e-Bike Theory

Peter Williams, The Prospectory, October 2009


 We have just finished a 6 month trial of two electrically assisted pedal bikes in the Brecon Beacons National Park.  Hundreds of people tried the bikes, and we surveyed about 60 of them.  Electric bikes appear to work as a routine transport for many people on round trips of up to 20 miles.  You need to be active, but you do not need to be fit.  These findings are reported in detail in the trial report.

 This note provides technical background to the final report.  It covers the basic principles of electrically assisted pedal bikes, how and why they work the way they do.


 Bicycles are extremely efficient – certainly more so than walking, and faster.  They are more efficient even when you take into account that the cyclist is also moving the bike.

 How do we measure the effort required to pedal a bike?  The smart answer is “in Watts”.  Watts are the international units of power or ability to do work. Physicists and electricians speak in Watts, but older people in the UK and US tend to measure mechanical power in horsepower (HP). They think of Watts only when talking about electrical power, and find it hard comparing a 40 Watt light bulb to a 40 Watt central heating pump, let alone the 40 Watts of mechanical power it takes to pedal a bike at 7 to 10 mph on a flat road in still air.  Nevertheless, these are all measures of the capacity to do work, and the 40 Watt electric motor in a central heating pump would indeed be able to power a bike (though only on the flat, and you’d take a while to get going).

To propel a bike at 13 mph (still on the flat, and still with no wind) takes about 80 watts and to do so for 12½ hours would need 1000 (that’s 80 times 12½) Watt-hours of work – generally written as 1 kWh, or one kilowatt hour.  Electricity and gas bills are expressed in kilowatt hours, and 1 kWh of electricity currently costs about 12p in 2009, so it’s not a great rate of pay for a cyclist.

We express the power of cars in horsepower (bhp  – or PS  in parts of Europe), but the European standard measure is also the kilowatt or kW.   My car’s engine, for example, develops 96kW – enough to propel over a thousand bikes at 13 mph on the flat, and nearly 3 times the power produced by the Talybont-on-Usk turbine at maximum output.  Fortunately, my car doesn’t need all of its 96 kW to do 13 mph on the flat, but it is sobering to think that when accelerating very hard it is working hard enough to boil 32 electric kettles.

 My car’s diesel tank holds about 60 litres of fuel, enough to travel over 600 miles.  Since the engine does about 10kWh of work for every litre of diesel it burns, a full tank is enough for 600kWh of work.  Bought from an electricity board, that would cost £72 – which is actually a little bit more than the local supermarket charges me for diesel.  A kWh can move a bike well over 150 miles, but my car only about 10 miles, albeit much faster.  The bottom line is that a push bike is so efficient that a fairly small motor and battery can improve the speed and effort required to propel it.  And that’s basically what electric bikes provide.


 Although 80 watts will push a bike along at 13mph on the flat, it takes over 320 Watts to go at that speed up a 1 in 20 (5%) hill.  That’s because on the flat, at low speeds, the cyclist is working mainly to counter the frictional and rolling resistance of the bike and the wind resistance of himself and his bike.  Once he starts to climb a hill, however, he is also lifting his weight and that of the bike against gravity.  320 watts is getting on for ½ a horsepower, so not surprisingly, you need to be pretty fit and strong to generate that amount of power for any length of time.  Most of us slow down, so the climb takes longer but the effort is reduced.

 If you are prepared to tackle the 5% slope at 4½ mph (about brisk walking pace) you only need around 100 Watts, which most active people can manage (for a short while, at least).  And of course, if the bike has gears you change to a lower gear to make it easier.  Understanding why it is easier helps us understand how and why power assistance works.

 First, we need to distinguish work and effort.  The work needed to propel a bike on a given slope (or lack of it) at a given speed is completely independent of the gear you use.  To go up that hill at 4 mph is going to take 100 watts whether you try it in top gear or bottom gear.  At first sight that doesn’t make sense – you couldn’t possibly climb a steep hill (even at 4 mph) in top gear.  That is certainly true, but it isn’t because of the work that needs to be done, but because of how hard you would need to press on the pedals – i.e. the force you can generate.  Force and work are related, but are not the same thing.

 Physicists define work as force times distance moved.  The work rate needed to move a bike and rider weighing a total of 100kg up a 5% hill at 4.5 mph or 2 metres a second is about 100 Watts.  The force needed to do that at the rear wheel is therefore about 100W/2 m/s = 50 Newtons.  Newtons are the international unit of force, defined as the force needed to accelerate a 1 kg mass by 1 meter per second, per second.  Back wheel force is created by force on the pedals, transmitted through the chain via the gears.  If the pedal crank is half the radius of the wheel, you would have to push twice as hard on the pedal as the bike tyre pushes against the road, if the pedals went round once for every revolution of the wheel.  In fact, in bottom gear on my bike, the pedals only go round 0.73 times for each revolution of the wheel, so the force on the pedals is actually about 137 Newtons.  A force of 137 Newtons is equivalent to 1/6th of my weight.

But if I tried to pedal at the same speed in top gear, I would need a massive 370 Newtons, or getting on for half my weight, and since pedalling isn’t a continuous force, the peak force on the pedals would be close to my full weight.  And this, don’t forget, is required to move the bike at just above “wobble” speed.  To go faster than 4.5 mph up this hill would require substantial force eventually exceeding my weight, and while it is possible to press down with more than your full weight on a pedal, it’s very hard.

So that’s why we change down to pedal up hills.  Although the work we do is fixed by the speed, the force we have to exert on the pedals isn’t.  It is this force that ultimately determines what our “normal” cycling speed is, and the best gear to use in any situation.


 Electric motors (and their batteries) are like cyclists in some respects: they have a “comfortable” force they can exert and a work rate they can sustain.  They also have a peak force, and peak power, and just like cyclists these do not occur at the same speed.  An electric motor exerts its maximum force at or close to rest, but the amount of electrical energy converted into mechanical work is much less at low speeds.  Until the motor is spinning at its optimum speed, more of the electricity going through its coils is converted to heat.  If you forcibly stop an electric motor, but continue to put current through it, all the energy will be converted to heat and you can “burn out” the motor.

 At the other end of the scale, you can also run an electric motor faster than its optimum speed, but it will not do very much work and will not take very much current.  This is quite like pedalling downhill on a normal bike – there comes a speed at which you might as well not bother.

So having an electric motor on your bike reduces the amount of work the cyclist has to do.  The cyclist can use that help in two ways:

 1.     To go the same speed for less effort.

2.     To go more quickly for the same effort.

 … or anything in between.  How hard you have to push down on the pedal, and the speed at which the pedals go round, should vary linearly with the speed.  But twice the effort or pedalling rate (sometimes called the cadence) doesn’t actually make you go twice as fast.  This is because air resistance goes up with the square of the speed – there is roughly 4 times as much of it holding you back at 20mph as at 10mph.  And air resistance is a significant counter force at any speed above about 10 or 12 mph.  When you hurtle down a hill, it is air resistance that will ultimately keep your speed from increasing indefinitely.  Most of us would be too scared to find out what that speed actually is on a steep hill, so we brake.  Tour de France cyclists, on the other hand, don’t.  Not only that, they crouch down and tuck their elbows in to reduce the air resistance.

 Coming back to power assisted cycling, the amounts of power needed to make a noticeable difference to the cycling experience are actually not that great.  To pedal my bike on the flat at 10mph requires about 60 watts.  In middle gear on my bike, that means pedalling at a little less than 60 rpm, pressing down on the pedals with an average force of 57 Newtons, which is the equivalent of about 13 pounds (5.8 kg) weight.

 But suppose I have an electric bike which will do half the work for me.  Pedalling on the flat at 10mph still needs 60 watts, but now I only have to provide 30 of them.  So I can stay in the same gear and press down half as hard on the pedals, or press down the same amount but at half the speed.  That will require me to change up, in my case to top gear.

 In practice, I do something in between, and the reason for that is to do with how efficiently I operate – as distinct from how efficiently the electric motor does.  If I stay in middle gear, I’ll be pressing half as hard but pedalling just as fast as before.  Some of the muscular effort of pedalling goes into raising and lowering your knees, and the more times per minute you do that, the more of your energy it uses.  Professional cyclists pedal at high rates, so in fact they waste more energy pumping their legs.  But they choose to do that because they can generate more power at a higher pedal rate, so that the wastage as a proportion is lower than it would be at a more leisurely rate.  They burn more calories riding the way they do than you or I would.

 So in practice we tend to home in on a combination of force and cadence that we’re comfortable with.  And it’s likely that that will prove to be our most sustainable rate.  The trick is to find a cadence and gear that combine the most sustainable rate for the cyclist with the most sustainable rate for the electric motor and the battery.  And that is particularly true when it comes to hill climbing.

 Climbing hills uses a lot more power and requires a lot more work because on top of everything else you are raising the bike and rider against the force of gravity.  And the faster you choose to do that, the higher the work rate (although remember – modulo the effects of non-linear forces like air resistance – it takes the same amount of work (watt-hours) to climb to the top of a hill no matter how quickly or slowly you choose to do it).  The work is the same, but the quicker you do it, the greater the force you have to exert or the faster you have to pedal, or both.

 So if I only want to pedal as hard as if I was doing 10mph on the flat, I can’t expect to climb a hill at 10 mph.  Since I have gears, there should be a lower gear where I am still pedalling at 60rpm and pushing down with a force of 57 Newtons.   That produces 60 watts and 60 watts will get barely me up a 1 in 40 hill at 4 mph.  But if I had a motor putting 60 watts in as well, I could climb a 1 in 20 (5%) hill at a steady 5 mph.

What this means is that, for the effort it takes to pedal a bike without power assistance on the flat at 10 mph, with an electric bike I can climb a 1 in 20 hill at about 5mph.  In practice, I may be prepared to put in a bit more effort to climb the hill faster.  If it isn’t too long, I might be willing to input 100 watts of pedal power.  If the motor puts in the same, with 200 watts between us we can climb that same hill at 8mph, as long as the battery lasts.

If I do run out of battery, then I’d have trouble going up a 5% slope at all on just my 100 watts.  I’d be reduced to half speed, perilously close to wobble.  But if the battery holds around 250 Watt-hours, you can pedal about 20 miles up a 5% slope before it goes flat.  You would be pretty tired at the end of the 2½ hours it would take you, but nothing like as tired as you’d be if you had to do it in 5 hours unaided.

 This brings us neatly to the notion of battery range.  How far can an electric bike go on one charge?  As we have just seen, only about 20 miles up a 5% slope at 8mph, but theoretically as much as 70 to 80 miles on the flat at 10mph with no headwind.  In practice, of course, no road will ever be that flat for that long, and although you use less power going down a hill than you do going up, any uphill slope will reduce your range considerably. In the trial the bikes typically managed between 30 and 40 miles.


 Our trial featured two electric bikes with different designs.  One had a hub motor on the front wheel, controlled directly by a twist “throttle” of a type familiar to any motor-cyclist.  The other had a motor driving an extra cog on the chain.  These are the two main types of electric bike propulsion, though there are many more bikes with hub motors than chain motors.  Nearly all the chain-motored ones use a self-contained unit manufactured by Panasonic, but found in several different makes of bike.

The basic principles of biking and power assistance apply to both types, although it was possible to operate the hub motored bike without pedalling at all.  This greatly reduces the range and the steepness of hill that can be tackled, but it is also literally effortless, so proved popular with some triallists.

 With the hub motored bike, the rider chooses the level of power by twisting the throttle, and the amount of power he gets from the motor is independent of the amount he pedals.   There are different ways of using this kind of power system.  You can elect to use it only for climbing hills or accelerating from rest, otherwise pedalling without assistance.  This will maximise range.  At the other extreme, you can have full motor power on all the time, and top it up with pedal power only when going up hills.   This makes for an effortless and fast ride, but much shorter range.

With a hub motor you must learn how much help the motor needs to climb steeper hills.  If the motor runs too slowly to produce its maximum power, you may not be able to climb the hill at all, but if you pedal harder, you accelerate the bike to a speed where the motor can provide (paradoxically) enough power for you to be able to reduce your contribution. The unhelpful advice from an e-bike hill-climbing expert is that you should go up hills as fast as you possibly can, if you want to maximise the range of your bike.

 The chain motored bike has a different model of power assistance.  You can set 3 (well ok, 4) levels of power: none, half, equal, and 1.3 times the pedal input.  And although it’s not quite clear how the controller manages this, by “half” they mean half the effort (the force) you put in yourself.  So if you don’t pedal hard, neither does the motor!  This arrangement, which I personally prefer, is more like a servo-mechanism of the kind you find on the steering or brakes of a modern car.  You don’t have to make any decision about the amount of power you want from the motor, the bike simply amplifies your own effort by the amount you previously set.

 This difference in approach is most obvious when climbing hills.   As we have seen, this requires much more power for the same speed.  On the hub motored bike, up to a point, you can get all that extra power from the motor.  You yourself can pedal along at the same speed, calling for boost as and when you need it.  On the chain motored bike, when you hit a hill, you will slow down if you only make the same effort as before.  Most users agree that for most journeys, a hub motored bike will be faster than a chain motored bike of the same power.  But they also agree that a chain motored bike will go up much steeper hills on less power, and have a longer range than a hub motored bike.  It isn’t immediately obvious why this should be.  There are some clear technical reasons, but the psychological ones may be more important.

 If the cyclist is happy to put in X Watts, and the maximum power of the motor is Y Watts, then a hub motor has better performance in all conditions except climbing a hill faster than can be climbed with an effort of just X + Y Watts.  At that point either the rider must slow down until he finds the speed that can be achieved, or he must increase his own input to maintain his chosen speed.  That’s the physics of the situation, but the psychology is more complex.

 If he normally operates the throttle at full power, the hub powered rider quickly loses track of the relationship between his own effort and the speed it achieves.  When he hits a hill, the motor cannot provide more power so the bike slows.  If it slows too much, motor efficiency can drop to a point where it appears to run out of puff.  And the hub motor itself has no gears to change down to.

The first time this happens, he might conclude that this hill is too steep for his bike.  This could be true, of course, but hub motors are powerful and it is likely that if you change down the chain gears and pedal a bit harder yourself, you can reduce the power needed from the motor to a level that will enable it to take you up the hill at a speed you can manage between you.  But this may be counter-intuitive, because while you may have changed to a lower gear to increase the pedal power this has no direct effect on the motor power, which you probably boosted with the throttle, as you would in a car.  What you need to do is reduce motor power and increase pedal power until you find a speed you can maintain while climbing the hill.

 A chain powered bike offers a gentler introduction to hill climbing.  Just like an ordinary bike, when you reach a hill you slow down and change gear to find a speed at which you can successfully climb it.  This directly affects the power provided by the motor, but the effect is quite transparent.  If you can climb the hill between you, you will quickly and naturally find the speed at which you can do it, just as you would without power.  The only psychological hurdle you have to jump is to realise that despite normally bowling along at a much higher speed, you have to slow down to climb a hill.  Once you figure that out, there’s nothing you can’t climb with a chain powered bike, partly because you naturally increase your contribution, and partly because the motor is also taking advantage of the chain gears to run at closer to its optimum speed.

 So as far as performance is concerned, the psychological difference between the two may be more significant than the mechanical difference.  Hub powered bikes offer an experience closer to that of a car, where you tend to maintain speed independently of terrain until you encounter a gradient the car can’t manage.  You can normally use a hub motored bike, especially a powerful one, in this way.  But when you can’t it requires a bit of thought to work out the optimum way to proceed.  Chain powered bikes offer an experience much closer to that of a regular bike, they just make your legs seem much more powerful.


 The law governing electrically assisted pedal bikes in the UK is arcane.  Since there is no global standard, and most e-bikes are manufactured abroad, the e-bikes you can buy in the UK – a relatively small e-bike market – tend to reflect the regulations in their country of origin.  But they also reflect foreign cycling culture.

 In the UK, cycling is now mainly a leisure activity for enthusiasts – it is no longer a primary mode of transport.  People in the UK cycle for fun, tend to go quickly – 20mph is a common average – on sports bikes, wearing clothes designed for cycling.  In Holland and China, where bikes (and electric bikes) are very much more common, people cycle as a convenient way of getting around.  Their bikes tend to be sturdy reliable load carriers, ridden by people wearing normal clothes.  One consequence of this is that while Dutch or Chinese cyclists appreciate e-bikes and buy them in huge numbers, UK cyclists regard e-bikes as cheating and don’t buy them at all!

 Electric bikes are motorised vehicles, so the law restricts the performance of e-bikes whose owners want them treated as bikes.  In the EU the law restricts the power and the maximum speed to which the motor can take the bike.  A bike can be pedalled at higher speeds than this, but the motor must, by law, not operate beyond it.  These speed limits are set for safety, but also to fit in with the prevailing culture of a cycle-commuting rather than leisure or racing market.

 So an e-bike motor in the the EU cannot legally assist the rider above about 15 mph on public roads.  In Japan, where the Panasonic chain drive motors are made, an e-bike cannot provide full power above 10mph, and to conform to both laws, a Panasonic chain motor progressively reduces its output from 10 mph so that beyond 15mph it provides none.  Hub motors are subject to the same law, but implement it differently.  The one we used in the trial will, if requested, provide full power at all speeds up to 15mph but none beyond that.  Although this could be disconcerting, the effect is mitigated by progressively reducing the throttle to provide only as much top-up power as is required to maintain the chosen speed.  This preserves the battery, and means that ideally the rider will not call for power above the speed limit, anyway.


 Electric bikes are currently a “niche” product in the UK, because most UK cyclists are leisure and sports oriented.  They prefer high tech racing bikes, or off road bikes, and ride them for fun.  Many of them regard powered pedal cycling with contempt.  Almost everyone can ride a bicycle, but unless they become “cyclists”, most people don’t cycle very much after buying their first car or motor bike.  It’s just too much effort, unless you enjoy it.  It’s easy enough on the flat in still conditions, but climbing hills on a bike is a struggle for most people, and not enjoyable.

But a modest 250W motor and a 250Wh battery can make pedal cycling significantly easier for anyone.  It can flatten shallow hills and make quite steep ones doable without becoming breathless or sweaty.  Your average speed goes up from around 10mph to more like 14mph to 15mph.  And you can go to work or the shops, if they are less than 3 miles away, in much the same time as going by car if you take into account the time taken to find a parking space!

 So electric bikes might appeal to people who aren’t cyclists but occasionally wish they were.  The average annual mileage of an electric bike in the UK is said to be about 1200 miles, which is 4 or 5 times the average for un-powered bikes – a statistic which unfortunately reflects the fact that most UK bikes, most of the time, are gathering dust in garages.


11 Responses to e-Bike Theory

  1. Pingback: are the newer crank motors different to the earlier models ?

  2. john says:

    Say you have a crazy hill, about (20% – climbing 100m every 500m) and you want to carry 30 lbs worth of groceries in addition to yourself and the bike. But assume the hill is short, less than a third of a mile. What types of electric bikes can get you comfortably up the hill (at say about 7mph or faster) if you are only willing to put in enough effort via pedaling to sustain 10mph on the flat. Presumably no pedal assist bike would work for this, but can a throttle bike provide enough energy?

    • bcebicycle says:

      I regularly carry over 30 lbs on my Pedego City Commuter and have not issues climbing significant hills up and down for 5 miles. On some occasions I don’t even put in a 10mph flat land effort and can easily climb at over 10Mph. I weigh 180lbs. I have a 48v. 500watt motor. Your results may vary.

      • Prospector says:

        And I’m afraid mine do, because here in the EU 500 watt motors on electric bikes are illegal on roads. The authorities take the not unreasonable view that if you want a motor bike you should be subject to the relevant motor bike legislation. As it is, in the EU electric bikes must be pedal assist, and motor assistance must top out at 15 mph, but in the UK you can ride one when you’re 14 and you don’t HAVE to wear a helmet, pass a test, or carry insurance.

      • bcebicycle says:

        Prospector that is good info. I don’t agree that the view is reasonable, and this is why. There are to many variables when we discuss motor size to blanket all bikes with that a motor that large as motorbikes. On my 500 watt motor I cannnot keep up with the most simple of motorcycles. In my location anything under 50cc is considered a moped, and also does not require a helmet, age limit, or test to drive one. That is what attracts me to them is the simplicity and minimal impact on resources, both my own and the environments. The US is just now starting to see growth in the ebike sector and the way it all plays out is still a mystery. Clearly the UK has taken a stance compared to what we are seeing in some locations in the US. Are your electric bikes allowed on multi use paths?

      • Prospector says:

        You have a point, and the e-bike industry in Europe lobbied for the legislative restriction to simply apply to the speed up to which you’re allowed motor assist, rather than the power of the motor. This would, they said, the further development of low environmental impact “cargo” bikes and tandems. Still, you have to draw the line somewhere, and the motorbike industry would complain if their market was undercut by a class of vehicle not subject to the expensive rules and regulations they have to live with. In the UK, I think you need a helmet, a driving test, and insurance even to drive a 50cc “moped” (most of which no longer bother with the “ped”).

        But yes, you can ride an e-bike on any designated cycle path and indeed on many so-called “bridleways” – paths in mostly rural areas which are open to horse-riders but not to motorised vehicles which tend to be very restricted in the public off-road routes they can use.

    • Prospector says:

      It depends where you live. In the EU, technically, you are limited to a 250 watt motor which can only assist you up to 15mph and if you are yourself pedalling. Some recent chain or crank powered motors are more efficient and sophisticated than the ones we tested, but even if they put in their full 250 watts the physics is the same – with a full 250 watts of motor and your 50 watts of pedal you’ll have trouble making 7mph up a 1 in 5 hill. If you weigh as much as I do, and you have 15kg of cargo on top of that, you’re moving 100kg plus the weight of the bike, and to do that at 7mph up a 20% slope by my reckoning will need 750 watts from somewhere, and you’re only up for about 50 of them! If you’re lucky enough to live in the U.S., you should be able to do it with a 1kW motor.

      • Samuel Frade says:

        The maximum power that a vehicle’s electric motor can output such that it is considered a bicycle in the US is 750W and the top speed when using the motor alone is 20mph. Whether the motor can be engaged to reach higher speeds when used in tandem with pedaling is, to my understanding, a legal gray area.

  3. Pat whitton says:

    Thank you for … So far.. The most practical and applicable article on this subject i have found… I am designing an enclosed two wheeled vehicle.. Electric.. And human powered… The electrical load requirements you provide are from reality.. Exactly what i need to calculated my capacities.. Thank you again

    • Prospector says:

      Thank you. It sounds like an interesting project. Are you enclosing the vehicle for weather-proofing or for speed and energy efficiency? Most cycling speed world records use bikes with some kind of faring, because a regular bike and rider have a horrible drag (CD) factor. But at 10mph to 15mph, air resistance isn’t – as far as I know – a critical factor in energy use. Weight, on the other hand, certainly is because it affects the energy needed to accelerate the bike and to climb hills. Good luck with your design.

  4. Robert Ranger says:

    Very good paper and I have read quite a few. I am from Quebec where the BionX hub motor kit was created, I think the hub motor can be very efficient depending on how it is used. I ride mine between 26 and 32 + km/h usually. Yesterday I rode 60 km with half the charge of an 4 years old + second hand battery (37 V, 9,6 Ah) on a trail made on an ancient railroad track. I ride low on a cyclocross Norco bike, 80 + RPM cadence and use cycling clothing.

    Robert (rangeraudy@hotmail.com

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