# A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Principles of MFL Testing

#### Flux Density Distribution of MFL System

_{main}represents the main flux, forming the necessary flux loop for the testing process. Figure 3b shows that the main flux Φ

_{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.

_{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.

_{hl}

_{1}) and the lower part horizontal leakage flux (Φ

_{hl}

_{2}). In the outer-slot region, winding leakage flux (Φ

_{eal}) is dominant. This flux is shown with the black dashed line frame in Figure 3a. 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.

_{main}, winding leakage flux Φ

_{eal}, and horizontal leakage flux Φ

_{hl}, are defined, as shown in Figure 4. The horizontal leakage flux Φ

_{hl}representation in two parts will be 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.

## 3. Designing MEC Model for MFL System

_{yoke}), the pole region (P

_{ub}, P

_{lb,}and P

_{mb}), and the specimen (P

_{sp}) as its magnetic elements. Air permeances account for both leakage flux types and flux passing through the air gap. Specifically, they include the winding leakage flux permeance (P

_{eal}), upper and lower part permeances of the horizontal leakage flux (P

_{hl}

_{1}and P

_{hl}

_{2}), and the air gap permeance (P

_{gap}). Figure 5b shows the dimensional parameters of the MFL system. Also, descriptions of these parameters are given in Table 1.

#### 3.1. Calculation of the MEC Model Permeances

#### 3.1.1. Calculation of the Horizontal Leakage Flux Permeances

_{hl}is related to the total inner-slot magnetic energy and can be calculated by the Sudhoff method [22]. The following equation is obtained by neglecting the MMF drops in the magnetic regions and applying Ampere’s law to the path of horizontal leakage flux.

_{x}is the magnitude of the horizontal flux intensity in the winding slots, and y is the vertical position of the H

_{x}.

_{hl}

_{1}and P

_{hl}

_{2}permeances, respectively. The magnetic strength of these parts can be defined as follows:

_{h}is the depth of the upper part of the horizontal leakage flux Φ

_{hl}

_{1}, ensuring that both regions’ energy is equal. The magnetic field energy can be calculated as follows:

_{0}and dv = w

_{s}l

_{c}d

_{y}in Equation (5), the upper and lower part energies are defined by the following equations:

_{h}parameter is the first step in calculating P

_{hl}

_{1}and P

_{hl}

_{2}permeances. By equating Equations (5) and (6), the d

_{h}relation is obtained as follows:

_{h}parameter in Equations (5) and (6), the E

_{1}and E

_{2}energies can now be solved. Additionally, magnetic energy can be calculated in terms of permeance as follows:

_{1}and E

_{2}equations are separately equated to Equation (8) and the P

_{hl}

_{1}and P

_{hl}

_{2}permeances are obtained as follows:

_{hl}

_{1}and P

_{hl}

_{2}have been revealed in the operations performed so far. Additionally, in the MEC solution, the d

_{h}

_{1}and d

_{h}

_{2}distances that define the vertical positions of these permeances need to be determined. The d

_{h}

_{1}and d

_{h}

_{2}values are required to determine the lengths of the magnetic branches of the pole region. Using the energy approach described above at this stage would be appropriate. To determine the d

_{h}

_{1}and d

_{h}

_{2}values, we need to again divide each upper and lower part into two sections with equal energy. This solution, which enables the calculation of d

_{h}

_{1}for the upper part, is defined by the following equations:

_{1}

^{U}and E

_{1}

^{L}show the energies of the upper and lower parts of the upper region, respectively. The parameter d

_{h}

_{1}can be shown as follows:

_{h}

_{2}parameter is calculated by applying the same method for the lower region. This process is carried out using the following Equations.

_{2}

^{U}and E

_{2}

^{L}show the energies of the upper and lower parts of the lower region, respectively. The parameter d

_{h}

_{2}can be shown as follows:

#### 3.1.2. Calculation of the Magnetic Branches Permeances

_{br}is the length of a branch, A

_{br}is the cross-sectional area of a branch, and µ

_{B}(.) refers to the variable permeability of a branch. The calculation method for the variable permeabilities will be explained in Section 4 in detail. The permeance equations of the magnetic branches can be presented as follows:

#### 3.1.3. Calculation of the Winding Leakage Flux Permeance

_{eal}. This flux consists of two parts: interior and exterior. The reluctance of the external part of the winding leakage flux can be calculated by Equation (22).

_{c}+ 2w

_{e}.

_{eali}) is shown in Equation (23).

#### 3.1.4. Calculation of the Permeances of the Air Gap and Fringing Effect

## 4. Nonlinear Solution of the MEC Model

_{new}and new flux density B

_{new}between node i and node j can be calculated by the following Equations.

_{new}flows.

_{new}value is calculated using the B-H table data with the following equation:

^{(k)}is the permeability of the kth iterative process, and μ

^{(k−1)}is the permeability of the (k − 1)th iterative process. If the error condition is met, the result is considered convergent and saved for use.

## 5. Testing the MEC Model with FEA

#### 5.1. FEA Studies

_{e}and w

_{s}parameters was taken as a basis for obtaining different MFL system models. Due to their nonlinear effect on the main and leakage flux, w

_{e}and w

_{s}have a higher impact on the system operation than other parameters. The system height w

_{w}parameter can be accepted within the effect caused by changes in w

_{s}. Therefore, the w

_{w}value was taken as fixed in the analyses. The l

_{g}distance affects the horizontal leakage flux almost linearly. Therefore, this parameter was also fixed in the analyses. The w

_{e}and w

_{s}values selected for analyses are given in Table 2. The fixed parameters are defined as w

_{m}= 5 mm, w

_{w}= 40 mm, l

_{c}= 60 mm, d

_{w}= 8 mm, and l

_{g}= 2 mm.

_{e}and w

_{s}parameters, and corresponding FEA studies were conducted. In these studies, the necessary MMF values were adjusted for each model to form the flux density within the material at a level of 1.8T. The adjustment of MMF values, defined by the equation F = Ni, was performed by keeping the number of turns N fixed and assigning different current values i. Figure 10 illustrates the variation in MMF values obtained from FEA studies for different combinations of w

_{e}and w

_{s}. As seen from Figure 10, the MMF values required to generate 1.8T within the tested material vary directly with w

_{s}and inversely with w

_{e}.

_{e}and w

_{s}parameters on the horizontal leakage flux is illustrated in Figure 11. As observed, the variation in w

_{e}has a more dominant effect than w

_{s}. This is because the w

_{e}dimension directly influences the main flux of the system. This effect becomes more pronounced when the w

_{e}value approaches the material’s thickness w

_{m}. As a result, the flux densities in the pole regions increase, leading to an exponential increase in the horizontal leakage flux within the slot. A similar effect emerges with the variation in the w

_{s}parameter. Since the w

_{s}dimension determines the distance between the poles, it changes the air reluctance of the horizontal leakage flux. While a decrease in w

_{s}contributes positively to the flux density in the test material, it also increases the horizontal leakage flux, depending on the pole flux densities. Forming near saturation flux density in the tested sample can be seen as the main reason for this dilemma. Ultimately, the variation in w

_{e}and w

_{s}dimensions alters the character of the leakage flux within the slot, which is more critical for the system. An increase in horizontal leakage flux particularly affects the sensor measurement region directly, leading to adverse effects such as an increase in the DC level in the defect signal, shape distortions, and signal noise. Figure 1c illustrates an example of these distortions.

#### 5.2. MEC Model Verification

_{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.

_{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

_{e}has a more dominant effect, particularly as it approaches the value of w

_{m}, compared to w

_{s}.

_{e}and w

_{s}on the relative error values. It provides the average relative error (MRE) resulting from variations in one parameter while keeping the other parameter fixed.

_{s}and decreasing w

_{e}. Additionally, the variation in w

_{e}has a greater impact on the error rates than w

_{s}.

_{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. On the other hand, in the MEC model, the mentioned flux distributions are represented only by three permeances (P

_{ub}, P

_{mb}, and P

_{lb}) along the vertical direction for the pole region and only by two permeances (P

_{hl}

_{1}, P

_{hl}

_{2}) for the inner-slot leakage flux.

#### 5.3. Elapsed Times

## 6. Conclusions

_{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.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic illustration of the MFL testing. (

**b**) Normal defect signal. (

**c**) Distorted defect signal.

**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.

**Figure 11.**The behavior of the horizontal leakage flux according to varying w

_{e}and w

_{s}parameters.

**Figure 12.**The flux density results of the pole region based on changes in the w

_{e}and w

_{s}parameters.

Parameter Symbol | Description |
---|---|

w_{s} | Length of the slot |

l_{g} | Length of the air gap |

w_{e} | Width of the yoke |

w_{m} | Thickness of the specimen |

d_{w} | Width of the windings |

w_{w} | Height of the windings and yoke |

l_{c} | Depth of the yoke and specimen |

Parameter | Values (mm) | ||||
---|---|---|---|---|---|

w_{e} | 5 | 10 | 15 | 20 | 25 |

w_{s} | 30 | 40 | 50 | 60 | 70 |

w_{e} (mm) | MAE_w_{e} (T) | w_{s} (mm) | MAE_w_{s} (T) |
---|---|---|---|

(For w_{s} 30 to 70) | (For w_{e} 5 to 25) | ||

5 | 0.17 | 30 | 0.05 |

10 | 0.11 | 40 | 0.06 |

15 | 0.07 | 50 | 0.07 |

20 | 0.04 | 60 | 0.09 |

25 | 0.03 | 70 | 0.1 |

w_{e} (mm) | MRE_w_{e} (%) | w_{s} (mm) | MRE_w_{s} (%) |
---|---|---|---|

(For w_{s} 30 to 70) | (For w_{e} 5 to 25) | ||

5 | 8.7 | 30 | 3.5 |

10 | 6.0 | 40 | 4.5 |

15 | 4.6 | 50 | 5.4 |

20 | 3.7 | 60 | 6.2 |

25 | 3.1 | 70 | 6.7 |

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**MDPI and ACS Style**

Kara, O.; Çelik, H.H.
A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System. *Appl. Sci.* **2024**, *14*, 4071.
https://doi.org/10.3390/app14104071

**AMA Style**

Kara O, Çelik HH.
A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System. *Applied Sciences*. 2024; 14(10):4071.
https://doi.org/10.3390/app14104071

**Chicago/Turabian Style**

Kara, Okan, and Hasan Hüseyin Çelik.
2024. "A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System" *Applied Sciences* 14, no. 10: 4071.
https://doi.org/10.3390/app14104071