## 4 Finite-Element Analysis (FEA)  
3-D FEM is employed to analyze the magnetic circuit. The investigation provides an overall depiction of the different parts of the suggested generator saturation levels, and excerpts their characteristics. The fundamental FEA equations are presented as Equation (29)-(31) [30].  
\[\
abla\times\overset{\rightarrow}{B}=\mu\overset{\rightarrow}{J_{s}} \tag{29}\]  
\[\
abla\times\overset{\rightarrow}{J_{s}}=-\sigma\frac{d\overset{\rightarrow} {B}}{dt} \tag{30}\]  
\[\overset{\rightarrow}{B}=\
abla\times\overset{\rightarrow}{\Psi} \tag{31}\]  
where \(\overset{\rightarrow}{B},\overset{\rightarrow}{J_{s}},\overset{\rightarrow} {\Psi},\sigma\), and \(\mu\), are the magnetic flux density, current density, magnetic vector potential electrical conductivity, and magnetic permeability, respectively. If \(\overset{\rightarrow}{H}\) is taken as the magnetic field intensity, previous equations result in Equation (32):  
\[\
abla\times\left(\frac{1}{\sigma}\
abla\times\overset{\rightarrow}{H}\right) +\frac{\partial\overset{\rightarrow}{B}}{\partial t}=0 \tag{32}\]  
where the vector fields are represented by first-order edge elements, and the scalar fields by second-order nodal unknowns. Field equations are combined with circuit equations for conductors. The numerical solutions of Equations (29)-(31) involve a finite element discretization methodology. Due to the complexity of magnetic field distribution in the axial and radial spaces of TFPMDEIG, the 3-D FEM is employed to analyze its electromagnetic characteristics. Three-dimensional FEM perfectly calculates the different components of the flux density. The design was simulated using the JMAG Designer 10.5 software package (JSOL Corporation, Tokyo, Japan).  
The first step in modeling by means of 3-D FEM is to draw the geometry of the problem. This can be done by tools in the software, or by a drawing technical software, such as Solid Works. Figure 11 shows the drawn geometry of a pair of poles in this generator.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Row** & **Definition** & **Parameter** & **Value** \\ \hline
1 & Inner radius of the rotor disk & \(R_{in}[\mathrm{mm}]\) & 40.1 \\
2 & Outer radius of the rotor disk & \(R_{out}[\mathrm{mm}]\) & 120 \\
3 & Thickness of permanent magnet & \(l_{pm}[\mathrm{mm}]\) & 7 \\
4 & Size of the core of E-shaped side arm & \(l_{p}[\mathrm{mm}]\) & 11 \\
5 & Thickness of E-shaped core & \(b_{p}[\mathrm{mm}]\) & 25 \\
6 & The length between the middle and the side arm of E-shaped core & \(b_{u}[\mathrm{mm}]\) & 11 \\
7 & Turns of winding & \(N_{p}\) & 1450 \\
8 & Main harmonic of induced voltage & \(E_{f}[\mathrm{V}]\) & 115.28 \\
9 & Resistance & \(R_{a}[\Omega]\) & 46.54 \\
10 & Synchronous Reactance & \(X_{s}[\Omega]\) & 50.03 \\
11 & Power angle & \(\delta^{\circ}\) & 19.84 \\ \hline \end{tabular}
\end{table}
Table 4: Machine’s dimensions and computational parameters.  
The proposed generator in this paper has 8 poles, and the magnetic and geometric pattern of a pair of poles is repeated according to the number of paired poles. Therefore, modeling only a pair of poles is enough for balanced performance analysis and achieving the desired goals. After geometric drawing, the material for each section must be determined. This is done considering Table 5.  
It must be noted that it is possible to use SMC, instead of laminated steel, for construction of the E and I cores. In this research, however, SMC was not used for the following reasons:  
1. The low frequency and the rotor's low rotation speed,
2. The severe difficulties with acquiring SMC cores in a small quantity.