# Development of a Magnetostrictive FeNi Coated Surface Acoustic Wave Current Sensor
## 2 Simulation on the Sensor Response  
Prior to sensor device development, the sensor response was simulated theoretically to extract the optimal design parameters. In the simulation, the SAW sensor was simplified as a structure composed of a semi-infinite piezoelectric substrate with IDT pattern, a SiO\({}_{2}\) coating, and a magnetostrictive FeNi thin film as shown in Figure 2. Also, the theoretical simulation is referring to the way of an analytical resolution of acoustic motion equations in layered structure considering the magnetostrictive effect described in [12], so the influences of different parameters on sensor performances appear clearly. The corresponding coordinate system is shown in Figure 2. Acoustic wave propagates along the \(x_{1}\) axis on the \(x_{1}-x_{2}\) plane at \(x_{3}=0\). All material parameters of the mediums are transformed into the coordinate system. Each material in the layered structure is defined by its physical parameters: stiffness constants (\(c^{I}_{i\beta t}\)), piezoelectric constant (\(\epsilon_{kij}\)), permittivity constant (\(\epsilon_{ij}\)), and density (\(\rho^{I}\)) for LiNbO\({}_{3}\), stiffness constants ((\(c^{II}_{i\beta t}\))) and density (\(\rho^{II}\)) for SiO\({}_{2}\), Young's modulus (\(E\)) and density (\(\rho^{III}\)) for FeNi. \(h_{1}\) and \(h_{2}\) express the thicknesses of SiO\({}_{2}\) and FeNi, respectively. The corresponding physical parameters for piezoelectric substrate, SiO\({}_{2}\), and FeNi are listed in Table 1[20]. The SAW velocity relating to the electromagnetic field intensity can be calculated by solving the elastic wave equations of each media utilizing corresponding boundary conditions. The following is the specific theoretical simulation. Figure 1: The scheme and working principle of the surface acoustic wave (SAW) current sensor. Figure 2: The corresponding coordinate system in this work. It is well known that the constitutive equations in a piezoelectric medium are expressed as in [21]. \[\left\{ \begin{array}{l}\rho\frac{\partial^{2}u_{i}}{\partial t^{2}}-\
abla_{iK}c _{KL}^{I}\
abla_{Lj}u_{j}-\
abla_{iK}e_{Kj}\
abla_{j}\Phi=0\\ \
abla_{i}e_{iL}\left(\
abla_{Lj}u_{j}\right)-\
abla_{i}e_{ij}\
abla_{j}\Phi=0 \end{array} \right. \tag{1}\]  
where \(i,j\) = 1, 2, 3, \(K\), \(L\) = 1, 2,... 6, \(\rho\), \(u_{i}\) and \(\Phi\) are the density, displacement and electric potential in YZ-LiNbO\({}_{3}\), respectively. The field space of the frequency domain and the frequency slowness field (\(s\)) can be characterized by Fourier transform pairs, hence, a general solution of the coupled wave Equation (1) was assumed in the slowness (\(s\)) region as  
\[\left\{ \begin{array}{l}\widetilde{u}_{i}^{I}(s,x_{3})=A_{i}^{I}\exp\bigl{(}- j\alpha^{I}\omega sx_{3}\bigr{)}\\ \widetilde{\Phi}^{I}(s,x_{3})=A_{4}^{I}\exp\bigl{(}-j\alpha^{I}I\omega sx_{3} \bigr{)}\end{array} \right. \tag{2}\]  
where, \(s\) is the SAW slowness, and defined by the reciprocal of the SAW velocity, \(\alpha\) is the acoustic attenuation factor in \(x_{3}\) direction. Substituting Equation (2) into the coupling wave Equation (1), and the Christoffel equation of the piezoelectric substrate in the slowness field is obtained, which is a quadratic polynomial of \(\alpha\), and expressed by  
\[\left((\alpha)^{2}\mathbf{R}_{1}^{I}+\alpha\mathbf{R}_{2}^{I}+\mathbf{R}_{3}^{I}\right) \left( \begin{array}{c}A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\end{array} \right)=0 \tag{3}\]  
\[\mathbf{R}_{1}^{I}=\left( \begin{array}{cccc}c_{55}&c_{54}&c_{53}&e_{25}\\ c_{45}&c_{44}&c_{43}&e_{24}\\ c_{35}&c_{34}&c_{33}&e_{23}\\ e_{25}&e_{24}&e_{23}&-e_{2z}\end{array} \right)\mathbf{R}_{2}^{I}=\left( \begin{array} []{cccc}c_{15}+c_{51}&c_{14}+c_{56}&c_{13}+c_{55}&e_{21}+e_{55}\\ c_{65}+c_{41}&c_{64}+c_{46}&c_{63}+c_{45}&e_{26}+e_{x4}\\ c_{55}+c_{31}&c_{54}+c_{36}&c_{53}+c_{35}&e_{25}+e_{33}\\ e_{21}+e_{5}&e_{26}+e_{x4}&e_{25}+e_{x3}&-e_{2x}-e_{xz}\end{array} \right) \tag{4}\]