## 2 Conception of the Characteristic Modes  
The theory of characteristic modes is a very promising branch of research. The characteristic modes form a very convenient basis for the decomposition of currents and fields of scattering or radiation by the object without losses under harmonic excitation. It is essential that they possess the orthogonality property not only on the object surface but also on the surface of the sphere of infinite radius and show interesting mathematical properties that can be interpreted physically in terms of radiated and stored powers.  
More particularly, for the given ideally conducting object with sufficiently smooth surface \(S\), the electric current density \(\mathbf{J}\) on its surface can be represented as the sum of the characteristic modes (or eigencurrents [17; 20; 22]):  
\[\mathbf{J}=\sum_{n}\alpha_{n}\mathbf{J}_{n}=\sum_{n}\frac{V_{n}^{i}\mathbf{J}_ {n}}{1+j\lambda_{n}}, \tag{1}\]  
where \(\lambda_{n}\) is the eigenvalues and \(V_{n}^{i}\) is the excitation coefficients of individual modes defined by the scalar products of the modes \(\mathbf{J}_{n}\) and the applied electric field strength \(\mathbf{E}^{i}\):  
\[V_{n}^{i}=(\mathbf{J}_{n},\mathbf{E}^{i})=\iint_{S}\mathbf{J}_{n}\cdot \mathbf{E}^{i}ds. \tag{2}\]  
It should be emphasized that each mode \(\mathbf{J}_{n}\) represents the surface current density that can really be excited on the object. From (1), it can be seen that the structure of the weighting coefficient  
\[\alpha_{n}=\frac{V_{n}^{i}}{1+j\lambda_{n}} \tag{3}\]allows one to consider the parameter defining the degree of radiation efficiency for each mode:  
\[MS_{n}=\bigg{|}\frac{1}{1+j\lambda_{n}}\bigg{|}. \tag{4}\]  
For this important parameter, it is customary to use the term mode significance.  
It is most convenient to investigate the resonant properties of the modes using the characteristic angle parameter \(\beta_{n}\). This parameter is related with the eigenvalue by the formula  
\[\beta_{n}=180^{\circ}-\arctan(\lambda_{n}). \tag{5}\]  
From the above formulas it can be seen that the mode \(\mathbf{J}_{n}\) is resonant when \(\lambda_{n}=0\). In this case, \(\beta_{n}=180^{\circ}\) and \(MS_{n}=1\). Thus, the mode is completely radiating when \(\beta_{n}=180^{\circ}\). The excitation mode will radiate more effectively, the closer the \(\beta_{n}\) value is to \(180^{\circ}\) or the \(\lambda_{n}\) value is to \(0\). At the same time, from Equation (1), it can be seen that the excitation quality of the mode \(\mathbf{J}_{n}\) depends on the weighting coefficient \(\alpha_{n}\). In turn, this coefficient depends on the eigenvalue \(\lambda_{n}\) and the mode excitation coefficient \(V_{n}^{i}\). In particular, under the conditions of resonance, \(\lambda_{n}\) is equal to \(0\), and then  
\[\alpha_{n}=V_{n}^{i}.\]  
The eigenvalues of some modes in the present frequency band can be large. This means that the weighting coefficient \(\alpha_{n}\) is small regardless of the excitation method; as a result, the contribution of such modes is not essential. The essential contribution is given only by the significant modes, the characteristic angles of which are close to \(180^{\circ}\). The weighting coefficients \(\alpha_{n}\) depend also on the mode excitation coefficient \(V_{n}^{i}\). Therefore, when attempting to increase a role in this or that significant mode, excitation should be applied to the antenna part where the mode eigencurrent is maximum. In this regard, we emphasize that the characteristic modes are also numerically calculated for the antenna without a power supply unit.  
Concepts of the characteristic modes of current, directional patterns, stored energies, radiated powers, and radiation Q-factors of these modes can be found in works [20; 22].