# Analysis of Inductive Displacement Sensors with Large Range and Nanoscale Resolution
## 1 Introduction
The above displacement sensors (capacitive sensors, eddy current sensors, LVDIs and inductive sensors) are inevitably facing the problem of small range when miniaturization and nanoscale resolution are required, so they need to be improved or be replaced with better alternatives to adapt to the compact and large stroke multi-degree-of-freedom nanopositioning systems. Typical commercial products corresponding to these sensors are shown in Table 1, in which the sensor probe diameter must be less than \(\diameter 10\) mm. It is desired that the sensor probe be as small as possible and meet large range and nanoscale resolution requirements. The inductive sensors (LVDIs) of our interest are difficult to be found corresponding commercial products in the market, but they can be smaller in size than the advanced products shown in the table. However, from the current literatures [5; 8], the measuring range of the inductive sensor meeting the requirements of miniaturization and nanoscale resolution has not exceeded 0.2 mm. This is because of the significant nonlinearity of inductive sensors operating in large range. If the range of inductive sensors can be extended on the basis of miniaturization, such sensors may be more competitive than the commercial displacement sensors listed in Table 1. With the development of modern digital linearization techniques, such as polynomial compensation by software, good original linearity is not so important in many situations, thus the range of inductive sensors can be extended by using the nonlinearity characteristic. On this basis, a reliable design method of large range inductive sensors is needed. It is necessary to analyze the inductance vs. measured displacement characteristic and the measuring circuit factors of inductive sensors, comprehensively consider the effects of leakage inductance, fringing effect and core loss, and select the appropriate excitation frequency. The size of the sensor core and the number of coil turns should also be considered. In this way, the error source can be suppressed to the greatest extent and the good performance of the inductive sensor in a large range of operation can well be guaranteed. In this paper, in order to obtain an appropriate design method and ensure the large range operation performance of the inductive sensor with miniaturized size, extensive analysis is carried out by analytical analysis, finite element method and experiments. This paper is organized as follows: Section 2 discusses the structure of inductive sensors and its current performance limitations. Consequently, a novel leakage inductance model of inductive sensors is established by using FEM in Section 3. By considering the fringing effect, the influence of armature size on the inductance of inductive sensors is studied, and the design rule of armature size is established. The important role of \(Q\) value in suppress the complex error terms of differential inductive sensors and improving their performance for large range operation is analyzed. In Section 4, based on the requirements of high \(Q\) value and high bandwidth, the corresponding design rule is formulated, and the selection method of the optimal excitation frequency is proposed. The content of Section 5 is the experimental results of the prototype sensors to support the research of analysis and design methods in this paper. In Section 6, the conclusion is made.