Voltage = V Volts Capacitance = C Farads Total energy = E = ½ C V2 Joules Effective = Ee = ¼ C V2 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.
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) A/Z (Farads) c = Capacitance/Volume = C / A / Z = (4 Pi Ke) 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 / ZA 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 = 5.2918e-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
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 100If 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 = 5.3The "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 = 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/MdroIf 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/MdroIf 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 1.0Supercapacitors 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 = 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 = .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 Energy E/M Power P/M Price Energy/$ C Voltage kg kJoule kJ/kg kWatt kW/kg $ kJoule/$ Farads Volts PM-5R0V105-R .000454 .0062 13.8 5.7 .0011 1 5.0 Maxwell BCAP0350 .060 .638 10.6 .459 7.65 16 .040 350 2.7 Adafruit .135 .984 7.3 20 .049 630 2.5 BMOD0006E160B02 5.2 37.1 7.1 2.08 .40 1170 .032 5.8 160 XLM-62R1137-R 15 125.3 8.4 124.2 8.3 1396 .090 130 62.1