From the monthly archives:

August 2010

Calculating the energy use and maximum range of ebikes

by Electric bike guru on August 9, 2010

Using some of the physics and battery science that were introduced in the previous two articles, we are now in a position to measure the energy use and maximum range of a given EBike motor and battery combination.
The first step is to estimate how much force is required to move a person on a non-powered bicycle. This force, called vehicle resistance is the force required to keep the bike moving at steady speed on a level road. If you were pushing a bike at steady speed on a level road (with the bicyclist not pedaling) then the vehicle resistance is just the force you would apply with your hands to the bike as you pushed it.

Electric consumption of ebikes is easy to measure

Electric consumption of ebikes is easy to measure

Total Vehicle Resistance
Total vehicle resistance has two components: it’s the sum of the rolling resistance and aerodynamic drag. Rolling resistance is just the force needed to overcome tire friction at very low speed. Aerodynamic drag is the wind force due to the forward motion thru still air. It you’ve ever stuck your hand out of an auto window and felt the wind force against it you’ve felt aerodynamic drag.
Rolling resistance for bicycles (and other rubber-tire vehicles) is just equal to the weight of the vehicle times a constant, known as the rolling resistance coefficient, which depends on the tire design, inflation pressure, and road surface roughness. Aerodynamic drag is approximately equal to a constant times the square of the velocity. This constant depends on the aerodynamic characteristics of the bike and rider.

At very slow speed where there is little wind resistance (aerodynamic drag), resistance is mostly rolling resistance. For a bicycle with properly inflated tires, the specific rolling resistance is roughly 1% of vehicle weight. If we estimate that the weight of the loaded bike and rider at 220 pounds, the rolling resistance is just 1% of 220 pounds, or 2.2 lb.
The aerodynamic drag must be added to rolling resistance to get the total resistance and energy used. Unfortunately bicycles have rather high aerodynamic drag. The typical resistance force for bicycles is:
F = 2.2 + 16.5 v2 pounds-force (v in miles/hour)

As speeds increase, the aerodynamic drag begins to dominate (since it’s proportional to v2). A typical bicycle speed on the level is about 12.5 mi/hr. At this moderate speed, aerodynamic resistance is 2.3 pounds, slightly more than the rolling resistance, and the total resistance goes up to 4.7 lb.

Mechanical Energy
Total vehicle resistance is also the mechanical energy required to travel a given distance. Now that we know the force required to push an EBike along, we can calculate the work required, or energy expended. Remember from the previous article, that work, W = f * d, where f is force and d is distance.

There are 5,280 feet in a mile. Therefore, the work required to travel one mile would be 4.7 lb X 5,280 feet, or 24,816 foot-pounds. For this to be useful as a measure of battery capacity, we will need to convert this to Watt-seconds (joules). The conversion factor can be found from a wide range of sources: 1 foot-pound = 1.356 watt-seconds.
Therefore, in the example above, 24,816 ft-lbs X 1.356 watt-seconds/ft-lb = 33,650 watt-seconds. Since there are 3,600 seconds in an hour, in the above example, 33,650 watt-seconds of mechanical energy converts to 9.4 watt-hours.

Battery Capacity
To compute the battery’s maximum range, we want to know how much of the battery’s total energy is consumed by traveling one mile. For an electrochemical battery, the total energy available before 100% discharge (assuming 100% efficiency, which, of course, is an inexact over- simplification) is the product of the battery voltage times the ampere-hours rating, which is the total current flow available. For example a 36-volt EBike battery that has an 8 ampere-hour rating has a total energy content of 8 X 36 = 288 watt-hours.
Thus, in moving our Electric Bike one mile, we have used 9.4/288 or 3.2% of the battery capacity. Further, based on the existing battery capacity, we can predict a maximum range of 100/3.2 or 30.8 miles. These same calculations can be repeated or modified for different situations, such as rider weight, speed, grade, etc. While some of these calculations may seem a little difficult to grasp at first, actually, the mathematics is quite simple, involving primarily multiplication and division.

I hope I have been able to break down here some of the concepts involved in understanding the energy use and maximum range of EBikes. Becoming educated in the physics and mathematics of electric Bikes can pay dividends in the long-run, saving time and money, eliminating confusion, and providing a means for making quantitative comparisons of the performance claims of competing EBike manufacturers.


EBikes Physics 101

by Electric bike guru on August 2, 2010

First, Some Basic Physics
Work is defined as the transfer of energy.  In physics, they say that work is done on an object when you transfer energy to that object.

For example, if a golfer uses a club and gets a stationary golf ball moving when he or she hits the ball, the club does work on the golf ball as it strikes the ball. Energy leaves the club and enters the ball. This is a transfer of energy.  And, before the ball was struck, the golfer did work on the club. The club was initially standing still, and the golfer got it moving when he or she swung the club.  So, the golfer did work on the club, transferring energy into the club, making it move, and the club did work on the ball, transferring energy into the ball, getting it moving..

Formula For Work
In the previous golf example the club places a force on the ball, and this force acts on the ball over the short displacement, or distance through which the club and the ball are in contact as the ball is being hit. Energy is transferred as the force acts over this displacement.  The amount of work is calculated by multiplying the force times the displacement. That formula looks like this:
W = F * d

Work is measured in a unit known as the joule.  One joule is the amount of energy required to move an object one meter (about three feet) using a force of one Newton, or about 0.225 pounds.  (Work can also be measured in foot-pounds.  One foot-pound is the energy required to move an object one foot using a force of one pound.)

Ampere’s Law of Magnetism
On an EBike, the force originates from an electric motor.  In an electric motor, coils of wire pushing against each other push a moving central hub around in a continuous circle.  Although the parts in a motor move in a circle, the physics is the same as what has been described above.
How do these coils of wire generate the force to do work by pushing an object, such as your Ebike?  Well, to answer this question, we need to go a little deeper down the rabbit hole.

an electrical current generates a magnetic field

an electrical current generates a magnetic field

Around 1820, Hans Christian Oersted discovered that an electrical current generates a magnetic field encircling it.  Then, in 1826, Andre-Marie Ampere published his Circuital Law, relating the magnetic field around a closed loop to the electric current passing through the loop.

Right hand rule magnetic force

Right hand rule magnetic force

This is often summarized as the “right-hand rule,” where the fingers of the right hand are curled, and the thumb is extended.  When a current travels around a loop in the direction of the curled fingers, a magnetic force is generated in the direction of the extended thumb.

To amplify this effect and make it practical to use in a motor, the wire “loop” is replaced by a coil, or multiple coils, each of which may contain several hundred such loops.  These coils are then rotated next to a magnet or magnets, and the two magnetic fields interact, producing forces of attraction and repulsion.

But magnets are heavy, and coils of wire are light.  So couldn’t the magnets be replaced by coils of wire which have a current passing through them?  Excellent suggestion!  And, in fact that’s just how modern, light-weight motors for EBikes are manufactured.

BionX Electric Drive Motor (cutaway view)

BionX Electric Drive Motor (cutaway view)

The intensity of the magnetic field produced by the loop coils is proportional to the electric current passing through them.  More current produces more a stronger field, and a stronger field produced more force.  But wait a minute.  Where does this current come from?  Well. the current for these coils comes, of course, from the battery which powers your EBike.  The two main types of batteries are SLA (Sealed Lead-Acid) and Li (Lithium Ion).

But we still haven’t explained just exactly what an electrical current actually consists of?  That’s also a good question, but unfortunately, we have used up our available space for this article.  To avoid keeping you in suspense unnecessarily, the answer to the question above is, in a word, “electrons.”  And in the next article, we will explore a little more about what is an electron, and find out how batteries manage to get these little things moving around in a circuit (and doing work.)  We will also see how the formula for work above (W = f *d) can be used to measure the amount of work done by an EBike motor, and to predict the maximum range of the combination of motor and battery that are on your EBike.