In this section, electromagnetic performances of the 20-slot/22-pole double-layer winding configurations are investigated by FEA. Aiming to focus on the analysis of armature flux density, PMs are removed from the FEA models. The MMF distribution, air-gap flux density and their corresponding harmonic analysis of the double-layer winding configuration are shown in Figure 2.

As shown in Figure 2, the MMF distribution indicates that an obvious 2-pole harmonic exists in the air gap, which is consistent with the calculated FEA results, and it verifies that the MMF analysis method is effective in the qualitative analysis process of armature flux distribution. The harmonic analysis shows that the 2-pole, 18-pole, and 22-pole space harmonics are dominant harmonics, but only the 22-pole harmonic interacts with the 22-pole PM field and generates constant torque. Generally, the armature space harmonic which interacts with the PM field and produces constant torque is named fundamental harmonic, and the space harmonics whose pole numbers are less than that of the fundamental harmonic are called sub harmonics. For the 20-slot/22-pole windings, the 22-pole space harmonic is fundamental harmonic, the 2-pole and 18-pole harmonics are sub harmonics.

The sub harmonics induce eddy current losses in the PMs, and the eddy-current losses are determined by the relative velocity (versus the fundamental harmonic) and amplitude of the harmonic. The relative velocities of the harmonics are in inverse proportion to their orders. Thus, among all of the sub harmonics, the 2-pole sub harmonic has the highest relative velocity with respect to the fundamental harmonic (eleven-fold rotating speed of the machine), and the 18-pole sub harmonic has a relative low speed but high amplitude, so both 2-pole and 18-pole sub harmonics generate considerable effect on rotor losses. In following section we focus on the investigation of the 2-pole sub harmonic.

With only one phase activated, the MMF and air-gap flux density of single-layer windings are shown in Figure 3.

For the 20-slot single-layer winding configuration, there are only two coils of one phase, and they have opposite winding directions, so it is inevitable that a 2-pole harmonic exists in the air-gap MMF, and this is consistent with the FEA calculated air-gap flux density.

For the 20-slot double-layer winding configuration, there are four coils of one phase, the adjacent two coils have opposite winding directions, and one coil side of the adjacent two coils shares the same slot. With only one phase activated, the MMF of double-layer windings is shown in Figure 4.

From the MMF point of view, double-layer windings can be equivalently considered as superposition of two single-layer windings whose winding directions are opposite, as shown in Figure 4, so the 2-pole harmonics in the two equivalent single-layer windings will suppress each other, then the 2-pole harmonic in double-layer windings is reduced. Since the adjacent two coils are wound on different teeth, there is a certain phase shift between the two equivalent single-layer windings, so the 2-pole harmonic cannot be reduced to zero in conventional double-layer windings, which is consistent with the analysis results of Section 3.1.

From the above analysis, we know that the 2-pole sub harmonic in conventional double-layer winding configuration cannot be reduced to zero. For the 2-pole sub harmonic, the wave crest of the 2-pole harmonic locates in the center of Area 1, and the wave trough of the 2-pole harmonic locates in the center of Area 2, as shown in Figure 5a.

In order to reduce the 2-pole harmonic to zero, a “negative” 2-pole MMF is introduced to cancel the 2-pole sub harmonic, as shown in Figure 5b. Due to the introduced “negative” 2-pole MMF, the wave crest in Area 1 and wave trough in Area 2 can be canceled. The introduced “negative” 2-pole MMF can be realized by adopting unequal-turn winding topology, as shown in Figure 5c and Figure 6. Different from conventional winding configurations, the number of turns per coil side is different for the unequal-turn winding configuration. The manufacturing of unequal-turn winding is a little different from that of conventional winding, and the winding coils with different turns per coil side can be realized by winding only one side of the last turn. N₁ donates turn number of coil sides in the slots which contain coils of the same phase, N₂ donates turn number of coil sides in the slots which contain coils of different phases, and the relation between N₁ and N₂ is [31]: (1) N 2 = N 1 + 1 and 50 % ≤ N 1 / N 2 ≤ 100 %

The winding factor of the unequal-turn winding topology cannot be calculated using the classic formula like the conventional winding topology. An improved slot-vector diagram is developed to calculate the winding factor of the unequal-turn windings. Slot-vector diagram represents the electromotive force (EMF) induced in the conductors of each slot, and it is useful to study the harmonics of different orders, i.e., sub harmonics, fundamental harmonic, and high-order harmonics. For the studied 20-slot/22-pole winding configuration, the slot-vector diagram is shown in Figure 7a.

The resultant phasor of the fundamental harmonic is given by the sum of the phasors 1, −2, 3, −11, 12, −13, the amplitudes of the phasors 1, 3, 11, and 13 are all N₂, the amplitudes of both the phasors 2 and 12 are 2N₁. Winding factor of the fundamental harmonic equals the ratio between the amplitude of the resultant phasor and the sum of amplitude of the phasors 1, 2, 3, 11, 12, and 13.

The winding factor of other harmonics can also be investigated by the slot-vector diagram. Since the winding connection has been fixed by the fundamental harmonic (as shown in Figure 7a), the resultant phasor of the v-th harmonic is given by the sum of the phasors 1, −2, 3, −11, 12 and −13, as shown in Figure 7b. Winding factors of the other harmonics equal the ratios between the amplitude of the resultant phasor and the sum of amplitudes of the phasors 1, 2, 3, 11, 12 and 13. In fact, Figure 7b is a universal slot-vector diagram for calculating winding factors of the unequal-turn winding configuration. For example, when v = 11, Figure 7b is the same as Figure 7a; when v = 1, winding factor of the 2-pole sub-harmonic can be calculated.

Then winding factor for the unequal-turn winding topology can be described as: (2) k w ν = N 2 · cos ( π ⋅ ν / 10 ) − N 1 N 2 + N 1

The winding factors of the 2-pole harmonic and the fundamental harmonic vary with the ratio between turns of different coil sides, as shown in Figure 8.

It is shown that the 2-pole harmonic is reduced to zero at the optimal point with N₁/N₂ = 0.95, and the fundamental harmonic is reduced by 0.06%. Also the relation between N₁ and N₂ meets Equation (1), so N₁ is 19, and N₂ is 20.

With the optimum turn ratio of different coil sides, the winding factors and amplitudes of the air-gap flux densities for the unequal-turn windings are calculated, as shown in Table 2. The winding factors and amplitudes of the air-gap flux densities for conventional winding topologies are also listed.

For the single-layer winding configuration, the amplitude of the 2-pole harmonic is 1.56 times of the fundamental harmonic, and for the conventional double-layer winding configuration, the 2-pole harmonic reduces to 24% of the fundamental harmonic. With the optimum turn ratio of different coil sides, the 2-pole harmonic is cancelled in the unequal-turn winding configuration. The air-gap flux densities of the conventional double-layer and unequal-turn winding configuration are compared, as shown in Figure 9. The FEA results indicate that the lower-order harmonic with 2 poles is cancelled and the fundamental harmonic nearly unchanged, which verifies the effectiveness of the unequal-turn winding configuration on reducing the 2-pole sub harmonic.

For multiphase fault-tolerant PMSM, low mutual inductance between phases is a key requirement. In order to compare the mutual inductances of the unequal-turn winding configuration with those of the conventional winding topologies, the static magnetic fields of the conventional and unequal-turn winding configurations with PMs removed and only one phase windings activated (Phase A in this paper) are calculated, the flux distributions are shown in Figure 10.

For single-layer winding configuration, non-wound teeth are named fault-tolerant teeth, the flux linkage in the fault-tolerant teeth is leakage flux of Phase A which links through slot openings. Since the flux which is activated by Phase A does not link to the other four phase windings, the magnetic isolation between different phases is quite effective, and it meets the low mutual inductance demand of fault-tolerant PMSM.

For double-layer winding configuration, all teeth are wound, the slot-opening flux leakage of Phase A couples with Phases C and D, so it can be predicable that the mutual inductances of double-layer winding configuration are larger than those of single-layer winding configuration. And for double-layer winding configuration, the mutual inductances between Phase A and Phases C/D are larger than those between Phase A and Phases B/E.

For the unequal-turn winding configuration, parts of the flux of Phase A couples with Phases C and D, the coupled flux composes of slot-opening flux leakage and flux linked through the air gap, and there is only slot-opening flux leakage coupled with other phases in conventional double-layer winding configuration, so it can be predicable that the mutual inductances of the unequal-turn winding configuration are larger than those of the conventional double-layer winding configuration.

The inductance parameters of the conventional and unequal-turn winding configurations are calculated by FEA, as shown in Table 3.

L_AA is self-inductance of Phase A, M_AB, M_AC, M_AD, and M_AE are mutual inductances between Phase A and Phases B, C, D, and E, respectively.

It can be seen from Table 2 that the mutual inductances between Phase A and Phases B and E (M_AB and M_AE) are almost zero for all of the three investigated schemes. For the single-layer, double-layer and unequal-turn winding configurations, the mutual inductances between Phase A and Phases C and D (M_AC and M_AD) are 0.73%, 7.9% and 8.3% of the self-inductances (L_AA), respectively.

The single-layer winding configuration features high self-inductance and low mutual inductances, the mutual inductances of double-layer winding configuration are larger than those of the single-layer winding configuration. Compared with the double-layer winding configuration, the mutual inductances of the unequal-turn winding configuration are a little larger, but the 2-pole sub harmonic in the unequal-turn winding configuration is completely eliminated, so the unequal-turn winding configuration is very promising.

For the conventional and unequal-turn winding configurations, the radial force density distributions under both no-load and load conditions, and the corresponding harmonic analysis omitting the dc component are calculated, as shown in Figures 11 and 12. The no-load curves of conventional and unequal-turn winding configurations are the same. The radial force harmonic distribution in Figure 12b is normalized to the amplitude of the 22nd-order force harmonic in Figure 12a.

The no-load radial force density contains harmonics of orders 2, 20, 22, 44, etc., and the strongest 22nd and 44th order harmonics are consequences of the 11th working harmonic, i.e., the 22-pole PM field. For the conventional single- and double-layer windings, the dominant harmonics of load radial force density are with orders 2, 10, 12, 20, 22, 44, etc. Compared with the conventional single- and double-layer windings, the amplitudes of load radial force density harmonics of unequal-turn windings are lower, such as the 2nd, 10th, 12th and 20th order harmonics, etc. It is worth noting that the load radial force density contains a strong 2nd-order harmonic, and this strong harmonic can be clearly observed in Figure 11b–d. Since the machine structure is stiffer to short-wave distortion which corresponds to high order vibration mode, so the 2nd-order harmonic is the most detrimental to the machine strength, and compared with the conventional winding topologies, the 2nd-order harmonic of unequal-turn winding is reduced, which alleviates the harm of this harmonic.

Three five-phase 20-slot/22-pole PMSM schemes featuring single-layer, double-layer and unequal-turn winding configurations have been designed and calculated using the commercial FEA software Ansoft Maxwell. For the three machine schemes, the stator lamination, efficient length, air gap length, width and thickness of magnet, and electromagnetic power are the same. Hence, direct electromagnetic performance comparison for the three schemes is possible. The magnet conductivity is σ = 625,000 s/m. In order to calculate the impact of skin effect on PM losses, high mesh density in magnets are adopted. Based on the 2-D FEA models, the PM losses are calculated without consideration of axial segmentation, circumferential segmentation, and 3D effect. The machine performances under the rated speed of 450 rpm are compared, as shown in Table 4.

All of the three winding configurations compared in Table 4 have smooth torques, which is beneficial to improve the driving comfort of EVs. Compared with the single-layer winding configuration, the core loss and PM loss of the double-layer winding configuration are reduced by 9.1% and 5.8%, and its efficiency is enhanced. By comparison with the double-layer winding configuration, the core loss and PM loss of the unequal-turn winding configuration are reduced by 1.8% and 4.2%, respectively. Among the three investigated schemes, the unequal-turn winding configuration has the highest efficiency, and it is more suitable for EV applications.

Assume Phase E is open-circuited, as shown in Figure 13.

The electromagnetic power of PMSM equals the sum of instantaneous power of each of the stator windings. The instantaneous power of each of the stator windings can be calculated by the phase current and the EMF induced in the windings. Through maintaining the sum of instantaneous power unchanged, the output electromagnetic power under open-circuit fault conditions can be the same as that under healthy conditions. For a certain speed, the output torque in open-circuit fault conditions can be kept the same as that before the fault happened. Under restrictive condition of keeping the output torque unchanged, the optimal currents for healthy phase windings can be calculated by minimizing a cost function of copper loss: (3) P w = ∑ j = 0 3 ( i j ) 2 + λ [ T d − 1 ω ∑ j = 0 3 E 0 cos ( θ − j 2 π 5 ) i j ]i_j is instantaneous current in the healthy phase windings, j = 0, 1, 2 and 3 represent Phases A, B, C and D, respectively. E₀ is peak value of back EMF, T_d is the demanded torque, and ω is speed, λ is Lagrange multiplier.

The problem dealt with open-circuit fault-tolerant control of the unequal-turn winding PMSM is transformed into a conditional extreme problem. Based on the Lagrange multiplier method, Equation (4) is obtained: (4) [ 2 0 0 0 − E 0 ω cos θ 0 2 0 0 − E 0 ω cos ( θ − 2 π 5 ) 0 0 2 0 − E 0 ω cos ( θ − 4 π 5 ) 0 0 0 2 − E 0 ω cos ( θ + 4 π 5 ) E 0 ω cos θ E 0 ω cos ( θ − 2 π 5 ) E 0 ω cos ( θ − 4 π 5 ) E 0 ω cos ( θ + 4 π 5 ) 0 ] [ i a i b i c i d λ ] = [ 0 0 0 0 T d ]

Through solving Equation (4), the optimal currents for healthy phase windings are obtained, as shown in Equation (5): (5) [ i a i b i c i d ] = [ ω T d cos ( θ ) 2 E 0 [ 1 + cos ( π / 5 ) cos ( 2 π / 5 ) cos ( 2 θ − π / 5 ) ] ω T d cos ( θ − 2 π / 5 ) 2 E 0 [ 1 + cos ( π / 5 ) cos ( 2 π / 5 ) cos ( 2 θ − π / 5 ) ] ω T d cos ( θ − 4 π / 5 ) 2 E 0 [ 1 + cos ( π / 5 ) cos ( 2 π / 5 ) cos ( 2 θ − π / 5 ) ] ω T d cos ( θ + 4 π / 5 ) 2 E 0 [ 1 + cos ( π / 5 ) cos ( 2 π / 5 ) cos ( 2 θ − π / 5 ) ] ]

From Equation (5), we can find that the peak values of the calculated currents in fault conditions are larger than those in healthy conditions, which indicates an increased cost and converter capacity. For a five-phase fault-tolerant PMSM system used for EV applications, the operation under fault-conditions is an occasional case and small-probability event, and it is a waste to prevent occasional faults at the cost of increasing cost and converter capacity. Thus, in order not to improve the converter capacity, peak values of the calculated phase currents for healthy phase windings are scaled in the same proportion to achieve the same peak current values as the ones before faults happened. In fact, part of the output electromagnetic power is sacrificed. The currents for healthy phase windings and the torque performances are calculated and verified by FEA, as shown in Figure 14 and Table 5.

Under healthy conditions, the rated torque is 254.6N·m with a torque ripple of 0.44%. When Phase E is open-circuited, the output torque reduces to 80% of the rated torque, and the torque ripple dramatically increases to 23.38%, which is detrimental to the system performance. After adopting the proposed fault-tolerant control strategy, the machine efficiency improves slightly, the output torque decreases to 68.68% of the rated torque with a torque ripple of 4.22% and this will enable the EV applications to smoothly operate under reduced load conditions.

Assume Phases D and E are open-circuited, as shown in Figure 15.

Like the one-phase-open fault condition, the optimal currents for healthy phase windings can be calculated by minimizing a cost function of copper loss: (6) P w = ∑ j = 0 2 ( i j ) 2 + λ [ T d − 1 ω ∑ j = 0 2 E 0 cos ( θ − j 2 π 5 ) i j ]where j = 0, 1, 2 represent Phases A, B and C, respectively. The problem dealt with fault-tolerant control of two adjacent-phases-open of the unequal-turn winding PMSM is transformed into a conditional extreme problem. Based on the Lagrange multiplier method, Equation (7) is obtained: (7) [ 2 0 0 − E 0 ω cos θ 0 2 0 − E 0 ω cos ( θ − 2 π 5 ) 0 0 2 − E 0 ω cos ( θ − 4 π 5 ) E 0 ω cos θ E 0 ω cos ( θ − 2 π 5 ) E 0 ω cos ( θ − 4 π 5 ) 0 ] [ i a i b i c λ ] = [ 0 0 0 T d ]

Through solving Equation (7), the optimal currents for healthy phase windings are obtained: (8) [ i a i b i c ] = [ 2 ω T d cos ( θ ) E 0 [ 3 + 2 cos ( 2 π / 5 ) cos ( 2 θ − π / 5 ) + cos ( 2 θ + π / 5 ) ] 2 ω T d cos ( θ − 2 π / 5 ) E 0 [ 3 + 2 cos ( 2 π / 5 ) cos ( 2 θ − π / 5 ) + cos ( 2 θ + π / 5 ) ] 2 ω T d cos ( θ + 2 π / 5 ) E 0 [ 3 + 2 cos ( 2 π / 5 ) cos ( 2 θ − π / 5 ) + cos ( 2 θ + π / 5 ) ] ]

In this case, peak values of the calculated phase currents for healthy phase windings are also adjusted proportionally so that they can be the same as the ones before faults happened. The currents for healthy phase windings and the torque behaviors are calculated and verified by FEA, as shown in Figure 16 and Table 6.

As Phases D and E are open-circuited, the output torque reduces to 60.5% of the rated torque, and the torque ripple increases to 20.76%, which is detrimental to the system performance. After adopting the proposed fault-tolerant control strategy, the machine efficiency exceeds 95%, and the output torque further reduces to 49.28% of the rated torque with a torque ripple of 9.15%. In fact, part of the output electromagnetic power is sacrificed to obtain a smooth output torque, but the driving comfort of the EVs is significantly improved.

Assume Phases C and E are open-circuited, as shown in Figure 17.

It is similar to the two adjacent-phases-open fault condition that the optimal currents for healthy phase windings can be calculated by minimizing a cost function of copper loss: (9) P w = ∑ j = 0 , 1 , 3 ( i j ) 2 + λ [ T d − 1 ω ∑ j = 0 , 1 , 3 E 0 cos ( θ − j 2 π 5 ) i j ]where j = 0, 1, 3 represent Phases A, B and D, respectively. The problem dealt with fault-tolerant control of two non-adjacent-phases-open of the unequal-turn winding PMSM is transformed into a conditional extreme problem. Based on the Lagrange multiplier method, Equation (10) is obtained: (10) [ 2 0 0 − E 0 ω cos θ 0 2 0 − E 0 ω cos ( θ − 2 π 5 ) 0 0 2 − E 0 ω cos ( θ + 4 π 5 ) E 0 ω cos θ E 0 ω cos ( θ − 2 π 5 ) E 0 ω cos ( θ + 4 π 5 ) 0 ] [ i a i b i d λ ] = [ 0 0 0 T d ]

Through solving Equation (10), the optimal currents for healthy phase windings are obtained: (11) [ i a i b i d ] = [ 2 ω T d cos ( θ ) E 0 [ 3 + 2 cos ( 2 π / 5 ) cos ( 2 θ − π / 5 ) + cos ( 2 θ − π / 5 ) ] 2 ω T d cos ( θ − 2 π / 5 ) E 0 [ 3 + 2 cos ( 2 π / 5 ) cos ( 2 θ − π / 5 ) + cos ( 2 θ − π / 5 ) ] 2 ω T d cos ( θ + 4 π / 5 ) E 0 [ 3 + 2 cos ( 2 π / 5 ) cos ( 2 θ − π / 5 ) + cos ( 2 θ − π / 5 ) ] ]

In this case, peak values of the calculated phase currents for healthy phase windings are also adjusted proportionally so that they can be the same as the ones before faults happened. The currents for healthy phase windings and the torque behaviors are calculated and verified by FEA, as shown in Figure 18 and Table 7.

As Phases C and E are open-circuited, the output torque reduces to 60.1% of the rated torque, and the torque ripple dramatically increases to 51.43%, which is detrimental to the system performance. After adopting the proposed fault-tolerant control strategy, the machine efficiency slightly decreases compared with the healthy condition, the output torque reduces to 42.46% of the rated torque, and it is 108.1N·m with a torque ripple of 6.86%, the torque ripple is significantly reduced, and this will enable the EV applications to smoothly operate under reduced load conditions. The comprehensive comparison of the machine performances under different operating conditions are listed, as shown in Table 8. The per unit values of Table 8 are shown in Table 9.

Since the electromagnetic power of PMSM equals the sum of instantaneous power of each of the stator windings, so when the machine operates under open-circuits of one or two phases, the output torques drop to 0.8 and 0.6 p.u., respectively, as shown in Tables 8 and 9. For one-phase-open, two adjacent-phases-open, and two non-adjacent-phases-open faults, after adoption of the proposed fault-tolerant control strategies, the torques decrease to 0.687, 0.493 and 0.425 p.u., respectively, but the torque ripples of all conditions are significantly reduced, and the machine efficiencies are even improved in some cases, which indicates that the proposed fault-tolerant control strategies are quite acceptable, with some sacrifice of the electromagnetic power.