## 3 Numerical Simulation
### Simulation Verification and Factor Analysis  
The typical dynamic magnetic field distributions of MPTs with and without the SMPs at 85 kHz in wave generation and detection are shown in Figure 7. With the SMP, the magnetic flux lines can be restricted in the narrow area around the coils instead of dispersing in the air region. Figure 8 shows the amplitudes of the magnetic field intensities along the \(x\)-axis from Line 1 and Line 2, respectively. The dynamic magnetic field existing in the magnetostrictive patch are right below the coils with currents. In wave detection, the high peaks are magnetic intensities in the coils. The dynamic magnetic field in the magnetostrictive patch was enhanced more than three times with the SMP and the induced magnetic fields in the coils were also stronger. The results shown in Figure 9 confirmed the enhancement of the SMPs at other frequencies. It should be mentioned that the sharp fall in Figure 9 may have been caused by different current loadings as shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material Name** & **Air** & **Aluminum** & **Copper** & **Fe–Co Alloy** & **Mn–Zn Ferrite** \\ \hline Relative permeability & 1 & 1 & 1 & 2000 & 2800 \\ \hline Conductivity (S/m) & 0 & 3.774 \(\times\) 10\({}^{7}\) & 5.988 \(\times\) 10\({}^{7}\) & 2.5 \(\times\) 10\({}^{6  
For the proper selection of the SMP, several properties, including the relative permeability, thickness (t\({}_{\text{s}}\)) and lift-off distance (g\({}_{\text{sc}}\)), were studied with the other parameters of the model having fixed values. The typical results at 55 kHz, 85 kHz, 115 kHz and 145 kHz are shown in Figures 10-12. The magnetic fields get stronger with an increase in the relative permeability of the SMP, as shown in Figure 10a, which matches the theoretical prediction in Section 2.2. However, when the relative permeability is larger than 2000, the influence of relative permeability changes is subtle. The lower limit of the relative permeability for maximum magnetic field intensity in wave detection is 100. The dynamic magnetic field attenuates exponentially during the propagation in a good conductor so it is only distributed near the surface of the conductor (i.e., skin effect [37]). With this effect, the thickness of the SMP can be very small (0.01\(\sim\)0.6 mm in the simulation). The improvement of magnetic fields by an increase in the thickness was verified from the curves of 55 kHz and 85 kHz in Figure 11a, which did not increase significantly when the thickness was over 0.15 mm. For other cases at higher frequencies or in wave detection, it reaches the maximum for higher frequencies at the minimum thickness. The distance between the SMP and coils (i.e., the lift-off distance in this paper) will affect the improvement of the SMP if something necessarily has to be put in between (e.g., a second insulation layer). The influence of the SMP gets smaller naturally when the SMP gets further away from the coils. It can be explained with the increment of the magnetic resistance in the magnetic circuits over the lift-off distance as described in Section 2.2. Figure 12 shows the results for different lift-off distances. It could be found that during the lift-off of the SMP, the influence attenuation of the SMP continued until a long distance (2.5 mm) in wave generation, but the influence remained constant when the SMP was 0.5 mm away in wave detection.  
Figure 10: MPTs with the SMPs of different relative permeabilities: (**a**) The magnetic field intensity in Line 1 in wave generation; and (**b**) the magnetic field intensity in Line 2 in wave detection.  
Figure 9: Comparisons of MPTs with and without the SMPs at 55–150 kHz: (**a**) The magnetic field intensity in Line 1 in wave generation; and (**b**) the magnetic field in Line 2 in wave detection.