# Analysis of Inductive Displacement Sensors with Large Range and Nanoscale Resolution
## 3 Principle Analysis
### Measurement Circuit Factors  
The measurement circuit of differential inductive sensors is simple, as shown in Figure 11. AC bridge is used as the front circuit. A pair of inductive sensor probes with identical geometrical and electrical parameters act as arms of the bridge. Their impedances are \(Z_{1}=\mathrm{j}\cdot 2\pi\ell_{1}+r_{1}\), \(Z_{2}=\mathrm{j}\cdot 2\pi\ell_{2}+r_{2}\), respectively. The input voltage of the bridge is \(\tilde{U}_{\mathrm{in}}\), and the output voltage of the bridge is given by,  
\[\widetilde{U}_{\mathrm{out}}=\frac{Z_{1}-Z_{2}}{Z_{1}+Z_{2}}\frac{\widetilde{ U}_{\mathrm{in}}}{2}=-\frac{\mathrm{j}\cdot 2\pi f(L_{1}-L_{2})+(r_{1}-r_{2})}{2 \mathrm{j}\cdot 2\pi f(L_{1}+L_{2})+(r_{1}+r_{2})]}\widetilde{U}_{\mathrm{in}} \tag{13}\]  
Ideally, when the armature is at zero position, the two sensor probes have the same inductance and resistance values. Hence, at zero, \(L_{1}=L_{2}=L_{0}+L_{5}=L_{0}^{*}\), \(r_{1}=r_{2}=r_{0}\). In a small range near zero position, \(L_{1}+L_{2}=2L_{0}^{*}\), \(r_{1}+r_{2}=2r_{0}\), and the \(Q\) value of each probe is \(Q_{0}=2\pi fL_{0}^{*}/r_{0}\) when the armature at zero position, then Equation (13) can be converted to,  
\[\widetilde{U}_{\mathrm{out}}=\frac{\widetilde{U}_{\mathrm{in}}}{4}\left[ \frac{\mathrm{Q}_{0}}{1+\mathrm{Q}_{0}}\frac{\Delta L}{L_{0}^{*}}+\frac{1}{1+ \mathrm{Q}_{0}}\frac{\Delta r}{r_{0}}+\mathrm{j}\frac{\mathrm{Q}_{0}}{1+ \mathrm{Q}_{0}{}^{2}}\left(\frac{\Delta L}{L_{0}^{*}}-\frac{\Delta r}{r_{0}} \right)\right] \tag{14}\]  
where \(\Delta L=L_{1}-L_{2}\), \(\Delta r=r_{1}-r_{2}\). \(L_{0}^{*}\) is the total zero inductance including the leakage inductance (\(L_{5}\)). This equation contains two error terms with \(\Delta r\), which are in phase and orthogonal to \(\tilde{U}_{\mathrm{in}}\), respectively. For the differential inductive sensor with large range, the error terms will be more complex. This is because the nonlinearity in the large range makes \(L_{1}+L_{2}\) and \(r_{1}+r_{2}\) not ideal constant values. These error terms will lead to large zero residual voltage, because it is difficult to ensure that the in-phase component and orthogonal component of the output voltage are zero at the same time when the armature is at the zero position. In addition, the existence of complex error terms will make it difficult to find a calibration curve whose mapping error meets the accuracy requirements. For example, for the calibration of the sensor with complex error terms, if the polynomial fitting curve is used as the calibration curve, higher order will be required, and even so, there may still be large mapping errors. Moreover, due to the existence of the error terms, the change of sensor resistance caused by temperature change will affect the stability of the sensor.  
Therefore, the large range inductive sensor is required to have a \(Q\) value as high as possible, so that the error terms can be suppressed. As long as the \(Q\) value is large enough, Equation (13) can be simplified to,  
\[\widetilde{U}_{\mathrm{out}}=-\frac{\widetilde{U}_{\mathrm{in}}}{2}\frac{ \Delta L}{L_{1}+L_{2}} \tag{15}\]  
According to the previous analysis, \(\Delta L\) contains the displacement information of the armature, so the change of bridge output voltage \(\tilde{U}_{\mathrm{out}}\) reflects the armature displacement.  
Figure 11: Simplified AC bridge measuring circuit of differential inductive sensors.