# An ECCD--Electronic Charge Compensation Device--As a Quantum Dissipative System
## 3 Simulation Study of the Electromagnetic Radiofrequency Field in an ECCD
### Resonant Modes
Additionally, some spatial composition with 3D symmetry--spherical or quasi-spherical symmetry--means that the qualitative behaviors of the electromagnetic field remain analogous to its resonant confinement. It will also mean the existence of an evanescent electromagnetic field. The analytical mathematical description in this 3D case is further complicated, as we need to resort to vector spherical harmonic functions that are dependent on Bessel functions and Legendre Polynomials [9]. But in the 2D case of cylindrical revolution symmetry, the analogy persists and the external field is evanescent and exponential in character. By breaking the 2D symmetry when passing to 3D in quasi-spherical conditions, the evanescent electromagnetic field will present azimuthal dependence; i.e., it will vary according to the angle of the direction of the electromagnetic field with the axis of revolution in 2D--axis OZ. All those reasons justify resorting to numerical solutions based on finite element calculations, which allows the results to be obtained faster and to be in accordance with geometric structures that are more complicated but more realistic. However, it should be noted that both the calculation tools used in the simulation and the analytical solutions which we have expressly mentioned in the fields of analysis--1D planes, cylindrical 2D symmetry, and spherical or quasi spherical 3D symmetry--are based on the Maxwell equations, and more directly, on the well-known Helmholtz spatial equation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Phase** & **Field E [V/m]** & **FIELD H [A/m]** & **Current J [A/m\({}^{2}\)]** \\ \hline
0\({}^{\circ}\) approximately & & \\
40\({}^{\circ}\) approximately & & \\
80\({}^{\circ}\) approximately & & \\
120\({}^{\circ}\) approximately & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Electric and magnetic fields and surface current density at the resonant frequency 1.70 GHz. Therefore, it is crucial to access the interpretation of the results arising via simulation based on the support that can be found in knowledge of the different possible analytical solutions in ideal fields close to the geometric structures of real resonators, such as the ECCD case. This is because even the \"language\" designating modes, quality factor, resonant frequency, etc., are concepts derived from the analytical study of the different existing ideal models (1D, 2D, and 3D) based on the accessible symmetry they present. Following the analysis of the simulation results, it can be concluded that:  
* The ECCD--DINTECO DDCE 100 model--is an electromagnetic field resonant structure that has a main resonant mode at 1.7 GHz and a quasi-continuous oblique mode from 0.549 to 2.42 GHz. The main mode corresponds effectively to the central dimension in the intra-cavity axis of d = 83.79 mm in the so-called order mode m = 1 (called half-wave, \(\uplambda\)/2 = d, empty). The prominent secondary or lateral mode at 1.153 GHz seems attributable to a confocal structure of the non-axial lateral parts that would be associated with a 4 dn \(\approx\uplambda\) (considering two straight sections: round-trip and two slightly oblique). The confocal condition is satisfied and maintained in the structures despite severe variations due to dimensional changes (derived from strong temperature variations) or changes in the refractive index of the medium. They can undoubtedly occur with a low-density plasma, such as the one produced during discharge by breakage of the dielectric rigidity inside the ECCD or the external cavity's resonant absorption. The secondary mode, therefore, acts as an external discharge insurance due of the strong evanescent field in the mode in relation to exogenous or endogenous external circumstances that can affect the ECCD. * The quasi-continuous background modes are clearly angular modes, and they have an important contribution to external evanescent fields. * Figure 7 shows results of the simulations.