## 2 Comparative Electromagnetic Analysis  
### Back-EMF and Torque  
In the case of PM machines, a decrease in total harmonic distortion (THD) of back-EMF leads to high torque density because the input current decreases to generate an equal torque. In addition, it is not proportional to the size of the THD because it affects the torque ripple. Therefore, a PM machine  
Figure 1: The analysis model of permanent magnet machines with two different pole/slot combinations: (**a**) electromagnetic analysis model and (**b**) prototype model.  
should be designed to accommodate sinusoidal waveforms. The phase back-EMF and the current are described in Equation (1).  
\[\begin{split} e_{phase}=\omega_{r}\sum\limits_{n=1,\omega dd}^{ \infty}K_{n}\sin(np\omega_{r}t)\\ i_{phase}=I_{1}\sin(p\omega_{r}t)\end{split} \tag{1}\]  
where \(\omega_{r}\) is the rotational speed, \(t\) is the time, \(K_{n}\) is the back-EMF coefficient, \(I_{1}\) is the peak of current, and \(np\) indicates the \(n\)th-order harmonic and the number of pole pairs, respectively. The electromagnetic torque is expressed as the product of the back-EMF and the current (Equation (2)).  
\[\begin{split}& T=\frac{3}{2}K_{1}I_{1}\\ &+\frac{1}{2}I_{1}\sum\limits_{n=1,\omega dd}^{\infty}K_{n}[1+2 \cos[(n-1)\cdot 2n/3]]\times\cos(n-1)p\omega_{r}t\\ &+\frac{1}{2}I_{1}\sum\limits_{n=1,\omega dd}^{\infty}K_{n}[1+2 \cos[(n+1)\cdot 2n/3]]\times\cos(n+1)p\omega_{r}t\end{split} \tag{2}\]  
Here, the first term represents the average electromagnetic torque, and the second and third terms represent the torque ripple produced by the THD of the back-EMF and the current, respectively. Thus, the amplitude of the harmonics of the back-EMF and the current affects the magnitude of the torque ripple. Figure 2a,b compare the back-EMF analysis results of the two models and indicate that the 9-slot model has better THD than the 12-slot model. The THD of the 9- and 12-slot models is 20.10% and 16.7726, respectively, and it can be seen that the harmonics of the 9-slot model is larger than that of the 12-slot model.  
Figure 2: Comparison of the analysis results of permanent magnet machines with two different pole/slot combinations: (**a**) back-EMF, (**b**) THD of back-EMF, (**c**) torque ripple, and (**d**) cogging torque.  
Previous studies have presented the goodness factor equation of a pole/slot combination, which is derived using the pole number, the slot number, and the lowest common multiple (LCM) of the pole and slot. This means that as the goodness factor increases, the cogging torque increases. The goodness factor equation is expressed in Equation (3). Here, \(N_{LCM}\) is the LCM of the pole and slot numbers. \(Q_{s}\) and \(p\) indicate the number of the slot and the number of the pole pair, respectively.  
According to Equation (3), the magnitude of the cogging torque of the 9-slot model is higher than that of the 12-slot model.  
\[C_{t}=\frac{2 pQ_{s}}{N_{LCM}} \tag{3}\]  
Figure 2c compares the results of the torque ripple analysis, indicating that the 9-slot model has better torque ripple characteristics than that of the 12-slot model. Figure 2d compares the results of the cogging torque analysis, indicating that the 12-slot model has better cogging torque characteristics than that of the 9-slot model. The torque ripple values of the 9- and 12-slot models are 20.1% and 16%, respectively; the cogging torques of the 9- and 12-slot models are 1.5379 and 311.5311 mNm, respectively, and it can be confirmed that the torque and EMF characteristics of the 9-slot model are better than that of the 12-slot model.