A Useful Mathematical Lemma: Shifted Coordinate Transformations for Analytical Modeling of Plasma Multi-Ring Antennas

چکیده مقاله

Transformations of scalar and vector quantities from one Cartesian coordinate system to a shifted Cartesian frame constitute a powerful tool in linear algebra with wide applicability in solving physical problems. In this work, we employ such coordinate transformations to construct a unified analytical formulation for multi-ring plasma antennas. By expressing the vector and scalar potentials in systematically shifted Cartesian coordinates, the electromagnetic fields and radiation impedance of single-ring and multi-ring structures are derived in closed form. The method clarifies how radiation resistance and field distributions depend on ring number, radii, and vertical displacements, while showing that symmetric arrangements lead to notable mathematical simplifications. Because the framework is built upon generalizable coordinate transformations, it naturally extends to a broad class of antenna geometries composed of repetitive or modular elements. This highlights the strength of the mathematical approach: many complex antenna configurations become analytically tractable once reformulated through shifted Cartesian transformation techniques