## 2 Static Properties Analysis  
### Effect of Structure on Static Properties  
In the considered case of the IDE capacitive humidity sensor, the electric potential and electric field can be well interpreted by the superposition principle if the charge distribution is known [36]. When there is an applied electric field, the electric potential distribution \(V(r)\) can be described by the following equations,  
\[V(\overrightarrow{r})=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{- \infty}^{+\infty}\frac{\sigma(\overrightarrow{r})}{4\pi\varepsilon\left| \overrightarrow{r}-\overrightarrow{r^{\prime}}\right|}d^{3}\overrightarrow{r^ {\prime}} \tag{1}\]  
where \(\sigma(r)\) denotes the distribution of the surface charges, \(r\) denotes 3D model sphere radius, and \(\varepsilon\) denotes the dielectric coefficient. For the IDE with a long periodic structure, the edge effect at the end of the IDE is negligible, and its potential expression can be written as  
\[V(x,y,z)=V(x,y,0) \tag{2}\]  
Note that the entire potential distribution can be obtained by expressing the potential expression on the \(x-y\) plane. Therefore, for IDE with longer lengths, it is sufficient to discuss the potential \(V(x,y)\) in the \(x-y\) plane. The two models used to describe IDE structure are shown in Figure 1.  
Figure 1 depicts two IDE models at different positions for the sensitive layer. Where \(g\) denotes the distance between the two electrode fingers, and \(w\) denotes the width of the fingers. Take the interdigital electrode made of indium tin oxide(ITO) as an example. The configuration in Figure 1b has the simplest distribution of electric potential and electric field, which can be regarded as a parallel plate. The electric potential \(V_{b}(x,y)\) and the electric field \(E_{b}(x,y)\) of Figure 1b are shown in Equations (3) and (4),  
\[V_{b}(x,y)=\frac{V_{0}}{g}X \tag{3}\]  
\[E_{b}(x,y)=\frac{\partial V_{b}(x,y)}{\partial X}=\frac{V_{0}}{g} \tag{4}\]  
In comparison, and when the sensitive layer is thicker than the electrode, the difference in capacitance is mainly caused by the structural change of (a). Using the Fourier series and applying the boundary conditions, the electric potential \(V_{a}(x,y)\) of (a) can be expressed as  
\[V_{a}(x,y)=\frac{4V_{0}}{\pi}\sum_{n=1}^{\infty}\frac{1}{2n-1}J_{0}[\frac{(2n- 1)\pi g}{2(w+g)}]\sin[\frac{(2n-1)\pi x}{(w+g)}]\exp[-\frac{(2n-1)\pi|y|}{(w+g )}] \tag{5}\]  
where \(V_{0}\) denotes the applied voltage, \(J_{0}\) denotes the zeroth bessel function of the first kind, \(x\) denotes the distance along the horizontal axis in Figure 1, and \(y\) denotes the thickness of the sensitive layer. The electric field \(E_{a}(x,y)\) and the surface charge distribution \(\sigma(x)\) are shown in Equations (6) and (7),  
\[E_{a}(x,y)=-\
abla V_{a}(x,y) \tag{6}\]  
\[\sigma(x)=-2\epsilon(\frac{\partial^{2}V_{a}(x,y)}{\partial y^{2}}),y=0 \tag{7}\]  
where \(\epsilon\) denotes the dielectric coefficient of the sensitive layer. Considering an applied voltage \(V_{0}\) = 1 V, the \(g\), and \(w\) are 10 \(\upmu\)m and 50 \(\upmu\)m, respectively. The electric potential, electric field, and the surface charge distributions in the \(x\)-direction under different thicknesses of the sensitive layer are analyzed, as shown in Figure 2.  
Figure 1: A cross-section view of the \(x-y\) plane with a pair of IDE placed (a) at the bottom with an applied voltage of \(V_{0}\) and (b) at the side of the sensitive layer. The current fraction can effectively predict the capacitance of the sensing layer. Maximum capacitance is considered to be achieved when all currents pass through the sensitive layer, the current fraction is shown in Equation (12),  
\[V(x,y,z)=V(x,y,0) \tag{2}\]0  
where \(I\) denotes the total current flowing through the sensitive layer. The relationship between the sensitivity and the thickness of the sensitive layer is theoretically calculated and simulated.