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Tony4t3
If you charge a capacitor to 12V and discharge it at a current of 1A so that after 10sec the voltage is (say) 7V, then
I*t = C*ΔV = Q
C = I*t/ΔV = 1*10/(12-7) = 10/5 = 2Farads (Have you heard of Supercapacitors?)
If you get a step-up switcher that converts 4V to 12V, then ask how long it will take to charge a 2Farad capacitor (with no load during the charging phase)?
I*t = Q = C*ΔV = 2*5 = 10Coulombs
Suppose that the 4V source is constrained to supplying 1A, and the switcher is 80% efficient. The switcher requires 10/0.8 = 12.5C to be transferred from the 4V source to the 2F capacitor.
I*t = Q
t = Q/I = 12.5/1 = 12.5seconds.