# Investigation of Electromagnetic Angle Sensor Integrated in FR4-Based Scanning Micromirror
## 2 Principle and Modeling
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Parameters** & \(I\) & \(w\) & \(N_{d}\) & \(N_{s}\) & \(a\) & \(b\) & \(s\) & \(Q\) & \(K\) & \(B\) \\ \hline Value & 12 & 12 & 11 & 9 & 4 & 4 & 0.5 & 59 & 0.016 & 100 \\ Unit & mm & mm & - & - & mil & mil & mm & - & N\(\cdot\)m/rad & mT \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the FR4 micromirror integrated with angle sensor. By substituting Equation (5) to Equation (7), the moving induced electromotive force \(\varepsilon_{s}\) in the sensing coil can be expressed as  
\[\varepsilon_{s}=\theta_{0}\omega_{n}BM_{s}\sin\omega t. \tag{9}\]  
According to Equation (9), we can see that the amplitude of the moving induced electromotive force is proportional to that of the deflection angle. Moreover, it has a phase difference of \(\pi/2\) with the deflection angle of the micromirror, and is in phase with the driving voltage. However, the angle sensor output voltage consists of not only the moving-induced electromotive force but, also, the coupling voltage coupled from the driving signal. The coupling effect of the driving and sensing coils can be described as a coupled inductor model, and its equivalent circuit model is shown in Figure 2a. The resistances of the driving coil and sensing coil are 3.7 \(\Omega\) and 2.4 \(\Omega\), respectively. In order to better analyze the coupling effect and obtain the corresponding parameters, electromagnetic field finite element analysis (FEA) of the device is performed by using the Ansoft Maxwell simulation tool. All the material properties used in the simulation are from the material library of the software. The finite element model of the mirror consisting of the FR4 substrate, driving and sensing coils, and an aluminum-coated silicon mirror plate, is shown in Figure 2b, as well as the parameters obtained through magnetostatic analysis. We can see that the coupling effect is weak, due to the small coupling coefficient (\(K_{c}=0.23606\)). According to our previous work, the current in the driving coil is much larger than that in the sensing coil. Thus, the coupling effect of the sensing coil on the driving coil can be neglected. We only need to analyze the coupling voltage in the sensing coil coupled from the driving signal. Figure 2: **(a)** An equivalent coupled inductor circuit model to describe the coupling effect between the driving and sensing coils. **(b)** Electromagnetic field finite element model of the mirror consisting of the FR4 substrate, driving and sensing coils, and an aluminum-coated silicon mirror plate, and the parameters obtained by magnetostatic analysis. \(R_{d}\) and \(R_{s}\) are the resistances, and \(L_{d}\) and \(L_{s}\) are the inductances of the driving coil and sensing coil, respectively. \(M\) is the mutual inductance between the driving and sensing coils, \(K_{c}\) is the coupling coefficient, and \(U_{sc}\) is the coupling voltage in the sensing coil coupled from the driving signal. The coupling voltage \(U_{sc}\) in the sensing coil coupled from the driving signal can be expressed as  
\[U_{sc}=M\dot{I}_{d}. \tag{10}\]  
By substituting Equation (2) to Equation (10), the coupling voltage \(U_{sc}\) in the sensing coil can be given as a function of the driving voltage:  
\[U_{sc}=M\frac{U_{d0}\omega}{\sqrt{{R_{d}}^{2}+\left(\omega{L_{d}}\right)^{2}}} \sin\Bigl{(}\omega t+\frac{\pi}{2}\Bigr{)}. \tag{11}\]  
According to Equation (11), we can obtain that the amplitude of the coupling voltage in the sensing coil is proportional to that of the driving voltage. The coupling voltage in the sensing coil has a phase difference of \(\pi/2\) with the driving voltage.