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The Effects of Permanent Magnet Segmentations on Electromagnetic Performance in Ironless Brushless DC Motors

: This paper investigates the electromagnetic torque by considering back electromagnetic force (back-EMF) trapezoidal degrees of ironless brushless DC (BLDC) motors through the two-dimensional ﬁnite element method (2-D FEM). First, the change percentages of the electromagnetic torque with back-EMF trapezoidal degrees, relative to those of PMs without segments, are investigated on the premise of the same back-EMF amplitude. It is found that both PM symmetrically and asymmetrically segmented types inﬂuence back-EMF trapezoidal degrees. Second, the corresponding electromagnetic torque, relative to that of PMs without segments, is studied in detail. The results show that the electromagnetic torque can be improved or deteriorated depending on whether the back-EMF trapezoidal degree is lower or higher than that of PMs without segments. Additionally, the electromagnetic torque can easily be improved by increasing the number of PMs’ symmetrical segments. In addition, the electromagnetic torque in PMs with asymmetrical segments is always higher than that of PMs without segments. Finally, two ironless PM BLDC motors with PMs symmetrically segmented into three segments and without segments are manufactured and tested. The experimental results show good agreement with those of the 2-D FEM method. This approach provides signiﬁcant guidelines to electromagnetic torque improvement without much increase in manufacturing costs and process complexity.


Introduction
With the merits of simple structure, easy control, fast transient response, and high efficiency, permanent magnet brushless DC (PM BLDC) motors have been widely used in different areas and applications, such as electric vehicles, energy storage flywheels, robots, household appliances, etc. [1][2][3][4][5][6][7]. However, with the increasing requirements of energy efficiency and high performance of equipment, conventional PM BLDC motors cannot meet the demands of high efficiency and high torque density any longer [8,9]. The lightweight ironless PM BLDC motor eliminates the stator iron loss and, therefore, has a great potential in torque density improvement [10][11][12]. Additionally, in order to obtain the maximum output torque, the back electromagnetic force (back-EMF) waveform of the PM BLDC motor should be square or rectangular [13,14]. However, it is found that the back-EMF waveform of the designed ironless BLDC motor is much more sinusoidal in practice [15]. It has been proved in [16] that, in conditions of the same ideal square wave input currents and same back-EMF amplitudes, the electromagnetic torque with square back-EMF waveform is 15.5% higher than that with sinusoidal waveforms. This indicates that the more trapezoidal degree the back-EMF has, the higher the load torque will be, and the lower the resulting torque ripple.
In [15,17], it is highlighted that, due to the nonideality of magnetic material, design considerations, and manufacturing limitations, the real back-EMF waveform is not exactly trapezoidal, which deteriorates the electromagnetic torque. The trapezoidal degree of back-EMF can be obtained through different control strategies [15,[17][18][19][20][21][22][23][24]. In [15], a phase current injection method is proposed to compensate for the nonideal back-EMF. In [17], a current optimization control method is proposed to reduce the torque ripple of PM BLDC motors, which is caused by the nonideal trapezoidal back-EMF. It is declared in [22] that the torque ripple can be reduced by the optimization of the phase rectification error, which is not inevitable in motor sensorless control. In [23], an average torque control strategy with a single period is proposed to minimize the torque ripple, and back-EMF is nonideal. In [24], a second-order sensorless control strategy for a BLDC motor system with low inductance and nonideal back-EMF is adopted to improve the reliability of current commutation during high-speed operation and reduce the power consumption of high-speed steady-state operation. In [13], the PM thickness and migration angle of a double-layer brushless DC motor are optimized to make the back-EMF waveform of the inside and outside stator winding consistent with each other, and the torque ripple is decreased dramatically. The electromagnetic torque performance of the PM BLDC motor can be improved with some adopted control strategies. However, it will increase the complexity of the motor control system.
As is known, the back-EMF waveform depends directly on the magnetic field distribution in the coil region, which is excited from the rotor PMs. Thus, trapezoidal back-EMF can be obtained through PM segmentation with different magnetic field intensities. However, to the best of the authors' knowledge, few papers focus on this aspect [2,25,26]. In [27], a method for the comprehensive optimal design of a slotless PM BLDC motor with surfacemounted magnets is proposed, to find the optimal geometries of the assumed motor. In [28], the dynamic response of a BLDC motor is determined using artificial intelligence, which provides guidance for multiparameter optimization of electromagnetic torque improvement. In this paper, a method for improving the back-EMF trapezoidal degree by PM's different segmentation types is proposed. Additionally, in order to keep the amplitude of the back-EMF constant, the combination of different intensities of PM magnetization is adopted. Firstly, the modeling of the ironless BLDC motor and PMs with symmetrical and asymmetrical segments are established. Secondly, the cases with the same back-EMF amplitudes are calculated through a two-dimensional finite element method (2-D FEM) multiobjective optimization. Thirdly, the change percentages of the electromagnetic torque with the back-EMF trapezoidal degree, relative to PMs without segments, are investigated. Finally, two ironless BLDC motor prototypes, one with PMs symmetrically segmented into three segments and the other without segments, are manufactured and tested to validate the simulated results.

Modeling
The 14-pole and 12-slot ironless BLDC motor equipped with an outer rotor is presented in Figure 1. And the specifications of all the symbols used in this paper, are tabulated in Nomenclature. The winding support frame is made of nonmagnetic material, such as epoxy resin or polyimide to reduce the weight. Additionally, the stator iron loss is eliminated, which is beneficial for torque density improvement. As the slot number per pole per phase is 2/7, concentrated windings with all teeth wounded were adopted. The outer rotor structure provides higher inertia than the inner rotor configuration under the same conditions. Additionally, it can be easily connected with external interfaces. The main magnetic field is excited by PMs. The solid arrows on the PMs show the corresponding radial magnetizing direction. As evident, the magnetization directions of each adjacent two PMs are inverse with each other. The embrace equals 1, which means the inner side of the rotor back iron is covered by PMs. two PMs are inverse with each other. The embrace equals 1, which means the inner side of the rotor back iron is covered by PMs. In order to obtain trapezoidal back-EMF, PMs were segmented symmetrically and asymmetrically, as illustrated in Figure 2. For the case of PMs segmented symmetrically, PMs were segmented into several equal sub-PMs. Here, the number of the symmetrically segmented PMs is denoted as 2k + 1 (k = 1, 2, 3). The sum of the sub-PM center angles is π/p. The remanence of the middle PM (mid-PM for short) is recorded as B1, and then taking the mid-PM as a reference, the remanence of the left or right side PMs (side-PMs for short) is denoted as B2, B3, …, Bk (k = 2, 3, 4) in sequence. The remanences of the side-PMs at the symmetric position are consistent with each other. While for the other case of PMs segmented asymmetrically, PMs were segmented into three sub-PMs. The central angle of the middle PM is set as θ, and that of the sub-PMs on both sides is assumed as β1 and β2, respectively. As shown in Figure 1, the inner side of the outer rotor is full of PMs without segmentations. Hence, restricted conditions were made to keep the embrace of PMs segmented into several sub-PMs to be 1. For symmetrically segmented PMs, θ = β1 = β2 = 60/7°, while for those with asymmetrical segmentation, β1 = β2 = 60/7°-α, θ = 60/7 + 2α. Additionally, the varied range of α is (0°, 4°).
It should be noted that the magnetization intensity for the case of PMs segmented symmetrically is the same for all the segmented sub-PMs. However, for the case of PMs segmented asymmetrically, the remanence of the mid-PM and side-PMs are defined as B1 and B2 (B1 ≠ B2), respectively.

PMs Segmented Symmetrically into Several Segments
As mentioned before, for the case of PMs segmented symmetrically, the segment number is denoted as 2k + 1 (k = 1, 2, 3). Hence, the magnetic density variables of the sub-PMs are 2, 3, and 4, respectively. According to the actual performance of market-owned magnetic steel, the magnetic density of each sub-PM ranges from 0.  Figure 1. Schematic diagram of the ironless BLDC half model.
In order to obtain trapezoidal back-EMF, PMs were segmented symmetrically and asymmetrically, as illustrated in Figure 2. For the case of PMs segmented symmetrically, PMs were segmented into several equal sub-PMs. Here, the number of the symmetrically segmented PMs is denoted as 2k + 1 (k = 1, 2, 3). The sum of the sub-PM center angles is π/p. The remanence of the middle PM (mid-PM for short) is recorded as B 1 , and then taking the mid-PM as a reference, the remanence of the left or right side PMs (side-PMs for short) is denoted as B 2 , B 3 , . . . , B k (k = 2, 3, 4) in sequence. The remanences of the side-PMs at the symmetric position are consistent with each other. While for the other case of PMs segmented asymmetrically, PMs were segmented into three sub-PMs. The central angle of the middle PM is set as θ, and that of the sub-PMs on both sides is assumed as β 1 and β 2 , respectively. As shown in Figure 1, the inner side of the outer rotor is full of PMs without segmentations. Hence, restricted conditions were made to keep the embrace of PMs segmented into several sub-PMs to be 1. For symmetrically segmented PMs, θ = β 1 = β 2 = 60/7 • , while for those with asymmetrical segmentation, β 1 = β 2 = 60/7 • −α, θ = 60/7 + 2α. Additionally, the varied range of α is (0 • , 4 • ).
two PMs are inverse with each other. The embrace equals 1, which means the inner side of the rotor back iron is covered by PMs. In order to obtain trapezoidal back-EMF, PMs were segmented symmetrically and asymmetrically, as illustrated in Figure 2. For the case of PMs segmented symmetrically, PMs were segmented into several equal sub-PMs. Here, the number of the symmetrically segmented PMs is denoted as 2k + 1 (k = 1, 2, 3). The sum of the sub-PM center angles is π/p. The remanence of the middle PM (mid-PM for short) is recorded as B1, and then taking the mid-PM as a reference, the remanence of the left or right side PMs (side-PMs for short) is denoted as B2, B3, …, Bk (k = 2, 3, 4) in sequence. The remanences of the side-PMs at the symmetric position are consistent with each other. While for the other case of PMs segmented asymmetrically, PMs were segmented into three sub-PMs. The central angle of the middle PM is set as θ, and that of the sub-PMs on both sides is assumed as β1 and β2, respectively. As shown in Figure 1, the inner side of the outer rotor is full of PMs without segmentations. Hence, restricted conditions were made to keep the embrace of PMs segmented into several sub-PMs to be 1. For symmetrically segmented PMs, θ = β1 = β2 = 60/7°, while for those with asymmetrical segmentation, β1 = β2 = 60/7°-α, θ = 60/7 + 2α. Additionally, the varied range of α is (0°, 4°).
It should be noted that the magnetization intensity for the case of PMs segmented symmetrically is the same for all the segmented sub-PMs. However, for the case of PMs segmented asymmetrically, the remanence of the mid-PM and side-PMs are defined as B1 and B2 (B1 ≠ B2), respectively.

PMs Segmented Symmetrically into Several Segments
As mentioned before, for the case of PMs segmented symmetrically, the segment number is denoted as 2k + 1 (k = 1, 2, 3). Hence, the magnetic density variables of the sub-PMs are 2, 3, and 4, respectively. According to the actual performance of market-owned magnetic steel, the magnetic density of each sub-PM ranges from 0.1 T to 1.5 T, with the step of 0.2T. Then, the solution numbers of the case with PMs symmetrically segmented  It should be noted that the magnetization intensity for the case of PMs segmented symmetrically is the same for all the segmented sub-PMs. However, for the case of PMs segmented asymmetrically, the remanence of the mid-PM and side-PMs are defined as B 1 and B 2 (B 1 = B 2 ), respectively.

PMs Segmented Symmetrically into Several Segments
As mentioned before, for the case of PMs segmented symmetrically, the segment number is denoted as 2k + 1 (k = 1, 2, 3). Hence, the magnetic density variables of the sub-PMs are 2, 3, and 4, respectively. According to the actual performance of market-owned magnetic steel, the magnetic density of each sub-PM ranges from 0.1 T to 1.5 T, with the step of 0.2T. Then, the solution numbers of the case with PMs symmetrically segmented is 8 k (k = 2, 3, 4). The back-EMF amplitude of the ironless BLDC motor without PM segmentation is 6 V. As for the PM segmented into three and five segments, Cases of the back-EMF amplitude within 6 ± 0.05 V were extracted and are tabulated in Tables 1 and 2, respectively. Additionally, for the PM segmented into seven segments, the back-EMF amplitude within 6 ± 0.05 V are numerous, thus Cases within 6 ± 0.01 V were extracted and are listed in Table 3. It is worth noting that the back-EMF amplitude difference ratio for the PMs symmetrically segmented into three and five segments to PMs without segments is about 0.83%. Additionally, when the PMs symmetrically segmented into seven segments, the back-EMF amplitude difference ratio to the PMs without segments is about 0.17%. This indicates that, with the increase in the number of PM segments, more cases that can meet the restricted condition can be obtained. As 120 • commutation control is generally used for BLDC motors, the ideal back-EMF should be trapezoidal. Thus, in order to compare the back-EMF trapezoidal degrees of the ironless BLDC motor with PM segmented into 2k + 1 (k = 1, 2, 3) segments, the real and normalized back-EMF waveforms with different Cases are illustrated in denotes PMs without segments. The waveforms during the 120 • electrical degree period were cut out and normalized while considering the 120 • commutation control. During the normalization process, for each Case, the values of back-EMF at 90 • electrical degrees were chosen as base values, correspondingly. It can be seen that the fitting degree between the normalized real shape and the ideal shape of the back-EMF of the three symmetrically segmented types are different. In order to evaluate the quality of the back-EMF trapezoidal degree, an evaluation factor was adopted [29]. The evaluation factor for PMs symmetrically segmented into 2k + 1 (k = 1, 2, 3) segments ζ 2k+1 e is defined as Equation (1). Additionally, the smaller the evaluation factor, the better the trapezoidal degree of the back-EMF.
where α n is the nth harmonics of the normalized back-EMF, and β n is the nth harmonics of the ideal waveform.  As 120° commutation control is generally used for BLDC motors, the ideal back-EMF should be trapezoidal. Thus, in order to compare the back-EMF trapezoidal degrees of the ironless BLDC motor with PM segmented into 2k + 1 (k = 1, 2, 3) segments, the real and normalized back-EMF waveforms with different Cases are illustrated in Figures 3-5, respectively. The Cases of the three figures correspond to Tables 1-3, respectively. Case-0 denotes PMs without segments. The waveforms during the 120° electrical degree period were cut out and normalized while considering the 120° commutation control. During the normalization process, for each Case, the values of back-EMF at 90° electrical degrees were chosen as base values, correspondingly. It can be seen that the fitting degree between the normalized real shape and the ideal shape of the back-EMF of the three symmetrically segmented types are different. In order to evaluate the quality of the back-EMF trapezoidal degree, an evaluation factor was adopted [29]. The evaluation factor for PMs symmetrically segmented into 2k + 1 (k = 1, 2, 3) segments 21 k e  + is defined as Equation (1). Additionally, the smaller the evaluation factor, the better the trapezoidal degree of the back-EMF.   As 120° commutation control is generally used for BLDC motors, the ideal back-EMF should be trapezoidal. Thus, in order to compare the back-EMF trapezoidal degrees of the ironless BLDC motor with PM segmented into 2k + 1 (k = 1, 2, 3) segments, the real and normalized back-EMF waveforms with different Cases are illustrated in Figures 3-5, respectively. The Cases of the three figures correspond to Tables 1-3, respectively. Case-0 denotes PMs without segments. The waveforms during the 120° electrical degree period were cut out and normalized while considering the 120° commutation control. During the normalization process, for each Case, the values of back-EMF at 90° electrical degrees were chosen as base values, correspondingly. It can be seen that the fitting degree between the normalized real shape and the ideal shape of the back-EMF of the three symmetrically segmented types are different. In order to evaluate the quality of the back-EMF trapezoidal degree, an evaluation factor was adopted [29]. The evaluation factor for PMs symmetrically segmented into 2k + 1 (k = 1, 2, 3) segments 21 k e  + is defined as Equation (1). Additionally, the smaller the evaluation factor, the better the trapezoidal degree of the back-EMF.    for all PMs symmetrically segmented into three, five, and seven segments, respectively. Case-0 (i.e., PMs without segments) is taken as a reference, and the corresponding evaluation factor is 0.2102. It can be found when PMs are symmetrically segmented into three, five, and seven segments, the number of cases for which the evaluation factor be smaller than that of the reference is one, two, and six. This indicates that a better trapezoidal back-EMF can easily be obtained with the increasing number of PM segments. It should be also noted that the back-EMF amplitude difference ratio between PMs symmetrically segmented into seven segments and PMs without segments is lower than the other two symmetrically segmented types.
where αn is the nth harmonics of the normalized back-EMF, and βn is the nth harmonics of the ideal waveform. Figure 6a-c show the variations in the evaluation factors for all PMs symmetrically segmented into three, five, and seven segments, respectively. Case-0 (i.e., PMs without segments) is taken as a reference, and the corresponding evaluation factor is 0.2102. It can be found when PMs are symmetrically segmented into three, five, and seven segments, the number of cases for which the evaluation factor be smaller than that of the reference is one, two, and six. This indicates that a better trapezoidal back-EMF can easily be obtained with the increasing number of PM segments. It should be also noted that the back-EMF amplitude difference ratio between PMs symmetrically segmented into seven segments and PMs without segments is lower than the other two symmetrically segmented types.

PM Asymmetrical Segmentation
Here, PMs were asymmetrically segmented into three sub-PMs, as shown in Figure 2.
As the remanence of the mid-PM is much lower than that of the two side-PMs, the volume of the mid-PM increases with a linear increase in the variation of α, which leads to a small-ranging decrease in the whole remanence of the PM correspondingly. Hence, from Figure 7, it can be inferred that the back-EMF amplitude decreases with a linear increase in the variation α, but the extent of the increase is gradually reduced.
On the premise of the same back-EMF amplitude, as in PMs with symmetrical segmentations, the cases with the back-EMF amplitude ratio of each certain α of Figure 7 to that of PMs without segments, which is about 0.83%, were selected. Additionally, the corresponding back-EMF waveforms were simulated and normalized, as illustrated in Figure 8. It can be seen that four, three, and three cases satisfy the precondition for α = 1, α = 2, and α = 3, respectively. This normalized process is similar to PMs symmetrically segmented. For α = 4, there is only one case that satisfies the precondition; the real and normalized waveforms are not shown here. show the distribution of the back-EMF amplitude with different magnetic intensities of and B2 for α = 1°, 2°, 3°, 4°, respectively. As the remanence of the mid-PM is much low than that of the two side-PMs, the volume of the mid-PM increases with a linear increa in the variation of α, which leads to a small-ranging decrease in the whole remanence the PM correspondingly. Hence, from Figure 7, it can be inferred that the back-EMF am plitude decreases with a linear increase in the variation α, but the extent of the increase gradually reduced. On the premise of the same back-EMF amplitude, as in PMs with symmetrical se mentations, the cases with the back-EMF amplitude ratio of each certain α of Figure 7 that of PMs without segments, which is about 0.83%, were selected. Additionally, the co responding back-EMF waveforms were simulated and normalized, as illustrated in Figu 8. It can be seen that four, three, and three cases satisfy the precondition for α = 1, α = and α = 3, respectively. This normalized process is similar to PMs symmetrically se mented. For α = 4, there is only one case that satisfies the precondition; the real and no malized waveforms are not shown here.
Similarly, in order to effectively evaluate the trapezoidal degree of the back-EMF, th evaluation factors of the normalized back-EMF waveforms were calculated and are tab lated in Tables 4-6, respectively. It can be seen that (1) the evaluation factor ranges fro 0.12 to 0.166 for PMs asymmetrically segmented into three segments under the precond tion; (2) the evaluation factor of case-1 almost keep unchanged with α = 1°, 2° and 3°, whic means the evaluation factor of case-1 is not sensitive to the variation α; (3) the back-EM Similarly, in order to effectively evaluate the trapezoidal degree of the back-EMF, the evaluation factors of the normalized back-EMF waveforms were calculated and are tabulated in Tables 4-6, respectively. It can be seen that (1) the evaluation factor ranges from 0.12 to 0.166 for PMs asymmetrically segmented into three segments under the precondition; (2) the evaluation factor of case-1 almost keep unchanged with α = 1 • , 2 • and 3 • , which means the evaluation factor of case-1 is not sensitive to the variation α; (3) the back-EMF trapezoidal degree decreases with the improvement in the variation of α and the remanence of the two sub-PMs. However, when the variation of α and the remanence of the two sub-PMs increase to a certain extent, the evaluation factor of back-EMF will not decrease but increase, which indicates the back-EMF waveform trapezoidal will deteriorate. trapezoidal degree decreases with the improvement in the variation of α and the remanence of the two sub-PMs. However, when the variation of α and the remanence of the two sub-PMs increase to a certain extent, the evaluation factor of back-EMF will not decrease but increase, which indicates the back-EMF waveform trapezoidal will deteriorate.
(a) (b) (c)   Table 6. Evaluation factor of the back-EMF trapezoidal degree with α = 3°.  trapezoidal degree decreases with the improvement in the variation of α and the remanence of the two sub-PMs. However, when the variation of α and the remanence of the two sub-PMs increase to a certain extent, the evaluation factor of back-EMF will not decrease but increase, which indicates the back-EMF waveform trapezoidal will deteriorate.

Symmetrically Segmented PMs
The load torque was obtained under the same three-phase input currents, and the rated current value is 1A. Figure 9a-c illustrate the load torque with PMs symmetrically segmented into three, five, and seven segments, respectively.
Average torque and torque ripple are two key elements to evaluate load torque performance. Figure 10a-c illustrate the torque performance with PMs symmetrically segmented to three, five, and seven segments, respectively. The torque ripple is defined as Equation (2). After comparing Figure 10 with Figure 6, it can be found the average torque change situation is reverse to that of the back-EMF evaluation factor, which is consistent with theoretical analysis. The smaller the back-EMF trapezoidal degree evaluation factor is, the higher the load torque will be, and vice versa.

Symmetrically Segmented PMs
The load torque was obtained under the same three-phase input currents, and the rated current value is 1A. Figure 9a-c illustrate the load torque with PMs symmetrically segmented into three, five, and seven segments, respectively.
Average torque and torque ripple are two key elements to evaluate load torque performance. Figure 10a-c illustrate the torque performance with PMs symmetrically segmented to three, five, and seven segments, respectively. The torque ripple is defined as Equation (2). After comparing Figure 10 with Figure 6, it can be found the average torque change situation is reverse to that of the back-EMF evaluation factor, which is consistent with theoretical analysis. The smaller the back-EMF trapezoidal degree evaluation factor is, the higher the load torque will be, and vice versa.
where ξ 2k+1 T is the torque ripple of PMs symmetrically segmented into 2k + 1 segments; T 2k+1 max is the maximum torque; T 2k+1 min is the minimum torque; T 2k+1 avg is the average torque; 2k + 1 (k = 1, 2, 3) is the number of the segment number of PMs symmetrically segmented.

Symmetrically Segmented PMs
The load torque was obtained under the same three-phase input currents, and t rated current value is 1A. Figure 9a-c illustrate the load torque with PMs symmetrica segmented into three, five, and seven segments, respectively.
Average torque and torque ripple are two key elements to evaluate load torque p formance. Figure 10a-c illustrate the torque performance with PMs symmetrically se mented to three, five, and seven segments, respectively. The torque ripple is defined Equation (2). After comparing Figure 10 with Figure 6, it can be found the average torq change situation is reverse to that of the back-EMF evaluation factor, which is consiste with theoretical analysis. The smaller the back-EMF trapezoidal degree evaluation fac is, the higher the load torque will be, and vice versa.  for PMs symmetrically segmented into 2k + 1 (k = 1, 2, 3) segments can be defined in Equation (3) and Equation (4), respectively.
where T

(2k+1)
Case−n is the load torque at Case-n (n = 1, 2, 3, . . . ) for PMs symmetrically segmented into 2k + 1 (k = 1, 2, 3) segments; T 0 is the load torque of PMs without segment.  Figure 11a-c illustrate the back-EMF trapezoidal degree evaluation factor and load torque change percentages of PMs symmetrically segmented into 2k + 1 (k = 1, 2, 3) segments with respect to those of PMs without segments, respectively. For the back-EMF trapezoidal degree evaluation factor change percentage, the negative value means the trapezoidal degree of the back-EMF for a certain Case is good than that of the PMs without segments, and vice versa. Meanwhile, for the load torque change percentage, the positive value indicates the average load torque is higher than that of PMs without segments, and vice versa.  is the load torque at Case-n (n = 1, 2, 3, …) for PMs symmetrically into 2k + 1 (k = 1, 2, 3) segments; T0 is the load torque of PMs without segment. Figure 11a-c illustrate the back-EMF trapezoidal degree evaluation facto torque change percentages of PMs symmetrically segmented into 2k + 1 (k = 1 ments with respect to those of PMs without segments, respectively. For the trapezoidal degree evaluation factor change percentage, the negative value trapezoidal degree of the back-EMF for a certain Case is good than that of the PM segments, and vice versa. Meanwhile, for the load torque change percentage, t It can be seen from Figure 11 that (1) the change percentage of the load torque with respect to the situation of PMs without segment is always reversed to that of back-EMF trapezoidal degree; (2) when all the PMs are symmetrically segmented into three segments, load torque of only Case-1 shows 2.87% higher than that of the PMs without segment; (3) when all the PMs are symmetrically segmented into five segments, the load torque of Case-1 and Case-2 are 5.76% and 0.81% higher than that of PMs without segments; (4) when all PMs are symmetrically segmented into seven segments, there are seven cases' load torque higher than that of PMs without segments, and the largest value is 4.36%. However, it should be noted that for PMs symmetrically segmented into seven segments, cases were selected in which the back-EMF amplitude difference to PMs without segments is 0.17%, which is five times smaller than the other situations of PMs symmetrically segmented into three and five segments. Hence, it can be deduced that there would be more cases with higher load torque than that of PMs without segments if the cases were selected based on the same back-EMF amplitude difference ratio of 0.83%. It can be seen from Figure 11 that (1) the change percentage of the load torque w respect to the situation of PMs without segment is always reversed to that of back-E trapezoidal degree; (2) when all the PMs are symmetrically segmented into three s ments, load torque of only Case-1 shows 2.87% higher than that of the PMs without s ment; (3) when all the PMs are symmetrically segmented into five segments, the l torque of Case-1 and Case-2 are 5.76% and 0.81% higher than that of PMs without s ments; (4) when all PMs are symmetrically segmented into seven segments, there seven cases' load torque higher than that of PMs without segments, and the largest va is 4.36%. However, it should be noted that for PMs symmetrically segmented into se segments, cases were selected in which the back-EMF amplitude difference to PMs w out segments is 0.17%, which is five times smaller than the other situations of PMs sy metrically segmented into three and five segments. Hence, it can be deduced that th would be more cases with higher load torque than that of PMs without segments if cases were selected based on the same back-EMF amplitude difference ratio of 0.83%.

Asymmetrically Segmented PMs
Figure 12a-c illustrate the load torque with different cases of the PMs asymmetric segmented when α equals 1°, 2°, and 3°, respectively. Additionally, case-0 is the PMs w out segments.
Similarly, in order to evaluate the load torque performance with PMs in differ asymmetrical segments, the average torque and torque ripple with different cases w calculated. The results are illustrated in Figure 13. The torque ripple of PMs asymme cally segmented is defined as Equation (5).

Asymmetrically Segmented PMs
Figure 12a-c illustrate the load torque with different cases of the PMs asymmetrically segmented when α equals 1 • , 2 • , and 3 • , respectively. Additionally, case-0 is the PMs without segments.
Similarly, in order to evaluate the load torque performance with PMs in different asymmetrical segments, the average torque and torque ripple with different cases was calculated. The results are illustrated in Figure 13. The torque ripple of PMs asymmetrically segmented is defined as Equation (5).
where ξ α n T is the torque ripple of PMs asymmetrically segmented with different α; T α n max is the maximum torque; T α n min is the minimum torque; T α n avg is the average torque; α n (n = 1, 2, 3) is the variation angle.
After comparing Figure 13a-c with Tables 4-6, it can be found that the average torque varies reversely with the back-EMF trapezoidal degree evaluation factor, which means the more trapezoidal degree the back-EMF has, the higher the average torque obtained. Figure 14a-c quantitatively illustrate the back-EMF trapezoidal degree evaluation factor and load torque change percentages of PMs asymmetrically segmented, when α equals to 1 • , 2 • , and 3 • , respectively. Similarly, compared with PMs without segments, the change percentage of the back-EMF trapezoidal degree η (α n ) ζ and load torque η (α n ) T for PMs asymmetrically segmented with different α is defined as Equations (6) and (7), respectively.
It can be seen from Figure 14 that (1) the back-EMF trapezoidal degree evaluation factor change percentage is always negative, which means the back-EMF trapezoidal degree of different cases is better than that of the PMs without segments; (2) the average torque sign of different cases continues to be positive, which is the result from the good trapezoidal degree of back-EMF waveforms; (3) the smaller the back-EMF trapezoidal degree evaluation factor change percentage, the greater the average torque change percentage, and vice versa; (4) with the adoption of asymmetrically segmented PM type here, the load torque increase 9.23% higher than that of PMs without segments, while the corresponding back-EMF trapezoidal degree evaluation factor change factor decreases by 42.9%, compared with PMs without segments. After comparing Figure 13a-c with Table 4-6, it can be found that the averag varies reversely with the back-EMF trapezoidal degree evaluation factor, which the more trapezoidal degree the back-EMF has, the higher the average torque obt  PMs asymmetrically segmented with different α is defined as Equation (6) and (7), respectively.
where ( ) case n k   is the back-EMF trapezoidal degree evaluation factor at casek (k = 1, 2, 3, …) of PMs asymmetrically segmented; ζ0 is the back-EMF trapezoidal degree evaluation factor of PMs without segments.
where ( ) case n k T  is the load torque at casek (k = 1, 2, 3, …) of PMs asymmetrically segmented; T0 is the load torque of PMs without segment.

Percentage (%)
casek Figure 14. The relationship between the back-EMF trapezoidal degree variation with the load torque of PMs asymmetrically segmented, compared with those of PMs without segmentation: (a) α = 1 • ;

Discussion
PM segmentation is often used to restrain PM eddy current loss for high-speed PM motors. The ironless PM BLDC motor investigated in this paper is an outer rotor type, which indicates the outer rotor core can compensate for the heavy centrifugal force of the PMs mounted in the inner surface of the outer rotor core. As proved in [16], the electromagnetic force can be improved 15.5% with trapezoidal back-EMF, compared with sinusoidal back-EMF. Hence, the effect of PM segmentations with different magnetic intensities on no-load back-EMF and load torque improvement was studied.
As for PM symmetrical segmentations, the minimum evaluation factor (as shown in Equation (1)) for PMs segmented into three, five, and seven segments, are 0.182, 0.157, and 0.1582, respectively. It should be noted that the smaller the evaluation factor is, the better the trapezoidal degree of the back-EMF. However, the restricted condition for the extracted Cases of PM segment into seven segments is much stricter than the other two segment types. However, it seems that the trapezoidal degree of PMs segmented into five segments present better results than that of PMs segmented into seven segments. Likewise, the load torque is presented in Section 4.1. Considering the different deviation set for different PM symmetrical segmentations, it can be concluded that trapezoidal back-EMF can be easier obtained with the increase in the number of PM symmetrical segments.
From the comprehensive analysis of the results, it can be found that PM asymmetrical segmentations show better results in back-EMF and load torque performance. In this paper, the load torque for PMs asymmetrical segmentation showed 60% higher than that of PM symmetrical segmentations.
Compared with monoblock PM, segment processing may deteriorate the strength, especially with high rotational speed. However, under the protection of the outer rotor core, the situation of segmented PMs may be well. As for the inner rotor with surface-mounted PMs, well-behaved sleeves or other protective procedures for the PMs should be mounted on the PMs. The effect of PM trapezoidal magnetization on electromagnetic performance may be carried out in future works.

Experiment
For the sake of validating the viability of the proposed method, and the correctness of the results, two ironless PM BLDC motor prototypes were manufactured and tested. One of the prototypes was without segmentation, while the other was with PMs symmetrically segmented into 3 segments. Additionally, the magnetic field intensity of the segmented sub-PMs corresponds to Case-2 in Table 1. The schematic diagram of the experimental setup and the real test platform are shown in Figures 15 and 16, respectively. The DC source was used to provide input power for the driving motor. The host computer controlled the driving motor to rotate at the speed of 3800 rpm, through the motor controller. The test motor and the driving motor were connected through a coupler. Then, the test motor was driven by the driving motor. Considering the symmetry of the three-phase motor, only the no-load back-EMF of phase A was captured by the oscilloscope. controlled the driving motor to rotate at the speed of 3800 rpm, through the motor controller. The test motor and the driving motor were connected through a coupler. Then, the test motor was driven by the driving motor. Considering the symmetry of the three-phase motor, only the no-load back-EMF of phase A was captured by the oscilloscope.    Figure 17 shows the waveforms of the no-load back-EMF of 2-D FEM and ex mental results with PMs without segment and PMs asymmetrically segmented into t segments. It can be seen that the back-EMF amplitude of the 2-D FEM results of the segmented into 3 segments is 2.8% higher than that of the PMs without segments. Th mainly because the flux density excited by the PMs segmented into 3 segments is hi than that of the PMs without segments. The fittings degrees between the 2-D FEM experimental results are 12.9% and 9.7% for the cases of PM without segments and asymmetrically segmented into three segments, respectively. The difference may ma be due to the deficiency of the PMs' magnetic field intensity in real motor topology the ignorance of the axial end effect in 2-D FEM calculation. Overall, the experim results are in good agreement with those of the FEM method.   Figure 17 shows the waveforms of the no-load back-EMF of 2-D FEM and experimental results with PMs without segment and PMs asymmetrically segmented into three segments. It can be seen that the back-EMF amplitude of the 2-D FEM results of the PMs segmented into 3 segments is 2.8% higher than that of the PMs without segments. This is mainly because the flux density excited by the PMs segmented into 3 segments is higher than that of the PMs without segments. The fittings degrees between the 2-D FEM and experimental results are 12.9% and 9.7% for the cases of PM without segments and PMs asymmetrically segmented into three segments, respectively. The difference may mainly be due to the deficiency of the PMs' magnetic field intensity in real motor topology and the ignorance of the axial end effect in 2-D FEM calculation. Overall, the experimental results are in good agreement with those of the FEM method.

FEM results
Experimental results

FEM results
PMs without segment: PMs asymmetrically segmented into 3 segments: Figure 17. Simulation and experimental results of PMs without segment and PMs asymmetrically segmented into 3 segments.

Conclusions
In this paper, a method of improving the electromagnetic performance of the ironless permanent magnet (PM) BLDC motor in terms of no-load back-EMF and load torque was investigated through PM segmentation with different intensities of magnetizations for the segmented sub-PMs. The results showed that (1) PMs with both symmetrical and asym-

Conclusions
In this paper, a method of improving the electromagnetic performance of the ironless permanent magnet (PM) BLDC motor in terms of no-load back-EMF and load torque was investigated through PM segmentation with different intensities of magnetizations for the segmented sub-PMs. The results showed that (1) PMs with both symmetrical and asymmetrical segmentations can obtain trapezoidal back-EMF waveforms, which is beneficial for torque improvement; (2) PMs with asymmetrical segmentations present better results on back-EMF and load torque characteristics than those of symmetrical segmentations; (3) specifically, as for the ironless PM BLDC motor investigated in this paper, the load torque for PMs asymmetrically segmented into three segments showed 60% higher than that of PMs symmetrically segment into five and seven segments. Moreover, trapezoidal back-EMF can be easier obtained with the increase in the number of PMs' symmetrical segments. Finally, two ironless PM BLDC motor prototypes, one with PMs symmetrically segmented into three segments and the other without segments, were manufactured and tested. The experimental and FEM results were compared, and the results showed good agreement.