## 2 Sensor Structure and Measurement Principle  
### Single-Row Sensor Measurement Principle  
The structure of the single-row capacitive displacement sensor is illustrated in Figure 1a. The sensing structure consists of independent plate capacitor arrays, where the sequence of four adjacent rectangular excitation electrodes labeled \(S+\), \(C+\), \(S-\), and \(C-\) are fixed in stationary positions to provide a small period of \(W\). Four orthogonal sinusoidal AC excitation signals of amplitude \(A\) and angular frequency \(\omega\), denoted as \(U_{S+}=A_{m}\)sin(\(\omega t\)), \(U_{C+}=A_{m}\)cos(\(\omega t\)), \(U_{S-}=-A_{m}\)sin(\(\omega t\)), and \(U_{C-}=-A_{m}\)cos(\(\omega t\)) are respectively applied to the electrodes labeled \(S+\), \(C+\), \(S-\), and \(C-\). Here, the length of the excitation electrodes in the \(y\) direction is \(H\) and the width is \((W/4-I)\), where \(I\) is the interval between two adjacent excitation electrodes in the \(x\) direction. The stationary excitation electrodes are overlaid by an array of sinusoidal-shaped induction electrodes of length \(H\) and maximum width \(W/2\) that are movable in the \(x\) direction and have relative positions fixed with a periodicity of \(W\), where the interval between two adjacent induction electrodes in the \(x\) direction is \(W/2\). As illustrated in Figure 1b, the moveable induction electrodes are aligned parallel with the stationary excitation electrodes with a distance of separation \(d_{0}\) in the \(z\) direction. The sequence of four excitation electrodes labeled \(S+\), \(C+\), \(S-\), and \(C-\) and one induction electrode thereby form four planar capacitors with capacitance values denoted as \(C_{1}\), \(C_{3}\), \(C_{2}\), and \(C_{4}\), respectively. In addition, the values of \(C_{1}\) and \(C_{3}\), and \(C_{2}\) and \(C_{4}\) vary oppositely as the induction electrode moves along the \(x\) axis, as illustrated by the equivalent circuit shown in Figure 1d. Here, R represents load resistance. According to the superposition theorem of the equivalent circuit, the output \(U_{o}\) is obtained as  
\[U_{o}=U_{S+}\frac{Z_{L1}}{Z_{L1}+Z_{1}}+U_{S-}\frac{Z_{L3}}{Z_{L3}+Z_{3}}+U_{C +}\frac{Z_{L2}}{Z_{L2}+Z_{2}}+U_{S-}\frac{Z_{L4}}{Z_{L4}+Z_{4}}, \tag{1}\]  
where \(Z_{Lj}=-1/(\sum\limits_{i=1}^{4}\frac{1}{C_{i}}+\frac{1}{R})\) is equivalent capacitive-reactance for \(i\
eq j,j=1,2,3,4\), when \(S+\), \(C+\), \(S-\), and \(C-\) work in dependence; \(Z_{1}\), \(Z_{2}\), \(Z_{3}\) and \(Z_{4}\) are the capacitive-reactance of \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\), changing with the movement of the moveable induction electrodes owing to the variation of the value of \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\). Thus, the final output can be given as follows [22]:  
\[U_{o}=\frac{U_{S+}\Delta S_{S+}}{2S_{0}}+\frac{U_{S-}\Delta S_{S-}}{2S_{0}}+ \frac{U_{C+}\Delta S_{C+}}{2S_{0}}+\frac{U_{C-}\Delta S_{C-}}{2S_{0}}, \tag{2}\]  
where the value \(S_{0}=(HW/\pi)\) cos (\(\pi\)/\(W\)) represents the maximum area of overlap between an excitation electrode and an induction electrode, and \(\Delta S_{S+}\), \(\Delta S_{C+}\), \(\Delta S_{S-}\), and \(\Delta S_{C-}\) are the efficient variations in the overlapping area between the \(S+\), \(C+\), \(S-\), and \(C-\) excitation electrodes and a single induction electrode, respectively, when \(d_{0}\) and the dielectric coefficient \(\varepsilon_{0}\) remain Thus, the final output is given as follows:  
\[U_{o}=~{}A\sin(\omega t-\frac{2\pi x}{W}), \tag{4}\]  
where \(A\) is the coupling coefficient, and \(U_{o}\) is a travelling wave signal whose frequency and amplitude are determined by the excitation signals \(U_{S+}\), \(U_{C+}\), \(U_{S-}\), and \(U_{C-}\), but its phase varies proportionately with respect to \(x\), and changes with a periodicity of \(W\).