圆盘线圈自、互感的闭式解及其自感最优值

Closed Form Expressions of Self and Mutual Inductance of Disk Coils and the Optimal Value of Self-Inductance

圆盘线圈是近年来兴起的无线能量传输系统中能量耦合的核心元件。通过引入广义超几何函数pFq 将圆盘线圈的自感积分表达式化为闭式。类似地,通过建立完全椭圆积分与广义超几何函数之间的联系将同轴共面圆盘线圈的互感化为闭式。这些闭式解使上述电感计算问题不再需要数值积分。进一步,基于所得闭式解,分析了圆盘线圈的GPW(given piece of wire)优化问题,得到了给定导线所能绕制的圆盘线圈自感最大值,且达到最大值的圆盘线圈外半径与内半径之比是一个与导线几何参数无关的常数。数值计算表明所得公式与已有方法所得结果一致性较好。

The disk coils are the core element of energy coupling of the wireless power transmission system blooming in recent years. By introducing generalized hypergeometric functions pFq, the integral expression of self-inductance of disk coils is transformed into a closed form. Similarly, the closed form expression of mutual inductance of coaxial coplanar disk coils is obtained by establishing a relationship between the complete elliptic integrals and generalized hypergeometric functions. These closed form solutions can bypass the numerical integration of traditional methods for the inductance calculations. Furthermore, based on the obtained closed form solution the GPW (given piece of wire) optimization problem is analyzed and the maximum value of self-inductance is found for disk coils which are wound by a piece of given wire. It is also revealed that the ratio of the outer and inner radii of the optimal disk coils is independent of the geometric parameters of winding wires. Numerical results show the high consistency between existing methods and methods proposed in this paper.