A High Precision Capacitive Linear Displacement Sensor with Time-Grating That Provides Absolute Positioning Capability Based on a Vernier-Type Structure

Nanometer-scale measurement devices with high accuracy and absolute long-range positioning capability are increasingly demanded in the field of computer numerical control machining. To meet this demand, the present report proposes a capacitive absolute linear displacement sensor with time-grating that employs a vernier-type structure based on a previously proposed single-row capacitive sensing structure. The novel proposed vernier-type absolute time-grating (VATG) sensor employs two capacitor rows, each with an equivalent measurement range. The first capacitor row is designed with n periods to realize fine measurement, while the second capacitor row is designed with n − 1 periods, and the phase difference between the second row and the first row is employed to obtain absolute positioning information. A prototype VATG sensor with a total measurement range of 600 mm and n = 150 is fabricated using printed circuit board manufacturing technology, and its measurement performance is evaluated experimentally. Harmonic analysis demonstrates that the measurement error mainly consists of first-harmonic error, which is mostly caused by signal crosstalk. Accordingly, an optimized prototype VATG sensor is fabricated by adding a shielding layer between the two capacitor rows and designing a differential induction structure. Experimental results demonstrate that the measurement error of the optimized prototype sensor is ±1.25 μm over the full 600 mm range and ±0.25 μm over a single 4 mm period.


Introduction
Absolute position displacement measurement sensors play an irreplaceable role in the field of high-precision measurements and as a component in the closed loop feedback control of computer numerical control (CNC) machine tool systems.The rapid development of high-precision positioning technology in CNC machine tool systems has generated an increasing demand for robust absolute displacement measurement sensors with high positioning accuracy and excellent repetitive positioning precision in harsh environments, such as under the conditions of electromagnetic interference, mechanical vibration, extreme temperature variation, and dust contamination [1][2][3][4][5][6][7].Meanwhile, many types of absolute grating displacement measurement sensors with high precision and high resolution have been developed, including capacitive grating sensors [1,2], magnetic grating sensors [8], and optical grating sensors [9].
Absolute optical grating (AOG) sensors are based on an evaluation of the intensity of incident light from a known illumination source, and represent an efficient means for conducting closed loop position control in CNC machine tool systems [10][11][12][13].The measurement error of AOG sensors has been reduced to values as small as ±0.275 µm over a 10 mm range and ±5 µm over a full 3040 mm range.However, mechanical vibration, temperature variation, and dust contamination affect the light intensity and other characteristics of the illumination source incident on the sensor, all of which detract from the measurement performance of AOG sensors.It has a weak anti-interference capability, particularly in the harsh environments associated with many CNC machining applications [13,14].
In contrast to AOG sensors, both magnetic and capacitive grating sensors have high tolerance for mechanical vibration and contamination due to dust and oil and, therefore, are commonly employed in servo drive systems [2,8,15].The measurement error of absolute magnetic grating sensors can achieve values as low as ±5 µm over a 1000 mm range and can obtain a measurement resolution of 0.75 µm [15][16][17].However, the large size of the magnetic head and poor resistance to electromagnetic interference restrict the use of such sensors for high accuracy applications.In comparison, capacitive grating sensors provide a fuller range of environmental advantages, which make them ideally suited to harsh machining environments [2,[18][19][20][21].Moreover, the measurement error of absolute capacitive grating sensors can achieve values as low as ±0.3 µm over a 6 mm range, with an error of less than 0.01% over the full measurement range [2].However, the measurement range of absolute capacitive grating sensors is limited, which restricts their use in practical applications [2].
The present work addresses the limitations of absolute capacitive grating sensors by developing a novel capacitive absolute linear displacement sensor that employs a vernier-type structure with time-grating based on a previously proposed single-row capacitive sensing structure [22].The proposed vernier-type absolute time-grating (VATG) sensor employs two capacitor rows.According to the design of vernier calipers, the auxiliary scale, or vernier scale, includes n gradations over the total caliper length, while the main scale includes n − 1 gradations over the same total caliper length.Similarly, one capacitor row of the VATG sensor is composed of n periods over the total sensor length and serves as a fine measurement scale to ensure high measurement resolution, which is analogous to the auxiliary scale of vernier calipers.The other row of the VATG sensor is composed of n − 1 periods over the total sensor length as a coarse measurement scale used to find the phase difference of two rows to realize absolute positioning, which is analogous to the main scale of vernier calipers.A prototype VATG sensor is fabricated using printed circuit board (PCB) manufacturing technology with a capacitor row composed of 150 periods to realize fine measurement, and the second capacitor row composed of 149 periods to realize course measurement.Based on an analysis of the measurement performance of the prototype VATG sensor, the design is optimized by increasing the distance between the two rows and adding a shielding layer between them to reduce the influence of signal crosstalk, and a differential induction structure is adopted to eliminate common-mode interference, which significantly improves the measurement performance of the modified prototype VATG sensor.The remainder of this paper is organized as follows.Section 2 introduces the structures and measurement principles of the single-row capacitive sensor and the proposed VATG sensor.Section 3 presents an analysis of the limit of error of the VATG sensor.The experimental results and optimization methodology are presented and discussed in Section 4. Finally, Section 5 presents the main conclusions of the work.

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 ω, denoted as U S+ = A m sin(ωt), U C+ = A m cos(ωt), U S− = −A m sin(ωt), and U C− = −A m cos(ωt) 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 where ) is equivalent capacitive-reactance for i = 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]: where the value S 0 = (HW/π) cos (πI/W) represents the maximum area of overlap between an excitation electrode and an induction electrode, and ∆S S+ , ∆S C+ , ∆S S− , and ∆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 ε 0 remain constant.However, Figure 1a,c illustrates the case where the sinusoidal-shaped induction electrode moving along the x axis with respect to the stationary rectangular excitation electrodes can be equivalent to a rectangular induction electrode moving along the x axis with respect to a set of four stationary half-sinusoidal-shaped excitation electrodes if the single-row sensing structure is divided into two halves along the x axis.Here, the latter case can be regarded as an uncertain limit integral of a cosine or sine function [3, [21][22][23][24].Therefore, the effective area variation ∆S can be formulated according to the displacement x along the x direction as follows [22]: the output as follows: where T is the time period of the travelling wave signal determined by the excitation signals.
Accordingly, a single-row capacitive structure cannot obtain absolute position measurements without first knowing the starting position.However, absolute position measurements can be obtained by adopting a double-row capacitive structure.

Absolute Measurement Principle
The double-row capacitive structure of the VATG sensor is illustrated in Figure 2a, where stationary excitation electrodes serve as the stator, and moveable induction electrodes serve as the mover.Here, the stator consists of two-row stationary excitation electrodes, and each row occupies an equivalent overall length of L. In the same way, the mover consists of two-row corresponding inductive electrodes, and each row occupies an equivalent length of L/12.Meanwhile, a multilayer thin-film sensing structure is employed, where the electrodes and the lead wires are separated by a thin-film insulating layer, and the electrodes and lead wires are connected by a connecting column.Thus, the final output is given as follows: 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. Because the travelling wave signal represents a relationship between the spatial displacement and a time standard, we express the output as follows: where T is the time period of the travelling wave signal determined by the excitation signals.
Accordingly, a single-row capacitive structure cannot obtain absolute position measurements without first knowing the starting position.However, absolute position measurements can be obtained by adopting a double-row capacitive structure.

Absolute Measurement Principle
The double-row capacitive structure of the VATG sensor is illustrated in Figure 2a, where stationary excitation electrodes serve as the stator, and moveable induction electrodes serve as the mover.Here, the stator consists of two-row stationary excitation electrodes, and each row occupies an equivalent overall length of L. In the same way, the mover consists of two-row corresponding inductive electrodes, and each row occupies an equivalent length of L/12.Meanwhile, a multilayer thin-film sensing structure is employed, where the electrodes and the lead wires are separated by a thin-film insulating layer, and the electrodes and lead wires are connected by a connecting column.As shown in Figure 2b, the VATG sensor includes two single-row capacitive sensing structures denoted as row 1 and row 2 with different periodicities of Wrow1 and Wrow2, respectively.Because the value of I is equivalent for both rows, the widths of the excitation electrodes in row 1 and row 2 are (Wrow1/4 − I) and (Wrow2/4 − I), respectively.In addition, L/Wrow1 and L/Wrow2 are both integers, and L/Wrow1 − L/Wrow2 = 1.According to (5), the corresponding output voltage signals Urow1 and Urow2 of row 1 and row 2 can be given as where Arow1 and Arow2 are coupling coefficients of row 1 and row 2, respectively.We then define a reference signal Uref with an equivalent frequency as Urow1 and Urow2.Accordingly, we can define the time difference Δtrow1 along the time axis as representing the phase difference between Urow1 and Uref; As shown in Figure 2b, the VATG sensor includes two single-row capacitive sensing structures denoted as row 1 and row 2 with different periodicities of W row1 and W row2 , respectively.Because the value of I is equivalent for both rows, the widths of the excitation electrodes in row 1 and row 2 are (W row1 /4 − I) and (W row2 /4 − I), respectively.In addition, L/W row1 and L/W row2 are both integers, and L/W row1 − L/W row2 = 1.According to (5), the corresponding output voltage signals U row1 and U row2 of row 1 and row 2 can be given as where A row1 and A row2 are coupling coefficients of row 1 and row 2, respectively.We then define a reference signal U ref with an equivalent frequency as U row1 and U row2 .Accordingly, we can define the time difference ∆t row1 along the time axis as representing the phase difference between U row1 and U ref ; and the time difference ∆t as representing the phase difference between U row1 and U row2 [25].As shown in Figure 2b, U row1 , U row2 , and U ref are routed to a shaping circuit to convert them into square wave signals.The respective rising edges of the square wave signals, which are collectively denoted as P s for convenience, are then routed to a phase-comparing circuit constructed using a field-programmable gate array (FPGA).High-frequency clock pulses P t with a period T p are also input into the phase-comparing circuit as the time standard to interpolate ∆t row1 and ∆t.The interpolation process is illustrated in Figure 2c, where the rising edge of the converted U ref initiates the counting of pulses P f until the rising edge of the converted U row1 is encountered.Then, pulses P c are counted until the next rising edge of the converted U ref is encountered after a total period T. Accordingly, if n pulses P f are counted, ∆t row1 = nT p to realize the fine measurement of row 1, and, if m pulses P c are counted, ∆t = mT p to realize the coarse measurement of row 2. We also note from Figure 2c that T = NT p , and mT p can be employed to calculate the number of periods, M, that the induction electrode array has moved relative to the stationary excitation electrodes as follows: where the operator int(•) returns the integer value of its argument.Therefore, the relationship between x and ∆t can be defined as follows: Figure 3 presents the phase curves of U row1 and U row2 within their periods W row1 and W row2 with respect to displacement x obtained for L = 60 mm, W row1 = 10 mm, W row2 = 12 mm.According to Figure 3a, the phases of the two signals are distributed in a sawtooth form according to the value of x in each period.We also note that the phase curve of U row1 becomes increasingly advanced relative to that of U row2 with increasing x.Additionally, until x = 600 mm, the phase of row 2 differs by exactly a period from that of row 1.From Figure 3b, we note that the phase difference arising between the two signals with increasing x is monotonic and very nearly linear.As such, positioning can be realized according to the phase difference.
and the time difference Δt as representing the phase difference between Urow1 and Urow2 [25].As shown in Figure 2b, Urow1, Urow2, and Uref are routed to a shaping circuit to convert them into square wave signals.The respective rising edges of the square wave signals, which are collectively denoted as Ps for convenience, are then routed to a phase-comparing circuit constructed using a field-programmable gate array (FPGA).High-frequency clock pulses Pt with a period Tp are also input into the phase-comparing circuit as the time standard to interpolate Δtrow1 and Δt.The interpolation process is illustrated in Figure 2c, where the rising edge of the converted Uref initiates the counting of pulses Pf until the rising edge of the converted Urow1 is encountered.Then, pulses Pc are counted until the next rising edge of the converted Uref is encountered after a total period T. Accordingly, if n pulses Pf are counted, Δtrow1 = nTp to realize the fine measurement of row 1, and, if m pulses Pc are counted, Δt = mTp to realize the coarse measurement of row 2. We also note from Figure 2c that T = NTp, and mTp can be employed to calculate the number of periods, M, that the induction electrode array has moved relative to the stationary excitation electrodes as follows: int(( ) / ) where the operator int(•) returns the integer value of its argument.Therefore, the relationship between x and Δt can be defined as follows: Figure 3 presents the phase curves of Urow1 and Urow2 within their periods Wrow1 and Wrow2 with respect to displacement x obtained for L = 60 mm, Wrow1 = 10 mm, Wrow2 = 12 mm.According to Figure 3a, the phases of the two signals are distributed in a sawtooth form according to the value of x in each period.We also note that the phase curve of Urow1 becomes increasingly advanced relative to that of Urow2 with increasing x.Additionally, until x = 600 mm, the phase of row 2 differs by exactly a period from that of row 1.From Figure 3b, we note that the phase difference arising between the two signals with increasing x is monotonic and very nearly linear.As such, positioning can be realized according to the phase difference.

Analysis of the Limit of Error
As discussed, row 1 and row 2 of the VATG sensor correspond to the auxiliary ruler and the main ruler of a vernier caliper, respectively.In a similar manner, we can denote the magnitude of a single scale division on row 2 of the VATG sensor (corresponding to the main ruler of a vernier caliper) as W row2 and the magnitude of a single scale division on row 1 (corresponding to the auxiliary ruler of a vernier caliper) as W row1 .Therefore, the limit of error of the VATG sensor is given as follows.
This relationship is illustrated in Figure 4.According to the measurement principle of the VATG sensor, the realization of absolute position measurement is defined completely according to E limit , that is, E < E limit , where E is the error in absolute position measurements.

Analysis of the Limit of Error
As discussed, row 1 and row 2 of the VATG sensor correspond to the auxiliary ruler and the main ruler of a vernier caliper, respectively.In a similar manner, we can denote the magnitude of a single scale division on row 2 of the VATG sensor (corresponding to the main ruler of a vernier caliper) as Wrow2 and the magnitude of a single scale division on row 1 (corresponding to the auxiliary ruler of a vernier caliper) as Wrow1.Therefore, the limit of error of the VATG sensor is given as follows.This relationship is illustrated in Figure 4.According to the measurement principle of the VATG sensor, the realization of absolute position measurement is defined completely according to Elimit, that is, E < Elimit, where E is the error in absolute position measurements.

Results and Discussion
A prototype VATG sensor like that illustrated in Figure 2a 5.It consists of an Aerotech ABL2000 air-bearing linear stage with a positioning accuracy of ±1.5 × 10 −6 over a 120 mm traveling range, a Physik Instrumente H-824 6-axis miniature hexapod stage with a repeatability of ±0.1 μm, a Renishaw XL80 laser interferometer with a ±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 d0 = 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.

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 µm [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) ≈ 4.02684564 mm, and E limit = W row2 − W row1 ≈ 26.845 µm.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 ±1.5 × 10 −6 over a 120 mm traveling range, a Physik Instrumente H-824 6-axis miniature hexapod stage with a repeatability of ±0.1 µm, a Renishaw XL80 laser interferometer with a ±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 µm.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 µm at least, and achieves a measurement error of 3.5 µm at most over the full displacement range, which is much less than the E limit value of 26.845 µm.Therefore, the prototype VATG sensor can realize absolute displacement measurements.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 μm.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 μm at least, and achieves a measurement error of 3.5 μm at most over the full displacement range, which is much less than the limit E value of 26.845 μm.Therefore, the prototype VATG sensor can realize absolute displacement measurements.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 μm.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 μm at least, and achieves a measurement error of 3.5 μm at most over the full displacement range, which is much less than the limit E value of 26.845 μm.Therefore, the prototype VATG sensor can realize absolute displacement measurements.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 ±0.75 μm 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.The figure indicates that the primary harmonic component includes the first-order harmonic error component with an amplitude greater than 0.5 μm 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 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 ±0.75 µm 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.The figure indicates that the primary harmonic component includes the first-order harmonic error component with an amplitude greater than 0.5 µm 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.
where δ 1 and δ 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.
where δ 1 and δ 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 5.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 µm.Moreover, Figure 6c indicates that the error obtained over a 4 mm period was reduced to no more than ±0.25 µm.Finally, Figure 6d indicates that the first-order harmonic error was greatly reduced from greater than 0.5 µm to about 0.05 µm.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 5.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 μm.Moreover, Figure 6c indicates that the error obtained over a 4 mm period was reduced to no more than ±0.25 μm.Finally, Figure 6d indicates that the first-order harmonic error was greatly reduced from greater than 0.5 μm to about 0.05 μm.increasing the distance between the two rows and adding a shielding layer between them to reduce the influence of signal crosstalk.In addition, a differential induction structure is adopted to reduce common mode disturbance.

Conclusions
The present work addressed the limitations of absolute capacitive grating sensors by developing a novel VATG sensor that employs a double-row capacitive structure based on a previously proposed single-row capacitive sensing structure.One capacitor row is composed of n periods over the total sensor length and serves as a fine measurement scale analogous to the auxiliary scale of vernier calipers.The other row is composed of n − 1 periods over the total sensor length as a coarse measurement scale analogous to the main scale of vernier calipers.A prototype VATG sensor with a total length of 600 mm was fabricated using PCB manufacturing technology with a fine measurement capacitor row composed of 150 periods and course measurement capacitor row composed of 149 periods.Based on an analysis of the measurement performance of the prototype VATG sensor, the 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, and a differential induction structure was adopted to reduce common mode disturbance, which significantly improved the measurement performance of the sensor.Experimental results demonstrated that the measurement error of the optimized prototype sensor was ±1.25 µm over the full 600 mm range, ±0.25 µm over a single 4 mm period, and the first-order harmonic error was very small.

Figure 1 .
Figure 1.Schematic diagrams of a single-row capacitive displacement sensing structure and its measurement principle.Top view (a) and side view (b) schematic diagrams of the single-row sensing structure.(c) Schematic diagram presents a geometric representation of the effective area variation of the selected region in (a).(d) Equivalent capacitor circuit based on (b).

Figure 1 .
Figure 1.Schematic diagrams of a single-row capacitive displacement sensing structure and its measurement principle.Top view (a) and side view (b) schematic diagrams of the single-row sensing structure.(c) Schematic diagram presents a geometric representation of the effective area variation of the selected region in (a).(d) Equivalent capacitor circuit based on (b).

Figure 2 .
Figure 2. (a) Schematic diagram of a double-row Vernier-type capacitive structure for conducting absolute position measurements with time-grating.(b) Schematic diagram illustrating the absolute position measurement principle of double-row capacitive structure.(c) High-frequency clock pulse interpolation.

Figure 2 .
Figure 2. (a) Schematic diagram of a double-row Vernier-type capacitive structure for conducting absolute position measurements with time-grating.(b) Schematic diagram illustrating the absolute position measurement principle of double-row capacitive structure.(c) High-frequency clock pulse interpolation.

Figure 3 .
Figure 3. (a) Phase of outputs of row 1 (Urow1) and row 2 (Urow2) within their periods Wrow1 and Wrow2, respectively, versus the displacement of the induction electrode array.(b) Phase difference with respect to the displacement of the induction electrode array.

Figure 3 .
Figure 3. (a) Phase of outputs of row 1 (U row1 ) and row 2 (U row2 ) within their periods W row1 and W row2 , respectively, versus the displacement of the induction electrode array.(b) Phase difference with respect to the displacement of the induction electrode array.

Figure 4 .
Figure 4. Schematic illustrating the determination of the limit of error for the proposed VATG sensor.

Figure 4 .
Figure 4. Schematic illustrating the determination of the limit of error for the proposed VATG sensor.

Figure 5 .
Figure 5. Experimental setup employed for evaluating the measurement performance of the prototype VATG sensors manufactured using printed circuit board technology.

Figure 5 .
Figure 5. Experimental setup employed for evaluating the measurement performance of the prototype VATG sensors manufactured using printed circuit board technology.

Figure 5 .
Figure 5. Experimental setup employed for evaluating the measurement performance of the prototype VATG sensors manufactured using printed circuit board technology.

Figure 6 .
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.

Figure 6 .
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.

2 δ
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 Figure7.

Figure 7 .
Figure 7. Schematic diagram illustrating the optimized structure of the VATG sensor, which includes increasing the distance between the two rows and adding a shielding layer between them to reduce the influence of signal crosstalk.In addition, a differential induction structure is adopted to reduce common mode disturbance.

Figure 7 .
Figure 7. Schematic diagram illustrating the optimized structure of the VATG sensor, which includes increasing the distance between the two rows and adding a shielding layer between them to reduce the influence of signal crosstalk.In addition, a differential induction structure is adopted to reduce common mode disturbance.