# Gearing a mid motor

Go back to Middrive

A big part in designing a middrive is the gearing. The base rotational speed in a middrive is the human crank speed or cadence. Depending on the drivers needs it is between 75 (grandmother) and 105rpm (time trial driver). The motor to crank reduction must fit the motor speed, letting it spin at it's most power and good efficiency. I will show how this can be computed.

## Method 1: Gearing a given Motor and Battery setup

For this, relevant numbers are:

• the battery current limit [1] set the controller software
• the speed coonstant (KV) of the motor
• the motor phase resistance
• the motor eddy current losses

### Method 1.1: From this you can simulate the speed of highest motor power within a given battery and controller setup(motor_rpm@max_power)

A brushless motor would have its most power at half of it's unloaded rpm, if there was no current limit set in the controller. The point where it can possibly reach its most power is though between 50% and 90% of its no load rpm, depending on the controllers battery current limit. This point can be calculated, but it's not easy. There is at least a bunch of motor data out there. You can calculate it using PeakEff [2] for example or find it at ebikes.ca [3] for hubmotors. There are many other sources out there not listed yet.

now you can calculate the optimal reduction to the cranks: motor_rpm@max_power/cadence

Here is an example graph using ebikes.ca driving calculator with the motor_rpm@max_power point marked:

• speed@max_power: 29kph
• wheel circumference for 26" wheel: 26 * 2.54 *pi = 2.07meters
• motor_rpm@max_power = 29kph * 2.07meters * 1000 / 60 = 233.5rpm
• optimal reduction to the cranks: 233.5rpm / 85rpm = 2.747

In this example we could use a 16T motor sprocket if using a 44T sprocket on the cranks:

• 44T / 2.747 = 16.02T

### Method 1.2: Measuring the speed of highest output power with a test setup

Another way is to measure it using a cycle analyst or an equal measurement system, using the motor in a single speed or hub config: When accelerating with full throttle you should reach the speed at which the battery current starts to drop. Now you drive at (or at least very near) the speed of maximum output power - while accelerating further above that.

now you can calculate the optimal reduction to the cranks: motor_rpm@max_power/cadence

If your test setup is geared too fast for this test (current will just does not drop), just drive downhill and faster till the current drops. Or - if you can - shift down and repeat.