Inductance calculation method for transcranial magnetic stimulation figure-8 coils Accounting for spatial mutual inductance_Journal of Biomedical Engineering

Authors：

FAN Yuansheng ^1,2 , WU Nianshuang ^1,2 , LI Jiawei ³ , MENG Weiyu ^1,2 , WU Zhen ^1,2 , WANG Penghao ^1,2 , ZHANG Cheng ^1,2 , WU Changzhe ¹ , HUO Xiaolin ^1,2 ,  ZHANG Guanghao ^1,2

1. Department of Bioelectromagnetic Technology, Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, P. R. China;
2. School of Electronics, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, P. R. China;
3. School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, P. R. China;

Corresponding?author：

ZHANG Guanghao, Email: zhangguanghao@mail.iee.ac.cn

Keywords：

Transcranial magnetic stimulation; Figure-8 coil; Inductance; Mutual inductance

DOI：

10.7507/1001-5515.202601070

Video：

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Transcranial magnetic stimulation (TMS) is widely used in the treatment of neuropsychiatric disorders, and the stimulation coil constitutes a critical component of TMS devices. Existing inductance calculations for figure-8 coils suffer from three notable shortcomings: they are mostly limited to single-layer geometries, they still rely on elliptic integrals, and they systematically overlook spatial mutual coupling among windings. To address these issues, we propose a high-precision analytical method that accommodates both flat spiral and multi-layer stacked coil configurations. Instead of adopting a conventional lumped-equivalent model, the proposed approach decoupled the total inductance into four independent contributions: self-inductance, coaxial layer-to-layer mutual inductance, coplanar mutual inductance, and non-coplanar mutual inductance. The self-inductance term was computed through a modified magnetic circuit procedure. The coaxial mutual inductance was obtained from parametric curves that incorporated the wire cross-section, built upon the geometric mean distance theory. The coplanar and non-coplanar mutual inductances were derived by combining the Maxwell–Wien series expansion with geometric perturbation theory. The overall coil inductance was then determined by analyzing the circuit connection. Finite element simulations and experimental measurements verified the accuracy and robustness of the method at both the individual-component level and the system level, laying a theoretical foundation for the optimal design of high-performance TMS coils.

Citation： FAN Yuansheng, WU Nianshuang, LI Jiawei, MENG Weiyu, WU Zhen, WANG Penghao, ZHANG Cheng, WU Changzhe, HUO Xiaolin, ZHANG Guanghao. Inductance calculation method for transcranial magnetic stimulation figure-8 coils Accounting for spatial mutual inductance. Journal of Biomedical Engineering, 2026, 43(4): 678-685. doi: 10.7507/1001-5515.202601070 Copy

1.	Trapp N T, Purgianto A, Taylor J J, et al. Consensus review and considerations on TMS to treat depression: a comprehensive update endorsed by the National Network of Depression Centers, the Clinical TMS Society, and the International Federation of Clinical Neurophysiology[J]. Clin Neurophysiol, 2025, 170: 206-233.
2.	Kishi T, Ikuta T, Sakuma K, et al. Theta burst stimulation for depression: a systematic review and network and pairwise meta-analysis[J]. Mol Psychiatry, 2024, 29: 3893-3899.
3.	Manna A, Donato F, Nuzzo M, et al. Clinical updates and perspectives on transcranial magnetic stimulation (TMS) in neurological rehabilitation[J]. J Clin Med, 2024, 13(13): 3794.
4.	Arumugham S S, Narayanaswamy J C, Balachander S, et al. Clinical practice guidelines for obsessive-compulsive disorder: 2025 update[J]. Indian J Psychiatry, 2026, 68(1): 44-67.
5.	Koch G, Altomare D, Benussi A, et al. The emerging field of non-invasive brain stimulation in Alzheimer's disease[J]. Brain, 2024, 147(12): 4003-4016.
6.	Siebner H R, Funke K, Aberra A S, et al. Transcranial magnetic stimulation of the brain: what is stimulated? - a consensus review[J]. Clin Neurophysiol, 2022, 140: 59-97.
7.	Steuber E R, McGuire J F. A meta-analysis of transcranial magnetic stimulation in obsessive-compulsive disorder[J]. Biol Psychiatry Cogn Neurosci Neuroimaging, 2023, 8(11): 1145-1155.
8.	Ueno S, Sekino M. Figure-eight coils for magnetic stimulation: from focal stimulation to deep stimulation[J]. Front Hum Neurosci, 2021, 15: 805971.
9.	Deng Z D, Lisanby S H, Peterchev A V. Electric field depth-focality tradeoff in transcranial magnetic stimulation: simulation comparison of 50 coil designs[J]. Brain Stimul, 2013, 6(1): 1-13.
10.	Al-Sultan F S, Al-Zahrani A, Al-Kahtani F, et al. Engineering and technological advancements in repetitive transcranial magnetic stimulation (rTMS): a five-year review[J]. Brain Sci, 2024, 14(11): 1092.
11.	Halawa I, Reichert K, Aberra A S, et al. Effect of pulse duration and direction on plasticity induced by 5 Hz repetitive transcranial magnetic stimulation in correlation with neuronal depolarization[J]. Front Neurosci, 2021, 15: 773792.
12.	Qi Z, Liu H, Jin F, et al. A wearable repetitive transcranial magnetic stimulation device[J]. Nat Commun, 2025, 16(1): 2731.
13.	Ma K, Vlasov A, Simsek Z B, et al. Optimal asymmetric electric field pulses for selective transcranial magnetic stimulation with minimised power and coil heating[J]. Brain Stimul, 2025, 18(6): 1918-1929.
14.	Maccabee P J, Eberle L, Amassian V E, et al. Spatial distribution of the electric field induced in volume by round and figure '8' magnetic coils: relevance to activation of sensory nerve fibers[J]. Electroencephalogr Clin Neurophysiol, 1990, 76(2): 131-141.
15.	Roth Y, Tendler A, Arikan M K, et al. Real-world efficacy of deep TMS for obsessive-compulsive disorder: post-marketing data collected from twenty-two clinical sites[J]. J Psychiatr Res, 2021, 137: 667-672.
16.	Lu M, Ueno S. Comparison of the induced fields using different coil configurations during deep transcranial magnetic stimulation[J]. PLoS One, 2017, 12(6): e0178422.
17.	Zangen A, Roth Y, Voller B, et al. Transcranial magnetic stimulation of deep brain regions: evidence for efficacy of the H-coil[J]. Clin Neurophysiol, 2005, 116(4): 775-779.
18.	Zhang Jinshui, Peterchev A V, Goetz S M. Asymmetric modular pulse synthesizer as a high-fidelity solution for transcranial magnetic stimulation with practically any pulse shape[J]. Brain Stimul, 2025, 18(6): 1935-1938.
19.	Tan X, Guo A, Tian J, et al. A novel pulse-current waveform circuit for low-energy consumption and low-noise transcranial magnetic stimulation[J]. Front Neurosci, 2025, 18: 1500619.
20.	Gomez-Tames J, Asamoah B, Miyaguchi H, et al. Design of transcranial magnetic stimulation coils with optimized stimulation depth[J]. IEEE Access, 2024, 12: 1330-1340.
21.	Nguyen H, Makaroff S N, Li C Q, et al. High inductance magnetic-core coils have enhanced efficiency in inducing suprathreshold motor response in rats[J]. Phys Med Biol, 2023, 68(24): 244001.
22.	Masalehdan T. Deep brain stimulation through engineered micro-coils[D]. Glasgow: University of Glasgow, 2024: 50-52, 66-69.
23.	Rissanen I J, Souza V H, Nieminen J O, et al. Advanced pipeline for designing multi-locus TMS coils with current density constraints[J]. IEEE Trans Biomed Eng, 2023, 70(7): 2025-2034.
24.	Bagherzadeh H, Meng Q, Deng Z D, et al. Angle-tuned coils: attractive building blocks for TMS with improved depth-spread performance[J]. J Neural Eng, 2022, 19(2): 026059.
25.	Liu Shuang, Yoshioka H, Kuwahata A, et al. Optimizing transcranial magnetic stimulator coils for minimal influence of individual variability in head geometry[J]. AIP Adv, 2024, 14(2): 025325.
26.	Park T Y, Franke L, Pieper S, et al. A review of algorithms and software for real-time electric field modeling techniques for transcranial magnetic stimulation[J]. Biomed Eng Lett, 2024, 14(3): 393-405.
27.	Makaroff S N, Nguyen H, Meng Q, et al. Modeling transcranial magnetic stimulation coil with magnetic cores[J]. J Neural Eng, 2023, 20(1): 016028.
28.	Gomez L J, Murphy D L K, Koponen L M, et al. Optimization, implementation, and performance of TMS coils with maximum focality and various stimulation depths[J]. J Neural Eng, 2026, 23(2): 026001.
29.	Babic S, Smith M, Fokas N, et al. The analytical formula for calculating the self-inductance for the circular coil of the rectangular cross-section with a non-uniform current density[J]. Prog Electromagn Res M, 2021, 103: 15-26.
30.	Mohan S S, Del Mar Hershenson M, Boyd S P, et al. Simple accurate expressions for planar spiral inductances[J]. IEEE J Solid-State Circuits, 1999, 34(10): 1419-1424.
31.	Faria A, Marques L, Ferreira C, et al. A fast and precise tool for multi-layer planar coil self-inductance calculation[J]. Sensors, 2021, 21(14): 4864.
32.	Attar H, Hekmati A, Moradnouri A, et al. An accurate method for calculation of self-inductance of HTS coils with and without turn-to-turn insulation[J]. IEEE Trans Appl Supercond, 2023, 33(7): 4603308.
33.	Umetani K, Kawahara S, Acero J, et al. Analytical formulation of copper loss of Litz wire with multiple levels of twisting using measurable parameters[J]. IEEE Trans Ind Appl, 2021, 57(3): 2407-2420.
34.	Yi Jiliang, Yang Peng, Li Zhongqi, et al. Mutual inductance calculation of circular coils for an arbitrary position with a finite magnetic core in wireless power transfer systems[J]. IEEE Trans Transp Electrif, 2023, 9(1): 1950-1959.
35.	Hussain I, Woo D K. Simplified mutual inductance calculation of planar spiral coil for wireless power applications[J]. Sensors, 2022, 22(4): 1537.
36.	Babic S I, Akyel C. Calculating mutual inductance between circular coils with inclined axes in air[J]. IEEE Trans Magn, 2008, 44(7): 1743-1750.
37.	Quercio M, Parise M, Milillo D, et al. Analytical modeling of mutual inductance in superconducting pancake coils for wireless power transfer systems[J]. IEEE Access, 2025, 13: 90698-90705.
38.	Babic S, Sirois F, Akyel C, et al. Mutual inductance calculation between circular filaments arbitrarily positioned in space: alternative to Grover’s formula[J]. IEEE Trans Magn, 2010, 46(9): 3591-3600.
39.	Hussain I, Woo D K. Calculation of mutual inductance between arbitrarily positioned planar spiral coils for wireless power applications[J]. Int J Appl Electromagn Mech, 2024, 74(3): 235-249.
40.	Maxwell J C. A treatise on electricity and magnetism[M]. Vol. 2. Oxford: Clarendon Press, 1873.
41.	Grover F W. Inductance calculations: working formulas and tables[M]. New York: D. Van Nostrand Company, 1946.
42.	Wu Nianshuang, Wu Zhen, Zhang Cheng, et al. Probe design for measurement and verification of electric field induced by figure-8 coil in transcranial magnetic stimulation[J]. IEEE Trans Instrum Meas, 2024, 73: 3534511.

1. Trapp N T, Purgianto A, Taylor J J, et al. Consensus review and considerations on TMS to treat depression: a comprehensive update endorsed by the National Network of Depression Centers, the Clinical TMS Society, and the International Federation of Clinical Neurophysiology[J]. Clin Neurophysiol, 2025, 170: 206-233.
2. Kishi T, Ikuta T, Sakuma K, et al. Theta burst stimulation for depression: a systematic review and network and pairwise meta-analysis[J]. Mol Psychiatry, 2024, 29: 3893-3899.
3. Manna A, Donato F, Nuzzo M, et al. Clinical updates and perspectives on transcranial magnetic stimulation (TMS) in neurological rehabilitation[J]. J Clin Med, 2024, 13(13): 3794.
4. Arumugham S S, Narayanaswamy J C, Balachander S, et al. Clinical practice guidelines for obsessive-compulsive disorder: 2025 update[J]. Indian J Psychiatry, 2026, 68(1): 44-67.
5. Koch G, Altomare D, Benussi A, et al. The emerging field of non-invasive brain stimulation in Alzheimer's disease[J]. Brain, 2024, 147(12): 4003-4016.
6. Siebner H R, Funke K, Aberra A S, et al. Transcranial magnetic stimulation of the brain: what is stimulated? - a consensus review[J]. Clin Neurophysiol, 2022, 140: 59-97.
7. Steuber E R, McGuire J F. A meta-analysis of transcranial magnetic stimulation in obsessive-compulsive disorder[J]. Biol Psychiatry Cogn Neurosci Neuroimaging, 2023, 8(11): 1145-1155.
8. Ueno S, Sekino M. Figure-eight coils for magnetic stimulation: from focal stimulation to deep stimulation[J]. Front Hum Neurosci, 2021, 15: 805971.
9. Deng Z D, Lisanby S H, Peterchev A V. Electric field depth-focality tradeoff in transcranial magnetic stimulation: simulation comparison of 50 coil designs[J]. Brain Stimul, 2013, 6(1): 1-13.
10. Al-Sultan F S, Al-Zahrani A, Al-Kahtani F, et al. Engineering and technological advancements in repetitive transcranial magnetic stimulation (rTMS): a five-year review[J]. Brain Sci, 2024, 14(11): 1092.
11. Halawa I, Reichert K, Aberra A S, et al. Effect of pulse duration and direction on plasticity induced by 5 Hz repetitive transcranial magnetic stimulation in correlation with neuronal depolarization[J]. Front Neurosci, 2021, 15: 773792.
12. Qi Z, Liu H, Jin F, et al. A wearable repetitive transcranial magnetic stimulation device[J]. Nat Commun, 2025, 16(1): 2731.
13. Ma K, Vlasov A, Simsek Z B, et al. Optimal asymmetric electric field pulses for selective transcranial magnetic stimulation with minimised power and coil heating[J]. Brain Stimul, 2025, 18(6): 1918-1929.
14. Maccabee P J, Eberle L, Amassian V E, et al. Spatial distribution of the electric field induced in volume by round and figure '8' magnetic coils: relevance to activation of sensory nerve fibers[J]. Electroencephalogr Clin Neurophysiol, 1990, 76(2): 131-141.
15. Roth Y, Tendler A, Arikan M K, et al. Real-world efficacy of deep TMS for obsessive-compulsive disorder: post-marketing data collected from twenty-two clinical sites[J]. J Psychiatr Res, 2021, 137: 667-672.
16. Lu M, Ueno S. Comparison of the induced fields using different coil configurations during deep transcranial magnetic stimulation[J]. PLoS One, 2017, 12(6): e0178422.
17. Zangen A, Roth Y, Voller B, et al. Transcranial magnetic stimulation of deep brain regions: evidence for efficacy of the H-coil[J]. Clin Neurophysiol, 2005, 116(4): 775-779.
18. Zhang Jinshui, Peterchev A V, Goetz S M. Asymmetric modular pulse synthesizer as a high-fidelity solution for transcranial magnetic stimulation with practically any pulse shape[J]. Brain Stimul, 2025, 18(6): 1935-1938.
19. Tan X, Guo A, Tian J, et al. A novel pulse-current waveform circuit for low-energy consumption and low-noise transcranial magnetic stimulation[J]. Front Neurosci, 2025, 18: 1500619.
20. Gomez-Tames J, Asamoah B, Miyaguchi H, et al. Design of transcranial magnetic stimulation coils with optimized stimulation depth[J]. IEEE Access, 2024, 12: 1330-1340.
21. Nguyen H, Makaroff S N, Li C Q, et al. High inductance magnetic-core coils have enhanced efficiency in inducing suprathreshold motor response in rats[J]. Phys Med Biol, 2023, 68(24): 244001.
22. Masalehdan T. Deep brain stimulation through engineered micro-coils[D]. Glasgow: University of Glasgow, 2024: 50-52, 66-69.
23. Rissanen I J, Souza V H, Nieminen J O, et al. Advanced pipeline for designing multi-locus TMS coils with current density constraints[J]. IEEE Trans Biomed Eng, 2023, 70(7): 2025-2034.
24. Bagherzadeh H, Meng Q, Deng Z D, et al. Angle-tuned coils: attractive building blocks for TMS with improved depth-spread performance[J]. J Neural Eng, 2022, 19(2): 026059.
25. Liu Shuang, Yoshioka H, Kuwahata A, et al. Optimizing transcranial magnetic stimulator coils for minimal influence of individual variability in head geometry[J]. AIP Adv, 2024, 14(2): 025325.
26. Park T Y, Franke L, Pieper S, et al. A review of algorithms and software for real-time electric field modeling techniques for transcranial magnetic stimulation[J]. Biomed Eng Lett, 2024, 14(3): 393-405.
27. Makaroff S N, Nguyen H, Meng Q, et al. Modeling transcranial magnetic stimulation coil with magnetic cores[J]. J Neural Eng, 2023, 20(1): 016028.
28. Gomez L J, Murphy D L K, Koponen L M, et al. Optimization, implementation, and performance of TMS coils with maximum focality and various stimulation depths[J]. J Neural Eng, 2026, 23(2): 026001.
29. Babic S, Smith M, Fokas N, et al. The analytical formula for calculating the self-inductance for the circular coil of the rectangular cross-section with a non-uniform current density[J]. Prog Electromagn Res M, 2021, 103: 15-26.
30. Mohan S S, Del Mar Hershenson M, Boyd S P, et al. Simple accurate expressions for planar spiral inductances[J]. IEEE J Solid-State Circuits, 1999, 34(10): 1419-1424.
31. Faria A, Marques L, Ferreira C, et al. A fast and precise tool for multi-layer planar coil self-inductance calculation[J]. Sensors, 2021, 21(14): 4864.
32. Attar H, Hekmati A, Moradnouri A, et al. An accurate method for calculation of self-inductance of HTS coils with and without turn-to-turn insulation[J]. IEEE Trans Appl Supercond, 2023, 33(7): 4603308.
33. Umetani K, Kawahara S, Acero J, et al. Analytical formulation of copper loss of Litz wire with multiple levels of twisting using measurable parameters[J]. IEEE Trans Ind Appl, 2021, 57(3): 2407-2420.
34. Yi Jiliang, Yang Peng, Li Zhongqi, et al. Mutual inductance calculation of circular coils for an arbitrary position with a finite magnetic core in wireless power transfer systems[J]. IEEE Trans Transp Electrif, 2023, 9(1): 1950-1959.
35. Hussain I, Woo D K. Simplified mutual inductance calculation of planar spiral coil for wireless power applications[J]. Sensors, 2022, 22(4): 1537.
36. Babic S I, Akyel C. Calculating mutual inductance between circular coils with inclined axes in air[J]. IEEE Trans Magn, 2008, 44(7): 1743-1750.
37. Quercio M, Parise M, Milillo D, et al. Analytical modeling of mutual inductance in superconducting pancake coils for wireless power transfer systems[J]. IEEE Access, 2025, 13: 90698-90705.
38. Babic S, Sirois F, Akyel C, et al. Mutual inductance calculation between circular filaments arbitrarily positioned in space: alternative to Grover’s formula[J]. IEEE Trans Magn, 2010, 46(9): 3591-3600.
39. Hussain I, Woo D K. Calculation of mutual inductance between arbitrarily positioned planar spiral coils for wireless power applications[J]. Int J Appl Electromagn Mech, 2024, 74(3): 235-249.
40. Maxwell J C. A treatise on electricity and magnetism[M]. Vol. 2. Oxford: Clarendon Press, 1873.
41. Grover F W. Inductance calculations: working formulas and tables[M]. New York: D. Van Nostrand Company, 1946.
42. Wu Nianshuang, Wu Zhen, Zhang Cheng, et al. Probe design for measurement and verification of electric field induced by figure-8 coil in transcranial magnetic stimulation[J]. IEEE Trans Instrum Meas, 2024, 73: 3534511.

Journal of Biomedical Engineering

Inductance calculation method for transcranial magnetic stimulation figure-8 coils Accounting for spatial mutual inductance

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