Selecting the best pole and slot combination for a BLDC (PMSM) motor with concentrated windings

The full spreadsheet used in this post is available for download. To modify the sheet just choose the 'Make a copy option' from the drop-down file menu.

Introduction 

Concentrated windings have a single coil per tooth and are commonly used on 'hobbyist' style BLDC (PMSM) out-runner motors.

A key advantage of concentrated windings is that they can be quickly and cheaply wound by machine as seen in the video below.

Even motors with very thick conductors can be wound by machine.

Other advantages include short end turns, which do not contribute torque to the motor and only increase the winding resistance, and space for effective air cooling. A major drawback of concentrated windings is that without careful consideration of the stator slot number and rotor pole number the performance of the motor will be poor.

This post will examine the advantages and disadvantages of different slot and pole combinations. The information is based off the paper 'Distribution, coil-span and winding factors for PM machines with concentrated windings' by S.E Skaar et al. and the book 'Design of Brushless Permanent-magnet Machines' by J. R. Hendershot and T. Miller. See the 'recommended reading' list above for more information on this book. The emotor.com glossary page is also useful for reference.

The winding factor

An electric motors 'winding factor' (not to be confused with its copper fill factor) is a number between 0 and 1 which represents the fraction of the armature current which is used to produce torque.

From emotor.com's glossary page the winding factor is defined as the following:

The winding factor for a specific winding expresses the ratio of flux linked by that winding compared to flux that would have been linked by a single-layer full-pitch non-skewed integer-slot winding with the same number of turns and one single slot per pole per phase. The torque of an electric motor is proportional to the fundamental winding factor.

Despite being so fundamental, the calculation of the winding factor is not often discussed in textbooks. I found that the paper mentioned above has the easiest to follow description, although I had to rely on the emotor.com description of an unbalanced winding as the solution provided in the paper appears to not cover all scenarios.

I will not be going into detail about the calculation of the winding factor. However, you can find the spreadsheet used to calculate the values shown in this post here. This spreadsheet includes up to 108 slots and 70 poles and is easily extended further where needed.

The winding factor for a 3 phase machine with a non-skewed rotor can be seen listed in the table below where the top row is the number of poles and the left-hand column is the number slots. Since this is a three-phase machine the number of slots increases by three and since a magnet has two poles (a pole pair) the pole number increases by two.

It can be seen that some slot and pole combination have a winding factor of one while others are approaching zero, or even negative. However, not all of these slot and pole combinations can be used as is described below.

1. Exclude windings where q < 0.25

The slot to pole ratio with consideration for the number of phases is designated by the variable q. If q is less than 0.25 then the arc covered by a rotor pole is now less than half a stator tooth. This results in multiple north and south magnet poles interacting with each stator tooth and so the torque generated by the motor is reduced. Therefore, q values of less than 0.25 are generally not considered feasible and can be eliminated. The images below were created with this winding layout tool. Note that in reality, the gap between each stator tooth would be much smaller.

Example winding layouts that give a q value that is not considered feasible.

The calculated q-values for different slot and plot combinations.

The winding factor of each slot and pole combination with those combinations that give q<0.25 removed.

2. Exclude windings where q > 0.5

Alternatively, if q is greater than 0.5 then it no longer makes sense to use a concentrated winding as a single rotor pole will span over multiple teeth. Instead, a distributed windings would be used. Therefore, these combinations can also be excluded for our purposes.

3. Exclude windings where Ns = Nm

If the number of slots (Ns) is equal to the number of poles (Nm) then the motor will produce large cogging torque and will no longer be self stating. This combination can therefore also be eliminated.

4. Removal of unbalanced windings and motors without symmetry

A motor with balanced windings will have the same number of coils for each phase per repeating segment of the motor. A motor must have balanced windings for operation. More information here. 

Also, it is ideal to have a motor that has a symmetry of at least 2 (i.e. the motor has two repeating sections) as this helps avoid unbalanced radial forces and noisy operation. In other words, a motor without any symmetry will produce its torque on only one side of the rotor. If a motor has a symmetry of two then its torque will be produced on opposite sides of the rotor, balancing the forces.

The slot and pole combinations that produced unbalanced windings and no symmetry can, therefore, be eliminated.

Based on the table above it can be seen that 12s10p and 12s14p are attractive combinations. This explains why these slot and pole combinations are so popular for 'hobbyist' out-runner electric motors.

5. Consideration of cogging torque

In addition to the winding factor, cogging torque is also an important consideration. Cogging torque creates vibration and noise during operation and acts to disturb the motor away from its desired position when used in servo applications. The cogging torque frequency is also closely correlated with the production of rotor losses both with and without current supplied to the armature.

The cogging torque frequency is given by the least common multiple (LCM) of the stator slots and the rotor poles. The figure below displays the possible slot and pole combinations with the winding factor replaced with the cogging torque frequency. 

Selecting as high a cogging frequency as possible is desirable as it reduces the amplitude of the cogging torque. With this in mind, 24s22p would be an attractive option with its 264 cogging steps per rotation and 0.949 winding factor as opposed to only 84 cogging steps per rotation for 12s14p.

6. Higher order harmonics

So far we have only been considering the fundamental winding factor. However, different slot and pole combinations also affect the winding space harmonics. This topic will be covered in more detail in a future post once I have a better understanding of the topic.

7. Additional considerations

From the last two tables above its easy to conclude that a 24s22p motor is 'better' than a 12s14p motor since it has both a higher winding factor and a higher cogging frequency. However, there are other factors that also need to be considered.

7. 1 Maximum electrical frequency

The electrical frequency (current sine-wave supplied to each phase) scales linearly with the number of pole pairs (Nm/2) of a motor. Therefore, doubling the pole count of a motor will increase its core losses by a factor of four since it scales with the square of the electrical frequency. Doubling the pole count will also double the back EMF produced and so twice the required voltage will be needed to drive the motor at the same RPM. However, rewinding of the motor with a lower number of conductors per tooth can be used to offset this increase in the back EMF and will not impact the efficiency of the motor. In addition, reducing the lamination thickness will help minimise eddy current losses, which scale with the square of the lamination thickness. See this spreadsheet for more details. At the extreme, the switching frequency of the motor controller used to drive the motor may also need to be increased in order to maintain a good approximation of a sine wave. This will incur additional dead time losses in the motor controller. 

The electrical frequency required to operate the motor at 10,000 RPM is shown below.

7. 2 Mechanical winding considerations

Trying to squeeze more slots into a small motor is not always possible. Increasing the number of slots while keeping the motor diameter the same typically means that multiple smaller conductors must be used in order to bend around the tighter radii and make use of the available space. Smaller conductors have a larger fraction of their total cross section taken up by insulation and so the copper fill factor of the motor can be reduced even if the winding factor is increased. A lower copper fill factor means a higher current density in the winding and so higher I^2R losses.

7. 3 Cooling considerations

Effective cooling is critical for a high power density electric motor. Having a large number of slots and small conductors may congest the air flow path and reduce air cooling effectiveness. This, in combination with higher core losses due to an increase in the electrical frequency, can reduce the power density of a motor.

7. 4 Tooth-tip leakage

Squeezing more slots into a motor may also require that the gaps between the teeth be reduced. Reducing the size of this gap can allow some flux to jump (or zig-zag) from one tooth to the next, reducing the effective torque produced by the motor.

7. 5 Rotor skewing

If you plan on skewing your motor in an effort to reduce its cogging torque then it is important that the required skew does not lower the winding factor too much. The reduction in the winding factor is given by the skew factor discussed here. This topic will be covered in more detail in a future post.

7. 6 Reduced rotor and stator back-iron (yoke) mass

Increasing the pole and slot count of an electric motor has the advantage of reducing the flux density in the back-iron. This is because the higher number of stator slots and rotor poles means that flux does not need to travel as far around the stator or rotor to make a closed magnetic path. For every doubling of the pole or slot number, the flux density is halved and so the thickness, and therefore mass, of the back-iron can also be halved. This acts to increase the power density of the motor. 

For example, Siemens appears to have increased the gravimetric torque density (torque per unit mass) by 50% for their 260 kW light aircraft electric motor by increasing its pole and slot count from an already high 36s30p to 72s60p. Note that this assumption is based solely on the images from their promotional material.

Conclusion

The tables above provide guidance when selecting the number of slots and poles for a BLDC (PMSM) motor with concentrated windings. If the diameter of your motor is less than approximately 60 mm and if your maximum RPM is less than 10,000 then 10s14p or 12s14p may be a good choice. For larger diameter and/or motors that rotate more slowly a higher slot and pole count can be used, such as 24s26p with a reduced number of turns per tooth and thinner Fe-Si laminations. Higher pole count motors will have a higher electrical frequency, increasing losses, but will also have a lower mass since a reduced back-iron thickness can be used while also having a smaller cogging torque amplitude.