## 3 Finite Element Modeling and Probe Parameter Determination
### Determination of Probe Parameters
#### 3.2.2 Determination of Detection Coil Parameters
Therefore, in the orthogonal experiment, only three independent factors need to be considered (\(D_{\rm r}\), \(H_{\rm r}\), and \(K_{\rm r}\)), and the objective function is as shown in Equation (13):  
\[maxC(D_{\rm r},H_{\rm r},K_{\rm r}) \tag{13}\]  
Similar to the excitation coil, in the process of determining the parameters of the detection coil, an orthogonal experiment with four (\(2^{2}\)) levels is chosen for each factor. Considering the geometric structure of the probe, the width of the detection coil winding must be larger than the outer diameter of the excitation coils. To ensure that the detection coil extracts a sufficiently large signal and is not overwhelmed by noise, the selection of four levels for each factor of the detection coil is based on the orthogonal experimental results of the excitation coils. This means that the selection is made by slightly expanding the range based on the outer diameter of the excitation coils. The four levels chosen for each factor of the detection coil are shown in Table 3.  
\(X_{\rm r}\) is chosen as the reference factor. An orthogonal experiment with three factors and four levels, L16 (\(3^{4}\)), is conducted, requiring a total of 16 trials. The results are shown in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**NO.**} & \multicolumn{3}{c}{**Evaluated Parameters**} & **Reference** & **Results** \\ \cline{2-5}  & \(D_{\rm r}\)/mm & \(H_{\rm r}\)/mm & \(K_{\rm r}\)/mm & \(X_{\rm r}\) & \(C\) \\ \hline \#1 & 9 & 0.6 & 0.6 & \(X_{\rm r1}\) & 0.05874 \\ \#2 & 9 & 0.8 & 0.8 & \(X_{\rm r2}\) & 0.05785 \\ \#3 & 9 & 1 & 1 & \(X_{\rm r3}\) & 0.05762 \\ \#4 & 9 & 1.2 & 1.2 & \(X_{\rm r4}\) & 0.05683 \\ \#5 & 10 & 0.6 & 0.8 & \(X_{\rm r3}\) & 0.05396 \\ \#6 & 10 & 0.8 & 0.6 & \(X_{\rm r4}\) & 0.05547 \\ \#7 & 10 & 1 & 1.2 & \(X_{\rm r1}\) & 0.05259 \\ \#8 & 10 & 1.2 & 1 & \(X_{\rm r2}\) & 0.05774 \\ \#9 & 11 & 0.6 & 1 & \(X_{\rm r4}\) & 0.04989 \\ \#10 & 11 & 0.8 & 1.2 & \(X_{\rm r3}\) & 0.04877 \\ \#11 & 11 & 1 & 0.6 & \(X_{\rm r2}\) & 0.05067 \\ \#12 & 11 & 1.2 & 0.8 & \(X_{\rm r1}\) & 0.04972 \\ \#13 & 12 & 0.6 & 1.2 & \(X_{\rm r2}\) & 0.04260 \\ \#14 & 12 & 0.8 & 1 & \(X_{\rm r1}\) & 0.04622 \\ \#15 & 12 & 1 & 0.8 & \(X_{\rm r4}\) & 0.04649 \\ \#16 & 12 & 1.2 & 0.6 & \(X_{\rm r3}\) & 0.04684 \\ \hline \(A_{\rm Z1}\) & 0.03340 & 0.05130 & 0.05293 & 0.05182 & \\ \(A_{\rm Z2}\) & 0.05494 & 0.05208 & 0.05201 & 0.05222 & \\ \(A_{\rm Z3}\) & 0.04976 & 0.05184 & 0.05287 & 0.05180 & \\ \(A_{\rm Z4}\) & 0.04554 & 0.05278 & 0.05020 & 0.05217 & \\ \hline \(T_{\rm Z}\) & 0.02153 & 0.00148 & 0.00273 & 0.00041 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Orthogonal experiment table for detection coil. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Impact Factors** & **Level 1** & **Level 2** & **Level 3** & **Level 4** \\ \hline \(D_{\rm r}\)/mm & 9 & 10 & 11 & 12 \\ \(H_{\rm r}\)/mm & 0.6 & 0.8 & 1 & 1.2 \\ \(K_{\rm r}\)/mm & 0.6 & 0.8 & 1 & 1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Factors and levels of orthogonal tests of the detection coil. In Table 4, the \(T_{Z}\) of each factor is greater than the \(T_{Z}\) of the reference factor \(X_{r}\). This indicates that each factor at the selected levels has a significant effect on the detection voltage change rate, proving the rationality of the experiment. By comparing the range \(T_{Z}\), we can analyze the extent to which each factor affects the detection voltage change rate. A larger \(T_{Z}\) indicates a more significant influence of factor \(Z\) on the evaluation metric. Based on the \(T_{Z}\) values in Table 4, it can be seen that \(D_{r}\) is the primary factor, while \(H_{r}\) and \(K_{r}\) are secondary factors. The number of levels with the largest A value for every factor is selected to enhance the rate of change in detection voltage. Thus, the combination \(D_{r2}H_{r4}K_{r3}\) is determined as the geometry parameters of the detection coil. In summary, the parameters of the excitation coils and detection coil have been determined, as shown in Table 5.