# A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System
## 2 Principles of MFL Testing
#### Flux Density Distribution of MFL System  
The flux types of the MFL system can be divided into two major groups: main flux and leakage flux. \(\Phi_{main}\) represents the main flux, forming the necessary flux loop for the testing process. Figure 2(b) shows that the main flux \(\Phi_{main}\) completes its cycle by passing through the air gap and the test material over the yoke. This flux forms the desired flux density level within the material being tested.  
During the transition of the main flux from the air gap to the test material, an additional fringing effect arises from the edges. Due to the significantly lower permeability of air compared to the magnetic material, the air gap exhibits high magnetic reluctance. The main flux \(\Phi_{main}\) is highly sensitive to these reluctance changes. In conjunction with this change, flux forms arcs in the air, entering the material and increasing the main flux. Therefore, neglecting the fringing effect in the MEC model calculations will reduce the degree of representation of the MFL system.  
In the MFL system, the leakage flux completes its cycle through air. Although this flux typically forms at low levels, it reaches non-negligible high levels in saturation conditions. This is because saturated regions can no longer pass more flux and flux flows through the air medium. Moreover, the cross-sectional area of the air is much larger than the cross-sectional area of the system, leading to an exponential increase in this effect.  
The leakage flux, which cannot be ignored in the MFL system, can be examined under two categories: inner-slot and outer-slot. Due to its orthogonal feature, the vectorial leakage flux generated at the inner-slot can be separated into horizontal and vertical components [22]. However, due to the magnetic sources' strong inter-pole interaction, the inner-slot region's horizontal flux component is much more dominant. Therefore, the vertical component of the flux in this region can be neglected. The horizontal leakage flux component also tends to follow the main flux direction. This generally causes horizontal leakage flux to have a different directional distribution in the upper and lower inner-slot parts. Thus, it becomes crucial to represent the inner-slot leakage flux with at least two different flux types: the upper part horizontal leakage flux (\(\Phi_{H1}\)) and the lower part horizontal leakage flux (\(\Phi_{hl2}\)). In the outer-slot region, winding leakage flux (\(\Phi_{ml}\)) is dominant. This flux is shown with the black dashed line frame in Figure 2(a). Winding leakage flux symmetrically forms around the magnetic sources in both inner- and outer-slot regions. However, the effect within the inner-slot is generally neglected.  
Considering the behaviors mentioned above, the flux types of the MFL system, namely main flux \(\Phi_{main}\), winding leakage flux \(\Phi_{all}\), and horizontal leakage flux \(\Phi_{hl}\), are defined, as shown in Figure 4. The horizontal leakage flux \(\Phi_{hl}\) representation in two parts will be  
Figure 3: The direction of the flux densities (the arrows) and flux density distribution (color change), (**a**) the flux density distribution of the air, (**b**) the distribution of the main flux.  
explained in detail later. Moreover, studies have been carried out under the assumption that the flux densities occurring in the magnetic regions are uniformly distributed.