# A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System
## 2 Principles of MFL Testing  
The MFL method is based on the principle of subjecting the material to be inspected to a closed magnetic flux loop. For this purpose, a U-shaped yoke made of high-permeability material is utilized. In the MFL system, magnetic flux sources are positioned on the poles of the yoke close to the material. The mechanical system created in this way is moved over the material to conduct the MFL inspection. Permanent magnets or coils can be used as magnetic flux sources. Using coils enables the adjustment of magnetic flux, providing flexible solutions in the system design [21]. A magnetic neutral point occurs at the midpoint between the system's poles, and magnetic sensors are placed at this point. The flux passing through the tested material exhibits a leaking effect from the defect region into the air. Finally, the sensors detect this effect, and MFL signals are obtained. The general representation of the MFL test and tangential (\(x\) axis) component of sample defect signals are shown in Figure 1.  
In the MFL method, the flux density level formed in the test material is of critical importance. A flux density under saturation level produces weak and unstable signals, and minor defects cannot be detected. If the material is saturated, the leakage flux into the air is highly triggered. However, saturation conditions may lead to deformations in the defect signal. Therefore, it is necessary to form a flux density level defined as near saturation in the tested material [9]. This preferred approach aims to generate effective and efficient MFL signals. Figure 2 depicts an example of a _B-H_ curve and its flux density zones. While there are no exact limits, the zones, namely under saturation, near saturation, and saturation, can be approximately defined based on the behavior of the material's _B-H_ curve.  
Another interaction is that, with forming near saturation in the tested material, high flux densities and even saturation states can also occur in the pole regions of the MFL system. The first noticeable cost of these situations is the exponentially increasing energy consumption. This is because a much higher flux transition occurs from the system to the air in the high flux density or saturation states. However, what is more important is that the horizontal leakage flux and magnetic noise occurring at the inner-slot region increase significantly. This effect influences the sensor measurement area, resulting in distortions in the defect signal, increased signal noise, and a rise in the DC component. The defect signal in Figure 1c illustrates an example of these distortions. Another noteworthy point is that shortening the distance between the poles positively contributes to increasing the flux density in the material. On the other hand, it also causes a significant increase in leakage flux occurring at the inner-slot region. This effect leads to erroneous and/or incomplete interpretations in defect characterization, negatively impacting test efficiency. Even small amplitude defect signals may be lost below the noise level.  
Variation in the system dimensions has complex effects on the flux densities of the tested material and pole regions. Further, in real MFL applications, the flux densities of these regions cannot be directly measured. Therefore, an optimization process that will eliminate the aforementioned adverse effects becomes critical to achieving an efficient  
Figure 1: (**a**) Schematic illustration of the MFL testing. (**b**) Normal defect signal. (**c**) Distorted defect signal.  
Figure 2: A _B-H_ curve and its zones.  
system design for actual application purposes. The requirement of such an MEC model that ensures sufficient accuracy becomes significant for such an optimization process, which can be carried out through multiple parameter changes.