## 3 Synthesized Transmitting Coil
### Optimization of the STC Parameters Using FoM  
In order to optimize the STC parameters, we made a simulation to analyze the MF performance using the STC, MTC, and no-MF. Figure 3 shows the simulation of the longitudinal component of the magnetic field \(Hz\) with different winding schemes using the magnetic module of COMSOL 6.0. A standard 5 in casing was used with a wall thickness of 7.52 mm and inner radius \(r=55.98\) mm, where \(t=0\) corresponds to the time when the electrical field was turned off. A three-section STC with 300 turns in total was considered. The radius and wire diameter of the transmitting coils were 12 and 0.33 mm, respectively. The second section (S\({}_{2}\)) of the STC was a P-section (with a clockwise winding direction), and S\({}_{1}\) and S\({}_{3}\) were N-sections (with an anticlockwise winding direction) with the same number of turns (\(n_{1}=n_{3}\)). The proportions for the P- and N-sections of the STC are representedby gray and black rectangular blocks, respectively. The longitudinal distribution of the magnetic field along the inner wall of the casing was simulated when \(n_{2}=240\) (80% of the total number of turns), 180 (60%), 120 (40%), and 60 (20%). The no-MF and MTC-based MF [17] cases are shown for comparison. The simulation clearly showed that the magnetic field distribution can be changed according to the winding direction of the STC. The MTCs were set to a total number of 300 turns; although the STC had almost the same performance as the MTC-based MF, there should be some performance loss because the MTCs can adjust the transmitting current more finely in theory. To analyze the MF performance, we followed the lead of [PERSON] et al. [14] and defined the main lobe width as the distance between the two nearest intersections of the \(-3\) dB line and the curves. Figure 3a shows that the STC obtained a narrower main lobe than the no-MF case when \(n_{2}=240\). As \(n_{2}\) was decreased, the main lobe became much narrower, but side lobes appeared, as shown in Figure 3b. At \(n_{2}=120\), the height of the side lobe exceeded that of the main lobe, as shown in Figure 3c. At \(n_{2}=60\), the main lobe became wider than that of the no-MF case and was even more deformed, as shown in Figure 3d. Therefore, the proposed STC can obtain a narrower main lobe than the no-MF case, but the appropriate winding parameters need to be determined. Similar to [PERSON] et al. [15], we defined an FoM to evaluate the MF performance. We then determined the optimal STC design according to the combination of winding parameters that realized the largest FoM. The MF performance is related to the width of the main lobe and height of the side lobe. The ratios of the side lobe height, \(h_{\text{s}}\), and reduction in the main lobe width, \(w_{\text{r}}\), can be defined as follows:  
\[h_{\text{s}}=\frac{h_{\text{side lobe}}}{h_{\text{main lobe}}} \tag{10}\]  
Figure 3: Longitudinal magnetic field distribution at \(r=55.98\) mm and \(t=0\) of the three-section STC: (a) \(n_{2}=240\), (b) \(n_{2}=180\), (c) \(n_{2}=120\), and (d) \(n_{2}=60\). \[w_{\rm r}=\frac{w\big{|}_{\text{non-focusing}}-w\big{|}_{\text{focusing}}}{w\big{|}_{ \text{non-focusing}}} \tag{11}\]  
where \(h\) denotes the height of the main lobe or side lobe, and \(w\) denotes the width of the main lobe in the no-MF, MTC-based MF, or STC-based MF cases. The MF performance is optimized by decreasing \(h_{\rm s}\) and increasing \(w_{\rm r}\). Therefore, the following FoM was defined:  
\[\text{FoM}=\sqrt{w_{\rm r}}\times(1-h_{\rm s}) \tag{12}\]  
where a square root is applied to \(w_{\rm r}\) to increase its contribution to the FoM because this parameter directly affects the longitudinal distribution of the magnetic field. Note that when the main lobe width is widened (i.e., \(w_{\rm r}\) is negative, as shown in Figure 3d), the MF performance is poor. This was outside the scope of this study, and the FoM was set to zero in this case.