# A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System
## 5 Testing the MEC Model with FEA  
### FEA Studies  
The accuracy test of the designed MEC model was conducted with reference to FEA studies. For this purpose, a series of 3D MFL models with varying dimensions were established in the ANSYS Maxwell [27] simulation environment. Figure 9 illustrates the FEA solutions of the MFL system. Our fundamental approach in the FEA studies is to obtain results by forming a fixed flux density level in the tested material for different MFL models and to evaluate the MEC model using these results as a reference. Here, adjusting the MMF values is accomplished by forming a fixed flux density in the material. This enables a more effective evaluation of the proposed MEC model's performance. This is because changes in the dimensions of different MFL models impact the overall system dynamics. This testing approach, combined with ensuring near saturation in the test material, offers valuable insights into achieving the most efficient system design. Moreover,  
Figure 8: Iteration algorithm of the MEC model. the effects of cycle length brought by different system dimensions are also included in the problem solution. The magnetostatic solver type was applied in FEA studies. The boundary parameter in the simulation model was set at a distance of 50 cm from the system to avoid boundary effects. Since our testing approach is not aimed at a specific test material or application, using different \(B\)-\(H\) curves for yoke and test material will not affect the character of the observed results. Therefore, in FEA and MEC studies, the required \(B\)-\(H\) data for the yoke and test material have been derived from the \(B\)-\(H\) curve of the X-52 steel [8] material shown in Figure 2.  
The variation in the \(w_{e}\) and \(w_{\mathrm{s}}\) parameters was taken as a basis for obtaining different MFL system models. Due to their nonlinear effect on the main and leakage flux, \(w_{e}\) and \(w_{\mathrm{s}}\) have a higher impact on the system operation than other parameters. The system height \(w_{w}\) parameter can be accepted within the effect caused by changes in \(w_{\mathrm{s}}\). Therefore, the \(w_{w}\) value was taken as fixed in the analyses. The \(l_{g}\) distance affects the horizontal leakage flux almost linearly. Therefore, this parameter was also fixed in the analyses. The \(w_{e}\) and \(w_{\mathrm{s}}\) values selected for analyses are given in Table 2. The fixed parameters are defined as \(w_{m}\) = 5 mm, \(w_{w}\) = 40 mm, \(l_{e}\) = 60 mm, \(d_{w}\) = 8 mm, and \(l_{g}\) = 2 mm. Twenty-five different MFL models were produced by combinational changes in the \(w_{e}\) and \(w_{\mathrm{s}}\) parameters, and corresponding FEA studies were conducted. In these studies, the necessary MMF values were adjusted for each model to form the flux density within the material at a level of 1.8T. The adjustment of MMF values, defined by the equation \(F\) = \(Ni\), was performed by keeping the number of turns \(N\) fixed and assigning different current values \(i\). Figure 10 illustrates the variation in MMF values obtained from FEA studies for different combinations of \(w_{e}\) and \(w_{\mathrm{s}}\). As seen from Figure 10, the MMF values required to generate 1.8T within the tested material vary directly with \(w_{\mathrm{s}}\) and inversely with \(w_{e}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \multicolumn{5}{c}{**Values (mm)**} \\ \hline \(w_{e}\) & 5 & 10 & 15 & 20 & 25 \\ \(w_{\mathrm{s}}\) & 30 & 40 & 50 & 60 & 70 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Varied parameters’ values. Figure 9: FEA solutions of the MFL model. (**a**) Mesh model. (**b**) Flux density distribution. The nonlinear change effect of the \(w_{e}\) and \(w_{s}\) parameters on the horizontal leakage flux is illustrated in Figure 11. As observed, the variation in \(w_{e}\) has a more dominant effect than \(w_{s}\). This is because the \(w_{e}\) dimension directly influences the main flux of the system.