A bit of a look into the future for the first post of this year. I don't have any immediate plans to do anything electric vehicle, but the Atomic Duck could make a good lightweight EV platform, so I've been thinking about how the bits might fit in; specifically: What is the best size for a capacitor bank of a lightweight electric vehicle?

Now I have very little experience with electric drivetrains, so I would love to hear any answer from anyone with more knowledge; but I've also had a go at reasoning it out, so you should also tell me if I'm way off base too!

A capacitor bank is a large matrix of capacitors that act like a buffer between the battery and motor/generator on an electric vehicle. Capacitors have low energy capacity compared to batteries, so you wouldn't want to use them for the only energy storage, but they do have very high charge/discharge rates that don't affect their energy storage capacity.

As such, by placing a bank of capacitors between the motor and the batteries, the capacitor bank can discharge quickly for acceleration and then be recharged from the motor to provide regenerative braking, while the batteries supply (and receive) much lower currents to drive the motor in steady state operation and supply the energy that is lost from the moving vehicle system (e.g. through aerodynamic drag, heat, friction and noise). This lower current, constant charge and discharge keeps the batteries in much better condition than high current operation, meaning that the batteries will have longer operational lives.

I reasoned that capacitor bank that can hold the kinetic energy of the vehicle at maximum normal cruising speed would be good for a lightweight vehicle.

\[ E_{kinetic} = mv^2 \]

Therefore, a nominal 350kg vehicle with a cruise speed of 60mph (28.82 m/s) would have to have capacitors that can hold 0.25 MJ (251759 J or ≈70 W·h).

The required capacitor size is dependant on voltage:

\[ E_{storage} = \frac{1}{2}CV^2 \]

So capacitance can be calculated with:

\[ C = \frac{2E}{V} \]

At 12V, 0.25 MJ needs a capacitance of 3497 F.

Using seven 500 F (16V) ultracapacitor modules in series would give 3500 F of capacitance at a total mass of almost 40 kg (5.5 kg per module).

When I said capacitors had lower energy density, it really shows here. The ultracapacitors I've used for reference have an energy density of 11.5 kJ/kg (3.2 W·h/kg) compared to 108-144 kJ/kg (30-40 W·h/kg) for lead acid batteries. For instance two 30Ah, 12V sealed lead acid batteries can store ten times the energy of this capacitor bank, 2.59 MJ (720 W·h), for just under 15kg.

On the other hand though, the ultracapacitor can operate at up to 7600A (short circuit, max 60 sec) giving a maximum of 91 kJ/s @ 12 V operation. Lead acid batteries can operate at either 7A (slow discharge) or 100A (fast discharge, surge) meaning that they normally only release energy at 0.084-1.2 kJ/s.

As you can see, a capacitor bank that can hold energy required to push a vehicle to it's cruising speed is very large, and could even have greater mass than the battery storage, but being able to regenerate 'all' of a vehicle's kinetic energy in the capacitors would simplify the control strategy for recharging the battery array by making it unnecessary in normal operation.

Capacitance scales linearly with vehicle mass, but it increases with the square of speed, so lighter and lower speed vehicles have more efficiently sized capacitor bank with this approach.

The required battery storage is more difficult to calculate though, as it's heavily dependant on vehicle use and efficiency; and I've not even taken into account any efficiencies, or having to regulate capacitor voltage.