Main site of science textbooks
Crowdfunding site for the free
online science textbooks project

Electric vehicles

Electric vehicles outperform gasoline vehicles in all regards except range, and if you splurge on the battery you can have the range (and ludicrous power). Electric vehicles are more powerful, quieter, simpler, more flexible, and cheaper than gasoline vehicles, and you can put an electric motor on anything, even a rollerblade. Electric power is ideal for compact and cheap city cars.

Air drag

Air drag determines a vehicle's top speed and energy usage, and this determines the minimum battery size.

Air density        =  D  =  1.22 kg/meter3
Air drag area      =  A
Speed              =  V
Air drag force     =  F  =  ½ A D V2
Air drag power     =  P  =  ½ A D V3  =  F V
Range              =  X
Energy used        =  E  =  F X
Battery mass       =  M
Battery cost       =  S
Battery energy/mass=  e  =  E/M  =    .8  MJoules/kg
Battery power/mass =  p  =  P/M  =  1600  Watts/kg
Battery energy/$   =  s  =  S/M  =  .010  MJoules/$
A compact car designed for city speeds doesn't need much power. Example values for various electric vehicles:
                        Speed    Power     Force   Force/prsn  People   Range  Drag area
                         m/s     kWatt     Newton    Newton              km     m2

Skate                      10       .18     18         18         1       5       .3
Kick scooter               10       .18     18         18         1       5       .3
Bike                       15       .82     55         55         1       8       .4
Car, small, city speed     20      4.9     244        244         1      10      1
Car, large, freeway speed  30     33      1100       1100         1      15      2
Bus, freeway speed         30     99      3290         46        72      15      6
Train car, freeway speed   30     99      3290         27       120      15      6
Airbus A380               251 251000   1000000       1840       544   10000    160

1 Horsepower  =  746 Watts
Energy usage is proportional to the drag force per person. If a bus is full it is 5 times more efficient than a compact car, but buses are rarely full and usually slow.

Buses and trains are substantially more efficient than planes and they should be favored over short flights.

"Power" is the minimum power required for the given speed.

We assume a minimalist battery -- the smallest battery that can provide the given power. We then calculate the energy for this battery using the battery parameters and we caculate a range using this energy. Larger range can be achieved with a larger battery. Since a minimalist battery is cheap, a larger battery is usually feasible.

The drag parameter is obtained from an analysis of commercial vehicles. Data

Battery cost

Battery cost as a function of power is 20 Watts/$. A 1 kWatt bike battery costs $50, a 10 kWatt city car battery costs $500, and a 100 kWatt freeway car battery costs $5000. For city vehicles the battery is a small fraction of the vehicle cost and for freeway vehicles it's a significant cost.

Flying electric cars

A flying car powered by lithium-ion batteries can fly for 45 minutes and cover 100 km. The minimum price of the car is set by the battery. The smallest battery capable of powering a 1-person car costs $8000.

Flying cars will be capable of vertical takeoff and landing and will have 2, 3, or 4 rotors. The rotor number is determined by a tradeoff between efficiency (fewer rotors is better) vs. stability and failsafe (more rotors is better). The car will also have a wing to help with horizontal flight.

The properites of flying cars are determined by the properties of lithium-ion batteries and rotors. In the sections below we use these to construct a concrete design for a flying car.

Number of rotors

The design of the car depends on the physics of rotors. For a rotor,

Power required to hover  =  Constant * LiftForce3/2 / RotorRadius
The larger the rotor radius the better, so long as it's not so large as t dominate the mass of the car. We choose the design so that the total mass in rotors is half the mass of the pilot.

The most efficient copter has one lift rotor (a "monocopter"). Increasing the number of rotors while preserving the total rotor mass means that each rotor becomes smaller, hence it takes more power to fly.

Increasing the rotor number increases stability and redundancy. Most drones use 4, 6, or 8 rotors. 4 rotors offers good stability and failsafe and there is no point to a flying car with more than 4 rotors. Flying cars can be expected to have 2, 3, or 4 rotors.

The flight time is proportional to the battery mass, hence the battery should be as large as possible but not so large so as to dominate the car mass. We choose a design with a battery mass equal to the pilot mass. With this mass, the battery power is twice that required to hover, and so power isn't a problem.

State-of-the-art lithium-ion batteries have an energy/mass of .8 MJoules/kg and can fly a car for 44 minutes. In the future, lithium-sulfur batteries will take over with an energy/mass of 1.4 MJoules/kg.

We outline a design using 2 large lift rotors plus a few small stability rotors, with the following masses:

Flying car mass =  120 kg    (Excluding battery and pilot)
Battery mass    =  100 kg
Pilot mass      =   80 kg
Total car mass  =  300 kg

Total aircraft mass =  M        = 300    kg        (Includes passenger)
# of large rotors   =  N        =   2
Rotor radius        =  R        =   1.5  meters
Gravity constant    =  g        =   9.8  meters/second2
Rotor force         =  F = Mg/N =1470    Newtons
Rotor quality       =  q        =   1.02
Air density         =  D        =   1.22 kg/meter3
Rotor power         =  Pr=(qDR)-1F3/2= 30.2  kWatts
Hover power         =  Ph= N Pr =  60.4  kWatts
Hover power/mass    =      P/M  = 101  Watts/kg
battery mass        =  m        = 100  kg
Battery power/mass  =  p        =1200    Watts/kg
Battery power       =  Pb= p m  = 120    kWatts
Battery energy/mass =  e        =    .8  MJoules/kg
Battery energy      =  E = e m  =  80    MJoules
Battery $/energy    =  c =        100    $/MJoule
Battery cost        =  C = c E  =8000    $
Hover time          =  T = E/Ph =2650    seconds  =  44 minutes
The properties of propellers are discussed in the
propeller section. The rotor tip speed is
Rotor lift/drag  =  Q  =           5.5
Rotor tip speed  =  V  =  PQ/F  =  113 m/s
The ideal horizontal cruise speed is around 1/3 of the rotor tip speed. If we assume a cruise speed of 40 meters/second and a flight time of 44 minutes then the range is 106 km.
Battery pack strategy

Electric bike motors use either 36, 48, or 72 Volts. The following table shows how to build a battery pack for each motor power.

Power  Volts  Cells  Series  Parallel  Current   Cell      Cell    Cell  Cell  Cell
kWatt                                  Amperes  Amperes  Amphours   $    type  ID#

   .5   36     10
   .75  36     10      10       1         21      25       2.1      4     A    LG HD4
  1.5   48     13      13       1         31      30       2.0      4.5   A    Sony VTC4
  3     72     20      20       1         42      60       4.5      4.5   C    Basen
  6     72     40      20       2         83     120       4.5      4.5   C    Basen
 12     72     80      20       3        167     180       4.5      4.5   C    Basen

Cells     Total number of cells, equal to the number of cells connected in series
          times the number of cells connected in parallel.
Series    Number of cells connected in series. For example, 20 batteries
          with 3.6 volts each connected in series produces a voltage of 72 Volts.
Parallel  Number of cells connected in parallel.
Current   Current required to provide given power
Cell      Maximum current of a cell

Main page

Support the free online science textbooks project

© Jason Maron, all rights reserved.