# Capacitive Angle Sensor Research Using COMSOL Multiphysics
## 1 Introduction  
The measurement of absolute angle plays an important role in positioning and mechanical engineering; it is also widely used in the automotive industry and aerospace and machine manufacturing [1; 2; 3]. To measure an absolute angle using non-contact sensing, full circle measurement range, high resolution and high precision, rotary encoders are needed [4; 5; 6]. Rotary encoders can be divided into several types according to working principle, such as optical, electromagnetic and capacitive rotary encoders [7; 8; 9]. Given the optical rotary encoder principle, absolute encoders are generally used for optical sensors, which are more complex than the incremental angle encoders used for other types rotary encoders [10]. However, its performance deteriorates in harsh environments [11]. Compared with optical rotary encoders, electromagnetic rotary encoders are more reliable, but they usually have permanent magnets set inside, which makes electromagnetic sensors vulnerable to interference from external magnetic fields [12]. Additionally, optical and electromagnetic rotary encoders are relatively expensive, and usually used in high-end application scenarios [13; 14]. In contrast, capacitive encoders are more attractive because they can be easily manufactured as small units using simpler construction and lower power consumption, thus making it easier to achieve high precision measurement [15; 16]. These advantages make capacitive encoders more suitable for widespread use in industrial applications. Furthermore, the sensitive regions of optical and electromagnetic encoders are located only in a strictly confined section of the coding disc, whereas in capacitive encoders, the entire overlapping region of the electrode plane contributes to its output, thus providing a stronger output signal [17]. In recent years, due to the above advantages, capacitive angle sensor technology has become widely used in non-contact measurement of angular displacement, which has been widely studied and applied in key components of industrial robots and automotive applications [18]. In 1997, [PERSON] presented a high-performance intelligent capacitive angular position sensor with a full-circle measurement range. Optimizing the structure of the sensing element reduced the effect of electric field bending and mechanical error. However, the electrodes were still fan-shaped, which was not conducive to high-precision measurement; the study did not mention the accuracy of measurement with a nonlinearity of approximately 0.03\({}^{\circ}\)[19]. Later, [PERSON] developed a high-precision and high-dynamic modern capacitive rotary encoder based on previous work in 2015. Improving the electrode structure by changing the fan-shaped sensitive electrode to a petal-form sensitive electrode achieved a high-precision test result, with precision better than 0.006\({}^{\circ}\) and dynamic nonlinearity evaluated at \(\pm\)0.4\({}^{\circ}\)[16]. In 2018, [PERSON] reported a sensor with a single excitation designed to realize the phase differentiation of sine and cosine signals through a new petal-form sensitive electrode. The design overcame the poor repeatability of error caused by the matching error of multiple excitation signals in amplitude and phase, which made it difficult to achieve high precision measurement using the harmonic compensation method. The reported accuracy could reach 0.008\({}^{\circ}\) (after compensation) [20]. The causes of nonlinearity in practical circuits have been discussed. In 2021, [PERSON] researched the presence of chaos observed in configurations in which passive resistors and integrated devices are mainly considered. This approach was based on hidden dynamics and verified the cause of nonlinear dynamics [21]. Although capacitive angle sensors have been extensively studied, their performance simulation is rarely reported. Without this simulation, when designing a capacitive angle sensor, some parameter adjustments require repeated processing, which increases design time and cost.