Analytical Calculation for Self and Mutual Inductance of Coaxial Multi Layer Cylindrical Hollow Coils
Chen Bin1,2, Jiang Pengfei1, Wan Nina3, Feng Yuzhang4, Huang Li1,2
1. College of Electrical Engineering and New Energy China Three Gorges University Yichang 443002 China; 2. Hubei Provincial Engineering Technology Research Center for Power Transmission Line China Three Gorges University Yichang 443002 China; 3. Yichang Electric Company State Grid Hubei Electric Power Company Yichang 443000 China; 4. Wuhan Electric Company State Grid Hubei Electric Power Company Wuhan 430000 China
Abstract：In the electromagnetic coupling of wireless power transmission systems, accurate calculation of the coil inductance parameters is an important theoretical basis. The existing analytical calculations for coil inductance mostly focus on planar plate coils to study the relationship between coil inductance and coil structure, ignoring the influence of frequency. However, the coaxial multilayer cylindrical structure of the air-core transformer (ACT) has a higher coupling coefficient and transmission efficiency than the planar plate coil under the same quality conditions. However, since the ACT has no magnetic conductive material, it has no specific magnetic flux path, and the magnetic field cannot be bound to a certain part. Usually, it is calculated using the three-dimensional finite element method, which requires a high computer configuration, a large amount of storage space, and sufficiently fast computing speed, resulting in high computational costs. Therefore, this paper proposes an analytical calculation method for the self and mutual inductance of coaxial multilayer cylindrical hollow coils considering high-frequency eddy current effects. Firstly, the magnetic chain method derives the analytical equations for the self and mutual inductance of a single-turn coaxial parallel charged circular ring. Secondly, the cross-section of the single-turn coil is discretized into grid-charged circular ring elements with the same shape. The analytical equations for the single-turn coil’s self-inductance and the coils' mutual inductance are derived using the discrete equivalent summation method. Based on the first and second types of complete elliptic integrals and Kelvin functions, analytical formulas are derived to calculate the self-inductance of the primary and secondary coils and the total mutual inductance. Finally, taking two different arrangements of cylindrical hollow coils as examples, the proposed analytical method is verified by comparing analytical calculations, finite element calculations, and experimental measurements. This method considers the effect of frequency on coil inductance parameters and significantly reduces the computational cost. Within the frequency range of 1~100 kHz, the inductance's average relative errors of the proposed method and the simulated values are 2.779% and 4.118% for the cylindrical hollow coils without cross-transposition, and 2.278% and 2.200% for the partially cross-transposition cylindrical hollow coils. The proposed method can calculate the self-inductance and total mutual inductance of the primary and secondary coils of cylindrical hollow coils with different arrangements and frequencies. Therefore, the electromagnetic coupling of the cylindrical hollow coil can be analyzed, which provides a reference for the parametric design of the coil.
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