## 4 Electromagnetic Torque Modeling and Torque Density Optimization  
### Electromagnetic Torque Model  
The electromagnetic torque of axial flux permanent magnet motor can be calculated by the Maxwell tensor method, and the electromagnetic torque of the motor is as follows:  
\[T_{\text{ele}}=\frac{1}{\mu_{0}}\int_{R_{1}}^{R_{0}}r^{2}\int_{0}^{2\pi}\left( B_{\text{nl},x}^{\rightarrow}+B_{\text{arm},x}^{\rightarrow}\right)\left(B_{ \text{nl},y}^{\rightarrow}+B_{\text{arm},y}^{\rightarrow}\right)\text{d} \theta\text{d}r \tag{17}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Loss Name** & **Parameter Value** & **Unit** \\ \hline Stator core loss & 0.63 & W \\ \hline Stator winding loss & 33.31 & W \\ \hline Eddy current loss of rotor core & 0.11 & W \\ \hline Eddy current loss of permanent & 0.42 & W \\ magnet & & \\ \hline Mechanical loss & 7.5 & W \\ \hline \hline \end{tabular}
\end{table}
Table 2: Motor electromagnetic component loss table.  
Figure 25: Eddy current loss in permanent magnet.  
The above formula is further expanded to the following:  
\[T_{\text{ele}}=\frac{1}{\mu_{0}}{\int_{R_{\text{i}}}^{R_{\text{o}}}r^{2}}{\int_{0} ^{2\pi}{B_{\text{tmp}}}\text{d}\theta\text{d}r} \tag{18}\]  
In Formula (18), \(B_{\text{tmp}}\) is \(\left(\overset{\rightarrow}{B_{\text{nl}_{\text{s}}}}\overset{\rightarrow}{B_ {\text{nl}_{\text{y}}}}+\overset{\rightarrow}{B_{\text{arm}_{\text{s}}}} \overset{\rightarrow}{B_{\text{arm}_{\text{y}}}}+\overset{\rightarrow}{B_{ \text{nl}_{\text{,x}}}}\overset{\rightarrow}{B_{\text{arm}_{\text{y}}}}+ \overset{\rightarrow}{B_{\text{arm}_{\text{,x}}}}\overset{\rightarrow}{B_{ \text{nl}_{\text{,y}}}}\right)\), \(\overset{\rightarrow}{B_{\text{nl}_{\text{,x}}}}\) is the circumferential component of the no-load magnetic field density, \(\overset{\rightarrow}{B_{\text{nl}_{\text{,x}}}}\) is the circumferential component of the armature reaction magnetic field density, and \(\overset{\rightarrow}{B_{\text{nl}_{\text{,y}}}}\) is the axial component of the armature reaction magnetic field density. Among them are the following:  
\[\overset{\rightarrow}{B_{\text{nl}_{\text{,x}}}}\overset{\rightarrow}{B_ {\text{nl}_{\text{,y}}}}=0 \tag{19}\]  
\[\overset{\rightarrow}{B_{\text{arm}_{\text{,x}}}}\overset{\rightarrow}{B_ {\text{arm}_{\text{,y}}}}=0 \tag{20}\]  
The electromagnetic torque analytical model of the axial flux permanent magnet motor is obtained by introducing Equations (19) and (20) into Equation (18):  
\[T_{\text{ele}}=\frac{1}{\mu_{0}}{\int_{R_{\text{i}}}^{R_{\text{o}}}r^{2}}{\int_ {0}^{2\pi}{\left(\overset{\rightarrow}{B_{\text{nl}_{\text{,x}}}}B_{\text{arm }_{\text{,y}}}+B_{\text{arm}_{\text{,x}}}\overset{\rightarrow}{B_{\text{nl}_ {\text{,y}}}}\right)}\text{d}\theta\text{d}r} \tag{21}\]  
It can be concluded that the electromagnetic torque of axial flux permanent magnet motor is related to the air gap length \((L-t_{\text{m}}/2)\), permanent magnet remanence \(\overset{\rightarrow}{B_{\text{r}}}\), permanent magnet relative permeability \(\mu_{\text{r}}\), permanent magnet magnetic field space harmonic number \(n\), permanent magnet pole arc coefficient \(\alpha_{\text{p}}\), permanent magnet thickness \(t_{\text{m}}\), motor pole distance \(\tau_{\text{p}}\), motor stator slot width \(b_{0}\), motor slot distance \(\tau_{\text{s}}\), motor stator core average radius \(R_{\text{avg}}\), distance between permanent magnet center and stator surface \(L\), phase number \(m\), motor stator slot number \(Z\), each harmonic current amplitude \(I_{\text{s}}\), motor pole pair \(p\), motor series turns per phase \(N_{\text{ph}}\), time harmonic number \(s\), and space harmonic number \(v\).