1 Introduction

As the demand for wind power generators has steadily increased over the years, generators with higher power and torque densities are required. The size of a generator can be reduced using a gearbox that lowers the required torque of the generator and increases the speed. However, gearbox failure is a common reason for wind turbine failure and requires preventative and periodic maintenance. Moreover, the mechanical loss in the gearbox reduces the efficiency of the wind turbine. Recently, direct-driven generators without gearboxes have been widely studied and developed for wind power generation to solve the aforementioned problems. In direct-driven generators, a higher torque density than that of geared generators is required at low speeds because the high torque of the wind turbine is transmitted directly [1, 2]. Accordingly, high-voltage generating systems, winding methods for improving the high slot filling factor, and new topologies utilizing flux modulation effects have been extensively studied to improve torque density [3,4,5].

Due to their higher torque density at low speeds than permanent-magnet synchronous machines (PMSMs), permanent-magnet vernier machines (PMVMs) are widely used in direct-driven applications such as in-wheel type electric vehicles, collaborative robots with robotic joints, and wind turbines without gearboxes [6,7,8]. Compared with PMSMs, PMVMs produce a higher torque at the same size by utilizing both the fundamental and harmonic components of the air-gap magnetic flux distributions (flux modulation effects). However, PMVMs have certain drawbacks, such as high rotor iron loss and low power factor, that result in lower efficiency and higher requirement for inverter capacity [9,10,11]. These disadvantages can be overcome by applying fractional slot and distributed winding methods to PMVMs [12].

Vernier machines are classified as synchronous machines integrated with magnetic gears such as flux reversal, flux switching, and transverse-flux machines [13]. Among magnetic-geared synchronous machines, PMVMs are preferred because of their simple and robust structures with high torque at low speeds. The high torque density characteristics of the PMVMs at low speeds were inspected using an air-gap permeance function and a power equation and subsequently verified using finite element analysis [14].

To improve the performance in terms of torque, power density, power factor, and torque ripple, new topologies in PMVMs have been investigated, including multilayer stators and rotors [11, 15, 16], consequent poles [17], Halbach array magnets [18], flux modulation poles (FMPs) [19], and flux barriers [20,21,22]. Despite the usefulness of these new topologies, they result in structural complexity, low manufacturability, and high cost. Although consequent poles can be effectively used to reduce the number of permanent magnets in the PMVMs, performance degradation is inevitable owing to the reduced number of permanent magnets. Halbach-array magnets and flux barriers can be used to reduce torque ripple in a PMVM. However, these magnets and flux barriers can only slightly increase the torque density of the PMVMs. Therefore, the aforementioned topologies are difficult to apply to a direct-driven wind power generator that requires a high torque density. FMPs can be effectively used to increase the gear ratio and torque density of a PMVM. However, low power factor and structural complexity are major drawbacks of the FMP.

Investigating pole-slot combinations by evaluating winding methods such as integer and fractional slots, concentrated and distributed windings, slot pitches, and winding layers is an alternative approach for enhancing the performance of PMVMs. The winding method significantly affects the electromagnetic characteristics and performance of PMVMs. In prior studies [23, 24], various winding methods and their effects on PMVMs were investigated and simulated using finite element analysis. However, few studies have been conducted on the fractional slot distributed winding PMVMs (FSDW PMVMs) compared to fractional slot concentrated winding PMVMs (FSCW PMVMs) because the distributed winding increases the height of the coil end owing to overlapping between coils. The high coil-end parts of the motors cause an increase in the size of the system. However, in recent years, as various studies and mass production of hairpin windings have been implemented, the disadvantages of the distributed winding can be overcome. Consequently, the coil end can be reduced and slot filling factor can be increased. This can lead to an increase in demand for a distributed winding motor that is applied to a hairpin. Thus, distributed winding, which is mainly applied to integer slots, requires research to apply to fractional slot motors.

In this study, eight models were designed for direct-driven wind power generation, including FSCW PMSMs, FSCW PMVMs, FSDW PMVMs, and FSCW PMVMs with FMP. For the high torque required in the wind turbine, each model was selected to define criteria for a high winding factor of more than 0.9 among various pole-slot combinations. In addition, to focus on comparing the effects of the winding methods, all models were unified in the number of effective slots as 24. Subsequently, the electromagnetic characteristics and their performance in terms of torque density, torque ripple, iron and permanent-magnet (PM) losses, and power factor were studied and compared using finite element analysis. Finally, the best model for direct-driven wind power generators (FSDW PMVMs) was proposed in terms of torque density, efficiency, and power factor [25].

2 Flux Modulation Principle of PMVM

2.1 Air-Gap Permeance Method

The air-gap permeance method analyzes the characteristics of PMVMs by calculating the flux density in the air gap using the PM magnetomotive force (MMF) and slot performance. This method does not reflect the effects of nonlinear characteristics, iron loss, and PM eddy current loss owing to magnetic field saturation. This is widely used as it can explain the driving principle of a PMVM using an equation [14]. With the essential harmonic components, the permeance distribution in the air gap, \(P\left(\theta \right)\), and MMF of the PM, \({F}_{PM}\left(\theta ,{\theta }_{m}\right)\), are expressed as (1) and (22), respectively.

$$P\left(\theta \right)\approx {P}_{0}-{P}_{{Z}_{s}}cos({Z}_{s}\theta )$$
(1)
$${F}_{PM}\left(\theta ,{\theta }_{m}\right)\approx {F}_{PM1}cos[{Z}_{r}\left(\theta -{\theta }_{m}\right)]$$
(2)

Here \({Z}_{r}\) denotes the number of rotor magnet pole pairs, and \({Z}_{s}\) denotes the effective slot number based on the geometry of the stator. \({P}_{0}\) denotes the average value; \({P}_{{Z}_{s}}\) denotes the \({Z}_{s}\) th harmonic component; \({F}_{PM1}\) denotes the MMF amplitude produced by the rotor PMs; and \(\uptheta\) and \({\theta }_{m}\) denote the MMF position and mechanical angle of the rotor, respectively.

The air-gap flux density, \(B\left(\uptheta \right)\), is obtained using the product of (1) and (2), as expressed in (3), where \({B}_{PM0}\) denotes the MMF of the PM, and \({B}_{PM1}\) denotes the space harmonic component of the air-gap magnetic flux density. The rotational directions of the two terms are generated.

$$\begin{aligned} B\left( {\theta ,\theta_{m} } \right) & = F_{PM} \left( {\theta ,\theta_{m} } \right)P\left( \theta \right) \\ & \approx B_{PM0} cosZ_{r} \left( {\theta - \theta_{m} } \right) \\ & - B_{PM1} \left[ {\begin{array}{*{20}c} {cos\left[ {\left( {Z_{r} - Z_{s} } \right)\theta - Z_{r} \theta_{m} } \right]} \\ {cos\left[ {\left( {Z_{r} + Z_{s} } \right)\theta - Z_{r} \theta_{m} } \right]} \\ \end{array} } \right] \\ \end{aligned}$$
(3)
$${B}_{PM0}={F}_{PM1}{P}_{0}$$
(4)
$${B}_{PM1}={\frac{1}{2}F}_{PM1}{P}_{{Z}_{s}}$$
(5)

The coil flux linkage, \({\lambda }_{ph}\left({\theta }_{m}\right)\), of the PMVM is expressed by (6), where \({r}_{g}\) and \({d}_{g}\) denote the air-gap radius and diameter, respectively. \({N}_{ph}\) denotes the number of series turns per phase, \({L}_{stk}\) denotes the stack length, \(q\) denotes the slots/phase/pole, \({\theta }_{ph}\) denotes the position of each phase, \({\alpha }_{T}\) denotes the slot pitch, and \({\theta }_{c}\) denotes the coil pitch. \({k}_{w0}\) and \({k}_{w1}\) are the fundamental and modulated harmonic winding factor components, respectively, that are replaced by \({\theta }_{c}\) terms. For a conventional PMSM, the \({B}_{PM1}\) term is not modulated and occurs as a harmonic component of the back electromotive force (EMF) and torque.

$$\begin{aligned} \lambda _{{ph}} \left( {\theta _{m} } \right) & = \frac{{r_{g} N_{{ph}} L_{{stk}} }}{q}\mathop \sum \limits_{{i = 0}}^{{q - 1}} \mathop \int \limits_{{\theta _{{ph}} + i\alpha _{T} }}^{{\theta _{{ph}} + i\alpha _{T} + \theta _{c} }} B\left( {\theta ,\theta _{m} } \right)d\theta \\ & = \frac{{d_{g} N_{{ph}} l_{{st}} }}{Q}\mathop \sum \limits_{{i = 0}}^{{q - 1}} \left| {~\frac{{B_{{PM0}} }}{{Z_{r} }}\sin \left[ {Z_{r} \left( {\theta - \theta _{m} } \right)} \right]} \right. \\ & - \frac{{B_{{PM1}} }}{{Z_{r} - Z_{s} }}\sin \left[ {\left( {Z_{r} - Z_{s} } \right)\left( {\theta - Z_{r} \theta _{m} } \right)} \right] \\ & \quad - \frac{{B_{{PM1}} }}{{Z_{r} - Z_{s} }}\sin \left[ {\left( {Z_{r} + Z_{s} } \right)\left( {\theta - Z_{r} \theta _{m} } \right)} \right]|_{{\theta _{{ph}} + i\alpha _{T} }}^{{\theta _{{ph}} + i\alpha _{T} + \theta _{c} }} \\ & = d_{g} N_{{ph}} L_{{stk}} (\frac{{B_{{PM0}} }}{{Z_{r} }}k_{{w0}} \mp \frac{{B_{{PM1}} }}{{Z_{r} - Z_{s} }}k_{{w1}} )\sin \left[ {Z_{r} \theta _{m} \mp \left( {Z_{r} - Z_{s} } \right)\theta _{{ph}} } \right] \\ \end{aligned}$$
(6)

2.2 Combination of Poles and Slots

To generate the flux modulation effect, the cases in which interaction occurs according to \({Z}_{r}\) and \({Z}_{s}\) should be calculated to design the winding method, as given in (7).

$${Z}_{s}-{Z}_{r} =p$$
(7)

When the condition for the relationship between \({Z}_{r}\) and \({Z}_{s}\) is unequal, as in (9), the \({B}_{PM1}\) terms are activated by the number of winding pole pairs, \(p\), in (3), which represents the flux modulation effect [9]. Hence, additional pole-slot combinations are obtained. The \(B\left(\theta ,{\theta }_{m}\right)\) frequency relation of each term represents the gear ratio, as expressed in (8).

$${G}_{r}={Z}_{r}/p$$
(8)

For PMSMs, \(p\) is made equal to \({Z}_{r}\) to synchronize the speed and generate the output torque. By employing various techniques for tuning each parameter presented in (9), PMVMs of various pole–slot combinations with \({G}_{r}\) greater than 1 can be designed.

3 Analysis and Design of PMVM

3.1 Configuration of Pole-Slot Combinations

The design is carried out to determine the stacking length that delivers the target performance, satisfying the conditions of the current density and outer diameter presented in Table 1 and a winding coefficient greater than 0.9. The models that satisfy the condition are classified into the winding method, utilization of the vernier effect and FMP, and gear ratio, resulting in FSCW PMSM, FSCW PMVM, FSDW PMVM, and FSCW PMVM with FMP. The PMSM and PMVM have the following relationships between the pole and slot.

Table 1 Design requirements of a wind power generator machine
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First, FSCW PMSMs are 24-slot models without an FMP and can apply p as 11 or 13 according to the number of slots. FSCWs with p values of 11 and 13 are denoted as FSCW-A and FSCW-B, respectively. In addition, they are synchronously operated because the pole pairs of the rotor and stator are identical.

Second, FSCW PMVMs are also 24-slot models without FMP and can apply \(p\) as 11 or 13 according to the number of slots owing to FSCW-A and FSCW-B. Additionally, if we select the pole pair of the rotor by applying (9), a flux modulation effect is generated.

Third, FSDW PMVMs using FSDW-A and FSDW-B reduce \(p\) by applying to 24 slots stator without an FMP. Because the fifth- and seventh-harmonic amplitudes of the armature MMF are high, \({Z}_{r}\) can be 17 or 19, and the resultant gear ratio is higher.

Finally, FSCW PMVMs with FMP determine the number of effective slots according to the number of FMPs per tooth. As \({Z}_{s}\) is twice the number of slots in the case of one FMP per tooth, 24 effective slots were designed by applying FMP to the 12 slots for comparison under identical conditions. If FSCW-C is applied to 12 slots, the third, fifth, and seventh are activated, of which only the fifth and seventh are orders with a winding factor greater than 0.9. Thus, as in the FSDW PMVM, \({Z}_{r}\) can be 19 and 17, and \({G}_{r}\) can be selected as high.

Five winding methods were used: FSCW-A (p = 10), FSCW-B (p = 11), FSDW-A (p = 7), FSDW-B (p = 5), and FSCW-C (p = 5, 7). The flux density distributions with only the stator and winding methods are shown in Fig. 1. Figure 1a, b, and e depict concentrated windings with one slot pitch, and (c) and (d) depict distributed windings with two slot pitches. This clearly shows the working harmonics through flux linkage of the effective winding MMF.

Fig. 1
figure 1

PMVM configurations with U-phase winding and magnetic flux density distribution: a FSCW-A (p = 10); b FSCW-B (p = 11); c FSDW-A (p = 7); d FSDW-B (p = 5); and e FSCW-C with FMP (p = 5, 7)

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3.2 Winding MMF Harmonic Analysis

Figure 2 shows the air-gap flux density spectra from the winding MMF and slotting effect. After selecting \(p\), which is greater than 0.9 of the winding factor, Zr was determined through a fast Fourier transform (FFT) analysis of the winding MMF. For the FSCW-A method, \(p\) was selected as the 10th, where \({Z}_{r}\) was expected to be 10 and 14. For the FSCW-B method, \(p\) was selected as the 11th, where the expected \({Z}_{r}\) values were 11 and 13. Using (9), the possible pole-slot combinations of the PMVM were observed to be 28–24 and 26–24, whereas the PMSM combinations were 20–24 and 22–24. Similarly, \(p\) was prominently selected as the seventh in FSDW-A and fifth in FSDW-B. \({Z}_{r}\) was determined to be 17 and 19. In the case of a distributed winding, the winding MMF waveforms were amplified as the coils of each phase overlap. The possible pole–slot combinations of the PMVM were verified to be 34–24 and 38–24. The number of FMP structures applied to a tooth determines \({Z}_{s}\). If one FMP is applied to the tooth, then \({Z}_{s}\) is twice the number of slots. For an equivalent comparison of the designed models, the last combinations were made of 24 effective slots from 12-slot with one FMP. By applying the FSCW-C to the 12-slot stator, the third, fifth, and seventh winding MMFs were engaged. Among them, the orders of the pole-slot combination with the winding factor condition were the fifth and seventh. Thus, the pole-slot combinations of PMVM were possible with 38–12 and 34–12.

Fig. 2
figure 2

Winding MMF spectra by the winding method

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3.3 Design Specification of Wind Power Generator Machine

The main requirements of direct-driven wind power generator machines are listed in Table 1. The size that generated 236 Nm at the rated power was limited to 115 mm of stack length, at 260 mm of outer diameter. The specifications of the eight models designed according to the constraints are listed in Table 2. Although the FSDW PMVM is on the side with a low winding factor, the stack length limit from M-2 to M-8 is satisfied. A high \({G}_{r}\) generates a higher torque density among the designed models with an equivalent current density and specific operating point. Although M-7 has \({G}_{r}=2.4\) and \({k}_{w0}=0.933\), the low amplitude of the seventh MMF results in a low torque density. Compared with the FSCW PMSM, the stack length of the FSDW PMVM was reduced by a maximum of 23%. The number of turns was tuned to the voltage limit without derating.

Table 2 Parameters of PMSM and PMVM according to pole–slot combinations
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4 Results and Discussion

4.1 Back-EMF and Torque Characteristics Analysis

The amplitude of the back-EMF, \({\varepsilon }_{ph}\), in (6) has a working harmonic, including the modulated harmonic term given by.

$$\varepsilon_{ph} \approx d_{g} N_{ph} L_{stk} (B_{PM0} k_{w0} \mp \frac{{Z_{r} }}{{Z_{r} - Z_{s} }}B_{PM1} k_{w1} )cos\left[ {Z_{r} \omega_{m} t \mp \left( {Z_{r} - Z_{s} } \right)\theta_{ph} } \right].$$
(9)

When the current phase, which is equivalent to the back-EMF phase, is applied, the average torque, \(\tau\), of the PMVM is expressed as follows:

$$\tau = \frac{{\varepsilon_{ph} i_{ph} }}{{\omega_{m} }} \propto \left( {B_{PM0} k_{w0} \mp \frac{{Z_{r} }}{{Z_{r} - Z_{s} }}B_{PM1} k_{w1} } \right)I_{\max } .$$
(10)

Figure 3 shows the FEA result graph in terms of the open-circuit and on-load. At the rated load, the phase difference of the coil flux linkage was caused by the modulation of the stator MMF. In Fig. 3b, c, the on-load flux linkage peak points of the high \({G}_{r}\) models moved far from the 180-degree point of the rotor position. Models designed to generate the same torque by applying a specific current density generally have a reduced size and low back-EMF according to \({G}_{r}\). The \({Z}_{r}-{Z}_{s}\) and \({Z}_{r}+{Z}_{s}\) order harmonics did not affect the torque ripple that tended to decrease as \({G}_{r}\) increased. These results verified the validity of the analysis, as listed in Table 3.

Fig. 3
figure 3

FEA results: a open-circuit flux linkage; b open-circuit back-EMF waveforms; c flux linkage on-load; and d electromagnetic torque at rated condition

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Table 3 Performance of models through the FEA results at rated operating point
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4.2 Loss Analysis

The cases of the FSDW PMVM and PMSM with FMP have ample modulated asynchronous harmonics that cause considerable rotor losses. Therefore, the rotor loss must be analyzed. The PM eddy and core iron losses are expressed as follows:

$$P_{eddy} = \frac{{D^{3} \cdot L_{stk} \cdot T \cdot \omega_{m}^{2} \cdot B_{mag,k}^{2} \cdot \sigma }}{24} \cdot \left( {1 - \frac{192}{{\pi^{5} }} \cdot \frac{D}{L} \cdot \mathop \sum \limits_{n = 0}^{\infty } \frac{{tanh\left( {\frac{{\left( {2n + 1} \right)\pi L}}{2D}} \right)}}{{\left( {2n + 1} \right)^{5} }}} \right),$$
(11)
$${P}_{core}=\sum_{k}\left({A}_{e}{f}_{k}^{2}{B}_{core,k}^{2}+{A}_{h}{f}_{k}{B}_{core,k}^{2}\right),$$
(12)

where \(D\), \(T\), and \(\upsigma\) denote the diameter, thickness, and conductivity of the PMs, respectively. \({\upomega }_{m}\) denotes the rotational speed of the \(k\) order harmonic. \({B}_{mag,k}\) and \({B}_{core,k}\) denote the \(k\) order harmonic amplitudes of the PM and core flux, respectively. \({A}_{e}\) and \({A}_{h}\) denote the coefficients of the eddy current loss and hysteresis loss, respectively. \({f}_{k}\) denotes the \(k\) order harmonic frequency. The flux density amplitude and frequency of each harmonic have a significant effect on the PM eddy loss.

Figure 4 depicts the total losses by FEA and efficiency at rated power. The machines designed to generate high torque at low speed have a large proportion of copper loss. With an increase in the \({G}_{r}\), the rotor loss and PM eddy loss increased owing to modulated harmonics. The PM eddy losses of the M-5 and M-8 were approximately four and eleven times higher than that of M-1, respectively. Therefore, the efficiencies were reduced by 81% and 90%, respectively. Compared to the FSCW-FMP PMVM having an equivalent \({G}_{r}\), the FSDW PMVM had low losses except copper loss, and thus had high efficiency. In addition, the prospect of improving the PM eddy loss existed. Despite low-speed operation, the PM eddy loss of the PMVM was very high. This should be considered in the design process using ferrite magnets or the manufacturing technique of split neodymium magnets.

Fig. 4
figure 4

FEA results of electromagnetic loss and efficiency

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4.3 Efficiency Characteristics

The efficiency tendency graph based on the rotational speed at the rated torque is shown in Fig. 5. Due to the loss of the rotor, the efficiency tendency of \({G}_{r}\) varied according to the speed. The peak points of the efficiency gradually moved to a low speed depending on the \({G}_{r}\). The efficiency of the FSCW machine increased with the rotation speed. However, that of the FSDW and FMCW PMVM with FMP (from M-5 to M-8) had peak values in the range of 200 r/min to 300 r/min. This result proved that a PMVM with a high \({G}_{r}\) was advantageous for low-speed operations in terms of efficiency.

Fig. 5
figure 5

Efficiency tendency (at rated torque) by rotation speed

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4.4 Power Factor Characteristics

The phasor diagram of the PMVM is illustrated in Fig. 6, where \({\varphi }_{wa}\) and \({\varphi }_{na}\) denote the working and nonworking armature flux linkages, respectively. \({\varphi }_{\sigma }\) and \({\varphi }_{m}\) represent the leakage flux and PM flux linkage, respectively. \({E}_{0}\), \(V\), and \({\omega }_{r}\) denote the phasors of the back-EMF, phase voltage, and rotation speed of the PMVM, respectively [22].

Fig. 6
figure 6

Phasor diagram of PMVM

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The phasor diagram shows the relationship between the power factor and output density of the PMVM. The gray area represents the working flux density that produces the torque. However, the high amplitude of the working harmonic can enhance the power factor. As shown in Fig. 3c, the phase of the flux linkage on the load shifted with increasing torque density. The power factor, \(PF\), can be expressed as in (13).

$$PF=cos\theta =\frac{{E}_{0}}{V}={\left(1+{\left(\frac{{\omega }_{r}({\varphi }_{wa}+{\varphi }_{na}+{\varphi }_{\sigma })}{{\omega }_{r}{\varphi }_{m}}\right)}^{2}\right)}^{-\frac{1}{2}}$$
(13)

Each model was tuned for the number of turns close to the voltage limit; therefore, \(V\) was at the same level. The magnitude of the back-EMF, presented in Table 3, determines the power factor. The relationship between the torque density and power factor is shown in Fig. 7. The torque density increased by up to 23%, whereas the power factor decreased by up to 40%. Among them, the power factor of M-5 of FSDW PMVM decreased by only 12%.

Fig. 7
figure 7

FEA Results of torque density and power factor

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5 Conclusion

In this study, both FSCW and FSDW PMVMs were investigated in terms of gear ratio, torque density, electromagnetic losses, and power factor. FSCW PMVMs have been widely used because of their high power factor and structural simplicity. Because the gear ratio of FSCW PMVMs cannot be higher than that of PMVMs with FMP and FSDW PMVMs, achieving a high torque density performance is difficult. FSDW PMVMs and FSCW PMVMs with FMP had higher torque densities than FSCW PMVMs, and FSDW PMVMs had a higher power factor than FSCW PMVMs with FMP. Therefore, we concluded that FSDW PMVMs were more suitable for direct-driven wind power generators than FSCW PMSMs, FSCW PMVMs, and PMVMs with FMP, in terms of high torque density and power factor. Finally, in the case of distributed winding, although the structure is simple, the slot filling factor is rather low and the coil end length is higher; therefore, considerable research has not been conducted compared to other PMVMs. However, by applying a hairpin to distributed winding, the low slot filling factor and high coil end can be alleviated to a certain extent; therefore, FSDW PMVMs are expected to have a higher torque density while maintaining a high power factor.