Characteristic analysis of flux switching PM machine with various pole and slot combinations

Abstract

This paper examines the influence of pole and slot combinations on the electromagnetic performance of the flux-switching permanent magnet (FSPMs) machine. For this purpose, first the production of air gap flux density in machine analytically discussed. Next, five different pole and slot combination are selected by using the knowledge of air-gap flux density expression. The performances of the different combinations are analyzed and compared using the finite element analysis (FEA), and optimal pole and slot combination is suggested. It is found that the pole and slot combination has a dominant effect on machine performance and needs to be considered during the design stage.

Keywords

Flux switching permanent magnet machine air gap flux density pole and slot combination performance analysis

1. Introduction

In recent decades, flux-switching permanent magnet (FSPM) motor has gained significant attention due to vigorous rotor structure, high torque density, and good heat dissipation capabilities. FSPM motor has a similar structure as a conventional machine but uses a different working principle. Since all the excitation sources are placed on the stator, which provides excellent mechanical stability. FSPM motor applies the modulation flux and is a right candidate for direct-drive wind turbines where low speed and high-torque is required.

The history of FSPM machine back over 50 years [1], and more recently, this topic is selected by many researchers worldwide. The FSPMs machines accomplished many goals [2]. The analytical expressions of the FSPMs machine are derived using lumped magnetic reluctance networks [3], and the nature of back electromotive force (EMF) production and geometrical parameters that influence the machine performance are discussed in [4]. The performance of flux switching machine along with field winding and permanent magnet are compared in [5]. In reference [6], the torque production phenomena based on spatial harmonic was addressed. The irreversible demagnetization characteristics of magnets in FSPMs machines is described [7], and provide a guideline for improvement techniques.

However, all these prior investigations were restricted to the FSPM machines with 12 stator slots and 10 rotor pole numbers although other combinations can be possible for this machine. In this paper, first, the nature of the air-gap flux density production of an FSPM machine is examined. Next, using the air gap flux density information, the five different slot-pole combinations, 24-slots/20-poles, 12-slots/10-poles, 12-slots/8-poles, 6-slots/5-poles, 6-slots/4-poles are designed to analyze the back-EMF characteristics in detail with emphasis on the dominant influence of pole-slot combinations. The comparative study establishes that the choice of slot and pole has a meaningful effect on the performance and the better performance can be attained by selecting the proper slot and pole combination. The comparison draws the conclusion that the 6-slot/5-pole FSPM machine has a higher back-EMF. The experimental validation with different prototypes is difficult at this stage; therefore, the performance comparison is made by using a two-dimensional finite element analysis (FEA).

2. Air gap flux density of an FSPM machine

In the topology of FSPMs machine, the stator core is divided, and magnets are sandwiched between the core. The armature windings are installed in stator slots by overlapping the permanent magnets. Since all the excitation sources are housed in the stator side and rotor is mechanically stable. The structure of FSPMs machine is shown in Fig. 1. The negative and positive flux linkage is obtained when the rotor pole is aligned with the stator pole, called flux-switching action.

Fig. 1.

Topology of flux-switching PMs machin.

Fig. 2.

The waveforms when θ_m = 0° and 3° showing the movements.

Table 1

Magnitude and speed of working harmonics of Air gap flux density

Magnitude	Speed	Pole pairs
${\displaystyle \frac{g}{{\mu}_{0}}}F_{m}{\Lambda}_{r1}\left({\Lambda}_{s0}-{\displaystyle \frac{{\Lambda}_{s1}}{2}}\right)$	$\frac{Z_{r}}{\|Z_{m}-Z_{r}\|}}{\omega}_{m$	\|Z_m − Z_r \|
${\displaystyle \frac{g}{{\mu}_{0}}}F_{m}{\Lambda}_{r1}\left({\Lambda}_{s0}-{\displaystyle \frac{{\Lambda}_{s1}}{2}}\right)$	$\frac{Z_{r}}{\|Z_{m}+Z_{r}\|}}{\omega}_{m$	\|Z_m + Z_r \|
${\displaystyle \frac{g}{{\mu}_{0}}}F_{m}{\Lambda}_{r1}\left({\displaystyle \frac{{\Lambda}_{s1}}{2}}\right)$	$\frac{Z_{r}}{\|3Z_{m}+Z_{r}\|}}{\omega}_{m$	\|3Z_m + Z_r \|
${\displaystyle \frac{g}{{\mu}_{0}}}F_{m}{\Lambda}_{r1}\left({\displaystyle \frac{{\Lambda}_{s1}}{2}}\right)$	$\frac{Z_{r}}{\|3Z_{m}-Z_{r}\|}}{\omega}_{m$	\|3Z_m − Z_r \|

The air gap magnetic flux density due to magnets can be computed as the product of airgap permeance function and Magnet Magnetomotive force (MMF). If the magnetic potential of the stator core pieces is considered constant, the MMF expression can be given as $\begin{eqnarray}\displaystyle F_{g}({\theta})=F_{m}\cos (Z_{m}{\theta}) & & \displaystyle\end{eqnarray}$ (1) where F_m, Z_m, and θ are the MMF magnitude in airgap, the number of magnet pole pairs and the angular position in the air gap, respectively. The FSPMs machine has double saliency because of rotor teeth and stator slot; therefore, the total permeance can be achieved by obtaining the stator side permeance and rotor side permeance and combining the two permeances [8]. The permeance because of stator slots is given in (2), when the reference angle is chosen at the center of stator slot. The permeance as seen from the stator looking into rotor teeth is given in (3). The “+” sign represents the location of the reference axis at the center of rotor teeth. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Lambda}_{s}({\theta})={\Lambda}_{s0}-{\Lambda}_{s1}\cos (Z_{s}{\theta})\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Lambda}_{r}({\theta},{\theta}_{m})={\Lambda}_{r0}+{\Lambda}_{r1}\cos Z_{r}({\theta}-{\theta}_{m})\end{eqnarray}$ (3) where 𝛬_s0 and 𝛬_s1 are the average permeance and fundamental term due to stator slots, similarly 𝛬_r0 and 𝛬_r1 are the average permeance and fundamental term because of rotor slots, and these coefficients can be determined with the geometry of machine [8]. Herein, Z_s, Z_r, and θ_m are the number of stator slots, number of rotor poles and rotor position. The total permeance is estimated using the method of double saliency structure [8] and given in (4). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\Lambda}_{g}({\theta},{\theta}_{m})\approx {\displaystyle \frac{g{\Lambda}_{s}({\theta}){\Lambda}_{r}({\theta}-{\theta}_{m})}{{\mu}_{0}}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \approx \underbrace{{\displaystyle \frac{g}{{\mu}_{0}}}{\Lambda}_{s0}{\Lambda}_{r0}}_{G_{0}}-\underbrace{{\displaystyle \frac{g}{{\mu}_{0}}}{\Lambda}_{r0}{\Lambda}_{s1}\cos (Z_{s}{\theta})}_{G_{1}({\theta})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \underbrace{+{\displaystyle \frac{g}{{\mu}_{0}}}{\Lambda}_{r1}\left\{{\Lambda}_{s0}\cos Z_{r}({\theta}-{\theta}_{m})-{\displaystyle \frac{{\Lambda}_{s1}}{2}}(\cos ((Z_{s}+Z_{r}){\theta}-Z_{r}{\theta}_{m})+\cos ((Z_{s}-Z_{r}){\theta}+Z_{r}{\theta}_{m}))\right\}}_{G_{2}.({\theta},{\theta}_{m})}\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$ (4) The air gap magnetic flux density is the product of the magnet MMF F_g(θ) and the airgap permeance function 𝛬_g(θ, θ_m) as given in (5). $\begin{eqnarray}\displaystyle B_{g}({\theta},{\theta}_{m})\text{} & =\text{} & \displaystyle F_{g}({\theta}){\Lambda}_{g}({\theta},{\theta}_{m})\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle F_{g}({\theta})\{G_{0}-G_{1}({\theta})+G_{2}({\theta},{\theta}_{m})\}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle \underbrace{G_{0}F_{g}({\theta})-G_{1}({\theta})F_{g}({\theta})}_{B_{stat.}({\theta})}+\underbrace{G_{2}({\theta},{\theta}_{m})F_{g}({\theta})}_{B_{mov.}({\theta},{\theta}_{m})}.\end{eqnarray}$ (5) The (5) illustrates that the air gap flux density of an FSPMs machine consists of the three term, since the first two terms G₀F_g(θ) and G₁(θ)F_g(θ) does not depend on the rotor movement θ_m, they are denoted as B_stat.(θ). By using the general relation Z_s = 2Z_m for conventional FSPMs machine , B_stat.(θ) can be summed as $\begin{eqnarray}\displaystyle B_{stat.}({\theta})\text{} & {\approx}\text{} & \displaystyle G_{0}F_{g}({\theta})-G_{1}({\theta})F_{g}({\theta})\nonumber\\ \displaystyle \text{} & {\approx}\text{} & \displaystyle {\displaystyle \frac{g}{{\mu}_{0}}}{\Lambda}_{r0}F_{m}\left(\left({\Lambda}_{s0}-{\displaystyle \frac{{\Lambda}_{s1}}{2}}\right)\cos (Z_{m}{\theta})-{\displaystyle \frac{{\Lambda}_{s1}}{2}}\cos (3Z_{m}{\theta})\right)\end{eqnarray}$ (6) The third term G₂(θ, θ_m)F_g(θ) denoted as B_mov.(θ, θ_m) is dependent on the rotor position θ_m and concise in (7). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{mov.}({\theta},{\theta}_{m})=G_{2}({\theta},{\theta}_{m})F_{g}({\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \approx {\displaystyle \frac{g}{{\mu}_{0}}}F_{m}{\Lambda}_{r1}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \left\{\begin{array}{@{}l@{}}\left({\Lambda}_{s0}-\frac{{\Lambda}_{s1}}{2}\right)\cos ((Z_{m}-Z_{r}){\theta}+Z_{r}{\theta}_{m})+\left({\Lambda}_{s0}-{\displaystyle \frac{{\Lambda}_{s1}}{2}}\right)\cos ((Z_{m}+Z_{r}){\theta}-Z_{r}{\theta}_{m})\\ -{\displaystyle \frac{{\Lambda}_{s1}}{2}}\cos ((3Z_{m}+Z_{r}){\theta}-Z_{r}{\theta}_{m})-{\displaystyle \frac{{\Lambda}_{s1}}{2}}\cos ((3Z_{m}-Z_{r}){\theta}+Z_{r}{\theta}_{m})\end{array}\right\}\end{eqnarray}$ (7) The expression (5) shows that the term B_stat.(θ) is the stationary flux density and term B_mov.(θ, θ_m) is the rotating flux density of an FSPM machine. Thus, the contribution to the back-Emf production is only by the rotating flux density B_mov.(θ, θ_m). To clarify the concept, both waves are plotted for each instance θ_m =0° and 3° in Fig. 2, and it is evident that the only rotating flux wave B_mov.(θ, θ_m) is moving with the rotor.

Fig. 3.

Conception of winding and fastest flux wave.

3. Selection of different pole and slot combinations

As aforementioned, the expression (5) shows that the air gap flux density of an FSPMs machine comprises of two flux waves; stationary and rotating flux wave, while only rotating flux wave B_mov.(θ, θ_m) will produce the back-Emf. Therefore, for an FSPM machine, one must focus on the B_mov.(θ, θ_m) term, the magnitude and speed of each working harmonic component of the air gap flux density are given in Table 1.

Table 2
Pole and Slot Combinations Z_m − Z_r = −p

Number of stator Number of magnet Number of rotor Winding pole

slots (Z_s) pole pairs (Z_m ) poles (Z_r ) pairs p

Machine I 24 12 20 8

Machine II 12 6 10 4

Machine III 12 6 8 2

Machine IV 6 3 5 2

Machine V 6 3 4 1

	Number of stator	Number of magnet	Number of rotor	Winding pole
Machine I	24	12	20	8
Machine II	12	6	10	4
Machine III	12	6	8	2
Machine IV	6	3	5	2
Machine V	6	3	4	1

Table 3

Specifications of the FE analysis

Parameters	Value	Parameters	Value
Stator inner diameter	406 [mm]	Turns per phase	200 [Turns]
Stator outer diameter	620 [mm]	Air-gap length	2 [mm]
Rotor outer diameters	402 [mm]	Total Magnet area	12420(mm)²
Stack length	50 [mm]	Magnet type	Neodymium (NdFe)

Fig. 4.

Structures of Flux switching PM machines and Phase back-EMF. (a) 24-slots/20-poles. (b) 12-slots/10-poles. (c) 12-slots/8-poles. (d) 6-slots/5-poles. (e) 6-slots/4-poles. (f) Phase back-Emf waveforms (g) Phase Back-Emf Spectra.

From Table 1, it can be seen that |Z_m − Z_r| pole pair has the most significant magnitude and fastest speed among these coefficients, which means this main flux component will offer the maximum contribution to the generation of backEMF. Consequently, to achieve better performance it is essential to place a winding considering this flux component.

For the slot concentrated three phase winding, the three slots constitute one winding pole pair p as shown in Fig. 3. In order to get the maximum flux linkage from main |Z_m − Z_r| flux component, it is necessary to apply the same number of winding pole pair, so the relation in (8) can be attained. $\begin{eqnarray}\displaystyle Z_{m}-Z_{r}=-p. & & \displaystyle\end{eqnarray}$ (8) The relation in (8) illustrates that there is over one combination can be utilized to achieve the main flux component, rotating flux and back-EMF. On the other hand, the previous study focuses on the 12-slots/10-poles combinations that have the Z_m =6 and p = 4. Thus, from the equation in (8), the five different combinations are selected as shown in Table 2, to investigate the suitable combination for an FSPMs machine.

The choice of the slot and pole is made up considering expression (8) and the practical possibility for an FSPMs machine. The selection of magnet pole pairs Z_m is directly related to the number of stator slots (Z_s = 2Z_m) and winding pole pairs. For instance, for the machine IV and V, the number of Z_m is 3, it should be noted that Z_m cannot be selected 4 or 5 because it will make an even number of stator slots and three phase slot concentrated winding is not possible in this fashion. Moreover, when Z_m is 3, the possible winding pole pairs p is 1 or 2 in 6 slots, thus; to fulfill the expression (8) condition only the 4 or 5 number of rotor pole can be possible as given in Table 2.

4. Simulations and analysis

To investigate the influence of different slot and pole combination, the five combinations 24-slots/20-poles, 12 slots/10-poles, 12-slots/8-poles, 6-slots/5-poles, 6-slots/4-poles are designed. In order to check the effect of pole and slot combination on machine performance it is necessary to make a comparison on fair conditions.

Therefore, the total magnet area, turn per phase and dimensions including the air gap, stator outer diameter and stack length are designed with the same specifications in all structures. The detailed specifications are given in Table 3. In addition, the ratio of rotor slot opening to slot pitch has important effect on the machine performance and maximum back-EMF can be achieved when this ratio is 0.6 [5]. Therefore; the ratio of rotor slot opening to slot pitch is kept the 0.6 in all structures, in order to examine the effect of slot pole combination. Moreover, the double-layer slot concentrated winding has been employed.

At no-load condition, the phase back-EMF and spectra of all five structures are calculated at 1200 rpm by finite element analysis (FEA) and shown in Fig. 4. From the results, it can be seen that the 6-slot/5-pole FSPM machine has better performance than the commonly used 12-slot/10-pole structure. The investigation shows that higher slot-pole combination decreases the magnet thickness which in results MMF drop in the air gap. Besides, the higher slot-pole reduces the stator teeth width that increases the flux concentration in teeth and cause saturation that weaken the back-EMF. Therefore, the combination having small number of slots and poles is better for the FSPMs machine. The cogging torque is caused by interaction between magnets and poles so a large number of slot and pole combination will produce more cogging torque.

5. Conclusions

In this paper, the investigation of conventional FSPMs with different slot and pole combination is carried out, and the better combination is proposed. To accomplish this, first, a simplified analytical model is developed. From the expression of flux density, it is found that FSPMs machine has two flux waves in the air gap, while only one flux wave is rotating with rotor position and contribute to the back-EMF. The rotating flux wave comprises several flux components, however, the main contribution due to |Z_m − Z_r| pole pair and winding should be assigned considering this flux component. The previous studies used 12 slots and 10 poles to achieve this main flux component, but there are five more combinations has been applied to get this flux wave and back-EMF. It is found that the 6-slot/5-pole combination has the best electromagnetic performance for FSPMs machine. The selection of slot and pole combination is a very significant factor during the design stage of an electrical machine and this paper provides the approach to select the optimal slot and pole combination for FSPMs machine.

Footnotes

Acknowledgements

This work was supported in part by the Korea Electric Power Corporation (R18XA04) and in part by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under Grant NRF-2016R1A6A1A03013567.

Conflict of interest

The authors declare no conflict of interest.

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Magnitude	Speed	Pole pairs
${\displaystyle \frac{g}{{\mu}_{0}}}F_{m}{\Lambda}_{r1}\left({\Lambda}_{s0}-{\displaystyle \frac{{\Lambda}_{s1}}{2}}\right)$	$\frac{Z_{r}}{\|Z_{m}-Z_{r}\|}}{\omega}_{m$	\|Z_m − Z_r \|
${\displaystyle \frac{g}{{\mu}_{0}}}F_{m}{\Lambda}_{r1}\left({\Lambda}_{s0}-{\displaystyle \frac{{\Lambda}_{s1}}{2}}\right)$	$\frac{Z_{r}}{\|Z_{m}+Z_{r}\|}}{\omega}_{m$	\|Z_m + Z_r \|
${\displaystyle \frac{g}{{\mu}_{0}}}F_{m}{\Lambda}_{r1}\left({\displaystyle \frac{{\Lambda}_{s1}}{2}}\right)$	$\frac{Z_{r}}{\|3Z_{m}+Z_{r}\|}}{\omega}_{m$	\|3Z_m + Z_r \|
${\displaystyle \frac{g}{{\mu}_{0}}}F_{m}{\Lambda}_{r1}\left({\displaystyle \frac{{\Lambda}_{s1}}{2}}\right)$	$\frac{Z_{r}}{\|3Z_{m}-Z_{r}\|}}{\omega}_{m$	\|3Z_m − Z_r \|