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This article only applies to e-bike motors up to 250W

Abstract

This article describes how to simulate e-bike motors up to 250W, at different speeds and road inclinations: hub motors, 2-speed hub motors and mid-drive motors. The mid-drive motor can be simulated at a constant cadence. 

Notes

  • The simulations are not valid for the more powerful motors above 250W that are commonly used in the USA. At higher motor power there are serious differences in performance between different motor types.
  • The simulations are only about the motor itself, without the software. At mid drive motors, the software has a major influence on the motor behavior.
  • Direct drive hub motors have not been simulated since these are outdated concepts and far too heavy.
  • Driving in a headwind has the same effect as driving uphill.

Simulation notes

All small permanent magnetic motors have equal characteristics, so basically any motor can be used as base for the simulations. Here the simulations are done with the Cute Q85-SX motor. This motor is used for both the hub motor and the mid-drive motor simulation.

In the article "Permanent magnet DC electric motor tuning" we see that the efficiency of a motor increases with the speed. 

When driving uphill, the speed is low but the required power is high. A hub motor has a low-speed uphill and that is why the efficiency is very bad uphill. Because the speed of a mid-drive motor is constant, it performs just as well uphill as it does on a flat road.

Note that de hubmotor is taken as a whole unit, for example the motor constant k applies to the entire motor.

How to simulate a mid-drive motor in Excel

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. By switching between the gears, the cyclist achieves that the cadence is always optimal, for recreational cyclist this is about 80rpm. 

When simulating a mid-drive motor, you actually have to shift between gears to keep the cadence constant. In the Excel sheet I solved this conveniently with a trick: we pretend that an automatic continuously variable gear is used. So, we keep simply the motor speed constant in Excel at all speeds.

The gear ratio (motor-n / cadence) 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.

Limit indication with vertical red lines

If a maximum value is exceeded, this wiil be indicated by vertical red lines. The maximum values are entered in the tab Motor graph:

Tmax (Imax)     20 Nm
Umax       36 V
Pin_max 300?     300 W

Simulation examples

You can download the Excel sheet on GitHub here.

These values have to be entered into the Excel sheet:

Tab Motor graph

Motor parameters incl. gear      
R25 25°C       0,6 Ω @ 25°C
Mosfet 1x     0
T 50°C       50 °C
k       1,57  
Tf       0,82 Nm
effG gearbox efficiency   100 %
U (only needed for motor graph) 36 V
Tmax (Imax)     20 Nm
Umax       36 V
Pin_max 300?     300 W

Tab Power required

Slope       4 [%]
Assistance above   0 [W]
           
Bike parameters      
Total mass (90)   90 [kg]
Rolling resistance Cr (0,005) 0,005  
Air density ρ (1,23)   1,23  
Drag coefficient Cw (1) 1  
Reference area (0,5)   0,5 [m²]
Transmission eff.   100 [%]
Wheel diameter (0,7 / 0,65) 0,7 [m]

Tab Efficiency mid drive

Cadence (80)   80 [rpm]
gear ratio (n/cadence) 2  

Slope 0%

Here are the graphs for a flat road with full motor support, without pedaling.
Two motors are slightly limited at maximum speed.
Note that we have to look at the graphs above 100W, here all 3 motors have almost equal efficiency.
We see major differences below 50W, but these are not relevant because of the low power.

Simulation 250W hub motor slope 0pr assistance above 0W
Simulation 250W hub motor slope 0% assistance above 0W
Simulation 250W mid drive motor slope 0pr assistance above 0W cadence 80rpm
Simulation 250W mid drive motor slope 0% assistance above 0W cadence 80rpm
Simulation Xiongda 2 speed slope 0pr assistance above 0W
Simulation Xiongda 2 speed slope 0% assistance above 0W

Slope 2%

Here are the graphs for 2% uphill with full motor support, without pedaling.

Simulation 250W hub motor slope 2pr assistance above 0W
Simulation 250W hub motor slope 2% assistance above 0W

Simulation 250W mid drive motor slope 2pr assistance above 0W cadence 80rpm
Simulation 250W mid drive motor slope 2% assistance above 0W cadence 80rpm

Simulation Xiongda high speed slope 2pr assistance above 0W
Simulation Xiongda high speed slope 2% assistance above 0W
Simulation Xiongda low speed slope 2% assistance above 0W
Simulation Xiongda low speed slope 2% assistance above 0W

Slope 4%

Here are the graphs for 4% uphill with full motor support, without pedaling.
A mid drive motor has a higher efficiency than a hub motor, about 80% over a wide range.

Simulation 250W hub motor slope 4pr assistance above 0W
Simulation 250W hub motor slope 4% assistance above 0W
Simulation 250W mid drive motor slope 4pr assistance above 0W 80rpm
Simulation 250W mid drive motor slope 4% assistance above 0W cadence 80rpm

We clearly see that we obviously must not use the Xiongda motor on the high speed mode when cycling uphill.

Xiongda used at a wrong speed uphil 4%
Xiongda used at a wrong speed uphil 4%

At a speed of 8 - 15km/h the performance of the Xiongda 2-speed motor is almost equal to a 250W mid drive motor.

Simulation Xiongda low speed slope 4% assistance above 0W
Simulation Xiongda low speed slope 4% assistance above 0W

Slope 8%

Here are the graphs for 8% uphill with full motor support, without pedaling.
We see that the efficiency of the Xiongda decreases at low speeds, but that is only the case if we do not pedal. Normally, we always pedal too on steep slopes. On the graph of 12% we see that the efficiency of the Xiongda is better at all speeds when we pedal too.

Simulation 250W mid drive motor slope 8pr assistance above 0W 80rpm
Simulation 250W mid drive motor slope 8% assistance above 0W cadence 80rpm
Simulation Xiongda low speed slope 8pr assistance above 0W
Simulation Xiongda low speed slope 8% assistance above 0W

Slope 12%

Here are the graphs for a steep hill of 12% where we pedal ourselves with a power of 100W.
The hub motor has a hard time going uphill, the loss is 50% and the motor may overheat.

Simulation 250W hub motor slope 12pr assistance above 100W
Simulation 250W hub motor slope 12pr assistance above 100W

The Xiongda is only a bit at a disadvantage on steep slopes like this one. Even then, the efficiency of the Xiongda is still 70% at a 12% slope.

Simulation 250W mid drive motor slope 12pr assistance above 100W cadence 80rpm
Simulation 250W mid drive motor slope 12% assistance above 100W cadence 80rpm
Simulation Xiongda low speed slope 12% assistance above 100W
Simulation Xiongda low speed slope 12% assistance above 100W

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