# Analysis of Inductive Displacement Sensors with Large Range and Nanoscale Resolution
## 2 Structure of Inductive Sensors  
Inductive sensors are mainly composed of sensor core (sensor probe), armature and coil, as shown in Figure 1. Both core and armature are made of magnetically permeable materials with high permeability. When the air gap length (\(\delta\)) between the armature and sensor probe changes, the reluctance in the air gap changes, so as the inductance of the sensor. By measuring the change of inductance (\(L\)), the displacement of the armature can be obtained. Inductive sensors usually adopt differential measurement mode with a pair of two electronically matched sensor probes, which are positioned on opposite sides or ends of the armature for displacement measurement. The advantages of this approach are improved sensitivity and linearity of the transfer characteristic, improved immunity to environmental changes, and a simplified electronic interface. The structure of differential inductive sensors is shown in Figure 2. The sensor probe to armature relationship is such that as the armature moves toward to one sensor probe, it moves away from the other an equal amount. As a result, the inductance of one probe increases and the other one decreases. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sensor** & **Principle** & **Manufactures** & **Size (mm)** & **Range (mm)** &
\begin{tabular}{c} **RMS Resolution** \\ **(@100 Hz)** \\ \end{tabular} \\ \hline DIT-5200-15N & Eddy current & KAMAN & \(\mathcal{O}4.78\times 28.6\) & \(\pm 0.25\) & \(3.5\) mm \\ CS02 & Capacitive & Micro-Epsilon & \(\mathcal{O}6\times 12\) & \(0.2\) & \(0.15\) nm (@2 Hz) \\ ES-U1 & Eddy current & Micro-Epsilon & \(\mathcal{O}6\times 30\) & \(1\) & \(20\) nm (@20 Hz) \\ KRS719(01) & LVDT & Micro-Epsilon & \(\mathcal{O}10\times 16\) & \(\pm 1\) & \(1.4\) μm \\ D-510.021 & Capacitive & Physik Instrumente & \(\mathcal{O}8\times 30\) & \(0.15\) & \(1.5\) nm (@20 Hz) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Typical commercial miniaturized nanometer displacement sensors. Figure 1: Structure of an inductive sensor. Differential inductive sensors are commonly used for measuring small displacement for their high precision. However, there will be great nonlinearity if they are operated in a large range. We define the armature displacement as \(x\), and the armature displacement at the center of the two probes is zero. \(\Delta L=L_{1}-L_{2}\) is defined as the differential inductance. Figure 3 shows \(\Delta L\) vs. \(x\) nonlinear relationship of the differential inductive sensor operated in large range. The curve can be divided into 3 zones:  
1. Zone1: Linear workspace zone in differential working mode;
2. Zone2: Nonlinear workspace zone in differential working mode;
3. Zone3: Nonlinear workspace zone in single-probe-dominated mode. When the displacement to be measured is in a small range within the Zone1, the inductive sensor is working in the ideal differential mode with high linearity. As the displacement increases to Zone2, the nonlinearity increases. As further displacement increases, the inductance value of one probe away from the armature decreases to a very small value. The differential effect is weakened. The working mode of the inductive sensor in Zone3 changes into a single-probe-dominated mode. There are several difficulties in the design of inductive sensors with large range and high precision:  
1. The inaccurate traditional calculation model of inductance vs. displacement characteristic can not meet the design requirements of large range inductive sensors;
2. In terms of circuit factors, there are complex error terms present in the transfer characteristic of differential inductive sensors;
3. Due to the existence of core loss, it is difficult to effectively improve the \(Q\) value;
4. There is a lack of a design rule to select the optimal excitation frequency for the inductive sensor. These difficulties are the main reasons that limit the expected performance of inductive sensors in large range operation, which will be analyzed in detail in Sections 3 and 4.