# Analysis of Inductive Displacement Sensors with Large Range and Nanoscale Resolution
## 4 Q Value and Excitation Frequency Analysis
### Selection Rule of Excitation Frequency  
Figure 14a shows that appropriately increasing the excitation frequency \(f\) can increase the \(Q\) value. However, when the \(f\) increases to a certain value, the increase rate of core loss becomes faster, which makes the sensor resistance \(r\) increase quickly, resulting in the decrease of \(Q\) value. Therefore, in order to ensure a large \(Q\) value, it is necessary to select an appropriate excitation frequency. However, the excitation frequency corresponding to the maximum \(Q\) value in the \(Q\) vs. \(f\) curve may not be optimal, which will be analyzed in detail below. Figure 14: A design rule considering \(Q\) value and excitation frequency \(f\): (**a**) \(Q\) vs. \(f\) relationship; (**b**) Selection rule of excitation frequency \(f\). In fact, for the inductive sensor with large core loss, the resistance \(r\) of the inductive sensor changes greatly with the air gap, as shown in the FEM results (Figure 15) and experimental results (Figure 21). Ideally, the operating characteristics of inductive sensors are completely determined by \(L\) vs. \(\delta\) relationship. If \(r\) vs. \(\delta\) relationship is significant, it would greatly reduce the accuracy of the sensor. The significance of \(r\) vs. \(\delta\) can be expressed as \(Q\) vs. \(\delta\). As can be seen from Equation (18), if the significance of \(r\) vs. \(\delta\) is low, \(Q\) vs. \(\delta\) and \(L\) vs. \(\delta\) have the same change trend, and \(Q\) decreases with the increase of \(\delta\). However, if the significance of \(r\) vs. \(\delta\) is very high, \(Q\) vs. \(\delta\) and \(L\) vs. \(\delta\) have the opposite trend, and \(Q\) increases with the increase of \(\delta\). For the latter, the change of \(r\) is very large with \(\delta\). In this case, even if \(Q\) value increases, it is difficult to suppress the adverse effect of \(r\) on the sensor with the change of \(\delta\). Therefore, a selection rule of excitation frequency \(f\) is proposed, as shown in Figure 14. Meaning of each parameter in Figure 14a: \(Q_{\rm m}\) refers to the maximum value of the \(Q\) vs. \(f\) curve, and \(f_{\rm m}\) is the excitation frequency corresponding to \(Q_{\rm m}\). \(Q_{\rm r}\) is the minimum \(Q\) value required by the sensor. If \(Q_{\rm m}\geq Q_{\rm r}\), there will be an excitation frequency corresponding to \(Q_{\rm r}\), which is called \(f_{\rm r}\) (\(f_{\rm r}>f_{\rm m}\)). When the excitation frequency \(f\) increases to a certain value (defined as \(f_{\rm t}\)), the \(Q\) value begins to increase with the increase of air gap, and \(Q_{\rm t}\) is the defined value corresponding to \(f_{\rm t}\). The frequency named \(f_{\rm a}\) is the minimum excitation frequency that meets the specific bandwidth requirement of the sensor. The selection of excitation frequency \(f\) can also be simply summarized into the following three steps:  
1. [label=(0)]
2. Ensure the excitation frequency must not exceed the frequency at which the \(Q\) value begins to increase with the air gap (\(f\leq f_{\rm t}\));
3. Ensure the excitation frequency must be greater than the minimum excitation frequency required for the sensor bandwidth (\(f\geq f_{\rm a}\));
4. Determine whether the maximum \(Q\) value, corresponding to the frequency \(f_{\rm a}\leq f\leq f_{\rm t}\), satisfies the required minimum \(Q\) value. If satisfied, the frequency corresponding to this maximum \(Q\) value is the optimum excitation frequency to be selected. If not, the sensor will need to be redesigned. Two prototype sensor probes tested in this paper belong to Case1 of the design rule shown in Figure 14. However, their corresponding \(Q_{\rm t}\) values are both less than 10, which is far from meeting our requirements, as shown in Figure 23. Therefore, the current prototype sensors can be improved by increasing their \(Q\) value, so as to better ensure the performance of large range operation.