# Analysis of Inductive Displacement Sensors with Large Range and Nanoscale Resolution
## 5 Experimental Results  
Two types prototype sensor probes (A1.5N1000 and A1.0N600) with different sizes based the structure shown in Figure 1 were used for experiments to support the previous analysis. The geometry and material specifications of them are shown in Table 2. The corresponding 3D model of the prototype sensor is shown in Figure 18a. The sensor probe size is within 6 mm \(\times\) 6 mm \(\times\) 3 mm, which ideally meets the miniaturization requirements. An experimental platform is built as shown in Figure 18b. The target is driven by a spiral microprobe with a resolution of 1 \(\upmu\)m. A capacitive sensor (Type: CSH05) supplied by MICRO-EPSILON is used as a reference standard, which has nanoscale resolution. A high-precision impedance analyzer is used to test the impedance of prototype sensors. In addition, a phase lock-in amplifier (Type: LF2 LI) provided by Zurich Instruments is used to demodulate the analog signal from the differential inductive sensor. Experiments are conducted mainly to obtain the impedance characteristics (inductance, resistance, \(Q\)) of prototype inductive sensor probes and the characteristic of differential inductive sensor. The obtained experimental results are directly used to support the analysis in this paper. An impedance analyzer is used to measure the impedance-related quantity of the prototype sensor probes (A1.5N1000 and A1.0N600). Figures 19 and 20 are mainly used to verify the correctness of the proposed leakage inductance model of the inductive sensor. Figures 21-23 are mainly used in the analysis of \(Q\) value and excitation frequency. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **A1.5N1000** & **A1.0N600** \\ \hline \(a\) & 1.5 mm & 1.0 mm \\ Coil turns, N & 1000 & 600 \\ Wire diameter of coil, d & 0.05 mm & 0.05 mm \\ Core material & \multicolumn{1}{c}{Laminated Ni-Fe alloy (50\% Ni)} & Ni-Fe alloy (50\% Ni) \\ Coil material & Copper & Copper \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometry and material specifications of prototype sensors. Figure 18: 3D model of the inductive sensor and the experimental platform: (**a**) Corresponding 3D model of the prototype sensor; (**b**) Experimental platform. Figure 19: Inductance vs. air gap characteristics of prototype sensors with the excitation frequency of 16 kHz: (**a**) \(L\)-\(\delta\) characteristic of A1.5N1000; (**b**) \(L\)-\(\delta\) characteristic of A1.0N600. Figure 20: Comparison of \(L\)-\(\delta\) characteristics of prototype sensors with the excitation frequency of 16 kHz obtained by experiment and modified analytical model: (**a**) A1.5N1000; (**b**) A1.0N600. The bridge circuit shown in Figure 11 and the lock-in amplifier (Type: MF2 LI) are used to calibrate the differential inductive sensor composed of a pair of prototype sensor probes (Type: A1.5N1000). The input voltage of the bridge is 7 Vrms. In addition, according to the proposed excitation frequency selection method, the excitation frequency of 16 kHz is selected. Set the amplifier's magnification of measurement circuit to 1. The experimental results are shown in Figure 24 and Table 3. RMS resolution was obtained by dividing the output peak to peak noise by the zero position sensitivity and dividing by a further 6.6. The sensor has a high resolution within a large range of \(\pm\)0.5 mm, which proves the feasibility  
Figure 23: The relationship curve between \(Q\) value and excitation frequency of two prototype sensor probes under different air gap: (**a**) \(Q\) vs. \(f\) of A1.5N1000; (**b**) \(Q\) vs. \(f\) of A1.0N600. Figure 21: Influence of air gap \(\delta\) and excitation frequency \(f\) on the resistance \(r\) of inductive sensors: (**a**) Relationship between \(r\) and \(\delta\) of A1.5N1000 with the excitation frequency of 16 kHz; (**b**) Relationship between \(r\) and \(f\) of A1.5N1000 (\(\delta\) = 0.7 mm); (**c**) Relationship between \(r\) and \(\delta\) of A1.0N600 with the excitation frequency of 16 kHz; (**d**) Relationship between \(r\) and \(f\) of A1.0N600 (\(\delta\) = 0.3 mm).