# Understanding e-bike power/torque

In the UK e-bikes are currently limited to a maximum of 250W assistance if you wish to use them on public roads. However, it seems that the motors do vary in terms of the available torque - from around 40Nm on some of the lighter e-bikes to 80Nm on those that are described as more "powerful".

Now clearly they are not more powerful as they putting out the same 250W max, but what I don't quite understand is what difference that extra torque makes.

I'm used to measuring my power output in watts and thought there was a linear relationship between power and speed, so I don't quite understand where torque comes into it, and what difference this makes in practise.

• The relationship between power and speed isn't linear at all -- it's approximately cubic. To double speed requires about 8x as much power; to increase speed 25% from, say, 12mph to 15mph requires about a doubling of power. Torque tells you about acceleration, not top speed. Feb 14, 2019 at 15:37
• Yep I shouldn't have said linear - what I meant was that speed would be determined by power output (and aerodynamics of course) Feb 14, 2019 at 19:31

Very quick answer - the torque numbers that are quoted are maximum torque values which do not correspond to maximum power.

Power = torque × rotational speed, so, at slow speeds the motor unit can provide more torque (and hence more acceleration) while staying within the power limit.

A bit more:

There a decent page here on the characteristics of DC motors. The essential takeaway is that DC motors can provide less torque the faster they spin.

Re:

[I] thought there was a linear relationship between power and speed.

The power at constant velocity is that needed to balance the forces resisting forward motion - rolling resistance and aerodynamic drag for a bicycle or other vehicle. Those resistance forces generally increase with the cube of velocity.

If you want to play with the numbers there is an interactive calculator here.

• Thanks - I should have paid more attention in physics class :-D Feb 14, 2019 at 19:19
• @JohnM YouTube can help you catch up. Power is work over time. youtube.com/watch?v=u-MH4sf5xkY Feb 15, 2019 at 4:13

Let's assume you ride an electric bike with wheel motor.
- You find the perfect grade on which you climb at 62.5W and 20Nm torque.
- The grade doubles. You continue your climb at the same speed, with 125W and 40Nm torque.
- The grade doubles again. You continue your climb with 125W and 40Nm torque at half speed on the "less powerful" bike.
- The grade doubles again. You continue your climb with 62.5W and 40Nm torque at a quarter speed on the "less powerful" bike.

Now, riding on the "more powerful bike" (250W with 80Nm torque), in the third case you will continue climbing at 250W and 80Nm torque at normal speed, and would need another doubling of the grade to go to 125W and 80Nm at half speed.

A motor with higher torque (all else being equal) will give you more power at low motor rpm, even if it is limited at the same power at higher rpm.

I'd like to add that there is some major handwaving about wattage in the ebike industry.

I explain, of course :

• The maximum wattage isn't really a technical limitation but a legal limitation
• customers want maximum power that fit the (legal) bill
• so we're in the very case of customer getting more than what's on the paper because "woaa that drivetrain is more powerful than that other one" even if, on the paper, they're both 250W
• there is no standardized method to determine the 250w legal limit
• sooo... most, if not all, manufacturer provide "continuous max power : 250w" that can be much much lower than the "peak power" (that could easily be 36v*20A=720W).
• while the torque number provided on paper are, indeed, max torque. (50-80NM in most case)

So, with that said, what now...

• You're on a hill, at max 50NM torque, with the engine running at 100RPM = 250w.
• Same hill, 50NM torque but at 200RPM = 500w
• more watt, you go faster at same torque.

Easy stuff in an ideal world. In reality it's a bit nightmareish because it's not linear at all. Plus add a few tons of torque and power limitation (eg : max current, heat, flawed/marketing rating, voltage drop, ...).

To be honest i don't even know if the 250w limitation is the energy consumption of the motor, or the work produced, which can be radically different : you block an engine (with whatever, brake, a wall, who care) and switch it ON, it will suck up the battery at whatever maximum Current (Amps) it can get, overheat, and so on, but effectively produce 0W of work since it will turn at 0RPM. Weird, huh ?

• I can't be completely sure that those writing the regulations haven't done something odd, but electric motor power is (in almost every case I've encountered) a measure of input power. That's an easy calculation for a simple DC motor. But of course input power under what circumstances is the question. It's certainly not stalled power consumption. Closer would be free running at the rated voltage. But a full battery may deliver more than that. Mar 21, 2019 at 6:49
• Thx for the editing @Criggie, i learned most of my english on internet and my vocabulary can be a little harsh ... "major handwaving" sound nice too :D Mar 21, 2019 at 9:11