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Batman gadgets
Dr. Jay Maron


Electric propulsion

Tesla Model S

Electric propulsion is better than gasoline propulsion in all categories except range. Electric motors are quiter, simpler, more powerful, and more flexible than gasoline motors and they can be put on anything. In the future we can expect electric vehicles such as:

             Power   Max speed   Mass
             kWatts     mph       kg

Roller skate      .1     15        6
Kick scooter      .4     25        7
Bike             3       45       30
Car             60       85     1000
Flying car     150      120      400
Each kind of electric vehicle is expanded on below, each with a complete design based on current technology.
Energy and power

Energy          =  E          Joules
Time            =  T          seconds
Power           =  P  =  E/T  Watts
Mass            =  M          kilograms
Energy/Mass     =  e  =  E/M  Joules/kilogram
Power/Mass      =  p  =  P/M  Watts/kilogram

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
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

The performance of flying cars is determined chiefly by the battery energy per mass. For modern lithium-ion batteries, a car with the following performance is possible:

Flying time   =   44 minutes
Range         =   80 km
battery cost  = 8000 $        (Sets the minimum cost of the 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-sulfure 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        (Mass of the lightest commercial flying cars)
Battery mass     =  100 kg
Passenger mass   =   80 kg
Total 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.
Electric bikes

Electric bikes are easy to make. All you have to do is replace a conventional wheel with an electric wheel and attach a battery pack. Electric wheels come in kits and you can make the battery pack yourself. Example configurations for various motor powers:

Power    Max    Range   Motor   Battery   Battery
        speed           cost     cost     energy
kWatt    mph    miles    $         $      MJoule

   .75   30      10     160       40       .5
  1.5    35      20     240       60      1.2
  3      45      40     570      100      1.8
  6      55      80    1150      200      3.6
The bikes have one electric wheel and one conventional wheel except for 6 kWatt bike, which has 2 electric wheels with 3 kWatt each.

Electric wheel prices are from Amazon.com.


Electric bike speed limits
             Speed   Power   License
              mph    kWatt   required?

Connecticut    30    1.5     Yes
California     28     .75    No
Massachusetts  25     .75    Yes
Oregon         20    1.0     No
Washington     20    1.0     No
Pennsylvania   20     .75    No
Delaware       20     .75    No
Maryland       20     .5     No
DC             20     ?      No

Flight time

The flight time of a drone is determined chiefly by the battery energy per mass. Modern lithium-ion batteries

The flight time of a drone is determined by:
Typical parameters for a drone are:

Drone mass         =  M          =  1.0 kg
Battery mass       =  Mbat        =  .5 kg           (The battery is the most vital component)
Battery energy     =  E          =  .4  MJoules
Battery energy/mass=  ebat= E/Mbat=  .8 MJoules/kg
Drone energy/mass  =  e  =  E/M  =  .4  MJoules/kg
Drone power/mass   =  p  =  P/M  =   60 Watts/kg    (Practical minimum to hover. Independent of mass)
Drone power        =  P  =  p M  =   60 Watts       (Power required to hover)
Flight time        =  T  =  E/P  = 6250 seconds  =  104 minutes
The flight time in terms of component parameters is
T  =  (ebat/p) * (Mbat/M)

Fuel
Black: Carbon    White: Hydrogen    Red: Oxygen

Methane (Natural gas)
Ethane
Propane
Butane (Lighter fluid)
Octane (gasoline)
Dodecane (Kerosene)

Hexadecane (Diesel)
Palmitic acid (fat)
Ethanol (alcohol)

Glucose (sugar)
Fructose (sugar)
Galactose (sugar)
Lactose = Glucose + Galactose
Starch (sugar chain)
Leucine (amino acid)

ATP (Adenosine triphosphate)
Phosphocreatine
Nitrocellulose (smokeless powder)
TNT
HMX (plastic explosive)

Lignin (wood)
Coal

Medival-style black powder
Modern smokeless powder
Capacitor
Lithium-ion battery
Nuclear battery (radioactive plutonium-238)
Nuclear fission
Nuclear fusion
Antimatter


Vehicle power

Tesla Roadster

The energy sources that can be used by vehicles are:

              Energy/Mass   Power/mass   Energy/$   Rechargeable   Charge   Maximum charging
               MJoule/kg     Watt/kg     MJoule/$                  time          cycles

Gasoline            45                   60
Battery, aluminum    4.6       130                      No
Battery, lithium-ion  .8      1200         .010         Yes        1 hour      1000
Supercapacitor        .026   14000         .0005        Yes        Instant     Infinite
Aluminum capacitor    .010   50000         .0001        Yes        Instant     Infinite

Lithium-ion batteries

The properties of the best commercial lithium ion batteries are:

Energy/Mass     =    .8  Joule/kg
Power/Mass      =  1200  Watt/kg
Energy/$        =    .01 MJoule/kg
Density         =   3.5  gram/cm3
Recharges       =1000
Shelf life      =   1.0  year
Voltage         =   3.7  Volt
Energy/Mass and Power/Mass are an engineering tradeoff. One can be increased at the expense of the other.
Battery energy and power

Battery energy is often given in "Watt hours" or "Ampere hours".

Voltage          =  V         Volts
Charge           =  C         Coulombs    (1 Amphour = 3600 Coulombs)
Electric current =  I         Amperes
Electric power   =  P  =  VI  Watts
Time             =  T         seconds
Energy           =  E  =  PT  Joules
                       =  CV  Joules
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


Battery packs

A single battery is a "cell" and a set of cells is a "pack". Packs are used to multiply the energy and power of cells.

Battery packs are notorous for catching fire, but cell technology has reached the point where it's now possible to make safe battery packs, and the design is simple enough so that anyone can construct their own packs.

Cells can be combined in series and/or parallel. Connecting in series multiples voltage, and voltage is helpful for achieving high power in a motor.

Connecting in series is easier than in parallel. If it's possible to achieve the required power without parallelization then one should do so, and this is usually possible with modern cells.

Series packs have the advantage that the cells can easily be extracted and charged individually, and cells can be interchanged between packs. One can also construct a set of series packs and swap them in like gun clips.

High power electric bikes use a voltage of 72 Volts. If we use one series array of C cells then a pack provides 4440 Watts and 1.2 MJoules. Any electric device requiring less than this much power can be powered by a series pack.

The properties of a modern high-power cell are:

Type         =  "C"
Voltage      =   3.7 Volts
Energy       =  60   kJoules
Power        = 155   Watts
Mass         =  92   grams
Energy/mass  = 650   kJoules/kg
Power/mass   =1680   Watts/kg
Current      =  42   Amperes
Manufacturer = "Basen"
When the cells are connected in series the values for voltage and power are:
Cells   Voltage    Power
         Volts     kWatts

   1      3.7        .15
   2      7.4        .30
   3     11          .45     Electric kick scooter
   4     15          .60
   6     24          .90     Electric bike
  10     36         1.5
  20     72         3.0      Compact electric car
  96    356        15.0      Large electric car

Commercial lithium batteries

           Size   Charge  Current  Price
                 Amphours  Amps      $

Basen        C     4.5      60     8.0
Panasonic    B     4.0      15     8.0
Sony VTC6    A     3.0      30     8.0
Panasonic    A     3.5      10     5.5
Efest IMR    AA     .65      6.5   3.5
Efest IMR    AAA    .35      3     3.0
Prices from www.liionwholesale.com
Capacitors
Voltage          =  V             Volts
Capacitance      =  C             Farads
Total energy     =  E  =  ½ C V2  Joules
Effective        =  Ee =  ¼ C V2  Joules
Not all of the energy in a capacitor is harnessable because the voltage diminishes as the charge diminishes, hence the effective energy is less than the total energy.
Acceleration

Acceleration depends on the size of the battery, and supercapacitors can add an extra boost. For a typical car that accelerates from 0 to 100 km/h (37.8 m/s) in 8 seconds, the size of the battery required is:

Car mass          =  M             =  1200 kg
Target speed      =  V             =  27.8 m/s           (100 km/h. Speed at end of acceleration)
Kinetic energy    =  E  =  ½ M V2  =  464000 Joules
Time              =  T             =     8 seconds   (Time to accelerate from rest to speed V)
Engine efficiency =  Q             =    .8
Power             =  P  =  E/T     = 58000 Watts
Battey power/mass =  p             =   750 Watts/kg
Battery mass      =  MB =  P / (Q p)   =   112 kg
Battery cost/mass =  c             =    86 $/kg
Battery cost      =  C             =  9600 $

Recovering breaking energy

Supercapacitors are ideal for recovering breaking energy because they can be charged/discharged more times than batteries. To capture the energy from breaking from freeway speed, on order of 27 kg of supercapacotors are required.

Car mass                   =  M   =  1200 kg
Car velocity               =  V   =  27.8 m/s
Car kinetic energy         =  E   =464000 Joules
Supercapacitor energy/mass =  e   = 16000 Joules/kg
Supercapacitor mass        =  E/e =    29 kg

Lasers
       Power    $  Diameter  Length  Beam  Beam
       Watts          mm       mm    mrad   mm

Violet   .075   70   16.5     170     .5    4   wicked nano
Violet   .1     10   16.2                       lasers-pointers
Violet   .2     20   20       112               laserpointerpro
Violet   .5     30   24       148               laserpointerpro
Violet  1.0    100   24       180               laserpointerpro
Blue     .2     65                              freemascot.com
Blue    1.0     70                              freemascot.com


Color   Wavelength (nm)

Violet    405
Blue      445
Green     532
Yellow    589
Red       635

Flashlights
                   $  Lumens  Diameter  Mass    Lumens per
                                inch    ounce     inch2

Thrunite Ti4T       36   300    .55               1260
Thrunite Ti4        24   252    .55
ThorFire PF4        20   210    .6
Nitecore MT06       23   165    .55
Revtronic pocket    14   105    .6
Thrunite Archer 2A  36   500    .87     2.1        840
Revtronic 650       35   650   1.0
Fenix UC35          90   960   1.0
Barska TC1200      106  1200   1.0                1530
Fenix TK16          92  1000   1.3
Streamlight HL3     78  1100   1.6      7.1
Litecore TM03      158  2800   1.6                1390

MicroSD memory

Sandisk microsd cards on Amazon.com

GigaBytes   $

  32       13
  64       25
 128       45
 256      160

Appendix

Engine efficiency

Photoelectric cell
Thermoelectric generator
Stirling engine
Stirling engine

Electric car engine  .80
Gasoline engine      .15
Diesel engine        .20
Human muscles        .22
Biomass plant        .25
Natural gas plant    .35
Solar cell           .20     Crystalline type
Solar cell           .40     Multilayer type
Turboprop, Mach .4   .80     Turboprops work up to Mach .5
Turbojet,  Mach .4   .40
Turbofan,  Mach .4   .68
Turbojet,  Mach .9   .77
Turbofan,  Mach .9   .90
For an electric vehicle the overall efficiency is similar to that of a diesel engine.
Overall efficiency  =  Power plant efficiency  *  Vehicle efficiency  =  .35 * .80 =  .28

Conductivity

White: High conductivity
Red:   Low conductivity

Electric and thermal conductivity
         Electric  Thermal  Density   Electric   C/Ct     Heat   Heat      Melt   $/kg  Young  Tensile Poisson  Brinell
         conduct   conduct            conduct/            cap    cap                                   number   hardness
        (e7 A/V/m) (W/K/m)  (g/cm^3)  Density   (AK/VW)  (J/g/K) (J/cm^3K)  (K)         (GPa)  (GPa)             (GPa)

Silver      6.30   429      10.49       .60      147       .235   2.47     1235    590    83   .17      .37      .024
Copper      5.96   401       8.96       .67      147       .385   3.21     1358      6   130   .21      .34      .87
Gold        4.52   318      19.30       .234     142       .129   2.49     1337  24000    78   .124     .44      .24
Aluminum    3.50   237       2.70      1.30      148       .897   2.42      933      2    70   .05      .35      .245
Beryllium   2.5    200       1.85      1.35      125      1.825   3.38     1560    850   287   .448     .032     .6
Magnesium   2.3    156       1.74      1.32      147      1.023   1.78      923      3    45   .22      .29      .26
Iridium     2.12   147      22.56       .094     144       .131   2.96     2917  13000   528  1.32      .26     1.67
Rhodium     2.0    150      12.41       .161     133       .243   3.02     2237  13000   275   .95      .26     1.1
Tungsten    1.89   173      19.25       .098     137       .132   2.54     3695     50   441  1.51      .28     2.57
Molybdenum  1.87   138      10.28       .182     136       .251            2896     24   330   .55      .31     1.5
Cobalt      1.7    100       8.90       .170               .421            1768     30   209   .76      .31      .7
Zinc        1.69   116       7.14                          .388             693      2   108   .2       .25      .41
Nickel      1.4     90.9     8.91                          .444            1728     15
Ruthenium   1.25   117      12.45                                          2607   5600
Cadmium     1.25    96.6     8.65                                           594      2    50   .078     .30      .20
Osmium      1.23    87.6    22.59                          .130            3306  12000
Indium      1.19    81.8     7.31                                           430    750    11   .004     .45      .009
Iron        1.0     80.4     7.87                          .449            1811          211   .35      .29      .49
Palladium    .95    71.8                                                   1828
Tin          .83    66.8                                                    505     22    47   .20      .36      .005
Chromium     .79    93.9                                   .449            2180
Platinum     .95                                           .133            2041
Tantalum     .76                                           .140            3290
Gallium      .74                                                            303
Thorium      .68
Niobium      .55    53.7                                                   2750
Rhenium      .52                                           .137            3459
Vanadium     .5     30.7                                                   2183
Uranium      .35
Titanium     .25    21.9                                   .523            1941
Scandium     .18    15.8                                                   1814
Neodymium    .156                                                          1297
Mercury      .10     8.30                                  .140             234
Manganese    .062    7.81                                                  1519
Germanium    .00019                                                        1211

Dimond iso 10    40000
Diamond     e-16  2320                                     .509
Tube       10     3500                                                Carbon nanotube. Electric conductivity = e-16 laterally
Tube bulk          200                                                Carbon nanotubes in bulk
Graphene   10     5000
Graphite    2      400                                     .709       Natural graphite
Al Nitride  e-11   180
Brass       1.5    120
Steel               45                                                Carbon steel
Bronze       .65    40
Steel Cr     .15    20                                                Stainless steel (usually 10% chromium)
Quartz (C)          12                                                Crystalline quartz.  Thermal conductivity is anisotropic
Quartz (F)  e-16     2                                                Fused quartz
Granite              2.5
Marble               2.2
Ice                  2
Concrete             1.5
Limestone            1.3
Soil                 1
Glass       e-12      .85
Water       e-4       .6
Seawater    1         .6
Brick                 .5
Plastic               .5
Wood                  .2
Wood (dry)            .1
Plexiglass  e-14      .18
Rubber      e-13      .16
Snow                  .15
Paper                 .05
Plastic foam          .03
Air        5e-15      .025
Nitrogen              .025                                1.04
Oxygen                .025                                 .92
Silica aerogel        .01

Siemens:    Amperes^2 Seconds^3 / kg / meters^2     =   1 Ohm^-1
For most metals,
Electric conductivity / Thermal conductivity  ~  140  J/g/K

Magnetic field magnitudes
                                     Teslas

Field generated by brain             10-12
Wire carrying 1 Amp                  .00002     1 cm from the wire
Earth magnetic field                 .0000305   at the equator
Neodymium magnet                    1.4
Magnetic resonance imaging machine  8
Large Hadron Collider magnets       8.3
Field for frog levitation          16
Strongest electromagnet            32.2         without using superconductors
Strongest electromagnet            45           using superconductors
Neutron star                       1010
Magnetar neutron star              1014

Dielectric strength

The critical electric field for electric breakdown for the following materials is:


              MVolt/meter
Air                3
Glass             12
Polystyrene       20
Rubber            20
Distilled water   68
Vacuum            30        Depends on electrode shape
Diamond         2000

Relative permittivity

Relative permittivity is the factor by which the electric field between charges is decreased relative to vacuum. Relative permittivity is dimensionless. Large permittivity is desirable for capacitors.

             Relative permittivity
Vacuum            1                   (Exact)
Air               1.00059
Polyethylene      2.5
Sapphire         10
Concrete         4.5
Glass          ~ 6
Rubber           7
Diamond        ~ 8
Graphite       ~12
Silicon         11.7
Water (0 C)     88
Water (20 C)    80
Water (100 C)   55
TiO2         ~ 150
SrTiO3         310
BaSrTiO3       500
Ba TiO3     ~ 5000
CaCuTiO3    250000

Magnetic permeability

A ferromagnetic material amplifies a magnetic field by a factor called the "relative permeability".

                Relative    Magnetic   Maximum    Critical
              permeability  moment     frequency  temperature
                                       (kHz)      (K)
Metglas 2714A    1000000                100               Rapidly-cooled metal
Iron              200000      2.2                 1043
Iron + nickel     100000                                  Mu-metal or permalloy
Cobalt + iron      18000
Nickel               600       .606                627
Cobalt               250      1.72                1388
Carbon steel         100
Neodymium magnet       1.05
Manganese              1.001
Air                    1.000
Superconductor         0
Dysprosium                   10.2                   88
Gadolinium                    7.63                 292
EuO                           6.8                   69
Y3Fe5O12                      5.0                  560
MnBi                          3.52                 630
MnAs                          3.4                  318
NiO + Fe                      2.4                  858
CrO2                          2.03                 386

Effect of temperature on conductivity

Resistivity in 10^-9 Ohm Meters

              293 K   300 K   500 K

Beryllium     35.6    37.6     99
Magnesium     43.9    45.1     78.6
Aluminum      26.5    27.33    49.9
Copper        16.78   17.25    30.9
Silver        15.87   16.29    28.7

Wire gauges
Gauge  Diameter  Continuous  10 second  1 second  32 ms    Resistance
          mm      current    current    current   current
                  Ampere     Ampere     Ampere    Ampere   mOhm/meter
 
 0        8.3      125        1900      16000     91000       .32
 2        6.5       95        1300      10200     57000       .51
 4        5.2       70         946       6400     36000       .82
 6        4.1       55         668       4000     23000      1.30
12        2.0       20         235       1000      5600      5.2
18        1.02      10          83        250      1400     21.0
24         .51       3.5        29         62       348     84
30         .255       .86       10         15        86    339
36         .127       .18        4         10        22   1361
40         .080                  1          1.5       8   3441

Metal
      Conductivity   Melt   Hardness  Hardness  Stiffness  Strength  Density  Price/kg
      MAmps/Volt/m  Kelvin    Mohs      GPa        GPa       GPa                $/kg

Silver    63.0       1235     2.5       .24        83        .17      10.5       590
Copper    59.6       1358     3         .87        30        .21       9.1         6
Gold      45.2       1337     2.5       .24        78        .12      19.3     24000
Aluminum  35.0        933     2.8       .24        70        .05       2.7         2
Beryllium 25         1560     5.5       .6        287        .45       1.85      850
Magnesium 23          923     2.5       .26        45        .22       1.74        3
Iridium   21.2       2917     6.5      1.67       528       1.32      22.6     13000
Tungsten  18.9       3695     7.5      2.57       441       1.51      19.2        50
Zinc      16.9        693     2.5       .41       108        .2        7.1         2
Cadmium   12.5        594     2.0       .20        50        .078      8.6         2
Indium    11.9        430     1.2       .009       11        .004      7.3       750
Tin        8.3        505     1.5       .005       47        .20                  22
Osmium                        7.0

Battery types
              Energy/Mass  Power/Mass  Recharge  Year  Anode  Cathode   Market fraction of
               MJoule/kg    Watt/kg                                     Lithium-ion batteries

Lithium air          6.12               N      Future  Li    O2
Aluminum air         4.68     200       N      1970    Al    O2
Lithium thionyl      2.00     700       N      1973    Li    SOCl2
Zinc air             1.59               N      1932    Zn    O2
Lithium-ion sulfur   1.44     670       Y      Future  Li    S               0
Lithium metal        1.01     400       N      1976    Li    MnO2
Lithium-ion CoNiAlO2  .79               Y      1999    Li    CoNiAlO2         .10
Lithium-ion CoNiMnO2  .74    1200       Y      2008    Li    CoNiMnO2         .29
Lithium-ion CoO2      .70     200       Y      1991    Li    CoO2             .29
Lithium-ion Mn2O4     .54    1200       Y      1999    Li    Mn2O4            .10
Lithium-ion FePO4     .47    1200       Y      1996    Li    FePO4            .22
Alkaline              .40               Y      1992    Zn    MnO2
NiMH                  .34    1000       Y      1990    MH    NiO(OH)
Lead acid             .15     180       Y      1881    Pb    PbO2
NiCd                  .14     200       Y      1960    Cd    NiO(OH)

Battery internal resistance

Suppose a battery is connected to a load with resistance R. The load resistance and the battery internal resistance are in series.

Load resistance     =  R
Battery resistance  =  r
Battery voltage     =  V
Current             =  I  =  V / (R+r)
Load power          =  P  =  R I2  =  V2 R / (R+r)2
Battery power       =  p
Motor efficiency    =  e  =  P/(P+p)  =  1/(1+r/R)
The load power is maximized when R=r.

Electric motors typically have an efficiency of .8 for converting battery energy to mechanical energy. If e=.8 then R/r=4.


Capacitance
A   =  Plate area
Z   =  Plate spacing
Ke  =  Electric force constant  =  8.9876e9 N m2 / C2
Q   =  Max charge on the plate     (Coulombs)
Emax=  Max electric field       =  4 Pi Ke Q / A
V   =  Voltage between plates   =  E Z     =  4 Pi Ke Q Z / A
En  =  Energy                   =  .5 Q V  =  .5 A Z E2 / (4 π Ke)
e   =  Energy/Volume            =  E / A Z =  .5 E2 / (4 π Ke)
q   =  Charge/Volume            =  Q / A / Z
C   =  Capacitance              =  Q/V     =  (4 Pi Ke)-1 A/Z   (Farads)
c   =  Capacitance/Volume       =  C / A / Z =  (4 Pi Ke)-1 Emax2 / V2
Eair=  Max electric field in air=  3 MVolt/meter
k   =  Dielectric factor        =  Emax / Eair


Continuum                                                 Macroscopic

Energy/Volume  =  .5 E2  / (4 Pi Ke)           <->        Energy = .5 C V2
               =  .5 q V                                         =  .5 Q V
c              =  (4 Pi Ke)-1 Emax2  / V2      <->        C      = (4 Pi Ke)-1 A / Z

A capacitor can be specified by two parameters:
*)   Maximum energy density or maximum electric field
*)   Voltage between the plates

The maximum electric field is equal to the max field for air times a dimensionless number characterizing the dielectric

Eair =  Maximum electric field for air before electical breakdown
Emax =  Maximum electric field in the capacitor
Rbohr=  Bohr radius
     =  Characteristic size of atoms
     =  5.2918e-11 m
     =  hbar2 / (ElectronMass*ElectronCharge2*Ke)
Ebohr=  Bohr electric field
     =  Field generated by a proton at a distance of 1 Bohr radius
     =  5.142e11 Volt/m
Maximum energy density  =  .5 * 8.854e-12 Emax2


                         Emax (MVolt/m)   Energy density
                                            (Joule/kg)
Al electrolyte capacitor     15.0            1000
Supercapacitor               90.2           36000
Bohr limit               510000            1.2e12            Capacitor with a Bohr electric field

Inductance

A solenoid is a wire wound into a coil.

N  =  Number of wire loops
Z  =  Length
A  =  Area
Mu =  Magnetic constant  =  4 π 10-7
I  =  Current
It =  Current change/time
F  =  Magnetic flux      =  N B A        (Tesla meter2)
Ft =  Flux change/time                   (Tesla meter2 / second)
B  =  Magnetic field     =  Mu N I / Z
V  =  Voltage            =  Ft =  L It  =  N A Bt  =  Mu N2 A It / Z
L  =  Inductance         =  Ft / It  =  Mu N2 A / Z      (Henrys)
E  =  Energy             =  .5 L I2
Hyperphysics: Inductor
Propeller lift and power

The properties of a propeller are:

Rotor radius     =  R
Air density      =  D  =  1.22 kg/meter3
Rotor tip speed  =  V
Rotor lift force =  Fl =  D W R2 V2
Rotor drag force =  Fd
Rotor lift param =  W  =  Fl D-1 R-2 V-2
Rotor lift/drag  =  Q  =  Fl / Fd
Rotor power      =  P  =  Fd V  =  F V / Q
Rotor quality    =  q  =  Q W½ D½
                       =  Fl3/2 P-1 R-1
Rotor force/power=  Z  =  Fl/ P
                       =  Q / V
                       =  D½ W½ Q R F
                       =  q R Fl
The physical parameters of a propeller are {R,Q,W,q}, with typical values of
Q  = 5.5
W  =  .045
q  = 1.29
Most propellers have 2 blades and some have 3. If there are 4 or more blades then q declines.

A measurement of Fl and V determines W.
A measurement of P, Fl, and V determines Q.
A measurement of Fl, P, and r determines q.

Q and W are not independent. They are related to the blade aspect ratio.

Q  ≈  Aspect ratio
W  ≈  Q
q  ≈  Q½

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
Hover force       =  F  =  M g
Hover power       =  P
Force/Power ratio =  Z  =  F/P
Power/Mass ratio  =  p  =  P/M  =  g/Z

Drone power system

One has to choose a wise balance for the masses of the motor, battery, fuselage, and payload. The properties of the electrical components are:

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

Electric motor          -         10.0        -        .062     160
Lithium-ion battery     .75        1.5        .009     .0142    106
Lithium supercapacitor  .008       8          .0010    .09       90
Aluminum capacitor      .0011    100
If the battery and motor have equal power then the battery has a larger mass than the motor.
Mass of motor            =  Mmot
Mass of battery          =  Mbat
Power                    =  P             (Same for both the motor and the battery)
Power/mass of motor      =  pmot  =  P/Mmot  =   8.0 kWatt/kg
Power/mass of battery    =  pbat  =  P/Mbat  =   1.5 kWatt/kg
Battery mass / Motor mass=  R    =Mbat/Mmot  =  pmot/pbat  =  5.3
The "sports prowess" of a drone is the drone power divided by the minimum hover power. To fly, this number must be larger than 1.
Drone mass               =  Mdro
Motor mass               =  Mmot
Motor power/mass         =  pmot =  8000 Watts/kg
Hover minimum power/mass =  phov =    60 Watts/kg
Drone power              =  Pdro =  pmot Mmot
Hover minimum power      =  Phov =  phov Mdro
Sports prowess           =  S   =  Pdro/Phov  =  (pmot/phov) * (Mmot/Mdro)  =  80 Mmot/Mdro
If S=1 then Mmot/Mdro = 1/80 and the motor constitutes a negligible fraction of the drone mass. One can afford to increase the motor mass to make a sports drone with S >> 1.

If the motor and battery generate equal power then the sports prowess is

S  =  (pbat/phov) * (Mbat/Mdro)  =  25 Mbat/Mdro
If Mbat/Mdro = ½ then S=12.5, well above the minimum required to hover.

Suppose a drone has a mass of 1 kg. A squash racquet can have a mass of as little as .12 kg. The fuselage mass can be much less than this because a drone doesn't need to be as tough as a squash racquet, hence the fuselage mass is negligible compared to the drone mass. An example configuration is:

              kg

Battery       .5
Motors        .1   To match the battery and motor power, set motor mass / battery mass = 1/5
Rotors       <.05
Fuselage      .1
Camera        .3
Drone total  1.0
Supercapacitors can generate a larger power/mass than batteries and are useful for extreme bursts of power, however their energy density is low compared to batteries and so the burst is short. If the supercapacitor and battery have equal power then
Battery power/mass         =  pbat  =  1.5 kWatts/kg
Supercapacitor power/mass  =  psup  =  8.0 kWatts/kg
Battery power              =  P
Battery mass               =  Mbat  =  P / pbat
Supercapacitor mass        =  Msup  =  P / psup
Supercapacitor/Battery mass=  R     =Msup/ Mbat  =  pbat/psup  =  .19
The supercapacitor is substantially ligher than the battery. By adding a lightweight supercapacitor you can double the power. Since drones already have abundant power, the added mass of the supercapacitor usually makes this not worth it.

If a battery and an aluminum capacitor have equal powers,

Aluminum capacitor mass  /  Battery mass  =  .015
If a battery or supercapacitor is operating at full power then the time required to expend all the energy is
Mass          =  M
Energy        =  E
Power         =  P
Energy/Mass   =  e  =  E/M
Power/Mass    =  p  =  P/M
Discharge time=  T  =  E/P  =  e/p

                     Energy/Mass  Power/Mass   Discharge time   Mass
                      MJoule/kg    kWatt/kg       seconds        kg

Lithium battery         .75          1.5          500           1.0
Supercapacitor          .008         8.0            1.0          .19
Aluminum capacitor      .0011      100               .011        .015
"Mass" is the mass required to provide equal power as a lithium battery of equal mass.
Commercial electric vehicles

                       Mass      Battery  Battery  Battery  Power   Flight  Price
                                 energy    mass                      time
                        kg       MJoule     kg      MJ/kg   kWatt   minutes   $

Drone   Jetjat Nano       .011     .00160                  .0033       8      40
Drone   ByRobot Fighter   .030     .0040                   .0067      10     120
Drone   XDrone Zepto      .082     .0067                   .0046      24      40
Drone   Walkera QRY100    .146     .0213    .0413   .52    .018       20     100
Drone   DJI Mavic Pro     .725     .157     .24     .65    .11        24    1000
Drone   DJI Phantom 4    1.38      .293     .426    .69    .17        28    1000
Drone   JYU Spider X     2.1       .360     .812    .44    .20        30     155
Drone   MD4-1000         2.65     1.039                    .20        88    2000
Drone   Walkera QRX800   3.9       .799    1.134    .70    .22        60    2700
Drone   AEE F100         6.0      1.598                    .38        70   58000
Drone   Ehang 184      200       51.8                    37.50        23  300000
Skate   Hammacher        6.4                               .10               700
Scooter Zero             7.0       .899                    .45               500
Bike    Revelo          15        1.35                     .25
Bike    Seagull         26.3      2.25                    1.0               2000
Bike    Wolverine       38.6      8.64                    7.0              10450
Car     Mitsu. MiEV   1080       58      201        .29  47                16300
Car     Tesla S P85D  2239      306      540        .57 568               115000
Light   Barska TC1200     .41      .032     .45     .71    .015              120
Laser   Violet laser      .182     .009     .0152   .62    .0001              20
Battery RAVPower          .590     .414     .590    .70                       60
Phone   Samsung S5        .145     .039     .038   1.03
"Flight time" is the maximum hover time for drones.
"Drone power" is the power to hover for drones and the maximum engine power for cars and bikes.
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