# Analysis of Inductive Displacement Sensors with Large Range and Nanoscale Resolution
## 4 Q Value and Excitation Frequency Analysis
### Analysis for Increasing \(Q\) Value and Excitation Frequency  
From the above analysis, we have known that \(\cos\) loss is a culprit to limit the \(Q\) value and excitation frequency. Therefore, it is necessary to analyze the factors that determine the core loss, so as to help designers select appropriate methods to improve \(Q\) value and excitation frequency. The hysteresis loss of the iron core is directly proportional to the area of the hysteresis loop. Therefore, the coercivity of the core shall be as low as possible, and the induction \(B\) should work at a small value. The eddy current loss is mainly related to the conductivity of the core. The higher the conductivity, the greater the eddy current loss. Thus, the soft magnetic materials with as low conductivity as possible, such as ferrite, can be used to reduce the eddy current loss. However, the disadvantage of ferrite is its lower Curie temperature and its brittleness as a ceramic material. Soft magnetic alloys (such as Fe-Ni alloy) have perfect mechanical properties and high temperature stability, but the conductivity is too high (generally greater than 2 \(\times\) 10\({}^{6}\) S/m). For such materials, laminated core is generally used to suppress the eddy current loss. From the point of view of suppressing the eddy current loss, the influences of material conductivity on resistance and \(Q\) value of the inductive sensor have been studied by FEM, as shown in Figures 15 and 16. Here, we ignore the hysteresis loss and assume that there is only eddy current loss of inductive sensors due to the complexity of modeling and calculation of the hysteresis loss (Note: Because the hysteresis loss is ignored, the real value \(Q\) will be much lower than the FEM results. In addition, it is impossible for the \(Q\) value to increase indefinitely with increasing the excitation frequency). The FEM results show that with the increase of the core conductivity, the resistance of the inductive sensor increases, while the \(Q\) value decreases. Moreover, the excitation frequency corresponding to the maximum \(Q\) value of \(Q\) vs. \(f\) curve decreases as the core conductivity increases. The reduction of the allowable excitation frequency will hinder the improvement of the bandwidth of the sensor. We also find that when the core conductivity decreases to 1 \(\times\) 10\({}^{4}\) S/m, the eddy current loss resistance changes slowly with the excitation frequency. At the same time, as the excitation frequency increases, the \(Q\) value rises to a height and is no longer affected by the excitation frequency, and almost unchanged. In this case, the only factor limiting the actual \(Q\) value and the allowable excitation frequency is the hysteresis loss. Therefore, we come to a conclusion: If the conductivity of the selected core material is less than 1 \(\times\) 10\({}^{4}\) S/m, the eddy current losses are small enough that the core does not need to be laminated. Otherwise, the laminated core consisting of thin sheets must be used to suppress core loss. The selection rule of the thickness of sheets can refer to the calculation formula of the skin depth [13; 14]:  
\[\delta_{\rm e}=\sqrt{\frac{2}{\omega\mu\sigma}} \tag{23}\]  
where \(\delta_{\rm e}\) denotes the skin depth; \(\mu\) and \(\sigma\) are the permeability and conductivity of the core, respectively; \(\omega\) = 2\(\pi\)\(f\). In theory, the thinner the sheet, the better the effect of suppressing eddy current loss. In addition, if the thickness of the sheet were much less than \(\delta_{\rm e}\), the eddy current loss can be ignored. The thickness of the sheet is limited by the manufacturing process, which usually makes the core material with high conductivity unable to reach the sheet thickness required to suppress the eddy current loss. The sheet thickness required for soft magnetic materials with low conductivity is easier to obtain, so that the inductive sensor can achieve higher \(Q\) value and larger allowable excitation frequency.