Comparative Study of Dual PM Vernier Machines

: In this paper, two types of dual permanent magnet (PM) machines, i.e., stator slot dual-PM (SSDPM) machine and split-tooth dual-PM (STDPM) machine, are investigated and compared. Both machines have consequent pole structure with Halbach array PMs. Their difference lies in the position of stator PM. The SSDPM machine has Halbach array PMs in the stator slots, while the STDPM machine has PMs between the split teeth. Torque characteristics, i.e., average torques and torque ripples, of different slot/pole number combinations of the two machines are compared. The 24 stator slots/20 rotor slots/4 armature pole pair (24S20R4Pa) SSDPM machine with distributed windings and the 24 stator slots/10 rotor slots/4 armature pole pair (12S20R4Pa) STDPM machine with concentrated windings are compared under both open-circuit and on-load conditions. The results show that the SSDPM machine is more competitive by delivering higher torque density and higher power density.

Stator PM FM machines include FSPM machines, FRPM machines, and DSPM machines [10][11][12][13][14]. FSPM machines have better flux-focusing effect than the other two types of machines. Stator PM machines have easier heat management if a forced liquid cooling is employed. This means that the electric loading can be further increased to improve the torque density. Rotor PM FM machines mainly refer to VPM machines, which can produce high torque at low speed, albeit with poor power factor [15,16].
In this paper, two types of dual-PM machines, i.e., stator slot dual-PM (SSDPM) machine and split-tooth dual-PM (STDPM) machine, together with consequent pole rotors, are investigated and compared. On one hand, they can deliver very high torque. More importantly, since there is only one airgap, they have much simpler structures than dual-stator/dual-rotor machines. The working principle, slot/pole number combination, together with comparison of two dual-PM machines, will be discussed. Figure 1 shows the SSDPM machine with 24 stator slots/20 rotor slots/4 armature pole pair (24S20R4Pa) and split-tooth dual-PM (STDPM) machine with (12S20R4Pa).

Machine Topology
In [5], the dual-PM machine employs radially magnetized PMs in both the rotor slots and the stator slots. Article [20] improves the dual-PM machine by replacing the radially magnetized stator PMs with Halbach PMs. In this paper, to further enhance the torque density, dual Halbach PMs are employed in the SSDPM machine, as shown in Figure 1a. Both the rotor PM and stator PM are Halbach PM arrays.
The STDPM machines also employ Halbach PM array in the rotor slots, but with radially magnetized PMs between the split teeth [17].

Working Principle
For the dual-PM machines, the torque can be attributed to two parts: rotor PM and stator PM. The 12S11R1Pa SSDPM machine and 12S23R1Pa STDPM machine are taken as examples to explain the machine decomposition, as shown in Figures 2 and 3. The key design dimensions are shown in Figure 4. The design details are given in Tables 1 and 2.   On one hand, when the stator PMs are set as air, they become vernier PM machines and split-tooth vernier PM machines, and the slot/pole number combinations comply with the Equation (1) [31] and Equation (2) [32], respectively.
in which N s is the stator slot number, N r is the rotor slot number, P a is the armature pole pair number, n is the split tooth number which is larger than 1.
On the other hand, when the rotor PMs are set as air, they become flux reversal PM machines, and the slot/pole number combinations also comply with the above-mentioned equations. Hence, the slot/pole number combinations of dual-PM machines comply with 4 of 16 (1) and (2). To calculate the torque decomposition of dual-PM machines, linear material is employed, and the relative permeability is set as 50, considering the overload capability.
Before discussing detailed results, it should be mentioned that the torque, power, iron loss, power factor, back EMF, etc., are all normalized due to confidentiality. As shown in Figure 5, for the 12S11R1Pa SSDPM machine, the rotor PM machine and the stator PM machine produce 0.515 p.u. and 0.335 p.u. torque, respectively. The dual-PM machine delivers 1.7% lower torque than the sum of the rotor PM machine and the stator PM machine. This is mainly due to that the flux leakage increases when the rotor PM and stator PM are working together. Similarly, for the 12S23R1Pa STDPM machine, the dual-PM machine produces 1.5% lower torque than the sum of the corresponding rotor PM machine and stator PM machine, as shown in Figure 6.

Influence of Rotor Slot Number
As shown in (1) and (2), when the stator slot number and armature pole pair number are fixed, the rotor slot number can be chosen as N s + P a and N s − P a for SSDPM machine, and nN s + P a and nN s − P a for STDPM machine. To compare the effect of rotor slot number selection, 12S1Pa SSDPM machines with 11R and 13R, 12S1Pa STDPM machine with 23R and 25R are investigated, respectively, as shown in Figures 7 and 8.
For fair comparison, all the investigated machine topologies are globally optimized for maximum torque under the same stator outer diameter, stack length, copper loss (120 • C) (copper loss is set according to the thermal restriction). All the optimizations are based on Genetic Algorithm (GA), and 30 individuals in each population with 35 generations have been employed. Tables 1 and 3 list the key design dimensions. In addition, the end winding length is calculated by the following equation: in which y is the slot pitch, r 3 is the stator outer radius, y k is the stator yoke width, s h is slot height. Figures 9 and 10 show that when the rotor slot number is selected lower than the stator slot number, higher torque can be produced. More importantly, higher power factor, lower iron loss, and higher efficiency can also be achieved with lower rotor slot number, as listed in Table 4, due to lower electrical frequency.

Influence of Different Slot/Pole Number Combinations
Apart from the influence of stator yoke width, stator/rotor PM width, rotor diameter, stator/rotor PM height, for SSDPM machines with different slot/pole number combinations, the torques are mainly affected by pole ratio (pole ratio is an important index for flux modulation machines, and can be defined as N r /P a .) [33], flux leakage [33], end winding length, and winding factor. All the machine topologies are globally optimized for maximum torque under the aforementioned conditions, and the design details are all listed in Tables 1-3. Figure 11 shows the influence of stator slot number and armature pole pair number on the average torques of SSDPM machines. The peak torques of 12S, 18S, 24S, 30S, and 36S appear when the armature pole pair numbers are 2, 3, 2, 2, and 4, respectively. On one hand, these combinations have shorter end winding length compared with those having 1 armature pole pair number. On the other hand, they have higher pole ratio than those having larger armature pole pair number. The 12S7R5Pa SSDPM machine exhibits higher torque than the 12S8R4Pa one due to higher winding factor. It should be noted that the maximum torque values also increase with the increase of stator slot number, as shown in Figure 12. Dual-PM machines with stator slot number larger than 36 are not calculated because: (a) The increase of electrical frequency significantly affects other indexes, e.g., increasing the losses, decreasing the power factor, increasing the carrier frequency. All of these will deteriorate the overall performances. Figure 12 shows that the torque/electrical frequency decreases with the increase of stator slot number. The torque ripples of SSDPM machines are shown in Figure 13. The 12S8R4Pa, 24S16R8Pa, and 30S20R10Pa exhibit significantly higher torque ripple than other combinations. This is due to the fact that these combinations have the smallest least common multiple (LCM) of the rotor slot number and the stator slot number. Torques and torque ripples of STDPM machines are shown in Figures 14 and 15, respectively. Apart from the aforementioned influence factors, split-tooth number will increase the pole ratio, and thus affect the torque characteristics. When the stator slot number is 6, increasing the split tooth number will increase the torque. However, when the stator slot number is 12 and the armature pole number is 1, increasing the split tooth number does not increase the torque, due to the increased flux leakage, as shown in Figure 16.

Comparison of Stator Slot Dual-PM Machine and Split Tooth Dual-PM Machine
In this section, the 24S20R4Pa SSDPM machine with distributed windings and the 12S20R4Pa STDPM machine with concentrated windings, as shown in Figure 17, are compared. Employing split-tooth mainly helps to ease manufacturing and decrease end winding length. These two combinations are chosen for comparison since they have the same armature pole pair number, rotor tooth number, pole ratio, stator outer diameter, and stack length. The design details of these two machines are listed in Tables 1 and 2.  Figure 18 compares the torque waveforms of the two machines under the same copper loss (120 • C). The SSDPM machine can deliver 48% higher torque with 63.2% lower torque ripple compared with the STDPM machine. In actual operation, both DC voltage and phase current are limited. Here, space vector pulse width modulation (SVPWM) control strategy is employed. The turn number per slot is adjusted as 10. The parallel branch is adjusted to 2 and 1, for the SSDPM and STDPM machines, respectively. Figure 19 shows the flux density distributions of the two compared dual-PM machines at open circuit condition and load condition (I d = 0 and I q = I max ). For the SSDPM and STDPM machines, the maximum flux density appears in the tooth and the split teeth, respectively, no matter what the condition is. In addition, slightly higher flux density can be observed in the yoke of the SSDPM machine at load condition. Despite this, the SSDPM exhibits better overload capability, which will be shown later. Hence, for both machines, the iron core material is fully utilized. Figure 20 compares the back EMFs of the two dual-PM machines. The SSDPM machine exhibits 40.4% higher fundamental harmonic, albeit with much higher 5th harmonic. The STDPM machine has obvious 2nd harmonic due to unbalanced PM magnetomotive force (MMF). Figure 21 shows the cogging torques of the two compared machines. The cogging torque of the STDPM machine is 3.6 times larger than the SSDPM machine, which explains the higher torque ripple in the STDPM machine. For the SSDPM and STDPM machines, the fluctuation frequency in one electrical period is 6 and 3, respectively. The higher amplitude and lower fluctuation frequency of the cogging torque of the STDPM machine are mainly due to its smaller least common multiple (LCM) of the stator slots and the rotor poles. Figure 22 shows that the SSDPM machine produces 46% higher torque than the STDPM machine, together with 51.7% lower torque ripple. This is due to the fact that the rotor PM and stator PM in the SSDPM machine produce 26% and 122% higher torque than their counterparts, as shown in Figure 23. It can also be concluded that the stator PM machine with Halbach array PMs in the stator slots can deliver much higher torque than that with PMs between split teeth. Figure 24 shows that the SSDPM machine has better overload capability than the STDPM machine. The overload capability can be indicated by inductance. The former has smaller inductance than the latter. This is due to the fact that the inductance is inversely proportional to the length of the flux path and the SSDPM has a longer flux path, as shown in Figure 19a,b. The torques/powers versus speed are calculated according to [33], and shown in Figure 25. The two dual-PM machines have similar corner speed. The SSDPM machine can produce 50.3% higher power than its counterpart.     Table 5 compares the electromagnetic performances of the dual-PM machines. The SSDPM machine has higher torque, higher torque per PM volume, higher power factor, and higher efficiency. Hence, the SSDPM machine is preferred.

Conclusions
Two types of dual-PM machines, i.e., SSDPM machine and STDPM machine, are analyzed in terms of working principle, slot/pole number combinations, along with the comparison of electromagnetic performances. The results show that: (a) Dual-PM machines can be decomposed into stator PM machines and rotor PM machines that share the same slot/pole number combinations. (b) There exists one optimum armature pole pair number for maximum torque. (c) The SSDPM machines exhibits better electromagnetic performances than the STDPM machines.