# Validating the Characteristic Modes Solvers

## TL;DR

This paper introduces analytical formulas for characteristic modes of spherical shells, providing benchmarks to validate computational methods and analyze their performance across various parameters.

## Contribution

It presents closed-form formulas for characteristic modes, enabling precise validation of numerical solvers and assessment of mode tracking algorithms.

## Key findings

- Commercial and academic solvers reliably identify dominant modes.
- Higher-order modes are difficult to accurately identify with current methods.
- Mode tracking routines perform poorly despite recent advancements.

## Abstract

Characteristic modes of a spherical shell are found analytically as spherical harmonics normalized to radiate unitary power and to fulfill specific boundary conditions. The presented closed-form formulas lead to a proposal of precise synthetic benchmarks which can be utilized to validate the method of moments matrix or performance of characteristic mode decomposition. Dependence on the mesh size, electrical size and other parameters can systematically be studied, including the performance of various mode tracking algorithms. A noticeable advantage is the independence on feeding models. Both theoretical and numerical aspects of characteristic mode decomposition are discussed and illustrated by examples. The performance of state-of-the-art commercial simulators and academic packages having been investigated, we can conclude that all contemporary implementations are capable of identifying the first dominant modes while having severe difficulties with higher-order modes. Surprisingly poor performance of the tracking routines is observed notwithstanding the recent ambitious development.