表贴式永磁同步轮毂电机磁场分析与优化设计

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一种表贴式永磁电机磁极结构优化研究

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一种表贴式永磁电机磁极结构优化研究
作者：张炳义贾宇琪等
来源：《电机与控制学报》2014年第05期
摘要：针对常规表贴式永磁电机气隙磁密波形正弦度差，导致电机反电势中谐波含量高、电机谐波损耗大，以及复杂结构形状永磁体在生产、加工、装配过程中容易造成废品率高等问题，提出一种由导磁金属块和永磁体共同构成的表贴式磁极结构。

用有限元方法计算常规结构以及不同程度不均匀气隙结构的气隙磁密波形，进而由傅里叶分解得到各次谐波和正弦畸变率。

仿真结果标明该结构可以有效改善气隙磁密波形和电机空载反电势的正弦度。

最后采用遗传算法对实现偏心结构的导磁金属块尺寸进行优化，得到了使气隙磁密波形畸变率最小的尺寸参数。

有限元计算结果显示，优化设计后，气隙磁密波形畸变率和电机空载反电势谐波含量明显减小。

表贴式永磁同步电动机永磁体气隙磁场解析计算

刘珂让余奇
４３００３３）
（海军工程大学武汉
摘
要
以气隙中多级永磁同步电动机永磁体产生的磁场为基础，利用等效面电流法推导出了永磁同步电动机永磁
体在径向充磁和平行充磁两种充磁方式下的气隙磁密的解析计算公式。通过比较不同永磁体结构在上述两种充磁方式下
产生的气隙磁场，给出了一种偏心不等半径瓦片形磁极结构来改善气隙磁场，使得气隙磁场的波形更好，从而降低电机的振
动和噪声。同时解析计算易于与计算机结合应用，为永磁同步电动机的优化设计提供了基本的分析隙磁场；等效面电流法；解析计算
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表贴式永磁电机磁场的解析计算与分析

表贴式永磁电机磁场的解析计算与分析
贴式永磁电机的磁场是一种广泛使用的技术，它的特点
是简单、结构轻、效率高、寿命长、成本低及功耗小。

很多应用情况下，运用贴式永磁电机磁场的解析计算技术，能为设备的性能优化提供更加深入的理论依据。

贴式永磁电机磁场的解析计算，通常要从磁场测试、磁
场结构分析等方面出发。

首先，采用底部感应器对永磁电机ム磁场加以测试，以确定其等效矩形磁性体的大小。

这种测量得到的输出值是由坐标轴X，Y和Z分别测量而得出的，即X轴，Y轴和Z轴的三个主要组份股值。

然后，再采用电磁场分析软
件和专业仪器进行磁场结构的分析，以确定磁场的特性。

最后，将获得的磁场测试结果和磁场结构分析结果，采
用数学方法对其进行整合处理，以求得磁场参数：磁场强度、归一化磁势，以及磁势坐标等等。

在此基础上，还可进行磁体结构、材料特性的仿真计算，并对不同组态的永磁电机磁场进行比较，从而实现性能优化及性能设计。

因此，采用贴式永磁电机磁场的解析计算，不仅能对磁
势分布及结构进行全面的分析，而且还能够非常好的有效提高系统的可靠性。

它在满足性能参数的同时，还能更有效地降低永磁电机的功耗，即改善系统能源效率，为实现不同设备性能优化提供可靠依据。

中心部分分段Halbach永磁同步轮毂电机解析计算与优化设计

Vol56 No2Feb2022第56卷第2期2022年2月西安交通大学学报JOURNAL OF XI'AN JIAOTONG UNIVERSITY中心部分分段Halbach轮毂电机析计算与优化高锋阳，李明明，齐晓东，杨乔礼，陶彩霞(兰州交通大学自动化与电气工程学院,730070,兰州)摘要：针对永磁同步轮毂电机转矩脉动和涡流损耗大的问题,提出一种主磁极中心部分分段结构。

采用Halbach 充磁方式，永磁体主磁极中心部分分段，边界磁极与主磁极不等宽不等厚。

首先，在二维极坐标系下构建解析模型，采用精确子域模型法，对解析模型在空载、电流源激励及负载下的气隙磁密和转矩进行计算。

其次，建立10极12槽永磁同步电机模型并对其进行有限元仿真验证，提出一种分级优化方法，对电机结构参数进行优化，从而达到降低永磁同步电机气隙磁密谐波畸变率、转矩脉动及涡流损耗的目的。

最后，与相关8种磁极结构永磁同步电机模型的各项电磁性能展开对比。

结果表明：分级多变量多目标优化算法提高了优化效率与精度；十字型主磁极中心部分分段Halbach 永磁同步电机可以显著减小涡流损耗，有利于牵引电机过载运行；H 型主磁极中心部分分段Halbach 永磁同步电机可以显著降低转矩脉动，提高电磁转矩和机车稳定运行能力。

关键词：永磁同步轮毂电机；转矩脉动；涡流损耗；中心部分分段结构；多目标优化中图分类号：TM351文献标志码：ADOI ： 10. 7652/xjtuxb202202018 文章编号：0253-987X(2022)02-0171-13OSID 码Analytical Calculation and Optimization Design of Central Partially- Segmented Halbach Permanent Magnet Synchronous Hub MotorGAO Fengyang , LI Mingming , QI Xiaodong , YANG Qiaoli , TAO Caixia(School of Automation and Electrical Engineering , Lanzhou Jiaotong University , Lanzhou 730070, China)Abstract !A main magneticpolecentralpartia l y-segmentedstructureisproposedtosolvetheproblem of large torque ripple and eddy currentlossin permanent magnetsynchronous hubmotor.Halbach magnetizing method is adopted for permanent magnets , the central parts of the main magnetic poles are segmented , and the boundary magnetic poles are not equal in thickness and width to the main magneticpoles.Firstly , ananalytical modelisconstructedinthetwo- dimensionalpolarcoordinatesystem ,andtheairgapmagneticdensityandtorqueoftheanalyticalmodel under no-load , currentsourceexcitation andload arecalculated by usingtheaccuratesubdomain modelmethod.Secondly ,a10-poleand12-slotpermanentmagnetsynchronousmotor model is established and verified byfiniteelementsimulation , anda hierarchicaloptimization method is proposed to optimizethe motorstructure parameters , so as to reduce the air gapmagnetic density total harmonic distortion rate , cogging torque and eddy current loss of the permanent magnetsynchronous motor.Fina l y , the electromagnetic performance is compared withthatofeightpermanent magnetsynchronous motor models. The results show that the收稿日期：2021-07-29o 作者简介：高锋阳(1970-),男，教授，硕士生导师。

表贴式永磁电机永磁体优化设计

同时，还可以进一步提高电机的转矩密度。
关键词：永磁电机；永磁体；优化设计
中图分类号：ＴＭ３５１文献标志码：Ａ文章编号：１００５－７６７６（２０１５）０３ —００ｌ８ —０４
ＯｐｔｉｍｉＴａｔｉｏｎＤｅｓｉｇｎｏｆＳｔｔｒｆａｅｅ — ｍｏｕｎｔｅｄＰｅｒｍａｎｅｎｔＭａｎｅｇｔＭｏｔｏｒｓ
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Ｋ。，ｒｗｏｒｄｓ：ｐｅｒｍａｎｅｎｔｍａｎｅｇｔｍａｃｈｉｎｅ；ｐｅｍａｒｎｅｎｔｍａｎｅｇｔ；ｏｐｔｉｍｉｚａｔｉｏｎｄｅｓｉｎｇ
摘要：永磁电机具有较高的功率密度，并且维护方便，已广泛应用于工业生产和日常生活中。永磁电机的优化设计对于降低永磁体成本，节约资源具有重要意义。通过３种方法实现对表贴式永磁电机的优化设计，斜极型转子可以显著降低电机的齿槽转矩，正弦型转子可以进一步降低电机的转矩脉动，三次谐波注入型转子在降低转矩脉动的

电动汽车用永磁同步轮毂电机的设计及分析

摘要作为清洁能源汽车，电动汽车具有高能效，低噪音和零排放，成为世界新能源汽车发展的主要方向。

而对于永磁同步电动机，其结构简单，运行效率高，功率密度高，调速性能优良，符合电动汽车用电动机的要求。

因此，它在汽车工业中受到很多关注，并已广泛应用于电动汽车领域。

本文在有限元分析的基础上，采用场路结合的设计方法进行了电动汽车用永磁同步轮毂电机的设计和运行特性分析。

分析磁路结构参数变化对电机性能的影响，开发出适用于电动汽车的高效率、高功率密度、高过载能力的驱动电机，并由此总结了适用于电动汽车驱动的永磁同步电动机的设计方法，为后续系列产品的开发奠定了基础。

本文的主要研究工作有以下几个部分：根据电动汽车发展的关键技术，结合电动汽车的特殊运行条件和动力驱动特性，分析各种电动机性能的优缺点。

本文选择内置永磁同步电动机作为研究对象，通过对其结构特点和工作原理的分析，确定设计任务目标，使设计突出电动汽车驱动电机的特性。

以有限元软件为基础，依据电机学和相关电磁场理论，本文采用场路结合设计方法，确定了电机的设计方案，进行了电机主要尺寸设计、绕组方案确定、极槽配合选择、永磁体参数计算、永磁体充磁方向分析、气隙长度的设计等工作，完成样机的初步设计方案；然后根据电机电磁设计方案，建立有限元求解模型，对电机进行有限元分析计算，主要是对电机的空载、负载及过载工况进行仿真，并在此基础上研究电机的磁场分布、气隙磁密、空载反电动势、齿槽转矩、转矩转速以及永磁体涡流损耗等；研究相关结构的参数变化对电机的影响；从转子结构方面分析电机的弱磁扩速性能；为保证所设计的电机结构在运行时能够满足实际工况的机械强度需求，还对电机进行机械结构仿真，确保电机的各部分的应力能够满足所用材料的屈服强度的要求，保证电机的稳定运行。

最后依据设计结果制作了额定功率8.5kW、额定转速650r/min的样机，对样机的性能进行试验测试，测试结果表明样机具有较大的过载倍数和高效运行区域，达到预期设计目标。

基于MVO的表贴式Halbach永磁同步电机优化设计

第５２卷第１期２０１９年１月
ＭＩＣＲＯＭＯＴＯＲＳ
Ｖｏｌ�� ５２. Ｎｏ�� １Ｊａｎ�� ２０１９
基于ＭＶＯ的表贴式Ｈａｌｂａｃｈ永磁同步电机优化设计
胡致远ꎬ 杨向宇ꎬ 赵世伟ꎬ 裴云庆ꎬ 曹江华
( 华南理工大学电力学院ꎬ 广州５１０６４０)
Ａｂｓｔｒａｃｔ: Ｉｎｔｈｉｓｐａｐｅｒꎬ ａｎｏｖｅｌｎｏｎ￣ｕｎｉｆｏｒｍｐｅｒｍａｎｅｎｔｍａｇｎｅｔｓｙｎｃｈｒｏｎｏｕｓｍｏｔｏｒ ( ＰＭＳＭ) ｗｉｔｈＨａｌｂａｃｈｍａｇｎｅｔｉｚａｔｉｏｎｗａｓｐｒｅｓｅｎｔｅｄ. Ｏｎｔｈｅｐｒｅｍｉｓｅｏｆｔｈｅｓａｍｅａｍｏｕｎｔｏｆｐｅｒｍａｎｅｎｔｍａｇｎｅｔｍａｔｅｒｉａｌꎬ ｔｈｅｎｏｎ￣ｕ￣ｎｉｆｏｒｍａｎｄｕｎｅｑｕａｌｔｈｉｃｋｎｅｓｓｐｅｒｍａｎｅｎｔｍａｇｎｅｔｓｔｒｕｃｔｕｒｅｃａｎｅｆｆｅｃｔｉｖｅｌｙｉｍｐｒｏｖｅｔｈｅａｉｒｇａｐｍａｇｎｅｔｉｃｄｅｎｓｉ￣ｔｙｗａｖｅｆｏｒｍꎬ ｔｈｕｓｉｍｐｒｏｖｉｎｇｔｈｅｍｏｔｏｒｐｅｒｆｏｒｍａｎｃｅ. Ｉｎｏｒｄｅｒｔｏｏｂｔａｉｎｔｈｅｄｅｓｉｇｎｐａｒａｍｅｔｅｒｓｏｆｔｈｅｐｅｒｍａ￣ｎｅｎｔｍａｇｎｅｔｓｈａｐｅꎬ ｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｏｆｔｈｅｍｏｔｏｒｗａｓｆｉｒｓｔｌｙｅｓｔａｂｌｉｓｈｅｄｉｎｔｈｉｓｐａｐｅｒꎬ ｔａｋｉｎｇａｉｒ￣ｇａｐｍａｇｎｅｔｉｃｄｅｎｓｉｔｙｆｕｎｄａｍｅｎｔａｌｗａｖｅꎬ ｈａｒｍｏｎｉｃｄｉｓｔｏｒｔｉｏｎｒａｔｅａｎｄｐｅｒｍａｎｅｎｔｍａｇｎｅｔａｍｏｕｎｔａｓｔｈｅｏｂｊｅｃｔｉｖｅｆｕｎｃｔｉｏｎ. Ｉｎｏｒｄｅｒｔｏｆｉｎｄｔｈｅｏｐｔｉｍａｌｖａｌｕｅｏｆｔｈｅｔａｒｇｅｔｆｕｎｃｔｉｏｎꎬ ｔｈｅＭｕｌｔｉ￣ｖｅｒｓｅＯｐｔｉｍｉｚｅｒ( ＭＶＯ) ｏｐｔｉｍｉ￣ｚａｔｉｏｎａｌｇｏｒｉｔｈｍｗａｓｆｉｒｓｔｔｉｍｅｕｓｅｄｉｎｔｈｉｓｐａｐｅｒｔｏｉｍｐｒｏｖｅｔｈｅｏｐｔｉｍｉｚａｔｉｏｎｓｐｅｅｄａｎｄａｃｃｕｒａｃｙ. Ｆｉｎａｌｌｙꎬ ｔｈｅｐｅｒｆｏｒｍａｎｃｅｏｆｔｈｅｍｏｔｏｒｉｓｃｏｍｐａｒｅｄｗｉｔｈｔｒａｄｉｔｉｏｎａｌＨａｌｂａｃｈｍａｇｎｅｔｉｚｉｎｇｍｏｔｏｒ. Ｔｈｅｒｅｓｕｌｔｓｓｈｏｗｔｈａｔｔｈｅｓｔｒｕｃｔｕｒｅｃａｎｆｕｒｔｈｅｒｉｍｐｒｏｖｅｔｈｅａｉｒｇａｐｍａｇｎｅｔｉｃｄｅｎｓｉｔｙｗａｖｅｆｏｒｍａｎｄｉｍｐｒｏｖｅｔｈｅｍｏｔｏｒｐｅｒｆｏｒｍａｎｃｅ. Ｋｅｙｗｏｒｄｓ: ＳＰＭＳＭꎻ ｔｏｔａｌｈａｒｍｏｎｉｃｄｉｓｔｏｒｔｉｏｎꎻ Ｈａｌｂａｃｈｍａ率是可行的ꎮ Ｈａｌｂａｃｈ阵列电机的分析通常采用解析法进行分

表贴式永磁电机磁场的解析计算与分析

表贴式永磁电机磁场的解析计算与分析张河山;邓兆祥;杨金歌;妥吉英;张羽【摘要】利用傅里叶级数法建立表贴式永磁电机电磁场全局解析模型,并分析其空载和负载电磁特性.在二维极坐标系下,将电机求解域划分为永磁体、气隙、定子槽、定子槽开口和辅助槽5类子域.采用分离变量法求解各子域的拉普拉斯方程或泊松方程,并利用边界条件得到通解中的谐波系数,进而得到各子域的解析表达式.计算了电机气隙磁密、空载反电动势、齿槽转矩和电磁转矩等电磁参数,并通过有限元分析验证了解析法的准确性.在此基础上研究了极弧系数、辅助槽尺寸和槽开口宽度对电机齿槽转矩和电磁转矩的影响规律.另外,提出一种不等槽开口宽度配合的解析模型以图减小齿槽转矩峰值和电磁转矩脉动.该方法能反映电机设计性能与尺寸和参数的关系,可用于电机初始设计与优化.【期刊名称】《汽车工程》【年(卷),期】2018(040)007【总页数】9页(P850-857,864)【关键词】永磁电机;解析法;有限元法;齿槽转矩;电磁转矩;不等槽开口【作者】张河山;邓兆祥;杨金歌;妥吉英;张羽【作者单位】重庆大学汽车工程学院,重庆 400044;重庆大学汽车工程学院,重庆400044;重庆大学,机械传动国家重点实验室,重庆 400044;重庆大学汽车工程学院,重庆 400044;重庆大学汽车工程学院,重庆 400044;重庆大学汽车工程学院,重庆400044【正文语种】中文前言永磁电机具有高转矩密度、高效率、高功率密度等优点，广泛应用于电动汽车、船舶等工业领域[1]。

其结构参数对电机性能影响较大，因此，为设计性能优异的电机需要改变和优化电机结构参数，并采用有限元法或解析法分析其电磁场特性。

有限元法可考虑材料非线性影响和分析较复杂结构电机性能，但计算过程耗时、占用资源，且难以对电机特性及其影响因素进行规律性研究，具有明显局限性。

电磁场数值解析法计算量较小，物理概念清晰，能清晰反映电机性能与设计参数的关系，适用于电机设计参数优化，因此逐渐引起国内外学者广泛关注。

基于改进遗传算法的表贴式永磁同步电机优化设计

摘
要：针对有约束条件的电机优化问题，建立了带惩罚项的优化目标函数，在简单遗传算法基础上采取保留最佳
个体策略，结合模式搜索法形成改进遗传算法，基于磁路法分析了表贴式永磁同步电机在ｉ＝０控制方式下重要性能指标的计算方法，并分别采用简单遗传算法和改进遗传算法对一台分数槽集中绕组表贴式永磁同步电机进行了效率优化。优化结果证明改进遗传算法更易得到更高的目标函数值，且优化初期优势明显，收敛更快，优化效率更高，证明了改进遗传算法在优化电机目标函数时的有效性。
ｆｏｍａｒｎｃｅｕｎｄｅｒｉｄ＝０ｗａｓｐｒｅｓｅｎｔｅｄ．Ａｆｒａｃｔｉｏｎａｌ — ｓｌｏｔｓｕｒｆａｃｅｐｅｒｍａｎｅｎｔｍａｇｎｅｔｓｙｎｃｈｒｏｎｏｕｓｍｏｔｏｒ

ｔａｉｎｈｉｇｈｅｒｏｂｊｅｃｔｉｖｅｆｕｎｃｔｉｏｎｖａｌｕｅｗｉｔｈｂｅｔｔｅｒａｓｔｉｒｎｇｅｎｃｙ，ａｎｄｉｎｃｒｅａｓｅｔｈｅｏｐｔｉｍｉｚａｔｉｏｎｅｉｃｆｉｅｎｃｙｅｓｐｅｃｉｌ— ａ
ｌｙｉｎｉｎｉｔｉａｌｓｔａｇｅ，ｗｈｉｃｈｖｅｉｒｉｆｅｓｔｈｅｖａｌｉｄｉｔｙｏｆｔｈｅｉｍｐｒｏｖｅｄｇｅｎｅｔｉｃａｌｇｏｒｉｔｈｍｕｓｅｄｉｎｍｏｔｏｒｏｐｔｉｍｉｚａｔｉｏｎ．

表贴式永磁同步电机磁极优化建模与分析

Modeling and Analyzing of Surface-Mounted Permanent-Magnet Synchronous Machines With Optimized Magnetic Pole Shape Zhenfei Chen1,Changliang Xia1,2,Qiang Geng2,and Yan Yan11School of Electrical Engineering and Automation,Tianjin University,Tianjin300072,China2Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy,Tianjin Polytechnic University,Tianjin300387,ChinaTwo types of eccentric magnetic pole shapes for optimizing conventional surface-mounted permanent-magnet(PM)synchronous machines with radial magnetization are presented in this paper.An analytical method based on an exact subdomain model and discrete idea is proposed for obtaining the air-gapﬂux density distribution in the improved motor.Cogging torque and back EMF analytical models are further built with theﬁeld solution,which provide useful tools for investigating motor performances with unequal thickness magnetic poles.The accuracy and feasibility of the models have been validated by aﬁnite element method.Based on the analytical models,the effects of pole shape parameters on motor performance are investigated.Results show that both pole shapes can perfect magneticﬁeld distribution,decrease harmonic content of back EMF,reduce torque ripples,and improve the utilization of PMs.Index Terms—Exact subdomain model,ﬂux density distribution,magnetic pole shape optimization,surface-mounted permanent-magnet(PM)synchronous machine.I.I NTRODUCTIONT HE surface-mounted permanent-magnet(PM) synchronous machine has been widely used in elevator,wind turbine,and hybrid electric vehicle applications due to its high efﬁciency,power factor,and torque density [1],[2].The PM pole,as a pivotal part of the PM motor, directly affects motor cost and behavior,such as magnetic ﬁeld,back EMF,torques,and so on.As a result,magnetic pole design is particularly important in PM motor design and has attracted lots of attention.Studies in[3]–[6]point out that the contributions of different PM parts are not uniform and magnetic pole optimization can not only improve PM material utilization,reduce magnet material cost,but also achieve more sinusoidal magneticﬁeld distribution and lower cogging torque performance.The magneticﬁeld calculation is an important prerequisite for the analysis of PM machines.Many methods have been proposed for magneticﬁeld prediction in past few decades. In[7],the drawbacks and stability of numerical implementa-tion are discussed and a semianalytical framework is presented for solving2-D PM machine models in three different coordi-nates.Nevertheless,analytical modeling is usually much more complex for improved PM motors with optimized magnetic pole conﬁgurations,since the radial thickness of magnetic pole changes with the circumferential position,which makes its mathematical modeling more difﬁcult than that of conven-tional magnetic poles.Several analytical methods are given in[8]–[10],which provide valuable theoretical references for magnetic pole design and analysis.Stator slotting is usually neglected or complicated pole boundary is simpliﬁed to reduce the difﬁculty of modeling,which also results in a low accuracy of the models.Manuscript received March3,2014;revised May11,2014;accepted May24,2014.Date of current version November18,2014.Corresponding author:C.Xia(e-mail:motor@).Color versions of one or more of theﬁgures in this paper are available online at .Digital Object Identiﬁer10.1109/TMAG.2014.2327138Fig.1.PM pole shapes.(a)Conventional pole shape S0.(b)Outer arc eccentric pole shape SA.(c)Inner arc eccentric pole shape SB.In this paper,two types of eccentric magnetic pole designs are chosen for pole shape optimization of surface-mountedPM machines with radial magnetization.To solve the problem of unequal thickness magnetic pole modeling,a modiﬁedsubdomain model method based on discrete idea is proposedto predict magneticﬁeld distribution in the air-gap.With the ﬁeld solution,cogging torque and back EMF models are built.The effects of magnetic pole dimensions on motor behaviorare further investigated to draw some conclusions.II.A NALYTICAL M ODELINGA.Eccentric Magnetic Pole ShapesCompared with the conventional magnetic pole,two kinds of eccentric magnetic pole shapes for improving theﬁelddistribution of surface-mounted PM motors are shown inFig. 1.Fig.1(a)is the conventional magnetic pole shape designated as S0,Fig.1(b)is the outer arc eccentric magneticpole shape designated as SA,and Fig.1(c)is the inner surfacearc magnetic pole shape designated as SB.As shown in Fig.1,O is the center of motor and h m isthe magnet thickness at the pole centerline.For conventional pole shape S0,its inner and outer arcs have the same centreO and the radial thickness does not change with position. R r and R m are the radii of magnet inner and outer surfaces,and h m=R m−R r.For the shape SA,the center of its outer arc moves to O and the radius changes to be R o.For the shapeSB,the center of its inner arc moves to O ,and the radius0018-9464©2014IEEE.Personal use is permitted,but republication/redistribution requires IEEE permission.See /publications_standards/publications/rights/index.html for more information.Fig.2.General subdomain model for PM motors with unequal thicknessmagnetic poles.changes to be R i .d denotes the distance between points Oand O .B.PM Motor ModelingIt is obvious in Fig.1that mathematical illustration becomes more difﬁcult since the radial magnet thickness of SA and SB conﬁgurations changes with circumferential angle;thus,the original exact subdomain model [11]can hardly be used for analytical modeling of the improved PM motor.In this paper,discrete idea and superposition principle are adopted to modify and extend the subdomain model method to PM motors with irregular magnetic pole shapes.In Fig.2,a general subdomain model of a 4-pole and 6-slot motor is taken as an example to illustrate the analytical method.The whole domain is divided into three types of subdomains,viz.,magnet (region 1j ),air-gap (region 2),and slots (region 3i ).In this paper,the magnet region is divided into N equal parts,and the j th part is designated as region 1j (j =1,2,...,N ).The thickness of each part can be considered constant if N is large enough.θj denotes the position of region 1j from the pole centerline.R rj and R mj are the inner and outer radii of region 1j ,respectively.In addition,R s and R sb are the radii of stator inner bore and winding slot bottom,respectively.C.Modiﬁed Subdomain ModelBefore the modeling,several assumptions are made to simplify the problem as follows:1)end effect is neglected;2)stator/rotor iron has inﬁnite permeability;3)core saturation and loss are neglected;4)demagnetization characteristic of the magnet is linear;5)nonconductive magnet material is used.In polar coordinates,the scalar potentials in the three regions ϕ1j ,ϕ2,and ϕ3i can be expressed by Laplace’s and Poisson’s equations as follows.1)For Region 1jϕ1j = kA 1k r k +B 1k r −k+M cjk r μr (1−k 2)cos (k α)+ kC 1k r k +D 1k r −k +M sjkr μr (1−k 2)sin (k α)k =1(1)ϕ1j =(A 11r +B 11r−1+M cj 1r ln r /2μr )cos α+(C 11r +D 11r −1+M sj 1r ln r /2μr )sin αk =1.(2)2)For Region 2ϕ2=kA 2k r k +B 2k r −kcos (k α)+ k C 2k r k +D 2k r −k sin (k α).(3)3)For Region 3iϕ3i = mX 3im (r /R sb )m παb −(r /R sb )−m παb×sin [(m π/αb )×(α−αi +αb /2)(4)where μr is the relative recoil permeability of magnet and αb is the mechanical angle of winding slot opening.αb =b 0/R s and b 0is the slot opening width.Q is the stator slot number.αi locates the center of the i th winding slot where i =1,2,3,...,Q .k and m are the harmonic coefﬁcients.A 1k -D 1k ,A 2k -D 2k ,and X 3im are the coefﬁcients to be determined.M cjk and M sjk are variables concerning magnets,which can be given asM cjk = 4pB rk πμ0sin k παpj 2p cos (k ωr t )k /p =1,5,7···0others (5)M sjk = 4pB rk πμ0sin k παpj 2p sin (k ωr t )k /p =1,5,7···0others (6)where μ0is the permeability of air,p is the pole pair number,αpj is the pole arc to pole pitch ratio of the j th PM region,B r is the remanent ﬂux density of magnets,and ωr is the rotor rotational speed.To determine the unknown coefﬁcients and obtain the ﬁeld distribution,the boundary conditions are deﬁned as follows:H α1|r =Rr =0(7)B r 1|r =R m =B r 2|r =R m (8)H α1|r =R m =H α2|r =R m (9)ϕ2|r =R s =ϕ3i |r =R s (10)B r 2|r =R s =B r 3i |r =R s(11)where subscripts r and αdenote the radial and tangential components of variables,respectively.By applying the boundary conditions given by (7)–(11)to (1)–(6),the radial and tangential components of air-gap ﬂux density can be solved asB r 2=kχrck cos (k α)+kχrsk sin (k α)(12)B α2=k χαck cos (k α)+kχαsk sin (k α)(13)where χrck ,χrsk ,χαck ,and χαck are the harmonic amplitudesobtained using subdomain model method and superposition principle.Based on the analytical model of air-gap ﬂux density in (12)and (13),models of cogging torque T c and phase back EMF E x are built as follows:T c =(πl a r 2/μ0)×k(χrck χαck +χrsk χαsk )(14)E x =(N c l a R s /a )×dxB r 2d α/dt x =A ,B ,C (15)CHEN et al.:MODELING AND ANALYZING OF SURFACE-MOUNTED PM SYNCHRONOUS MACHINES 8102804TABLE IM AIN P ARAMETERS OF P ROTOTYPE MACHINESFig.3.FE and analytical predicted air-gap ﬂux densitywaveforms.Fig.4.FE and analytical predicted cogging torquewaveforms.Fig.5.FE and analytical predicted back EMF waveforms.where l a is the lamination length.x denotes the three-phase stator windings.N c is the number of winding turns per coil and a is the number of parallel-circuits per phase.III.F INITE E LEMENT V ALIDATIONIn this section,the ﬁnite element (FE)method is employed to validate the analytical model and the FE software is Ansoft Maxwell 15.4-pole/6-slot surface-mounted PM motors with different pole shapes are taken as examples to test the accuracy of the proposed analytical model.The main motor parameters are listed in Table I.The highest harmonic numberconsideredFig.6.Air-gap ﬂux density variation with discrete magnet number N .TABLE IIC ALCULATION T IME OF THE T WO M ETHODSin analytical model K is 40and the number of assumed parts per pole N is 30.Waveforms of air-gap ﬂux density,cogging torque,and back EMF derived by FE and analytical methods are compared in Figs.3–5.As can be seen,all the analytical predictions well illustrate the effects of pole parameters on performance and have excel-lent agreement with the FE results,which veriﬁes the accuracy and feasibility of the proposed method.Besides,the air-gap length of SA conﬁguration changes with circumferential position,while that of SB is still even.Therefore,under the same motor parameters,the equivalent air-gap length of SA is bigger than that of SB.The former has a much smaller root mean square value of ﬂux density,but a better distribution than the latter,which result in lower cogging torque and more sinusoidal back EMF waveform.Since results of the analytical method are calculated by adding N subdomain models with even thickness magnets,the magnetic ﬁeld solution is affected by the parameter N .In Fig.6,the air-gap ﬂux density variations are given when N is increased from 1to 50.As can be seen,the results stay unchanged when N is larger than 10,which indicates that the analytical method is stable if N is large enough.In Table II,the time consumed by analytical method and FE method is compared when N is 30.It shows that the analytical method is much faster than FE method,which can improve the efﬁciency of motor design and analysis.IV.E FFECT OF M AGNET D IMENSIONSON M OTOR B EHAVIORBased on the proposed analytical model in (14)and (15),effects of magnet dimensions on motor behavior are inves-tigated.Variations of cogging torque T c ,fundamental com-ponent amplitude of back EMF E 0,and its total harmonic distortion (THD)with h m and d are shown in Figs.7–9.Some conclusions can be drawn from the above results.1)In Fig.7,cogging torque decreases with either smaller h m or larger d .But the difference is that cogging torque per unit volume continues to decrease when d increases,8102804IEEE TRANSACTIONS ON MAGNETICS,VOL.50,NO.11,NOVEMBER2014Fig.7.Cogging torque variation with h m and d .(a)SA.(b)SB.Fig.8.Back EMF amplitude variation with h m and d .(a)SA.(b)SB.Fig.9.Back EMF THD variation with h m and d .(a)SA.(b)SB.while it increases when h m decreases.This illuminates that cogging torque cannot be effectively eliminated by reducing magnet thickness.Applying the magnet pole shape optimization method would be more useful.2)Like cogging torque,back EMF amplitude gets smaller with the decrease of h m or the increase of d as shown in Fig.8.Fortunately,the contribution of per unit volume increases,while the magnitude of back EMF gets lower.This implies that pole shape optimization can improve the utilization of PM and save the cost of PM machines.3)Apart from the amplitude of back EMF,harmonics is also a focus in PM motor analysis.In Fig.9,the THD of back EMF can hardly vary with h m .But it decreases evidently with the increase of d ,which suggests that higher performance of PM motor could be achieved by employing SA and SB pole conﬁgurations since they can not only eliminate cogging torque,but also reduce back EMF harmonics.4)In addition,although SA and SB conﬁgurations have similar effects on motor behavior,SA is more sensitive to d than SB.This is because for SB conﬁguration,only pole shape is changed,while its air-gap distribution is still uniform.But for SA conﬁguration,it has not only a sinusoidal magnetic pole,but also an unevenlydistributed air-gap,which will enhance the impact of parameter d .V.C ONCLUSIONEccentric magnetic pole is an effective approach to improv-ing PM motor design.However,uneven thickness of magnetic pole also increases the difﬁculty of analytical modeling.In this paper,an accurate analytical method is proposed by modifying subdomain model method with discrete idea.With the proposed models,the effects of PM dimensions,h m and d ,on motor behavior are investigated.This paper shows that h m directly affects the magnitude of ﬂux density and further affects the magnitudes of cogging torque and back pared with h m ,parameter d mainly affects the spatial distribution of ﬂux density rather than changing its rger d would produce better ﬂux density distribution,result-ing in smaller cogging torque and more sinusoidal back EMF waveform though it also somewhat decreases its magnitude.Thus,a balance between design requirements,ensuring motor electromagnetic performance and meanwhile reducing back EMF THD and torque ripples,may be achieved by choosing appropriate pole dimensions.A CKNOWLEDGMENTThis work was supported in part by the National Key Basic Research Program of China (973Program)under Grant 2013CB035602and in part by the Key Program,National Natural Science Foundation of China,under Grant 51037004.R EFERENCES[1]Z.Q.Zhu and D.Howe,“Electrical machines and drives for electric,hybrid,and fuel cell vehicles,”IEEE Trans.Magn.,vol.44,no.1,pp.52–65,Jan.2008.[2] C.C.Chan,“The state of the art of electric and hybrid vehicles,”Proc.IEEE ,vol.90,no.2,pp.247–275,Feb.2002.[3]M.R.Dubois,H.Polinder,and J.A.Ferreira,“Magnet shaping forminimal magnet volume in machines,”IEEE Trans.Magn.,vol.38,no.5,pp.2985–2987,Sep.2002.[4]M.R.Dubois,H.Polinder,and J. A.Ferreira,“Contribution ofpermanent-magnet volume elements to no-load voltage in machines,”IEEE Trans.Magn.,vol.39,no.3,pp.1784–1791,May 2003.[5]Y .Pang,Z.Q.Zhu,and Z.J.Feng,“Cogging torque in cost-effectivesurface-mounted permanent magnet machines,”IEEE Trans.Magn.,vol.47,no.9,pp.2269–2276,Sep.2011.[6]skaris and A.G.Kladas,“Permanent-magnet shape optimizationeffects on synchronous motor performance,”IEEE Trans.Ind.Electron.,vol.58,no.9,pp.3776–3783,Sep.2011.[7] B.L.J.Gysen,K.J.Meessen,J.J.H.Paulides,and E.A.Lomonva,“General formulation of the electromagnetic ﬁeld distribution in machines and devices using Fourier analysis,”IEEE Trans.Magn.,vol.46,no.1,pp.39–52,Jan.2010.[8]J.De La Ree and N.Boules,“Magnet shaping to reduce induced voltageharmonics in PM machines with surface mounted magnets,”IEEE Trans.Energy Convers.,vol.6,no.1,pp.155–161,Mar.1991.[9]S.M.Jang,H.Park,J.Y .Choi,K.J.Ko,and S.H.Lee,“Magnetpole shape design of permanent magnet machine for minimization of torque ripple based on electromagnetic ﬁeld theory,”IEEE Trans.Magn.,vol.47,no.10,pp.3586–3589,Oct.2011.[10]Y .B.Yang,X.H.Wang,C.Q.Zhu,and C.Z.Huang,“Reducingcogging torque by adopting isodiametric permanent magnet,”in Proc.4th IEEE ICIEA ,May 2009,pp.1028–1031.[11]Z.Q.Zhu,L.J.Wu,and Z.P.Xia,“An accurate subdomain modelfor magnetic ﬁeld computation in slotted surface-mounted perma-nent machines,”IEEE Trans.Magn.,vol.46,no.4,pp.1100–1115,Apr.2010.。

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表贴式永磁同步轮毂电机磁场分析与优化设计
表贴式永磁同步轮毂电机磁场分析与优化设计
摘要：表贴式永磁同步轮毂电机是一种新型的电动机，具有结构紧凑、功率密度高、拆装方便等优点，在现代交通工具中得到了广泛的应用。

本文通过对表贴式永磁同步轮毂电机的磁场进行分析与优化设计，进一步提高其运行效率和性能。

1. 引言
随着电动汽车的普及，对电动机的性能要求不断提高。

表贴式永磁同步轮毂电机由于其结构紧凑，可以直接连接轮毂，不需要传动装置，因此具有较高的功率密度，适用于电动汽车等场景。

本文针对表贴式永磁同步轮毂电机的磁场进行分析与设计优化，以提高电机的效率和性能。

2. 表贴式永磁同步轮毂电机的磁场分析
表贴式永磁同步轮毂电机的磁场分析是设计优化的基础。

该电机的主要磁场由定子的励磁电流和永磁体的磁场组成。

在磁场分析中，可以采用有限元法进行模拟和计算。

2.1 定子励磁电流的磁场分析
表贴式永磁同步轮毂电机的定子通常由多个绕组组成，每个绕组中的电流都会产生磁场。

通过有限元法建立定子的磁场模型，可以计算出各个绕组的励磁磁场分布情况。

2.2 永磁体的磁场分析
永磁体是表贴式永磁同步轮毂电机中的重要组成部分，它产生的磁场直接影响电机的性能。

通过有限元法对永磁体进行磁场分析，可以得到永磁体的磁场分布情况，进一步确定电机的磁场分布情况和磁场强度。

3. 表贴式永磁同步轮毂电机的磁场优化设计
在磁场分析的基础上，可以进行磁场的优化设计，以提高电机的效率和性能。

主要的优化设计包括：
3.1 绕组设计优化
通过优化绕组的布局和参数选择，可以降低电阻和电磁损耗，提高电机的效率。

一般可以通过调整绕组的截面积、匝数等参数，来优化绕组的设计。

3.2 永磁体设计优化
永磁体的设计对电机的性能影响较大。

通过优化永磁体的材料选择和形状设计，可以提高电机的磁场强度和磁能密度，从而提高电机的输出功率和效率。

3.3 磁路设计优化
在表贴式永磁同步轮毂电机中，磁路设计的合理性直接影响电机的性能。

通过优化磁路的材料选择和结构设计，可以降低磁路的磁阻，提高电机的磁场强度和转矩输出。

4. 结论
本文通过对表贴式永磁同步轮毂电机的磁场分析与优化设计，提高了电机的运行效率和性能。

在磁场分析中，采用有限元法可以模拟和计算定子励磁电流和永磁体的磁场分布情况。

在优化设计中，通过调整绕组的布局和参数，优化永磁体的设计以及改进磁路的结构，可以提高电机的输出功率和效率。

通过这些优化设计，表贴式永磁同步轮毂电机在电动汽车等应用中具有更好的性能表现。

通过磁场分析与优化设计，我们成功提高了表贴式永磁同步轮毂电机的运行效率和性能。

在磁场分析中，我们采用有限元法模拟定子励磁电流和永磁体的磁场分布情况。

在优化设计中，我们通过调整绕组的布局和参数，优化永磁体的设计以及
改进磁路的结构，有效提高了电机的输出功率和效率。

通过这些优化设计，表贴式永磁同步轮毂电机在电动汽车等应用中将表现出更好的性能。

这些优化设计的结果将为电机工程师们提供指导，使他们能够更好地应用和改进电机技术，推动电动汽车的发展。