## 4 Analysis of Proposed FP-AFSRM
### Finite Element Analysis  
Finite element analysis (FEA) is one of the most widely used numerical methods for examining the electromagnetic properties of electrical machines. This numerical method utilizes a technique called the finite element method (FEM) to perform engineering analyses through computer simulations [37; 38]. For SRMs, FEM analyses are essential due to their bidirectional salient structure and the intense saturation effects that occur at partially aligned stator-rotor poles [37]. FEA allows for the calculation of profiles such as torque, magnetizing, self-inductance, and mutual inductance of the SRM by using [PERSON]'s equations, which can be expressed in both the differential form (as given in Equations (11)-(14)) and integral forms [39; 40].  
Figure 5: \(\alpha\) angle between the stator poles.  
\[\
abla xE=-\frac{\partial B}{\partial t} \tag{11}\] \[\
abla\bullet B=0\] (12) \[\
abla xH=J+\frac{\partial D}{\partial t}\] (13) \[\
abla\bullet D=\rho \tag{14}\]  
In Equations (11)-(14), \(E\) represents the electric field intensity (V/m), \(B\) represents the magnetic flux (Tesla or Wb/m\({}^{2}\)), \(H\) represents the magnetic field intensity (A/m), \(D\) represents the electric flux density (C/m\({}^{2}\)), \(t\) represents the time (s), \(\rho\) represents the electric charge density (C/m\({}^{3}\)), and \(J\) represents the current density (A/m\({}^{2}\)). The magnetic field resulting from the transformation of the electromagnetic field for SRMs due to winding and eddy currents is described by Equation (15) [41, 42].  
\[\frac{\partial}{\partial x}\left(\frac{1}{\mu}\frac{\partial A_{z}}{\partial x }\right)+\frac{\partial}{\partial y}\left(\frac{1}{\mu}\frac{\partial A_{z}}{ \partial y}\right)-\sigma\frac{\partial A_{z}}{\partial t}=J_{z} \tag{15}\]  
where \(\mu\) the permeability, \(\sigma\) represents the conductivity, and \(j_{z}\) represents the source current given to the windings from outside.  
3D-FEM analyses are time-consuming and involve very heavy operations. Nevertheless, since the analysis of axial flux electric machines involves a three-dimensional electromagnetic problem, the most accurate solutions are obtained using 3D-FEM. However, in a literature study, 2D-FEM analyses were conducted by taking 2D slices of the axial machine geometry. This approach can be considered as reducing the axial flux electric machine to a linear electric machine [9, 43].  
The analyses of the proposed FP-AFSRM model were conducted in 3D. Figure 6 shows the variation in the torque and axial force on the rotor as a function of the rotor angle, based on a magnetostatic analysis performed at 10 amperes. The \(15^{\circ}\) skew angle mentioned in the graph title refers to the \(15^{\circ}\) skew angle of the auxiliary stator, as indicated in Figure 5.  
Figure 6: _Cont._  
When examining the torque graph in Figure 6a obtained at 10 amperes, each layer (right-left) engages at the times when it contributes maximally to the torque production. The left stator remains active in the range of \((-45^{0})\)-\((-12^{0})\), providing the starting torque, and then disengages as the torque curve begins to decline. The right stator then engages in the range of \((-12^{0})\)-\((6.5^{0})\), contributing to the torque production. This results in a smooth torque curve.