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Electric vehicles
Dr. Jay Maron

Tesla Model S

Electric vehicles are simpler and quieter than gasoline vehicles. Gasoline vehicles have better range and better power/$.

Quietness is important for small vehicles, because there are cities with lots of small vehicles. Small GVs are loud.

GVs come with baggage like gears, a powertrain, combustion, a flywheel, a muffler, etc., none of which are present in EVs. In an EV, the powertrain is wires, and the motors can be on the wheels. The motors can be independent.

Sports car manufacturers have embraced EVs, such as Tesla, Porsche, Mercedes Benz, and Jaguar.

In an EV, the battery can be on the bottom of the car, which helps with stability. It can also be suspended.


Power

For small motors (less than 5 kWatt), GVs have a higher power/$ than EVs. GVs are 18 Watts/$ and EVs are 5 Watts/$.

            Power/Mass  Power/$  Energy/$  Energy/mass
             Watt/kg    Watt/$   MJoule/$   MJoule/kg

Gasoline motor      250     18      -         -
Electric motor      200     10      -         -
Battery, li-ion     500     10      .01       .6        Energy/$ is for the battery, not the electricity
Battery, lead acid  180     10      .02       .18       Energy/$ is for the battery, not the electricity
Generator           100     10      -         -         Gasoline to electric
Gasoline            100     10    45        47

For an ensemble of a li-ion battery and an electric motor, the overall power/mass is 5 Watts/$.


Speed

Driving a mid-sized car at 75 mph requires 40 kWatts, which are with reach of Lithium-ion batteries and gasoline generators.

The relationship power and speed is:

Power ∼ Speed3 * CrossSection

The challenge is range. Battery energy is easily enough to cross cities but not enough for long-distance freeway driving.

BatteryEnergy ∼ Range * Speed2 * CrossSection

For a typical electric car driving at city speed,

Car speed          =  V                  = 20    meters/second
Air density        =  D                  =  1.22 kg/meter3
Air drag area      =  K                  =  1.5  meters2
Air drag force     =  F  =  ½ K D V2     =366    Newtons
Air drag power     =  P  =  ½ K D V3     =  7.3  kWatts
Battery energy     =  E  =  F X          =100    MJoules
Range              =  X  =  E/(½ K D V2) =272    km



      Drag area (meters2)

Kick scooter     .4
Bike             .5
Compact car     1.0
Mid-sized car   1.5
SUV             2
Bus             5

An electric generator can extend range. Generators have poor power/mass but for ground vehicles, mass isn't a problem. If we equip electric vehicles with generators,

             Speed  Generator   Generator  Generator
              m/s   power (kW)  cost ($)   mass (kg)

Kick scooter   10       .25         25         1
Bike           15      1.0         100         5
Trike          20      4           400        20
Compact car    30     16          1600        80
Mid-sized car  30     25          2500       125
The generator is always a small fraction of the mass and cost of the vehicle.

If a lithium-ion battery and a generator have equal power then the battery is cheaper and lighter than the generator.

Battery cost  /  Generator cost  =  .4
Battery mass  /  Generator mass  =  .17
Batteries are good for short-term bursts of power and generators are good for long-term cruising. A vehicle should have both.

Car stats

Tesla Model S
BMW i8
Jaguar I-Pace
Mercedes ECQ
Formula-E
Lamborghini Diablo
Hummer H2

The best electric cars that are commercially available are Teslas and the Porsche Taycan.

                    Motor   100kph  Top  Battery  Battery  Battery  Battery   Car    Car    Year
                    power    time  speed energy    mass                       mass   cost
                    kWatt     s     m/s  MJoule     kg    MJoule/kg kWatt/kg   kg     $

Rimac C Two          1408    2.0    115   432                                 1950 2400000  2019  Concept car
NIO EP9              1000    2.7     87   324      635      .51     1.54      1735 1200000  2016  Concept car
Rimac Concept One     913    2.5     94   324                                 1850  980000  2013  Concept car
Lucid Air Dream       805    2.7     75   396                                 2300  230000  2021
GM Hummer EV          746                 720                                 4103          2021
Tesla Model S         615                 360      635      .57      .97      2250
Tesla Cybertruck Tri  600    3.0     58   720                                 3000   91000  2021
Porsche TaycanTurboS  560    2.8     72   301                                 2370  250000  2021
Porsche TaycanTurbo   500    3.2     72   336                                 2380  208000  2021
Chevy Silverado EV+   495    4.4          720                                               2023
Audi e-tron GT RS     475    3.3     69   336                                 2350  186000  2021
Ford F-150 Lightning+ 420    4.5          472                                 2995   90000  2022
Audi e-tronGTquattro  390    4.1     68   336                                 2300  135000  2021
Chevy Silverado EV    380                 720                                               2023
Ford F-150 Lightning  318                 472                                 2805   40000  2022
Porsche Taycan        300    5.4     64   285                                 2125  112000  2021
Jaguar I-Pace EV400   294    4.8     56   324                                 2208  104000  2021
Hyundai Ioniq 5       230    5.2     50   209                                 1800   48000  2021
Formula-E racecar     200    3.0     62   194      250      .78      .80       725          2018
Ford Mustang Mach-E   198    4.9     50   273                                 1993   85000  2021
Nissan Leaf e+        160    7.3     44   202                                 1756   52000  2021
BMW i3s               135    6.9     44   152                                 1365   57000  2021
Nissan Leaf           110    7.1     40   130                                 1850   40000  2021
Volkswagen ID.3       110    8.9     44   173                                 1725   40000  2021
Citroen e-C4          100    9.7     42   180                                 1650   47000  2021
Honda e-Advance       113    8.3     40   103                                 1595   53000  2021
Opel Corsa-e          100    8.1     42   180                                 1530   39000  2021
Fiat 500e hatchback    70    9.5     38    86                                 1350   32000  2021
Smart EQ fortwo        60   11.6     36    63                                 1095   25000  2021
Tesla Semi                   5.2                 11800                              150000  2020
Tesla Cybertruck Bi          5.0     54   432                                 2750   50000  2021
Tesla Cybertruck One         7.0     50   360                                 2600   55000  2021

Power sources

A major factor for performance is the battery. For high-performance batteries,

Power/mass  = 1200 Watts/kg
Energy/mass =  0.8 MJoules/kg.
Energy/$    = 0.01 MJoules/$

The top speed is almost always electronically limited.

Power sources tend to have a tradeoff between energy/mass and power/mass. The power sources for a car are:

                       Energy/Mass  Power/Mass   Ragone   Time
                        MJoule/kg   kWatts/kg   MJ*kW/kg2  seconds

Battery, Lithium polymer    1.0        1.0        1.0     1000
Battery, Lithium ion         .8        1.2         .96     667
Battery, Lithium titanate    .4        4          1.6      667
Supercapacitor               .05      20          1.0        2.5
Aluminum capacitor           .01     100          1.0         .1
Gasoline combustion motor   -          8          -          -
Electric generator          -           .2        -          -
Electric motor              -          8          -          -
Rocket, Kerosene + H2O2     -       2000          -          -


Energy/Mass   =  e          Joules/kg
Power/Mass    =  p          Watts/kg
Ragone number =  r  =  e p  Joule*Watt/kg2
Ragone time   =  t  =  e/p  second

Features:

NIO EP9              Adjustable downforce wing capable of 3.0 G cornering
                     The wing can produce 2450 kg of downforce at 67 meters/second
                     Active suspension, adjustable road clearance
                     Carbon fibre chassis and exterior
Tesla Roadster       Gas thrusters to enhance maneuverability
Porsche Taycan       Advanced motor cooling system
Rimac Concept One    Carbon-ceramic breaks

Top speed

The relationship between top speed and engine power is determined by drag.

Top speed        =  V
Fluid density    =  D  =  1.22 kg/meter2          (air at sea level)
Cross section    =  A
Drag coef, fluid =  C
Fluid drag force =  F  =  ½ C A D V2
Power            =  P  =  F V
Drag area        =  K  =  C A


                Top speed   Power  Drag area
                   m/s      kWatt   meters2

Bugatti Veyron      119.7    883      .84
Lamborghini SV       97.2    559     1.00
LaFerrari            96.9    708     1.28
Porsche 918          94.4    661     1.29
Lamborghini Diablo   90.3    362      .81
Nissan GTR           87.2    357      .88
Saab 900             58.3    137     1.13
Aptera 2             38.1     82     1.39

Bike, streamlined    38.7      1.4    .040
Bike, sprint         18        1.4    .40
Bike, cruise         10         .30   .49

eScooter Zoomair      7.2       .25  1.10
eSkate                5.3       .11  1.21

Sub, human power      4.1      1.4    .041
Sub, Virginia nuke   17.4  30000    11.4        Virginia Class nuclear submarine
Blue Whale           13.9

Skydive, min speed   40       30      .77       Mass = 75 kg
Skydive, max speed  124      101      .087      Mass = 75 kg

*:  The top speed is electronically limited

For human-powered vehicles we assume an athlete with:

Power/mass  =   20 Watts/kg
Mass        =   70 kg
Power       = 1400 Watts

Energy efficiency

The energy expended per person is proportional to the vehicle cross section divided by the number of passengers.

People            =  N
Distance traveled =  X
Air drag force    =  F
Energy            =  E  =  F X
Energy efficiency =  Q  =  E/(NX)  =  F/N


                        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
A full bus 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 should be favored over short flights.

Trains are not substantially more efficient than buses and they are far less flexible.


Acceleration

A hard acceleration requires more power than cruise driving. The power is delivered over a short timescale and can be suplied by can be supplied by capacitors or flywheels. These devices can also recover breaking energy, and they're better suited for this than batteries because they can be charged millions of times whereas batteries can only be charged thousands of times.

                                  Power/Mass  Power  Energy/Mass  Energy  Car mass
                                   Watt/kg    kWatt  (MJoules/kg)   MJ       kg

Car, Accel from 0-100 kph in 8 sec     97      145     .000386     1.16     1500
Car, cruise at 100 kp                  17       25     .000386

Flying electric cars

Flying electric cars are easy because lithium-ion batteries have a good power to mass ratio. For 1-person flying car, the minimum power/mass required to hover is 200 Watts/kg and a lithium-ion battery can produce 1000 Watts/kg.

Electric motors contribute neglibly to the mass of the car because they have a power/mass much larger than the battery (7000 Watts/kg). The propeller weighs even less than the motor.

Because motors are easy, a flying vehicle can have many motors and propellers, which helps with safety. Also, the more propellers a vehicle has, the easier it is to fly, because it reduces the minimum power/mass required to fly.

Electric motors are simpler and safer than gasoline motors. Gasoline motors come with baggage like gears, powertrains, combustion, flywheels, mufflers, etc., none of which are present in electric motors.

For a 1-person flying car,

                             Power/Mass
                              Watts/kg

Minimum for fixed-wing flight   120
Minimum for hovering flight     200
Minimum vehicle power/mass      300  = pveh     Should be easily able to hover. Use 1.5 times the minimum to hover
Lithium-ion battery            1000  = pbat

To hover, the battery mass has to be at least 3/10 the vehicle mass to provide enough power. The minimum battery mass fraction is pbat/pveh = 3/10.

Since a lithium-ion battery can easily power a hoovering vehicle, vertical takeoff is easy. There is no niche for a runway-based car.

The battery can be made larger to increase range. For fixed-wing flight, an electric flying car has a cruising speed of 50 meters/second and a range of 150 km.

Lithium-ion battery energy/mass=  e            =  1.0  MJoule/kg
Minimum for fixed-wing flight  =  pfix         =  100  Watts/kg
Battery mass fraction          =  f            =   .3
Flying time                    =  T = f e/pfix = 3000  seconds
Fixed-wing cruising speed      =  V            =   50  meter/second
Range                          =  X  =  V T    =  150  km

An electric generator has a power/mass of 200 Watts/kg, too small to fly. A small detachable electric generator could be potentially added for recharging off grid.


Mass scalings

For flight at constant velocity,

Number of propellers           =  N
Hovering minimum power/mass    =  p  ~  M1/6 N-1/6
Fixed-wing cruising speed      =  V  ~  M1/6
Fixed-wing cruising flight time=  T  ~  M-1/6
Fixed-wing cruising range      =  X  ~  M0

A single-person aircraft has a mass of ~ 400 kg and a hovering minimum power/mass of 200 Watts/kg. A 1 kg drone has a hovering minimum power/mass of 74 Watts/kg.


Battery energy and power

Voltage          =  V         Volts
Charge           =  C         Coulombs
Time             =  T         seconds
Electric current =  I  = C/T  Amperes (Amps)
Electric power   =  P  =  VI  Watts
Energy           =  E  =  PT  Joules
                       =  CV  Joules
Battery energy is often given in "Watt hours" or "Ampere hours".

1 Watt hour = 3600 Joules = 1 Watt * 3600 seconds

1 Amp hour = 3600 Coulombs = 1 Coulombs/second * 3600 seconds

A battery with a voltage of 3.7 Volts that delivers 1 Ampere for 1 hour has an energy of
Energy = 1 Ampere * 3.7 Volts * 3600 seconds = 13320 Joules


Hovering flight

Hovering propeller

For propellers,

Rotor radius     =  R
Air density      =  D  =  1.22 kg/meter3 at sea level
Rotor tip speed  =  V
Rotor width param=  Cr
Rotor lift force =  F =  D Cr R2 V2
Rotor drag force =  F
Rotor lift/drag  =  Qr =  F / F
Rotor power      =  P  =  F V  =  F V / Qr
Rotor force/power=  Z  =  F/ P
                       =  Qr / V
                       =  R F D½ Cr½ Qr
                       =  R F D½ qr
Rotor quality    =  qr =  Qr Cr½
The physical parameters of a propeller are {Qr,Cr,qr}, with typical values of
Qr = 5.5
Cr =  .045
qr = 1.17
Most propellers have 2 blades and some have 3. If there are 4 or more blades then qr declines.

The parameters are not independent. They're related through the blade aspect ratio.

K  ≈  Aspect ratio
Cr ≈  K
Qr ≈  K
qr ≈  K½

Hovering time
Aircraft mass        =  M
Gravity              =  g
Aircraft force       =  F =  M g
Rotor radius         =  R                  ~  M1/3
Hovering force/power =  Z  =  qr D½ R F  ~  M-1/6
Hovering power/mass  =  p  =  g / Z        ~  M1/6
Aircraft energy/mass =  e                  ~  M0
Hovering time        =  T  =  e / p        ~  M-1/6

Drive propeller

A drive propeller has to move substantially faster than the aircraft to be effective. This distinguishes it from a hovering propeller, which is designed to minimze propeller speed.

Rotor radius      =  R
Air density       =  D  =  1.22 kg/meter3
Aircraft speed    =  U
Rotor speed coef. =  s
Rotor tip speed   =  V  =  s U
Rotor lift force  =  F
Rotor drag force  =  F
Rotor lift/drag   =  Qr =  F / F
Rotor power       =  P  =  F V  =  F V / Q
Rotor force/power =  Z  =  Q / V
Typically, Q ~ 5.5 and s ~ 3.
Power/Mass ratio

A commonly-appearing quantity is the power/mass ratio, which is inversely proportional to the force/power ratio.

Mass              =  M
Gravity           =  g
Rotor quality     =  q
Hover force       =  F  =  M g
Hover power       =  P
Force/Power ratio =  Z  =  F/P
Power/Mass ratio  =  p  =  P/M  =  g/Z

Typical parameters
Air density       =  Dair=  1.22
Seawater density  =  Dwater= 1025
Gravity           =  g   =  9.8     meters/second2
Wing drag coef.   =  Cw  =   .03
Wing Lift/drag    =  Qw  =  7
Rotor lift/drag   =  Qr  =  5.5
Rotor width param =  Cr  =   .045
Rotor quality     =  qr  =  1.17  =  Qr Cr½
Rotor force/power =  Zr
Rotor agility     =  pr  =  g/Zr
Wing agility      =  pw

Propeller-driven level flight
Aircraft mass        =  M
Gravity              =  g
Air density          =  D  =  1.22 kg/meter3
Aircraft speed       =  U
Rotor speed coef.    =  s
Rotor tip speed      =  V  =  s U
Aircraft lift force  =  F  =  M g
Rotor lift force     =  F
Wing lift/drag       =  Qw =  F / F
Rotor drag force     =  F
Rotor lift/drag      =  Qr =  F / F
Rotor power          =  P  =  F V  =  F V / Qr  =  F V / (Qr Qw)
Aircraft force/power =  Z  =  F / P  =  [Qr Qw / s] / U
There is a tradeoff between Qr and s.
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