Analysis of Miniature Metamaterial and Magnetodielectric Arbitrary-Shaped Patch Antennas Using Characteristic Modes: Evaluation of the $Q$ Factor

Evaluating the Q factor of antennas using metamaterial substrates is a challenge and requires new analysis techniques. This challenge is even more important for arbitrary-shaped antennas. In this paper, we address the problem of analysis and evaluation of metamaterial-based antennas through the use of characteristic modes. We propose a new expression for the total Q factor, which is independent of the excitation and based on the eigenvalues of the structure. The proposed expression is used to compute the Q factor of an arbitrary-shaped resonant antenna and shows excellent agreement with other expressions in the literature such as the ones proposed by Yaghjian, Best, and Gustafsson. The approach is then used to analyze small patch antennas over metamaterial and magnetodielectric substrates. It is shown using the defined Q factor expression that a magnetic substrate offers a significant bandwidth enlargement for the first active modes, and a dielectric substrate presents the opposite effect. This agrees with other works based on analytical approaches. Furthermore, we demonstrate that a reactive impedance metamaterial substrate presents identical modal significances that are identical to a conventional magnetodielectric substrate when associated with a patch antenna. This analysis is suitable for structures presenting noncanonical geometries and arbitrary shapes such as metamaterial inclusions.