# Analysis of Inductive Displacement Sensors with Large Range and Nanoscale Resolution
## 3 Principle Analysis
### Finite Element Analysis and High-Precision Modeling of Leakage Inductance
Combining Equations (8) and (9), yields \(m=1.67\) and \(n=3.34\). In addition, we can acquire an expression as follows,  
\[G_{\text{S}}=\mu_{0}(1.67a+3.34) \tag{10}\]  
Inserting Equation (10) into (7) yields,  
\[L_{\text{S}}=\mu_{0}N^{2}\times(1.67a+3.34) \tag{11}\]  
Equation (3) is the traditional analytical model for calculating the inductance sensor, and the inductance obtained does not consider the leakage inductance. By adding the Equation (11) to Equation (3), we can obtain a modified analytical model of inductance vs. air gap characteristic:  
Figure 6: Inductance vs. air gap \(\delta\) relationships obtained by FEM and analytical model, respectively. Figure 6.0 Relationships among \(L_{\text{S}}\), \(G_{\text{S}}\), \(N^{2}\) and \(\pi\). (a) \(L_{\text{FEM}}\) vs. \(N^{2}\); (b) \(L_{\text{S}}\) vs. \(N^{2}\); (c) \(G_{\text{S}}\) vs. \(a\). Note: \(a\) is the side length of the core leg (whose cross section is square) of the inductive sensor. Figure 6.1  
where \(L^{*}\) is the total inductance of inductive sensors, including the leakage inductance; \(a^{2}\) is the cross-sectional area (\(S=a^{2}\)) of the air gap. The effects of different permeability and conductivity of sensor core and different excitation frequency on the inductance of inductive sensors are analyzed by FEM. Figure 8 demonstrates the above modified model for estimating leakage inductance and total inductance is applicable to the inductive sensors with cores made of different permeable materials and with different excitation frequencies, as long as the cores have high permeability and low conductivity. For permeable materials with high conductivity, the laminated core can reduce the equivalent conductivity, so that it can also meet the requirements of the model. According to the above analysis, combined with other basic electromagnetic theories, we summarize the applicable conditions of the above estimation model of leakage inductance of the inductive sensor:  
1. [label=(0)]
2. Ensure the magnetic induction \(B\) works in the linear segment of the \(B\)-\(H\) curve  
2. It is suitable for the inductive sensor with the core of different permeable materials. However, the core materials with high permeability and low conductivity should be used whenever possible;
3. It is suitable for different excitation frequencies below 100 kHz;
4. The core must be the same as the structure proposed in this paper, and the core legs can also be cylindrical (correspondingly, \(a\) will represent the diameter of the cylindrical section of the core);
5. If the sizes of the core other than the dimension \(a\) have a significant change, the values of \(m\), \(n\) in Equation (9) need to be corrected. Using the above model can avoid tedious and time-consuming numerical calculations and help us to quickly calculate the leakage inductance. Based on the leakage inductance model, we can accurately predict the inductance vs. armature displacement characteristic of the inductive sensor, and better ensure the correct selection of the geometry parameters (such as the core size and the coil turns). Moreover, the sensitivity of the differential inductive sensor is closely related to the leakage inductance, which will be analyzed in Section 3.5. The calculation model of the leakage inductance can help designers to design the parameters of the sensor accurately according to the sensitivity requirements.