## 2 Mathematical Modeling  
### Static Sensitivity Modeling  
The relationship between sensor structure and performance can be well analyzed by theoretical modeling. Figure 1a shows the basic structure of the interdigital capacitive humidity sensor. A pair of interdigitated electrodes cross each other to form an electrode layer, with the moisture-sensitive layer evenly covering the surface of the electrode layer. To eliminate extreme environmental influences, a heating layer is placed between the electrode layer and the substrate layer. When the relative humidity of the environment changes, the dielectric constant of the moisture-sensitive layer will change, and the electric field of the sensor will also change accordingly, as shown in Figure 1b. The relative humidity of the environment can be calculated by detecting the change in the capacitance value of the sensor. Figure 1c demonstrates several key structural parameters of the sensor. Wherein, the length and width of the interdigitated structure are represented by \(L\) and \(b\), respectively, and the distance between two interdigitated fingers is represented by \(a\). The thickness of the moisture-sensitive layer, the thickness of the electrode, and the thickness of the substrate layer are represented by \(h_{1}\), \(h_{2}\), and \(h_{3}\), respectively. Since the capacitance value of a pair of interdigitated structures is very small, multiple pairs of interdigitated structures are often connected in parallel to expand the basic capacitance value of the sensor. In the process of modeling the electric field of the sensor, to simplify the description of the problem, only the electric field of a pair of interdigitated structures is considered first, and other complicate structures are calculated as the parallel connection of capacitances. Considering the complex and changeable interdigital structure, a pair of electrodes is often on different planes during processing. To comprehensively ana lyze various situations, we choose the conformal mapping method to model the irregular and complex electrode structure. The cross-sectional structure of a pair of interdigitated electrodes in the z-plane is shown in Figure 1d. The angle between a pair of electrode plates AB and CD is \(\phi\), and the length is half of the side length \(l_{1}=l_{2}=b/2\). The extension line of the plate intersects at point O, and the distance from point O to point A and point C is half the distance between the capacitors \(r_{1}=r_{2}=a/2\). Sufficiently longitudinal dimensions of the electrodes ensure that the problem can be viewed as a two-dimensional structure in cross-section. The interdigital structure of the sensor is complex, and it is very complicated to solve it directly on the Z plane. In order to simplify it to a simple parallel plate structure, we transform the Z plane to the W plane of the uniform field region with a simple rectangular boundary by means of coordinate transformation. The mapping from the Z plane to the W plane can be expressed as:  
\[\left\{ \begin{array}{l}w=I\int_{0}^{t}\frac{dt}{\sqrt{[t-t(z_{\rm{A}})][t-t(z _{\rm{D}})][t-t(z_{\rm{C}})]}}+J\\ t(z)=\frac{2z^{\pi/\varphi}+\left(\sqrt{\left(\frac{\theta}{2}\right)^{2}+l_{1 }^{2}+\frac{b}{2}}\right)^{\pi/\varphi}-\left(\frac{\theta}{2}+\frac{b}{2} \right)^{\pi/\varphi}}{2\left(\frac{\theta}{2}+\frac{b}{2}\right)^{\pi/ \varphi}}\end{array} \right. \tag{1}\]  
where \(h_{i}\) is the thickness of the electrode material of the \(i\)-th layer (\(i=1,2,3,\ldots\)). \(I\) and \(J\) are constants related to the correspondence between planes. Points A, B, C, and D can be represented in the Z coordinate system as \(z_{A}=(r_{1}+l_{1})\mathrm{e}^{\mathrm{i}\varphi}\), \(z_{B}=r_{1}\mathrm{e}^{\mathrm{i}\varphi}\), \(z_{C}=r_{2}\mathrm{e}^{\mathrm{i}\theta}\), and \(z_{D}=(r_{2}+l_{2})\mathrm{e}^{\mathrm{i}\theta}\).