# A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System
## 5 Testing the MEC Model with FEA
### MEC Model Verification  
The main goal of the MFL system design is to achieve the appropriate flux density in the tested material. In addition, the fact that magnetic sources are located on the poles causes these regions to play a decisive role in the system's main flux and leakage flux. Thus,  
Figure 11: The behavior of the horizontal leakage flux according to varying \(w_{e}\) and \(w_{s}\) parameters. Figure 10: The MMFs determined based on varying \(w_{e}\) and \(w_{s}\) values. the pole region flux densities, along with the flux density of the test material, were taken as a basis in the performance analysis of the designed MEC model. By its nature, the MEC calculates its measurements as the average of the branch cross-sectional area. Therefore, the flux densities of the FEA studies were also measured as volumetric average values. In addition, the pole region in the MEC model is represented by three separate permeances, \(P_{ub}\), \(P_{mb}\), and \(P_{lb}\). Therefore, in the MEC model, the flux density of the pole region value was obtained by averaging the flux densities calculated for these three permeances. The MMF values derived from the FEA studies served as inputs for the MEC algorithm to perform analyses. The first performance criterion for verifying the MEC model is the consistency between the pole region flux densities calculated by the MEC and the FEA results. Figure 12 shows the pole region flux densities obtained from both FEA and MEC studies comparatively for all tests. The graphs in Figure 12 demonstrate that despite variations in \(w_{e}\) and \(w_{s}\), the MEC model tracks the FEA results. To observe the individual effects of changes based on \(w_{e}\) and \(w_{s}\) on this result, Table 3 is also generated. Table 3 presents the mean absolute error (MAE) resulting from variations in one parameter while keeping the other fixed. Based on the findings from Figure 12 and Table 3, it is observed that as \(w_{e}\) decreases and \(w_{s}\) increases, the discrepancies between MEC and FEA increase. Additionally, it is noted that \(w_  
According to the results shown in Figure 13, the errors in flux density produced by the MEC model range from a maximum of 10.2% to a minimum of 1.6%, with an average of 5.2%. These results indicate the MEC model's capability to represent the MFL system at a high level. From Table 4, it is understood that the MRE values rise with increasing \(w_{s}\) and decreasing \(w_{e}\). Additionally, the variation in \(w_{e}\) has a greater impact on the error rates than \(w_{s}\). In the flux density calculations, the primary factor contributing to differences between the FEA and MEC model results is that the flux density results must be measured as the average value in both methods. However, the flux densities throughout the system actually exhibit a non-uniform distribution. This is because the magnetic flux tends to flow predominantly through the inner parts of the system due to its preference for the shortest path. This effect results in high flux densities in the inner parts. In the results, the error values reaching their maximum, particularly as \(w_{e}\) approaches the material's thickness \(w_{m}\), can be interpreted in this context. Because, in this case, the flux densities of the inner parts of the pole region dramatically increase, and the direction and magnitude of the leakage flux within the slot change. This primarily diminishes the adequacy of the MEC model permeances employed to represent the leakage flux within the slot. Additionally, the increasing impact of other neglected leakage fluxes also becomes a factor that escalates errors. A similar effect emerges due to the rising MMF values with the increase in \(w_{s}\). However, this effect remains at a lower level compared to \(w_{e}\). The FEA method can model these effects perfectly because it analyzes the system by dividing it into tiny discrete elements using the mesh method.