# Capacitive Angle Sensor Research Using COMSOL Multiphysics
## 2 Basic Principle and Design  
The structure of the simulated capacitor angle sensor is shown in Figure 1. As COMSOL Multiphysics has drawing defects in its polar coordinate system, the patterns were first created using a GeoGebra graphing calculator, converted into DXF format files and then imported to COMSOL Multiphysics. The structure comprises a petal-form sensitive electrode for the rotor, which receives the signal (Figure 1a), and a fan-shaped electrode for the stator, which acts as the induction signal (Figure 1b). In the above study, the differences in sine and cosine signals were realized by subtracting the signals obtained from the fan-shaped electrode [16; 20]. Figure 1 shows that the petal-form sensitive electrode and the fan-shaped electrode have the same outer and inner ring. Figure 1: Structure of the capacitor angle sensor: (**a**) petal-shaped rotor and (**b**) fan-shaped stator. Figure 2 shows the structural schematic diagram; the electrode material is copper, the rest is air domain. As shown in Figure 2a, the petal-form sensitive electrodes are sine-wave shapes in polar coordinates, sorted by X number of cycles (in this design, X = 8). The corresponding cycle angle is \(2\pi/\)X = 45\({}^{\circ}\). The construction of the petal-form sensitive electrode is shown in Figure 1a with a blue area boundary. The red dotted ring (with radius R) is the centerline of the boundary; R\({}_{1}\) and R\({}_{2}\) represent the inner and outer contour lines, respectively. The green dotted lines are the centers of amplitude of the cosine functions R\({}_{1}\) and R\({}_{2}\), and \(\tau\) is the amplitude. The \(\theta\) is the mechanical angle and the X: \(\theta\) is the electric angle, which means that the electric angle is X-times the mechanical angle. The mechanical angle \(\theta\) is a variable of integration. In polar coordinates, the equations of the two boundary functions are expressed as follows:  
\[\left\{ \begin{aligned} &\text{R}_{1}=(\text{R}+\tau)+\tau\cdot \cos(\text{X}\cdot\theta)\\ &\text{R}_{2}=(\text{R}-\tau)-\tau\cdot\cos(\text{X}\cdot\theta) \end{aligned} \right. \tag{1}\]  
In this design, R = 50 cm and \(\tau\) = 5 cm; substituting R, \(\tau\) and X into (1):  
\[\left\{ \begin{aligned} &\text{R}_{1}=55+5\cdot\cos(8\cdot\theta)\\ &\text{R}_{2}=45-5\cdot\cos(8\cdot\theta)\end{aligned} \right. \tag{2}\]  
The solid red line area in Figure 2a indicates induction electrodes composed of four fan-shaped electrodes in one group, corresponding to a petal-form sensitive electrode; the four fan-shaped electrodes within a cycle, labelled A, B, C and D, are cycles in turn. Given the capacitance values of fan-shaped electrodes A, B, C and D, with corresponding petal-form electrode areas (C\({}_{\text{A}}\), C\({}_{\text{B}}\), C\({}_{\text{C}}\) and C\({}_{\text{D}}\)), the differences between A and C, and B and D (C\({}_{\text{A}}\)-C\({}_{\text{C}}\) and C\({}_{\text{B}}\)-C\({}_{\text{D}}\)) were used to obtain two sine and cosine signals with phase differences of 90 degrees, thus realizing the high-precision measurement of the angle [20]. According to the capacitive coupling principle, the capacitances between the sensitive electrode on the rotor and the four collection electrodes on the stator could be obtained and presented as:  
\[\left\{\begin{aligned} &\text{C}_{\text{A}}=\frac{\epsilon_{0}\epsilon}{ \text{d}}\text{S}_{\text{A}}=\frac{\epsilon_{0}\epsilon}{\text{d}}\cdot\int_{( \frac{\pi}{\pi}+\theta)/\text{X}}^{\theta/\text{X}}\frac{1}{2}\left(\text  
converted into DXF format files and then imported to COMSOL Multiphysics; the fan-shaped electrodes were built in COMSOL Multiphysics and used its Boolean operations. All electrode material was made of copper, and a sphere with a zero charge boundary condition applied to its outer surface truncated the air space surrounding the device. A DC voltage was applied to the petal-form sensitive electrode to determine the capacitance. Under the assumption of electrostatic conditions, the surface of each electrode was at the same potential.