# A Novel Nonlinear Magnetic Equivalent Circuit Model for Magnetic Flux Leakage System
## 3 Designing MEC Model for MFL System
### Calculation of the MEC Model Permeances
#### 3.1.1 Calculation of the Horizontal Leakage Flux Permeances  
The permeance of the horizontal leakage flux \(\Phi_{hl}\) 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 [PERSON]'s law to the path of horizontal leakage flux. \[H_{x}w_{s}=(2 Ni)\frac{y}{w_{w}},\ \ \ \ \ 0\leq y\leq w_{w} \tag{1}\]  
where \(H_{x}\) is the magnitude of the horizontal flux intensity in the winding slots, and \(y\) is the vertical position of the \(H_{x}\). To represent horizontal leakage flux using two separate permeances, the inner-slot region needs to be divided into upper and lower parts, as shown in Figure 6. The upper and lower parts will be represented by \(P_{hl1}\) and \(P_{hl2}\) permeances, respectively. The magnetic strength of these parts can be defined as follows:  
\[H_{1x}w_{s}=(2 Ni)\frac{y}{w_{w}},\ \ \ \ \ 0\leq y\leq d_{h} \tag{2}\]  
\[H_{2x}w_{s}=(2 Ni)\frac{y}{w_{w}},\ \ \ \ \ \ d_{h}\leq y\leq w_{w} \tag{3}\]  
where \(d_{h}\) is the depth of the upper part of the horizontal leakage flux \(\Phi_{hl1}\), ensuring that both regions' energy is equal. The magnetic field energy can be calculated as follows:  
\[E=\frac{1}{2}\mu\int_{V}H^{2}dv \tag{4}\]  
where \(\mu\) is the permeability coefficient. Taking \(\mu=\mu_{0}\) and \(dv=w_{s}l_{c}d_{y}\) in Equation (5), the upper and lower part energies are defined by the following equations:  
\[E_{1}=\frac{1}{2}\mu_{0}\int\limits_{y=0}^{d_{h}}\left(\frac{(2 Ni)y}{w_{s}w_{ w}}\right)^{2}w_{s}l_{c}d_{y} \tag{5}\]  
\[E_{2}=\frac{1}{2}\mu_{0}\int\limits_{y=d_{h}}^{w_{w}}\left(\frac{(2 Ni)y}{w_{s} w_{w}}\right)^{2}w_{s}l_{c}d_{y} \tag{6}\]  
Determining the \(d_{h}\) parameter is the first step in calculating \(P_{hl1}\) and \(P_{hl2}\) permeances. By equating Equations (5) and (6), the \(d_{h}\) relation is obtained as follows:  
\[d_{h}=\frac{w_{w}}{\sqrt[3]{2}} \tag{7}\]By substituting the calculated \(d_{h}\) 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:  
\[E=\frac{1}{2}P_{hl}.(2 Ni)^{2} \tag{8}\]  
The obtained \(E_{1}\) and \(E_{2}\) equations are separately equated to Equation (8) and the \(P_{hl1}\) and \(P_{hl2}\) permeances are obtained as follows:  
\[P_{hl1}=P_{hl2}=\frac{\mu_{0}l_{c}w_{w}}{6w_{s}} \tag{9}\]  
The permeance equations for \(P_{hl1}\) and \(P_{hl2}\) have been revealed in the operations performed so far. Additionally, in the MEC solution, the \(d_{h1}\) and \(d_{h2}\) distances that define the vertical positions of these permeances need to be determined. The \(d_{h1}\) and \(d_{h2}\) 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 \(d_{h1}\) and \(d_{h2}\) values, we need to again divide each upper and lower part into two sections with equal energy. This solution, which enables the calculation of \(d_{h1}\) for the upper part, is defined by the following equations:  
Figure 6.0  
Figure 6.1  
where \(E_{1}^{\,\,\,\,U}\) and \(E_{1}^{\,\,\,L}\) show the energies of the upper and lower parts of the upper region, respectively. The parameter \(d_{h1}\) can be shown as follows:  
Figure 6.2  
Similarly, the \(d_{h2}\) parameter is calculated by applying the same method for the lower region. This process is carried out using the following Equations. Figure 6.3  
Figure 6.4 The vertical position parameters and the energy parts. Figure 6.5  
where \(E_{2}\textsl{i}\textsl{d}\) and \(E_{2}\textsl{i}\textsl{d}\) show the energies of the upper and lower parts of the lower region, respectively. The parameter \(d_{h2}\) can be shown as follows:  
Figure 6.6