# A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System
## 1 Introduction  
Magnetic flux leakage (MFL), one of the nondestructive testing methods, is an effective and reliable technique that detects anomalies such as cracks, metal loss, and corrosion in the structure of magnetic materials [1, 2]. Critical materials of industrial importance, such as oil and gas pipes [3], tank floors [4], steel ropes [5], and railways [6], are inspected by the MFL method. Safe material usage is ensured by conducting this inspection process periodically. One of the essential steps in the MFL method is the defect characterization process [7]. Defect characterization refers to estimating the physical properties of a defect, such as length, width, and depth, by evaluating the MFL signal. At this stage, depth information is a vital parameter that needs to be identified [8]. The main challenge in determining the depth is that the relationship between the physical depth and the signal amplitude is not linear [9]. This relationship involves varying complexity depending on factors such as the magnetic and physical properties of the MFL system components and the flux density level of the material under test. The inappropriate combination of these factors will make MFL signals difficult to interpret and make it challenging to estimate defect depth information. Therefore, a system design that considers the mentioned factors becomes crucial for an effective defect characterization. The finite element analysis (FEA) method stands out as a tool that provides accurate and precise solutions in the design of electromagnetic systems [10, 11, 12]. FEA can analyze complex designs incorporating magnetic materials by considering the nonlinear behaviorof these materials [13]. On the other hand, in studies such as design optimization and rapid prototyping, excessive time-consuming processes arise due to the high computational volume. This problem is escalated by large numbers of mesh assignments and leads to costly hardware requirements. Therefore, it seems impossible to overcome this problem using the FEA method. Magnetic equivalent circuit (MEC) analysis based on Kirchhoff's law offers a faster alternative solution. MEC is a permeance network in which the permeances corresponding to the primary flux paths of an electromagnetic system are modeled as a set of interconnected lumped elements [14]. Although the MEC method has less accuracy than FEA, it has a lower computational cost due to reduced model complexity [15; 16; 17]. These advantages make it an effective means of optimization for electromagnetic systems [13; 16]. Additionally, the desired level of accuracy can be achieved by modeling the sufficient number of permeances corresponding to the flux paths. There are few studies in the literature about the magnetic circuit of the MFL system. One of the studies [18] is about optimizing the system parameters using the magnetic circuit method. Another study [19] presents multi-objective optimization of the system's magnetic circuit using the genetic algorithm. In the study of [PERSON] et al. [20], the effect of the eccentricity value of the magnets on the inspection is examined by using a simplified equivalent magnetic circuit of the system. However, in these studies, critical factors such as leakage flux and the material's nonlinear behavior are not considered when designing the magnetic circuit. However, an efficient magnetic circuit design can only be achieved by taking into account the leakage flux and the nonlinear behavior of the magnetic material. In this study, an efficient and reliable MEC model that can be used in the design of any MFL system is presented. In this context, our primary goal is to design an MEC model that not only models a specific MFL system but also adapts to systems with different dimensions and parameters. For this purpose, in addition to the main flux of the magnetic system, factors such as leakage flux and nonlinear behavior of the magnetic material are taken into consideration when designing the MEC model.