
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 400Each kind of electric vehicle is expanded on below, each with a complete design based on current technology.
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 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 determines a vehicle's top speed and energy usage, and this determines the minimum battery size.
Air density = D = 1.22 kg/meter^{3} Air drag area = A Speed = V Air drag force = F = ½ A D V^{2} Air drag power = P = ½ A D V^{3} 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/$A compact car designed for city speeds doesn't need much power. Example values for various electric vehicles:
Speed Power Force Force/prsn People Range Drag area m/s kWatt Newton Newton km m^{2} Skate 10 .18 18 18 1 5 .3 Kick scooter 10 .18 18 18 1 5 .3 Bike 15 .82 55 55 1 8 .4 Car, small, city speed 20 4.9 244 244 1 10 1 Car, large, freeway speed 30 33 1100 1100 1 15 2 Bus, freeway speed 30 99 3290 46 72 15 6 Train car, freeway speed 30 99 3290 27 120 15 6 Airbus A380 251 251000 1000000 1840 544 10000 160 1 Horsepower = 746 WattsEnergy usage is proportional to the drag force per person. If a bus is full it is 5 times more efficient than a compact car, but buses are rarely full and usually slow.
Buses and trains are substantially more efficient than planes and they should be favored over short flights.
"Power" is the minimum power required for the given speed.
We assume a minimalist battery  the smallest battery that can provide the given power. We then calculate the energy for this battery using the battery parameters and we caculate a range using this energy. Larger range can be achieved with a larger battery. Since a minimalist battery is cheap, a larger battery is usually feasible.
The drag parameter is obtained from an analysis of commercial vehicles. Data
Battery cost as a function of power is 20 Watts/$. 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.
The performance of flying cars is determined chiefly by the battery energy per mass. For modern lithiumion 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)
The design of the car depends on the physics of rotors. For a rotor,
Power required to hover = Constant * LiftForce^{3/2 / 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.
Stateoftheart lithiumion batteries have an energy/mass of .8 MJoules/kg and can fly a car for 44 minutes. In the future, lithiumsulfure 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/second^{2} Rotor force = F = Mg/N =1470 Newtons Rotor quality = q = 1.02 Air density = D = 1.22 kg/meter^{3} Rotor power = P_{r}=(qDR)^{1}F^{3/2}= 30.2 kWatts Hover power = P_{h}= N P_{r} = 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 = P_{b}= 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/P_{h} =2650 seconds = 44 minutesThe 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/sThe 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 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 3.6The 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.
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
The flight time of a drone is determined chiefly by the battery energy per mass. Modern lithiumion batteries
The flight time of a drone is determined by:
Typical parameters for a drone are:
Drone mass = M = 1.0 kg Battery mass = M_{bat} = .5 kg (The battery is the most vital component) Battery energy = E = .4 MJoules Battery energy/mass= e_{bat}= E/M_{bat}= .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 minutesThe flight time in terms of component parameters is
T = (e_{bat}/p) * (M_{bat}/M)
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, lithiumion .8 1200 .010 Yes 1 hour 1000 Supercapacitor .026 14000 .0005 Yes Instant Infinite Aluminum capacitor .010 50000 .0001 Yes Instant Infinite
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/cm^{3} Recharges =1000 Shelf life = 1.0 year Voltage = 3.7 VoltEnergy/Mass and Power/Mass are an engineering tradeoff. One can be increased at the expense of the other.
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 Joules1 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
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 highpower 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
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 3.0Prices from www.liionwholesale.com
Voltage = V Volts Capacitance = C Farads Total energy = E = ½ C V^{2} Joules Effective = E_{e} = ¼ C V^{2} JoulesNot 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 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 V^{2} = 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 = M_{B} = P / (Q p) = 112 kg Battery cost/mass = c = 86 $/kg Battery cost = C = 9600 $
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
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 laserspointers 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
$ Lumens Diameter Mass Lumens per inch ounce inch^{2} 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
Sandisk microsd cards on Amazon.com
GigaBytes $ 32 13 64 25 128 45 256 160
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 .90For an electric vehicle the overall efficiency is similar to that of a diesel engine.
Overall efficiency = Power plant efficiency * Vehicle efficiency = .35 * .80 = .28
White: High conductivity Red: Low 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 e16 2320 .509 Tube 10 3500 Carbon nanotube. Electric conductivity = e16 laterally Tube bulk 200 Carbon nanotubes in bulk Graphene 10 5000 Graphite 2 400 .709 Natural graphite Al Nitride e11 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) e16 2 Fused quartz Granite 2.5 Marble 2.2 Ice 2 Concrete 1.5 Limestone 1.3 Soil 1 Glass e12 .85 Water e4 .6 Seawater 1 .6 Brick .5 Plastic .5 Wood .2 Wood (dry) .1 Plexiglass e14 .18 Rubber e13 .16 Snow .15 Paper .05 Plastic foam .03 Air 5e15 .025 Nitrogen .025 1.04 Oxygen .025 .92 Silica aerogel .01 Siemens: Amperes^2 Seconds^3 / kg / meters^2 = 1 Ohm^1For most metals,
Electric conductivity / Thermal conductivity ~ 140 J/g/K
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 10^{10} Magnetar neutron star 10^{14}
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 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
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 Rapidlycooled metal Iron 200000 2.2 1043 Iron + nickel 100000 Mumetal 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
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
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
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
Energy/Mass Power/Mass Recharge Year Anode Cathode Market fraction of MJoule/kg Watt/kg Lithiumion 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 Lithiumion sulfur 1.44 670 Y Future Li S 0 Lithium metal 1.01 400 N 1976 Li MnO2 Lithiumion CoNiAlO2 .79 Y 1999 Li CoNiAlO2 .10 Lithiumion CoNiMnO2 .74 1200 Y 2008 Li CoNiMnO2 .29 Lithiumion CoO2 .70 200 Y 1991 Li CoO2 .29 Lithiumion Mn2O4 .54 1200 Y 1999 Li Mn2O4 .10 Lithiumion 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)
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 I^{2} = V^{2} 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.
A = Plate area Z = Plate spacing Ke = Electric force constant = 8.9876e9 N m^{2} / C^{2} 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 E^{2} / (4 π Ke) e = Energy/Volume = E / A Z = .5 E^{2} / (4 π Ke) q = Charge/Volume = Q / A / Z C = Capacitance = Q/V = (4 Pi Ke)A capacitor can be specified by two parameters:
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 = 5.2918e11 m = hbar^{2} / (ElectronMass*ElectronCharge^{2}*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 * 8.854e12 Emax^{2} 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
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 meter^{2}) Ft = Flux change/time (Tesla meter^{2} / second) B = Magnetic field = Mu N I / Z V = Voltage = Ft = L It = N A Bt = Mu N^{2} A It / Z L = Inductance = Ft / It = Mu N^{2} A / Z (Henrys) E = Energy = .5 L I^{2}Hyperphysics: Inductor
Rotor radius = R Air density = D = 1.22 kg/meter^{3} Rotor tip speed = V Rotor lift force = F_{l} = D W R^{2} V^{2} Rotor drag force = F_{d} Rotor lift param = W = F_{l} D^{1} R^{2} V^{2} Rotor lift/drag = Q = F_{l} / F_{d} Rotor power = P = F_{d} V = F V / Q Rotor quality = q = Q W^{½} D^{½} = F_{l}^{3/2} P^{1} R^{1} Rotor force/power= Z = F_{l}/ P = Q / V = D^{½} W^{½} Q R F^{½} = q R F_{l}^{½}The physical parameters of a propeller are {R,Q,W,q}, with typical values of
Q = 5.5 W = .045 q = 1.29Most propellers have 2 blades and some have 3. If there are 4 or more blades then q declines.
A measurement of F_{l} and V determines W.
A measurement of P, F_{l}, and V determines Q.
A measurement of F_{l}, P, and r determines q.
Q and W are not independent. They are related to the blade aspect ratio.
Q ≈ Aspect ratio W ≈ Q^{½} q ≈ Q^{½}
A commonlyappearing 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
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 Lithiumion battery .75 1.5 .009 .0142 106 Lithium supercapacitor .008 8 .0010 .09 90 Aluminum capacitor .0011 100If the battery and motor have equal power then the battery has a larger mass than the motor.
Mass of motor = M_{mot} Mass of battery = M_{bat} Power = P (Same for both the motor and the battery) Power/mass of motor = p_{mot} = P/M_{mot} = 8.0 kWatt/kg Power/mass of battery = p_{bat} = P/M_{bat} = 1.5 kWatt/kg Battery mass / Motor mass= R =M_{bat}/M_{mot} = p_{mot}/p_{bat} = 5.3The "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 = M_{dro} Motor mass = M_{mot} Motor power/mass = p_{mot} = 8000 Watts/kg Hover minimum power/mass = p_{hov} = 60 Watts/kg Drone power = P_{dro} = p_{mot} M_{mot} Hover minimum power = P_{hov} = p_{hov} M_{dro} Sports prowess = S = P_{dro}/P_{hov} = (p_{mot}/p_{hov}) * (M_{mot}/M_{dro}) = 80 M_{mot}/M_{dro}If S=1 then M_{mot}/M_{dro} = 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 = (p_{bat}/p_{hov}) * (M_{bat}/M_{dro}) = 25 M_{bat}/M_{dro}If M_{bat}/M_{dro} = 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 1.0Supercapacitors 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 = p_{bat} = 1.5 kWatts/kg Supercapacitor power/mass = p_{sup} = 8.0 kWatts/kg Battery power = P Battery mass = M_{bat} = P / p_{bat} Supercapacitor mass = M_{sup} = P / p_{sup} Supercapacitor/Battery mass= R =M_{sup}/ M_{bat} = p_{bat}/p_{sup} = .19The 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 = .015If 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.
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 MD41000 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.