## 4 Unbalanced Model  
In conventional studies, a three-phase linear machine is divided into three domains and their positions are rearranged to generate a phase difference in the pulsation force between each domain [21]. Similarly, a cross-section of the optimal 8-slot 8-pole model in Chapter IV and its three separated domains are shown in Figure 13a.  
Region 1 and region 3 are symmetric with region 2. Because the pole pitch is \(\tau\)p = 23 mm, when region 1 is shifted to the left and region 3 is shifted to the right by 45\({}^{\circ}\), the width of slots 3 and 6 are increased by 0.125 \(\tau\)p = 2.875 mm. In this section, while region 2 is fixed, regions 1 and 3 were shifted in the opposite directions by three amounts, 0.125 \(\tau\)p (45\({}^{\circ}\) or 2.875 mm), 0.25 \(\tau\)p (90\({}^{\circ}\) or 5.75 mm), and 0.375 \(\tau\)p (135\({}^{\circ}\) or 8.625 mm).  
Figure 13b presents an example, when regions 1 and 3 were reversely shifted by 5.75 mm. This means that the widths of slots 3 and 6 were extended by 5.75 mm as well. Hence, the axial length of region 2, the slot area, and the number of turns per slot were increased accordingly. In this case, the number of turns in slots 1, 2, 4, 5, 7, and 8 was 440, while it was 468 turns in slots 3 and 6.  
A comparison of the detent force and the electromagnetic force is presented in Figure 14. It should be noted that the detent force is an interaction between the PM and iron core when there is no applied current, the so-called no-load condition. It is intuitively perceivable that the detent force and electromagnetic force are dramatically reduced by increasing shifting lengths of regions 1 and 3. When the shifting length is 8.625 mm (or 135\({}^{\circ}\)), the maximum values are only around 206.3 and 715.2 N, respectively, which are 10.9 and 4.5 times smaller than those in the optimal model. This effective reduction can be best explained by examining the force waveforms produced by each region.  
Considering the best case, when regions 1 and 3 are reversely shifted by 8.625 mm (or 135\({}^{\circ}\)), the detent force on each region is as shown in Figure 15. Because region 2 retained the configuration of the optimal model, the detent force on this region is the highest. However, the phase difference of the detent force on region 2 and the sum of the force on regions 1 and 3 is mostly 180\({}^{\circ}\), so the total force  
Figure 14: Detent and electromagnetic force for different shifting lengths.  
Figure 13: Cross-section of the: (**a**) Optimal 8-slot 8-pole machine; (**b**) proposed machine with teeth arrangement (5.75 mm of shifting length).  
on the whole machine is dramatically reduced. The effect of this approach can be even greater if an optimization process is applied.  
Despite the great reduction in force, the proposed machine has to satisfy other performance criteria as well. According to a report in [21], when a permanent magnet linear synchronous machine is designed with the teeth arrangement approach (or unbalanced model), only the relative position of the moving teeth are modified without any performance sacrifice. However, that machine was operated by linear motion or one direction motion, and its characteristics are different from a short-stroke, vibration generator.  
The evidence from Figure 16 proves not only that the detent force is mostly reduced, but output power is also significantly decreased from 435.4 W to only 136.9 W, which is much lower than the design target. This is because of the relative position between winding slots 1, 2, 3, 6, 7, and 8, and because the excited flux density is shifted to an unaligned position. Only slots 4 and 5 remained in aligned position with the maximum excited flux density, and produced maximum power.  
In general, unbalanced model (or teeth arrangement) configuration is extremely effective to reduce force without any notable increase in volume and mass. However, same as skewing approach, there is a conflict of interest between output power and force.