A rigorous mathematical solution to the problem of synthesizing wire antennas using the multipole expansion method (MEM) is presented. The input current and required size of the antenna are estimated using the knowledge of the required field pattern. This solution is based on three steps. First, obtaining the expansion’s weighting coefficients that carry both the signature of the current and size of the antenna. In essence, the analysis of the direct modeling needs to be addressed first in order to define these coefficients. Second, extracting the coefficients from the required field pattern using the natural orthogonality property of the spherical harmonic functions. Finally, performing a numerical search for the argument of the spherical Bessel functions to satisfy the ratio between two successive moments. Once the size of the antenna is found, a recursive expression is used to obtain the input current. For the current distributions assumed herein, the odd order coefficients of the MEM are shown to be the only necessary coefficients to construct the radiation far-field. Also, it is found that the estimated model parameters are in excellent agreement with the analytical ones.

中文翻译：

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使用多极展开法 (MEM) 合成线状天线

提出了使用多极扩展方法 (MEM) 合成线天线问题的严格数学解决方案。使用所需场图的知识来估计天线的输入电流和所需尺寸。此解决方案基于三个步骤。首先，获得携带电流特征和天线尺寸的扩展权重系数。本质上，为了定义这些系数，需要首先解决直接建模的分析。其次，使用球谐函数的自然正交特性从所需的场模式中提取系数。最后，对球面贝塞尔函数的自变量进行数值搜索，以满足两个连续矩之间的比率。一旦找到天线的尺寸，递归表达式用于获得输入电流。对于这里假设的电流分布，MEM 的奇数阶系数被证明是构建辐射远场唯一必要的系数。此外，还发现估计模型参数与分析参数非常吻合。