## 4 Results and Discussion  
A prototype VATG sensor like that illustrated in Figure 2a was fabricated using PCB manufacturing technology with row 1 composed of \(n=150\) periods and row 2 composed of 149 periods, where the accuracy of PCB manufacturing technology can reach 10 \(\upmu\)m [23]. The parameters of the sensor were \(L=600\) mm, \(W_{row1}=L/n=600/150=4\) mm, \(W_{row2}=L/(n-1)=(600/149)\approx 4.02684564\) mm, and \(E_{limit}=W_{row2}-W_{row1}\approx 26.845\)\(\upmu\)m. The measurement performance of the VATG sensor was evaluated experimentally. The experimental setup is shown in Figure 5. It consists of an Aerotech ABL2000 air-bearing linear stage with a positioning accuracy of \(\pm 1.5\times 10^{-6}\) over a 120 mm traveling range, a Physik Instrumente H-824 6-axis miniature hexapod stage with a repeatability of \(\pm 0.1\)\(\upmu\)m, a Renishaw XL80 laser interferometer with a \(\pm 0.5\) ppm precision as a measurement standard, a signal processing system, and a sensor loaded with shells. Here, the shell of the stator is 655 mm long, 37 mm wide, and 15 mm high, while that of the mover is 96 mm long, 37 mm wide, and 26 mm high. The stator was fixed on the air-bearing linear stage. The mover was mounted on the six-axis miniature hexapod stage for adjusting the relative spatial position between the excitation and induction electrodes with sub-micrometer precision, and maintaining a gap width \(d_{0}=0.8\) mm. As such, the stator was actually translated during the experiments, and the mover was held stationary. The data was acquired and processed by the signal processing system, and the results were compared with the data obtained from the laser interferometer. The displacement measurement errors of the two rows of the prototype VATG sensor individually are shown in Figure 6a as a function of the displacement at 145 points along the entire 600 mm range of the sensor. Here, the error is observed to increase linearly with increasing displacement, and the minimum error is around 6.5 \(\upmu\)m. This type of error is characteristic of all sensors using a repeated periodic structure and that employ an incremental measurement method. However, the double-row capacitive structure can eliminate measurement error accumulated over a small period, as indicated by the measurement error of the prototype VATG sensor over the full range shown by the curve in Figure 6b obtained before optimization. Compared with the measurement errors of the single capacitive rows in Figure 6a, the measurement error has been reduced by 3 \(\upmu\)m at least, and achieves a measurement error of 3.5 \(\upmu\)m at most over the full displacement range, which is much less than the \(E_{limit}\) value of 26.845 \(\upmu\)m. Therefore, the prototype VATG sensor can realize absolute displacement measurements. Figure 4: Schematic illustrating the determination of the limit of error for the proposed VATG sensor. Figure 5: Experimental setup employed for evaluating the measurement performance of the prototype VATG sensors manufactured using printed circuit board technology. Figure 6: (**a**) Measurement errors of two rows of the initial prototype VATG sensor over the full measurement range. (**b**) Measurement errors of the VATG sensor obtained over the full measurement range before optimization and after optimization. (**c**) Measurement errors obtained over a single 4 mm period before optimization and after optimization. (**d**) Frequency spectra of measurement errors for a single 4 mm period before optimization and after optimization. The measurement performance of the prototype VATG sensor was further evaluated by plotting the measurement error with respect to displacement taken at 42 points over a single 4 mm period. The results presented in Figure 6c before optimization demonstrate that the sensor exhibits a periodic measurement error of \(\pm\)0.75 \(\upmu\)m over the single period. Figure 6d presents the results of harmonic analysis based on the fast Fourier transform of the periodic measurement error data obtained before optimization.