芯材料處于低頻磁場(chǎng)的情況則測(cè)量條件和現(xiàn)實(shí)工況差距較大，為了進(jìn)一步模擬現(xiàn)實(shí)中電氣設(shè)備的運(yùn)行狀況，試驗(yàn)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(jì)算值和測(cè)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(cè)量值的11.55 %、8.03 %和6.56 %。其中絕對(duì)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(duì)誤差的逐漸減小說(shuō)明本文提出的模型計(jì)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(jì)算值和測(cè)量值的對(duì)比。圖5a、5b為50和200 Hz下本文模型、經(jīng)典鐵芯損耗模型、考慮單一因素的改進(jìn)模型的計(jì)算值和測(cè)量值的比較。圖5c為200 Hz時(shí)樣品在激磁方向?yàn)椴牧系那邢?、軋制方向時(shí)的交變鐵芯損耗和旋轉(zhuǎn)磁化下鐵芯損耗的對(duì)比。從圖5a、5b中可以看出，經(jīng)典鐵芯損耗公式在頻率較高時(shí)的計(jì)算誤差相比本文提出的改進(jìn)模型更大，50 Hz時(shí)本文模型鐵芯損耗的計(jì)算值與測(cè)量值的誤差比經(jīng)典鐵芯損耗模型減小9.21%，而200 Hz時(shí)則減小39.76%。由圖5b可知，在低磁密時(shí)只考慮單一因素的改進(jìn)模型與本文模型在精度方面差距不明顯，當(dāng)磁密較高時(shí)本文模型的精確度更高：在磁密為0.95 T時(shí)對(duì)比只考慮趨膚效應(yīng)、局部磁滯作用、動(dòng)態(tài)磁滯回環(huán)的鐵芯損耗模型，本文模型的計(jì)算精度分別精確提高4.34%、15.2%、7.95%。而在磁密為0.38 T時(shí)分別提高1.94%、3.88%、7.37%。

1.三維磁特性測(cè)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(hào)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(cè)量系統(tǒng)各部分設(shè)備a. Equipment component of 3D magnetic properties measurement system1.勵(lì)磁線圈 2.鐵軛 3.內(nèi)置被測(cè)鐵芯材料的傳感箱 4.勻場(chǎng)保護(hù)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(cè)量裝置主磁路和被測(cè)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(jié)構(gòu)c. Structure of sensing box

圖4 硅鋼片35WW270鐵芯損耗測(cè)量值和計(jì)算值對(duì)比

圖5c中交變激勵(lì)下的鐵芯損耗大于旋轉(zhuǎn)磁化的鐵芯損耗，這是由于在交變激勵(lì)中，鐵芯處于不停換向的磁場(chǎng)中，鐵芯材料中磁疇的不可逆換向和疇壁移動(dòng)一直發(fā)生，致使磁滯損耗一直隨磁密增加而增加；而在旋轉(zhuǎn)磁場(chǎng)磁化作用下，由于外磁場(chǎng)為方向固定的矢量，磁密接近飽和（0.76T）后由于材料內(nèi)部磁化強(qiáng)度逐漸和外界磁場(chǎng)強(qiáng)度一致，疇壁逐漸消失，而由疇壁可逆和不可逆運(yùn)動(dòng)帶來(lái)的磁滯損耗和額外損耗也逐漸減小至0，總損耗主要表現(xiàn)為經(jīng)典渦流損耗。圖4中50 Hz時(shí)旋轉(zhuǎn)鐵芯損耗的下降趨勢(shì)不如200 Hz時(shí)的明顯，主要是由于材料在低頻時(shí)磁化激勵(lì)的周期較長(zhǎng)，磁滯損耗減少速度比200 Hz時(shí)的慢。在磁密逐漸飽和的過(guò)程中50 Hz激勵(lì)下的鐵芯總損耗中磁滯損耗的占比比200 Hz激勵(lì)下的高。以磁密為0.8 T時(shí)為例，50 Hz時(shí)的鐵芯總損耗比200 Hz時(shí)高了33.54%。而200 Hz時(shí)，在磁密由0.2 T增大到0.9 T過(guò)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(jīng)典渦流損耗。由于高頻時(shí)的渦流損耗遠(yuǎn)高于低頻，所以對(duì)外呈現(xiàn)的鐵芯總損耗數(shù)值更大。

圖5 硅鋼片35WW270鐵芯損耗測(cè)量值和計(jì)算值對(duì)比 Fig.5 Core loss comparison of silicon steel sheet 35WW270 between measured value and calculated value

4 結(jié) 論

本文根據(jù)農(nóng)用電器的實(shí)際使用情況，綜合考慮趨膚效應(yīng)、局部磁滯作用和動(dòng)態(tài)磁滯回環(huán)，構(gòu)建了適用于無(wú)取向硅鋼片材料的旋轉(zhuǎn)鐵芯損耗模型，并以農(nóng)用電氣設(shè)備中常用的無(wú)取向硅鋼片35WW270為對(duì)象進(jìn)行試驗(yàn)驗(yàn)證。得到如下結(jié)論：

1）與經(jīng)典鐵芯損耗模型相比，本文提出的改進(jìn)模型考慮了實(shí)際應(yīng)用中存在旋轉(zhuǎn)磁化的條件，綜合了高頻諧波的趨膚效應(yīng)、局部磁滯作用和動(dòng)態(tài)磁滯回環(huán)的情況，計(jì)算精度較高，誤差不超過(guò)11.55%。與經(jīng)典鐵芯損耗模型和只考慮單一因素的改進(jìn)模型相比，本文模型在高頻和高磁密下的鐵芯損耗計(jì)算精度分別提高了25.32%和9.16%。而且本文改進(jìn)模型隨激磁頻率的增加而更接近于測(cè)量值。當(dāng)激磁頻率為50 Hz時(shí)本文模型對(duì)比經(jīng)典鐵芯損耗模型的誤差減小9.21 %，而200 Hz時(shí)誤差則減小39.76 %。

2）無(wú)取向硅鋼片的磁密尚未飽和時(shí)，交變激勵(lì)下的鐵芯損耗隨磁密增加而增加，而旋轉(zhuǎn)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(zhuǎn)磁化下的鐵芯損耗在磁密增加到0.76 T處開始出現(xiàn)減少，而交變激勵(lì)下的鐵芯損耗則一直隨磁密增加而增加。

3）交變激勵(lì)情況下和旋轉(zhuǎn)磁化時(shí)鐵芯損耗隨磁密增加的不同變化趨勢(shì)主要是由兩者磁滯損耗的不同變化趨勢(shì)造成的。而磁滯損耗的不同變化趨勢(shì)是由鐵芯材料中磁疇壁運(yùn)動(dòng)的不同方式所導(dǎo)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(guó)家重點(diǎn)研發(fā)計(jì)劃（2017YFB0903904）；河北省百名優(yōu)秀創(chuàng)新人才支持計(jì)劃項(xiàng)目（SRLC2017031）

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

李永建，教授，博士，博士生導(dǎo)師，主要從事工程電磁場(chǎng)與磁技術(shù)。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(zhǎng)庚，劉 洋. 考慮趨膚效應(yīng)和動(dòng)態(tài)磁滯效應(yīng)的電機(jī)旋轉(zhuǎn)鐵芯損耗模型[J]. 農(nóng)業(yè)工程學(xué)報(bào)，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