Recent advances in additive direct-write circuit technology have enabled the fabrication of inductive winding geometries previously considered impossible by conventional methods. In this letter, a winding fabricated around a rectangular cross-section toroidal core is considered. As an extension of research embedding toroidal cores within printed circuit board, this letter considers the minimization of the winding length when the vertical path, along the toroidal z-direction, is not restricted to be perfectly vertical, which is only possible with direct-write techniques. It will be shown that the exact analysis results in a transcendental equation for the optimized vertical midpoints of the winding turns. However, a few reasonable approximations allow an analytical expression for this optimized midpoint to be found. Comparisons of this approximate analytical solution to the exact solution found by numerical methods will be presented for typical toroid dimensions. The accuracy of this approximate solution with changing turn density will also be presented. For typical toroid dimensions, it was found that the choice of midpoint can change the total winding length by as much as 10%. The analytical approximate solution for the midpoint was found to deviate by 2.75% of angular gap from the exact solution, which resulted in an error in the minimum winding length of only 0.07%, a result which will be discussed in this letter.

中文翻译：

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用于绕组长度优化的矩形截面环形螺旋几何分析

增材直写电路技术的最新进展已经能够制造出以前被传统方法认为不可能的电感绕组几何形状。在这封信中，考虑了围绕矩形横截面环形磁芯制造的绕组。作为在印刷电路板中嵌入环形磁芯的研究的延伸，这封信考虑了当沿环形 z 方向的垂直路径不限于完全垂直时绕组长度的最小化，这只能通过直接写入来实现技术。将表明，精确的分析导致了绕组匝的优化垂直中点的超越方程。然而，一些合理的近似值允许找到这个优化中点的解析表达式。对于典型的环形尺寸，将介绍这种近似解析解与通过数值方法找到的精确解的比较。还将介绍这种近似解决方案随着转弯密度变化的准确性。对于典型的环形尺寸，发现中点的选择可以将总绕组长度改变多达 10%。发现中点的解析近似解与精确解偏离了 2.75% 的角间隙，这导致最小绕组长度的误差仅为 0.07%，这一结果将在本信中讨论。结果发现，中点的选择可以使总绕组长度改变多达 10%。发现中点的解析近似解与精确解偏离了 2.75% 的角间隙，这导致最小绕组长度的误差仅为 0.07%，这一结果将在本信中讨论。结果发现，中点的选择可以使总绕组长度改变多达 10%。发现中点的解析近似解与精确解偏离了 2.75% 的角间隙，这导致最小绕组长度的误差仅为 0.07%，这一结果将在本信中讨论。