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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.
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
=  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
```

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 supercapacitors
```                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
```

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