# Development of a Magnetostrictive FeNi Coated Surface Acoustic Wave Current Sensor
## 2 Simulation on the Sensor Response
\begin{array}{l}\widetilde{T}_{3j}^{III}=0\\ \widetilde{\Phi}^{III}-\widetilde{\Phi}^{G}=0\\ \widetilde{D}_{3}^{III}-\widetilde{D}_{3}^{G}=0\end{array} \right. \tag{12}\]  
To sum up, there are 21 undetermined unknowns in the Christoffel equation expressed by \(C_{n}^{I}(n=1,2,3,4)\), \(C_{p}^{II}(p=1,2,3,4,5,6)\), \(C_{p}^{III}(p=1,2,3,4,5,6)\), \(B_{1}^{II}\), \(B_{2}^{II}\), \(B^{\mathbf{G}}\), \(B_{1}^{III}\) and \(B_{2}^{III}\). It means that there are 21 equations to determine the unknowns. To simply the calculation, the approach of effective dielectric constant was used. The effective dielectric constant at the boundary of \(x_{3}=0\) is defined by [22]  
\[\varepsilon_{s}(s)=\frac{\widetilde{\sigma}(s)}{2\pi f|s|\widetilde{\Phi}(s,x_ {3})|_{x_{3}=0}} \tag{13}\]Accordingly, the \(\varepsilon_{s}(s)\) value corresponding different slowness \(s\) can be deduced, and also the surface effective dielectric constant \(\varepsilon_{s}(s)\) curve can be pictured. Usually, \(\varepsilon_{s}(s)\) is a complex number. The slowness \(s_{m}\) corresponding to the maximum value of the real part shows the case of metallization at \(x_{3}=0\). While the slowness \(s_{l}\) corresponding to the zero-crossing of the real part shows the free surface at \(x_{3}=0\). According to the reciprocal relationship, the corresponding SAW velocity in metalized surface \(v_{m}\), and velocity in free surface, \(v_{l}\), at \(x_{3}=0\), can be obtained, respectively. Usually, the magnetostriction of FeNi film will be changed by its aspect ratio [23],  
\[\beta=\beta_{0}\big{(}1+p\cdot k_{p}\big{)} \tag{14}\]  
where \(\beta\) and \(\beta_{0}\) are the perturbed and unperturbed magnetostriction of the FeNi film, respectively, and \(p=l/w\) is the aspect ratio, \(l\) and \(w\) are the length and width of FeNi film, respectively. \(k_{p}\) is the correlation coefficient, and is set to 7.13 \(\times\) 10\({}^{-4}\)[23]. Owing to the magnetostrictive nature, the FeNi film would be magnetized under the electromagnetic field, resulting in the changes in FeNi film thickness \(h\) and density \(\rho\), which can be written by  
\[\left\{ \begin{array}{l}h=h_{2}(1-\frac{\beta}{2})\\ \rho=m/[a_{0}\times(1+\frac{\beta}{2})\times b_{0}\times(1-\frac{\beta}{2}) \times h_{2}\times(1-\frac{\beta}{2})]\end{array} \right. \tag{15}\]  
where \(m\) is the mass of the FeNi film, \(a_{0}\), \(b_{0}\), \(h_{2}\) are the initial length, width and thickness of unperturbed FeNi, respectively, and obviously \(m=\rho^{III}\times(a_{0}\times b_{0}\times h_{2})\). Moreover, the young's modulus (\(E\)) of the FeNi will also be modulated by the electromagnetic field intensity (\(H\)), and can be taken experimentally by using the laser resonance method [24]. Correspondingly, the elastic coefficient of the FeNi will also be expressed as  
\[\left\{ \begin{array}{l}c_{11}{}^{F}=\frac{E\times(1-u_{F})}{(1+u_{F})(1-2u_ {F})}\\ c_{12}{}^{F}=\frac{E\times u_{F}}{(1+u_{F})(1-2u_{F})}\\ c_{44}{}^{F}=\frac{E\times(1-2u_{F})}{2(1+u_{F})(1-2u_{F})}\end{array} \right. \tag{16}\]  
where the Poissoncoefficient of FeNi, \(u_{F}\), was considered as constants irrelevant to the electromagnetic field, and it was set to 0.3. Hence, using above equations and the measured relationship between the young's modulus of FeNi and the magnetic field (Figure 3), the shift in SAW velocity shift depending on the applied electromagnetic field can be calculated. Also, considering the relationship between the electromagnetic field intensity (\(H\)) and current (A) described by the Biot-Savart law [25],  
\[H=\mu_{0}I/(4\pi l) \tag{17}\]  
where, \(\mu_{0}\) denotes the permeability in vacuum. \(l\) expresses the distance between the sensor chip and wires, and is set to 1 cm in this calculation, the sensitivity in current sensing can be evaluated theoretically.