**3.1 Magnetic circuit model**

In order to derive the sizing equation of FRPMMs, the magnetic circuit model should be built at first; then, based on the model, the analytical equations of airgap flux density, back-EMF, and torque will be deduced.

The equivalent magnetic circuit model can be plotted as **Figure 4**. At No.1 stator tooth, its magnetic field distribution corresponds to the position shown in **Figure 2(b)**, that is, the rotor tooth is closer to the S-pole magnet. The S-pole magnetic generates two paths of magnetic flux, one is pole leakage flux *Φpl*, which goes through the adjacent N-pole magnet, the other is main flux *Φm*, which goes through the stator tooth, stator yoke, rotor tooth, and rotor yoke, thus can provide winding flux linkage and back-EMF. At No. 2 stator tooth, its magnetic field distribution corresponds to the position shown in **Figure 2(c)**, that is, the rotor axis is at the same distance from the S-pole and N-pole magnets. Thus, at this time, the two magnets can only generate one magnetic flux path, that is, the pole leakage flux *Φpl*. At No. 3 stator tooth, its magnetic field distribution corresponds to the position shown in **Figure 2(d)**, that is, the rotor tooth is closer to the N-pole magnet. The N-pole magnetic generates two paths of magnetic flux, one is pole leakage flux *Φpl*, which goes through the adjacent S-pole magnet, the other is main flux *Φm*, which goes through the stator tooth, stator yoke, rotor tooth, and rotor yoke, thus can provide winding flux linkage and back-EMF. It should be noted that the magnetic flux path of No. 1 stator tooth is just opposite to that of No. 3 stator tooth, so winding flux polarity in these two cases is just opposite to each other.

As mentioned above, **Figure 4** provides the magnetic circuit of FRPMMs, which can help analyze the flux distribution of FRPMMs at different rotor positions. However, the magnetic circuit requires the establishment of the whole FRPMM

**Figure 4.** *Equivalent magnetic model of FRPMMs.*

after obtaining the bipolar flux linkage, as shown in **Figure 3**, the winding can produce a bipolar back-electromagnetic motive force (EMF). If the armature windings are injected with currents having the same frequency and phase with the back-

*No-load flux lines of the FRPMM excited by the PMs: (a) rotor position =0 elec. degree; (b) rotor position =90*

*elec. degree; (c) rotor position =180 elec. degree; (d) rotor position =270 elec. degree.*

*Direct Torque Control Strategies of Electrical Machines*

EMF, a steady torque can be yielded.

*Variation of flux linkage of phase a winding at different rotor positions.*

**Figure 2.**

**Figure 3.**

**70**

magnetic path, which is rather complex. Besides, the pole leakage flux, main flux, and the reluctance at each rotor positions should be calculated, which needs high workload. Therefore, a simplified magnetic circuit should be built. Observing **Figure 2**, it can be seen that a small rotor displacement brings a large rotation in stator flux field. This phenomenon is called as flux modulation effect, i.e. a highpole slow-speed magnetic field becomes a low-pole high-speed magnetic field through the modulation effect of iron teeth. Therefore, the physical nature of FRPMM is indeed the flux modulation effect. The research of some flux modulation machines are usually based on the PM magnetic motive force (MMF)-airgap permeance model, such as the Vernier machine in [27]. So, this chapter will use this model to analyze FRPMMs.

In PM MMF-airgap permeance model, the no-load airgap flux density *B*(*θs*,*θ*) can be written as the product of PM MMF *FPM*(*θs*) and specific airgap permeance Λ(*θs*,*θ*):

$$B(\theta\_s, \theta) = F\_{\rm PM}(\theta\_s) \Lambda(\theta\_s, \theta) \tag{1}$$

MMF and airgap permeance, the no-load airgap flux density can be obtained. Then, the stator flux linkage *λph*(*θ*) can be deduced using winding function theory:

0

where *N*(*θs*) is the phase winding function. After that, the phase back-EMF

*dλph*ð Þ*θ*

*Eph*ðÞ¼ *t*

*Te* <sup>¼</sup> <sup>3</sup> 2

where *Iph* is the peak value of phase current. Therefore, from Eqs. (1–4), it can be found that if the torque equation need to be calculated, the key is to obtain the equation of airgap flux density *B*(*θs*,*θ*), which is further determined by the PM MMF *FPM*(*θs*) and specific airgap permeance Λ(*θs*,*θ*). Therefore, in the next parts, the equations of the PM MMF *FPM*(*θs*) and specific airgap permeance Λ(*θs*,*θ*) will be

As aforementioned, to derive the torque equation, the no-load airgap flux density *B*(*θs*,*θ*) should firstly be known, whose equation can be given as Eq. (1). Then, the next step is to derive the expressions of *FPM*(*θs*) and Λ(*θs*,*θ*). The PM MMF waveform excited by the magnets is shown in **Figure 7**, which can be given as:

> *FC*; 0≤*θ<sup>s</sup>* <ð Þ 1 � *SO π=Zs* 0; 1ð Þ � *SO π=Zs* ≤*θ<sup>s</sup>* <ð Þ 1 þ *SO π=Zs FC* ; 1ð Þ þ *SO π=Zs* ≤*θ<sup>s</sup>* <2*π=Zs* �*FC* ; 2*π=Zs* ≤*θ<sup>s</sup>* <ð Þ 3 � *SO π=Zs* 0; 3ð Þ � *SO π=Zs* ≤*θ<sup>s</sup>* <ð Þ 3 þ *SO π=Zs* �*FC* ; 3ð Þ þ *SO π=Zs* ≤*θ<sup>s</sup>* <4*π=Zs*

*B θ<sup>s</sup>* ð Þ , *θ N*ð Þ *θ<sup>s</sup> dθ<sup>s</sup>* (2)

*dt* (3)

(5)

*EphIph* (4)

*<sup>λ</sup>ph*ð Þ¼ *<sup>θ</sup> rglstk*ð2*<sup>π</sup>*

*Eph*(*t*) and average torque *Te* can be calculated as:

deduced in detail.

*Flux Reversal Machine Design*

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

**Figure 7.**

**73**

*Magnet MMF waveform.*

**3.2 Airgap flux density equation**

*FPM*ð Þ¼ *θ<sup>s</sup>*

8 >>>>>>>><

>>>>>>>>:

where the definitions of angles *θ<sup>s</sup>* and *θ* are shown in **Figure 5**. Then, the simplified magnetic circuit model can be given in **Figure 6**. Once knowing the PM

**Figure 5.** *Definitions of different angles in FRPMM.*

**Figure 6.** *Simplified equivalent magnetic model of FRPMMs.*

magnetic path, which is rather complex. Besides, the pole leakage flux, main flux, and the reluctance at each rotor positions should be calculated, which needs high workload. Therefore, a simplified magnetic circuit should be built. Observing **Figure 2**, it can be seen that a small rotor displacement brings a large rotation in stator flux field. This phenomenon is called as flux modulation effect, i.e. a highpole slow-speed magnetic field becomes a low-pole high-speed magnetic field through the modulation effect of iron teeth. Therefore, the physical nature of FRPMM is indeed the flux modulation effect. The research of some flux modulation machines are usually based on the PM magnetic motive force (MMF)-airgap permeance model, such as the Vernier machine in [27]. So, this chapter will use this

In PM MMF-airgap permeance model, the no-load airgap flux density *B*(*θs*,*θ*) can be written as the product of PM MMF *FPM*(*θs*) and specific airgap permeance

where the definitions of angles *θ<sup>s</sup>* and *θ* are shown in **Figure 5**. Then, the simplified magnetic circuit model can be given in **Figure 6**. Once knowing the PM

*B θ<sup>s</sup>* ð Þ¼ , *θ FPM*ð Þ *θ<sup>s</sup>* Λ *θ<sup>s</sup>* ð Þ , *θ* (1)

model to analyze FRPMMs.

*Direct Torque Control Strategies of Electrical Machines*

Λ(*θs*,*θ*):

**Figure 5.**

**Figure 6.**

**72**

*Definitions of different angles in FRPMM.*

*Simplified equivalent magnetic model of FRPMMs.*

MMF and airgap permeance, the no-load airgap flux density can be obtained. Then, the stator flux linkage *λph*(*θ*) can be deduced using winding function theory:

$$\lambda\_{ph}(\theta) = r\_{\rm g} l\_{stk} \int\_0^{2\pi} B(\theta\_s, \theta) N(\theta\_s) d\theta\_s \tag{2}$$

where *N*(*θs*) is the phase winding function. After that, the phase back-EMF *Eph*(*t*) and average torque *Te* can be calculated as:

$$E\_{ph}(t) = \frac{d\lambda\_{ph}(\theta)}{dt} \tag{3}$$

$$T\_{\epsilon} = \frac{\mathfrak{Z}}{2} E\_{ph} I\_{ph} \tag{4}$$

where *Iph* is the peak value of phase current. Therefore, from Eqs. (1–4), it can be found that if the torque equation need to be calculated, the key is to obtain the equation of airgap flux density *B*(*θs*,*θ*), which is further determined by the PM MMF *FPM*(*θs*) and specific airgap permeance Λ(*θs*,*θ*). Therefore, in the next parts, the equations of the PM MMF *FPM*(*θs*) and specific airgap permeance Λ(*θs*,*θ*) will be deduced in detail.

#### **3.2 Airgap flux density equation**

As aforementioned, to derive the torque equation, the no-load airgap flux density *B*(*θs*,*θ*) should firstly be known, whose equation can be given as Eq. (1). Then, the next step is to derive the expressions of *FPM*(*θs*) and Λ(*θs*,*θ*). The PM MMF waveform excited by the magnets is shown in **Figure 7**, which can be given as:

$$F\_{\rm PM}(\theta\_{i}) = \begin{cases} F\_{C}; & 0 \le \theta\_{i} < (1 - \text{SO})\pi/\mathcal{Z}\_{i} \\ \mathbf{0}; & (1 - \text{SO})\pi/\mathcal{Z}\_{i} \le \theta\_{i} < (1 + \text{SO})\pi/\mathcal{Z}\_{i} \\\ F\_{C}; & (1 + \text{SO})\pi/\mathcal{Z}\_{i} \le \theta\_{i} < 2\pi/\mathcal{Z}\_{i} \\\ -F\_{C}; & 2\pi/\mathcal{Z}\_{i} \le \theta\_{i} < (3 - \text{SO})\pi/\mathcal{Z}\_{i} \\\ \mathbf{0}; & (3 - \text{SO})\pi/\mathcal{Z}\_{i} \le \theta\_{i} < (3 + \text{SO})\pi/\mathcal{Z}\_{i} \\\ -F\_{C}; & (3 + \text{SO})\pi/\mathcal{Z}\_{i} \le \theta\_{i} < 4\pi/\mathcal{Z}\_{i} \end{cases} \tag{5}$$

**Figure 7.** *Magnet MMF waveform.*

where *FC* is:

$$F\_C = \frac{B\_r h\_m}{\mu\_r \mu\_0} \tag{6}$$

where *b*<sup>o</sup> is the rotor slot opening width and *t* is the rotor slot pitch, as shown in **Figure 8**. Combining Eq. (1), Eqs. (5–13), the no-load airgap flux density *B*(*θs*,*θ*)

*Bi* sin *iZs*

2 � *Zr* � �*θ<sup>s</sup>* � *Zr<sup>θ</sup>*

As can be seen in Eq. (14), the number of pole pairs in the air gap flux density is *iZs*/2 � *Zr*, *i* = 1,3,5 … Then, in order to make the flux density induce EMF in the armature windings, the pole pair number of the armature windings *P* should be equal to *iZs*/2 � *Zr*, *i* = 1,3,5 … Besides, for three phase symmetry, the winding pole

All in all, the slot-pole combination of three-phase FRPMMs is ruled by the

� *Zr*;

where min means to select the minimum number of these qualified harmonic orders so as to obtain a maximal pole ratio of FRPMMs. Therefore, the feasible slotpole combinations can be summarized as **Table 2**. Non-overlapping windings (i.e., concentrated windings) are usually used in FRPMMs because of the higher fault tolerance and easier manufacture than regular overlapping windings. However, some FRPMMs are suggested to employ overlapping windings in order to have a larger winding factor and thus a higher torque density. Therefore, both winding factors, that is, *kwn* (using non-overlapping winding) and *kwr* (using overlapping winding) are calculated for each FRPMM so as to see the difference of using

Once the stator winding pole pair is selected, the stator flux linkage can be deduced using winding function theory, just as mentioned in Eq. (2). The winding

> X∞ *i*¼1, 3, 5

*Ni* <sup>¼</sup> <sup>2</sup> *iπ Ns*

*Ni* cosð Þ *iPθ<sup>s</sup>* (18)

*<sup>P</sup> kwi* (19)

*N*ð Þ¼ *θ<sup>s</sup>*

2

*i* ¼ 1, 3, 5 … *k* ¼ 1, 2, 3 …

� � (14)

*Fi*Λ1*r*, *i* ¼ 1, 3, 5 … (15)

*GCD Zs* ð Þ , *<sup>P</sup>* <sup>¼</sup> <sup>3</sup>*k*, *<sup>k</sup>* <sup>¼</sup> 1, 2, 3 … (16)

*Zs GCD Zs* ð Þ , *<sup>P</sup>* <sup>¼</sup> <sup>3</sup>*<sup>k</sup>*

(17)

� �

can be finally calculated as:

*Flux Reversal Machine Design*

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

where the magnitude *Bi* is

**3.3 Slot-pole combinations**

following equation:

different winding types.

function *N*(*θs*) in Eq. (2) can be written as:

**3.4 Torque equation**

**75**

*<sup>B</sup> <sup>θ</sup><sup>s</sup>* ð Þ¼ , *<sup>θ</sup>* <sup>X</sup><sup>∞</sup>

pair number must also meet the following requirement:

*<sup>P</sup>* <sup>¼</sup> min *<sup>P</sup>* <sup>¼</sup> *iZs*

*Zs*

*i*¼1, 3

*Bi* <sup>¼</sup> <sup>1</sup> 2

Then, it can be written in Fourier series as follows:

$$F\_{\rm PM}(\theta\_s) = \sum\_{i=1,3,5}^{\infty} F\_i \sin\left(\frac{iZ\_s}{2}\theta\_s\right) \tag{7}$$

where the magnitude *Fi* is

$$F\_i = \frac{4}{\pi} \frac{1}{i} \frac{B\_r h\_m}{\mu\_0 \mu\_r} \left[ 1 + (-1)^{\frac{i+1}{2}} \sin\left(\frac{i\pi}{2} \text{SO}\right) \right] \tag{8}$$

Then, the next step is to derive the specific airgap permeance Λ(*θs*,*θ*) in Eq. (1). Since the stator slotting effect has already been considered in Eqs. (5–8), the specific airgap permeance Λ(*θs*,*θ*) can be replaced by the airgap permeance with smoothed stator and slotted rotor Λ*r*(*θs*,*θ*). The model of smoothed stator and slotted rotor is shown in **Figure 8**. Then, the Λ*r*(*θs*,*θ*) can be expressed by:

$$
\Lambda\_r(\theta\_s, \theta) \approx \Lambda\_{0r} + \Lambda\_{1r} \cos \left[ Z\_r(\theta\_s - \theta) \right] \tag{9}
$$

The coefficients of the airgap permeance function Λ0*<sup>r</sup>* and Λ1*<sup>r</sup>* in Eq. (9) can be obtained using the conformal mapping method [28, 29]:

$$
\Lambda\_{0r} = \frac{\mu\_0}{\mathcal{g}'} \left( \mathbf{1} - \mathbf{1}.6 \beta \frac{b\_o}{t} \right) \tag{10}
$$

$$\mathbf{g'} = \mathbf{g} + h\_m/\mu\_r \tag{11}$$

$$\Lambda\_{1r} = \frac{\mu\_0}{\mathcal{g}'} \frac{4}{\pi} \beta \left[ 0.5 + \frac{\left(b\_o/t\right)^2}{0.78125 - 2\left(b\_o/t\right)^2} \right] \sin\left(1.6\pi \frac{b\_o}{t}\right) \tag{12}$$

$$\beta = 0.5 - \frac{1}{2\sqrt{\mathbf{1} + \left(\frac{b\_o}{2t}\frac{t}{g'}\right)^2}}\tag{13}$$

**Figure 8.** *Schematic of single-side salient structure on rotor.*

where *b*<sup>o</sup> is the rotor slot opening width and *t* is the rotor slot pitch, as shown in **Figure 8**. Combining Eq. (1), Eqs. (5–13), the no-load airgap flux density *B*(*θs*,*θ*) can be finally calculated as:

$$B(\theta\_s, \theta) = \sum\_{i=1,3}^{\infty} B\_i \sin \left[ \left( \frac{iZ\_s}{2} \pm Z\_r \right) \theta\_s - Z\_r \theta \right] \tag{14}$$

where the magnitude *Bi* is

where *FC* is:

where the magnitude *Fi* is

*FC* <sup>¼</sup> *Brhm μrμ*<sup>0</sup>

X∞ *i*¼1, 3, 5

<sup>1</sup> þ �ð Þ<sup>1</sup> *<sup>i</sup>*þ<sup>1</sup>

Then, the next step is to derive the specific airgap permeance Λ(*θs*,*θ*) in Eq. (1). Since the stator slotting effect has already been considered in Eqs. (5–8), the specific airgap permeance Λ(*θs*,*θ*) can be replaced by the airgap permeance with smoothed stator and slotted rotor Λ*r*(*θs*,*θ*). The model of smoothed stator and slotted rotor is

The coefficients of the airgap permeance function Λ0*<sup>r</sup>* and Λ1*<sup>r</sup>* in Eq. (9) can be

*<sup>g</sup>*<sup>0</sup> <sup>1</sup> � <sup>1</sup>*:*6*<sup>β</sup> bo*

<sup>0</sup>*:*<sup>78125</sup> � <sup>2</sup>ð Þ *bo=<sup>t</sup>* <sup>2</sup>

" #

*<sup>β</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>5</sup> � <sup>1</sup> 2

� �

*Fi* sin *iZs*

2 *θs* � �

<sup>2</sup> sin *<sup>i</sup><sup>π</sup>*

Λ*<sup>r</sup> θ<sup>s</sup>* ð Þ , *θ* ≈Λ0*<sup>r</sup>* þ Λ1*<sup>r</sup>* cos½ � *Zr*ð Þ *θ<sup>s</sup>* � *θ* (9)

*t*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *bo* 2*t t g*0

*g*<sup>0</sup> ¼ *g* þ *hm=μ<sup>r</sup>* (11)

sin 1*:*6*<sup>π</sup> bo*

� �<sup>2</sup> <sup>r</sup> (13)

*t* � �

� � � �

<sup>2</sup> *SO*

Then, it can be written in Fourier series as follows:

*Direct Torque Control Strategies of Electrical Machines*

*Fi* <sup>¼</sup> <sup>4</sup> *π* 1 *i Brhm μ*0*μ<sup>r</sup>*

shown in **Figure 8**. Then, the Λ*r*(*θs*,*θ*) can be expressed by:

obtained using the conformal mapping method [28, 29]:

<sup>Λ</sup>1*<sup>r</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup> *g*0 4

*Schematic of single-side salient structure on rotor.*

**Figure 8.**

**74**

<sup>Λ</sup>0*<sup>r</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

*<sup>π</sup> <sup>β</sup>* <sup>0</sup>*:*<sup>5</sup> <sup>þ</sup> ð Þ *bo=<sup>t</sup>* <sup>2</sup>

*FPM*ð Þ¼ *θ<sup>s</sup>*

(6)

(7)

(8)

(10)

(12)

$$B\_i = \frac{1}{2} F\_i \Lambda\_{1r}, \ i = 1, 3, 5 \dots \tag{15}$$

#### **3.3 Slot-pole combinations**

As can be seen in Eq. (14), the number of pole pairs in the air gap flux density is *iZs*/2 � *Zr*, *i* = 1,3,5 … Then, in order to make the flux density induce EMF in the armature windings, the pole pair number of the armature windings *P* should be equal to *iZs*/2 � *Zr*, *i* = 1,3,5 … Besides, for three phase symmetry, the winding pole pair number must also meet the following requirement:

$$\frac{Z\_s}{GCD(Z\_s, P)} = 3k, \ k = 1, 2, 3 \dots \tag{16}$$

All in all, the slot-pole combination of three-phase FRPMMs is ruled by the following equation:

$$\begin{aligned} P &= \min \left\{ P = \frac{iZ\_s}{2} \pm Z\_r; \quad \frac{Z\_s}{GCD(Z\_s, P)} = 3k \right\} \\ i &= 1, 3, 5 \dots \quad k = 1, 2, 3 \dots \end{aligned} \tag{17}$$

where min means to select the minimum number of these qualified harmonic orders so as to obtain a maximal pole ratio of FRPMMs. Therefore, the feasible slotpole combinations can be summarized as **Table 2**. Non-overlapping windings (i.e., concentrated windings) are usually used in FRPMMs because of the higher fault tolerance and easier manufacture than regular overlapping windings. However, some FRPMMs are suggested to employ overlapping windings in order to have a larger winding factor and thus a higher torque density. Therefore, both winding factors, that is, *kwn* (using non-overlapping winding) and *kwr* (using overlapping winding) are calculated for each FRPMM so as to see the difference of using different winding types.

#### **3.4 Torque equation**

Once the stator winding pole pair is selected, the stator flux linkage can be deduced using winding function theory, just as mentioned in Eq. (2). The winding function *N*(*θs*) in Eq. (2) can be written as:

$$N(\theta\_t) = \sum\_{i=1,3,5}^{\infty} N\_i \cos\left(i P \theta\_t\right) \tag{18}$$

$$N\_i = \frac{2}{i\pi} \frac{N\_s}{P} k\_{wi} \tag{19}$$


**Table 2.** *Slot-pole combinations of three-phase FRPMM.*

where *Ni* is the *i*th harmonics of the winding function and *kwi* is the winding factor of the *i*th harmonics. As can be seen in Eq. (17), the pole pair number is *iZs*/2 � *Zr* (*i* = 1,3,5 … ). So, the sum or difference of any two pole pair harmonics *Pi*<sup>1</sup>

j j¼ *Pi*<sup>1</sup> � *Pi*<sup>2</sup> *kZs*, *k* ¼ 1, 2, 3 …

Therefore, all the flux density harmonics are tooth harmonics of each other, that is, they have the same absolute values of winding factors, and their absolute wind-

Then, combining Eq. (2), Eq. (3), Eqs. (18–21), the back-EMF can be finally

X∞ *i*¼1

�1, winding factor of ð Þ *iZs=*<sup>2</sup> � <sup>Z</sup>*<sup>r</sup>* th harmonic equals � *kw*<sup>1</sup>

Since the reluctance torque of FRPMM is negligible, the electromagnetic torque

X∞ *i*¼1

So far, the general torque equation has been obtained as Eq. (24), but in this equation, some parameters such as *Bi*, *Iph* cannot be determined in the initial design stage of FRPMMs, so it is desirable that Eq. (24) can be transformed to a combination of several basic parameters, such as electric loading, magnetic loading, which

As known for electrical machines, the electric loading *Ae* can be written as:

*Ae* <sup>¼</sup> <sup>6</sup>*NsIph* 2 ffiffi 2 <sup>p</sup> *<sup>π</sup>rg*

Then, the equivalent magnetic loading of three-phase FRPMM *Bm* is defined as:

sgn <sup>∗</sup> *Bi iZs* <sup>2</sup> � *Zr*

under *id* = 0 control can be expressed as Eq. (4). Then, combining Eq. (4) and

*kwPi*<sup>1</sup> j j¼ *kwPi*<sup>2</sup> j j¼ *kw*<sup>1</sup> (21)

� �*=<sup>P</sup>* (22)

� �*=<sup>P</sup>* (24)

� �*=<sup>P</sup>* (26)

*<sup>g</sup> lstkkwZrAeBm* (27)

sgn <sup>∗</sup> *Bi iZs* <sup>2</sup> � *Zr*

sgn <sup>∗</sup> *Bi iZs* <sup>2</sup> � *Zr* (20)

(23)

(25)

*Pi*<sup>1</sup> ¼ *i*1*Zs=*2 � *Zr Pi*<sup>2</sup> ¼ *i*2*Zs=*2 � *Zr*

and *Pi*<sup>2</sup> is a multiple of stator slot number, that is,

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

*Flux Reversal Machine Design*

8 ><

>:

ing factor equals the fundamental winding factor *kw*1:

*Eph* ¼ 2*ωmrg lstkNsZrkw*<sup>1</sup>

Eq. (22), the average torque *Te* is able to be calculated as:

can be easily determined in the initial design stage.

*Te* ¼ 3*Iphrg lstkNsZrkw*<sup>1</sup>

*Bm* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

So, the torque expression in Eq. (24) can be rewritten as:

*Te* <sup>¼</sup> ffiffi 2 <sup>p</sup> *<sup>π</sup><sup>r</sup>* 2

*i*¼1

sgn <sup>¼</sup> 1, winding factor of ð Þ *iZs=*<sup>2</sup> � <sup>Z</sup>*<sup>r</sup>* th harmonic equals *kw*<sup>1</sup>

obtained as:

where

**77**

(

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

where *Ni* is the *i*th harmonics of the winding function and *kwi* is the winding factor of the *i*th harmonics. As can be seen in Eq. (17), the pole pair number is *iZs*/2 � *Zr* (*i* = 1,3,5 … ). So, the sum or difference of any two pole pair harmonics *Pi*<sup>1</sup> and *Pi*<sup>2</sup> is a multiple of stator slot number, that is,

$$\begin{cases} P\_{i1} = i\_1 Z\_s / 2 \pm Z\_r \\ P\_{i2} = i\_2 Z\_s / 2 \pm Z\_r \\ |P\_{i1} \pm P\_{i2}| = kZ\_s, \ k = 1, 2, 3 \dots \end{cases} \tag{20}$$

Therefore, all the flux density harmonics are tooth harmonics of each other, that is, they have the same absolute values of winding factors, and their absolute winding factor equals the fundamental winding factor *kw*1:

$$|k\_{wP\_{i1}}| = |k\_{wP\_{i2}}| = k\_{w1} \tag{21}$$

Then, combining Eq. (2), Eq. (3), Eqs. (18–21), the back-EMF can be finally obtained as:

$$E\_{\rm ph} = 2a\rho\_m r\_g l\_{stk} N\_s Z\_r k\_{w1} \sum\_{i=1}^{\infty} \text{sgn} \, \ast \, \frac{B\_i}{\left(\frac{iZ\_r}{2} \pm Z\_r\right)/P} \tag{22}$$

where

$$\text{sgn} = \begin{cases} \text{ 1, winding factor of } (i\mathbf{Z}\_{\text{s}}/2 \pm \mathbf{Z}\_{\text{r}})^{\text{th}} \text{ harmonic equals } k\_{w1} \\ -\text{1, winding factor of } (i\mathbf{Z}\_{\text{s}}/2 \pm \mathbf{Z}\_{\text{r}})^{\text{th}} \text{ harmonic equals } -k\_{w1} \end{cases} \tag{23}$$

Since the reluctance torque of FRPMM is negligible, the electromagnetic torque under *id* = 0 control can be expressed as Eq. (4). Then, combining Eq. (4) and Eq. (22), the average torque *Te* is able to be calculated as:

$$T\_{\epsilon} = \Im l\_{ph} r\_{\bar{g}} l\_{\imath k} N\_{\imath} Z\_r k\_{w1} \sum\_{i=1}^{\infty} \text{sgn} \, \* \, \frac{B\_i}{\left(\frac{iZ\_r}{2} \pm Z\_r\right)/P} \tag{24}$$

So far, the general torque equation has been obtained as Eq. (24), but in this equation, some parameters such as *Bi*, *Iph* cannot be determined in the initial design stage of FRPMMs, so it is desirable that Eq. (24) can be transformed to a combination of several basic parameters, such as electric loading, magnetic loading, which can be easily determined in the initial design stage.

As known for electrical machines, the electric loading *Ae* can be written as:

$$A\_{\epsilon} = \frac{\mathfrak{G} N\_s I\_{ph}}{2\sqrt{2}\pi r\_{\mathfrak{g}}} \tag{25}$$

Then, the equivalent magnetic loading of three-phase FRPMM *Bm* is defined as:

$$B\_m = \sum\_{i=1}^{\infty} \text{sgn} \ast \frac{B\_i}{\left(\frac{iZ\_r}{2} \pm Z\_r\right)/P} \tag{26}$$

So, the torque expression in Eq. (24) can be rewritten as:

$$T\_e = \sqrt{2}\pi r\_g^2 l\_{stk} k\_w Z\_r A\_e B\_m \tag{27}$$

*Zs*

**76**

6

*P* *SPP*

*PR* *kwn* *kwr*

> 12

*P* *SPP*

*PR* *kwn* *kwr*

> 18

> *P*

*SPP*

*PR* *kwn* *kwr*

PS:

kwn *and* kwr *are fundamental*

**Table 2.** *Slot-pole* 

*combinations*

 *of three-phase*

 *FRPMM.*

Non-overlapping

 winding is

 *winding factors calculated based on* 

0.902

 0.866

 0.945 recommended.

*non-overlapping*

 *winding type and* 

*recommended*

 *winding types, respectively.*

 0.945

 1

 0.945

 0.96

 0.96

 0.945

> Other:

Overlapping

 winding is

recommended.

 1

 0.945

 0.945

 0.866

 0.902

0.902

 0.866

 0.735

 0.617

 0.5

 0.492

 0.167

 0.167

 0.492

 0.5

 0.617

 0.735

 0.866

 0.902

2/7

 0.5

 0.8

 1.25

 2

 3.5

 8

 10

 5.5

 4

 3.25

 2.8

 2.5

 16/7

3/7

 0.5

 0.6

 0.75

 1

 1.5

 3

 3

 1.5

 1

 0.75

 0.6

 0.5

 3/7

7

 6

 5

432

0.866

0.866

0.5

0.5

4

1

0.5

2

1

1

*Zr*

**2**

 **3**

 **4**

1

4

0.5

1

1

2

0.5

1

 0.966

 0.25

 5

 2

21

 0.866

 0.866

 2.5

 0.5

12

 **5**

 **6**

 **7**

0.5

3.5

0.866

0.866

2

7

0.25

0.966

 1

 1

 1

234

 0.866

 0.933

 0.5

 0.866

 0.933

 4

 2.5

 2.2

 1

 0.5

 0.4

1245

 1

 1

 0.866

 0.5

 0.5

 0.866

 8

 10

 5.5

 1

 1

 0.5

 **8**

 **10** 2112

 **11**

 **12**

 **13**

0.5

6.5

0.866

0.866

0.4

2.6

0.933

0.933

 0.866

 5

 6

 7

 0.866

 3.5

 0.5

54

 1

 0.5

 14

 1

21

 **14**

 **15**

 **16**

 1 1

16

0.5

1

 2

*Direct Torque Control Strategies of Electrical Machines*

1

8

0.5

1

Thus, the rotor volume *Vr*, which equals *πlstkr* 2 *g*, can be obtained:

$$V\_r = \frac{T\_\epsilon}{\sqrt{2}k\_w Z\_r A\_\epsilon B\_m} \tag{28}$$

and then the airgap radius *rg* and the stack length *lstk* can be derived as:

$$r\_{\mathfrak{g}} = \sqrt[3]{V\_r/(\pi k\_{lr})}\tag{29}$$

$$l\_{stk} = \sqrt[3]{V\_r k\_{lr}^2 / \pi} \tag{30}$$

*kw*\**Zr*\**Bm*. Since the machine volume and PM usage are kept the same, the equivalent magnet loading *Bm* is mainly determined by the pole ratio (PR). So, the variation trend of torque is similar to that of *kw*\**Zr*\*PR. It can be seen that the torque achieves the maximum value when the rotor slot number is 8, 14, and 21 for 6 stator slots, 12 stator slots, and 18 stator slots, respectively. When the recommend windings are used, which means that the winding factor are maximized, the main factor that affects the torque is the *Zr*\*PR*.* As shown in **Table 2**, the variation of PR is irregular, hence the variation of torque with rotor slot number is irregular. As can be seen, for recommended winding types, the torque achieves the maximal value when the rotor slot number is 8, 10 and 17 for 6 stator slots, 12 stator slots, and 18

*Effect of combinations of stator slots and rotor slots on torque: (a) non-overlapping windings; (b) recommended*

As shown in Eq. (27), the airgap radius *rg* is also very important for the output torque. **Figure 10** investigates the effect of optimal rotor slot number *Zr* at different *rg*. For 6, 12, 18 stator slots, their rotor slot numbers are selected as 8, 14, and 21,

stator slots, respectively.

*Effect of* rg *on optimal* Zr *when split ratio is 0.6.*

**Figure 9.**

*Flux Reversal Machine Design*

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

*windings.*

**Figure 10.**

**79**

*4.1.2 Influence of airgap radius on average torque*

where *klr* is the aspect ratio, equals to the ratio of *rg* to *lstk*. It can be found in Eq. (27) that the key parameters affecting the torque density are the airgap radius *rg*, stack length *lstk*, winding factor *kw*, rotor slot number *Zr*, electric loading *Ae,* and equivalent magnetic loading *Bm*, among which the stack length *lstk* can be determined by the volume requirement, and winding factor *kw* is approximate to 1. So, the remaining parameters *rg*, *Zr*, *Ae*, *Bm* should be determined at the initial stage of the design process. Thus, the influences of the above key parameters on important performances, such as average torque, pulsating torque, power factor, PM demagnetization performance, will be investigated in the following parts.

### **4. Influence of design parameters on key performances**

#### **4.1 Average torque performances**

#### *4.1.1 Influence of slot-pole combinations on average torque*

As aforementioned, the rotor slot number *Zr* is one of key parameters that should be determined in the first design stage. How to determine the rotor slot number is a question. In this part, the influence of *Zr* on the torque performance will be investigated, giving instruction on how to select *Zr*. The parameters of the FRPMM models are listed in **Table 3**. These parameters are kept the same for the FRPMMs in order to have a reasonable comparison of their torque performance. That is to say, the airgap radius *rg*, stack length *lstk*, and electric loading *Ae* are the same.

**Figure 9** shows the influence of rotor slot number on the output torque when non-overlapping windings and recommended windings are used respectively. For **Figure 9(a)**, when non-overlapping windings are adopted, the average torque is mainly related to the product of winding factor and rotor slot number, that is,


#### **Table 3.**

*Parameters of the three-phase FRPMM models.*

**Figure 9.**

Thus, the rotor volume *Vr*, which equals *πlstkr*

*Direct Torque Control Strategies of Electrical Machines*

2

*Vr* <sup>¼</sup> *Te* ffiffi 2 <sup>p</sup> *kwZrAeBm*

and then the airgap radius *rg* and the stack length *lstk* can be derived as:

*lstk* ¼

netization performance, will be investigated in the following parts.

**4. Influence of design parameters on key performances**

*4.1.1 Influence of slot-pole combinations on average torque*

**4.1 Average torque performances**

same.

**Table 3.**

**78**

*Parameters of the three-phase FRPMM models.*

*rg* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Vr=*ð Þ *πklr*

where *klr* is the aspect ratio, equals to the ratio of *rg* to *lstk*. It can be found in Eq. (27) that the key parameters affecting the torque density are the airgap radius *rg*, stack length *lstk*, winding factor *kw*, rotor slot number *Zr*, electric loading *Ae,* and equivalent magnetic loading *Bm*, among which the stack length *lstk* can be determined by the volume requirement, and winding factor *kw* is approximate to 1. So, the remaining parameters *rg*, *Zr*, *Ae*, *Bm* should be determined at the initial stage of the design process. Thus, the influences of the above key parameters on important performances, such as average torque, pulsating torque, power factor, PM demag-

As aforementioned, the rotor slot number *Zr* is one of key parameters that should be determined in the first design stage. How to determine the rotor slot number is a question. In this part, the influence of *Zr* on the torque performance will be investigated, giving instruction on how to select *Zr*. The parameters of the FRPMM models are listed in **Table 3**. These parameters are kept the same for the FRPMMs in order to have a reasonable comparison of their torque performance. That is to say, the airgap radius *rg*, stack length *lstk*, and electric loading *Ae* are the

**Figure 9** shows the influence of rotor slot number on the output torque when non-overlapping windings and recommended windings are used respectively. For **Figure 9(a)**, when non-overlapping windings are adopted, the average torque is mainly related to the product of winding factor and rotor slot number, that is,

**Parameter Value Parameter Value** Stator outer diameter 170 mm Stator inner diameter 105 m Stator slot opening ratio 0.25 Remanent permeability 1.065 Stack length 100 mm PM thickness 2.5 mm Series turns per phase 80 Airgap length 0.5 mm Rotor slot opening ratio 0.65 Rated current 5.3A Rated speed 600 rpm Magnet remanence 1.21 T

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Vrk*<sup>2</sup> *lr=π* <sup>3</sup> q

*g*, can be obtained:

p<sup>3</sup> (29)

(28)

(30)

*Effect of combinations of stator slots and rotor slots on torque: (a) non-overlapping windings; (b) recommended windings.*

**Figure 10.** *Effect of* rg *on optimal* Zr *when split ratio is 0.6.*

*kw*\**Zr*\**Bm*. Since the machine volume and PM usage are kept the same, the equivalent magnet loading *Bm* is mainly determined by the pole ratio (PR). So, the variation trend of torque is similar to that of *kw*\**Zr*\*PR. It can be seen that the torque achieves the maximum value when the rotor slot number is 8, 14, and 21 for 6 stator slots, 12 stator slots, and 18 stator slots, respectively. When the recommend windings are used, which means that the winding factor are maximized, the main factor that affects the torque is the *Zr*\*PR*.* As shown in **Table 2**, the variation of PR is irregular, hence the variation of torque with rotor slot number is irregular. As can be seen, for recommended winding types, the torque achieves the maximal value when the rotor slot number is 8, 10 and 17 for 6 stator slots, 12 stator slots, and 18 stator slots, respectively.

#### *4.1.2 Influence of airgap radius on average torque*

As shown in Eq. (27), the airgap radius *rg* is also very important for the output torque. **Figure 10** investigates the effect of optimal rotor slot number *Zr* at different *rg*. For 6, 12, 18 stator slots, their rotor slot numbers are selected as 8, 14, and 21,

respectively. Moreover, non-overlapping windings are used in these models because non-overlapping winding is simple and has the same end winding length. It can be seen that when the airgap radius is small, the optimal rotor slot number is small. This is because when the airgap radius is small, the leakage flux between adjacent rotor teeth occupies a large percent, so the optimal rotor slot number should be small to reduce the leakage flux as much as possible. When the airgap radius gets larger and larger, the leakage flux decreases gradually. Hence, the optimal rotor slot number increases.

Then, keeping the stator outer diameter as a constant, that is, 170 mm, the effects of airgap radius of average torque are analyzed in **Figure 11**. It indicates that when the airgap radius increases, the output torque goes up. This is because the torque is proportional to the square of airgap radius. The larger the airgap radius, the higher the torque. However, the torque is not only influenced by the airgap radius, but also the electric loading *Ae*. With the increase of airgap radius, the inner diameter of the stator increases, and thus the slot area decreases, leading to the decrease of winding turns per slot and the electric loading. Therefore, as the airgap radius keeps increasing, the output torque decreases afterwards.

#### *4.1.3 Influence of magnetic loading and equivalent electric loading on average torque*

In addition to the rotor slot number *Zr*, airgap radius *rg*, the rest of key parameters affecting the torque in Eq. (27) are the electric loading *Ae* and the equivalent magnetic loading *Bm*. **Figure 12** analyzes the influence of *Ae* and *Bm* on the average torque at different stator slot number. For these models, the airgap radius is fixed as 55 mm and their rotor slot number is chosen as their corresponding optimal value. Also, non-overlapping windings are adopted. As can be seen, the output torque increases with the electric loading. This reason is very simple, that is, a larger current, a higher torque. But for the equivalent magnetic loading, the variation trend of torque does not monotonically increase with the equivalent magnetic loading. This is due to the saturation effect of the iron core. Moreover, it can be seen that the knee point of the equivalent magnet loading increases with the stator slot number. Since the winding pole pair of the 18-stator-slot FRPMM is 6, which is larger than 1-winding-pole-pair of the 6-stator-slot and 4-winding-pole-pair of the

12-stator-slot FRPMM, the stator iron of the 18-stator-slot FRPMM is less likely to

*Effect of equivalent magnetic loading and electric loading on torque: (a)* Zs *= 6; (b)* Zs *= 12; (c)* Zs *= 18.*

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

saturate than the others.

*Flux Reversal Machine Design*

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

**Figure 12.**

**81**

**4.2 Pulsating torque performances**

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

**Figure 11.** *Effect of* rg *on torque when stator outer diameter is 170 mm.*

respectively. Moreover, non-overlapping windings are used in these models because non-overlapping winding is simple and has the same end winding length. It can be seen that when the airgap radius is small, the optimal rotor slot number is small. This is because when the airgap radius is small, the leakage flux between adjacent rotor teeth occupies a large percent, so the optimal rotor slot number should be small to reduce the leakage flux as much as possible. When the airgap radius gets larger and larger, the leakage flux decreases gradually. Hence, the optimal rotor slot

Then, keeping the stator outer diameter as a constant, that is, 170 mm, the effects of airgap radius of average torque are analyzed in **Figure 11**. It indicates that when the airgap radius increases, the output torque goes up. This is because the torque is proportional to the square of airgap radius. The larger the airgap radius, the higher the torque. However, the torque is not only influenced by the airgap radius, but also the electric loading *Ae*. With the increase of airgap radius, the inner diameter of the stator increases, and thus the slot area decreases, leading to the decrease of winding turns per slot and the electric loading. Therefore, as the airgap

*4.1.3 Influence of magnetic loading and equivalent electric loading on average torque*

In addition to the rotor slot number *Zr*, airgap radius *rg*, the rest of key parameters affecting the torque in Eq. (27) are the electric loading *Ae* and the equivalent magnetic loading *Bm*. **Figure 12** analyzes the influence of *Ae* and *Bm* on the average torque at different stator slot number. For these models, the airgap radius is fixed as 55 mm and their rotor slot number is chosen as their corresponding optimal value. Also, non-overlapping windings are adopted. As can be seen, the output torque increases with the electric loading. This reason is very simple, that is, a larger current, a higher torque. But for the equivalent magnetic loading, the variation trend of torque does not monotonically increase with the equivalent magnetic loading. This is due to the saturation effect of the iron core. Moreover, it can be seen that the knee point of the equivalent magnet loading increases with the stator slot number. Since the winding pole pair of the 18-stator-slot FRPMM is 6, which is larger than 1-winding-pole-pair of the 6-stator-slot and 4-winding-pole-pair of the

radius keeps increasing, the output torque decreases afterwards.

*Direct Torque Control Strategies of Electrical Machines*

number increases.

**Figure 11.**

**80**

*Effect of* rg *on torque when stator outer diameter is 170 mm.*

**Figure 12.** *Effect of equivalent magnetic loading and electric loading on torque: (a)* Zs *= 6; (b)* Zs *= 12; (c)* Zs *= 18.*

12-stator-slot FRPMM, the stator iron of the 18-stator-slot FRPMM is less likely to saturate than the others.
