## 4 Results and Discussion
The figure indicates that the primary harmonic component includes the first-order harmonic error component with an amplitude greater than 0.5 \(\upmu\)m with much smaller fourth-order and fifth-order harmonic component contributions and relatively minor contributions from the remaining components. Here, the first-order harmonic error component is faintest because the experimental environment is controlled by high-precision instruments. However, physical non-uniformities in the four excitation electrode arrays, as well as non-uniformities in the electric fields of the electrodes, can introduce first-order harmonic error [24; 25]. Moreover, non-uniformities in the electric fields of the electrodes can be exacerbated by signal crosstalk among the electrodes of two rows. That is to say, a certain quantity of excitation signals (briefly summarized as two types of excitation signals, sinusoidal and cosine signals) are introduced into the inductive output signal. To analyze the effect of crosstalk among electrodes on the measurement accuracy of the VATG sensor, we take row 1 as an example and consider the following mathematical model of a capacitive time-grating sensor. \[U_{row1} = A_{row1}\sin(\omega t-\tfrac{2\pi x}{\text{W}})+\delta_{1}\sin( \omega t)+\delta_{2}\cos(\omega t) \tag{10}\] \[= A_{row1}\sin(\omega t-\tfrac{2\pi x}{\text{W}})+\sqrt{\delta_{1} {}^{2}+\delta_{2}{}^{2}}\sin\Bigl{(}\omega t+\arctan\tfrac{\delta_{2}}{ \delta_{1}}\Bigr{)},\]  
where \(\delta_{1}\) and \(\delta_{2}\) are the relational coefficients of crosstalk signals caused by a certain quantity of excitation signals. Thus, signal crosstalk can result in first-order harmonic error. To address this issue, the VATG sensor design was optimized by increasing the distance between the two rows and adding a shielding layer between them to reduce the influence of signal crosstalk among the electrodes of two rows. In addition, we adopted a differential induction structure to reduce common mode disturbance. To analyze the effect of the differential induction structure for reducing common mode disturbance, the following mathematical model can be established. \[U_{row1} = \bigl{[}A_{row1}\sin(\omega t-\tfrac{2\pi x}{\text{W}})+\delta^{ \prime}_{1}\sin(\omega t)+\delta^{\prime}_{2}\cos(\omega t)\bigr{]} \tag{11}\] \[-\bigl{[}A_{row1}\sin(\omega t-\tfrac{2\pi x}{\text{W}}+\pi)+ \delta^{\prime}_{1}\sin(\omega t)+\delta^{\prime}_{2}\cos(\omega t)\bigr{]}\] \[= 2A_{row1}\sin(\omega t-\tfrac{2\pi x}{\text{W}}),\]  
where \(\delta^{\prime}_{1}\) and \(\delta^{\prime}_{2}\) are the relational coefficients of common mode disturbance caused by physical non-uniformities in the four excitation electrode arrays and by non-uniformities in the electric fields of the electrodes. These changes are illustrated in Figure 7.  
A new prototype VATG sensor employing the optimized structure illustrated in Figure 7 was then fabricated, and its measurement performance was evaluated using the experimental setup shown in Figure 4:0 The experimental results obtained after optimization in Figure 6b-d demonstrate that the optimized sensor structure provides greatly enhanced measurement performance. Figure 6b indicates that the measurement error obtained over the full range of the optimized sensor was reduced to no more than 2.5 \(\upmu\)m. Moreover, Figure 6c indicates that the error obtained over a 4 mm period was reduced to no more than \(\pm\)0.25 \(\upmu\)m. Finally, Figure 6d indicates that the first-order harmonic error was greatly reduced from greater than 0.5 \(\upmu\)m to about 0.05 \(\upmu\)m.