本研究提出模擬電動馬達(EM)熱場特性之工程方法，特別關注其在穩態和變載荷/變速情況下定子繞組平均溫度與馬達各構件最高溫度間之關聯性，進而將構件最高溫度回歸成定子繞組線圈電阻率之函數式，藉由馬達電控系統估算定子繞組線圈瞬時電阻率，換算成馬達各構件之最高溫度，將上述工程方法(簡稱電阻法)植入馬達電控系統，進行過熱防護。本熱場模擬方法改進線圈繞組之非均勻熱模型，除評估繞組槽充填率和銅線、浸漬材料之熱導率外，並依據繞組軸向兩端彎曲繞組之配置角度，計算繞組軸向兩端之等效熱導率。另外馬達中各夾層氣室空氣球溫度(fluid bulk temperature)之設定，對馬達熱場模擬之準確性扮演關鍵角色。本研究依據能量守恆原理，發展迭代全熱場模擬之方案，計算馬達中各夾層氣室之氣球溫度，此數值模型於穩態及暫態實驗條件所得結果與18組馬達內部量測溫度比對，最大差異為±7％，確認本模擬方法之準確性。由於本模擬同時探討馬達於穩態和變載荷/變速情況下之馬達熱場特徵，馬達定子與轉子間氣隙之泰勒渦流受到轉子變速及(或)變轉向之影響，改變其對流熱傳係數，遂進行實驗研究，量測轉子變速及(或)變轉向之泰勒渦流對流熱傳係數，並彙整熱傳實驗數據，推導其紐賽數(Nusselt number)之實驗公式，整合至模擬馬達熱場之模型，計算馬達於變轉速(轉向)之暫態熱場，評估上述電阻法之可行性。研究結果發現，轉子變速及(或)變轉向之泰勒渦流對流熱傳係數隨轉子角加速度變化之情形，與一階自由度動態系統特徵類似，於轉子臨界角加速度，出現紐賽數峰值，當轉子角加速度偏離此臨界角加速度，泰勒渦流紐賽數隨之降低。彙整各種穩態/非穩態條件所得之熱場模擬結果，馬達各構件之最高溫度均與定子繞組線圈平均溫度及其電阻率呈收斂之線性相關，進而以定子繞組線圈電阻率為控制參數，推導出估算馬達各構件最高溫度之回歸公式，提供台達電運用，實現藉由量測馬達電氣訊號進行各構件最高溫度線上監測之可能性。

The present study proposes a simulation method for predicting the thermal field of an electric motor with the focus to unravel the correlative relationship between the maximum component temperatures (MCT) and the mean temperature of stator winding (MTSW) in the operating conditions with constant/variable loads/speeds. By converting the MTSW into the resistivity of the copper coil (RCC), the MCTs are correlated into the functions of RCC in stator winding. With the instant RCC of the stator winding measured by the controlled unit of an electric motor, the instant MCTs are acquired using the MCT versus RCC correlations devised by present study for overheat protection. The abovementioned engineering approach is referred to as the RCC method.
The present simulation method refines the non-homogenous thermal conductivity model for the composite components in an electric motor. In addition to the fill ratio of copper wires in a slot, material properties of copper wire and impregnated materials, the effect of angular location on the effective thermal conductivity of the stator windings at their two axial ends is included in the model. As the precision of the air bulk temperature in each air chamber of an electric motor plays a significant role for simulation accuracy, the present study develops an iterative scheme that complies with the energy conservation principle for calculating the air bulk temperatures and the convective heat transfer coefficients which vary with wall-to-fluid temperature differences. Compared with the eighteen thermocouple measurements installed in the motor with the steady and unsteady operating conditions, the maximum discrepancies between the measured and predicted temperatures are less than +7％, leading to the accuracy assurance of the present numerical model.
As the present study also explores the applicability of the RCC method in the operating conditions with varying loads/speeds, the influence of varying rotational speed/direction on the heat convection of the Taylor vortices in the annual air gap between stator and rotor is experimentally studied. The experimental study detects the convective heat transfer coefficients of the Taylor vortices under the test condition with varying rotational speed/direction of the rotor. The measured Nusselt numbers is correlated into the empirical correlation that integrated with the present simulation model for predicting the thermal fields of the electric motor with varying rotational speed/direction. The simulation results acquired in these unsteady conditions also confirm the applicability of the RCC method for predicting the MCTs. The research results reveal the heat transfer impacts on Taylor vortices due to the varying rotational speed/direction of the rotor. The heat transfer characteristic of the Taylor vortices with varying rotational speed/direction of rotor are similar to that with a SDOF dynamic system. The Nusselt number peak of the Taylor vortices emerges at a critical rotational acceleration. Apart from the critical rotational acceleration, the Nusselt numbers of the Taylor vortices are decayed from the Nusselt number peak. Based on all the simulation results in the steady and unsteady conditions, all the MCTs are linearly correlated with MTSW and hence RCC. With RCC as the controlling parameter, a set of MCT correlations are generated for predicting MCTs using the on-line measured RCC to enhance the practical applications of Delta INC. The engineering strategy for on-line monitoring the MCTs of an electric motor based on the measured RCC by the controller is accomplished.

[1] X. Wang, B. Li, D. Gerada, K. Huang, I. Stone, S. Worrall, Y. Yan, A critical review on thermal management technologies for motors in electric cars, Applied Thermal Engineering 201 (2022) 117758.
[2] M.-S. Kim, K.-S. Lee, S. Um, Numerical investigation and optimization of the thermal performance of a brushless DC motor, Int. J. Heat Mass Transfer 52 (2009) 1589-1599
[3] M. Fénot, Y. Bertin, E. Dorignac, G. Lalizel, A review of heat transfer between concentric rotating cylinders with or without axial flow, Int. J. Thermal Sciences 50 (2011) 1138-1155.
[4] D.A. Howey, P.R.N. Childs, A.S. Holmes, Air-gap convection in rotating electrical machines, IEEE Transactions on Industrial Electronics 59 (2012) 1367-1375.
[5] M.L. Hosain, R.B. Fdhila, K. Rönnberg, Taylor-Couette flow and transient heat transfer inside the annulus air-gap of rotating electrical machines, Applied Energy 207 (2017) 624-633.
[6] C. Kim, K.-S. Lee, Numerical investigation of the air-gap flow heating phenomena in large-capacity induction motors, Int. J. Heat and Mass Transfer 110 (2017) 746-752.
[7] P.-S. Wu, M.-F. Hsieh, W. L. Cai, J.-H. Liu, Y.-T. Huang, J.F. Caceres, S.W. Chang, Heat transfer and thermal management of interior permanent magnet synchronous electric motor, Inventions 4 (2019) 4040069.
[8] N. Simpson, R. Wrobel, P.H. Mellor, Estimation of equivalent thermal parameters of impregnated electrical windings, IEEE Transactions on Industry Applications 49 (2013) 2505-2515.
[9] A. Boglietti, E. Carpaneto, M. Cossale, S. Vaschetto, M. Popescu and D. Staton, Stator winding thermal conductivity evaluation: An industrial production assessment, 2015 IEEE Energy Conversion Congress and Exposition (ECCE), Montreal, QC, Canada, 2015, pp. 4865-4871.
[8] J.-F. Trigeol, Y. Bertin, P. Lagonotte, Thermal modeling of an induction machine through the association of two numerical approaches, IEEE Transactions on Energy Conversion 21 (2006) 314-323.
[9] A. Boglietti, A. Cavagnino, D. Staton, Determination of critical parameters in electrical machine thermal models, IEEE Transactions on Industry Applications 44 (2008) 1150-1159.
[10] C. Kral, A. Haumer, T. Bauml, Thermal model and behavior of a totally-enclosed-water-cooled squirrel-cage induction machine for traction applications, IEEE Transactions on Industrial Electronics 55 (2008) 3555-3565.
[11] S. Nategh, O. Wallmark, M. Leksell, S. Zhao, Thermal analysis of a PMaSRM using partial FEA and lumped parameter modeling, IEEE Transactions on Energy Conversion 27 (2012) 477-488.
[12] S. Nategh, Z. Huang, A. Krings, O. Wallmark, M. Leksell, Thermal modeling of directly cooled electric machines using lumped parameter and limited CFD analysis, IEEE Transactions on Energy Conversion 28 (2013) 979-990.
[13] C. Kral, A. Haumer, S.B. Lee, A practical thermal model for the estimation of permanent magnet and stator winding temperatures, IEEE Transactions on power electronics 29 (2014) 455-464.
[14] Y. Alexandrova, R.S. Semken, J. Pyrhönen, Permanent magnet synchronous generator design solution for large direct-drive wind turbines Thermal behavior of the LC DD-PMSG, Applied Thermal Engineering 65 (2014) 554-563.
[15] H. Li and Y. Shen, Thermal analysis of the permanent-magnet spherical motor, IEEE Transactions on Energy Conversion 30 (2015) 991-998.
[16] X. Hao, B. Peng, Y. Chen, G. Xie, Transient thermal model of a permanent magnet synchronous planar motor considering spreading thermal resistance, Applied Thermal Engineering 81 (2015) 1-9.
[17] J.A. Malumbres, M. Satrustegui, I. Elosegui, J.C. Ramos, M. Martínez-Iturralde, Analysis of relevant aspects of thermal and hydraulic modeling of electric machines. Application in an open self ventilated machine, Applied Thermal Engineering 75 (2015) 277-288.
[18] X. Huang, Q. Tan, L. Li, J. Li and Z. Qian, Winding temperature field model considering void ratio and temperature rise of a permanent-magnet synchronous motor with high current density, IEEE Transactions on Industrial Electronics 64 (2017) 2168-2177.
[19] C. Kim, K.-S. Lee, Thermal nexus model for the thermal characteristic analysis of an open-type air-cooled induction motor, Applied Thermal Engineering 112 (2017) 1108-1116.
[20] V. Madonna, A. Walker, C. Gerada, M. Galea, Improved thermal management and analysis for stator end-windings of electrical machines IEEE Transactions on Industrial Electronics 66 (2018) 5057-5069.
[21] I. Smolyanov, F. Sarapulov, F. Tarasov, Calculation of linear induction motor features by detailed equivalent circuit method taking into account non-linear electromagnetic and thermal properties, Computers & Mathematics with Applications 78 (2019) 3187-3199.
[22] M. Cavazzuti, G. Gaspari, S. Pasquale, E. Stalio, Thermal management of a Formula E electric motor: Analysis and optimization, Applied Thermal Engineering 157 (2019) 113733.
[23] D. Wang, Y. Liang, C. Li, P. Yang, C. Zhou, L. Gao, Thermal equivalent network method for calculating stator temperature of a shielding induction motor, Int. J. Thermal Sciences 147 (2020) 106149.
[24] X. Chen, J. Wang, A. Griffo, A. Spagnolo, Thermal modeling of hollow conductors for direct cooling of electrical machines, IEEE Transactions on Industrial Electronics 67 (2020) 895-905.
[25] X. Sun and M. Cheng, Thermal analysis and cooling system design of dual mechanical port machine for wind power application, IEEE Transactions on Industrial Electronics 60 [2013] 1724-1733.
[26] Z. Xu, M. Galea, C. Tighe, T. Hamiti, C. Gerada, S.J. Pickering, Mechanical and thermal management design of a motor for an aircraft wheel actuator, 17th International Conference on Electrical Machines and Systems (ICEMS), Oct. 22-25, 2014, Hangzhou, China.
[27] X. Liu, H. Yu, Z. Shi, L. Huang, T. Xia, R. Guo, Porous metal model for calculating slot thermal conductivity coefficient of electric machines, Applied Thermal Engineering 111 (2017) 981-988.
[28] X. Liu, H. Yu, Z. Shi, T. Xia, M. Hu, Electromagnetic-fluid-thermal field calculation and analysis of a permanent magnet linear motor, Applied Thermal Engineering 129 (2018) 802-811.
[29] Y. Xu, Y. Jia, M. Ai, Y. Wang, Heat transfer characteristics of external ventilated path in compact high-voltage motor, Int. J. Heat Mass Transfer 124 (2018) 1136-1146.
[30] B. Melka, J. Smolka, J. Hetmanczyk, Z. Bulinski, D. Makiela, A. Ryfa, Experimentally validated numerical model of thermal and flow processes within the permanent magnet brushless direct current motor, Int. J. Thermal Sciences 130 (2018) 406-415.
[31] Z. Rehman, K. Seong, Three-D numerical thermal analysis of electric motor with cooling jacket, Energies 11 (2018) 11010092.
[32] Y. Sun, S. Zhang, W. Yuan, Y. Tang, J. Li, K. Tang, Applicability study of the potting material based thermal management strategy for permanent magnet synchronous motors, Applied Thermal Engineering 149 (2019) 1370-1378.
[33] W. Chen, Y. Ju, D. Yan, L. Guo, Q. Geng, T. Shi, Design and optimization of dual-cycled cooling structure for fully-enclosed permanent magnet motor, Applied Thermal Engineering 152 (2019) 338-349.
[34] B. Melka, J. Smolka, J. Hetmanczyk, P. Lasek, Numerical and experimental analysis of heat dissipation intensification from electric motor, Energy 182 (2019) 269-279.
[35] Y. Sun, S. Zhang, G. Chen, Y. Tang, F. Liang, Experimental and numerical investigation on a novel heat pipe based cooling strategy for permanent magnet synchronous motors, Applied Thermal Engineering 170 (2020) 114970.
[36] P.-S. Wu, M.-F. Hsieh, Y.E. Lu, W.L. Cai, S.W. Chang, Thermal performance improvement by rotating thermosyphon loop in rotor of an interior permanent magnet synchronous electric motor, Inventions 7 (2022) 7020037.
[37] S.D. Wilson, P. Stewart, B.P. Taylor, Methods of resistance estimation in permanent magnet synchronous motors for real-time thermal management, IEEE Transaction on energy conversion 25 (2010) 698-707.
[38] E.C. Cobb, O.A. Saunders, Heat transfer from a rotating disc, Proc. R. Soc. Lond. Ser. A 220 (1956) 343-351.
[39] W.M. Kays, I.S. Bjorklund, Heat transfer from a rotating cylinder with and with and without cross flow. Trans. ASME Ser. C 80 (1958) 70-78.
[40] S.W. Churcill and H.H.S. Chu, Correlation equatious for laminar and turbulent free convection from a vertical plant, Int. J. Heat Mass Transfer, 18 (1975) 1323-1329.
[41] E. Radziemska, W.M. Lewandowski, Heat transfer by natural convection from an isothermal downward-facing round plate in unlimited space, Applied Energy 68 (2001) 347-366.
[42] R.C. Birkebak and A. Abdulkadir, Heat Transfer by natural convection from the lower side of a finite horizontal heated surface, 4th Int. Heat Transfer Conference, Paris, France, August 31 - September 5, 1970, IHTC4.3450 1-10.
[43] S. Churchill, H. Chu, Correlating equations for laminar and turbulent free convection from a horizontal cylinder. Int. J. Heat Mass Transfer 18 (1975) 1049–1053.
[44] M. Avila, Stability and angular-momentum transport of fluid flows between corotating cylinders, Physical Review Letters, 108 (2012), 124501.
[45] X.-Y. Leng, J.-Q. Zhong, Aspect-ratio dependence of heat and angular momentum transport in turbulent Taylor-Couette flows with axial thermal forcing, Int. J. Heat Mass Transfer, 195 (2022) 123194.
[46] S.J. Kline, F.A. McClintock, Describing uncertainties in single sample experiments, Mech. Eng. 75 (1953) 3-8.