# Optimal Shape Design of Direct-Drive Permanent Magnet Generator for 1 kW-Class Wind Turbines
## 3 Electromagnetic Analysis
### Magnetic Flux Linkage and Induced Electromotive Force  
The initial design model of the generator was evaluated for its electromagnetic characteristics using commercial numerical analysis software: ANSYS RM-Expert and Maxwell [21]. The governing equations for the magnetic fields used in the numerical analysis are presented by Equation (1) to (3). Equation (1) represents [PERSON]'s law for the electric current and magnetic field, while Equation (2) shows the electric potential based on [PERSON]'s law and the relationship between electric and magnetic fields. [PERSON]'s law for a magnetic field is expressed by Equation (3) [22]:  
\[\
abla\times\mathrm{H}=\mathrm{J} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Parameters** & **Magnet Thickness** & **Air Gap** & \(D_{i}\) & \(l\) & \(b_{1}\) & \(b_{2}\) & \(h_{1}\) & \(h_{2}\) & \(N_{ph}\) \\ \hline Values & 10 & 1 & 65 & 35 & 2 & 8 & 2 & 16 & 200 \\ Unit & & & & mm & & & & & Turn \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specification of shape parameters and the number of winding coil turns at initial design stage of the generator.  
Figure 5: A principle how to transform the 3-D geometry of an AFPM generator to a 2-D geometry.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parts** & **Materials** \\ \hline Permanent Magnet & NdFe30 \\ Stator & D21-50 \\ Bearing & Stainless steel \\ Coil & Copper \\ Rotor & D21-50 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Materials used in the generator.  
\[\
abla\times\mathrm{E_{e}}=-\frac{\partial}{\partial\mathrm{t}}\mathrm{B} \tag{2}\]  
\[\
abla\cdot\mathrm{B}=0 \tag{3}\]  
where \(\mathrm{B}=(\
abla\cdot\mathrm{A})\), \(\mathrm{H}\) is magnetic field strength (\(\mathrm{A}/\mathrm{m}\)), \(\mathrm{J}\) is the current density (\(\mathrm{A}/\mathrm{m}^{2}\)), \(\mathrm{t}\) is time (s), \(\mathrm{E_{e}}\) is [PERSON]'s electric field, \(\mathrm{A}\) is the magnetic vector potential and \(\mathrm{B}\) is the magnetic flux density field.  
Figure 6 presents the mesh generation scheme used for the two-dimensional electromagnetic analysis of the AFPM generator. As the generator possesses a symmetrical feature, a half model was utilized to expedite the analysis. To balance computational efficiency and simulation accuracy, the mesh generation was optimized through a trade-off between the mesh density and the computation time. Fine meshes tend to increase simulation time, while coarser meshes yield poorly resolved outputs. Therefore, length-based meshes were adopted with a mesh size of 0.5 mm for the air-gap and coarse meshes with a larger size of 3 mm for the remaining regions, to ensure both computational efficiency and simulation accuracy [23].  
To evaluate the generator's performance, an electromagnetic analysis under no-load conditions was conducted to investigate the flux linkage and electromotive force. The obtained results are shown in Figure 7, which provides the waveforms of flux linkage and electromotive force \(E_{F}\) for the three phases, \(\mathrm{A}\), \(\mathrm{B}\) and \(\mathrm{C}\), with a peak value of 0.32 Wb and a maximum value of 15.7 V. Figure 7 presents the three-dimensional analysis results for the proposed generator model, which reveals a good agreement of the flux linkage and electromotive force obtained from the two- and three-dimensional models.