This work introduces a theoretical extension of the characteristic mode formulation for analyzing the vertical electric dipole lying above a lossy dielectric half-space. As the conventional characteristic formulation fails to maintain the orthogonality of the characteristic field modes over the infinite sphere, an alternate modal formulation is proposed here to maintain the orthogonality for both the current and field modes. The modal results are found to match closely with its method of moment counterparts. Later, the modes of an isolated dipole with no ground plane have been used to predict the role of the lossy ground plane through a theory of the linear combination of the eigenvectors. The proposed formulations have been studied with different heights from the ground plane and are compared with the direct modal solutions to validate its accuracy. It helps to provide a thorough understanding of how the isolated modes interact among each other to constitute the perturbed modes in the presence of the lossy half-space. It can find application to include the lossy earth effect in the study of the lightning fields and the path loss modeling of the antennas over the lossy ground.

中文翻译：

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有损介质半空间上方的垂直偶极子——特征模式分析

这项工作引入了特征模式公式的理论扩展，用于分析位于有损介电半空间上方的垂直电偶极子。由于传统的特征公式不能在无限球面上保持特征场模式的正交性，这里提出了一种替代的模态公式来保持电流和场模式的正交性。发现模态结果与其对应的矩方法密切匹配。后来，没有地平面的孤立偶极子的模式已被用于通过特征向量的线性组合理论来预测有损地平面的作用。所提出的公式已经在距地平面不同高度的情况下进行了研究，并与直接模态解决方案进行了比较以验证其准确性。它有助于全面了解孤立模式之间如何相互作用以在存在有损半空间的情况下构成扰动模式。它可以在闪电场的研究和天线在有损地面上的路径损耗建模中找到应用，包括有损大地效应。