Mid drive motor simulation

A mid drive motor has, as opposed to a wheel hub motor, an almost constant speed. Since a mid drive motor drives the crank, the motor speed is proportional to the cadence, which is the pedal speed. By switching between the gears, the cyclist achieves that the cadence is always optimal, for recreational cyclist this is about 80rpm. But note that many people on electric bicycles have a very low cadence. In the article "Permanent magnet DC electric motor tuning" we see that the efficiency of a motor increases with the speed. Since the speed of a mid drive motor can be high at all travel speeds, its efficiency is higher than a wheel hub motor. But as we will see later, the theoretical advantage of a hub motor is disappointing in practice. 

In this Excel sheet on GitHub, we can simulate a mid drive motor and a wheel hub motor. Take the worksheet tab "Efficiency mid drive η(kmh)".

This simulation is done with the Cute Q85-SX motor. This motor is used for both the hub motor and the mid-drive motor simulation, otherwise we cannot compare fairly. Note that de hubmotor is taken as a whole unit, for example the motor constant k applies to the entire motor. The motor has to have a higher speed than the bottom bracket because the motor is not designed for mid-drive of course, but that does not matter. This is now called the gear ratio (motor-n / cadence). The gear ratio should be chosen in such a way that the motor performs best over a wide range of speeds and slopes. I have figured out that in our case a gear ratio of 2 is optimal.

In the simulation, I assume the ideal situation, that the bicycle has an infinite number of gears. So, by switching, the mid drive motor speed remains constant at all travel speeds. These values must be entered into the Excel sheet:

• Cadence. Fill in your optimal cadence.
• Mid motor drive gear ratio. This determines the motor speed.

Try different gear ratios to get the best efficiency over a wide travel speed range. These graphs show the little higher efficiency of a mid drive motor:

In the next example, the cyclist pedals too; the Excel value "Assistance above" = 100W. Here, the wheel hub motor works in a range where its efficiency is higher; there is not much difference between the efficiency between both configurations:

Efficiency mid-drive motor vs hub motor 

Here we will see that the theoretical advantage of a hub motor is disappointing in practice. It is remarkable that a mid-drive motor has advantages only on steep slopes. 

These values are used in the Excel sheet:
Assistance above 50W
Cadence 70rpm

Slope 0°

Both graphs are practically the same.

Slope 4°

At a slope of 4°, the efficiency of a mid-drive motor is slightly better.

Slope 6°

At a slope of 6°, the efficiency is clearly better.At a slope of 6°, the efficiency is clearly better.
So we can conclude that the efficiency of a mid-drive motor is only better with steeper slopes.