# Analysis of Inductive Displacement Sensors with Large Range and Nanoscale Resolution
## 3 Principle Analysis
### Finite Element Analysis and High-Precision Modeling of Leakage Inductance  
The finite element model of the inductive sensor is established by using COMSOL Multiphysics, as shown in Figure 4. The inductance vs. air gap characteristic of the inductive sensor can be accurately calculated by finite element method (FEM). To analyze the leakage inductance, it is necessary to know the distribution of the leakage flux. It is clearly illuminated in Figure 4 that the leakage flux is mainly concentrated in the slot between the two legs of the sensor probe, as well as the outer side of the leg. The flux density at several points (m0, m1, m2, m3, m4, c0, c1) are obtained by FEM, as Figure 5 shows. It is difficult to build an analytical model to accurately predict the leakage inductance of an inductive sensor. Therefore, we try to estimate the leakage inductance with the help of FEM. \(L_{\text{Analytical}}\), \(L_{\text{FEM}}\) are applied to express the inductance of an inductive sensor calculated by the traditional analytical model (Equation (3)) and by FEM, respectively, as shown in Figure 6. \(L_{\text{FEM}}\) considers the leakage inductance \(L_{\text{S}}\) while \(L_{\text{Analytical}}\) does not, hence, \(L_{\text{S}}\) can be approximated by (\(L_{\text{FEM}}\)-\(L_{\text{Analytical}}\)). Figure 6 shows the change trend of \(L_{\text{Analytical}}\) vs. \(\delta\) curve and \(L_{\text{FEM}}\) vs. \(\delta\) curve is basically the same, which indicates that the leakage inductance is not sensitive to the change of air gap. In particular, when the air gap becomes larger, the leakage inductance approaches a constant that does not change with the air gap. In order to estimate the leakage inductance accurately in the design process, it is necessary to establish the functional relationship between the leakage inductance and the design parameters, such as the number of coil turns, the geometric size of core, etc. Parameter scanning using FEM analysis shows the relationship curves among \(L_{\text{FEM}}\), \(L_{\text{S}}\), and \(N^{2}\), as shown in Figure 7a,b. In addition, we have found that \(L_{\text{S}}\) is proportional to \(N^{2}\), thus, \(L_{\text{S}}\) can be defined by,  
\[L_{\text{S}}=G_{\text{S}}N^{2} \tag{7}\]  
where \(G_{\text{S}}\) is a defined coefficient to calculate the leakage inductance, and \(N\) is the coil turns of the inductive sensor. \(G_{\text{S}}\) is the slope of the fitting line of \(L_{\text{S}}\) vs. \(N^{2}\) curve in Figure 7b, and \(G_{\text{S}}=7.4\times 10^{-6}\) here. \(G_{\text{S}}\) of inductive sensors with different size can be obtained by the same method. Figure 4: The magnetic flux of inductive sensors given by FEM. Figure 5: The flux density \(B\) at different points. The calculation of leakage inductance with FEM is cumbersome and time-consuming, and the FEM model cannot be established before the preliminary design of geometric parameters is completed. Therefore, the FEM model can only be used for analysis, not for design. We expect that there is an expression that can capture the relationship between \(G_{\text{S}}\) and the size of the sensor core. The cross section of sensor core can be chosen as a square with the side length of \(a\), and \(a\) can be used to quickly and accurately estimate \(G_{\text{S}}\) in the design process. Then, by Equation (7) and the known coil turns \(N\), the leakage inductance can be quickly estimated. The relationship between \(G_{\text{S}}\) and \(a\) is calculated, which has a good linearity, as shown in Figure 7c. An approximate expression of \(G_{\text{S}}\) is obtained by linear fitting, that is,  
\[G_{\text{S}}=2.1\times 10^{-6}a+4.2\times 10^{-6} \tag{8}\]  
Further analysis shows that the coefficient in front of \(a\) is close to the order of magnitude of the vacuum permeability \(\mu_{0}=4\pi\times 10^{-7}\) H/m. Hence, we assume that \(G_{\text{S}}\) is related to \(\mu_{0}\), and \(G_{\text{S}}\) can be expressed as,  
\[G_{\text{S}}=\mu_{0}(ma+n) \tag{9}\]  
where \(m\), \(n\) are two parameters for inductive sensors.