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## Magnetic field and inductance of filament conductor segment model with current continuity

Cui Xiang
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• #### 摘要

传统的载流细导体段模型是分析导体闭合回路磁场的基本模型, 尽管不满足电流连续性定律, 但适用于导体闭合回路的磁场分析. 然而, 对于工程中只关注导体闭合回路中某一局部的多分支导体段并联的电流分配问题, 传统模型将不能完整地反映各分支导体段之间磁场的相互作用. 为此, 现有文献提出的位移电流模型, 满足了电流连续性定律, 较好地解决了上述问题, 但是, 仍然存在理论不完整、不自洽以及计算公式复杂等问题. 本文提出载流细导体段的传导电流模型, 确保了载流细导体段在段内、段端及段外的电流连续性. 推导出物理内涵更加深刻的总磁场微分方程和矢量磁位计算公式. 提出载流细导体段传导电流模型磁场能量和电感的计算公式, 极大地降低了计算复杂度, 弥补了现有文献的不足. 本文算例从模型、公式、计算等方面验证了本文理论和计算公式的正确性.

#### Abstract

Traditional current-carrying model of filament conductor segment is the basic model for analyzing the magnetic field in a closed conductor loop. Although it does not satisfy the law of current continuity, it is still suitable for analyzing the magnetic field in the closed conductor loop. However, for the current distribution problem of some local multi-branch parallel-connection conductor segments in a closed conductor loop, the traditional model cannot fully reflect the interaction between magnetic fields of the local multi-branch parallel-connection conductor segments. For this reason, the displacement current model proposed in the existing literature satisfies the law of current continuity and solves the above problem well. But, there remain some problems to be solved, such as incomplete theory, inconsistency and complex formulae in the existing literature. In this paper, a conductive current-carrying model of filament conductor segment is proposed, which ensures current continuity inside, outside, and on two terminals of the filament conductor segment. The differential equation of total magnetic field with more profound physical connotation and the formulae of vector magnetic potential are derived in detail. Further, the formulae for calculating magnetic field energy and inductance of the conductive current-carrying model of filament conductor segment are proposed, which greatly reduces the computational complexity and makes up for the shortcoming of model in the existing literature. An example given in this paper verifies the correctness of the above theory and formulae.

#### 作者及机构信息

###### 通信作者: 崔翔, x.cui@ncepu.edu.cn
• 基金项目: 国家级-国家自然科学基金委员会-国家电网公司智能电网联合基金(Grant No. U1766219)

#### 参考文献

 [1] Stratton J A 1941 Electromagnetic Theory (New York: McGraw-Hill Book Company) p232 [2] Ogura K, Steinmetz C P 1907 Phys. Rev. 25 184 [3] Rosa E B 1908 Bulletin of the Bureau of Standards 4 301Google Scholar [4] Grover F W 1947 Inductance Calculations: Working Formulas and Tables (New York: D. Van Nostrand Company Inc.) pp31−247 [5] Kaiser K L 2005 Electromagnetic Compatibility Handbook (New York: CRC Press) Chapter 15 pp168−204 [6] Paul C R 2011 Inductance: Loop and Partial (New York: John Wiley & Sons) pp117−306 [7] Ni C W, Zhao Z B, Cui X 2018 IEEE Trans. Electromagn. Compat. 60 803Google Scholar [8] Wakeman F, Li G, Golland A 2005 Proceedings of PCIM Europe Nuremberg, Germany, June 7−9, 2005 p137 [9] Ruehli A E 1972 IBM J. Res. Develop. 16 470Google Scholar [10] Holloway C L, Kuester E F, Ruehli A E, Antonini G 2013 IEEE Trans. Electromagn. Compat. 55 600Google Scholar [11] Kamon M, Tsuk M, White J 1994 IEEE Trans. Microw. Theory Tech. 42 1750Google Scholar [12] FastHenry Package http://www.fastfieldsolvers.com/links.htm [2019-10-12] [13] Ruehli A E, Antonini G, Esch J, Ekman J, Mayo A, Orlandi A 2003 IEEE Trans. Electromagn. Compat. 45 167Google Scholar [14] Yu W J, Yan C H, Wang Z Y 2007 Eng. Anal. Boundary Elem. 31 812Google Scholar [15] Holloway C L, Kuester E F 2009 IEEE Trans. Electromagn. Compat. 51 338Google Scholar [16] Antonini G, Orlandi A, Paul C R 1999 IEEE Trans. Microw. Theory Tech. 47 979Google Scholar [17] Ni C W, Zhao Z B, Cui X 2018 IEEE Trans. Magn. 53 1Google Scholar [18] Kalhor H A 1988 IEEE Trans. Educ. 31 236Google Scholar [19] 倪筹帷 2018 博士学位论文 (北京: 华北电力大学) Ni C W 2018 Ph. D. Dissertation (Beijing: North China Electric Power University) (in Chinese) [20] Kalhor H A 1990 IEEE Trans. Educ. 33 2365Google Scholar [21] 倪筹帷, 赵志斌, 崔翔 2017 中国电机工程学报 37 5181Google Scholar Ni C W, Zhao Z B, Cui X 2017 Proc. CSEE 37 5181Google Scholar

#### 施引文献

• 图 1  传统的载流细导体段模型　(a) 单个载流细导体段; (b) 两个载流细导体段

Fig. 1.  Classical current models of filament conductor segments: (a) One filament conductor segment; (b) two filament conductor segments.

图 2  多导体段并联及其等效电路　(a) 三导体段并联[7]; (b) 压接型IGBT器件内部多凸台芯片并联[8]; (c) 多导体段并联的等效电路

Fig. 2.  Multiconductor parallel connection and its equivalent circuit: (a) Three conductors in parallel connection[7]; (b) multi-protrusion chips parallel connection in Press-Pack IGBTs[8]; (c) equivalent circuit for multiconductor parallel connection.

图 3  载流细导体段的传导电流模型

Fig. 3.  Conductive current model of filament conductor segment.

图 4  两根细导体段

Fig. 4.  Two filament conductor segments.

图 5  两根细导体段不同端点之间的距离

Fig. 5.  Distances between different terminals of two filament conductor segments.

图 6  两根半径相同的圆柱截面的平行细导体段[19]

Fig. 6.  Two parallel filament conductor segments with same length and radius[19].

图 7  两根平行细导体段自电感与互电感[19]　(a) 单位长度自电感; (b) 单位长度互电感

Fig. 7.  Inductances of two parallel filament conductor segments[19]: (a) Unit-length self-inductance; (b) unit-length mutual-inductance.

•  [1] Stratton J A 1941 Electromagnetic Theory (New York: McGraw-Hill Book Company) p232 [2] Ogura K, Steinmetz C P 1907 Phys. Rev. 25 184 [3] Rosa E B 1908 Bulletin of the Bureau of Standards 4 301Google Scholar [4] Grover F W 1947 Inductance Calculations: Working Formulas and Tables (New York: D. Van Nostrand Company Inc.) pp31−247 [5] Kaiser K L 2005 Electromagnetic Compatibility Handbook (New York: CRC Press) Chapter 15 pp168−204 [6] Paul C R 2011 Inductance: Loop and Partial (New York: John Wiley & Sons) pp117−306 [7] Ni C W, Zhao Z B, Cui X 2018 IEEE Trans. Electromagn. Compat. 60 803Google Scholar [8] Wakeman F, Li G, Golland A 2005 Proceedings of PCIM Europe Nuremberg, Germany, June 7−9, 2005 p137 [9] Ruehli A E 1972 IBM J. Res. Develop. 16 470Google Scholar [10] Holloway C L, Kuester E F, Ruehli A E, Antonini G 2013 IEEE Trans. Electromagn. Compat. 55 600Google Scholar [11] Kamon M, Tsuk M, White J 1994 IEEE Trans. Microw. Theory Tech. 42 1750Google Scholar [12] FastHenry Package http://www.fastfieldsolvers.com/links.htm [2019-10-12] [13] Ruehli A E, Antonini G, Esch J, Ekman J, Mayo A, Orlandi A 2003 IEEE Trans. Electromagn. Compat. 45 167Google Scholar [14] Yu W J, Yan C H, Wang Z Y 2007 Eng. Anal. Boundary Elem. 31 812Google Scholar [15] Holloway C L, Kuester E F 2009 IEEE Trans. Electromagn. Compat. 51 338Google Scholar [16] Antonini G, Orlandi A, Paul C R 1999 IEEE Trans. Microw. Theory Tech. 47 979Google Scholar [17] Ni C W, Zhao Z B, Cui X 2018 IEEE Trans. Magn. 53 1Google Scholar [18] Kalhor H A 1988 IEEE Trans. Educ. 31 236Google Scholar [19] 倪筹帷 2018 博士学位论文 (北京: 华北电力大学) Ni C W 2018 Ph. D. Dissertation (Beijing: North China Electric Power University) (in Chinese) [20] Kalhor H A 1990 IEEE Trans. Educ. 33 2365Google Scholar [21] 倪筹帷, 赵志斌, 崔翔 2017 中国电机工程学报 37 5181Google Scholar Ni C W, Zhao Z B, Cui X 2017 Proc. CSEE 37 5181Google Scholar
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##### 出版历程
• 收稿日期:  2019-08-09
• 修回日期:  2019-10-30
• 刊出日期:  2020-02-05

## 电流连续的细导体段模型的磁场及电感

• 华北电力大学, 新能源电力系统国家重点实验室, 北京　102206
• ###### 通信作者: 崔翔, x.cui@ncepu.edu.cn
基金项目: 国家级-国家自然科学基金委员会-国家电网公司智能电网联合基金(Grant No. U1766219)

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