# Investigation of Electromagnetic Angle Sensor Integrated in FR4-Based Scanning Micromirror
## 2 Principle and Modeling  
Figure 1a shows the schematic drawing of the assembled FR4 micromirror integrated with angle sensor. The device comprises a FR4 platform, a 11.7 mm \(\times\) 10.3 mm aluminum-coated silicon mirror plate on it, a pair of permanent magnets assembled in parallel, and the package structure (i.e., the coverplate and baseplate). The thicknesses of the FR4 platform and silicon mirror plate are 0.4 mm and 0.5 mm, respectively. The layout of the FR4 platform is illustrated in Figure 1b. The inner sensing coil (double-layer planar Cu coil) and outer driving coil (single-layer planar Cu coil) are simultaneously integrated into the platform. When a driving signal with the resonant frequency (361.8 Hz) of the micromirror is applied in the driving coil, the generated Lorentz force will actuate the mirror to oscillate around the torsion bars at resonance. At the same time, an electromotive force is induced in the sensing coil. Therefore, the induced electromotive can be read out as a sensor signal for monitoring the variation of the deflection angle. The motion (mechanical half deflection angle \(\theta\)) of the oscillating micromirror can be expressed by the following dynamical equation:  
\[J_{m}\ddot{\theta}+C\dot{\theta}+K\theta=T, \tag{1}\]  
Figure 1: (**a**) Schematic drawing of the assembled FR4-based electromagnetic scanning micromirror integrated with angle sensor. (**b**) Layout of the integrated sensing coil in the FR4 platform. where \(J_{m}\), \(C\), \(K\), and \(T\) represent the moment of inertia of the platform with Si mirror plate, the damping coefficient, the torsion stiffness of torsion bars, and the driving torque, respectively. As a sinusoidal driving voltage signal, \(U_{d}=U_{d0}\text{sin}\omega t\) is applied in the driving coil, and the corresponding driving current \(I_{d}\) is  
\[I_{d}=\frac{U_{d}}{\sqrt{R_{d}{}^{2}+(\omega L_{d})^{2}}}=\frac{U_{d0}}{\sqrt{ R_{d}{}^{2}+(\omega L_{d})^{2}}}\sin\omega t, \tag{2}\]  
where \(R_{d}\) and \(L_{d}\) represent the resistance and inductance of the driving coil, respectively. Thus, the driving torque can be described as  
\[T=I_{d}BM_{d}, \tag{3}\]  
where \(B\) is the magnetic flux density produced by the magnets; \(M_{d}\) is identified as the area sum of all the driving coil, and it can be calculated from  
\[ \begin{array}{l}M_{d}=\sum\limits_{n=1}^{N_{d}}\Big{(}\frac{w-2s-b}{2}-(n-1) (a+b)\Big{)}\cdot((2l-4s-3a-5b)-4(n-1)(a+b))\\ -\Big{(}\frac{w-2s-b}{2}-(N_{d}-1)(a+b)\Big{)}\cdot((l-2s-2a-3b)-2(N_{d}-1)(a+ b)),\end{array}  \tag{4}\]  
where \(N_{d}\) is the number of the driving coil turns, \(l\) and \(w\) are the length and width of the FR4 platform, respectively, \(a\) is the spacing of adjacent coils, \(b\) is the width of coil, and \(s\) is the width of the exterior border zone of the platform. Therefore, by solving Equation (1), the mechanical half deflection angle \(\theta\) at resonant frequency \(\omega_{n}\) is  
\[\theta=\theta_{0}\sin\Big{(}\omega_{n}t-\frac{\pi}{2}\Big{)}, \tag{5}\]  
where \(\theta_{0}\) is the amplitude of deflection angle and is calculated as  
\[\theta_{0}=\frac{U_{d0}BM_{d}Q}{K\sqrt{R_{d}{}^{2}+(\omega L_{d})^{2}}}. \tag{6}\]  
Meanwhile, the moving induced electromotive force \(\varepsilon_{s}\) in the sensing coil can be described as  
\[\varepsilon_{s}=BM_{s}\hat{\theta}, \tag{7}\]  
where \(M_{s}\) is identified as the area sum of all the sensing coil, and it can be calculated from  
\[ \begin{array}{l}M_{s}=2\sum\limits_{n=1}^{N_{s}}\big{(}(a+\frac{3}{2}b)+(n- 1)(a+b)\big{)}\cdot((5a+7b)+4(n-1)(a+b))\\ -\big{(}(a+\frac{3}{2}b)+(N_{s}-1)(a+b)\big{)}\cdot((3a+4b)+2(N_{s}-1)(a+b))\\ +\big{(}\frac{w}{2}-s\big{)}\cdot\Big{(}(a+\frac{3}{2}b)+(N_{s}-1)(a+b)+\Big{(} \frac{l}{2}-s\Big{)}\Big{)},\end{array}  \tag{8}\]  
where \(N_{s}\) is the number of the sensing coil turns. All the aforementioned parameters are listed in Table 1.