# How should one wind the coils for a BLDC motor to maximize torque, if nothing else matters?

I'm trying to build a brushless DC motor with six winding arms, pretty typical design. If I'm trying to maximize the torque output of my motor, what would improve the motor best:

• Extending the length of the arms, so more coils per arm?
• Keeping the lengths the same and increasing coil density?
• Having more arms, say 12 winding arms rather than 6?

Or something else?

• Have you checked any textbooks about the theory used in motors? Jan 12 at 8:06

The motor torque equation can be written as: $$\tau = \ell D B N i = K_t i$$

• $$N$$: effective number of windings
• $$\ell$$: rotor longitudinal length
• $$B$$: magnetic flux density
• $$D$$: rotor diameter
• $$i$$: winding current
• $$K_t$$: motor torque constant

Your question can be rephrased as "How to maximize $$K_t$$?". It makes mores sense if we take the weight of the motor into consideration. You can always use a bigger motor to get more torque. It is a common practice to design motors to maximize torque density, $$\tau/m$$, where $$m$$ is the motor mass. Using the torque equation, there are three main properties to consider:

## 1. Motor geometry ($$\ell D$$):

Assuming axial motor configuration, $$\frac{\tau}{m} \propto \frac{\ell D}{\ell D^2} = \frac{1}{D}.$$ Therefore, long skinny motors lead to higher torque density. Different motor configurations can affect torque density.

## 2. Flux density ($$B$$):

$$B = \frac{\Phi}{A} \quad\left(\frac{\text{magnetic flux}}{\text{area of the flux path}}\right)$$ Assuming the area of the flux path $$A$$, depends on the motor geometry and is fixed, we would like to maximize $$\Phi$$. The magnetic flux can be modeled in terms of the magnetomotive force $$M$$, and the magnetic reluctance $$R$$, very much like electrical circuits: $$\Phi = \frac{M}{R}$$ a. To maximize $$M$$, you should:

• Use a magnet with more field strength. This depends on the magnet material for permanent magnets.
• Use a thicker magnet, which is limited by your motor geometry.

b. To minimize $$R$$, you should:

• Form a closed magnetic circuit using ferrous materials, e.g. iron.
• Minimize the thickness of air gaps (or any materials other than iron) in the flux path.
• Consider using a ferrous rotor, but doing this will increase the rotational inertia of the rotor and the inductance of the windings.

## 3. Amp-turns ($$N i$$):

Assuming the motor geometry is kept the same, more turns ($$N$$) usually means smaller winding wire diameter. Smaller wire diameter decreases the maximum (continuous) winding current ($$i$$) before the wiring insulation melts. This trade-off means $$N i$$ approximately stays the same for different wire diameters, as long as you avoid too thick or too thin wires to ensure good packing. Insulation thickness is also important here.

Note: Different wire diameters lead to different voltage and current requirements, because it changes the winding electrical resistance.

This was a gross simplification of considerations that goes into designing BLDC motors. A very important factor that we have not discussed are the thermal properties of the motor, which is usually the main limitation on the instantaneous and continuous torque characteristics of the motor. If it would not melt, you could always apply more current to get more torque.

## Summary:

Q: Extending the length of the arms, so more coils per arm?

A: The length of the arms should be decided based on the geometrical constraints of your design. In general, doing this you will increase the torque but it will decrease the torque density of your motor (See 1).

Q: Keeping the lengths the same and increasing coil density?

A: Assuming good packing, the only way to increase the coil density without changing the motor dimensions is to reduce the wire diameter which reduces the winding current, canceling out the benefits (See 3).

Q: Having more arms, say 12 winding arms rather than 6?

A: The number of arms should be decided based on the manufacturing difficulty and costs, and the winding density, along with other factors. More arm volume could mean less wire volume. Similar winding densities will lead to similar performance. For a smooth operation, more than two winding arms are preferred.