# Development of a Magnetostrictive FeNi Coated Surface Acoustic Wave Current Sensor
## 2 Simulation on the Sensor Response
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Materials Parameters**} & \multicolumn{4}{c}{**Values**} \\ \cline{2-4}  & & **LiNbO\({}_{3}\)** & **SiO\({}_{2}\)** & **FeNi** \\ \hline  & \(C_{11}\) & 2.03 & 0.785 & 1.077 \\  & \(C_{12}\) & 0.53 & & 0.46 \\  & \(C_{13}\) & 0.75 & & \\ Stiffness Coefficients (\(10^{11}\) N m\({}^{-2}\)) & \(C_{14}\) & 0.09 & & \\  & \(C_{33}\) & 2.45 & & \\  & \(C_{44}\) & 0.6 & 0.312 & 0.307 \\ \hline  & \(e_{31}\) & 0.2 & & \\ Piezoelectric Modules (C m\({}^{-2}\)) & \(e_{15}\) & 3.7 & & \\  & \(e_{33}\) & 1.3 & & \\  & \(e_{22}\) & 2.5 & & \\ \hline Permittivity Constants (\(10^{-11}\) F m\({}^{-1}\)) & \(\begin{array}{c}\varepsilon_{11}\\ \varepsilon_{33}\end{array}\) & 84 & & \\  & \(\varepsilon_{33}\) & 29 & & \\ \hline Density (kg/m\({}^{3}\)) & & 7450 & 2651 & 8.38 \\ \hline Magnetosrictive Constant (\(10^{-6}\)) & & & 400 \\ \hline Young’s Modulus (\(10^{9}\) N/m\({}^{2}\)) & & & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of the piezoelectric substrate, SiO\({}_{2}\), and FeNi. \[\mathbf{R}_{3}^{I}=\left( \begin{array}{cccc}c_{11}-\frac{\rho}{s^{2}}&c_{16}&c_{ 15}&e_{x1}\\ c_{61}&c_{66}-\frac{\rho}{s^{2}}&c_{65}&e_{x6}\\ c_{51}&c_{56}&c_{55}-\frac{\rho}{s^{2}}&e_{x5}\\ e_{x1}&e_{x6}&e_{x5}&-e_{xx}\end{array} \right)\]  
where, \(\alpha^{I}\) is the characteristic root, and \(A^{I}_{1,\,2,\,3,\,4}\) are undetermined constants, i.e., eigenvectors. The necessary condition for the existence of the non-trivial solution in Christoffel equation is \(\det(\Gamma)=0\), and an eight degree equation about \(\alpha^{I}\) was obtained. \(\Gamma\) is the determinant of the coefficient matrix. To ensure the decrease of the acoustic displacement into the substrate, four roots of \(\alpha^{I}\) with imaginary part more than zero were picked, that is, \(\alpha^{I}_{\,n}\) (\(n=1,\,2,\,3,\,4\)). Then, Substituting \(\alpha^{I}_{\,n}\) into the Christoffel equation, a set of eigenvectors corresponding to each \(\alpha^{I}_{\,n}\) are deduced. It is clear that only the linear superposition of the solutions as Equation (2) corresponding to \(\alpha^{I}_{\,n}\) (\(n=1,\,2,\,3,\,4\)) can satisfy the Christoffel equation and the boundary conditions. Thus, the integral solution was formed as the linear superposition of the wavelet solution, that is,  
\[\left( \begin{array}{c}\vec{u}_{1}^{I}(s,x_{3})\\ \vec{u}_{2}^{I}(s,x_{3})\\ \vec{u}_{3}^{I}(s,x_{3})\\ \vec{\phi}^{I}(s,x_{3})\end{array} \right)=\left( \begin{array}{cccc}A_{11}^{I }&A_{12}^{I}&A_{13}^{I}&A_{14}^{I}\\ A_{21}^{I}&A_{22}^{I}&A_{23}^{I}&A_{24}^{I}\\ A_{31}^{I}&A_{32}^{I}&A_{33}^{I}&A_{34}^{I}\\ A_{41}^{I}&A_{42}^{I}&A_{43}^{I}&A_{44}^{I}\end{array} \right)\left( \begin{array} []{c}C_{1}^{I}\exp(-j\alpha_{1}\omega sx_{3})\\ C_{2}^{I}\exp(-j\alpha_{2}\omega sx_{3})\\ C_{3}^{I}\exp(-j\alpha_{3}\omega sx_{3})\\ C_{4}^{I}\exp(-j\alpha_{4}\omega sx_{3})\end{array} \right) \tag{4}\]  
where \(C_{n}^{I}\) (\(n=1,\,2,\,3,\,4\)) is the weighted factor which determined by the boundary conditions. Usually, the SiO\({}_{2}\) and FeNi can be considered as isotropic media, and their typical matrix representation of acoustic wave equations are defined by  
\[\
abla_{iK}c_{KL}^{E}\
abla_{Lj}u_{j}=\rho\frac{\partial^{2}u_{i}}{\partial t ^{2}},\,i,j=1,\,2,\,3,\,K,L=1,\,2,\ldots,6, \tag{5}\]  
So, the integral solution in acoustic equation of the isotropic SiO\({}_{2}\) film is assumed as  
\[u_{i}^{II}(x_{1},x_{3})=A_{i}^{II}\exp\left(-j\alpha_{p}^{II}\omega sx_{3}\right) \tag{6}\]  
Then, substituting Equation (6) into the wave Equation (5), the corresponding Christoffel equation is obtained by  
\[\left(\left(\alpha^{II}\right)^{2}\mathbf{R}_{1}^{II}+\alpha^{II}\mathbf{R}_{2 }^{II}+R_{3}^{II}\right)\left( \begin{array}{c}A_{11}^{II}\\ A_{2}^{II}\\ A_{3}^{II}\end{array} \right)=0 \tag{7}\]  
\[\mathbf{R}_{1}^{II}=\left( \begin{array}{ccc}c_{44}&0&0\\ 0&c_{44}&0\\ 0&0&c_{11}\end{array} \right)\mathbf{,}\,\mathbf{R}_{2}^{II}=\left( \begin{array} []{ccc}0&0&c_{44}\\ 0&0&0\\ c_{44}&0&0\end{array}