Numerical aspects of the characteristic modes theory

Abstract

Nowadays, the characteristic modes theory becomes more and more popular in the electromagnetic field as a very efficient tool for analysing and solving scattering and radiation problems. Unlike the existing analysis tools, the characteristic modes analysis is based on the eigenvalues and the eigenvectors. Actually, these two key factors help enormously in gaining useful physical insights into the scatterer’s modes, the modes contributions to the radiation field, and reveals which mode is dominant for each frequency sample of the frequency band being examined. Normally, the eigenvalues and the eigenvectors are derived directly from the modal decomposition of the impedance matrix of the scatterer using the real and imaginary parts of the matrix. Unfortunately, it is found that the real part of the impedance matrix doesn’t fully fill full the theoretical specifications of the characteristic modes theory, which harms the numerical accuracy of the characteristic modes and reduces tremendously the total number of the obtained modes. Therefore, in order to respond to the theoretical requirements of the characteristic modes theory, a new method is proposed to treat this issue by reconstructing the real part matrix using the expansion of the Green dyadic in spherical vector waves. The new approach proved its efficiency and resulted in higher numerical accuracy, an increased number of the characteristic modes, and improved the computational speed in comparison to the conventional approach. However, after obtaining the accurate characteristic modes, the issue of how to connect the modes with each other is raised. The choice of a suitable method to perform the modes tracking is very critical to the accuracy of the analysis of the characteristic mode. Thus, the modal tracking problem is investigated also, with a main focus on the modes swapping and degenerated modes phenomena encountered in the eigenvectors correlation method. Consequently, an enhanced modal tracking method is proposed based on the difference in magnitudes between successive eigenvalues, as a criterion to establish the associations between the modes along the frequency axis. The tracking method is validated using several numerical applications, and satisfactory results are obtained. At times, to achieve a higher accuracy of the modal tracking, it is mandatory to increase the number of frequency samples resulting in a larger size of the impedance matrix. The computation of the impedance matrix at discrete frequency samples is a very time-consuming process, particularly for wideband frequency applications, therefore, the necessity to apply the impedance matrix interpolation method is highlighted. Finally, relevant concepts used throughout this dissertation are detailed in the appendices section, together with the source codes used to perform the modal decomposition, the modal tracking, and the impedance matrix interpolation.