Optimum lowest input energy for first-order circuits in transient state

This paper presents optimum lowest input energy for first-order circuits in transient state. A resistors network with a capacitor is chosen for derivation. The derivation is based on calculus of variations theory or more specifically, the Euler-Lagrange's differential equation. Essential parameters are introduced and the most energy-efficient input source function is obtained. The lowest input energy of a capacitive circuit and an inductive circuit are derived and stated as corollaries. Using the corollaries with a circuit of a supercapacitor is illustrated. Speed and energy trade-off is shown that it is not always held by inspecting the corollaries. Finally, optimum transient duration is determined as well as optimum lowest input energy.