# A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System
## 6 Conclusions  
This study proposes a novel MEC model to achieve an efficient MFL system design. This model is designed considering the nonlinear behaviors of the magnetic elements, winding leakage flux, horizontal leakage flux within the slot, and the fringing effect. The proposed MEC model can comprehensively represent the system's behavior with these features. The main requirement in the MFL system design is to form a level of flux density near saturation in the test material with the change in system dimensions. Additionally, it is also crucial to consider the undesired effects of slot leakage flux on the sensor for an efficient system design. In this study, a reverse engineering-like test approach is adopted to observe the mentioned effects, keeping the flux density of the test material fixed. This approach enables us to effectively and comparably observe other system parameters' influence on common objectives.  
The accuracy test of the proposed MEC model is carried out based on the FEA results, considering the variation in the \(w_{e}\) and \(w_{s}\) parameters. In the flux density results of the pole region, as \(w_{e}\) decreases from 25 mm to 5 mm, the average absolute error MAE_w\({}_{e}\) increases by 0.167T. The rise in the \(w_{s}\) from 30 mm to 70 mm results in the MAE_w\({}_{s}\) value increasing by 0.05T. These results show the error variation in the MEC model in tracking the FEA. In terms of the flux density calculation of the test material, the maximum relative error is obtained as 10.2%, and the average relative error is 5.2%. Moreover, with the change in \(w_{e}\) from 25 mm to 5 mm, the average relative error MRE_w\({}_{e}\) increases by 5.6%. With the increase in \(w_{s}\) from 30 mm to 70 mm, the MRE_w\({}_{s}\) error value rises by 3.2%. These results show that the change in \(w_{e}\) significantly affects the errors compared to \(w_{s}\). This is because as \(w_{e}\) approaches the \(w_{m}\) value, the increase in the flux density of the pole region is much greater than the increase associated with the change in \(w_{s}\). Ultimately, the results indicate that despite changes in critical system dimensions, the MEC model produces errors at a reasonable level and successfully tracks the FEA results.  
The primary purpose of the designed MEC model is to generate fast solutions in the system design process. In this study, the average processing time is 5.2 s for the MEC model, whereas for FEA, this time is determined as 21.5 min. The proposed MEC model offers a significant time advantage compared to the FEA method while generating acceptable error levels. With this feature, the necessary MFL system design for rapid prototyping tasks requiring system optimization can be accomplished using the proposed MEC model.  
The performance of the proposed MEC model can be enhanced through model refinement. In this regard, it is necessary to consider the naturally occurring inhomogeneous distribution effect in the flux densities of magnetic elements first. This process can be achieved by modeling the system's magnetic yoke by dividing it into segments. Secondly, it is essential to define additional permeances in the MEC model to represent leakage fluxes more accurately. With these improvements, the MEC model can provide more accurate results.  
Conceptualization, O.K. and H.H.C.; methodology, O.K. and H.H.C.; formal analysis, O.K. and H.H.C.; software, O.K.; writing--original draft preparation, O.K.; validation H.H.C.; writing--review and editing, O.K. and H.H.C. All authors have read and agreed to the published version of the manuscript.  
The research presented in this study has been supported by the Marmara University Scientific Research Projects Coordination Unit, with financial support provided under Grant number FDK-2021-10373.  
The data presented in this study are available from the corresponding author upon reasonable request.