*4.2.1 Influence of slot-pole combinations on pulsating torque*

Apart from the torque density, pulsating torque is also very important because a large pulsating torque will increase the vibration and noise of machines. **Figure 13** shows the cogging torque and ripple torque waveforms of 13-, 14-, 16-, 17-, and 19-rotor-slot FRPMMs. The stator slot number of these models is all chosen as 12. For the rated torque, we can see in **Figure 13(b)** that the 14-rotor-slot FRPMM yields the largest among the five models. As for the pulsating torque, we can see that the cogging torque and ripple torque of 16-rotor-slot FRPMM are the largest, and that of 19-rotor-slot FRPMM is the least. This phenomenon is related to the least common multiple of stator slot number and rotor slot number. The larger least common multiple, the lower pulsating torque. The least common multiples of the 13-, 14-, 16-, 17-, and 19-rotor-slot FRPMMs are 156, 84, 48, 204, and 228, respectively. Therefore, the 19-rotor-slot FRPMM exhibit the lowest cogging torque and ripple torque. However, attentions should be paid to use odd rotor number because it will cause other problems such as eccentricity stress. **Figure 14** compares the radial stress of the five FRPMM models. It can be seen that for the even rotor slot number FRPMMs, that is, 14 and 16 rotor slots, the stress harmonics only have even orders, which will not lead to eccentricity. However, for the odd rotor slot number

FRPMMs, that is, 13, 17, and 19 rotor slots, there are many odd stress harmonics. Since the first-order harmonic is dominant for the eccentricity, the 13-rotor-slot

recommended. The first-order stress harmonic for 17 and 19 rotor slots are very

The influences of split ratio and PM thickness on cogging torque and ripple torque of FRPMMs are also analyzed in **Figure 15**. This figure is plotted based on the 14-rotor-slot, which is chosen because it has the largest torque density and a relatively low pulsating torque, as shown in **Figure 13**. It can be found in **Figure 15(a)** that the cogging torque increases with the PM thickness and the split ratio. When the PM thickness increases, the airgap flux density increases, and thus the interaction between the PMs and slot-teeth becomes greater, which leads to a higher cogging torque. As the split ratio increases, the airgap radius increases, hence the cogging torque increases with the split ratio [30]. As for the ripple torque, the ripple torque has the maximum value when the split ratio is around 0.66. This is because the ripple torque is not only related to the slot structure but also influenced by the electric loading. As aforementioned, the pulsating torque resulting from the slot structure is increased with the split ratio. However, as the split ratio increases, the slot area is reduced and the electric loading gets smaller and smaller, so the ripple torque resulting from the electric loading becomes lower. Considering these two impacts, the ripple torque has a maximal value when the split ratio changes.

As we know, the airgap structure is significant for the pulsating torque because

the pulsating torque results from the interaction between the two sides of the airgap, that is, stator and rotor. Therefore, this chapter also analyzes the influences of stator slot opening ratio and rotor slot opening ratio on cogging torque and ripple torque. Here, the stator/rotor slot opening ratio is defined as the ratio of stator/rotor slot opening width to the stator/rotor slot pitch. **Figure 16** shows the variation of cogging torque and ripple torque with the two slot opening ratios. It can be seen that the cogging torque increases with the stator slot opening ratio. The reason is that a larger stator slot opening ratio reduces the PM width and the smoothness of PM MMF, thus the changing of the PM MMF along the tangential direction increases

*Effect of split ratio and PM thickness on pulsating torque performances: (a) cogging torque (%); (b) ripple*

FRPMM has a large eccentricity stress. Therefore, 13-rotor-slot is not

*4.2.2 Influence of PM thickness and split ratio on pulsating torque*

*4.2.3 Influence of slot opening ratios on pulsating torque*

**Figure 15.**

*torque (%).*

**83**

small, so their eccentricity can be neglected.

*DOI: http://dx.doi.org/10.5772/intechopen.92428*

*Flux Reversal Machine Design*

**Figure 13.** *Effect of slot-pole combination on pulsating torque performances: (a) cogging torque waveforms (%); (b) rated torque waveforms.*

**Figure 14.**

*Radial stress analysis of the FRPMMs: (a) 13-rotor-slot; (b) 14-rotor-slot; (c) 16-rotor-slot; (d) 17-rotor-slot; (e) 19-rotor-slot.*

#### *Flux Reversal Machine Design DOI: http://dx.doi.org/10.5772/intechopen.92428*

FRPMMs, that is, 13, 17, and 19 rotor slots, there are many odd stress harmonics. Since the first-order harmonic is dominant for the eccentricity, the 13-rotor-slot FRPMM has a large eccentricity stress. Therefore, 13-rotor-slot is not recommended. The first-order stress harmonic for 17 and 19 rotor slots are very small, so their eccentricity can be neglected.

#### *4.2.2 Influence of PM thickness and split ratio on pulsating torque*

The influences of split ratio and PM thickness on cogging torque and ripple torque of FRPMMs are also analyzed in **Figure 15**. This figure is plotted based on the 14-rotor-slot, which is chosen because it has the largest torque density and a relatively low pulsating torque, as shown in **Figure 13**. It can be found in **Figure 15(a)** that the cogging torque increases with the PM thickness and the split ratio. When the PM thickness increases, the airgap flux density increases, and thus the interaction between the PMs and slot-teeth becomes greater, which leads to a higher cogging torque. As the split ratio increases, the airgap radius increases, hence the cogging torque increases with the split ratio [30]. As for the ripple torque, the ripple torque has the maximum value when the split ratio is around 0.66. This is because the ripple torque is not only related to the slot structure but also influenced by the electric loading. As aforementioned, the pulsating torque resulting from the slot structure is increased with the split ratio. However, as the split ratio increases, the slot area is reduced and the electric loading gets smaller and smaller, so the ripple torque resulting from the electric loading becomes lower. Considering these two impacts, the ripple torque has a maximal value when the split ratio changes.

#### *4.2.3 Influence of slot opening ratios on pulsating torque*

As we know, the airgap structure is significant for the pulsating torque because the pulsating torque results from the interaction between the two sides of the airgap, that is, stator and rotor. Therefore, this chapter also analyzes the influences of stator slot opening ratio and rotor slot opening ratio on cogging torque and ripple torque. Here, the stator/rotor slot opening ratio is defined as the ratio of stator/rotor slot opening width to the stator/rotor slot pitch. **Figure 16** shows the variation of cogging torque and ripple torque with the two slot opening ratios. It can be seen that the cogging torque increases with the stator slot opening ratio. The reason is that a larger stator slot opening ratio reduces the PM width and the smoothness of PM MMF, thus the changing of the PM MMF along the tangential direction increases

*Effect of split ratio and PM thickness on pulsating torque performances: (a) cogging torque (%); (b) ripple torque (%).*

**Figure 13.**

**Figure 14.**

**82**

*(e) 19-rotor-slot.*

*torque waveforms.*

*Direct Torque Control Strategies of Electrical Machines*

*Effect of slot-pole combination on pulsating torque performances: (a) cogging torque waveforms (%); (b) rated*

*Radial stress analysis of the FRPMMs: (a) 13-rotor-slot; (b) 14-rotor-slot; (c) 16-rotor-slot; (d) 17-rotor-slot;*

**Figure 16.**

*Effect of stator slot opening ratio and rotor slot opening ratio on pulsating torque: (a) cogging torque (%); (b) ripple torque (%).*

the cogging torque. As for the rotor slot opening ratio, which simultaneously influences all the harmonic contents of the airgap permeance, it has great and nonlinear impact on the pulsating torque. Since the pulsating torque results from the interaction of multi permeance harmonics, the variation of pulsating torque changes nonlinearly with the rotor slot opening ratio. It can be seen in **Figure 16** that the optimal cogging torque and ripple torque can be achieved when the stator slot opening ratio and rotor slot opening ratio are around 0.25 and 0.7, respectively.

#### **4.3 Power factor performances**

#### *4.3.1 Influence of stator inner diameter and PM thickness on power factor*

Since the power factor of FRPMMs is usually low, which is around 0.4–0.7, meanwhile a low power factor will increase the converter capacity and cost, the influences of key parameters on the power factor should be also analyzed to achieve a relatively high power factor. The power factor can be given as:

$$PF = \mathbf{1} / \sqrt{\mathbf{1} + \left(\frac{L\_t I\_s}{\mu\_m}\right)^2} \tag{31}$$

*4.3.2 Influence of slot opening ratio on power factor*

*Effect of PM thickness/airgap length on power factor.*

*Effect of stator inner diameter/airgap length on power factor.*

**Figure 17.**

*Flux Reversal Machine Design*

*DOI: http://dx.doi.org/10.5772/intechopen.92428*

**Figure 18.**

**85**

optimal value for the stator slot opening ratio.

Another important parameter that influences the airgap structure is the slot opening ratio. Hence, **Figures 19** and **20** analyzes the effect of stator slot opening ratio and rotor slot opening ratio on pulsating torque performances, respectively. It can be seen in **Figure 19** that the maximum power factor can be obtained when the stator slot opening ratio is approximately to 0.3. The explanation is as follows. When the stator slot opening ratio is too small, the slot leakage flux between the stator tips is large, thus the main flux is reduced, and the back-EMF is lowered, resulting in smaller back-EMF. And when the stator slot opening ratio is too large, the PM width will be narrower. Although the slot leakage flux is reduced, the main flux is not high due to the narrower PMs, thus the back-EMF is lowered. Therefore, the stator slot opening ratio cannot be too small or too large, that is, there is an

Then, the influences of rotor slot opening ratio on power factor can be seen in **Figure 20**. It indicates that when the rotor slot opening ratio is around 0.7, the

where *Is* is the winding current, *Ls* is the synchronous inductance (because the saliency ratio is approximate to 1, *Ld* ≈ *Lq*), and *ψ<sup>m</sup>* is the PM flux linkage. Then, the effect of stator inner diameter on power factor is shown in **Figure 17**. Here, the stator outer diameter is kept as 124 mm, and the airgap length is fixed as 0.5 mm. It can be found that with the increase of stator inner diameter, the power factor increases continuously. The reason is that with the increase of stator inner diameter, the slot area decreases, so the winding turns per phase decreases, thus leading to the reduction of the synchronous inductance *Ls*. The lower *Ls*, the higher power factor, as shown in Eq. (31). Apart from the stator inner diameter, another important parameter affecting the power factor is the PM thickness *hm*. **Figure 18** investigates the variation of power factor with respect to the PM thickness. It indicates that the power factor initially increases with the PM thickness but then decreases. The reason is explained as follows. As the PM thickness increases, the PM flux linkage *ψ<sup>m</sup>* becomes larger, so the power factor increases. However, the synchronous inductance *Ls* also increases with the PM thickness, which leads to the reduction of power factor afterwards. Therefore, there is an optimal PM thickness for a maximum achievable power factor.

*Flux Reversal Machine Design DOI: http://dx.doi.org/10.5772/intechopen.92428*

**Figure 17.** *Effect of stator inner diameter/airgap length on power factor.*

**Figure 18.** *Effect of PM thickness/airgap length on power factor.*

#### *4.3.2 Influence of slot opening ratio on power factor*

Another important parameter that influences the airgap structure is the slot opening ratio. Hence, **Figures 19** and **20** analyzes the effect of stator slot opening ratio and rotor slot opening ratio on pulsating torque performances, respectively. It can be seen in **Figure 19** that the maximum power factor can be obtained when the stator slot opening ratio is approximately to 0.3. The explanation is as follows. When the stator slot opening ratio is too small, the slot leakage flux between the stator tips is large, thus the main flux is reduced, and the back-EMF is lowered, resulting in smaller back-EMF. And when the stator slot opening ratio is too large, the PM width will be narrower. Although the slot leakage flux is reduced, the main flux is not high due to the narrower PMs, thus the back-EMF is lowered. Therefore, the stator slot opening ratio cannot be too small or too large, that is, there is an optimal value for the stator slot opening ratio.

Then, the influences of rotor slot opening ratio on power factor can be seen in **Figure 20**. It indicates that when the rotor slot opening ratio is around 0.7, the

the cogging torque. As for the rotor slot opening ratio, which simultaneously influences all the harmonic contents of the airgap permeance, it has great and nonlinear impact on the pulsating torque. Since the pulsating torque results from the interaction of multi permeance harmonics, the variation of pulsating torque changes nonlinearly with the rotor slot opening ratio. It can be seen in **Figure 16** that the optimal cogging torque and ripple torque can be achieved when the stator slot opening ratio and rotor slot opening ratio are around 0.25 and 0.7, respectively.

*Effect of stator slot opening ratio and rotor slot opening ratio on pulsating torque: (a) cogging torque (%);*

Since the power factor of FRPMMs is usually low, which is around 0.4–0.7, meanwhile a low power factor will increase the converter capacity and cost, the influences of key parameters on the power factor should be also analyzed to achieve

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*LsIs ψ <sup>m</sup>* � �<sup>2</sup>

(31)

1 þ

where *Is* is the winding current, *Ls* is the synchronous inductance (because the saliency ratio is approximate to 1, *Ld* ≈ *Lq*), and *ψ<sup>m</sup>* is the PM flux linkage. Then, the effect of stator inner diameter on power factor is shown in **Figure 17**. Here, the stator outer diameter is kept as 124 mm, and the airgap length is fixed as 0.5 mm. It can be found that with the increase of stator inner diameter, the power factor increases continuously. The reason is that with the increase of stator inner diameter, the slot area decreases, so the winding turns per phase decreases, thus leading to the reduction of the synchronous inductance *Ls*. The lower *Ls*, the higher power factor, as shown in Eq. (31). Apart from the stator inner diameter, another important parameter affecting the power factor is the PM thickness *hm*. **Figure 18** investigates the variation of power factor with respect to the PM thickness. It indicates that the power factor initially increases with the PM thickness but then decreases. The reason is explained as follows. As the PM thickness increases, the PM flux linkage *ψ<sup>m</sup>* becomes larger, so the power factor increases. However, the synchronous inductance *Ls* also increases with the PM thickness, which leads to the reduction of power factor afterwards. Therefore, there is an optimal PM thickness for a maxi-

s

*4.3.1 Influence of stator inner diameter and PM thickness on power factor*

a relatively high power factor. The power factor can be given as:

*PF* ¼ 1*=*

**4.3 Power factor performances**

*Direct Torque Control Strategies of Electrical Machines*

**Figure 16.**

*(b) ripple torque (%).*

mum achievable power factor.

**84**

chapter. Since the magnetic properties of PM materials are sensitive to temperature,

**Figure 21** shows the demagnetization curve of the magnets. The upper half is a straight line, and lower half under the knee point *Bknee* is a curved line. When the FRPMM works on the straight line (such as point *P*1), the return line coincides with the demagnetization curve, and the magnetic performance of the magnets will not be lost. However, when the armature equivalent MMF *Ha*´ is too large at load condition, or the knee point is too high, the working point *Bknee* is moved to *P*2. At this time, the recovery line does not coincide with the original demagnetization line, thus the intersection of the *B*-axis changes from *Br* to *Br*1, causing the irreversible demagnetization. Then, the PM properties and machine performances will no longer return to the original. So, the PM flux density should be examined in order to check the risk of irreversible demagnetization. As we know, the PM flux density is determined by the design parameters such as electric loading *Ae*, PM thickness *hm*, rotor slot opening ratio, etc. So, in this chapter, the effects of electric loading *Ae*, PM thickness *hm*, rotor slot opening ratio *bo/t* on PM demagnetization performances of FRPMMs will be studied. For instance, the PM material is selected as N38SH, and

**Figure 22** shows the PM flux density of a 12-stator-slot/14-rotor-slot FRPMM when the electric loading *Ae* is 1600A/cm, the PM thickness *hm* is 3 mm, rotor slot opening ratio *bo/t* is 0.65. It can be seen that the PM flux density distribution varies with the rotor position. When the rotor position is 140°, the PM does not demagnetize, while at 0° and 340°, the PM will demagnetize. Hence, in the following analysis, the PM flux density at the most severe moment of demagnetization is

**Figure 23** studies the magnetic flux density distribution in the PMs under dif-

ferent electric loadings. It can be found that the larger electric loading *Ae*, the smaller minimum flux density. This is because the larger electric loading, the higher armature MMF *Ha*´, and the more left operating point *P*2, so the lower flux density

. When

and the temperature coefficient of NdFeB magnet is as high as 0.126%K<sup>1</sup>

tization performances of FRPMMs at different conditions.

*Flux Reversal Machine Design*

*DOI: http://dx.doi.org/10.5772/intechopen.92428*

knee point of the PM flux density at 100°C is 0.35 T.

selected.

**Figure 21.**

**87**

*PM demagnetization curve.*

the current of FRPMMs is large, the winding heating can easily affect the PMs attached to the stator teeth surface, causing the decrease of PM magnetic performances. On the other hand, when the winding current is large, the demagnetizing effect of the armature field is enhanced, and thus the PMs have the possibility to be demagnetized. Therefore, it is of great importance to investigate the PM demagne-

**Figure 19.** *Effect of stator slot opening ratio on power factor.*

**Figure 20.** *Effect of rotor slot opening ratio on power factor.*

power factor reaches the maximal value. This is because the power factor is mainly influenced by the back-EMF. When the rotor slot opening ratio increases, the effective airgap length becomes smaller, thus the main flux is increased and the back-EMF is improved. As a result, the power factor is increased. When the rotor slot opening ratio keeps increasing, the flux modulation effect of the rotor teeth becomes weaker and weaker, thus the smaller modulated flux, and the lower back-EMF. Therefore, there is also an optimal value for rotor slot opening ratio when a high power factor is demanded.

#### **4.4 PM demagnetization performances**

For PM machines, PM demagnetization performances are very important because it is highly related to the safe operation and machine reliability. Therefore, the PM demagnetization performances of FRPMMs should be analyzed in this

#### *Flux Reversal Machine Design DOI: http://dx.doi.org/10.5772/intechopen.92428*

chapter. Since the magnetic properties of PM materials are sensitive to temperature, and the temperature coefficient of NdFeB magnet is as high as 0.126%K<sup>1</sup> . When the current of FRPMMs is large, the winding heating can easily affect the PMs attached to the stator teeth surface, causing the decrease of PM magnetic performances. On the other hand, when the winding current is large, the demagnetizing effect of the armature field is enhanced, and thus the PMs have the possibility to be demagnetized. Therefore, it is of great importance to investigate the PM demagnetization performances of FRPMMs at different conditions.

**Figure 21** shows the demagnetization curve of the magnets. The upper half is a straight line, and lower half under the knee point *Bknee* is a curved line. When the FRPMM works on the straight line (such as point *P*1), the return line coincides with the demagnetization curve, and the magnetic performance of the magnets will not be lost. However, when the armature equivalent MMF *Ha*´ is too large at load condition, or the knee point is too high, the working point *Bknee* is moved to *P*2. At this time, the recovery line does not coincide with the original demagnetization line, thus the intersection of the *B*-axis changes from *Br* to *Br*1, causing the irreversible demagnetization. Then, the PM properties and machine performances will no longer return to the original. So, the PM flux density should be examined in order to check the risk of irreversible demagnetization. As we know, the PM flux density is determined by the design parameters such as electric loading *Ae*, PM thickness *hm*, rotor slot opening ratio, etc. So, in this chapter, the effects of electric loading *Ae*, PM thickness *hm*, rotor slot opening ratio *bo/t* on PM demagnetization performances of FRPMMs will be studied. For instance, the PM material is selected as N38SH, and knee point of the PM flux density at 100°C is 0.35 T.

**Figure 22** shows the PM flux density of a 12-stator-slot/14-rotor-slot FRPMM when the electric loading *Ae* is 1600A/cm, the PM thickness *hm* is 3 mm, rotor slot opening ratio *bo/t* is 0.65. It can be seen that the PM flux density distribution varies with the rotor position. When the rotor position is 140°, the PM does not demagnetize, while at 0° and 340°, the PM will demagnetize. Hence, in the following analysis, the PM flux density at the most severe moment of demagnetization is selected.

**Figure 23** studies the magnetic flux density distribution in the PMs under different electric loadings. It can be found that the larger electric loading *Ae*, the smaller minimum flux density. This is because the larger electric loading, the higher armature MMF *Ha*´, and the more left operating point *P*2, so the lower flux density

**Figure 21.** *PM demagnetization curve.*

power factor reaches the maximal value. This is because the power factor is mainly influenced by the back-EMF. When the rotor slot opening ratio increases, the effective airgap length becomes smaller, thus the main flux is increased and the back-EMF is improved. As a result, the power factor is increased. When the rotor slot opening ratio keeps increasing, the flux modulation effect of the rotor teeth becomes weaker and weaker, thus the smaller modulated flux, and the lower back-EMF. Therefore, there is also an optimal value for rotor slot opening ratio when a

For PM machines, PM demagnetization performances are very important because it is highly related to the safe operation and machine reliability. Therefore, the PM demagnetization performances of FRPMMs should be analyzed in this

high power factor is demanded.

*Effect of rotor slot opening ratio on power factor.*

**Figure 19.**

**Figure 20.**

**86**

*Effect of stator slot opening ratio on power factor.*

*Direct Torque Control Strategies of Electrical Machines*

**4.4 PM demagnetization performances**

**Figure 22.**

*PM demagnetization at different rotor positions: (a) rotor position = 0°; (b) rotor position = 140°; (c) rotor position = 340°.*

**Figure 23.** *Influence of* Ae *on PM demagnetization.*

in the magnets. When the electric loading *Ae* is 1400A/cm, the PM irreversible demagnetization just occurs. In addition, it can be seen that the entire magnetic flux density map is skewed to the right. This is because the N-pole magnet is intercepted in this analysis, and there is an S-pole magnet next to the N-pole magnet. There is PM pole leakage flux between the S-pole magnet (negative axis) and the N-pole magnet (positive axis), so the magnetic flux density around the 0 position is lower, and away from the 0 position, the magnetic flux density gradually rises.

and the PM thickness is selected as five times the airgap length, that is, 3 mm. The larger rotor slot opening ratio, the narrower rotor teeth, thus the more saturated rotor teeth, and the smaller magnetic reluctance. As shown in **Figure 20**, when the magnetic gets smaller, the more left operating point *P*2, and thus the lower PM flux density. It can be seen in **Figure 25** that when the rotor slot opening ratio *bo*/*t* is 0.9, the irreversible PM demagnetization just occurs. In Ref. [28], it is claimed that the maximum back-EMF can be achieved when the rotor slot opening ratio *bo*/*t* is around 0.6. So, during the design process, the optimal rotor slot opening ratio can

be directly applied without consideration of the PM demagnetization risk.

The geometrical parameters of stator and rotor are shown in **Figure 26**. The

**5. Geometric design of stator and rotor**

no-load flux of each winding pole could be calculated as:

**5.1 Stator design**

**89**

**Figure 24.**

**Figure 25.**

*Influence of* hm *on PM demagnetization.*

*Flux Reversal Machine Design*

*DOI: http://dx.doi.org/10.5772/intechopen.92428*

*Influence of* bo*/*t *on PM demagnetization.*

**Figure 24** analyzes the effect of PM thickness *hm* on the PM demagnetization performances. At this time, the electric loading is chosen as 800 A/cm, and the rotor slot opening ratio is selected as 0.65. It can be seen in **Figure 24** that when the PM thickness *hm* is less than 2.5 mm, the irreversible demagnetization will happen, while when the PM thickness *hm* is larger than 2.5 mm, the irreversible demagnetization will not. In this model, the airgap length is 0.5 mm. Therefore, in the design stage, the PM thickness should be better to set as five times or more the airgap length. Considering the back-EMF, it is claimed in [3] that when the PM thickness is about three times the airgap length, the back-EMF will reach the maximum. But considering both back-EMF and PM demagnetization risk, it is safer to set the PM thickness as about five times airgap length.

**Figure 25** shows the influences of rotor slot opening ratio *bo*/*t* on the flux density distribution inside the PMs. At this time, the electric loading is chosen as 800 A/cm,

**Figure 24.** *Influence of* hm *on PM demagnetization.*

**Figure 25.** *Influence of* bo*/*t *on PM demagnetization.*

and the PM thickness is selected as five times the airgap length, that is, 3 mm. The larger rotor slot opening ratio, the narrower rotor teeth, thus the more saturated rotor teeth, and the smaller magnetic reluctance. As shown in **Figure 20**, when the magnetic gets smaller, the more left operating point *P*2, and thus the lower PM flux density. It can be seen in **Figure 25** that when the rotor slot opening ratio *bo*/*t* is 0.9, the irreversible PM demagnetization just occurs. In Ref. [28], it is claimed that the maximum back-EMF can be achieved when the rotor slot opening ratio *bo*/*t* is around 0.6. So, during the design process, the optimal rotor slot opening ratio can be directly applied without consideration of the PM demagnetization risk.

### **5. Geometric design of stator and rotor**

#### **5.1 Stator design**

The geometrical parameters of stator and rotor are shown in **Figure 26**. The no-load flux of each winding pole could be calculated as:

in the magnets. When the electric loading *Ae* is 1400A/cm, the PM irreversible demagnetization just occurs. In addition, it can be seen that the entire magnetic flux density map is skewed to the right. This is because the N-pole magnet is intercepted in this analysis, and there is an S-pole magnet next to the N-pole magnet. There is PM pole leakage flux between the S-pole magnet (negative axis) and the N-pole magnet (positive axis), so the magnetic flux density around the 0 position is lower,

*PM demagnetization at different rotor positions: (a) rotor position = 0°; (b) rotor position = 140°; (c) rotor*

*Direct Torque Control Strategies of Electrical Machines*

**Figure 24** analyzes the effect of PM thickness *hm* on the PM demagnetization performances. At this time, the electric loading is chosen as 800 A/cm, and the rotor slot opening ratio is selected as 0.65. It can be seen in **Figure 24** that when the PM thickness *hm* is less than 2.5 mm, the irreversible demagnetization will happen, while when the PM thickness *hm* is larger than 2.5 mm, the irreversible demagnetization will not. In this model, the airgap length is 0.5 mm. Therefore, in the design stage, the PM thickness should be better to set as five times or more the airgap length. Considering the back-EMF, it is claimed in [3] that when the PM thickness is about three times the airgap length, the back-EMF will reach the maximum. But considering both back-EMF and PM demagnetization risk, it is safer to set the PM

**Figure 25** shows the influences of rotor slot opening ratio *bo*/*t* on the flux density distribution inside the PMs. At this time, the electric loading is chosen as 800 A/cm,

and away from the 0 position, the magnetic flux density gradually rises.

thickness as about five times airgap length.

**Figure 22.**

**Figure 23.**

**88**

*Influence of* Ae *on PM demagnetization.*

*position = 340°.*

*Direct Torque Control Strategies of Electrical Machines*

$$
\phi\_m = \mathcal{Z}\lambda\_w l\_{\text{stk}} B\_m / \pi \tag{32}
$$

Then, next step is to calculate the stator outer radius *ro*. Firstly, the total slot area of all the stator slots *Aslot* can be written based on the winding electric loading *Ae*

where *Sfg* is the slot fill factor. Meanwhile, the total slot area of all the stator slots

Combining the Eqs. (39) and (40), the slot depth *hs* can be determined. Then,

Defining the average flux density of each rotor yoke and middle of rotor tooth as *Bry*, and *Brt*, respectively, the rotor yoke thickness *hry* and rotor tooth width *wrt* are able to be achieved using the similar derivation procedure as Eq. (35) and Eq. (36).

Based on the analytical equations and the investigations of key performances in the former parts, a quick and accurate analytical design of a FRPMM can be realized

1.Based on the performance investigations in **Figures 9–24**, the initial design values, including combination of stator slot and rotor slot number, electric loading, equivalent magnetic loading, airgap length, materials of active parts,

2.Then, assuming an appropriate aspect ratio *klr*, the airgap radius *rg,* and the

3.Based on Eqs. (32–44), the detailed geometric parameters of the stator core and the rotor core are able to be obtained. Therefore, the stator outer diameter

4.After that, check if the stator outer diameter and the machine total length satisfy the required design specifications. If so, proceed to the FEA

stack length *lstk* can be worked out using Eqs. (29) and (30).

*ro* and machine total length *lo* can be finally determined.

*Ae=JeSfg* (39)

<sup>2</sup> � *Zswths* (40)

*ro* ¼ *rg* þ *hm* þ *h*<sup>1</sup> þ *hs* þ *hy* (41)

*hry* ¼ *rgBm=ZrBrykstk* (42) *wrt* ¼ 4*rgBm=ZrkstkBrt* (43)

*hrs* ¼ *rs* þ *hry* (44)

*Aslot* ¼ 2*π rg* þ *hm*

<sup>2</sup> � *<sup>π</sup> rg* <sup>þ</sup> *hm* <sup>þ</sup> *<sup>h</sup>*<sup>1</sup>

*Aslot* can be also derived out using the structural parameters:

*Aslot* ¼ *π rg* þ *hm* þ *h*<sup>1</sup> þ *hs*

the stator outer radius *ro* can be given as:

Finally, the *hry* and *wrt* are given as:

Then, the rotor slot depth *hrs* is determined as:

by following these procedures (as depicted in **Figure 27**):

**6. Design methodology and evaluations**

etc. can be firstly selected.

and the current density *Je*:

*Flux Reversal Machine Design*

*DOI: http://dx.doi.org/10.5772/intechopen.92428*

**5.2 Rotor design**

**6.1 Design procedure**

**91**

where *λ<sup>w</sup>* is the winding pitch. If full-pitch winding is adopted, the winding pitch is able to be written as:

$$
\lambda\_w = 2\pi r\_\text{g} / 2P \tag{33}
$$

Then, the no-load flux of each winding pole *ϕ<sup>m</sup>* in Eq. (32) could change to:

$$
\phi\_m = 2r\_\text{g} l\_{stk} B\_m / P \tag{34}
$$

Defining the average flux density at the stator yoke as *By*, the stator yoke thickness *hy* can therefore be deduced as:

$$h\_{\gamma} = \frac{\phi\_m}{2B\_{\gamma}k\_{stk}l\_{stk}} = \frac{r\_{\text{g}}B\_m}{PB\_{\gamma}k\_{stk}}\tag{35}$$

Similarly, defining the average flux density at the middle of stator tooth as *Bt*, the stator tooth width is able to be worked out:

$$w\_t = \frac{\phi\_m}{\text{3SPP}k\_{stk}l\_{stk}B\_t} = \frac{4r\_{\text{g}}B\_m}{Z\_s k\_{stk}B\_t} \tag{36}$$

Moreover, in order to simultaneously maintain a relatively large torque density as well as reduce the risk of PM demagnetization, the PM thickness is recommended to be:

$$h\_m = 4\text{g} \sim 6\text{g} \tag{37}$$

where *g* is the airgap length. Since the optimal torque density is often obtained when the slot opening ratio is approximate to 0.25 [28], the stator slot opening width *wo* could be written as:

$$
\omega\_o = \pi (r\_\text{g} + h\_m) / 2Z\_s \tag{38}
$$

**Figure 26.** *Geometry of stator and rotor.*

*Flux Reversal Machine Design DOI: http://dx.doi.org/10.5772/intechopen.92428*

Then, next step is to calculate the stator outer radius *ro*. Firstly, the total slot area of all the stator slots *Aslot* can be written based on the winding electric loading *Ae* and the current density *Je*:

$$A\_{\rm slot} = 2\pi (r\_{\rm g} + h\_m) A\_{\rm e} / J\_{\rm e} \text{S}\_{\rm f\bar{\rm g}} \tag{39}$$

where *Sfg* is the slot fill factor. Meanwhile, the total slot area of all the stator slots *Aslot* can be also derived out using the structural parameters:

$$A\_{slat} = \pi \left(r\_{\rm g} + h\_m + h\_1 + h\_t\right)^2 - \pi \left(r\_{\rm g} + h\_m + h\_1\right)^2 - Z\_s w\_l h\_s \tag{40}$$

Combining the Eqs. (39) and (40), the slot depth *hs* can be determined. Then, the stator outer radius *ro* can be given as:

$$r\_o = r\_\text{g} + h\_m + h\_1 + h\_s + h\_y \tag{41}$$

#### **5.2 Rotor design**

*ϕ<sup>m</sup>* ¼ 2*λwlstkBm=π* (32)

*λ<sup>w</sup>* ¼ 2*πrg=*2*P* (33)

*ϕ<sup>m</sup>* ¼ 2*rg lstkBm=P* (34)

*hm* ¼ 4*g* � 6*g* (37)

*=*2*Zs* (38)

(35)

(36)

where *λ<sup>w</sup>* is the winding pitch. If full-pitch winding is adopted, the winding pitch

Then, the no-load flux of each winding pole *ϕ<sup>m</sup>* in Eq. (32) could change to:

Defining the average flux density at the stator yoke as *By*, the stator yoke

Similarly, defining the average flux density at the middle of stator tooth as *Bt*,

Moreover, in order to simultaneously maintain a relatively large torque density as well as reduce the risk of PM demagnetization, the PM thickness is recommended

where *g* is the airgap length. Since the optimal torque density is often obtained when the slot opening ratio is approximate to 0.25 [28], the stator slot opening

*wo* ¼ *π rg* þ *hm*

3*SPPkstklstkBt*

<sup>¼</sup> *rgBm PBykstk*

> <sup>¼</sup> <sup>4</sup>*rgBm ZskstkBt*

*hy* <sup>¼</sup> *<sup>ϕ</sup><sup>m</sup>* 2*Bykstklstk*

*wt* <sup>¼</sup> *<sup>ϕ</sup><sup>m</sup>*

is able to be written as:

to be:

**Figure 26.**

**90**

*Geometry of stator and rotor.*

width *wo* could be written as:

thickness *hy* can therefore be deduced as:

*Direct Torque Control Strategies of Electrical Machines*

the stator tooth width is able to be worked out:

Defining the average flux density of each rotor yoke and middle of rotor tooth as *Bry*, and *Brt*, respectively, the rotor yoke thickness *hry* and rotor tooth width *wrt* are able to be achieved using the similar derivation procedure as Eq. (35) and Eq. (36). Finally, the *hry* and *wrt* are given as:

$$h\_{\gamma\gamma} = r\_{\rm g} B\_m / Z\_r B\_{r\rm ry} k\_{\rm stk} \tag{42}$$

$$
\omega\_{rt} = 4r\_{\rm g}B\_m / Z\_r k\_{stk} B\_{rt} \tag{43}
$$

Then, the rotor slot depth *hrs* is determined as:

$$h\_{\pi\circ} = r\_{\circ} + h\_{r\circ} \tag{44}$$

### **6. Design methodology and evaluations**

#### **6.1 Design procedure**

Based on the analytical equations and the investigations of key performances in the former parts, a quick and accurate analytical design of a FRPMM can be realized by following these procedures (as depicted in **Figure 27**):


verifications of machine performances. If not, reset the initial values such as combination of stator slot and rotor slot number, electric loading, equivalent magnetic loading, etc.

5.Conducting FEA simulations, the electromagnetic performances such as back-EMF, average torque, pulsating torque, power factor, efficiency, etc. can be obtained. Check if all the performances satisfy the design specifications. If not, adjust the design parameters in the former steps and iterate the design flow

In order show the effectiveness of the introduced analytical method, a FRPMM is designed based on the method. **Table 4** shows the specifications of the FRPMM, which mainly includes the rated torque, machine volume, cooling method, rated power, and speed. According to the rated torque, a design margin of 5% is suggested so as to make sure the torque output. Therefore, the requirement of the torque is 8.4 Nm for this design. Then, the combination of stator slots and rotor slots is determined in the first place. This combination is selected due to its high torque density and low pulsating torque, as shown in **Figure 13**. Then, since the cooling method is natural cooling, the electric loading and the equivalent magnetic loading are chosen as 300A/cm and 0.2 T, respectively. After that, based on the output torque value 8.4 Nm and Eq. (29), the airgap radius is determined as 38.5 mm. Furthermore, assuming the yoke flux density of stator core and rotor core as 1.0 T,

and the teeth flux density of stator core and rotor core as 1.2 T, the detailed

**Parameter Value Parameter Value** Rated torque 8 Nm Rated speed 300 rpm Rotor inner diameter 32 mm PM material N38SH Stator outer diameter 130 mm Stack length 120 mm Airgap length 0.6 mm Iron material 50WW470 Cooling method Natural cooling Rated power 0.25 kW

Stator Outer diameter 124 mm Inner diameter 79 mm

Magnet PM thickness 3 mm Magnet width 7.8 mm Rotor Outer diameter 77.8 mm Slot depth 10.4 mm

**Parameter Value Parameter Value**

Turns per phase 300 Teeth width 11.5 mm Slot number 12 Yoke thickness 6 mm Slot depth 13.5 mm Yoke flux density 1.0 T Winding pole pair 1 Teeth flux density 1.1 T

Teeth flux density 1.2 T Yoke thickness 12.5 mm Inner diameter 32 mm Yoke flux density 1.0 T Teeth width 4 mm Slot number 17

until every output meets the requirement.

6.Finally, it is the result output.

*DOI: http://dx.doi.org/10.5772/intechopen.92428*

*Flux Reversal Machine Design*

**6.2 Case study**

**Table 4.**

**Table 5.**

**93**

*Design specifications of a three-phase FRPMM.*

*Design parameters of the FRPMM using the design method.*

**Figure 27.** *Design flow of FRPMM.*

### *Flux Reversal Machine Design DOI: http://dx.doi.org/10.5772/intechopen.92428*

