Design and Optimization of a Multi-Coil Structure for Dual-Zone Uniform Magnetic Fields
Liu Xufei1, Fu Jiqing2, Peng Lisha1, Huang Songling1,3
1. Department of Electrical Engineering Tsinghua University Beijing 100084 China; 2. National Institute of Metrology Beijing 100029 China; 3. State Key Laboratory of Power System Operation and Control Beijing 100084 China
Abstract：This study presents a novel coil design and optimization approach for generating high-uniformity magnetic fields in two spatially separated regions, addressing the limitations of conventional coil systems in compact, multi-sensor calibration scenarios. Traditional Helmholtz and modified multi-coil configurations often require large physical dimensions to maintain field uniformity over extended volumes. When scaled down, they tend to suffer from performance degradation and electromagnetic crosstalk, particularly when used for simultaneous calibration of multiple magnetometers. To resolve these issues, we propose a dual-zone magnetic field system based on a coaxial seven-segment coil structure, specifically optimized to generate two discrete yet highly uniform magnetic field regions with minimized interference and reduced footprint. The optimization method combines a continuous nonlinear solver with a mixed-integer programming strategy. By jointly adjusting the axial positions and relative number of turns in each coil segment, the system minimizes high-order magnetic field nonuniformity in the target regions. The optimization is guided by a Taylor-series-based field analysis that focuses on reducing spatial derivatives of the magnetic field near the center, thereby directly improving uniformity. A continuous optimization is first performed under physical constraints such as coil spacing, overall length, and region separation. The resulting fractional solutions are then discretized into feasible integer turns, followed by local adjustment to refine uniformity. Simulation results demonstrate the effectiveness of the proposed method across varying coil segment counts, with seven segments offering the best tradeoff between design complexity and field performance. The dual-zone system achieves comparable or slightly lower field uniformity than an optimized single-zone six-coil system, but it uniquely enables two 60 mm-long uniform zones spaced more than 70 cm apart. This dual-region capability is particularly advantageous for calibrating multiple sensors simultaneously, minimizing crosstalk and enabling more efficient workflows. Finite-element simulation using Ansys Maxwell confirms the analytical optimization results, validating the coil positions and field distribution. The designed system maintains field uniformity within 4.8×10-5 across each 60 mm zone. Sensitivity analysis further indicates that axial coil placement tolerances within ±0.01 mm are essential for preserving high uniformity, while deviations of ±0.1 mm can degrade performance by an order of magnitude, highlighting the critical importance of precise mechanical fabrication. To validate the design experimentally, a physical prototype was fabricated by the National Institute of Metrology. The device, measuring approximately 1.3 m by 0.6 m, was constructed with axial placement precision better than 10 µm. Field measurements show a uniformity of 9×10-6 over a 6 cm scanning range in both regions, closely aligning with simulation predictions and confirming the effectiveness of the dual-zone design under real-world conditions. In summary, the proposed method provides a practical and effective solution for dual-zone uniform magnetic field generation in compact devices. The combination of structured coil segmentation and hybrid optimization achieves field performance previously limited to larger systems, with potential applications in multi-sensor calibration, low-field magnetic experiments, and portable magnetic measurement platforms. Future work will explore alternative winding geometries and global optimization techniques to further enhance robustness and manufacturing flexibility.
刘旭菲, 伏吉庆, 彭丽莎, 黄松岭. 异区均匀磁场线圈结构的优化设计[J]. 电工技术学报, 2026, 41(6): 1817-1827.
Liu Xufei, Fu Jiqing, Peng Lisha, Huang Songling. Design and Optimization of a Multi-Coil Structure for Dual-Zone Uniform Magnetic Fields. Transactions of China Electrotechnical Society, 2026, 41(6): 1817-1827.
[1] 董庆亮, 上飞飞, 刘翀, 等. 海洋磁力仪绝对准确度的测试与分析[J]. 海洋测绘, 2021, 41(6): 73-77. Dong Qingliang, Shang Feifei, Liu Chong, et al.The test and analysis of the marine magnetometer absolute accuracy[J]. Hydrographic Surveying and Charting, 2021, 41(6): 73-77. [2] 裴彦良, 梁瑞才, 刘晨光, 等. 海洋磁力仪的原理与技术指标对比分析[J]. 海洋科学, 2005, 29(12): 4-8. Pei Yanliang, Liang Ruicai, Liu Chenguang, et al.Principle of marine magnetometer and specification comparative analysis[J]. Marine Sciences, 2005, 29(12): 4-8. [3] Wang Jin, Li Guofeng, Liang Ke, et al.The theory of field parameters for Helmholtz coil[J]. Modern Physics Letters B, 2010, 24(2): 201-209. [4] 黄松涛, 张伟, 张鹏飞, 等. 基于亥姆霍兹线圈的大尺寸均匀电磁场模拟[J]. 北京航空航天大学学报, 2015, 41(2): 203-208. Huang Songtao, Zhang Wei, Zhang Pengfei, et al.Large scale uniform electromagnetic field simulation based on Helmholtz coil[J]. Journal of Beijing University of Aeronautics and Astronautics, 2015, 41(2): 203-208. [5] 汤云东, 丁宇彬, 金涛. 基于亥姆霍兹线圈装置的磁热疗优化方法[J]. 电工技术学报, 2023, 38(5): 1248-1260. Tang Yundong, Ding Yubin, Jin Tao.Research on optimization method of magnetic hyperthermia based on Helmholtz coil device[J]. Transactions of China Electrotechnical Society, 2023, 38(5): 1248-1260. [6] Thabuis A, Rivera M C, Ren Xiaotao, et al.Topology optimization of an electromagnet manipulator for 3-D uniform magnetic field generation[J]. IEEE Transactions on Magnetics, 2023, 59(5): 8000404. [7] 汤云东, 丁宇彬, 金涛. 考虑磁场均匀性优化的开放式磁粒子成像检测装置改进方法[J]. 电工技术学报, 2025, 40(6): 1718-1728. Tang Yundong, Ding Yubin, Jin Tao.Research on the improved method of open detection device for magnetic particle imaging considering magnetic field uniformity optimization[J]. Transactions of China Electrotechnical Society, 2025, 40(6): 1718-1728. [8] 刘佳琪, 王国强. 一种高度均匀磁场梯度的产生方法[J]. 仪表技术与传感器, 2024(10): 122-126. Liu Jiaqi, Wang Guoqiang.Method for generating highly uniform magnetic field gradient[J]. Instrument Technique and Sensor, 2024(10): 122-126. [9] Zhu Xuehua, Ye Juntao, Ren Ziruo, et al.Design of high uniform magnetic field with four coils based on improved wolf pack algorithm[J]. Engineering Science and Technology, an International Journal, 2025, 61: 101944. [10] Kędzia P, Czechowski T, Baranowski M, et al.Analysis of uniformity of magnetic field generated by the two-pair coil system[J]. Applied Magnetic Resonance, 2013, 44(5): 605-618. [11] Baranov P, Baranova V.Modeling axial 8-coil system for generating uniform magnetic field in COMSOL[J]. MATEC Web of Conferences, 2016, 48: 03001. [12] 李鑫, 张伟, 李艳, 等. 磁强计校准装置的建立[J]. 电测与仪表, 2012, 49(增刊1): 33-35. Li Xin, Zhang Wei, Li Yan, et al.The establishment of magnetometer calibration device[J]. Electrical Measurement & Instrumentation, 2012, 49(S1): 33-35. [13] 石铭, 王羚, 程泓勋, 等. 基于激光偏振调制的全光Cs原子磁力仪研究[J]. 红外与激光工程, 2018, 47(9): 0920001. Shi Ming, Wang Ling, Cheng Hongxun, et al.Research of all-optical Cs atomic magnetometer by using modulation of laser’s polarization[J]. Infrared and Laser Engineering, 2018, 47(9): 0920001. [14] 王士良, 邱隆清, 王永良, 等. 航空超导全张量磁梯度仪的串扰研究[J]. 低温物理学报, 2017, 39(1): 36-40. Wang Shiliang, Qiu Longqing, Wang Yongliang, et al.Study on crosstalk of an airborne magnetic full-tensor SQUID gradiometer system[J]. Chinese Journal of Low Temperature Physics, 2017, 39(1): 36-40. [15] Shifrin V Y, Chikvadze T I, Kalabin V N, et al.Transportable calibration system for scalar geomag-netometers[J]. Metrologia, 2005, 42(5): 394-399. [16] Turner R.A target field approach to optimal coil design[J]. Journal of Physics D: Applied Physics, 1986, 19(8): L147. [17] 黄清明, 陈珊珊, 张建青, 等. 基于目标场点法和流函数的磁共振有源匀场线圈设计方法[J]. 物理学报, 2019, 68(19): 268-274. Huang Qingming, Chen Shanshan, Zhang Jianqing, et al.Method of designing magnetic resonance active shimming coil based on target field point method and flow function[J]. Acta Physica Sinica, 2019, 68(19): 268-274. [18] 闫孝姮, 淡新贤, 陈伟华, 等. 基于改进目标场法梯度线圈的感应式磁声磁粒子浓度成像研究[J]. 电工技术学报, 2024, 39(14): 4305-4316. Yan Xiaoheng, Dan Xinxian, Chen Weihua, et al.Magneto-acoustic magnetic particle concentration imaging based on improved target field method gradient coil[J]. Transactions of China Electrotechnical Society, 2024, 39(14): 4305-4316. [19] Cao Qian, Liu Ying, Zhao Tian, et al.Design of highly uniform magnetic field cylinder coils based on grey wolf optimizer algorithm in atomic sensors[J]. IEEE Sensors Journal, 2021, 21(18): 19922-19929. [20] Wu Wenfeng, Zhou Binquan, Liu Zhanchao, et al.Design of highly uniform magnetic field coils based on a particle swarm optimization algorithm[J]. IEEE Access, 2019, 7: 125310-125322. [21] 吕志峰, 张金生, 王仕成, 等. 基于智能优化算法和有限元法的多线圈均匀磁场优化设计[J]. 北京航空航天大学学报, 2019, 45(5): 980-988. Lyu Zhifeng, Zhang Jinsheng, Wang Shicheng, et al.Optimal design of multi-coil system for generating uniform magnetic field based on intelligent optimization algorithm and finite element method[J]. Journal of Beijing University of Aeronautics and Astronautics, 2019, 45(5): 980-988. [22] Thabuis A, Ren Xiaotao, Duong T, et al.Exploring beyond the Helmholtz coils for uniform magnetic field generation with topology optimization[J]. IEEE Transactions on Magnetics, 2022, 58(9): 7001404. [23] 张伟. 高均匀度磁场线圈的设计[J]. 计量学报, 2010, 31(5): 404-407. Zhang Wei.The design of magnetic coil with high homogeneity[J]. Acta Metrologica Sinica, 2010, 31(5): 404-407. [24] Wang Shuying, Lu Jixi, Qian Yujie, et al.Magnetic field coil design based on distributed Taylor expansion: toward large uniform area with high orthogonality and self-shielding feature[J]. IEEE Transactions on Instrumentation and Measurement, 2024, 73: 9004010. [25] 何祥瑞, 荣灿灿, 刘明海. 基于无线电能传输系统多线圈结构参数优化设计[J]. 电工技术学报, 2021, 36(增刊2): 404-411. He Xiangrui, Rong Cancan, Liu Minghai.Optimization design of multi-coil structure parameters based on wireless power transfer system[J]. Transactions of China Electrotechnical Society, 2021, 36(S2): 404-411. [26] 张杰, 郑云耀, 刘生春, 等. 基于解耦内点法与混合整数规划法的区域电网动态无功优化算法[J]. 中国电力, 2023, 56(1): 112-118. Zhang Jie, Zheng Yunyao, Liu Shengchun, et al.A dynamic reactive power optimization algorithm for regional power grid based on decoupling interior point method and mixed integer programming method[J]. Electric Power, 2023, 56(1): 112-118. [27] 赵昌龙, 吕起印, 王旭旭, 等. 基于Maxwell的磁场发生装置设计及分析[J]. 制造技术与机床, 2022(1): 29-33. Zhao Changlong, Lü Qiyin, Wang Xuxu, et al.Design and analysis of magnetism installation based on Maxwell[J]. Manufacturing Technology & Machine Tool, 2022(1): 29-33.
2026, Vol. 41 
