Capacitor is used in AC circuits where DC power is unintended.
Batteries are used in DC circuits where constant DC power is required.
These two can't be interchanged. They are for different functions.
A battery’s voltage stays nearly constant as it discharges at a constant current. Thus the power output is nearly constant. A capacitor's voltage drops linearly over time at a constant current. Thus, power output drops linearly. In order to use the capacitor as a battery replacement, you need a voltage regulation circuit that can step up voltage as it drops, which means you need a linear increase in current output to deliver a constant power output.
The other problem is that modern ultracapacitors have much lower specific energy than batteries. The best ultracaps on the market manage 8-10 Wh/kg, most are more like 5 Wh/kg. The best Li-ion batteries deliver close to 200 Wh/kg, many formulations can reach over 100 Wh/kg. So you need about 20x the weight to use ultracaps. But possibly more, since at some point during discharge, depending on application, the voltage will drop too low to be usable, leaving power unused.
Then there's self-discharge: how quickly does power “leak” from a storage device. The only NiMh cells are rugged, but self discharge as high as 20–30% per month. Li-ion cells reduce this to more like <2% per month depending on the specific Li-ion technology, maybe 3% in some systems depending on battery monitoring overhead. Today's Ultracapacitors drop as much as 50% of charge in the first month. That may not matter in a device that's recharged daily, but it absolutely limits the use cases for caps vs batteries, at least until better designs are created.
And because you need so many, the current cost of ultracapacitors can be 6x-20x the cost of batteries. If your application needs a very small power output, particulary with very short high current surges, the ultracap may be an option. Otherwise, it’s not going to be a battery replacement in the near future.
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