# Analysis of Inductive Displacement Sensors with Large Range and Nanoscale Resolution
## 1 Introduction  
The nanometer displacement sensor is an important part of the nanopositioning system, which provides feedback signal for the motion control and greatly affects the performance of the system [1, 2, 3]. There are six main requirements for displacement sensors in the nanopositioning applications of our interest: (1) Nanoscale resolution (0.1-10 nm); (2) Large measuring range (for example, hundreds of microns or even few millimeters); (3) Miniaturization (for example, the size of each of the three coordinate directions is several microns); (4) High bandwidth; (5) Immunity to harsh environments; (6) Multi-degree-of-freedom measurement can be realized by configuring multiple sensors in different positions. The most commonly used measurement methods that can potentially meet the above conditions are capacitive, eddy current and inductive sensors [4]. Among these types of sensors, capacitive sensors have the highest resolution. However, the dielectric between capacitive plates is sensitive to environmental conditions, which makes capacitive sensors unsuitable for harsh environments. The eddy current sensor is one of the most common position sensors in nanopositioning systems because of its high resolution and immunity to environmental changes, as well as super-high bandwidth of tens of kHz. One possible drawback of eddy current sensor is that it is easy to be disturbed by external magnetic field. There are many types of inductive sensors, of which the two most commonly used inductance techniques for nanoscale measurement are linear variable differential transformers (LVDTs) and linear variable differential inductors (LVDIs) [5]. LVDTs aresensitive to lateral motion, thus not suited to multi-degree-of-freedom applications [4]. Generally, LVDTs are defined independently, and they are distinguished from the inductive sensors we will analysis below in this paper. Compared with LVDTs, LVDIs are more suitable for the nanopositioning applications of our interest. In some literature [6; 7], LVDIs are also referred to as variable reluctance transducers/sensors, in the following the appearance of \"inductive sensor\" all refer to this type of sensors. Further discussion of such inductive sensors will be carried out below. Inductive sensors have been introduced in a review [5], which states that such sensors (with total dimensions of around 5 mm \(\times\) 5 mm \(\times\) 5 mm) can reach a range of 0.2 mm with a signal-to-noise ratio of about 10,000:1 and achieve a resolution better than 50 pm RMS in the range of 50-100 nm. This indicates that the inductive sensor has ultra-high resolution and compactness required by nanopositioning applications. In early 1980s, [PERSON] et al. [8] described a simple scheme for the continuous measurement of displacement using the differential inductive sensor and its application in reaction torque measurement of a rotating machine delivering or receiving power. Their prototype instrument can reach a resolution of about 80 nm RMS for the measuring range of \(\pm\)0.1 mm. In recent 10 years, inductive sensors are still concerned by researchers. [PERSON] et al. [7] have proposed a simple, and high accuracy dual slope inductance-to-digital converter (DSIDC) suitable for differential inductive sensors. This converter not only provides a direct digital output, but also improves the linear range of operation significantly. [PERSON] et al. [9] have studied the differential inductive sensor for application in the position detection of the active magnetic bearing (AMB). They designed an inductive sensor that has higher linearity, higher sensitivity and wider bandwidth than traditional inductive displacement sensors and the suspension accuracy of the magnetic suspension molecular pump can reach 2.4 um when applying this sensor. It is difficult to simultaneously satisfy the miniaturized, large measuring range, nanoscale resolution for displacement sensors.