## 4 Theoretical Analysis and Field Distribution
### Comparison of the Typical Performances  
In this research, four different stimulation coils were compared. Figure 10 shows a comparison of the distribution of the magnetic flux densities, respectively, for the proposed coil, a figure-8 quad coil, a figure-8 coil, and a regular round coil, where the four coils had the same diameter and excitation signal. The results showed that the proposed coil achieves a higher stimulation intensity and depth.  
In terms of the induced electric field, the distributions of the electric field are simulated by Equation (7). As shown in Figure 11, the electrical field strength of the proposed coil at 8 cm beyond the coil plane is better than that of the other three coils.  
Figure 10: Comparison of the magnetic flux density distribution between the proposed coil and the typical coils. (**a**) The proposed coil, (**b**) figure-8 quad coil, (**c**) figure-8 coil, and (**d**) round coil.  
Figure 9: Comparison of the measurements and the simulated calculations of the magnetic flux density.  
Another comparison of the four coils (8 cm beyond the coil plane) when it comes to the magnetic flux density was carried out. As illustrated in Figure 12, the maximum value of the magnetic flux density of the proposed coil was 26.1 mT and the figure-8 quad coil, figure-8 coil, and round coil were 20.4 mT, 13.0 mT, and 13.9 mT, respectively.  
The facility is evaluated by the full width at half maximum (FWHM) [14]. FWHM is a parameter that is commonly used to describe the level of power concentration. It is given by the distance between the points on the curve, which is greater than \(1/\sqrt{2}\) (3 dB decay point) of its maximum magnitude.  
As shown in Figure 13, the FWHM of the electrical field of the different coils beyond the coil plane close to the cortex is proposed. With the increase in the depth inside the brain, the proposed coil has the smallest FWHM (lower is better).  
Figure 11: Comparison of the electrical field between the four coils.  
Figure 12: Comparison of the magnetic flux density between the four coils.  
A comparison of the electric fields at different stimulation depths is shown in Figure 14. For the same stimulation depth, the proposed coil could induce a higher electric field.  
The comparison between FWHM, electric field, and magnetic flux density of the four types of coil at the same depth is shown in Table 3. According to the performed simulations and comparisons, the results show that the proposed coil has the highest field strength and best focality.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Electrical Field** & **Magnetic Flux Density** & **FWHM** \\  & **(V/m)** & **(mT)** & **(cm)** \\ \hline Proposed coil & 120.6 & 26.1 & 11.7 \\ Figure-8 quad coil & 97.9 & 20.4 & 12.3 \\ Figure-8 coil & 58.7 & 13.0 & 11.9 \\ Round coil & 3.3 & 13.9 & 14.7 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The comparison of the four types of the coil at the same depth (8 cm beyond the coil plane).  
Figure 14: Comparison of the electric fields at different stimulation depths.  
Figure 13: Comparison of the full width at half maximum (FWHM) between the four coils.