BLDC (PMSM) end turns and torque production

Jan 2020 update: I have removed the discussion of end turns in iron-core motors from this post as I have since learned that the arguments used were flawed. I will address this topic in my next post.

End turns are not considered when calculating the torque of a hobbyist grade BLDC (PMSM) electric motor. This post explores why.

1.End turns and BLDC motors

End turns are the region of a BLDC motor's windings which extend out from the slots at either end of a motor.

Current supplied to a motor passes from one slot to the next via the end turns. However, end turns are not typically considered when calculating the torque produced by a BLDC motor in 2D FEA software packages such as FEMM.

Although end turns can be ignored when it comes to calculating torque, they do contribute to the resistance of the windings. It is therefore ideal to have the end turns be as short as possible when designing a motor so as to improve motor efficiency by minimising ohmic losses in the windings.

Electric motors with concentrated windings, like those found in most hobbyist outrunners, have shorter end turns than those used in motors with distributed windings. Motors that use concentrated windings are therefore more efficient in some use cases, such as in high torque density, low-speed applications. However, as only a limited number of slot and pole combinations are available for concentrated windings they are not suitable for all use cases.

2. Torque generation and end turns in core-less motors

To understand why end turns are not considered when calculating the torque output of a BLDC motor let us consider a simplified scenario. In the diagram below a single turn copper wire is placed inside a homogenous magnetic field provided by permanent magnets placed above and below the wire. The magnets are arranged so that the applied field direction is different for either side of the wire.

This situation is an analogous to core-less electric motors which have no ferromagnetic material in the stator. If a current flows through the wire then a Lorentz force is produced on that wire as described by:

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

where the magnetic field (B) is supplied from the magnets, the current (I) is that flowing through the wire of a given length (L). Looking at this setup from the top down we see the following.

It is clear that a Lorentz force is only generated on the wire which is located between the magnets, which is analogous to the wire located within the slots of a BLDC motor. The parts of the wire which extend outside of the magnets are not exposed to a magnetic field and so the force generated is zero. This situation is the same for end turns in a BLDC motor, and so is ignored by FEA software packages.

However, for the sake of argument, let us also consider a situation where the magnets could be made long enough that they also covered the end turns. In this situation, a force is generated by the end turns. However, this force is tangential to that generated by the wires in the 'slots' and so would not contribute to the production of useful torque within a motor. This force would also be mostly cancelled out as can be seen from below.

Another arrangement I have seen used by DIY motor makers is to take round coils and magnets and arrange them to make an axial flux motor or generator.

While this approach will work, it is also inefficient since the force produced on the wires is always tangential to the flow of current and so a significant fraction of the force is not available to useful work.

Note that although all of the above examples are for core-less motors, the situation is unchanged for iron cored motors.

In my next post, the concept of flux linkage will be explored and used to understand why end turns still do not contribute to flux in conventional iron-core motors.

4. Conclusion

End turns in air-core motor designs are not considered in the calculation of torque as they are either outside of the region exposed to the permanent magnet's magnetic field or the force generated is in the wrong direction.

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.