## 2 Materials and Methods
### Magnetstrictive Coupling Modeling  
MP-SAWS is capable of measuring magnetic fields by coupling with magnetostrictive materials. During testing, MP-SAWS can detect changes in pressure to conduct the sensing test. The magnetostrictive material is affixed to the side of the SAWS base. When an external magnetic field is present, the magnetostrictive material undergoes deformation and extrudes the LiNbO\({}_{3}\) substrate. After SAWS detects different degrees of extrusion, the output results expressed in their characteristic frequency are utilized to obtain the corresponding sensing relationship between the magnetic field and characteristic frequency. Within the magnetostrictive module, the relationship between stress and strain with the magnetic field is represented by the following [60]:  
\[\mathrm{S}=C_{H}:(\varepsilon-\varepsilon_{me}(M)) \tag{8}\]  
\[\varepsilon_{me}=\frac{3\lambda_{s}}{2M_{s}^{2}}dev(M\otimes M) \tag{9}\]  
The non-linear magnetization behavior observed in the magnetostrictive material arises from its inherent non-linear constitutive relationship, expressed as follows:  
\[\mathrm{M}=M_{s}L(\left|H_{eff}\right|)\ \frac{H_{eff}}{\left|H_{eff}\right|} \tag{10}\]  
\[\mathrm{L}=\coth\left(\frac{3\chi_{0}\left|H_{eff}\right|}{M_{s}}\right)- \frac{M_{s}}{3\chi_{0}\left|H_{eff}\right|} \tag{11}\]  
\[H_{eff}=\mathrm{H}+\frac{3\lambda_{s}}{\mu_{0}M_{s}^{2}}dev(S)M \tag{12}\]  
S represents the stress, the elastic tensor \(C_{H}\) is determined by Young's modulus and Poisson's ratio, M represents the magnetization field, \(\lambda_{s}\) is the saturation magnetostriction coefficient, \(M_{s}\) is the saturation magnetization, L is the Langevin function, \(\chi_{0}\) is magnetic susceptibility in the initial linear region, and \(H_{eff}\) is the effective magnetic field in the material. In the simulation, only the response of the magnetostrictive material to a unidirectional magnetic field is considered; although Terfenol-D is not an isotropic material, the model can be simplified and treated as isotropic. In practical use, the easy-axis direction should be the one actually applied for the measurement. For potential errors, the actual value of the material's \(\lambda_{s}\) may be lower, which could affect the final sensitivity. The magnetostrictive module uses these equations to obtain the device's response to physical quantities such as magnetic field and stress through finite element analysis methods. The extra symbols are explained in the following equations:  
\[\mathrm{A}:\mathrm{B}=\sum_{i=1}\sum_{j=1}A_{ij}B_{ij} \tag{13}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Electrode Group** & **Al (IDT)** \\ \hline Odd & +1 V \\ Even & Ground \\ \hline \hline \end{tabular}
\end{table}
Table 6: Electrode group settings.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Boundary** & **Mechanical Condition** & **Electrical Condition** \\ \hline \(\Gamma_{1}\) & Free & Zero-charged \\ \(\Gamma_{2}\) & Fixed & Grounded \\ \hline \hline \end{tabular}
\end{table}
Table 5: Boundary settings.  
\[\left(\mathrm{M}\otimes\mathrm{M}\right)_{ij}=M_{i}M_{j} \tag{14}\]  
The parameters of the magnetostrictive material are listed in Table 7.