Optimal charging of capacitors

Charging a capacitor from a voltage source with internal resistor is one of the basic problems in circuit theory. In recent years, this simple problem has attracted some interest in the area of low-power digital circuits. The efficiency, i.e., the energy stored in the capacitor versus the energy delivered by the source is one of the key measures. The common believe that the source has to deliver twice the capacitor energy holds true only for a linear circuit with step function as source voltage. In this paper we compute several optimal solutions with respect to time and efficiency. The case of nonlinear capacitors is discussed in some detail. The source voltage can depend on time in a rather involved manner. However, replacing the voltage source by a current source simplifies the problem significantly since the current has to be constant over time regardless of the characteristic of the capacitance. This result should have implications on the circuit technique employed for low-power circuits. Furthermore, if leak conductances in parallel to the capacitor are taken into account, the achievable minimal energy dissipation is limited and, if ramps are used for charging, an optimal charging time different from infinity occurs.