# Development of a Magnetostrictive FeNi Coated Surface Acoustic Wave Current Sensor
## 2 Simulation on the Sensor Response
\right)\mathbf{,}\,\mathbf{R}_{3}^{II}=\left( \begin{array} []{ccc}c_{11}-\frac{\rho^{L}}{s^{2}}&0&0\\ 0&c_{44}-\frac{\rho^{L}}{s^{2}}&0\\ 0&0&c_{44}-\frac{\rho^{L}}{s^{2}}\end{array} \right)\]  
Similarly, the conditions for existence of non-trivial solution in Christoffel equation is that the determinant of the coefficient matrix (\(\Gamma\)) should be zero, that is, \(\det(\Gamma)=0\). A six degree equation about \(\alpha^{I}\) can be deduced. Since the thickness of the SiO\({}_{2}\) film is limited, the six roots of \(\alpha_{p}^{L}\) (\(p=1,\,2,\,\ldots,\,6\)) all satisfy the Christoffel equation, and the integral form of the displacement in the frequency slowness domain is expressed by  
\[\vec{u}_{i}^{II}(s,x_{3})=\sum_{p=1}^{6}C_{p}^{II}\left(A_{i}^{II}\right)^{(p)} \exp\left(-j\alpha_{p}^{II}\omega sx_{3}\right) \tag{8}\]  
where \(i=1,\,2,\,3\). \(A_{i}^{II}\) is eigenvectors corresponding to \(a_{p}^{II}\), \(C_{p}^{II}\) (\(p=1,\,2,\,\ldots,\,6\)) are the weighted coefficients, and determined by the boundary conditions. The solution in FeNi thin film is similar to SiO\({}_{2}\). And the corresponding integral solution of the displacement in the frequency slowness domain of the FeCo film can be expressed as  
\[\widetilde{u}_{i}^{III}(x_{1},x_{3})=\sum_{p=1}^{6}C_{p}^{III}\left(A_{i}^{III} \right)^{(p)}\exp\left(-j\alpha_{p}^{III}\omega sx_{3}\right) \tag{9}\]  
where, the corresponding six eigen values and eigenvectors in FeNi film are expressed as \(a_{p}^{III}\) and \(A_{p}^{III}\) (\(p=1\), \(2\), \(3\), \(4\), \(5\), \(6\)), and \(C_{n}^{III}\) (\(n=1\), \(2\), \(3\), \(4\)) are the weighted coefficients. There is no free charge distribution in the region of \(0<x_{3}<h_{1}\), \(h_{1}<x_{3}<h_{1}+h_{2}\), and \(h_{1}+h_{2}<x_{3}\), which are marked by _II, III_ and \(G\), respectively. Then the potentials \(\Phi^{II}\), \(\Phi^{III}\) and \(\Phi^{G}\) satisfying the Laplace equation \(\
abla^{2}\Phi(x_{1},x_{3})=0\) are introduced to meet the electrical boundary conditions, and the electric displacement components \(D_{3}\) in the \(x_{3}\) direction are determined in the corresponding regions. \(\Phi^{II}\), \(\Phi^{III}\), \(\Phi^{G}\), \(D_{3}^{II}\), \(D_{3}^{III}\) and \(D_{3}^{G}\) in the frequency slowness field are expressed by  
\[\left\{ \begin{array}{l}\widetilde{\Phi}^{II}=B_{1}^{II}\exp(-\omega sx_{3})+ B_{2}^{II}\exp(\omega sx_{3})\\ \widetilde{\[PERSON]}^{III}=B_{1}^{III}\exp(-\omega sx_{3})+B_{2}^{III}\exp(\omega sx _{3})\\ \widetilde{\[PERSON]}^{G}=B^{G}\exp(-\omega sx_{3})\\ \widetilde{D}_{3}^{II}=-jk_{s}\left[je^{II}B_{1}^{II}\exp(-\omega sx_{3})-je^ {II}B_{2}^{II}\exp(\omega sx_{3})\right]\\ \widetilde{D}_{3}^{III}=-jk_{s}\left[je^{II}B_{1}^{III}\exp(-\omega sx_{3})-je^ {II}B_{2}^{III}\exp(\omega sx_{3})\right]\\ \widetilde{D}_{3}^{G}=-jk_{s}\left[je_{\epsilon 0}B^{G}\exp(-\omega sx_{3}) \right]\end{array} \right. \tag{10}\]  
where, \(\epsilon^{II}\), \(\epsilon^{III}\) and \(\epsilon_{0}\) are the dielectric constant of the SiO\({}_{2}\) layer, FeNi layer and the air dielectric constant, respectively, \(\epsilon_{0}=8.854\times 10^{-12}\) (F/m). The boundary conditions for each surface of the layered media structure shown in Figure 2 are described as below. Firstly, the stress, displacement, and electric potential are continuous at \(x_{3}=0\) and \(x_{3}=h_{1}\), that is,  
\[\left\{ \begin{array}{l}\widetilde{T}_{3j}-\widetilde{T}_{3j}^{L}=0\\ \widetilde{u}_{i}-\widetilde{u}_{i}^{L}=0\\ \widetilde{\Phi}-\widetilde{\Phi}^{L}=0\\ \widetilde{D}_{3}-\widetilde{D}_{3}^{L}=-\widetilde{\sigma}(s)\\ \widetilde{T}_{3j}^{II}-\widetilde{T}_{3j}^{III}=0\\ \widetilde{u}_{i}^{II}-\widetilde{u}_{i}^{III}=0\\ \widetilde{\Phi}^{II}-\widetilde{\Phi}^{III}=0\\ \widetilde{D}_{3}^{II}-\widetilde{D}_{3}^{III}=0\end{array} \right.,\,i,j=1,\, 2,\,3 \tag{11}\]  
The normal component of the stress at \(x_{3}=h_{1}+h_{2}\) is considered as zero, and the corresponding displacement and electric potential are continuous as  
\[\left\{