It is easy to derive the efficiency for loading a capacitor from a constant voltage or a constant current source, basically because exponential functions and constants are very, well, integration-friendly.
However, what is the efficiency for charging a capacitor with a fixed capacitance C and a fixed ESR R from a constant power source to a defined voltage U within a defined time T, and how do I derive that?
Best Answer
You are asking about efficiency and I am not able to answer that, but maybe it helps to answer the question of the capacitor voltage U(t) for constant power charging without considering R_ESR.
With constant power P, energy E over time in the cap is $$ E(t)=P*t=\frac{1}{2}CU(t)^{2} $$
This can be rewritten as $$ U(t)=\sqrt{\frac{2Pt}{C}} $$
We can also find I(t) with following equation... $$ I(t)=\frac{d}{dt}CU(t)=\frac{C}{2}\sqrt{\frac{2P}{C}}*\frac{1}{\sqrt{t}} $$ And here is the graph for U(t) [green] and I(t) [red] with constant power P=1W and C=1F.
This is the voltage and current of the capacitor when it is charged with constant power source.