## 4 Finite Elements Method (FEM) Analysis Results  
Based on the conceptual plot in Figure 4, the design conditions were set as boundary conditions for the intervals directly affected by the coil (copper). Frequencies ranging from 20 to 550 kHz are mainly used for inspection of the steam generator at nuclear power plants. To simulate the curved part of the steam generator's electric tube, six sections were defined according to the location of the curved tube shown in Figure 5 and modeled by extruding a curvature (Figure 6). A total of six types of curved piping were designed, and a coil probe was placed in the angular position to analyze the current density and electric field at each location. Figure 7 shows the analysis of the distribution of current density in the cross-sectional area at 20 kHz when the coil probe was located at 0\({}^{\circ}\), 29\({}^{\circ}\), and 45\({}^{\circ}\) of the U-bend of the electric tube. Since 0\({}^{\circ}\) was in the lowest position of ovality and 29\({}^{\circ}\) was 5.2%, the highest position of 45\({}^{\circ}\) was the outermost angle of the modeled curved pipe. Coil probes were designed to always be located in the middle of the tube, and the frequency 20 kHz was relatively low, so the effects from the pipe's exterior could be reflected. As shown in Figure 7, increasing the ovality brought the coil probe closer to the inside of the pipe as the pipe deformed. Although the deformation of the piping was mostly concentrated in the extrados region depending on the location, it also partially affected the intrados. The curve begins at 0\({}^{\circ}\), and due to the presence of residual stress during the manufacturing process, we found that the current density increased slightly compared to the extrados due to the slight deformation of the intrados. However, at the 29\({}^{\circ}\) and 45\({}^{\circ}\) positions of the curve, we found that the current density increased due to the concentration of deformation in the extrados position. Most of the curved sections exhibited a similar overall pattern because of this deformation of the extrados, which can lead to changes in reactance and resistance in the eddy-current test. Figure 8 shows a three-dimensional analysis of changes in the electric field depending on the coil position at a frequency of 300 kHz. The ovality of the pipe is close to 0.3% at 0\({}^{\circ}\), and the field shown in the coil and pipe is distributed evenly in the direction around the pipe. However, the higher ovality of the pipe resulted in deformation of the inner/external section area, which also partially concentrated the electric field. This is caused by the coil probe entering and leaving the curved section where the electric tube begins to change shape, and by the close distance between the coil and pipe in the inner and outer surface areas. A close area is a sign that that the outer arc surface has contacted the concave section surrounding the coil, and the inner arc surface has contacted the coil in a convex and relatively small area. We predicted that this phenomenon is mainly caused by the curved part of the tube, and would affect the electric field in the coil. Figure 7: The results of the simulations of current density at angle positions: (**a**) 0\({}^{\circ}\), (**b**) 29\({}^{\circ}\), and (**c**) 45\({}^{\circ}\). Figures 7 and 8 show that it is difficult to directly compare the amplitude effects on each section because only the current density and electric field effects on the electric tubes generated by the coils are mainly identified. Therefore, it was necessary to analyze the effects of noise at each location to extract the values of the effect and obtain analysis results. Figure 9 shows the reactance change according to the position of the coil in the pipe. The frequency of the general test, between 20 and 300 kHz, showed little change in the value of the reactance due to the difference in ovality, while a difference of about 0.25 was found at a given frequency (550 kHz).