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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

Power

To give a sense for the strength of human and electric power,

                          kiloWatts

Human unstrenuous cycling     .1
Human strenuous cycling       .3
Human sprint cycling         1
Typical electric bike        1
Monster electric bike        6
Typical car                100
Electric power opens the way for light cheap city vehicles. Electric power easily has the speed and range for city driving.
Speed

To give a sense for the relationship between power and top speed,

        Power   Speed   Speed
        kWatt    m/s     mph

Bike      .12     7      17     Human unstrenuous cycling
Bike      .25     9      21     Human strenuous cycling
Bike     1       15      33     Human sprint cycling
Bike     2       19      42
Bike     4       24      53

Trike    1       12      26
Trike    4       19      42
Trike   16       30      67

Car     16       26      58     Minimum power for cities
Car     32       33      73     Minimum power for highways
Car     64       41      92     Minimum power for freeways
Car    128       52     116     Typical car
Car    512       82     184     Sports car

Battery

Typical values for a 16 kWatt car battery are:

Energy/Mass   =  e  =  E/M  =   .8   MJoules/kg
Power/Mass    =  p  =  P/M  =  1.2   kWatts/kg
Energy/Cost   =  c  =  E/C  =   .01  MJoules/$
Power/Cost    =  d  =  P/C  =   .012 kWatts/$
Energy/Power  =  D  =  E/P  =   .67  MJoules/kWatt
Mass          =  M  =       = 13     kg
Energy        =  E  =       = 11     MJoules
Power         =  P  =       = 16     kWatts
Cost          =  C  =       = 1330   $

Range

Air drag determines a vehicle's top speed, energy use, and minimum battery power. A 16 kWatt car with a minimalist battery has a range of 30 km when driving at a speed of 20 meters/second.

Air density        =  D                        =  1.22 kg/meter3
Air drag coef.     =  K                        =   .75 meters2
Car speed          =  V                        = 20    meters/second
Air drag force     =  F  =  K D V2  =366    Newtons
Air drag power     =  P  =  K D V3  =  7.3  kWatts
Battery energy     =  E                        = 11    MJoules
Distance traveled  =  X  =  E/F                = 30    km

Energy efficiency

The goal is to minimize the energy cost per person. For an N-person vehicle,

People            =  N
Distance traveled =  X
Air drag force    =  F
Energy            =  E  =  F X
Energy efficiency =  Q  =  E/(NX)  =  F/N
The energy efficiency is equal to the air drag force divided by the number of passengers. Example values for various 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
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.


Flying electric cars

The properties of a flying car are determined by the properties of propellers and lithium-ion batteries. Typical parameters for a 1-person car are:

Hovering time      =   25 minutes
Cruise speed       =  100 meters/second
Range              =  155 km
Hovering power     =   40 kWatts
Vehicle mass       =  320 kg
Battery energy/mass=   .8 MJoules/kg
Battery power/mass = 1200 Watts/kg
Battery cost/MJoule=  100 $/MJoule
Battery mass       =   80 kg
Battery energy     =   64 MJoules
Battery power      =   96 kWatts
Battery cost       = 6400 $

For hovering, the more rotors the better. The hovering time scales as rotor number to the 1/6 power. Adding rotors also increases stability and failsafe.


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

A electric propeller-driven aircraft can hover for more than an hour. The hovering time is determined by the battery energy per mass and by the rotor radius. Example values:

Drone mass         =  M          =  1.0  kg
Battery mass       =  m          =   .5  kg
Battery energy/mass=  e  =  E/m  =   .8  MJoules/kg
Battery energy     =  E          =   .4  MJoules
Hover power/mass   =  p  =  P/M  =   94  Watts/kg     (Hover power for a 1 kg drone with a 1/4 meter radius rotor)
Hover power        =  P  =  p M  =   94  Watts
Flight time        =  T  =  E/P  = 3990  seconds  =  66 minutes
The flight time is
T  =  (e/p)⋅(m/M)

Hovering power per mass

The power per mass required to hover is determined by the physics of rotors. For a 1 kg vehicle with a 1/4 meter radius rotor,

Mass               =  M  =  1    kg
Gravity constant   =  g  =  9.8  meters/second
Rotor radius       =  R  =   .25 meters
Rotor quality      =  q  =  1.3
Hover power/mass   =  p  =  M½ g3/2 q-1  R-1  =  94 Watts
The rotor radius scales as M1/3 and the hover power/mass scales as M1/6. If we scale the above vehicle from 1 kg up to 300 kg (the mass of a 1-person vehicle) the hovering power/mass is 240 Watts/kg and the total power is 73 kWatts, or 98 horsepower.
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 sizes

Energies and powers are for lithium batteries, which have a voltage of 3.7 Volts. The "ID #" is often used instead of cell size.

Cell   Energy  Power  Current  Mass  Diameter  Length  Charge   Price    ID #
size   kJoule  Watt   Ampere   gram     mm       mm    AmpHour    $

D       107     220     60     138      32       67     8.0      13      32650
C        67     220     60      92      26       50     5.0       8      26650, 25500
B        58     160     45      72      22       60     4.5       5      21700, 20700
A        47     110     30      49      18       50     3.5       3      18650
AA        9      22      6      15      14       53      .70      1      14500
AAA       4.7    11      3       7.6    10       44      .35       .5    10440
AAAA      2.3     6      1.5     3.8     8       42      .17       .25   75400

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               No     Future  Li    O2
Aluminum air         4.68     200       No     1970    Al    O2
Lithium thionyl      2.00     700       No     1973    Li    SOCl2
Zinc air             1.59               No     1932    Zn    O2
Lithium-ion sulfur   1.44     670       Yes    Future  Li    S               0
Lithium metal        1.01     400       No     1976    Li    MnO2
Lithium-ion CoNiAlO2  .79               Yes    1999    Li    CoNiAlO2         .10
Lithium-ion CoNiMnO2  .74    1200       Yes    2008    Li    CoNiMnO2         .29
Lithium-ion CoO2      .70     200       Yes    1991    Li    CoO2             .29
Lithium-ion Mn2O4     .54    1200       Yes    1999    Li    Mn2O4            .10
Lithium-ion FePO4     .47    1200       Yes    1996    Li    FePO4            .22
Alkaline              .40               Yes    1992    Zn    MnO2
NiMH                  .34    1000       Yes    1990    MH    NiO(OH)
Lead acid             .15     180       Yes    1881    Pb    PbO2
NiCd                  .14     200       Yes    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
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.
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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