## 3 Parameter Characteristics of the Sensor  
Through the finite element simulation, MP-SAWS's admittance curve is illustrated and shown in Figure 4a. In circuit theory, admittance refers to the reciprocal of impedance. It describes the resistance to the flow of an electric current through the electric system. During AC scanning, admittance serves as an indicator of the system's operational mode. In the case of SAWSs, there are two characteristic frequencies at short circuits: the resonant frequency with minimum impedance and the anti-resonant frequency with maximum impedance. The admittance curve in Figure 4a includes information on the vibration mode and deformation degree of the SAWS at the resonant and anti-resonant frequencies. According to Figure 4a, the maximum value of admittance occurs at 1.8852 GHz, which corresponds to the resonant frequency. Conversely, the anti-resonant frequency is found to be 1.9510 GHz. The correspondence among the phase velocity, wavelength, and resonant frequency conforms to Equation (15) as follows:  
\[v=f\cdot\lambda \tag{15}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Material** & **Terfenol-D** \\ \hline Density, \(\rho\) [kg/m\({}^{3}\)] & 9250 \\ Saturation magnetostriction, \(\lambda_{s}\) [ppm] & 2000 \\ Poisson’s ratio, \(\mu\) & 0.45 \\ Electric conductivity, \(\rho\) [S/m] & \(1.7\times\ 10^{6}\) \\ Young’s modulus, E [Pa] & 60 \(\times\ 10^{9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameters of magnetostrictive materials. Figure 4: Frequency-domain response curve of the MP-SAWS. (**a**) Admittance curve and corresponding deformation pattern at resonance and anti-resonance frequencies; (**b**) wave velocity curve; (**c**) S11 reflection coefficient plotted in decibels; and (**d**) mechanical quality factor (Qm) curve. Thus, the initial phase velocity of the SAWS was determined to be 3770.4 m/s, as shown in Figure 4b. The scattering parameters are used to describe the frequency-domain characteristics of transmission channels, also known as S-parameters. They can be used to calculate various indicators, such as input impedance and frequency response, by measuring the ratio of incident and reflected waves. Thus, S-parameters play a crucial role in studying the performance of SAW resonators. In a two-port network, the S-parameters are categorized into S11, S21, S12, and S22. For multi-port networks, the S-parameters include additional combinations and can form an S-parameter matrix. The S11 parameter curve of MP-SAWS is shown in Figure 4c. S11 is the input reflection coefficient, which is an important parameter for describing the performance of a single-port resonator. The smoothness of the S11 parameter curve provides a qualitative analysis of the reflectivity of the resonator. As the curve represents the reflection loss (return loss), smaller values indicate lower levels of reflection and higher transmission coefficients. The relationship between the S11 coefficient and the reflectivity of MP-SAWS is as follows [61]:  
\[S11=\frac{V_{1}^{-}}{V_{1}^{+}},\ P=\frac{V^{2}}{R} \tag{16}\]  
\[\frac{P_{R}}{P_{I}}(dB)=10 lg\frac{P_{R}}{P_{I}} \tag{17}\]  
\[S11(dB)=10 lgS11^{2}=20 lgS11 \tag{18}\]  
In this set of equations, R represents the resistance value; \(V_{1}^{+}\) and \(V_{1}^{-}\) denote the input voltage signal and output voltage signal (reflection signal) at the resonator port; and \(P_{R}\) and \(P_{I}\) represent the reflection power and the incident power of SAW. Conversion from numerical values to decibels is achieved by taking the logarithm. The S11 parameter curve indicates that the MP-SAWS exhibits a strong response near 1.8852 GHz, displaying a low valley peak. In this frequency range with high-transmission capability, the curve exhibits minimal side lobes and noise, and the amplitude can reach \(-100\) dB. There is also a strong correspondence between the admittance curve and the S11 curve.