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.

Understanding BLDC (PMSM) electric motor constants - Optimal magnet length for high torque density

In the last post it was shown that the torque density of a motor can be improved by making the flux gap as small as possible. It was also seen that the rotor magnets are considered part of the flux gap. Therefore, it would appear that an ideal motor will always have rotor magnets that are as thin as possible.

However, it is also well known that the longer you make a permanent magnet, the larger the flux density at its surface. This raises an important question: Are long magnets and a large flux gap better than short magnets and a small flux gap when it comes to producing the most torque?

Small flux gap and magnets on the left, large magnets and flux gap on the right.

The short answer: 

For a 'hobby grade' out-runner electric motor the optimal magnet length will depend on your exact motor design, but in general, it will likely be around 1 to 4mm in length. Shorter magnets see a rapid fall off in their flux contribution to the airgap while longer magnets increase the magnetic reluctance of the magnetic circuit, reducing the stator contribution. Very long magnets will also cause the stator and rotor back iron to saturate which reduces performance.

Read on if you would like a more detailed understanding.

Permanent magnet self demagnetisation

Below are four magnets  modelled in FEMM with a length of 1, 2, 4 and 8 mm 

Despite each magnet being made of the same material there is a clear difference in the flux density present at the surface poles. The reason for this effect is that shorter magnets have a higher demagnetisation factor in that direction. The demagnetisation factor reduces the B field inside the magnet and is dependent upon the magnet geometry. The concept of a demagnetisation factor also applies to soft magnetic materials, not just permanent magnets. Long magnets will have a lower demagnetisation factor than shorter magnets. Unfortunately, there is no simple equation that can be used to describe the demagnetisation factor for something as basic as a cube. However, there is a simple relationship for an ellipsoid. Note that if you have a magnetic circuit that makes a closed circuit then the demagnetisation factor is zero but there are also no magnetic poles.

If we draw a line projecting out from the surface pole of each magnet and measure the flux density at each point we get the following plot.

Here we can see that the flux density in air for the 8 x 8 mm magnet at a distance of ~ 10 mm is the same as the surface (0 mm distance) for the 1 x 8 mm. This trend of increasing flux density with magnet length does not continue forever. The flux density in air at a distance of 0.5 mm from the surface of the magnets is plotted vs magnet length below.

As the magnet is made longer, and its demagnetisation factor in that direction decreases, the field produced at the surface poles would eventually approach that of the magnets remanent magnetisation. The plot above will look quite different if the same magnet was instead placed into the rotor of a motor since you then also have high magnetic permeability material in the stator and rotor helping guide the flux from the magnet, reducing its demagnetisation factor. However the overall concept remains the same.

Effect of magnet length on the torque produced by a simple motor

As in the previous post, we can use a simple model of a motor to test how different magnet lengths impact the torque produced for a fixed winding current. Below are four different scenarios. Each motor has a gap between the stator and the magnets of 0.5 mm. Therefore, the total flux gap is given by the magnet length + 0.5 mm. 

1 mm rotor magnet

2 mm rotor magnet

4 mm rotor magnet

8 mm rotor magnet

We can see a few things right away. First, the flux density in the rotor 'back iron' increases considerably as you make the magnets longer to the point that the back iron begins to saturate. This can also be seen in the stator teeth. Secondly, we can see that more flux escapes the rotor and fringes into the surrounding air. This is due to the saturation of the back iron. 

The flux density in the flux gap is plotted below with different length magnets. The flux contribution from only the stator windings was estimated by removing the magnets and simulating the motor. The same was done for the flux density contribution for the magnets, this time with the stator winding current set to zero.

It is clear that as you make the magnets longer the flux contribution from the stator becomes smaller due to the increase in the flux gap size. On the other hand, the flux density contribution from the magnets increases as they are made longer. Based on this plot we would expect that the torque will continue to rise as the magnets are increased in length. However, when the rotor torque is plotted with respect to the magnet length we can see that maximum torque is reached for magnets that are about 4 mm long. Further increasing the magnet length sees the torque slowly fall off.

This fall off in torque for magnets longer than about 4 mm is likely due to the stator core and rotor back iron beginning to saturate. Perhaps more interestingly is if we plot the specific torque density (torque per unit volume) and gravimetric torque density (torque per unit mass).  

When the magnets are made longer they are adding mass and volume to the entire motor while the torque gradually decreases. Therefore, there is a sharp fall off in the specific torque density for magnets longer than about 2 mm. Note that in the above example only the magnet length was changed. If more than one parameter was refined for, such as the thickness of the back iron, then the results will differ from those above.

In addition to just the torque output there are many other factors which need to be considered when you change the length of the magnets contained in a motor. A few that come to mind may include:

• Core losses are likely to increase when magnet length is increased as the stator and rotor iron is operating closer to saturation. Stray flux from the magnets may also cause more eddy current losses in the windings.
• Magnets are easily the most expensive part of a hobby grade electric motor. Therefore, motor cost will increase considerably if you were to use longer magnets, even if a redesigned motor did see a slight increase in torque with magnet length.
• Increasing the magnet length will add more mass to the rotor which increases its moment of inertia, reducing the dynamic response time of the motor.
• Cogging torque generated by the salient poles of the stator will likely be much worse with more powerful magnets
• There are many different grades of magnetic materiel. Using more or less powerful magnets would likely change the optimum magnet length.

Conclusion

Increasing the length of the permanent magnets used in a 'BLDC' (PMSM) motor will increase the torque produced only up to a point. The optimum magnet length will depend on many factors, but as a general rule of thumb, this length will be between 1 and 4 mm for 'hobby grade' out-runner electric motors constructed with back iron in the rotor. Further increasing the magnet length will only reduce the motor performance and increase the motors production costs.

If you have noticed any errors in the above article then please let me know. If you would like to play around with any of the models shown in this post in FEMM you can find the files hosted here. This tutorial gives you enough information to get started if you have never used FEMM before.

How to estimate the torque of a BLDC (PMSM) electric motor using only its Kv and current draw

There exists a fundamental relationship between an electric motors velocity constant `(K_{V}),` armature* current `(I_{A})` and torque `(\tau)`. This relationship is as follows:

`\tau \approx  \frac{8.3 \times I_{A} }{K_{V}}`

were `\tau` is in N.m, `I_{A}` is in A and  `K_{V}` is in RPM/V. This relationship is extremely useful since most 'hobby grade' BLDC/PMSM manufacturers do not publish the usual motor constants you would expect from an industrial product while `K_{V}` and peak `I_{A}` is normally specified on sites like hobbyking.com.

The above relationship works for two reasons:

1.  A motors `K_{V}` and its torque constant `(K_{\tau})` are fundamentally the same thing.
2.  The torque generated by a motor for a given current is governed by `K_{\tau}`.

Of course, there are limitations to this approach. If a motor is close to saturation or if the current waveform supplied by a motor controller does not exactly match a motors back EMF waveform (i.e. the current is not exactly on the q-axis at all times) then your torque will be less than that suggested above. However, my own testing shows that this approximation is quite accurate when a 'hobby grade' PMSM motor is driven using field oriented control (FOC). Even when not using FOC or a PMSM this approach should still provide a good starting point.

Read on if you are interested in a more detailed understanding of why this relationship works and the limitations of this approach.

Fundamental torque production

Electric motors do useful work by producing torque and rotation. The amount of steady state torque produced by a well optimised and non-salient machine** with a specific volume (specific torque density) is ultimately dependent upon three factors:

• The average flux density acting upon the armature. For a BLDC/PMSM motor this 'flux' is provided by the rotor's permanent magnets.
• The average current that can be maintained by the armature without overheating**. For a BLDC/PMSM motor the armature consists of the copper windings in the stator.
• The total length of the armature windings which has 'flux' acting upon it. For a BLDC/PMSM motor this length represents the number of turns of wire in the stator armature which interact with the 'flux' provided by the rotor.

In other words, all the complexity of an electric motor ultimately boils down to the "BIL" Lorentz force law:

Force = Magnetic field `\times`Current`\times` Conductor length

Increase one of the above terms without negatively impacting another and you have increased the torque output of your system. When discussing motor constants I find it useful to keep this simple relationship in mind.

Velocity constant  = Torque constant

In the 'hobby' community most people seem comfortable with the concept of a motors velocity constant `(K_{V})`. `K_{V}` is easily measured and is given by half the peak to peak back EMF generated line to line (across any two motor leads) for a given mechanical frequency with a unit RPM/V. 

What seems less well understood is that an electric motors torque constant `(K_{\tau})` is equal to `K_{V}` so that

`K_{\tau}^* = K_{V}^*`

where here `K_{\tau}^*` is a motors per-phase torque constant and `K_{V}^*` is a motors per-phase velocity constant given by half the peak to peak back EMF generated line to neutral (from the centre tap point on a star wound motor) for a given electrical frequency (Elec. Freq. = Mech. Freq. / No. Pole Pairs) with the units `\frac{V \cdot S}{rad}`.

When first learning about electric motors it took me a while to accept that `K_{\tau}^*` does indeed equal to `K_{V}^*`. The mathematics is clear and their SI units are equivalent but something about it just didn't feel right. To overcome this I find it helps to keep in mind that all aspects of an electric motor which impact its `K_{V}^*` (e.g. flux gap size, magnet strength, winding turn number, rotor length etc.) also impact the amount of torque a motor can produce for a given current.

 `K_{V}^*` is not especially useful since we are normally interested in a motors 'total torque constant' produced by all three phases and not that of a single phase. The 'total torque constant' of a 3 phase PMSM as estimated using its line-line `K_{V}` (RPM/V) is instead given by

`K_{\tau} = \frac{3}{2} \times \frac{1}{\sqrt{3}}\times \frac{60}{2\pi} \times \frac{1}{K_{V}}`

As is discussed by Oskar Weigl here, the factor `\frac{3}{2}` is derived from eq. 3.2 in this paper, the `\frac{1}{\sqrt{3}}` is for converting the line-line voltage (which is what is commonly used by hobby motor manufacturers in the determination of the `K_{V}`) to phase voltage (example here) and `\frac{60}{2\p}` is to convert from rpm to rad/s, which is needed to estimate the torque constant from the voltage constant, due to the voltage constant being specified in units of RPM/V. 

Torque constant determines torque output for a given current

`K_{\tau}` is also defined as

`K_{\tau} = \frac{\tau}{I_{A}}`

where  `I_{A}` was the armature current. Therefore the 'overall torque' output of a 3 phase PMSM is given by

`\tau = \frac{3}{2} \times \frac{1}{\sqrt{3}}\times \frac{60}{2\pi} \times \frac{1}{K_{V}}\times I_{A}`

This finally leads us to our 'conversion constant' of approximately 8.3 mentioned in the introduction

`\tau \approx 8.3 \times \frac{1}{K_{V}}  \times I_{A} \approx \frac{8.3 \times I_{A} }{K_{V}}`

and so the 'conversion constant' of 8.3 is just a simplified approximation for converting `K_{V}` to `K_{V}^*`.

It is important to point out here that this relationship is true for any three PMSM. Star, delta, big, small, high `K_{V}`,  low `K_{V}`, in-runner, out-runner, cored or core-less, 'weak' or 'strong' magnets, it doesn't make any difference. Equally important, the torque referenced above is the 'electromagnetic' torque produced by the motor. This torque represents a 100% efficient conversion from electrical energy to mechanical energy. The 'shaft' torque, which is the usable output torque of the motor, will always be less than the electromagnetic torque due losses in the system. If you would like to go deeper into all the topics described above than this then I highly recommend James Mevey's Master's thesis from Kansas State which which was recommended and summarised by Shane Colton in his blog.

All this theory is great, but lets see if it actually works in practice.

Measuring the motor constants of some real motors

In order to put all this theory to the test I have measured the motor constants for the following collection of 'hobby grade' out-runner motors.

`K_{V}` was first measured using the method described here. I took a photo of my oscilloscope at around 500 RPM for each motor as shown below.

1000 Kv Racerstar BR2212

190 Kv Keda 6364

150 Kv Odrive 6374

280 kV Turnigy SK3 5055

270 Kv Odrive N5065

All of these motors are 12N14P and of a similar construction. Its clear that the shape of the back EMF is sinusoidal and not trapezoidal. Therefore, these motors are technically permanent magnet synchronous motors (PMSM) and not brushless DC motors. Overall, the measured `K_{V}` of each motor was quite close to their labelled values as will be shown in a table later.

To estimate `K_{\tau}` the output torque of each motor was measured with respect to the armature current. This was done by winding a string around the rotor of each motor, attaching that string to a lever arm that then pulled down onto a laboratory balance. This balance then measured the weight, and therefore force, produced by the motor. This method is only possible thanks to the use of a high resolution encoder (8192 counts per rotation) and an Odrive motor controller which uses field oriented control (FOC) to place all current on the motors q-axis for maximum torque, even when stationary. 

The setup is crude and would have been much cleaner if I had just attach an arm to each motors rotor. However, by using a string I didn't need to make a new arm adaptor for each motor and could instead just consider the diameter of the rotor in the final calculations. This was enough to get the job done.

Using this setup and a simple script I slowly stepped up the commanded current for each motor  while manually recording the weight on the scale. After some back calculations the torque output for each motor with commanded current was found.

No surprises so far with the bigger motors producing more torque per amp and the torque increasing linearly with current. 

A summary of the motor parameters can be seen below and the raw data can be found here. 

		Racerstar BR2212	Turnigy SK3 5055	Odrive D5065	Keda 6364	Odrive D6374
Rated kV	rpm/V	1000	280	270	190	150
Measured kV	rpm/V	1058	276	259	182	151
Phase Resistance	Ohm	0.128	0.032	0.039	0.039	0.039
Phase Inductance	H	1.84E-05	1.33E-05	2.02E-05	2.13E-05	2.81E-05
Weight	kg	0.045	0.389	0.411	0.647	0.885
Price	$ USD	6	52	69	47	99
Torque constant (Kt)	N·m/A	0.008	0.029	0.030	0.042	0.053
Kt/kg	N·m/A/kg	0.172	0.073	0.073	0.065	0.060
Kt/$ USD	N·m/A/$ USD	0.00129	0.00055	0.00043	0.00090	0.00054
Motor constant (Km)	N⋅m/sqrt(W)	0.02	0.16	0.15	0.21	0.27
Km/kg	N⋅m/sqrt(W)/kg	0.485	0.417	0.369	0.329	0.308
Conversion constant		8.1	8.2	8.2	8.2	8.2

I have calculated the 'conversion constant' based on an average of a few different current vs torque measurements. The calculated 'conversion constant'  is actually very close to the theoretical value, with measured values between 8.1 and 8.2. Any error is likely due to the friction in the system and my less than ideal measurement setup. Also note that these values are based on the manufacture rated `K_{V}` of the motors and they are little lower if my own `K_{V}` is used instead. I'm not sure why this is the case but it may have something to do with the 'fudge factor' each manufacture assumes when estimating `K_{V}`.

Also listed is `K_{\tau}` with respect to motor weight and motor cost. Surprisingly, the smallest motor (1000 `K_{V}` BR2212) came out on top by a considerable margin in both cases, with a `K_{\tau}` more than double that of the other motors. This suggests that the electrical loading (current density in the copper windings) is much higher in the smallest motor when compared to the others. This same result could be achieved for the other motors by re-winding them to have a lower `K_{V}`. However, since the 'torque efficiency' (torque produced per Watt) is the same no matter how a motor is wound provided the amount of copper remains the same, and that the mass fraction of copper is likely to be about the same for these motors, the smallest motor will produce no more torque per unit weight than the rest when thermally limited. This assumption is backed up by the fact that the motor constant `(K_{M})` per weight is about the same for all motors tested. Its likely that the motor manufacture decided to wind the smallest motor this way so that its base speed matched that required by an appropriately sized prop and the typical battery voltage used on model aircraft and drones.

Limitations of this approach

Using `K_{V}` and a motors current draw to estimate its torque output only works provided that: 

• A motor does not produce any useful reluctance torque (i.e. its a  non-salient machine**). This is true for essentially all 'hobby grade' electric motors. 
• A motors torque increase linearly with current. This is not true if your motor is close to saturation. However, most 'hobby grade' electric motors are designed to operate with a current limit well below that needed to saturate them and so this is generally a safe assumption for stead state use.
• A motor is operated below its base speed. Operating a motor above its base speed by field weakening effectively lowers a motors `K_{V}` in that region and so unless you know by how much `K_{V}` is reduced you can not calculate the torque output. However, since no 'hobby grade' motor controllers (ESC's) that I know of utilise field weakening this is also not an issue.
• The current waveform supplied by a motor controller exactly matches a motors back EMF waveform (i.e. the current is not exactly on the q-axis at all times). This is generally true when using field oriented control (FOC) on a PMSM motor that has inbuilt position sensors (hall effect, encoder etc.). Operating a motor with 'six step 120 degree' commutation at low speed without a position sensor will result in less torque being produced than that predicted using the 'conversion constant' while high speed operation should be pretty close.

Conclusion

The torque produced by brushless permanent magnet synchronous motor can be easily estimated so long as its `K_{V}` and armature current is known. This relationship works because a motors `K_{V}` is fundamentally the same thing as its motor torque constant provided the right units are used, which is where a 'conversion constant' of ~8.3 is required.

* The armature is considered the winding in which a rotational 'back emf' would be generated if the motor were used as a generator. In some motor designs the armature is on the rotor (e.g. brushed DC motor) or in the stator (e.g. brushless DC motor).

** A non-salient machine in this context is any motor which does not derive useful torque from reluctance torque. A motor can have salient poles on the rotor or stator and still be considered a non-salient machine with this definition.

Equations were produced in this post with the help of arachnoid.com. If you have noticed any errors in the above article then please let me know.

Understanding BLDC electric motor constants - The Kv torque fallacy

It is a common misconception that if you have two otherwise identical electric motors, one with a low Kv and one with a high Kv, the lower Kv motor will be capable of producing more torque with less waste heat.

This assumption is incorrect.

The specific torque density of an electric motor (torque per unit volume) is independent of its Kv. Similarly, the heat generated by an electric motor while producing a given torque value is also independent of Kv. Read on to see why.

Torque produced by a BLDC motor for a fixed current density

The torque capability of a  BLDC motor is determined by the average magnetic field strength produced by the stator which acts on the rotor, the average magnetic field strength produced by the rotor magnets which act on the stator and the dimensions of the rotor itself.

If we have two otherwise identical motors, one with a low Kv and one with a high Kv, then we can assume that the average magnetic field produced by the rotor magnets and the dimensions of the rotor itself (i.e. its radius and length) are the same. This leaves only the average magnetic field strength produced by the stator as a possible difference.

The average magnitude of flux provided by the stator which acts over the entire surface of the rotor is determined by many factors (flux gap size, stator core material, the geometry of the motor etc.) but we can once again assume that these are all the same between our high and low Kv motors. Therefore, the only possible difference between our two motors can come from the average current density in the stator windings.

Looking at a cross-sectional view of the stator we can see that there is only a fixed area available to place the copper windings.

Let's look at just a single stator tooth and the impact that a different turn number will have on the applied magnetic field strength when placed in the available winding area.

The motor with fewer turns of wire will have a lower induced voltage produced by the rotor magnets as they pass by the tooth, giving it its high Kv rating when compared to the motor with more turns.

The high Kv motor has 4 turns of wire each at 10 A for a combined total of 40 A/tooth. The low Kv motor has 10 turns of wire each at 4 A, for the same total of 40A/tooth. Therefore these two motors will provide the same magnetic field strength and have the same torque output. 

Yes, you could increase the current in the low Kv motor to be the same as the high Kv motor at 10A and produce more torque. However, this is fundamentally no different than increasing the current in the low Kv motor with the same end result. Therefore, rewinding a motor to increase its Kv only makes sense when you wish to match the motor current draw to the current limit of your existing motor controller (ESC). You could just as easily achieve a higher torque output by purchasing a new motor controller with a higher current limit and keeping your existing motor unchanged. Alternatively, if you have a motor with a very poor copper fill factor (area in the stator slot filled with copper vs empty air) then it may also make sense to rewind your motor.

Note that for the purposes of this argument we are ignoring the production of any useful reluctance torque (like that used by a reluctance motor) which will be true for almost all motor you encounter as a hobbyist.

Now let's consider waste heat generation for our high and low Kv motors.

Waste heat produced by a BLDC motor for a fixed torque

The power dissipated by a motor winding is given by:

`P=I^2R` 

where I is the current in the windings and R is the resistance of the windings. As the power dissipation in the motor scales with the square of the stator current, it feels only natural to assume that the low Kv motor, with its 4A current draw, will produce less heat than our high Kv motor with its 10A current draw. However, this assumption fails to take into consideration that the total area of the copper windings is fixed and therefore the current density remains the same.

The DC resistance of a wire is given by:

`R_{DC}=\frac{l\rho}{A}`

where l is the wire length, `\rho` is the conductivity of the conductor and A is the conductor area. In order to simplify this argument lets assume we are using square cross-section conductors.

In order to fit more turns into the same area, we had to reduce the cross-section of each individual conductor, which reduced its area and therefore increased its resistance. If we assume that each turn of wire has a length of 1 and that the total conductor cross-sectional area is also 1, then enter the current and turn numbers listed above we find:

`P=I^2R=I^2\frac{l\rho}{A}=10^2\frac{4\rho}{1/4} = 4^2\frac{10\rho}{1/10} = 1600\rho`

Therefore, for a given torque (fixed current density), copper fill factor and copper winding area, the power dissipation is not changed by altering the motor Kv. 

If you do wish to increase the specific torque density of an existing BLDC motor then you have a few options:

1. Rewinding the motor to increase the copper fill factor by more efficiently packing the conductors.
2. Replace the permanent magnets in the rotor with higher energy density magnets.
3. Reduce the flux gap distance between the rotor and the stator.

Increasing your peak current output of your motor controller so that the motor windings run with a higher current density will of course also increase your peak torque output. However, this will require additional cooling to handle the extra waste heat and you run the risk of saturating the core material

Note that in the above example we assumed a DC resistance. In reality, a motor will operate with an AC current. At very high frequencies it may make sense to rewind a motor to use many parallel small conductors rather than singular thick conductors in order to minimise the skin effect.

Conclusion

You will not improve the specific torque density or lower the power dissipation for a given torque output by rewinding a motor to have a lower Kv. However, it can make sense to rewind a motor so that its peak current draw will be better matched with an existing motor controller. 

Equations were produced in this post with the help of arachnoid.com and are based on those found in the book Electric Motors and Drives: Fundamentals, types and applications by Austin Hughes. If you have noticed any errors in the above article then please let me know.

January 2021 addendum

An anonymous commenter has pointed out that the above argument does not consider the impact that changing the wire diameter has on the lengths of wire between each wound tooth or to the ESC. See the comment below for more details. In short, if you were to decrease the Kv of a motor by doubling the number of turns and halving the conductor area you may think that the total length of wire from one phase terminal to the next is also doubled. However, this turns out not to be the case because the length of wire from the ESC to the motor and from one wound tooth to the next does not actually change. Therefore, the total increase in the length of wire is slightly less than double, making the lower Kv motor technically more efficient at producing the same amount of torque. However, this effect is likely to be very small in most scenarios. 

Thanks, Anonymous!