## Friday, November 23, 2018

### Why you don't see tape wound foil conductors used on electric motors

Most electric motors are wound using conductors (wires) with a circular cross section. Some motors are wound using square or rectangle cross section conductors. However, I have never seen a motor with conductors that have a flat, tape-like cross section made from thin foils which are then wound like a roll of sticky tape.

 A comparison of the different winding topologies on a stator (black)
That was until this blog post by Shane Colton come to my attention.

This raised a question: Why is it that this style of winding is not used for electric motors?

The short answer: Unless you are working with extremely large motors, or motors that uses very large currents and low turn numbers, the copper fill factor for tape wound foil conductors will be much less than an equivalent round or square cross section conductor.

### Potential benefits for using a tape wound foil conductor

First, why use this style of conductors? The benefits may include:

1. A high copper fill factor. Perfectly stacked round conductors will have a maximum fill factor of ~0.9. Tape wound conductors may improve on this. (This turns out to not be true in most cases)
2. Ease of winding. A single wide tape may be easier to stack than many small conductors.
3. High conductor cross section while maintaining a small minimum bend radius. For an equal conductor area a flat foil will have a much smaller minimum bend radius than a round conductor.
4. Improved conductor utilisation at high frequency. Due to the skin effect, the centre of a round conductor will carry little current when operating at high electrical frequencies.
5. Possibility of improved cooling capacity. The large uniform area of a tape wound conductor may prevent hot spot formation and facilitate better heat conduction to the surface of a winding stack when compared to round wires.

### Potential negative consequences for using a tape wound foil conductor

Potential drawbacks to this style of conductors include:
1. Difficulty in winding a tape wound foil conductor due to the surrounding stator teeth getting in the way.
2. Stray flux crossing the conductor at right angles (normal to the flat surface of the foil) may significantly increase eddy current losses in a motor.
3. A high aspect ratio may mean that the electrical insulation occupies a significant fraction of the available area, reducing the copper fill factor.
Lets look at each of these drawbacks one by one:

### 1. Difficulty in winding around stator teeth

For small radial flux motors it may be difficult to automate the winding process. However, segmented radial flux motors or yokeless and segmented armature (YASA) axial flux motors, such as those made by Magnax, Emrax and Yasa, and used by Shane in his blog post, are becoming increasingly common.

 A YASA motor with segmented stator teeth.
A YASA axial flux motor has a stator constructed from individual core sections which can each be wound separately before final assembly of the motor. These motors could therefore be easily wound with a tape style conductor.

 Stator segments wound and ready for assembly of a Magnax motor
Therefore for segmented core motors winding a tape wound foil should pose no problem.

### 2. Induced eddy current losses

Lets consider the worse case scenario. Below is a 2D FEMM simulation of a open ended 30 mm diameter, 80 mm long mild steel rod with 10 turns of conductors added to one end with a current of 6A. Side note: Drawing lines in FEMM is a pain. I recommend using a dedicated CAD program like like F360 and then just exporting your model as a DXF and importing that into FEMM

Its quite clear that a large portion of the flux is crossing the windings placed at one end and so it would be expected that a foil conductor would have large induced eddy currents. To test if this is really a problem when operating at electrical typical electrical frequencies you would find in an electric motor I wound two rods with the same dimensions as mentioned above. One with round conductors and one with a foil conductor. The foil was cur to a width such that its CSA was the same as the round conductor and both had ten turns applied.

Using an AC BH tracer I thin applied a 1 kHz and 10 kHz sinusoidal current to the windings. The result:
 AC BH loops
Its clear that at 1 kHz there is little difference in the shape of the BH loop while at 10 kHz they are quite distinct. This indicates that even under this worse case scenario a 10 kHz frequency, which is well above the typical electrical frequency used by electric motors, is required to produce significant eddy currents in the foil conductors.

This suggest that for an electric motor, which will likely operate at a lower frequency and has less stray flux than this example, will not suffer from significant eddy currents in the conductors.

### 3. Copper fill factor

It is desirable to have as high a copper fill factor as possible when constructing an electric motor so as to reduce the current density in the conductors so as to minimise I^2R losses. The copper fill factor is impacted by:
• The stacking factor ( P_{d}): How efficiently a conductor is arranged in the slot.
• The copper to insulation area ratio: A thick layer of insulation gives a low area ratio and a low fill factor.
As previously mentioned, round conductors have a fundamental stacking factor limit of ~0.9. Square and tape conductors can have a theoretical 100% stacking factor.

I imagine that it is this desire for a high stacking factor without the need to purchase expensive square conductors that convinced Shane to use tape wound conductors.

Even though square and tape wound conductors can have a theoretical 100% fill factor, the copper fill factor (ratio of copper to total area) can never be 100% due to the need for an insulation layer in-between each winding. The thicker this insulation layer, the lower the copper (Cu) fill factor. Therefore the copper stacking factor is given by:

Cu fill factor =\frac{CSA_{Cu}}{(CSA_{Cu}+CSA_{Ins})}\times Stacking factor

According to this site, typical 24 gauge magnet wire has an insulation layer thickness of ~0.025 mm. For Shane's motor he used a single polyimide (Kapton) tape, which has a typical thickness of 0.03mm, to insulate just one side of his copper foil.

Using this information I have estimated the copper fill factor when winding a conductor with a cross sectional area of 13.1 mm^2 which is 2.54 mm wide, like that used by Shane's design. The spreadsheet can be found here.

Here are the results:
• Round conductor: 88.5 % Copper fill factor
• Square conductor: 97.3 % Copper fill factor
• Tape conductor: 94.5 % Copper fill factor
This is quite a respectable copper fill factor and would only be superseded by the use of square conductors. Shane is therefore justified in his choice of a tape wound conductor.

However, one thing to note here is that the currents used by Shane's motor are unusually high, and his winding turn number is quite low. You will more typically encounter motors that have dozens to hundreds of turns and that operate at well less than 50 A. In this situation a tape wound conductor may make less sense.

 A more typical motor that operates at < 50A with many turns per tooth

For example, if you wound a small motor with a tape conductor that has a cross section equal to an 24 gauge conductor (0.5 mm diameter, 0.205mm^2) onto a 10 mm stator tooth your fill factor is as follows:
• Round conductor: 75 % Copper fill factor
• Square conductor: 80.8 % Copper fill factor
• Tape conductor: 40.6 % Copper fill factor
All of the fill factors listed above are relatively low since the insulation layer is fixed while the motor size is reduced. The tape fill factor is also unacceptably small since the area per turn of the insulating Kapton layer at ~0.1 mm^2 is now comparable to that of the copper foil itself at 0.2 mm^2.

To really drive this point home lets consider the copper fill factor of Shane's motor if he were to use lower currents and a higher number of turns, and therefore smaller cross section conductors.

If you were to halve the peak current of Shane's motor while maintaining the same current density (i.e. also halve the copper cross section) then it no longer makes sense to use a tape wound conductor. At his point you are probably better off purchasing flat magnet wire or just sticking with round wire.

User Kindiana over on the Odrive Robotics discord channel also pointed out that motor manufacturers are even less likely to be interested in tape wound conductors for the following reasons:

• The extra weight and cost of copper is likely not worth the efficiency improvements at low torque for RC applications and is likely also why they don't stack windings closer together.
• The tooling for machine winding foil may be far more complicated for a one piece stator compared to round wires or bundles of round wires.
• There may be no readily available source of pre-insulated copper foils with custom dimensions, so manufacturers  may need to do their own insulation.

## Conclusion

At first glance a tape wound conductor may appear as an attractive means to increase your electric motors copper fill factor without the need for expensive square conductors. However, when you look at the numbers more closely it becomes apparent that the high aspect ratio of a tape wound conductor will likely result in a worse fill factor for most applications. Therefore careful consideration is required before you decide on a conductor geometry for your electric motor.

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.

## Saturday, November 17, 2018

### 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 assumptions 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, geometary 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.

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

The motor with less turns of wire will have a lower induced voltage produced by the rotor magents 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 high Kv motor to be the same as the low 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 lets 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 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.