# High-Accuracy Calibration Based on Linearity Adjustment for Eddy Current Displacement Sensor
## 2 Original Output Characteristics Test  
### Relational Model of Displacement and Acquisition Voltage  
The basic principle of ECDS to measure the displacement \(x\) of target metal materials is shown in Figure 1a. According to [PERSON]'s law, when an alternating current is applied to the sensor coil, an eddy current is generated in the target material, as shown in Figure 1b. The generated eddy current can be regarded as another induced coil, which has an influence on the voltage and current in the sensor coil at a different degree in accordance with \(x\). Considering the induced coil as a reduced impedance, the equivalent circuit is shown in Figure 1c. In Figure 1b,c, \(U\), \(I\), \(R\), \(L\) denote the voltage, current, resistance and inductance, respectively. Subscript '_Coil_', '_Tar_', 'r' represent the sensor coil, target material, and the reduced impedance, respectively; \(M\) represents their inductance. Then, the reduced impedance \(\Delta Z\) can be:  
\[\Delta Z=\widetilde{M}(R_{Tar}-j\omega L_{Tar}), \tag{1}\]  
where \(\widetilde{M}=\omega^{2}M^{2}/[R_{Tar}^{2}+(\omega L_{Tar})^{2}]\). The following relationship exists between the current in the sensor coil and the one in target material:  
\[\dot{I}_{Tar}=\dot{I}_{Coil}[1-\frac{x/r}{\sqrt{(x/r)^{2}+1}}]\approx\dot{I}_{ Coil}(1-\frac{x}{r}), \tag{2}\]  
where \(r\) is the radius of the sensor coil. Meanwhile, ignoring the minor terms \((x/r)^{2}\) only when \(x/r\ll 1\), the output curve of the sensor has a good linearity. In Figure 2a, according to Kirchhoff's law, there is the following relationship:  
\[-j\omega M\dot{I}_{Coil}+(R_{Tar}+j\omega L_{Tar})\dot{I}_{Tar}=0. \tag{3}\]  
Combining Equations (2) and (3), Equation (4) can be obtained:  
\[\widetilde{M}=\frac{\omega^{2}M^{2}}{[R_{Tar}^{2}+(\omega L_{Tar})^{2}]}=(1- \frac{x}{r})^{2}=1-2\frac{x}{r}+(\frac{x}{r})^{2}. \tag{4}\]  
Substituting Equation (4) into Equation (1) and ignoring the minor terms \((x/r)^{2}\), then Equation (5) can be obtained:  
\[\Delta Z=(1-2\frac{x}{r})(R_{Tar}-j\omega L_{Tar}). \tag{5}\]  
Figure 1: Principle of displacement measurement based on eddy current displacement sensor (ECDS): (**a**) principle model; (**b**) simplified circuit; (**c**) equivalent circuit. Equation (5) shows that the displacement values measured by the sensor is approximately linear to the reduced impedance of the target material. In order to calculate the displacement change, a high-accuracy LC oscillation circuit needs to be established to convert the reduced impedance into voltage for subsequent acquisition. Measurement circuit is shown on the left side of Figure 2, and the simplified equivalent circuit is shown on the right. The monitored voltage can be expressed as:  
\[ \begin{split} U_{2}&=U_{0}-\frac{U_{0}}{R_{1}+Z_{1} }\cdot R_{1},\\ U_{1}&=U_{0}-\frac{U_{0}}{R_{2}+Z_{2}}\cdot R_{2}, \end{split}  \tag{6}\]  
where the impedance of the reference coil \(Z_{1}=1/(j\omega C_{3}+j\omega C_{2}-j(\omega L_{2})^{-1})\), and the impedance of the active coil \(Z_{2}=1/(j\omega C_{4}+j\omega C_{1}-j(\omega L_{1})^{-1})\). The reference coil is set up to serve as the comparative value of the active coil and to facilitate subsequent adjustment of the voltage at the displacement \(x=0\), which is completely shielded from the active coil and insensitive to displacement changes. The active coil is the sensing coil and its impedance changes as the displacement \(x\) changes. Then, the voltage difference is recorded as \(\Delta U=U_{2}-U_{1}\). When \(x\) changes, \(Z_{2}\) changes to \(Z_{2}^{\prime}=Z_{2}+\Delta Z_{2}(\Delta Z_{2}\ll Z_{2})\). Thus, the variation of \(\Delta U\) (ignoring the minor items) can be written as:  
\[d(\Delta U)=\Delta U^{\prime}-\Delta U=-\frac{U_{0}R_{2}}{(R_{2}+Z_{2})^{2}} \Delta Z_{2}. \tag{7}\]  
By Equation (7), we can get that the voltage difference \(d(\Delta U)\) is linear to the reduced impedance \(\Delta Z_{2}\).