## 3 Theory and Modeling
### Principal and Modeling of Electret Transducer Component
of the electret directly opposite the plates, and \(C\) represents its equivalent capacitance. The induced charge density \(\delta_{1}\) on plate 1 can be concluded as Equation (3):  
\[\delta_{1}=\frac{\delta}{\frac{v_{F}d}{dt}+1} \tag{3}\]  
The electret and the electrode have a sector angle of \(\pi/10\). During one cycle of rotating based the hypothesis of the model, the total induced charge \(Q_{1}\) on electrode 1 varies as Equation (4):  
\[ \begin{array}{ll}Q_{1}=\delta_{1}S\\ \\ =\frac{\omega_{r}tr^{2}\delta}{\frac{2\pi\omega}{\alpha}-2}&0\leq\omega_{r}t< \frac{\pi}{10}\\ \\ =\frac{(\pi-5\omega_{r}t)^{2}\delta}{\frac{10\omega}{\alpha}-10}&\frac{\pi}{10} \leq\omega_{r}t<\frac{\pi}{5}\end{array}  \tag{4}\]  
where \(S\) represents the area of the sector-shaped plate directly opposite the electret, and \(r\) is the radius of the sector. When the electret moves at a speed of \(\omega_{r}\), the induced charge quantity varies, initially increasing and then decreasing, repeating this cyclical variation with the arrival of the next electret. It can be considered that the induced charge \(Q_{2}\) on plate 2 follows a similar pattern but is \(\pi/10\) phase advance. In this model, the movement of the electret will lead to changes in the induced charge on the plates. Since there is a pathway between the plates, the induced charge can result in changes through charge transfer in the circuit, thereby generating output current. The open-circuit voltage between the plates is determined by the charge difference between them, while the output short-circuit current is determined by the amount of charge transferred between the plates [12]. Therefore, increasing the induced charge difference between the plates and its variation amplitude can enhance the output capability of the component. By using electrets with the same surface charge but opposite polarities arranged in intervals, the amplitude of the induced charge variation on plates that are close enough can be increased within one cycle, expanding the range of charge variation beyond a single polarity. When employing a unipolar electret with alternating positive and negative charges, the structural variation is illustrated in Figure 6.  
The induced charge on the electrode plate is simultaneously influenced by the electret with surface charge densities of \(\delta\) and \(-\delta\). Continuing to analyze the right electrode plate 1, the induced charge generated by the negatively polarized electret directly opposite the  
Figure 6: Electrostatic induction and output method of unipolar electret structures, with alternating positive \(\delta\) and negative \(-\delta\) polarities. area of \(S^{\prime}\) is denoted as \(\delta^{\prime}_{1}\). Then, Equation (4) could be modified as following Equation (5) for the optimized model, based on the same assumptions:  
\[ \begin{array}{l}Q_{1}=\delta_{1}S+\delta^{\prime}_{1}S^{\prime}\\ \\ =\frac{\omega_{\rm r}\tau t\tau^{2}\delta}{\frac{\omega_{\rm r}\tau}{\alpha t}-2 }-\frac{(\frac{\tau}{10\omega_{\rm r}t})\tau^{2}\delta}{\frac{\omega_{\rm r} \tau}{\alpha t}-10}\\ \\ =Q_{A}(\omega_{\rm r}t)\ \ \ \ \ [0,\frac{\pi}{10}]\\ \\ =\frac{(\frac{\tau}{10\omega_{\rm r}t})\tau^{2}\delta}{\frac{\omega_{\rm r} \tau}{\alpha t}-10}-\frac{(10\omega_{\rm r}t-\pi)\tau^{2}\delta}{\frac{\omega_ {\rm r}\tau}{\alpha t}-20}\\ \\ =Q_{B}(\omega_{\rm r}t)\ \ \ \ \ [\frac{\pi}{10},\frac{\pi}{5}]\\ \end{array}  \tag{5}\]  
Similarly, \(Q_{2}\) leads by \(\pi/10\), half a cycle. If a structure with alternating positive and negative charges is not used, the number of unipolar electrets arranged at equal intervals must be less than the number of electrodes for the induced charge on the plates to vary.