## 1 Introduction  
Recently, very much attention has been paid to the development of antennas providing highly efficient radiation of ultra-wideband (UWB) electromagnetic signals. The field of application of these antennas is extremely wide: high-speed transfer of big data volumes, UWB sensing of media and materials, security systems, etc. The antenna designs are constantly modified and improved. To some extent, this is promoted by a fairly clear idea of the radiation features of the antennas that have long been used in practice. In a somewhat different position are the so-called three-dimensional combined antennas [1]. The concept of their design has a reliable, physically realistic basis from the viewpoint of general theoretical considerations already proposed a long time ago [2]. It is proposed to combine in a special way in the circuit design the radiating antennas of electric and magnetic types. Among the characteristic features of these radiators are that the stored electric energy prevails in the near-field zone of the electric-type radiators, whereas the stored magnetic energy prevails for the magnetic radiators. The key idea of the concept is that a decrease in the stored reactive energy component takes place in the near zone of the antenna during equalization of the stored energies, thereby leading to the expansion of the working frequency band. At the same time, a comprehensive electrodynamic description of the radiation process of these antennas, giving a clear idea of their radiation mechanism, has not reached the proper level yet. It is of interest to note that the related concept based on a number of theoretical works (for example, see [3, 4]) has been successfully used [5, 6, 7, 8, 9], providing various wireless communication systems. In particular, it was reported that some modifications of highly effective antennas had been developed for cellular and Wi-Fi networks operating at frequencies of 850 MHz and 2.4 GHz. Practically simultaneously, papers [10, 11, 12, 13] devoted to the so-called magneto-electric dipole antennas were published. As a matter of fact, the authors embodied the above-mentioned concept of expanding the working frequency band of the antenna while the width of the directional pattern in the E- and H-planes remained unchanged. The number of publications in this direction turned out to be so large that it was proposed to use a new generalizing term--Huygens dipole antennas [14]. Interesting results were obtained in the process of a detailed study of the characteristics of three-dimensional UWB antennas with a special configuration of their elements: asymmetrical loop TEM horns and polyconical radiators [15; 16]. A distinguishing feature of these antennas is a very wide matching band. However, the authors of developments [5; 6; 7; 8; 9] and works [10; 11; 12; 13; 14; 15; 16] either rely on tools for computer optimization of the antenna parameters and characteristics or make use of earlier accumulated experience and heuristic considerations when studying antenna radiation features. To calculate the parameters and characteristics of different antennas and scatterers, the method of moments [17] has been proven to be extremely efficient. Moreover, during the subsequent period, the theory of characteristic modes (TCM) was formulated in the convenient form within the framework of this method [18], the initial variant of which was proposed in work [19]. Unfortunately, this theory did not find wide application within several decades owing to high requirements to computer modeling means. However, the publication of significant works [20; 21; 22] not only has revealed its actual potential but also has forced many foreign researchers to turn to this theory. In the present work, we rely on the TCM in a theoretical study of the three-dimensional UWB antenna. The physical content of the main steps of mathematical modeling of a combined antenna, called snail-shaped in work [23] because of its characteristic shape, is consistently illustrated within the framework of the TCM.