TW202101862A - Rotor lamination and rotor assembly using same - Google Patents

Abstract

A rotor lamination applied to a motor is disclosed. The motor has a motor air gap width value, and the rotor lamination includes a main-body portion, a plurality of edges and a plurality of magnet-receiving slots. The center of the main-body portion is located at the axis of the motor. The plurality of edges are disposed around an outside of the main-body. The plurality of magnet-receiving slots are configured to accommodate a plurality of magnets of the motor and disposed around the center of the main-body portion. Each magnet-receiving slot is configured to accommodate the corresponding magnet. The magnet-receiving slot has a slot width value in an outward direction from the axis. The magnet-receiving slot and the corresponding edge at outside of the main-body portion form a magnet depth value. The magnet depth value is greater than the value of a first rate constant multiplied by the slot width value and then subtracted the motor air gap width value, and is less than the value of the first rate constant multiplied by the slot width value and then plus the motor air gap width value.

Description

Rotor laminations and applicable rotor components

This case is about a rotor assembly of a motor, especially a rotor lamination and its applicable rotor assembly.

Generally speaking, the structure of a permanent magnetic electric machine or permanent magnetic motor includes a rotor (Rotor) and a stator (Stator). The stator is equipped with windings, and the rotor is equipped with permanent magnets. And the rotor can be formed by stacking rotor laminations such as silicon steel sheets. Among them, the magnetic force generated between the stator and the rotor interacts to make the rotor rotate.

In order to improve the efficiency or performance of a motor, it is necessary to increase the torque ratio that a unit current can generate. This value is also called the Torque Constant (KT), which is often used to evaluate the efficiency or performance of the motor. When the motor has a larger torque constant KT, it only needs a lower current under the same torque demand, which can effectively reduce the copper wire loss and achieve the effect of improving efficiency.

Traditional permanent magnet motors mostly adopt a flower-petal-shaped rotor design, which opens a plurality of slots near the outer diameter of the rotor to arrange the magnetic flux and achieve the effect of increasing the motor torque or reducing the torque of the rotation. However, when designing petal-shaped rotors, in order to ensure that the output torque performance (larger torque) can still be maintained under the conditions of optimized torque ripple, it is necessary to set the arc depth (Arc depth) ), the placement of the magnet (Magnet) and the size of the rib (Rib) are balanced. When relying on simulation analysis software, it often takes a very long time to find the design value to balance the performance due to the extremely large number of variable factors, and the relationship between the rotor size parameters are coupled with each other, which also causes the calculation of the optimal rotor size The difficulty rises.

In view of this, it is really necessary to provide a rotor lamination and a suitable rotor assembly to solve the problems faced by the prior art.

The purpose of this case is to provide a rotor lamination and a suitable rotor assembly. Through the motor air gap width value and the magnet accommodating groove width value, the magnet depth value between the magnet accommodating groove and the outer periphery of the main body is designed to obtain the smallest torque ripple in the maximum output torque range, thereby improving the motor The anti-demagnetization endurance of the output torque of motor achieves the effect of improving the efficiency of the motor.

Another objective of this case is to provide a rotor lamination and a suitable rotor assembly. According to the width of the motor air gap and the width of the magnet accommodating slot, between two adjacent magnet accommodating slots, the arcing depth is designed concavely from the outer periphery of the rotor to the rotor shaft center to be in the range of the maximum output torque. Minimal torque ripple is obtained in the motor, thereby improving the mutual influence of magnet leakage/flux leakage and air-gap flux distribution/density in the motor, achieving the effect of improving motor efficiency .

Another objective of this case is to provide a rotor lamination and a suitable rotor assembly. According to the motor air gap width value and the magnet accommodating groove width value, the first rib width value between the magnet accommodating groove and the trimming part and the second rib width value between two adjacent magnet accommodating grooves are designed to Under the condition of avoiding flux weakening control to demagnetize the magnet, it can effectively reduce the leakage of magnetic flux caused by the rotor ribs, ensure that the rotor assembly provides the best output torque performance, and achieve the effect of improving the motor efficiency.

Another objective of this case is to provide a rotor lamination and a suitable rotor assembly. By optimizing the size and parameters, the difficulty of design can be simplified and the speed of product development can be accelerated.

To achieve the foregoing objective, the present application provides a rotor lamination suitable for a motor. The motor has a motor air gap width. The rotor lamination includes a main body, a plurality of peripheral edges, and a plurality of magnet accommodating slots. The center group of the main body is located on the axis of the motor. A plurality of peripheral edges are arranged around the outer side of the main body. A plurality of magnet accommodating grooves are arranged in the groove to accommodate a plurality of magnets of the motor, and a plurality of magnet accommodating grooves are arranged on the main body relative to the shaft ring, wherein each magnet accommodating groove accommodates a corresponding magnet, and the magnet accommodates The slot has a slot width value in the direction outward from the axis, and there is a magnet depth value between the magnet slot and the outer periphery of the main body, wherein the magnet depth value is greater than a first rate constant times the slot width After the value, subtract the calculated total value of the motor air gap width value, and the magnet depth value is less than the first multiplying constant multiplied by the accommodating slot width value plus the calculated total value of the motor air gap width value.

In order to achieve the foregoing objective, this case provides another rotor assembly suitable for a motor. The motor has a motor air gap width value. The rotor assembly includes a plurality of magnets and a plurality of rotor laminations. A plurality of rotor laminations are stacked along the direction of one axis of the motor. The rotor lamination includes a main body, a plurality of peripheral edges, and a plurality of magnet accommodating slots. The center of the main body is located on the axis. A plurality of peripheral edges are arranged around the outer side of the main body. A plurality of magnet accommodating grooves relatively accommodate a plurality of magnets in the grooves, which are ringed on the main body with the axis as the center, wherein each magnet accommodating groove accommodates a corresponding magnet, and the magnet accommodating groove faces the rotor from the axis The outer direction of the component has a accommodating slot width value, and there is a magnet depth value between the magnet accommodating slot and the outer periphery of the main body, wherein the magnet depth value is greater than a first multiplying constant multiplied by the accommodating slot width value and then subtracted The calculated sum value of the motor air gap width value, and the magnet depth value is less than the first multiplying constant multiplied by the accommodating slot width value plus the calculated sum value of the motor air gap width value.

Some typical embodiments embodying the features and advantages of this case will be described in detail in the following description. It should be understood that this case can have various changes in different aspects, all of which do not depart from the scope of the case, and the descriptions and drawings therein are essentially for illustrative purposes, not for limiting the case.

Please refer to Figures 1 to 5 first. The first figure is a three-dimensional structure view of the motor of the preferred embodiment of the present invention, the second figure is a plan sectional view of the motor structure of the first figure, and the third figure is a three-dimensional view of the rotor laminations of the preferred embodiment of the present invention. Fig. 4 shows the top view of the rotor lamination of the preferred embodiment of the present invention, and Fig. 5 shows the partial enlarged view of the rotor lamination of the preferred embodiment of the present invention. In this embodiment, the motor 1 at least includes a rotor assembly 2 and a stator assembly 3. Among them, the combination of the rotor assembly 2 and the stator assembly 3 is completed by adopting an outer stator and an inner rotor. In this embodiment, the stator assembly 3 has a hollow portion 31 and a plurality of windings 32 corresponding to a plurality of teeth portions thereof, and the rotor assembly 2 is disposed in the hollow portion 31 of the stator assembly 3. Among them, the rotor assembly 2 and the stator assembly 3 of the motor 1 are assembled to form a motor air gap width g. Preferably, the motor air gap width g is between 0.25 mm and 1.0 mm, but this case is not limited to this. In this embodiment, the rotor assembly 2 includes a plurality of magnets 27 and a plurality of rotor laminations 20. The plurality of rotor laminations 20 can be made of silicon steel material, but this case is not limited to this. A plurality of rotor laminations 20 are stacked along an axis C of the motor 1. The axis C is the center of the rotor assembly 2, and the rotor assembly 2 rotates around the axis C, which is also the center of the motor 1. In addition, in this embodiment, the number of arrangement of the rotor laminations 20, arrangement spacing, and thickness of a single piece... etc. can be adjusted and changed according to actual application requirements. Each rotor lamination 20 includes a main body 21, a plurality of peripheral edges 23 and a plurality of magnet accommodating slots 22. The center assembly of the main body 21 is located at the axis C of the motor 1. A plurality of peripheral edges 23 surround the outer side of the main body 21. A plurality of magnet accommodating grooves 22 are symmetrically arranged around the main body 21 centered on the axis C and penetrate the main body 21. After a plurality of rotor laminations 20 are correspondingly stacked along the axis C, the aforementioned plurality of magnet accommodating slots 22 can relatively accommodate a plurality of magnets 27 in space. In detail, a plurality of magnet accommodating grooves 22 respectively hold a corresponding plurality of magnets 27 in their respective grooves. The aforementioned magnet 27 may be, for example, a long bar-shaped cylindrical permanent magnet, but the present case is not limited to this. In this embodiment, the number of the plurality of magnet accommodating slots 22 is relative to the number of the plurality of magnets 27, that is, the number of both is the same, for example, both are 8. Each of the magnet accommodating grooves 22 sets the corresponding magnets 27, that is, the 8 magnet accommodating grooves 22 and the corresponding 8 magnets 27 respectively correspond to the center of the circle of about 45 degrees with the axis C as the center. Angular symmetry and one-to-one distribution and configuration, but not limited to this. In other embodiments, the number of the aforementioned plurality of magnet accommodating grooves 22 and the number of the plurality of magnets 27 may be 6, 10, 12, for example. In other words, the number of magnet accommodating slots 22 and the number of magnets 27 in this case can be expressed as 2N, where N is an integer, and N is greater than or equal to 3. In this way, the rotor assembly 2 can provide a 2N pole design, which will not be repeated here. In addition, the magnet accommodating slot 22 can also accommodate the magnets 27 one-to-many, but it is not limited to this.

(1)It is worth noting that, as shown in FIGS. 4 and 5, in this embodiment, the magnet accommodating groove 22 is in the direction from the axis C toward the outside of the rotor assembly 2 (or toward the peripheral edge 23 of the main body 21 The top) has a accommodating slot width value T, wherein the accommodating slot width value T is equal to or slightly larger than the thickness of the magnet 27, so that the magnet 27 can be firmly embedded in the corresponding magnet accommodating slot 22 without being separated. Preferably, the accommodating groove width value T is less than 15 times the motor air gap width value g, but this case is not limited to this. In this embodiment, the plurality of peripheral edges 23 are adjacent to each other and surround the main body portion 21, that is, the plurality of peripheral edges 23 are disposed around the outer side of the main body portion 21. In addition, there is a magnet depth value Md between the magnet accommodating groove 22 and the outer periphery 23 of the main body 21. In an embodiment, the magnet depth value Md is greater than a first multiplying constant K1 multiplied by the accommodating slot width value T and then subtracted from the motor air gap width value g, and the magnet depth value Md is less than the first multiplying constant K1 multiplied by the volume After the slot width value T is added, the calculated sum value of the motor air gap width value g. The relationship can be shown in the following formula (1):

(1)

In this embodiment, the first rate constant K1 is between 1.4 and 1.5. Table 1 is the output torque and torque ripple obtained by simulating different magnet depth values: Table 1 Magnet depth value Md Output torque [N. M] Torque ripple [%] K1×T-2g 28.6 1.7 K1×T-1.5g 29.19 1.65 K1×T-1g 29.99 1.6 K1×T-0.5g 31.01 1.55 K1×T+0g 31.89 1.7 K1×T+0.5g 31.47 1.85 K1×T+1g 31.16 2.12 K1×T+1.5g 30.38 1.73 K1×T+2g 29.81 1.56

Figure 6 shows the relationship between different magnet depth values and output torque. Figure 7 is a graph showing the relationship between different magnet depth values and torque ripple. As shown in Figures 6 and 7, when the magnet depth value Md designed for the rotor lamination 20 is within the range of formula (1), that is, the magnet depth value Md is greater than a first multiplying constant K1 times the accommodating slot width value After T is subtracted from the calculated total value of the motor air gap width value g, and the magnet depth value Md is less than the first multiplying constant K1 multiplied by the accommodating slot width value T and added to the calculated total value of the motor air gap width value g, the motor 1 can In the range of maximum output torque, the minimum torque ripple is obtained at the same time. Thereby, the resistance to demagnetization of the output torque of the motor can be improved, thereby achieving the effect of improving the efficiency of the motor.

(2)In addition, please refer to Figure 1 to Figure 5 and Figure 8. Fig. 8 is an enlarged view showing another partial structure of the rotor lamination of the preferred embodiment of the present invention. In this embodiment, an outer peripheral edge 23a may be further defined between every two adjacent peripheral edges 23, wherein the rotor lamination 20 further includes a plurality of arc trimming portions 24, which are respectively located between two adjacent magnet accommodating grooves 22 (Or between every two adjacent peripheries 23 mentioned above), a plurality of outer peripheries 23a on the outer side of the main body portion 21 are recessed toward the axis C, and there is a gap between the outer peripheries 23a and the bottom edge 24a of the trimming portion 24 The value of arc trimming depth Pd. The arc repair depth value Pd is greater than a second multiplying constant K2 times the accommodating slot width value T and then the motor air gap width value g is subtracted from the calculated sum value, and the arc repairing depth value Pd is less than the second magnification constant K2 times the accommodating slot After the width value T, add the calculated sum value of the motor air gap width value g. The relationship can be shown in the following formula (2):

(2)

In this embodiment, the second rate constant K2 is between 0.5 and 0.6. Table 2 is the output torque and torque ripple obtained by simulating different arc repair depth values: Table 2 Trimming depth value Pd Output torque [N. M] Torque ripple [%] K2×T-2g 29.4 2.21 K2×T-1.5g 30.65 2.14 K2×T-1g 31.78 2.05 K2×T-0.5g 32.63 1.85 K2×T+0g 31.89 1.7 K2×T+0.5g 30.98 1.85 K2×T+1g 30.28 2.03 K2×T+1.5g 28.8 1.73 K2×T+2g 27.5 1.65

Figure 9 shows the relationship between different arc trimming depth values and output torque. Figure 10 shows the relationship between different arc trimming depth values and torque ripple. As shown in Figures 9 and 10, when the arc trimming depth Pd of the rotor lamination 20 is within the range of formula (2), that is, the arc trimming depth Pd is greater than a second multiplying constant K2 times the accommodating slot After the width value T is subtracted from the calculated total value of the motor air gap width g, and the arc repair depth value Pd is less than the second multiplying constant K2 multiplied by the accommodating slot width value T and added to the calculated total value of the motor air gap width g, The motor 1 can achieve the minimum torque ripple within the maximum output torque range. In this way, the mutual influence of the magnet leakage flux and the air gap flux distribution/density in the motor can be improved, thereby achieving the effect of improving the motor efficiency.

(3)In addition, please refer to Figures 1 to 5 and Figures 8 and 11. FIG. 11 is an enlarged view showing another partial structure of the rotor lamination of the preferred embodiment of the present invention. In this embodiment, there is a first rib 25 between each magnet accommodating groove 22 and the corresponding arc trimming portion 24, which is located at the boundary of the bottom 24a of the arc trimming portion 24 from 23b to the magnet accommodating groove 22 The edge 22a has a first rib width value Rr. Wherein the rib width value Rr is greater than the accommodating slot width value T divided by the accommodating slot width value T and the motor air gap width value g, minus 0.5 times the motor air gap width value g, and is less than the total The groove width value T is divided by the sum of the accommodating groove width value T and the motor air gap width value g and then plus 0.25 times the motor air gap width value g. The relationship can be shown in the following formula (3):

(3)

(4)In addition, in this embodiment, there is a second rib 26 between every two adjacent magnet accommodating grooves 22, and the second rib 26 has a second rib width value Rt. Wherein the second rib width value Rt is greater than the accommodating slot width value T divided by the accommodating slot width value T and the motor air gap width value g plus 0.25 times the motor air gap width value g, and is less The accommodating slot width value T is divided by the sum of the accommodating slot width value T and the motor air gap width value g and then plus 1.25 times the motor air gap width value g. The relationship can be shown in the following formula (4):

(4)

The first rib width Rr between the magnet accommodating groove 22 and the trimming portion 24 and the second rib width Rt between the two adjacent magnet accommodating grooves 22 are thus designed to prevent the magnet from being weakened. Under the condition of demagnetization, the leakage magnetic flux phenomenon caused by the rotor ribs can be effectively reduced, and the rotor assembly 2 can provide the best output torque performance, thereby achieving the effect of improving the motor efficiency.

In summary, this case provides a rotor lamination and a suitable rotor assembly. According to the width of the motor air gap and the width of the magnet accommodating groove, design the magnet depth value between the magnet accommodating groove and the outer periphery of the main body, so as to obtain the smallest torque ripple within the maximum output torque range, thereby improving the motor Resistance to demagnetization of output torque. According to the width of the motor air gap and the width of the magnet accommodating slot, between two adjacent magnet accommodating slots, the arc trimming depth is designed concavely from the outer periphery to the axis to be within the maximum output torque range. Obtain the smallest torque ripple, thereby improving the mutual influence of the magnet leakage flux and the air gap flux distribution/density in the motor. Through the motor air gap width value and the magnet accommodating groove width value, the first rib width value between the magnet accommodating groove and the trimming part and the second rib width value between two adjacent magnet accommodating grooves are designed. In order to avoid the demagnetization of the magnet due to the weakening control, it can effectively reduce the leakage of magnetic flux caused by the rotor ribs, ensure that the rotor assembly provides the best output torque performance, and achieve the effect of improving the motor efficiency. By optimizing the size and parameters, the difficulty of design can be simplified and the speed of product development can be accelerated.

This case can be modified in many ways by those who are familiar with this technology, but it is not deviated from the protection of the patent application.

1: motor 2: Rotor assembly 20: Rotor lamination 21: Body part 22: Magnet storage slot 22a: At the edge 23: Perimeter 23a: outer periphery 23b: At the border 24: Trim the arc 24a: bottom edge 25: first rib 26: second rib 27: Magnet 3: stator assembly 31: Hollow part 32: winding g: Motor air gap width value C: axis Md: Depth value of magnet Pd: arc repair depth value Rr: first rib width value Rt: The second rib width value T: Width of the containing slot

Fig. 1 is a three-dimensional structural diagram of the motor of the preferred embodiment of the present invention. Figure 2 is a plan sectional view of the motor structure in Figure 1. Figure 3 is a three-dimensional structural view of the rotor laminations of the preferred embodiment of the present invention. Figure 4 shows a top view of the rotor laminations of the preferred embodiment of the present invention. Fig. 5 is an enlarged view showing the partial structure of the rotor lamination in the preferred embodiment of the present invention. Figure 6 shows the relationship between different magnet depth values and output torque. Figure 7 is a graph showing the relationship between different magnet depth values and torque ripple. Fig. 8 is an enlarged view showing another partial structure of the rotor lamination of the preferred embodiment of the present invention. Figure 9 shows the relationship between different arc trimming depth values and output torque. Figure 10 shows the relationship between different arc trimming depth values and torque ripple. Fig. 11 is an enlarged view showing another partial structure of the rotor lamination of the preferred embodiment of the present invention.

20: Rotor lamination

21: Body part

22: Magnet storage slot

23: Perimeter

24: Trim the arc

25: first rib

26: second rib

C: axis

Md: Depth value of magnet

T: Width of the containing slot

Claims (14)

A rotor lamination is suitable for a motor, the motor has a motor air gap width value, and the rotor lamination includes: A body part, and the center assembly of the body part is located at a shaft center of the motor; A plurality of peripheral edges are arranged around the outer side of the main body part; and A plurality of magnet accommodating grooves, the plurality of magnet accommodating grooves relatively accommodating a plurality of magnets of the motor in the groove, the plurality of magnet accommodating grooves are arranged on the main body part relative to the shaft ring, and each of the magnets The magnet corresponding to the accommodating groove, the magnet accommodating groove has a accommodating groove width value in a direction outward from the axis, and there is a gap between the magnet accommodating groove and the peripheral edge corresponding to the outer side of the main body A magnet depth value, Wherein the magnet depth value is greater than a first multiplying constant multiplied by the accommodating slot width value and then subtracted from the motor air gap width value of the operation sum, and the magnet depth value is less than the first multiplying constant multiplied by the accommodating slot width value Then add the calculated sum of the motor air gap width value. The rotor lamination of claim 1, wherein the first rate constant is between 1.4 and 1.5. The rotor lamination as described in claim 1, further comprising a plurality of arc trimming parts, which are respectively located between two adjacent magnet accommodating slots, and define an outer peripheral edge direction between every two adjacent peripheral edges The shaft center is recessed and has an arcing depth value, wherein the arcing depth value is greater than a second multiplying constant multiplied by the accommodating slot width value and then subtracting the calculated sum of the motor air gap width value, and the arcing The depth value is less than the second multiplying constant multiplying the accommodating slot width value and adding the motor air gap width value to the calculated sum value. The rotor lamination of claim 3, wherein the second rate constant is between 0.5 and 0.6. The rotor lamination according to claim 3, wherein a first rib is provided between each of the magnet accommodating grooves and the corresponding arc trimming portion, the first rib has a first rib width value, and The first rib width value is greater than the accommodating groove width value divided by the sum of the accommodating groove width value and the motor air gap width value minus 0.5 times the calculated sum of the motor air gap width value, and the first The rib width value is less than the accommodating slot width value divided by the accommodating slot width value and the motor air gap width value plus 0.25 times the motor air gap width value of the calculated sum. The rotor lamination of claim 5, wherein there is a second rib between two adjacent magnet accommodating slots, the second rib has a second rib width, and the second rib The width value is greater than the accommodating slot width value divided by the accommodating slot width value and the motor air gap width value plus 0.25 times the motor air gap width value of the calculated sum value, and the second rib width value The sum of the value of the width of the accommodating slot divided by the width of the accommodating slot and the width of the motor air gap plus 1.25 times the value of the motor air gap. The rotor lamination according to claim 1, wherein the number of the plurality of magnet accommodating slots is the same as the number of the plurality of magnets, the number is 2N, where N is an integer, and N is greater than or equal to 3. A rotor assembly is suitable for a motor, the motor has a motor air gap width, and the rotor assembly includes: A plurality of magnets; and A plurality of rotor laminations are stacked along the direction of an axis of the motor, wherein each of the rotor laminations includes: A body, the center of the body is located on the axis A plurality of peripheral edges are arranged around the outer side of the main body part; and A plurality of magnet accommodating grooves, the plurality of magnet accommodating grooves relatively accommodating the plurality of magnets in the grooves, and are arranged around the main body with the axis as the center, wherein each of the magnet accommodating grooves corresponds to For the magnet, the magnet accommodating groove has a accommodating groove width from the shaft center toward the outer side of the rotor assembly, and there is a magnet depth between the magnet accommodating groove and the peripheral edge corresponding to the outer side of the main body value, Wherein the magnet depth value is greater than a first multiplying constant multiplied by the accommodating slot width value and then subtracted from the motor air gap width value of the operation sum, and the magnet depth value is less than the first multiplying constant multiplied by the accommodating slot width value Then add the calculated sum of the motor air gap width value. The rotor assembly according to claim 8, wherein the first rate constant is between 1.4 and 1.5. The rotor assembly according to claim 8, wherein each of the rotor laminations further includes a plurality of arc trimming parts, which are respectively located between two adjacent magnet accommodating slots, and between every two adjacent peripheral edges It is defined that an outer periphery is recessed toward the shaft center and has a trimming depth value, wherein the trimming depth value is greater than a second multiplying constant multiplied by the accommodating slot width value and then subtracted by the motor air gap width value. Value, and the arc trimming depth value is less than the second multiplying constant multiplied by the accommodating slot width value and then plus the motor air gap width value. The rotor assembly according to claim 10, wherein the second rate constant is between 0.5 and 0.6. The rotor assembly according to claim 10, wherein a first rib is provided between each of the magnet accommodating slots and the corresponding arc trimming portion, the first rib has a first rib width value, and the first rib A rib width value is greater than the accommodating groove width value divided by the sum of the accommodating groove width value and the motor air gap width value minus 0.5 times the calculated sum of the motor air gap width value, and the first rib The part width value is smaller than the sum of the accommodating slot width divided by the accommodating slot width value and the motor air gap width value plus 0.25 times the motor air gap width value. The rotor assembly according to claim 12, wherein there is a second rib between two adjacent magnet accommodating grooves, the second rib has a second rib width, and the second rib width The value is greater than the accommodating slot width value divided by the accommodating slot width value and the motor air gap width value plus 0.25 times the motor air gap width value of the calculated sum, and the second rib width value is less than The accommodating slot width value is divided by the sum of the accommodating slot width value and the motor air gap width value, and then 1.25 times the motor air gap width value is added to the calculated sum value. The rotor assembly according to claim 8, wherein the number of the plurality of magnet accommodating slots is the same as the number of the plurality of magnets, and the number is 2N, where N is an integer, and N is greater than or equal to 3.

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